The Formation of Shocks in 3-Dimensional Fluids (EMS Monographs in Mathematics)

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EMS Monographs in Mathematics Edited by Ivar Ekeland (Pacific Institute, Vancouver, Canada) Gerard van der Geer (University of Amsterdam, The Netherlands) Helmut Hofer (Courant Institute, New York, USA) Thomas Kappeler (University of Zürich, Switzerland) EMS Monographs in Mathematics is a book series aimed at mathematicians and scientists. It publishes research monographs and graduate level textbooks from all fields of mathematics. The individual volumes are intended to give a reasonably comprehensive and self-contained account of their particular subject. They present mathematical results that are new or have not been accessible previously in the literature.

Previously published in this series: Richard Arratia, A.D. Barbour, Simon Tavaré, Logarithmic combinatorial structures: a probabilistic approach

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Demetrios Christodoulou

The Formation of Shocks in 3-Dimensional Fluids M

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European Mathematical Society

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Author: Prof. Demetrios Christodoulou Department of Mathematics ETH-Zentrum 8092 Zürich Switzerland

2000 Mathematical Subject Classification (primary; secondary): 35L67; 35L65, 35L70, 58J45, 76L05, 76N15, 76Y05

ISBN 978-3-03719-031-9 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliographie; detailed bibliographic data are available in the Internet at http://dnb.ddb.de. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2007 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Printed in Germany 987654321

This work is dedicated to the memory of my father L AMBROS C HRISTODOULOU born Alexandria 1913 deceased Athens 1999 whose kindness is the fondest memory I have

Contents

Prologue and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1

Relativistic Fluids and Nonlinear Wave Equations. The Equations of Variation . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2

The Basic Geometric Construction . . . . . . . . . . . . . . . . . . . . . .

39

3

The Acoustical Structure Equations . . . . . . . . . . . . . . . . . . . . .

53

4

The Acoustical Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

5

The Fundamental Energy Estimate . . . . . . . . . . . . . . . . . . . . . .

99

6

Construction of the Commutation Vectorfields . . . . . . . . . . . . . . . .

139

7

Outline of the Derived Estimates of Each Order . . . . . . . . . . . . . . .

169

8

Regularization of the Propagation Equation for d/trχ. Estimates for the Top Order Angular Derivatives of χ . . . . . . . . . . . .

203

Regularization of the Propagation Equation for
/ µ. Estimates for the Top Order Spatial Derivatives of µ . . . . . . . . . . . . .

275

9 10

11

12

Control of the Angular Derivatives of the First Derivatives of the x i . Assumptions and Estimates in Regard to χ . . . . . . . . . . . . . . . . . . Part 1: Control of the angular derivatives of the first derivatives of the x i . . . . . . . . . . . . . . . . . . . Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl . . . . . . . . . .

329 329 403

Control of the Spatial Derivatives of the First Derivatives of the x i . Assumptions and Estimates in Regard to µ . . . . . . . . . . . . . . . . . . Part 1: Control of the spatial derivatives of the first derivatives of the x i . . . . . . . . . . . . . . . . . . . . . . . .  Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l . . . . . . .

473 589

Recovery of the Acoustical Assumptions. Estimates for Up to the Next to the Top Order Angular Derivatives of χ and Spatial Derivatives of µ . . .

665

473

viii

13

Contents

The Error Estimates Involving the Top Order Spatial Derivatives of the Acoustical Entities. The Energy Estimates. Recovery of the Bootstrap Assumptions. Statement and Proof of the Main Theorem: Existence up to Shock Formation . . . . . . . . . . . . . . . . . . . . Part 1: Derivation of the properties C1, C2, C3 . . . . . . . . . . . . Part 2: The error estimates of the acoustical entities . . . . . . . . . Part 3: The energy estimates . . . . . . . . . . . . . . . . . . . . . . Part 4: Recovery of assumption J. Recovery of the bootstrap assumption. Proof of the main theorem . . . . . . . . . . . .

. . . .

741 741 757 831

. . .

874

Sufficient Conditions on the Initial Data for the Formation of a Shock in the Evolution . . . . . . . . . . . . . . . . . . . . . . . . . .

893

The Nature of the Singular Hypersurface. The Invariant Curves. The Trichotomy Theorem. The Structure of the Boundary of the Domain of the Maximal Solution . . . . . . . . . . . . . . . . . . . . .

927

Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

977

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

987

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

989

14 15

. . . .

. . . .

Prologue and Summary

The equations describing the motion of a perfect fluid were first formulated by Euler in 1752 (see [Eu1], [Eu2]), based, in part, on the earlier work of D. Bernoulli [Be]. These equations were among the first partial differential equations to be written down, preceded, it seems, only by D’Alembert’s 1749 formulation [DA] of the one-dimensional wave equation describing the motion of a vibrating string in the linear approximation. In contrast to D’Alembert’s equation however, we are still, after the lapse of two and a half centuries, far from having achieved an adequate understanding of the observed phenomena which are supposed to lie within the domain of validity of the Euler equations. The phenomena displayed in the interior of a fluid fall into two broad classes, the phenomena of sound, the linear theory of which is acoustics, and the phenomena of vortex motion. The sound phenomena depend on the compressibility of a fluid, while the vortex phenomena occur even in a regime where fluid may be considered to be incompressible. The formation of shocks, the subject of the present monograph, belongs to the class of sound phenomena, but lies in the nonlinear regime, beyond the range covered by linear acoustics. The phenomena of vortex motion include the chaotic form called turbulence, the understanding of which is one of the great challenges of science. Let us make a short review of the history of the study of the phenomena of sound in fluids, in particular the phenomena of the formation and evolution of shocks in the nonlinear regime. At the time when the equations of fluid mechanics were first formulated, thermodynamics was in its infancy, however it was already clear that the local state of a fluid as seen by a comoving observer is determined by two thermodynamic variables, say pressure and temperature. Of these, only pressure entered the equations of motion, while the equations involve also the density of the fluid. Density was already known to be a function of pressure and temperature for a given type of fluid. However in the absence of an additional equation, the system of equations at the time of Euler, which consisted of the momentum equations together with the equation of continuity, was underdetermined, except in the incompressible limit. The additional equation was supplied by Laplace in 1816 [La] in the form of what was later to be called the adiabatic condition, and allowed him to make the first correct calculation of the speed of sound. The first work on the formation of shocks was done by Riemann in 1858 [Ri]. Riemann considered the case of isentropic flow with plane symmetry, where the equations of fluid mechanics reduce to a system of conservation laws for two unknowns and with two independent variables, a single space coordinate and time. He introduced for such

2

Prologue and Summary

systems the so-called Riemann invariants, and with the help of these showed that solutions which arise from smooth initial conditions develop infinite gradients in finite time. Riemann also realized that the solutions can be continued further as discontinuous solutions, but here there was a problem. Up to this time the energy equation was considered to be simply a consequence of the laws of motion, not a fundamental law in its own right. On the other hand, the adiabatic condition was considered by Riemann to be part of the main framework. Now as long as the solutions remain smooth it does not matter which of the two equations we take to be the fundamental law, for each is a consequence of the other, modulo the remaining laws. However this is no longer the case once discontinuities develop, so one must make a choice as to which of the two equations to regard as fundamental and therefore remains valid thereafter. Here Riemann made the wrong choice. For, only during the previous decade, in 1847, had the first law of thermodynamics been formulated by Helmholtz [He], based in part on the experimental work of Joule on the mechanical equivalence of heat, and the general validity of the energy principle had thereby been shown. In 1865 the concept of entropy was introduced into theoretical physics by Clausius [Cl2], and the adiabatic condition was understood to be the requirement that the entropy of each fluid element remains constant during its evolution. The second law of thermodynamics, involving the increase of entropy in irreversible processes, had first been formulated in 1850 by Clausius [Cl1] without explicit reference to the entropy concept. After these developments the right choice in Riemann’s dilemma became clear. The energy equation must remain at all times a fundamental law, but the entropy of a fluid element must jump upward when the element crosses a hypersurface of discontinuity. The formulation of the correct jump conditions that must be satisfied by the thermodynamic variables and the fluid velocity across a hypersurface of discontinuity was begun by Rankine in 1870 [Ra] and completed by Hugoniot in 1889 [Hu]. With Einstein’s discovery of the special theory of relativity in 1905 [Ei], and its final formulation by Minkowski in 1908 [Mi] through the introduction of the concept of spacetime with its geometry, the domain of geometry being thereby extended to include time, a unity was revealed in physical concepts which had been hidden up to this point. In particular, the concepts of energy density, momentum density or energy flux, and stress, were unified into the concept of the energy-momentum-stress tensor and energy and momentum were likewise unified into a single concept, the energy-momentum vector. Thus, when the Euler equations were extended to become compatible with special relativity, it was obvious from the start that it made no sense to consider the momentum equations without considering also the energy equation, for these two were parts of a single tensorial law, the energy-momentum conservation law. This law together with the particle conservation law (the equation of continuity of the non-relativistic theory), constitute the laws of motion of a perfect fluid in the relativistic theory. The adiabatic condition is then a consequence for smooth solutions. A new basic physical insight on the shock development problem was reached first, it seems, by Landau in 1944 [Ln]. This was the discovery that the condition that the entropy jump be positive as a hypersurface of discontinuity is traversed from the past to the future, should be equivalent to the condition that the flow is evolutionary, that is, that conditions

Prologue and Summary

3

in the past determine the fluid state in the future. More precisely, what was shown by Landau was that the condition of determinism is equivalent, at the linearized level, to the condition that the tangent hyperplane at a point on the hypersurface of discontinuity, is on one hand contained in the exterior of the sound cone at this point corresponding to the state before the discontinuity, while on the other hand intersects the sound cone at the same point corresponding to the state after the discontinuity, and that this latter condition is equivalent to the positivity of the entropy jump. This is interesting from a general philosophical point of view, because it shows that irreversibility can arise, even though the laws are all time-reversible, once the solution ceases to be regular. To a given state at a given time there always corresponds a unique state at any given later time. If the evolution is regular in the associated time interval, then the reverse is also true: to a given state at a later time there corresponds a unique state at any given earlier time, the laws being time-reversible. This reverse statement is however false if there is a shock during the time interval in question. Thus determinism in the presence of hypersurfaces of discontinuity selects a direction of time and the requirement of determinism coincides, modulo the other laws, with what is dictated by the second law of thermodynamics which is in its nature irreversible. This recalls the interpretation of entropy, first discovered by Boltzmann in 1877 [Bo], as a measure of disorder at the microscopic level. An increase of entropy was thus understood to be associated to an increase in disorder or to loss of information, and determinism can only be expected in the time direction in which information is lost, not gained. An important mathematical development with direct application to the equations of fluid mechanics in the physical case of three space dimensions, was the introduction by Friedrichs of the concept of a symmetric hyperbolic system in 1954 [F] and his development of the theory of such systems. It is through this theory that the local existence and domain of dependence property of solutions of the initial value problem associated to the equations of fluid mechanics are established. Another development in connection to this was the general investigation by Friedrichs and Lax in 1971 [F-L] (see also [Lx1]) of nonlinear first order systems of conservation laws which for smooth solutions have as a consequence an additional conservation law. This is the case for the system of conservation laws of fluid mechanics, which consists of the particle and energy-momentum conservation laws, which for smooth solutions imply the conservation law associated to the entropy current. It was then shown that if the additional conserved quantity is a convex function of the original quantities, the original system can be put into symmetric hyperbolic form. Moreover, for discontinuous solutions satisfying the jump conditions implied by the integral form of the original conservation laws, an inequality for the generalized entropy was derived. The problem of shock formation for the equations of fluid mechanics in one space dimension, and more generally for systems of conservation laws in one space dimension, was studied by Lax in 1964 [Lx2], and 1973 [Lx3], and John [J1] in 1974. The approach of these works was analytic, the strategy being to deduce an ordinary differential inequality for a quantity constructed from the first derivatives of the solution, which showed that this quantity must blow up in finite time, under a certain structural assumption on the system called genuine nonlinearity and suitable conditions on the initial data. The gen-

4

Prologue and Summary

uine nonlinearity assumption is in particular satisfied by the non-relativistic compressible Euler equations in one space dimension provided that the pressure is a strictly convex function of the specific volume. A more geometric approach in the case of systems with two unknowns was developed by Majda in 1984 [Ma1] based in part on ideas introduced by Keller and Ting in 1966 [K-T]. In this approach, which is closer in spirit to the present monograph, one considers the evolution of the inverse density of the characteristic curves of each family and shows that under appropriate conditions this inverse density must somewhere vanish within finite time. In this way, not only were the earlier blow-up results reproduced, but, more importantly, insight was gained into the nature of the breakdown. Moreover Majda’s approach also covered the case where the genuine nonlinearity assumption does not hold, but we have linear degeneracy instead. He showed that in this case, global-in-time smooth solutions exist for any smooth initial data. The problem of the global-in-time existence of solutions of the equations of fluid mechanics in one space dimension was treated by Glimm in 1965 [Gl] through an approximation scheme involving at each step the local solution of an initial value problem with piecewise constant initial data. The convergence of the approximation scheme then produced a solution in the class of functions of bounded variation. Now, by the previously established results on shock formation, a class of functions in which global existence holds must necessarily include functions with discontinuities, and the class of functions of bounded variation is the simplest class having this property. Thus, the treatment based on the total variation, the norm in this function space, in itself an admirable investigation, would be insuperable if the development of the one-dimensional theory was the goal of scientific effort in the field of fluid mechanics. However that goal can only be the mathematical description of phenomena in real three-dimensional space and one must ultimately face the fact that methods based on the total variation do not generalize to more than one space dimension. In fact it is clear from the study of the linearized theory, acoustics, which involves the wave equation, that in more than one space dimension only methods based on the energy concept are appropriate. The first and thus far the only general result on the formation of shocks in threedimensional fluids was obtained by Sideris in 1985 [S]. Sideris considered the compressible Euler equations in the case of a classical ideal gas with adiabatic index γ > 1 and with initial data which coincide with those of a constant state outside a ball. The assumptions of his theorem on the initial data were that there is an annular region bounded by the sphere outside which the constant state holds, and a concentric sphere in its interior, such that a certain integral in this annular region of ρ − ρ0 , the departure of the density ρ from its value ρ0 in the constant state, is positive, while another integral in the same region of ρvr , the radial momentum density, is non-negative. These integrals involve kernels which are functions of the distance from the center. It is also assumed that everywhere in the annular region the specific entropy s is not less than its value s0 in the constant state. The conclusion of the theorem is that the maximal time interval of existence of a smooth solution is finite. The chief drawback of this theorem is that it tells us nothing about the nature of the breakdown. Also the method relies on the strict convexity of the pressure as

Prologue and Summary

5

a function of the density displayed by the equation of state of an ideal gas, and does not extend to more general equations of state. The other important work on shocks in three space dimensions was the 1983 work of Majda [Ma2], [Ma3], on what he calls the shock front problem. In this problem we are given initial data in 3 which is smooth in the closure of each component of 3 \ S, where S is a smooth surface in 3 . The data is to satisfy the condition that there exists a function σ on S such that the jumps of the data across S satisfy the Rankine–Hugoniot jump conditions as well as the entropy condition with σ in the role of the shock speed. The higher order compatibility conditions associated to an initial boundary value problem are also required to be satisfied. We are then required to find a time interval [0, τ ], a smooth hypersurface K in the spacetime slab [0, τ ] × 3 and a solution of the compressible Euler equations which is smooth in the closure of each component of [0, τ ] × 3 \ K , and satisfies across K the Rankine–Hugoniot jump conditions as well as the entropy condition. We may think of this problem as the local-in-time shock continuation problem. Majda solved this problem under an additional condition on the initial data which seems to be necessary for the stability of the linearized problem. The additional condition follows from the other conditions in the case of a classical ideal gas, but it does not follow for a general equation of state. The present monograph considers the relativistic Euler equations in three space dimensions for a perfect fluid with an arbitrary equation of state. We consider regular initial data on a space-like hyperplane 0 in Minkowski spacetime which outside a sphere coincide with the data corresponding to a constant state. We consider the restriction of the initial data to the exterior of a concentric sphere in 0 and we consider the maximal classical development of this data. Then, under a suitable restriction on the size of the departure of the initial data from those of the constant state, we prove certain theorems which give a complete description of the maximal classical development, which we call maximal solution. In particular, the theorems give a detailed description of the geometry of the boundary of the domain of the maximal solution and a detailed analysis of the behavior of the solution at this boundary. A complete picture of shock formation in threedimensional fluids is thereby obtained. Also, sharp sufficient conditions on the initial data for the formation of a shock in the evolution are established and sharp lower and upper bounds for the temporal extent of the domain of the maximal solution are derived. The reason why we consider only the maximal development of the restriction of the initial data to the exterior of a sphere is in order to avoid having to treat the long time evolution of the portion of the fluid which is initially contained in the interior of this sphere. For, we have no method at present to control the long time behavior of the pointwise magnitude of the vorticity of a fluid portion, the vorticity satisfying a transport equation along the fluid flow lines. Our approach to the general problem is the following. We show that given arbitrary regular initial data which coincide with the data of a constant state outside a sphere, if the size of the initial departure from the constant state is suitably small, we can control the solution for a time interval of order 1/η0 , where η0 is the sound speed in the surrounding constant state. We then show that at the end of this interval a thick annular region has formed, bounded by concentric spheres, where the flow is irrotational and isentropic, the constant state holding outside the outer sphere. We then

6

Prologue and Summary

study the maximal classical development of the restriction of the data at this time to the exterior of the inner sphere. In the irrotational isentropic case there is a function φ which we call a wave function, the differential of which at a point determines the state of the fluid at that point, and the fluid equations reduce to a nonlinear wave equation for φ, as is shown in Chapter 1. The order of presentation in this monograph is however the reverse of that just outlined. After the first four chapters which set up the general framework, we confine attention to the irrotational isentropic problem up to Chapter 13, where the main theorem, Theorem 13.1, is proved. We return to the general problem in Chapter 14, after establishing a theorem, Theorem 14.1, which, in the irrotational isentropic context, gives sharp sufficient conditions on the initial data for the formation of a shock in the evolution. It is at this point where our treatment of the general problem resumes, and we analyze the solution of the general problem during the initial time interval. In fact, our analysis allows us to find which conditions on the data at the beginning of the time interval result in data at the end of the time interval verifying the assumptions of Theorem 14.1. In this way we are able to establish a theorem, Theorem 14.2, which, in the general context of fluid mechanics, gives sharp sufficient conditions on the initial data for the formation of a shock in the evolution. We should emphasize at this point that if we were to restrict ourselves from the beginning to the irrotational isentropic case, we would have no problem extending the treatment to the interior region, thereby treating the maximal solution corresponding to the data on the complete initial hyperplane 0 . In fact, it is well known that sound waves decay in time faster in the interior region and our constructions can readily be extended to cover this region. It is only our present inability to achieve long time control of the magnitude of the vorticity along the flow lines of the fluid, that prevents us from treating the interior region in the general case. The geometry of the boundary of the domain of the maximal solution is studied in Chapter 15, the main results being expressed by Theorem 15.1 and Propositions 15.1, 15.2, and 15.3. The boundary consists of a regular part and a singular part. Each component of the regular part C is an incoming characteristic hypersurface with a singular past boundary. The singular part of the boundary of the domain of the maximal solution is the
locus of points where the inverse density of the wave fronts vanishes. It is the union ∂− H H , where each component of ∂− H is a smooth embedded surface in Minkowski spacetime, the tangent plane to which at each point is contained in the exterior of the sound cone at that point. On the other hand each component of H is a smooth embedded hypersurface in Minkowski spacetime, the tangent hyperplane to which at each point is contained in the exterior of the sound cone at that point, with the exception of a single generator of the sound cone, which lies on the hyperplane itself. The past boundary of a component of H is the corresponding component of ∂− H . The latter is at the same time the past boundary of a component of C. As is explained in the Epilogue, the maximal classical solution is
the physical solution of the problem up to C ∂− H , but not up to H . The problem of the physical continuation of the solution is set up in the Epilogue as the shock development problem. This problem is associated to each component of ∂− H and its solution requires the construction of a hypersurface of discontinuity K , lying in the past of the cor-

Prologue and Summary

7

responding component of H , but having the same past boundary as the latter, namely the given component of ∂− H . Thus, although the notion of maximal classical solution is not physically appropriate up to H , it does provide the basis for constructing the physical continuation, the solution of the
shock development problem, by providing not only the right boundary conditions at C ∂− H , but also a barrier at H which is indispensable for controlling the physical continuation. The actual treatment of the shock development problem and the subsequent shock interactions shall be the subject of a subsequent monograph. The present monograph concludes with a derivation of a formula for the jump in vorticity across K , which shows that while the flow is irrotational ahead of the shock, it acquires vorticity immediately behind, which is tangential to the shock front and is associated to the gradient along the shock front of the entropy jump. We have chosen to work in this monograph with the relativistic Euler equations rather than confining ourselves to their non-relativistic limit, for three reasons. The first is the obvious reason that there is a class of natural phenomena, those of relativistic astrophysics, which lie beyond the domain of the non-relativistic equations. The second reason is that there is a substantial gain in geometric insight in considering the relativistic equations. At a fundamental level, the picture looks simpler from the relativistic perspective, because of the aforementioned unity of physical concepts brought about by the spacetime geometry viewpoint of special relativity. As an example we give the equation (1.51) of Chapter 1: (1) i u ω = −θ ds. Here ω is the vorticity 2-form. According to the definitions of Chapter 1, ω = dβ where β is the 1-form defined, relative to an arbitrary system of coordinates, by: √ βµ = − σ u µ , u µ = gµν u ν ,

(2)

(3)

√ σ being the relativistic enthalpy per particle, u µ the fluid velocity and gµν the Minkowski metric. In (1), θ is the temperature and s the entropy per particle, while i u denotes contraction on the left by the vectorfield u. Equation (1) is arguably the simplest explicit form of the energy-momentum equations. Our derivation in the Epilogue of the jump in vorticity behind a shock relies on this equation. The 1-form β plays a fundamental role in this monograph. In the irrotational isentropic case it is given by β = dφ, where φ is the wave function. The third reason why we have chosen to work with the relativistic equations is that no special care is needed to extract information on the non-relativistic limit. This is due to the fact that the non-relativistic limit is a regular limit, obtained by letting the speed of light in conventional units tend to infinity, while keeping the sound speed fixed. To allow the results in the non-relativistic limit to be extracted from our treatment in a straightforward manner, we have chosen to avoid summing quantities having different physical dimensions when such sums would make sense only when a unit of velocity has been chosen, even though we follow the natural choice within the framework of special

8

Prologue and Summary

relativity of setting speed of light equal to unity in writing down the relativistic equations of motion. We shall presently give an example to illustrate what we mean. Consider the vectorfield K 0 defined by equation (5.15) of Chapter 5: K 0 = (η0−1 + α −1 κ)L + L,

L = α −1 κ L + 2T.

(4)

The terms here have not yet been defined, but the reader may return to this example after assimilating the appropriate definitions. In any case, η0 is, as mentioned above, the sound speed in the surrounding constant state. The function α is the inverse density of the hyperplanes t corresponding to the constant values of the time coordinate t, with respect to the acoustical metric h µν : h µν = gµν + (1 − η2 )u µ u ν ,

u µ = gµν u ν ,

(5)

gµν being again the Minkowski metric, η the sound speed, and u µ the fluid velocity. This is a Lorentzian metric on spacetime, the null cones of which are the sound cones. The function α has the physical dimension of velocity. The function κ is the inverse spatial density of the wave fronts with respect to the acoustical metric, a dimensionless quantity. Thus in the sum η0−1 +α −1 κ, which is the coefficient of L in the first term of (4), each term has the physical dimension of inverse velocity. The vectorfield L is the tangent vectorfield of the bicharacteristic generators, parametrized by t, of a family of outgoing characteristic sets of an acoustical function u. The wave fronts St,u are the hypersurfaces Cu , the level  surfaces of intersection Cu t . The physical dimension of L is inverse time. Thus the first term in (4) has the physical dimension of inverse length. The vectorfield T defines a flow on each of the t , taking each wave front onto another wave front, the normal, relative to the induced acoustical metric h, the flow of the foliation of t by the surfaces St,u . It has the physical dimension of an inverse length. The first term in the second part of (4) also has the same physical dimension, hence the physical dimension of the vectorfield L is inverse length as well. We conclude that each term in (4) has the physical dimension of inverse length, thus the physical dimension of K 0 is inverse length. Denoting, as above, by σ the square of the relativistic enthalpy per particle, we have: √ σ = e + pv (6) where e is the relativistic energy per particle, p the pressure and v is the volume per particle. Let H be the function defined by: 1 − η2 = σ H.

(7)

The derivative of H with respect to σ at constant s plays a central role in shock theory . This quantity is expressed by (see equation (E.47) in the Epilogue):       dH d 2v 3v dv (8) = −a +√ dσ s d p2 σ dp s s

where a is the positive function: a=

η4 . 2σ v 3

Prologue and Summary

9

The sign of (d H /dσ )s in the state ahead of a shock determines, as is shown in the Epilogue, the sign of the jump in pressure in crossing the shock to the state behind. The jump in pressure is positive if this quantity is negative, the reverse otherwise. The value of (d H /dσ )s in the surrounding constant state is denoted by  in this monograph. This constant determines the character of the shocks for small initial departures from the constant state. In particular when  = 0, no shocks form and the domain of the maximal classical solution is complete. Consider the function (d H /dσ )s as a function of the thermodynamic variables p and s. Suppose that we have an equation of state such that at some value s0 of s we have (d H /dσ )s=s0 = 0, that is, the function (d H /dσ )s vanishes everywhere along the adiabat s = s0 . We show in Chapter 1 that in this case the irrotational isentropic fluid equations corresponding to the value s0 of the entropy are equivalent to the minimal surface equation, the wave function φ defining a minimal graph in a Minkowski spacetime of one more spatial dimension. Thus the minimal surface equation defines a dividing line between two different types of shock behavior. Now, the relativistic enthalpy is dominated by the term mc2 , the contribution of the particle rest mass m, to the energy per particle, c being the speed of light. We note here that the particle rest mass may be taken to be unity, so that all quantities per particle are quantities per unit rest mass. Thus in the non-relativistic limit the second term in parenthesis in (8) vanishes and the expression in parenthesis reduces simply to: (d 2 v/d p2 )s . Now, the case where (d 2 v/d p2 )s > 0, the adiabats being convex curves in the p, v plane, so that (d H /dσ )s < 0, is the more commonly found in nature, however the reverse case does occur in the gaseous region near the critical point in the liquid-to-vapor phase transition and in similar transitions at higher temperatures associated to molecular dissociation and to ionization (see [Z-R]). One of the basic concepts on which our approach relies is the general concept of variation, or variation through solutions, on which our treatment not only of the irrotational isentropic case but also of the general equations of motion is based. This concept has been discussed in the general context of Euler–Lagrange equations, that is, systems of partial differential equations arising from an action principle, in the monograph [Ch]. It was shown there that to a variation is associated a linearized Lagrangian, and it was also shown how energy currents are in general constructed on the basis of this linearized Lagrangian. It is through energy currents and their associated integral identities that the estimates essential to our approach are derived. Here the first order variations correspond to the one-parameter subgroups of the Poincar´e group, the isometry group of Minkowski spacetime, extended by the one-parameter scaling or dilation group, which leave the surrounding constant state invariant. The higher order variations correspond to the oneparameter groups of diffeomorphisms generated by a set of vectorfields, the commutation fields, to be discussed below. The construction in [Ch] of an energy current requires a multiplier vectorfield which at each point belongs to the closure of the positive component of the inner characteristic core in the tangent space at that point. In the irrotational isentropic case the characteristic in the tangent space at a point consists only of the sound cone at that point and this requirement becomes the requirement that the multiplier vectorfield be non-space-like and future-directed with respect to the acoustical metric (5). We use two multiplier vectorfields in our analysis of the isentropic irrotational problem.

10

Prologue and Summary

The first is the vectorfield K 0 defined by (4) and the second is the vectorfield K 1 defined by equation (5.16) of Chapter 5: K 1 = (ω/ν)L.

(9)

Here ν is the mean curvature of the wave fronts St,u , sections of the outgoing characteristic hypersurfaces Cu , relative to their characteristic normal L, the tangent vectorfield to the bi-characteristic generators of Cu , parametrized by t. However ν is defined not relative to the acoustical metric h µν but rather relative to a conformally related metric h˜ µν : h˜ µν = h µν .

(10)

It turns out that there is a choice of conformal factor  such that in the isentropic irrotational case a first order variation φ˙ of the wave function φ satisfies the wave equation relative to the metric h˜ µν . This is shown in Chapter 1 and this choice defines  in the remainder of the monograph. The definition makes  the ratio of a function of σ to the value of this function in the surrounding constant state, thus  is equal to unity in the constant state. It turns out moreover that  is bounded above and below by positive constants. The function ω appearing in (9) is required to satisfy certain conditions (conditions D1–D5 of Chapter 5) and it is shown in Proposition 13.4 that the function ω = 2η0 (1 + t) does satisfy these requirements. A similar analysis to the one done above in the case of the multiplier field K 0 shows, taking into account the fact that the physical dimension of ν is inverse time, that the multiplier field K 1 has the physical dimension of length. The vectorfield K 1 corresponds to the generator of inverted time translations, which are proper conformal transformations of the Minkowski spacetime with its Minkowskian metric gµν . The latter was first used by Morawetz [Mo] to study the decay of solutions of the initial boundary value problem for the classical wave equation outside an obstacle. The vectorfield K 1 is an analogue of the multiplier field of Morawetz for the acoustical spacetime which is the same underlying manifold but equipped with the acoustical metric h µν . The energy currents associated to K 0 and K 1 are defined by equations (5.18) and (5.19) of Chapter 5, respectively. The energy current associated to K 1 contains certain additional lower order terms, defined through the function ω. Analogous terms were present in the work of Morawetz. The general structure of these terms has been investigated, in the general context of Euler–Lagrange equations, in [Ch]. To each variation ψ, of any order, there are energy currents associated to ψ and to K 0 and K 1 respectively. These currents define the energies E u0 [ψ](t), E1u [ψ](t), and fluxes F0t [ψ](u), F1t [ψ](u). For given t and u the energies are integrals over the exterior of the surface St,u in the hyperplane t , while the fluxes are integrals over the part of the outgoing characteristic hypersurface Cu between the hyperplanes 0 and t . To obtain the energy and flux associated to K 1 , certain integrations by parts are performed. This construction is presented in Chapter 5. The precise choice of the factor ω/ν in (9) is dictated by the need to eliminate certain error integrals which would otherwise be present. It is these energy and flux integrals, together with a spacetime integral K [ψ](t, u) associated to K 1 , to be discussed below, which are used to control the solution. It is evident from the above that the means by which the solution is controlled depend on the choice of the acoustical function u, the level sets of which are the outgoing

Prologue and Summary

11

characteristic hypersurfaces Cu . The function u is determined by its restriction to the initial hyperplane 0 . The divergence of the energy currents, which determines the growth of the energies and fluxes, itself depends on (K 0 ) π, ˜ in the case of the energy current associated to K 0 , and (K 1 ) π, ˜ in the case of the energy current associated to K 1 . Here for any vectorfield X in spacetime, we denote by (X ) π˜ the Lie derivative of the conformal acoustical metric h˜ with respect to X. We may call (X ) π˜ the deformation tensor corresponding to X. In the case of higher order variations, the divergences of the energy currents depend ˜ for each of the commutation fields Y to be discussed below. also on the (Y ) π, All these deformation tensors ultimately depend on the acoustical function u, or, which is the same, on the geometry of the foliation of spacetime by the outgoing characteristic hypersurfaces Cu , the level sets of u. The most important geometric property of this foliation from the point of view of the study of shock formation is the density of the packing of its leaves Cu . One measure of this density is the inverse spatial density of the wave fronts, that is, the inverse density of the foliation of each spatial hyperplane t by the surfaces St,u . This is the function κ which appears in (4) and is given in arbitrary coordinates on t by: −1 κ −2 = (h )i j ∂i u∂ j u (11) where h i j is the induced acoustical metric on t . Another measure is the inverse temporal density of the wave fronts, the function µ given in arbitrary coordinates in spacetime by: 1 = −(h −1 )µν ∂µ t∂ν u. µ

(12)

The two measures are related by: µ = ακ

(13)

where α is the inverse density, with respect to the acoustical metric, of the foliation of spacetime by the hyperplanes t . The function α also appears in (4) and is given in arbitrary coordinates in spacetime by: α −2 = −(h −1 )µν ∂µ t∂ν t.

(14)

It is expressed directly in terms of the 1-form β in the general case, or dφ in the irrotational isentropic case. It turns out moreover, that it is bounded above and below by positive constants. Consequently µ and κ are equivalent measures of the density of the packing of the leaves of the foliation of spacetime by the Cu . Shock formation is characterized by the blowup of this density or equivalently by the vanishing of κ or µ. The above and the basic geometric construction are discussed in Chapter 2. The other entity, besides κ or µ which describes the geometry of the foliation by the Cu , is the second fundamental form of the Cu . Since the Cu are null hypersurfaces with respect to the acoustical metric h, their tangent hyperplane at a point is the set of all vectors at that point which are h-orthogonal to the generator L, and L itself belongs to the tangent hyperplane, being h-orthogonal to itself. Thus the second fundamental form χ of Cu is intrinsic to Cu and in terms of the metric h/ induced by the acoustical metric on the St,u sections of Cu , it is given by: L / L h/ = 2χ

(15)

12

Prologue and Summary

where L / X ϑ for a covariant St,u tensorfield ϑ denotes the restriction of L X ϑ to T St,u . The acoustical structure equations such as the propagation equation for χ along the generators / χ, the divergence of χ intrinsic to St,u , in of Cu , the Codazzi equation which expresses div terms of d/trχ, the differential on St,u of trχ, and a component of the acoustical curvature and of k, the second fundamental form of the t relative to h, are presented in Chapter 3. Also included in the acoustical structure equations is the Gauss equation which expresses /) in terms of χ and a component of the acoustical curvature the Gauss curvature of (St,u , h and of k, and an equation which expresses L /T χ in terms of the Hessian of the restriction of µ to St,u and another component of the acoustical curvature and of k. In the same chapter the components of k are analyzed. These acoustical structure equations contain terms which blow up as κ or µ tend to zero. The regular form of these equations is given in the next chapter, Chapter 4, the subject of which is the analysis of the acoustical curvature. It is there shown that the terms which blow up as κ or µ tend to zero cancel. The most important acoustical structure equation from the point of view of the formation of shocks is the propagation equation for µ along the generators of Cu , equation (3.96): Lµ = m + µe. (16) An equivalent equation, (3.99), is satisfied by κ. This equation is derived in Chapter 3 only in the irrotational isentropic case, in contrast to the other structure equations which hold in general. The function m is in this case given by: m=

dH 1 (Lφ)2 (T σ ) 2 dσ

(17)

while the function e, given by (3.98), depends only on the derivatives along L of the ψα , α = 0, 1, 2, 3, the first variations corresponding to the spacetime translations. A similar propagation equation holds in the general case with the function m given by:   1 2 dH m = (β L ) (T σ ) (18) 2 dσ s and the function e depending only on the derivatives of the βα , the rectangular components of β, tangential to the Cu . We shall presently indicate why this has to be the case without performing the calculations, which in any case parallel those of Chapter 3, with the general formulas (3.4) and (3.10) for the vectorfield V i and the metric h i j induced on the t , in place of the corresponding formulas (3.6) and (3.11) in the irrotational isentropic case. The fluid velocity being transversal to Cu , by virtue of the adiabatic condition u · ds = 0, any derivative of s can be expressed as a derivative tangential to Cu . On the other hand, according to (1) for any vector X we have: ω(u, X) = −θ X · ds. It follows that the differential of the entropy and the vorticity do not contribute to the function m, hence an equation of the form (16) results with m given by (18). It is the function m which determines shock formation, when being negative, causing µ to decrease to zero.

Prologue and Summary

13

The path we have followed in attacking the problem of shock formation in 3-dimensional fluids illustrates the following approach in regard to quasilinear hyperbolic systems of partial differential equations. That is, the quantities which are used to control the solution must be defined using the causal, or characteristic, structure of spacetime determined by the solution itself, not an artificial background structure. The original system of equations must then be considered in conjunction with the system of equations which this structure obeys, and it is only through the study of the interaction of the two systems that results are obtained. The work [C-K] on the stability of the Minkowski space in the framework of general relativity was the first illustration of this approach. In the present case however, the structure, which is here the acoustical structure, degenerates as shocks begin to form, and the precise way in which this degeneracy occurs must be guessed beforehand and established in the course of the argument of the mathematical proof. The fact that the underlying structure degenerates implies that our estimates are no longer even locally equivalent to standard energy estimates, which would of necessity have to fail when shocks appear. Chapter 5 establishes the fundamental energy estimate, Theorem 5.1. This applies to a solution of the homogeneous wave equation in the acoustical spacetime, in particular to any first order variation. Now the higher order variations satisfy inhomogeneous wave equations in the acoustical spacetime, the source functions depending on the deformation tensors of the commutation fields. These source terms give rise to error integrals, that is to spacetime integrals of contributions to the divergence of the energy currents, which are written down but not estimated in Chapter 5. The remaining error integrals however, are all estimated in the proof of Theorem 5.1, and since these estimates apply to any variation of any order, Chapter 5 contributes in an essential way to the main theorem of Chapter 13. The proof of Theorem 5.1 relies on certain bootstrap assumptions on the acoustical entities. The most crucial of these assumptions concern the behavior of the function µ. These are the assumptions C1, C2, and C3, which are established in the first part of Chapter 13, by Propositions 13.1, 13.2, and 13.3, respectively, on the basis of the final set of bootstrap assumptions, which consists only of pointwise estimates for the variations up to certain order. To give an idea at this point of the nature of these assumptions, the assumption C2 required in Chapter 5 to obtain the fundamental energy estimate up to time s is (modulo assumption C1): µ−1 (T µ)+ ≤ Bs (t) : for all t ∈ [0, s] where Bs (t) is a function such that: s (1 + t)−2 [1 + log(1 + t)]4 Bs (t)dt ≤ C

(19)

(20)

0

with C a constant independent of s. Here T is the vectorfield defined above and we denote by f+ and f − , respectively the positive and negative parts of an arbitrary function f . This assumption is then established by Proposition 13.2 with Bs (t) the following function:

(1 + τ ) Bs (t) = C δ0 √ + Cδ0 (1 + τ ) σ −τ

(21)

14

Prologue and Summary

where τ = log(1 + t), σ = log(1 + s), and δ0 is a small positive constant appearing in the final set of bootstrap assumptions. Now, the spacetime integral K [ψ](t, u) mentioned above, is defined by (5.169). It is essentially the integral of 1 − (ω/ν)(Lµ)− |d/ψ|2 2 in the spacetime exterior to Cu and bounded by 0 and t . Assumption C3 on the other hand states that there is a positive constant C independent of s such that in the region below s where µ < η0 /4 we have: Lµ ≤ −C −1 (1 + t)−1 [1 + log(1 + t)]−1

(22)

In view of this assumption, the integral K [ψ](t, u) gives effective control of the derivatives of the variations tangential to the wave fronts in the region where shocks are to form. The same assumption, which is established by Proposition 13.3, also plays an essential role in the study of the singular boundary in Chapter 15. The final stage of the proof of Theorem 5.1 is the analysis of a system of integral inequalities in two variables t and u satisfied by the five quantities E0u [ψ](t), E1u [ψ](t), F0t [ψ](u), F1t [ψ](u), and K [ψ](t, u). This analysis is reflected at analogous stages in the course of the proof of Theorem 13.1. The commutation fields Y are defined in Chapter 6. They are five: the vectorfield T which is transversal to the Cu , the field Q = (1 + t)L along the generators of the Cu and the three rotation fields Ri : i = 1, 2, 3 which are tangential to the St,u sections. ◦

◦

The latter are defined to be  Ri : i = 1, 2, 3, where the Ri i = 1, 2, 3 are the generators of spatial rotations associated to the background Minkowskian structure, while  is the h-orthogonal projection to the St,u . Note that the commutation field T has the physical dimension of inverse length while the other commutation fields are dimension˜ (Q) π, ˜ and (Ri ) π˜ : i = 1, 2, 3 are less. Expressions for the deformation tensors (T ) π, then derived, which show that these depend on the acoustical entities µ and χ. The last however depend in addition on the derivatives of the restrictions to the surfaces St,u of the spatial rectangular coordinates x i : i = 1, 2, 3, as well as on the derivatives of the x i with respect to T and L, that is, on the rectangular components T i and L i of the vectorfields T and L (note that L 0 = 1, T 0 = 0). The estimates of these and their derivatives with respect to the commutation fields in terms of the acoustical entities occupies a major part of Chapters 10 and 11 as shall be discussed below. In Chapter 7 a recursion formula is obtained for the source functions associated to the higher order variations, on the basis of which an explicit formula for these source functions is obtained in Chapter 13. Then the error integrals arising from the contributions to the source functions containing the top and next to the top order derivatives of the variations are estimated. Chapters 8 and 9 are crucial for the entire work because it is here that the estimates for the top order derivatives of the acoustical entities are derived. The expressions for the source functions and the associated error integrals from Chapter 7 show that the error integrals corresponding to the energies of the n +1st order variations contain the nth order derivatives of the deformation tensors, which in turn contain the nth order derivatives of

Prologue and Summary

15

χ and n + 1st order derivatives of µ. Thus to achieve closure, we must obtain estimates for the latter in terms of the energies of up to the n + 1st order variations. Now, the propagation equations of Chapter 3 for χ and µ, appearing in regular form in Chapter 4, give appropriate expressions for L / L χ and Lµ. However, if these propagation equations, which may be thought of as ordinary differential equations along the generators of the Cu , are integrated with respect to t to obtain the acoustical entities χ and µ themselves, and their spatial derivatives are then taken, a loss of one degree of differentiability would result and closure would fail. We overcome this difficulty in the case of χ in Chapter 8 by considering the propagation equation for µtrχ. We show that, by virtue of a wave equation for σ , which follows from the wave equations satisfied by the first variations corresponding to the spacetime translations, the principal part on the right-hand side of this propagation equation can be put into the form −L fˇ of a derivative of a function − fˇ with respect to L. This function is then brought to the left-hand side and we obtain a propagation equation for µtrχ + fˇ. In this equation χ, ˆ the trace-free part of χ enters, but the propagation equation in question is considered in conjunction with the Codazzi equation, which constitutes an elliptic system ˆ given trχ. We thus have an ordinary differential equation along the on each St,u for χ, generators of Cu coupled to an elliptic system on the St,u sections. More precisely, the propagation equation which is considered at the same level as the Codazzi equation is a propagation equation for the St,u 1-form µd/trχ + d/ fˇ, which is a consequence of the equation just discussed. To obtain estimates for the angular derivatives of χ of order l we similarly consider a propagation equation for the St,u 1-form: (i1 ...il )

xl = µd/(Ril . . . Ri1 trχ) + d/(Ril . . . Ri1 fˇ)

In the case of µ the aforementioned difficulty is overcome in Chapter 9 by considering the propagation equation for µ
/ µ, where
/ µ is the Laplacian of the restriction of µ to the St,u . We show that by virtue of a wave equation for T σ , which is a differential consequence of the wave equation for σ , the principal part on the right-hand side of this propagation equation can again be put into the form L fˇ of a derivative of a function fˇ with respect to L. This function is then likewise brought to the left-hand side and we /ˆ2 µ, the trace-free part of obtain a propagation equation for µ
/ µ − fˇ . In this equation D the Hessian of the restriction of µ to the St,u enters, but the propagation equation in question is considered in conjunction with the elliptic equation on each St,u for µ, which the specification of
/ µ constitutes. Again we have an ordinary differential equation along the generators of Cu coupled to an elliptic equation on the St,u sections. To obtain estimates of the spatial derivatives of µ of order m + l + 2 of which m are derivatives with respect to T , we similarly consider a propagation equation for the function: (i1 ...il )  x m,l

= µRil . . . Ri1 (T )m
/ µ − Ril . . . Ri1 (T )m fˇ .

This allows us to obtain estimates for the top order spatial derivatives of µ of which at least two are angular derivatives. A remarkable fact, which is shown in Chapter 13, is that the missing top order spatial derivatives do not enter the source functions, hence do not contribute to the error integrals. In fact it is shown that the only top order spatial

16

Prologue and Summary

derivatives of the acoustical entities entering the source functions are those in the 1-forms (i1 ...il ) x and the functions (i1 ...il ) x  . l m,l The paradigm of an ordinary differential equation along the generators of a characteristic hypersurface coupled to an elliptic system on the sections of the hypersurface as the means to control the regularity of the entities describing the geometry of the characteristic hypersurface and the stacking of such hypersurfaces in a foliation, was first encountered in [C-K]. It is interesting to note that this paradigm does not appear in space dimension less than three. In the case of the work on the stability of the Minkowski space however, in contrast to the present case, the gain of regularity achieved in this treatment is not essential for obtaining closure, because there is room for one degree of differentiability. This is because of the fact that the Einstein equations arise from a Lagrangian which is quadratic in the canonical velocities, that is, in the derivatives of the unknown functions, in contrast to the equations of fluid mechanics, or more generally of continuum mechanics, which in the Lagrangian picture are equations for a mapping of spacetime into the material manifold, the Lagrangian not depending quadratically on the differential of this mapping (see [Ch]). As a consequence, the metric determining the causal structure depends in continuum mechanics on the derivatives of the unknowns, rather than only on the unknowns themselves. In the present work, the appearance of the factor of µ, which vanishes where shocks / µ in the definitions of (i1 ...il ) xl originate, in front of d/ Ril . . . Ri1 trχ and Ril . . . Ri1 (T )m
 (i ...i ) l 1 and x m,l above, makes the analysis far more delicate. This is compounded with the difficulty of the slower decay in time which the addition of the terms −d/ Ril . . . Ri1 fˇ and Ril . . . Ri1 (T )m fˇ forces. The analysis requires a precise description of the behavior of µ itself, given by Proposition 8.6, and a separate treatment of the condensation regions, where shocks are to form, from the rarefaction regions, the terms referring not to the fluid density but rather to the density of the stacking of the wave fronts. To overcome the difficulties the following weight function is introduced:  µm,u (t) µm,u (t) = min , 1 , µm,u (t) = min µ (23) η0 tu where tu is the exterior of St,u in t , and, in Chapter 13, the quantities E0u [ψ](t), E1u [ψ](t), F0t [ψ](u), F1t [ψ](u), and K [ψ](t, u) corresponding to the highest order variations are weighted with a power, 2a, of this weight function. Lemma 8.11, then plays a crucial role in Chapters 8 and 9 as well as in Chapter 13 where everything comes together. We present this lemma here in an imprecise manner to indicate what is involved. Let:

 −1 −µ , Mu (t) = max (Lµ) − u t

Ia,u = 0

t

   µ−a m,u (t )Mu (t )dt .

(24)

Then under certain bootstrap assumptions in the past of s , for any constant a ≥ 2, there is a positive constant C independent of s, u and a such that for all t ∈ [0, s] we have: Ia,u (t) ≤ Ca −1 µ−a m,u (t).

(25)

Prologue and Summary

17

As mentioned in the discussion of Chapter 6 above, estimates for the derivatives of the spatial rectangular coordinates x i with respect to the commutation fields must also be obtained, the derivative of the x i with respect to the vectorfields Tˆ and L being the spatial rectangular components Tˆ i and L i of these vectorfields. Here Tˆ = κ −1 T is the vectorfield of unit magnitude with respect to h corresponding to T . Thus, although the argument depends mainly on the causal structure of the acoustical spacetime, the underlying Minkowskian structure, to which the rectangular coordinates belong, has a role to play as well, and it is the estimates in question which analyze the mutual relationship of the two structures. The major part of Chapters 10 and 11 is the derivation of estimates for the spatial derivatives of the first derivatives of the x i , in terms of the acoustical entities. In particular, Propositions 10.1 and 10.2 give estimates for the angular derivatives of the x i and of the Tˆ i = Tˆ x i , that is, their derivatives with respect to the rotation fields R j , while Propositions 11.1 and 11.2 give estimates for the spatial derivatives of the Tˆ i , that is the derivatives with respect to T and the R j , of which at least one is a T -derivative. The corollaries of these propositions provide the remaining estimates, including the required estimates for the deformation tensors of the commutation fields in terms of the acoustical entities. In particular, Corollaries 10.1.e and 10.2.e give, through Lemma 10.6, estimates for the iterated commutators of the set of rotation fields, Corollaries 10.1.i and 10.2.i give estimates for the angular derivatives of the commutators [L, Ri ] = (Ri ) Z (see Lemma 8.2), while Lemma 10.24 gives estimates for the angular derivatives of the commutators [T, Ri ] = (Ri )  (see Lemma 10.22). On the other hand, Corollaries 11.1.c and 11.2.c give estimates for the spatial derivatives of the commutators [L, T ] = . The remainder of Chapters 10 and 11 deduce the required estimates for the quantities appearing in the final estimates of Chapters 8 and 9 for the 1-forms (i1 ...il ) xl and the functions (i1 ...il−m ) x  m,l−m , respectively. Chapter 12 contains the recovery of the acoustical bootstrap assumptions used in the previous chapters, in particular in Chapters 10 and 11. That is, these acoustical assumptions are established, using the method of continuity, on the basis of the final set of bootstrap assumptions, which consists only of pointwise estimates for the variations up to certain order. In the same chapter the estimates for up to the next to the top order angular derivatives of χ and spatial derivatives of µ are derived. These, when substituted in the estimates of Chapters 10 and 11, give control of all quantities involved in terms of estimates for the variations. A fundamental role in Chapter 12 is played by Propositions 12.2, 12.4, 12.5, and 12.7 which establish the coercivity hypotheses H0, H1, H2 and propositions roughly speaking show H2 on which the previous chapters depend. These / Ri ϑ|2 bounds pointwise |D /ϑ|2 . that for any covariant St,u tensorfield ϑ, the sum i |L Proposition 12.8, which shows that if X is any St,u -tangential vectorfield and ϑ any co/ X ϑ in terms of the L / Ri ϑ and the variant St,u tensorfield then we can bound pointwise L L / Ri X = [Ri , X], also plays an important role. Chapter 13 begins by establishing the basic assumptions C1, C2, and C3, on the behavior of the function µ on which the energy estimates rely. We then formulate the final bootstrap assumption, to which all other assumptions have been reduced, and which consists only of pointwise estimates for the variations up to certain order. After that we deduce an explicit formula for the source functions, using the recursion formula derived

18

Prologue and Summary

in Chapter 7, and analyze the structure of the terms containing the top order spatial derivatives of the acoustical entities, showing that these can be expressed in terms of the 1-forms (i1 ...il ) x and the functions (i1 ...il−m ) x  l m,l−m . These terms are shown to contribute borderline error integrals, the treatment of which is the main source of difficulties in the problem. These borderline integrals are all proportional to the constant  mentioned above, the value of (8) in the surrounding constant state, hence are absent in the case  = 0. We should make clear here that the only variations which are considered up to this point are the variations arising from the first order variations corresponding to the group of spacetime translations. In particular the final bootstrap assumption involves only variu (t), F t u t ations of this type, and each of the five quantities E0,[n] 0,[n] (u), E1,[n] (t), F1,[n] (u), and K [n] (t, u), which together control the solution, is defined to be the sum of the corresponding quantity E0u [ψ](t), F0t [ψ](u), E1u [ψ](t), F1t [ψ](u), and K [ψ](t, u), over all variations ψ of this type, up to order n. To estimate the borderline integrals however, we introduce an additional assumption, assumption J, which concerns the first order variations corresponding to the scaling or dilation group and to the rotation group, and the second order variations arising from these by applying the commutation field T . This assumption is later established through energy estimates of order 4 arising from these first order variations and derived on the basis of the final bootstrap assumption, just before the recovery of the final bootstrap assumption itself. It turns out that the borderline integrals all contain the factor T ψα , where ψα : α = 0, 1, 2, 3 are the first variations corresponding to spacetime translations and J is used to obtain an estimate   assumption   for suptu µ−1 |T ψα | in terms of suptu µ−1 |Lµ| , which involves on the right the factor ||−1 (see (13.198)). Upon substituting this estimate in the borderline integrals, the factors involving  cancel, and the integrals are estimated using (25). The above is an outline of the main steps in the estimation of the borderline integrals associated to the vectorfield K 0 . The estimation of the borderline integrals associated to the vectorfield K 1 , is however still more delicate. In this case we first perform an integration by parts on the outgoing characteristic hypersurfaces Cu , obtaining hypersurface integrals over tu and 0u and another spacetime volume integral. In this integration by parts the terms, including those of lower order, must be carefully chosen to obtain appropriate estimates, because here the long time behavior, as well as the behavior as µ tends to zero, is critical. Another integration by parts, this time on the surfaces St,u , is then performed to reduce these integrals to a form which can be estimated. The estimates of the hypersurface integrals over tu are the most delicate (the hypersurface integrals over 0u only involve the initial data) and require separate treatment of the condensation and rarefaction regions, in which the properties of the function µ, in particular those established by Proposition 8.6, come into play. In proceeding to derive the energy estimates of top order, n = l + 2, the power 2a of the weight µm,u (t) is chosen suitably large to allow us to transfer the terms contributed by the borderline integrals to the left-hand side of the inequalities resulting from the integral identities associated to the multiplier fields K 0 and K 1 . The argument then proceeds along the lines of that of Chapter 5, but is more complex because account must be taken of the terms corresponding to the estimates of Chapter 7 and of all the other terms contributed by the source functions, and also because of the fact that here we are dealing with weighted

Prologue and Summary

19

quantities. Once the top order energy estimates are established, we revisit the lower order energy estimates, using at each order the energy estimates of the next order in estimating the error integrals contributed by the highest spatial derivatives of the acoustical entities at that order. We then establish a descent scheme, which yields, after finitely many steps, u (t), F t u t estimates for the five quantities E0,[n] 0,[n] (u), E1,[n] (t), F1,[n] (u), and K [n] (t, u), for n = l + 1 − [a], where [a] is the integral part of a, in which weights no longer appear. It is these unweighted estimates which are used to close the bootstrap argument by recovering the final bootstrap assumption. This is accomplished by the method of continuity through the use of the isoperimetric inequality on the wave fronts St,u , and leads to the main theorem, Theorem 13.1. This theorem shows that there is another differential structure, that defined by the acoustical coordinates t, u, ϑ introduced in Chapter 2, such that relative to this structure the maximal classical solution extends smoothly to the boundary of its domain. This boundary contains however a singular part where the function µ vanishes, hence, in these coordinates, the acoustical metric h degenerates. With respect to the standard differential structure induced by the rectangular coordinates x α : α = 0, 1, 2, 3 in Minkowski spacetime, the solution is continuous but not differentiable on the singular part of the boundary, the derivative Tˆ µ Tˆ ν ∂µ βν blowing up as we approach the singular boundary. Thus, with respect to the standard differential structure, the acoustical metric h is everywhere in the closure of the domain of the maximal solution non-degenerate and continuous, but not differentiable on the singular part of the boundary of this domain, while with respect to the differential structure induced by the acoustical coordinates h is everywhere smooth, but degenerate on the singular part of the boundary. We have not sought to obtain an optimal lower bound for exponent a. This lower bound is significantly reduced by observing that only one among the first order variations actually contributes borderline integrals, namely the variation corresponding to time translations. As has already been mentioned, the first part of Chapter 14 establishes a theorem, Theorem 14.1, which gives sharp sufficient conditions on the initial data for the formation of a shock in the evolution, in the case of irrotational isentropic initial data. The proof is through Proposition 8.6 and is based on the study of the evolution with respect to t of the mean value on the sections St,u of each outgoing characteristic hypersurface Cu of the quantity: τ = (1 − u + η0 t)Lψ0 − (ψ0 − k).

(26)

Here ψ0 is the first variation corresponding to time translations and k is its value in the surrounding constant state. The proof of Theorem 14.1 uses the estimate provided by the spacetime integral K [ψ0 ](t, u) associated to ψ0 . Theorem 14.1 is followed by the analysis of the solution of the problem with general initial data during the initial time interval of order 1/η0 , as has already been discussed above. The last part of Chapter 14 establishes a theorem, Theorem 14.2, which extends Theorem 14.1 to the general case, removing the irrotational and isentropic restrictions on the initial data. The proof of Theorem 14.2 is based on the 1-form: (27) ξµ = β˙µ + θ s˙ u µ

20

Prologue and Summary

corresponding to any first order variation ( p, ˙ s˙ , u) ˙ of a general solution ( p, s, u), through solutions of the general equations of motion, and the associated functions: i = L µ ξµ ,

i = L µ ξµ .

(28)

We then study the evolution of the mean value on the St,u sections of each Cu of the quantity: τ  = (1 − u + η0 t)i − v0 ( p − p0 ) (29) where v0 and p0 are respectively the volume per particle and pressure in the surrounding constant state. Here certain crucial integrations by parts on the St,u sections as well as on Cu itself are performed, in which the structure of Cu as a characteristic hypersurface comes into play. We remark that Theorems 14.1 and 14.2 also give a sharp upper bound on the time interval required for the onset of shock formation. The contents of Chapter 15 have already been briefly described above. Proposition 15.1 describes the singular part of the boundary of the domain of the maximal classical solution from the point of view of the acoustical spacetime. It shows that this singular part has the intrinsic geometry of a regular null hypersurface in a regular spacetime and, like the latter, is ruled by invariant curves of vanishing arc length. On the other hand, the extrinsic geometry of the singular boundary is that of a space-like hypersurface which becomes null at its past boundary. The invariant curves are then used to define canonical acoustical coordinates. Theorem 15.1 is the main result of the chapter. This theorem shows that at each point q of the singular boundary, the past sound cone in the cotangent space at q degenerates into two hyperplanes intersecting in a 2-dimensional plane. We thus have a trichotomy of the bi-characteristics, or null geodesics of the acoustical metric, ending at q, into the set of outgoing null geodesics ending at q, which corresponds to one of the hyperplanes, the set of incoming null geodesics ending at q, which corresponds to the other hyperplane, and the set of the remaining null geodesics ending at q, which corresponds to the 2dimensional plane. The intersection of the past characteristic cone of q with any t in the past of q similarly splits into three parts, the parts corresponding to the outgoing and to the incomings sets of null geodesics ending at q being embedded discs with a common boundary, an embedded circle, which corresponds to the set of the remaining null geodesics ending at q. All outgoing null geodesics ending at q have the same tangent vector at q. This vector is then an invariant characteristic vector associated to the singular point q. This striking result is in fact the reason why the considerable freedom in the choice of the acoustical function does not matter in the end. For as is shown in Proposition 15.2, which considers the transformation from one acoustical function to another, the foliations corresponding to different families of outgoing characteristic hypersurfaces have equivalent geometric properties and degenerate in precisely the same way on the same singular boundary. Finally, Proposition 15.3 gives a detailed description of the boundary of the domain of the maximal classical solution from the point of view of Minkowski spacetime. The contents of the Epilogue have already been adequately described above.

Prologue and Summary

21

In concluding this introduction we remark that shocks develop not only in the context of fluid mechanics but also in magnetohydrodynamics, that is, the mechanics of a perfectly electrically conducting fluid in the presence of a magnetic field, in the nonlinear regime of the theory of elasticity, that is, the mechanics of deformable solids, isotropic or crystalline, as well as in the electrodynamics of continuous nonlinear media, in particular in the propagation of electromagnetic waves in electrically insulating fluids or solids with a nonlinear relationship between the electromagnetic field and the electromagnetic displacement. The pioneering work on shock formation in the theory of elasticity, in the spherically symmetric case, has been done by John [J2]. It is hoped that the present monograph will provide a springboard for those wishing to attack the general problem of shock formation in any of these fields. The present work relies more heavily on differential geometric concepts and methods than previous works on the same subject. For those prospective readers whose background is mainly in the fields of fluid mechanics or partial differential equations, but who may not have acquired an equally strong background in differential geometry, we recommend as a reference the book by Bishop and Crittenden [B-C] for the basic differential geometric concepts, and the book by Schoen and Yau [S-Y] as an excellent introduction to geometric analysis. In regard to notational conventions, Latin indices take the values 1,2,3, while Greek indices take the values 0,1,2,3. Repeated indices are meant to be summed, unless otherwise specified.

Chapter 1

Relativistic Fluids and Nonlinear Wave Equations. The Equations of Variation The mechanics of a perfect fluid is described in the framework of the Minkowski spacetime of special relativity by a future-directed unit time-like vectorfield u, the fluid 4velocity, and two positive functions n and s, the number of particles per unit volume (in the local rest frame of the fluid) and the entropy per particle, respectively. In terms of a system of rectangular coordinates (x 0 , x 1 , x 2 , x 3 ), with x 0 a time coordinate and (x 1 , x 2 , x 3 ) space coordinates, the metric components gµν , µ, ν = 0, 1, 2, 3, are given by, g00 = −1, g11 = g22 = g33 = 1, gµν = 0 : if µ = ν.

(1.1)

The conditions on the 4-velocity components u µ , µ = 0, 1, 2, 3, are then: gµν u µ u ν = −1, u 0 > 0.

(1.2)

Here, and throughout this monograph, we follow the summation convention, according to which repeated upper and lower indices are summed over their range. The mechanical properties of a perfect fluid are specified once we give the equation of state, which expresses the mass-energy density ρ as a function of n and s: ρ = ρ(n, s).

(1.3)

According to the laws of thermodynamics, the pressure p and the temperature θ are then given by: 1 ∂ρ ∂ρ − ρ, θ = . (1.4) p=n ∂n n ∂s The functions ρ, p, θ are assumed positive. Moreover, it is assumed that p is an increasing function of n at constant s and θ is an increasing function of s at constant n. The particle current is the vectorfield I whose components are given by: I µ = nu µ .

(1.5)

24

Chapter 1. Relativistic Fluids and Nonlinear Wave Equations

The energy-momentum-stress tensor is the symmetric 2-contravariant tensorfield T whose components are: (1.6) T µν = (ρ + p)u µ u ν + p(g −1 )µν . Here (g −1 )µν , µ, ν = 0, 1, 2, 3, are the components of the reciprocal metric, (g −1 )00 = −1, (g −1 )11 = (g −1 )22 = (g −1 )33 = 1, (g −1 )µν = 0 : if µ = ν.

(1.7)

The equations of motion of a perfect fluid are the conservation laws: ∂µ I µ = 0

(1.8)

∂ν T µν = 0.

(1.9)

Here, and throughout this monograph the symbol ∂µ =

∂ ∂xµ

denotes partial derivative with respect to the rectangular coordinate x µ . Taking the component of equation (1.9) along u by contracting with u µ = gµν u ν yields the equation:

u µ ∂µ ρ + (ρ + p)∂µ u µ = 0.

(1.10)

Now, according to equation (1.8), 1 ∂µ u µ = − u µ ∂µ n. n Substituting in (1.10) reduces that equation to the form: u µ ∂µ ρ =

(ρ + p) µ u ∂µ n. n

(1.11)

On the other hand by virtue of the equation of state (1.3) and the definitions (1.4), ∂µ ρ =

(ρ + p) ∂µ n + nθ ∂µ s. n

(1.12)

Comparing (1.11) and (1.12) we conclude that u µ ∂µ s = 0.

(1.13)

That is, the entropy per particle is constant along the flow lines, that is the integral curves of the vectorfield u. This conclusion holds as long as we are dealing with a solution of the equations of motion in the classical sense, that is the variables u µ , µ = 0, 1, 2, 3; n, s are C 1 functions of the rectangular coordinates. A portion of a fluid is called isentropic if s is constant throughout this portion. It follows from the preceding that if a portion of

Chapter 1. Relativistic Fluids and Nonlinear Wave Equations

25

the fluid is isentropic at time x 0 = 0 then the same portion, as given by the flow of u, is isentropic for all later time x 0 > 0, as long as the solution remains C 1 . Let us denote by  the 1-covariant, 1-contravariant tensorfield, which is at each point x in spacetime the operator of projection to x , the local simultaneous space of the fluid at x, namely the orthogonal complement in the tangent space at x of the linear span of u. The components of  are given by: µ µ µ ν = δν + u u ν

(1.14)

µ

where δν is the Kronecker symbol. The projection of equation (1.9) to x reads, at each point, (1.15) (ρ + p)u ν ∂ν u µ + µν ∂ν p = 0 where

µ

µν = λ (g −1 )λν = (g −1 )µν + u µ u ν .

Let us introduce now the positive function  σ = The function

ρ+p n

2 .

√ (ρ + p) σ = n

is called enthalpy per particle. By virtue of equations (1.3) and (1.4), ered to be a function of p and s, and its differential is given by: √ 1 d σ = d p + θ ds. n

(1.16)

√

σ can be consid-

(1.17)

We shall use in the following p and s instead of n and s as the basic thermodynamic variables. The sound speed η is defined by:   ∂p 2 η = (1.18) ∂ρ s a fundamental thermodynamic assumption being that the right-hand side of (1.18) is positive. Then η is defined to be positive. Another condition on η in the present framework of special relativity is that η < 1, namely that the sound speed is less than the universal constant represented by the speed of light in vacuum. By virtue of (1.13) and the definition (1.18) we have:      ∂ρ ∂ρ µ µ ∂µ p + ∂µ s u ∂µ ρ = u ∂p s ∂s p =

uµ ∂µ p. η2

(1.19)

26

Chapter 1. Relativistic Fluids and Nonlinear Wave Equations

The equations of motion of a perfect fluid are then seen to be equivalent to the following system in terms of the variables ( p, s, u µ : µ = 0, 1, 2, 3): u µ ∂µ s = 0 u µ ∂µ p + η2 (ρ + p)∂µ u µ = 0

(ρ + p)u ν ∂ν u µ + µν ∂ν p = 0.

(1.20)

Let ( p, s, u) be a given solution of the equations of motion (1.20) and let {( pt , st , u t ) : t ∈ I }, I an open interval of the real line containing 0, be a differentiable 1-parameter family of solutions such that: ( p0, s0 , u 0 ) = ( p, s, u). Then

  ( p, ˙ s˙ , u) ˙ = (d pt /dt)t =0 , (dst /dt)t =0 , (du t /dt)t =0

(1.21)

is a variation of ( p, s, u) through solutions. Note that the constraint (1.2) on u implies the following constraint on u: ˙ (1.22) u µ u˙ µ = 0. Differentiating the equations of motion for ( pt , st , u t ) with respect to t at t = 0 we obtain the equations of variation: u µ ∂µ s˙ = −u˙ µ ∂µ s µ

µ

µ

u ∂µ p˙ + η (ρ + p)∂µ u˙ = −u˙ ∂µ p − q∂ ˙ µu 2

(1.23) µ

˙ µν ∂ν p. (ρ + p)u ν ∂ν u˙ µ + µν ∂ν p˙ = −[(ρ˙ + p)u ˙ ν + (ρ + p)u˙ ν ]∂ν u µ −  Here q is the function: q = η2 (ρ + p) and for any function f of the thermodynamic variables alone:     ∂f ∂f ˙ f = p˙ + s˙ . ∂p s ∂s p Also:

˙ µν = u˙ µ u ν + u µ u˙ ν . 

(1.24)

(1.25)

(1.26)

In equations (1.23) we have placed on the left the principal terms, which are linear in the first derivatives of the variation, and on the right the lower order terms which are linear in the variation itself, with coefficients depending linearly on the first derivatives of the background solution. To the variation ( p, ˙ s˙ , u) ˙ and the system (1.23) is associated the energy current J˙, a vectorfield with components: J˙µ = u µ s˙2 +

1 u µ p˙ 2 + 2u˙ µ p˙ + (ρ + p)u µ gνλ u˙ ν u˙ λ + p)

η2 (ρ

(1.27)

Chapter 1. Relativistic Fluids and Nonlinear Wave Equations

27

(where u˙ ν = gµν u˙ µ ). We have: −u µ J˙µ = s˙ 2 +

1 p˙ 2 + (ρ + p)gνλ u˙ ν u˙ λ . + p)

η2 (ρ

(1.28)

In view of the fact that u˙ is subject to the constraint (1.22), so that gνλ u˙ ν u˙ λ = νλ u˙ ν u˙ λ µ

(where νλ = gµν λ ), this is a positive definite quadratic form in the variation ( p, ˙ s˙, u). ˙ Consider for any covector ξµ in the cotangent space at a given point x the quadratic form ˙ s˙, u), ˙ ( p˙  , s˙ , u˙  ) the corresponding symmetric ξµ J˙µ (x). For any pair of variations ( p, bilinear form is: ˙ s˙ , u), ˙ ( p˙  , s˙ , u˙  )) (1.29) ξµ J˙µ (( p,  1 = ξµ u µ s˙ s˙ + 2 u µ p˙ p˙  + u˙ µ p˙  + p˙ u˙ µ + (ρ + p)u µ gνλ u˙ ν u˙ λ . η (ρ + p) Consider now the set of all non-zero covectors ξ at x such that the symmetric bilinear form (1.29) is degenerate at x, that is, there is a non-zero variation ( p, ˙ s˙ , u) ˙ such that: ˙ s˙ , u), ˙ ( p˙  , s˙ , u˙  ))(x) = 0 ξµ J˙(( p,

: for all variations ( p˙  , s˙ , u˙  ).

(1.30)

This defines the characteristic subset of the cotangent space at x. Taking into account the constraint (1.22) we see that the condition (1.30) is equivalent to the following linear system for the variation ( p, ˙ s˙ , u): ˙ (ξµ u µ )˙s = 0

It follows that either: in which case: or:

(ξµ u µ ) p˙ + ξµ u˙ µ = 0 η2 (ρ + p)   ˙ λ + (ρ + p)(ξµ u µ )u˙ λ = 0. λν pξ

(1.31)

ξµ u µ = 0

(1.32)

p˙ = 0 and ξµ u˙ µ = 0

(1.33)

(h −1 )µν ξµ ξν = 0

(1.34)

in which case: s˙ = 0 and u˙ ν = −

νλ ξλ p˙ . (ρ + p)(ξµ u µ )

(1.35)

In (1.34) h −1 is the reciprocal acoustical metric (non-degenerate quadratic form in the cotangent space at each point), given by: 1 µ ν u u η2   1 −1 µν = (g ) − − 1 uµuν . η2

(h −1 )µν = µν −

(1.36)

28

Chapter 1. Relativistic Fluids and Nonlinear Wave Equations

It is the reciprocal of the acoustical metric h (non-degenerate quadratic form in the tangent space at each point), given by: h µν = gµν + (1 − η2 )u µ u ν

(1.37)

which plays a fundamental role in this monograph. The subset of the cotangent space at x defined by the condition (1.32) is a hyperplane Px∗ while the subset of the cotangent space at x defined by the condition (1.34) is a cone C x∗ . The dual to Px∗ characteristic subset of the tangent space at x is the linear span of u, while the dual to C x∗ characteristic subset C x is the set of all non-zero vectors X at x satisfying: h µν X µ X ν = 0.

(1.38)

Now we can construct an orthonormal basis for the tangent space at x by taking u at x to be the time-like element of the basis and complementing it with an orthonormal basis for the space-like hyperplane x , the orthogonal complement of u in the tangent space at x. The dual to such a basis is then a basis for the cotangent space at x. In other words, we may construct a new system of rectangular coordinates (x 0 , x 1 , x 2 , x 3 ) by translating the origin to the point x and choosing the positive x 0 axis in the direction of the vector u at x. In such a system the fluid at x appears to be at rest and we have: u 0 = 1, u 1 = u 2 = u 3 = 0 : at x while the components of the metric g and its reciprocal g −1 have the standard form (see (1.1) and (1.7)), hence also: u 0 = −1, u 1 = u 2 = u 3 = 0 : at x. Equations (1.34) and (1.38) reduce in such a system to the form: −η−2 (ξ0 )2 + (ξ1 )2 + (ξ2 )2 + (ξ3 )2 = 0

(1.39)

−η2 (X 0 )2 + (X 1 )2 + (X 2 )2 + (X 3 )2 = 0.

(1.40)

The requirement that η < 1 is seen to be the physical requirement that in the tangent at each point x the sound cone (1.40) lies within the light cone. This is equivalent to the requirement that in the cotangent space at each point x the light cone lies within the sound cone (1.39). The latter implies that a covector ξ at x, defining a hyperplane Hx in the tangent space at x which is space-like relative to the Minkowski metric g, and such that ξ has positive evaluation on the future-directed normal to Hx , belongs to the interior of the positive component of C x∗ . It follows from the above that the set of all covectors ξ at x such that the quadratic form ξ · J˙(x) is positive definite is the interior of the positive component of C x∗ .

Chapter 1. Relativistic Fluids and Nonlinear Wave Equations

29

Let us now calculate the divergence of J˙. Substituting from the equations of variation (1.23), the principal terms cancel and we find: 1 µ u ∂µ q p˙ 2 + ∂µ ((ρ + p)u µ )gνλ u˙ ν u˙ λ q2 p˙ −2˙s u˙ µ ∂µ s − 2 (u˙ µ ∂µ p + q∂ ˙ µuµ) q ˙ λν ∂λ p. −2u˙ ν [(ρ˙ + p)u ˙ λ + (ρ + p)u˙ λ ]∂λ u ν − 2u˙ ν 

∂µ J˙µ = ∂µ u µ s˙2 −

(1.41)

The right-hand side is quadratic in the variation with coefficients which depend linearly on the first derivatives of the background solution. A distinguished class of solutions among all solutions of the equations of motion (1.20) are the solutions of the form, in rectangular coordinates: uµ = aµ,

p = p0 , s = s0 , hence n = n( p0 , s0 ) = n 0

(1.42)

where the a µ , µ = 0, 1, 2, 3 are constants, the components of a unit future-directed timelike vector. Moreover p0 and s0 , hence also n 0 , are constants. The solutions of the form (1.43) are the constant states. By an appropriate choice of rectangular coordinates we can set: (1.43) a 0 = 1, a i = 0 : i = 1, 2, 3. We now introduce the 1-form β whose components are given by: √ βµ = − σ u µ .

(1.44)

We calculate Lu β, the Lie derivative of β with respect to the vectorfield u. It is given by: (Lu β)µ = u ν ∂ν βµ + βν ∂µ u ν an expression valid in any system of coordinates. Taking into account condition (1.2) and then using equation (1.15) we obtain: √ (Lu β)µ = −u ν ∂ν ( σ u µ ) √  √  √ σ σ ν ∂µ p + u µ u ∂ν p − ∂ν σ . = (ρ + p) (ρ + p) Now by (1.17) the expression in the last parenthesis is equal to −θ ∂ν s, therefore by virtue of equation (1.13) the last term vanishes. Taking again account of (1.17) then yields: √ (Lu β)µ = ∂µ σ − θ ∂µ s which we may write simply as: √ Lu β = d σ − θ ds.

(1.45)

The fluid vorticity ω is the 2-form which is the exterior derivative of the 1-form β: ω = dβ.

(1.46)

30

Chapter 1. Relativistic Fluids and Nonlinear Wave Equations

In terms of components we have: ωµν = ∂µ βν − ∂ν βµ an expression valid in an arbitrary system of coordinates. We should note here that our vorticity 2-form does not correspond to the classical notion of vorticity. What corresponds to the classical notion of vorticity is the vectorfield: µ =

1 −1 µαβγ ( ) u α ωβγ . 2

(1.47)

Here ( −1 ) is the reciprocal volume form of the Minkowski metric g or volume form in the cotangent space at each point. We denote by  the volume form of g. If (E 0 ,E 1 ,E 2 ,E 3 ) is a positive basis for the tangent space at x which is orthonormal with respect to g and (ϑ0 , ϑ1 , ϑ2 , ϑ3 ) is the dual basis for the cotangent space at x, we have:  −1 (ϑ0 , ϑ1 , ϑ2 , ϑ3 ) = (E 0 , E 1 , E 2 , E 3 ) = 1. The components of  and  −1 are given in an arbitrary system of coordinates by: αβγ δ =



−detg[αβγ δ],

[αβγ δ] ( −1 )αβγ δ = √ −detg

(1.48)

where [αβγ δ] is the 4-dimensional fully antisymmetric symbol. In rectangular coordinates (1.48) reduces to: αβγ δ = ( −1 )αβγ δ = [αβγ δ]. The vectorfield  is the obstruction to integrability of the distribution of local simultaneous spaces {x }. We call  the vorticity vector . At each point x, the vector  (x) belongs to x . Let us recall at this point the following general relation between the exterior derivative and the Lie derivative with respect to a vectorfield X, as applied to an exterior differential form ϑ of any rank: (1.49) L X ϑ = i X dϑ + di X ϑ. Here i X denotes contraction on the left by X. Taking X = u and ϑ to be the 1-form β, we have from (1.44): √ iu β = σ . (1.50) Comparing then (1.49) with (1.45) we conclude that: i u ω = −θ ds.

(1.51)

In the case of an isentropic fluid portion equation (1.51) reduces to: i u ω = 0.

(1.52)

Since by the definition (1.46) we have, in general: dω = 0

(1.53)

Chapter 1. Relativistic Fluids and Nonlinear Wave Equations

31

it follows, taking in (1.49) X = u and ϑ to be the 2-form ω, that in an isentropic fluid portion we have: (1.54) Lu ω = 0. Thus in an isentropic portion of the fluid the vorticity is Lie transported along the flow lines. A portion of the fluid is called irrotational if ω vanishes throughout this portion. Consider now a fluid portion which is isentropic as well as irrotational at time x 0 = 0. Then according to the preceding the same portion, as given by the flow of u, remains isentropic and irrotational for all later time x 0 > 0, as long as the solution remains C 1 in terms of the basic variables u, n and s, or u, p and s. If the fluid portion is simply connected, we can then introduce a function φ, determined up to an additive constant, such that β = dφ (1.55) that is, in components, βµ = ∂µ φ. It follows from the definition of β, equation (1.44), that the derivative of φ is positive along any future-directed non-space-like vector, so φ has the property that φ(y) > φ(x) whenever the point y belongs to the causal future of the point x. Equations (1.44) and (1.55) allow us to express: ∂ µφ σ = −(g −1 )µν ∂µ φ∂ν φ, u µ = − √ σ

(∂ µ = (g −1 )µν ∂ν ).

(1.56)

With (1.55) and (1.56), equation (1.45) is identically satisfied, and this is equivalent to equation (1.15). Moreover, for an isentropic fluid portion equation (1.13) is likewise identically satisfied. It follows that for an irrotational isentropic fluid portion the whole content of the equations of motion is contained in the particle current conservation law (1.8), which takes the form of a nonlinear wave equation: ∂µ (G∂ µ φ) = 0

(1.57)

G = G(σ )

(1.58)

n n2 G=√ = . ρ+p σ

(1.59)

where is given by:

Note that G can be considered to be a function of σ alone by virtue of the fact that s is constant. Summarizing, given the equation of state (1.3) expressing ρ as a function of n at some fixed value of s, equation (1.16) together with the first of equations (1.4) allows us to express n, ρ and p as functions of σ at that value of s, following which equation (1.59) gives us G as a function of σ and equation (1.57) expresses the equations of motion of an irrotational isentropic fluid, with entropy per particle equal to the given value, as a

32

Chapter 1. Relativistic Fluids and Nonlinear Wave Equations

nonlinear wave equation for the function φ, the fluid variables being determined in terms of this function by equations (1.56). Equation (1.57) is the Euler–Lagrange equation corresponding to a Lagrangian density function of the form L = L(σ ) (1.60) where we have, simply, L = p.

(1.61)

For, the Euler–Lagrange equation corresponding to a Lagrangian density of the form (1.60) is:   dL µ ∂ φ = 0. ∂µ 2 dσ This coincides with (1.57) since, by (1.17) at constant s and (1.59), 2

dL dp 1 dp n =2 = √ √ = √ = G. dσ dσ σd σ σ

Now, not all Lagrangian densities of the form (1.60) give rise to Euler–Lagrange equations which have a fluid interpretation. First the range of σ must be restricted to the positive real line and L √ must be positive since p is to be positive. Next, L must be an increasing function of σ since dp √ =n d σ √ must be positive; thus G √ must be positive. In fact, the stronger condition that L/ σ be an increasing function of σ must hold, since (1.16) √ dp √ d σ √ (L/ σ ) = σ √ − p = ρ d σ d σ √ must be positive. Finally L must be a convex function of σ so that dn √ d σ will be positive as required by the positivity of d p/dn (at constant s). A particular Lagrangian density of the form (1.60) which does satisfy the requirements for a fluid interpretation if the range of σ is appropriately restricted is: √ L = 1 − 1 − σ. (1.62) The range of σ is a priori (−∞, 1). The requirements for a fluid interpretation are met if we restrict the range of σ to the interval (0, 1). This Lagrangian has a very simple geometric interpretation. Introducing another rectangular spatial coordinate x 4 we consider in Minkowski spacetime of four spatial dimensions the graph x 4 = φ(x),

x = (x 0 , x 1 , x 2 , x 3 )

(1.63)

Chapter 1. Relativistic Fluids and Nonlinear Wave Equations

33

of a function φ defined on a domain √ in the standard Minkowski spacetime of three spatial dimensions. Then the integral of 1 − σ over a subdomain U of the domain of definition of φ is simply the area of the graph over U , a time-like hypersurface, and the corresponding Euler–Lagrange equation is the minimal surface equation for such a graph. We shall see in the sequel that the minimal surface equation is of an exceptional character among all equations of the form (1.57). The nonlinear wave equation (1.57) can be written in the form: (h −1 )µν ∂µ ∂ν φ = 0

(1.64)

where h −1 is the reciprocal acoustical metric (1.36), which in the irrotational isentropic case is given by: (1.65) (h −1 )µν = (g −1 )µν − F∂ µ φ∂ ν φ. Here F = F(σ )

(1.66)

is given by: 2 dG . (1.67) G dσ √ The condition that the Lagrangian density L be a convex function of σ implies that: F=

1+σF =

1 > 0. η2

(1.68)

The acoustical metric h, given in general by (1.37), is in the irrotational isentropic case given by: h µν = gµν + H ∂µ φ∂ν φ (1.69) where H = H (σ ) is the function: H=

F 1+σF

(1.70) (1.71)

and we have: 1 − σ H = η2 .

(1.72)

In particular, if L is the Lagrangian density (1.62) then H =1 and h is the induced metric on the graph (1.63). In the √ general context of fluid mechanics we define the function H according to (1.72), with σ being, according to the above, the enthalpy. We also define in general the function F by (1.71). The physical requirement that the sound cone (1.40) lies within the light cone is then expressed by the condition F >0

(1.73)

34

Chapter 1. Relativistic Fluids and Nonlinear Wave Equations

In the isentropic irrotational case this is equivalent to the condition that the Lagrangian density L be a convex function of σ (1.67). Modulo the other conditions, this condition √ in fact implies the earlier condition that L be a convex function of σ . Conditions (1.68) and (1.73) on the function F are equivalent to the following condition on the function H : 0 < σ H < 1.

(1.74)

The function H or, more precisely, its differential, plays a central role in the problem of shock formation in fluid mechanics. In the isentropic case H is a function of σ and we have: 1 √ dη2 dH = σ √ + 1 − η2 . (1.75) −σ 2 dσ 2 d σ Now, from (1.16) and (1.18) we have, in the isentropic case: √ n d σ = η2 √ σ dn

(1.76)

hence, substituting in (1.75) we obtain: −σ 2

n dη dH = + 1 − η2 . dσ η dn

(1.77)

A distinguished class of solutions among all solutions of the nonlinear wave equation (1.57) are the solutions of the form, in rectangular coordinates, φ(x) = kµ x µ

(1.78)

where the kµ , µ = 0, 1, 2, 3, are constants. These solutions correspond to the constant states (1.43). The required property of φ is then equivalent to the condition that −k µ = −(g −1 )µν kν be the components of a future-directed time-like vector. For these we have: σ = k 2 where k 2 = −gµν k µ k ν > 0 and the correspondence with the constant states (1.43) is: k µ = −ka µ ,

√

σ0 =

ρ0 + p0 = k. n0

By an appropriate choice of rectangular coordinates we can set: k0 = −k 0 = k > 0, ki = k i = 0 : i = 1, 2, 3 so that φ(x) = kx 0 .

(1.79)

Let φ be a given solution of the nonlinear wave equation (1.57) and let {φt : t ∈ I }, I an open interval of the real line containing 0, be a differentiable 1-parameter family of solutions such that φ0 = φ.

Chapter 1. Relativistic Fluids and Nonlinear Wave Equations



Then ψ=

dφt dt

35

 (1.80) t =0

is a variation of φ through solutions. Differentiating ∂µ (G(σt )∂ µ φt ) = 0, σt = −∂ µ φt ∂µ φt with respect to t at t = 0 we obtain the equation of variation: ∂µ (G(σ )(h −1 )µν ∂ν ψ) = 0.

(1.81)

Here h −1 is the reciprocal acoustical metric (1.65) corresponding to the solution φ. We can write the equation of variation in a more intelligible form in terms of the conformal ˜ given by: acoustical metric h, (1.82) h˜ µν = h µν where the conformal factor  is given by: =

G/G 0 η/η0

(1.83)

where G 0 and η0 are the constant values of G and η in a reference constant state. Taking into account that by equation (1.37) deth = −1 + σ H = −η2 we obtain: deth˜ =



G/G 0 η/η0

Since also (h˜ −1 )µν = it follows that:



4 deth = − 

η/η0 G/G 0



(G/G 0 )4 2 η . (η/η0 )2 0

(1.84)

(1.85)

(h −1 )µν ,

˜ h˜ −1 )µν = η0 G (h −1 )µν . −deth( G0

Therefore the equation of variation (1.81) can be written in the form  ˜ h˜ −1 )µν ∂ν ψ) = 0. ∂µ ( −det h( ˜ This is simply the linear wave equation corresponding to the metric h:
h˜ ψ = 0.

(1.86)

Now if φ is a solution of the nonlinear wave equation (1.57) and { f t } is a 1-parameter subgroup of the isometry group of Minkowski spacetime (4 , g), then for each t, (1.87) φt = φ ◦ f t

36

Chapter 1. Relativistic Fluids and Nonlinear Wave Equations

is also a solution. It follows that  ψ=

dφ ◦ f t dt

 t =0

= Xφ

(1.88)

satisfies the linear wave equation (1.86). Here X is the vectorfield generating { f t } (considered as a differential operator). The isometry group of Minkowski spacetime is the 10dimensional Poincar´e group, consisting of the 4-dimensional normal Abelian subgroup of translations together with the 6-dimensional Lorentz group O(3, 1). The generators of the translations are the vectorfields Tµ =

∂ : µ = 0, 1, 2, 3 ∂xµ

(1.89)

corresponding to the rectangular coordinates x µ : µ = 0, 1, 2, 3, while the generators of the Lorenz group are the vectorfields µν = x µ

∂ ∂ − x ν µ : µ < ν = 0, 1, 2, 3; ∂xν ∂x

x µ = gµν x ν .

(1.90)

Thus, for any solution φ of the nonlinear wave equation, each of the 10 functions Tµ φ : µ = 0, 1, 2, 3; µν φ : µ < ν = 0, 1, 2, 3 satisfies the linear wave equation (1.86) with the metric h˜ being that corresponding to φ. Consider now the 1-parameter group of dilations of the Minkowski spacetime, given by: x → et x : t ∈  where x = (x 0 , x 1 , x 2 , x 3 ) is the position vector in linear, in particular rectangular coordinates. Let again φ be a given solution of the nonlinear wave equation (1.57) and let us define for each t ∈  the function φt by: φt (x) = e−t φ(et x) in terms of linear coordinates. We then have (∂µ φt )(x) = (∂µ φ)(et x) and, with σt = −∂ µ φt ∂µ φt , hence, if also:

σt (x) = σ (et x)

I µ = G(σ )∂ µ φ, µ

µ

It = G(σt )∂ µ φt ,

It (x) = I µ (et x).

(1.91)

Chapter 1. Relativistic Fluids and Nonlinear Wave Equations

37

µ

Therefore ∂µ I µ = 0 implies ∂µ It = 0, that is φt is, for each t, also a solution of the nonlinear wave equation. It follows that   dφt = Sφ (1.92) ψ= dt t =0 satisfies the linear wave equation (1.86). Here S is the differential operator: S = D − 1, D = x µ

∂ . ∂xµ

(1.93)

Finally we remark that analogous considerations apply to the general equations of motion (1.20). This leads to the following conclusion. If ( p, s, u) is a solution of the equations of motion (1.20) and V is any one of the vectorfields: Tµ : µ = 0, 1, 2, 3, µν : µ < ν = 0, 1, 2, 3,

D

(1.94)

then ( p, ˙ s˙ , u), ˙ defined by p˙ = V p, s˙ = V s, u˙ = LV u : for V = D, u˙ = L D u + u : for V = D, is a solution of the equations of variation (1.23). Here: LV u = [V, u] is the Lie derivative of the vectorfield u with respect to the vectorfield V .

(1.95)

Chapter 2

The Basic Geometric Construction Let us choose a time function t in Minkowski spacetime, equal to the coordinate x 0 of some rectangular coordinate system. We shall denote by t an arbitrary level set of the function t. The t are parallel space-like hyperplanes. The initial data for the equations of motion (1.20) is to be given on the hyperplane 0 and consists in specification of the triplet ( p, s, u). We assume that there is a sphere S0,0 in 0 outside which the initial data coincide with those of a constant state (see (1.42), (1.43)), that is we have: p = p0 s = s0 , u 0 = 1, u i = 0 : i = 1, 2, 3.

(2.1)

By suitable translation and scaling we can then take S0,0 to be the unit sphere centered at the origin in 0 . We consider an annular interior neighborhood of S0,0 in 0 : ε

00 = {x ∈ 0 : 1 − ε0 ≤ r (x) ≤ 1}

(2.2)

where r is the Euclidean distance function from the origin in 0 and ε0 is a positive constant, subject throughout this monograph only to the condition: ε0 ≤

1 . 2

(2.3)

We define in 0 the function: u = 1−r

(2.4)

which on 0 minus the origin is a smooth function without critical points, vanishing on S0,0 and increasing inward. For each value of u in the closed interval [0, ε0], the corresponding level set S0,u of u is a sphere of radius 1 − u in the interval [1 − ε0 , 1], and we have:  ε 00 = S0,u (2.5) u∈[0,ε0 ]

40

Chapter 2. The Basic Geometric Construction

In the case that the initial data is irrotational and isentropic outside S0,ε0 , the data in the exterior of S0,ε0 in 0 specify the pair (φ, ∂0 φ) in this region. This is initial data for the nonlinear wave equation (1.57) in the region in question and we have: φ = 0, ∂0 φ = k : in the exterior of S0,0 in 0 .

(2.6)

To any given initial data set as above there corresponds a unique maximal solution of the equations of motion (1.20), of the nonlinear wave equation (1.57) in the irrotational isentropic case. The notion of maximal solution or maximal development of an initial data set is the following. Given an initial data set, the local existence theorem asserts the existence of a development of this set, namely of a domain D in Minkowski spacetime, whose past boundary is the domain of the initial data, and of a solution defined in D and taking the given data at the past boundary, such that if we consider any point p ∈ D and any curve issuing at p with the property that its tangent vector at any point q − belongs to I q , the closure of the past component of the open double cone defined by h q , the acoustical metric at q, then the curve terminates in the past at a point of the domain of the initial data. The local uniqueness theorem asserts that if (D1 , ( p1 , s1 , u 1 )) and (D2 , ( p2 , s2 , u 2 )) are two developments of the same initial data set ((D1 , φ1 ) and (D2 , φ2 ) in the irrotational isentropic case),  then ( p1 , s1 , u 1 ) coincides with ( p2 , s2 , u 2 ) in D1 D2 (φ1 coincides with φ2 in D1 D2 in the irrotational isentropic case). It follows that the union of all developments of a given initial data set is itself a development, the unique maximal development of the initial data set. (See [Ch] for a discussion of the above notions in the general context of equations derivable from an action principle. The notion of maximal development in the context of general relativity was introduced by ChoquetBruhat and Geroch [C-G]). We consider, in the domain of the maximal solution, the family {Cu : u ∈ [0, ε0 ]} of outgoing characteristic hypersurfaces corresponding to the family {S0,u : u ∈ [0, ε0 ]}:  Cu (2.7) 0 = S0,u : ∀u ∈ [0, ε0 ]. Each bicharacteristic generator of each Cu is to extend in the domain of the maximal solution as long at it remains on the boundary of the domain of dependence of the exterior of the surface S0,u in the initial hyperplane 0 . If we denote by Wε0 the spacetime domain:  Cu (2.8) Wε0 = u∈[0,ε0 ]

then the domain Mε0 of the maximal solution corresponding to the given initial data set is the union of Wε0 with the domain in Minkowski spacetime bounded by the exterior of the unit sphere S0,0 in the initial hyperplane 0 and by the outgoing characteristic hypersurface C0 corresponding to S0,0 . By the domain of dependence theorem the solution coincides in Mε0 \ Wε0 with the constant state (1.79). This implies that C0 is a complete cone, each of its bicharacteristic generators extending to infinity in the parameter t. We extend the function u to Wε0 by requiring that its level sets are precisely the outgoing characteristic hypersurfaces Cu . The function u is then a solution of the equation: (h −1 )µν ∂µ u∂ν u = 0.

(2.9)

Chapter 2. The Basic Geometric Construction

41

We shall call such a function an acoustical function. The vectorfield Lˆ given by: Lˆ µ = −(h −1 )µν ∂ν u

(2.10)

is then a future-directed null geodesic vectorfield with respect to the Lorentzian metric h. Its integral curves are the bicharacteristic generators of each Cu . Now the parametrization of these given by Lˆ is affine; however, for reasons which shall become apparent in the following we wish the generators to be parametrized by the function t instead. For this reason we choose to work with the collinear vectorfield L = µ Lˆ

(2.11)

where the proportionality factor µ, a positive function, is chosen so that

Thus

Lt = 1.

(2.12)

1 = −(h −1 )µν ∂µ t∂ν u. µ

(2.13)

The function µ plays a fundamental role in the present monograph. For each u ∈ [0, ε0 ] there is a greatest lower bound t∗ (u) of the extent of the generators of Cu , in the parameter t, in the domain of the maximal solution. This t∗ (u) is either a positive real number or ∞. According to the above we have t∗ (0) = ∞. Let us denote t∗ε0 =

inf t∗ (u).

(2.14)

u∈[0,ε0 ]

In the following we shall confine attention to the spacetime domain Wε∗0 consisting of all the points in Wε0 whose time coordinate t is less than t∗ε0 . For each (t, u) ∈ [0, t∗ε0 ) × [0, ε0 ] we define the closed surface:  St,u = Cu (2.15) t . We have:

Wε∗0 =



St,u .

(2.16)

(t,u)∈[0,t∗ε0 )×[0,ε0 ]

We define in Wε∗0 the vectorfield T by the conditions that it be tangential to the hypersurfaces t , orthogonal with respect to the metric h to the family of surfaces {St,u : u ∈ [0, ε0 ]} in each t , and that it verify: T u = 1.

(2.17)

 = [L, T ].

(2.18)

Consider the commutator: By (2.12), (2.17) and also the fact that by (2.9), (2.10), (2.11), Lu = 0

(2.19)

42

Chapter 2. The Basic Geometric Construction

as well as the fact that, by virtue of the tangentiality of T to the t , Tt = 0

(2.20)

u = t = 0.

(2.21)

we have: Therefore the vectorfield  is tangential to the surfaces St,u . From (2.9), (2.10), (2.11) we have: h(L, L) = 0

(2.22)

that is, L is a null vector with respect to the acoustical metric h. Moreover, from (2.10), (2.11), (2.17), ˆ T ) = −µT u = −µ. h(L, T ) = µh( L, (2.23) Now, the requirement (see Chapter 1) that the sound cone lie within the light cone implies that the hypersurfaces t are also space-like with respect to the acoustical metric h. Therefore T is a space-like vector with respect to h and we can write: h(T, T ) = κ 2

(2.24)

where κ is a positive function. Let now N be any vector tangent to one of the characteristic hypersurfaces Cu at a point: Nu = 0. By (2.10), (2.11), this is equivalent to: h(L, N) = 0. In particular any vector X tangent to one of the surfaces St,u at a point is h-orthogonal to L: (2.25) h(L, X) = 0 : ∀X ∈ T St,u . Since, by definition, X is also h-orthogonal to T : h(T, X) = 0 : ∀X ∈ T St,u

(2.26)

it follows that at each point p on each surface St,u , the plane  p spanned by L and T at p, is h-orthogonal to T p St,u , the tangent space at p to St,u , and we have: T p Wε∗0 =  p ⊕ T p St,u .

(2.27)

The metric h in  p is given by (2.22), (2.23), (2.24) in terms of the functions µ and κ. By virtue of the h-orthogonal decomposition (2.27) the metric h is then completely specified once we give the metric h / induced on the surfaces St,u : /(X, Y ) = h(X, Y ) : ∀X, Y ∈ T p St,u . h

(2.28)

Chapter 2. The Basic Geometric Construction

43

For each u ∈ [0, ε0 ] the generators of Cu define a smooth one-to-one mapping of S0,u onto St,u for each t ∈ [0, t∗ε0 ). Moreover, each S0,u is diffeomorphic to the standard sphere S 2 ⊂ 3 . Therefore, given for each u ∈ [0, ε0] a diffeomorphism ϕu of S 2 onto S0,u , we can assign to any point p on any surface St,u a point ϑ of S 2 , namely the pre-image of a point q on S0,u , when p lies along the generator of Cu issuing at q. If local coordinates (ϑ 1 , ϑ 2 ) are chosen on S 2 this assignment defines local coordinates on St,u for every (t, u) ∈ [0, t∗ε0 ) × [0, ε0]. In so far as the family of diffeomorphisms {ϕu : u ∈ [0, u 0 ]} is arbitrary, these coordinates are arbitrary as they can be subjected to transformations of the form: ˜ ϑ → ϑ˜ = ϑ(u, ϑ). (2.29) The local coordinates (ϑ 1 , ϑ 2 ), together with the functions (t, u) define a complete system of local coordinates (t, u, ϑ 1 , ϑ 2 ) for Wε∗0 . We shall call these acoustical coordinates and we shall derive an expression for the acoustical metric h in Wε∗0 in these coordinates. First, the integral curves of L are the lines of constant ϑ and u, parametrized by t. Therefore the vectorfield L is given in our coordinates simply by: L=

∂ . ∂t

(2.30)

Next, by (2.17) the vectorfield T has unit u-component and by (2.20) vanishing t-component. Therefore T is given by an expression of the form: T =

∂ − ∂u

(2.31)

where  is a vectorfield which is tangential to the surfaces St,u . Thus  can be expanded in terms of the coordinate frame field   ∂ : A = 1, 2 (2.32) ∂ϑ A and we have:

∂ . ∂ϑ A By equations (2.30), (2.31) and (2.18) we have:  = A

(2.33)

[L, ] = −

(2.34)

or, in terms of components, ∂ A (2.35) = − A . ∂t By an appropriate transformation of the form (2.29) we can set  = 0 along any hypersurface H with the property that each integral curve of L intersects H at a single point. In particular, we can set  = 0 along any one of the hypersurfaces t . However the non-vanishing of  forbids setting  = 0 everywhere.

44

Chapter 2. The Basic Geometric Construction



With / AB h

∂ ∂ = /h , A ∂ϑ ∂ϑ B

 : A, B = 1, 2

(2.36)

the components of the metric h / induced by h on the surfaces St,u in the coordinate frame (2.32), the metric h, according to the above, takes in our acoustical coordinate system the form: (2.37) h = −2µdtdu + κ 2 du 2 + h/ AB (dϑ A +  A du)(dϑ B +  B du). We define in Wε∗0 the vectorfield B by the conditions that it be orthogonal to the hypersurfaces t with respect to the metric h and that it verify: Bt = 1

(2.38)

The hypersurfaces t being space-like relative to h, the vectorfield B is future-directed time-like relative to h. Thus, there is a positive function α such that

In fact, we have: and:

h(B, B) = −α 2 .

(2.39)

h µν B ν = −α 2 ∂µ t

(2.40)

α −2 = −(h −1 )µν ∂µ t∂ν t.

(2.41)

In particular, in the class of rectangular coordinate systems with time coordinate x 0 fixed as above, we have simply: α −2 = −(h −1 )00 = 1 + (η−2 − 1)(u 0 )2 = 1 + F(β0 )2 in general (see (1.36)),

α −2 = 1 + F(∂0 φ)2

(2.42) (2.43)

in the irrotational isentropic case (see (1.65)). The function α is the lapse function of the foliation {t } relative to the metric h. That is, the integral t2 αdt t1

along an integral curve of B, is the arc length, with respect to h, of the segment of the curve between the hypersurfaces t1 and t2 . Thus α measures the normal separation of the leaves of the foliation {t }. Note that α < 1, so the lapse function relative to the acoustical metric h of the foliation {t } is less than that relative to the Minkowski metric g. Now B at any given point p belongs to the plane  p , a time-like plane relative to the metric h. Therefore B is a linear combination of L and T . In fact, in view of the conditions (2.12), (2.20) and (2.38) we have: B = L+ fT

(2.44)

Chapter 2. The Basic Geometric Construction

45

where the coefficient f must be positive since L is outgoing. Taking the h-inner product of equation (2.44) with T , yields, in view of (2.23), (2.24) and the fact that h(B, T ) = 0, µ = f κ 2.

(2.45)

On the other hand, substituting for L from (2.44) in (2.22) yields, in view of (2.39), (2.24) and the orthogonality of B and T relative to h, 0 = h(B, B) + f 2 h(T, T ) = −α 2 + f 2 κ 2 .

(2.46)

Thus, in view of the positivity of f we have: f =

α . κ

(2.47)

Substituting in (2.45) then yields the following relation between the metric coefficients κ and µ: µ = ακ. (2.48) Consider now the metric h induced by the acoustical metric h on the hypersurfaces t . From (1.37), (1.69), (1.72), it is given in rectangular coordinates (x 1 , x 2 , x 3 ), in general, by: h i j = gi j + σ H u i u j = gi j + Hβi β j (2.49) and in the irrotational isentropic case by: h i j = g i j + H ∂i φ∂ j φ

: i, j = 1, 2, 3.

(2.50)

Here g is the metric induced by the Minkowski metric g on the t , namely the Euclidean metric: g i j = δi j (2.51) in rectangular coordinates. Note that h dominates g. Let u be the g-orthogonal projection of the velocity vectorfield u to t . The components of u in rectangular coordinates are simply u i : i = 1, 2, 3. If at a point p on t we have u( p) = 0, then h coincides with g at p. Otherwise u( p) = 0 and h has eigenvalues equal to 1 in the plane in T p t which is g-orthogonal to u, and the eigenvalue: 1 + σ |u|2g H in the linear span of u. In the irrotational isentropic case the preceding considerations take the following form. Let us denote by d f the restriction to T t of d f , the differential of a function f defined on spacetime. If at a point p ∈ t we have dφ( p) = 0, then h coincides with g at p. Otherwise dφ( p) = 0 and h has eigenvalues equal to 1 in the plane tangent to the level set of φ in t through p, and the eigenvalue 1 + ρH

46

Chapter 2. The Basic Geometric Construction

in the line which is g-orthogonal to this plane. Here ρ is the non-negative function: ρ = |dφ|2g = (g −1 )i j ∂i φ∂ j φ =

3 

(∂i φ)2

(2.52)

i=1

(in rectangular coordinates). This is the expression for ρ in the irrotational isentropic case. In the general case we have: (2.53) σ = −(g −1 )µν βµ βν and we define the function ρ by: ρ = σ |u|2g = |β|2g = (g−1 )i j βi β j =

3 

(βi )2 .

(2.54)

i=1

In the general case the reciprocal h (h

−1 i j

) = (g−1 )i j −

−1

of the induced metric is given by:

σH H u i u j = (g−1 )i j − βi β j 1 + ρH 1 + ρH

(2.55)

and in the isentropic irrotational case by: (h

−1 i j

) = (g−1 )i j −

H ∂ i φ∂ j φ 1 + ρH

: i, j = 1, 2, 3.

(2.56)

Let us introduce the vectorfield Tˆ , collinear and in the same sense as T and of unit magnitude with respect to the acoustical metric h: Tˆ = κ −1 T.

(2.57)

By (2.44) and (2.36) we can express L in the form: L = B − α Tˆ .

(2.58)

In terms of the rectangular coordinate system we have: B= where, by (2.40),

∂ + V, ∂x0

V = Vi

∂ ∂xi

V i = −α 2 (h −1 )i0 = α 2 (η−2 − 1)u i u 0

in general (see (1.36)). By (2.42), substituting for

η2

(2.59)

(2.60)

from (1.72):

η2 = 1 − σ H, we obtain, in view of the definition (2.55), α2 =

1−σH 1 + ρH

(2.61)

Chapter 2. The Basic Geometric Construction

47

in general. Substituting from (2.61) in (2.60), we obtain the following expression for V i (rectangular coordinates): Vi =

Hσ Hβ0βi ui u0 = − 1 + ρH 1 + ρH

in general, Vi = −

H ∂0φ∂i φ 1 + ρH

(2.62)

(2.63)

in the irrotational isentropic case. Writing: ∂ Tˆ = Tˆ i i ∂x

(2.64)

we arrive, in view of (2.58), (2.59) and (2.62) at the following general expression for the vectorfield L in the rectangular coordinate system:     0u i H σ u ∂ Hβ0β i ∂ ∂ ∂ i i ˆ ˆ − αT − = 0 − αT + . (2.65) L= 1 + ρ H ∂xi 1 + ρ H ∂xi ∂x0 ∂x In the irrotational isentropic case this becomes:   ∂ ∂ H ∂0φ∂ i φ i ˆ L= − αT + . 1 + ρH ∂xi ∂x0

(2.66)

Now, the vectorfield T can be expressed in terms of the metric h induced on t as (2.24): h i j T j = κ 2 ∂i u. (2.67) Hence, we have: κ −2 = (h

−1 i j

) ∂i u∂ j u.

Thus κ can be viewed as the lapse function relative to h of the foliation of  ε t 0 = t Wε∗0

(2.68)

(2.69)

given by the family of surfaces {St,u : u ∈ [0, ε0 ]}. Let λ be the lapse function of the same foliation relative to the Euclidean metric g: λ−2 = (g −1 )i j ∂i u∂ j u =

3 

(∂i u)2 .

(2.70)

i=1

The above-mentioned comparison of the metrics h and g then implies that:

λ ≤ κ ≤ λ 1 + ρH.

(2.71)

48

Chapter 2. The Basic Geometric Construction

We may also introduce a vectorfield S analogous to T but defined relative to the Minkowski metric g rather than the acoustical metric h. Thus S is defined by the conditions that it be tangential to the hyperplanes t , orthogonal with respect to g to the family of surfaces {St,u : u ∈ [0, ε0 ]} in each t , and that it verify: Su = 1

(2.72)

S = T −Y

(2.73)

It follows that: where Y is a vectorfield which is tangential to the surfaces St,u . Moreover, we have: g(S, S) = λ2

(2.74)

and the vectorfield S can be expressed as (2.67): gi j S j = λ2 ∂i u.

(2.75)

We shall presently derive an expression for the ratio λ/µ. Let us denote: ν = β µ ∂µ u, β µ = (g −1 )µν βν

(2.76)

where β is the 1-form defined by (1.44). Since (g −1 )µν ∂µ t∂ν u = −∂0 u and ∂ µ φ∂ ν φ∂µ t∂ν u = −νβ0 substituting in (2.13) the expression (1.36) for (h −1 )µν we obtain: 1 = ∂0 u − Fνβ0 . µ On the other hand, since

(2.77)

β µ β ν ∂µ u∂ν u = ν 2

equation (2.9) reads: 0 = (g −1 )µν ∂µ u∂ν u − Fν 2 = −(∂0 u)2 + |du|2g − Fν 2 .

(2.78)

In view of the fact that according to (2.70) |du|2g = λ−2 we then obtain: (∂0 u)2 + Fν 2 =

1 . λ2

(2.79)

Let Sˆ be the vectorfield which is collinear to and in the same sense as S and of unit magnitude with respect to the Euclidean metric. Thus Sˆ is the interior unit normal to the surfaces St,u in a given hyperplane t with respect to the Euclidean metric. By (2.74), Sˆ = λ−1 S

(2.80)

Chapter 2. The Basic Geometric Construction

and by (2.75)

49

∂i u = gi j λ−1 Sˆ j

(2.81)

hence the definition (2.76) can be written in the form: ν = −β0 ∂0 u + λ−1 β Sˆ .

(2.82)

Substituting this expression in (2.79) the latter takes the form of a quadratic equation for the quantity (2.83) x = λ∂0 u namely: ax 2 − 2bx + c = 0

(2.84)

where the coefficients are given by: a = 1 + F(β0 )2 b = F(β0 )(β Sˆ ) c = F(β Sˆ )2 − 1

ˆ = βµ Sˆ µ . where β Sˆ = β( S)

The discriminant is: δ = b2 − ac = 1 + F((β0 )2 − (β Sˆ )2 ). Denoting by β the restriction to T t of β and by β/ the restriction to T St,u of β, we have: |β|2g = (β Sˆ )2 + |β/|2g/ where g/ is the metric induced by g on St,u . Thus, in view of the fact that from (1.44): σ = −(g −1 )µν βµ βν = (β0 )2 − |β|2g as well as the fact that, from (1.68), 1 + σ F = η−2 , we arrive at the following expression for δ: δ = η−2 + F|β/|2g/ .

(2.85)

Now, by (2.77),

λ = λ(∂0 u − Fνβ0 ). µ Substituting from (2.82), (2.83), we obtain: λ = ax − b. µ

√ By (2.84) this is equal to: ± δ. The negative root is impossible since both λ, µ > 0. We conclude that: √ λ = δ (2.86) µ with δ given by (2.85).

50

Chapter 2. The Basic Geometric Construction

We finally calculate the Jacobian of the mapping: (t, u, ϑ 1 , ϑ 2 ) → (x 0 , x 1 , x 2 , x 3 )

(2.87)

namely the transformation from acoustical to rectangular coordinates. Since x 0 = t we have: ∂x0 ∂x0 ∂x0 = 1, = = 0 : A = 1, 2. (2.88) ∂t ∂u ∂ϑ A Also, ∂xµ = Lµ, ∂t the rectangular components of the vectorfield L. Thus, ∂xi = L i : i = 1, 2, 3. ∂t

(2.89)

Next, applying (2.31) to the rectangular coordinates x i , i = 1, 2, 3, yields: ∂xi = T i + ξi ∂u

(2.90)

ξ i =  A X iA

(2.91)

where and:

∂xi : i = 1, 2, 3; A = 1, 2. (2.92) ∂ϑ A From (2.88), (2.89), (2.90) and (2.92), the Jacobian determinant of the transformation (2.87) is given by:   1 0 0 0   1 1  L T + ξ1 X1 X1  1 2
=  2 2 2 2 2.  L T + ξ X1 X2   L3 T 3 + ξ 3 X 3 X 3  1 2 X iA =

We have,

where:

 1   T + ξ1 X1 X1  1 2  2
=  T + ξ 2 X 12 X 22  =
 +
  T 3 + ξ3 X3 X3  1 2  1 1 1 T X X  1 2  
=  T 2 X 12 X 22   T 3 X3 X3  1 2

and

 1 1 1 ξ X X  1 2  
=  ξ 2 X 12 X 22  .  ξ3 X3 X3  1 2

Now, by virtue of (2.91),  1 1 1 X X X   A 1 2
 =  A ·  X 2A X 12 X 22  = 0.  X3 X3 X3  A=1 A 1 2 2 

Chapter 2. The Basic Geometric Construction

51

Thus
reduces to
 , which we can conveniently write as:
= (T, X 1 , X 2 )

(2.93)

the triple product of the vectors T , X 1 , X 2 in the 3-dimensional Euclidean space represented by each of the hyperplanes t . The vectors X 1 , X 2 at a point of t span the tangent plane to the surface St,u through that point. Since the difference T − S = Y lies in that plane (2.73) we can also write:
= (S, X 1 , X 2 ).

(2.94)

Now, the outer product X 1 ∧ X 2 of the vectors X 1 and X 2 is orthogonal to the tangent plane with respect to the Euclidean metric, therefore collinear to S and we can arrange the orientation so that it is in the same sense as S. Consequently, we have simply:
= |S||X 1 ∧ X 2 |

(2.95)

where | | = | |g denotes the magnitude with respect to the Euclidean metric g. Let us denote by g/ the metric induced on the surfaces St,u by the Euclidean metric. We have, g/ = g/ AB dϑ A dϑ B

(2.96)

g/ AB = (X A , X B ).

(2.97)

where: Here ( , ) = g( , ) denotes the Euclidean inner product. We then have: 

|X 1 ∧ X 2 | = |X 1 |2 |X 2 |2 − (X 1 , X 2 )2 = detg/.

(2.98)

In view of (2.74) and (2.98) we arrive at the following expression for the Jacobian determinant:

(2.99)
= λ detg/. Note that by the comparison of the metrics h and g discussed above, we have:





detg/ ≤ deth/ ≤ 1 + ρ H detg/.

(2.100)

In fact we can obtain an alternative expression for the Jacobian determinant
, directly from expression (2.93). For, let e be the volume form of the induced acoustical metric h on t . Then, since the determinant of the matrix of components h i j in rectangular coordinates is: (2.101) deth = 1 + ρ H √ the acoustical volume form e is 1 + ρ H times the Euclidean volume form ( , , ). Therefore (2.93) is equivalent to: e(T, X 1 , X 2 )
= √ . 1 + ρH

(2.102)

52

Chapter 2. The Basic Geometric Construction

On the other hand, since Tˆ is the unit normal to St,u relative to h, we have: e(Tˆ , X 1 , X 2 ) = /e(X 1 , X 2 ) (2.103) √ where /e is the area form of h /. Since also /e(X 1 , X 2 ) = deth/, while T = κ Tˆ , we obtain the following alternative expression for
: √ κ deth/ . (2.104)
= √ 1 + ρH A main theme of the present monograph is that, provided that the initial departure from the constant state is suitably small, the mapping (2.87) extends smoothly to the future boundary of the maximal development , however the lapse function λ vanishes there, therefore the differential of the inverse mapping √ becomes singular at this boundary. On the other hand, the St,u surface area element detg/ remains positive at the boundary of the maximal development. Moreover, the functions p, s, u µ , where u µ are the rectangular components of the velocity vectorfield, or s, βµ , where βµ are the rectangular components of the 1-form β defined by (1.44), or, in the irrotational isentropic case ψµ = ∂µ φ, namely the 1st partial derivatives of the wave function with respect to the rectangular coordinates, extend smoothly to the boundary as functions of the acoustical coordinates (t, u, ϑ 1 , ϑ 2 ). However, since the transformation to the rectangular coordinates √ (x 0 , x 1 , x 2 , x 3 ) becomes singular at the future boundary, the partial derivatives of σ and the u µ , or of the βµ , of order greater than or equal to 1, or, in the irrotational isentropic case, the partial derivatives of the wave function with respect to the rectangular coordinates of order greater than or equal to 2, blow up at this boundary. The prospect of the vanishing of λ, or equivalently of κ or µ, at the future boundary of the maximal development is thus to guide all considerations which follow.

Chapter 3

The Acoustical Structure Equations In this chapter we shall formulate the basic equations which govern the geometry of the 2-parameter foliation of the spacetime manifold Wε∗0 , endowed with the acoustical metric h, given by the surfaces {St,u : t ∈ [0, t∗ε0 ), u ∈ [0, ε0 ]}. We begin with the geometry of the 1-parameter foliation of (Wε∗0 , h) given by the hypersurfaces {t : t ∈ [0, t∗ε0 )}. In the previous chapter we introduced the normal vectorfield B generating a 1-parameter group mapping these hypersurfaces onto each other, as well as the lapse function α and the induced metric h. We now define k, the second fundamental form of the t (relative always to h) by: 2αk = L B h.

(3.1)

Here L B h denotes the restriction to T t of the Lie derivative of h with respect to B. (We can say that L B h is the Lie derivative with respect to B of the induced metric h). If X, Y are any two vectors tangent to t at a point we then have: αk(X, Y ) = h(D X B, Y ) = h(DY B, X)

(3.2)

where D is the covariant derivative operator associated to h. In terms of the rectangular coordinate system we have (see (2.59)): B=

∂ + V, ∂x0

where, by (2.62), Vi = −

V = Vi

Hβ0βi 1 + ρH

∂ ∂xi

(3.3)

(3.4)

with ρ the function defined by (2.54): ρ = βi β i =

3  (βi )2 . i=1

(3.5)

54

Chapter 3. The Acoustical Structure Equations

In the irrotational isentropic case we have, by (2.63), Vi = −

H ψ0 ψi 1 + ρH

and ρ is given by (2.52): ρ = ψi ψ i =

(3.6)

3  (ψi )2

(3.7)

i=1

(rectangular coordinates). Here and in the following we denote by ψµ , µ = 0, 1, 2, 3, the partial derivatives of the wave function φ with respect to the rectangular coordinates: ψµ = ∂µ φ.

(3.8)

The components ki j , i, j = 1, 2, 3, of k in the rectangular coordinate system are given, in general, according to (3.1) by: 2αki j =

∂h i j + (LV h)i j = Bh i j + h im ∂ j V m + h m j ∂i V m ∂x0

(3.9)

where h i j are the components of the induced metric on t in rectangular coordinates, given in general by (2.49): h i j = δi j + Hβi β j (3.10) and in the isentropic irrotational case by (2.50): h i j = δi j + H ψi ψ j .

(3.11)

We shall now derive a more detailed expression for ki j in the irrotational isentropic case which will play an important role in the sequel. Differentiating (3.6) yields: dH 1 H2 ψ0 ψm ∂i σ + ψ0 ψm ∂i ρ 2 (1 + ρ H ) dσ (1 + ρ H )2 H (ψm ∂i ψ0 + ψ0 ∂i ψm ). − (1 + ρ H )

∂i V m = −

(3.12)

Now, h j m ψ m = (δ j m + H ψ j ψm )ψ m = (1 + ρ H )ψ j and: h j m ∂i ψ m = (δ j m + H ψ j ψm )∂i ψ m = ∂i ψ j +

1 H ψ j ∂i ρ 2

hence: h j m ∂i V

m

1 ψ0 =− (1 + ρ H )



1 dH ψ j ∂i σ − H 2ψ j ∂i ρ + H ∂i ψ j dσ 2

 − H ψ j ∂i ψ0 . (3.13)

Chapter 3. The Acoustical Structure Equations

55

On the other hand we have: Bh i j = B(H ψi ψ j ) (3.14) dH ψi ψ j Bσ + H (ψ j ∂0 ψi + ψi ∂0 ψ j + V m (ψ j ∂m ψi + ψi ∂m ψ j )). = dσ Substituting from (3.6) and taking into account the fact that: ∂µ ψν = ∂ν ψµ = ∂µ ∂ν φ

(3.15)

we obtain, in view of (3.7), V m (ψ j ∂m ψi + ψi ∂m ψ j ) = −

1 H ψ0 (ψ j ∂i ρ + ψi ∂ j ρ). 2 (1 + ρ H )

Substituting in (3.14) then yields: dH ψi ψ j Bσ + H (ψ j ∂0 ψi + ψi ∂0 ψ j ) dσ 1 H 2ψ0 (ψ j ∂i ρ + ψi ∂ j ρ). − 2 (1 + ρ H )

Bh i j =

(3.16)

Substituting finally from (3.16) and (3.13) in (3.9) and taking into account the symmetry (3.15) we obtain the following formula for ki j in the irrotational isentropic case (rectangular coordinates):  dH ψ0 (ψi ∂ j σ + ψ j ∂i σ ) 2αki j = ψi ψ j Bσ − dσ (1 + ρ H ) 2H ψ0 ∂i ψ j . − (3.17) (1 + ρ H ) We now return to the general case. The components R i j mn of the curvature tensor of the induced metric in rectangular coordinates can be calculated directly from (3.11). They satisfy the Gauss equations: R i j mn + kim k j n − k j m kin = Ri j mn

(3.18)

where Ri j mn are the corresponding components of the curvature tensor of (Wε∗0 , h) (along t ). Since t is a 3-dimensional manifold the R i j mn can be expressed in terms of the components S i j of the corresponding Ricci tensor: S i j = (h

−1 mn

)

R im j n .

(3.19)

We have, R i j mn = h im S j n + h j n S im − h j m S in − h in S j m − (1/2)(h im h j n − h j m h in )S

(3.20)

56

Chapter 3. The Acoustical Structure Equations

where S is the scalar curvature of (t , h): S = (h

−1 i j

) Si j .

(3.21)

We turn to the geometry of a given characteristic hypersurface Cu , as described by the 1-parameter foliation corresponding to the family of sections {St,u : t ∈ [0, t∗ε0 )} by the hyperplanes t . The induced metric h/ on St,u has already been introduced in Chapter 2, as has the null (relative to h) vectorfield L whose integral curves are the bicharacteristic generators of Cu , null geodesics of (Wε∗0 , h). Any vector X tangent to Cu at a point p can be uniquely decomposed into a vector collinear to L and a vector tangent to the St,u section through p which we denote by X: X = c X L + X.

(3.22)

Thus we have at each p ∈ Cu a projection  of T p Cu onto T p St,u . If X, Y is a pair of vectors tangent to Cu at p, then: h(X, Y ) = h/(X, Y ). Let XA =

∂ ∂ϑ A

A = 1, 2

(3.23)

(3.24)

be the local vectorfields corresponding to the local coordinates (ϑ 1 , ϑ 2 ) introduced in the previous chapter. Given any Z ∈ T p Cu we can expand: Z = Z A X A .

(3.25)

Taking the h /-inner product with X B then yields: /(X B , Z ) = h/ AB Z A . h

(3.26)

The vectorfield L is normal in Cu to the family of sections of Cu and generates a 1-parameter group mapping these sections onto each other. We now define χ, the second fundamental form of St,u relative to Cu by: 2χ = L / L h.

(3.27)

Here L / L h denotes the restriction to T St,u of the Lie derivative of h with respect to L. (We can say that L / L h is the Lie derivative with respect to L of the induced metric h/.) If X, Y are any two vectors tangent to St,u at a point, we then have: χ(X, Y ) = h(D X L, Y ) = h(DY L, X).

(3.28)

Now, since the vectorfield Lˆ = µ−1 L is geodesic (see Chapter 2, equations (2.9)– (2.13)) , we have: D L L = µ−1 (Lµ)L. (3.29)

Chapter 3. The Acoustical Structure Equations

57

Denoting, χ AB = χ(X A , X B )

(3.30)

let us derive a propagation equation for χ AB along the generators of Cu . In view of formula (3.28) we have: Lχ AB = h(D L D X A L, X B ) + h(D X A L, D L X B ).

(3.31)

D L X A − D X A L = [L, X A ] = 0.

(3.32)

Now, Thus, by the definition of the curvature transformation of h, D L D X A L − D X A D L L = R(L, X A )L.

(3.33)

Substituting for D L L from (3.29) and taking the h-inner product with X B we obtain, in view of (3.28), (3.34) h(D L D X A L, X B ) = µ−1 (Lµ)χ AB − α AB where α AB are the curvature tensor components: α AB = R(X A , L, X B , L).

(3.35)

Recall that the curvature tensor is obtained from the curvature transformation as follows. If X, Y, Z , W are any four vectors at a point, R(W, Z , X, Y ) = h(W, R(X, Y )Z ). We turn to the second term on the right in (3.31). By (3.33) this term is h(D X A L, D X B L). Now, the fact that L is a null vectorfield with respect to the metric h, implies that for any vectorfield X the vectorfield D X L is h-orthogonal to L, therefore tangential to Cu . Thus, setting (3.36) W A = D X A L : A = 1, 2 equation (3.23) applies: h(W A , W B ) = h/(W A , W B ). By (3.26), we obtain, in view of (3.28), W A = χ AB X B ,

χ AB = (h/−1 ) BC χ AC

(3.37)

hence: /(W A , W B ) = χ AC χ BC . h This is the second term on the right in (3.31). We thus conclude that χ AB satisfies the following propagation equation along the generators of Cu : Lχ AB = µ−1 (Lµ)χ AB + χ AC χ BC − α AB .

(3.38)

58

Chapter 3. The Acoustical Structure Equations

We shall now deduce from (3.38) a propagation equation for trχ. Taking the trace of the curvature tensor we obtain the Ricci tensor: Sµν = (h −1 )κλ Rκµλν .

(3.39)

The reciprocal acoustical metric h −1 is expressed in terms of the frame L, T, X 1 , X 2 by: µ

(h −1 )µν = −α −2 L µ L ν − µ−1 (L µ T ν + T µ L ν ) + (h/−1 ) AB X A X νB .

(3.40)

This readily follows from the expressions for the inner products of the frame vectorfields with respect to h given in Chapter 2. In view of the expression (3.40) and of the symmetries of the curvature tensor, we have: µ

trα = (h /−1 ) AB Rµκνλ X A L κ X νB L λ = (h −1 )µν Rµκνλ L κ L λ = Sκλ L κ L λ or, simply, trα = S(L, L).

(3.41)

Taking account of the fact that Ltrχ = L((h /−1 ) AB χ AB ) = (h/−1 ) AB Lχ AB + χ AB L(h/−1 ) AB −1 AB = (h / ) Lχ AB − χ AB (h/−1 ) AC (h/−1 ) B D Lh/C D = (h /−1 ) AB Lχ AB − 2χ AB (h/−1 ) AC (h/−1 ) B D χC D when taking the trace of (3.38), we obtain the desired propagation equation for trχ: Ltrχ = µ−1 (Lµ)trχ − |χ|2h/ − S(L, L).

(3.42)

In terms of the acoustical coordinates (t, u, ϑ 1 , ϑ 2 ) L is given by (2.30), and the definition (3.27) becomes simply: 2χ AB =

∂h/ AB . ∂t

(3.43)

We shall now write down the Gauss and Codazzi equations of the embedding of St,u in the acoustical spacetime, that is, in the spacetime manifold endowed with the acoustical metric h. First, we consider the surface St,u as a member of the 1-parameter family {St,u : u ∈ [0, ε0 ]} defining a local foliation of the hypersurface t . The normal vectorfield which generates the 1-parameter group mapping the members of this family onto each other is the vectorfield T , introduced in the previous chapter, and the corresponding lapse function is the function κ. The second fundamental form θ of St,u relative to (t , h) is defined by: (3.44) 2κθ = L /T h where L /T h denotes the restriction to T St,u of the Lie derivative with respect to T of h. Then, if X, Y are any two vectors tangent to St,u at a point, κθ (X, Y ) = h(D X T, Y ) = h(D Y T, X).

(3.45)

Chapter 3. The Acoustical Structure Equations

59

Let us denote by k/ the restriction of k, the second fundamental form of t , to T St,u . Then, since by (2.44) and (2.47), L = α(α −1 B − κ −1 T ), we have: χ = α(k/ − θ ).

(3.46)

This allows us to express θ in terms of χ and k/. The Gauss equation of the embedding of St,u in t is expressed in terms of the local frame (X 1 , X 2 ) by: R / ABC D − θ AC θ B D + θ BC θ AD = R ABC D .

(3.47)

Here R / ABC D are the components of the curvature tensor of the induced metric h/. Since / ABC D in terms of the Gauss curvature K and the St,u is 2-dimensional we can express R components of the induced metric: R / ABC D = K (h/ AC h/ B D − h/ BC h/ AD ).

(3.48)

j

Multiplying equations (3.18) with X iA X B X Cm X nD (and summing repeated upper and lower indices) yields: R ABC D + k/ AC k/ B D − k/ BC k/ AD = R ABC D . (3.49) Substituting for R ABC D in terms of R ABC D from (3.49) in (3.47) we obtain: K (h / AC h /B D − h / BC h / AD ) + k/ AC k/ B D − k/ BC k/ AD − θ AC θ B D + θ BC θ AD = R ABC D . (3.50) In view of the symmetries of the curvature tensor we can express: R ABC D = ρ AB C D where  AB are the components of the area form of (St,u , h/):

 AB = deth/[ AB].

(3.51)

(3.52)

Here [AB] is the fully antisymmetric 2-dimensional symbol. Contracting (3.50) with /−1 ) B D and taking into account the fact that (1/2)(h /−1 ) AC (h (h /−1 ) B D  AB C D = h/ AC we obtain:

 1 (trh/ k/)2 − |k/|2h/ − (trh/ θ )2 + |θ |2h/ = ρ. (3.53) 2 This is the Gauss equation of the embedding of St,u in the acoustical spacetime. It contains the whole content of (3.50), in view of the symmetries involved. We shall presently derive the Codazzi equations of the embedding of St,u in the acoustical spacetime. Let X, Y, Z be arbitrary vectorfields tangent to a surface St,u . Extend them to the corresponding hypersurface Cu by the condition that they commute with K+

60

Chapter 3. The Acoustical Structure Equations

L. They are then tangential to each of the sections St,u , t ∈ [0, ε0 ), and by formula (3.28) we have: X (χ(Y, Z )) = D X (h(DY L, Z )) = h(D X DY L, Z ) + h(DY L, D X Z ). On the other hand, by the definition of the operator D /, (D / X χ)(Y, Z ) = X (χ(Y, Z )) − χ(D / X Y, Z ) − χ(Y, D / X Z ). Hence, / X Y, Z ) − χ(Y, D / X Z ). (D / X χ)(Y, Z ) = h(D X DY L, Z ) + h(DY L, D X Z ) − χ(D Subtracting a similar formula with X and Y interchanged and noting that: D X DY L − DY D X L = R(X, Y )L + D[X,Y ] L,

h(R(X, Y )L, Z ) = R(Z , L, X, Y )

and: D /XY − D / Y X = [X, Y ] we obtain: (D / X χ)(Y, Z ) − (D / Y χ)(X, Z ) = R(Z , L, X, Y ) +h(DY L, D X Z ) − h(D X L, DY Z ) − χ(Y, D / X Z ) + χ(X, D / Y Z ).

(3.54)

Let us define the 1-form ζ on St,u by: ζ (X) = h(D X L, T )

(3.55)

for any vector X tangent to St,u at a point. We shall find an expression for ζ below. For now we note that from (3.36), (3.37), and (2.23), we have: D X L = −µ−1 ζ(X)L + χ · X

(3.56)

for any vector X tangent to St,u at a point. Here χ · X is the vector tangent to St,u defined by: /(χ · X, Y ) = χ(X, Y ) h for any vector Y tangent to St,u at the same point. Substituting in the case of the St,u tangential vectorfield X the expression (3.56) for D X L we deduce that: /Y Z ) h(D X L, DY Z ) = µ−1 ζ(X)χ(Y, Z ) + χ(X, D

(3.57)

because −h(L, DY Z ) = h(DY L, Z ) = χ(Y, Z ). Substituting (3.57) and a similar expression with X and Y interchanged in (3.54) we arrive at the equation: (D / X χ)(Y, Z ) − (D / Y χ)(X, Z ) = R(Z , L, X, Y ) − µ−1 (ζ(X)χ(Y, Z ) − ζ(Y )χ(X, Z )). (3.58)

Chapter 3. The Acoustical Structure Equations

61

Let us set X = X A , Y = X B , Z = X C in this equation. Noting that by virtue of the symmetries of the curvature tensor we can express: R(X C , L, X A , X B ) =  AB βC

(3.59)

we obtain the Codazzi equations: D / A χ BC − D / B χ AC =  AB βC − µ−1 (ζ A χ BC − ζ B χ AC ).

(3.60)

Denoting by curl / χ the 1-form on St,u with components: curl / χC = where

1 AB / A χ BC − D / B χ AC )  (D 2

/−1 ) AC (h/−1 ) B D C D = ( deth/)−1 [ AB]  AB = (h

the whole content of the Codazzi equations is, in view of the symmetries involved, contained in the equation: (3.61) curl / χ = β − µ−1 ζ ∧ χ. Here, we denote by ζ ∧ χ the 1-form on St,u with components: (ζ ∧ χ)C =

1 AB  (ζ A χ BC − ζ B χ AC ). 2

In particular, equation (3.61) implies, is in fact equivalent to, the contraction of equations (3.60) with (h /−1 ) AC , namely the equation: div / χ − d/trχ = β ∗ − µ−1 (ζ · χ − ζ trχ).

(3.62)

Here d/trχ is the differential of the function trχ on St,u , ζ · χ is the 1-form on St,u with components: (ζ · χ) B = (h/−1 ) AC ζ A χ BC and β ∗ is the dual of β:

β B∗ = (h/−1 ) AC βC  AB .

We shall now derive an expression for the 1-form ζ on St,u defined by (3.55). Substituting in this definition the expression for L just preceding (3.46), we obtain, for any vector X tangent to St,u at a point, ζ (X) = h(D X B, T ) − ακ −1 h(D X T, T ) − h(T, T )X (ακ −1 ). Recalling that h(T, T ) = κ 2 we have h(D X T, T ) = κ Xκ, hence this reduces to: ζ (X) = h(D X B, T ) − κ Xα. In view of the formula (3.2) we have: h(D X B, T ) = αk(X, T ).

62

Chapter 3. The Acoustical Structure Equations

Let us define the 1-form  on St,u by: κ(X) = k(X, T )

(3.63)

for any vector X tangent to St,u at a point. The above formula for ζ can then be expressed in the form: ζ = κ(α − d/α). (3.64) We now define on each St,u section of Cu the 1-form: η = ζ + d/µ.

(3.65)

Since h(L, T ) = −µ and ζ is defined by (3.55) we have: η(X) = −h(D X T, L)

(3.66)

for every vector X tangent to St,u at a point. We denote: η A = η(X A ).

(3.67)

We can express the commutator  of the vectorfields L and T , a vectorfield tangential to the surfaces St,u (see Chapter 2), in terms of the 1-forms ζ and η. We have:  = A X A, Ah / AB = h(X B , D L T ) − h(X B , DT L) and, h(X B , D L T ) = −h(D L X B , T ) = −h(D X B L, T ) = −ζ B .

(3.68)

ˆ which, To compute h(X B , DT L) we express L in terms of the geodesic vectorfield L, since it is the gradient of a function (see equation (2.10)), satisfies: ˆ = h(Y, D X L) ˆ h(X, DY L)

(3.69)

for arbitrary vectors X, Y tangent to the spacetime manifold at a point, by virtue of the symmetry of the Hessian of any function. Thus, we have: ˆ = µh(X B , DT L) ˆ h(X B , DT L) = h(X B , DT (µ L)) −1 ˆ = µh(T, D X B L) = µh(T, D X B (µ L)) = h(T, D X B L) − µ−1 (X B µ)h(T, L) = ζB + X B µ = ηB .

(3.70)

Substituting (3.68) and (3.70) above yields:  A = −(h/−1 ) AB (ζ B + η B ).

(3.71)

Equation (3.70) together with the facts that  h(T, DT L) = h(T, D L T ) = L

1 2 κ 2



Chapter 3. The Acoustical Structure Equations

63

and h(L, DT L) = 0 (as L is null with respect to h), implies that: DT L = (h /−1 ) AB η B X A − α −1 (Lκ)L

(3.72)

(see (2.48)). Moreover, since D L T − DT L =  is given by (3.71) we also obtain: /−1 ) AB ζ B X A − α −1 (Lκ)L. D L T = −(h

(3.73)

Next, let us compute D X A T . Expanding in the frame L, T, X 1 , X 2 , we can write: D X A T = a A L + b A T + c AB X B . Taking the h-inner product with L gives: −µb A = h(D X A T, L) = −η A by (3.66). Taking the h-inner product with T gives:  −µa A + κ 2 b A = h(D X A T, T ) = X A

 1 2 κ . 2

Finally, taking the h-inner product with X C gives, in view of (3.45), c AB h / BC = h(D X A T, X C ) = h(D X A T, X C ) = κθ AC . The above imply: D X A T = α −1 κ A L + µ−1 η A T + κθ AB (h/−1 ) BC X C

(3.74)

where we have used the fact that by (3.64) and (3.65) α −1 η − d/κ = κ. Next, we compute DT T . Since T is tangential to the hypersurfaces t we may decompose: DT T = D T T + a B. Taking the h-inner product with B we obtain, substituting from (2.44), (2.47), −α 2 a = h(DT T, B) = h(DT T, L) + ακ −1 h(DT T, T )   1 2 −1 κ = −T µ − h(T, DT L) + ακ T 2 = −T µ − κ Lκ + αT κ = −κ(T α + Lκ)

(3.75)

64

Chapter 3. The Acoustical Structure Equations

where we have substituted from (3.72) and (2.48). From (2.67), (D T T )i = T j D j T i = κ 2 (h 2

−1 j k

) ∂k u D j {κ 2(h

−1 j k

−1 il

) ∂l u}

−1 il

−1

) ∂l u∂ j (κ 2 ) + κ 4 (h ) j k ∂k u(h 1 −1 −1 = T i T j κ −2 ∂ j (κ 2 ) + κ 4 (h )il ∂l {(h ) j k ∂ j u∂k u}. 2

= κ (h

) ∂k u(h

−1 il

) D l (∂ j u)

From (2.68) the last expression in parentheses is κ −2 , hence we obtain: 1 −1 (D T T )i = κ −2 T i T j ∂ j (κ 2 ) − (h )i j ∂ j (κ 2 ). 2 We conclude that: DT T =

1 −2 1 κ T (κ 2 )T − (h/−1 ) AB X B (κ 2 )X A . 2 2

(3.76)

Combining with the former result (3.75) for the component along B yields, in view of (2.44), (2.47), DT T = κα −2 (T α + Lκ)L +(κ −1 T κ + α −1 Lκ + α −1 T α)T 1 − (h/−1 ) AB X B (κ 2 )X A . 2

(3.77)

Note that since by (2.39) and (3.2), h(DT T, B) = −h(T, DT B) = −αk(T, T ). (3.75) yields at the same time the following propagation law for κ along the generators of Cu :   1 2 κ = −κ T α + αk(T, T ). (3.78) L 2 In the present monograph we have chosen µ instead of κ to describe the stacking of the hypersurfaces Cu . The two quantities are of course equivalent, being related by equation (2.48): µ = ακ so we could equally well have chosen to work with κ. By (3.78) we obtain the following propagation equation for µ: Lµ = κ Lα − αT α + αµk(Tˆ , Tˆ ). Here Tˆ is the unit vectorfield: (see (2.57)).

Tˆ = κ −1 T

(3.79)

Chapter 3. The Acoustical Structure Equations

65

We shall now derive an explicit form of the propagation equation for µ in the irrotational isentropic case, using the expression (3.17) for ki j in this case and the expression (2.61) for α (which is valid in general). Expanding B as in (2.44), we have, in view of (2.47), B = L + ακ −1 T.

(3.80)

Bσ = Lσ + ακ −1 T σ.

(3.81)

Thus,

Substituting in (3.17) yields:   2αψ0 ψTˆ 1 dH 2 2 ˆ ˆ αµk(T , T ) = α (ψTˆ ) − (T σ ) 2 dσ (1 + ρ H ) α H ψ0 ˆ i 1 dH (ψ ˆ )2 (Lσ ) − T (T ψi ) + µ 2 dσ T (1 + ρ H )

(3.82)

where: ψTˆ = Tˆ i ψi .

(3.83)

To calculate the terms in equation (3.79) involving derivatives of α, we deduce from (2.61) a formula for dα, the differential of α. Noting that σ + ρ = (ψ0 )2 , we obtain: −d(α 2 ) =

H (ψ0 )2 d H dσ + (dσ + α 2 dρ). 2 (1 + ρ H ) dσ (1 + ρ H )

(3.84)

Again, using the formulas σ = (ψ0 )2 − ρ, we obtain:

1 − α2 =

(ψ0 )2 H , 1 + ρH

ρ = ψ i ψi ,

  H ψ0 ψ i dσ + α 2 dρ = 2ψ0 dψ0 − dψi . 1 + ρH

(3.85)

Substituting in (3.84) then yields: −αdα =

1 (ψ0 )2 d H dσ 2 (1 + ρ H )2 dσ   H ψ0 H ψ0ψ i dψi . + dψ0 − (1 + ρ H ) 1 + ρH

(3.86)

66

Chapter 3. The Acoustical Structure Equations

It follows that: κ Lα − αT α =

1 (ψ0 )2 d H (T σ ) 2 (1 + ρ H )2 dσ 1 µ (ψ0 )2 d H − 2 (Lσ ) 2 α (1 + ρ H )2 dσ   H ψ0 H ψ0ψ i + T ψi T ψ0 − (1 + ρ H ) 1 + ρH   µ H ψ0 H ψ0ψ i Lψi . − 2 Lψ0 − α (1 + ρ H ) 1 + ρH

(3.87)

Substituting (3.87) and (3.82) in (3.79), the latter takes the form: 2 dH ψ0 − αψTˆ (T σ ) 1 + ρH dσ   2 1 µ ψ0 2 dH (Lσ ) − 2 − (αψTˆ ) 2α 1 + ρH dσ     H ψ0 H ψ0 ψ i + T ψ0 − + α Tˆ i T ψi (1 + ρ H ) 1 + ρH   H ψ0 ψ i µ H ψ0 Lψ0 − Lψi . − 2 1 + ρH α (1 + ρ H )

1 Lµ = 2



Recalling expression (2.66) for the vectorfield L, namely:   ∂ H ψ0 ψ i ∂ i ˆ + αT L= − ∂x0 1 + ρH ∂xi and defining:

(3.89)

ψ L = L µ ψµ

we have: ψL =

(3.88)

(3.90)

ψ0 − αψTˆ . 1 + ρH

(3.91)

This is the factor in parentheses in the first term on the right-hand side in (3.88). By (3.91) the factor in parentheses in the second term is: 

ψ0 1 + ρH

2 − (αψTˆ )2 = ψ L (ψ L + 2αψTˆ ).

(3.92)

Again by (3.89) the coefficient of the factor H ψ0 /(1 + ρ H ) in the third term on the right-hand side of (3.88) is: L µ (T ψµ ) = L µ T ν ∂ν ψµ = L µ T ν ∂µ ψν = T i (Lψi ) =

µ ˆi T (Lψi ) α

(3.93)

Chapter 3. The Acoustical Structure Equations

67

using (3.15). Thus, again in view of (3.89), this coefficient combines with the coefficient of the same factor in the fourth term to: −

µ µ L (Lψµ ). α2

(3.94)

Since also, by (1.71), (2.61), H 1 =F (1 + ρ H ) α 2

(3.95)

it follows that the sum of the last two terms on the right-hand side of (3.88) is equal to: −µFψ0 L µ (Lψµ ) We conclude that (3.88) reduces to: Lµ = m + µe where: m=

dH 1 (ψ L )2 (T σ ) 2 dσ

(3.96)

(3.97)

and:

dH 1 (Lσ ) − Fψ0 L µ (Lψµ ). ψ L (ψ L + 2αψTˆ ) (3.98) 2α 2 dσ Equation (3.96) is the desired propagation equation for µ. We can then express Lκ in terms of Lµ to obtain: (3.99) Lκ = m  + κe e=−

where:

m  = α −1 m,

e = e − α −1 Lα.

(3.100)

Using (3.86) evaluated on L, (3.91), (3.95), as well as the expression (3.89), we find: e =

dH 1 (ψTˆ )2 (Lσ ) + Fψ0 Tˆ i (Lψi ). 2 dσ

(3.101)

The propagation equations (3.96), (3.99) play a fundamental role in this monograph. We now return again to the general case. To complete the set of connection coefficients of the frame L, T, X 1 , X 2 , we compute D X A X B . We decompose: DX A X B = D / X A X B + a AB L + b AB T. Taking the h-inner product with L gives: −µb AB = h(L, D X A X B ) = −h(D X A L, X B ) = −χ AB . Taking the h-inner product with T gives: −µa AB + κ 2 b AB = h(T, D X A X B ) = −h(D X A T, X B ) = −κθ AB .

68

Chapter 3. The Acoustical Structure Equations

Hence, in view of (3.46), we obtain: b AB = µ−1 χ AB ,

a AB = α −1 k/ AB

We collect the expressions which we have obtained for the connection coefficients of the frame L, T, X 1 , X 2 in the following table: D L L = µ−1 (Lµ)L DT L = −α −1 (Lκ)L + η A X A D X A L = −µ−1 ζ A L + χ AB X B D L T = −α −1 (Lκ)L − ζ A X A DT T = κα −2 (T α + Lκ)L + (µ−1 T µ + α −1 Lκ)T 1 −1 AB / ) X B (κ 2 )X A − (h 2 D X A T = α −1 κ A L + µ−1 η A T + κθ AB X B DL X A = DX A L / X A X B + α −1 k/ AB L + µ−1 χ AB T. DX A X B = D

(3.102)

Here, /−1 ) AB ζ B , ζ A = (h

η A = (h /−1 ) AB η B , χ AB = (h/−1 ) BC χ AC ,

θ AB = (h/−1 ) BC θ AC .

Note that we have not obtained an expression for DT X A . This reflects the arbitrariness in the coordinates ϑ 1 , ϑ 2 . It shall not play any role in the following. We now investigate the connection between the derivative of χ relative to T and the derivative of η tangentially to the surfaces St,u . The proper notion of derivative of χ relative to T is that of the Lie derivative. However, as χ is defined on each surface St,u , we first extend χ to a symmetric 2-covariant tensorfield on spacetime by the conditions: χ(X, T ) = χ(X, L) = 0 for any vectorfield X. We then define L /T χ to be the restriction to St,u of LT χ, the Lie derivative with respect to T of χ thus extended. We then have: (L /T χ)(X A , X B ) = (DT χ)(X A , X B ) + χ(X A , D X B T ) + χ(X B , D X A T ) = (DT χ)(X A , X B ) + κθ BC χ AC + κθ AC χ BC

(3.103)

where we have substituted from table (3.102). To compute (DT χ)(X A , X B ) we write: (DT χ)(X A , X B ) = T (χ AB ) − χ(DT X A , X B ) − χ(X A , DT X B ).

(3.104)

Here and in the following we denote by  the h-orthogonal projection to the surfaces St,u . Now, T (χ AB ) = T (h(D X A L, X B )) = h(DT D X A L, X B ) + h(D X A L, DT X B ).

(3.105)

Chapter 3. The Acoustical Structure Equations

69

We have, DT D X A L − D X A DT L − D[T ,X A ] L = R(T, X A )L hence: h(DT D X A L, X B ) = h(D X A DT L, X B ) +h(D[T ,X A ] L, X B ) + R(X B , L, T, X A ).

(3.106)

Substituting for DT L from table (3.102) we obtain: /XA X B) h(D X A DT L, X B ) = −α −1 (Lκ)χ AB + X A (η B ) − η(D / AηB = −α −1 (Lκ)χ AB + D

(3.107)

where we denote: (D / X A η)(X B ) = D / AηB . From Chapter 2 (see (2.31)) the commutator [T, X A ] is tangential to the surfaces St,u . Therefore, h(D[T ,X A ] L, X B ) = χ([T, X A ], X B ) = χ(DT X A , X B ) − χ(D X A T, X B ) (3.108) = χ(DT X A , X B ) − κθ AC χ BC where we have substituted for D X A T from Table 3.102. Also, substituting for D X A L from Table 3.102 yields: h(D X A L, DT X B ) = −µ−1 ζ A h(L, DT X B ) + χ(X A , DT X B ) = = µ−1 ζ A η B + χ(X A , DT X B )

(3.109)

in view of the fact that −h(L, DT X B ) = h(DT L, X B ) = η B . Substituting (3.107) and (3.108) in (3.106), and the result together with (3.109) in (3.105), and the result in turn in (3.104), we obtain: / A η B + µ−1 ζ A η B (3.110) (DT χ)(X A , X B ) = D −1 C −α (Lκ)χ AB − κθ AC χ B − R(X A , T, X B , L) The symmetric part of this equation is: (DT χ)(X A , X B ) =

1 1 / B η A ) + µ−1 (ζ A η B + ζ B η A ) (D / AηB + D 2 2 1 −α −1 (Lκ)χ AB − κ(θ AC χ BC + θ BC χ AC ) 2 1 − (R(X A , T, X B , L) + R(X B , T, X A , L)). 2

(3.111)

The antisymmetric part is an equation for curl / η = curl / ζ , a component of the Codazzi equations of the embedding of the hypersurfaces t in the acoustical spacetime. Since ki j , the 2nd fundamental form of the t , is expressed directly by equations (3.9) in terms

70

Chapter 3. The Acoustical Structure Equations

of the derivatives of the wave function, this equation contains no new information. Substituting finally (3.111) in (3.103) and denoting (L /T χ)(X A , X B ) = LT χ AB we obtain the desired equation: L /T χ AB =

1 1 (D / AηB + D / B η A ) + µ−1 (ζ A η B + ζ B η A ) 2 2 1 −1 −α (Lκ)χ AB + κ(θ AC χ BC + θ BC χ AC ) − γ AB 2

(3.112)

where we have defined: γ AB =

1 (R(X A , T, X B , L) + R(X B , T, X A , L)). 2

(3.113)

In the sequel we shall use, in addition to the frame field L, T, X 1 , X 2 , the frame field L, L, X 1 , X 2 , obtained by replacing T with the incoming null vectorfield L, given by: L = α −1 κ L + 2T.

(3.114)

At each point p ∈ Wε∗0 , L( p) and L( p) are, respectively, outgoing and incoming futuredirected null normals at p to the surface St,u through p, with L normalized by the condition Lt = 1 (2.12) and L normalized by the condition: h(L, L) = −2µ.

(3.115)

The reciprocal acoustical metric is expressed in terms of the null frame by: µ

(h −1 )µν = −(1/2µ)(L µ L ν + L µ L ν ) + (h/−1 ) AB X A X νB .

(3.116)

From Table 3.102 we can deduce in a straightforward manner a table for the connection coefficients of the null frame L, L, X 1 , X 2 : D L L = µ−1 (Lµ)L D L L = −L(α −1 κ)L + 2η A X A D X A L = −µ−1 ζ A L + χ AB X B D L L = −2ζ A X A D L L = (µ−1 Lµ + L(α −1 κ))L − 2µ(h/−1 ) AB X B (α −1 κ)X A D X A L = µ−1 η A L + χ AB X B DL X A = DX A L 1 1 DX A X B = D / X A X B + µ−1 χ AB L + µ−1 χ AB L. 2 2 Here,

χ AB = (h/−1 ) BC χ AC

(3.117)

Chapter 3. The Acoustical Structure Equations

71

and, in analogy with (3.46), χ = κ(k/ + θ )

(3.118)

Recapitulating, the geometry of a characteristic hypersurface Cu is described in terms of the family of sections {St,u : t ∈ [0, t∗ε0 )}, namely their induced metric h/, Gauss curvature K , and second fundamental form χ relative to Cu . The last satisfies the propagation equation (3.38) along the generators of Cu as well as the Codazzi equation (3.61) on each section. The manner of the stacking of the hypersurfaces Cu : u ∈ [0, ε0 ], forming a local foliation of the spacetime manifold, is described by the function µ which satisfies the propagation equation (3.96) along the generators of each Cu . We shall now derive expressions for the operators
h and
h˜ acting on functions f defined on the spacetime domain Wε∗0 . We have:
h f = trw

(3.119)

where w is the Hessian of f with respect to the metric h: w = D(d f )

(3.120)

(D denotes as in the preceding the covariant derivative operator corresponding to h, and d f denotes the differential of f ). It is a symmetric 2-covariant tensorfield on Wε∗0 . In (3.119) tr denotes the trace with respect to h. In view of the expression (3.116) for h −1 we have: trw = (h −1 )µν wµν = −µ−1 w L L + (h/−1 ) AB w AB . (3.121) Let also w / be the Hessian of the restriction of f to St,u with respect to the induced metric h /: w /=D /(d/ f ) (3.122) (D / denotes as in the preceding the covariant derivative operator on St,u corresponding to h /, and d/ f denotes the differential of the restriction of f to St,u ). It is a symmetric / =
h/ is given by: 2-covariant tensorfield on each St,u . Then the operator

/ f = trw /.

(3.123)

(Here tr is the trace with respect to h /.) In terms of the frame, we have, w AB = w(X A , X B ) = X A (X B f ) − (D X A X B ) f

(3.124)

/(X A , X B ) = X A (X B f ) − (D / X A X B ) f. w / AB = w

(3.125)

and, By the last line of Table 3.117 we then obtain:

Hence:

1 1 w AB = w / AB − µ−1 χ AB (L f ) − µ−1 χ AB (L f ). 2 2

(3.126)

1 1 (h /−1 ) AB w AB = trw / − trχ (L f ) − µ−1 trχ(L f ). 2 2

(3.127)

72

Chapter 3. The Acoustical Structure Equations

Also, by the fourth line of Table 3.117, w L L = w(L, L) = L(L f ) − (D L L) f = L(L f ) + 2ζ · d/ f.

(3.128)

Substituting (3.128) and (3.127) in (3.121) we obtain, in view of (3.119) and (3.123) the following formula for the operator
h : 1 1
h f =
/ f − µ−1 trχ(L f ) − µ−1 trχ(L f ) 2 2 −1 −1 − µ L(L f ) − 2µ ζ · d/ f.

(3.129)

The fact that the conformal acoustical metric h˜ is related to the acoustical metric h by: h˜ µν = h µν implies that for an arbitrary function f :
h˜ f = −1
h f + −2 (h −1 )µν ∂µ ∂ν f d −1 µν (h ) ∂µ σ ∂ν f. = −1
h f + −2 dσ

(3.130)

In view of the expression (3.116) for h −1 we have: 1 1 (h −1 )µν ∂µ σ ∂ν f = − µ−1 (Lσ )(L f ) − µ−1 (Lσ )(L f ) + d/σ · d/ f 2 2 Thus, setting:

  1 d (Lσ ) trχ + −1 2 dσ   d 1 ν= (Lσ ) trχ + −1 2 dσ ν=

(3.131) (3.132)

we obtain the following formula for the operator
h˜ : / f − µ−1 L(L f ) − µ−1 (ν L f + ν L f ) 
h˜ f =
d d/σ · d/ f. − 2µ−1 ζ · d/ f + −1 dσ

(3.133)

Consider now the functions L i and Tˆ i , i = 1, 2, 3, namely the components of the vectorfields L and Tˆ in rectangular coordinates. Recall that L 0 = 1 and Tˆ 0 = 0. We shall now derive expressions for the derivatives of these functions with respect to L and T , as well as expressions for the differentials of the restrictions of these functions to each surface St,u . These expressions shall allow us to draw the conclusion that these derivatives and differentials remain regular as µ → 0.

Chapter 3. The Acoustical Structure Equations

73

In the following we shall denote by ∇ the operator of covariant differentiation in spacetime with respect to the Minkowski metric g. Since Minkowski spacetime has a linear structure, for any tensorfield T the components of ∇T in any linear system of coordinates (in particular in rectangular coordinates) are simply the partial derivatives of the corresponding components of T . The operator of covariant differentiation in spacetime with respect to the acoustical metric h shall be denoted by D, as above. For an arbitrary vectorfield W in spacetime, we have, in an arbitrary coordinate system, ∂ W ν hν + !µλ W λ ∂xµ

Dµ W ν =

(3.134)

h

ν are the connection coefficients of the acoustical metric h where !µλ µν in the given coordinate system. On the other hand, in the same coordinate system we have:

∇µ W ν =

g ∂Wν ν λ + ! µλ W ∂xµ

(3.135)

g

ν are the connection coefficients of the Minkowskian metric g where !µλ µν in that coordinate system. Therefore, Dµ W ν = ∇µ W ν +
νµλ W λ (3.136)

where
is the difference of the two connections, a tensorfield, whose components in any given coordinate system are: g

h

ν ν − !µλ .
νµλ =!µλ

(3.137)

Let us now fix the coordinate system to be a rectangular coordinate system of the Minkowskian metric g. We then have: g

ν !µλ = 0,

hence: ∇µ W ν =

∂Wν ∂xµ

and (3.137) reduces to: h

ν = (h −1 )νκ !µλκ
νµλ =!µλ

(3.138)

where:

1 (∂µ h λκ + ∂λ h µκ − ∂κ h µλ ). (3.139) 2 For the remainder of this chapter and throughout the succeeding chapters up to Chapter 13 we shall confine ourselves to the irrotational isentropic case. The components of the acoustical metric in rectangular coordinates are then given by (1.69), and their 1st partial derivatives with respect to these coordinates are: !µλκ =

∂κ h µν =

dH ∂κ σ ∂µ φ∂ν φ + H (∂µ φ∂κ ∂ν φ + ∂ν φ∂κ ∂µ φ). dσ

(3.140)

74

Chapter 3. The Acoustical Structure Equations

Substituting in (3.139) we obtain: !µλκ =

1 dH (ψλ ψκ ∂µ σ + ψµ ψκ ∂λ σ − ψµ ψλ ∂κ σ ) + H ψκ ∂µ ψλ . 2 dσ

(3.141)

Consider now the vectorfield: L(L µ )

∂ ∂xµ

(rectangular coordinates)

(3.142)

Since L 0 = 1, we have L(L 0 ) = 0, therefore the vectorfield (3.142) is tangential to the hypersurfaces t . Hence, it can be expanded as: a L Tˆ + b L

(3.143)

where b L is an St,u - tangential vectorfield. On the other hand, the vectorfield (3.142) is also ∇ L L, the covariant derivative of L with respect to itself in the background flat Minkowski connection. Thus, ∇ L L = a L Tˆ + b L . (3.144) Taking the h-inner product with Tˆ we obtain, since h(Tˆ , Tˆ ) = 1, h(Tˆ , b L ) = 0, a L = h(∇ L L, Tˆ ).

(3.145)

b L = b LA X A

(3.146)

Writing: and taking the h-inner product with X B we obtain, since h(X A , X B ) = h/ AB , / AB = h(∇ L L, X B ). b LA h

(3.147)

Now, from (3.136), (3.138) we have: h(∇ L L, Tˆ ) = h(D L L, Tˆ ) − !αβν L α L β Tˆ ν .

(3.148)

Substituting for D L L from Table 3.102 we obtain: h(D L L, Tˆ ) = −κ −1 (Lµ) = −κ −1 m − αe

(3.149)

where we have used the propagation equation (3.96). Also, by (3.141),  1 d H  −1 !αβν L α L β Tˆ ν = −κ (ψ L )2 (T σ ) + 2ψ L ψTˆ (Lσ ) + H ψTˆ L µ (Lψµ ). (3.150) 2 dσ Substituting (3.149) and (3.150) in (3.148), by virtue of equation (3.97) the terms involving κ −1 cancel and we obtain: h(∇ L L, Tˆ ) = −αe −

dH ψ L ψTˆ (Lσ ) − H ψTˆ L µ (Lψµ ). dσ

(3.151)

Chapter 3. The Acoustical Structure Equations

75

Substituting for e from (3.98) and taking account of the fact that by (3.95) and (3.91), α Fψ0 =

α −1 H ψ0 = H (α −1 ψ L + ψTˆ ) 1 + ρH

(3.152)

we obtain, in view of (3.145), aL = α

−1

 ψL

1 dH µ ψ L (Lσ ) + H L (Lψµ ) . 2 dσ

(3.153)

Also, since h(D L L, X B ) = 0, we have, using (3.141), (3.154) h(∇ L L, X B ) = −!αβν L α L β X νB   1 dH 2 ψ B ψ L (Lσ ) − (ψ L )2 (d/ B σ ) − H ψ B L µ (Lψµ ) =− 2 dσ where we denote: ψ A = ψi X iA .

(3.155)

We thus obtain, in view of (3.147):   1 dH  2 ψ B ψ L (Lσ ) − (ψ L )2 (d/ B σ ) + H ψ B L µ (Lψµ ) . b LA = −(h /−1 ) AB 2 dσ (3.156) Consider next the vectorfield: L(Tˆ µ )

∂ ∂xµ

(rectangular coordinates).

(3.157)

Since Tˆ 0 = 0, we have L(Tˆ 0 ) = 0, therefore the vectorfield (3.157) is tangential to the hypersurfaces t . Hence, it can be expanded as: p L Tˆ + q L

(3.158)

where q L is an St,u -tangential vectorfield. On the other hand, the vectorfield (3.142) is also ∇ L Tˆ , the covariant derivative of Tˆ with respect to L in the background flat Minkowski connection. Thus, (3.159) ∇ L Tˆ = p L Tˆ + q L . Taking the h-inner product with Tˆ we obtain, since h(Tˆ , Tˆ ) = 1, h(Tˆ , q L ) = 0, p L = h(∇ L Tˆ , Tˆ ).

(3.160)

q L = q LA X A

(3.161)

Writing:

76

Chapter 3. The Acoustical Structure Equations

and taking the h-inner product with X B we obtain, since h(X A , X B ) = h/ AB , q LA h / AB = h(∇ L Tˆ , X B ).

(3.162)

Since h(D L Tˆ , Tˆ ) = (1/2)L(h(Tˆ , Tˆ )) = 0, we have, using (3.141), h(∇ L Tˆ , Tˆ ) = −!αβν L α Tˆ β Tˆ ν 1 dH (ψ ˆ )2 (Lσ ) − H ψTˆ Tˆ i (Lψi ). =− 2 dσ T Hence, in view of (3.160) we obtain:  1 dH p L = −ψTˆ ψTˆ (Lσ ) + H Tˆ i (Lψi ) . 2 dσ

(3.163)

(3.164)

Also, we have: h(∇ L Tˆ , X B ) = h(D L Tˆ , X B ) − !αβν L α Tˆ β X νB

(3.165)

h(D L Tˆ , X B ) = κ −1 h(D L T, X B ) = −κ −1 ζ B .

(3.166)

From Table 3.102,

On the other hand, by (3.141), !αβν L α Tˆ β X νB =

 1 d H  −1 κ ψ L ψ B (T σ ) + ψTˆ ψ B (Lσ ) − ψ L ψTˆ (d/ B σ ) 2 dσ (3.167) +H ψ B Tˆ i (Lψi ).

Substituting (3.166) and (3.167) in (3.165) and defining: ζ A = ζ A +

1 dH ψ L ψ A (T σ ) 2 dσ

(3.168)

we obtain, in view of (3.162), 1 dH ψ ˆ ( ψ B (Lσ ) − ψ L (d/ B σ )) 2 dσ T − H ψ B Tˆ i (Lψi ).

q LA h / AB = −κ −1 ζ B −

(3.169)

We shall show that κ −1 ζ  is regular as µ → 0. By (3.64) and (3.168) we have: κ −1 ζ A = α A − d/ A α

(3.170)

where:  A =  A +

1 dH ψ L ψ A (T σ ). 2ακ dσ

(3.171)

Chapter 3. The Acoustical Structure Equations

77

By the definition (3.63) and the formula (3.17) for ki j ,  A = ki j Tˆ i X A  1 dH ψ0 −1 ψTˆ ψ A (Bσ ) − (ψ ˆ (d/ A σ ) + κ ψ A (T σ )) = 2α dσ (1 + ρ H ) T 1 H ψ0 ˆ i − (3.172) T (d/ A ψi ). α (1 + ρ H ) j

Here we have used the fact that by (3.15): j j Tˆ i X A ∂i ψ j = Tˆ i X A ∂ j ψi = Tˆ i (X A ψi ) = Tˆ i (d/ A ψi ).

Substituting we then obtain:

Bσ = Lσ + ακ −1 T σ   ψ0 1 dH − αψTˆ ψ A (T σ ) 2ακ dσ 1 + ρ H   ψ0 1 dH ψTˆ ψ A (Lσ ) − (d/ A σ ) + 2α dσ 1 + ρH 1 H ψ0 ˆ i T (d/ A ψi ). − α (1 + ρ H )

A = −

(3.173)

By (3.91) the factor in parentheses in the first term on the right-hand side of (3.173) is simply ψ L , therefore in view of the definition (3.171) we obtain:   1 dH ψ0 1 H ψ0 ˆ i  A = ψTˆ ψ A (Lσ ) − (d/ A σ ) − T (d/ A ψi ). (3.174) 2α dσ 1 + ρH α (1 + ρ H ) This is regular as µ → 0. It then follows from (3.170) that so is κ −1 ζ  . In fact, by (3.86), −αd/ A α =

1 (ψ0 )2 d H d/ A σ 2 (1 + ρ H )2 dσ   H ψ0ψ i H ψ0 d/ A ψ0 − d/ A ψi . + (1 + ρ H ) 1 + ρH

Substituting this together with (3.174) in (3.170) yields:    1 dH ψ0 ψ0 −1  κ ζA = − αψTˆ (d/ A σ ) ψTˆ ψ A (Lσ ) + 2 dσ α(1 + ρ H ) 1 + ρ H  H ψ0 H ψ0 ψ i + d/ A ψi − α Tˆ i d/ A ψi . d/ A ψ0 − α(1 + ρ H ) 1 + ρH In view of (3.91) and (3.89) this reduces to, simply:  1 dH ψTˆ ψ A (Lσ ) + (ψTˆ + α −1 ψ L )ψ L (d/ A σ ) κ −1 ζ A = 2 dσ + H (ψTˆ + α −1 ψ L )L µ (d/ A ψµ ).

(3.175)

78

Chapter 3. The Acoustical Structure Equations

Consider next the vectorfield: T (L µ )

∂ ∂xµ

(rectangular coordinates).

(3.176)

Since L 0 = 0, we have T (L 0 ) = 0, therefore the vectorfield (3.176) is tangential to the hypersurfaces t . Hence, it can be expanded as: aT Tˆ + bT

(3.177)

where bT is an St,u -tangential vectorfield. On the other hand, the vectorfield (3.176) is also ∇T L, the covariant derivative of L with respect to T in the background flat Minkowski connection. Thus, ∇T L = aT Tˆ + bT . (3.178) Taking the h-inner product with Tˆ yields: aT = h(∇T L, Tˆ )

(3.179)

bT = bTA X A

(3.180)

and, writing and taking the h-inner product with X B yields:

We have:

bTA h / AB = h(∇T L, X B ).

(3.181)

h(∇T L, Tˆ ) = h(DT L, Tˆ ) − !αβν T α L β Tˆ ν .

(3.182)

h(DT L, Tˆ ) = Lκ

(3.183)

From Table 3.102, while from (3.141), !αβν T α L β Tˆ ν = κ



1 dH (ψTˆ )2 (Lσ ) + H ψTˆ Tˆ i (Lψi ) . 2 dσ

(3.184)

Here we have used the fact that by (3.15): Tˆ α L β ∂α ψβ = Tˆ α L β ∂β ψα = Tˆ i (Lψi ). Substituting (3.183) and (3.184) in (3.182) we obtain, in view of (3.179),  1 dH ψTˆ (Lσ ) + H Tˆ i (Lψi ) aT = Lκ − κψTˆ 2 dσ

(3.185)

and Lκ is expressed by (3.99). We have: h(∇T L, X B ) = h(DT L, X B ) − !αβν T α L β X νB

(3.186)

Chapter 3. The Acoustical Structure Equations

79

and from Table 3.102, h(DT L, X B ) = η B = ζ B + d/ B µ

(3.187)

while from (3.141), !αβν T α L β X νB =

 1 dH  ψ L ψ B (T σ ) + κψTˆ ( ψ B (Lσ ) − ψ L (d/ B σ )) 2 dσ (3.188) + κ H ψ B Tˆ i (Lψi ).

Substituting (3.187) and (3.188) in (3.186) we obtain, in view of (3.181), 1 dH ψ L ψ B (T σ ) (3.189) 2 dσ dH 1 ψˆ ( ψ B (Lσ ) − ψ L (d/ B σ )) + H ψ B Tˆ i (Lψi ) . −κ 2 T dσ

bTA h / AB = η B −

Consider finally the vectorfield: T (Tˆ µ )

∂ ∂xµ

(rectangular coordinates)

(3.190)

which is tangential to the hypersurfaces t . It can be expanded as: pT Tˆ + qT

(3.191)

where qT is an St,u -tangential vectorfield. On the other hand, the vectorfield (3.190) is also ∇T Tˆ , the covariant derivative of Tˆ with respect to T in the background flat Minkowski connection. Thus, (3.192) ∇T Tˆ = pT Tˆ + qT . Taking the h-inner product with Tˆ yields: pT = h(∇T Tˆ , Tˆ )

(3.193)

qT = qTA X A

(3.194)

and, writing and taking the h-inner product with X B yields: qTA h / AB = h(∇T Tˆ , X B ).

(3.195)

Since h(DT Tˆ , Tˆ ) = (1/2)T (h(Tˆ , Tˆ )) = 0, we have, from (3.141), h(∇T Tˆ , Tˆ ) = −!αβν T α Tˆ β Tˆ ν 1 dH (ψ ˆ )2 (T σ ) − H ψTˆ Tˆ i (T ψi ) =− 2 dσ T

(3.196)

80

Chapter 3. The Acoustical Structure Equations

hence, in view of (3.193) we obtain: pT = − We have:

1 dH (ψ ˆ )2 (T σ ) − H ψTˆ Tˆ i (T ψi ). 2 dσ T

h(∇T Tˆ , X B ) = h(DT Tˆ , X B ) − !αβν T α Tˆ β X νB

(3.197)

(3.198)

and from Table 3.102, h(DT Tˆ , X B ) = κ −1 h(DT T, X B ) = −d/ B κ

(3.199)

 1 dH 2ψTˆ ψ B (T σ ) − κ(ψTˆ )2 (d/ B σ ) !αβν T α Tˆ β X νB = 2 dσ + H ψ B Tˆ i (T ψi ).

(3.200)

while from (3.141),

Substituting (3.199) and (3.200) in (3.198), we finally obtain, in view of (3.195),   1 dH ψTˆ 2 ψ B (T σ ) − κψTˆ (d/ B σ ) 2 dσ − H ψ B Tˆ i (T ψi ).

qTA h / AB = −d/ B κ −

(3.201)

We proceed to derive expressions for the differentials d/(L i ) and d/(Tˆ i ) of the restrictions of the functions L i and Tˆ i , i = 1, 2, 3, to each surface St,u . Let X be an arbitrary element of Tx St,u , the tangent plane to St,u at a point x. Then ∇ X L, the covariant derivative of the vectorfield L with respect to X in the background flat Minkowski connection is given in rectangular coordinates by: ∇ X L = X (L µ )

∂ ∂xµ

(rectangular coordinates).

(3.202)

Since L 0 = 1, we have X (L 0 ) = 0, hence this vector is tangential to the hypersurface t . It can therefore be expanded as: ∇ X L = a/ X Tˆ + b/ X

(3.203)

where b/ X ∈ Tx St,u . Now, a / X and b/ X depend linearly on X. Therefore there is an element /a(x) ∈ Tx∗ St,u (the cotangent plane to St,u at x) and an element b/(x) ∈ L(Tx St,u , Tx St,u ) (the set of linear transformations of Tx St,u ) such that: /(x) /X = X · a a

and

b/ X = X · b/(x).

(3.204)

Thus, a /(x) is the value at x of a 1-form /a on St,u and b/(x) is the value at x of a type T11 -type tensorfield b/ on St,u . Setting X = X A (x), A = 1, 2, in (3.202), (3.203), and letting x vary over St,u , we obtain: d/ A (L i )

∂ = ∇ X A L = a/ A Tˆ + b/ A ∂xi

(3.205)

Chapter 3. The Acoustical Structure Equations

81

where: /A = a a /(X A )

and

b/ A = b/(X A ) = b/ BA X B .

(3.206)

Taking the h-inner product with Tˆ we obtain, since h(Tˆ , Tˆ ) = 1 and h(b/ A , Tˆ ) = 0, / A = h(∇ X A L, Tˆ ). a

(3.207)

Also, taking the h-inner product with X C we obtain, since h(Tˆ , X C ) = 0 and h(X B , X C ) = h / BC , b/ BA h / BC = h(∇ X A L, X C ). (3.208) We have:

h(∇ X A L, Tˆ ) = h(D X A L, Tˆ ) − !αβν X αA L β Tˆ ν .

(3.209)

h(D X A L, Tˆ ) = κ −1 ζ A .

(3.210)

From Table 3.102, By (3.141), !αβν X αA L β Tˆ ν =

 1 dH ψ L ψTˆ (d/ A σ )+ ψ A ψTˆ (Lσ ) − κ −1 ψ A ψ L (T σ ) 2 dσ (3.211) +H ψTˆ L µ (d/ A ψµ ).

Substituting (3.210) and (3.211) in (3.209) and recalling the definition (3.168), we obtain, in view of (3.207), 1 dH ψ ˆ (ψ L (d/ A σ )+ ψ A (Lσ )) 2 dσ T − H ψTˆ L µ (d/ A ψµ ).

/ A = κ −1 ζ A − a

Moreover, substituting for κ −1 ζ  from (3.175) this reduces to, simply:  1 dH ψ L (d/ A σ ) + H L µ (d/ A ψµ ) . / A = α −1 ψ L a 2 dσ

(3.212)

(3.213)

Also, we have: h(∇ X A L, X C ) = h(D X A L, X C ) − !αβν X αA L β X Cν

(3.214)

and from Table 3.102, h(D X A L, X C ) = χ AC

(3.215)

while from (3.141), !αβν X αA L β X Cν =

1 dH {ψ L ψC (d/ A σ )+ ψ A ψC (Lσ ) − ψ L ψ A (d/C σ )} 2 dσ (3.216) +H ψC L µ (d/ A ψµ ).

82

Chapter 3. The Acoustical Structure Equations

Substituting (3.215) and (3.216) in (3.214) we obtain, in view of (3.208), 1 dH {ψ L ψC (d/ A σ )+ ψ A ψC (Lσ ) − ψ L ψ A (d/C σ )} 2 dσ (3.217) − H ψC L µ (d/ A ψµ ).

b/ BA h / BC = χ AC −

Let again X be an arbitrary element of Tx St,u and consider ∇ X Tˆ , the covariant derivative of the vectorfield Tˆ with respect to X in the background flat Minkowski connection. This is given in rectangular coordinates by: ∂ ∇ X Tˆ = X (Tˆ µ ) µ ∂x

(rectangular coordinates).

(3.218)

Since Tˆ 0 = 0, we have X (Tˆ 0 ) = 0, hence the vector (3.218) is tangential to t . It can therefore be expanded as: ∇ X Tˆ = p/ X Tˆ + q/ X (3.219) where q/ X ∈ Tx St,u . Since p/ X and q/ X depend linearly on X, there is an element p/(x) ∈ Tx∗ St,u and an element q/(x) ∈ L(Tx St,u , Tx St,u ) such that: p/ X = X · p/(x)

and

q/ X = X · q/(x).

(3.220)

Thus, p/(x) is the value at x of a 1-form p/ on St,u and q/(x) is the value at x of a type T11 -type tensorfield q/ on St,u . Setting X = X A (x), A = 1, 2, in (3.218), (3.219), and letting x vary over St,u , we obtain: d/ A (Tˆ i )

∂ = ∇ X A Tˆ = p/ A Tˆ + q/ A ∂xi

(3.221)

where: p/ A = p/(X A )

and

q/ A = q/(X A ) = q/ BA X B .

(3.222)

Taking the h-inner product with Tˆ we obtain, since h(Tˆ , Tˆ ) = 1 and h(q/ A , Tˆ ) = 0, p/ A = h(∇ X A Tˆ , Tˆ ).

(3.223)

Also, taking the h-inner product with X C we obtain, since h(Tˆ , X C ) = 0 and / BC , h(X B , X C ) = h q/ BA h / BC = h(∇ X A Tˆ , X C ). (3.224) Since h(D X A Tˆ , Tˆ ) = (1/2)X A (h(Tˆ , Tˆ )) = 0, we have, by (3.141), h(∇ X A Tˆ , Tˆ ) = −!αβν X αA Tˆ β Tˆ ν 1 dH (ψ ˆ )2 (d/ A σ ) − H ψTˆ Tˆ i (d/ A ψi ) =− 2 dσ T

(3.225)

hence, in view of (3.223), also: p/ A = −

1 dH (ψ ˆ )2 (d/ A σ ) − H ψTˆ Tˆ i (d/ A ψi ). 2 dσ T

(3.226)

Chapter 3. The Acoustical Structure Equations

We have:

h(∇ X A Tˆ , X C ) = h(D X A Tˆ , X C ) − !αβν X αA Tˆ β X Cν

and from Table 3.102,

h(D X A Tˆ , X C ) = θ AC

83

(3.227) (3.228)

while from (3.141),  1 dH ψTˆ ψC (d/ A σ ) + κ −1 ψ A ψC (T σ ) − ψTˆ ψ A (d/C σ ) 2 dσ (3.229) + H ψC Tˆ i (d/ A ψi ).

!αβν X αA Tˆ β X Cν =

Substituting (3.228) and (3.229) in (3.227) we obtain, in view of (3.224),  / BC = θ AC − q/ BA h

1 dH ψ ˆ { ψC (d/ A σ )− ψ A (d/C σ )} − H ψC Tˆ i (d/ A ψi ) 2 dσ T

(3.230)

where we have defined:  θ AB = θ AB −

1 dH ψ A ψ B (T σ ). 2κ dσ

(3.231)

We shall show that θ  is regular as µ → 0. From (3.46) we have:  = −α −1 χ AB + k/AB θ AB

where: k/AB = k/ AB −

1 dH ψ A ψ B (T σ ). 2κ dσ

(3.232)

(3.233)

By the formula (3.17) for ki j , j

k/ AB = ki j X iA X B  1 dH ψ0 ψ A ψ B (Bσ ) − = ( ψ A (d/ B σ )+ ψ B (d/ A σ )) 2α dσ (1 + ρ H ) 1 H ψ0 − ω / AB α (1 + ρ H )

(3.234)

where (3.15): j

ω / AB = X iA X B ∂i ψ j = X iA (d/ B ψi ) = ω /B A. Substituting

(3.235)

Bσ = Lσ + ακ −1 T σ

we then obtain, in view of the definition (3.233),  1 dH ψ0 ψ A ψ B (Lσ ) − k/AB = ( ψ A (d/ B σ )+ ψ B (d/ A σ )) 2α dσ (1 + ρ H ) 1 H ψ0 − ω / AB . α (1 + ρ H ) This is regular as µ → 0. It then follows from (3.232) that so is θ  .

(3.236)

Chapter 4

The Acoustical Curvature The acoustical structure equations involve components of the curvature tensor of the acoustical metric h in the frame (L, T, X 1 , X 2 ). These are obtained from curvature tensor components in rectangular coordinates, Rµναβ , by contraction with the appropriate frame vectorfields. Thus, the curvature components α AB occurring in the propagation equations (3.38) for χ AB are given by (see (3.35)): µ

Rµναβ X A L ν X αB L β = α AB

(4.1)

the curvature components β A occurring in the Codazzi equations (3.60) are given by (see (3.59)): µ β (4.2) Rµναβ X C L ν X αA X B = βC  AB and the curvature component ρ occurring in the Gauss equation (3.53) is given by (see (3.51)): µ β (4.3) Rµναβ X A X νB X Cα X D = ρ AB C D . / Aη B + D / B η A ) we have the Moreover, in equation (3.112) relating L /T χ AB to (1/2)(D curvature components γ AB , given by (3.113): γ AB =

1 µ µ (Rµναβ X A T ν X αB L β + Rµναβ X B T ν X αA L β ). 2

(4.4)

In the present chapter we shall investigate the structure of the acoustical curvature tensor, analyzing in more detail the components above. To begin with, the curvature components in rectangular coordinates are expressed in the standard way in terms of the metric components and their partial derivatives up to 2nd order with respect to the same coordinates. We have: (2)

(1)

Rµναβ = R µναβ + R µναβ

(4.5)

(2)

where R µναβ contains and is linear in the partial derivatives of the 2nd order: (2) R µναβ =

1 (∂α ∂ν h βµ + ∂β ∂µ h αν − ∂α ∂µ h βν − ∂β ∂ν h αµ ) 2

(4.6)

86

Chapter 4. The Acoustical Curvature (1)

and R µναβ contains and is quadratic in the partial derivatives of 1st order: (1) R µναβ =

−(h −1 )κλ (!αµκ !βνλ − !βµκ !ανλ ).

(4.7)

Here (see (3.139)), λ !µνκ = h κλ !µν =

1 (∂µ h νκ + ∂ν h µκ − ∂κ h µν ). 2

(4.8)

Actually, we find it convenient to introduce, for each pair α, β, the components Hµναβ of the covariant – with respect to h – Hessian of the rectangular component h αβ , considered as a scalar function, namely: κ ∂κ h αβ . Hµναβ = ∂µ ∂ν h αβ − !µν

(4.9)

(2)

[2]

We then define R µναβ in analogy with R µναβ by replacing in (4.6) each ∂µ ∂ν h αβ by the corresponding Hµναβ : [2] R µναβ =

1 (Hανβµ + Hβµαν − Hαµβν − Hβναµ). 2

(4.10)

The decomposition: [2]

[1]

Rµναβ = R µναβ + R µναβ

(4.11)

then results, where: (1) [1] R µναβ = R µναβ

1 κ κ κ κ + (!αν ∂κ h βµ + !βµ ∂κ h αν − !αµ ∂κ h βν − !βν ∂κ h αµ ). 2

(4.12)

Using the fact that ∂κ h µν = !κµν + !κνµ we find: [1] R µναβ

=

1 −1 κλ  (!κµβ + !µβκ + !βκµ )!ανλ (h ) 2 + (!κνα + !νακ + !ακν )!βµλ − (!κνβ + !νβκ + !βκν )!αµλ

 −(!κµα + !µακ + !ακµ )!βνλ .

(4.13)

Denoting: ψµ = ∂µ φ,

ωµν = ∂µ ψν = ∂ν ψµ = ωνµ ,

τµ = ∂µ σ

(4.14)

1 dH (τα ψβ ψγ + τβ ψα ψγ − τγ ψα ψβ ) + H ωαβ ψγ 2 dσ

(4.15)

we have (see (3.141)): !αβγ =

Chapter 4. The Acoustical Curvature

87

hence: !αβγ + !βγ α + !γ αβ =

1 dH (τα ψβ ψγ + τβ ψα ψγ + τγ ψα ψβ ) 2 dσ +H (ωαβ ψγ + ωβγ ψα + ωαγ ψβ ).

(4.16)

Substituting in (4.13) and noting that: (h −1 )κλ ψκ ψλ = ((g −1 )κλ − Fψ κ ψ λ )ψκ ψλ = −σ (1 + σ F) = −

σ 1−σH

(4.17)

(ψ κ = (g −1 )κλ ψλ ) and: (h −1 )κλ ψλ ωκµ =

ψ κ ωκµ τµ 1 =− 1−σH 2 (1 − σ H )

(4.18)

[1]

we find the following expression for R µναβ : [1] R µναβ

σ H2 Aµναβ (1 − σ H )   1 dH dH 1 σ +H Bµναβ − 4 (1 − σ H ) dσ dσ   1 1 dH + H H Cµναβ . − 2σ 4 (1 − σ H ) dσ

=−

(4.19)

Here, Aµναβ = ωµβ ωνα − ωµα ωνβ

(4.20)

Bµναβ = (τµ ψν − τν ψµ )(τα ψβ − τβ ψα )

(4.21)

Cµναβ = τµ ξναβ − τν ξµαβ + τα ξβµν − τβ ξαµν

(4.22)

ξµαβ = ωαµ ψβ − ωβµ ψα .

(4.23)

and: where: The expressions Aµναβ , Bµναβ , Cµναβ , all possess the algebraic properties of the Riemann tensor, namely the antisymmetry in the second as well as the first pair of indices and the cyclic property, which imply the symmetry under exchange of the first with the second pair of indices. In particular the cyclic property of Cµναβ follows from that of ξµαβ : ξµαβ + ξαβµ + ξβµα = 0. [2]

We turn to R µναβ , given by (4.10). Using (see (3.140)) ∂µ h αβ =

dH τµ ψα ψβ + H (ωµα ψβ + ωµβ ψα ), dσ

88

Chapter 4. The Acoustical Curvature

we deduce the following decomposition for Hµναβ : Hµναβ =

4 

(i)

(4.24)

H µναβ

i=0

where:

(0) H µναβ =

dH vµν ψα ψβ dσ

(4.25)

H (wµνα ψβ + wµνβ ψα )

(4.26)

 dH  τµ (ωνα ψβ + ωνβ ψα ) + τν (ωµα ψβ + ωµβ ψα ) dσ

(4.27)

d2 H τµ τν ψα ψβ dσ 2

(4.28)

H (ωµα ωνβ + ωµβ ωνα ).

(4.29)

(1) H µναβ = (2) H µναβ =

(3) H µναβ = (4) H µναβ =

In equation (4.25) vµν is the covariant – relative to h – Hessian of the function σ :

We have:

vµν = Dµ Dν σ = vνµ .

(4.30)

κ τκ = Dν τµ vµν = Dµ τν = ∂µ τν − !µν

(4.31)

Also, in equation (4.26) wµνα is the covariant – relative to h – Hessian of the rectangular component ψα , considered as a scalar function:

and we have:

wµνα = Dµ Dν ψα = wνµα

(4.32)

κ ωκα = Dν ωµα . wµνα = Dµ ωνα = ∂µ ωνα − !µν

(4.33) [2]

Corresponding to the decomposition (3.33) of Hµναβ , we have a decomposition of R µναβ : [2] R µναβ =

4 

[2,i] R µναβ

(4.34)

i=0

where: [2,i] R µναβ =

(i) (i) (i) 1 (i) ( H ανβµ + H βµαν − H αµβν − H βναµ ). 2

(4.35)

We find: [2,2] R µναβ =

−

1 dH Cµναβ 2 dσ

(4.36)

Chapter 4. The Acoustical Curvature

89

(where Cµναβ is given by (4.22)) 1 d2 H Bµναβ 2 dσ 2

(4.37)

−H Aµναβ

(4.38)

1 H (ναβ ψµ + µβα ψν + ανµ ψβ + βµν ψα ) 2

(4.39)

[2,3] R µναβ =

−

(where Bµναβ is given by (4.21)) [2,4] R µναβ =

(where Aµναβ is given by (4.20)). On the other hand, [2,1] R µναβ =

where: µαβ = wµαβ − wµβα .

(4.40)

From (4.33), in view of the second of equations (4.14), we obtain: κ κ µνα = !µα ωκν − !µν ωκα

= (h −1 )κλ (!µαλ ωκν − !µνλ ωκα ) 1 d H −1 κλ = (h ) [(τµ ψα ψλ + τα ψµ ψλ − τλ ψµ ψα )ωκν 2 dσ − (τµ ψν ψλ + τν ψµ ψλ − τλ ψµ ψν )ωκα ] + H (h −1)κλ (ωµα ωκν − ωµν ωκα )ψλ .

(4.41)

Taking account of (4.18) this becomes: 1 d H −1 κλ (h ) τλ ψµ (ψα ωκν − ψν ωκα ) 2 dσ 1 dH 1 τµ (τν ψα − τα ψν ) − 4 (1 − σ H ) dσ 1 H − (ωµα τν − ωµν τα ). 2 (1 − σ H )

µνα = −

(4.42)

Substituting in (4.39) the contribution of the first term on the right-hand side of (4.42) is seen to vanish and we obtain:   [2,1] H 1 dH Bµναβ + H Cµναβ . (4.43) R µναβ = 4 (1 − σ H ) dσ The sum:

4  [2,i] R µναβ i=1

90

Chapter 4. The Acoustical Curvature [1]

combines with R µναβ , given by (4.19), to form the lower order part of the curvature tensor, which we denote by Nµναβ : [1]

Nµναβ = R µναβ +

4  [2,i] R µναβ .

(4.44)

i=1

We obtain: Nµναβ = −F Aµναβ − Here,

1 1 F2 Bµναβ − F1 Cµναβ . 2 2

 1 2 dH + H dσ 2   2 σ d H 1 dH 2 + F2 = 2 (1 − σ H ) dσ dσ 2 1 F1 = (1 − σ H )

(4.45)



(4.46) (4.47)

and Aµναβ , Bµναβ , Cµναβ are given by (4.20), (4.21), (4.22), respectively. Also, recall that H . F= 1−σH [2,0] µναβ ,

The principal part of the curvature tensor is then R [2,0] µναβ

Pµναβ = R

which we denote Pµναβ :

.

(4.48)

By (4.25) and (4.35) it is given by: 1 dH (vαν ψβ ψµ + vβµ ψα ψν − vαµ ψβ ψν − vβν ψα ψµ ). (4.49) 2 dσ The decomposition of the curvature tensor into its principal part and its lower order part is then: Rµναβ = Pµναβ + Nµναβ . (4.50) Pµναβ =

Note that the principal part vanishes if and only if H is constant, in which case the original nonlinear wave equation reduces to the minimal surface equation (see Chapter 1). In the following we shall analyze in detail the curvature components α AB , β A , ρ and γ AB which enter the acoustical structure equations (see (4.1)–(4.4)). We denote the ( P)

corresponding components of the principal part by α (N)

lower order part by α

( P)

α

AB ,

(N)

AB =

(N) (N)

β A, ρ , γ

AB .

AB ,

( P)

( P) ( P) AB ,

β A, ρ , γ

Thus,

P(X A , L, X B , L)

(4.51) ( P)

P(X C , L, X A , X B ) =  AB β

(4.52)

C ( P)

P(X A , X B , X C , X D ) =  AB C D ρ ( P) 1 γ AB = (P(X A , T, X B , L) + P(X B , T, 2

and of the

(4.53) X A , L))

(4.54)

Chapter 4. The Acoustical Curvature

91

and, (N)

α

AB =

N(X A , L, X B , L)

(4.55) (N)

N(X C , L, X A , X B ) =  AB β

(4.56)

C (N)

N(X A , X B , X C , X D ) =  AB C D ρ 1 γ AB = (N(X A , T, X B , L) + N(X B , T, X A , L)). 2

(N)

(4.57) (4.58)

We begin with an analysis of the components of the principal part (4.49). These contain the components of the Hessian – with respect to h – of σ (see (4.30)). The com( P)

ponents of the latter which occur in α

( P) AB

( P)

( P)

β A , ρ are:

v(X A , L),

v(L, L), while in γ

AB ,

v(X A , X B )

(4.59)

there occur also the components: v(X A , T ),

v(L, T ).

(4.60)

The component v(T, T ) does not occur in the components of the curvature which enter the acoustical structure equations. Now, our approach shall yield estimates for the derivatives of σ with respect to T and L and for the derivatives – covariant with respect to h/ – of the restrictions of these to each St,u . We must therefore express (4.59) and (4.60) in terms of these. By (4.30), (4.31), we have: v(L, L) = L µ L ν Dµ ∂ν σ = L(Lσ ) − (D L L)σ Substituting for D L L from Table 3.102 we then obtain: v(L, L) = L(Lσ ) − µ−1 (Lµ)(Lσ ).

(4.61)

Next, we have: µ

v(X A , L) = X A L ν Dµ ∂ν σ = X A (Lσ ) − (D X A L)σ. Substituting for D X A L from Table 3.102 we obtain: v(X A , L) = d/ A (Lσ ) + µ−1 ζ A (Lσ ) − χ AB (d/ B σ ).

(4.62)

Next, we have: µ

v(X A , X B ) = X A X νB Dµ ∂ν σ = X A (X B σ ) − (D X A X B )σ. Substituting for D X A X B from Table 3.102 and noting that: X A (X B σ ) − (D / X A X B )σ = (D / 2 σ )(X A , X B ) = (D / 2 σ ) AB

(4.63)

92

Chapter 4. The Acoustical Curvature

where D / 2 σ is the Hessian – with respect to h/ – of the restriction of σ to each St,u , we obtain: / 2 σ ) AB − α −1 k/ AB (Lσ ) − µ−1 χ AB (T σ ). (4.64) v(X A , X B ) = (D Also, again by (4.30), (4.31) we have: µ

v(X A , T ) = X A T ν Dµ ∂ν σ = X A (T σ ) − (D X A T )σ. Substituting for D X A T from Table 3.102 we obtain: v(X A , T ) = d/ A (T σ ) − α −1 κ A (Lσ ) − µ−1 η A (T σ ) − κθ AB (d/ B σ ). Also, we have:

(4.65)

v(L, T ) = L µ T ν Dµ ∂ν σ = L(T σ ) − (D L T )σ

and substituting for D L T from Table 3.102 we obtain: v(L, T ) = L(T σ ) + α −1 (Lκ)(Lσ ) + ζ A (d/ A σ ).

(4.66)

Now, by (4.51), (4.49), we have: ( P)

α

AB

=

1 dH (v(X B , L)ψ L ψ A + v(X A , L)ψ L ψ B 2 dσ − v(X A , X B )(ψ L )2 − v(L, L) ψ A ψ B ).

(4.67)

We substitute in (4.67) for v(X A , L) from (4.62), expressing ζ A in terms of ζ A by (3.168), that is: 1 dH ψ L ψ A (T σ ) ζ A = ζ A − (4.68) 2 dσ where κ −1 ζ A is given by (3.175). We substitute in (4.67) for v(X A , X B ) from (4.64), expressing k/ AB in terms of k/AB by (3.233), that is: k/ AB = k/AB +

1 dH ψ A ψ B (T σ ) 2κ dσ

(4.69)

where k/AB is given by (3.236). Finally, we substitute for v(L, L) from (4.61), expressing Lµ by equation (3.96). In view of (3.97) we have: µ−1 (Lµ) =

dH 1 −1 µ (ψ L )2 (T σ ) + e 2 dσ

(4.70) ( P)

where e is given by (3.98). Then, in the resulting expression for α tional to the product (T σ )(Lσ ) cancel and we obtain: ( P)

α

AB =

AB ,

the terms propor-

( P) dH 1 (χ C ψ B + χ BC ψ A )(d/C σ )+ α AB µ−1 mχ AB − ψ L 2 dσ A

(4.71)

Chapter 4. The Acoustical Curvature ( P)

where α

AB ,

93

given by: ( P)

α

=

AB

1 dH [ψ L ( ψ A d/ B (Lσ )+ ψ B d/ A (Lσ )) 2 dσ −(ψ L )2 (D / 2 σ ) AB − ψ A ψ B L(Lσ ) +



1 d H −1 α ψ L ( ψ A (κ −1 ζ B )+ ψ B (κ −1 ζ A )) 2 dσ  +α −1 (ψ L )2 k/AB + ψ A ψ B e (Lσ )

(4.72)

is regular as µ → 0. By (4.52), (4.49), we have: ( P)∗

β

( P)

A

= −(h /−1 ) BC  AB β C 1 d H −1 BC (h / ) [(v(X A , X C ) ψ B − v(X B , X C ) ψ A )ψ L = 2 dσ +( ψ A v(X B , L)− ψ B v(X A , L)) ψC ] .

(4.73)

We substitute in (4.73) for v(X A , X B ) from (4.64), expressing k/ AB in terms of k/AB as in (4.69). Also, we substitute in (4.73) for v(X A , L) from (4.62), expressing ζ A in terms of ( P)∗

ζ A as in (4.68). Then, in the resulting expression for β product (T σ )(Lσ ) cancel and we obtain: ( P)∗

β

( P)∗

where β ( P)∗

β

A

=

A,

A

A,

the terms proportional to the

dH 1 ψ L (T σ )(χ AB ψ B − trχ ψ A ) = − µ−1 2 dσ ( P)∗ 1 dH B − ψ ( ψ A χ BC − ψ B χ AC )(d/C σ )+ β A 2 dσ

(4.74)

given by:

 1 dH ψ L [(D / 2 σ ) AB ψ B − (
/ σ ) ψ A ]+ ψ A ψ B d/ B (Lσ ) − | ψ|2 d/ A (Lσ ) 2 dσ (4.75)  1 d H −1 −ψ L ( ψ B k/AB − ψ A trk/ )+ ψ A ψ B (κ −1 ζ B ) − | ψ|2 (κ −1 ζ A ) (Lσ ) α + 2 dσ

is regular as µ → 0. By (4.53), (4.49), we have: ( P)

( P) 1 −1 AC −1 B D (h / ) (h / )  AB C D ρ (4.76) 2 1 d H −1 AC −1 B D (h / ) (h / ) [v(X C , X B ) ψ D ψ A + v(X D , X A ) ψC ψ B = 4 dσ −v(X C , X A ) ψ D ψ B − v(X D , X B ) ψC ψ A ] .

ρ =

94

Chapter 4. The Acoustical Curvature

We substitute in (4.76) for v(X A , X B ) from (4.64), expressing k/ AB in terms of k/AB as ( P)

in (4.69). Then, in the resulting expression for ρ , the terms proportional to the product (T σ )(Lσ ) again cancel and we obtain:   ( P) 1 dH ρ = − µ−1 (T σ ) ψ · χ· ψ − trχ| ψ|2 + ρ 2 dσ

( P)

(4.77)

( P)

where ρ , given by: ( P)

ρ =

 1 dH (D / 2 σ ) AB ψ A ψ B − (
/ σ )| ψ|2 2 dσ   1 d H −1 α (Lσ ) k/AB ψ A ψ B − trk/ | ψ|2 . − 2 dσ

(4.78)

We turn finally to (4.54). By (4.49) we have: ( P) γ AB

=

1 dH {(1/2)( ψ A v(X B , T )+ ψ B v(X A , T ))ψ L 2 dσ + (1/2)( ψ A v(X B , L)+ ψ B v(X A , L))ψT − v(X A , X B )ψ L ψT − v(L, T ) ψ A ψ B }.

(4.79)

We substitute in (4.79) for v(X A , T ) and v(L, T ) from (4.65) and (4.66) respectively, expressing θ in terms of χ through (3.46), that is: θ = k/ − α −1 χ.

(4.80)

Also, we substitute in (4.79) for v(X A , L) and v(X A , X B ) from (4.62) and (4.64) respectively. We then obtain: ( P) γ AB

dH 1 (T σ )ψ L ( ψ A η B + ψ B η A ) = − µ−1 4 dσ 1 d H −1 α ψ L ψTˆ (T σ )χ AB + 2 dσ 1 d H −1 α κ(ψ L − αψTˆ )( ψ A χ BC + ψ B χ AC )(d/C σ ) + 4 dσ ( P) AB

+ γ

.

(4.81)

Here, ( P) γ AB

=

1 dH {(1/2)ψ L [ ψ A d/ B (T σ )+ ψ B d/ A (T σ )] 2 dσ + (1/2)κψTˆ [ ψ A d/ B (Lσ )+ ψ B d/ A (Lσ )] − κψ L ψTˆ (D / 2 σ ) AB − ψ A ψ B L(T σ )} +

1 dH κl AB 2 dσ

(4.82)

Chapter 4. The Acoustical Curvature

95

where: κl AB = (1/2)α −1 [ ψ A (ψTˆ ζ B − ψ L (κ B ))+ ψ B (ψTˆ ζ A − ψ L (κ A ))

(4.83)

−2 ψ A ψ B (Lκ) + 2ψ L ψTˆ (κk/ AB )](Lσ ) −(1/2)[ψ L ( ψ A (κk/ BC )+ ψ B (κk/ AC )) + 2 ψ A ψ B ζ C ](d/C σ ). ( P)

The γ AB are regular as µ → 0. In fact, expressing ζ A in terms of ζ A by (4.68),  A in terms of  A by (3.171), k/ AB in terms of k/AB by (4.69), and substituting for Lκ from equation (3.99), we find: l AB = (1/2)α −1 [ ψ A (ψTˆ (κ −1 ζ B ) − ψ L  B )+ ψ B (ψTˆ (κ −1 ζ A ) − ψ L  A )

(4.84)



− 2 ψ A ψ B e + 2ψ L ψTˆ k/AB ](Lσ ) −(1/2)[ψ L ( ψ A k/B C + ψ B k/A C ) + 2 ψ A ψ B (κ −1 ζ C )](d/C σ )

which shows that the l AB are themselves regular as µ → 0. We turn to the analysis of the components (4.55)–(4.58) of the lower order part ( A) ( A) ( A) AB , β A , ρ , γ AB , be the cor(B) (B) (B) (B) responding components of Aµναβ , given by (4.20). Let α AB , β A , ρ , γ AB , be the (C) (C) (C) (C) corresponding components of Bµναβ , given by (4.21). Let also α AB , β A , ρ , γ AB , be ( A)

Nµναβ of the curvature tensor, given by (4.45). Let α

the corresponding components of Cµναβ , given by (4.22). We then have: (N)

( A)

1 (B) 1 (C) − F2 α AB − F1 α AB (4.85) 2 2 (N) ( A) 1 (B) 1 (C) β A = −F β A − F2 β A − F1 β A (4.86) 2 2 (N) ( A) 1 (B) 1 (C) ρ = −F ρ − F2 ρ − F1 ρ (4.87) 2 2 (N) ( A) (B) 1 1 (C) γ AB = −F γ AB − F2 γ AB − γ AB . (4.88) 2 2 The above components involve the components of the symmetric tensor ωµν (see /, with components: (4.14)). On each St,u , we have the symmetric tensor ω α

AB

= −F α

AB

j

ω / AB = X iA X B ∂i ψ j = X iA (d/ B ψi )

(4.89)

/Tˆ , with components: the 1-forms ω / L and ω (ω / L ) A = L µ X νA ∂µ ψν = L µ (d/ A ψµ ) = X iA (Lψi ) ˆi

(ω /Tˆ ) A = T

j X A ∂i ψ j

ˆi

(4.90)

= T (d/ A ψi )

(4.91)

ω L L = L µ L ν ∂µ ψν = L µ (Lψµ )

(4.92)

and the functions ω L L and ω L Tˆ , where: µ

ˆν

ˆi

ω L Tˆ = L T ∂µ ψν = T (Lψi ) (the component ωTˆ Tˆ does not occur in (4.85)–(4.88)).

(4.93)

96

Chapter 4. The Acoustical Curvature

We have (see (4.20)–(4.23)): ( A)

α

(B)

α

(C)

α

AB

AB =

AB =

(ω / L ) A (ω /L )B − ω / AB ω L L

(4.94)

[ψ L (d/ A σ )− ψ A (Lσ )][ψ L (d/ B σ )− ψ B (Lσ )]

(4.95)

= (d/ A σ )[ψ L (ω / L ) B − ψ B ω L L ] + (d/ B σ )[ψ L (ω / L ) A − ψ A ω L L ] −(Lσ )[2ψ L ω / AB − ψ A (ω / L ) B − ψ B (ω /L ) A ] ( A)∗

β

(B)∗

β

A=

(C)∗

β

A

(4.96)

ω / AB (ω / L ) B − trω /(ω /L ) A

(4.97)

[ ψ B (Lσ ) − ψ L (d/ B σ )][ ψ B (d/ A σ )− ψ A (d/ B σ )]

(4.98)

A=

= (Lσ )[ω / AB ψ B − trω / ψ A ] / L ) B ψ B − trω /ψ L ] +(d/ A σ )[(ω +(d/ B σ )[ω / AB ψ L − 2(ω / L ) A ψ B + (ω / L ) B ψ A ] ( A)

ρ = (1/2)[|ω /|2 − (trω /)2 ]

(B)

(4.99) (4.100)

ρ = | ψ|2 |d/σ |2 − ( ψ · d/σ )2

(4.101)

ρ = 2[ ψ · ω / · d/σ − ( ψ · d/σ )trω /].

(4.102)

(C)

Also, we have: ( A) γ AB

= (1/2)κ[(ω / L ) A (ω /Tˆ ) B + (ω / L ) B (ω /Tˆ ) A − 2ω / AB ω L Tˆ ]

(4.103)

(B) γ AB

= (1/2)[κψTˆ (d/ A σ )− ψ A (T σ )][ψ L (d/ B σ )− ψ B (Lσ )]

(4.104)

+ (1/2)[κψTˆ (d/ B σ )− ψ B (T σ )][ψ L (d/ A σ )− ψ A (Lσ )] (C) γ AB

= (1/2)(d/ A σ )κ[ψ L (ω /Tˆ ) B + ψTˆ (ω / L ) B − 2 ψ B ω L Tˆ ] /Tˆ ) A + ψTˆ (ω / L ) A − 2 ψ A ω L Tˆ ] + (1/2)(d/ B σ )κ[ψ L (ω /Tˆ ) B + ψ B (ω /Tˆ ) A − 2ψTˆ ω / AB ] + (1/2)(Lσ )κ[ ψ A (ω + (1/2)(T σ )[ ψ A (ω / L ) B + ψ B (ω / L ) A − 2ψ L ω / AB ].

(4.105)

The above formulas show that all the components (4.85)–(4.88) of the lower order part of the curvature tensor are regular as µ → 0. We now revisit the propagation equations (3.38) for χ AB . Substituting the decomposition ( P)

α AB = α

(N)

AB

+ α

AB

Chapter 4. The Acoustical Curvature

97 ( P)

and then substituting the formula (4.71) for α AB , and also substituting the propagation equation for µ, equation (3.96), the propagation equations for χ AB take the form: dH 1 (χ C ψ B + χ BC ψ A )(d/C σ ) Lχ AB = χ AC χ BC + eχ AB + ψ L 2 dσ A ( P)

− α

(N)

AB

− α

AB

.

(4.106)

In contrast to the original form (3.38), the right-hand side is now manifestly regular as µ → 0. Next, we revisit the Codazzi equations in the divergence form (3.62). Substituting the decomposition ( P)

βA = β

(N)

A

+ β

A

( P)∗

and then substituting the formula (4.74) for β A , and also substituting the expression for ζ A in terms of ζ A , equation (4.68), the Codazzi equations become: (div / χ) A − d/ A trχ = −µ−1 (χ AB ζ B − trχζ A ) 1 dH B ψ ( ψ A χ BC − ψ B χ AC )(d/C σ ) − 2 dσ ( P)∗

+ β

(N)∗

A

+ β

A

.

(4.107)

Since κ −1 ζ A is regular as µ → 0 (see (3.175)), so is the right-hand side. Next, we revisit the Gauss equation (3.53). Here we first substitute for θ in terms of χ and k/ from (4.80) and for k/ in terms of k/ from (4.69) to obtain: 1 1 {(trθ )2 − |θ |2 − (trk/)2 + |k/|2 } = α −2 {(trχ)2 − |χ|2 } 2 2 −α −1 (trχtrk/ − χ · k/ ) 1 dH (T σ ){trχ| ψ|2 − ψ · χ· ψ}. − µ−1 2 dσ We then substitute the decomposition ( P)

(N)

ρ=ρ + ρ ( P)

and the formula (4.77) for ρ . The Gauss equation then takes the form: K =

1 −2 α {(trχ)2 − |χ|2 } − α −1 (trχtrk/ − χ · k/ ) 2 ( P)

(N)

+ ρ + ρ which is regular as µ → 0.

(4.108)

98

Chapter 4. The Acoustical Curvature

Finally, we revisit equations (3.112) relating L /T χ AB to (1/2)(D / AηB + D / B η A ). Substituting the decomposition ( P) AB

γ AB = γ

(N)

+ γ

AB

( P) γ AB ,

and then substituting the formula (4.81) for and also substituting the expression for ζ A in terms of ζ A , equation (4.68), equations (3.112) become, in view also of (3.99), (4.80), and (4.69), L /T χ AB =

1 1 / B η A ) + µ−1 (ζ A η B + ζ B η A ) (D / AηB + D 2  2  1 −2 1 d H −α (T σ ) ψ L (ψ L + αψTˆ )χ AB + ( ψ A χ BC + ψ B χ AC ) ψC 2 dσ 2 1 −α −1 κ(e χ AB + χ AC χ BC ) + κ(k/AC χ BC + k/BC χ AC ) 2 1 −1 d H (ψ L − αψTˆ )( ψ A χ BC + ψ B χ AC )(d/C σ ) − α κ 4 dσ ( P) AB

− γ

(N)

− γ

AB

.

The right-hand side is now manifestly regular as µ → 0.

(4.109)

Chapter 5

The Fundamental Energy Estimate In this chapter we consider the wave equation
h˜ ψ = 0

(5.1)

˜ As we have seen in the spacetime whose metric is the conformal acoustical metric h. in Chapter 1, this is the equation (1.86) satisfied by a variation of the wave function φ through solutions. The conformal acoustical metric h˜ is related to the acoustical metric h by equation (1.82), namely: (5.2) h˜ µν = h µν where the conformal factor  is given by: =

G/G 0 η/η0

(5.3)

where the constants G 0 and η0 refer to the surrounding constant state. We consider, more generally, the inhomogeneous wave equation:
h˜ ψ = ρ.

(5.4)

We shall be led to equations of this form, in the next chapter, when considering higher order variations. The metric h˜ is defined in the domain Mε0 of the maximal solution, discussed in Chapter 2. By virtue of the linear character of equations (5.1) and (5.4), with ρ a given function defined on Mε0 , we may assume that ψ is defined on the whole of Mε0 , as is indeed the case when ψ is a first- or higher order variation of φ. In applications to first order estimates we shall take ψ to each be one of: ψ1 = T0 φ − k; Ti φ : i = 1, 2, 3; i j φ : i < j = 1, 2, 3; Sφ.

(5.5)

For the first of the above, we appeal to the fact that if ψ is a solution of (5.1), so is ψ − c, for any constant c. Then each of the ψ1 vanishes in the exterior of the outgoing

100

Chapter 5. The Fundamental Energy Estimate

characteristic hypersurface C0 . In applications to higher order estimates we shall take each ψ to be one of: (5.6) ψn = Yi1 · · · Yin−1 ψ1 where ψ1 is any one of the above first order variations and the indices i 1 , . . . , i n−1 take values in the set {1, 2, 3, 4, 5}, with {Yi : i = 1, 2, 3, 4, 5} the set of vectorfields to be discussed in the next chapter. Each of the ψn also vanishes in the exterior of C0 . We shall therefore assume in general that ψ vanishes in the exterior of C0 . We may then confine attention to the spacetime domain Wε0 defined in Chapter 2. For any s ∈ (0, t∗ε0 ), we set: Wεs0 = {x ∈ Wε0 : x 0 ∈ [0, s]} (5.7) or, equivalently, Wεs0 = In particular we have:



St,u .

(5.8)

(t,u)∈[0,s]×[0,ε0]

 s∈(0,t∗ε0 )

Wεs0 = Wε∗0

Recall that here and throughout this monograph the positive constant ε0 is subject to the restriction: 1 ε0 ≤ 2 but is otherwise arbitrary. Everything which follows depends on the geometric construction of Chapter 3. We remark here that the characteristic hypersurfaces Cu depend only on the conformal class of the metric h. Thus, the 2-parameter foliation of Wε∗0 given by the surfaces St,u likewise depends only on the conformal class of h. The geometric properties of this foliation may ˜ Since we are studying the be referred either to the metric h or to the conformal metric h. ˜ it would appear more natural if we refer these properties to wave equation in the metric h, ˜ However, our bootstrap argument is constructed relative to h, therefore we shall refer h. the said properties to h, and introduce additional assumptions referring to the conformal factor . We shall presently list the bootstrap assumptions on the geometric properties of the 2-parameter foliation which we shall need in this chapter. First we have the basic assumptions: There is a positive constant C independent of s such that in Wεs0 , A1: C −1 ≤  ≤ C A2: C −1 ≤ α/η0 A3: µ/η0 ≤ C[1 + log(1 + t)] We remark that by equation (2.43) we have: α≤1

(5.9)

Chapter 5. The Fundamental Energy Estimate

101

Next, we have the assumptions: B1: C −1 (1 + t)−1 ≤ ν ≤ C(1 + t)−1 B2: |ν| ≤ C(1 + t)−1 [1 + log(1 + t)]4 B3: |χ| ˆ ≤ C(1 + t)−1 [1 + log(1 + t)]−2 B4: |χ| ˆ ≤ C(1 + t)[1 + log(1 + t)]−6 Here, 1 1 ˜ = (trχ + L log ) trχ 2 2 1 1 ˜ = (trχ + L log ). ν = trχ 2 2

ν=

(5.10) (5.11)

Moreover, χˆ and χˆ are the trace-free parts of χ and χ respectively, and the pointwise norms of tensors on St,u are with respect to the induced metric h/. Next, we have the assumptions: B5: |L log | ≤ C(1 + t)−1 [1 + log(1 + t)]−2 B6: |L log | ≤ C(1 + t)[1 + log(1 + t)]−6 and the assumptions: B7: |ζ + η| ≤ C(1 + t)−1 [1 + log(1 + t)] B8: |d/(α −1 κ)| ≤ C(1 + t)−1 [1 + log(1 + t)] B9: |Lα| ≤ C(1 + t)−1 [1 + log(1 + t)]2 B10: |L(α −1 κ)| ≤ C(1 + t)−1 [1 + log(1 + t)]3 B11: |L(α −1 κ)| ≤ C(1 + t)[1 + log(1 + t)]−2 as well as the assumptions: B12: |Lν + ν 2 | ≤ C(1 + t)−2 [1 + log(1 + t)]−2 B13: |Lν| ≤ C(1 + t)−2 [1 + log(1 + t)]3 B14: |d/ν| ≤ C(1 + t)−2 [1 + log(1 + t)]1/2 The next set of bootstrap assumptions are the crucial assumptions regarding the behavior of the function µ. In the following we denote by f + and f − respectively the positive and negative parts of the function f : f+ (x) = max{ f (x), 0},

f − (x) = min{ f (x), 0}

C1: µ−1 (Lµ)+ ≤ (1 + t)−1 [1 + log(1 + t)]−1 + A(t) where A(t) is a non-negative function such that: s A(t)dt ≤ C (independent of s) 0

C2: µ−1 (Lµ + Lµ)+ ≤ B(t)

102

Chapter 5. The Fundamental Energy Estimate

where B(t) is a non-negative function such that: s (1 + t)−2 [1 + log(1 + t)]4 B(t)dt ≤ C (independent of s). 0

Moreover, denoting by U the region: U = {x ∈ Wε∗0 : µ < η0 /4} we have: Lµ ≤ −C −1 (1 + t)−1 [1 + log(1 + t)]−1 in U

C3:

(5.12) 

Wεs0

The final set of bootstrap assumptions concerns the existence of a function ω verifying the following conditions: D1: C −1 (1 + t) ≤ ω/η0 ≤ C(1 + t) D2: |Lω − νω| ≤ C[1 + log(1 + t)]−2 D3: |Lω| ≤ C[1 + log(1 + t)]3 D4: |d/ω| ≤ C[1 + log(1 + t)]1/2 and:

! s ! ε0

D5:

0

0

 sup St,u (µ|
h˜ ω|)du dt ≤ C[1 + log(1 + s)]4

The aim of the present chapter is the proof of: ˜ defined in Mε0 Theorem 5.1 Let ψ be a solution of the wave equation in the metric h, and vanishing in the exterior of C0 . Suppose that assumptions A1–A3, B1–B14, C1–C3, D1–D5 hold in Wεs0 , for some s ∈ (0, tε0 ]. Let us denote:  ε0  −2 2 2 2 [η0 (Lψ) + (Lψ) + |d/ψ| ]dµh/ du. D0 = 0

S0,u

Then, there exist constants C independent of s such that:  ε0  −2 −2 2 2 2 (i) sup [η0 µ(η0 + µ)(η0 (Lψ) + |d/ψ| ) + (Lψ) ]dµh/ du ≤ C D0 t ∈[0,s] 0

St,u

(ii)

St,u

(iii)

ψ 2 dµh/ ≤ Cε0 D0 s 



sup u∈[0,ε0 ] 0

St,u

[η0−2 (η0 + µ)(Lψ)2 + µ|d/ψ|2 ]dµh/ dt ≤ C D0

(iv) sup [1 + log(1 + t)]−4 (1 + t)2 t ∈[0,s]  ε0

× 0

(v)

St,u

sup [1 + log(1 + s)] u∈[0,ε0 ]



µ[η0−2 (Lψ + νψ)2 + |d/ψ|2 ]dµh/ −4



s 0

du ≤ C D0

 (1 + t)



2

(Lψ + νψ) dµh/ dt ≤ C D0 . 2

St,u

Chapter 5. The Fundamental Energy Estimate

103

Here, the pointwise norms of tensors on St,u are with respect to the induced metric h/. Moreover, there is a constant C independent of s such that: (1 + t)[1 + log(1 + t)]−1 |d/ψ|2 dµh/ dudt ≤ C D0 [1 + log(1 + s)]4 (vi)  U

Wεs

0

where U is the region where µ < η0 /4 (see (5.12)). Proof. We begin with the energy-momentum-stress tensor T˜µν associated to the function ˜ ψ through the metric h: T˜µν := ∂µ ψ∂ν ψ − (1/2)h˜ µν (h˜ −1 )κλ ∂κ ψ∂λ ψ = ∂µ ψ∂ν ψ − (1/2)h µν (h −1 )κλ ∂κ ψ∂λ ψ := Tµν . We have:

(5.13)

D˜ µ T˜µν := (h˜ −1 )µλ D˜ µ T˜λν = ∂ν ψ
h˜ ψ

where D˜ is the covariant derivative operator associated to the metric h˜ µν . Thus, for a ˜ while for a solution of solution of 5.1 T˜µν is divergence-free with respect to the metric h, the inhomogeneous wave equation 5.4, D˜ µ T˜µν = ρ∂ν ψ.

(5.14)

We consider the future-directed, time-like with respect to the acoustical metric h, vectorfield K 0 : (5.15) K 0 = (η0−1 + α −1 κ)L + L. Also, given a function ω satisfying assumptions D1–D5 we consider the future-directed, null with respect to the acoustical metric h, vectorfield K 1 : K 1 = (ω/ν)L.

(5.16)

We denote by π˜ 0 and π˜ 1 the Lie derivatives of the conformal acoustical metric h˜ with respect to the vectorfields K 0 and K 1 respectively: π˜ 0 = L K 0 h˜

˜ π˜ 1 = L K 1 h.

(5.17)

To K 0 we associate the vectorfield P˜0 : µ P˜0 = −T˜ µν K 0ν

where:

(5.18)

T˜ µν = (h˜ −1 )µκ Tκν .

To K 1 we associate the vectorfield P˜1 , given by: µ P˜1 = −T˜ µν K 1ν − (h˜ −1 )µν (ωψ∂ν ψ − (1/2)ψ 2 ∂ν ω).

(5.19)

104

Chapter 5. The Fundamental Energy Estimate

Let:

µ T˜ µν = (h˜ −1 )νλ T˜ λ = (h˜ −1 )µκ (h˜ −1 )νλ T˜κλ .

For any vectorfield X, we have, by virtue of the symmetry of T˜ µν , D˜ µ (T˜ µν X ν ) = ( D˜ µ T˜ µν )X ν + T˜ µλ (h˜ λν D˜ µ X ν ) = ρ Xψ + (1/2)T˜ µλ (h˜ λν D˜ µ X ν + h˜ µν D˜ λ X ν ) = ρ Xψ + (1/2)T˜ µλ L X h˜ µλ where we have used equation (5.14). Thus, the divergence of the vectorfield P˜0 with respect to the metric h˜ is given by: µ D˜ µ P˜0 = −ρ K 0 ψ − (1/2)T˜ µν π˜ 0,µν := Q˜ 0 .

(5.20)

Also, the divergence of the vectorfield P˜1 with respect to the metric h˜ is given by: µ D˜ µ P˜1 = −ρ(K 1 ψ + ωψ) − (1/2)T˜ µν π˜ 1,µν −ω(h˜ −1 )µν ∂µ ψ∂ν ψ + (1/2)ψ 2
˜ ω. h

Taking into account the fact that: ˜ := h˜ µν T˜ µν = −(h˜ −1 )µν ∂µ ψ∂ν ψ trT and introducing:

π˜ 1 = π˜ 1 − 2ωh˜

(5.21)

we can write the above in the form: µ  D˜ µ P˜1 = −ρ(K 1 ψ + ωψ) − (1/2)T˜ µν π˜ 1,µν + (1/2)ψ 2
h˜ ω := Q˜ 1 .

Consider now the equation

D˜ µ P˜ µ = Q˜

(5.22)

(5.23)

˜ In arbitrary local coordinates we have: for an arbitrary vectorfield P˜ and function Q.   1 1 µ µ 2 ˜ ˜ ˜ ˜

∂µ ( −deth P ) = ∂µ ( −deth˜ P˜ µ ) = −2 Dµ P µ Dµ P =

−deth˜ 2 −deth˜ where: Hence, with:

P µ = 2 P˜ µ .

(5.24)

Q = 2 Q˜

(5.25)

equation (5.23) is equivalent to the equation: Dµ P µ = Q for the vectorfield P and function Q.

(5.26)

Chapter 5. The Fundamental Energy Estimate

105

We may express equation (5.26) in acoustical coordinates (t, u, ϑ 1 , ϑ 2 ) (see Chapter 3). Expanding P in the associated coordinate frame field,   ∂ ∂ ∂ ∂ , , , , ∂t ∂u ∂ϑ 1 ∂ϑ 2 P = Pt

 ∂ ∂ ∂ + Pu + (P ϑ ) A A , ∂t ∂u ∂ϑ A

and noting that, by the expression (2.37) for the metric h in acoustical coordinates, we have:

√ −deth = µ deth/ (5.27) equation (5.26) takes the form: 

∂

∂ 1 (µ deth (µ deth/ P u ) + div √ /Pt ) + / M = µQ ∂u deth / ∂t

(5.28)

where M is the vectorfield on S 2 given in the local coordinates (ϑ 1 , ϑ 2 ) by: M =µ

 ∂ (P ϑ ) A A . ∂ϑ A

This follows noting that: div / M=√

∂

1 ( deth/ M A ). deth/ ∂ϑ A

We integrate equation (5.28) on S 2 with respect to the measure:

dµh/ = deth/dϑ 1 dϑ 2

(5.29)

to obtain the equation:     ∂ ∂ t u µP dµh/ + µP dµh/ = µQdµh/ . ∂t ∂u St,u St,u St,u

(5.30)

Replacing (t, u) by (t  , u  ) and integrating with respect to (t  , u  ) on [0, t] × [0, u], we obtain, under the hypothesis that P vanishes in the closure of the exterior of C0 , E u (t) − E u (0) + F t (u) = Qdµh . (5.31) Wut

Here E u is the “energy”: E (t) = u

tu



µP dµh/ du = t



u 0



 µP dµh/ du  t

St,u 

(5.32)

106

Chapter 5. The Fundamental Energy Estimate

and F t is the “flux”:



F (u) = t



µP dµh/ dt = u

C ut

t 

 St  ,u

0

µP dµh/ dt  . u

(5.33)

In obtaining (5.31) we have used the fact that by virtue of (5.27),   Qdµh = µQdµh/ du  dt  .

(5.34)

In the above formulas tu is the annular region:  tu = St,u 

(5.35)

Wut

St  ,u 

[0,t ]×[0,u]

u  ∈[0,u]

in the space-like hyperplane t , Cut is the characteristic hypersurface Cu truncated from above by t :  Cut = St  ,u (5.36) t  ∈[0,t ]

and

Wut

is the spacetime domain:



Wut =

St  ,u 

(5.37)

(t  ,u  )∈[0,t ]×[0,u]

bounded by the characteristic hypersurfaces C0 and Cu and the space-like hyperplanes 0 and t . Now, the vectorfield P can also be expanded in the frame field L, L, X 1 , X 2 (see Chapter 3):  PAXA P = PL L + PL L + A

From equations (2.30), (2.31), (3.114), (3.24), L=

∂ ∂t

L = α −1 κ

(5.38) 

∂ ∂ ∂ +2 − A A ∂t ∂u ∂ϑ



∂ . ∂ϑ A Substituting and equating coefficients in the expansion of P yields: XA =

(5.40)

P t = P L + α −1 κ P L

(5.41)

P = 2P

(5.42)

u

and:

(5.39)

L

(P ϑ ) A = P A − 2 A P L .

We shall write down the expressions for the energy and flux integrals corresponding to the vectorfields P0 and P1 defined above.

Chapter 5. The Fundamental Energy Estimate

107

We begin with the table of components of the covariant form of the energy-momentum-stress tensor Tµν (see (5.13)) in the null frame L, L, X 1 , X 2 : TL L = (Lψ)2 TL L = (Lψ)2 TL L = µ|d/ψ|2 TL A = (Lψ)(d/ A ψ) TL A = (Lψ)(d/ A ψ) T AB = (d/ A ψ)(d/ B ψ) − (1/2)h/ AB (−µ−1 (Lψ)(Lψ) + |d/ψ|2 ).

(5.43)

Here, d/ A ψ = X A ψ and the pointwise norms of tensors on St,u are with respect to the induced metric h/. We also give here, for future reference, the table of the components of the associated contravariant tensor T µν = (h −1 )µκ (h −1 )νλ Tκλ = 2 T˜ µν . From the expression 3.116 for h −1 and the above table we obtain: (Lψ)2 (Lψ)2 |d/ψ|2 LL LL T = T = 4µ2 4µ2 4µ 1 1 = − (Lψ)(d/ A ψ) T L A = − (Lψ)(d/ A ψ) 2µ 2µ 1 −1 AB (h / ) (Lψ)(Lψ) + {(d/ A ψ)(d/ B ψ) − (1/2)(h/−1 ) AB |d/ψ|2 } (5.44) = 2µ

T LL = T LA T AB where:

d/ A ψ = (h/−1 ) AB d/ B ψ.

We consider first P0 . From equations (5.24) and (5.18) we have: µ

P0 = −T µν K 0ν . Here,

(5.45)

T µν = (h −1 )µκ Tκν = T˜ µν .

The components of P0 which are h-orthogonal to the surfaces St,u are: 1 h(P0 , L) = 2µ 1 P0L = − h(P0 , L) = 2µ L

P0 = −

 ((η−1 + α −1 κ)TL L + TL L ) 2µ 0  ((η−1 + α −1 κ)TL L + TL L ). 2µ 0

Substituting from Table 5.43 we obtain:  ((η−1 + α −1 κ)(Lψ)2 + µ|d/ψ|2 ) 2µ 0  P0L = ((η−1 + α −1 κ)µ|d/ψ|2 + (Lψ)2 ). 2µ 0 L

P0 =

(5.46)

108

Chapter 5. The Fundamental Energy Estimate

Substituting these expressions in (5.41) and (5.42), and those in turn in (5.32) and (5.33) we obtain the following expressions for the energy and flux integrals associated to the vectorfield K 0 :  −1 α κ(η0−1 + α −1 κ)(Lψ)2 + (Lψ)2 E0u (t) = tu 2  + (η0−1 + 2α −1 κ)µ|d/ψ|2 dµh/ du  (5.47) 

F0t (u) =  (η0−1 + α −1 κ)(Lψ)2 + µ|d/ψ|2 dµh/ dt  . (5.48) C ut

We now consider P1 . From (5.19) and (5.24) we have: µ

P1 = −{T µν K 1ν + (h −1 )µν (ωψ∂ν ψ − (1/2)ψ 2 ∂ν ω)}.

(5.49)

The components of P1 which are h-orthogonal to the surfaces St,u are: 1 h(P1 , L) = 2µ 1 P1L = − h(P1 , L) = 2µ L

P1 = −

 {ων −1 TL L + ωψ(Lψ) − (1/2)ψ 2 (Lω)} 2µ  {ων −1 TL L + ωψ(Lψ) − (1/2)ψ 2 (Lω)}. 2µ

Substituting from Table 5.43 yields:  {ων −1 (Lψ)2 + ωψ(Lψ) − (1/2)ψ 2 (Lω)} 2µ  {ων −1 µ|d/ψ|2 + ωψ(Lψ) − (1/2)ψ 2 (Lω)}. P1L = 2µ L

P1 =

Substituting these in (5.41), (5.42) and those in turn in (5.32), (5.33) yields:  −1 −1 E1u (t) = ων [α κ(Lψ)2 + µ|d/ψ|2 ] tu 2

(5.50)

(5.51)

 +ωψ[α −1 κ(Lψ) + (Lψ)] − (1/2)ψ 2 [α −1 κ(Lω) + (Lω)] dµh/ du 

 t F1 (u) =  ων −1 (Lψ)2 + ωψ(Lψ) − (1/2)ψ 2 (Lω) dµh/ dt  . (5.52) C ut

Consider first the flux integral (5.52). We actually define the flux integral associated to the vectorfield K 1 to be: t ων −1 (Lψ + νψ)2 dµh/ dt  . (5.53) F1 (u) = C ut

We then have: F1t (u) − F1t (u)

=−

C ut

(1/2)[L(ωψ 2 ) + 2νωψ 2 ]dµh/ dt  .

(5.54)

Chapter 5. The Fundamental Energy Estimate

109

Now, by the definition of χ, the 2nd fundamental form of St,u relative to Cu , equation (3.27), we have: (5.55) L / L dµh/ = trχdµh/ . It follows, in view of equation (5.10) that for an arbitrary function f defined in the spacetime domain Wε∗0 we have:   ∂  f dµh/ = (L f + 2ν f )dµh/ . (5.56) ∂t St,u St,u Setting f = (1/2)ωψ 2 and comparing with (5.54) we deduce:   t ∂ (1/2)ωψ 2 dµh/ dt  F1t (u) − F1t = −  St  ,u 0 ∂t =− (1/2)ωψ 2 dµh/ + (1/2)ωψ 2 dµh/ . St,u

(5.57)

S0,u

Consider next the energy integral (5.51). We actually define the energy integral associated to the vectorfield K 1 to be:  −1 −1 E1u (t) = ων {α κ(Lψ + νψ)2 + µ|d/ψ|2 }dµh/ du  . (5.58) u 2 t Taking into account the fact that, by (3.114), L − α −1 κ L = 2T we find: E1u (t)

− E1u (t)

=

tu

 2ωψ(T ψ) 2

(5.59)

(5.60)

 −(1/2)[α −1 κ Lω + Lω + 2α −1 κνω]ψ 2 dµh/ du  .

Now, from the definition of θ , the 2nd fundamental form of St,u relative to t , equation (3.44), we have: (5.61) L /T dµh/ = κtrθ dµh/ . It follows that for an arbitrary function f defined on the spacetime domain Wε∗0 :   ∂ f dµh/ = {T f + [κtrθ + T (log )] f }dµh/ . ∂u St,u St,u Setting f = (1/2)ωψ 2 yields:   2ωψ(T ψ) + (T ω)ψ 2 + [κtrθ + T (log )]ωψ 2 dµh/ du  tu 2   u ∂  2  du ωψ = dµ = (1/2)ωψ 2 dµh/ . h/  St,u  2 St,u 0 ∂u

(5.62)

110

Chapter 5. The Fundamental Energy Estimate

Here we have used the fact that ψ vanishes on St,0 ⊂ C0 . Comparing with (5.60) we obtain: (1/2)ωψ 2 dµh/ − I (5.63) E1u (t) − E1u (t) = St,u

where I is the integral: (1/2){2T ω + α −1 κ[Lω + (ν + αtrθ )ω] + ωT (log )}ψ 2 dµh/ du  . I = tu

Taking account of (5.59), equation (5.10), equations (3.46) and (3.118) which give: trχ = α(trk/ − trθ ) trχ = κ(trk/ + trθ )

(5.64)

as well as equation (5.11), we find that the coefficient of (1/2)ψ 2 in the integrant of I is Lω + νω, hence I =

tu

(1/2)(Lω + νω)ψ 2 dµh/ du 

and (5.63) takes the form: u u 2 E1 (t) − E1 (t) = (1/2)ωψ dµh/ − St,u

tu

(1/2)(Lω + νω)ψ 2 dµh/ du  .

(5.65)

Substituting (5.65) and (5.57) in the identity (5.31) for K 1 , we see that the surface integral St,u

(1/2)ωψ 2 dµh/

cancels and the identity takes the form: u t E1 (t) + F1 (u) = (1/2)(Lω + νω)ψ 2 dµh/ du  tu − (1/2)(Lω + νω)ψ 2 dµh/ du  0u

+ E1u (0) +

Wut

Q 1 dµh

(5.66)

with E1u and F1t given by (5.58) and (5.53) respectively. Also, the identity (5.31) for K 0 reads: E0u (t) + F0t (u) = E0u (0) +

Wut

with E0u and F0t given by (5.47) and (5.48) respectively.

Q 0 dµh

(5.67)

Chapter 5. The Fundamental Energy Estimate

111

By virtue of the basic assumptions A1 and A2 there is a positive constant C such that: C −1 E0u (t)



u

≤



0

 St,u 

≤ CE0u (t) t  C −1 F0t (u) ≤ 0

[η0−2 µ(η0

+

µ)(η0−2 (Lψ)2

+ |d/ψ| ) + (Lψ) ]dµh/ du  2

2



St  ,u

(5.68)

[η0−2 (η0 + µ)(Lψ)2 + µ|d/ψ|2 ]dµh/ dt 

≤ CF0t (u).

(5.69)

Moreover, by virtue of the assumptions A1, A2, B1 and D1 there is a positive constant C such that:  u  −2 −1 u 2 2 2 C E1 (t) ≤ (1 + t) µ[η0 (Lψ + νψ) + |d/ψ| ]dµh/ du  0

≤ C −1 F1t (u) ≤

CE1u (t) t

St,u 

(1 + t  )2



0

 St  ,u

(5.70)

(Lψ + νψ)2 dµh/ dt 

≤ CF1t (u).

(5.71)

We shall now show that there is a positive numerical constant C such that for all u ∈ [0, ε0 ]: ψ 2 dµh/ ≤ ε0 CE0u (t). (5.72) St,u

To prove this we make use of the acoustical coordinates(t, u, ϑ), ϑ = (ϑ 1 , ϑ 2 ). Now, on a given hyperplane t we may set  = 0, taking the coordinate lines ϑ = const. on the given t to be the integral curves of the vectorfield T (see Chapter 3). We then have: T =

∂ ∂u

hence, in view of the fact that ψ vanishes on C0 , u ψ(t, u, ϑ) = (T ψ)(t, u  , ϑ)du  .

(5.73)

0

It follows that:

St,u

ψ 2 dµh/˜ = =

S2

ψ 2 (t, u, ϑ)dµh/˜ (t, u, ϑ) 

S2



≤ ε0

S2

u





2

(T ψ)(t, u , ϑ)du dµh/˜ (t, u, ϑ) 0  u 2   (T ψ) (t, u , ϑ)du dµh/˜ (t, u, ϑ). 0

(5.74)

112

Chapter 5. The Fundamental Energy Estimate

Now, in view of the fact that h /˜ = h /, (5.61) implies: L /T dµh/˜ = (κtrθ + T log )dµh/˜

(5.75)

while from (5.64) and the definitions (5.10), (5.11), we have:

Thus:

κtrθ + T log  = −α −1 κν + ν.

(5.76)

L /T dµh/˜ = (−α −1 κν + ν)dµh/˜ .

(5.77)

By virtue of assumptions B1, B2 (and A1, A2) we have |α −1 κν + ν| ≤ C(1 + t)−1 [1 + log(1 + t)]4 ≤ C  .

(5.78)

Integrating (5.77) along the integral curves of T on t , using (5.79) and recalling that ε0 is subject to the restriction ε0 ≤ 1/2, we obtain: C −1 ≤

dµh/˜ (t, u, ϑ) dµh/˜ (t, 0, ϑ)

: for all (u, ϑ) ∈ [0, ε0 ] × S 2 .

≤C

(5.79)

In view of (5.79), the right-hand side of (5.74) is bounded by:

u

Cε0 0





S2









(T ψ) (t, u , ϑ)dµh/˜ (t, u , ϑ) du = Cε0 2

u



 (T ψ) dµh/ du  . 2

St,u 

0

(5.80) By (5.59),

T ψ = (1/2)((Lψ) − α −1 κ(Lψ))

hence,

(T ψ)2 ≤ (1/4)((Lψ)2 + α −2 κ 2 (Lψ)2 ).

Comparing with (5.47), the result (5.72) then follows, in view of the basic assumption A1. We now introduce the following quantities which are, by their definition, nondecreasing functions of t at each u: u

u E 1 (t)

E 0 (t) = supt  ∈[0,t ]E0u (t  )

(5.81)

F0t (u)

(5.82)

= supt  ∈[0,t ][1 + log(1 + t  )]−4 E1u (t  )

t F 1 (u) = u quantities E 0 (t),

t 

supt  ∈[0,t ] [1 + log(1 + t  )]−4 F1 (u).

(5.83) (5.84)

u E 1 (t)

are also non-decreasing functions of u at each t. In Note that the view of inequalities (5.68)–(5.72) statements (i)–(v) of Theorem 5.1 are equivalent to the statement that, under the assumptions of the theorem, ε

E 00 (s),

ε

sup F0s (u), E 1 0 (s),

u∈[0,ε0 ]

s

sup F 1 (u) ≤ C D0 .

u∈[0,ε0 ]

(5.85)

Chapter 5. The Fundamental Energy Estimate

113

In view of the identities (5.66), (5.67), this will follow if we succeed in estimating, under the said assumptions, the spacetime integrals Q 0 dµh , Q 1 dµh Wut

Wut

u

u

t

in terms of the four quantities E 0 (t), F0t (u), E 1 (t), F 1 (u). From (5.20), (5.25), we have: Q 0 = −2 ρ K 0 ψ − (1/2)T µν π˜ 0,µν = Q 0,0 + Q 0,1 + Q 0,2 + Q 0,3 + Q 0,4 + Q 0,5 + Q 0,6 + Q 0,7

(5.86)

where (see Table 5.44): Q 0,0 = −2 ρ K 0 ψ Q 0,1 = −(1/2)T L L π˜ 0,L L = −(1/8)µ−2 (Lψ)2 π˜ 0,L L

(5.87) (5.88)

Q 0,2 = −(1/2)T L L π˜ 0,L L = −(1/8)µ−2 (Lψ)2 π˜ 0,L L

(5.89)

Q 0,3 = −T

LL

π˜ 0,L L = −(1/4)µ

−1

|d/ψ| π˜ 0,L L

−1

LA

2

A

Q 0,4 = −T π˜ 0,L A = (1/2)µ (Lψ)(d/ ψ)π˜ 0,L A Q 0,5 = −T L A π˜ 0,L A = (1/2)µ−1 (Lψ)(d/ A ψ)π˜ 0,L A

(5.90) (5.91) (5.92)

and, −(1/2)T AB π˜ 0,AB = Q 0,6 + Q 0,7 where: Q 0,6 = −(1/2){d/ A ψd/ B ψ − (1/2)(h/−1 ) AB |d/ψ|2 }π /ˆ˜ 0,AB Q 0,7 = −(1/4)µ−1(Lψ)(Lψ)trπ /˜ 0 .

(5.93) (5.94)

Here π /˜ 0 denotes the restriction to St,u of π˜ 0 , π /˜ 0,AB = π˜ 0,AB , /˜ 0 and we have made use of the trace-free nature of and π /ˆ˜ 0 denotes the trace-free part of π the factor d/ A ψd/ B ψ − (1/2)(h/−1 ) AB |d/ψ|2 . We have (see (5.17)): π˜ 0 = π0 + (K 0 )h and the components of π0 = L K 0 h can be directly calculated from Table 3.117 of connection coefficients of the null frame L, L, X 1 , X 2 , as the vectorfield K 0 is expressed in

114

Chapter 5. The Fundamental Energy Estimate

this frame by (5.15). We find in this way: π˜ 0,L L = 0 π˜ 0,L L = −4µ{L(α −1 κ) − (η0−1 + α −1 κ)L(α −1 κ)} π˜ 0,L L = −2µ{µ−1 (K 0 µ) + (K 0 log ) + 2L(α −1 κ)} π˜ 0,L A = −2(ζ A + η A ) π˜ 0,L A = 2{(η0−1 + α −1 κ)(ζ A + η A ) − µd/ A (α −1 κ)} π /˜ˆ 0,AB = 2{(η0−1 + α −1 κ)χˆ AB + χˆ AB } trπ /˜ 0 = 4{(η0−1 + α −1 κ)ν + ν}.

(5.95)

From (5.22), (5.25) we have:  Q 1 = −2 ρ K 1 ψ − (1/2)T µν π˜ 1,µν + (1/2)2 ψ 2
h˜ ω

(5.96)

= Q 1,0 + Q 1,1 + Q 1,2 + Q 1,3 + Q 1,4 + Q 1,5 + Q 1,6 + Q 1,7 + Q 1,8 where (see Table 5.44): Q 1,0 = −2 ρ(K 1 ψ + ωψ) Q 1,1 = Q 1,2 = Q 1,3 = Q 1,4 = Q 1,5 = and,

 −2 2  −(1/2)T π˜ 1,L ˜ 1,L L L = −(1/8)µ (Lψ) π LL  −2 2  −(1/2)T π˜ 1,L L = −(1/8)µ (Lψ) π˜ 1,L L  −1  −T L L π˜ 1,L /ψ|2 π˜ 1,L L = −(1/4)µ |d L LA  −1 A  −T π˜ 1,L A = (1/2)µ (Lψ)(d/ ψ)π˜ 1,L A LA  −1 A  −T π˜ 1,L A = (1/2)µ (Lψ)(d/ ψ)π˜ 1,L A LL

(5.97) (5.98) (5.99) (5.100) (5.101) (5.102)

 −(1/2)T AB π˜ 1,AB = Q 1,6 + Q 1,7

where: Q 1,6 = −(1/2){d/ A ψd/ B ψ − (1/2)(h/−1 ) AB |d/ψ|2 }π /ˆ˜ 1,AB Q 1,7 =

 −(1/4)µ−1 (Lψ)(Lψ)trπ /˜ 1 .

(5.103) (5.104)

Also, Q 1,8 = (1/2)2 ψ 2
h˜ ω. Here

 π /˜ 1

(5.105)

denotes the restriction to St,u of π˜ 1 ,   , π /˜ 1,AB = π˜ 1,AB

 π /ˆ˜ 1 denotes the trace-free part of π /˜ 1 and we have again made use of the trace-free nature of the factor d/ A ψd/ B ψ − (1/2)(h/−1 ) AB |d/ψ|2 .

Chapter 5. The Fundamental Energy Estimate

115

By (5.21) and the fact that π˜ 1 = π1 + (K 1 )h we have:

π˜ 1 = {π1 + (K 1 log  − 2ω)h}

and the components of π1 = L K 1 h can be directly calculated from Table 3.117, as the vectorfield K 1 is expressed by (5.16). We find in this way:  π˜ 1,L L =0  −1 −1 −1 π˜ 1,L L = −4µ{L(ν ω) − ν ωL(α κ)}  −1 −1 π˜ 1,L L = −2µ{µ K 1 µ + L(ν ω) − 2ω + K 1 log }  π˜ 1,L A =0  −1 π˜ 1,L / A (ν −1 ω)} A = 2{ν ω(ζ A + η A ) − µd

π /ˆ˜ 1,AB = 2ν −1 ωχˆ AB 

trπ /˜ 1 = 0.

(5.106)

We now consider the spacetime integral of Q 0 . In view of the formula (5.27), we have, for an arbitrary function f ,     Wut

t

µ−1 f dµh =

tu

0

f

u

dt  =

0

f

C ut 

du 

(5.107)

where:

tu

u

f =



0

 St  ,u 

f dµh/ du



C ut 

f =

t  0

 St  ,u 

f dµh/ dt  .

(5.108)

Now, we are considering the case that the source function ρ vanishes. Thus, Q 0,0 = 0.

(5.109)

Q 0,1 = 0.

(5.110)

µ−1 |π˜ 0,L L | ≤ C(1 + t)[1 + log(1 + t)]−2

(5.111)

µ|Q 0,2 | ≤ C(1 + t)[1 + log(1 + t)]−2 (Lψ)2 .

(5.112)

By the first of formulas (5.95), also:

By assumptions B10, B11 and A1,

hence, from (5.89),

116

Chapter 5. The Fundamental Energy Estimate

Writing: (Lψ)2 ≤ 2(Lψ + νψ)2 + 2(νψ)2

it then follows that:

Wut

where:



t

J0 =

|Q 0,2 |dµh ≤ C(J0 + J1 )

(1 + t  )[1 + log(1 + t  )]−2



and:

J1 =

t

(1 + t  )[1 + log(1 + t  )]−2



(νψ)2 dt 

(5.114)

 tu

0

 tu

0

(5.113)

(Lψ + νψ)2 dt  .

(5.115)

Now by assumption B1,

t

J0 ≤ C

 −1

(1 + t )



[1 + log(1 + t )]

−2



 tu

0

ψ

2

dt  .

Substituting the estimate (5.72) we then obtain: t J0 ≤ Cε02 (1 + t  )−1 [1 + log(1 + t  )]−2 E0u (t  )dt  .

(5.116)

0

On the other hand,



t

J1 =

f 0 (t  )

0

where

g(t) =

t

 2

dg(t  )  dt dt 



(1 + t )

 tu

0

(Lψ + νψ)

2

dt 

(5.117)

and f 0 (t) is the function: f 0 (t) = (1 + t)−1 [1 + log(1 + t)]−2

(5.118)

Integrating by parts we obtain (g(0) = 0): J1 = f 0 (t)g(t) −

t

g(t  )

0

d f 0 (t  )  dt . dt 

Now, by (5.69) we have: t 0

hence:

t 0

tu

St  ,u

(Lψ)2 dµh/ dt  ≤ CF0t (u)

(Lψ)2 dµh/ dt  ≤ C



u 0

F0t (u  )du  .

(5.119)

(5.120)

Chapter 5. The Fundamental Energy Estimate

117

Also, by inequality (5.72) and assumption B1, t 0

(νψ) dµh/ dt ≤ Cε0

t

2

tu



Let us set:

(1 + t )

t 0

sup

t

g(t) =

ψ dµh/ dt  (5.121)

1/2 tu



We then have:

St  ,u 

(1 + t  )−2 E0u (t  )dt  .

 F(t, u) =

 2

u  ∈[0,u]

0

≤ Cε02



 −2

(Lψ + νψ)2

.

(5.122)

(1 + t  )2 F 2 (t  , u)dt 

(5.123)

0

while: f 0 (t)g(t) ≤ [1 + log(1 + t)]−2 

t

≤

F 2 (t  , u)dt 



t

(1 + t  )F 2 (t  , u)dt 

0 1/2 

[1 + log(1 + t)]−4

(5.124)

0

t

(1 + t  )2 F 2 (t  , u)dt 

1/2 .

0

Now, by (5.120) and (5.121), t F 2 (t  , u)dt  ≤ C 0

u 0

F0t (u  )du  + Cε02



t 0

On the other hand, by (5.71) we have: t (1 + t  )2 F 2 (t  , u)dt  ≤ C 0

(1 + t  )−2 E0u (t  )dt  .

u 0

(5.125)

F1t (u  )du  .

(5.126)

Substituting (5.125) and (5.126) in (5.124) then yields: 1/2  u t f 0 (t)g(t) ≤ C F0t (u  )du  + ε02 (1 + t  )−2 E0u (t  )dt  0

0

 · [1 + log(1 + t)]−4

u 0

Moreover, since

F1t (u  )du 

1/2 .

(5.127)

   d f0  −2 −2    dt  ≤ C(1 + t) [1 + log(1 + t)] ,

in view of (5.123), (5.126) we have:    u t t      d f 0 (t )    −2  −2 t    g(t )  (1 + t ) [1 + log(1 + t )] F1 (u )du dt  . dt ≤ C dt   0 0 0 (5.128)

118

Chapter 5. The Fundamental Energy Estimate

The bounds (5.127), (5.128) together imply: 

u

J1 ≤ C 0

F0t (u  )du 

+

ε02

t

(1 +

0

t  )−2 E0u (t  )dt 

1/2

1/2 F1t (u  )du  0  u t  +C (1 + t  )−2 [1 + log(1 + t  )]−2 F1t (u  )du  dt  .  · [1 + log(1 + t)]−4

u

0

(5.129)

0

Substituting the estimates (5.116) and (5.129) in (5.113) then yields: t |Q 0,2 |dµh ≤ Cε02 (1 + t  )−1 [1 + log(1 + t  )]−2 E0u (t  )dt  Wut

0



u

+C

F0t (u  )du  + ε02

0

0

t

(1 + t  )−2 E0u (t  )dt 

 −4 · [1 + log(1 + t)]

u 0



t

+C

 −2

(1 + t )

(5.130)



[1 + log(1 + t )]

F1t (u  )du 

−2



0

0

u

1/2

1/2

 F1t (u  )du 



dt  .

We skip for the moment the terms Q 0,3 , Q 0,4 , Q 0,5 and we turn to the estimation of the spacetime integrals of Q 0,6 and Q 0,7 . Now π /ˆ˜ 0 is given in Table 5.95. By assumptions B3, B4, A1 and A3 (recall that −1 κ = α µ) we have: (5.131) |π /ˆ˜ 0 | ≤ C(1 + t)[1 + log(1 + t)]−6 . In view of (5.70) and (5.83) we can then estimate: t |Q 0,6 |dµh ≤ C (1 + t  )−1 [1 + log(1 + t  )]−6 E1u (t  )dt  Wut

0

≤C

0

t

u

(1 + t  )−1 [1 + log(1 + t  )]−2 E 1 (t  )dt  .

(5.132)

Also, trπ /˜ 0 is given in Table 5.95. By assumptions B1, B2, A1 and A3 we have: |trπ /˜ 0 | ≤ C(1 + t)−1 [1 + log(1 + t)]4 .

(5.133)

Writing |Lψ| ≤ |Lψ + νψ| + |νψ| we can then estimate:

Wut

|Q 0,7 |dµh ≤ C(J2 + J3 )

(5.134)

Chapter 5. The Fundamental Energy Estimate

119

where: J2 =

t





f 1 (t )

|νψ||Lψ|

(5.135)

|Lψ + νψ||Lψ|

(5.136)

f 1 (t) = (1 + t)−1 [1 + log(1 + t)]4 .

(5.137)

tu

0

J3 =

t



f 1 (t  )

tu

0

and: We estimate:

1/2 1/2

J2 ≤ I0 I1

(5.138)

where:

t

I0 =





(1 + t ) f 1 (t )

tu

0



t

I1 =





(1 + t  )−1 f 1 (t  )

(νψ)



0

2

dt 

(5.139)

 tu

(Lψ)2 dt  .

(5.140)

By assumption B1 and the estimate (5.72) we have, with f 2 (t) = (1 + t)−1 f 1 (t) = (1 + t)−2 [1 + log(1 + t)]4

t

I0 ≤ Cε0

f 2 (t  )E0u (t  )dt  .

0

Also, by (5.70),



t

I1 ≤ C We thus obtain: 1/2



J2 ≤ Cε0

I2 = 0

t



(5.143)

f2 (t  )E0u (t  )dt  .

(5.144)

1 (I1 + I2 ) 2

J3 ≤ where:

t 0

We estimate:

(5.142)

f 2 (t  )E0u (t  )dt  .

0



(1 + t ) f 1 (t )

(5.141)

(5.145)



 tu

(Lψ + νψ)

Recalling the definition (5.117) we can write: t dg(t  )  f 2 (t  ) dt . I2 = dt  0

2

dt  .

(5.146)

(5.147)

120

Chapter 5. The Fundamental Energy Estimate

Now the integral I2 is similar to the integral J1 but with the function f 2 in the role of the function f0 . Since f 2 decays faster than f 0 (and the same is true for their derivatives), the integral I2 satisfies an estimate similar to (5.129): 

u

I2 ≤ C 0

F0t (u  )du  + ε02 



t 0

(1 + t  )−2 E0u (t  )dt 

1/2

1/2 F1t (u  )du  0  u t  +C (1 + t  )−2 [1 + log(1 + t  )]−2 F1t (u  )du  dt  . · [1 + log(1 + t)]−4



u

0

Combining the above estimates we conclude that: t |Q 0,7 |dµh ≤ C f2 (t  )E0u (t  )dt  Wut

(5.148)

0

0



u

+C 0

F0t (u  )du 

(5.149)

+ ε02

0

t

(1 + t  )−2 E0u (t  )dt 

 · [1 + log(1 + t)]−4

1/2

1/2 F1t (u  )du  0  u t   −2  −2 +C (1 + t ) [1 + log(1 + t )] F1t (u  )du  dt  . u

0

0

We turn to the spacetime integral of Q 1 . Since we are considering the case that the source function ρ vanishes, we have: Q 1,0 = 0.

(5.150)

Q 1,1 = Q 1,4 = Q 1,7 = 0.

(5.151)

In view of Table 5.106, also:

 Consider then Q 1,2 (5.99). The coefficient π˜ 1,L L is given in Table 5.106. By assumptions B1, B10, B13, D1, D3 (and A1),  3 µ−1 |π˜ 1,L L | ≤ C(1 + t)[1 + log(1 + t)]

(5.152)

µ|Q 1,2 | ≤ C(1 + t)[1 + log(1 + t)]3 (Lψ)2 .

(5.153)

hence: Writing again, (Lψ)2 ≤ 2(Lψ + νψ)2 + 2(νψ)2 , it then follows that:

Wut

|Q 1,2 |dµh ≤ C(J4 + J5 )

(5.154)

Chapter 5. The Fundamental Energy Estimate

121

where:

t

J4 =

(1 + t  )[1 + log(1 + t  )]3

0



t

J5 =





(1 + t )[1 + log(1 + t )]

0

 tu

 (νψ)2 dt 

(5.155)

 3

tu

 (Lψ + νψ)

2

dt  .

(5.156)

Now, by assumption B1 and the estimate (5.72), t J4 ≤ Cε02 (1 + t  )−1 [1 + log(1 + t  )]3 E0u (t  )dt  0 u

≤ Cε02 E 0 (t)[1 + log(1 + t)]4 .

(5.157)

On the other hand, appealing to the definition (5.117) we can write: t dg(t  )  J5 = f 3 (t  ) dt dt  0

(5.158)

where f 3 is the function: f 3 (t) = (1 + t)−1 [1 + log(1 + t)]3 . Thus, the integral J5 can be estimated in a similar way to that in which the integral J1 was estimated above (see (5.117)–(5.129)), and we obtain, taking into account the fact that the function g(t) is non-decreasing,   t t  d f 3 (t  )   d f 3 (t  )    J5 = f 3 (t)g(t) − g(t  ) dt ≤ g(t) f (t) + 3  dt   dt dt  0 0 u ≤C F1t (u  )du  (5.159) 0

the factor in parentheses being bounded by a numerical constant. In conclusion, u u |Q 1,2 |dµh ≤ C F1t (u  )du  + Cε02 E 0 (t)[1 + log(1 + t)]4 . (5.160) Wut

0

Consider next Q 1,6 (5.103). The coefficient π /ˆ˜ 1 is given in Table 5.106. By assumptions A1, B1, B3 and D1, |π /ˆ˜ 1 | ≤ C(1 + t)[1 + log(1 + t)]−2 . Hence, by (5.70),  t   −2 |Q 1,6 |dµh ≤ C (1 + t )[1 + log(1 + t )] Wut



0 t

≤C 0

(5.161)  tu

µ|d/ψ|

(1 + t  )−1 [1 + log(1 + t  )]−2 E1u (t  )dt  .

2

dt  (5.162)

122

Chapter 5. The Fundamental Energy Estimate

To estimate the spacetime integral of the term Q 1,8 , we use assumption D5 and the estimate (5.72). We have:    t  u 2  |Q 1,8 |dµh ≤ C sup St  ,u  (µ|
h˜ ω|) · ψ dµh/ du dt  Wut

0

≤ Cε0

0

St  ,u 

t  0

u 0

sup St  ,u  (µ|
h˜ ω|)du  E0u (t  )dt  .

Appealing to assumption D5 we then conclude that: u |Q 1,8 |dµh ≤ Cε0 [1 + log(1 + t)]4 E 0 (t).

(5.163)

Wut

We now turn to the crucial terms Q 0,3 and Q 1,3 .  Consider first Q 1,3 . The coefficient π˜ 1,L L is given in Table 5.106. We have:  −1 −1 π˜ 1,L L = −2µ(ων )(µ Lµ + r1 )

(5.164)

where r1 is the remainder: r1 = ω−1 (Lω − νω) − ν −1 (Lν + ν 2 ) + L log . Thus, Wut

Q 1,3 dµh =

t  0

tu

  −1 −1 2 (ων )(µ Lµ + r1 )µ|d/ψ| dt  . 2

(5.165)

(5.166)

Decomposing µ−1 Lµ into its positive and negative parts, µ−1 Lµ = µ−1 (Lµ)+ + µ−1 (Lµ)− we write: t  Q 1,3 dµh = Wut

0

tu

  (ων −1 ){µ−1 (Lµ)+ + µ−1 (Lµ)− + r1 }µ|d/ψ|2 dt  . 2 (5.167)

Thus, by virtue of assumption C1 and the formula (5.58): t Q 1,3 dµh ≤ {(1 + t  )−1 [1 + log(1 + t  )]−1 + A(t  ) + sup |r1 |}E1u dt  − K (t, u) Wut

0

tu

where K (t, u) is the – non-negative – spacetime integral:  −1 −1 K (t, u) = − ων µ (Lµ)− |d/ψ|2 dµh . t 2 Wu

(5.168)

(5.169)

Chapter 5. The Fundamental Energy Estimate

123

Moreover, by assumptions B1, D1, B12, D2 and B5 we have: sup |r1 | ≤ C(1 + t)−1 [1 + log(1 + t)]−2 . ε

(5.170)

t 0

Consider next Q 0,3 . The coefficient π˜ 0,L L is given in Table 5.95. We have: π˜ 0,L L = −2µ(µ−1(Lµ + Lµ) + r0 )

(5.171)

where r0 is the remainder: r0 = α −2 Lµ + 2L(α −1 κ) + K 0 log  = 3L(α −1 κ) + 2α −2 κ Lα + (1 + α −1 κ)L log  + L log  recalling that µ = ακ. Thus, t  Q 0,3 dµh = Wut

0

tu

  −1 (µ (Lµ + Lµ) + r0 )µ|d/ψ|2 dt  . 2

(5.172)

(5.173)

Decomposing µ−1 (Lµ + Lµ) into its positive and negative parts, µ−1 (Lµ + Lµ) = µ−1 (Lµ + Lµ)+ + µ−1 (Lµ + Lµ)− we then have: t  Q 0,3 dµh ≤ Wut

0

tu

  −1 2 {µ (Lµ + Lµ)+ + |r0 |}µ|d/ψ| dt  . 2

(5.174)

Thus, by virtue of assumption C2, the formula (5.58), as well as assumptions B1, D1, t Q 0,3 dµh ≤ (1 + t  )−2 {B(t  ) + sup |r0 |}E1u dt  . (5.175) Wut

0

tu

Moreover, by assumptions B5, B6, B9 and B10 we have: sup |r0 | ≤ C(1 + t)[1 + log(1 + t)]−6 . ε

(5.176)

t 0

We finally have the terms Q 0,4 , Q 0,5 , Q 1,5 . Consider first Q 1,5 . From (5.102) and (5.106) we have: |Q 1,5 |dµh ≤ M1 + R1

(5.177)

Wut

where:

M1 = R1 =

Wut

Wut

(ων −1 )|Lψ||d/ψ||ζ + η|µ−1 dµh

(5.178)

|Lψ||d/ψ||d/(ων −1 )|dµh .

(5.179)

124

Chapter 5. The Fundamental Energy Estimate

We first estimate the main integral M1 . We decompose: M1 = M1 + M1

where: M1

=

M1 =



U



Uc

Wut



Wut

(5.180)

(ων −1 )|Lψ||d/ψ||ζ + η|µ−1 dµh

(5.181)

(ων −1 )|Lψ||d/ψ||ζ + η|µ−1 dµh

(5.182)

and U is the region defined by (5.12). According to assumption C3 we have: −(Lµ)− ≥ C −1 (1 + t  )−1 [1 + log(1 + t  )]−1 in U



Wut .

(5.183)

Comparing with the definition (5.169) we conclude that:  −1 −1 ων µ (Lµ)− |d/ψ|2 dµh K ≥−  U Wut 2 1 ≥ ων −1 (1 + t  )−1 [1 + log(1 + t  )]−1 µ−1 |d/ψ|2 dµh . (5.184) 2C U  Wut We can thus estimate: where:



N1 =

Wut

M1 ≤ C K 1/2 N1

1/2

(5.185)

(ων −1 )(1 + t  )[1 + log(1 + t  )]|ζ + η|2 |Lψ|2 µ−1 dµh .

(5.186)

By virtue of assumption B7 we have: N1 ≤ C (ων −1 )(1 + t  )−1 [1 + log(1 + t  )]3 |Lψ|2 µ−1 dµh .

(5.187)

Wut

Thus, N1 ≤ C(N1,0 + N1,1 ) where:



N1,0 = N1,1 =

Wut

Wut

(5.188)

(ων −1 )(1 + t  )−1 [1 + log(1 + t  )]3 (νψ)2 µ−1 dµh

(5.189)

(ων −1 )(1 + t  )−1 [1 + log(1 + t  )]3 (Lψ + νψ)2 µ−1 dµh .

(5.190)

Now, by assumptions B1, D1, and A1,

t

N1,0 ≤ C 0

 −1

(1 + t )



[1 + log(1 + t )]

 3 0

u



 ψ dµh/ du 2

St  ,u 

 

dt 

Chapter 5. The Fundamental Energy Estimate

125

hence, by the estimate (5.72), N1,0 ≤ ≤

Cε02

t

(1 + t  )−1 [1 + log(1 + t  )]3 E0u (t  )dt 

0 u Cε02 E 0 (t)[1 + log(1 + t)]4 .

Also, trivially, in view of the formula (5.53),  u

N1,1 ≤ C

C ut 

0

We conclude that:



N1 ≤ C

ων

−1

(Lψ + νψ)

(5.191)

 2

u ε02 E 0 (t)[1 + log(1 + t)]4

du  = C



u 0



u

+ 0

F1t (u  )du  .

F1t (u  )du 

(5.192)

(5.193)

hence from (5.185): M1

≤ C K (t, u)

Since in U c

1/2

 2 u 4 ε0 E 0 (t)[1 + log(1 + t)] +



u 0

F1t (u  )du 

1/2

Wut it holds that µ ≥ η0 /4, 2 M1 ≤ √ ων −1 |Lψ||d/ψ||ζ + η|µ−1/2dµh η0 U c  Wut 2 ≤ √ ων −1 |Lψ||d/ψ||ζ + η|µ−1/2 dµh η0 Wut  1/2 ων −1 |d/ψ|2 2 1/2 dµh ≤ √ · N1   η0 Wut (1 + t )[1 + log(1 + t )]

where N1 is given by (5.186). Thus, by formula (5.58),  t 1/2 E1u (t  )dt  1/2 M1 ≤ C N1  )[1 + log(1 + t  )] (1 + t 0 and N1 has already been estimated in (5.193) above. We conclude that: 1/2  t E1u (t  )dt   M1 ≤ C   0 (1 + t )[1 + log(1 + t )] 1/2  u u · ε02 E 0 (t)[1 + log(1 + t)]4 + F1t (u  )du  .

.

(5.194)

(5.195)

(5.196)

(5.197)

0

We finally estimate the remainder integral R1 of (5.179). Now, assumptions B14 and D4 (together with B1, D1) imply that: ω−1 ν|d/(ων −1 )| ≤ C(1 + t)−1 [1 + log(1 + t)]1/2

(5.198)

126

Chapter 5. The Fundamental Energy Estimate

hence, taking into account assumption A3, the factor µ1/2 ω−1 ν|d/(ων −1 )| is bounded in the same way as |ζ + η|, namely by C(1 + t)−1 [1 + log(1 + t)]. It follows that R1 is bounded in the same way as M1 . Collecting the above results we conclude that:  1/2   t u   E (t )dt 1 |Q 1,5 |dµh ≤ C K (t, u)1/2 +   0 (1 + t )[1 + log(1 + t )] Wut 1/2  u 2 u 4 t   F1 (u )du . (5.199) · ε0 E 0 (t)[1 + log(1 + t)] + 0

We now consider the term Q 0,4 . From (5.91) and (5.95) we have: |Q 0,4 |dµh ≤ M0

(5.200)

Wut



where: M0 =

Wut

Again, we decompose: where:

(5.201)

M0 = M0 + M0

(5.202)



M0 =

U



M0

|Lψ||d/ψ||ζ + η|µ−1 dµh .

=



Uc

Wut



|Lψ||d/ψ||ζ + η|µ−1 dµh

Wut

(5.203)

|Lψ||d/ψ||ζ + η|µ−1 dµh .

(5.204)

Let us define, for 0 ≤ t0 < t1 ≤ s, the region: Wut0 ,t1 = {x ∈ Wus : t0 ≤ x 0 ≤ t1 }.

(5.205)

Now, by (5.184) and assumptions B1, D1, 1 K (t, u) ≥ (1 + t  )[1 + log(1 + t  )]−1 µ−1 |d/ψ|2 dµh . C U  Wut

(5.206)

We can thus estimate: /ψ||ζ + η|µ−1 dµh  t ,t |Lψ||d U

Wu0

1

≤ C K (t1 , u)

 1/2 t ,t1

 ≤ C K (t1 , u)1/2

t ,t1

 ≤ C K (t1 , u)1/2

Wu0

Wu0 t1

t0

 −1

(1 + t )



2 −1

[1 + log(1 + t )]|Lψ| |ζ + η| µ 2

(1 + t  )−3 [1 + log(1 + t  )]3 |Lψ|2 µ−1 dµh

(1 + t  )−3 [1 + log(1 + t  )]3 E0u (t  )dt 

using assumption B7 and recalling formula (5.47).

1/2 dµh

1/2

1/2 (5.207)

Chapter 5. The Fundamental Energy Estimate

127

Let us define: K (t, u) = sup [1 + log(1 + t  )]−4 K (t  , u)

(5.208)

t  ∈[0,t ]

a non-decreasing function of t as well as u. Then, provided that there is a numerical constant C such that: 1 + log(1 + t1 ) ≤C (5.209) 1 + log(1 + t0 ) we may conclude from the above that: /ψ||ζ + η|µ−1 dµh  t ,t |Lψ||d U

Wu0

(5.210)

1



≤ C K (t1 , u)1/2

t1 t0

u

(1 + t  )−3 [1 + log(1 + t  )]7 E 0 (t  )dt 

1/2 .

Let the non-negative integer N be the integral part of log t/ log 2. Then: log t = N + r, log 2

0 ≤ r < 1.

(5.211)

We set: t−1 = 0,

tn = 2n+r : n = 0, 1, . . . , N.

(5.212)

Then t N = t and we have the partition: Wut =

N 

t

,t

Wun−1 n .

(5.213)

n=0

Hence, M0

=

N 

 M0,n

(5.214)

n=0

where, for n = 0, 1, . . . , N,  = M0,n

U



t

Wun−1

,tn

|Lψ||d/ψ||ζ + η|µ−1 dµh .

(5.215)

Now with (tn−1 , tn ) in the role of (t0 , t1 ), condition (5.209) holds for a fixed numerical constant C. Therefore (5.210) applies with (tn−1 , tn ) in the role of (t0 , t1 ):  ≤ C K (tn , u)1/2 An M0,n

1/2

where:

An =

tn tn−1

(5.216) u

(1 + t  )−3 [1 + log(1 + t  )]7 E 0 (t  )dt  .

(5.217)

128

Chapter 5. The Fundamental Energy Estimate

Since K (tn , u) ≤ K (t, u), substituting in (5.214) we obtain: M0 ≤ C K (t, u)1/2

N 

1/2

An

n=0

≤ C K (t, u)

1/2

 N 

1/2  (n + 1)

−2

n=0

≤ C K (t, u)1/2

 N 

1/2 (n + 1)2 An

N  (n + 1)2 An

1/2

n=0

.

(5.218)

n=0

Now, for n = 0, 1, . . . , N and t ∈ [tn−1 , tn ], [1 + log(1 + t  )]2 ≥ [1 + log(1 + tn−1 )]2 ≥ C −1 (n + 1)2 . It follows that: N N   (n + 1)2 An ≤ C n=0

u

n=0 tn−1 t

=C

tn

0

(1 + t  )−3 [1 + log(1 + t  )]9 E 0 (t  )dt  u

(1 + t  )−3 [1 + log(1 + t  )]9 E 0 (t  )dt  .

Substituting in (5.218) we conclude that:  t 1/2 u M0 ≤ C K (t, u)1/2 (1 + t  )−3 [1 + log(1 + t  )]9 E 0 (t  )dt  . 0

To estimate the integral M0 (5.204) we note that since µ ≥ η0 /4 in U c : 2  |Lψ||d/ψ||ζ + η|µ−1/2 dµh M0 ≤ √ η0 U c  Wut 2 ≤ √ |Lψ||d/ψ||ζ + η|µ−1/2 dµh η0 Wut  t  2 1/2 = √ |Lψ||d/ψ||ζ + η|µ dt  . η0 0  u t

Hence: M0

1/2  1/2 t  2 2 2 2 ≤ √ µ||d/ψ| · |Lψ| |ζ + η| dt  η0 0 tu tu t u  1/2 [1 + log(1 + t  )]E0u (t  )1/2  E1 (t ) · dt ≤C  (1 + t  ) 0 (1 + t )

(5.219)

(5.220)

Chapter 5. The Fundamental Energy Estimate



t

≤C

 −2

(1 + t )

0



[1 + log(1 + t t

·

129

0



u )]3 E 1 (t  )dt 

1/2

u

(1 + t  )−2 [1 + log(1 + t  )]3 E 0 (t  )dt 

1/2 .

(5.221)

From (5.200), (5.202), (5.220), (5.221) we conclude that:  t 1/2 u |Q 0,4 |dµh ≤ C K (t, u)1/2 (1 + t  )−3 [1 + log(1 + t  )]9 E 0 (t  )dt  Wut

0



t

+C 0



t

·

u

(1 + t  )−2 [1 + log(1 + t  )]3 E 1 (t  )dt   −2

(1 + t )

[1 + log(1 + t

0



u )]3 E 0 (t  )dt 

1/2

1/2 .

We finally consider the term Q 0,5 . From (5.92) and (5.95) we have: |Q 0,5 |dµh ≤ M˜ 0 + R0

(5.222)

(5.223)

Wut

where: M˜ 0 =



R0 =

Wut

Wut

|Lψ||d/ψ|(η0−1 + α −1 κ)|ζ + η|µ−1 dµh

(5.224)

|Lψ||d/ψ||d/(α −1 κ)|dµh .

(5.225)

Decomposing: M˜ 0 = M˜ 0 + M˜ 0 M˜ 0 = |Lψ||d/ψ|(η0−1 + α −1 κ)|ζ + η|µ−1 dµh 

(5.226)

M˜ 0 =

(5.228)



U

Uc

(5.227)

Wut



Wut

|Lψ||d/ψ|(η0−1 + α −1 κ)|ζ + η|µ−1 dµh

we handle M˜ 0 in a way similar to that in which M0 was handled. By virtue of inequality (5.206) we have: /ψ|(η0−1 + α −1 κ)|ζ + η|µ−1 dµh  t ,t |Lψ||d U

Wu0

1

≤ C K (t1 , u)



1/2 t ,t1

Wu0

(1 + t  )−1 [1 + log(1 + t  )]

(Lψ)2 (η0−1 + α −1 κ)2 |ζ + η|2 µ−1 dµh

1/2

130

Chapter 5. The Fundamental Energy Estimate

 ≤ C K (t1 , u)1/2

t ,t1

Wu0

(1 + t  )−3 [1 + log(1 + t  )]4

(η0−1 + α −1 κ)(Lψ)2 µ−1 dµh

1/2

where we have used assumptions A2, A3 and B7. It follows that under the condition 5.209 (see definition 5.208): /ψ|(η0−1 + α −1 κ)|ζ + η|µ−1 dµh  t ,t |Lψ||d U

Wu0

1



≤ C K (t1 , u)

1/2 t ,t1

Wu0

(1 + t  )−3 [1 + log(1 + t  )]8

(η0−1 + α −1 κ)(Lψ)2 µ−1 dµh

1/2

.

(5.229)

In reference to (5.211)–(5.213) we have: M˜ 0 =

N 

 M˜ 0,n

(5.230)

n=0

where,  M˜ 0,n =

U



t

Wun−1

,tn

|Lψ||d/ψ|(η0−1 + α −1 κ)|ζ + η|µ−1 dµh .

(5.231)

The estimate (5.229) then applies with (tn−1 , tn ) in the role of (t0 , t1 ): 1/2  M˜ 0,n ≤ C K (tn , u)1/2 A˜ n

where: A˜ n =

t

Wun−1

,tn

(5.232)

(1 + t  )−3 [1 + log(1 + t  )]8 (η0−1 + α −1 κ)(Lψ)2 µ−1 dµh .

(5.233)

We thus obtain, as in (5.218),  M˜ 0 ≤ C K (t, u)1/2

N  (n + 1)2 A˜ n

1/2

n=0

and we have: N N   2 ˜ (n + 1) An ≤ C

t

,tn

(1 + t  )−3 [1 + log(1 + t  )]10

n−1 n=0 Wu (η0−1 + α −1 κ)(Lψ)2 µ−1 dµh

n=0

=C

Wut

(1 + t  )−3 [1 + log(1 + t  )]10

(η0−1 + α −1 κ)(Lψ)2 µ−1 dµh .

(5.234)

Chapter 5. The Fundamental Energy Estimate

131

Now the factor (1 +t  )−3 [1 +log(1 +t  )]10 is bounded by a numerical constant. It follows that: N  (n + 1)2 A˜ n ≤ C (η0−1 + α −1 κ)(Lψ)2 µ−1 dµh . Wut

n=0

In view of formula (5.48) we then obtain: N  2 ˜ (n + 1) An ≤ C

u 0

n=0

F0t (u  )du 

(5.235)

and (5.234) then yields: M˜ 0 ≤ C K (t, u)1/2

 0

u

F0t (u  )du 

1/2 .

(5.236)

 Next we estimate M˜ 0 . Since µ ≥ η0 /4 in U c Wut , we have, by assumptions A2, A3 and B7, 2 |Lψ||d/ψ||ζ + η|(η0−1 + α −1 κ)µ−1/2dµh M˜ 0 ≤ √ η0 U c  Wut ≤C (1 + t  )−1 [1 + log(1 + t  )]3/2|Lψ||d/ψ|(η0−1 + α −1 κ)1/2 µ−1/2 dµh . Wut

Since the factor (1 + t  )−1 [1 + log(1 + t  )]3/2 is bounded by a numerical constant we obtain: M˜ 0 ≤ C {(η−1 + α −1 κ)(Lψ)2 + µ|d/ψ|2 }µ−1 dµh . Wut

0

In view of formula (5.48) we then conclude that: u F0t (u  )du  . M˜ 0 ≤ C

(5.237)

0

Finally, the remainder integral R0 is under assumption B8 bounded by: (1 + t  )−1 [1 + log(1 + t  )]|Lψ||d/ψ|(η0−1 + α −1 κ)1/2 µ−1/2 dµh R0 ≤ C Wut

noting that by (5.9), (η0−1 + α −1 κ)1/2µ−1/2 = ((η0−1 + α −2 µ)/µ)1/2 ≥ α −1 ≥ 1. Since the factor (1 + t  )−1 [1 + log(1 + t  )] is bounded by a numerical constant we again obtain: {(η0−1 + α −1 κ)(Lψ)2 + µ|d/ψ|2 }µ−1 dµh R0 ≤ C ≤C

Wut u

0

F0t (u  )du  .

(5.238)

132

Chapter 5. The Fundamental Energy Estimate

Putting together the above results we conclude that: 

Wut

u

|Q 0,5 |dµh ≤ C K (t, u)1/2 0

F0t (u  )du 

1/2



u

+C 0

F0t (u  )du  .

(5.239)

We now focus attention on the integral identity (5.66). By (5.96) and (5.150), (5.151), (5.160), (5.162), (5.163), (5.168) (and (5.170)) and (5.199), the spacetime integral in the right-hand side of (5.66) is bounded from above by: t ˜  )E u (t  )dt  Q 1 dµh ≤ C M(t, u) + L(t, u) + A(t 1 Wut

0

−K (t, u) + C(K (t, u)1/2 + L(t, u)1/2 )M(t, u)1/2 .

(5.240)

Here, u u M(t, u) = E 0 (t)[1 + log(1 + t)]4 + F1t (u  )du  0 t L(t, u) = (1 + t  )−1 [1 + log(1 + t  )]−1 E1u (t  )dt 

(5.241) (5.242)

0

and, ˜ = A(t) + C(1 + t)−1 [1 + log(1 + t)]−2 . A(t)

(5.243)

Note that by virtue of assumption C1 we have: t ˜  )dt  ≤ C (independent of t). A(t

(5.244)

0

We now apply the inequalities: C2 1 −K + C K 1/2 M 1/2 ≤ − K + M, 2 2

C L 1/2 M 1/2 ≤

C2 1 L+ M 2 2

to the last terms on the right-hand side of (5.240). Substituting in (5.240) yields: t 3 1 ˜  )E u (t  )dt  . Q 1 dµh ≤ − K (t, u) + C M(t, u) + L(t, u) + A(t 1 2 2 0 Wut

(5.245)

Here and in the following we shall denote by C various numerical constants. We have: t u L(t, u) ≤ (1 + t  )−1 [1 + log(1 + t  )]3 E 1 (t  )dt  0

1 u ≤ [1 + log(1 + t)]4 E 1 (t). 4

(5.246)

Chapter 5. The Fundamental Energy Estimate

133

Also, since (see (5.84)): t

F1t (u) ≤ [1 + log(1 + t)]4 F 1 (u) defining: V1 (t, u)

we have:

u 0

= 0

u

t

F 1 (u  )du 

(5.247)

(5.248)

F1t (u  )du  ≤ [1 + log(1 + t)]4 V1 (t, u)

(5.249)

hence (see (5.241)): u

M(t, u) ≤ [1 + log(1 + t)]4 (E 0 (t) + V1 (t, u)).

(5.250)

Note that V1 (t, u) is a non-decreasing function of t at each u as well as a non-decreasing function of u at each t. Also, t t  u   4   ˜ ˜  )E u A(t )E1 (t )dt ≤ [1 + log(1 + t)] A(t (5.251) 1 (t )dt . 0

0

By assumptions A1, B2, D1, D3 and the estimate (5.72), the space-like hypersurface integrals on the right-hand side of (5.66) are bounded by:     u  2 2 (1/2)(Lω + νω)ψ − (1/2)(Lω + νω)ψ  ≤ CE 0 [1 + log(1 + t)]4 .  u   tu 0 (5.252) Also, by (5.68) and (5.70) at t = 0, the remaining term on the right-hand side of (5.66) is bounded by: (5.253) E1u (0) ≤ CE0u (0). In view of (5.245)–(5.253) the integral identity (5.66) implies: 1 K (t, u) (5.254) 2 t 3 u u   ˜  )E u ≤ [1 + log(1 + t)]4 E (t) + C(E 0 (t) + V1 (t, u)) + A(t 1 (t )dt . 8 1 0

E1u (t) + F1t (u) +

Keeping only the term E1u (t) on the left we have: [1 + log(1 + t)]−4 E1u (t) ≤

3 u u E (t) + C(E 0 (t) + V1 (t, u)) + 8 1

(5.255)



t 0

  ˜  )E u A(t 1 (t )dt .

The same holds with t replaced by any t  ∈ [0, t]. Now the right-hand side of (5.255) is a non-decreasing function of t at each u. The inequality corresponding to t  thus holds a

134

Chapter 5. The Fundamental Energy Estimate

fortiori if we again replace t  by t on the right-hand side. Taking then the supremum over all t  ∈ [0, t] on the left-hand side we obtain: t 3 u u u   ˜  )E u A(t E 1 (t) ≤ E 1 (t) + C(E 0 (t) + V1 (t, u)) + 1 (t )dt 8 0 which implies: u

u

E 1 (t) ≤ C(E 0 (t) + V1 (t, u)) + C



t 0

  ˜  )E u A(t 1 (t )dt

(5.256)

(for new constants C). This is a linear integral inequality, with respect to t, for the function u u E 1 (t). In view of (5.244) and the fact that E 0 (t) + V1 (t, u) is an non-decreasing function of t at each u, (5.256) implies: u

u

E 1 (t) ≤ C(E 0 (t) + V1 (t, u)) (for a new constant C), hence also: t u   ˜  )E u (t )dt ≤ E (t) A(t 1 1 0

t

(5.257)

˜  )dt  A(t

0

u

≤ C(E 0 (t) + V1 (t, u))

(5.258)

(for a new constant C). Substituting (5.257) and (5.258) on the right-hand side of (5.254) and keeping only the term F1t (u) on the left-hand side we obtain: u

[1 + log(1 + t)]−4 F1t (u) ≤ C(E 0 (t) + V1 (t, u))

(5.259)

(for a new constant C). The same holds with t replaced by any t  ∈ [0, t]. Now the right-hand side of (5.259) is a non-decreasing function of t at each u. The inequality corresponding to t  thus holds a fortiori if we again replace t  by t on the right-hand side. Taking then the supremum over all t  ∈ [0, t] on the left-hand side we obtain: t

u

F 1 (u) ≤ C(E 0 (t) + V1 (t, u)).

(5.260)

Recalling the definition (5.248) of V1 (t, u), this is a linear integral inequality, with respect t to u, for the function F 1 (u): u t u t F 1 (u) ≤ CE 0 (t) + C F 1 (u  )du  . (5.261) 0

u E 0 (t)

In view of the fact that is a non-decreasing function of u at each t while [0, ε0 ] is a bounded interval, (5.261) implies: t

u

F 1 (u) ≤ CE 0 (t)

(5.262)

(for a new constant C), hence also: u

V1 (t, u) ≤ Cε0 E 0 (t).

(5.263)

Chapter 5. The Fundamental Energy Estimate

135

Substituting this in (5.257) we obtain: u

u

E 1 (t) ≤ CE 0 (t)

(5.264)

(for a new constant C). Also, substituting the estimates (5.257), (5.258), (5.263), in (5.254) and keeping only the term (1/2)K (t, u) on the left-hand side we obtain: u

[1 + log(1 + t)]−4 K (t, u) ≤ CE 0 (t)

(5.265)

(for a new constant C), which implies: u

K (t, u) ≤ CE 0 (t).

(5.266)

We now turn our attention to the integral identity (5.67). By (5.86) and (5.109), (5.110), (5.130), (5.132), (5.149), (5.175) (and (5.176)), (5.222), (5.239), the spacetime integral on the right-hand side of (5.67) is bounded from above by, in view of (5.249): Q 0 dµh Wut



≤

t

u

(1 + t  )−2 [1 + log(1 + t  )]4 B(t  )E 1 (t  )dt  0 t u u +C (1 + t  )−1 [1 + log(1 + t  )]−2 (E 1 (t  ) + E 0 (t  ))dt  0

+ C V0 (t, u)  1/2 t u + C V0 (t, u) + (1 + t  )−1 [1 + log(1 + t  )]−2 E 0 (t  )dt  (V1 (t, u))1/2 0 t  −2 +C (1 + t ) [1 + log(1 + t  )]2 V1 (t  , u)dt  0

+ C K (t, u)



t

1/2

 −1

(1 + t )

[1 + log(1 + t

0



u )]−2 E 0 (t  )dt 

+ C K (t, u)1/2 (V0 (t, u))1/2 .

(5.267)

Here we have defined:

1/2

u

V0 (t, u) = 0

F0t (u  )du  .

(5.268)

We are now to substitute on the right-hand side of (5.267), the estimates (5.263), (5.264), (5.266), just derived. In doing this we estimate the fourth term on the right in (5.267) by:  1/2 t u u (1 + t  )−1 [1 + log(1 + t  )]−2 E 0 (t  )dt  (5.269) CE 0 (t)1/2 V0 (t, u) + 0

 t C2 δ u  −1  −2 u   (1 + t ) [1 + log(1 + t )] E 0 (t )dt ≤ E 0 (t) + V0 (t, u) + 2 2δ 0

136

Chapter 5. The Fundamental Energy Estimate

the sixth term by: u CE 0 (t)1/2

≤



t

 −1

(1 + t )

0

δ u C2 E 0 (t) + 2 2δ



t 0

[1 + log(1 + t



u )]−2 E 0 (t  )dt 

1/2

u

(1 + t  )−1 [1 + log(1 + t  )]−2 E 0 (t  )dt 

 (5.270)

and the seventh term by: u

CE (t)1/2 (V0 (t, u))1/2 ≤

δ u C2 V0 (t, u). E 0 (t) + 2 2δ

(5.271)

The above hold for any positive constant δ (we shall choose δ below). After these substitutions, (5.267) reduces to:     t 3δ u 1 1 ˜  )E u0 (t  )dt  Q 0 dµh ≤ E 0 (t) + C 1 + B(t V0 (t, u) + C 1 + 2 δ δ Wut 0 (5.272) (for new constants C, which are independent of δ). Here: ˜ B(t) = (1 + t)−2 [1 + log(1 + t)]4 B(t) + C(1 + t)−1 [1 + log(1 + t)]−2 . Note that by assumption C2: t

˜  )dt  ≤ C B(t

(independent of t).

(5.273)

(5.274)

0

In view of (5.272) the identity (5.67) yields:   t 3δ u 1 ˜  )E u0 (t  )dt  . B(t E0u (t) + F0t (u) ≤ E0u (0) + E 0 (t) + C 1 + V0 (t, u) + 2 δ 0 (5.275) We now set:

1 . 3 Keeping only the term E0u (t) on the left in (5.275) we then have: t 1 u ˜  )E u0 (t  )dt  E0u (t) ≤ E0u (0) + E 0 (t) + C V0 (t, u) + C B(t 2 0 δ=

(5.276)

(5.277)

(for new constants C). The same holds with t replaced by any t  ∈ [0, t]. The right-hand side of (5.277) being a non-decreasing function of t at each u, the inequality corresponding to t  holds a fortiori if we again replace t  by t on the right. Taking then the supremum over all t  ∈ [0, t] on the left we obtain: t 1 u u u ˜  )E u0 (t  )dt  B(t E 0 (t) ≤ E0 (0) + E 0 (t) + C V0 (t, u) + C 2 0

Chapter 5. The Fundamental Energy Estimate

137

which implies:

u

E (t) ≤ E0u (0) + C V0 (t, u) + C

t

0

˜  )E u0 (t  )dt  B(t

(5.278)

(for new constants C). This is a linear integral inequality, with respect to t, for the function u E 0 (t). In view of (5.274) and the fact that V0 (t, u) is a non-decreasing function of t at each u, (5.278) implies: u E (t) ≤ C(E0u (0) + V0 (t, u)) (5.279) (for a new constant C), hence also: t ˜  )E u0 (t  )dt  ≤ E u0 (t) B(t 0

t

˜  )dt  B(t

0

≤ C(E0u (0) + V0 (t, u))

(5.280)

(for a new constant C). Substituting (5.279) and (5.280) on the right-hand side of (5.275) (noting (5.276)) and keeping only the term F0t (u) on the left-hand side we obtain: F0t (u) ≤ C(E0u (0) + V0 (t, u))

(5.281)

(for a new constant C). Recalling the definition (5.268) of V0 (t, u), this is a linear integral inequality, with respect to u, for the function F0t (u): u t u F0t (u  )du  . (5.282) F0 (u) ≤ CE0 (0) + C 0

In view of the fact that E0u (0) is a non-decreasing function of u while [0, ε0] is a bounded interval, (5.282) implies: (5.283) F0t (u) ≤ CE0u (0) (for a new constant C), hence also: V0 (t, u) ≤ Cε0 E0u (0).

(5.284)

Substituting this in (5.279) we obtain: u

E 0 (t) ≤ CE0u (0).

(5.285)

Substituting finally (5.285) in (5.262), (5.264), (5.266), yields: u

t

E 1 (t), F 1 (u), K (t, u) ≤ CE0u (0).

(5.286)

In view of the inequalities (5.68)–(5.72) (and the definitions (5.81)–(5.84)) the first five conclusions of the theorem readily follow from (5.283), (5.285), (5.286), while the sixth conclusion follows from the estimate on K (t, u) of (5.286) together with the lower bound (5.206) (and the definition (5.208)). The proof of Theorem 5.1 is thus complete.

Chapter 6

Construction of the Commutation Vectorfields In the present chapter we shall construct the vectorfields Yi : i = 1, 2, 3, 4, 5 mentioned at the beginning of the previous chapter. These vectorfields are used to define the higher order variations of the wave function φ. That is, an nth order variation is of the form (see (5.6)): (6.1) ψn = Yi1 · · · Yin−1 ψ1 where ψ1 is a first order variation, namely one of the functions (5.5). Here the indices i 1 , . . . , i n−1 take values in the set {1, 2, 3, 4, 5}. Since ψ1 is a solution of the homoge˜ neous wave equation corresponding to the conformal acoustical metric h,
h˜ ψ1 = 0.

(6.2)

ψn satisfies an inhomogeneous wave equation
h˜ ψn = ρn

(6.3)

where the source function ρn is obtained by successively commuting each of the vectorfields Yi1 , . . . , Yin−1 with the operator
h˜ . For this reason the vectorfields Yi : i = 1, 2, 3, 4, 5 shall be called commutation fields. One condition that we require of the set of commutation vectorfields is that at each point x in the spacetime domain Wε0 under consideration the corresponding set of vectors at x spans the tangent space to the spacetime manifold at x. Thus the set of all ψn for fixed n corresponding to a given ψ1 contains all derivatives of that ψ1 of order n − 1. We take the commutation field Y1 to be the vectorfield T : Y1 = T

(6.4)

Since this is transversal to the characteristic hypersurfaces Cu , we require each of the commutation fields Yi : i = 2, 3, 4, 5 to be tangential to the Cu . Moreover we require that for each u ∈ [0, ε0 ] and each point x ∈ Cu , the set of vectors {Yi (x) : i = 2, 3, 4, 5} span the tangent space to Cu at x.

140

Chapter 6. Construction of the Commutation Vectorfields

Next we take the vectorfield Y1 to be collinear to the vectorfield L whose integral curves are the bicharacteristic generators of the Cu . More precisely, we set: Y2 = Q

(6.5)

Q = (1 + t)L

(6.6)

where:

As this is transversal to the surfaces St,u , the sections of the Cu by the space-like hyperplanes t , we require each of the commutation fields Yi : i = 3, 4, 5 to be tangential to the St,u and moreover, for each t and u and each point x ∈ St,u the set of vectors {Yi (x) : i = 3, 4, 5} to span the tangent plane to St,u . More precisely, we set: Yi+2 = Ri : i = 1, 2, 3

(6.7)

where: ◦

Ri =  Ri : i = 1, 2, 3

(6.8)

◦

Here the Ri are defined relative to the background Euclidean metric g on each t . They are the generators of rotations about the three rectangular coordinate axes: ◦

Ri = i j k x j

∂ 1 = i j k  j k ∂xk 2

(6.9)

(rectangular coordinates), where i j k is the fully antisymmetric 3-dimensional symbol (see (1.90)). In the definition (6.8),  is the orthogonal projection to the tangent plane to the surfaces St,u with respect to the induced acoustical metric h on t . Now since
h˜ is a geometric differential operator on functions, defined solely ˜ its commutator with an arbitrary vectorfield Y , considered also through the metric h, as a differential operator on functions, involves just (Y ) π˜ , the Lie derivative with respect ˜ The precise formula will be given in the sequel. What we wish to note at this to Y of h. point is that control of the commutator [
h˜ , Y ] depends on controlling the deformation tensor (Y ) π˜ corresponding to the commutation field Y . The (Y ) π˜ refer to the conformal ˜ Since h˜ = h, these are related to the (Y ) π, the deformation tensors acoustical metric h. relative to the acoustical metric h, by: (Y )

π˜ = 

(Y )

π + (Y )h

(6.10)

For any pair Z 1 , Z 2 of vectors at a point we have: (Y )

π(Z 1 , Z 2 ) = h(D Z 1 Y, Z 2 ) + h(D Z 2 Y, Z 1 )

(6.11)

Chapter 6. Construction of the Commutation Vectorfields

141

The components of the deformation tensors of the commutation fields T and Q in the null frame L, L, X 1 , X 2 can then be directly computed from Table 3.117. We find: (T ) (T ) (T )

π˜ L L = 0 π˜ L L = 4µT (α −1 κ) π˜ L L = −2(T µ + µT log )

(T )

π˜ L A = −(ζ A + η A ) π˜ L A = −α −1 κ(ζ A + η A ) (T ) ˜ˆ π / AB = (χˆ AB − α −1 κ χˆ AB )

(T )

tr

(T ) ˜

π / = 2(ν − α −1 κν)

(6.12)

and: (Q)

π˜ L L = 0

(Q)

π˜ L L = 4µ{Q(α −1 κ) − α −1 κ}

(Q)

π˜ L L = −2{Qµ + µQ log  + µ}

(Q)

π˜ L A = 0

(Q)

π˜ L A = 2(1 + t)(ζ A + η A ) (Q) ˆ˜ π / AB = 2(1 + t)χˆ AB (Q) ˜ tr π / = 4(1 + t)ν

(6.13)

/˜ the restriction of (Y ) π˜ to St,u , and by (Y ) π /ˆ˜ the trace-free part Here we denote by (Y ) π (Y ) ˜ of π /. Noting that the definition (6.8) of the commutation fields Ri : i = 1, 2, 3 is intrinsic to each space-like hypersurface t , we shall derive expressions for the components of (Ri ) π in the frame L, T, X , X , taking advantage of the fact that T, X , X is a frame 1 2 1 2 field for each t . For the following we refer to Table 3.102. We have: (Ri )

π L L = 2h(D L Ri , L) = −2h(Ri , D L L) = −2µ−1 (Lµ)h(Ri , L) = 0

(6.14)

L being h- orthogonal to the Ri , since L is h- orthogonal to the St,u while the Ri are tangential to the St,u . For the same reason T is h- orthogonal to the Ri , hence: (Ri )

πT T = 2h(DT Ri , T ) = −2h(Ri , DT T )

Noting that since the Ri are tangential to the St,u we can expand Ri = RiA X A , we obtain (see Table 3.102):

(Ri )

πT T = 2κ Ri κ

(6.15)

142

Chapter 6. Construction of the Commutation Vectorfields

Next, (Ri )

π LT = h(D L Ri , T ) + h(DT Ri , L) = −h(Ri , D L T ) − h(Ri , D L T ) = −η A RiA + ζ A RiA

hence, by (3.65):

(Ri )

Next, we have:

(Ri )

π LT = −Ri µ

(6.16)

π L A = h(D L Ri , X A ) + h(D X A Ri , L)

Now, h(D X A Ri , L) = −h(Ri , D X A L) = −χ AB RiB On the other hand, from the definition (6.8), ◦

◦

D L Ri = (D L ) Ri +(D L Ri ) Noting that for any vectorfield Z h(Z , X A ) = h(Z , X A ), we obtain:

◦

◦

h(D L Ri , X A ) = h((D L ) Ri , X A ) + h(D L Ri , X A ) Thus, (Ri )

Next,

◦

◦

π L A = h((D L ) Ri , X A ) + h(D L Ri , X A ) − χ AB RiB (Ri )

(6.17)

πT A = h(DT Ri , X A ) + h(D X A Ri , T )

By (3.45), h(D X A Ri , T ) = −h(Ri , D X A T ) = −h(Ri , D X A T ) = −κθ AB RiB Using also the fact that by definition (6.8): ◦

◦

DT Ri = (DT ) Ri +(DT Ri ) we obtain: (Ri )

◦

◦

πT A = h((DT ) Ri , X A ) + h(DT Ri , X A ) − κθ AB RiB

Finally, we have: (Ri )

and:

π AB = h(D X A Ri , X B ) + h(D X B Ri , X A ) ◦

◦

D X A Ri = (D X A ) Ri +(D X A Ri )

(6.18)

Chapter 6. Construction of the Commutation Vectorfields

143

Thus, (Ri )

◦

◦

π AB = h((D X A ) Ri , X B ) + h((D X B ) Ri , X A ) ◦

◦

+h(D X A Ri , X B ) + h(D X B Ri , X A ) ◦

(6.19) ◦

◦

The formulas (6.17)–(6.19) involve (D) Ri as well as D Ri . Since the Ri are tangential to the t we can expand: ◦

Ri = RiA X A + λi Tˆ , for some functions λi . Since

Tˆ = κ −1 T

(6.20)

T = 0,

for any vectorfield Z we have: (D Z )T = D Z (T ) − (D Z T ) = −(D Z T ) Setting Z equal to L, T, X A : A = 1, 2 successively, we obtain, using Table 3.102, (D L )T = ζ A X A (DT )T = κ(d/ A κ)X A (D X A )T = −κθ AB X B

(6.21)

Since X A = X A , for any vectorfield Z we have: (D Z )X A = D Z (X A ) − (D Z X A ) = D Z X A − (D Z X A ) In particular, from Table 3.102, (D L )X A = −µ−1 ζ A L Next, we can expand: DT X A − (DT X A ) = a A L + b A T Taking the h- inner products with L and T successively yields: h(DT X A , L) = −µb A

and

h(DT X A , T ) = −µa A + κ 2 b A

On the other hand we have: h(DT X A , L) = −h(X A , DT L) = −η A and: h(DT X A , T ) = −h(X A , DT T ) = κd/ A κ

(6.22)

144

Chapter 6. Construction of the Commutation Vectorfields

Substituting and solving for a A and b A yields: a A = −α −1 d/ A κ + α −2 η A

b A = µ−1 η A

and

We thus obtain: (DT )X A = (−α −1 d/ A κ + α −2 η A )L + µ−1 η A T

(6.23)

Finally, we have: (D X A )X B = D X A X B − (D X A X B ) Since /XA X B, (D X A X B ) = D the last entry of Table 3.102 gives: (D X A )X B = α −1 k/ AB L + µ−1 χ AB T

(6.24)

◦

Substituting in (D) Ri the expansion (6.20), that is: ◦

(D) Ri = RiA (D)X A + κ −1 λi (D)T, and using the formulas (6.21)–(6.24) for (D)T and (D)X A , we obtain the following ◦

results for the expressions in (6.17)–(6.19) involving (D) Ri : ◦

h((D L ) Ri , X A ) = κ −1 λi ζ A ◦

h((DT ) Ri , X A ) = λi d/ A κ ◦

h((D X A ) Ri , X B ) = −λi θ AB

(6.25)

◦

The expressions (6.17)–(6.19) involve also D Ri . Now, in an arbitrary coordinate system we have: ◦ Dµ Riν =

◦

∂ Riν h ν ◦λ + !µλ Ri ∂xµ

h

ν are the connection coefficients of the acoustical metric h where !µλ µν in the given coordinate system. On the other hand, in the same coordinate system we have: ◦ ∇µ Riν =

◦

◦ g ∂ Riν ν + !µλ Riλ µ ∂x

g

ν are the connection coefficients of the Minkowskian metric g where !µλ µν in that coordinate system. Therefore, ◦

◦

◦

Dµ Riν = ∇µ Riν +
νµλ Riλ

(6.26)

Chapter 6. Construction of the Commutation Vectorfields

145

where
is the difference of the two connections, a tensorfield, whose components in any given coordinate system are: g

h

ν ν − !µλ
νµλ =!µλ

(6.27)

Let us now fix the coordinate system to be a rectangular coordinate system of the Minkowskian metric g. We then have: g

ν !µλ = 0,

hence: ◦ ∇µ Riν = ◦

◦

∂ Riν ∂xµ

and the components Riν are given by (6.9): ◦

◦

Ri0 = 0, Thus,

◦

Rin = i j n x j ◦

∇0 Riν = 0,

∇µ Ri0 = 0,

(6.28) ◦

∇m Rin = imn

(6.29)

(In the above formulas the Greek indices take the values 0,1,2,3 while the Latin indices take the values 1,2,3.) In rectangular coordinates of g (6.27) reduces to: h

ν = (h −1 )νκ !µλκ
νµλ =!µλ

(6.30)

where, as in Chapter 3 (see equation (3.139)), 1 (∂µ h λκ + ∂λ h µκ − ∂κ h µλ ) 2 According to equation (3.141) we have: !µλκ =

1 dH (∂µ σ ψλ ψκ + ∂λ σ ψµ ψκ − ∂κ σ ψµ ψλ ) + H ψκ ∂µ ψλ 2 dσ

!µλκ =

(6.31)

From (6.26), (6.29)–(6.31) we conclude that the expressions ◦

◦

◦

h(D L Ri , X A ), h(DT Ri , X A ), h(D X A Ri , X B ), which appear in the formulas (6.17), (6.18), (6.19) respectively, are given by: ◦ µ

◦

h(D L Ri , X A ) = h µν L λ Dλ Ri X νA = =

◦ µ +!λµν Ri )X νA ◦ h mn L l ilm X nA + H ψ A Rim (Lψm ) ◦ ◦ 1 dH {(Lσ )( Rim ψm ) ψ A + ψ L [( Ri +

(6.32)

◦ µ L λ (h µν ∇λ Ri

2 dσ

◦

σ ) ψ A − (d/ A σ )( Rim ψm )]}

146

Chapter 6. Construction of the Commutation Vectorfields ◦ µ

◦

h(DT Ri , X A ) = h µν T λ Dλ Ri X νA

(6.33)

◦ µ T λ (h µν ∇λ Ri

◦ µ +!λµν Ri )X νA ◦ h mn T l ilm X nA + H ψ A Rim (T ψm ) ◦ ◦ 1 dH {(T σ )( Rim ψm ) ψ A + κψTˆ [( Ri +

= =

2 dσ

◦

σ ) ψ A − (d/ A σ )( Rim ψm )]}

◦ µ

◦

h(D X A Ri , X B ) = h µν X λA Dλ Ri X νB = = +

(6.34)

◦ µ X λA (h µν ∇λ Ri

◦ µ +!λµν Ri )X νB ◦ h mn X lA ilm X nB + H ψ B Rim (d/ A ψm ) ◦ ◦ 1 dH {(d/ A σ )( Rim ψm ) ψ B + ( Ri σ ) ψ A

2 dσ

◦

ψ B − (d/ B σ )( Rim ψm ) ψ A }

The above formulas are expressed in rectangular coordinates of the Euclidean metric g on t . We now substitute in (6.17) from (6.25) and (6.32) to obtain: (Ri )

π L A = −χ AB RiB + ilm L l h mn X nA

◦ +κ −1 λi ζ A + H ψ A Rim (Lψm ) ◦ ◦ 1 dH {(Lσ )( Rim ψm ) ψ A + ψ L [( Ri +

2 dσ

(6.35) ◦

σ ) ψ A − (d/ A σ )( Rim ψm )]}

Next we substitute in (6.18) from (6.25) and (6.33) to obtain: (Ri )

πT A = −κθ AB RiB + ilm T l h mn X nA

◦ +λi d/ A κ + H ψ A Rim (T ψm ) ◦ ◦ 1 dH {(T σ )( Rim ψm ) ψ A + κψTˆ [( Ri +

2 dσ

(6.36) ◦

σ ) ψ A − (d/ A σ )( Rim ψm )]}

Finally, we substitute in (6.19) from (6.25) and (6.34). Since h mn = δmn + H ψm ψn (rectangular coordinates of the Euclidean metric g on t ), and ilm δmn = iln is antisymmetric in l, n while X lA X nB + X lB X nA is symmetric, only the second part of h mn contributes to ilm h mn (X lA X nB + X lB X nA ), hence: ilm h mn (X lA X nB + X lB X nA ) = H ilm ( ψ A X lB + ψ B X lA )ψm

Chapter 6. Construction of the Commutation Vectorfields

147

We thus obtain: (Ri )

π AB = −2λi θ AB

◦

+H ilm ( ψ A X lB + ψ B X lA )ψm + H ( ψ A X lB + ψ B X lA ) Rim ∂l ψm dH ◦ ( Ri σ ) ψ A ψ B + (6.37) dσ In the formulas (6.35), (6.36), (6.37) for ◦ Ri

(Ri ) π

L A,

(Ri ) π

RiA X A σ

(Ri ) π

T A,

AB ,

respectively,

κ −1 λi T σ

we are to substitute for σ the expression + corresponding to the expansion (6.20). Now, the coefficient λi in the expansion (6.20) can only be assumed to be bounded as µ → 0. Therefore, to obtain (Ri ) π L A , (Ri ) πT A , (Ri ) π AB , bounded as µ → 0, we must show that the coefficient of λi in each of the formulas (6.35), (6.36), (6.37), is in fact bounded as µ → 0. The coefficient of λi in (6.37) is: −2θ AB +

1 dH  ψ A ψ B (T σ ) = −2θ AB κ dσ

 is bounded as µ → 0. In fact, we have (see (3.232)): (see (3.231)) where θ AB  = −α −1 χ AB + k/AB θ AB

(6.38)

where k/AB , defined by (see (3.233)): k/AB = k/ AB −

1 dH ψ A ψ B (T σ ) 2κ dσ

(6.39)

and given by (3.236), is bounded as µ → 0. The coefficient of λi in (6.37) is then: 2(α −1 χ AB − k/AB ) and this is bounded as µ → 0. The formula (6.37) takes the form: (Ri )

π AB = 2λi (α −1 χ AB − k/AB ) +H ilm ( ψ A X lB + ψ B X lA )ψm

◦

+H { ψ A (d/ B ψm )+ ψ B (d/ A ψm )} Rim dH ψ A ψ B (Ri σ ) + dσ

(6.40)

The coefficient of λi in (6.36) is: d/ A κ +

1 dH ψ ˆ ψ A (T σ ) 2 dσ T

This coefficient is indeed bounded as µ → 0. Recalling from Chapter 2 (see (2.57)) that T = κ Tˆ

(6.41)

148

Chapter 6. Construction of the Commutation Vectorfields

let us define the functions y i by setting: Tˆ i = −

xi + yi 1 − u + η0 t

(6.42)

Using (6.42) and the fact that: ◦

ilm x l h mn X nA = h mn Rim X nA = h/ AB RiB we re-write first two terms in (6.36) in the form:  B l n −κθ AB Ri + ilm T h mn X A = −κ θ AB +

/h AB 1 − u + η0 t

(6.43)

 RiB + κilm y l h mn X nA

Then formula (6.36) takes the form:   / AB h (Ri ) RiB + κilm y l h mn X nA πT A = −κ θ AB + (6.44) 1 − u + η0 t   ◦ 1 dH +λi d/ A κ + ψTˆ ψ A (T σ ) + H ψ A Rim (T ψm ) 2 dσ ◦ ◦ 1 dH {(T σ )( Rim ψm ) ψ A + κψTˆ [(Ri σ ) ψ A − (d/ A σ )( Rim ψm )]} + 2 dσ Finally, the coefficient of λi in (6.35) is: κ −1 ζ A +

1 dH ψ L ψ A (T σ ) = κ −1 ζ A 2κ dσ

(see (3.168)) where κ −1 ζ  is bounded as µ → 0. In fact, we have (see (3.170)): κ −1 ζ A = α A − d/ A α

(6.45)

where  A , defined by (see (3.171)):  A =  A +

1 dH ψ L ψ A (T σ ) 2ακ dσ

(6.46)

and given by (3.174), is bounded as µ → 0. Let us also define the functions z i by setting: Li =

η0 x i + zi 1 − u + η0 t

(6.47)

Comparing (6.42) and (6.47) with equation (2.66) of Chapter 2, we conclude that the functions z i are related to the functions y i by: z i = −αy i +

H ψ0ψi (α − η0 )x i + 1 − u + η0 t 1 + ρH

(6.48)

Chapter 6. Construction of the Commutation Vectorfields

149

Using (6.47) and (6.43) we re-write the first two terms in (6.35) in the form:   η0 h/ AB B l n −χ AB Ri + ilm L h mn X A = − χ AB − RiB + ilm z l h mn X nA 1 − u + η0 t The formula (6.35) takes the form:   / AB η0 h (Ri ) RiB + ilm z l h mn X nA π L A = − χ AB − 1 − u + η0 t

(6.49)

◦

+λi (κ −1 ζ A ) + H ψ A Rim (Lψm ) ◦ ◦ 1 dH + {(Lσ )( Rim ψm ) ψ A + ψ L [(Ri σ ) ψ A − (d/ A σ )( Rim ψm )]} 2 dσ In the remainder of the present chapter we shall show how the (Y ) π, ˜ the deformation tensors of the commutation fields Y , are to be controlled in terms of χ, the second fundamental form of the surfaces St,u with respect to the characteristic hypersurfaces Cu , the function µ, lapse function of the characteristic foliation, and the ψµ = ∂µ φ, the derivatives of the wave function with respect to the translations of the underlying Minkowski spacetime. The estimates of the present chapter are only preliminary and are intended to give the reader a feeling for the sizes of the various quantities and their interdependence. The actual estimates shall be derived in Chapters 10, 11, and 12. In the following the basic bootstrap assumptions A1, A2, A3 of Chapter 5 are assumed to hold. For convenience we restate these assumptions below. The set of three basic bootstrap assumptions shall be referred to as assumptions A. There is a positive constant C independent of s such that in Wεs0 , A1: C −1 ≤  ≤ C A2: C −1 ≤ α/η0 A3: µ/η0 ≤ C[1 + log(1 + t)] We recall that:

∂φ (6.50) ∂xµ (rectangular coordinates). To obtain the estimates which follow we shall make use of the following bootstrap assumptions on the ψµ and their first derivatives. ψµ =

In the following, we denote by δ0 a positive constant, which is by definition less than or equal to unity. This constant is used to keep track of the relative size of various quantities. It is to be chosen suitably small at various points in the remainder of this monograph. On the other hand, no smallness condition is assumed on ε0 , other than the condition: 1 ε0 ≤ 2 There is a positive constant C independent of s such that in Wεs0 , E1: |ψ0 − k|, |ψi | ≤ Cδ0 (1 + t)−1 E2: |T ψµ | ≤ Cδ0 (1 + t)−1 ,

|Lψµ |, |d/ψµ | ≤ Cδ0 (1 + t)−2

150

Chapter 6. Construction of the Commutation Vectorfields

The constant k in the first of E1 is the positive constant characterizing the surrounding constant state (see Chapter 1, last paragraph). We remark here that the second of E2 shall, in the actual estimates of Chapters 10, 11, and 12, follow from the bound: |Qψµ | ≤ Cδ0 (1 + t)−1 while the third shall follow from the bounds: |Ri ψµ | ≤ Cδ0 (1 + t)−1

: i = 1, 2, 3

We shall also make use of the following bootstrap assumptions on the first derivatives of the function µ on the hypersurfaces t and on the quadratic form χ in the tangent plane at each point on each of the surfaces St,u . There is a positive constant C independent of s such that in Wεs0 , |T µ| ≤ Cδ0 [1 + log(1 + t)], |d/µ| ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]     η0 h /  ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] F2: χ − 1−u+η t

F1:

0

Here and in the following, η0 denotes the sound speed in the surrounding constant state. In addition to the above bootstrap assumptions, we make the following assumption on the initial data. 

There is a positive constant C such that on 00 we have: I0:

|κ − 1| ≤ Cδ0

Under the assumptions A, E, F, and I0, the estimates which we shall derive shall hold in Wεs0 with constants which are independent of s. First, on the basis of assumptions A and E alone we shall derive estimates on k/ and , components of k, the second fundamental form of the hypersurfaces t with respect to the acoustical metric h, and on the first derivatives of α, the lapse function of the foliation of the acoustical spacetime by the t . We begin by noting that since σ is given by: σ = (ψ0 )2 −

3  (ψi )2 i=1

the assumptions E1, E2 imply the following bounds on σ and its first derivatives: |σ −k 2 | ≤ Cδ0 (1+t)−1 ,

|T σ | ≤ Cδ0 (1+t)−1 , |Lσ |, |d/σ | ≤ Cδ0 (1+t)−2 (6.51)

Recall that Tˆ is the unit normal to the surfaces St,u in t relative to the induced acoustical metric h. Since for each i = 1, 2, 3, |Tˆ i | is bounded by the Euclidean magnitude of the vector Tˆ which is in turn bounded by its magnitude with respect to h, namely by unity, we have the following elementary bound on Tˆ i , i = 1, 2, 3: |Tˆ i | ≤ 1

(6.52)

|ψTˆ | ≤ Cδ0 (1 + t)−1

(6.53)

It follows by assumption E1 that:

Chapter 6. Construction of the Commutation Vectorfields

151

Next, we have the following elementary bound on L i , i = 1, 2, 3: |L i | ≤ 1

(6.54)

This is deduced as follows. We first define the t -tangent vector: L = Li

∂ ∂xi

(6.55)

Then we have:

∂ +L (6.56) ∂x0 Now in the tangent space at each point in spacetime, the sound cone is contained in the light cone. Thus the vector L is future-directed, time-like or null with respect to the background Minkowskian structure. Hence the Euclidean magnitude of L is bounded by L 0 = 1. The bound then follows as |L i | is in turn bounded by the Euclidean magnitude of L. The bound (6.54) together with assumption E1 implies: L=

|ψ L − k| ≤ Cδ0 (1 + t)−1

(6.57)

The induced acoustical metric h on t dominates the Euclidean metric g (see Chap−1 ter 2). Hence h is dominated by g −1 . In fact in rectangular coordinates of the Euclidean metric we have: (g−1 )i j = δi j

g i j = δi j ,

h i j = δi j + H ψi ψ j ,

(h

−1 i j

) = δi j −

H ψi ψ j (1 + ρ H )

(6.58)

−1

Note also that h , the reciprocal of the induced acoustical metric on the t is expressed in terms of the frame Tˆ , X 1 , X 2 by: (h

−1 i j

j ) = Tˆ i Tˆ j + (h/−1 ) AB X iA X B

(6.59)

It follows by virtue of assumption E1 that: /−1 ) AB ψ A ψ B = (h /−1 ) AB X iA X B ψi ψ j ≤ (h | ψ|2 = (h j

≤

−1 i j

3 

) ψi ψ j

(ψi )2 ≤ Cδ02 (1 + t)−2

i=1

that is:

| ψ| ≤ Cδ0 (1 + t)−1

(6.60)

According to (6.39), k/ is given by: k/ AB = k/AB +

1 dH ψ A ψ B (T σ ) 2κ dσ

(6.61)

152

Chapter 6. Construction of the Commutation Vectorfields

and k/ is given by (3.236). For a bilinear form w in the tangent space to the surface St,u at a point, the magnitude |w| of w is given by: |w|2 = (h /−1 ) AC (h/−1 ) B D w AB wC D By (6.60) and the bounds (6.51), the second term on the right in (6.61), multiplied by κ, is bounded in magnitude by Cδ03 (1 + t)−3 . The estimate of the first term in the expression (3.236) for k/ is similar, the bound in magnitude being by Cδ02 (1 + t)−3 , using again (6.60), assumption E1 and the bounds (6.51). To estimate the second term, the square of the magnitude of ω / AB = X iA (d/ B ψi ) is: /−1 ) AC (h /−1 ) B D ω / AB ω /C D = (h/−1 ) AC (h/−1 ) B D X iA X C (d/ B ψi )(d/ D ψ j ) |ω /|2 = (h j

≤ (h

−1 i j

) (h /−1 ) B D (d/ B ψi )(d/ D ψ j ) ≤

3 

|d/ψi |2 ≤ Cδ02 (1 + t)−4

(6.62)

i=1

using assumption E2. Hence (using also E1) the second term in the expression (3.236) is bounded in magnitude by Cδ0 (1 + t)−2 . We thus obtain the estimate: |k/ | ≤ Cδ0 (1 + t)−2

(6.63)

Combining with the previous estimate for the second term on the right in (6.61), multiplied by κ, and taking also account of the fact that by virtue of assumptions A we have:

we conclude that:

κ = α −1 µ ≤ C[1 + log(1 + t)]

(6.64)

κ|k/| ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]

(6.65)

According to (6.46)  is given by:  A =  A +

1 dH ψ L ψ A (T σ ) 2ακ dσ

(6.66)

and   is given by (3.174). For a linear form v in the tangent space to St,u at a point, the magnitude |v| of v is given by: |v|2 = (h/−1 ) AB v A v B By (6.53), (6.60), assumption E1, and the bounds (6.51), we conclude that the first term on the right in (3.174) is bounded in magnitude by Cδ02 (1+t)−3 . Also, by (6.52) and assumption E2 the second term on the right in (3.174) is bounded in magnitude by Cδ0 (1 + t)−2 . We thus obtain the estimate: (6.67) |  | ≤ Cδ0 (1 + t)−2 By assumption E1 and the estimates (6.60) and (6.51), the second term on the right in (6.66), multiplied by κ, is bounded in magnitude by Cδ02 (1 + t)−2 . Taking account of (6.64) we then conclude that κ|| ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]

(6.68)

Chapter 6. Construction of the Commutation Vectorfields

153

Next we shall estimate, on the basis of assumptions A and E, the function α and its first derivatives. Now, the function α is expressed by equation (2.43) as: α −2 = 1 + F(ψ0 )2

(6.69)

In the surrounding constant state we have: α = η0 Thus, assumption E1 implies: |α − η0 | ≤ Cδ0 (1 + t)−1

(6.70)

Differentiating (6.69) tangentially to St,u we obtain: −2α −3 d/α =

dF (ψ0 )2 d/σ + 2Fψ0 d/ψ0 dσ

Using assumptions E1, E2 and the bounds (6.51) yields: |d/α| ≤ Cδ0 (1 + t)−2

(6.71)

Differentiating (6.69) with respect to T , −2α −3 T α =

dF (ψ0 )2 T σ + 2Fψ0 T ψ0 dσ

−2α −3 Lα =

dF (ψ0 )2 Lσ + 2Fψ0 Lψ0 dσ

and with respect to L,

and using assumptions E1, E2 and the bounds (6.51) yields: |T α| ≤ Cδ0 (1 + t)−1 ,

|Lα| ≤ Cδ0 (1 + t)−2

(6.72)

The estimates (6.71) and (6.67) yield an estimate for κ −1 ζ  through the expression (6.45): (6.73) |κ −1 ζ  | ≤ Cδ0 (1 + t)−2 while the estimate (6.68) together with (6.71) (and (6.64)) yields the following estimate for ζ : (6.74) |ζ | ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] The assumptions E1, E2 and A, allow us to bound also Lµ, given by equation (3.96): Lµ = m + µe (6.75) where m and e are given by (3.97) and (3.98) respectively. By (6.57) and (6.51): |m| ≤ Cδ0 (1 + t)−1

(6.76)

154

Chapter 6. Construction of the Commutation Vectorfields

while by (6.57), (6.53), (6.51), (6.54), and assumptions E1 and E2:

Hence, we obtain: In view of the relation κ =

α −1 µ

|e| ≤ Cδ0 (1 + t)−2

(6.77)

|Lµ| ≤ Cδ0 (1 + t)−1

(6.78)

and the second of (6.72) we also obtain: |Lκ| ≤ Cδ0 (1 + t)−1

(6.79)

Integrating this inequality along the integral curves of L, generators of the Cu , we obtain: |κ − κ0 | ≤ Cδ0 log(1 + t)

(6.80)

Here the value of κ0 at a point in Wε∗0 is the value of κ at the corresponding point on 0 along the same integral curve of L. Combining (6.80) with the assumption I0 on the initial data, we obtain the following estimate in Wεs0 : |κ − 1| ≤ Cδ0 [1 + log(1 + t)]

(6.81)

which improves the estimate (6.64). We now combine assumptions F1 with (6.71) and the first of (6.72) to estimate the first derivatives of κ on the hypersurfaces t : |T κ| ≤ Cδ0 [1 + log(1 + t)],

|d/κ| ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.82)

Next, we combine assumptions F1 with the estimate (6.74) for ζ to obtain an estimate for η through equation (3.65), that is,

We obtain:

η = ζ + d/µ

(6.83)

|η| ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.84)

We now combine assumption F2 with the estimate (6.65) and the bounds (6.51) to obtain estimates for χ, ˆ trχ, ˆ and the functions ν, ν, defined by (5.10), (5.11):   1 d log  trχ + Lσ 2 dσ   1 d log  Lσ ν= trχ + 2 dσ ν=

(6.85) (6.86)

If δ0 is assumed suitably small, assumption F2 implies: trχ ≥ C −1 (1 + t)−1

(6.87)

Chapter 6. Construction of the Commutation Vectorfields

155

This restriction on the size of δ0 is imposed from now on. According to equations (3.46), (3.118), χ = α(k/ − θ )

(6.88)

χ = κ(k/ + θ )

(6.89)

We thus have: trχ = 2κtrk/ − α −1 κtrχ

and

χˆ = 2κ k/ˆ − α −1 κ χˆ

The estimate (6.65) together with (6.87), assumption F2 and the basic assumptions A then yield: ˆ ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]2 |trχ| ≤ C(1 + t)−1 [1 + log(1 + t)], |χ|

(6.90)

Similarly, we obtain: ˆ ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]2 κ|trθ | ≤ C(1 + t)−1 [1 + log(1 + t)], κ|θ|

(6.91)

Finally, combining assumption F2 and (6.87) with the bounds (6.51) we obtain:     C −1 (1 + t)−1 ≤ ν ≤ C(1 + t)−1 , ν − η0 (1 + η0 t)−1  ≤ C(1 + t)−2 [1 + log(1 + t)] (6.92) and combining the first of the estimates (6.90) with the bounds (6.51) we obtain: |ν| ≤ C(1 + t)−1 [1 + log(1 + t)]

(6.93)

The above results allow us to bound the deformation tensors of the vectorfields T and Q, given by Tables 6.12 and 6.13 respectively. / L and (Y ) π / L the In the following, for any commutation field Y we denote by (Y ) π 1-forms on each surface St,u with components: (Y )

π / L (X A ) =

(Y )

πL A,

(Y )

π / L (X A ) =

(Y )

πL A

/˜ L and (Y ) π /˜ L . and similarly for (Y ) π By the first of the estimates (6.82) and (6.72) together with the bounds (6.51) we obtain, in reference to the Table 6.12, µ−1 |

(T )

π˜ L L | ≤ Cδ0 [1 + log(1 + t)]

(6.94)

|

(T )

π˜ L L | ≤ Cδ0 [1 + log(1 + t)]

(6.95)

By the estimates (6.74) and (6.84) (and the basic assumptions A), | −1 µ |

(T ) ˜

π / L | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] (T ) ˜ π / L | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.96) (6.97)

156

Chapter 6. Construction of the Commutation Vectorfields

By the second of the estimates (6.90) and assumption F2 (and the basic assumptions A), | (T ) π (6.98) /ˆ˜ | ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]2 Also, by the estimates (6.92) and (6.93), |tr

(T ) ˜

π /| ≤ C(1 + t)−1 [1 + log(1 + t)]

(6.99)

By (6.79) and the second of the estimates (6.72), and by (6.78) together with the bounds (6.51), we obtain, in reference to Table 6.13, µ−1 |

(Q)

π˜ L L | ≤ C[1 + log(1 + t)]

(6.100)

|

(Q)

π˜ L L | ≤ C[1 + log(1 + t)]

(6.101)

By the estimates (6.74) and (6.84), |

(Q) ˜

π / L | ≤ Cδ0 [1 + log(1 + t)]

(6.102)

Finally, by assumption F2 and the estimate (6.92) we obtain: (Q) ˆ˜

π /| ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] |tr (Q)π /˜ | ≤ C |

We can obtain a more precise estimate on tr (Q) π /˜ by writing (see (6.13)),   1 − η0 (Q) ˜ π / − 4 = −4 + 4( − 1)(1 + t)ν tr 1 + η0 t   η0 +4(1 + t) ν − 1 + η0 t

(6.103) (6.104)

(6.105)

Then by (6.92), together with the fact that since (k 2 ) = 1 by virtue of the first of (6.51) we have | − 1| ≤ Cδ0 (1 + t)−1 we obtain: |tr

(Q) ˜

π / − 4| ≤ C(1 + t)−1 [1 + log(1 + t)]

(6.106) (6.107)

To bound the deformation tensors of the vectorfields Ri we must derive estimates for the functions λi , defined by equation (6.20), which enter the expressions (6.40), (6.44), (6.49), for the components (Ri ) π AB , (Ri ) πT A , (Ri ) π L A , respectively, as well as the functions y i , which enter (6.44) and are defined by equation (6.42), and the functions z i , which enter (6.49) and are defined by equation (6.47).

Chapter 6. Construction of the Commutation Vectorfields

157

Let us denote by r the Euclidean radial coordinate: " # 3 # r = $ (x i )2

(6.108)

i=1

The estimates which follow require upper and lower bounds for the function r on each surface St,u . We derivethese bounds as follows. First,  since the function r achieves its maximum value in t Wε∗0 at the outer boundary t C0 = St,0 where it is equal to 1 + η0 t, we have the upper bound: r ≤ 1 + η0 t

: in Wε∗0

(6.109)

This suffices for the time being. Another upper bound for r will be derived later. We turn to the lower bound. We have: 3 3   xi T i x i Tˆ i =κ Tr = r r i=1

(6.110)

i=1

In view of the fact that the Euclidean magnitude of the vector Tˆ is bounded by unity, this implies: |T r | ≤ κ (6.111) Hence, by the estimate (6.81), |T r | ≤ 1 + Cδ0 [1 + log(1 + t)] (6.112)  Integrating (6.112) along the integral curves of T from t C0 = St,0 where r = 1+η0 t, to St,u , yields the lower bound: r ≥ 1 − u + η0 t − Cδ0 u[1 + log(1 + t)]

(6.113)

In view of the fact that, in Wε∗0 , u takes values in [0, ε0 ], taking δ0 suitably small we conclude that: (6.114) r ≥ C −1 (1 + η0 t) We proceed to derive a suitable estimate for the coefficient functions λi . According to (6.20) we have: ◦ (6.115) λi = h( Ri , Tˆ ) As we remarked earlier Tˆ is the unit normal to the surfaces St,u in t relative to the induced acoustical metric h. From (3.11) we have: ◦

◦

◦

h( Ri , Tˆ ) =  Ri , Tˆ  + H Rim ψm ψTˆ

(6.116)

Here and in the following we shall denote by  ,  the inner product with respect to the Euclidean metric on t :  ,  = g( , )

158

Chapter 6. Construction of the Commutation Vectorfields

We shall also denote by   the magnitude of tensors at a point in t with respect to the Euclidean metric g, reserving the notation | | for the corresponding magnitude with respect to the induced acoustical metric h (or simply the absolute value of real numbers or functions at a point). Since the metric h dominates the Euclidean metric g, we have: Tˆ  ≤ 1

(6.117)

Also, from (6.28), (6.109), ◦

 Ri  =

r 2 − (x i )2 ≤ r ≤ 1 + η0 t

(6.118)

From (6.117), (6.118) and assumption E1 we have: ◦

| Rim ψm ψTˆ | ≤ Cδ02 (1 + t)−1

(6.119)

We remark that |Ri |, the magnitude of Ri with respect to the metric h is bounded ◦

◦

by | Ri |, the corresponding magnitude of Ri . Since the maximal eigenvalue of h with respect to the Euclidean metric is 1 + ρ H (see (6.58)), we have: ◦

|Ri |2 ≤ (1 + ρ H ) Ri 2

(6.120)

In particular, by (6.118) and assumptions E1 |Ri | ≤ C(1 + t)

(6.121)

◦

Let us introduce the functions λi by: ◦

◦

λi =  Ri , Tˆ 

(6.122)

By (6.115), (6.116), we then have: ◦

◦

λi =λi +H Rim ψm ψTˆ

(6.123) ◦

Thus, in view of (6.119) we shall arrive at an estimate for λi once we bound λi . Our ◦

approach is to derive an ordinary differential equation for λi along the integral curves of the vectorfield L, generators of Cu , and use the following additional assumption on the initial data. ε

There is a positive constant C such that on 00 : I1:

◦

| λi | ≤ Cδ0

Now, we have:

◦

L λi =

 m

◦

(L Rim )Tˆ m +

 m

◦

Rim L Tˆ m

(6.124)

Chapter 6. Construction of the Commutation Vectorfields

159

According to equation (2.66) of Chapter 2:   ∂ ∂ H ψ0ψ i i ˆ L= − αT + 0 1 + ρ H ∂xi ∂x

(6.125)

Hence, by (6.140) and in view of the fact that according to (6.28), ◦

Rim = i j m x j , we have:

◦

L Rim = −

H ψ0ψ j i j m 1 + ρH

(6.126)

Moreover, L Tˆ m is given by equation (3.159) of Chapter 3: L Tˆ m = p L Tˆ m + q Lm

(6.127)

where p L is given by (3.164) and: q Lm = q LA X m A

(6.128)

is given through (3.169). Substituting (6.126) and (6.127) in (6.124) we obtain, in view of the definition (6.122), ◦

◦

◦

L λi = p L λi + Ri , q L  −

H ψ0 i j m ψ j Tˆ m 1 + ρH

(6.129)

Now, according to (3.164)  p L = −ψTˆ

1 dH ψ ˆ (Lσ ) + H Tˆ i (Lψi ) 2 dσ T

(6.130)

Using (6.53), (6.51), and assumption E2, we obtain: | p L | ≤ Cδ02 (1 + t)−3

(6.131)

According to (3.169): 1 dH ψ ˆ ( ψ B (Lσ ) − ψ L (d/ B σ )) 2 dσ T − H ψ B Tˆ i (Lψi )

/ AB = −κ −1 ζ B − q LA h

(6.132)

Using the estimate (6.73) together with (6.53), (6.60), (6.51), and assumption E2, we obtain: (6.133) |q L | ≤ Cδ0 (1 + t)−2 In view also of (6.118) it follows that: ◦

◦

◦

| Ri , q L | ≤  Ri q L  ≤  Ri |q L | ≤ Cδ0 (1 + t)−1

(6.134)

160

Chapter 6. Construction of the Commutation Vectorfields

Finally, the last term on the right in (6.129) is also bounded by: Cδ0 (1 + t)−1 by virtue of assumption E1. In view of the above results, equation (6.129) implies the following ordinary differential inequality along the integral curves of L: ◦

◦

|L λi | ≤ Cδ02 (1 + t)−3 | λ +Cδ0 (1 + t)−1

(6.135)

ε

Integrating this inequality from 00 and taking into account the assumption I1 on the initial data, we obtain the following estimate on Wεs0 : ◦

| λi | ≤ Cδ0 [1 + log(1 + t)]

(6.136)

which, through (6.12) implies, in view of assumption E1, that on Wεs0 : |λi | ≤ Cδ0 [1 + log(1 + t)]

(6.137)

This is the desired estimate for the functions λi . ◦

The derivatives of the functions λi with respect to L have been estimated above. We ◦

shall presently derive expressions for the remaining first derivatives of the λi , which show ◦

that all first order derivatives of the λi can be directly estimated. We have: ◦

T λi =



◦

(T Rim )Tˆ m +

m



◦

Rim T (Tˆ m )

(6.138)

m

According to (6.28),

◦

Rim = i j m x j , hence:

◦

Tˆ Rim = i j m Tˆ j and:



◦

(T Rim )Tˆ m = i j m Tˆ j Tˆ m = 0

(6.139) (6.140)

m

that is, the first term on the right in (6.138) vanishes. The equation in question then simplifies to:  ◦ ◦ Rim T (Tˆ m ) (6.141) T λi = m

Now

T (Tˆ m )

is given by (3.192): T (Tˆ m ) = pT Tˆ m + qTm

(6.142)

where pT is given by (3.197), while qTm = qTA X m A

(6.143)

Chapter 6. Construction of the Commutation Vectorfields

161

(see (3.194)) is given through (3.201). Substituting (6.142) in (6.141) we obtain, in view of the definition (6.122), ◦

◦

◦

T λi = pT λi + Ri , qT  Next, we have:

(6.144)

◦

j

d/ A Rim = i j m d/ A x j = i j m X A hence:

◦

j d/ A λi = i j m X A Tˆ m +



◦

Rim d/ A Tˆ m

(6.145)

(6.146)

m

Moreover, d/ A Tˆ m is given by equation (3.221) of Chapter 3: d/ A Tˆ m = p/ A Tˆ m + q/mA

(6.147)

where p/ A is given by (3.226) and: q/mA = q/ BA X m B

(6.148)

is given through (3.230). Substituting (6.147) in (6.146) we obtain, in view of the definition (6.122), ◦

◦

◦

ˆj d/ A λi = p/ A λi −i j m X m / A A T +  Ri , q

(6.149)

Note that for any vector V in Euclidean space, we have:  ◦  Ri , V 2 = r 2 V 2

(6.150)

i

where  is the Euclidean projection operator to the Euclidean coordinate spheres:  ij = δ ij − r −2 x i x j This fact readily follows using the identities:  i j m ikn = δ j k δmn − δ j n δkm ,

(6.151)



i

i m ni = nm

i

Taking in (6.150) V to be Tˆ we obtain, in view of the definition (6.122),  Tˆ 2 = r −2



◦

(λi )2

(6.152)

i

The estimate (6.136) together with the lower bound (6.114) then yields:  Tˆ  ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.153)

162

Chapter 6. Construction of the Commutation Vectorfields

We shall now use (6.153) to obtain another upper bound for r , complementing the bound (6.109). Let us denote by N the Euclidean outward unit normal to the Euclidean coordinate spheres: xi ∂ (6.154) N= r ∂xi Then (6.110) takes the form: T r = κN, Tˆ  (6.155) We decompose:

Tˆ = NN, Tˆ  +  Tˆ

(6.156)

This is the Euclidean orthogonal decomposition of Tˆ relative to the Euclidean coordinate spheres. Hence, (6.157) Tˆ 2 = N, Tˆ 2 +  Tˆ 2 and, since:

h i j Tˆ i Tˆ j = 1,

we have:

h i j = δi j + H ψi ψ j ,

Tˆ 2 = 1 − H (ψTˆ )2

(6.158)

1 − N, Tˆ 2 = H (ψTˆ )2 +  Tˆ 2

(6.159)

Substituting in (6.157) yields:

The estimates (6.53) and (6.153) then imply: 1 − N, Tˆ 2 ≤ Cδ02 (1 + t)−2 [1 + log(1 + t)]2

(6.160)

If δ0 is suitably small, this does not exceed 1/2, say. Thus the Euclidean angle between Tˆ and N lies either in the sector [0, π/4] or in the sector [3π/4, π]. Now, Tˆ is the inward unit normal to the surfaces St,u in t with respect to the induced acoustical metric h, while St,0 is itself a coordinate sphere, the inner boundary of the exterior Euclidean region. Thus Tˆ = −N and the angle is π on St,0 . It follows by continuity that the angle in question lies in the sector [3π/4, π], where N, Tˆ  < 0, 1 − N, Tˆ  > 1. Then (6.160) implies that: 0 ≤ 1 + N, Tˆ  ≤ Cδ02 (1 + t)−2 [1 + log(1 + t)]2

(6.161)

Going back now to (6.155), writing: T r + 1 = κ(1 + N, Tˆ ) − (κ − 1)

(6.162)

and using the estimates (6.161) and (6.81) we obtain: |T r + 1| ≤ Cδ0 [1 + log(1 + t)] Since r = 1 + η0 t on St,0 , we have, on St,u :



u

r − 1 − η0 t + u = 0

(T r + 1)du 

(6.163)

(6.164)

Chapter 6. Construction of the Commutation Vectorfields

163

where the integration is along an integral curve of T , from St,0 to St,u . Using the estimate (6.163) we then obtain: |r − 1 − η0 t + u| ≤ Cδ0 u[1 + log(1 + t)]

(6.165)

This estimate gives an upper bound as well as a lower bound for r . The lower bound coincides with the lower bound (6.113), but the upper bound does not coincide with the upper bound (6.109). It is stronger than the upper bound (6.109) when Cδ0 [1 + log(1 + t)] < 1, weaker otherwise. We note finally that the estimate (6.165) implies:   1  1  −  ≤ Cδ0 u(1 + t)−2 [1 + log(1 + t)] (6.166) r 1 − u + η0 t  Consider now the vectorfield: y  = Tˆ + N =  Tˆ + (1 + N, Tˆ )N

(6.167)

The estimates (6.153) and (6.161) imply: y   ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.168)

Let y i : i = 1, 2, 3 be the rectangular components of the vectorfield y  . Comparing the definitions (6.42), (6.154), and (6.167), we see that the functions y i are expressed in terms of the y i by:   1 1 − y i = y i − x i (6.169) r 1 − u + η0 t The bounds (6.168), (6.166), together with the elementary bound (6.109), imply the following estimate on Wεs0 : |y i | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.170)

Using this estimate together with (6.70) and assumption E1, we then obtain through (6.47) the following estimate on Wεs0 : |z i | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.171)

We turn to estimate the deformation tensors of the vectorfields Ri . Since L = α −1 κ L + 2T , expressions (6.14)–(6.16) are equivalent to: (Ri )

πL L = 0

(Ri )

π L L = −2Ri µ

(Ri )

π L L = 4µRi (α

(6.172) (6.173) −1

κ)

(6.174)

The bootstrap assumption F1 and the estimates (6.71) and (6.82) imply: (Ri )

π L L | ≤ Cδ0 [1 + log(1 + t)]

(6.175)

−1 (Ri )

π L L | ≤ Cδ0 [1 + log(1 + t)]

(6.176)

| µ

|

164

Chapter 6. Construction of the Commutation Vectorfields

Next, we estimate the St,u 1-form (Ri ) π / L , whose components are (Ri ) π L A , given by (6.49). We have five terms on the right-hand side, each of which represents an St,u 1-form. By assumption F2 and (6.121) the first term is bounded in magnitude by: Cδ0 (1 + t)−1 [1 + log(1 + t)] The square magnitude of the second term is: (h /−1 ) AB h mn h pq X nA X B ilm z l ir p z r q

Here all repeated indices except i are taken to be summed. Substituting the expansion (6.59) we conclude that this does not exceed: h mp ilm z l ir p z r Substituting from (6.58) this becomes:  l =i

 (z l )2 + H 

3 

2 ilm z l ψm 

l,m=1

which does not exceed: (1 + Hρ)

 (z l )2 l =i

From assumption E1 and the estimate (6.171) we then conclude that the second term on the right in (6.49) is bounded in magnitude by: Cδ0 (1 + t)−1 [1 + log(1 + t)] By the estimates (6.73) and (6.137), the third term is bounded in magnitude by: Cδ02 (1 + t)−2 [1 + log(1 + t)] By assumptions E1, E2 and the bound (6.118), the fourth term is bounded in magnitude by: Cδ02 (1 + t)−2 Lastly, by assumption E1, and the bounds (6.51), (6.118), (6.121), the fifth term is bounded in magnitude by: Cδ02 (1 + t)−2 Combining the above results we obtain the estimate: |

(Ri )

π / L | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.177)

/T whose components (Ri ) πT A are given by Next we estimate the St,u 1-form (Ri ) π (6.44). We have again five terms on the right-hand side, each of which represents an St,u

Chapter 6. Construction of the Commutation Vectorfields

165

1-form. In regard to the first term we express (see (6.38)):     1 dH / h h/  = κ θ AB + − ψ A ψ B (T σ ) κ θ AB + 1 − u + η0 t 1 − u + η0 t 2 dσ   η0 h/ (α − η0 )h/ = −α −1 κ χ − − α −1 κ 1 − u + η0 t 1 − u + η0 t 1 dH ψ A ψ B (T σ ) +κk/AB − 2 dσ By assumption F2 and the estimates (6.70), (6.63), (6.60) and (6.51), the first term is bounded in magnitude by: Cδ0 (1 + t)−1 [1 + log(1 + t)]2 The second term is handled in a similar way to that in which the second term on the right in (6.49) was handled. Thus, its magnitude does not exceed )  (1 + Hρ) (y l )2 l =i

which by the estimate (6.170) and assumption E1 is bounded by: Cδ0 (1 + t)−1 [1 + log(1 + t)]2 The third term on the right in (6.44) is:   1 dH ψTˆ ψ A (T σ ) λi d/ A κ + 2 dσ From the estimate (6.137) together with the bounds (6.51), (6.53), (6.60), (6.82), we conclude that this term is bounded in magnitude by: Cδ02 (1 + t)−1 [1 + log(1 + t)]2 By assumptions E1, E2 and the bound (6.118), the fourth term ◦

H ψ A Rim (T ψm ), is bounded in magnitude by:

Cδ02 (1 + t)−1

Lastly, by assumption E1, and the bounds (6.64), (6.51), (6.118), (6.121), the fifth term is bounded in magnitude by: Cδ02 (1 + t)−2 Combining the above results we obtain the estimate: |

(Ri )

π /T | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]2

(6.178)

166

Chapter 6. Construction of the Commutation Vectorfields

Since L = α −1 κ L + 2T , it follows from (6.177) and (6.178) that: |

(Ri )

π / L | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]2

Finally, we estimate the quadratic form point. The components of this quadratic form, (Ri )

(Ri ) π / in

π /(X A , X B ) =

(Ri )

(6.179)

the tangent space to St,u at each

π AB ,

are given by equation (6.40). Here we have four terms on the right. By assumption F2 and the estimates (6.63) and (6.137), the trace-free part of the first term is bounded in magnitude by: Cδ02 (1 + t)−2 [1 + log(1 + t)] while by assumption F2, (6.87), and the estimates (6.63) and (6.137) the trace of this term is bounded in absolute value by: Cδ0 (1 + t)−1 [1 + log(1 + t)] The second term on the right in (6.40) is of the form: H ( i w AB + i w B A ) where: i

w AB = X kA X lB ψk i ωl ,

i

ωl = ilm ψm

The square of the magnitude of the second term is thus: 2H 2(h /−1 ) AC (h /−1 ) B D ( i w AB i wC D + i w AB i w DC ) and, writing

n /−1 )mn (h /−1 ) AB X m A X B = (h

and substituting the expansion (6.59) we have, in view of the fact that h by g−1 ,

−1

is dominated

(h /−1 ) AC (h /−1 ) B D i w AB i wC D = (h/−1 )kp ψk ψ p (h/−1 )lq i ωl i ωq −1

−1

≤ (h )kp ψk ψ p (h )lq i ωl i ωq           ≤ (ψk )2 ( i ωl )2 ≤ (ψk )2  (ψl )2  k

l

l =i

k

Similarly, 2  (h /−1 ) AC (h /−1 ) B D i w AB i w DC = (h/−1 )kq ψk i ωq      ≤ (h /−1 )kp ψk ψ p (h /−1 )lq i ωl i ωq ≤ (ψk )2  (ψl )2  k

l =i

Chapter 6. Construction of the Commutation Vectorfields

167

We thus conclude by assumption E1 that the second term on the right in (6.40) is bounded in magnitude by: Cδ02 (1 + t)−2 By assumptions E1, E2 and the bound (6.118), the third term ◦

H Rim (X nA ψn (d/ B ψm ) + X nB ψn (d/ A ψm )), is bounded in magnitude by:

Cδ02 (1 + t)−2

Lastly, by assumption E1, and the bounds (6.51) and (6.121), the fourth term is bounded in magnitude by: Cδ03 (1 + t)−3 Combining the above results we conclude that the following estimates hold: |

(Ri )

trπ /| ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] | (Ri ) π /ˆ | ≤ Cδ02 (1 + t)−2 [1 + log(1 + t)]2

(6.180) (6.181)

˜ which refer to the metric Since according to (6.10) the deformation tensors (Y ) π, ˜h = h, are related to the deformation tensors (Y ) π, which refer to the metric h, by (Y )

π˜ = 

(Y )

π+

d (Y σ )h dσ

and by the bounds (6.51) and (6.121) we have |Ri σ | ≤ Cδ0 (1 + t)−1 , (6.172) and the estimates (6.175)–(6.177), (6.179)–(6.181) imply the following for the deformation tensors (Ri ) π: ˜ (Ri )

π˜ L L = 0

(6.182)

(Ri )

π˜ L L | ≤ Cδ0 [1 + log(1 + t)]

(6.183)

−1 (Ri )

≤ Cδ0 [1 + log(1 + t)]

| µ

|

| | |

π˜ L L | (Ri ) ˜ π /L | (Ri ) ˜ π /L | (Ri ) ˜

≤ Cδ0 (1 + t)

−1

[1 + log(1 + t)]

≤ Cδ0 (1 + t)

−1

[1 + log(1 + t)]

(6.184) (6.185) 2

(6.186)

trπ /| ≤ Cδ0 (1 + t) [1 + log(1 + t)] | (Ri ) π /˜ˆ | ≤ Cδ02 (1 + t)−2 [1 + log(1 + t)]2

(6.187)

−1

In concluding the present chapter we shall estimate the functions (Y )

δ=

1 tr˜ 2

(Y )

π˜ − µ−1 Y µ − 2−1 Y 

(6.188) (Y ) δ,

defined by: (6.189)

168

Chapter 6. Construction of the Commutation Vectorfields

These functions play an important role in the next chapter. Here tr˜ denotes the trace with ˜ Since respect to the metric h. h˜ −1 = −1 h −1 we have:

tr˜

(Y )

π˜ = −1 tr

(Y )

π˜ = −1 (−µ−1

(Y )

π˜ L L + tr

(Y ) ˜

π /)

Hence, (Y )

1 δ = − −1 µ−1 (Y ) π˜ L L − µ−1 Y µ 2   d −1 1 (Y ) ˜ tr Yσ + π /−2 2 dσ

From Tables 6.12 and 6.13 we then find:  1 (T ) tr δ = −1 2  1 (Q) tr δ = −1 2

 d Tσ dσ  d (Q) ˜ Qσ + 1 π /− dσ (T ) ˜

π /−

(6.190)

(6.191) (6.192)

Hence, by the estimates (6.99), (6.104) and the bounds (6.51) we obtain: |

(T )

δ| ≤ C(1 + t)−1 [1 + log(1 + t)]

(6.193)

|

(Q)

δ| ≤ C

(6.194)

Finally, from the formulas (6.173) and (6.10), (Ri )

π˜ L L = −2Ri µ − 2µRi 

We thus have: (Ri )

δ = −1



1 tr 2

(Ri ) ˜

π /−

d Ri σ dσ

 (6.195)

The estimate (6.187) together with the bounds (6.51) then yields: |

(Ri )

δ| ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(6.196)

Chapter 7

Outline of the Derived Estimates of Each Order We begin with the following proposition. Proposition 7.1 Let ψ be a solution of the inhomogeneous wave equation
h˜ ψ = ρ and let Y be an arbitrary vectorfield. Then ψ = Y ψ satisfies the inhomogeneous wave equation:
h˜ ψ  = ρ  where the source function ρ  is related to the source function ρ by: ˜ ρ  = div

(Y )

1 J˜ + Yρ + tr˜ 2

(Y )

πρ ˜

Here (Y ) J˜ is the commutation current associated to ψ and Y , a vectorfield with components, with respect to an arbitrary local frame field, (Y ) ˜µ

J =

and

(Y ) π ˜

1 ˜ −1 µα ˜ −1 νβ ((h ) (h ) + (h˜ −1 )να (h˜ −1 )µβ − (h˜ −1 )µν (h˜ −1 )αβ ) 2

(Y )

π˜ αβ ∂ν ψ

is the Lie derivative of the metric h˜ with respect to the vectorfield Y .

Proof. Let f s be the local 1-parameter group of diffeomorphisms generated by Y . We denote by f s∗ the corresponding pullback. We then have:
f s∗ h˜ ( f s∗ ψ) = f s∗ (
h˜ ψ) = f s∗ ρ

(7.1)

170

Chapter 7. Outline of the Derived Estimates of Each Order

Now, in an arbitrary system of local coordinates,
h˜ ψ =

and:
fs∗ h˜ ( f s∗ ψ) = 

1 −deth˜

1 −det f s∗ h˜

 −1 µν ˜ ˜ ν ψ) −deth∂ ∂µ ((h )

 −1 µν ˜ ˜ ν ( f s∗ ψ)) −det f s∗ h∂ ∂µ ((( f s∗ h) )

(7.2)

(7.3)

Let us differentiate the expression (7.3) with respect to s at s = 0. In view of the facts that:   d ˜ f s∗ h = LY h˜ = (Y ) π, ˜ ds s=0     d 1 −det f s∗ h˜ = −deth˜ tr˜ (Y ) π, ˜ ds 2 s=0   d ˜ −1 )µν (( f s∗ h) = −(h˜ −1 )µα (h˜ −1 )νβ (Y ) π˜ αβ ds s=0 and: 

d fs∗ ψ ds



 = s=0

d ψ ◦ fs ds

 = Yψ s=0

we obtain:    1 d 1 ˜ ν ψ)
fs∗ h˜ ( f s∗ ψ) =−

˜ µ ((h˜ −1 )µν −deth∂ (7.4) tr˜ (Y ) π∂ ds 2 −deth˜ s=0  1 ˜ +

∂µ −deth(−( h˜ −1 )µα (h˜ −1 )νβ (Y ) π˜ αβ −deth˜  ˜ ν (Y ψ) +(1/2)(h˜ −1)µν tr˜ (Y ) π)∂ ˜ ν ψ + (h˜ −1 )µν −deth∂ On the other hand, by (7.1),       d d d = = = Yρ
˜ ( f s∗ ψ) f s∗ ρ ρ ◦ fs ds fs∗ h ds ds s=0 s=0 s=0

(7.5)

Comparing (7.4) and (7.5) and in view of the expression (7.2) and a similar expression with Y ψ in the role of ψ, the proposition follows. Let X be an arbitrary vectorfield. In an arbitrary system of local coordinates the divergence of X with respect to the acoustical metric h is expressed by: divX = Dµ X µ = √

∂ √ ( −deth X µ ) −deth ∂ x µ 1

Chapter 7. Outline of the Derived Estimates of Each Order

171

while its divergence with respect to the conformal acoustical metric h˜ is expressed by:  1 ∂ ˜ ( −deth˜ X µ ) divX = D˜ µ X µ =

µ ∂ x ˜ −deth 

Since h˜ µν = h µν , we have:

√ −deth˜ = 2 −deth,

D˜ µ X µ = −2 Dµ (2 X µ ) (Y ) J˜,

If we apply this to the vectorfield

˜ div where:

(Y )

we obtain:

J˜ = −2 div

(Y )

(7.6)

J = 2

(Y )

(Y )

J

(7.7)

J˜

(7.8)

With respect to an arbitrary local frame field we have: (Y ) µ

J =

1 −1 µα −1 νβ ((h ) (h ) + (h −1 )να (h −1 )µβ − (h −1 )µν (h −1 )αβ ) 2

Setting

(Y ) µν

π˜

we can write: (Y ) µ

J =



= (h −1 )µα (h −1 )νβ

(Y ) µν

π˜

(Y )

1 − (h −1 )µν tr 2

π˜ αβ

(Y )

 π˜ ∂ν ψ

(Y )

π˜ αβ ∂ν ψ (7.9) (7.10) (7.11)

We now consider the nth order variations ψn of the wave function φ, as defined in the beginning of the previous chapter, by applying to a first order variation ψ1 a string of commutation vectorfields Yi : i = 1, 2, 3, 4, 5 of length n − 1 (see equation (6.1)). We shall use Proposition 7.1 to derive a recursion formula for the corresponding source functions ρn ,
h˜ ψn = ρn (7.12) (see equation (6.3)), starting from the fact that the source function ρ1 corresponding to a first order variation vanishes (see equation (6.2)): ρ1 = 0

(7.13)

Now, an nth order variation ψn results by applying one of the commutation vectorfields to an n − 1th order variation ψn−1 . Denoting by Y one of the Yi : i = 1, 2, 3, 4, 5, we have: (7.14) ψn = Y ψn−1 Thus Proposition 7.1 directly applies to ψn−1 and ψn yielding the following relation between ρn and ρn−1 : ˜ ρn = div

(Y )

1 J˜n−1 + Yρn−1 + tr˜ 2

(Y )

πρ ˜ n−1

(7.15)

172

Chapter 7. Outline of the Derived Estimates of Each Order

Here,

(Y ) J˜ n−1

(Y ) ˜µ Jn−1

=

is the commutation current associated to ψn−1 and Y :

1 ˜ −1 µα ˜ −1 νβ ((h ) (h ) + (h˜ −1 )να (h˜ −1 )µβ − (h˜ −1 )µν (h˜ −1 )αβ ) 2

(Y )

π˜ αβ ∂ν ψn−1 (7.16) Equation (7.15) is a recursion formula for the sources ρn , but it is not quite in the form which can be used in our estimates. To obtain the appropriate form we consider instead the re-scaled sources: (7.17) ρ˜n = 2 µρn Then by (7.15), (7.16), and (7.7)–(7.11), the ρ˜n satisfy the recursion formula: ρ˜n = Here: (Y ) µ Jn−1

and

(Y ) δ



(Y )

σn−1 + Y ρ˜n−1 +

(Y )

δ ρ˜n−1

(Y )

=

σn−1 = µdiv (Y ) Jn−1  1 −1 µν (Y ) (Y ) µν π˜ − (h ) tr π˜ ∂ν ψn−1 2

(7.18) (7.19) (7.20)

are the functions defined by equation (6.189): (Y )

δ=

1 tr˜ 2

(Y )

π˜ − µ−1 Y µ − 2−1 Y 

(7.21)

Moreover, by (7.13), ρ˜1 = 0

(7.22)

Since ψn is a solution of the inhomogeneous wave equation (7.12), the argument of Theorem 5.1 can be applied to ψn to yield estimates analogous to those of that theorem with ψn in the role of ψ, provided that we can appropriately estimate the error integrals contributed by the source function ρn . These are the spacetime integrals Q 0,0,n dµh , Q 1,0,n dµh (7.23) Wut

Wut

where Q 0,0,n and Q 1,0,n are the terms Q 0,0 and Q 1,0 , associated to the vectorfields K 0 and K 1 respectively, given by (5.87) and (5.97) with ψn and ρn in the role of ψ and ρ: Q 0,0,n = −2 ρn K 0 ψn Q 1,0,n = −2 ρn (K 1 ψn + ωψn )

(7.24) (7.25)

Now, by virtue of equation (5.27) we have: dµh = µdtdudµh/ Thus, in view of (7.17) the error integrals (7.23) take the form: − ρ˜n (K 0 ψn )dt  du  dµh/

(7.26)

(7.27)

Wut



and −

Wut

ρ˜n (K 1 ψn + ωψn )dt  du  dµh/

(7.28)

Chapter 7. Outline of the Derived Estimates of Each Order

173

We shall first consider the contribution of the term (Y ) σn−1 in ρ˜n (see (7.18)–(7.20)) to the above error integrals. Let V be an arbitrary vectorfield defined in the spacetime domain Wε0 . We decompose: /, where V / = V = V A X A V = V LL + V LL + V is tangential to the surfaces St,u . We have: V L = −(1/2µ)h(V, L) V L = −(1/2µ)h(V, L) V A = (h/−1 ) AB h(V, X B ) The divergence of the vectorfield V is then expressed as: divV = (D L V ) L + (D L V ) L + (D X A V ) A Replacing V above by D L V , D L V , D X A V , respectively, we obtain: (D L V ) L = −(1/2µ)h(D L V, L) (D L V ) L = −(1/2µ)h(D L V, L) (D X A V ) A = (h/−1 ) AB h(D X A V, X B ) Moreover, substituting the decomposition of V we obtain, appealing to Table 3.117, h(D X A V, X B ) = V L h(D X A L, X B ) + V L h(D X A L, X B ) + h(D X A V /, X B ) /XAV /, X B ) = χ AB V L + χ AB V L + h/(D hence: (D X A V ) A = trχ V L + trχ V L + div / V / We thus obtain: divV = −(1/2µ)(h(D L V, L) + h(D L V, L)) + trχ V L + trχ V L + div / V / We can express the first term on the right-hand side in terms of: VL = h(V, L),

VL = h(V, L)

Appealing again to Table 3.117 we obtain: h(D L V, L) = L(h(V, L)) − h(V, D L L) = L(VL ) + h(V, 2ζ A X A ) = L(VL ) + 2ζ A V A h(D L V, L) = L(h(V, L)) − h(V, D L L) = L(VL ) − h(V, −L(α −1 κ)L + 2η A X A ) = L(VL ) + L(α −1 κ)VL − 2η A V A

174

Chapter 7. Outline of the Derived Estimates of Each Order

Substituting and noting that according to equation (3.65) we have: η A − ζ A = d/ A µ we obtain the following formula for the divergence of an arbitrary spacetime vectorfield V : / (µV /) −2µdivV = L(VL ) + L(VL ) − 2div +L(α −1 κ)VL + trχ VL + trχ VL

(7.29)

We shall apply the above formula to the commutation current (Y ) Jn−1 , given by (7.20). We introduce the vectorfields (Y ) Z˜ and (Y ) Z˜ , associated to the commutation vectorfield Y , and tangential to the surfaces St,u , by the conditions that: h( Z˜ , V ) =

(Y )

π(L, ˜ V ),

h( Z˜ , V ) =

(Y )

π(L, ˜ V )

(7.30)

for any vector V ∈ T Wε0 . In terms of the null frame we have: (Y )

where:

(Y )

Z˜ A =

Z˜ = (Y )

Z˜ A X A ,

(Y )

Z˜ =

π˜ L B (h /−1 ) AB ,

(Y )

A Z˜ =

(Y )

(Y )

A Z˜ X A

(Y )

π˜ L B (h/−1 ) AB

(7.31)

Note that the St,u - tangential vectorfields (Y ) Z˜ and (Y ) Z˜ correspond, through the in/ L and (Y ) π / L , respectively, which were introduced in duced metric h /, to the 1-forms (Y ) π the previous chapter. Taking into account these definitions as well as the fact that: tr we bring the components of (Y ) (Y )

(Y )

π˜ = −(1/µ)

(Y ) J n−1

Jn−1,L = −(1/2)tr Jn−1,L

µ

π˜ L L + tr

(Y ) ˜

π /

(Y ) ˜ (Y )

Z˜ · d/ψn−1 (Y ) ˜ Z · d/ψn−1 (Y )

π˜ L L (Lψn−1 )

A Z˜ A (Lψn−1 ) − (1/2) (Y ) Z˜ (Lψn−1 ) +(1/2)( (Y ) π˜ L L − µtr (Y ) π /˜ )(h/−1 ) AB d/ B ψn−1 +µ (Y ) π /˜ BC (h/−1 ) AB (h/−1 )C D d/ D ψn−1

= −(1/2)

(7.32)

to the following form:

π /(Lψn−1 ) + (Y ) ˜ = −(1/2)tr π /(Lψn−1 ) + −(1/2µ)

(Y ) A /Jn−1

(Y )

(Y )

(7.33)

We note here the absence of a term involving (1/µ) (Y ) π˜ L L in the expressions for (Y ) J (Y ) J n−1,L , n−1,L , despite the presence of such a term in (7.32). This is related to the fact that the operator
g on a 2-dimensional manifold M with a metric g is conformally covariant. Here, in the role of such a 2-dimensional manifold we have the 2-dimensional distribution of time-like planes spanned by the vectors L and L at each point. This distribution is not integrable, the obstruction to integrability being [L, L] = 2,

 = −(h/−1 ) AB (ζ B + η B )X A

(7.34)

Chapter 7. Outline of the Derived Estimates of Each Order

175

(see (3.71) and Table 3.117). However, the conformal covariance is still reflected by the fact that the restriction of the commutation current to the plane spanned by L and L depends only on the trace-free, relative to this plane, part of the restriction of π˜ to the plane, therefore not on π˜ L L . Applying formula (7.29) to (Y ) Jn−1 we obtain: (Y )

σn−1 = −(1/2)L(

(Y )

−(1/2)L(α

−1

Jn−1,L ) − (1/2)L( κ)

(Y )

(Y )

Jn−1,L ) + div / (µ

Jn−1,L − (1/2)trχ

(Y )

(Y )

/Jn−1 )

Jn−1,L − (1/2)trχ

(Y )

(7.35) Jn−1,L

The first line on the right in (7.35) contains the principal terms, the derivatives of products of components of (Y ) π˜ with derivatives of ψn−1 , while the second line contains lower order terms of the form of triple products of connection coefficients of the null frame with components of (Y ) π˜ and derivatives of ψn−1 . To estimate the contribution of (Y ) σn−1 to the error integrals we shall decompose: (Y )

σn−1 =

(Y )

σ1,n−1 +

(Y )

σ2,n−1 +

(Y )

σ3,n−1

(7.36)

where (Y ) σ1,n−1 is to contain all the terms which are products of components of (Y ) π˜ with 2nd derivatives of ψn−1 , (Y ) σ2,n−1 is to contain all the terms which are products of 1st derivatives of (Y ) π˜ with 1st derivatives of ψn−1 , and (Y ) σ3,n−1 is to contain only lower order terms. Now, in view of the expressions (7.33), the first two terms on the right in (7.35) contain L( (Y ) Z˜ · d/ψn−1 ) and L( (Y ) Z˜ · d/ψn−1 ). In decomposing each of these into a term which is a product of a component of (Y ) π˜ with a 2nd derivative of ψn−1 , a term which is a product of a 1st derivative of (Y ) π˜ with a 1st derivative of ψn−1 , and a lower order term, we shall make use of the following lemma. Lemma 7.1 Let V be a vectorfield defined on the spacetime domain Wε∗0 and tangential to the surfaces St,u . Also, let f be a function on Wε∗0 . We then have: L(V · d/ f ) = V · d/ L f + L / L V · d/ f L(V · d/ f ) = V · d/ L f + L / L V · d/ f − (V · d/(α −1 κ))L f Here, we denote: L / L V = L L V = [L, V ],

L / L V = L L V = [L, V ]

Proof. Since V is tangential to the St,u we have: V · d/ f = V · d f Now, the exterior derivative commutes with the Lie derivative. Hence: L(V · d f ) = L L V · d f + V · d L f

(7.37)

L(V · d f ) = L L V · d f + V · d L f

(7.38)

and:

176

Chapter 7. Outline of the Derived Estimates of Each Order

We decompose: / L V + (L L V ) L L + (L L V ) L L LL V = L and we have: (L L V ) L = −(1/2µ)h(L L V, L) (L L V ) L = −(1/2µ)h(L L V, L) Taking account of the fact that V is h-orthogonal to L and L, and appealing to Table 3.117, h(L L V, L) = h([L, V ], L) = h(D L V − DV L, L) = −h(V, D L L) − h(DV L, L) = 2ζ (V ) − 2ζ (V ) = 0 and, h(L L V, L) = h([L, V ], L) = h(D L V − DV L, L) = h(D L V, L) = −h(V, D L L) = 0 We conclude that /L V LL V = L

(7.39)

that is, L L V is tangential to the surfaces St,u . (This follows also by considering the fact that the 1-parameter group "s generated by L maps St,u onto St +s,u while the 1-parameter group generated by V maps St,u onto itself.) We decompose: / L V + (L L V ) L L + (L L V ) L L LL V = L and we have: (L L V ) L = −(1/2µ)h(L L V, L) (L L V ) L = −(1/2µ)h(L L V, L) Taking again account of the fact that V is h- orthogonal to L and L, and appealing to Table 3.117, h(L L V, L) = h([L, V ], L) = h(D L V − DV L, L) = h(D L V, L) = −h(V, D L L) = 2µV (α −1 κ) and, h(L L V, L) = h([L, V ], L) = h(D L V − DV L, L) = −h(V, D L L) − h(DV L, L) = −2η(V ) + 2η(V ) = 0 We conclude that:

LL V = L / L V − (V (α −1 κ))L

In view of (7.39), (7.40) and the formulas (7.37), (7.38), the lemma follows.

(7.40)

Chapter 7. Outline of the Derived Estimates of Each Order (Y ) Z ˜,

Applying Lemma 7.1 to the case V = L(

(Y )

Z˜ · d/ψn−1 ) =

(Y )

Z˜ · d/ψn−1 ) =

f = ψn−1 , we obtain:

/L Z˜ · d/ Lψn−1 + L

(Y )

while applying the lemma to the case V = L(

177

(Y ) Z˜ ,

(Y )

Z˜ · d/ψn−1

(7.41)

f = ψn−1 , yields:

/ L (Y ) Z˜ · d/ψn−1 Z˜ · d/ Lψn−1 + L − (Y ) Z˜ · d/(α −1 κ)Lψn−1 (Y )

(7.42)

We are now ready to substitute the expressions given by Table 7.33 for the components of the commutation current (Y ) Jn−1 into the expression (7.35) for (Y ) σn−1 . There results a decomposition of the form (7.36), with (Y ) σ1,n−1 , (Y ) σ2,n−1 , and (Y ) σ3,n−1 given by: (Y )

(Y ) ˜

π /(L Lψn−1 + ν Lψn−1 )

σ1,n−1 = (1/2)tr

−1 (Y )

+(1/2)(µ π˜ L L )L 2 ψn−1 − (Y ) Z˜ · d/ Lψn−1 − (Y ) Z˜ · d/ Lψn−1 (Y )

+(1/2) (Y )

σ2,n−1 = (1/4)L(tr

π˜ L L
/ ψn−1 + µ

(Y ) ˆ˜

π /· D / 2 ψn−1

(Y ) ˜

π /)Lψn−1 + (1/4)L(tr

+(1/4)L(µ

(7.43)

(Y ) ˜

π /)Lψn−1

−1 (Y )

π˜ L L )Lψn−1 ˜ Z · d/ψn−1 − (1/2)L −(1/2)L /L / L (Y ) Z˜ · d/ψn−1 −(1/2)div / (Y ) Z˜ Lψn−1 − (1/2)div / (Y ) Z˜ Lψn−1 /ˆ˜ ) · d/ψ +(1/2)d/ (Y ) π˜ · d/ψ + div / (µ (Y ) π (Y )

LL

n−1

n−1

(7.44)

and: (Y )

σ3,n−1 =

(Y ) L σ3,n−1 Lψn−1

+

(Y ) L σ3,n−1 Lψn−1

+

(Y )

/σ3,n−1 · d/ψn−1

(7.45)

where: (Y ) L σ3,n−1

π / + (1/4)trχ(µ−1

(Y )

Z˜ · d/(α

(Y )

π˜ L L )

−1

κ) = −(1/4)(L log )tr (Y ) π /˜ +(1/2)

(Y ) L σ3,n−1 (Y ) /σ3,n−1

(Y ) ˜

= (1/4)trχtr

(7.46) (7.47)

(Y ) ˜

= −(1/2)(tr π /) −(1/2)(trχ + L(α −1 κ))

The above expressions for (Y ) σ1,n−1 , (Y ) σ2,n−1 , Note the presence of the lower order term (1/2)tr

(Y ) ˜

(Y )

Z˜ − (1/2)trχ

(Y ) σ 3,n−1 ,

π /ν Lψn−1

(Y )

Z˜

(7.48)

are as mentioned earlier.

178

Chapter 7. Outline of the Derived Estimates of Each Order

in (Y ) σ1,n−1 , contributing to the first term in (7.43). This lower order term is subtracted from the second term of the second line in (7.35), yielding, in view of definition (5.10), ν = (1/2)(trχ + L log ), the coefficient (Y ) σ3,n−1 of Lψn−1 in (Y ) σ3,n−1 . Then (Y ) σ3,n−1 consists of the second line in (7.35), after the subtraction just mentioned, the commutator term: L

(1/4)tr

(Y ) ˜

π /[L, L]ψn−1 = −(1/4)tr

(Y ) ˜

π /(L(α −1 κ)Lψn−1 + 2 · d/ψn−1 )

which results from re-writing the second term of the first line in (7.35), and the contribution of the last term in (7.42). We shall begin our estimates of the contribution of (Y ) σn−1 to the error integrals (7.27), (7.28), with the estimates of the partial contribution of (Y ) σ1,n−1 . In these estimates we shall make use of the following assumption on the components of the deformation tensors of the commutation fields. G0: There is a positive constant C independent of s such that in Wεs0 , for all five commutation fields Y , we have: (Y )

µ

−1 (Y )

|

|

π˜ L L = 0 π˜ L L | ≤ C[1 + log(1 + t)]

(Y )

≤ C[1 + log(1 + t)]

| |

≤ C(1 + t)−1 [1 + log(1 + t)] ≤ C[1 + log(1 + t)]

π˜ L L | (Y ) ˜ π /L | (Y ) ˜ π /L |

| |

(Y ) ˆ˜

(Y )

π /| ≤ C(1 + t)−1 [1 + log(1 + t)] trπ /˜ | ≤ C

This assumption holds by virtue of the results of Chapter 6. We shall also make use of the following assumptions concerning the set of rotation fields {Ri : i = 1, 2, 3}. H0: There is a positive constant C independent of s such that for any function f differentiable on each surface St,u we have: |d/ f |2 ≤ C(1 + t)−2



(Ri f )2

i

H1: There is a positive constant C independent of s such that for any differentiable 1-form ξ on each surface St,u we have: |D /ξ |2 ≤ C(1 + t)−2

 i

|L / R i ξ |2

Chapter 7. Outline of the Derived Estimates of Each Order

179

Assumption H1 implies in particular, taking ξ = d/ f , where f is any function twice differentiable on each St,u , that:  |D / 2 f |2 ≤ C(1 + t)−2 |d/ Ri f |2 i

H2: There is a positive constant C independent of s such that for any differentiable traceless symmetric 2-covariant tensorfield ϑ on each surface St,u we have:  |D /ϑ|2 ≤ C(1 + t)−2 |L / Ri ϑ|2 i u (t), F t (u), E u (t), F t (u), the integrals (5.47), In the following we denote by E0,n 0,n 1,n 1,n (5.48), (5.58), (5.53), respectively, of Chapter 5, with ψn in the role of ψ:   u α −1 κ(η0−1 + α −1 κ)(Lψn )2 + (Lψn )2 E0,n (t) = tu 2  +(η0−1 + 2α −1 κ)µ|d/ψn |2 dµh/ du  (7.49) 

 t F0,n (u) =  (η0−1 + α −1 κ)(Lψn )2 + µ|d/ψn |2 dµh/ dt  (7.50) u (t) = E1,n t (u) = F1,n



C ut



tu

C ut

 −1 −1 ων {α κ(Lψn + νψn )2 + µ|d/ψn |2 }dµh/ du  2

(7.51)

ων −1 (Lψn + νψn )2 dµh/ dt 

(7.52)

In each of the above the sum is over the set of ψn of the form (7.14) as Y ranges over the set {Yi : i = 1, 2, 3, 4, 5}. The assumptions A1, A2, B1, D1, of Chapter 5, imply the analogues of (5.68)–(5.71):    u −2 −2 −1 u 2 2 2 [η0 µ(η0 + µ)(η0 (Lψn ) + |d/ψn | ) + (Lψn ) ]dµh/ du  C E0,n (t) ≤ 0

St,u 

u ≤ CE0,n (t) t C −1 F0,n (u) ≤

  t 0

St  ,u

[η0−2 (η0 + µ)(Lψn )2 + µ|d/ψn |2 ]dµh/ dt 

t ≤ CF0,n (u) u (t) ≤ (1 + t)2 C −1 E1,n

  u 0

 St,u 

u ≤ CE1,n (t)   t −1 t  2 (1 + t ) C F1,n (u) ≤ 0

t ≤ CF1,n (u)

(7.53)



µ[η0−2 (Lψn + νψn )2 + |d/ψn |2 ]dµh/ du  

St  ,u

(7.54)

(7.55)

(Lψn + νψn )2 dµh/ dt  (7.56)

180

Chapter 7. Outline of the Derived Estimates of Each Order

Again, in each of the above the sum is over the set of ψn of the form (7.14) as Y ranges over the set {Yi : i = 1, 2, 3, 4, 5}. Moreover, we consider, as in (5.81)–(5.84) the following quantities which are, by their definition, non-decreasing functions of t at each u: u

Note

u E 0,n (t) = supt  ∈[0,t ]E0,n (t  )

(7.57)

t F0,n (u)

(7.58)

u u E 1,n (t) = supt  ∈[0,t ] [1 + log(1 + t  )]−4 E1,n (t  ) t t  F 1,n (u) = supt  ∈[0,t ] [1 + log(1 + t  )]−4 F1,n (u) u that the quantities E 0,n (t), E 1,n are also non-decreasing functions of u The contribution of (Y ) σn−1 to the error integrals (7.27) and (7.28) is



−

Wut

(K 0 ψn )

(Y )



and −

Wut

(K 1 ψn + ωψn )

(7.59) (7.60) at each t.

σn−1 dt  du  dµh/ (Y )

σn−1 dt  du  dµh/

(7.61)

(7.62)

respectively. We shall first consider the partial contribution of (Y ) σ1,n−1 , given by (7.43), to each of these integrals, and we begin with the partial contribution to the integral (7.61), which is associated to the vectorfield (see (5.15)) K 0 = (η0−1 + α −1 κ)L + L namely, the error integral:

−

Wut

(K 0 ψn )

(Y )

σ1,n−1 dt  du  dµh/

(7.63)

(7.64)

The first term in (7.43), (1/2)tr

(Y ) ˜

π /(L Lψn−1 + ν Lψn−1 )

involves Lψn−1 . Now, by (3.114), (6.6), L is expressed in terms of the commutation fields by: (7.65) L = α −1 κ(1 + t)−1 Q + 2T Thus we have: (1/2)(L Lψn−1 + ν Lψn−1 ) = (LT ψn−1 + νT ψn−1 ) +(1/2)α −1 κ(1 + t)−1 (L Qψn−1 + ν Qψn−1 )

(7.66)

+(1/2)(1 + t)−1 (L(α −1 κ) − (1 + t)−1 α −1 κ)Qψn−1 The leading term is that on the first line on the right. We shall estimate below the contribution of this term, namely the integral (K 0 ψn )tr (Y ) π (7.67) − /˜ (LT ψn−1 + νT ψn−1 )dt  du  dµh/ Wut

Chapter 7. Outline of the Derived Estimates of Each Order

181

The contributions of the terms on the second and third lines in (7.65) are easier to estimate because of the presence of the beneficial factor (1 + t)−1 (the third line is a lower order term). By the last of assumptions G0, the integral (7.67) is bounded in absolute value by: C(M L + M L ) where:

L

M = ML =

Wut

Wut

(η0−1 + α −1 κ)|Lψn ||LT ψn−1 + νT ψn−1 |dt  du  dµh/

(7.68)

|Lψn ||LT ψn−1 + νT ψn−1 |dt  du  dµh/

(7.69)

Since T is one of the commutation fields, we have, by (7.56), (7.60), for all u ∈ [0, ε0 ], t (1 + t  )2 (LT ψn−1 + νT ψn−1 )2 dµh/ dt  ≤ CF 1,n (u)[1 + log(1 + t)]4 (7.70) C ut

To estimate M L we apply an argument similar to that of Chapter 5, following the definitions (5.211)–(5.213). That is, we define the non-negative integer N to be the integral part of log t/ log 2: log t = N + r, 0 ≤ r < 1 (7.71) log 2 and we set: t−1 = 0,

tm = 2m+r : m = 0, 1, . . . , N

(7.72)

Then t N = t and we have the partition: Wut =

N 

t

Wum−1

,tm

(7.73)

m=0

Hence, ML =

N 

L

Mm

(7.74)

m=0



where: L

Mm =

t ,tm Wum−1

|Lψn ||LT ψn−1 + νT ψn−1 |dt  du  dµh/

(7.75)

We have: L Mm

 ≤

t

Wum−1

,tm

(1 + t )

 ·

t

Wum−1

 3/2

,tm





1/2

(LT ψn−1 + νT ψn−1 ) dt du dµh/

 −3/2

(1 + t )

2





(Lψn ) dt du dµh/ 2

1/2 (7.76)

182

Chapter 7. Outline of the Derived Estimates of Each Order

Now, for t  ∈ [tm−1 , tm ] it holds that:

 −1/2 1 (1 + t  )−1/2 ≤ (1 + tm−1 )−1/2 = 1 + tm 2  −1/2 1 1 + tm ≤ = 21/2 (1 + tm )−1/2 2 2

Therefore, writing (1 + t  )3/2 = (1 + t  )−1/2 · (1 + t  )2 , we obtain: (1 + t  )3/2 (LT ψn−1 + νT ψn−1 )2 dt  du  dµh/ tm−1 ,tm Wu ≤ 21/2(1 + tm )−1/2 (1 + t  )2 (LT ψn−1 + νT ψn−1 )2 dt  du  dµh/ t

Wum

and,

t Wum

(1 + t  )2 (LT ψn−1 + νT ψn−1 )2 dt  du  dµh/  u

=

 2

t C um

0





(1 + t ) (LT ψn−1 + νT ψn−1 ) dµh/ dt 2



du 

u tm tm  F1,n (u )du  ≤ C F 1,n (u  )[1 + log(1 + tm )]4 du  0   0 u t ≤C F 1,n (u  )du  [1 + log(1 + tm )]4 ≤C

u

0

hence:

 3/2

t Wum−1

(1 + t ) ,tm

2







(LT ψn−1 + νT ψn−1 ) dt du dµh/ ≤ C Am

u 0

t

F 1,n (u  )du  (7.77)

where: Am = (1 + tm )−1/2 [1 + log(1 + tm )]4 = (1 + 2m+r )−1/2 [1 + log(1 + 2m+r )]4 On the other hand,  −3/2 2   (1 + t ) (Lψ ) dt du dµ ≤ n h/ t ,tm Wum−1

Wut

(1 + t  )−3/2 (Lψn )2 dt  du  dµh/



t

≤C 0

u (1 + t  )−3/2 E0,n (t  )dt 

(7.78)

It follows, in view of (7.76) that:  u 1/2  t 1/2 t L 1/2 u F 1,n (u  )du  (1 + t  )−3/2 E0,n (t  )dt  Mm ≤ C Am 0

0

and, by (7.74),   N  1/2  L M ≤C Am m=0

u 0

t F 1,n (u  )du 

1/2  0

t

u (1 + t  )−3/2 E0,n (t  )dt 

1/2 (7.79)

Chapter 7. Outline of the Derived Estimates of Each Order

183

Now, N 

 1/2 Am

≤

m=0

N 

1/2  (m + 1)

−2

m=0



≤C

N 

1/2 (m + 1) Am 2

m=0 1/2

N 

≤ C ,

(m + 1)2 Am

m=0

because the series

∞ 

(m + 1)2 Am

m=0

is convergent, the asymptotic form of the mth term being: m2 2(m+r)/2 We conclude that:



u

ML ≤ C 0

t

F 1,n (u  )du 

[(m + r ) log 2]4

1/2 

t 0

u (1 + t  )−3/2 E0,n (t  )dt 

1/2 (7.80)

We turn to the integral M L ((7.68)). We have (taking into account assumptions A): 1/2  −1 ML ≤ (η0 + α −1 κ)(Lψn )2 dt  du  dµh/ (7.81) Wut

 ·  ≤C 0

u

1/2 Wut

−1 (η0−1 + α −1 κ)(LT ψn−1 + νT ψn−1 )2 dt  du  dµh/

1/2  t F0,n (u  )du  ·

1/2 Wut

[1 + log(1 + t  )](LT ψn−1 + νT ψn−1 )2 dt  du  dµh/

The integral in the last factor can be expressed as the sum: N  [1 + log(1 + t  )](LT ψn−1 + νT ψn−1 )2 dt  du  dµh/ t ,tm m−1 m=0 Wu

Since for t  ∈ [tm−1 , tm ] it holds that: (1 + t  )−2 [1 + log(1 + t  )] ≤ (1 + tm−1 )−2 [1 + log(1 + tm )] ≤ 4(1 + tm )−2 [1 + log(1 + tm )] we obtain, in view of (7.70), [1 + log(1 + t  )](LT ψn−1 + νT ψn−1 )2 dt  du  dµh/ tm−1 ,tm Wu ≤ 4(1 + tm )−2 [1 + log(1 + tm )] (1 + t  )2 (LT ψn−1 + νT ψn−1 )2 dt  du  dµh/ t

Wum

184

Chapter 7. Outline of the Derived Estimates of Each Order

≤ (1 + tm ) ≤ C Bm 0

−2



u

[1 + log(1 + tm )]

 t

C um

0 u

tm F 1,n (u  )du 

where:

  2



u

≤ C Bm 0

(1 + t ) (LT ψn−1 + νT ψn−1 ) dµh/ dt 2



du 

t

F 1,n (u  )du 

Bm = (1 + tm )−2 [1 + log(1 + tm )]5

In view of the fact that the series

∞ 

Bm

m=0

is convergent, we then obtain, summing over m,  2   [1 + log(1 + t )](LT ψn−1 + νT ψn−1 ) dt du dµh/ ≤ C Wut

Substituting in (7.81) then yields:  u 1/2  L t   M ≤C F0,n (u )du 0

u 0

u 0

t F 1,n (u  )du 

t

F 1,n (u  )du 

(7.82)

1/2 (7.83)

We have thus completed the estimate of the integral (7.67). Next we shall estimate the contribution of the second term in (7.43) (1/2)(µ−1

(Y )

π˜ L L )L 2 ψn−1

to the error integral (7.64). Writing L 2 ψn−1 = L((1 + t)−1 Qψn−1 ) = (1 + t)−1 L Qψn−1 − (1 + t)−2 Qψn−1 the contribution in question takes the form: −(1/2) (K 0 ψn )(µ−1 (Y ) π˜ L L )(1 + t  )−1 (L Qψn−1 )dt  du  dµh/ +(1/2)

Wut

Wut

(K 0 ψn )(µ−1

(Y )

π˜ L L )(1 + t  )−2 (Qψn−1 )dt  du  dµh/

(7.84)

By the second of assumptions G0 and (7.63) the first of these integrals is bounded in absolute value by: L C(I1 + I1L ) where: L



I1 = I1L

=

Wut

Wut

|Lψn |(1 + t  )−1 [1 + log(1 + t  )]|L Qψn−1 |dt  du  dµh/

(7.85)

(η0−1 + α −1 κ)|Lψn |(1 + t  )−1 [1 + log(1 + t  )]|L Qψn−1 |dt  du  dµh/ (7.86)

Chapter 7. Outline of the Derived Estimates of Each Order

We estimate:  L I1

≤C

185

1/2  −2

(Lψn ) (1 + t ) 2

Wut

t

≤C





[1 + log(1 + t )] dt du dµh/ 2

1/2



× 





Wut



(L Qψn−1 ) dt du dµh/ 2

 −2

(1 + t )

[1 + log(1 + t



0

u )]2 E0,n (t  )dt 



and: I1L

≤C 0

u

1/2  0

u

t F0,n (u  )du 

1/2 (7.87)

t F0,n (u  )du 

(7.88)

Also, by the second of assumptions G0 and (7.63), the second of the integrals (7.84) is bounded by: C |Lψn ||Qψn−1 |(1 + t  )−2 [1 + log(1 + t  )]dt  du  dµh/ Wut



+C 

Wut

(η0−1 + α −1 κ)|Lψn ||Qψn−1 |(1 + t  )−2 [1 + log(1 + t  )]dt  du  dµh/ 1/2  −2

2

≤C

Wut

(Lψn ) (1 + t )

Wut

(Qψn−1 ) (1 + t ) +α

−1

Wu



≤C





κ)(Lψn ) dt du dµh/ 2

+α

−1

 −4

κ)(Qψn−1 ) (1 + t ) 2

 −2

(1 + t )

[1 + log(1 + t



0

·



[1 + log(1 + t )]dt du dµh/

Wu t



1/2

(η0−1 t

·



1/2

(η0−1 t

+C





1/2  −2

2





[1 + log(1 + t )]dt du dµh/

 ·



u )]E0,n (t  )dt 







[1 + log(1 + t )] dt du dµh/ 2

1/2

1/2  t u ε02 (1 + t  )−2 [1 + log(1 + t  )]E0,n (t  )dt  0



u

+C 0

t F0,n (u  )du 

1/2 

ε02

t 0

 −4

(1 + t )

[1 + log(1 + t

(7.89) 

u )]3 E0,n (t  )dt 

Here we have made use of inequality (5.72) with ψn in the role of ψ, namely:  u (t) ψn2 dµh/ ≤ ε0 CE0,n St,u

1/2

(7.90)

186

Chapter 7. Outline of the Derived Estimates of Each Order

the sum being over the set of ψn of the form (7.14) as Y ranges over the set {Yi : i = 1, 2, 3, 4, 5}. Next we shall estimate the contribution of the third and fourth terms in (7.43) −

(Y )

Z˜ · d/ Lψn−1 −

(Y )

Z˜ · d/ Lψn−1

(7.91)

to the error integral (7.64). Expressing L and L in terms of the commutation fields T and Q, L = α −1 κ(1 + t)−1 Q + 2T L = (1 + t)−1 Q (7.91) takes the form: Z˜ · d/T ψn−1 − (1 + t)−1 (α −1 κ −(1 + t)−1 (Y ) Z˜ · d/(α −1 κ)Qψn−1 −2

(Y )

(Y )

Z˜ +

(Y )

Z˜ ) · d/ Qψn−1 (7.92)

We shall derive below an estimate of the contribution of the first line in (7.92) to the error integral (7.64). The contribution of the second line, being a lower order term, is easier to handle. By the fourth and fifth of assumptions G0, the coefficients of d/T ψn−1 and d/ Qψn−1 in (7.92) are bounded in magnitude by: C(1 + t)−1 [1 + log(1 + t)] /˜ L |, | (Y ) Z˜ | = | (Y ) π /˜ L |.) Also, T ψn−1 and Qψn−1 are among (Note that | (Y ) Z˜ | = | (Y ) π the ψn . It follows that the contribution of the first line in (7.92) to the error integral (7.64) is bounded in absolute value by a constant multiple of a sum of two terms of the form: |K 0 ψn ||d/ψn |(1 + t  )−1 [1 + log(1 + t  )]dt  du  dµh/ (7.93) Wut

Substituting (7.63), this is in turn bounded by: |Lψn ||d/ψn |(1 + t  )−1 [1 + log(1 + t  )]dt  du  dµh/ Wut

+

Wut

(7.94)

(η0−1 + α −1 κ)|Lψn ||d/ψn |(1 + t  )−1 [1 + log(1 + t  )]dt  du  dµh/

We estimate the first integral of (7.94) by the product: 1/2    −2  2 2   (1 + t ) [1 + log(1 + t )] (Lψn ) dt du dµh/ Wut

1/2 Wut

Now, the integral in the first factor is bounded by: t u C (1 + t  )−2 [1 + log(1 + t  )]2 E0,n (t  )dt  0





|d/ψn | dt du dµh/ 2

Chapter 7. Outline of the Derived Estimates of Each Order

while the integral in the second factor decomposes into: 2   |d / ψ | dt du dµ + |d/ψn |2 dt  du  dµh/ n h/   U

Uc

Wut

187

(7.95)

Wut

where U is the region defined by (5.12): U = {x ∈ Wε∗0 : µ < η0 /4} Since 4µ/η0 ≥ 1 in U c , we can estimate: 4 2   |d / ψ | dt du dµ ≤ µ|d/ψn |2 dt  du  dµh/ n h/  t c t η 0 Wu Wu U u t ≤C F0,n (u  )du 

(7.96)

0

 To estimate the integral over U Wut we make use of the spacetime integral K n (t, u) defined by (5.169), with ψn in the role of ψ:  −1 −1 ων µ (Lµ)− |d/ψn |2 dµh K n (t, u) = − Wut 2 According to (5.206) (which relies on assumption C3 as well as B1, D1) we have: 1 K n (t, u) ≥ (1 + t  )[1 + log(1 + t  )]−1 |d/ψn |2 dt  du  dµh/ (7.97) C U  Wut Recalling the definition (5.208), that is, in the present context, K n (t, u) = sup [1 + log(1 + t  )]−4 K n (t  , u) t  ∈[0,t ]

and noting that for t  ∈ [tm−1 , tm ] it holds that: (1 + t  )−1 [1 + log(1 + t  )] ≤ 2(1 + tm )−1 [1 + log(1 + tm )], we then obtain:

U



W

tm−1 ,tm

|d/ψn |2 dt  du  dµh/ ≤ C Am K n (tm , u)

where: Am = (1 + tm )−1 [1 + log(1 + tm )]5 = (1 + 2m+r )−1 [1 + log(1 + 2m+r )]5 Hence, in view of the fact that the series ∞  m=0

Am

(7.98)

188

Chapter 7. Outline of the Derived Estimates of Each Order

is convergent (and the fact that K n (t, u) is a non-decreasing function of t) we conclude that: N  2   |d/ψn | dt du dµh/ = |d/ψn |2 dt  du  dµh/   U

m=0 U

Wut



≤C

N 

W tm−1 ,tm



Am

K n (t, u) ≤ C  K n (t, u)

(7.99)

m=0

Combining with (7.96) yields:  |d/ψn |2 dt  du  dµh/ ≤ C K n (t, u) + Wut

u 0

t F0,n (u  )du 

(7.100)

Thus, the first of the integrals (7.94) is bounded in absolute value by: 1/2  1/2  t u  −2  2 u   t   (1 + t ) [1 + log(1 + t )] E0,n (t )dt K n (t, u) + F0,n (u )du C 0

0

(7.101) Moreover, in view of (7.50), assumptions A, and the estimate (7.100), the second of the integrals (7.94) is bounded in absolute value by: 1/2  Wut

(1 + t  )−2 [1 + log(1 + t  )]2 (1 + α −1 κ)2 (Lψn )2 dt  du  dµh/ 

1/2 

2

·

Wut



u

≤C 0



|d/ψn | dt du dµh/

t F0,n (u  )du 

1/2 



u

K n (t, u) + 0

t F0,n (u  )du 

1/2 (7.102)

This concludes the estimate of the contribution of (7.91), the third and fourth terms in (7.43), to the error integral (7.64). Finally, we estimate the contribution of the last two terms in (7.43), (1/2)

(Y )

π˜ L L
/ ψn−1 + µ

(Y ) ˆ˜

π /· D / 2 ψn−1

to the error integral (7.64). By the third and sixth of assumptions G0, this contribution is bounded in absolute value by a constant multiple of |K 0 ψn ||D / 2ψn−1 |[1 + log(1 + t  )]dt  du  dµh/ (7.103) Wut

By assumption H1, we have: |D / 2 ψn−1 | ≤ C(1 + t)−1

 i

|d/ Ri ψn−1 | ≤ C(1 + t)−1



|d/ψn |

(7.104)

Chapter 7. Outline of the Derived Estimates of Each Order

189

since the Ri ψn−1 are among the ψn . Therefore (7.103) is bounded by a constant multiple of a sum of three terms of the form: |K 0 ψn ||d/ψn |(1 + t  )−1 [1 + log(1 + t  )]dt  du  dµh/ Wut

This is precisely the same form as (7.93), to which the previous estimate applies. We collect the above results, in simplified form, in the following lemma. Lemma 7.2 We have:       (K 0 ψn ) (Y ) σ1,n−1 dt  du  dµh/    Wut   t u u t ≤C (1 + t  )−3/2 E0,n (t  )dt  + F0,n (u  )du  0 0   1/2 u  1/2 t   +C F 1,n (u )du + K n (t, u) 0



t

× 0

u (1 + t  )−3/2 E0,n (t  )dt  +

0

u

t F0,n (u  )du 

1/2

We now consider the partial contribution of (Y ) σ1,n−1 to the error integral (7.62), which is associated to the vectorfield K 1 . In view of (5.16), namely K 1 = (ω/ν)L the error integral (7.62) is: (ω/ν)(Lψn + νψn ) − Wut

(Y )

(7.105)

σn−1 dt  du  dµh/

(7.106)

and we are here considering the partial contribution of (Y ) σ1,n−1 to this, namely: − (ω/ν)(Lψn + νψn ) (Y ) σ1,n−1 dt  du  dµh/ (7.107) Wut

Now, by virtue of assumptions B1 and D1 of Chapter 5 we have: C −1 (1 + t)2 ≤ ω/ν ≤ C(1 + t)2 So, the integral (7.107) is bounded in absolute value by a constant multiple of (1 + t  )2 |Lψn + νψn || (Y ) σ1,n−1 |dt  du  dµh/ Wut

and our task is to estimate this integral.

(7.108)

(7.109)

190

Chapter 7. Outline of the Derived Estimates of Each Order

We begin with the contribution of the first term in (7.43). Expressing this term using the expansion (7.66), we shall estimate explicitly the contribution of the leading term, namely the integral: Wut

(1 + t  )2 |Lψn + νψn ||tr

(Y ) ˜

π /||LT ψn−1 + νT ψn−1 |dt  du  dµh/

(7.110)

However, by the last of assumptions G0 and the fact that T ψn−1 is one of the ψn , this is simply bounded by: u t F1,n (u  )du  (7.111) C 0

To estimate the contribution of the second term in (7.43) to the integral (7.109), we write: L 2 ψn−1 = (1 + t)−1 L Qψn−1 − (1 + t)−2 Qψn−1 = (1 + t)−1 (L Qψn−1 + ν Qψn−1 ) − (1 + t)−2 (1 + (1 + t)ν)Qψn−1 In view of the second of assumptions G0 and assumption B1 of Chapter 5, as well as the fact that Qψn−1 is one of the ψn , the contribution of the second term in (7.43) to (7.109) is then bounded by: C

 Wut

+C

u

≤C 0

(1 + t  )[1 + log(1 + t  )]|Lψn + νψn |2 dt  du  dµh/  Wut

[1 + log(1 + t  )]|Lψn + νψn ||ψn |dt  du  dµh/

t F1,n (u  )du 



u

+C 0

t F1,n (u  )du 

(7.112) 1/2 

ε02

t

 −2

(1 + t )

[1 + log(1 + t

0



u )]2 E0,n (t  )dt 

1/2

Here we have used (7.90) in estimating the second integral. To estimate the contribution of the third and fourth terms in (7.43), namely of (7.91), to the integral (7.109), we use the expression (7.92). We shall estimate explicitly the contribution of the first line of (7.92), which contains the principal terms. The remarks which we made in considering the contribution of these terms to the integral (7.64) lead in the present case to the conclusion that the contribution of the same terms to the integral (7.109) is bounded by a constant multiple of a sum of two terms of the form: Wut

(1 + t  )[1 + log(1 + t  )]|Lψn + νψn ||d/ψn |dt  du  dµh/

(7.113)

Chapter 7. Outline of the Derived Estimates of Each Order

191

This is bounded by:   ×

1/2  2

Wut

Wut



2



(1 + t ) |Lψn + νψn | dt du dµh/ 1/2

(7.114)

[1 + log(1 + t  )]2 |d/ψn |2 dt  du  dµh/

Now, the integral in the first factor of (7.114) is bounded by: u t F1,n (u  )du  C 0

The integral in the second factor decomposes into:  2 2   [1 + log(1 + t )] |d / ψ | dt du dµ + n h/   U

Uc

Wut

Wut

[1 + log(1 + t  )]2 |d/ψn |2 dt  du  dµh/ (7.115)

In U c we have 4µ/η0 ≥ 1; we can thus estimate, in view of (7.55), [1 + log(1 + t  )]2 |d/ψn |2 dt  du  dµh/  Uc

Wut

4 ≤ [1 + log(1 + t  )]2 µ|d/ψn |2 dt  du  dµh/ η0 Wut t u ≤C (1 + t  )−2 [1 + log(1 + t  )]2 E1,n (t  )dt 

(7.116)

0

On the other hand, the integral over U is by (7.97) a fortiori bounded by: [1 + log(1 + t  )]2 |d/ψn |2 dt  du  dµh/ ≤ C K n (t, u)  U

(7.117)

Wut

Combining with (7.116), we obtain: [1 + log(1 + t  )]2 |d/ψn |2 dt  du  dµh/ Wut

 t u ≤ C K n (t, u) + (1 + t  )−2 [1 + log(1 + t  )]2 E1,n (t  )dt 

(7.118)

0

We conclude that (7.113) is bounded by: 

u

C 0

t F1,n (u  )du 

1/2 



t

K n (t, u) + 0

 −2

(1 + t )

[1 + log(1 + t



u )]2 E1,n (t  )dt 

1/2

(7.119)

192

Chapter 7. Outline of the Derived Estimates of Each Order

Finally, we estimate the contribution of the last two terms in (7.43) to the error integral (7.109). By the third and sixth of assumptions G0 and (7.104) this contribution is bounded in absolute value by a constant multiple of a sum of three terms of the form (7.113), which has already been estimated. We collect the above results in the following lemma. Lemma 7.3 We have:       (K 1 ψn + ωψn ) (Y ) σ1,n−1 dt  du  dµh/     Wut 1/2  u u t t ≤C F1,n (u  )du  + C F1,n (u  )du  0



0



t

· K n (t, u) +

 −2

(1 + t )

 2

[1 + log(1 + t )]

0

u u (E1,n (t  ) + ε02 E0,n (t  ))dt 

1/2

We now consider the partial contribution of (Y ) σ2,n−1 , given by (7.44) to each of the error integrals (7.61), (7.62). To estimate this contribution we must make an assumption on the Lie derivatives, with respect to the commutation fields, of the components of the deformation tensors of the commutation fields. This assumption is simply that these derivatives satisfy the same bounds as the components themselves, as given by assumption G0. That is, we assume: G1: There is a positive constant C independent of s such that in Wεs0 , for all five commutation fields we have: (Y )

Y( |Y (µ

−1 (Y )

|Y (

(Y )

|L /Y ( |L /Y (

π˜ L L ) = 0 π˜ L L )| ≤ C[1 + log(1 + t)]

π˜ L L )| (Y ) ˜ π / L )| (Y ) ˜ π / L )|

≤ C[1 + log(1 + t)] ≤ C(1 + t)−1 [1 + log(1 + t)] ≤ C[1 + log(1 + t)]

|L /Y ( (Y ) π /ˆ˜ )| ≤ C(1 + t)−1 [1 + log(1 + t)] /˜ )| ≤ C(1 + t)−1 [1 + log(1 + t)] |Y ( (Y ) trπ Here in each line Y occurs twice. In each of these occurrences Y is meant to range independently over the set of the five commutation fields {Yi : i = 1, 2, 3, 4, 5}. The last of the assumptions G1 seems stronger than what corresponds to the last of assumptions G0, however we recall from Chapter 6 (see (6.99), (6.187)) that we actually have /˜ | ≤ C(1 + t)−1 [1 + log(1 + t)] : i = 2 | (Yi ) trπ for all commutation fields except for Y2 = Q, while for Q we have ((6.107)): |tr

(Q) ˜

π / − 4| ≤ C(1 + t)−1 [1 + log(1 + t)]

Chapter 7. Outline of the Derived Estimates of Each Order

193

We see then that the last of assumptions G1 is in fact in accordance with the other assumptions. In the following we shall also make use of the bounds: u (Lψn−1 )2 dµh/ ≤ Cε0 E0,n (t)

St,u

S

u (Lψn−1 )2 dµh/ ≤ Cε0 (1 + t)−2 E0,n (t)

t,u

u |d/ψn−1 |2 dµh/ ≤ Cε0 (1 + t)−2 E0,n (t)

St,u

(7.120)

These bounds follow from inequality (7.90), the first two by expressing L and L in terms of the commutation fields T and Q, and the last by virtue of assumption H0. We consider now the error integral: (K 0 ψn ) (Y ) σ2,n−1 dt  du  dµh/ (7.121) − Wut

The first term in (7.44), (Y ) ˜

π /)Lψn−1 ,

(1/4)L(tr involves L(tr

(Y ) π /˜ ).

Writing L = (1 + t)−1 Q, we have, from the last of assumptions G1, |L(tr

(Y ) ˜

π /)| ≤ C(1 + t)−2 [1 + log(1 + t)]

(7.122)

It follows that the contribution of the first term to the integral (7.121) is bounded in absolute value by: L C(I2 + I2L ) where:



L

I2 = I2L

=

Wut

|Lψn ||Lψn−1 |(1 + t  )−2 [1 + log(1 + t  )]dt  du  dµh/

Wut

(η0−1 + α −1 κ)|Lψn ||Lψn−1 |(1 + t  )−2 [1 + log(1 + t  )]dt  du  dµh/ (7.124)

Using the first of (7.120) we can estimate: 1/2   L I2

(7.123)

t

≤

u

0

0

0

St  ,u 

(Lψn ) dµh/ du



0

u

1/2

2

St  ,u 

(Lψn−1 ) dµh/ du



· (1 + t  )−2 [1 + log(1 + t  )]dt   u  1/2 2 u  1/2 ε0 E0,n (t ) E0,n (t ) (1 + t  )−2 [1 + log(1 + t  )]dt 

t

≤C

2



= Cε0 0

t

u (1 + t  )−2 [1 + log(1 + t  )]E0,n (t  )dt 

(7.125)

194

Chapter 7. Outline of the Derived Estimates of Each Order

and (taking account also of assumptions A):  I2L

1/2 (η0−1 t

≤

+α

Wu



−1





κ)(Lψn ) dt du dµh/ 2

(7.126) 1/2

(η0−1 t

·

+α

−1

κ)(Lψn−1 ) (1 + t )

Wu



u

≤ Cε0 0

 −4

t F0,n (u  )du 

2

1/2 

t

 −4

(1 + t )







[1 + log(1 + t )] dt du dµh/

[1 + log(1 + t



0

2

u )]3 E0,n (t  )dt 

1/2

The second term in (7.44), (1/4)L(trπ /˜ )Lψn−1 , /˜ ). Writing L = 2T +α −1 κ(1+t)−1 Q, we have, from the last of assumptions involves L(trπ G1, /˜ )| ≤ C(1 + t)−1 [1 + log(1 + t)] |L(trπ

(7.127)

It follows that the contribution of the second term to the integral (7.121) is bounded in absolute value by: L

C(I3 + I3L ) where:

L

I3 = I3L =

Wut

Wut

|Lψn ||Lψn−1 |(1 + t  )−1 [1 + log(1 + t  )]dt  du  dµh/

(7.128)

(η0−1 + α −1 κ)|Lψn ||Lψn−1 |(1 + t  )−1 [1 + log(1 + t  )]dt  du  dµh/ (7.129)

Using the second of (7.120) we can estimate: L I3

≤

t  0

u

0

≤C 0

(Lψn ) dµh/ du 2

St  ,u 



1/2  0

u

1/2

(Lψn−1 ) dµh/ du 2

St  ,u 



· (1 + t  )−1 [1 + log(1 + t  )]dt  1/2 t 1/2 2 u u (t  ) (t  ) ε0 (1 + t  )−2 E0,n E0,n



· (1 + t  )−1 [1 + log(1 + t  )]dt 

= Cε0 0

t

u (1 + t  )−2 [1 + log(1 + t  )]E0,n (t  )dt 

(7.130)

Chapter 7. Outline of the Derived Estimates of Each Order

and (taking account also of assumptions A):  I3L ≤

Wut

195

1/2

(η0−1 + α −1 κ)(Lψn )2 dt  du  dµh/



·

(7.131) 1/2

Wut



u

≤ Cε0 0

(η0−1 + α −1 κ)(Lψn−1 )2 (1 + t  )−2 [1 + log(1 + t  )]2 dt  du  dµh/ t F0,n (u  )du 

1/2 

t 0

u (1 + t  )−4 [1 + log(1 + t  )]3 E0,n (t  )dt 

1/2

The third term in (7.44), (1/4)L(µ−1 involves L(µ−1 tions G1,

(Y ) π ˜

L L ). Writing

|L(µ−1

(Y )

(Y )

π˜ L L )Lψn−1 ,

L = (1 + t)−1 Q, we have, from the second of assump-

π˜ L L )| ≤ C(1 + t)−1 [1 + log(1 + t)]

(7.132)

It follows that the contribution of the third term to the integral (7.121) is also bounded in absolute value by: L C(I3 + I3L ) The fourth and fifth terms in (7.44), −(1/2)L /L involve L /L

(Y ) Z˜

and L /L

(Y )

(Y ) Z ˜.

Z˜ · d/ψn−1 − (1/2)L /L

(Y )

Z˜ · d/ψn−1 ,

Now, from (7.30) or (7.31), we have:

(Y ) ˜

/· π /L = h

(Y )

Z˜ ,

(Y ) ˜

(Y )

π / L = h/ ·

Z˜

hence: L /Y L /Y Since

(Y ) ˜

/Y π / L = h/ · L (Y ) ˜ π / L = /h · L /Y (Y )

π˜ = (

(Y )

Z˜ + (Y ) ˜ Z+ (Y )

(Y )

π /· (Y ) π /·

(Y ) (Y )

Z˜ Z˜

(7.133)

π + Y (log )h),

and by assumptions E1, E2 of Chapter 6 we have (see (6.51)):

which implies:

|Y σ | ≤ C(1 + t)−1

(7.134)

|Y log | ≤ C(1 + t)−1

(7.135)

for all five commutation fields Y , we then obtain, by virtue of assumptions G0, G1, |L /Y |L /Y

Z˜ | ≤ C(1 + t)−1 [1 + log(1 + t)] (Y ) ˜ Z | ≤ C[1 + log(1 + t)] (Y )

(7.136)

196

Chapter 7. Outline of the Derived Estimates of Each Order

These imply, writing L = 2T + α −1 κ(1 + t)−1 Q, L = (1 + t)−1 Q, that |L /L

(Y )

˜ |L Z|, /L

(Y )

Z˜ | ≤ C(1 + t)−1 [1 + log(1 + t)]

(7.137)

It follows that the contribution of the fourth and fifth terms in (7.44) to the error integral (7.121) is bounded in absolute value by: L

C(I4 + I4L ) where:



L

I4 = I4L =

|Lψn ||d/ψn−1 |(1 + t  )−1 [1 + log(1 + t  )]dt  du  dµh/

(7.138)

(η0−1 + α −1 κ)|Lψn ||d/ψn−1 |(1 + t  )−1 [1 + log(1 + t  )]dt  du  dµh/

(7.139)

Wut

Wut

Using the last of (7.120) we can estimate: L I4

≤

t  0

u

2

0



t

≤C 0

St  ,u 

(Lψn ) dµh/ du



1/2 

u

1/2



0

2

St  ,u 

|d/ψn−1 | dµh/ du



· (1 + t  )−1 [1 + log(1 + t  )]dt  1/2  u  1/2 2 u ε0 (1 + t  )−2 E0,n E0,n (t ) (t  ) (1 + t  )−1 [1 + log(1 + t  )]dt 



t

= Cε0

u (1 + t  )−2 [1 + log(1 + t  )]E0,n (t  )dt 

0

(7.140)

and (taking account also of assumptions A): 1/2

 I4L ≤

Wut

(η0−1 + α −1 κ)(Lψn )2 dt  du  dµh/



·

(7.141) 1/2

Wut



u

≤ Cε0 0

(η0−1 + α −1 κ)|d/ψn−1 |2 (1 + t  )−2 [1 + log(1 + t  )]2 dt  du  dµh/ t F0,n (u  )du 

1/2 

t 0

u (1 + t  )−4 [1 + log(1 + t  )]3 E0,n (t  )dt 

1/2

/ (Y ) Z˜ , respectively. The sixth and seventh terms in (7.44) involve div / (Y ) Z˜ and div ˜ ˜ Now, hypothesis H1 applied to the 1-forms π /L , π / L on each St,u yields: |D /

(Y )

˜ 2 = |D Z| /

(Y ) ˜

|D /

(Y )

˜ 2 = |D / Z|

(Y ) ˜

π / L |2 ≤ C(1 + t)−2



|L / Ri

(Y ) ˜

|L / Ri

(Y ) ˜

π / L |2

i

π / L | ≤ C(1 + t) 2

−2

 i

π / L |2

(7.142)

Chapter 7. Outline of the Derived Estimates of Each Order

197

(the Ri are tangential to the St,u ). The fourth and fifth of assumptions G1 then imply: |D /

(Y )

|D /

(Y )

Z˜ | ≤ C(1 + t)−2 [1 + log(1 + t)] Z˜ | ≤ C(1 + t)−1 [1 + log(1 + t)]

(7.143)

Z˜ | ≤ C(1 + t)−2 [1 + log(1 + t)] Z˜ | ≤ C(1 + t)−1 [1 + log(1 + t)]

(7.144)

thus also: |div /

(Y )

|div /

(Y )

It then follows (compare with (7.122) and (7.127)) that the contribution of the sixth term in (7.44), −(1/2)div / (Y ) Z˜ Lψn−1 , to the integral (7.121) is bounded in absolute value by: L

C(I2 + I2L ) while the contribution of the seventh term, (Y )

−(1/2)div /

Z˜ Lψn−1 ,

is bounded in absolute value by: L

C(I3 + I3L ) The eighth term in (7.44), (1/2)d/ involves d/ |d/

(Y )

(Y ) π ˜

LL.

(Y )

π˜ L L · d/ψn−1

By hypothesis H0 and the third of assumptions G1,

π˜ L L | ≤ C(1 + t)−1

) (Ri

(Y ) π ˜

LL)

2

≤ C(1 + t)−1 [1 + log(1 + t)]

(7.145)

i

It follows that the contribution of the eighth term to (7.121) is bounded, like the contributions of the fourth and fifth terms, in absolute value by: L

C(I4 + I4L ) Finally, the ninth (last) term in (7.44), div / (µ involves: div / (µ

(Y ) ˆ˜

(Y ) ˆ˜

π /) · d/ψn−1

π /) = µdiv /

(Y ) ˆ˜

π / + d/µ ·

(Y ) ˆ˜

π /

198

Chapter 7. Outline of the Derived Estimates of Each Order

Now, hypothesis H2 applied to the traceless symmetric tensorfield yields:  |D / (Y ) π |L / Ri (Y ) π /ˆ˜ |2 ≤ C(1 + t)−2 /ˆ˜ |2

(Y ) π /ˆ˜

on each St,u (7.146)

i

(the Ri are tangential to the St,u ). The sixth of assumptions G1 then yields: |D /

(Y ) ˆ˜

thus also: |div /

π /| ≤ C(1 + t)−2 [1 + log(1 + t)]

(7.147)

(Y ) ˆ˜

π /| ≤ C(1 + t)−2 [1 + log(1 + t)]

(7.148)

Taking into account the second of assumptions F1 of Chapter 6 (together with the basic bootstrap assumption A3), we then obtain: |div / (µ

(Y ) ˆ˜

π /)| ≤ C(1 + t)−2 [1 + log(1 + t)]2

(7.149)

Comparing the eighth and ninth terms and (7.145) and (7.149) we conclude that the contribution of the ninth (last) term in (7.44) to the error integral (7.121) is a fortiori bounded in absolute value by: L C(I4 + I4L ) We collect the above results, in simplified form, in the following lemma. Lemma 7.4 We have:       (Y )   (K 0 ψn ) σ2,n−1 dt du dµh/    Wut   t u (1 + t  )−2 [1 + log(1 + t  )]E0,n (t  )dt  + ≤ Cε0 0

We now consider the error integral (K 1 ψn + ωψn ) −

(Y )

Wut

u 0

t F0,n (u  )du 



σ2,n−1 dt  du  dµh/ ,

or (see (7.106)), −

Wut

(ω/ν)(Lψn + νψn )

(Y )

σ2,n−1 dt  du  dµh/

By (7.108) this integral is bounded in absolute value by a constant multiple of (1 + t  )2 |Lψn + νψn || (Y ) σ2,n−1 |dt  du  dµh/ Wut

and our task is to estimate this integral.

(7.150)

(7.151)

Chapter 7. Outline of the Derived Estimates of Each Order

199

In view of (7.122) the contribution of the first term in (7.44) to the integral (7.151) is bounded in absolute value by:



u

 |Lψn + νψn ||Lψn−1 |[1 + log(1 + t  )]dµh/ dt  du 

C 0

Cu 



u

≤C



0

1/2 Cu 

(1 + t  )2 (Lψn + νψn )2 dµh/ dt 



1/2

· Cu 



u

≤C 0

(1 + t  )−2 [1 + log(1 + t  )]2 (Lψn−1 )2 dµh/ dt 

1/2  t F1,n (u  )



u

≤ Cε0

 0

t F1,n (u  )du 

0

t

u ε0 E0,n (t  )(1 + t  )−2 [1 +

1/2 

t 0

du 



log(1 + t )] dt 2



(7.152)

1/2

u (1 + t  )−2 [1 + log(1 + t  )]2 E0,n (t  )dt 

du 

1/2

Here we have used the first of the inequalities (7.120). In view of (7.127) the contribution of the second term in (7.44) to the integral (7.151) is bounded in absolute value by:

u



 

|Lψn + νψn ||Lψn−1 (1 + t )|[1 + log(1 + t )]dµh/ dt

C 0

Cu 



u

≤C



0

Cu 

2

(1 + t ) (Lψn + νψn ) dµh/ dt



u



0

u

≤ Cε0 0



2

2

[1 + log(1 + t )] (Lψn−1 ) dµh/ dt

t (u  ) F1,n



du 

1/2 

Cu 





1/2  2

·

≤C



1/2



t F1,n (u  )du 

t 0



du 

u ε0 E0,n (t  )(1 + t  )−2 [1 + log(1 + t  )]2 dt 

1/2 

t 0

 −2

(1 + t )

[1 +

(7.153) 1/2

u log(1 + t  )]2 E0,n (t  )dt 

du 

1/2

Here we have used the second of the inequalities (7.120). In view of (7.132) the contribution of the third term in (7.44) to the integral 7.151 is also bounded in the same way.

200

Chapter 7. Outline of the Derived Estimates of Each Order

In view of (7.137) the contribution of the fourth and fifth terms in (7.44) to (7.151) is bounded in absolute value by:   u

|Lψn + νψn ||d/ψn−1 |(1 + t  )|[1 + log(1 + t  )]dµh/ dt  du 

C Cu 

0



u

≤C



1/2  2

Cu 

0

(1 + t ) (Lψn + νψn ) dµh/ dt



Cu 



u

0



[1 + log(1 + t )] |d/ψn−1 | dµh/ dt

t (u  ) F1,n



u

≤ Cε0 0



1/2 

· ≤C

2

1/2



t F1,n (u  )du 

t

2

2



du 

u ε0 E0,n (t  )(1 + t  )−2 [1 + log(1 + t  )]2 dt 

0 1/2  t 0

 −2

(1 + t )

[1 +

(7.154) 1/2

u log(1 + t  )]2 E0,n (t  )dt 

du 

1/2

Here we have used the last of the inequalities (7.120). In view of the bounds (7.144) the contribution of the sixth term in (7.44) to (7.151) is bounded in absolute value as in (7.152) while that of the seventh term is bounded in absolute value as in (7.153). Finally, in view of (7.145) and (7.149) the contributions of the eighth and ninth terms in (7.44) to the integral (7.151) are bounded in absolute value as in (7.154). We collect the above results in the following lemma. Lemma 7.5 We have:       (Y )   (K 1 ψn + ωψn ) σ2,n−1 dt du dµh/    Wut   u 1/2  t 1/2 t    −2  2 u   ≤ Cε0 F1,n (u )du (1 + t ) [1 + log(1 + t )] E0,n (t )dt 0

0

Finally, we consider the partial contribution of (Y ) σ3,n−1 , given by (7.45) to the error integrals (7.61) and (7.62). Now, assumptions E1, E2, F1, F2, of Chapter 6 imply (see (6.90), (6.74), (6.84), (6.72), (6.79)): |L log | ≤ C(1 + t)−2 |trχ| ≤ C(1 + t)−1 ,

|trχ | ≤ C(1 + t)−1 [1 + log(1 + t)]

|| ≤ C(1 + t)−1 [1 + log(1 + t)] |L(α −1 κ)| ≤ C(1 + t)−1 ,

|d/(α −1 κ)| ≤ C(1 + t)−1 [1 + log(1 + t)]

(7.155) (7.156) (7.157) (7.158)

Chapter 7. Outline of the Derived Estimates of Each Order

201

Together with assumptions G0 the above in turn imply that the coefficients (Y ) σ L 3,n−1 ,

(Y ) σ L 3,n−1 ,

given by (7.46), (7.47), (7.48), respectively, satisfy the bounds: | | |

(Y ) L σ3,n−1 | (Y ) L σ3,n−1 | (Y ) /σ3,n−1 |

≤ C(1 + t)−2 ≤ C(1 + t)−1 [1 + log(1 + t)] ≤ C(1 + t)−1 [1 + log(1 + t)]

(7.159)

Comparing with (7.122), (7.127), (7.137) it follows that the contributions of the second term in (7.45) to the error integrals (7.61), (7.62), are bounded as the corresponding contributions of the first term in (7.44), the contributions of the first term in (7.45) are bounded as the corresponding contributions of the second term in (7.44), and the contributions of the third term in (7.45) are bounded as the corresponding contributions of the fourth and fifth terms in (7.44). Thus, the contribution of (Y ) σ3,n−1 to each of the error integrals (7.61) and (7.62) is bounded in the same way as the corresponding contribution of (Y ) σ2,n−1 . Lemmas 7.2–7.5 together with the conclusion just reached yield the following lemma. Lemma 7.6 We have:    t u     (Y )    −3/2 u   t   ≤ C (K ψ ) σ dt du dµ (1 + t ) E (t )dt + F (u )du  0 n n−1 h/  0,n 0,n   Wut 0 0   1/2 u  1/2 t   +C F 1,n (u )du + K n (t, u) 0



t

× 0

u (1 + t  )−3/2 E0,n (t  )dt  +



u 0

t F0,n (u  )du 

1/2

and:       (Y )   (K 1 ψn + ωψn ) σn−1 dt du dµh/    Wut  u 1/2  t  t {K n (t, u)}1/2 ≤C F1,n (u  )du  + C F1,n (u )du  0

1/2  t  + C F1,n (u )du 



t 0

  1/2 u u dt  (1 + t  )−2 [1 + log(1 + t  )]2 E1,n (t  ) + ε02 E0,n

Recalling (see (7.18), (7.27), (7.28)) that the contributions of (Y ) σn−1 (through ρ˜n ) to the error integrals (7.23) are precisely the integrals estimated in the above lemma, and comparing with (5.240) and (5.267), with ψn in the role of ψ, we conclude that the contributions in question can be absorbed into the corresponding right-hand sides.

Chapter 8

Regularization of the Propagation Equation for d/trχ. Estimates for the Top Order Angular Derivatives of χ In the present chapter we shall deal with the problem of obtaining estimates for χ and its derivatives with respect to the set of rotation vectorfields {Ri : i = 1, 2, 3} of order n, given energy estimates of order n +1 for the ψµ : µ = 0, 1, 2, 3, the partial derivatives of the wave function φ with respect to the rectangular coordinates. That is, we shall estimate  , defined in Chapter 7, associated the former in terms of the quantities E0,n+1 and E1,n+1 to n + 1st order variations ψn+1 . These variations are obtained by applying to the ψµ a string of commutation vectorfields of length n. We call the derivatives with respect to the set of rotation vectorfields angular derivatives. We are thus requiring estimates for the nth order angular derivatives of χ in terms of estimates for the n + 1st order derivatives of the ψµ . The propagation equations (3.38) for χ AB involve on the right-hand side the curva( P)

ture components α AB , whose principal part α AB , given by (4.71), (4.72), contains the 2nd derivatives of σ , hence the 2nd derivatives of the ψµ . Now, the propagation equations are ordinary differential equations along the generators of the characteristic hypersurfaces Cu . Thus, in integrating these equations to obtain estimates for χ, no regularity is expected to be gained except along the generators. Consequently, in the estimates obtained in this way there will be a loss of one degree of differentiability along the St,u sections, and we can only estimate the n − 1st order angular derivatives of χ in terms of estimates for the n +1st order derivatives of ψµ . Such estimates, that is of the next-to-the-top order angular derivatives of χ in terms of the top order energies, are in fact also needed in our approach, and shall be derived in Chapter 12. In the present chapter however, we concentrate on the main problem, which is that of deriving estimates for the top order angular derivatives of χ in terms of the top order energies. To accomplish this aim we must avoid the loss of one degree of differentiability along the St,u sections.

Chapter 8. The Propagation Equation for d/trχ

204

Now a propagation equation of the type we are considering does not lead to such a loss of differentiability when – and only when – the principal part on the right-hand side can be put into the form of a derivative with respect to L. This is in fact the case for trace equation (3.42), the principal part on the right-hand side of which is contained in the Ricci tensor component S(L, L). The fact that the principal part of S(L, L) is of the form of a derivative with respect to L shall be demonstrated with the help of the following proposition. We denote, as in the previous chapters, τµ = ∂µ σ,

ωµν = ∂µ ψν = ∂ν ψµ = ωνµ ,

ωµλ = (g −1 )νλ ωµν

Proposition 8.1 The function σ satisfies the following inhomogeneous wave equation in the acoustical metric h: d
h σ = −−1 a − 2b dσ where a and b are the functions: a = (h −1 )µν τµ τν

and

b = (h −1 )µν ωµα ωνα

Proof. Each ψα , α = 0, 1, 2, 3 is the derivative of the wave function φ with respect to a translation of the underlying Minkowski spacetime, therefore in accordance with the results of Chapter 1 each of these functions is a solution of the linear wave equation (1.86) ˜ with respect to the conformal acoustical metric h:
h˜ ψα = 0

(8.1)

It follows that σ being given by: σ = −(g −1 )αβ ψα ψβ , satisfies the equation:
h˜ σ = −2(h˜ −1 )µν (g −1 )αβ ∂µ ψα ∂ν ψβ

(8.2)

the coefficients (g −1 )αβ , the components of the reciprocal Minkowski metric in rectangular coordinates, being constant. Moreover, the fact that the conformal acoustical metric h˜ is related to the acoustical metric h by: h˜ µν = h µν implies that for an arbitrary function f :
h˜ f = −1
h f + −2 (h −1 )µν ∂µ ∂ν f and we have:

(8.3)

d ∂µ σ dσ Considering equation (8.2) together with the relation (8.3) in the case f = σ , the proposition follows. ∂µ  =

Chapter 8. The Propagation Equation for d/trχ

205

We now consider the Ricci tensor Sµν = (h −1 )κλ Rκµλν ( P) µν

and its lower order part S

(N) µν ,

( P) µν

+ S

and decompose it into its principal part S

Sµν = S

(N) µν ,

(8.4)

according to the corresponding decomposition (4.50) of the curvature tensor: ( P) S µν =

(N) S µν =

(h −1 )κλ Pκµλν ,

(h −1 )κλ Nκµλν

(8.5)

Now, the principal part Pκµλν of the curvature tensor is given by (4.49): Pκµλν =

1 dH (ψκ ψν vλµ + ψλ ψµ vκν − ψµ ψν vλκ − ψλ ψκ vµν ) 2 dσ

where vµν is the covariant – relative to h – Hessian of σ (see (4.30), (4.31)): vµν = Dµ τν = Dµ Dν σ = vνµ Taking into account the fact that: (h −1 )κλ ψκ ψλ = ((g −1 )κλ − Fψ κ ψ λ )ψκ ψλ σ = −σ − Fσ 2 = − 1−σH we obtain: ( P) S µν =

1 dH 2 dσ

Here,

 ψµ ξν + ψν ξµ +

σ vµν − ψµ ψν trv 1−σH

(8.6)  (8.7)

ξµ = (h −1 )κλ ψκ vµλ

(8.8)

trv =
h σ

(8.9)

and, is expressed by Proposition 8.1 in terms of lower order terms. We now consider the Ricci tensor component S(L, L) = trα (see (3.41)) which appears in the propagation equation (3.42) for trχ. In analogy with (3.41) we have: ( P)

(N)

( P)

S (L, L) = tr α ,

(N)

and α

AB

(N)

S (L, L) = tr α

is given by (4.85), (4.94)–(4.96). From (8.7), ( P)

( P) µν

S (L, L) = S

Lµ Lν

(8.10)

Chapter 8. The Propagation Equation for d/trχ

206

is given by: ( P)

S (L, L) =

1 dH 2 dσ

where

 2ψ L ξ L +

σ v(L, L) − ψ L2 trv 1−σH

 (8.11)

ξ L = ξµ L µ ,

so that by the definition (8.8): ξ L = (h −1 )κλ ψκ (L µ vµλ ) = (h −1 )κλ ψκ D L τλ

(8.12)

Thus, ξ L is up to lower order terms equal to Ln where n is the function: n = (h −1 )κλ ψκ τλ

(8.13)

Also (see (4.61)) v(L, L) is up to a lower order term equal to L(Lσ ). We then conclude from (8.11), in view of the fact that by (8.9) and Proposition 8.1, trv = −−1

d a − 2b dσ

(8.14)

( P)

is of lower order, that in fact S (L, L) is equal, up to lower order terms, to the L derivative of a function f , where:   dH σ 1 f = τL (8.15) ψL n + dσ 2 (1 − σ H ) (N)

(N)

Since S (L, L) = tr α is (by definition) of lower order, it follows that we can express S(L, L) in the form: S(L, L) = L f + g (8.16) where the function g, which is actually defined by this equation, is of lower order, namely of the order of the 1st derivatives of the ψµ . The function g decomposes into the sum of four functions: g=

4  (i) g

(8.17)

i=1

where:

  dH dH ψL ξL − L ψL n dσ dσ   (2) 1 dH σ 1 dH σ g = v(L, L) − L τL 2 dσ 1 − σ H 2 dσ 1 − σ H (3) 1 dH 2 g =− ψ trv 2 dσ L (1)

g =

(4)

(N)

g = tr α

(8.18) (8.19) (8.20) (8.21)

Chapter 8. The Propagation Equation for d/trχ

207

(1)

The function g is the sum of three terms: (1)

g =

dH d2 H dH ψ L (ξ L − Ln) − n L(ψ L ) − ψ L n τL dσ dσ dσ 2

(8.22)

To calculate Ln we write: Ln = L((h −1 )αβ τα ψβ ) = L µ ∂µ ((h −1 )αβ τα ψβ ) = τα ψβ L µ ∂µ (h −1 )αβ + (h −1 )αβ ψβ L µ ∂µ τα + (h −1 )αβ τα L µ ∂µ ψβ Expressing ∂µ τα in terms of Dµ τα = vµα and writing ∂µ ψβ = ωµβ , this becomes: Ln = (h −1 )αβ ψβ L µ vµα + (h −1 )αβ τα L µ ωµβ ν +(h −1 )αβ ψβ L µ !µα τν + τα ψβ L µ ∂µ (h −1 )αβ

(8.23)

The last two terms are equal to: α L µ τα ψβ (h −1 )νβ !µν + ∂µ (h −1 )αβ



β = −L µ τα ψβ (h −1 )να (h −1 )λβ !µνλ = −L µ τα ψβ (h −1 )να !µν

Using the formula (3.141), that is: !µνλ =

1 dH (τµ ψν ψλ + τν ψµ ψλ − τλ ψµ ψν ) + H ψλ ωµν 2 dσ

as well as (8.6), the last is found to be equal to:   1 dH σ σH 2 (τ L n + ψ L a) + ψ L n + (h −1 )αβ τα L µ ωµβ 2 dσ 1 − σ H 1−σH Substituting in (8.23) we then obtain: 1 dH νL + Ln = ξ L + 1−σH 2 dσ Here,



σ (τ L n + ψ L a) + ψ L n 2 1−σH

νµ = (h −1 )αβ τα ωµβ ,



ν L = νµ L µ

(8.24)

(8.25)

To calculate L(ψ L ) we write: L(ψ L ) = L(L α ψα ) = L µ ∂µ (L α ψα ) = (L µ ∂µ L α )ψα + L µ L α ωµα Now an expression for L(L α ) has already been obtained in Chapter 3. However here it is simpler to calculate directly ψα L(L α ): α Lν ) ψα L(L α ) = ψα L µ (Dµ L α − !µν   = ψα µ−1 (Lµ)L α − (h −1 )αλ !µνλ L µ L ν

Chapter 8. The Propagation Equation for d/trχ

208

Substituting for !µνλ from the formula above and using (8.6), the last is found to be equal to:   dH σH 1 σ −1 ωL L + ψL τ L + nψ L µ (Lµ)ψ L + 1−σH dσ 1 − σ H 2 Hence we obtain: L(ψ L ) =

dH ωL L + µ−1 (Lµ)ψ L + ψ L 1−σH dσ



σ 1 τ L + nψ L 1−σH 2

 (8.26)

Substituting (8.24) and (8.26) in (8.22) we obtain the following expression for the (1)

function g :

(1)

g=−

  (1) dH dH −1 nψ L nψ L + µ (Lµ) + g dσ dσ

(8.27)

where: (1)

dH 1 (ν L ψ L + nω L L ) (1 − σ H ) dσ   dH 2 d2 H σ (aψ L + 3nτ L )ψ L − nτ L ψ L − 2(1 − σ H ) dσ dσ 2

g =−

(8.28)

Also, using (4.61) we obtain, in reference to (8.19), (2)

g =−

1 σ d H −1 1 d µ (Lµ)τ L − 2 1 − σ H dσ 2 dσ



σ dH 1 − σ H dσ

 τ L2

Moreover, substituting (8.14) in (8.20) we obtain:   (3) d 1 dH 2 g= ψ L −1 a + 2b 2 dσ dσ

(8.29)

(8.30)

Substituting in the definitions of the functions a and b (see Proposition 8.1) the expansion (3.116) of the reciprocal acoustical metric in terms of the null frame L, L, X 1, X 2: (8.31) (h −1 )κλ = −(1/2µ)(L κ L λ + L κ L λ ) + (h/−1 ) AB X κA X λB and recalling that τµ = ∂µ σ and ωµα = ∂µ ψα , we obtain the following expressions: a = −µ−1 (Lσ )(Lσ ) + |d/σ |2 b = −µ

−1

α

(8.32) α

(Lψ )(Lψα ) + d/ψ · d/ψα

(8.33)

which show that these functions are both O(µ−1 ) as µ → 0. Substituting in definition (8.13) the expansion (3.40) of the reciprocal acoustical metric h −1 in terms of the frame L, T, X 1 , X 2 : (h −1 )κλ = −α −2 L κ L λ − µ−1 (L κ T λ + T κ L λ ) + (h/−1 ) AB X κA X λB

(8.34)

Chapter 8. The Propagation Equation for d/trχ

209

we obtain the following expressions for the function n: n = −µ−1 ψ L (T σ ) − α −2 (ψ L + αψTˆ )(Lσ )+ ψ · d/σ

(8.35)

which shows that n = O(µ−1 ) as µ → 0. Substituting also the expansion (8.34) in definition (8.25) we obtain the following expression for the component ν L : / L · d/σ ν L = −µ−1 ω L L (T σ ) − α −2 (ω L L + αω L Tˆ )(Lσ ) + ω

(8.36)

which shows that ν L = O(µ−1 ) as µ → 0, as well. (See equations (4.89)–(4.93) in Chapter 4.) (2) (3)

In view of the above results, it follows from (8.27)–(8.30) that the functions g , g (1)

(1)

as well as g are all O(µ−1 ) as µ → 0, however g = O(µ−2 ) as µ → 0. On the other (4)

hand, g , given by (8.21) is O(1) as µ → 0. Now, by (8.35) we have:  dH dH dH 2 nψ L = −µ−1 ψ L (T σ ) + ψ L −α −2 (ψ L + αψTˆ )(Lσ )+ ψ · d/σ dσ dσ dσ The first term on the right is −2µ−1 m, where m, given by equation (3.97), is the first term in Lµ as given by equation (3.96). We thus have: dH nψ L = −2µ−1 (Lµ) + 2e˜ dσ

(8.37)

where,

 1 dH ψ L −α −2 (ψ L + αψTˆ )(Lσ )+ ψ · d/σ (8.38) 2 dσ and e is given by equation (3.98). On the other hand, according to definition (8.15) we have the following alternative expression for the left-hand side of (8.37): e˜ = e +

σ dH 1 dH nψ L = f − τL dσ 2 (1 − σ H ) dσ

(8.39)

In view of expressions (8.37) and (8.39), the first term on the right in (8.27) can be written in the form:   dH dH −1 nψ L nψ L + µ (Lµ) (8.40) − dσ dσ dH 1 σ dH τ L − 2e˜ nψ L = µ−1 (Lµ) f − µ−1 (Lµ) 2 (1 − σ H ) dσ dσ Thus, defining: (1)

(1)

dH g =g − dσ



σ 1 −1 µ (Lµ) τ L + 2enψ ˜ L 2 (1 − σ H )

 (8.41)

Chapter 8. The Propagation Equation for d/trχ

210 (1)

we have g = O(µ−1 ) as µ → 0, and: (1)

(1)

g = µ−1 (Lµ) f + g

(8.42)

fˇ = µf

(8.43)

Setting finally: and:

 gˇ = µ

 4  (i) g + g

(1)

(8.44)

i=2

the functions fˇ, gˇ are both bounded as µ → 0 and we have, by (8.16), (8.17) and (8.42): µS(L, L) = L fˇ + gˇ

(8.45)

We now substitute the expression (8.45) into the propagation equation (3.42) for trχ, multiplied by µ, and bring the term L fˇ to the left-hand side. We thus obtain the following regularized form of this equation: 1 ˆ 2 − gˇ L(µtrχ + fˇ) = 2(Lµ)trχ − µ(trχ)2 − µ|χ| 2

(8.46)

Here, decomposing χ into its trace-free part χˆ and its trace, we write: |χ|2 =

1 (trχ)2 + |χ| ˆ 2 2

(8.47)

Let us introduce the St,u 1-form:

We have:

x 0 = µd/trχ + d/ fˇ

(8.48)

x 0 = d/(µtrχ + fˇ) − (d/µ)trχ

(8.49)

We shall obtain a propagation equation for x 0 , by differentiating equation (8.46) tangentially to the St,u . We shall make use of the following lemma. Lemma 8.1 For an arbitrary function φ we have: L / L (d/φ) = d/(Lφ) Proof. This can most readily be established by using the (L, T, X 1 , X 2 ) frame. If we evaluate each side on L or T , then both sides vanish by definition. If we evaluate on X A , A = 1, 2, then the left-hand side is: L L (d/φ) · X A = L(d/φ · X A ) − d/φ · [L, X A ] = L(X A φ)

Chapter 8. The Propagation Equation for d/trχ

211

for, d/φ · X A = X A φ and [L, X A ] = 0. On the other hand, the right-hand side is: d/(Lφ) · X A = X A (Lφ) The two sides are then equal, in view of the fact that [L, X A ] = 0. Taking φ = µtrχ + fˇ in Lemma 8.1 we obtain, in view of (8.49) and (8.46), the following propagation equation for x 0 :     1 ˆ 2 ) − g0 (8.50) trχ − 2µ−1 (Lµ) d/ fˇ − µd/(|χ| L / L x 0 + trχ − 2µ−1 (Lµ) x 0 = 2 where:

1 g0 = d/gˇ − trχd/( fˇ + 2Lµ) + (d/µ)(Ltrχ + |χ|2 ) 2 In (8.51) we may substitute for Lµ from equation (3.96): Lµ = m + µe

(8.51)

(8.52)

By (8.37), (8.39) and (8.52), dH nψ L = −2µ−1 m + 2(e˜ − e) dσ σ dH 1 (Lσ ) = f − 2 (1 − σ H ) dσ Thus, defining: e = e − e˜ −

σ dH 1 (Lσ ) 4 (1 − σ H ) dσ

(8.53)

fˇ is given by, in view of (8.43), fˇ = −2(m + µe)

(8.54)

In (8.51) we may substitute for Ltrχ + |χ|2 from the original propagation equation (3.42), Ltrχ + |χ|2 = µ−1 (Lµ)trχ − trα, in which we write:

( P)

(N)

trα = tr α +tr α ( P)

and substitute for tr α from the trace of equation (4.71), that is: ( P)

tr α = µ−1 mtrχ − ψ L

( P) dH ψ · χ · d/σ + tr α dσ

In view of (8.52) we then obtain: Ltrχ + |χ|2 = f 0

(8.55)

Chapter 8. The Propagation Equation for d/trχ

212

where f 0 is the function: f 0 = etrχ + ψ L

( P) (N) dH ψ · χ · d/σ − tr α −tr α dσ

(8.56)

( P)

The function f 0 is bounded as µ → 0. The term tr α is obtained by taking the trace of equation (4.72): ( P)

tr α =

1 dH 2ψ L ψ · d/(Lσ ) − ψ L2
/ σ − | ψ|2 L(Lσ ) 2 dσ 

+ 2α −1 ψ L ψ · (κ −1 ζ  ) + α −1 ψ L2 trk/ + | ψ|2 e (Lσ )

 (8.57)

We conclude from the above that the St,u 1-form g0 defined by (8.51) is of the order of the 2nd derivatives of the ψµ and bounded as µ → 0. By (8.52), (8.54) and (8.55) it is given by: ˇ (8.58) g0 = d/gˇ − µtrχd/eˇ + (d/µ)( f 0 − trχ e) where we have defined: eˇ = e − e

(8.59)

To control the higher order angular derivatives of trχ, we introduce the St,u 1-forms: (i1 ...il )

xl = µd/(Ril . . . Ri1 trχ) + d/(Ril . . . Ri1 fˇ)

(8.60)

Thus, for a given positive integer l we have the multi-indices (i 1 , . . . , i l ) of length l, where each i k ∈ {1, 2, 3}, k = 1, . . . , l. To the multi-index (i 1 , . . . , i l ) there corresponds the string (Ri1 , . . . , Ril ) of rotation vectorfields. In deriving propagation equations for the xl , we shall make use of the following lemma. Lemma 8.2 Let Y be an arbitrary St,u - tangential vectorfield on the spacetime domain Wε∗0 . We have: [L, Y ] = (Y ) Z where (Y ) Z is an St,u - tangential vectorfield, associated to Y , and defined by the condition that for any vector V ∈ T Wε∗0 : h(

(Y )

Z, V ) =

(Y )

π(L, V )

In terms of the (L, T, X 1 , X 2 ) frame (or the null frame (L, L, X 1 , X 2 )), (Y )

Z=

(Y )

Z A X A,

(Y )

ZA =

(Y )

π L B (h/−1 ) AB

Proof. Let "s be the 1-parameter group generated by L and #s the 1-parameter group generated by Y . Then #s maps each surface St,u onto itself, while "s maps St,u onto St +s,u . Hence #−s ◦ "−s ◦ #s ◦ "s (8.61)

Chapter 8. The Propagation Equation for d/trχ

213

maps each St,u onto itself. Thus, the orbit by (8.61) of any given point q ∈ St,u lies in St,u . But this orbit must coincide to O(s 2 ) as s → 0 with the integral curve of the commutator [L, Y ] through q, parametrized by s 2 . It follows that [L, Y ] = (Y ) Z is a vectorfield which is tangential to the surfaces St,u , consequently we can expand: (Y )

Z=

(Y )

ZBXB

Taking the h- inner product with X A we obtain: / AB h

(Y )

Z B = h(

(Y )

Z , X A)

= h([L, Y ], X A ) = h(D L Y, X A ) − h(DY L, X A ) = (Y ) π L A − h(D X A Y, L) − h(DY L, X A ) Now, since h(L, X A ) = 0 we have: −h(DY L, X A ) = h(L, DY X A ) hence, substituting, / AB h

(Y )

ZB =

(Y )

π L A + h(L, [Y, X A ])

=

(Y )

πL A,

because the vectorfield [Y, X A ] is St,u -tangential, hence h-orthogonal to L, both Y, X A being St,u - tangential vectorfields. The lemma is thus proved. The following lemma, essentially a corollary of Lemma 8.2, is also used in the derivation of the propagation equations for the xl . Lemma 8.3 Let Y be an arbitrary St,u - tangential vectorfield on the spacetime domain Wε∗0 and let ξ be an arbitrary St,u 1-form on Wε∗0 . We have: /Y ξ − L /Y L /L ξ = L / (Y ) Z ξ L /L L Proof. Since both L, Y are tangential to the hypersurfaces Cu we can restrict ourselves to a given Cu . In defining L / L ξ, L /Y ξ we are considering the extension of ξ to T Cu by the condition ξ(L) = 0 We have: (L L ξ )(L) = L(ξ(L)) − ξ([L, L]) = 0 therefore: L /L ξ = LL ξ However, (LY ξ )(L) = Y (ξ(L)) − ξ([Y, L]) = ξ([L, Y ]) = ξ(

(8.62) (Y )

Z)

Chapter 8. The Propagation Equation for d/trχ

214

by Lemma 8.2. Since L /Y ξ is defined by restricting LY ξ to T St,u and then extending to /Y ξ )(L) = 0, it follows that, on the manifold Cu , T Cu by the condition (L L /Y ξ = LY ξ − ξ(

(Y )

Z )dt

(8.63)

in view of the fact that Lt = 1, d/t = 0. Consider now the evaluation of L /L L /Y ξ − L /Y L /L ξ on the frame vectorfields X A . This evaluation is: /Y ξ − LY L / L ξ )(X A ) (L L L

(8.64)

Substituting (8.62) and (8.63), (8.64) becomes: (L L LY ξ − L(ξ(

(Y )

Z ))dt − ξ(

(Y )

Z )L L (dt) − LY L L ξ )(X A )

(8.65)

Now, dt (X A ) = d/t (X A ) = 0, while: L L (dt) = d(Lt) = 0 therefore (8.65) reduces to: (L L LY ξ − LY L L ξ )(X A ) = (L[L ,Y ] ξ )(X A )

(8.66)

and by Lemma 8.2 this is: (L

(Y ) Z

ξ )(X A )

The lemma thus follows. Moreover, in the proof of the next proposition, as well as in the proofs of several of the propositions and lemmas which follow, we shall make use of a general elementary proposition on linear recursions, the proof of which is by a simple application of the principle of induction. Proposition 8.2 Let (yn : n = 1, 2, . . . ) be a given sequence in a space X and (An : n = 1, 2, . . . ) a given sequence of operators in X. Suppose that (x n : n = 0, 1, 2, . . . ) is a sequence in X satisfying the recursion: x n = An x n−1 + yn Then for each n = 1, 2, . . . we have: x n = An . . . A1 x 0 +

n−1  m=0

An . . . An−m+1 yn−m

Chapter 8. The Propagation Equation for d/trχ

To present the propagation equation for we introduce the functions:

215 (i1 ...il ) x

l

in as simple a form as possible,

fˇl = Ril . . . Ri1 fˇ   (i1 ...il ) h l = Ril . . . Ri1 |χ| ˆ 2 (i1 ...il )

(8.67) (8.68)

Proposition 8.3 For each non-negative integer l and each multi-index (i 1 , . . . , i l ), the St,u 1-form (i1 ...il ) xl satisfies the propagation equation:   L / L (i1 ...il ) xl + trχ − 2µ−1 (Lµ) (i1 ...il ) xl   1 trχ − 2µ−1 (Lµ) d/ (i1 ...il ) fˇl − µd/ (i1 ...il ) h l − (i1 ...il ) gl = 2 where the St,u 1-form (i1 ...il )

(i1 ...il ) g l

is given by:

gl = L / Ril . . . L / R i1 g 0 −

l−1 

L / Ril . . . L / Ril−k+1 L / (Ril−k )

(i1 ...il−k−1 ) Z

xl−k−1

k=0

+

l−1 

L / Ril . . . L / Ril−k+1

(i1 ...il−k )

yl−k

k=0

where the St,u 1-form g0 is given by (8.51), or, more explicitly, by (8.58). Here, for each j = 1, . . . , l, (i1 ...i j ) y j is the St,u 1-form: (i1 ...i j )

y j = (Ri j µ) (i1 ...i j −1 ) a j −1  + µRi j trχ − Ri j Lµ + 1 + (Ri j trχ)d/ 2

where

(i1 ...i j −1 ) a

(i1 ...i j −1 )

j −1

(i1 ...i j −1 )

(Ri j )

 Z µ d/(Ri j −1 . . . Ri1 trχ)

fˇj −1

is the St,u 1-form:

a j −1 = L / L d/(Ri j −1 . . . Ri1 trχ) + trχd/(Ri j −1 . . . Ri1 trχ) + d/

(i1 ...i j −1 )

h j −1

Proof. The propagation equation of the proposition reduces for l = 0 to the propagation equation (8.50) already established. Thus, by induction, assuming that the propagation equation of the proposition holds with l replaced by l − 1, that is, assuming that:   L / L (i1 ...il−1 ) xl−1 + trχ − 2µ−1 (Lµ) (i1 ...il−1 ) xl−1   1 −1 = (8.69) − 2µ (Lµ) d/ (i1 ...il−1 ) fˇl−1 − µd/ (i1 ...il−1 ) h l−1 − (i1 ...il−1 ) gl−1 2

Chapter 8. The Propagation Equation for d/trχ

216

holds for some St,u 1-form (i1 ...il−1 ) gl−1 , what we must show is that a propagation equation of the form given by the proposition holds true for l, where (i1 ...il ) gl is an St,u 1-form related to (i1 ...il−1 ) gl−1 by a certain recursion relation. This recursion relation shall then determine (i1 ...il ) gl , for each l, from the St,u 1-form g0 , given by (8.58). We begin by re-writing the term   2µ−1 (Lµ) (i1 ...il−1 ) xl−1 − d/ (i1 ...il−1 ) fˇl−1 in (8.69) as: 2(Lµ)d/(Ril−1 . . . Ri1 trχ) (see definitions (8.60), (8.67)) obtaining the equation: L /L

(i1 ...il−1 )

xl−1 + trχ

(i1 ...il−1 )

xl−1 1 = 2(Lµ)d/(Ril−1 . . . Ri1 trχ) + trχd/ 2

(8.70) (i1 ...il−1 )

fˇl−1 − µd/

(i1 ...il−1 )

h l−1 −

(i1 ...il−1 )

gl−1

We now apply L / Ril to this equation. Taking into account the fact that by Lemma 8.1 applied to the St,u - tangential vectorfield Ril and to the functions Ril−1 . . . Ri1 trχ,

(i1 ...il−1 )

fˇl−1 ,

(i1 ...il−1 )

h l−1 ,

we have: L / Ril d/(Ril−1 . . . Ri1 trχ) = d/(Ril . . . Ri1 trχ) L / Ril d/(

(i1 ...il−1 )

fˇl−1 ) = d/(

(i1 ...il )

fˇl )

L / Ril d/(

(i1 ...il−1 )

h l−1 ) = d/(

(i1 ...il )

hl )

(see definitions (8.67), (8.68)), we obtain: L / Ril L /L

(i1 ...il−1 )

xl−1 + trχL / Ril

(i1 ...il−1 )

(i1 ...il−1 )

xl−1

= 2(Lµ)d/(Ril . . . Ri1 trχ) + 2(Ril Lµ)d/(Ril−1 . . . Ri1 trχ) 1 1 + trχd/ (i1 ...il ) fˇl + (Ril trχ)d/ (i1 ...il−1 ) fˇl−1 2 2 − µd/ (i1 ...il ) h l − (Ril µ)d/ (i1 ...il−1 ) h l−1 −L / Ril

(i1 ...il−1 )

gl−1

(8.71)

Next, we apply Lemma 8.3 setting Y = Ril , ξ = /L L / Ril L

xl−1 + (Ril trχ)

(i1 ...il−1 )

xl−1 = L /L L / Ril

(i1 ...il−1 )

(i1 ...il−1 ) x

l−1 ,

xl−1 − L / (Ril )

to express: (i1 ...il−1 )

Z

xl−1

Now, definition (8.60) with l replaced by l − 1 reads: (i1 ...il−1 )

xl−1 = µd/(Ril−1 . . . Ri1 trχ) + d/(Ril−1 . . . Ri1 fˇ)

(8.72)

Chapter 8. The Propagation Equation for d/trχ

217

Applying L / Ril to this we obtain, in view of definition (8.60) and Lemma 8.1, L / Ril

(i1 ...il−1 )

xl−1 =

(i1 ...il )

xl + (Ril µ)d/(Ril−1 . . . Ri1 trχ)

(8.73)

Applying L / L to (8.73) and expressing: L Ril µ = Ril Lµ +

(Ril )

Zµ

using the fact that according to Lemma 8.2, [L, Ril ] =

(Ril )

Z

(8.74)

xl−1 = L / L (i1 ...il ) xl + (Ril µ)L / L d/(Ril−1 . . . Ri1 trχ)   + Ril Lµ + (Ril ) Z µ d/(Ril−1 . . . Ri1 trχ)

(8.75)

yields: / Ril L /L L

(i1 ...il−1 )

We then substitute (8.75) in (8.72) and the result in (8.71). Using (8.73) to re-write the second term on the left-hand side in (8.71), a propagation equation for (i1 ...il ) xl of the form given by the proposition results, with (i1 ...il ) gl expressed in terms of (i1 ...il−1 ) gl−1 by the recursion formula: (i1 ...il )

gl = L / Ril

(i1 ...il−1 )

gl−1 − L /

(Ri ) l Z

(i1 ...il−1 )

xl−1 +

(i1 ...il )

yl

(8.76)

The proposition then follows by applying Proposition 8.2 to this recursion, with the space of St,u 1-forms in the role of the space X, the operators L / Ril in the role of the operators (i ...i ) (i ...i ) ...i ) (i l l 1 1 1 l−1 An , and gl , yl − L / (Ril ) xl−1 , in the role of x n , yn , respectively. Z

We remark here that the St,u 1-form sition 8.3 reduces for j = 1 to

(i1 ...i j −1 ) a

j −1 defined in the statement of Propo-

a0 = Ld/trχ + trχd/trχ + d/h 0

(8.77)

Applying d/ to equation (8.55) we obtain, in view of Lemma 8.1 and the fact that |χ|2 =

1 (trχ)2 + h 0 , 2

simply: a0 = d/ f 0

(8.78)

/ Ri1 to (8.77) to obtain, in view of (8.78), the following For j ≥ 2, we apply L / Ri j −1 . . . L expression for

(i1 ...i j −1 ) a

j −1 :

(i1 ...i j −1 )

a j −1 = d/(Ri j −1 . . . Ri1 f0 ) +

(i1 ...i j −1 )

b j −1

(8.79)

Chapter 8. The Propagation Equation for d/trχ

218

where: (i1 ...i j −1 )

b j −1 = [L /L , L / Ri j −1 . . . L / Ri1 ]d/trχ +

j −1 

j −1 

m=1 k1 <···
(8.80)

(Rikm . . . Rik1 trχ)(d/ Ri j −1

ikm ...ik1 

...

Ri1 trχ)

Here, and in the following, the symbol   signifies that the enclosed indices are missing. We can obtain an explicit expression for the first term on the right-hand side of (8.80) by applying the following lemma, proved inductively, using Lemma 8.3. Lemma 8.4 For any positive integer l and any St,u 1-form ξ we have: [L /L , L / Ril . . . L / Ri1 ]ξ =

l−1 

L / Ril . . . L / Ril−k+1 L / (Ril−k ) L / Ril−k−1 . . . L / R i1 ξ Z

k=0

Applying this lemma to the St,u 1-form d/trχ, yields the following explicit expression for the first term on the right-hand side of (8.80): / Ri j −1 [L /L , L

j −2   ...L / Ri1 ]d/trχ = d/ Ri j −1 . . . Ri j −m

(Ri j −m−1 )

Z Ri j −m−2 . . . Ri1 trχ



m=0

(8.81) Let us investigate the order of the various terms in the propagation equation of Proposition 8.3. We agree to the convention that the order of the ψα : α = 0, 1, 2, 3 is 0, so that the order of σ is also 0. The metric h and the induced metrics h and h/ are also of order 0, as are the components L µ : µ = 0, 1, 2, 3 and Tˆ i : i = 1, 2, 3 of the vectorfields L and Tˆ , as well as ψ L , ψTˆ and ψ. The functions α, µ and therefore also κ are of order 0 as well. The ω /, ω / L , ω L L and ω L Tˆ are of order 1, as are the κ −1 ζ  and k/ . The χ, the focus of the present chapter, is of order 1 as well. It follows that (i1 ...il ) xl is of order l + 2. This should be the order of the principal terms in the propagation equation of Proposition 8.3. We are to set l = n − 1 in obtaining estimates for the nth order angular derivatives of χ in terms of the n + 1st order derivatives of the ψα , as announced at the beginning of this chapter, both entities of order n + 1 = l + 2. We shall distinguish the principal terms in the propagation equation of Proposition 8.3 from the lower order terms, and among the principal terms, the principal acoustical terms, namely those involving the l + 1st order angular derivatives of χ, rather than the l + 2 order derivatives of the ψα , which are in the role of given quantities in the present chapter. According to the above discussion, the function n defined by (8.35) is of order 1, its principal terms being 1st derivatives of σ . The function fˇ, defined through (8.43) and (8.15), then has the same properties. It follows that d/ (i1 ...il ) fˇl , which occurs on the righthand side of the propagation equation of Proposition 8.3 is of principal order l + 2 and its principal terms are l + 2 order derivatives of σ , angular derivatives of 1st derivatives. Also note that the function fˇ is linear in the function µ, hence d/ (i1 ...il ) fˇl contains l + 1st order angular derivatives of µ.

Chapter 8. The Propagation Equation for d/trχ

219

Next we shall investigate the last term in the propagation equation of Proposition 8.3, namely the term (i1 ...il ) gl . We begin this investigation with the function gˇ defined (1)

(i)

(1)

by (8.44) through the functions g and g : i = 2, 3, 4. The function g is defined by (1)

(8.41) through the function g , itself defined by (8.28). In view of the fact that by (8.32) and (8.36) a and ν L are of order 1, as they involve the 1st derivatives of the ψα (or of σ , (1)

which is the same), g is of order 1, its principal terms being 1st derivatives of the ψα . Now, Lµ, which appears in (8.41), is given by (8.52) with m and e given by (3.97) and (3.98). The latter show that m is of order 1, involving the 1st derivatives of σ , and e is ˜ given through e by also of order 1, as it involves the 1st derivatives of the ψα . Then e, (1)

(8.38), is also of order 1. It then follows from (8.41) that g has the same properties. The (2)

same is true of g , given by (8.29). In view of the fact that the function b, given by (8.33) (3)

is also of order 1, involving the 1st derivatives of the ψα , the same is true of g , given by (4)

(N)

(8.30). Finally, according to (8.21) the function g is tr α , obtained by taking the trace of (4.85), and of (4.94)–(4.96). This gives: (N)

( A)

tr α = −F α −

1 (B) 1 (C) α − F1 α 2 2

(8.82)

and: ( A)

/ L |2 − trω tr α = |ω / ωL L (B)

tr α = |ψ L (d/σ )− ψ(Lσ )|2 (C)

tr α = 2(d/σ ) · [ψ L ω / L − ψ ω L L ] − 2(Lσ )[ψ L trω /− ψ · ω /L ] (N)

(8.83) (8.84) (8.85)

(4)

It follows that tr α , that is g , is likewise of order 1 and involves the 1st derivatives of the ψα (or of σ , which is the same). We conclude the function gˇ as a whole has these properties. It follows that d/gˇ is of order 2, its principal terms being angular derivatives of 1st derivatives of the ψα . Also note that the function gˇ is linear in the function µ, hence d/gˇ contains the 1st angular derivatives of µ. Next we investigate the function f 0 defined by (8.56). Here the principal term is the ( P)

term tr α , given by (8.57). We see that this term is of order 2, its principal terms being 2nd derivatives of σ . Also note that the function f 0 does not involve the function µ. The St,u 1-form g0 is expressed in terms of d/g, ˇ f 0 , and d/e, ˇ by (8.58). The last is also of order 2, involving the angular derivatives of the 1st derivatives of σ or of the ψα . Thus g0 is of order 2, its principal terms being 2nd derivatives of the ψα , actually angular derivatives of 1st derivatives of the ψα . It follows that L / Ril . . . L / R i1 g 0

Chapter 8. The Propagation Equation for d/trχ

220

the first term in the expression for (i1 ...il ) gl given in the statement of Proposition 8.3, is of principal order l + 2 and its principal terms are l + 2 order derivatives of the ψα , angular derivatives of 1st derivatives. Also note that the 1-form g0 involves the 1st angular derivatives of µ, hence the term in question contains l +1st order angular derivatives of µ. We now consider the third (last) term in the expression for (i1 ...il ) gl , namely the sum l−1  L / Ril . . . L / Ril−k+1 (i1 ...il−k ) yl−k (8.86) k=0

Now

(i1 ...i j ) y

j

is also given in the statement of Proposition 8.3 as the sum of (i1 ...i j −1 )

a j −1 ,

d/(Ri j −1 . . . Ri1 trχ),

d/

(i1 ...i j −1 )

fˇj −1 ,

with coefficients Ri j µ, µRi j trχ −Ri j Lµ+

(Ri j )

Z µ = µRi j (trχ −e)−Ri j m+

(Ri j )

Z µ−e Ri j µ,

1 (Ri j trχ) 2

of order 1, 2, 2, respectively. Note that the first two of these coefficients involve the 1st angular derivatives of µ. The order of d/(Ri j −1 . . . Ri1 trχ) and d/ (i1 ...i j −1 ) fˇj −1 is j + 1. On the other hand, (i1 ...i j −1 ) a j −1 is given by (8.79). By the above discussion of f 0 , the first term on the right in (8.79) is of order j +2, its principal terms being angular derivatives of 2nd derivatives of σ . This is the principal term in (i1 ...i j −1 ) a j −1 . The second term on the right, (i1 ...i j −1 ) b j −1 , given by (8.80) is of lower order, namely j + 1. In fact only the first term in (8.80), given by (8.81), is of order j + 1 while the second term, the double sum, is of order j . Note also that (i1 ...i j −1 ) a j −1 does not involve the function µ. It follows from the above that it is the first term in the expression for (i1 ...i j ) y j given in the statement of Proposition 8.3 which is the principal term. We conclude that (i1 ...i j ) y j is of order j + 2 and its principal terms are j + 2 order derivatives of σ , angular derivatives of 2nd derivatives. Moreover, (i1 ...i j ) y j contains the 1st angular derivatives of µ. It follows that the sum (8.86) is of principal order l + 2 and its principal terms are l + 2 order derivatives of σ , angular derivatives of 2nd derivatives. Moreover, the sum in question contains lth order angular derivatives of µ. To complete the investigation of (i1 ...il ) gl , the second term in the expression given in the statement of Proposition 8.3 remains to be considered. This is the sum: l−1 

L / Ril . . . L / Ril−k+1 L /

) (Ri l−k Z

(i1 ...il−k−1 )

xl−k−1

(8.87)

xl−k−1

(8.88)

k=0

which we can write as: l−1 

L / (Ril−k ) L / Ril . . . L / Ril−k+1 Z

k=0

(i1 ...il−k−1 )

Chapter 8. The Propagation Equation for d/trχ

221

minus the commutator term: l−1  [L / (Ril−k ) , L / Ril . . . L / Ril−k+1 ]

(i1 ...il−k−1 )

Z

xl−k−1

(8.89)

k=1

We shall show that this commutator term is of lower order. In fact we can apply the following lemma: Lemma 8.5 For any St,u -tangential vectorfield X and any St,u 1-form ξ and positive integer l we have: [L /X , L / Ril . . . L / Ri1 ]ξ = − (ik1 ...ik j )

Here

l 

ik ...ik  j 1

l 

j =1 k1 <···
L / (ik1 ...ik j )

ξl− j

Y =L / R ik . . . L / R ik X j

(i1 ...im ) ξ m

Y

Y is the St,u -tangential vectorfield: (ik1 ...ik j )

and

(i1 . . . il )

1

the St,u 1-form: (i1 ...im )

ξm = L / R im . . . L / R i1 ξ

The proof is by induction, starting from the case l = 1, which is a standard result in differential geometry, in view of the fact that everything restricts to a given surface St,u . Applying Lemma 8.5 we see that the commutator term (8.89) is minus: l−1  k 

l 

L /

(im 1 ...im j ;il−k )

k=1 j =1 m 1 <···<m j =l−k+1

where

(im 1 ...im j ;il−k )

Z

L / Ril

im j ...im 1 

...

L / Ril−k+1

(i1 ...il−k−1 )

xl−k−1 (8.90)

Z is the St,u -tangential vectorfield: (im 1 ...im j ;il−k )

Z =L / R im . . . L / R im j

(Ril−k ) 1

Z

(8.91)

/ L and the St,u 2-covariant symmetric tenNow, the expressions for the St,u 1-form (Ri ) π sorfield (Ri ) π / obtained in Chapter 6 show these to be of order 1 and their principal acoustical terms involve χ. It follows that the vectorfield (Ri ) Z is also of order 1 and its prin(i ...i ;i ) cipal acoustical term involves χ. Hence the vectorfield m 1 m j l−k Z is of order j + 1 and its principal acoustical term is a j th order angular derivative of χ. In (8.90) we have, on St,u , the Lie derivative with respect to this vectorfield of the 1-form: L / Ril

im j ...im 1 

...

L / Ril−k+1

(i1 ...il−k−1 )

xl−k−1

(8.92)

Chapter 8. The Propagation Equation for d/trχ

222

This 1-form is of order l − j + 1 as it is an angular derivative of order k − j of xl−k−1 , which is itself of order l − k + 1. Now, as will be evident from the analysis which follows, if X is an St,u -tangential vectorfield of order m and ξ an St,u 1-form of order n, then the order of L / X ξ is max{m, n} + 1. Thus the order of the terms in the double sum (8.90) is max{ j + 1, l − j + 1} + 1 and the two extreme terms corresponding to j = 1 and j = l − 1 are of highest order, namely l + 1. This is one order less than principal for the propagation equation of Proposition 8.3, so the commutator term (8.89) is indeed a lower order term. We now return to the principal term (8.88). In view of the definition (8.60), taking into account the fact that, for any function φ, L / Ril . . . L / Ri1 d/φ = d/(Ril . . . Ri1 φ), we obtain: L / Ril . . . L / Ril−k+1 =

i



(i1 . l−k . . il )

(i1 ...il−k−1 )

xl−1 +

xl−k−1

k 

(8.93)

l 

j =1 m 1 <···<m j =l−k+1

(Rim j . . . Rim 1 µ)d/(Ril

im j ...im 1 il−k 

...

Ri1 trχ)

Here, the first term is the principal term and is of order l + 1, while the double sum is of order l. The contribution of the first term in (8.93) to (8.88) is then the principal part of (8.87). This contribution is: l−1 

L / (Ril−k )

i



. . il ) (i1 . l−k Z

xl−1

(8.94)

k=0

According to the above discussion, each term in this sum is of the principal order l + 2. We conclude that the second term in the expression for (i1 ...il ) gl given in the statement of Proposition 8.3 contains the principal acoustical terms and these are all the terms of the sum (8.94). Also, remark that the term in question does not involve the other acoustical entity, namely the function µ. We shall estimate each term in the sum (8.94) pointwise in terms of max | j

i



(i1 . l−k . . il j )

xl |

We shall accomplish this with the aid of the following lemma. Recall the hypothesis H0 of Chapter 7. Lemma 8.6 Let φ and ρ be arbitrary functions and µ a non-negative function. Let us denote by ξ the St,u 1-form: ξ = µd/φ + d/ρ

Chapter 8. The Propagation Equation for d/trχ

223

Let also: η j = µd/(R j φ) + d/(R j ρ)

: j = 1, 2, 3

Then for any St,u -tangential vectorfield X we have the pointwise estimate:    −1 |X| max |η j | + 2 max |d/(R j ρ)| + (1 + t)|d/µ||d/φ| |L / X ξ | ≤ C(1 + t) j j       / R j X (µ|d/φ| + |d/ρ|) + max L j

Proof. We have: L / X ξ = µd/(Xφ) + (Xµ)d/φ + d/(Xρ)

(8.95)

|L / X ξ | ≤ µ|d/(Xφ)| + |X||d/µ||d/φ| + |d/(Xρ)|

(8.96)

hence: Now, by hypothesis H0, |d/(Xφ)| ≤ C(1 + t)−1 max |R j (Xφ)|

(8.97)

R j (Xφ) = X (R j φ) + [R j , X]φ   / R j X · d/φ = X · d/(R j φ) + L

(8.98)

      / R j X |d/φ| max |R j (Xφ)| ≤ |X| max |d/(R j φ)| + max L

(8.99)

Substituting (8.99) in (8.97) yields:      −1   |X| max |d/(R j φ)| + max L |d/(Xφ)| ≤ C(1 + t) / R j X |d/φ|

(8.100)

j

and we have:

hence:

j

j

j

j

j

By the definition of the 1-forms η j , µ max |d/(R j φ)| = max |µd/(R j φ)| j

(8.101)

j

= max |η j − d/(R j ρ)| ≤ max |η j | + max |d/(R j ρ)| j

j

j

Moreover, the estimate (8.100) holds with the function ρ in place of the function φ, that is:      −1   / R j X |d/ρ| |X| max |d/(R j ρ)| + max L |d/(Xρ)| ≤ C(1 + t) (8.102) j

j

In view of (8.100)–(8.102) the lemma follows through (8.96). . .  Ri1 trχ and ρ to be the We now apply Lemma 8.6 taking φ to be the function Ril i.l−k il−k  il−k  . .  Ri1 fˇ = (i1 . . . il ) fˇl−1 . Then ξ is the St,u 1-form (i1 . . . il ) xl−1 and function Ril i.l−k

Chapter 8. The Propagation Equation for d/trχ

224

η j the St,u 1-form (i1 (Ril−k ) Z . Noting that

il−k  . . .

il j ) x

l . Moreover, we take

X to be the St,u -tangential vectorfield

µ|d/φ| + |d/ρ| ≤ |ξ | + 2|d/ρ|, we obtain:   L  / (Ril−k )

i



(i1 . l−k . . il ) Z

   xl−1  ≤ C(1 + t)−1 |

(Ril−k )

 Z | max |

i



(i1 . l−k . . il j )

j

xl |

 il−k  + 2 max |d/( (i1 . . . il j ) fˇl )| + |d/µ||d/(Ril i. l−k . . Ri1 trχ)| (8.103) j      il−k  il−k    | (i1 . . . il ) xl−1 | + 2|d/( (i1 . . . il ) fˇl−1 )| / R j (Ril−k ) Z  + max L j

Returning to the propagation equation for (i1 ...il ) xl given by Proposition 8.3, the term µd/( (i1 ...il ) h l ) remains to be considered. Since (i1 ...il ) h l given by (8.68) is an lth order angular derivative of |χ| ˆ 2 , this term is certainly a principal acoustical term. Moreover, it involves l + 1st order angular derivatives of χ, ˆ not trχ, which is what the propagation equation for (i1 ...il ) xl allows us to control. The term in question originates from the propagation equation (8.50) for x 0 = µd/trχ + d/ fˇ, from the term / χˆ µd/(|χ| ˆ 2 ) = 2µχˆ · D which appears on the right-hand side. Now, the propagation equation (8.50) must be considered in conjunction with the Codazzi equation (3.62), in the form (4.107), in order to control D / χ. ˆ Decomposing χ into its trace-free part χˆ and its trace (1/2)h/trχ, equation (4.107) takes the form: 1 (8.104) div / χˆ = d/trχ + i 2 where i is the St,u 1-form with components: i A = −α −1 (χ AB (κ −1 ζ B ) − trχ(κ −1 ζ A )) 1 dH B − ψ ( ψ A χ BC − ψ B χ AC )(d/C σ ) 2 dσ ( P)∗

+ β

(N)∗

A

+ β

A

(8.105)

which is regular as µ → 0. Equation (8.104) constitutes an elliptic system for χˆ on each surface St,u , section of Cu , which allows us to estimate on St,u χˆ and D / χˆ in terms of d/trχ and the derivatives of the ψµ of up to the 2nd order. The propagation equation (8.50) is an ordinary differential system for d/trχ along each generator of the characteristic hypersurface Cu , which allows us to estimate along each such generator d/trχ, in terms of / χ, ˆ and the derivatives of the ψµ of up to the 2nd order. d/(|χ|2 ), that is in terms of χˆ and D Thus, considering on Cu the propagation equation (8.50) in conjunction with the Codazzi equation (8.104) we shall be able to estimate χ and its 1st order angular derivatives in

Chapter 8. The Propagation Equation for d/trχ

225

terms of the ψµ and their derivatives of up to the 2nd order, avoiding in this way any loss of differentiability. Similar remarks apply to the higher order angular derivatives of χ. The propagation equation for (i1 ...il ) xl , defined by (8.60), is an ordinary differential system for d/(Ril . . . Ri1 trχ) along each generator of Cu . On the right-hand side of this propagation equation, given by Proposition 8.3, we have the aforementioned term µd/ (i1 ...il ) h l , whose principal part is: /ˆ Ri χ) ˆ 2µχˆ D /(L /ˆ Ri . . . L l

1

(see definition (8.68)). Here, if Y is an arbitrary St,u - tangential vectorfield we denote by L /ˆ Y the operator acting on trace-free 2-covariant symmetric tensorfields θ on St,u and ˆ/Y θ is the trace-free part of L /ˆ · /Y θ (we have trL /Y θ = (Y ) π defined by the statement that L (i ...i ) l 1 θ ). The propagation equation for xl must be considered in conjunction with an ˆ/ R χˆ on each St,u section, which allows us to estimate on St,u elliptic system for L /ˆ Ri . . . L i l 1 ˆ/ R . . . L ˆ/ R χ) D /(L ˆ in terms of d/(Ril . . . Ri1 trχ) and the derivatives of the ψµ of order up il

i1

to l + 2. The required elliptic system shall be derived from the Codazzi equation (8.104) with the help of the following lemma.

Lemma 8.7 Let (M, g) be a 2-dimensional Riemannian manifold, let X be an arbitrary vectorfield on M and let θ be a trace-free symmetric 2-covariant tensorfield on (M, g) satisfying the equation: divg θ = f for some 1-form f on M. Then Lˆ X θ , the trace-free part of L X θ , satisfies the equation: divg Lˆ X θ = where

(X )

f˙

f˙ is the 1-form given, in an arbitrary local frame, by: (X )

Here,

(X )

(X ) π

1 f˙a = (L X f )a + tr (X ) π fa 2 1 (X ) bc + πˆ (∇b θac + ∇c θab − ∇a θbc ) + (divg 2

(X )

π) ˆ b θab

= L X g.

Proof. Let φt be the local 1-parameter group of diffeomorphisms generated by X and let φt ∗ be the corresponding pullback. We then have: divφt∗ g (φt ∗ θ ) = φt ∗ (divg θ ) = φt ∗ f

(8.106)

Now, in an arbitrary system of local coordinates: (divg θ )a = (g −1 )bc ∇c θab   g g −1 bc ∂θab d d = (g ) − !ca θdb − !cb θad ∂xc

(8.107)

Chapter 8. The Propagation Equation for d/trχ

226 g

c are the connection coefficients of the metric g in the given coordinate system. where !ab Similarly, in the same coordinate system,   φt∗ g ∂(φt ∗ θ )ab φt∗dg −1 bc d (divφt∗ g (φt ∗ θ ))a = ((φt ∗ g) ) − !ca (φt ∗ θ )db − !cb (φt ∗ θ )ad ∂xc

(8.108) φt∗ g

c are the connection coefficients of φ g in the given coordinate system. We where !ab t∗ have:   g 1 −1 cd ∂gbd ∂gad ∂gab c !ab = (g ) + − (8.109) 2 ∂xa ∂xb ∂xd

and: φt∗ g c !ab =

1 ((φt ∗ g)−1 )cd 2

Since, by definition,





∂(φt ∗ g)ad ∂(φt ∗ g)ab ∂(φt ∗ g)bd + − ∂xa ∂xb ∂xd

d φt ∗ g dt

 t =0

= LX g =

(X )

 (8.110)

π

(8.111)

differentiating (8.110) with respect to t at t = 0 we obtain, in view of (8.109),   d φt∗c g 1 ! = (g −1 )cd (∇a (X ) πbd + ∇b (X ) πad − ∇d (X ) πab ) dt ab 2

(8.112)

t =0

Differentiating then (8.108) with respect to t at t = 0 and using (8.111), (8.112), and the fact that, by definition,   d φt ∗ θ = LX θ (8.113) dt t =0 we obtain:   d (divφt∗ g (φt ∗ θ ))a = (divg (L X θ ))a − (X ) π bc ∇c θab dt t =0 1 (∇a (X ) π bc )θbc + (2∇c (X ) πbc − ∇ b tr − 2 In reference to (8.106), noting that we also have, by definition,   d φt ∗ f = LX f dt t =0

(8.114) (X )

π)θab



(8.115)

we then conclude, equating the right-hand sides of (8.114) and (8.115), that L X θ satisfies the equation: divg (L X θ ) = (X ) f  (8.116)

Chapter 8. The Propagation Equation for d/trχ

227

where: (X )  fa

= (L X f )a + (X ) π bc ∇c θab 1 + (∇a (X ) π bc )θbc + (2∇c 2

(X )

πbc − ∇ b tr

(X )

π)θab

 (8.117)

In fact the above hold for an arbitrary n-dimensional manifold M, and irrespective of the trace-free nature of θ . In the present case where M is 2-dimensional, decomposing (X ) π into its trace-free part and its trace, (X )

π=

(X )

1 πˆ + gtr 2

(X )

π

(8.118)

we may write (8.117) as: (X )  fa

1 = (L X f )a + tr (X ) π f a + (X ) πˆ bc ∇c θab 2 1 (X ) bc + (∇a πˆ )θbc + (∇c (X ) πˆ bc )θab 2

(8.119)

Now, tr(L X θ ) = (g −1 )ab (L X θ )ab = −(L X g −1 )ab θab =

(X ) ab

π θab =

(X )

πˆ · θ

(8.120)

Therefore, Lˆ X θ , the trace-free part of L X θ , is given by: 1 Lˆ X θ = L X θ − g( 2

(X )

πˆ · θ )

(8.121)

and, noting that for any function ψ we have: divg (gψ) = dψ we obtain:

1 divg (Lˆ X θ ) = divg (L X θ ) − d( 2

(X )

πˆ · θ )

(8.122)

Hence, defining: 1 f − d( (X ) πˆ · θ ) (8.123) 2 we conclude that divg Lˆ X θ = (X ) f˙ and (X ) f˙ is given by the formula in the statement of the lemma. (X )

f˙ =

(X ) 

Proposition 8.4 For each non-negative integer l and each multi-index (i 1 , . . . , i l ), the trace-free symmetric 2-covariant tensorfield (i1 ...il )

ˆ/ R . . . L ˆ/ R χˆ χˆl = L i i l

1

on St,u satisfies the elliptic system: div / (

(i1 ...il )

χˆl ) =

1 d/(Ril . . . Ri1 trχ) + 2

(i1 ...il )

il

Chapter 8. The Propagation Equation for d/trχ

228 (i1 ...il ) i

where the St,u 1-form (i1 ...il )

l

is given by:

    1 1 il = L / Ril + tr (Ril ) π / ... L / Ri1 + tr (Ri1 ) π / i 2 2    l−1  1 1 L / Ril + tr (Ril ) π + / ... L / Ril−k+1 + tr 2 2

(Ril−k+1 )

 π /

(i1 ...il−k )

ql−k

k=0

Here i is the St,u 1-form given by (8.105) and for each j = 1, . . . , l, 1-form given by: (i1 ...i j )

q j,A =

(i1 ...i j ) q

j

is the St,u

1 (Ri j ) tr π /d/(Ri j −1 . . . Ri1 trχ) 4 1 (Ri j ) ˆ BC D / B ( (i1 ...i j −1 ) χˆ j −1) AC + D + / C ( (i1 ...i j −1 ) χˆ j −1 ) AB π / 2  −D / A ( (i1 ...i j −1 ) χˆ j −1 ) BC (Ri j ) ˆ B

+ (div /

π /) (

(i1 ...i j −1 )

χˆ j −1 ) AB

Proof. The elliptic system of the proposition reduces for l = 0, since i 0 = i , to the Codazzi equation (8.104). Thus, by induction, assuming that the elliptic system of the proposition holds with l replaced by l − 1, that is, assuming that: div / (

(i1 ...il−1 )

χˆl−1 ) =

1 d/(Ril−1 . . . Ri1 trχ) + 2

(i1 ...il−1 )

i l−1

holds for some St,u 1-form (i1 ...il−1 ) i l−1 , what we must show is that an elliptic system of the form given by the proposition holds true for l, where (i1 ...il )i l is an St,u 1-form related to (i1 ...il−1 ) i l−1 by a certain recursion relation. To accomplish this we apply Lemma 8.7, taking the manifold (M, g) to be (St,u , h/), the trace-free symmetric 2-covariant tensorfield θ to be (i1 ...il−1 ) χˆl−1 , the 1-form f to be 1 d/(Ril−1 . . . Ri1 trχ) + 2

(i1 ...il−1 )

i l−1

and the vectorfield X to be the rotation field Ril . Taking into account the fact that: L / Ril d/(Ril−1 . . . Ri1 trχ) = d/(Ril . . . Ri1 trχ) we then conclude that L /ˆ Ri

(i1 ...il−1 ) χˆ (i1 ...il ) χˆ satisfies the elliptic system of the l−1 = l 1-form (i1 ...il )i l related to the 1-form (i1 ...il−1 ) i l−1 by the recursion l

proposition with the relation: (i1 ...il )

il = L / Ril

(i1 ...il−1 )

1 i l−1 + tr 2

(Ril )

π /

(i1 ...il−1 )

i l−1 +

(i1 ...il )

ql

(8.124)

Chapter 8. The Propagation Equation for d/trχ

229

The proposition then follows by applying Proposition 8.2 to this recursion, with the space of St,u 1-forms in the role of the space X, the operators 1 L / Ril + tr 2

(Ril )

π /

in the role of the operators An , the second term being a multiplication operator, and (i1 ...il ) i , (i1 ...il ) q , in the role of x , y , respectively. l l n n Let us investigate the order of the terms in the expression for

(i1 ...il ) i

( P)∗

l

of Proposition

8.4. We begin with i , given by (8.105). The principal term is β , given by (4.75). We see that this is of order 2, its principal terms being angular derivatives of 1st derivatives of σ . It follows that the part / R i1 i L / Ril . . . L of the first term in the expression for (i1 ...il )i l is of principal order l + 2, and its principal terms are l + 1st order angular derivatives of 1st derivatives of σ . This is the principal part / are of order 1, and angular derivatives of (i1 ...il ) i l . For, as we noted previously, the (Ri ) π of order at most l − 1 of them are involved in the first term of the expression for (i1 ...il )i l . Thus, the remainder of the first term is of order l + 1. Moreover, the second term is of order l + 1, since (i1 ...i j ) q j is of order j + 1. Thus, of each term in the sum over k, the part / Ril−k+1 (i1 ...il−k ) ql−k L / Ril . . . L is of order l + 1, while the remainder is of order l. Also, remark that (i1 ...il ) i l does not involve the other acoustical entity, namely the function µ. As we remarked earlier, the elliptic system of Proposition 8.4 is to be considered in conjunction with the propagation equation of Proposition 8.3. The right-hand side of the /ˆ Ri χˆ is thus to be expressed in terms of (i1 ...il ) xl . Now, from elliptic system for L /ˆ Ri . . . L l 1 the definition (8.60) we have: µd/(Ril . . . Ri1 trχ) =

(i1 ...il )

xl − d/

(i1 ...il )

fˇl

We shall thus only be able to control the product of µ with the right-hand side of the elliptic system for L /ˆ Ri . . . L /ˆ Ri χˆ in terms of (i1 ...il ) xl . This shall lead to µ- weighted L 2 l 1 estimates on St,u . We shall develop these estimates in the setting of Lemma 8.7. Lemma 8.8 Let (M, g) be a compact 2-dimensional Riemannian manifold, and let θ be a trace-free symmetric 2-covariant tensorfield on (M, g) satisfying the equation: divg θ = f for some 1-form f on M. Let also µ be an arbitrary non-negative function on M. Then the following integral estimate holds on M:  2 1 2 2 2 2 |∇θ | + 2K |θ | dµg ≤ 3 µ µ | f | dµg + 3 |dµ|2 |θ |2 dµg 2 M M M where K is the Gauss curvature of (M, g).

Chapter 8. The Propagation Equation for d/trχ

230

Proof. Consider the 3-covariant tensorfield given in an arbitrary local frame by: ωabc = ∇a θbc − ∇b θac

(8.125)

For every vectorfield X, ωabc X c defines a 2-form on M. Since M is 2-dimensional, this 2-form must be a function φ, times the area 2-form  of (M, g). Since φ depends linearly on X, there is a 1-form e on M such that e · X = φ. Consequently, we can write: ωabc = ab ec

(8.126)

Contracting the right-hand side of (8.126) with (g −1 )ac we have: (g −1 )ac ab ec = ea ab =

∗

eb

(the Hodge dual of e). Contracting the left-hand side of (8.126) with (g −1 )ac and taking into account the fact that (g −1 )ac θac = trθ = 0 we obtain: (g −1 )ac ωabc = (divθ )b = f b We conclude that:

∗

e = f,

e=−

and (8.126) becomes: ωabc = −ab

∗

∗

f

(8.127)

fc

(8.128)

In view of the fact that ac b c = gab it follows that: 1 1 ωabc ωabc = ab 2 2

∗

f c  ab

∗

fc = |

∗

f |2 = | f |2

(8.129)

On the other hand, from the definition (8.125) we have: 1 1 ωabc ωabc = (∇a θbc − ∇b θac )(∇ a θ bc − ∇ b θ ac ) 2 2 = |∇θ |2 − ∇a θbc ∇ b θ ac

(8.130)

∇a θbc ∇ b θ ac = ∇ b (θ ac ∇a θbc ) − θ ac ∇ b ∇a θbc

(8.131)

We write: We have: ∇ b ∇a θbc − ∇a ∇ b θbc = Rb dba θdc + Rc dba θbd = S da θdc + Rc dba θbd

(8.132)

where Rabcd is the curvature tensor and Sab = (g −1 )cd Rcadb the Ricci tensor of (M, g). Since M is 2-dimensional, we have: Rabcd = K (gac gbd − gad gbc ),

Sab = K gab

Chapter 8. The Propagation Equation for d/trχ

231

Hence, the right-hand side of (8.132) is: K δad θdc + K (δcb δad − gca (g −1 )db )θbd = 2K θac in view of the fact that trθ = 0. Thus, (8.132) reduces to: ∇ b ∇a θbc − ∇a ∇ b θbc = 2K θac

(8.133)

Substituting in (8.131) we obtain: ∇a θbc ∇ b θ ac = ∇ b (θ ac ∇a θbc ) − θ ac ∇a ∇ b θbc − 2K |θ |2

(8.134)

In reference to the second term on the right, we write: θ ac ∇a ∇ b θbc = ∇a (θ ac ∇ b θbc ) − (∇a θ ac )(∇ b θbc ) = ∇a (θ ac ∇ b θbc ) − | f |2 Substituting in (8.134) then yields: ∇a θbc ∇ b θ ac = divg J − 2K |θ |2 + | f |2

(8.135)

where J is the vectorfield given by: J a = θ bc ∇b θ ac − θ ac ∇b θ bc = θ bc ∇b θ ac − θ ac f c

(8.136)

Substituting (8.135) in (8.130) we obtain: 1 ωabc ωabc = |∇θ |2 + 2K |θ |2 − | f |2 − divg J 2

(8.137)

Comparing (8.137) with (8.129) we conclude that: |∇θ |2 + 2K |θ |2 = 2| f |2 + divg J We now multiply this equation by µ2 and integrate over M to obtain:

 µ2 |∇θ |2 + 2K |θ |2 dµg M =2 µ2 | f |2 dµg + µ2 divg J dµg M

(8.138)

(8.139)

M

Writing µ2 divg J = divg (µ2 J ) − 2µ(J · dµ) and integrating over M, the integral of the first term on the right vanishes since M is a compact manifold, and we obtain: µ2 divg J dµg = −2 µ(J · dµ)dµg (8.140) M

M

Chapter 8. The Propagation Equation for d/trχ

232

In view of (8.136) we can estimate: |J | ≤ |θ |(|∇θ | + | f |) therefore:



(8.141)



−2

µ(J · dµ)dµg ≤ 2

µ|J ||dµ|dµg M

M

≤2

|dµ||θ |(µ|∇θ | + µ| f |)dµg M

Applying the inequalities: 1 2 µ |∇θ |2 + 2|dµ|2|θ |2 2 2|dµ||θ |µ| f | ≤ µ2 | f |2 + |dµ|2 |θ |2

2|dµ||θ |µ|∇θ | ≤

we then obtain:



−2

µ(J · dµ)dµg 1 2 2 2 2 ≤ µ |∇θ | dµg + µ | f | dµg + 3 |dµ|2 |θ |2 dµg 2 M M M

(8.142)

M

Finally, we substitute for the last integral on the right in (8.139) from (8.140) and we apply the inequality (8.142). Bringing the first integral on the right in (8.142) to the left-hand side of the resulting inequality then yields the lemma. The Gauss curvature of the surfaces St,u is given by equation (4.108). Let us recall the bootstrap assumptions A, E, F, of Chapter 6. We now introduce the following additional bootstrap assumption. There is a positive constant C independent of s such that for all (t, u) ∈ [0, s] × [0, ε0],  /2 : |D / 2 σ | ≤ Cδ0 (1 + t)−3 Lemma 8.9 Under the bootstrap assumptions A, E, F, of Chapter 6, and the additional bootstrap assumption  /2, the following estimate for the Gauss curvature of the surfaces St,u holds:     K − r −2  ≤ Cδ0 (1 + t)−3 [1 + log(1 + t)] In particular, taking δ0 suitably small, we have: K ≥ C −1 (1 + t)−2 Proof. Consider first the last term on the right-hand side of equation (4.108), namely the (N)

( A) (B) (C)

term ρ . This is expressed by (4.87) in terms of ρ , ρ , ρ , which are in turn given by (4.100), (4.101), (4.102), respectively. These depend on d/σ , ψ and ω /, which, by virtue of

Chapter 8. The Propagation Equation for d/trχ

233

the bootstrap assumptions A and E satisfy the bounds (6.51), (6.60) and (6.62) of Chapter 6, that is: |d/σ | ≤ Cδ0 (1 + t)−2

(8.143)

| ψ| ≤ Cδ0 (1 + t)−1

(8.144)

−2

(8.145)

|ω /| ≤ Cδ0 (1 + t)

In view of (8.143)–(8.145), we obtain from the expressions (4.100)–(4.102) the estimates: ( A)

| ρ | ≤ Cδ02 (1 + t)−4

(8.146)

(B)

| ρ | ≤ Cδ02 (1 + t)−6

(8.147)

(C)

| ρ | ≤ Cδ02 (1 + t)−5

(8.148)

(N)

It then follows from (4.87) that ρ , the fourth term on the right in (4.108) satisfies the estimate: (N) (8.149) | ρ | ≤ Cδ02 (1 + t)−4 ( P)

The third term on the right-hand side of (4.108), ρ , is given by (4.78). As was shown in Chapter 6, the bootstrap assumptions A and E imply the estimate (6.63) for k/ , that is: (8.150) |k/ | ≤ Cδ0 (1 + t)−2 The same assumptions imply the bounds (6.51), in particular: |Lσ | ≤ Cδ0 (1 + t)−2

(8.151)

From (8.150), (8.144) and (8.151) it follows that the second term on the right in (4.78) is bounded in absolute value by Cδ04 (1 + t)−6 . By virtue of the additional bootstrap assumption  /2 and (8.144), the first term on the right in (4.78) is bounded in absolute value by Cδ03 (1 + t)−5 . We conclude that: ( P)

| ρ | ≤ Cδ03 (1 + t)−5

(8.152)

We turn to the second term on the right-hand side of (4.108). By the estimate (8.150) and bootstrap assumptions F2 and A2, this term is bounded in absolute value by: Cδ0 (1 + t)−3 Finally, we consider the first term on the right-hand side of (4.108). Assumption F2 implies that:    2  1  (trχ)2 − |χ|2 − η0  ≤ Cδ0 [1 + log(1 + t)] (8.153) 2 r  (1 + t)3

Chapter 8. The Propagation Equation for d/trχ

234

To estimate appropriately the first term on the right in (4.108) we must also consider the difference α 2 − η02 . From formula (2.61) together with (1.72), that is: α2 =

1−σH , 1 + ρH

1 − σ H = η2

(8.154)

and assumption E1 we obtain:

which implies that also:

|α 2 − η02 | ≤ Cδ0 (1 + t)−1

(8.155)

|α −2 − η0−2 | ≤ Cδ0 (1 + t)−1

(8.156)

The bounds (8.153) and (8.156) yield the following estimate for the first term on the right-hand side of (4.108):     1 −2 1  [1 + log(1 + t)] 2 2  α ≤ Cδ0 (trχ) − − |χ| (8.157) 2 r2  (1 + t)3 In view of the above results the lemma follows. We now apply Lemma 8.8 to Proposition 8.4, taking the manifold (M, g) to be (St,u , h /), the trace-free symmetric 2-covariant tensorfields θ to be (i1 ...il ) χˆl , and the 1form f to be: 1 d/(Ril . . . Ri1 trχ) + (i1 ...il ) i l 2 In view of the fact that by Lemma 8.9, taking δ0 suitably small, we have K ≥ 0, we obtain: µD /

(i1 ...il )

χˆl  L 2 (St,u ) ≤ Cµd/(Ril . . . Ri1 trχ) L 2 (St,u ) + Cµ +Cd/µ L ∞ (St,u ) 

(i1 ...il )

(i1 ...il )

i l  L 2 (St,u )

χˆl  L 2 (St,u )

(8.158)

Now, by bootstrap assumption F1, |d/µ| ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] Also, µd/(Ril . . . Ri1 trχ) =

(i1 ...il )

xl − d/

(i1 ...il )

(8.159) fˇl

(8.160)

Substituting (8.160) and (8.159) in (8.158) yields the estimate: µD /

(i1 ...il )

χˆl  L 2 (St,u ) ≤ C

(i1 ...il )

+Cd/

xl  L 2 (St,u )

(i1 ...il )

+Cδ0 (1 + t)

(8.161)

fˇl  L 2 (St,u ) + Cµ −1

(i1 ...il )

[1 + log(1 + t)]

i l  L 2 (St,u )

(i1 ...il )

χˆl  L 2 (St,u )

Let us now return to the propagation equation of Proposition 8.3. Setting: (i1 ...il )

g˜l = µd/

(i1 ...il )

hl +

(i1 ...il )

gl

(8.162)

Chapter 8. The Propagation Equation for d/trχ

the propagation equation takes the form:   L / L (i1 ...il ) xl + trχ − 2µ−1 (Lµ) (i1 ...il ) xl   1 trχ − 2µ−1 (Lµ) d/ (i1 ...il ) fˇl − = 2

235

(i1 ...il )

g˜l

(8.163)

Recall that S0,0 is the unit sphere S 2 \3 (the initial hyperplane 0 is identified with We define a diffeomorphism "t,u of S 2 onto St,u as follows. First, at t = 0, "0,u is the diffeomorphism of S 2 onto S0,u defined by the flow of the vectorfield T on 0 . Then "t,u = "t ◦ "0,u where "t is the flow of the vectorfield L on Wε∗0 . That is, "t |S0,u is the diffeomorphism of S0,u onto St,u defined by the generators of Cu . Given an arbitrary St,u 1-form ξ , we may pull it back by "t,u to S 2 and consider the 1-form ξ(t, u) = "∗t,u ξ as a 1-form on S 2 depending on the parameters t and u. If ξ = L / L ζ , where ζ is an arbitrary St,u 1-form, then ξ(t, u) = ∂ζ(t, u)/∂t. If ξ = d/φ where φ is a function, then ξ(t, u) = d/φ(t, u), where φ(t, u) = φ ◦ "t,u is the corresponding function on S 2 , which depends on the parameters t and u, and d/φ(t, u) denotes its differential on S 2 . Also, we may pull back to S 2 the induced metric h/ on St,u and consider h/(t, u) = "∗t,u h/ as a metric on S 2 depending on the parameters t and u. We remark that the foregoing correspond simply to the description in terms of acoustical coordinates (t, u, ϑ 1 , ϑ 2 ), where (ϑ 1 , ϑ 2 ) ε are arbitrary local coordinates on S 2 and on 00 the coordinate lines corresponding to constant values of (ϑ 1 , ϑ 2 ) are orthogonal (with respect to h) to the surfaces S0,u (so that  = 0 on 0 ; see Chapter 2). In accordance with the above, the propagation equation (8.163) may be viewed as an equation for the 1-form (i1 ...il ) xl (t, u) on S 2 depending on t and u, an ordinary differential equation in the parameter t:   ∂ (i1 ...il ) −1 ∂µ (i1 ...il ) xl + trχ − 2µ xl ∂t ∂t   1 ∂µ = (8.164) trχ − 2µ−1 d/ (i1 ...il ) fˇl − (i1 ...il ) g˜l 2 ∂t 3 ).

In the following we denote by ( , ) the pointwise inner product of tensorfields on S 2 , depending on t and u, with respect to the metric h/(t, u). In particular, for 1-forms ξ(t, u), ζ (t, u), we have: (ξ(t, u), ζ (t, u)) = (h/−1 ) AB (t, u)ξ A (t, u)ζ B (t, u)

(8.165)

We also denote, as in the preceding, by | | the pointwise magnitude of tensorfields on S 2 , depending on t and u, with respect to the metric h/(t, u). Thus,

|ξ(t, u)| = (ξ(t, u), ξ(t, u)) (8.166) Now by the definition of χ (see Chapter 3) we have: ∂h / AB (t, u) ∂(h/−1 ) AB (t, u) = 2χ AB (t, u), = −2χ AB (t, u); ∂t ∂t /−1 ) AC (t, u)(h/−1 ) B D (t, u)χC D (t, u) χ AB (t, u) = (h

(8.167)

Chapter 8. The Propagation Equation for d/trχ

236

It follows from (8.165) and (8.167) that for 1-forms ξ(t, u), ζ(t, u), we have: ∂ ∂ξ ∂ζ (ξ, ζ ) = ( , ζ ) + (ξ, ) − 2ξ · χ · ζ ∂t ∂t ∂t

(8.168)

Here and in the following, for the sake of simplicity of notation, the dependence of everything on the parameters t and u is understood. Moreover, we denote by ξ · χ · ζ the function: (8.169) ξ · χ · ζ = ξ A χ AB ζ B In particular, taking ζ = ξ we obtain, in view of (8.166),   1 ∂ ∂ξ ∂ (ξ, ξ ) = ,ξ − ξ · χ · ξ |ξ | |ξ | = ∂t 2 ∂t ∂t

(8.170)

Moreover, decomposing χ into its trace-free part and its trace, 1 χ AB = χˆ AB + /h AB trχ, 2

1 χ AB = χˆ AB + (h/−1 ) AB trχ, 2

we have:

1 ξ · χ · ζ = ξ · χˆ · ζ + trχ(ξ, ζ ) 2 hence (8.170) takes the form: |ξ |

(8.171)

∂ ∂ξ 1 |ξ | = ( , ξ ) − trχ|ξ |2 − ξ · χˆ · ξ ∂t ∂t 2

(8.172)

We shall apply (8.172) taking ξ = (i1 ...il ) xl . From the propagation equation (8.164) we have:   ∂ (i1 ...il ) xl (i1 ...il ) 1 , xl − trχ| (i1 ...il ) xl |2 ∂t 2     ∂µ 3 ∂µ 1 + trχ | (i1 ...il ) xl |2 + −2µ−1 + trχ ( (i1 ...il ) xl , d/ (i1 ...il ) fˇl ) = − −2µ−1 ∂t 2 ∂t 2 −(

(i1 ...il )

xl ,

(i1 ...il )

g˜l )

(8.173)

To proceed we shall make the following assumption. In Wεs0 we have: AS: −2µ−1

∂µ 1 + trχ ≥ 0 ∂t 2

That is, the coefficient of ( (i1 ...il ) xl , d/ (i1 ...il ) fˇl ) in (8.173) is non-negative. This assumption shall be proved in the sequel. Since |(

(i1 ...il )

xl , d/

(i1 ...il )

fˇl )| ≤ |

(i1 ...il )

xl ||d/

(i1 ...il )

fˇl |,

Chapter 8. The Propagation Equation for d/trχ

and also |(

(i1 ...il )

xl ,

(i1 ...il )

237

g˜l )| ≤ |

(i1 ...il )

xl ||

(i1 ...il )

g˜l |,

the assumption AS implies through (8.173) that: ∂

(i1 ...il ) x

1 l (i1 ...il ) , xl ) − trχ| (i1 ...il ) xl |2 ∂t 2     3 1 −1 ∂µ (i1 ...il ) 2 −1 ∂µ + trχ | + trχ | ≤ − −2µ xl | + −2µ ∂t 2 ∂t 2

(

+|

(i1 ...il )

xl ||

(i1 ...il )

(i1 ...il )

xl ||d/

(i1 ...il )

g˜l |

fˇl |

(8.174)

Moreover, in reference to the last term on the right in (8.172) with role of ξ , ˆ (i1 ...il ) xl |2 | (i1 ...il ) xl · χˆ · (i1 ...il ) xl | ≤ |χ||

(i1 ...il ) x

l

in the (8.175)

(i1 ...il ) x

Applying then (8.172) taking ξ = l , and substituting (8.174) and (8.175) we deduce the following ordinary differential inequality for | (i1 ...il ) xl |: ∂|

(i1 ...il ) x

l|

∂t   3 −1 ∂µ + trχ − |χ| ˆ | ≤ − −2µ ∂t 2 +|

(i1 ...il )

(i1 ...il )

  1 −1 ∂µ + trχ |d/ xl | + −2µ ∂t 2

g˜l |

(i1 ...il )

fˇl |

(8.176)

The integrating factor here is:   t    3 µ(t, u) −2 −1 ∂µ   + trχ − |χ| ˆ (t , u)dt = exp (A(t, u))3/2e−S(t,u) −2µ ∂t 2 µ(0, u) 0 (8.177) Here,   t

A(t, u) = exp

trχ(t  , u)dt 

(8.178)

0

and,



t

S(t, u) =

|χ|(t ˆ  , u)dt 

(8.179)

0

Since, by (8.167), trχ = √

1 ∂

deth/ deth/ ∂t

(8.180)

√ deth/(t, u) (8.181) A(t, u) = √ deth/(0, u) that is, A(t, u) is the ratio of the area elements of St,u and S0,u at corresponding points along the same generator of Cu . Integrating the ordinary differential inequality (8.176) using the integrating factor (8.177) yields:

we have:

|

(i1 ...il )

xl (t, u)| ≤

(i1 ...il )

Fl (t, u) +

(i1 ...il )

G l (t, u)

(8.182)

Chapter 8. The Propagation Equation for d/trχ

238

where:

Fl (t, u) = e S(t,u)(A(t, u))−3/2(µ(t, u))2 · (µ(0, u))−2 | (i1 ...il ) xl (0, u)| (8.183)   t ∂µ 1  + trχ (t  , u)|d/ (i1 ...il ) fˇl (t  , u)| + (µ(t  , u))−2 (A(t  , u))3/2 e−S(t ,u) −2µ−1 ∂t 2 0 (i1 ...il )

and: (i1 ...il )

G l (t, u) = e S(t,u)(A(t, u))−3/2(µ(t, u))2 t  (µ(t  , u))−2 (A(t  , u))3/2 e−S(t ,u) | ·

(8.184) (i1 ...il )

g˜l (t  , u)|dt 

0

We begin our estimates with the area element ratio A(t, u) and the function S(t, u). By the bootstrap assumption F2 of Chapter 6 and equation (8.178):   log A(t, u) − 

t

0

Since



t 0

 t  2η0 [1 + log(1 + t  )]   ≤ Cδ dt dt ≤ C  δ0 0 1 − u + η0 t   (1 + t  )2 0   1 − u + η0 t 2η0  dt = 2 log , 1 − u + η0 t  1−u

it follows that: e

−Cδ0



1 − u + η0 t 1−u

2

 ≤ A(t, u) ≤ e

Cδ0

1 − u + η0 t 1−u

2 (8.185)

Also, by assumption F2 of Chapter 6 and equation (8.179) S(t, u) ≤ Cδ0 the integral



∞ 0

(8.186)

[1 + log(1 + t  )]  dt (1 + t  )2

being convergent. In view of the bounds (8.185) and (8.186) we obtain from (8.183) and (8.184): (i1 ...il )

Fl (t, u) ≤ e

Cδ0

+

(1 − u + η0 t)

(i1 ...il )

1

−3



Ml (t, u) +

(i1 ...il )

(i1 ...il )

0

Ml (t, u) 2 Ml (t, u)

(8.187)

Chapter 8. The Propagation Equation for d/trχ

239

where: (i1 ...il )

0

Ml (t, u)   µ(t, u) 2 (1 − u)3 | = µ(0, u)

(i1 ...il )

(8.188) (i1 ...il )

xl (0, u)|

1

Ml (t, u)      t µ(t, u) 2  3 −1 ∂µ  (1 − u + η0 t ) −2µ (t , u) · |d/ = µ(t  , u) ∂t − 0

(i1 ...il )

(8.189) (i1 ...il )

fˇl (t  , u)|dt 

2

Ml (t, u)    t µ(t, u) 2  3 1  trχ(t (1 − u + η t ) , u) |d/ = 0 µ(t  , u) 2 0

(8.190) (i1 ...il )

fˇl (t  , u)|dt 

Also: (i1 ...il )

G l (t, u)

≤ eCδ0 (1 − u + η0 t)−3 ·

t 0

µ(t, u) µ(t  , u)

2

(8.191) (1 − u + η0 t  )3 |

(i1 ...il )

g˜l (t  , u)|dt 

To proceed with the estimates of the above functions, we must first analyze the behavior of the function µ. This analysis shall be based on the following lemma. We introduce the bootstrap assumption: There is a positive constant C independent of s such that in Wεs0 , E / 30 : |
/ ψ0 | ≤ Cδ0 (1 + t)−3 Lemma 8.10 Under the bootstrap assumptions E1, E2 of Chapter 6, the additional bootstrap assumption E / 30 , the bootstrap assumption F2 of Chapter 6, as well as the basic bootstrap assumptions A, the following hold. Let us denote: Ps (u, ϑ) = (1 + s)(Lψ0 )(s, u, ϑ) We then have, for all t ∈ [0, s]: (Lψ0 )(t, u, ϑ) =

Ps (u, ϑ) + Rs (t, u, ϑ) (1 + t)

|Rs (t, u, ϑ)| ≤ Cδ0

[1 + log(1 + t)] (s − t) (1 + s) (1 + t)2

and:

where C is a constant independent of s.

Chapter 8. The Propagation Equation for d/trχ

240

Proof. The function ψ0 satisfies the linear wave equation in the conformal acoustical ˜ metric h:
h˜ ψ0 = 0 In view of the expression for the operator
h˜ given in Chapter 3, and the definitions of the functions ν and ν given in Chapter 5, this equation takes the form: L(Lψ0 ) + ν(Lψ0 ) = ρ0

(8.192)

where:

d log  d/σ · d/ψ0 (8.193) dσ Now, by assumption E / 30 and the basic bootstrap assumption A3 the first term on the right in (8.193) is bounded in magnitude by: ρ0 = µ
/ ψ0 − ν Lψ0 − 2ζ · d/ψ0 + µ

Cδ0 (1 + t)−3 [1 + log(1 + t)] By the estimate (6.93) on ν and bootstrap assumptions E2, the second term on the right in (8.193) is bounded in magnitude by: Cδ0 (1 + t)−3 [1 + log(1 + t)] as well. Moreover, by the estimate (6.74) on ζ and bootstrap assumptions E2, the third term on the right in (8.193) is bounded in magnitude by: Cδ02 (1 + t)−4 [1 + log(1 + t)] Finally, by the bounds (6.51) on σ and its first derivatives, bootstrap assumptions E2 and the basic bootstrap assumption A3, the last term on the right in (8.193) is bounded in magnitude by: Cδ02 (1 + t)−4 [1 + log(1 + t)] as well. We conclude that: |ρ0 | ≤ Cδ0 (1 + t)−3 [1 + log(1 + t)] Consider on S 2 the function: ˜ u) = A(t, u) A(t,



(t, u) (0, u)

(8.194)

 (8.195)

depending on the parameters t and u, where A(t, u) is defined by (8.181). We have (see (8.178) and the definition of ν): ˜ u) ∂ A(t, ˜ u), = 2ν(t, u) A(t, ∂t

˜ u) = 1 A(0,

(8.196)

˜ u))1/2 ρ0 (t, u) ρ˜0 (t, u) = ( A(t,

(8.197)

Thus, setting, in terms of the pullbacks by "t,u to S 2 , ˜ u))1/2 (Lψ0 )(t, u), τ0 (t, u) = ( A(t,

Chapter 8. The Propagation Equation for d/trχ

equation (8.192) is equivalent to:

241

∂τ0 = ρ˜0 ∂t

(8.198)

Integrating (8.196) we obtain: ˜ u) = e2N(t,u) , A(t,



t

N(t, u) =

ν(t  , u)dt 

(8.199)

0

Writing, ν(t, u) =

η0 + νˆ 1 − u + η0 t

(8.200)

we have,  N(t, u) = log

1 − u + η0 t 1−u



+ Nˆ (t, u),

Nˆ (t, u) =



t

νˆ (t  , u)dt 

(8.201)

0

hence: ˜ u) = A(t,



1 − u + η0 t 1−u

2

ˆ

e2 N (t,u)

(8.202)

Now, by the bootstrap assumption F2 and the bounds (6.51) |ˆν (t, u)| ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]

(8.203)

It follows that: | Nˆ (s, u) − Nˆ (t, u)| ≤



s







s

|ˆν (t , u)|dt ≤ Cδ0

t

(1 + t  )−2 [1 + log(1 + t  )]dt  (8.204)

t

The last integral is: s (1 + t  )−2 [1 + log(1 + t  )]dt  t     1 log(1 + t) log(1 + s) 1 + − − =2 (1 + t) (1 + s) (1 + t) (1 + s) [1 + log(1 + t)] (s − t) ≤2 (1 + t) (1 + s)

(8.205)

We thus obtain: | Nˆ (s, u) − Nˆ (t, u)| ≤ Cδ0

[1 + log(1 + t)] (s − t) (1 + t) (1 + s)

(8.206)

This implies in particular, replacing s, t by t, 0 respectively and noting that N(0, u) = 0, | Nˆ (t, u)| ≤ Cδ0

(8.207)

Chapter 8. The Propagation Equation for d/trχ

242

We now return to equation (8.198). Integrating on [t, s] we obtain: s τ0 (t, u) = τ0 (s, u) − ρ˜0 (t  , u)dt 

(8.208)

t

By (8.202), (8.207) and the estimate (8.194) (see 2nd of definitions (8.197)), |ρ˜0 (t, u)| ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] Hence, in view of (8.205),  s    [1 + log(1 + t)] (s − t)    ρ˜0 (t , u)dt  ≤ Cδ0  (1 + t) (1 + s) t Now, according to the 1st of definitions (8.197) and (8.202), we have:   1−u ˆ −1/2 ˜ (Lψ0 )(t, u) = ( A(t, u)) τ0 (t, u) = e− N (t,u) τ0 (t, u) 1 − u + η0 t

(8.209)

(8.210)

(8.211)

In particular this holds at t = s. On the other hand, according to the definition of the function Ps (u) in the statement of Lemma 8.2 (a function on S 2 ), (Lψ0 )(s, u) = It follows that: τ0 (s, u) =

Ps (u) (1 + s)

Ps (u) (1 − u + η0 s) Nˆ (s,u) e (1 − u) (1 + s)

(8.212)

Substituting (8.212) in (8.208) and the result in (8.211) yields: (Lψ0 )(t, u) =

Ps (u) (1 − u + η0 s) Nˆ (s,u)− N(t,u) ˆ e (1 + s) (1 − u + η0 t) s (1 − u) ˆ e− N (t,u) − ρ˜0 (t  , u)dt  (1 − u + η0 t) t

(8.213)

Thus, setting: (1 − u) ˆ e− N (t,u) Rs (t, u) = Ps (u)Bs (t, u) − (1 − u + η0 t) where: Bs (t, u) =



s

ρ˜0 (t  , u)dt 

(8.214)

1 (1 − u + η0 s) Nˆ (s,u)− N(t,u) 1 ˆ e − (1 + s) (1 − u + η0 t) (1 + t)

(8.215)

t

we obtain the desired expression: (Lψ0 )(t, u) =

Ps (u) + Rs (t, u) (1 + t)

(8.216)

Chapter 8. The Propagation Equation for d/trχ

243

Moreover, (8.215) can be written in the form:  1 (1 − u − η0 ) (s − t) Nˆ (s,u)− N(t,u) ˆ ˆ Nˆ (s,u)− N(t,u) Bs (t, u) = e −1− e (1 + t) (1 − u + η0 t) (1 + s) (8.217) Since from (8.206), ˆ

ˆ

e| N (s,u)− N(t,u)| ≤ eCδ0 ,   [1 + log(1 + t)] (s − t) ˆ  Nˆ (s,u)− N(t,u)  − 1 ≤ eCδ0 | Nˆ (s, u) − Nˆ (t, u)| ≤ Cδ0 , e (1 + t) (1 + s) it follows that: |Bs (t, u)| ≤ C

[1 + log(1 + t)] (s − t) (1 + s) (1 + t)2

(8.218)

Since by the bootstrap assumptions E2 we also have: |Ps (u)| ≤ Cδ0

(8.219)

it follows from (8.214), in view of (8.218) and (8.210), that the function Rs (t, u) satisfies the desired estimate: |Rs (t, u)| ≤ Cδ0

[1 + log(1 + t)] (s − t) (1 + t)2 (1 + s)

(8.220)

The proof of the lemma is thus complete. We now introduce the additional bootstrap assumptions: There is a positive constant C independent of s such that in Wεs0 , ELT 3 : |LT ψµ | ≤ Cδ0 (1 + t)−2 ELL 3 : |L Lψµ | ≤ Cδ0 (1 + t)−3 Proposition 8.5 Let the assumptions of Lemma 8.10 hold. Let the additional bootstrap assumptions ELT 3, ELL 3, hold as well. Denoting:   ∂µ (s, u, ϑ) E s (u, ϑ) = (1 + s) ∂t we then have, for all t ∈ [0, s]:   ∂µ E s (u, ϑ) + Q 1,s (t, u, ϑ) (t, u, ϑ) = ∂t (1 + t) and: |Q 1,s (t, u, ϑ)| ≤ Cδ0

[1 + log(1 + t)] (s − t) (1 + t)2 (1 + s)

Chapter 8. The Propagation Equation for d/trχ

244

Proof. Consider equation (8.52) for Lµ. The functions m and e are given by equations (3.97) and (3.98) respectively. We have: m=

dH 1 (ψ L )2 (T σ ) 2 dσ

(8.221)

Let us set: m 0 = k 3 (T ψ0 ),

m1 = m − m0,

(8.222)

where  is the constant:

dH 2 (k ) dσ As we have seen in Chapter 6, the bootstrap assumptions E1, E2, imply: =

|ψ L − k| ≤ Cδ0 (1 + t)−1 ,

|σ − k 2 | ≤ Cδ0 (1 + t)−1

and the second of these in turn implies:   d H  −1    dσ −  ≤ Cδ0 (1 + t) Moreover, since: T σ = 2ψ0 (T ψ0 ) − 2



(8.223)

(8.224)

(8.225)

ψi (T ψi )

i

the assumptions E1, E2 also imply: |T σ − 2k(T ψ0 )| ≤ Cδ0 (1 + t)−2

(8.226)

|m 1 | ≤ Cδ0 (1 + t)−2

(8.227)

It follows that: Also, m 1 is expressed by: 

m 1 = k  (ψ0 − k)(T ψ0 ) − 2



 ψi (T ψi )

i

+

   dH dH 1 + k2 −  (T σ ) (ψ L2 − k 2 ) 2 dσ dσ

(8.228)

Applying L to this we see that by virtue of bootstrap assumptions E1, E2 and ELT 3, which in particular imply: (8.229) |LT σ | ≤ Cδ0 (1 + t)−2 we have:

|Lm 1 | ≤ Cδ0 (1 + t)−3

Note in particular that L(ψ L ) = L(L µ ψµ ) = L(L µ )ψµ + L µ (Lψµ ),

(8.230)

Chapter 8. The Propagation Equation for d/trχ

and since by (3.144)

245

µ

L(L µ ) = a L Tˆ µ + b L ,

we have:

L(L µ )ψµ = a L ψTˆ + b L · ψ

(8.231)

Now a L and b L are given by (3.153) and (3.156) respectively. The bootstrap assumptions E1, E2 imply: |a L |, |b L | ≤ Cδ0 (1 + t)−2 (8.232) hence:

|L(L µ )ψµ | ≤ Cδ0 (1 + t)−3

(8.233)

Consider now the function e, given by expression (3.98). The bootstrap assumptions E1, E2, yield: |e| ≤ Cδ0 (1 + t)−2 (8.234) Moreover, applying L to the expression (3.98) we see that by virtue of bootstrap assumptions E1, E2 and ELL 3, which in particular imply:

we have:

|L Lσ | ≤ Cδ0 (1 + t)−3

(8.235)

|Le| ≤ Cδ0 (1 + t)−3

(8.236)

Note in particular that L(ψTˆ ) = L(Tˆ µ ψµ ) = L(Tˆ µ )ψµ + Tˆ µ (Lψµ ), and since by (3.159) we have:

µ

L(Tˆ µ ) = p L Tˆ µ + q L , L(Tˆ µ )ψµ = p L ψTˆ + q L · ψ

(8.237)

Now p L and q L are given by (3.164) and (3.169) respectively. The bootstrap assumptions E1, E2 imply: | p L | ≤ Cδ0 (1 + t)−3 , |q L | ≤ Cδ0 (1 + t)−2 (8.238) (see the estimate (6.73) for κ −1 ζ  ), hence: |L(Tˆ µ )ψµ | ≤ Cδ0 (1 + t)−3 Let us now set: m 0,1 =

1 3 k (Lψ0 ) 2

(8.239)

(8.240)

Then, since L = 2T + α −2 µL, we have: 1 m 0,1 = m 0 + k 3 α −2 (Lψ0 ) 2

(8.241)

Chapter 8. The Propagation Equation for d/trχ

246

Thus, setting also: 1 e1 = e − k 3 α −2 (Lψ0 ), 2

n 1 = m 1 + µe1

(8.242)

equation (8.52) takes the form: Lµ = m 0,1 + n 1

(8.243)

In view of (8.234), (8.227) and the basic bootstrap assumption A3 we have: |e1 | ≤ Cδ0 (1 + t)−2 ,

|n 1 | ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]

(8.244)

Moreover, by (8.236) and the assumption ELL 3, |Le1 | ≤ Cδ0 (1 + t)−3

(8.245)

which, together with (8.230) and the fact that by virtue of bootstrap assumptions E1 and E2 we have: (8.246) |Lµ| ≤ Cδ0 (1 + t)−1 , yields:

|Ln 1 | ≤ Cδ0 (1 + t)−3 [1 + log(1 + t)]

(8.247)

By (8.240) and Lemma 8.10 the function m 0,1 , considered as a function on S 2 depending on t and u, is given by: m 0,1 (t, u) =

1 1 3 Ps (u) k  + k 3 Rs (t, u) 2 (1 + t) 2

(8.248)

Let us define the function E s (u) on S 2 , depending on the parameters u and s, according to: E s (u) =

1 3 k Ps (u) + (1 + s)n 1 (s, u) 2

Then by the 2nd of (8.244),      E s (u) − 1 k 3 Ps (u) ≤ Cδ0 (1 + s)−1 [1 + log(1 + s)]   2

(8.249)

(8.250)

Moreover, from (8.243), (8.248), we have, in view of the fact that Rs (s, u) = 0,   ∂µ (s, u) = E s (u) (1 + s) (8.251) ∂t Defining then: Q 1,s (t, u) =

1 3 (s − t) k Rs (t, u) − n 1 (s, u) − (n 1 (s, u) − n 1 (t, u)) 2 (1 + t)

(8.252)

Chapter 8. The Propagation Equation for d/trχ

247

we obtain from (8.243), in view of (8.248) and the definitions (8.249) and (8.251),   ∂µ E s (u) (t, u) = + Q 1,s (t, u) (8.253) ∂t (1 + t) This is the desired formula. It remains to be shown that the function Q 1,s (t, u) satisfies the bound given in the statement of the proposition. This is so for the first term on the right in (8.252) by the bound for the function Rs (t, u) given in the statement of Lemma 8.10. By the 2nd of (8.244) the second term on the right in (8.252) also satisfies the required bound. As for the last term, we have, by the bound (8.247),  s s  ∂n 1      dt ≤ Cδ0 , u) (1 + t  )−3 [1 + log(1 + t  )]dt  |n 1 (s, u) − n 1 (t, u)| ≤ (t  ∂t  t t (8.254) The last integral is: s (1 + t  )−3 [1 + log(1 + t  )]dt  t     3 1 1 log(1 + t) log(1 + s) 1 = − − + 4 (1 + t)2 (1 + s)2 2 (1 + t)2 (1 + s)2 3 [1 + log(1 + t)] (s − t) (8.255) ≤ 4 (1 + s) (1 + t)2 We conclude that: |Q 1,s (t, u)| ≤ Cδ0

[1 + log(1 + t)] (s − t) (1 + t)2 (1 + s)

(8.256)

and the proof of the proposition is complete. We now replace t by t  in the formula for ∂µ/∂t given by Proposition 8.5 (that is, (8.253)), and integrate with respect to t  on [0, t] to obtain: µ(t, u) = µ(0, u) + E s (u) log(1 + t) +

t

Q 1,s (t  , u)dt 

(8.257)

0 ε

Recall that µ = ακ. Assumption E1 restricted to 00 implies: |α(0, u) − η0 | ≤ Cδ0

(8.258)

Also, according to the assumption I0 of Chapter 6 on the initial data, |κ(0, u) − 1| ≤ Cδ0

(8.259)

|µ(0, u) − η0 | ≤ Cδ0

(8.260)

It follows that:

Chapter 8. The Propagation Equation for d/trχ

248

The bound on the function Q 1,s (t, u) given in the statement of Proposition 8.5 implies: |Q 1,s (t, u)| ≤ Cδ0

[1 + log(1 + t)] (1 + t)2

(8.261)

Q 1,s (t, u)dt

(8.262)

Let us define on S 2 the function:

s

µ1,s (u) = 0

Substituting the bound (8.261) we obtain: |µ1,s (u)| ≤ Cδ0 the integral



∞ 0

(8.263)

[1 + log(1 + t  )]  dt (1 + t  )2

being convergent. We then define on S 2 the function: µ[1],s (u) = µ(0, u) + µ1,s (u)

(8.264)

|µ[1],s (u) − η0 | ≤ Cδ0

(8.265)

By (8.260) and (8.263), Taking δ0 suitably small, this implies: inf inf µ[1],s (u) ≥

u∈[0,ε0 ] S 2

η0 2

(8.266)

In view of the definitions (8.262) and (8.264), the formula (8.257) can be written in the form: µ(t, u) = µ[1],s (u) + E s (u) log(1 + t) + Q 0,s (t, u) (8.267)

where: Q 0,s (t, u) = −

s

Q 1,s (t  , u)dt 

(8.268)

t

Substituting in (8.268) the estimate (8.261) yields: s |Q 1,s (t  , u)|dt  |Q 0,s (t, u)| ≤ t s [1 + log(1 + t  )]  ≤ Cδ0 dt (1 + t  )2 t [1 + log(1 + t)] (s − t) ≤ 2Cδ0 (1 + t) (1 + s) (see (8.205)).

(8.269)

Chapter 8. The Propagation Equation for d/trχ

249

Finally, setting E s (u) , µ[1],s (u) Q 1,s (t, u) Qˆ 1,s (t, u) = µ[1],s (u)

µ(t, u) , µ[1],s (u) Q 0,s (t, u) Qˆ 0,s (t, u) = , µ[1],s (u)

Eˆ s (u) =

µˆ s (t, u) =

(8.270)

we collect the above results in the following proposition. Proposition 8.6 Let the assumptions of Proposition 8.5 as well as the assumption I0 of Chapter 6 hold. Then, there is a function µ[1],s (u, ϑ) on S 2 depending on the parameters u and s, satisfying: |µ[1],s (u, ϑ) − η0 | ≤ Cδ0 ,

inf µ[1],s (u, ϑ) ≥

inf

u∈[0,ε0 ] ϑ∈S 2

such that setting:

η0 2

µ(t, u, ϑ) µ[1],s (u, ϑ)

µˆ s (t, u, ϑ) = the following equalities hold:

µˆ s (t, u, ϑ) = 1 + Eˆ s (u, ϑ) log(1 + t) + Qˆ 0,s (t, u, ϑ) Eˆ s (u, ϑ) ∂ µˆ s (t, u, ϑ) = + Qˆ 1,s (t, u, ϑ) ∂t (1 + t) where: Qˆ 0,s (t, u, ϑ) = −



s

Qˆ 1,s (t  , u, ϑ)dt 

t

and Qˆ 0,s (t, u, ϑ), Qˆ 1,s (t, u, ϑ), satisfy the bounds: [1 + log(1 + t)] (s − t) | Qˆ 0,s (t, u, ϑ)| ≤ Cδ0 (1 + t) (1 + s) [1 + log(1 + t)] (s − t) | Qˆ 1,s (t, u, ϑ)| ≤ Cδ0 (1 + t)2 (1 + s) Let us now define: µm (t) = min µ= ε t 0

min

(u,ϑ)∈[0,ε0 ]×S 2

We set: µm (t) = min{

t 0

 max

(8.271)

µm (t) , 1} η0

Let us also define: M(t) = max {−µ−1 (Lµ)− } = ε

µ(t, u, ϑ)

(u,ϑ)∈[0,ε0]×S 2

−µ−1

(8.272) 

∂µ ∂t



 −

(t, u, ϑ)

(8.273)

Chapter 8. The Propagation Equation for d/trχ

250

The following lemma plays a crucial role in the sequel. Lemma 8.11 Let the assumptions of Proposition 8.6 hold on Wεs0 . Then for any constant a ≥ 4, there is a positive constant C independent of s and a such that for all t ∈ [0, s] we have: t    −1 −a µ−a Ia (t) := m (t )M(t )dt ≤ Ca µm (t) 0

provided that δ0 is suitably small depending on a. Proof. The proof is based on Proposition 8.6. We first note that for t ∈ [0, s] we can express M(t) as:    1 ∂ µˆ s − M(t) = max (t, u, ϑ) (8.274) µˆ s ∂t − (u,ϑ)∈[0,ε0 ]×S 2 Let us now denote:

Eˆ s,m =

min

(u,ϑ)∈[0,ε0 ]×S 2

Eˆ s (u, ϑ)

(8.275)

We have two cases to consider according as to whether Eˆ s,m is ≥ 0 or < 0. Case 1) Eˆ s,m ≥ 0. In this case, Eˆ s (u, ϑ) ≥ 0 : ∀(u, ϑ) ∈ [0, ε0 ] × S 2 From Proposition 8.6 we then have: ∂ µˆ s (t, u, ϑ) ≥ Qˆ 1,s (t, u, ϑ) ∂t hence:

 −

Also:

∂ µˆ s ∂t

 −

(t, u, ϑ) ≤ −( Qˆ 1,s )− (t, u, ϑ) ≤ Cδ0

[1 + log(1 + t)] (1 + t)2

µˆ s (t, u, ϑ) ≥ 1 + Qˆ 0,s (t, u, ϑ) ≥ 1 − Cδ0

(8.276)

(8.277)

Now (8.277) and (8.276) together imply: M(t) ≤ Cδ0

[1 + log(1 + t)] (1 + t)2

(8.278)

provided that δ0 does not exceed a fixed positive constant. On the other hand, we have: µ(t, u, ϑ) = µ[1],s (u, ϑ)µˆ s (t, u, ϑ) while from the statement of Proposition 8.6,    µ[1],s (u, ϑ)   − 1 ≤ Cδ0  η0

(8.279)

Chapter 8. The Propagation Equation for d/trχ

251

In view of the lower bound (8.277), it follows that: µm (t) ≥ 1 − Cδ0 η0

(8.280)

µm (t) ≥ 1 − Cδ0

(8.281)

hence also: Substituting the bounds (8.278) and (8.281) in the integral Ia (t) we obtain: t [1 + log(1 + t  )]  Ia (t) ≤ (1 − Cδ0 )−a Cδ0 dt (1 + t  )2 0 ≤ C  δ0 (8.282) provided that: δ0 a ≤ For then,

1 C

(8.283)

(1 − Cδ0 )−a ≤ (1 − a −1 )−a → e, as a → ∞

On the other hand, we have, in any case, by definition, µm (t) ≤ 1, hence µ−a m (t) ≥ 1 Therefore the inequality of the lemma holds in Case 1). Case 2) Eˆ s,m < 0. Let us set in this case: Eˆ s,m = −δ1 ,

δ1 > 0

(8.284)

In any case we have: δ1 ≤ Cδ0

(8.285)

In the following we denote: b(t, s) =

[1 + log(1 + t)] (s − t) , (1 + t) (1 + s)

c(t) =

[1 + log(1 + t)] (1 + t)2

(8.286)

We also denote: µˆ s,m (t) =

min

(u,ϑ)∈[0,ε0]×S 2

µˆ s (t, u, ϑ)

(8.287)

From Proposition 8.6 we have, in the present case, µˆ s (t, u, ϑ) ≥ 1 − δ1 log(1 + t) − Cδ0 b(t, s)

(8.288)

for all (u, ϑ) ∈ [0, ε0 ] × S 2 , hence: µˆ s,m (t) ≥ 1 − δ1 log(1 + t) − Cδ0 b(t, s)

(8.289)

Chapter 8. The Propagation Equation for d/trχ

252

On the other hand, evaluating the expression for µˆ s (t, u, ϑ) given in the statement of Proposition 8.6 at a minimum (u m , ϑm ) of the function Eˆ s (u, ϑ) on [0, ε0 ] × S 2 , we obtain: µˆ s,m (t) ≤ µˆ s (t, u m , ϑm ) = 1 − δ1 log(1 + t) + Q 0,s (t, u m , ϑm ) hence: µˆ s,m (t) ≤ 1 − δ1 log(1 + t) + Cδ0 b(t, s)

(8.290)

Moreover, again from Proposition 8.6 we have: 

∂ µˆ s − ∂t hence:

 −



(t, u, ϑ) ≤ −

∂ µˆ s − ∂t

( Eˆ s )− (u, ϑ) − ( Qˆ 1,s )− (t, u, ϑ) (1 + t)

 −

(t, u, ϑ) ≤

δ1 + Cδ0 c(t) (1 + t)

(8.291)

Consider now the integral Ia (t). Let us set: 1

t1 = e 2aδ1 − 1

(8.292)

We have two subcases to consider according as to whether t is ≤ t1 or > t1 . Subcase 2a) t ≤ t1 . Since t  ≤ t, where t  is the variable of integration in the integral Ia (t), in this subcase we have t  ≤ t1 , that is: 1 − δ1 log(1 + t  ) ≥ 1 −

1 2a

(8.293)

hence, by (8.289): µˆ s,m (t  ) ≥ 1 −

1 a

(8.294)

provided that: 1 (8.295) 2C where C is the constant appearing in (8.289). This is a smallness condition on δ0 of the form (8.283). In view of (8.279), the lower bound (8.294) implies: δ0 a ≤

2 µm (t  ) ≥1− η0 a hence also:

2 a provided again that a smallness condition on δ0 of the form (8.283) holds. µm (t  ) ≥ 1 −

(8.296)

Chapter 8. The Propagation Equation for d/trχ

253

Now, by (8.291) with t replaced by t  and (8.294) we have:    1 −1 δ1  M(t ) ≤ 1 − + Cδ0 c(t ) a (1 + t  ) 

(8.297)

The estimates (8.296) and (8.297) then yield:      2 −a t δ1 1 −1  + Cδ 1− Ia (t) ≤ 1 − c(t ) dt  0 a a (1 + t  ) 0 Here,



t 0

1 δ1 dt  = δ1 log(1 + t) ≤ δ1 log(1 + t1 ) = (1 + t  ) 2a

while,



t

Cδ0 c(t  )dt  ≤ C  δ0 ≤

0

1 2a

(8.298)

(8.299)

(8.300)

provided that a smallness condition on δ0 of the form (8.283) holds, the function c being integrable on [0, ∞). In view of the fact that the coefficient in (8.298):     2 −a 1 −1 → e2 , as a → ∞ 1− 1− a a it follows that for t ≤ t1 :

C (8.301) a In particular this holds for t = t1 . Since, in general, µm (t) ≤ 1 by definition, the inequality (8.301) implies the inequality of the lemma in Subcase 2a). Ia (t) ≤

Subcase 2b) t > t1 . In this subcase we write: t    µ−a Ia (t) = Ia (t1 ) + m (t )M(t )dt

(8.302)

t1

Substituting (8.301) with t replaced by t1 we obtain: t C    µ−a Ia (t) ≤ + m (t )M(t )dt a t1

(8.303)

Consider now the upper bound (8.290) evaluated at t = s. Since b(s, s) = 0, this reads: µˆ s,m (s) ≤ 1 − δ1 log(1 + s) (8.304) On the other hand, µ > 0 in the maximal domain Wε0 , in particular in Wεs0 . It follows that: 1 s ≤ t∗ where t∗ = e δ1 − 1 (8.305)

Chapter 8. The Propagation Equation for d/trχ

254

Let us introduce the variable:

τ  = log(1 + t  )

(8.306)

and correspondingly write: τ = log(1 + t),

σ = log(1 + s)

and:

1 , 2aδ1 Then in the integral in (8.303) we have: τ1 = log(1 + t1 ) =

τ∗ = log(1 + t∗ ) =

(8.307) 1 δ1

τ1 ≤ τ  ≤ τ ≤ σ ≤ τ∗

(8.308)

(8.309)

Expressing b(t, s) (see (8.286)) in terms of the new variables we have: b(t, s) = Since, 0≤

(1 + τ ) (eσ − eτ ) eτ eσ

(8.310)

(eσ − eτ ) = 1 − e−(σ −τ ) ≤ σ − τ, eσ

it follows that:

(1 + τ ) (σ − τ ) eτ Substituting (8.311) in (8.289) with t replaced by t  we obtain: 0 ≤ b(t, s) ≤

(1 + τ  ) µˆ s,m (t  ) ≥ 1 − δ1 τ  − Cδ0 (σ − τ  ) τ e  Cδ0 (1 + τ  ) = 1 − δ1 σ + δ1 1 − (σ − τ  )  δ1 eτ We have:

  1 (1 + τ  ) (1 + τ1 ) 1 − 2aδ 1 ≤ = 1 + e  eτ1 2aδ1 eτ

(8.311)

(8.312)

(8.313)

Hence, under the smallness condition Cδ0 ≤

1 2a

on δ0 (of the same form as (8.283)):

  1 1 Cδ0 (1 + τ  ) − 1 ≤ 1+ e 2aδ1  τ δ1 2aδ1 2aδ1 e

Now, in view of the fact that   1 1 −1 lim 1+ e x = 0, x→0+ x x

(8.314)

Chapter 8. The Propagation Equation for d/trχ

255

there is a positive constant K (a), depending on a, such that:   1 1 1 1 −1 : ∀x ≤ 1+ e x ≤ x x a K (a) Here we set: x = 2aδ1 ≤ 2aCδ0 We then conclude that the smallness condition: δ0 ≤ on δ0 , implies that:

1 2Ca K (a)

  1 1 1 − 1 1+ e 2aδ1 ≤ 2aδ1 2aδ1 a

(8.315)

(8.316)

It then follows from (8.314) that: 1−

Cδ0 (1 + τ  ) 1 ≥1−  τ δ1 a e

(8.317)

Therefore, in reference to (8.312), taking into account the fact that: 1 − δ 1 σ ≥ 1 − δ 1 τ∗ = 0

(8.318)

we obtain:  Cδ0 (1 + τ  ) 1 − δ1 σ + δ1 1 − (σ − τ  )  δ1 eτ   1 ≥ 1 − δ1 σ + 1 − δ1 (σ − τ  ) a     1 1 ≥ 1− (1 − δ1 σ ) + 1 − δ1 (σ − τ  ) a a   1 = 1− (1 − δ1 τ  ) a

(8.319)

From (8.312) we then conclude that the following lower bound holds:   1   µˆ s (t , u, ϑ) ≥ µˆ s,m (t ) ≥ 1 − (1 − δ1 τ  ) a : ∀(u, ϑ) ∈ [0, ε0] × S 2 , ∀t  ≥ t1

(8.320)

We now consider the bound (8.291). Since (see (8.286), (8.306)) (1 + t  )c(t  ) =

(1 + τ  )  eτ

(8.321)

Chapter 8. The Propagation Equation for d/trχ

256

the bound (8.291) with t replaced by t  reads:    Cδ0 (1 + τ  ) ∂ µˆ s −(1 + t  ) (t  , u, ϑ) ≤ δ1 1 +  ∂t − δ1 eτ

(8.322)

If the smallness condition (8.315) on δ0 holds, this implies:     1 ∂ µˆ s −(1 + t  ) (t  , u, ϑ) ≤ δ1 1 + ∂t − a : ∀(u, ϑ) ∈ [0, ε0 ] × S 2 , ∀t  ≥ t1

The bounds (8.320) and (8.323) together yield:   1 1 + δ1 a (1 + t  )M(t  ) ≤ 1 (1 − δ1 τ  ) 1− a Also, by (8.279), taking Cδ0 ≤ a −1 , and the lower bound (8.320),   2 µm (t  ) (1 − δ1 τ  ) ≥ 1− η0 a   2 µm (t ) ≥ 1 − (1 − δ1 τ  ) a

hence also:



(8.323)

(8.324)

(8.325)

(8.326)

We now consider the integral on the right in inequality (8.303). In terms of the variable τ  we have (see (8.306)–(8.308)): τ t −a        µm (t )M(t )dt = µ−a m (t )(1 + t )M(t )dτ τ1

t1

thus, the bounds (8.324) and (8.326) yield:   τ t 1 + a1 1 −a    µm (t )M(t )dt ≤ (1 − δ1 τ  )−a−1 δ1 dτ    a 1 2 1 − t1 τ1 1− a a Since:

τ

τ1

(1 − δ1 τ  )−a−1 δ1 dτ  =

1 1 [(1 − δ1 τ )−a − (1 − δ1 τ1 )−a ] ≤ (1 − δ1 τ )−a a a

while the coefficient: 

1+ 1−

1 a 1 a

(8.327)



1  a → e2 , as a → ∞ 2 1− a

Chapter 8. The Propagation Equation for d/trχ

we conclude that:



t t1

257

   µ−a m (t )M(t )dt ≤

C (1 − δ1 τ )−a a

(8.328)

On the other hand, by the upper bound (8.290) and (8.311), (1 + τ ) µˆ s,m (t) ≤ 1 − δ1 τ + Cδ0 (σ − τ ) τ  e Cδ0 (1 + τ ) (σ − τ ) = 1 − δ1 σ + δ1 1 + δ1 eτ

(8.329)

Now by (8.313) with τ  replaced by τ and (8.316), 1 Cδ0 (1 + τ ) ≤ δ1 eτ a provided that the smallness condition (8.315) on δ0 holds. Hence,   Cδ0 (1 + τ ) 1 δ1 1+ ≤ 1 + δ1 eτ a By virtue of (8.331) we obtain, in reference to (8.329),  Cδ0 (1 + τ ) 1 − δ1 σ + δ1 1 + (σ − τ ) δ1 eτ   1 δ1 (σ − τ ) ≤ 1 − δ1 σ + 1 + a     1 1 ≤ 1+ (1 − δ1 σ ) + 1 + δ1 (σ − τ ) a a   1 = 1+ (1 − δ1 τ ) a Substituting in (8.329) we then obtain the following upper bound:   1 µˆ s,m (t) ≤ 1 + (1 − δ1 τ ) a

(8.330)

(8.331)

(8.332)

(8.333)

which together with (8.279) (and a smallness condition on δ0 of the form (8.283)) yields:   2 µm (t) ≤ 1+ (8.334) (1 − δ1 τ ) η0 a hence also:

  2 µm (t) ≤ 1 + (1 − δ1 τ ) a

Therefore: µ−a m (t) ≥ 

1 1+

2 a

a (1 − δ1 τ )−a

(8.335)

(8.336)

Chapter 8. The Propagation Equation for d/trχ

258

Since the reciprocal of the coefficient:   2 a 1+ → e2 , as a → ∞ a we conclude that: µ−a m (t) ≥

1 (1 − δ1 τ )−a C

Comparison of (8.328) with (8.337) yields the conclusion: t C −a    µ−a µ (t) m (t )M(t )dt ≤ a m t1

(8.337)

(8.338)

In view of (8.303), (8.338) and the fact that µm (t) ≤ 1 the inequality of the lemma follows in Subcase 2b) as well. Case 2) is thus concluded. The proof of the lemma is now complete. The proof of Lemma 8.11 yields the following corollaries. Corollary 1 Under the assumptions of Lemma 8.11, there is a positive constant C independent of s such that the following holds. Suppose that at some (u, ϑ) ∈ [0, ε0 ] × S 2 we have Eˆ s (u, ϑ) < 0. Then we have: µˆ s (t, u, ϑ) ≤C µˆ s (t  , u, ϑ) for all t  ∈ [0, t] and all t ∈ [0, s]. Proof. The argument of Case 2) of the proof of Lemma 8.11 leading to upper and lower bounds on µˆ s,m (t) applies with [0, ε0 ] × S 2 replaced by any closed subset thereof, in particular the single point (u, ϑ) in which case we simply have: Eˆ s,m = Eˆ s (u, ϑ) = −δ1 , δ1 > 0,

µˆ s,m (t) = µˆ s (t, u, ϑ)

Then the lower and upper bounds (8.289) and (8.290) become: 1 − δ1 log(1 + t) − Cδ0 b(t, s) ≤ µˆ s (u, ϑ) ≤ 1 − δ1 log(1 + t) + Cδ0 b(t, s) (8.339) Fixing then a = 2 and setting, in accordance with (8.292), 1

t1 = e 4δ1 − 1

(8.340)

we have to consider the two subcases according as to whether t is ≤ t1 or > t1 . In the first subcase (8.294) holds, that is, in the present circumstances, µˆ s (t  , u, ϑ) ≥

1 2

(8.341)

Chapter 8. The Propagation Equation for d/trχ

259

while by the upper bound in (8.339), µˆ s (t, u, ϑ) ≤

3 2

(8.342)

thus the inequality of the corollary holds in the first subcase. In the second subcase, we have the upper bound (8.333), which in the present circumstances is: µˆ s (t, u, ϑ) ≤

3 (1 − δ1 τ ) 2

(8.343)

If then t  ≤ t1 , the lower bound (8.341) holds, so the inequality of the corollary holds, while if t  > t1 , then the lower bound (8.320) holds, which under the present circumstances is: 1 (8.344) µˆ s (t  , u, ϑ) ≥ (1 − δ1 τ  ) 2 thus, in view of the fact that (1 − δ1 τ ) ≤ (1 − δ1 τ  ), the inequality of the corollary again holds. Corollary 2 Under the assumptions of Lemma 8.11, there is a positive constant C independent of s and a such that:  −a µ−a m (t ) ≤ Cµm (t) holds for all t  ∈ [0, t] and all t ∈ [0, s]. Proof. We have the two cases of Lemma 8.11 to consider. Case 1) Eˆ s,m ≥ 0. In this case, the lower bound (8.281) holds with t replaced by t  : µm (t  ) ≥ 1 − Cδ0

(8.345)

On the other hand we have in general, by definition, µm (t) ≤ 1 It follows that:

 µ−a m (t ) ≤ (1 − Cδ0 )−a µ−a m (t)

(8.346)

and this is bounded by a constant independent of a provided that: Cδ0 ≤

1 a

Case 2) Eˆ s,m < 0. In this case we define δ1 as in (8.284) and t1 as in (8.292) and we consider the subcases 2a) t  ≤ t1 and 2b) t  ≥ t1 separately. In subcase 2a) the lower bound (8.296) holds. We thus have:    µ−a 2 −a m (t ) (8.347) ≤ 1− a µ−a m (t) which is bounded by a constant independent of a (for a ≥ 4).

Chapter 8. The Propagation Equation for d/trχ

260

In subcase 2b) the lower bound (8.326) as well as the upper bound (8.335) hold, hence: −a  −a     1 − a2 1 − a2 1 − δ1 τ a µ−a m (t ) ≤ (8.348) ≤ 1 − δ1 τ  µ−a 1 + a2 1 + a2 m (t) (for, τ  ≤ τ ) which is bounded by a constant independent of a (for a ≥ 4), as required. Corollary 3 The assumptions of Lemma 8.11 imply the assumption AS. Proof. We consider a given (u, ϑ) ∈ [0, ε0 ] × S 2 as in the proof of Corollary 1. Again, we have the two cases Eˆ s (u, ϑ) ≥ 0 and Eˆ s (u, ϑ) < 0 to consider. In the first case we have µˆ s (t, u, ϑ) ≥ 1 − Cδ0 as in Case 1 of the proof of Lemma 8.11 (see (8.277)). On the other hand from Proposition 8.6 we have: ∂ µˆ s Eˆ s (u, ϑ) 1 (t, u, ϑ) = + Qˆ 1,s (t, u, ϑ) ≤ Cδ0 ∂t (1 + t) (1 + t) Therefore: −2µ−1

∂µ ∂ µˆ s 1 = −2µ ˆ −1 ≥ −Cδ0 s ∂t ∂t (1 + t)

On the other hand, assumption F2 implies, as we have seen, trχ ≥ C −1

1 (1 + t)

The assumption AS thus follows in Case 1) if δ0 is suitably small. In the second case we set as in the proof of Corollary 1, Eˆ s (u, ϑ) = −δ1 , δ1 > 0, a = 2 and define t1 by (8.340). Then we have to consider the subcases t ≤ t1 and t > t1 . In the first subcase we have (see (8.341)) µˆ s (t, u, ϑ) ≥

1 2

while from Proposition 8.6 ∂ µˆ s [1 + log(1 + t)] (t, u, ϑ) ≤ Qˆ 1,s (t, u, ϑ) ≤ Cδ0 ∂t (1 + t)2 Therefore: −2µ−1

∂µ ∂ µˆ s [1 + log(1 + t)] = −2µˆ −1 ≥ −Cδ0 s ∂t ∂t (1 + t)2

and the assumption AS follows in subcase 2a) if δ0 is suitably small.

Chapter 8. The Propagation Equation for d/trχ

261

In the second subcase we have from Proposition 8.6, (1 + t)

∂ µˆ s (t, u, ϑ) = −δ1 + (1 + t) Qˆ 1,s (t, u, ϑ) ∂t  Cδ0 (1 + τ ) δ1 ≤ −δ1 1 − ≤− τ δ1 e 2

if δ0 is suitably small, according to (8.315) with a = 2. It follows that in subcase 2b) −2µ−1

∂µ >0 ∂t

and the assumption AS a fortiori holds. We now return to (8.187). What we wish to do is to obtain an estimate for the L 2 norm of (i1 ...il ) Fl (t) on [0, ε0] × S 2 . In general, if φ is a function defined on Wε∗0 and we consider φ(t, u), the corresponding function on S 2 depending on t and u, then the L 2 ε norm of φ on t 0 is, by definition: ) φ L 2 ( ε0 ) =

ε

t

) =

t 0

φ 2 dµh/ du

[0,ε0 ]×S 2

(φ(t, u))2 dµh/(t,u)du

(8.349)

On the other hand, the L 2 norm of φ(t, .) on [0, ε0 ] × S 2 is: ) φ(t) L 2 ([0,ε0 ]×S 2 ) =

[0,ε0 ]×S 2

(φ(t, u))2 dµh/(0,0)du

/(0, 0) being the standard metric on S 2 . We have: h √ √ deth /(t, u) deth/(0, u) dµh/(t,u) = √ dµh/(0,0) = A(t, u) √ dµh/(0,0) deth /(0, 0) deth/(0, 0)

(8.350)

(8.351)

ε

Now, by the definition of θ (see Chapter 3), we have on 00 , noting that  vanishes there, 1 ∂

deth/ κtrθ = √ deth/ ∂u The assumptions on the initial data imply:     κtrθ − 2  ≤ Cδ0  1−u

ε

: on 00

ε

: on 00

(8.352)

(8.353)

It then follows integrating (8.352) that: e

−Cδ0

√ deth/(0, u) (1 − u) ≤ √ ≤ eCδ0 (1 − u)2 deth/(0, 0) 2

(8.354)

Chapter 8. The Propagation Equation for d/trχ

262

Combining with (8.185) we obtain from (8.351): e−Cδ0 (1 − u + η0 t)2 dµh/(0,0) ≤ dµh/(t,u) ≤ eCδ0 (1 − u + η0 t)2 dµh/(0,0)

(8.355)

In view of (8.349), (8.350) it follows that: C −1 (1 + t)φ(t) L 2 ([0,ε0 ]×S 2 ) ≤ φ L 2 ( ε0 ) ≤ C(1 + t)φ(t) L 2 ([0,ε0]×S 2 ) t

(8.356)

Note moreover that if ξ is an St,u 1-form defined on Wε∗0 and we consider ξ(t, u), the corresponding 1-form on S 2 depending on t and u, then setting φ = |ξ |, φ(t, u) = |ξ(t, u)|, the above apply to the functions φ and φ(t, u) and we have: φ L 2 ( ε0 ) = ξ  L 2 ( ε0 ) , t

φ(t) L 2 ([0,ε0 ]×S 2 ) = |ξ(t)| L 2 ([0,ε0]×S 2 )

t

(8.357)

hence, by (8.356), C −1 (1 + t)|ξ(t)| L 2 ([0,ε0 ]×S 2 ) ≤ ξ  L 2 ( ε0 ) ≤ C(1 + t)|ξ(t)| L 2 ([0,ε0 ]×S 2 ) (8.358) t

0

1

2

We now consider the terms (i1 ...il ) M l (t, u), (i1 ...il ) M l (t, u), (i1 ...il ) M l (t, u) on the right-hand side of (8.187), given by (8.188), (8.189), (8.190) respectively. First, by the basic bootstrap assumption A3 and (8.358) at t = 0, 

(i1 ...il )

0

M l (t) L 2 ([0,ε0 ]×S 2 ) ≤ C[1 + log(1 + t)]2 

We turn to

(i1 ...il )

xl (0) L 2 ( ε0 )

(8.359)

0

1

(i1 ...il )

M l (t, u). We partition [0, ε0 ] × S 2 , into the open set:

Vs− = {(u, ϑ) ∈ [0, ε0 ] × S 2 : Eˆ s (u, ϑ) < 0}

(8.360)

and its complement, the closed set: Vs+ = {(u, ϑ) ∈ [0, ε0 ] × S 2 : Eˆ s (u, ϑ) ≥ 0}

(8.361)

We then have: 

(i1 ...il )

1

M l (t)2L 2 ([0,ε0 ]×S 2 )

= Now, for fixed t,

(i1 ...il )

(i1 ...il ) 1

2 M l (t) L 2 (Vs− ) + 

(i1 ...il ) 1

2 M l (t) L 2 (Vs+ )

(8.362)

1

M l (t) restricted to Vs− is a definite integral, with respect to t  , 1

of a function of t  with values in the space L 2 (Vs− ). Thus, the norm of (i1 ...il ) M l (t) in L 2 (Vs− ) does not exceed the integral of the L 2 (Vs− )-norm of the said function, with respect to t  : t 1 1  (i1 ...il ) M l (t) L 2 (Vs− ) ≤  (i1 ...il ) N l (t, t  ) L 2 (Vs− ) dt  (8.363) 0

Chapter 8. The Propagation Equation for d/trχ

263

where: 1

(i1 ...il )

N l (t, t  , u)       µ(t, u) 2  3 −1 ∂µ  = (1 − u + η0 t ) −2µ (t , u) |d/ µ(t  , u) ∂t −

(8.364) (i1 ...il )

fˇl (t  , u)|

We have: (i1 ...il )

1

N l (t, t  ) L 2 (Vs− )        µ(t) 2  3 −1 ∂µ  ≤ (1 + η0 t ) max max −2µ (t ) |d/ µ(t  ) ∂t − Vs− Vs−



(8.365) (i1 ...il )

fˇl (t  )| L 2 (Vs− )

Now, by Corollary 1 to Lemma 8.11 we have:   µ(t) max ≤C µ(t  ) Vs−

(8.366)

while by definition (8.273),     ∂µ (t  ) ≤ 2M(t  ) max −2µ−1 ∂t − Vs−

(8.367)

We thus obtain, substituting in (8.365), 

(i1 ...il )

1

N l (t, t  ) L 2 (Vs− ) ≤ C(1 + t  )3 M(t  )|d/

(i1 ...il )

fˇl (t  )| L 2 ([0,ε0]×S 2 )

(8.368)

Let us define: (i1 ...il )

Pl (t) = (1 + t)2 |d/

Suppose that, for non-negative quantities (i1 ...il )

Pl (t) ≤

(i1 ...il )

fˇl (t)| L 2 ([0,ε0]×S 2 )

(i1 ...il )

(0) (1) P l , (i1 ...il ) P l ,

(0) Pl

(i1 ...il )

(i1 ...il )

(t) +

Defining then the non-decreasing non-negative quantities (i1 ...il )

(0)

P l,a = sup {µam (t  )

(i1 ...il )

t  ∈[0,t ]

(i1 ...il )

(1)

P l,a = sup {(1 + t  )1/2 µam (t  ) t  ∈[0,t ]

(1) Pl

we have: (t)

(i1 ...il ) (0) Pl

(0)

P l,a ,

(t  )}

(i1 ...il )

(8.369)

(1) Pl

(8.370) (i1 ...il )

(1)

P l,a , by: (8.371)

(t  )}

(8.372)

Chapter 8. The Propagation Equation for d/trχ

264

Then, for t  ∈ [0, t] we have:  (i1 ...il )



Pl (t ) ≤

 µ−a m (t )

(0)

(i1 ...il )

 −1/2 (i1 ...il )

P l,a (t) + (1 + t )



(1)

P l,a (t)

(8.373)

Substituting in (8.368) we obtain, in view of the definition (8.369), (i1 ...il )





1

N l (t, t  ) L 2 (Vs− )

≤ C (1 + t)

(i1 ...il )

(0)

(8.374)



(1)

  P l,a (t) + (1 + t)1/2 P l,a (t) µ−a m (t )M(t )

for all t  ∈ [0, t]. Substituting this estimate in (8.363) we then obtain:  

(i1 ...il )

1

M l (t) L 2 (Vs− ) ≤ C (1 + t)

(i1 ...il )

(0)

P l,a (t) + (1 + t)

1/2



(1)

P l,a (t) Ia (t) (8.375)

At this point we apply Lemma 8.11 to conclude that: 

(i1 ...il )

1

M l (t) L 2 (Vs− ) 

≤ Ca

−1

(1 + t)

(i1 ...il )

(0)

P l,a (t) + (1 + t)

1/2

(1)



P l,a (t) µ−a m (t)

(8.376)

In analogy with (8.363) and (8.365), with Vs+ in the role of Vs− , we have: t 1 1 (i1 ...il )   (i1 ...il ) N l (t, t  ) L 2 (Vs+ ) dt  (8.377) M l (t) L 2 (Vs+ ) ≤ 0

and: (i1 ...il )

1

N l (t, t  ) L 2 (Vs+ )        µ(t) 2 −1 ∂µ  −2µ ≤ (1 + η0 t  )3 max max (t ) |d/ µ(t  ) ∂t − Vs+ Vs+



(8.378) (i1 ...il )

fˇl (t  )| L 2 (Vs+ )

To estimate the second and third factors on the right in (8.378) we can apply the argument of Case 1) of Lemma 8.11, replacing [0, ε0] × S 2 by its closed subset Vs+ . By the bound (8.278) we then have:     [1 + log(1 + t  )] −1 ∂µ  (t ) ≤ Cδ0 (8.379) max −2µ ∂t − (1 + t  )2 Vs+ and, in view of the lower bound (8.277),   µ(t) max ≤ C[1 + log(1 + t)] µ(t  ) Vs+

(8.380)

Chapter 8. The Propagation Equation for d/trχ

265

Substituting the estimates (8.379) and (8.380) as well as the definition (8.369) in (8.378), and the result in (8.377) we obtain: t 1 [1 + log(1 + t  )] (i1 ...il ) Pl (t  )dt   (i1 ...il ) M l (t) L 2 (Vs+ ) ≤ Cδ0 [1 + log(1 + t)]2 (1 + t  ) 0 (8.381) hence, by (8.373),   

(i1 ...il )

1

M l (t) L 2 (Vs+ ) ≤ Cδ0 [1 + log(1 + t)]2

t 0

(i1 ...il )

(0)

(1)

P l,a (t)+ P l,a (t) ·

[1 + log(1 + t  )] −a   µm (t )dt (1 + t  )

(8.382)

To estimate the integral in (8.382) we follow the proof of Corollary 2 of Lemma 8.11. We have the two cases Eˆ s,m ≥ 0 and Eˆ s,m < 0 to consider. In the first case the lower bound (8.345) holds, hence if 1 δ0 a ≤ , C we have:   1 −a µm (t  )−a ≤ 1 − ≤C (8.383) a and we obtain: t [1 + log(1 + t  )] −a   µm (t )dt ≤ C[1 + log(1 + t)]2 (8.384) (1 + t  ) 0 In the second case we define, as before, δ1 > 0 by Eˆ s,m = −δ1 , and t1 according to (8.292). Then for t  ≤ t1 the lower bound (8.296) holds, hence an estimate of the form (8.384) holds for all t ≤ t1 , in particular if we set t = t1 . It thus remains for us to estimate, when t > t1 , the integral: t [1 + log(1 + t  )] −a   µm (t )dt (8.385) (1 + t  ) t1 Here we have t  > t1 , so the lower bound (8.326) holds, under the smallness condition (8.315) on δ0 . Thus, the integral (8.385) is bounded by a constant multiple of the integral, in terms of the variable τ  (see (8.306)–(8.308)), τ τ (1 − δ1 τ )1−a (1 + τ  )(1 − δ1 τ  )−a dτ  ≤ (1 + τ ) (1 − δ1 τ  )−a dτ  ≤ (1 + τ ) · δ1 (a − 1) τ1 τ1 −1  1 =2 1− τ1 (1 + τ )(1 − δ1 τ )1−a (8.386) a The last equality is by virtue of the first equation of (8.308). By the upper bound (8.335) we then conclude that the last is bounded by a constant multiple of [1 + log(1 + t)]2 µ1−a m (t)

Chapter 8. The Propagation Equation for d/trχ

266

We conclude that, in general, t [1 + log(1 + t  )] −a   µm (t )dt ≤ C[1 + log(1 + t)]2 µ1−a m (t) (1 + t  ) 0

(8.387)

which, substituted in (8.382), yields: 

(i1 ...il )

1

M l (t) L 2 (Vs+ )

 (i1 ...il )

≤ Cδ0 [1 + log(1 + t)]4

(0)

(i1 ...il )

P l,a (t) +

(8.388)



(1)

P l,a (t) µ1−a m (t)

Combining finally the estimates (8.376) and (8.388) (see (8.362)) and taking into account the assumption that the product aδ0 ≤ C −1 , we obtain: 

(i1 ...il )

1

M l (t) L 2 ([0,ε0]×S 2 ) 

≤ Ca −1 (1 + t)



(i1 ...il )

(0)

P l,a (t) + (1 + t)1/2

(i1 ...il )

(1)

P l,a (t) µ−a m (t)

2

(i1 ...il )

We turn to

(i1 ...il )

(8.389)



M l (t, u). We have:

2

M l (t)2L 2 ([0,ε0 ]×S 2 ) = 

(i1 ...il )

2

M l (t)2L 2 (Vs− ) + 

(i1 ...il )

2

M l (t)2L 2 (Vs+ ) (8.390)

and, in analogy with (8.363) and (8.377), 

(i1 ...il )



2

M l (t) L 2 (Vs− ) ≤

and, 

(i1 ...il )

t

i1 ...il )

N l (t, t  ) L 2 (Vs− ) dt 



i1 ...il )

N l (t, t  ) L 2 (Vs+ ) dt 

0



2

M l (t) L 2 (Vs+ ) ≤

t

2



0

(8.391)

2

(8.392)

where: (i1 ...il )

2



N l (t, t , u) =



µ(t, u) µ(t  , u)

2

 3

(1 − u + η0 t )



 1  trχ(t , u) |d/ 2

(i1 ...il )

fˇl (t  , u)| (8.393)

We have: (i1 ...il )

2

N l (t, t  ) L 2 (Vs− )      µ(t) 2 1  trχ(t ≤ (1 + η0 t  )3 max max ) |d/ µ(t  ) 2 Vs− Vs−



(8.394) (i1 ...il )

fˇl (t  )| L 2 (Vs− )

Chapter 8. The Propagation Equation for d/trχ

267

Now, by Corollary 1 to Lemma 8.11, (8.366) holds, while by bootstrap assumption F2,   1 trχ(t  ) ≤ C(1 + t  )−1 max (8.395) [0,ε0 ]×S 2 2 We thus obtain, substituting in (8.394), 

2

(i1 ...il )

N l (t, t  ) L 2 (Vs− ) ≤ C(1 + t  )2 |d/

(i1 ...il )

fˇl (t  )| L 2 ([0,ε0 ]×S 2 )

(8.396)

or, in terms of the definition (8.369), 

2

(i1 ...il )

N l (t, t  ) L 2 (Vs− ) ≤ C



P l,a (t) + (1 + t  )−1/2

 ≤C



Pl (t  )

(0)

(i1 ...il )

≤C

(i1 ...il )

(i1 ...il )

(0)

(i1 ...il )

 −1/2 (i1 ...il )

P l,a (t) + (1 + t )



(1)

 P l,a (t) µ−a m (t )



(1)

P l,a (t) µ−a m (t)

(8.397)

Here, in the last two steps we have appealed to (8.373) and to Corollary 2 of Lemma 8.11. Substituting the estimate (8.397) in (8.391) yields: 

(i1 ...il )



2

M l (t) L 2 (Vs− )

≤ C (1 + t)

(i1 ...il )

(0)

P l,a (t) + (1 + t)1/2



(1)

(i1 ...il )

(8.398)

P l,a (t) µ−a m (t)

In analogy with (8.394) we have: (i1 ...il )

2

N l (t, t  ) L 2 (Vs+ )      µ(t) 2 1  trχ(t max ) |d/ ≤ (1 + η0 t  )3 max µ(t  ) 2 Vs+ Vs+



To estimate



µ(t) max ) µ(t Vs+



 =

max

(u,ϑ)∈Vs+

(8.399) (i1 ...il )

µˆ s (t, u, ϑ) µˆ s (t  , u, ϑ)

fˇl (t  )| L 2 (Vs+ )



we appeal to Proposition 8.6 to obtain:     1 + Cδ0 + Eˆ s (u, ϑ) log(1 + t) µ(t) max (8.400) ≤ max µ(t  ) Vs+ (u,ϑ)∈Vs+ 1 − Cδ0 + Eˆ s (u, ϑ) log(1 + t  ) √ √ We have two cases to distinguish √ according as to whether t  is < t or ≥ t. In the √ second case we have 1 + t  ≥ 1 + t ≥ 1 + t, hence: 1 Eˆ s (u, ϑ) log(1 + t  ) ≥ Eˆ s (u, ϑ) log(1 + t) 2

: ∀(u, ϑ) ∈ Vs+

Chapter 8. The Propagation Equation for d/trχ

268

and from (8.400) we obtain: 

µ(t) max µ(t  ) Vs+



: t ≥

≤2

√ t

(8.401)

provided that: 1 3 On the other hand, in the first case we have, in any case:   µ(t) max ≤ C[1 + log(1 + t)] µ(t  ) Vs+ Cδ0 ≤

: t <

√

t

(8.402)

In view of (8.395) and the definition (8.369) we then obtain:  

2

(i1 ...il )

N l (t, t  ) L 2 (Vs+ ) ≤ C[1 + log(1 + t)]2

(i1 ...il )

2

N l (t, t  ) L 2 (Vs+ ) ≤ C

(i1 ...il )

Pl (t  ) √ : if t  ≥ t

Pl (t  )

(i1 ...il )

: if t  <

√

t

(8.403)

Substituting the estimates (8.403) in (8.392), yields: 

2

(i1 ...il ) √



≤

M l (t) L 2 (Vs+ )

t

2

i1 ...il )



N l (t, t  ) L 2 (Vs+ ) dt  +

0

√



≤ C[1 + log(1 + t)]2 0

≤ C[1 + log(1 + t)] +C

t

√

t

0

  µ−a m (t )

+ C (1 + t)

(i1 ...il )

 (1 + t)

(i1 ...il )

(i1 ...il )



N l (t, t  ) L 2 (Vs+ ) dt  t

(i1 ...il )

√ t

 −1/2 (i1 ...il )

P l,a (t) + (1 + t )

P l,a (t) + (1 + t  )−1/2 1/2 (i1 ...il )

(i1 ...il )

(0)

(1)

1/2 (i1 ...il )

1/2 (i1 ...il )

(1)



(1)

P l,a (t) dt 



P l,a (t) dt 

P l,a (t) + (1 + t)

P l,a (t) + (1 + t)

P l,a (t) + (1 + t)

Pl (t  )dt 

(0)

(0)

(0)

(0)

2

i1 ...il )

t



(1 + t)

2



Pl (t  )dt  + C

 µ−a m (t )

(i1 ...il )



≤C

t

(i1 ...il )

t √



≤ C[1 + log(1 + t)]



√

2

t



1/4 (i1 ...il )

(1)



P l,a (t) µ−a m (t)



P l,a (t) µ−a m (t)

(1)



P l,a (t) µ−a m (t)

(8.404)

Here, in the last steps we have again appealed to (8.373) and to Corollary 2 of Lemma 8.11.

Chapter 8. The Propagation Equation for d/trχ

269

Combining finally the estimates (8.398) and (8.404) (see (8.390)), we obtain: 

(i1 ...il )



2

M l (t) L 2 ([0,ε0]×S 2 )

≤ C (1 + t)

(i1 ...il )

(0)

P l,a (t) + (1 + t)1/2



(1)

(i1 ...il )

(8.405)

P l,a (t) µ−a m (t)

The estimates (8.359), (8.376) and (8.405) together yield, through (8.187), the following estimate for the L 2 norm of (i1 ...il ) Fl (t) on [0, ε0] × S 2 : 

(i1 ...il )

Fl (t) L 2 ([0,ε0 ]×S 2 )

≤ C(1 + t)−3 [1 + log(1 + t)]2   + C(1 + t)

−2

(i1 ...il )

(i1 ...il )

xl (0) L 2 ( ε0 )



0

(0)

P l,a (t) + (1 + t)

−1/2 (i1 ...il )

(1)

P l,a (t) µ−a m (t)

(8.406)

We now consider the estimate (8.191) for the function (i1 ...il ) G l (t, u). By (8.366) and (8.380) we have, in general,   µ(t) ≤ C[1 + log(1 + t)] (8.407) max  [0,ε0 ]×S 2 µ(t ) Hence (8.191) implies: (i1 ...il )

G l (t, u) ≤ C(1 + t)−3 [1 + log(1 + t)]2



t

(1 + t  )3 |

(i1 ...il )

g˜l (t  , u)|dt  (8.408)

0

It follows that: 

(i1 ...il )

G l (t) L 2 ([0,ε0 ]×S 2 )

≤ C(1 + t)−3 [1 + log(1 + t)]2 ·

(8.409)



t

(1 + t  )3 |

(i1 ...il )

0

g˜ l (t  )| L 2 ([0,ε0 ]×S 2 ) dt 

Now the St,u 1-form (i1 ...il ) g˜l is given by (8.162). Here we must distinguish the principal acoustical terms . Defining, in reference to (8.68), the function: (i1 ...il ) ˙

hl =

(i1 ...il )

(i1 ...il )

h l − 2χˆ ·

χˆl

(8.410)

(see Proposition 8.4), the differential d/ (i1 ...il ) h˙ l of this function on St,u does not contain principal acoustical terms . Also, according to the discussion following Lemma 8.4, the principal acoustical part of the St,u 1-form (i1 ...il ) gl consists of the sum (8.94). Therefore the St,u 1-form: (i1 ...il )

g˙l =

(i1 ...il )

gl +

l−1  k=0

L / (Ril−k )

i



. . il ) (i1 . l−k Z

xl−1

(8.411)

Chapter 8. The Propagation Equation for d/trχ

270

does not contain principal acoustical terms . We conclude that, writing: (i1 ...il )

g˜l = 2µχˆ · D /

(i1 ...il )

l−1 

χˆl −

L / (Ril−k )

i



(i1 . l−k . . il ) Z

xl−1 +

(i1 ...il )

g¨l

(8.412)

k=0

the St,u 1-form

(i1 ...il ) g¨ , l (i1 ...il )

given by:

g¨l =

(i1 ...il )

g˙l + µ(2D / χˆ ·

(i1 ...il )

χˆl + d/

(i1 ...il ) ˙

hl )

(8.413)

does not contain principal acoustical terms. Let us define: X l (t) = max |

(i1 ...il )

i1 ...il

xl (t)| L 2 ([0,ε0]×S 2 )

(8.414)

We shall estimate | (i1 ...il ) g˜l (t)| L 2 ([0,ε0 ]×S 2 ) in terms of X l (t). Consider first the second term on the right in (8.412). By the estimate (8.103) we have, pointwise:   l−1   il−k    L / (Ril−k ) (i1 . . . il ) xl−1   Z   k=0   l−1     i  l−k  (Ril−k )  Z | (i1 . . . il j ) xl | + (i1 ...il ) z l (8.415) ≤ C(1 + t)−1    j

k=0

(i1 ...il ) z

is the non-negative function:   l−1   il−k  (i1 ...il ) zl = | (Ril−k ) Z |  |d/( (i1 . . . il j ) fˇl )| + (1 + t)|d/µ||d/(Ril i. l−k . . Ri1 trχ)|

where

l

k=0

+

j

l−1  

  /R j max L

k=0

j

(Ril−k )

   Z |

i



(i1 . l−k . . il )

xl−1 | + |d/(

i



(i1 . l−k . . il )

fˇl−1 )|



(8.416) Now, the assumptions of Lemma 8.11 together with the bootstrap assumption F1 of Chapter 6 imply the estimate (6.177), that is: |

(Ri )

Z | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

Substituting this in (8.415) it follows that: 0 l−1 0 0 0 il−k  0 0 L / (Ril−k ) (i1 . . . il ) xl−1 0 0 Z 0 0 k=0

(8.417)

(8.418)

L 2 ([0,ε0 ]×S 2 )

≤ Clδ0 (1 + t)−2 [1 + log(1 + t)]X l (t) + C(1 + t)−1 

(i1 ...il )

z l (t) L 2 ([0,ε0 ]×S 2 )

Chapter 8. The Propagation Equation for d/trχ

271

Consider next the first term on the right in (8.412). By the 2nd of bootstrap assumptions F2 we have: (i1 ...il )

2µ|χˆ · D /

≤ Cδ0 (1 + t)

−2

χˆl |(t) L 2 ([0,ε0 ]×S 2

[1 + log(1 + t)]µ|D /

(i1 ...il )

χˆl |(t) L 2 ([0,ε0]×S 2

(8.419)

We now consider the elliptic estimate (8.161) on St,u . Remarking that by (8.355) the inequalities C −1 (1 + t)|ξ(t, u)| L 2 (S 2) ≤ ξ  L 2 (St,u ) ≤ C(1 + t)|ξ(t, u)| L 2 (S 2)

(8.420)

hold with ξ being any St,u covariant tensorfield of any rank and ξ(t, u) the corresponding tensorfield on S 2 , depending on t and u, the estimate (8.161) is equivalent to: µ|D /

(i1 ...il )

≤ C|

χˆl |(t, u) L 2 (S 2)

(i1 ...il )

+ C|d/

xl (t, u)| L 2 (S 2)

(i1 ...il )

fˇl (t, u)| L 2 (S 2) + Cµ|

+ Cδ0 (1 + t)−1 [1 + log(1 + t)]|

(i1 ...il )

(i1 ...il )

i l |(t, u) L 2 (S 2)

χˆl (t, u)| L 2 (S 2)

(8.421)

Taking L 2 -norms on [0, ε0 ] then yields: µ|D /

(i1 ...il )

χˆl |(t) L 2 ([0,ε0]×S 2 ) ≤ C|

+ C|d/

(i1 ...il )

(i1 ...il )

xl (t)| L 2 ([0,ε0]×S 2 )

fˇl (t)| L 2 ([0,ε0]×S 2 ) + Cµ|

+ Cδ0 (1 + t)−1 [1 + log(1 + t)]|

(i1 ...il )

(i1 ...il )

i l |(t) L 2 ([0,ε0]×S 2 )

χˆl (t)| L 2 ([0,ε0 ]×S 2 )

(8.422)

In view of the estimates (8.418), (8.419) and (8.422), we obtain from (8.412) the estimate: |

(i1 ...il )

g˜l (t)| L 2 ([0,ε0 ]×S 2 )

≤ C(l + 1)δ0 (1 + t)−2 [1 + log(1 + t)]X l (t) +

(i1 ...il )

Q l (t)

(8.423)

where: (i1 ...il )

Q l (t) = Cδ02 (1 + t)−3 [1 + log(1 + t)]2 |

(i1 ...il )

χˆl (t)| L 2 ([0,ε0 ]×S 2 )

[1 + log(1 + t)]|d/

(i1 ...il )

fˇl (t)| L 2 ([0,ε0]×S 2 )

+Cδ0 (1 + t)−2 [1 + log(1 + t)]µ|

(i1 ...il )

i l |(t) L 2 ([0,ε0]×S 2 )

+Cδ0 (1 + t) +C(1 + t) +|

(i1 ...il )

−2

−1



(i1 ...il )

z l (t) L 2 ([0,ε0 ]×S 2 )

g¨l (t)| L 2 ([0,ε0]×S 2 )

(8.424)

We now return to the estimate (8.182). Taking L 2 -norms on [0, ε0] × S 2 yields: |

(i1 ...il )

≤

xl (t)| L 2 ([0,ε0]×S 2 )

(i1 ...il )

Fl (t) L 2 ([0,ε0 ]×S 2 ) + 

(i1 ...il )

G l (t) L 2 ([0,ε0]×S 2 )

(8.425)

Chapter 8. The Propagation Equation for d/trχ

272

Substituting the estimates (8.406), (8.409) and (8.423) then yields: |

(i1 ...il )

xl (t)| L 2 ([0,ε0 ]×S 2 ) ≤

(i1 ...il )

Bl (t) (8.426) t (1 + t  )[1 + log(1 + t  )]X l (t  )dt  + C(l + 1)δ0 (1 + t)−3 [1 + log(1 + t)]2 0

where: (i1 ...il )

Bl (t) = C(1 + t)−3 [1 + log(1 + t)]2   + C(1 + t)

−2

+ C(1 + t)

−3

(i1 ...il )

(i1 ...il )

xl (0) L 2 ( ε0 ) 0

(0)

P l,a (t) + (1 + t)

[1 + log(1 + t)]

t

2

−1/2 (i1 ...il )

(1 + t  )3

(i1 ...il )

(1)



P l,a (t) µ−a m (t)

Q l (t  )dt 

(8.427)

0

Taking in (8.426) the maximum over i 1 . . . i l and recalling the definition (8.414) we obtain the following ordinary integral inequality for X l : t (1 + t  )[1 + log(1 + t  )]X l (t  )dt  X l (t) ≤ Bl (t) + C(l + 1)δ0(1 + t)−3 [1 + log(1 + t)]2 0

where: Bl (t) = max

(i1 ...il )

i1 ...il



Let us set:

t

Yl (t) =

Bl (t)

(1 + t  )[1 + log(1 + t  )]X l (t  )dt 

(8.428) (8.429)

(8.430)

0

Then, since:

dYl (t) = (1 + t)[1 + log(1 + t)]X l (t) dt Yl satisfies by virtue of (8.428) the ordinary differential inequality: dYl (t) ≤ (1 + t)[1 + log(1 + t)]Bl (t) dt +C(l + 1)δ0 (1 + t)−2 [1 + log(1 + t)]3 Yl (t) The integrating factor here is:

(8.431)

e−Cl (t )

where:



[1 + log(1 + t  )]3  dt (8.432) (1 + t  )2 0 Integrating the ordinary differential inequality (8.431) from t = 0 where Yl vanishes, using the above integrating factor, yields: t  e−Cl (t ) (1 + t  )[1 + log(1 + t  )]Bl (t  )dt  (8.433) Yl (t) ≤ eCl (t ) Cl (t) = C(l + 1)δ0

0

t

Chapter 8. The Propagation Equation for d/trχ

273

Now, from (8.432) we have, since the integral

∞ 0

is convergent,

[1 + log(1 + t  )]3  dt (1 + t  )2

0 ≤ Cl (t) ≤ C  (l + 1)δ0

(8.434)

Hence, if δ0 satisfies the smallness condition: δ0 ≤

log 2 + 1)

C  (l

then (8.433) implies the estimate: t Yl (t) ≤ 2 (1 + t  )[1 + log(1 + t  )]Bl (t  )dt 

(8.435)

(8.436)

0

Substituting this estimate in (8.426), recalling the definition of Yl (t), we finally conclude that: |

(i1 ...il )

xl (t)| L 2 ([0,ε0]×S 2 ) ≤

(i1 ...il )

Bl (t)

+ 2C(l + 1)δ0 (1 + t)−3 [1 + log(1 + t)]2

0

(8.437) t

(1 + t  )[1 + log(1 + t  )]Bl (t  )dt 

Chapter 9

Regularization of the Propagation Equation for
/ µ. Estimates for the Top Order Spatial Derivatives of µ In the present chapter we shall deal with the problem of obtaining estimates for µ and its derivatives with respect to the set of commutation vectorfields {T, Ri : i = 1, 2, 3}. Since these commutation fields are tangential to the hypersurfaces t and span the tangent space to the t at each point, we call the corresponding derivatives spatial derivatives. The problem which shall concern us in the present chapter shall be that of obtaining estimates for the spatial derivatives of µ of order n + 1, given energy estimates of order n + 1 for the ψµ : µ = 0, 1, 2, 3, the partial derivatives of the wave function φ with respect to the rectangular coordinates. That is, we shall estimate the former in terms of the  , defined in Chapter 7, associated to n + 1st order variations quantities E0,n+1 and E1,n+1 ψn+1 . These variations are obtained by applying to the ψµ a string of commutation fields of length n. We are thus requiring estimates of the n + 1st order spatial derivatives of µ in terms of estimates for the n + 1st order derivatives of the ψµ . However not all n + 1st order spatial derivatives of µ can be so estimated. What we shall estimate is the 2nd order angular derivatives of the n − 1st order spatial derivatives of µ. In other words, at least two of the spatial derivatives shall be with respect to the set of rotation vectorfields {Ri : i = 1, 2, 3}. The 1st order angular derivatives of the nth order derivatives of µ with respect to T , and the n +1st order derivatives of µ with respect to T , cannot be estimated in terms of the n + 1st order derivatives of the ψµ , and are not required in our construction, as shall be seen in Chapter 13. The function µ satisfies the propagation equation (3.96): Lµ = m + µe

(9.1)

Chapter 9. The Propagation Equation for
/µ

276

On the right-hand side we have the functions m and e, given by (3.97) and (3.98) respectively: dH 1 (ψ L )2 (T σ ) 2 dσ

(9.2)

dH 1 ψ L (ψ L + 2αψTˆ ) (Lσ ) − Fψ0 L µ (Lψµ ) 2 dσ 2α

(9.3)

m= and: e=−

We see that e contains the 1st derivatives of the ψµ , but only with respect to L. However, m contains the 1st derivatives of the ψµ with respect to T , through the derivative of σ with respect to T . Now, the propagation equation (9.1) is an ordinary differential equation along the generators of the characteristic hypersurfaces Cu . Thus, in integrating this equation to obtain estimates for µ, no regularity is expected to be gained except along the generators. Consequently, in the estimates obtained in this way there will be a loss of one degree of differentiability along the transversal hypersurfaces t , and we can only estimate the nth order spatial derivatives of µ in terms of the n + 1st order derivatives of the ψµ . Such estimates, that is estimates of the next-to-the-top order spatial derivatives of µ in terms of the top order energies are in fact also needed in our approach, and shall be derived in Chapter 12. In the present chapter however, we concentrate on the main problem, which is that of deriving estimates for the top order spatial derivatives of µ, of which at least two are angular, in terms of the top order energies. To accomplish this aim we must avoid the loss of one degree of differentiability along the transversal hypersurfaces t . A propagation equation of the type we are considering does not lead to such a loss of differentiability when – and only when – the principal part on the right-hand side can be put into the form of a derivative with respect to L. As we already remarked, this is true for the function e, as is evident from the expression (9.3), but it is not true for the function m, because of the presence of the factor T σ . Now according to Proposition 8.1, σ satisfies ˜ Commuting the an inhomogeneous wave equation in the conformal acoustical metric h. vectorfield T with the operator
h˜ , it follows that T σ also satisfies an equation of this type. In view of the expression for the operator
h˜ given in Chapter 3, this implies that µ
/ (T σ ) is equal, up to lower order terms, to the derivative of L(T σ ) with respect to L. Thus, if we derive from (9.1) a propagation equation for µ
/ µ, that equation will indeed have the sought for property, namely the principal part on the right-hand side will be of the form of a derivative with respect to L. In deriving this equation we shall make use of the following lemma. Lemma 9.1 The following commutation formula holds: / χˆ · d/φ [L,
/ ]φ + trχ
/ φ = −2χˆ · D /ˆ2 φ − 2div Here, φ is an arbitrary function defined on Wε∗0 .

Chapter 9. The Propagation Equation for
/µ

277

Proof. Let us work in acoustical coordinates (t, u, ϑ 1 , ϑ 2 ). We have:
/ φ = (h /−1 ) AB D / A (d/ B φ)   ∂ 2φ −1 AB C ∂φ −! / AB C = (h / ) ∂ϑ A ∂ϑ B ∂ϑ

(9.4)

Since in acoustical coordinates L = ∂/∂t, we obtain:      2 ∂ ∂φ ∂φ ∂ L
/ φ = (h /−1 ) AB −! /CAB C ∂ϑ A ∂ϑ B ∂t ∂ϑ ∂t +

∂! /C ∂φ ∂ −1 AB (h / ) D / A (d/ B φ) − (h/−1 ) AB AB ∂t ∂t ∂ϑ C

We have:

∂ −1 AB (h / ) = −2χ AB ∂t

(9.5)

(9.6)

(see (8.167)), and:    ∂! /CAB 1 ∂ ∂h/ AD ∂h/ AB /B D −1 C D ∂h = + − (h / ) ∂t 2 ∂t ∂ϑ A ∂ϑ B ∂ϑ D   ∂χ AD ∂χ AB CD E −1 C D ∂χ B D = −χ ! / AB h / D E + (h/ ) + − ∂ϑ A ∂ϑ B ∂ϑ D / AχB D + D / B χ AD − D / D χ AB ) = (h −1 )C D (D that is:

∂! /CAB =D / A χ BC + D / B χ AC − D / C χ AB ∂t

(9.7)

hence:

∂! /CAB = 2div / χ C − d/C trχ = 2div / χˆ C (9.8) ∂t Substituting (9.8) and (9.6) in (9.5) and noting that the first term on the right-hand side of (9.5) is simply
/ (∂φ/∂t) =
/ Lφ we obtain: (h /−1 ) AB

/ A (d/ B φ) − 2(div / χ) ˆ A d/ A φ L
/φ =
/ Lφ − 2χ AB D

(9.9)

Writing then: / A (d/ B φ) = trχ
/ φ + 2χˆ AB D / A (d/ B φ) 2χ AB D the lemma follows. We now apply Lemma 9.1 to the function µ. Using the propagation equation (9.1), as well as the Codazzi equation (8.104) to express div / χˆ in terms of d/trχ, we then obtain: L(
/ µ) + (trχ − e)
/ µ = −2χˆ · D /ˆ2 µ − d/µ · (d/trχ + 2i − 2d/e) +
/ m + µ
/e (9.10)

Chapter 9. The Propagation Equation for
/µ

278

To proceed with the program outlined above, we must derive the inhomogeneous ˜ satisfied by T σ . According to Proposition wave equation, with respect to the metric h, ˜ 8.1 σ itself satisfies the following inhomogeneous wave equation with respect to h:
h˜ σ = τ where:

τ = −2(h˜ −1 )µν ωµα ωνα ;

(9.11)

ωµα = ∂µ ψα , ωµα = (g −1 )αβ ωµβ

(9.12)

We now apply Proposition 7.1 with σ and τ in the roles of ψ and ρ respectively, taking Y to be the vectorfield T . This yields the following inhomogeneous wave equation for T σ , ˜ in the metric h: (9.13)
h˜ (T σ ) = τ˙ where:

1 J˜ + T τ + tr˜ (T ) πτ ˜ 2 and (T ) J˜ is the commutation current associated to σ and T , given by: ˜ τ˙ = div

(T )

(9.14)

1 ˜ −1 µα ˜ −1 νβ ((h ) (h ) + (h˜ −1 )να (h˜ −1 )µβ − (h˜ −1 )µν (h˜ −1 )αβ ) (T ) π˜ αβ ∂ν σ (9.15) 2 Next, we consider equations (7.17)–(7.21) with σ and τ in the roles of ψn−1 and ρn−1 respectively. Then T σ and τ˙ play the roles of ψn and ρn respectively. Thus, defining: (T ) ˜µ

J =

τ˜˙ = 2 µτ˙

τ˜ = 2 µτ,

(9.16)

we have, in reference to the formula (7.18), τ˜˙ = v + T τ˜ +

(T )

δ τ˜

(9.17)

(T ) σ n−1 ,

where v plays the role of given by equation (7.36) and equations (7.43)–(7.48) with ψn−1 = σ , Y = T . We thus have: v = v1 + v2 + v3

(9.18)

where: v1 = (1/2)tr (T ) π /˜ (L Lσ + ν Lσ ) −1 (T ) π˜ L L )L 2 σ +(1/2)(µ − (T ) Z˜ · d/ Lσ − (T ) Z˜ · d/ Lσ +(1/2) v2 = (1/4)L(tr

(T )

π˜ L L
/σ + µ

(T ) ˜

(T ) ˆ˜

π /)Lσ + (1/4)L(tr

+(1/4)L(µ

−1 (T )

π /·D / 2σ (T ) ˜

π /)Lσ

π˜ L L )Lσ

Z˜ · d/σ − (1/2)L / L (T ) Z˜ · d/σ / (T ) Z˜ (Lσ ) −(1/2)div / (T ) Z˜ (Lσ ) − (1/2)div /ˆ˜ ) · d/σ / (µ (T ) π +(1/2)d/ (T ) π˜ · d/σ + div

−(1/2)L /L

(9.19)

(T )

LL

(9.20)

Chapter 9. The Propagation Equation for
/µ

279

and:

L

v3 = v3L Lσ + v3 Lσ + v/3 · d/σ

(9.21)

where: v3L = (1/4)trχtr

(T ) ˜

π / + (1/4)trχ(µ−1

+(1/2) Z˜ · d/(α −1 κ) L v3 = −(1/4)(L log )tr (T ) π /˜ (T ) ˜ π /) v/3 = −(1/2)(tr

(T )

π˜ L L )

(T )

−(1/2)(trχ + L(α −1 κ))

(9.22) (9.23) (T )

Z˜ − (1/2)trχ

(T )

Z˜

(9.24)

Here, the components of (T ) π˜ are given by Table 6.12 of Chapter 6. We see from this table that the components of (T ) π˜ involve χ and the 1st spatial derivatives of µ. Now, v1 consists of products of components of (T ) π˜ with 2nd derivatives of σ , while v2 consists of products of 1st derivatives of components of (T ) π˜ with 1st derivatives of σ . The term v3 is of lower order. Thus v2 contains the 2nd spatial derivatives of µ as well as the 1st spatial derivatives of χ. It turns out that the 2nd spatial derivatives of µ occur only as 2nd order angular derivatives. To see this we consider separately the terms proportional to Lσ , Lσ and d/σ in (9.20), writing: L (9.25) v2 = v2L Lσ + v2 Lσ + v/2 · d/σ where: v2L = (1/4)L(µ−1 +(1/4)L(tr L v2

= (1/4)L(tr

(T )

π˜ L L )

(T ) ˜

π /) − (1/2)div /

(T ) ˜

π /) − (1/2)div / (T ) ˜ Z − (1/2)L /L /L v/2 = −(1/2)L +(1/2)h /−1 · d/

(T )

(T )

Z˜

(9.26)

Z˜ (T ) ˜ Z (T )

(9.27)

π˜ L L + h/−1 · div / (µ

(T ) ˆ˜

π /)

(9.28)

(note that v/2 is, like v/3 , an St,u -tangential vectorfield). In v2L , v2 , v/2 , the components of (T ) π˜ which are differentiated with respect to L do not contribute principal acoustical terms, because the derivatives with respect to L of the acoustical entities are expressed by the propagation equations. Also, writing L

L = 2T + α −1 κ L the derivatives with respect to L are expressed in terms of derivatives with respect to T . Moreover, recalling that η = ζ + d/µ, equation (4.109) of Chapter 4 expresses L /T χ in terms of D / 2 µ. Thus, the part of v2L which contributes principal acoustical terms is: (1/2)T (tr

(T ) ˜

π /) − (1/2)div /

(T )

Z˜

Chapter 9. The Propagation Equation for
/µ

280

From Table 6.12 we see that the principal acoustical part of this is: α −1 κ(−T trχ + (1/2)
/ µ) Now, by the trace of equation (4.109), T trχ is equal to
/ µ to principal acoustical terms. We conclude that the principal acoustical part of v2L is simply: −(1/2)α −1 κ
/µ

(9.29)

L

The part of v2 which contributes principal acoustical terms is: −(1/2)div /

(T )

Z˜

and from Table 6.12 the principal acoustical part of this is: (1/2)
/µ

(9.30)

We turn to the St,u -tangential vectorfield v/2 . The part of v/2 which contributes principal acoustical terms is: Z˜ + (1/2)h/−1 · d/ /ˆ˜ ) +h /−1 · div / (µ (T ) π −L /T

(T )

(T )

π˜ L L (9.31)

Now, from Table 6.12, the principal acoustical part of −L /T (T ) Z˜ is the St,u -tangential vectorfield corresponding – through the induced metric h/ – to the St,u 1-form: d/T µ On the other hand, again from Table 6.12, the principal acoustical part of (1/2)d/ is: −d/T µ

(T ) π ˜

LL

Consequently, the principal acoustical parts of the first two terms of (9.31) cancel. Only the third term remains, the principal part of which is, by Table 6.12 the St,u tangential vectorfield corresponding to the St,u 1-form: −2κ 2div / χˆ and by the Codazzi equation (8.104) this is equal to: −κ 2 d/trχ

(9.32)

to principal acoustical terms. We conclude from the above that v is of order 2 in the sense of Chapter 8, and the principal acoustical terms in v are 2nd angular derivatives of µ and 1st angular derivatives of χ. In fact, the principal acoustical part of v is contained in v2 and is given by: {(
/ µ)(T σ ) − κ 2 (d/trχ) · (d/σ )}

(9.33)

Chapter 9. The Propagation Equation for
/µ

Thus, defining v  by:

281

/ µ)(T σ ) v  = v − (


(9.34)

the principal acoustical part of v  consists only of 1st angular derivatives of trχ, being in fact given by: (9.35) −κ 2(d/trχ) · (d/σ ) Moreover, v  is bounded as µ → 0. Consider now the function τ˜ , defined by (9.12) and (9.16). We have: τ = −2−1b, where b is the function:

τ˜ = −2µb

b = (h −1 )µν ωµα ωνα

(9.36) (9.37)

defined in the statement of Proposition 8.1. We have (see (8.33)): b = −µ−1 (Lψ α )(Lψα ) + d/ψ α · d/ψα hence:

τ˜ = −2{−(Lψ α )(Lψα ) + µd/ψ α · d/ψα }

(9.38)

We see that τ˜ is of order 1 and bounded as µ → 0. It follows that the term T τ˜ in (9.17) is of order 2 and bounded as µ → 0. It only contains the 1st spatial derivatives of µ. The term (T ) δ τ˜ in (9.17) is of lower order, the function (T ) δ being given by (6.191). Defining now the function τ˙  by: τ˙  = −1 τ˜˙ − (
/ µ)(T σ ) = −1 (v  + T τ˜ +

(T )

δ τ˜ )

(9.39)

we conclude from the above that τ˙  is of order 2 and its principal acoustical terms are 1st angular derivatives of trχ. Moreover, this function is bounded as µ → 0. By equations (9.13) and (9.16), / µ)}
h˜ (T σ ) = −2 µ−1 τ˜˙ = −1 µ−1 {τ˙  + (T σ )(


(9.40)

We now substitute for
h˜ (T σ ) from the general formula (3.133) of Chapter 3 for the / (T σ ) in the form: operator
h˜ , applied to the function T σ , to express µ


where:

µ
/ (T σ ) = L(L T σ ) + (T σ )(
/ µ) + n 0 + µn 1

(9.41)

n 0 = τ˙  + ν(LT σ ) + ν(LT σ ) − 2ζ · d/T σ

(9.42)

d d/σ · d/T σ (9.43) dσ The functions n 0 and n 1 are of order 2 and bounded as µ → 0. The principal acoustical part is contained in the term τ˙  , in fact in v  , and is given by (9.35) above. n 1 = −−1

Chapter 9. The Propagation Equation for
/µ

282

We now consider the function m, given by (9.2). We have:     1 1 2 dH 2 dH 2 dH
/ (T σ ) + d/ ψ L · d/T σ + (T σ )
/ ψL
/ m = ψL 2 dσ dσ 2 dσ

(9.44)

Substituting for
/ (T σ ) from (9.41) we obtain:
/m = where:

1 −1 2 d H µ ψL L(LT σ ) + µ−1 m(
/ µ) + µ−1 n 0 + n 1 2 dσ

dH 1 n 0 = ψ L2 n0 2 dσ     1 1 dH dH dH n 1 = ψ L2 n 1 + d/ ψ L2 · d/T σ + (T σ )
/ ψ L2 2 dσ dσ 2 dσ

(9.45)

(9.46) (9.47)

We then re-write (9.45) in the form:
/ m = L(µ−1 f 0 ) + ν(µ−1 f 0 ) + µ−1 (Lµ)(µ−1 f 0 ) + µ−1 m(
/ µ) + µ−1 n 0 + n 1 (9.48) where:

1 2 dH ψ (LT σ ) 2 L dσ   1    2 dH n0 = n0 − ν f0 − L ψL (LT σ ) 2 dσ f0 =

and:

(9.49)

(9.50)

Note that the functions n 0 , n 0 , n 1 , and f 0 , are all of order 2 and bounded as µ → 0. Substituting for n 0 from (9.42) in (9.46) and the result in (9.50) in fact yields:   1 dH n 0 = − L ψ L2 (LT σ ) 2 dσ    1 dH  + ν(LT σ ) − 2ζ · d/T σ + τ˙  (9.51) ψ L2 2 dσ The principal acoustical part of n 0 is contained in the term 1 2 dH  ψ τ˙ 2 L dσ and is given by: 1 dH 2 − ψ L2 κ d/σ · d/trχ 2 dσ Also, substituting for n 1 from (9.43) in (9.47) yields: d H d log  1 d/σ · d/T σ n 1 = − ψ L2 2  dσ dσ   1 2 dH 2 dH +d/ ψ L / ψL · d/T σ + (T σ )
dσ 2 dσ

(9.52)

(9.53)

Chapter 9. The Propagation Equation for
/µ

283

We note here that the last term on the right in (9.53) contains a principal acoustical part, / (L i ). This shall be shown below to depend on the 1st angular because
/ ψ L contains
derivatives of trχ. We turn to the function e, given by (9.3). Introducing the vectorfield:

we have:

Lˆ = µ−1 L

(9.54)

ˆ = −2 h(L, L)

(9.55)

and Lˆ is expressed in terms of L and Tˆ by:

Thus, setting:

Lˆ = α −2 (L + 2α Tˆ )

(9.56)

µ ψ Lˆ = Lˆ ψµ

(9.57)

(rectangular coordinates), (9.3) may be written more simply as: 1 dH e = − ψ L ψ Lˆ (Lσ ) − Fψ0 L µ (Lψµ ) 2 dσ

(9.58)

dH 1 L(
/ σ ) − Fψ0 L µ L(

/ e = − ψ L ψ Lˆ / ψµ ) + n 2 2 dσ

(9.59)

We then obtain:

where:

  dH n 2 = −d/ ψ L ψ Lˆ · d/(Lσ ) − 2d/(Fψ0 L µ ) · d/(Lψµ ) dσ   1 dH −
/ ψ L ψ Lˆ (Lσ ) −
/ (Fψ0 L µ )(Lψµ ) 2 dσ dH 1 [L,
/ ]σ + Fψ0 L µ [L,
/ ]ψµ + ψ L ψ Lˆ 2 dσ

(9.60)

and, by Lemma 9.1 and the Codazzi equation (8.104): [L,
/ ]σ = −trχ
/ σ − 2χˆ · D /ˆ2 σ − 2div / χˆ · d/σ ˆ = −trχ
/ σ − 2χˆ · D / 2 σ − (d/trχ + 2i ) · d/σ / ψµ − 2χˆ · D /ˆ2 ψµ − 2div / χˆ · d/ψµ [L,
/ ]ψµ = −trχ
= −trχ
/ ψµ − 2χˆ · D /ˆ2 ψµ − (d/trχ + 2i ) · d/ψµ

(9.61)

(9.62)

We see that n 2 is of order 2 and bounded as µ → 0. We note that the 3rd and 4th terms on the right in (9.60) contain principal acoustical parts corresponding to
/ (L i ) and
/ (Tˆ i ) / ψ Lˆ contains both). These shall be shown (for,
/ ψ L contains the first, while by (9.56)


Chapter 9. The Propagation Equation for
/µ

284

below to depend on the 1st angular derivatives of trχ. The principal acoustical terms in n 2 are, besides the terms just mentioned, the terms contributed by the last terms on the right in (9.61) and (9.62), which likewise depend on the 1st angular derivatives of trχ. We re-write (9.59) in the form:

where:

and: n 2

µ
/ e = L(µf 1 ) + n 2

(9.63)

dH 1 (
/ σ ) − Fψ0 L µ (
f 1 = − ψ L ψ Lˆ / ψµ ) 2 dσ

(9.64)

  dH 1
/ σ + L(µFψ0 L µ )
= µn 2 + L µψ L ψ Lˆ / ψµ 2 dσ

(9.65)

Note that the functions n 2 and f1 are both of order 2 and bounded as µ → 0. The function f 1 does not contain principal acoustical terms, while the principal acoustical part of n 2 is contained in the term µn 2 and corresponds to 1st angular derivatives of trχ as discussed above. We now substitute for
/ m from (9.48) and for µ
/ e from (9.63) into the propagation equation (9.10). Bringing the terms L(µ−1 f0 ) and L(µf 1 ) to the left-hand side, the propagation equation then takes the form: L(
/ µ − µ−1 f 0 − µf 1 ) = −(trχ − µ−1 (Lµ))(
/ µ) + (ν + µ−1 (Lµ))(µ−1 f 0 ) − 2χˆ · D /ˆ2 µ − d/µ · (d/trχ + 2i − 2d/e) + µ−1 n 0 + n 1 + n 2 (9.66) Let us define the functions: fˇ = f 0 + µ2 f 1

(9.67)

gˇ  = −µd/µ · (d/trχ + 2i − 2d/e) 1 d log  1 (Lσ ) f 0 − (µtrχ + 2(Lµ))µf 1 + 2 dσ 2 +n 0 + µ(n 1 + n 2 )

(9.68)

The functions fˇ and gˇ  are of order 2 and bounded as µ → 0. The function fˇ has no principal acoustical part, while the principal acoustical part of gˇ  consists of 1st angular derivatives of trχ. We also introduce, in place of
/ µ, the function: / µ − fˇ x  = µ
Then, recalling that

  1 d log  (Lσ ) , ν= trχ + 2 dσ

(9.69)

Chapter 9. The Propagation Equation for
/µ

285

equation (9.66) is seen to be equivalent to the following propagation equation for x  :   1  −1  −1 trχ − 2µ (Lµ) fˇ Lx + (trχ − 2µ (Lµ))x = − 2 −2µχˆ · D /ˆ2 µ + gˇ  (9.70) We now pause to derive expressions for
/ (L i ) and
/ (Tˆ i ). Now d/(L i ) and d/(Tˆ i ) are given by (3.205) and (3.221) respectively: d/(L i ) = a /Tˆ i + b/ · d/x i ,

d/(Tˆ i ) = p/Tˆ i + q/ · d/x i

(9.71)

Here a / and p/ are the St,u 1-forms given by (3.213) and (3.226) respectively, while b/ and q/ are the T11 -type St,u tensorfields given by (3.217) and (3.230) respectively. Note that the components of the St,u 1-forms d/x i , in an arbitrary local frame field (X A : A = 1, 2) for St,u , are simply the rectangular components of the vectorfields X A : d/ A x i = d/x i (X A ) = X A x i = X iA

(9.72)

Thus, we have: (b/ · d/x i )(X A ) = b/ BA d/ B x i = b/ BA X iB ,

(q/ · d/x i )(X A ) = q/ BA d/ B x i = q/ BA X iB

(9.73)

Taking the St,u divergence of each of (9.71) we obtain:
/ (L i ) = (div / a /+a / · p/)Tˆ i + a/ · (q/ · d/x i ) + (div / b/) · d/x i + b/ · D /2x i
/ (Tˆ i ) = (div / p/ + | p/|2 )Tˆ i + p/ · (q/ · d/x i ) + (div / q/) · d/x i + q/ · D /2x i

(9.74)

Here div / b/ and div / q/ are the St,u -tangential vectorfields with components: (div / b/) B = D / A b/ BA ,

(div / q/) B = D / A q/ BA

The principal acoustical part of
/ (L i ) is contained in the term (div / b/)·d/x i and is given by: (div / χ) · d/x i or, by the Codazzi equation, d/x i · d/trχ

(9.75)

while the principal acoustical part of
/ (Tˆ i ) is contained in the term (div / q/) · d/x i and is given by: −α −1 (div / χ) · d/x i or, by the Codazzi equation, − α −1 d/x i · d/trχ

(9.76)

Since we have (see (3.155), (9.72)): ψi d/x i = ψ

(9.77)

the principal acoustical part of
/ ψ L is: ψ · d/trχ

(9.78)

Chapter 9. The Propagation Equation for
/µ

286

and the principal acoustical part of
/ ψTˆ is: −α −1 ψ · d/trχ n 1 is:

(9.79)

It follows from the above results that the principal acoustical part of the function

dH (T σ ) ψ · d/trχ dσ and the principal acoustical part of the function n 2 is:  dH dH 1 −1 (Lσ ) ψ − 2Fψ0 ω d/σ · d/trχ −α ψTˆ / L − ψ L ψ Lˆ dσ 2 dσ ψL

(9.80)

(9.81)

We then conclude from (9.67), in view also of (9.52), that the principal acoustical part of the function gˇ  is given by: ξ · (µd/trχ) (9.82) where ξ is the St,u 1-form: /L + ξ = −d/µ − 2µFψ0 ω

 dH [ψ L (T σ ) − κψTˆ (Lσ )] ψ − κψ L (α −1 ψ L + ψTˆ )(d/σ ) dσ (9.83)

We now return to the propagation equation (9.70). To control the higher order spatial derivatives of µ, we introduce the functions: (i1 ...il )  x m,l

= µRil . . . Ri1 T m
/ µ − Ril . . . Ri1 T m fˇ

We have:

 = x x 0,0

In deriving propagation equations for the

 x m,l

(9.84) (9.85)

we shall make use of the following lemmas.

Lemma 9.2 Let α and β be St,u trace-free symmetric 2-covariant tensorfields defined in the spacetime domain Wε∗0 . Then we have: T (α · β) = (L /ˆT α) · β + α · (L /ˆT β) − tr

(T )

π /(α · β)

Proof. Since the vectorfield T is tangential to the hypersurfaces t , we can confine /ˆT α, L /ˆT β, we are considering the extensions of α and attention to a given t . In defining L β to T t by the conditions: α(V, T ) = β(V, T ) = 0

: ∀V ∈ T t

(9.86)

Since the reciprocal of the induced acoustical metric h on t is expressed in terms of the frame (Tˆ , X 1 , X 2 ) by: h

−1

= Tˆ ⊗ Tˆ + (h/−1 ) AB X A ⊗ X B

(9.87)

Chapter 9. The Propagation Equation for
/µ

287

we have: α · β = (h/−1 ) AC (h/−1 ) B D α AB βC D = (h

−1 ac

) (h

−1 bd

) αab βcd

(9.88)

where in the second line we have α, β extended to t . Then, T (α · β) = −

(T ) ab

π γab + (LT α)ab β ab + α ab (LT β)ab

Here, (T )

(T ) ab

π ab = (LT h)ab ,

π

= (h

−1 ac

) (h

−1 bd (T )

)

π cd

(9.89)

(9.90)

and γ is the symmetric 2-covariant tensorfield on t given by: γab = (h

−1 cd

) (αac βbd + βac αbd )

(9.91)

We have: γab T b = 0, therefore γ is the extension to T t of an St,u symmetric 2-covariant tensorfield which we also denote by γ . Now, for any two symmetric trace-free 2-dimensional matrices A and B we have: AB + B A − tr (AB)I = 0 In an orthonormal frame relative to h/ the components of α and β form such matrices A and B. The components of γ then form the matrix AB + B A. It follows that: γ AB = (α · β)h/ AB

(9.92)

Hence, the first term on the right in (9.89) is: (T )

π AB γ AB =

(T ) AB

π /

γ AB = tr

(T )

π /(α · β)

(9.93)

Also, since α and β satisfy (9.86), the second and third terms on the right in (9.89) are equal to β · L /T α and α · L /T β respectively. Moreover, since α and β are trace-free the last are in turn equal to β · L /ˆT α and α · L /ˆT β respectively. The lemma thus follows. Lemma 9.3 Let α and β be St,u trace-free symmetric 2-covariant tensorfields defined in the spacetime domain Wε∗0 . Then we have: /ˆRi α) · β + α · (L /ˆRi β) − tr Ri (α · β) = (L

(Ri )

π /(α · β)

The proof is along the lines of that of Lemma 9.2 but more straightforward, in view of the fact that the vectorfields Ri are tangential to the surfaces St,u . To present the propagation equation for we introduce the functions: (i1 ...il )

(i1 ...il ) x  m,l

in as simple a form as possible,

 = Ril . . . Ri1 T m fˇ fˇm,l

(9.94)

Chapter 9. The Propagation Equation for
/µ

288

Also, let µ /ˆ 2 be the St,u trace-free symmetric 2-covariant tensorfield: /ˆ2 µ µ /ˆ 2 = D

(9.95)

We introduce the St,u trace-free symmetric 2-covariant tensorfields: µ / 2,m,l = L /ˆ 2 / Rˆ il . . . L / Rˆ i1 (L /ˆT )m µ

(i1 ...il ) ˆ

(9.96)

 Proposition 9.1 For each non-negative integer m the function x m,0 satisfies the propagation equation:   1  −1  −1   − 2µχˆ · µ /ˆ 2,m,0 + gm,0 trχ − 2µ (Lµ) fˇm,0 Lx m,0 + (trχ − 2µ (Lµ))x m,0 = − 2  where the function gm,0 is given by:  gm,0 = T m gˇ  +

m−1  k=0

 T k x m−k−1,0 +

m−1 

 T k ym−k,0

k=0

where gˇ  is the function given by (9.68). Here, for each j = 1, . . . , m, y j,0 is the function: /µ y j,0 = −(T µ)a j −1,0 + (T Lµ − µT trχ − µ)T j −1
+

1 (T trχ) fˇj −1,0 − 2µ(L /ˆT χˆ − tr 2

(T )

π /χ) ˆ ·µ /ˆ 2, j −1,0

where a j −1,0 is the function: a j −1,0 = LT j −1
/ µ + trχ T j −1
/ µ + 2χˆ · µ /ˆ 2, j −1,0 Proof. The propagation equation of the proposition reduces for m = 0 to the propagation equation (9.70) already established. Thus, by induction in m, assuming that the propagation equation of the proposition holds with m replaced by m − 1, that is, assuming that:   1  −1  −1  trχ − 2µ (Lµ) fˇm−1,0 Lx m−1,0 + (trχ − 2µ (Lµ))x m−1,0 = − 2  −2µχˆ · µ /ˆ 2,m−1,0 + gm−1,0 (9.97)  holds for some function gm−1,0 , what we must show is that a propagation equation of the   form given by the proposition holds true for m, where gm,0 is a function related to gm−1,0  , for each by a certain recursion relation. This recursion relation shall then determine gm,0 m, from the function  = gˇ  (9.98) g0,0

given by (9.68), which appears in the propagation equation (9.70).

Chapter 9. The Propagation Equation for
/µ

289

We begin by rewriting the term   2µ−1 (Lµ)(x m−1,0 + fˇm−1,0 )

in (9.97) as: /µ 2(Lµ)T m−1
(see definitions (9.84), (9.94)) obtaining the equation: 1    + trχ x m−1,0 = 2(Lµ)T m−1
/ µ − trχ fˇm−1,0 Lx m−1,0 2  −2µχˆ · µ /ˆ 2,m−1,0 + gm−1,0

(9.99)

We now apply T to this equation. Since: / µ = T m
/µ T T m−1
  T fˇm−1,0 = fˇm,0

(see definition (9.94)) and by Lemma 9.2: /ˆT χ) ˆ ·µ /ˆ 2,m−1,0 + χˆ · µ /ˆ 2,m,0 − tr T (χˆ · µ /ˆ 2,m−1,0 ) = (L

(T )

π /(χˆ · µ /ˆ 2,m−1,0 )

(9.100)

(see definition (9.96)), we obtain:    T Lx m−1,0 + trχ T x m−1,0 + (T trχ)x m−1,0

= 2(Lµ)T m
/ µ + 2(T Lµ)T m−1
/µ 1 1   − trχ fˇm,0 − (T trχ) fˇm−1,0 2 2 − 2µχˆ · µ /ˆ 2,m,0 − 2µ(L /ˆT χ) ˆ ·µ /ˆ 2,m−1,0 + 2(µtr  + T gm−1,0

(T )

π / − (T µ))(χˆ · µ /ˆ 2,m−1,0 ) (9.101)

Next, using the fact that (see (3.71)):

we express:

[L, T ] =  = −(h/−1 ) AB (ζ B + η B )X A

(9.102)

   = LT x m−1,0 − x m−1,0 T Lx m−1,0

(9.103)

Now, definition (9.84) with l = 0 and m replaced by m − 1 reads:  = µT m−1
/ µ − T m−1 fˇ x m−1,0

Applying T to this we obtain, in view of definition (9.84),   T x m−1,0 = x m,0 + (T µ)T m−1
/µ

(9.104)

Chapter 9. The Propagation Equation for
/µ

290

Applying L to (9.104) and expressing: LT µ = T Lµ + µ yields:   LT x m−1,0 = Lx m,0 + (T µ)LT m−1
/µ

/µ +(T Lµ + µ)T m−1


(9.105)

We then substitute (9.105) in (9.103) and the result in (9.102). Using (9.104) to re-write  of the the second term on the left-hand side in (9.102), a propagation equation for x m,0   form given by the proposition results, with gm,0 expressed in terms of gm−1,0 by the recursion formula:     gm,0 = T gm−1,0 + x m−1,0 + ym,0 (9.106) The proposition then follows by applying Proposition 8.2 to this recursion, with the space of functions on Wε∗0 in the role of the space X, the operator T in the role of the operators  , y  An , and gm,0 m,0 + x m−1,0 , in the role of x n , yn , respectively. Proposition 9.2 For each pair of non-negative integers (m, l) and each multi-index  satisfies the propagation equation: (i 1 , . . . , i l ), the function (i1 ...il ) x m,l   1 (i1 ...il )  −1 (i1 ...il )  −1  trχ − 2µ (Lµ) (i1 ...il ) fˇm,l L x m,l + (trχ − 2µ (Lµ)) x m,l = − 2  −2µχˆ · (i1 ...il ) µ /ˆ 2,m,l + (i1 ...il ) gm,l where the function (i1 ...il )  gm,l

(i1 ...il ) g  m,l

is given by:

 = Ril . . . Ri1 gm,0 +

l−1 

Ril . . . Ril−k+1

(Ril−k )

Ril . . . Ril−k+1

(i1 ...il−k )  ym,l−k

Z

(i1 ...il−k−1 )  x m,l−k−1

k=0

+

l−1  k=0

 gm,0

where the function is given by Proposition 9.1. Here, for each j = 1, . . . , l, (i1 ...i j ) y  m, j is the function: (i1 ...i j )  ym, j

= −(Ri j µ)

(i1 ...i j −1 )  am, j −1

 + Ri j Lµ − µRi j trχ −

 Z µ (Ri j −1 . . . Ri1 T m
/ µ)   1 (R )  /χˆ · + (Ri j trχ) (i1 ...i j −1 ) fˇm, / Rˆ i j χˆ − tr i j π j −1 − 2µ L 2

where

(i1 ...i j −1 ) a  m, j −1

(Ri j )

(i1 ...i j −1 ) ˆ

µ / 2,m, j −1

is the function:

(i1 ...i j −1 )  am, j −1

= L(Ri j −1 . . . Ri1 T m
/ µ) + trχ(Ri j −1 . . . Ri1 T m
/ µ) + 2χˆ ·

(i1 ...i j −1 ) ˆ

µ / 2,m, j −1

Chapter 9. The Propagation Equation for
/µ

291

Proof. The propagation equation of the proposition reduces for l = 0 to the propagation equation of Proposition 9.1. Thus, by induction in l, assuming that the propagation equation of the proposition holds with l replaced by l − 1, that is, assuming that:  + (trχ − 2µ−1 (Lµ)) (i1 ...il−1 ) x m,l−1 (9.107)  1   trχ − 2µ−1 (Lµ) (i1 ...il−1 ) fˇm,l−1 /ˆ 2,m,l−1 + (i1 ...il−1 ) gm,l−1 − 2µχˆ · (i1 ...il−1 ) µ =− 2

L

(i1 ...il−1 )  x m,l−1



 holds for some function (i1 ...il−1 ) gm,l−1 , what we must show is that a propagation equa tion of the form given by the proposition holds true for l, where (i1 ...il ) gm,l is a function  (i ...i ) 1 l−1 related to gm,l−1 by a certain recursion relation. This recursion relation shall then  , for each l, from the function g  determine (i1 ...il ) gm,l m,0 given by Proposition 9.1. We begin by rewriting the term

2µ−1 (Lµ)(

(i1 ...il−1 )  x m,l−1

+

(i1 ...il−1 ) ˇ f m,l−1 )

in (9.107) as: /µ 2(Lµ)Ril−1 . . . Ri1 T m
(see definitions (9.84), (9.94)) obtaining the equation: L

(i1 ...il−1 )  x m,l−1

+ trχ

(i1 ...il−1 )  x m,l−1

1  = 2(Lµ)Ril−1 . . . Ri1 T m
/ µ − trχ (i1 ...il−1 ) fˇm,l−1 2  − 2µχˆ · (i1 ...il−1 ) µ /ˆ 2,m,l−1 + (i1 ...il−1 ) gm,l−1

(9.108)

We now apply Ril to this equation. Since: Ril

(i1 ...il−1 ) ˇ fm,l−1

(i1 ...il ) ˇ f m,l

=

(see definition (9.94)) and by Lemma 9.3: /ˆ 2,m,l−1 ) Ril (χˆ · µ ˆ · = (L / Rˆ il χ)

(9.109)

(i1 ...il−1 ) ˆ

µ / 2,m,l−1 + χˆ

(i1 ...il ) ˆ

µ / 2,m,l − tr

(Ril )

π /(χˆ ·

(i1 ...il−1 ) ˆ

µ / 2,m,l−1 )

(see definition (9.96)), we obtain: Ril L

(i1 ...il−1 )  x m,l−1

+ trχ Ril

(i1 ...il−1 )  x m,l−1

+ (Ril trχ)

(i1 ...il−1 )  x m,l−1 m

= 2(Lµ)Ril . . . Ri1 T m
/ µ + 2(Ril Lµ)Ril−1 . . . Ri1 T
/µ 1 1   − (Ril trχ) (i1 ...il−1 ) fˇm,l−1 − trχ (i1 ...il ) fˇm,l 2 2 /ˆ 2,m,l − 2µ(L /ˆ 2,m,l−1 − 2µχˆ · (i1 ...il ) µ / ˆRi χ) ˆ · (i1 ...il−1 ) µ l

+ 2(µtr +

(Ril )

π / − (Ril µ))(χˆ (i1 ...il−1 )  gm,l−1 Ril

·

(i1 ...il−1 ) ˆ

µ / 2,m,l−1 ) (9.110)

Chapter 9. The Propagation Equation for
/µ

292

Next, by Lemma 8.2 applied to Y = Ril , [L, Ril ] =

(Ril )

Z

(9.111)

hence: Ril L

(i1 ...il−1 )  x m,l−1

= L Ril

(i1 ...il−1 )  x m,l−1

−

(Ril )

Z

(i1 ...il−1 )  x m,l−1

(9.112)

Now, definition (9.84) with l replaced by l − 1 reads: (i1 ...il−1 )  x m,l−1

= µRil−1 . . . Ri1 T m
/ µ − Ril−1 . . . Ri1 T m fˇ

Applying Ril to this we obtain, in view of definition (9.84), Ril

(i1 ...il−1 )  x m,l−1

=

(i1 ...il )  x m,l

+ (Ril µ)(Ril−1 . . . Ri1 T m
/ µ)

(9.113)

Applying L to (9.113) and expressing: L Ril µ = Ril Lµ +

(Ril )

Zµ

yields: L Ril

(i1 ...il−1 )  x m,l−1

=L

(i1 ...il )  x m,l

+ (Ril µ)L(Ril−1 . . . Ri1 T m
/ µ)  / µ) + Ril Lµ + (Ril ) Z µ (Ril−1 . . . Ri1 T m


(9.114)

We then substitute (9.114) in (9.112) and the result in (9.110). Using (9.113) to re-write  the second term on the left-hand side in (9.110), a propagation equation for (i1 ...il ) x m,l  of the form given by the proposition results, with (i1 ...il ) gm,l expressed in terms of (i1 ...il−1 ) g  by the recursion formula: m,l−1 (i1 ...il )  gm,l

= Ril

(i1 ...il−1 )  gm,l−1

+

(Ril )

Z

(i1 ...il−1 )  x m,l−1

+

(i1 ...il )  ym,l

(9.115)

The proposition then follows by applying Proposition 8.2 to this recursion, with the space of functions on Wε∗0 in the role of the space X, the operators Ril in the role of the operators  , (i1 ...il ) y  + (Ril ) Z (i1 ...il−1 ) x  An , and (i1 ...il ) gm,l m,l m,l−1 , in the role of x n , yn , respectively.  The function (i1 ...i j −1 ) am, j −1 defined in the statement of Proposition 9.2 reduces for m = 0, j = 1, to the function:  a0,0 = L
/ µ + trχ
/ µ + 2χˆ · µ /ˆ 2

(9.116)

According to equation (9.10):  a0,0 = e
/ µ − 2d/µ · div / χˆ +
/ m + µ
/ e + 2d/µ · d/e

(9.117)

Chapter 9. The Propagation Equation for
/µ

293 (i1 ...il ) a  m,l

The following lemma gives us an expression for

 . in terms of a0,0

Lemma 9.4 For each non-negative integer m we have:    am,0 = T m a0,0 + bm,0

where:  = bm,0

m−1 

T k T m−k−1
/µ +

k=0

m−1 

 T k cm−k

k=0

and, for each j = 1, . . . , m, / µ − 2(L /ˆT χˆ − tr cj = −(T trχ)T j −1


(T )

π /χ) ˆ ·µ /ˆ 2, j −1,0

Moreover, for each pair of non-negative integers (m, l) and each multi-index (i 1 , . . . , i l ) we have: (i1 ...il )    am,l = Ril . . . Ri1 T m a0,0 + (i1 ...il ) bm,l where: (i1 ...il )  bm,l

 = Ril . . . Ri1 bm,0 +

l−1 

Ril . . . Ril−k+1

(Ril−k )

Ril . . . Ril−k+1

(i1 ...il−k )  cm,l−k

Z Ril−k−1 . . . Ri1 T m
/µ

k=0

+

l−1  k=0

and, for each j = 1, . . . , l and each multi-index (i 1 , . . . , i j ), (i1 ...i j )  cm, j

= −(Ri j trχ)(Ri j −1 . . . Ri1 T m
/ µ) −2(L / Rˆ i j χˆ − tr

(Ri j )

π /χ) ˆ ·

(i1 ...i j −1 ) ˆ

µ / 2,m, j −1

Proof. To prove the first part, we note that it holds trivially for m = 0 with:  b0,0 =0

(9.118)

 holds with m replaced by Thus, by induction in m, let us assume that the formula for am,0 m − 1, that is:    am−1,0 = T m−1 a0,0 + bm−1,0 (9.119)  holds, for some function bm−1,0 . Applying T to this we obtain:    = T m a0,0 + T bm−1,0 T am−1,0

(9.120)

On the other hand, from the definition in the statement of Proposition 9.1,  am−1,0 = LT m−1
/ µ + trχ T m−1
/ µ + 2χˆ · µ /ˆ 2,m−1,0

(9.121)

Chapter 9. The Propagation Equation for
/µ

294

and applying T to this yields, using Lemma 9.2,  T am−1,0 = T LT m−1
/ µ + trχ T m
/ µ + 2χˆ · µ /ˆ 2,m,0

+ (T trχ)T m−1
/ µ + 2(L /ˆT χˆ − tr

(T )

(9.122)

π /χ) ˆ ·µ /ˆ 2,m−1,0

Writing: T LT m−1
/ µ = LT m
/ µ − T m−1
/µ  from the statement of Proposition 9.1, takes the (9.122), in view of the definition of am,0 form:    = am,0 − T m−1
/ µ − cm (9.123) T am−1,0  is defined in the statement of the lemma. Equating the two expressions for where cm   if we set: T am−1,0 , that of (9.120) with that of (9.123), we obtain the formula for am,0    = T bm−1,0 + T m−1
/ µ + cm bm,0

(9.124)

 . We apply Proposition 8.2 to this reThis is a recursion relation for the functions bm,0 ∗ cursion, taking the space of functions on Wε0 as the space X, the operator T in the role   + T m−1
in the role of x n , and cm / µ in the role of yn . Since of the operators An , bm,0  here x 0 = 0 by (9.118), the formula for the bm,0 given in the statement of the lemma then results. To prove the second part we apply induction in l. Let us assume that the formula for  holds with l replaced by l − 1, that is: am,l   am,l−1 = Ril−1 . . . Ri1 T m a0,0 +

(i1 ...il−1 )  bm,l−1

(9.125)

 holds, for some function bm,l−1 . Applying Ril to this we obtain:   Ril am,l−1 = Ril . . . Ri1 T m a0,0 + Ril

(i1 ...il−1 )  bm,l−1

(9.126)

On the other hand, from the definition in the statement of Proposition 9.2, (i1 ...il−1 )  am,l−1

(9.127)

= L(Ril−1 . . . Ri1 T
/ µ) + trχ(Ril−1 . . . Ri1 T
/ µ) + 2χˆ · m

m

(i1 ...il−1 ) ˆ

µ / 2,m,l−1

and applying Ril to this yields, using Lemma 9.3, Ril

(i1 ...il−1 )  am,l−1

= Ril L(Ril−1 . . . Ri1 T m
/ µ) + trχ(Ril . . . Ri1 T m
/ µ) +2χˆ · (i1 ...il ) µ / µ) /ˆ 2,m,l + (Ril trχ)(Ril−1 . . . Ri1 T m
(R ) (i ...i ) i 1 l−1 +2(L / Rˆ i χˆ − tr l π /χ) ˆ · (9.128) µ /ˆ 2,m,l−1 l

Writing: Ril L Ril−1 . . . Ri1 T m
/ µ = L Ril . . . Ri1 T m
/µ−

(Ril )

Z Ril−1 . . . Ri1 T m
/µ

Chapter 9. The Propagation Equation for
/µ

295

 (9.128), in view of the definition of am,l from the statement of Proposition 9.2, takes the form:

Ril

(i1 ...il−1 )  am,l−1

=

(i1 ...il )  am,l

 − s (Ril ) Z Ril−1 . . . Ri1 T m
/ µ − cm,l

(9.129)

 is defined in the statement of the lemma. Equating the two expressions for where cm,l (i ...i  , that of (9.126) with that of (9.129), we obtain the formula for Ril 1 l−1 ) am,l−1 (i1 ...il ) a  if we set: m,l (i1 ...il )  bm,l

= Ril

(i1 ...il−1 )  bm,l−1

+

(Ril )

Z Ril−1 . . . Ri1 T m
/µ+

(i1 ...il )  cm,l

(9.130)

 , with respect to l. We apply PropoThis is a recursion relation for the functions bm,l sition 8.2 to this recursion, taking the space of functions on Wε∗0 as the space X, the   + operators Ril in the role of the operators An , bm,l in the role of x n , and (i1 ...il ) cm,l (Ril )  Z Ril−1 . . . Ri1 T m
/ µ in the role of yn . The formula for the bm,l given in the statement of the lemma then results.

Let us investigate the order of the various terms in the propagation equation of Proposition 9.2, following the analogous discussion in Chapter 8. In obtaining estimates for the n + 1st order spatial derivatives of µ, of which at least two are angular, as announced at the beginning of the present chapter, we are to set m +l = n −1 in Proposition 9.2. The principal terms in the propagation equation of Proposition 9.2 shall thus be the terms of order m + l + 2 = n + 1, and the principal acoustical terms shall be the principal terms in the spatial derivatives of the acoustical entities χ and µ. As we have seen above (see discussion following equations (9.67), (9.68)), the function fˇ is of order 2, but contains no acoustical part of order 2. It follows that the function (i1 ...il ) fˇ which occurs on the right-hand side of the propagation equation of Proposition m,l 9.2 is of principal order m + l + 2, but contains no principal acoustical part. We turn to the last term in the propagation equation of Proposition 9.2, namely the  . This function is expressed in the statement of Proposition 9.2, the function (i1 ...il ) gm,l first term in this expression being the term  Ril . . . Ri1 gm,0

(9.131)

 , given in the statement of ProposiWe must thus first investigate the function gm,0 tion 9.1. Here we concentrate on the terms of order m + 2, since it is these terms which will contribute principal terms to (9.131). As we have seen above (see discussion following equations (9.67), (9.68)), the function gˇ  is of order 2 and its principal acoustical part, (9.82), consists of 1st angular derivatives of trχ multiplied by µ. It follows that

T m gˇ 

(9.132)

 the first term in the expression for gm,0 given in the statement of Proposition 9.1, is of order m + 2 and its principal acoustical part consists of 1st angular derivatives of mth order T -derivatives of trχ, multiplied by µ:

ξ · (µd/T m trχ)

(9.133)

Chapter 9. The Propagation Equation for
/µ

296

or, in view of the trace of equation (4.109), ξ · (µd/trχ) ξ · (µd/T

m−1

: if m = 0 T
/ µ) : if m ≥ 1

(9.134)

to principal acoustical terms. In view of the definitions (8.48) and (9.84), (9.134) can be expressed to principal acoustical terms in terms of the St,u 1-form x 0 if m = 0, and in  terms of d/x m−1,0 if m ≥ 1: ξ · x0

: if m = 0

 ξ · d/x m−1,0 : if m ≥ 1

(9.135)

 given in the statement of Proposition The second term in the expression for gm,0 9.1, can be written as: m−1 

 T k x m−k−1,0 −

k=0

m−1 

 [, T k ]x m−k−1,0

(9.136)

k=0

The commutator sum is of lower order m + 1, while, in view of the definition (9.84),  = T k (µT m−k−1
/ µ − T m−k−1 fˇ ) T k x m−k−1,0 k    k  = x m−1,0 + / µ) (T j µ)(T m− j −1
j

(9.137)

j =1

The contribution of the 2nd term on the right in (9.137) to the first sum in (9.136) is of lower order m + 1, while the contribution of the 1st term is:  mx m−1,0

(9.138)

 given in the This is thus the principal part of the second term in the expression of gm,0 statement of Proposition 9.1.  The third (last) term in the expression for gm,0 is the sum: m−1 

 T k ym−k

(9.139)

k=0

Let us consider the expression for y j,0 given in the statement of Proposition 9.1. Here, we concentrate on the terms of order j + 2, since it is these terms which shall contribute terms of principal order m + 2 in the sum (9.139). Now, all terms in the expression for y j,0 except the first term:  −(T µ)a j −1,0 = −(T µ)(T j −1 a0,0 + b j −1,0)

(9.140)

Chapter 9. The Propagation Equation for
/µ

297

 , given by (9.117), is of order 3, but (see Lemma 9.4) are of lower order j + 1. Here, a0,0 contains no acoustical part of order 3, its principal terms being:


/ m + µ
/e It follows that T j −1 a0,0 is of principal order j + 2 but contains no principal acoustical part, its principal terms being: / m + µT j −1
/e T j −1
From Lemma 9.4, bj −1,0 =

j −2 

T k T j −k−2
/µ+

k=0

j −2 

T k cj −k−1

(9.141)

k=0

The first sum on the right in (9.141) is of order j + 1, while cj , given in the statement of Lemma 9.4, is of order j +1, hence the second sum on the right in (9.141) is only of order j . Thus bj −1,0 is of lower order j + 1. We conclude that the principal part of the sum  given in the statement of Proposition (9.139), the third term in the expression for gm,0 9.1, is:  −m(T µ)T m−1 a0,0 (9.142) This is of principal order m + 2, but contains no principal acoustical part. This completes  . our investigation of the function gm,0 We conclude from the above discussion that the principal part of the term (9.131)  . These results from the contributions of the terms (9.132), (9.138), and (9.142), in gm,0 contributions are all of principal order l + m + 2. The contribution of (9.132) to (9.131) is: Ril . . . Ri1 T m gˇ 

(9.143)

According to the above discussion (see (9.135)), the principal acoustical part of this can be expressed in terms of the St,u 1-form (i1 ...il ) xl (see definition (8.60)) if m = 0, in  terms of d/ (i1 ...il ) x m−1,l if m ≥ 1. It is given by: ξ·

(i1 ...il )

ξ · d/

: if m = 0

xl

(i1 ...il )  x m−1,l

: if m ≥ 1

(9.144)

where ξ is the St,u 1-form given by (9.83). By virtue of the hypothesis H0 of Chapter 7, we can estimate pointwise: |ξ · |ξ · d/

(i1 ...il )

xl | ≤ |ξ ||

(i1 ...il )  x m−1,l |

(i1 ...il )

≤ C(1 + t)

xl |

−1

|ξ | max | j

(i1 ...il j )  x m−1,l+1 |

(9.145)

The principal part of the contribution of (9.138) to (9.131) is, in view of the definition (9.84),  (9.146) m (i1 ...il ) x m−1,l

Chapter 9. The Propagation Equation for
/µ

298

By virtue of hypothesis H0, (9.146) can be estimated pointwise by: Cm(1 + t)−1 || max | j

(i1 ...il j )  x m−1,l+1 |

(9.147)

Finally, the principal part of the contribution of (9.142) to (9.131) is:  −m(T µ)Ril . . . Ri1 T m−1 a0,0

(9.148)

and this does not contain principal acoustical terms. This completes our investigation  of the first term in the expression for the function (i1 ...il ) gm,l given in the statement of Proposition 9.2.  . This is the sum: We turn to the second term in the expression for (i1 ...il ) gm,l l−1 

(Ril−k )

Z

(i1 ...il−k−1 )  x m,l−k−1

(9.149)

Z Ril . . . Ril−k+1

(i1 ...il−k−1 )  x m,l−k−1

(9.150)

Ril . . . Ril−k+1

k=0

which we write as: l−1 

(Ril−k )

k=0

minus the commutator term: l−1 

[

(Ril−k )

Z , Ril . . . Ril−k+1 ]

(i1 ...il−k−1 )  x m,l−k−1

(9.151)

k=0

The commutator term is readily seen to be of lower order l + m + 1, while in view of definition (9.84) the principal part of (9.150) is: l−1 

(Ril−k )

Z

i



(i1 . l−k . . il )  x m,l−1

(9.152)

k=0

This is a principal acoustical term. By virtue of hypothesis H0 it can be estimated pointwise by:  l   ik  (Rik ) −1  C(1 + t) | Z | max | (i1 . . . il j ) x m,l | (9.153) k=1

j

Finally, we consider the third term in the expression for l−1 

Ril . . . Ril−k+1

(i1 ...il−k )  ym,l−k

(i1 ...il ) g  . This is the sum: m,l

(9.154)

k=0  Let us consider the expression for (i1 ...i j ) ym, j given in the statement of Proposition 9.2. Here, we concentrate on the terms of order m + j + 2, since it is these terms which shall

Chapter 9. The Propagation Equation for
/µ

299

contribute terms of principal order m + l + 2 in the sum (9.154). Now, all terms in the  expression for (i1 ...i j ) ym, j except the first term: −(Ri j µ)

(i1 ...i j −1 )  am, j −1

 = −(Ri j µ)(Ri j −1 . . . Ri1 T m a0,0 +

(i1 ...i j −1 )  bm, j −1 )

(9.155)

 (see Lemma 9.4) are of lower order m + j + 1. Since a0,0 is of order 3, but contains no acoustical part of order 3, it follows that Ri j −1 . . . Ri1 T m a0,0 is of principal order  m + j + 2 but contains no principal acoustical part. The term (i1 ...i j −1 ) bm, j −1 is given by Lemma 9.4: (i1 ...i j −1 )  bm, j −1

 = Ri j −1 . . . Ri1 bm,0

+

j −2 

Ri j −1 . . . Ri j −k

(Ri j −k−1 )

Ri j −1 . . . Ri j −k

(i1 ...i j −k−1 )  cm, j −k−1

Z Ri j −k−2 . . . Ri1 T m
/µ

k=0

+

j −2 

(9.156)

k=0

By the discussion above, the first term on the right in (9.156) is of order m + j + 1.  , given in the The second term on the right is also of order m + j + 1, while (i1 ...i j ) cm, j statement of Lemma 9.4, is of order m + j + 1, hence the third on the right in (9.156)  is only of order m + j . Thus (i1 ...i j −1 ) bm, j −1 is of lower order m + j + 1. We conclude  that the principal part of the sum (9.154), the third term in the expression for (i1 ...il ) gm,l given in the statement of Proposition 9.2, is: −

l−1 

R



 (Ril−k µ)Ril . il−k . . Ri1 T m a0,0

(9.157)

k=0

This is of principal order m + l + 2, but contains no principal acoustical part. This com . pletes our investigation of the function (i1 ...il ) gm,l

 of Proposition 9.2, the term Returning to the propagation equation for (i1 ...il ) x m,l (i ...i ) (i ...i ) l l 1 1 ˆ ˆ 2µχˆ · µ / 2,m,l remains to be considered. Since µ / 2,m,l given by (9.96) is an lth order angular derivative of an mth order T derivative of µ /ˆ 2 , defined by (9.95) and itself of order 2, this term is certainly a principal acoustical term. Moreover, it involves the m + lth order spatial derivatives of µ /ˆ 2 , not
/ µ, which is what the propagation equation  allows us to control. The term in question originates from the propagation for (i1 ...il ) x m,l equation (9.70) for x  , defined by (9.69), from the term

/ˆ 2 2µχˆ · D /ˆ2 µ = 2µχˆ · µ which appears on the right-hand side. Now, the propagation equation (9.70), must be considered in conjunction with the definition (9.69), which written in the form: µ
/ µ = x  + fˇ

(9.158)

Chapter 9. The Propagation Equation for
/µ

300

constitutes an elliptic equation for µ on each surface St,u , section of Cu , which allows us /ˆ 2 in terms of x  and the derivatives of the ψµ of up to to estimate on St,u , d/µ as well as µ the 2nd order. The propagation equation (9.70) is an ordinary differential equation for x  along each generator of the characteristic hypersurface Cu , which allows us to estimate along each generator x  in terms of µ /ˆ 2 and the derivatives of the ψµ of up to the 2nd order. Thus, considering on Cu the propagation equation (9.70) in conjunction with the elliptic equation (9.158) we shall be able to estimate d/µ as well as D / 2 µ in terms of the ψµ and their derivatives of up to the 2nd order, avoiding in this way any loss of differentiability. Similar remarks apply to the higher order spatial derivatives of µ – of which at  of Proposition 9.2 is an least two are angular. The propagation equation for (i1 ...il ) x m,l ordinary differential equation along each generator of Cu . Now, on the right-hand side of this equation we have (i1 ...il ) µ /ˆ 2,m,l as noted above. This propagation equation must be considered in conjunction with an elliptic equation for the function: (i1 ...il )

µm,l = Ril . . . Ri1 T m µ

(9.159)

on each St,u section, which allows us to estimate D / 2 (i1 ...il ) µm,l , and through this (i1 ...il ) µ (i ...i )  /ˆ 2,m,l , in terms of 1 l x m,l and the derivatives of the ψµ of order up to m +l +2. The required elliptic equation shall be derived from the definition (9.84) written in the form:   / µ = (i1 ...il ) x m,l + (i1 ...il ) fˇm,l (9.160) µRil . . . Ri1 T m
with the help of the following lemmas. Lemma 9.5 Let (M, g) be a 2-dimensional Riemannian manifold, let X be an arbitrary vectorfield and f an arbitrary function on M. Then the following commutation formulas hold: X (
g f ) −
g (X f ) = − (X ) π ab (∇ 2 f )ab − tr (X ) π1b db f and:

1 (LˆX ∇ˆ2 f )ab − (∇ˆ2 (X f ))ab = − 2

(X )

πˆ ab
g f −

(X ) c πˆ 1,ab dc

f

Here, tr

(X ) b π1

(X ) c πˆ 1,ab

= ∇a =

(X )

1 πa b − ∇ b tr 2

(X )

π

1 (∇a (X ) πˆ b c + ∇b (X ) πˆ a c − ∇ c (X ) πˆ ab − gab ∇ d 2 1 + (δac db tr (X ) π + δbc da tr (X ) π − gab d c tr (X ) π) 4

and we denote L X g =

(X )

πˆ d c )

(X ) π.

Proof. Let φt be the local 1-parameter group of diffeomorphisms generated by X and let φt ∗ be the corresponding pullback. Consider an arbitrary 1-form α on M. We then have: g

φt∗ g

φt ∗ (∇ α) = ∇ (φt ∗ α)

(9.161)

Chapter 9. The Propagation Equation for
/µ

301

Now, in an arbitrary system of local coordinates we have: g

(∇ α)ab =

g ∂αb c − ! ab αc ∂xa

(9.162)

g

c are the connection coefficients of the metric g in the given coordinate system. where !ab Similarly, in the same coordinate system, φt∗ g

( ∇ (φt ∗ α))ab =

∂(φt ∗ α)b φt∗c g − !ab (φt ∗ α)c ∂xa

(9.163)

Differentiating (9.163) with respect to t at t = 0 we obtain:      g d ∂ d d φt∗ g c ( ∇ (φt ∗ α))ab (φ (φ = α) − ! α) t∗ b t∗ c ab dt ∂ x a dt dt t =0 t =0 t =0   φ g t∗ d c − ! αc (9.164) dt ab t =0

Now, by definition,



d (φt ∗ α) dt

 t =0

= LX α

and, by (9.161) the left-hand side of (9.164) is the ab-component of:   g d (φt ∗ (∇ α)) = L X (∇α) dt t =0 Thus, substituting in (9.164) the expression (8.112), that is:   d φt∗c g c ! = (X ) π1,ab dt ab

(9.165)

t =0

where: (X ) c π1,ab

=

1 (∇a 2

(X )

πb c + ∇b

(X )

πa c − ∇ c

we conclude that: (L X (∇α))ab = (∇(L X α))ab −

(X )

(X ) c π1,ab αc

πab )

(9.166) (9.167)

This holds for any 1-form α on M. In particular, in the case α = d f for some function f , since L X (d f ) = d(X f ), we obtain: (L X (∇ 2 f ))ab = (∇ 2 (X f ))ab −

(X ) c π1,ab dc

for any function f on M. Since:
g f = (g −1 )ab (∇ 2 f )ab

f

(9.168)

Chapter 9. The Propagation Equation for
/µ

302

it follows that: X (
g f ) = L X (
g f ) = − Since

(X ) ab

π (∇ 2 f )ab + (g −1 )ab (L X (∇ 2 f ))ab

(9.169)

(g −1 )ab (∇ 2 (X f ))ab =
g (X f ),

the first part of the lemma then follows substituting (9.168) in (9.169). In fact the above hold for an arbitrary n-dimensional manifold M. For the second part of the lemma we consider the case of a 2-dimensional manifold M. The trace-free part of ∇ 2 f is then: 1 ∇ˆ2 f = ∇ 2 f − g
g f 2 hence:

1 1 L X (∇ˆ2 f ) = L X (∇ 2 f ) − (X ) π
g f − g X (
g f ) 2 2 Substituting from (9.168) and the first part of the lemma we then obtain:

(9.170)

1 c dc f − (X ) πab
g f (9.171) (L X (∇ˆ2 f ))ab = (∇ 2 (X f ))ab − (X ) π1,ab 2 

1 − gab
g (X f ) − (X ) π cd (∇ 2 f )cd − tr (X ) π1c dc f 2 Decomposing

(X ) π

into its trace-free part and its trace: (X )

and similarly for

(X ) π

πab =

(X ) π ˆ

1

1 πˆ ab + gab tr 2

(X )

π

1: (X ) c π1,ab

then

(X )

=

(X ) c πˆ 1,ab

1 + gab tr 2

(X ) c π1

is given by the expression in the statement of the lemma, while, noting that 1 (∇ 2 (X f ))ab − gab
g (X f ) = (∇ˆ2 (X f ))ab , 2

(9.171) takes the form: c (L X (∇ˆ2 f ))ab = (∇ˆ2 (X f ))ab − (X ) πˆ 1,ab dc f 1 1 − (X ) πˆ ab
g f + gab (X ) πˆ cd (∇ˆ2 f )cd 2 2

(9.172)

Taking the trace-free part of this equality, the last term on the right drops out and we obtain the second part of the lemma.

Chapter 9. The Propagation Equation for
/µ

303

Lemma 9.6 Let f be an arbitrary function defined on a given hypersurface t . Then the following commutation formulas hold: T (
/ f)−
/ (T f ) = − and:

(T ) AB

π /

(D / 2 f ) AB − tr

1 /ˆ2 f ) AB − ( D /ˆ2 (T f )) AB = − (L /ˆT D 2

(T ) B π /1 d/ B

(T ) ˆ

π / AB
/f −

f

(T ) ˆ C π /1,AB d/C

f

Here,

 1 D / A (T ) π / BC + D / B (T ) π / AC − D / C (T ) π / AB 2 is the Lie derivative with respect to T of the induced connection on St,u , (T ) C π /1,AB

=

tr

(T ) C π /1

= (h/−1 ) AB

(T ) C π /1,AB

is its trace, and (T ) ˆ C π /1,AB

=

(T ) C π /1,AB

1 − /h AB tr 2

(T ) C π /1

its trace-free part. Proof. Since T is tangential to the t , we can confine attention to the given hypersurface t . We may then choose acoustical coordinates so that the coordinate lines on the given t corresponding to constant values of (ϑ1 , ϑ2 ) are h-orthogonal to the surfaces St,u on the given t . Then  = 0 on the given t (see Chapter 2) and we have: T =

∂ ∂u

: on the given hypersurface t

(9.173)

Consider an arbitrary St,u 1-form α defined on t . In terms of the coordinates (u, ϑ1 , ϑ2 ) we have: ∂α B −! /CAB αC (9.174) (D /α) AB = ∂ϑ A Differentiating (9.174) with respect to u we obtain: ∂ ∂ (D /α) AB = ∂u ∂ϑ A



∂α B ∂u

 −! /CAB

∂! /CAB ∂αC − αC ∂u ∂u

(9.175)

Now, in view of (9.173) we have: ∂α A = (L /T α) A , ∂u

∂ /T (D /α)) AB (D /α) AB = (L ∂u

(9.176)

Moreover, from Chapter 3 we have, in view of (9.173), ∂h / AB = ∂u

(T )

π / AB = 2κθ AB ,

∂ −1 AB (h/ ) = − ∂u

(T ) AB

π /

= −2κθ AB

(9.177)

Chapter 9. The Propagation Equation for
/µ

304

hence:    ∂! /CAB 1 ∂ ∂h/ AD ∂h/ AB ∂h /B D = + − (h /−1 )C D ∂u 2 ∂u ∂ϑ A ∂ϑ B ∂ϑ D   1 (T ) C D E /B D / AD / AB 1 −1 C D ∂ (T ) π ∂ (T ) π ∂ (T ) π =− π / ! / AB h / D E + (h/ ) + − 2 2 ∂ϑ A ∂ϑ B ∂ϑ D  1 −1 C D  D / A (T ) π / ) = (h /B D + D / B (T ) π / AD − D / D (T ) π / AB 2 that is:

∂! /CAB = ∂u

(T ) C π /1,AB

(9.178)

where:

 1 D / A (T ) π / BC + D / B (T ) π / AC − D / C (T ) π / AB 2 Substituting (9.178) in (9.175) we conclude in view of (9.176) that: (T ) C π /1,AB

=

(T ) C π /1,AB αC

/α)) AB = (D /(L /T α)) AB − (L /T (D

(9.179)

(9.180)

This holds for any St,u 1-form α defined on the hypersurface t . In particular, it holds in the case α = d/ f for some function f defined on t . In this case, we have, in terms of the coordinates (u, ϑ 1 , ϑ 2 ):   ∂ ∂f ∂f ∂α A = αA = hence: ∂ϑ A ∂u ∂ϑ A ∂u that is: L /T (d/ f ) = d/(T f )

(9.181)

Thus, (9.180) assumes the form: (L /T (D / 2 f )) AB = (D / 2 (T f )) AB −

(T ) C π /1,AB d/C

f

(9.182)

for any function f on t . Since:
/ f = (h/−1 ) AB (D / 2 f ) AB it follows in view of (9.177) that: T (
/ f) = L /T (
/ f) = − Since

(T ) AB

π /

(D / 2 f ) AB + (h/−1 ) AB (L /T (D / 2 f )) AB

(9.183)

(h /−1 ) AB (D / 2 (T f )) AB =
/ (T f ),

the first part of the lemma then follows substituting (9.182) in (9.183). To prove the second part of the lemma we note that the trace-free part of D / 2 f is: 1 D /ˆ2 f = D /f / 2 f − /h
2

Chapter 9. The Propagation Equation for
/µ

305

hence by (9.177): 1 1 L /T ( D / f) /ˆ2 f ) = L /T (D / 2 f ) − (T ) π /
/ f − h/ T (
2 2 Substituting from (9.182) and the first part of the lemma we then obtain:

(9.184)

1 (L /T ( D /ˆ2 f )) AB = (D / 2 (T f )) AB − (T ) π /C /C f − (T ) π / AB
/f (9.185) 1,AB d 2

 1 / (T f ) − (T ) π / AB
− h /C D (D / 2 f )C D − tr (T ) π /C /C f 1d 2 / into its trace-free part and its trace: Decomposing (T ) π (T )

and similarly for

π / AB =

1 π / AB + /h AB tr 2

(T ) ˆ

(T )

π /

(T ) π /: (T ) C π /1,AB

=

(T ) ˆ C π /1,AB

1 + /h AB tr 2

(T ) C π /1

then, noting that 1 (D / 2 (T f )) AB − /h AB
/ (T f ) = ( D /ˆ2 (T f )) AB , 2 (9.185) takes the form: C /ˆ 1,AB d/C f /ˆ2 f )) AB = ( D /ˆ2 (T f )) AB − (T ) π (L /T ( D 1 1 C D ˆ2 − (T ) π / f + h/ AB (T ) π / f )C D (9.186) /ˆ AB
/ˆ ( D 2 2 Taking the trace-free part of this equation, the last term on the right drops out and we obtain the second part of the lemma.

We shall now apply the above two lemmas to derive the following propositions. Let us recall the functions (i1 ...il ) µm,l defined by (9.159). Proposition 9.3 For each pair of non-negative integers (m, l) and each multi-index (i 1 , . . . , i l ) we have:
/

(i1 ...il )

µm,l − Ril . . . Ri1 T m
/µ =

(i1 ...il )

dm,l

where: (i1 ...il )

dm,l = Ril . . . Ri1 dm,0 +

l−1 

Ril . . . Ril−k+1 (

(Ril−k )

π /· D /2

(i1 ...il−k−1 )

µm,l−k−1 )

k=0

+

l−1 

Ril . . . Ril−k+1 (tr

(Ril−k )

π /1 · d/

(i1 ...il−k−1 )

µm,l−k−1 )

k=0

and: dm,0 =

m−1  k=0

T k(

(T )

π /· D / 2 µm−k−1,0 ) +

m−1  k=0

T k (tr

(T )

π /1 · d/µm−k−1,0 )

Chapter 9. The Propagation Equation for
/µ

306

Proof. We first consider the case l = 0. If also m = 0 we have trivially: d0,0 = 0

(9.187)

We shall derive a recursion formula for the functions dm,0 . Consider then: / µm−1,0 − T m−1
/µ dm−1,0 =
Applying T to this we obtain: T dm−1,0 = T (
/ µm−1,0 ) − T m
/µ

(9.188)

We now apply the first part of Lemma 9.6, setting f = µm−1,0 . This gives: (T )

/ µm,0 − T (
/ µm−1,0 ) =


π /· D / 2 µm−1,0 −

(T )

trπ /1 · d/µm−1,0

(9.189)

Substituting (9.189) in (9.188) and noting that: / µ = dm,0
/ µm,0 − T m
we obtain the recursion formula: (T )

dm,0 = T dm−1,0 +

π /· D / 2 µm−1,0 + tr

(T )

π /1 · d/µm−1,0

(9.190)

We apply Proposition 8.2 to this recursion taking the space of functions on Wε∗0 as the space X, the operator T in the role of the operators An , dm,0 in the role of x n , and (T )

π /· D / 2 µm−1,0 + tr

(T )

π /1 · d/µm−1,0

in the role of yn . Since here x 0 = 0 by (9.187), the formula for dm,0 given in the statement of the proposition then results. Next we consider the case l ≥ 1. We shall derive a recursion relation for the functions (i1 ...il ) dm,l with respect to l. Consider then: (i1 ...il−1 )

dm,l−1 =
/

(i1 ...il−1 )

µm,l−1 − Ril−1 . . . Ri1 T m
/µ

Applying Ril to this we obtain: Ril

(i1 ...il−1 )

dm,l−1 = Ril
/

(i1 ...il−1 )

µm,l−1 − Ril . . . Ril T m
/µ

(9.191)

We now apply the first part of Lemma 9.5 taking the manifold (M, g) to be (St,u , h/), the vectorfield X to be Ril and the function f to be (i1 ...il−1 ) µm,l−1 . Then (X ) π is (Ril ) π / and we obtain: / Ril


(i1 ...il−1 )

=
/

(i1 ...il )

µm,l−1

µm,l −

(9.192) (Ril )

2 (i1 ...il−1 )

π /· D /

µm,l−1 − tr

(Ril )

π /1 · d/

(i1 ...il−1 )

µm,l−1

Chapter 9. The Propagation Equation for
/µ

307

Substituting (9.192) in (9.191) and noting that: (i1 ...il )


/

µm,l − Ril . . . Ri1 T m
/µ =

(i1 ...il )

dm,l

we obtain the recursion formula: (i1 ...il )

= Ril

dm,l

(9.193)

(i1 ...il−1 )

(Ril )

dm,l−1 +

2 (i1 ...il−1 )

π /· D /

µm,l−1 + tr

(Ril )

π /1 · d/

(i1 ...il−1 )

µm,l−1

We apply Proposition 8.2 to this recursion taking the space of functions on Wε∗0 as the space X, the operators Ril in the role of the operators An , (i1 ...il ) dm,l in the role of x n , and (Ril ) π /· D / 2 (i1 ...il−1 ) µm,l−1 + tr (Ril ) π /1 · d/ (i1 ...il−1 ) µm,l−1 in the role of yn . The formula for then results.

(i1 ...il ) d

m,l

given in the statement of the proposition

Proposition 9.4 For each pair of non-negative integers (m, l) and each multi-index (i 1 , . . . , i l ) we have: D /ˆ2

(i1 ...il )

µm,l −

(i1 ...il ) ˆ

µ / 2,m,l =

(i1 ...il )

em,l

where: (i1 ...il )

em,l = L / Rˆ il . . . L / Rˆ i1 em,0 +

1 ˆ ˆ ( L / Ril . . . L / Ril−k+1 2 l−1

(Ril−k ) ˆ

π /
/

(i1 ...il−k−1 )

π /1 · d/

(i1 ...il−k−1 )

µm,l−k−1 )

k=0

+

l−1 

ˆ ( / Ril−k+1 L / Rˆ il . . . L

(Ril−k ) ˆ

µm,l−k−1 )

k=0

and: em,0 =

m−1 1 ˆ k L /T ( 2

(T ) ˆ

π /
/ µm−k−1,0 ) +

k=0

m−1 

k L /ˆT (

(T )

π /ˆ1 · d/µm−k−1,0 )

k=0

Proof. We first consider the case l = 0. If also m = 0 we have from the definitions (9.95), (9.96) and (9.159): (9.194) e0,0 = 0 We shall derive a recursion formula for the trace-free symmetric 2-covariant St,u -tensorfields em,0 . Consider then: em−1,0 = D /ˆ2 µm−1,0 − µ /ˆ 2,m−1,0 Applying the operator L /ˆT to this we obtain: /ˆT D /ˆ 2,m,0 L /ˆT em−1,0 = L /ˆ2 µm−1,0 − µ

(9.195)

Chapter 9. The Propagation Equation for
/µ

308

We now apply the second part of Lemma 9.6, setting f = µm−1,0 . This gives: 1 /ˆ2 µm−1,0 = D /ˆ2 µm,0 − L /ˆT D 2

(T ) ˆ

π /
/ µm−1,0 −

(T )

π /ˆ1 · d/µm−1,0

(9.196)

(T )

π /ˆ1 · d/µm−1,0

(9.197)

Substituting (9.196) in (9.195) and noting that: D /ˆ2 µm,0 − µ /ˆ 2,m,0 = em,0 we obtain the recursion formula: /ˆT em−1,0 + em,0 = L

1 2

(T ) ˆ

π /
/ µm−1,0 +

We apply Proposition 8.2 to this recursion taking the space of trace-free symmetric 2covariant St,u -tensorfields defined on Wε∗0 as the space X, the operator L /ˆT in the role of the operators An , em,0 in the role of x n , and 1 2

(T ) ˆ

π /
/ µm−1,0 +

(T )

π /ˆ1 · d/µm−1,0

in the role of yn . Since here x 0 = 0 by (9.194), the formula for em,0 given in the statement of the proposition then results. Next we consider the case l ≥ 1. We shall derive a recursion formula for the tracefree symmetric 2-covariant St,u -tensorfields (i1 ...il ) em,l with respect to l. Consider then: (i1 ...il−1 )

em,l−1 = D /ˆ2

(i1 ...il−1 )

µm,l−1 −

(i1 ...il−1 ) ˆ

µ / 2,m,l−1

Applying the operator L / Rˆ il to this we obtain: L / Rˆ il

(i1 ...il−1 )

/ˆ2 em,l−1 = L / Rˆ il D

(i1 ...il−1 )

µm,l−1 −

(i1 ...il ) ˆ

µ / 2,m,l

(9.198)

We now apply the second part of Lemma 9.5 taking the manifold (M, g) to be (St,u , h/), / the vectorfield X to be Ril and the function f to be (i1 ...il−1 ) µm,l−1 . Then (X ) π is (Ril ) π and we obtain: L / Rˆ il D /ˆ2

(i1 ...il−1 )

µm,l−1 = D /ˆ2 −

(i1 ...il )

1 2

µm,l

(Ril ) ˆ

π /
/

(9.199)

(i1 ...il−1 )

µm,l−1 −

(Ril ) ˆ

π /1 · d/

(i1 ...il−1 )

µm,l−1

Substituting (9.199) in (9.198) and noting that: D /ˆ2

(i1 ...il )

µm,l −

(i1 ...il ) ˆ

µ / 2,m,l =

(i1 ...il )

em,l

we obtain the recursion formula: (i1 ...il )

em,l = L / ˆRil

(i1 ...il−1 )

em,l−1 +

1 2

(Ril ) ˆ

π /
/

(i1 ...il−1 )

µm,l−1 +

(Ril ) ˆ

π /1 ·d/

(i1 ...il−1 )

µm,l−1 (9.200)

Chapter 9. The Propagation Equation for
/µ

309

We apply Proposition 8.2 to this recursion taking the space of trace-free symmetric 2/ Rˆ il in the role of covariant St,u -tensorfields defined on Wε∗0 as the space X, the operators L the operators An , (i1 ...il ) em,l in the role of x n , and 1 2

(Ril ) ˆ

π /
/

(i1 ...il−1 )

µm,l−1 +

in the role of yn . The formula for then results.

(i1 ...il ) e

(Ril ) ˆ

m,l

π /1 · d/

(i1 ...il−1 )

µm,l−1

given in the statement of the proposition

Let us investigate the order of the terms in the expression for (i1 ...il ) dm,l given by Proposition 9.3. First, consider an expression for dm,0 . Here, the leading terms are of order m + 1 and are contributed by L /kT (D / 2 µm−k−1 ) in the first sum, a 2nd angular derivative m−1 of T µ to principal terms, and by L /m−1 tr p in the last term of the second sum, a 1st T m−1 angular derivative of L /T θ to principal terms. The principal acoustical part of the latter is a 1st angular derivative of χ if m = 1, a 3rd angular derivative of T m−2 µ if m ≥ 2 (by equations (4.109)). It follows that the leading terms in Ril . . . Ri1 dm,0 , the first term in the expression for (i1 ...il ) dm,l , are of order m +l +1, the leading acoustical terms being l + 1st angular derivatives of χ and l + 2nd angular derivatives of µ if m = 1, l + 3rd angular derivatives of T m−2 µ and l + 2nd angular derivatives of T m−1 µ if m ≥ 2. The leading terms in l−1 

Ril . . . Ril−k+1 (

(Ril−k )

π /· D /2

(i1 ...il−k−1 )

µm,l−k−1 ),

k=0

the second term in the expression for (i1 ...il ) dm,l , are also of order m + l + 1, being l + 1st angular derivatives of T m µ, l ≥ 1. Finally, the leading term in l−1 

Ril . . . Ril−k+1 (tr

(Ril−k )

π /1 · d/

(i1 ...il−k−1 )

µm,l−k−1 ),

k=0

the third term in the expression for

(i1 ...il ) d

m,l ,

/ Ri2 tr (L / Ril . . . L

is:

(Ri1 )

π /1 ) · d/µm,0

contributed by the last term in the above sum. By the discussion in Chapter 8, this term involves the lth angular derivatives of χ and is thus of order l + 1, which coincides with the order of the leading terms in the first two terms of the expression for (i1 ...il ) dm,0 if m = 0. We conclude that (i1 ...il ) dm,l is of order m + l + 1 – one less than principal – and its leading acoustical terms are lth angular derivatives of χ and (for l ≥ 1) l + 1st angular derivatives of µ if m = 0 , l + 1st angular derivatives of χ and l + 2nd angular derivatives of µ and (for l ≥ 1) l + 1st angular derivatives of T µ if m = 1, l + 3rd angular

Chapter 9. The Propagation Equation for
/µ

310

derivatives of T m−2 µ and l + 2nd angular derivatives of T m−1 µ and (for l ≥ 1) l + 1st angular derivatives of T m µ if m ≥ 2. Let us also investigate the order of the terms in the expression for (i1 ...il ) em,l of Proposition 9.4. First, consider the expression for em,0 . Here, the leading terms are of or/ µm−k−1,0 ) in the first sum, a 2nd angular derivative der m +1 and are contributed by T k (
m−1 m−1 of T µ to principal terms, and by L /ˆT pˆ in the last term of the second sum, a 1st m−1 angular derivative of L /T θ to principal terms. The principal acoustical part of the latter is a 1st angular derivative of χ if m = 1, a 3rd angular derivative of T m−2 µ if m ≥ 2 (by equations (4.109)). It follows that the leading terms in L / Rˆ il . . . L / Rˆ i1 em,0 , the first term in the expression for (i1 ...il ) em,l , are of order m +l +1, the leading acoustical terms being l + 1st angular derivatives of χ and l + 2nd angular derivatives of µ if m = 1, l + 3rd angular derivatives of T m−2 µ and l + 2nd angular derivatives of T m−1 µ if m ≥ 2. The leading terms in 1 ˆ ˆ ( / Ril−k+1 L / Ril . . . L 2 l−1

(Ril−k ) ˆ

π /
/

(i1 ...il−k−1 )

µm,l−k−1 )

k=0

the second term in the expression for (i1 ...il ) em,l , are also of order m + l + 1, being l + 1st angular derivatives of T m µ, l ≥ 1. Finally, the leading term in l−1 

ˆ ( L / Rˆ il . . . L / Ril−k+1

(Ril−k ) ˆ

π /1 · d/

(i1 ...il−k−1 )

µm,l−k−1 )

k=0

the third term in the expression for

(i1 ...il ) e

/ Rˆ i2 (L / Rˆ il . . . L

m,l ,

is:

(Ri1 ) ˆ

π /1 ) · d/µm,0

contributed by the last term in the above sum. By the discussion in Chapter 8, this term involves the lth angular derivatives of χ and is thus of order l + 1, which coincides with the order of the leading terms in the first two terms of the expression for (i1 ...il ) em,0 if m = 0. We conclude that (i1 ...il ) em,l is of order m + l + 1 – one less than principal – and its leading acoustical terms are lth angular derivatives of χ and (for l ≥ 1) l + 1st angular derivatives of µ if m = 0 , l + 1st angular derivatives of χ and l + 2nd angular derivatives of µ and (for l ≥ 1) l + 1st angular derivatives of T µ if m = 1, l + 3rd angular derivatives of T m−2 µ and l + 2nd angular derivatives of T m−1 µ and (for l ≥ 1) l + 1st angular derivatives of T m µ if m ≥ 2. By (9.160) and Proposition 9.3 the function (i1 ...il ) µm,l satisfies on each St,u the elliptic equation:
/

(i1 ...il )

µm,l = µ−1 (

(i1 ...il )  x m,l

+

(i1 ...il ) ˇ f m,l )

+

(i1 ...il )

dm,l

(9.201)

Chapter 9. The Propagation Equation for
/µ

311

As we remarked earlier, this elliptic equation is to be considered in conjunction with the  . In view of (9.201), we shall then propagation equation of Proposition 9.2 for (i1 ...il ) x m,l only be able to control the product of µ with
/ (i1 ...il ) µm,l . Thus, we must develop µ2 weighted L estimates on St,u , appropriate to equation (9.201). These shall lead to µ/ˆ 2,m,l , to be used in the analysis of the propagation weighted L 2 estimates for (i1 ...il ) µ equation of Proposition 9.2. The required µ- weighted L 2 estimates on St,u are provided by the following lemma. Lemma 9.7 Let (M, g) be a compact 2-dimensional Riemannian manifold, and let φ be a function on (M, g) satisfying the equation:
g φ = ρ for some function ρ on M. Let also µ be an arbitrary non-negative function on M. Then the following integral estimate holds on M:  2 1 2 2 2 |∇ φ| + K |dφ| dµg ≤ 2 µ µ2 ρ 2 dµg + 3 |dµ|2 |dφ|2 dµg 2 M M M where K is the Gauss curvature of (M, g). Proof. Consider the 1-form: ψ = dφ

(9.202)

divg ψ =
g φ = ρ

(9.203)

We have: Taking the differential and multiplying by the function µ we obtain: µd(divg ψ) = d(µρ) − ρdµ

(9.204)

Now, in terms of components in an arbitrary local frame field d(divg ψ) is given by: ∇a (∇ b ψb ) and we have: ∇a (∇ b ψb ) = ∇ b (∇a ψb ) − Sa b ψb

(9.205)

(g −1 )bc

and Sac are the components of the Ricci curvature of (M, g). where Sa = Sac Since here M is 2-dimensional we have: b

Sa b = K δab Noting also that: ∇a ψb = ∇b ψa (9.205) reduces to: ∇a (∇ b ψb ) = ∇ b (∇b ψa ) − K ψa

(9.206)

Chapter 9. The Propagation Equation for
/µ

312

Substituting in (9.204) we obtain: µ∇ b (∇a ψb ) − µK ψa = da (µρ) − ρda µ

(9.207)

We multiply this equation by −µψ a and integrate over M. We have, integrating by parts, − µ2 ψ a ∇ b (∇b ψa ) = µ2 |∇ψ|2 dµg + 2µI a da µdµg (9.208) M

M

M

where I is the vectorfield with components: I a = ψ b ∇b ψ a

(9.209)

Moreover, we have, also integrating by parts,  µ2 ρ 2 + µρψ a da µ µψ a da (µρ)dµg = µρdivg (µψ)dµg = − M

M

(9.210)

M

In view of (9.208), (9.210), we then obtain the integral identity:

  µ2 ρ 2 + 2µρψ a da µ µ2 |∇ψ|2 + K |ψ|2 dµg + 2µI a da µdµg = M

M

M

(9.211)

Now, we have: 2µ|I a da µ| ≤ 2µ|I ||dµ| and from the definition (9.209), |I | ≤ |∇ψ||ψ| hence we can estimate: 2µ|I a da µ| ≤ 2µ|dµ||ψ||∇ψ| 1 ≤ µ2 |∇ψ|2 + 2|dµ|2|ψ|2 2

(9.212)

Also, we can estimate: 2µ|ρψ a da µ| ≤ µ2 ρ 2 + |dµ|2|ψ|2

(9.213)

In view of the estimates (9.212) and (9.213), the integral identity (9.210) implies:  1 |∇ψ|2 + K |ψ|2 dµg ≤ 2 µ2 µ2 ρ 2 dµg + 3 |dµ|2 |ψ|2 dµg (9.214) 2 M M M The lemma thus follows. We now apply Lemma 9.7 to equation (9.201), taking the manifold (M, g) to be (St,u , h /), the function φ to be (i1 ...il ) µm,l , and the function ρ to be: µ−1 (

(i1 ...il )  x m,l

+

(i1 ...il )

 )+ fˇm,l

(i1 ...il )

dm,l

Chapter 9. The Propagation Equation for
/µ

313

In view of the fact that by Lemma 8.9, taking δ0 suitably small, we have K ≥ 0, we obtain: µD /2

(i1 ...il )

µm,l  L 2 (St,u )

(i1 ...il )   x m,l  L 2 (St,u ) + C (i1 ...il ) fˇm,l  L 2 (St,u ) (i1 ...il ) + Cµ dm,l  L 2 (St,u ) + Cd/µ L ∞ (St,u ) d/ (i1 ...il ) µm,l  L 2 (St,u )

≤ C

(9.215)

Now, by bootstrap assumption F1, |d/µ| ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(9.216)

Substituting (9.216) in (9.215) yields the estimate: µD /2

(i1 ...il )

(i1 ...il )  x m,l  L 2 (St,u ) (9.217)  +C (i1 ...il ) fˇm,l  L 2 (St,u ) + Cµ (i1 ...il ) dm,l  L 2 (St,u ) +Cδ0 (1 + t)−1 [1 + log(1 + t)]d/ (i1 ...il ) µm,l  L 2 (St,u )

µm,l  L 2 (St,u ) ≤ C

In view of Proposition 9.4, we then obtain the following estimate for µ

(i1 ...il ) µ /ˆ 2,m,l

(i1 ...il )   x m,l  L 2 (St,u ) + C (i1 ...il ) fˇm,l  L 2 (St,u ) (9.218) +Cµ (i1 ...il ) dm,l  L 2 (St,u ) + Cµ (i1 ...il ) em,l  L 2 (St,u ) +Cδ0 (1 + t)−1 [1 + log(1 + t)]d/ (i1 ...il ) µm,l  L 2 (St,u )

(i1 ...il ) ˆ

µ / 2,m,l  L 2 (St,u ) ≤ C

Let us now return to the propagation equation of Proposition 9.2. Setting: (i1 ...il )  g˜ m,l

= −2µχˆ ·

(i1 ...il ) ˆ

µ / 2,m,l +

(i1 ...il )  gm,l

(9.219)

the propagation equation takes the form: L

(i1 ...il )  x m,l

+ (trχ − 2µ

−1

(Lµ))

(i1 ...il )  x m,l



1 trχ − 2µ−1 (Lµ) =− 2 +

(i1 ...il )  g˜ m,l



(i1 ...il ) ˇ f m,l

(9.220)

Defining as in Chapter 8 the diffeomorphisms "t,u of S 2 onto St,u , if ω is any St,u r covariant tensorfield defined in Wε∗0 , we consider the pullback ω(t, u) = "∗t,u ω, a r covariant tensorfield on S 2 depending on the parameters t and u. If in place of ω we have L / L ω, where again ω is any St,u r -covariant tensorfield defined in Wε∗0 , then the corresponding r -covariant tensorfield on S 2 is simply ∂ω(t, u)/∂t. The propagation equation  (t, u) on S 2 de(9.220) may then be viewed as an equation for the function (i1 ...il ) x m,l pending on t and u, an ordinary differential equation in the parameter t:     1 ∂µ (i1 ...il )  ∂µ (i1 ...il ) ˇ ∂ (i1 ...il )  trχ − 2µ−1 fm,l x m,l + trχ − 2µ−1 x m,l = − ∂t ∂t 2 ∂t +

(i1 ...il )  g˜ m,l

(9.221)

Chapter 9. The Propagation Equation for
/µ

314

Consider a function φ(t, u) on S 2 depending on the parameters t and u. We have: |φ|

1 ∂ 2 ∂ ∂φ |φ| = φ =φ ∂t 2 ∂t ∂t

(9.222)

 . From the propagation equation (9.221) We shall apply (9.222) in the case φ = (i1 ...il ) x m,l we have:   ∂ (i1 ...il )  ∂µ (i1 ...il )   x m,l x m,l + trχ − 2µ−1 )2 ( (i1 ...il ) x m,l ∂t ∂t   ∂µ (i1 ...il )  (i1 ...il ) ˇ 1 trχ − 2µ−1 =− x m,l fm,l 2 ∂t

+

(i1 ...il )   x m,l (i1 ...il ) g˜ m,l

(9.223)

Now, since |

(i1 ...il )   x m,l (i1 ...il ) fˇm,l |

≤|

(i1 ...il )   x m,l || (i1 ...il ) fˇm,l |

|

(i1 ...il )   x m,l (i1 ...il ) g˜ m,l |

≤|

(i1 ...il )   x m,l || (i1 ...il ) g˜ m,l |

and also

the assumption AS implies through (9.223) that:   ∂ (i1 ...il )  (i1 ...il )  −1 ∂µ  x m,l x m,l + trχ − 2µ |2 | (i1 ...il ) x m,l ∂t ∂t   1 −1 ∂µ   | (i1 ...il ) x m,l trχ − 2µ ≤ || (i1 ...il ) fˇm,l | 2 ∂t +|

(i1 ...il )   x m,l || (i1 ...il ) g˜ m,l |

Applying then (9.222) taking φ =  |: ential inequality for | (i1 ...il ) x m,l

(i1 ...il ) x  , m,l

(9.224) we deduce the following ordinary differ-

  ∂µ  + trχ − 2µ−1 | | (i1 ...il ) x m,l ∂t ∂t   1 ∂µ   ≤ trχ − 2µ−1 | (i1 ...il ) fˇm,l | + | (i1 ...il ) g˜ m,l | 2 ∂t

(i1 ...il ) x  | m,l

∂|

The integrating factor here is:  t     ∂µ µ(t, u) −2 A(t, u) exp trχ − 2µ−1 (t  , u)dt  = ∂t µ(0, u) 0

(9.225)

(9.226)

where A(t, u), defined by (8.178), is given by (8.181). Integrating the ordinary differential inequality (9.225) using the integrating factor (9.226) yields: |

(i1 ...il )  x m,l |

≤

(i1 ...il )

 Fm,l (t, u) +

(i1 ...il )

G m,l (t, u)

(9.227)

Chapter 9. The Propagation Equation for
/µ

where:

315

  Fl (t, u) = (A(t, u))−1 (µ(t, u))2 · (µ(0, u))−2 | (i1 ...il ) x m,l (0, u)| (9.228)   t ∂µ 1  trχ − 2µ−1 (t  , u)| (i1 ...il ) fˇm,l + (µ(t  , u))−2 A(t  , u) (t  , u)| 2 ∂t 0

(i1 ...il )

and: (i1 ...il )

G m,l (t, u) = (A(t, u))−1 (µ(t, u))2 t  · (µ(t  , u))−2 A(t  , u)| (i1 ...il ) g˜ m,l (t  , u)|dt 

(9.229)

0

Using the bounds (8.185) for A(t, u) we obtain from (9.228) and (9.229):  (i1 ...il )

 Fm,l (t, u) ≤ eCδ0 (1 − u + η0 t)−2

+

(i1 ...il )

(i1 ...il )

1

M  m,l (t, u) +

0

M  m,l (t, u)

(i1 ...il )



2

M  m,l (t, u)

(9.230)

where: (i1 ...il )

(i1 ...il )

(i1 ...il )

2

M



0

M  m,l (t, u) =

1

M  m,l (t, u) =

m,l

1 (t, u) = 2



µ(t, u) µ(0, u)

2 (1 − u)2 |

(i1 ...il )  x m,l (0, u)|

(9.231)

t

 µ(t, u) 2 (1 − u + η0 t  )2  , u) µ(t 0     ∂µ  × −2µ−1 (t  , u) · | (i1 ...il ) fˇm,l (t  , u)|dt  ∂t −

t 0

µ(t, u) µ(t  , u)

2

(1 − u + η0 t  )2 trχ(t  , u)|

(9.232)

(i1 ...il ) ˇ f m,l (t  , u)|dt 

(9.233)

Also: (i1 ...il )

G m,l (t, u) ≤ eCδ0 (1 − u + η0 t)−2  t µ(t, u) 2 (1 − u + η0 t  )2 | · µ(t  , u) 0

(9.234) (i1 ...il )  g˜ m,l (t  , u)|dt 

 (t) on [0, ε ] × S 2 . We shall estimate the L 2 norm of (i1 ...il ) Fm,l 0 First, by the basic bootstrap assumption A3 and inequalities (8.356) at t = 0,



(i1 ...il )

0

M  m,l (t) L 2 ([0,ε0]×S 2 ) ≤ C[1 + log(1 + t)]2 

(i1 ...il )  x m,l (0) L 2 ( ε0 ) 0

(9.235)

Chapter 9. The Propagation Equation for
/µ

316 1

We turn to (i1 ...il ) M  m,l . Partitioning [0, ε0] × S 2 into the open sets Vs− , Vs+ , defined by (8.360), (8.361), respectively, we have: 

(i1 ...il )

= Now, for fixed t,

1

M  m,l (t)2L 2 ([0,ε

(i1 ...il )

(i1 ...il )

0 ]×S

1

M  m,l (t)2L 2 (V

s−

(9.236)

2)

(i1 ...il )

+ )

1

M  m,l (t)2L 2 (V

s+ )

1

M  m,l (t) restricted to Vs− is a definite integral, with respect to t  , 1

of a function of t  with values in the space L 2 (Vs− ). Thus, the norm of (i1 ...il ) M  m,l (t) in L 2 (Vs− ) does not exceed the integral of the L 2 (Vs− )-norm of the said function, with respect to t  : 

(i1 ...il )

1

M



m,l

(t) L 2 (Vs− ) ≤

t



(i1 ...il )

0

1

N  m,l (t, t  ) L 2 (Vs− ) dt 

(9.237)

where: (i1 ...il )

1

N  m,l (t, t  , u)       µ(t, u) 2  2 −1 ∂µ  (1 − u + η t ) (t , u) | = −2µ 0 µ(t  , u) ∂t −

(9.238) (i1 ...il ) ˇ fm,l (t  , u)|

We have: (i1 ...il )

1

N  m,l (t, t  ) L 2 (Vs− )        µ(t) 2 −1 ∂µ  max (t )  −2µ ≤ (1 + η0 t  )2 max µ(t  ) ∂t − Vs− Vs−



(9.239) (i1 ...il ) ˇ f m,l (t  ) L 2 (Vs− )

Thus, in view of (8.366) and (8.367) (see definition (8.273)): 

(i1 ...il )

1

N  m,l (t, t  ) L 2 (Vs− ) ≤ C(1 + t  )2 M(t  )

(i1 ...il )

 (t  ) L 2 ([0,ε0 ]×S 2 ) (9.240) fˇm,l

Let us define: (i1 ...il )

 Pm,l (t) = (1 + t)

Suppose that, for non-negative quantities (i1 ...il )

 Pm,l (t) ≤

(i1 ...il )

(i1 ...il )

(i1 ...il ) (0) 

P

m,l

 fˇm,l (t) L 2 ([0,ε0]×S 2 )

(0) P

m,l ,

(t) +

(11 ...il )

(i1 ...il )

(1) P

(1) 

P

m,l ,

m,l

(9.241)

we have:

(t)

(9.242)

Chapter 9. The Propagation Equation for
/µ

317 (0) 

We define then the non-decreasing non-negative quantities (i1 ...il ) P by: (i1 ...il )

(i1 ...il )

(1) 

P

m,l,a

(0) 

P

m,l,a

(t) = sup {µam (t  )

(i1 ...il )

t  ∈[0,t ]

(t) = sup {(1 + t  )1/2 µam (t  )

(0) 

P

(i1 ...il )

t  ∈[0,t ]

m,l

(1) 

P

(1) 

(i ...il ) P m,l,a , 1

(t  )}

m,l

m,l,a ,

(9.243)

(t  )}

(9.244)

Then, for t  ∈ [0, t] we have: (i1 ...il )

 Pm,l (t  ) 

 ≤ µ−a m (t )

(i1 ...il )

(0) 

P

(t) + (1 + t  )−1/2

m,l,a

(i1 ...il )

(1) 

P

 m,l,a

(9.245)

(t)

Substituting in (9.240) we obtain, in view of the definition (9.241), (i1 ...il )





1

N  m,l (t, t  ) L 2 (Vs− )

≤ C (1 + t)

(0) (i1 ...il ) 

P

m,l,a

(t) + (1 + t)

(1) 1/2 

P

(9.246)

   µ−a m (t )M(t )

m,l,a

for all t  ∈ [0, t]. Substituting this estimate in (9.237) we then obtain: 

(i1 ...il )



1

M  m,l (t) L 2 (Vs− )

≤ C (1 + t)

(0) (i1 ...il ) 

P

(t) + (1 + t)

m,l,a

(1) 1/2 (i1 ...il ) 

P

(9.247)

 m,l,a

Ia (t)

At this point we apply Lemma 8.11 to conclude that: 

(i1 ...il )

1

M  m,l (t) L 2 (Vs− )  (0)

≤ Ca −1 (1 + t)

(i1 ...il )

P



m,l,a

(t) + (1 + t)1/2

(i1 ...il )

(1) 

P

 m,l,a

(9.248)

(t) µ−a m (t)

In analogy with (9.237) and (9.239), with Vs+ in the role of Vs− , we have: t 1 1 (i1 ...il )  M m,l (t) L 2 (Vs+ ) ≤  (i1 ...il ) N  m,l (t, t  ) L 2 (Vs+ ) dt  (9.249)  0

and: (i1 ...il )

1

N  m,l (t, t  ) L 2 (Vs+ )        µ(t) 2 −1 ∂µ  −2µ ≤ (1 + η0 t  )2 max max (t )  µ(t  ) ∂t − Vs+ Vs+



(9.250) (i1 ...il ) ˇ f m,l (t  ) L 2 (Vs+ )

Chapter 9. The Propagation Equation for
/µ

318

Substituting the estimates (8.379) and (8.380) as well as the definition (9.241) in (9.250) and the result in (9.249), we obtain: 

(i1 ...il )

1

M



m,l

(t) L 2 (Vs+ ) ≤ Cδ0 [1+log(1+t)]

t

2 0

[1 + log(1 + t  )] (1 + t  )

(i1 ...il )

 Pm,l (t  )dt 

(9.251)

hence, by (9.245), 

(i1 ...il )

1

M  m,l (t) L 2 (Vs+ ) 

≤ Cδ0 [1 + log(1 + t)]

t

· 0

2

(0) 

(i1 ...il )

P

m,l,a

(t) +

(i1 ...il )

(1) 

P

 m,l,a

(t)

[1 + log(1 + t  )] −a   µm (t )dt (1 + t  )

(9.252)

In view of the bound (8.387) for the integral on the right-hand side in (9.252), we conclude that: 

(i1 ...il )

1

M  m,l (t) L 2 (Vs+ ) 

(0) 

(i1 ...il )

≤ Cδ0 [1 + log(1 + t)]4

P

m,l,a

(t) +

(i1 ...il )



(1)

(9.253)

Pm,l,a (t) µ1−a m (t)

Combining finally the estimates (9.248) and (9.253) (see (9.236)) and taking into account the assumption that the product aδ0 ≤ C −1 , we obtain: 

(i1 ...il )

1

M  m,l (t) L 2 ([0,ε0 ]×S 2 )  (0)

≤ Ca −1 (1 + t)

We turn to 

(i1 ...il )

(i1 ...il )

=

(i1 ...il )

P



m,l,a

(t) + (1 + t)1/2

(i1 ...il )

(1) 

P

 m,l,a

(9.254)

(t) µ−a m (t)

2

M  m,l (t, u). We have: 2

M  m,l (t)2L 2 ([0,ε

(i1 ...il )

0 ]×S

2)

2

M  m,l (t)2L 2 (V

s− )

+

(i1 ...il )

2

M  m,l (t)2L 2 (V

s+ )

(9.255)

and, in analogy with (9.237) and (9.249), 

(i1 ...il )

2

M  m,l (t) L 2 (Vs− ) ≤



t 0



i1 ...il )

2

N  m,l (t, t  ) L 2 (Vs− ) dt 

(9.256)

Chapter 9. The Propagation Equation for
/µ

319

and, 

2

(i1 ...il )

M



(t) L 2 (Vs+ ) ≤

m,l

t



i1 ...il )

0

2

N  m,l (t, t  ) L 2 (Vs+ ) dt 

(9.257)

where: (i1 ...il )

2

N  m,l (t, t  , u) =

1 2



µ(t, u) µ(t  , u)

2

(1 − u + η0 t  )2 trχ(t  , u)|

(i1 ...il ) ˇ f m,l (t  , u)|

(9.258)

We have: 2

N  m,l (t, t  ) L 2 (Vs− )      µ(t) 2 1 max trχ(t  )  ≤ (1 + η0 t  )2 max  2 µ(t ) Vs− Vs−



(i1 ...il )

(9.259) (i1 ...il )

 fˇm,l (t  ) L 2 (Vs− )

Now, by Corollary 1 to Lemma 8.11 (8.366) holds, while by bootstrap assumption F2 (8.395) holds. Substituting these in (9.259) we obtain: 

2

(i1 ...il )

N  m,l (t, t  ) L 2 (Vs− ) ≤ C(1 + t  )2 

(i1 ...il ) ˇ fm,l (t  ) L 2 ([0,ε0]×S 2 )

(9.260)

or, in terms of the definition (9.241), 

2

(i1 ...il )

N  m,l (t, t  ) L 2 (Vs− ) ≤ C



(i1 ...il )

≤C  ≤C



(i1 ...il )

(0) 

P

m,l,a

(0) 

P

m,l,a

(i1 ...il )

 Pm,l (t  )

 −1/2 (i1 ...il )

(t) + (1 + t )

 −1/2 (i1 ...il )

(t) + (1 + t )

(1) 

P

 m,l,a

(1) 

P

 (t) µ−a m (t )

 m,l,a

(t) µ−a m (t)

(9.261)

Here, in the last two steps we have appealed to (9.245) and to Corollary 2 of Lemma 8.11. Substituting the estimate (9.261) in (9.256) then yields: 

(i1 ...il )



2

M  m,l (t) L 2 (Vs− )

≤ C (1 + t)

(i1 ...il )

(0) 

P

m,l,a

(t) + (1 + t)1/2

(i1 ...il )

(1) 

P

 m,l,a

(9.262)

(t) µ−a m (t)

In analogy with (9.259) we have: (i1 ...il )

2

N  m,l (t, t  ) L 2 (Vs+ )      µ(t) 2 1 ≤ (1 + η0 t  )2 max max trχ(t  )  2 µ(t  ) Vs+ Vs+



(9.263) (i1 ...il )

 (t  ) L 2 (Vs+ ) fˇm,l

Chapter 9. The Propagation Equation for
/µ

320

√ √ Here, we have two cases to distinguish according as to whether t  is < t or ≥ t. In the first case (8.402) holds, while in the second case (8.401) holds. In view of these, (8.395) and the definition (9.241), we obtain: 2



(i1 ...il )

N  m,l (t, t  ) L 2 (Vs+ ) ≤ C[1 + log(1 + t)]2



(i1 ...il )

N  m,l (t, t  ) L 2 (Vs+ ) ≤ C

2

(i1 ...il )

 Pm,l (t  )

(i1 ...il )

 Pm,l (t  )

: if t  ≥

: if t  <

√ t

√ t

(9.264)

Substituting the estimates (9.264) in (9.257), yields: 

2

(i1 ...il )

M  m,l (t) L 2 (Vs+ )

√ t



≤



i1 ...il )

0

2

N  m,l (t, t  ) L 2 (Vs+ ) dt  + √ t



≤ C[1 + log(1 + t)]

2

(i1 ...il )

0

≤ C[1 + log(1 + t)]2  √t −a  × µm (t ) 0





t

−a  √ µm (t ) t

+C

(i1 ...il )

(i1 ...il )

≤ C[1 + log(1 + t)]2  × (1 + t)

1/2 (i1 ...il )

 + C (1 + t)

(i1 ...il )

 ≤C



(1 + t)

(i1 ...il )

(0) 

P

(0) 

P

(0) 

P

(0) 

P

(0) 

P

m,l,a

m,l,a

t √ t

 Pm,l (t  )dt 

m,l,a

m,l,a

m,l,a





2

i1 ...il )

+C

N  m,l (t, t  ) L 2 (Vs+ ) dt 

t

√

(i1 ...il )

t

 Pm,l (t  )dt 

(1) 

 −1/2 (i1 ...il )

(t) + (1 + t )

 −1/2 (i1 ...il )

(t) + (1 + t )

(t) + (1 + t)

(t) + (1 + t)

(t) + (1 + t)

1/4 (i1 ...il )

1/2 (i1 ...il )

1/2 (i1 ...il )

(1) 

P

(1) 

P

(1) 

P

P

(1) 

P

 m,l,a

(t) dt  

m,l,a

(t) dt 

 m,l,a

(t) µ−a m (t) 

m,l,a

(t) µ−a m (t) 

m,l,a

(t) µ−a m (t)

(9.265)

Here, in the last steps we have again appealed to (9.245) and to Corollary 2 of Lemma 8.11. Combining finally the estimates (9.262) and (9.265) (see (9.255)), we obtain: 

(i1 ...il )



2

M  m,l (t) L 2 ([0,ε0]×S 2 )

≤ C (1 + t)

(i1 ...il )

(0) 

P

m,l,a

(t) + (1 + t)1/2

(i1 ...il )

(1) 

P

 m,l,a

(t) µ−a m (t)

(9.266)

Chapter 9. The Propagation Equation for
/µ

321

The estimates (8.279), (9.254) and (9.266) together yield, through (9.230), the fol (t) on [0, ε ] × S 2 : lowing estimate for the L 2 norm of (i1 ...il ) Fm,l 0 

(i1 ...il )

 Fm,l (t) L 2 ([0,ε0]×S 2 )

≤ C(1 + t)

−2

+ C(1 + t)

(9.267) 2

[1 + log(1 + t)]   (0)

−1

(i1 ...il )

P



m,l,a

(i1 ...il )  x m,l (0) L 2 ( ε0 ) 0

(t) + (1 + t)

−1/2 (i1 ...il )

(1) 

P

 m,l,a

(t) µ−a m (t)

We now consider the estimate (9.234) for the function (i1 ...il ) G m,l . In view of the bound (8.407), (9.234) implies: t (i1 ...il )   G m,l (t, u) ≤ C(1+t)−2 [1+log(1+t)]2 (1+t  )2 | (i1 ...il ) g˜ m,l (t  , u)|dt  (9.268) 0

It follows that: 

(i1 ...il )

G m,l (t) L 2 ([0,ε0 ]×S 2 ) ≤ C(1 + t)−2 [1 + log(1 + t)]2 (9.269) t  · (1 + t  )2  (i1 ...il ) g˜ m,l (t  ) L 2 ([0,ε0]×S 2 ) dt  0

(i1 ...il ) g˜  m,l

Now the function is given by (9.219). Here we must distinguish the principal acoustical terms. According to the discussion following Lemma 9.4, the principal  consists of the terms (9.144) and (9.146) and acoustical part of the function (i1 ...il ) gm,l  the sum (9.152). Let us define the function (i1 ...il ) g˙ m,l by: (i1 ...il )  g˙ 0,l

=

(i1 ...il )  g0,l − ξ · (i1 ...il ) xl l−1  il−k  (Ril−k )  − Z (i1 . . . il ) x 0,l−1 k=0

: if m = 0

(9.270)

and: (i1 ...il )  g˙ m,l

=

(i1 ...il )  gm,l

−m

− ξ · d/

(i1 ...il )  x m−1,l

(i1 ...il )  x m−1,l l−1  (R ) il−k

−

Z

i



(i1 . l−k . . il )  x m,l−1

k=0

: if m ≥ 1 Then the function

(i1 ...il ) g˙  m,l

(i1 ...il )  g˜ 0,l

(9.271)

does not contain principal acoustical terms and we have:

= −2µχˆ · +

l−1 

(i1 ...il ) ˆ

(Ril−k )

µ / 2,0,l + ξ ·

Z

i



(i1 ...il )

(i1 . l−k . . il )  x 0,l−1

xl

+

(i1 ...il )  g˙ 0,l

k=0

: if m = 0

(9.272)

Chapter 9. The Propagation Equation for
/µ

322

and: (i1 ...il )  g˜ m,l

= −2µχˆ · +m

(i1 ...il ) ˆ

µ / 2,m,l + ξ · d/

(i1 ...il )  x m−1,l

+

l−1 

(i1 ...il )  x m−1,l

(Ril−k )

Z

i



(i1 . l−k . . il )  x m,l−1

k=0

+

(i1 ...il )  g˙ m,l

: if m ≥ 1

(9.273)

Let us define:  = max  X m,l i1 ...il

(i1 ...il )  x m,l (t) L 2 ([0,ε0 ]×S 2 )

(9.274)

 (t)  We shall estimate  (i1 ...il ) g˜ m,l L 2 ([0,ε0 ]×S 2 ) in terms of X 0,l (t) and X l (t), defined by   (8.414), for m = 0, and in terms of X m,l (t) and X m−1,l+1 (t), for m ≥ 1. Consider first the second terms on the right in (9.272) and (9.273). Now the St,u 1-form ξ is given by (9.83). By the bootstrap assumptions E1, E2, as well as the second condition of F1, of Chapter 6, and the basic bootstrap assumptions A, we have:

|ξ | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(9.275)

Thus, by the pointwise estimates (9.145): ξ · ξ · d/

(i1 ...il )

xl  L 2 ([0,ε0]×S 2 ) ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]X l (t)

(i1 ...il )  x m−1,l  L 2 ([0,ε0]×S 2 )

(9.276)

 ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]X m−1,l+1 (t)

This is so because, writing max | j

(i1 ...il j )  x m−1,l+1 |

≤



|

(i1 ...il j )  x m−1,l+1 |

j

we have:  max | j

(i1 ...il j )  x m−1,l+1 | L 2 ([0,ε0 ]×S 2 )

≤





(i1 ...il j )  x m−1,l+1  L 2 ([0,ε0 ]×S 2 )

j  ≤ 3X m−1,l+1

Consider next the third term on the right in (9.273). From Chapter 6, the assumptions A, E1, E2, together with the second condition of F1, imply: || ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(9.277)

Thus, by the pointwise estimate (9.147): m

(i1 ...il )  x m,l  L 2 ([0,ε0 ]×S 2 )

 ≤ Cmδ0 (1 + t)−2 [1 + log(1 + t)]X m−1,l+1 (t) (9.278)

Chapter 9. The Propagation Equation for
/µ

323

Consider next the sum (9.152), which is the next to the last term on the right in (9.272) and (9.273). By the pointwise estimates (9.153) and (8.417): 0 l−1 0 0 0 0

(Ril−k )

Z

k=0

0 0

0 (i1 . l−k . . il )  x m,l−1 0 0 i



 ≤ Clδ0 (1 + t)−2 [1 + log(1 + t)]X m,l (t) L 2 ([0,ε0 ]×S 2 )

(9.279)

Finally, we consider the first term on the right in (9.272) and (9.273). By the bootstrap assumption F2 we have: 2µχˆ ·

(i1 ...il ) ˆ

≤ Cδ0 (1 + t)

µ / 2,m,l  L 2 ([0,ε0]×S 2 ) −2

[1 + log(1 + t)]µ|

(i1 ...il ) ˆ

µ / 2,m,l |(t) L 2 ([0,ε0 ]×S 2 )

(9.280)

We now appeal to the estimate (9.218) on St,u . In view of the inequalities (8.420), the estimate (9.218) is equivalent to: µ|

(i1 ...il ) ˆ

µ / 2,m,l |(t, u) L 2 (S 2) (i1 ...il )   ≤ C x m,l (t, u) L 2 (S 2 ) + C (i1 ...il ) fˇm,l (t, u) L 2 (S 2) (i1 ...il ) (i1 ...il ) + Cµ dm,l (t, u) L 2 (S 2 ) + Cµ| em,l |(t, u) L 2 (S 2) −1 (i1 ...il ) + Cδ0 (1 + t) [1 + log(1 + t)]|d/ µm,l |(t, u) L 2 (S 2)

(9.281)

Taking L 2 norms on [0, ε0 ] then yields: µ|

(i1 ...il ) ˆ

µ / 2,m,l |(t) L 2 ([0,ε0]×S 2 ) (i1 ...il )   ≤ C x m,l (t) L 2 ([0,ε0]×S 2 ) + C (i1 ...il ) fˇm,l (t) L 2 ([0,ε0 ]×S 2 ) + Cµ (i1 ...il ) dm,l (t) L 2 ([0,ε0 ]×S 2 ) + Cµ| (i1 ...il ) em,l |(t) L 2 ([0,ε0]×S 2 ) + Cδ0 (1 + t)−1 [1 + log(1 + t)]|d/ (i1 ...il ) µm,l |(t) L 2 ([0,ε0]×S 2 )

(9.282)

In view of the estimates (9.276), (9.278), (9.279) and (9.282) we obtain from (9.272), (9.273) the estimates: 

(i1 ...il )  g˜ 0,l (t) L 2 ([0,ε0 ]×S 2 )

 ≤ C(l + 1)δ0(1 + t)−2 [1 + log(1 + t)]X 0,l (t)

+ Cδ0 (1 + t)−1 [1 + log(1 + t)]X l (t) +

(i1 ...il )

Q 0,l (t)

: if m = 0 

(i1 ...il )  g˜ m,l (t) L 2 ([0,ε0 ]×S 2 )

 ≤ C(l + 1)δ0(1 + t)−2 [1 + log(1 + t)]X m,l (t)  (t) + C(m + 1)δ0 (1 + t)−2 [1 + log(1 + t)]X m−1,l+1

+

(i1 ...il )

Q m,l (t)

: if m ≥ 1

(9.283)

Chapter 9. The Propagation Equation for
/µ

324

where: (i1 ...il )

Q m,l (t) = Cδ02 (1 + t)−3 [1 + log(1 + t)]2 |d/

(i1 ...il )

µm,l |(t) L 2 ([0,ε0 ]×S 2 ) (i1 ...il ) ˇ f m,l (t) L 2 ([0,ε0]×S 2 ) + Cδ0 (1 + t) [1 + log(1 + t)] −2 (i1 ...il ) + Cδ0 (1 + t) [1 + log(1 + t)]µ dm,l (t) L 2 ([0,ε0]×S 2 ) + Cδ0 (1 + t)−2 [1 + log(1 + t)]µ| (i1 ...il ) em,l |(t) L 2 ([0,ε0]×S 2 )  +  (i1 ...il ) g˙ m,l (t) L 2 ([0,ε0 ]×S 2 ) (9.284) −2

We now return to the estimate (9.227). Taking L 2 -norms on [0, ε0] × S 2 yields: (i1 ...il )  x m,l (t) L 2 ([0,ε0]×S 2 ) (i1 ...il )  ≤ Fm,l (t) L 2 ([0,ε0]×S 2 )



+

(i1 ...il )

G m,l (t) L 2 ([0,ε0 ]×S 2 )

(9.285)

Substituting the estimates (9.267), (9.269) and (9.283) then yields: 

(i1 ...il )  x 0,l (t) L 2 ([0,ε0 ]×S 2 )

 B0,l (t) t (1 + t  )[1 + log(1 + t  )]X l (t  )dt  + Cδ0 (1 + t)−2 [1 + log(1 + t)]2 0 t  + C(l + 1)δ0 (1 + t)−2 [1 + log(1 + t)]2 [1 + log(1 + t  )]X 0,l (t  )dt 

≤

: if m = 0  (t) L 2 ([0,ε0 ]×S 2 ) ≤  (i1 ...il ) x m,l

(i1 ...il )

0

(i1 ...il )

 Bm,l (t)

t  [1 + log(1 + t  )]X m−1,l+1 (t  )dt  + C(m + 1)δ0 (1 + t)−2 [1 + log(1 + t)]2 0 t −2 2  + C(l + 1)δ0 (1 + t) [1 + log(1 + t)] [1 + log(1 + t  )]X m,l (t  )dt  0

: if m ≥ 1

(9.286)

where: (i1 ...il )

 Bm,l (t)

= C(1 + t)−2 [1 + log(1 + t)]2   (0) + C(1 + t)

−1

+ C(1 + t)

−2

(i1 ...il )

P



m,l,a

(i1 ...il )  x m,l (0) L 2 ( ε0 ) 0

(t) + (1 + t)

[1 + log(1 + t)]

t

2 0

(1 + t  )2

−1/2 (i1 ...il )

(i1 ...il )

(1) 

P

 m,l,a

Q m,l (t  )dt 

(t) µ−a m (t) (9.287)

Chapter 9. The Propagation Equation for
/µ

325

Taking in (9.286) the maximum over i 1 . . . i l and recalling the definition (9.274) we obtain: t   −2 2 (1 + t  )[1 + log(1 + t  )]X l (t  )dt  X 0,l ≤ B0,l (t) + Cδ0 (1 + t) [1 + log(1 + t)] 0 t −2 2  + C(l + 1)δ0 (1 + t) [1 + log(1 + t)] [1 + log(1 + t  )]X 0,l (t  )dt  0

: if m = 0   X m,l ≤ Bm,l (t) + C(m + 1)δ0 (1 + t)−2 [1 + log(1 + t)]2 t  × [1 + log(1 + t  )]X m−1,l+1 (t  )dt  0 t  + C(l + 1)δ0 (1 + t)−2 [1 + log(1 + t)]2 [1 + log(1 + t  )]X m,l (t  )dt  0

: if m ≥ 1

(9.288)

where:  Bm,l (t) = max

(i1 ...il )

i1 ...il

 Bm,l (t)

(9.289)

Setting: l =n−1−m  (9.288) is a set of ordinary integral inequalities for X m,n−1−m , m = 0, . . . , n −1. For m =  0, we have an integral inequality for X 0,n−1 containing on the right-hand side the quantity X n−1 which has been estimated in Chapter 8. For m ≥ 1, we have an integral inequality   containing on the right-hand side the quantity X m−1,n−1−(m−1) . Thus, the for X m,n−1−m integral inequalities, considered successively in the order of increasing m, depend only on quantities already estimated. Setting:  Yl,m (t) =



t 0

 [1 + log(1 + t  )]X m,l (t  )dt 

(9.290)

and recalling also the definition (8.430), since  (t) dYm,l

dt

 = [1 + log(1 + t)]X m,l (t)

 satisfy by virtue of (9.288) the ordinary differential inequalities: the Ym,l  (t) dY0,l

dt

 ≤ [1 + log(1 + t)]B0,l (t) + Cδ0 (1 + t)−2 [1 + log(1 + t)]3 Yl (t)  + C(l + 1)δ0 (1 + t)−2 [1 + log(1 + t)]3 Y0,l (t)

: if m = 0

Chapter 9. The Propagation Equation for
/µ

326  (t) dYm,l

dt

 ≤ [1 + log(1 + t)]Bm,l (t)  + C(m + 1)δ0(1 + t)−2 [1 + log(1 + t)]3 Ym−1,l+1 (t)  + C(l + 1)δ0 (1 + t)−2 [1 + log(1 + t)]3 Ym,l (t)

: if m ≥ 1

(9.291)

The integrating factor here is:

e−Cl (t )

where Cl (t) is of the form (8.432). Integrating the ordinary differential inequalities  vanish, using the above integrating factor, yields: (9.291) from t = 0 where the Ym,l  Y0,l (t)

≤e

Cl (t )



t



 e−Cl (t ) [1 + log(1 + t  )]B0,l (t  )dt 

0



t

+Cδ0

(1 + t  )−2 [1 + log(1 + t  )]3 Yl (t  )dt 

0

 Ym,l (t)

: if m = 0 t   ≤ eCl (t ) e−Cl (t ) [1 + log(1 + t  )]Bm,l (t  )dt  0 t  +C(m + 1)δ0 (1 + t  )−2 [1 + log(1 + t  )]3 Ym−1,l+1 (t  )dt  0

: if m ≥ 1

(9.292)

Now, C( t) satisfies a bound of the form (8.434), therefore if δ0 satisfies a smallness con dition of the form (8.435), then, taking into account the fact that Yl and the Ym−1,l+1 are non-decreasing functions of t while the integral

∞ 0

[1 + log(1 + t  )]3  dt (1 + t  )2

is convergent, (9.292) imply the estimates:  Y0,l (t) ≤ 2

0

t

 [1 + log(1 + t  )]B0,l (t  )dt  + Cδ0 Yl (t)

: if m = 0 t    (t) ≤ 2 [1 + log(1 + t  )]Bm,l (t  )dt  + C(m + 1)δ0 Ym−1,l+1 (t) Ym,l 0

: if m ≥ 1

(9.293)

It follows, setting l = n − 1 − m, m = 0, 1, . . . , n − 1, that:  Y0,n−1 (t) ≤ 2

0

t

 [1 + log(1 + t  )]B0,n−1 (t  )dt  + Cδ0 Yn−1 (t)

(9.294)

Chapter 9. The Propagation Equation for
/µ

327

and, for m = 1, . . . , n − 1:   (t) ≤ (m + 1)!(Cδ0 )m Y0,n−1 (t) (9.295) Ym,n−1−m m−1 t  m!  (Cδ0 )k +2 [1 + log(1 + t  )]Bm−k,n−1−(m−k) (t  )dt  (m − k)! 0 k=0

Substituting the bound (8.436) for Yn−1 (t), these imply that: t  Ym,n−1−m (t) ≤ 2 (1 + t  )[1 + log(1 + t  )]Bn−1 (t  )dt  0

+2

m 

t

k=0 0

 [1 + log(1 + t  )]Bk,n−1−k (t  )dt 

: for all m = 0, . . . , n − 1

(9.296)

provided that δ0 is suitably small depending on n. The bounds (8.436) and (9.296) in turn imply:  Yn−1 (t) + nY0,n−1 (t) t ≤ 2(n + 1) (1 + t  )[1 + log(1 + t  )]Bn−1 (t  )dt  0 t  + 2n [1 + log(1 + t  )]B0,n−1 (t  )dt  0

: for m = 0   (m + 1)Ym−1,n−1−(m−1) (t) + (n − m)Ym,n−1−m (t) t ≤ 2(n + 1) (1 + t  )[1 + log(1 + t  )]Bn−1 (t  )dt  0

+ 2(n + 1)

m−1  t k=0 t

+ 2(n − m)

0

0

 [1 + log(1 + t  )]Bk,n−1−k (t  )dt 

 [1 + log(1 + t  )]Bm,n−1−m (t  )dt 

: for m = 1, . . . , n − 1

(9.297)

 (t) These yield the sought-for estimates for  (i1 ...il ) x m,l L 2 ([0,ε0 ]×S 2 ) . For, recalling the  definitions of Yl and the Ym,l , the estimates (9.286) take upon setting l = n − 1 − m the form: (i1 ...in−1 )  x 0,n−1 (t) L 2 ([0,ε0 ]×S 2 ) 1 (i1 ...in−1 )  ≤ B0,n−1 (t) + Cδ0 (1 + t)−2 [1 + log(1 + t)]2 Yn−1 (t)



: for m = 0

 + nY0,n−1 (t)

2

Chapter 9. The Propagation Equation for
/µ

328

(i1 ...in−1−m )  x m,n−1−m (t) L 2 ([0,ε0 ]×S 2 ) (i1 ...in−1−m )  ≤ Bm,n−1−m (t) + Cδ0 (1 + t)−2 [1 + log(1 + t)]2



  × (m + 1)Ym−1,n−1−(m−1) (t) + (n − m)Ym,n−1−m (t)



: for m = 1, . . . , n − 1

(9.298)

Substituting then (9.297), we finally conclude that: (i1 ...in−1−m )  x m,n−1−m (t) L 2 ([0,ε0]×S 2 ) (i1 ...in−1−m )  ≤ Bm,n−1−m (t) + 2Cδ0 (1 + t)−2 [1 + log(1 + t)]2  t 



(1 + t )[1 + log(1 + t  )]Bn−1 (t  )dt 

× (n + 1)

0

+ (n + 1) +(n − m)

m−1  t k=0 t 0

: for all m = 0, . . . , n − 1

0

 [1 + log(1 + t  )]Bk,n−1−k (t  )dt 

 [1 + log(1 + t  )]Bm,n−1−m (t  )dt 



(9.299)

Chapter 10

Control of the Angular Derivatives of the First Derivatives of the x i . Assumptions and Estimates in Regard to χ Part 1: Control of the angular derivatives of the first derivatives of the x i The first part of the present chapter is concerned with derivation of estimates for angular derivatives of the spatial rectangular coordinates x i : i = 1, 2, 3 and for angular derivatives of the Tˆ x i = Tˆ i . By angular derivatives we mean derivatives with respect to the rotation fields R j : j = 1, 2, 3. The derivation of these estimates is based on bootstrap assumptions in regard to χ. These estimates are then used to obtain estimates for angular derivatives of the deformation tensors of the commutation fields. Before we proceed, we must discuss the general definition of the operator L / X , with /L , L /T , as applied to X an arbitrary St,u -tangential vectorfield, as well as the operators L q an arbitrary type T p St,u tensorfield. Consider first the case of an St,u 1-form ξ . Let X be an St,u -tangential vectorfield. Then L / X ξ , is simply defined by considering a given surface St,u , as the usual Lie derivative of ξ with respect to X, a notion intrinsic to St,u , which makes no reference to the ambient spacetime. To define L / L ξ however, we must consider the extension of ξ to a given Cu as an St,u 1-form, that is, by requiring ξ(L) = 0. We then define L / L ξ to be the restriction to T St,u of the usual Lie derivative of ξ with respect to L, a notion intrinsic to Cu . Note that in any case (L L ξ )(L) = 0. To define L /T ξ , we must similarly consider the extension of ξ to a given t as an St,u 1-form, that is, by requiring ξ(T ) = 0. We then define L /T ξ to be the restriction to T St,u of the usual Lie derivative of ξ with respect to T , a notion intrinsic to t . Note again that in any case (LT ξ )(T ) = 0. The case of any p-covariant St,u tensorfield in the role of ξ , is formally identical to the case of an St,u 1-form. Consider next the case of an St,u -tangential vectorfield Y . Let

330

Chapter 10. Control of the Angular Derivatives

X be another St,u tangential vectorfield. Then L / X Y is defined to be simply L X Y = [X, Y ], which is itself an St,u tangential vectorfield. According to Lemma 8.2, L L Y = [L, Y ] = (Y ) Z is an S -tangential vectorfield, so L / L Y is defined to be simply L L Y . From the t,u proof of Lemma 8.2 the fact that [L, Y ] is an St,u -tangential vectorfield is a consequence of the fact that the 1-parameter group of diffemorphisms "s generated by L maps St,u onto St +s,u , thus preserving the foliation of Cu by the St,u . Similarly, the fact that 1parameter group s generated by T maps St,u onto St,u+s , thus preserving the foliation of t by the St,u , implies that LT Y = (Y )  is an St,u -tangential vectorfield, as will be shown in the proof of Lemma 10.22. Thus also L /T Y is defined to be simply LT Y . The case of any q-contravariant St,u tensorfield W in the role of Y , is formally identical to the case of an St,u tangential vectorfield, such a tensorfield being expressible as the sum of tensor products of St,u -tangential vectorfields. Thus L /X W , L /L W , L /T W are defined to be simply L X W , L L W , LT W , respectively. q Consider finally the general case of an arbitrary type T p St,u tensorfield ϑ. We may consider ϑ as being, at each surface St,u and at each point x ∈ St,u , a p-linear form in Tx St,u with values in ⊗q Tx St,u . Then, as in the case of a p-covariant St,u tensorfield, / X ϑ is simply defined by considering a if X is an arbitrary St,u -tangential vectorfield, L given surface St,u , as the usual Lie derivative of ϑ with respect to X, a notion intrinsic to St,u , which makes no reference to the ambient spacetime. Moreover L / L ϑ is defined by considering ϑ on Cu extended to T Cu according to the condition that it vanishes if one of the entries is L and setting L / L ϑ equal to the restriction to T St,u of the usual Lie derivative with respect to L of this extension. Similarly L /T ϑ is defined by considering ϑ on t extended to T t according to the condition that it vanishes if one of the entries is T and setting L /T ϑ equal to the restriction to T St,u of the usual Lie derivative with respect / L ϑ, L /T ϑ, does not involve , the to T of this extension. Thus our definition of L / X ϑ, L h- orthogonal projection to St,u , and is therefore independent of the acoustical metric h. It follows that if ϑ and ϕ are any two St,u tensorfields of any, not necessarily the same, type, and we denote by ϑ · ϕ an arbitrary contraction of the tensor product ϑ ⊗ ϕ, then the Leibniz rule: / X ϑ) · ϕ + ϑ · (L / X ϕ) (10.1) L / X (ϑ · ϕ) = (L with X an arbitrary St,u -tangential vectorfield, holds, and so do the Leibniz rules: / L ϑ) · ϕ + ϑ · (L / L ϕ) L / L (ϑ · ϕ) = (L

(10.2)

/T ϑ) · ϕ + ϑ · (L /T ϕ) L /T (ϑ · ϕ) = (L

(10.3)

and: We also note that since the commutation field Q differs from L by a factor, 1 + t, which is constant on each St,u , the same results hold with Q in the role of L. As an illustration of the above consider the definition, in Chapter 3, of χ, according to which 2χ is the restriction to T St,u of L L h. Let X, Y be a pair of St,u -tangential vectorfields. Then (L L h)(X, Y ) = L(h(X, Y )) − h([L, X], Y ) − h(X, [L, Y ]) = L(h /(X, Y )) − h/([L, X], Y ) − h/(X, [L, Y ]) = (L / L h/)(X, Y )

Part 1: Control of the angular derivatives of the first derivatives of the x i

Thus indeed we have:

(L)

Similarly, we have:

331

π /=L / L h/ = 2χ

(10.4)

(T )

(10.5)

π /=L /T h/

Consider next the definition, in Chapter 3, of θ , according to which 2κθ is the restriction to T St,u of LT h. Let again X, Y be a pair of St,u -tangential vectorfields. Then (LT h)(X, Y ) = T (h(X, Y )) − h([T, X], Y ) − h(X, [T, Y ]) = T (h /(X, Y )) − h/([T, X], Y ) − h/(X, [T, Y ]) = (L /T h/)(X, Y ) Thus indeed we have: L /T h/ = 2κθ

(10.6)

Given now a positive integer l let us denote by E / l the bootstrap assumption that there is a constant C independent of s such that for all t ∈ [0, s]: E /l

: max max Ril . . . Ri1 ψα  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 i1 ...il

α

t

Also, for convenience of notation, let us denote by E / 0 the basic bootstrap assumption E1 of Chapter 6: E / 0 : |ψ0 − k|, max |ψi | ≤ Cδ0 (1 + t)−1 i

Given a positive integer l we then denote by E / [l] the bootstrap assumption: E / [l]

: E / 0 and . . . and E /l

Given a positive integer n we denote: n : if n is even 2 n ∗ = n−1 2 : if n is odd

(10.7)

Lemma 10.1 Let G be a smooth function of the (ψα : α = 0, 1, 2, 3). Suppose that the bootstrap assumption E / [l∗ ] holds for some positive integer l. Then there are constants C, Cl independent of s such that the following estimate holds: Ril ...i1 G L 2 ( ε0 ) t

≤C

 α

Ril . . . Ri1 ψα  L 2 ( ε0 ) + Cl δ0 (1 + t)−1

l−1 

t

k=1

max max Rik . . . Ri1 ψα  L 2 ( ε0 )

i1 ...ik

α

t

Proof. We have: Ril . . . Ri1 G =

l  k=1

(l,k) ∂kG H µ1 ...µk ∂ψµ1 . . . ∂ψµk

(10.8)

332

Chapter 10. Control of the Angular Derivatives (l,k)

Here, for each k = 1, . . . , l, H µ1 ...µk is defined by considering all partitions of the set {1, . . . , l} into k subsets s1 , . . . , sk , each consisting of at least one element. Each sm , m = 1, . . . , l is ordered, and the s1 , . . . , sk are ordered according to the order of their first elements. If sm = {n 1 , . . . , n p } where p = |sm | is the cardinality of sm , we denote by (R)sm the operator: (R)sm = Rin p . . . Rin1 Then:



(l,k) H µ1 ...µk =

(10.9)

((R)s1 ψµ1 ) . . . ((R)sk ψµk )

(10.10)

partitions

Note that since the coefficient:

∂k G ∂ψµ1 . . . ∂ψµk

(l,k)

of H µ1 ...µk in (10.8) is symmetric under arbitrary permutations of the ψµ1 , . . . , ψµk , the ordering of the s1 , . . . , sk is immaterial, so the ordering which we have adopted (namely according to the order of their first elements) is simply for the sake of definiteness. For k = 1 we have: (l,1) H µ1 =

Ril . . . Ri1 ψµ1

(10.11)

while for k = l we have: (l,l) H µ1 ...µl =

From (10.11):

(l,1) µ1

 H

(Ri1 ψµ1 ) . . . (Ril ψµl )

(10.12)

 L 2 ( ε0 ) ≤ max Ril . . . Ri1 ψα  L 2 ( ε0 ) α

t

t

(10.13)

(l,k)

Consider next H µ1 ...µk for k ≥ 2. Each term in the sum (10.10) corresponds to a partition {s1 , . . . , sk } and we have: k  |sm | = l (10.14) m=1

It follows that at most one of the |sm | may be greater than l∗ . For, if on the contrary two or more of the |sm | exceed l∗ (so they are ≥ l∗ + 1), then we would have: k 

|sm | ≥ 2l∗ + 2 ≥ l + 1

m=1

contradicting (10.14). Moreover, each |sm | ≤ l − 1. For, there are at least two terms in the sum (10.14) and each |sm | ≥ 1. Placing the factor (R)sm ψµm with the maximal |sm | in

Part 1: Control of the angular derivatives of the first derivatives of the x i ε

333

ε

L 2 (t 0 ), and each of the other factors in L ∞ (t 0 ), we obtain, by virtue of the bootstrap assumption E / [l∗ ] , since there is at least one such other factor, ((R)s1 ψµ1 ) . . . ((R)sk ψµk ) L 2 ( ε0 ) ≤ Cδ0 (1+t)−1 t

l−1  j =1

max max Ri j . . . Ri1 ψα  L 2 ( ε0 )

i1 ...i j

α

t

(10.15) Hence for k ≥ 2 we have: (l,k) µ1 ...µk

 H

 L 2 ( ε0 ) ≤ Nl,k ·Cδ0 (1+t)−1

l−1 

t

j =1

max max Ri j . . . Ri1 ψα  L 2 ( ε0 ) (10.16)

i1 ...i j

α

t

where Nl,k is the number of partitions as defined above. In view of the estimates (10.13) and (10.16), and the fact that l  Nl,k k=2

is a positive integer depending only on l, the lemma follows. Consider now the functions Tˆ j , the components of the vectorfield Tˆ in rectangular coordinates. By equations (3.221),(3.222), (3.226), (3.230) and (3.232), we have: Ri Tˆ j = p/i Tˆ j + q/i · d/x j

(10.17)

p/i = ψTˆ bi

(10.18)

Here, p/i are the functions: where bi are the functions: bi = −

1 dH ψ ˆ (Ri σ ) − H Tˆ j (Ri ψ j ) 2 dσ T

(10.19)

In (10.17), q/i are the St,u -tangential vectorfields given by: q/i = m · Ri + n i

(10.20)

Here, m is a T11 -type St,u tensorfield given by: /−1 m = mb · h

(in components: m BA = (m b ) BC (h/−1 )C A )

(10.21)

where m b is the 2-covariant St,u tensorfield: m b = −α −1 χ + k/ +

1 dH ψ ˆ ψ ⊗ d/σ 2 dσ T

(10.22)

Also, in (10.20) the n i are the St,u -tangential vectorfields: n i = bi ψ · h /−1

(in components: n iA = bi ψ B (h/−1 ) B A )

(10.23)

334

Chapter 10. Control of the Angular Derivatives

Let us define the functions y j , as in (6.42), by setting: Tˆ j = −

xj + yj 1 − u + η0 t

(10.24)

Then, since Ri x j = Ri · d/x j , we have: Ri y j = p/i Tˆ j + q/i · d/x j

(10.25)

where q/i are the St,u -tangential vectorfields: q/i = q/i + By (10.20)–(10.22) we have: where

m

is the

T11 -type

Ri 1 − u + η0 t

q/i = m  · Ri + n i

(10.26)

(10.27)

St,u tensorfield: m  = m b · h/−1

and m b = m b +

(10.28)

h/ 1 − u + η0 t

is the 2-covariant St,u tensorfield: m b

  / η0 (α −1 − η0−1 )h 1 dH η0 h/ −1 −α ψ ˆ ψ ⊗ d/σ (10.29) =− χ− + k/ + 1 − u + η0 t 1 − u + η0 t 2 dσ T ε

We first recall the L ∞ bound (6.170) of Chapter 6 for the functions y j on t 0 : y j  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] t

(10.30)

We consider next the angular derivatives of the rectangular coordinate functions x j . According to the definition (6.8) we have, in terms of components in rectangular coordinates, j

◦

(Ri ) j = k ( Ri )k where

j k

(10.31)

are the components of , the h-orthogonal projection to St,u on t : k = δk − h kl Tˆ l Tˆ j j

and (see (6.9)):

j

◦

( Ri ) j = im j x m

(10.32)

(10.33)

Recalling that according to (6.115): ◦

h( Ri , Tˆ ) = λi

(10.34)

Part 1: Control of the angular derivatives of the first derivatives of the x i

we then obtain:

335

◦

Ri x j = Ri x j − λi Tˆ j

(10.35)

hence:

◦ ∂ Ri = Ri −λi Tˆ j j ∂x Let us now define, for each non-negative integer k, the functions: (k) j x i1 ...ik

◦

◦

= Ri k . . . Ri 1 x j

(10.36)

(10.37)

These functions are linear functions of the rectangular coordinates, therefore the (k) j cl;i1 ...ik

=

∂

(k) x j i1 ...ik ∂xl

(10.38)

are constants. Let us then define for each non-negative integer k the functions by the equation: j j Rik . . . Ri1 x j = (k) x i1 ...ik − (k) δi1 ...ik In particular,

(0) j

δ =0

and by (10.35), (1) j δi

(k) δ j i1 ...i j

(10.39) (10.40)

= λi Tˆ j

(10.41)

Replacing k by k − 1 in (10.39) and applying Rik we obtain: Rik Rik−1 . . . Ri1 x j = Rik

(k−1) j x i1 ...ik−1

− Ri k

(k−1) j δi1 ...ik−1

(10.42)

Substituting for Rik from (10.36) we obtain, in view of the definitions (10.37) and (10.38), Ri k

(k−1) j x i1 ...ik−1

=

(k) j x i1 ...ik

−

(k−1) j cl;i1 ...ik−1 λik Tˆ l

(10.43)

Substituting this in (10.42) and comparing with (10.39), we conclude that the following recursion formula holds: (k) j δi1 ...ik

= Ri k

(k−1) j δi1 ...ik−1

+

(k−1) j cl;i1 ...ik−1 λik Tˆ l

(10.44)

We may apply Proposition 8.2 to this recursion taking the space X to be the real line, the j operators An to be the Rik , the x n to be the (k) δi1 ...ik and the yn to be: (k−1) j cl;i1 ...ik−1 λik Tˆ l

Since here by (10.40) x 0 = 0 we obtain the following expression for (k) j δi1 ...ik

=

k  m=1

(m−1) j cl;i1 ...im−1 Rik

(k) δ j i1 ...ik :

. . . Rim+1 (λim Tˆ l )

(10.45)

336

Since the (10.38),

Chapter 10. Control of the Angular Derivatives (k) x

are linear functions of the rectangular coordinates, we have, from (k) j x i1 ...ik

(k) j cl;i1 ...ik x l

=

(10.46)

From (10.37) and (10.33) we obtain: (0) j

(1) j xi

x = x j,

Moreover,

(2) j x ik

= il j x l

= δ j k x i − δik x j

(10.47)

(10.48)

For k ≥ 2 we have from (10.37) and (10.48): (k) j x i1 ...ik

◦

◦

◦

◦

(2) j x i1 i2

= Ri k . . . Ri 3

= Rik . . . Ri3 (δ j i2 x i1 − δi1 i2 x j ) (k−2) i1 x i3 ...ik

= δ j i2

− δi 1 i 2

(k−2) j x i3 ...ik

(10.49)

It follows from (10.49) that for k ≥ 2: max |

j ;i1 ...ik

(k) j x i1 ...ik |

=

(k−2) j x i1 ...ik−2 |

max |

j ;i1 ...ik−2

(10.50)

while from (10.47): max | j ;i

(1) j xi |

= max |

(0) j

j

x | = max |x j | j

(10.51)

It then follows that for each non-negative integer k we have: max |

j ;i1 ...ik

(k) j x i1 ...ik |

= max |x j | j

(10.52)

which by (6.109) implies: max 

j ;i1 ...ik

(k) j x i1 ...ik  L ∞ ( ε0 ) t

≤ 1 + η0 t

(10.53)

(In fact equality holds in (10.53).) Let us also recall from Chapter 6 (equations (6.122), (6.123)) that the functions λi can be expressed as: ◦

λi =λi +νi where:

◦

◦

λi =  Ri , Tˆ  and:

◦

νi = H Rim ψm ψTˆ

(10.54) (10.55) (10.56)

Part 1: Control of the angular derivatives of the first derivatives of the x i

337

◦

Substituting for Ri from (10.33) and expressing Tˆ in terms of y by (10.24), we obtain: ◦

Also,

λi = im j x m y j

(10.57)

νi = H inm x n Tˆ j ψm ψ j

(10.58)

Lemma 10.2 Let max Rik . . . Ri1 y j  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

j ;i1 ...ik

t

hold for k = 0, . . . , l. Let also assumptions E / [l] hold. Then if δ0 is suitably small (depending on l), we have: max Rik . . . Ri1 λi  L ∞ ( ε0 ) ≤ Cl δ0 [1 + log(1 + t)]

i;i1 ...ik

t

and: max 

j ;ii1 ...ik

(k+1) j δii1 ...ik  L ∞ ( ε0 ) t

≤ Cl δ0 [1 + log(1 + t)]

for all k = 0, . . . , l. Proof. We apply induction. For l = 0 the lemma reduces to the statement that the hypothesis: (10.59) max y j  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] t

j

together with the assumption E / 0 implies: max λi  L ∞ ( ε0 ) ≤ Cδ0 [1 + log(1 + t)] t

i

and: max  j ;i

(1) j δi  L ∞ ( ε0 ) t

≤ Cδ0 [1 + log(1 + t)]

(10.60)

(10.61)

Now, (10.59) and (6.109) imply through (10.57) that: ◦

| λi | ≤ Cδ0 [1 + log(1 + t)] while assumption E / 0 and (6.109) imply through (10.58) that: |νi | ≤ Cδ02 (1 + t)−1 The estimate (10.60) thus follows, in view of (10.54). The estimate (10.61) then also follows, in view of (10.41). Let now the lemma hold with l replaced by l−1, and consider the case l. In reference to (10.58), we can write Ril . . . Ri1 νi in the form:  ((R)s1 x n )((R)s2 Tˆ j )((R)s3 (H ψm ψ j )) (10.62) Ril . . . Ri1 νi = inm partitions

338

Chapter 10. Control of the Angular Derivatives

where we are considering all partitions of the set {1, . . . , l} into three subsets s1 , s2 , s3 , each of which may be empty, and the order of the subsets in the partition {s1 , s2 , s3 } matters (so that a permutation of the subsets of a given partition yields a partition which is considered distinct from the original). Each non-empty subset is ordered and the operators (R)sm are defined as in Lemma 10.1. We have: |s1 | + |s2 | + |s3 | = l

(10.63)

|(R)s3 (H ψm ψ j )| ≤ Cl δ02 (1 + t)−2

(10.64)

Assumption E / [l] then implies:

Next, expressing Tˆ j as in (10.24) we obtain: |(R)s2 Tˆ j | ≤ (1 − u + η0 t)−1 |(R)s2 x j | + |(R)s2 y j |

(10.65)

Let s2 = {n 1 , . . . , n p },

p = |s2 |

Then from (10.39), (R)s2 x j = Rin p . . . Rin1 x j =

( p) j x in ...in p 1

−

( p) j δin ...in p 1

(10.66)

By (10.53): |

( p) j x in ...in | p 1

≤ 1 + η0 t

while, since p ≤ l, by the inductive hypothesis: |

( p) j δin ...in | p 1

≤ Cl δ0 [1 + log(1 + t)]

It follows that (if δ0 is suitably small): |(R)s2 x j | ≤ C(1 + η0 t)

(10.67)

where C is a constant independent of l. Since by the hypothesis of the lemma in the case l the second term on the right in (10.65) is bounded by Cl δ0 (1 + t)−1 [1 + log(1 + t)] it then follows that (if δ0 is suitably small): |(R)s2 Tˆ j | ≤ C

(10.68)

where C is a constant independent of l. Finally, as in (10.67) we have: |(R)s1 x n | ≤ C(1 + η0 t)

(10.69)

Part 1: Control of the angular derivatives of the first derivatives of the x i

339

where C is a constant independent of l. The bounds (10.69), (10.68) and (10.64) together imply, through expression (10.62), |Ril . . . Ri1 νi | ≤ Cl δ02 (1 + t)−1

(10.70)

Next, in reference to (10.57) we can write: 

◦

Ril . . . Ri1 λi = im j

((R)s1 x m )((R) 2 y j )

(10.71)

partitions

where the sum is over all ordered partitions {s1 , s2 } of {1, . . . , l} into two subsets. Thus, |s1 | + |s2 | = l

(10.72)

The bound (10.69) together with the hypothesis of the lemma in the case l then yield through expression (10.71) the estimate: ◦

|Ril . . . Ri1 λi | ≤ Cl δ0 [1 + log(1 + t)]

(10.73)

The estimates (10.73) and (10.70) imply, in view of (10.54) the desired estimate for the lth angular derivatives of λi : |Ril . . . Ri1 λi | ≤ Cl δ0 [1 + log(1 + t)] We now consider from (10.46), (10.47), (l+1) j δii1 ...il

(l+1) δ j ii1 ...il . (0) c j = δ j ) k k

(10.74)

According to (10.45) this is given by: (noting that

l 

= Ril . . . Ri1 (λi Tˆ j ) +

(m) j ck;ii1 ...im−1 Ril

. . . Rim+1 (λim Tˆ k )

(10.75)

m=1

We consider the first term on the right in (10.75), which is the leading term. This term is expressed as:  ((R)s1 λi )((R)s2 Tˆ j ) : |s1 | + |s2 | = l (10.76) partitions

where the sum is again over all ordered partitions {s1 , s2 } of {1, . . . , l} into two subsets. The estimate (10.74) together with the inductive hypothesis and the bound (10.68) then imply that (10.76) is bounded by: Cl δ0 [1 + log(1 + t)] The lower order sum constituting the second term on the right in (10.75) is handled in a similar manner. We thus obtain the estimate: |

(l+1) j δii1 ...il |

≤ Cl δ0 [1 + log(1 + t)]

and the inductive step is complete. This completes the proof of the lemma.

(10.77)

340

Chapter 10. Control of the Angular Derivatives

Lemma 10.3 Let max Rik . . . Ri1 y j  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

j :i1 ...ik

t

/ [l∗ ] hold. Then if δ0 is suitably small hold for k = 0, . . . , l∗ . Let also assumptions E (depending on l) we have: max Rik . . . Ri1 λi  L 2 ( ε0 ) ≤ Cl {(1 + η0 t)Y[l] + δ0 W[l] }

i;i1 ...ik

t

and: max 

j ;ii1 ...ik

(k+1) j δii1 ...ik  L 2 ( ε0 ) t

≤ Cl {(1 + η0 t)Y[l] + δ0 W[l] }

for all k = 0, . . . , l. Here: Yk = max Rik . . . Ri1 y j  L 2 ( ε0 ) , j ;i1 ...ik

Also,

t

Y[l] =

l 

Yk .

k=0





W0 = max ψ0 − k L 2 ( ε0 ) , max ψi  L 2 ( ε0 ) , t

and for k = 0:

t

i

Wk = max Rik . . . Ri1 ψα  L 2 ( ε0 ) α;i1 ...ik

t

and: W[l] =

l 

Wk

k=0

Proof. We apply induction. For l = 0 the lemma reduces to the statement that the hypothesis (10.59) together with the assumption E / 0 implies: max λi  L 2 ( ε0 ) ≤ C{(1 + η0 t)Y0 + δ0 W0 } t

i

and: max  j ;i

(1) j δi  L 2 ( ε0 ) t

≤ C{(1 + η0 t)Y0 + δ0 W0 }

(10.78)

(10.79)

In fact, from (10.58), the assumptions E / 0 imply, putting one of the factors ψm ψ j in ε ε L 2 (t 0 ) and the other factor, as well as all remaining factors, in L ∞ (t 0 ), νi  L 2 ( ε0 ) ≤ Cδ0 W0 t

(10.80)

Also, from (10.57), in view of (6.109), ◦

 λi  L 2 ( ε0 ) ≤ C(1 + η0 t)Y0 t

(10.81)

Part 1: Control of the angular derivatives of the first derivatives of the x i

341

In view of (10.54), the result (10.78) follows immediately from (10.80) and (10.81), after ε which the result (10.79) follows through (10.41), putting the factor Tˆ j in L ∞ (t 0 ). Let now the lemma hold with l replaced by l − 1, and consider the case l. Let us first investigate the expression (10.62). Here, at most one of |s1 |, |s2 |, |s3 | may exceed l∗ . (Otherwise, |s1 | + |s2 | + |s3 | ≥ 2(l∗ + 1) ≥ l + 1, contradicting (10.63)). If it is |s3 | ε which may exceed l∗ , we put the factors (R)s1 x n , (R)s2 Tˆ j in L ∞ (t 0 ), and the factor ε 0 s 2 (R) 3 (H ψm ψ j ) in L (t ). We write:  ((R)s3,1 H )((R)s3,2 ψm )((R)s3,3 ψ j ) (10.82) (R)s3 (H ψm ψ j ) = partitions

where we are considering all ordered partitions {s3,1 , s3,2 , s3,3 } of the ordered set s3 into three ordered subsets. We have: |s3,1 | + |s3,2 | + |s3,3 | = |s3 | ≤ l so at most one of |s3,1 |, |s3,2 |, |s3,3 | may exceed l∗ . Putting the factor corresponding to ε ε the subset with the maximal cardinality in L 2 (t 0 ) and the other two factors in L ∞ (t 0 ) yields: (10.83) (R)s3 (H ψm ψ j ) L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−1 W[l] t

Thus in the case that only |s3 | may exceed l∗ we obtain: ((R)s1 x n )((R)s2 Tˆ j )((R)s3 (H ψm ψ j )) L 2 ( ε0 ) t

≤ (R)s1 x n  L ∞ ( ε0 ) · (R)s2 Tˆ j  L 2 ( ε0 ) · (R)s3 (H ψm ψ j ) L 2 ( ε0 ) t

t

t

≤ C(1 + η0 t) · C · Cl δ0 (1 + t)−1 W[l] ≤ Cl δ0 W[l]

(10.84)

where we have appealed to Lemma 10.2 with l replaced by l∗ and the resulting bounds (10.68), (10.69). In the case that only |s2 | may exceed l∗ we express the factor (R)s2 Tˆ j as: (R)s2 Tˆ j = (R)s2 y j − (1 − u + η0 t)−1 (R)s2 x j

(10.85)

Now, if s2 = {n 1 , . . . , n p }, p = |s2 |, we have, according to (10.39), (R)s2 x j = Rin p . . . Rin1 x j =

( p) j x in ...in p 1

−

( p) j δin ...in p 1

(10.86)

Thus (10.85) takes the form: (R)s2 Tˆ j = −(1 − u + η0 t)−1

( p) j x in ...in p 1

+(R)s2 y j + (1 − u + η0 t)−1

( p) j δin ...in p 1

(10.87)

To estimate the contribution of the first term on the right in (10.87) to the L 2 norm of the product ((R)s1 x n )((R)s2 Tˆ j )((R)s3 (H ψm ψ j ))

342

Chapter 10. Control of the Angular Derivatives ε

ε

on t 0 , we place this term in L ∞ (t 0 ): (1 − u + η0 t)−1

( p) j x in ...in  L ∞ ( ε0 ) p t 1

≤C

ε

by (10.53), the factor (R)s1 x n also in L ∞ (t 0 ) using (10.69), and the factor (R)s3 ε (H ψm ψ j ) in L 2 (t 0 ) using (10.83) with l replaced by l∗ . We then obtain that the contribution in question is bounded by: (10.88) Cl δ0 W[l∗ ] To estimate the contribution of the second term on the right in (10.87), we place this term ε in L 2 (t 0 ) using (R)s2 y j  L 2 ( ε0 ) ≤ Y[l] (10.89) t

and the factors (R)s1 x n , (R)s3 (H ψm ψ j ) in

ε L ∞ (t 0 ) using Lemma 10.2 and the resulting

bounds (10.69), (10.64). We then obtain that the contribution in question is bounded by: Cl δ02 (1 + t)−1 Y[l]

(10.90)

Finally, to estimate the contribution of the third term on the right in (10.87), we place this ε term in L 2 (t 0 ) using the fact that, since p ≤ l, by the inductive hypothesis we have: 

( p) j δin ...in  p 1

≤ Cl−1 {(1 + η0 t)Y[l−1] + δ0 W[l−1] }

(10.91)

ε

We then place the factors (R)s1 x n , (R)s3 (H ψm ψ j ) in L ∞ (t 0 ) as above. In this way we obtain that the contribution in question is bounded by: Cl δ02 (1 + t)−1 {Y[l−1] + δ0 (1 + t)−1 W[l−1] }

(10.92)

Combining the results (10.88), (10.90), (10.92), we conclude that in the case that only |s2 | may exceed l∗ : ((R)s1 x n )((R)s2 Tˆ j )((R)s3 (H ψm ψ j )) L 2 ( ε0 ) t

≤

Cl δ0 W[l−1] + Cl δ02 (1 + t)−1 Y[l]

(10.93)

Consider finally the case that only |s1 | may exceed l∗ . Let s1 = {n 1 , . . . , n p }, p = |s1 |. Then (10.86) holds with s2 replaced by s1 : (R)s1 x j = Rin p . . . Rin1 x j =

( p) j x in ...in p 1

−

( p) j δin ...in p 1

(10.94)

To estimate the contribution of the first term on the right in (10.94) to the L 2 norm of the product ((R)s1 x n )((R)s2 Tˆ j )((R)s3 (H ψm ψ j )) ε

ε

ε

on t 0 , we place this term in L ∞ (t 0 ) using (10.53), the factor (R)s2 Tˆ j also in L ∞ (t 0 ) ε using (10.68) and the factor (R)s3 (H ψm ψ j ) in L 2 (t 0 ) using (10.83) with l replaced by l∗ . We then obtain that the contribution in question is bounded by: Cl δ0 W[l∗ ]

(10.95)

Part 1: Control of the angular derivatives of the first derivatives of the x i

343

Finally, to estimate the contribution of the second term on the right in (10.94), we place ε ε this term in L 2 (t 0 ) using (10.91), and the factors (R)s2 Tˆ j , (R)s3 (H ψm ψ j ) in L ∞ (t 0 ) using Lemma 10.2 with l replaced by l∗ , and the resulting bounds (10.68), (10.64). We then obtain that the contribution in question is bounded by: Cl δ02 (1 + t)−1 {Y[l−1] + δ0 (1 + t)−1 W[l−1] }

(10.96)

Combining the results (10.95), (10.96) we conclude that in the case that only |s1 | may exceed l∗ : ((R)s1 x n )((R)s2 Tˆ j )((R)s3 (H ψm ψ j )) L 2 ( ε0 ) t

≤ Cl δ0 W[l−1] + Cl δ02 (1 + t)−1 Y[l−1]

(10.97)

Finally, combining the results of the three cases, (10.84), (10.93), (10.97), we conclude that: max Ril . . . Ri1 νi  L 2 ( ε0 ) ≤ Cl δ0 {W[l] + δ0 (1 + t)−1 Y[l] }

i;i1 ...il

(10.98)

t

Consider next the expression (10.71). At most one of |s1 |, |s2 | may exceed l∗ . In the ε case that s2 may exceed l∗ we put the factor (R)s2 y j in L 2 (t 0 ) using (10.89) and the ε factor (R)s1 x m in L ∞ (t 0 ) using (10.69). We then obtain: ((R)s1 x m )((R)s2 y j ) L 2 ( ε0 ) ≤ (R)s1 x m  L ∞ ( ε0 ) (R)s2 y j  L 2 ( ε0 ) t

t

t

≤ C(1 + η0 t)Y[l]

(10.99)

In the case that |s1 | may exceed l∗ we express (R)s1 x m as in (10.94). To estimate the contribution of the first term on the right in (10.94) to the L 2 norm of the product ((R)s1 x m )× ε ε ((R)s2 y j ) on t 0 , we place this term in L ∞ (t 0 ) using (10.53), and the factor (R)s2 y j ε in L 2 (t 0 ) using (10.89) with l replaced by l∗ . We then obtain that the contribution in question is bounded by: (10.100) C(1 + η0 t)Y[l∗ ] To estimate the contribution of the second term on the right in (10.94), we place this term ε ε in L 2 (t 0 ) using (10.91), and the factor (R)s2 y j in L ∞ (t 0 ) using the assumption of the lemma. The contribution in question is then seen to be bounded by: 

( p) m δin ...in p  L 2 ( ε0 ) t 1

· (R)s2 y j  L ∞ ( ε0 ) t

(10.101)

≤ Cl−1 {(1 + η0 t)Y[l−1] + δ0 W[l−1] } · Cl δ0 (1 + t)−1 [1 + log(1 + t)] Combining the results (10.100), (10.101) we obtain that in the case that only |s1 | may exceed l∗ : ((R)s1 x m )((R)s2 y j ) L 2 ( ε0 ) t

≤ Cl {(1 + η0 t)Y[l−1] + δ02 (1 + t)−1 [1 + log(1 + t)]W[l−1] }

(10.102)

344

Chapter 10. Control of the Angular Derivatives

Finally, combining the results of the two cases (10.99), (10.102), we conclude that: ◦

max Ril . . . Ri1 λi  L 2 ( ε0 )

i;i1 ...il

(10.103)

t

≤ Cl {(1 + η0 t)Y[l] + δ02 (1 + t)−1 [1 + log(1 + t)]W[l−1] } In view of (10.54), the estimates (10.98) and (10.103) yield the desired result: max Ril . . . Ri1 λi  L 2 ( ε0 ) ≤ Cl {(1 + η0 t)Y[l] + δ0 W[l] }

i;i1 ...il

t

(10.104)

The result: max 

j ;ii1 ...il

(l+1) j δii1 ...il  L 2 ( ε0 ) t

≤ Cl {(1 + η0 t)Y[l] + δ0 W[l] }

(10.105)

follows in a similar way. This completes the inductive step and thus the proof of the lemma. We turn to the task of deriving estimates for the angular derivatives of the Tˆ j , or equivalently of the functions y j (see (10.24)). The 1st angular derivatives of the y j are given by (10.25), with the functions p/i given by (10.18), (10.19) and the St,u -tangential vectorfields q/i given by (10.27)–(10.29) and (10.23). The higher order angular derivatives are then given by the following lemma. Lemma 10.4 For every non-negative integer k we have: Ri k . . . Ri 1 Ri y j = (k) p /ii1 ...ik

Here the

(k) q /ii1 ...ik

(k)  q/ii1 ...ik

· d/x j +

(k) j r/ii1 ...ik

p/ii1 ...ik = (Rik + p/ik ) . . . (Ri1 + p/i1 ) p/i

are the St,u -tangential vectorfields:

(k)  q/ii1 ...ik

+

p/ii1 ...ik Tˆ j +

are the functions: (k)

the

(k)

k−1 

=L / R ik . . . L / Ri1 q/i

L / R ik . . . L / Rik−m+1 {

(k−m−1)

p/ii1 ...ik−m−1 (q/ik−m − (1 − u + η0 t)−1 Rik−m )}

m=0

and the

(k) r/ j ii1 ...ik

are the functions:

(k) j r/ii1 ...ik

=

k−1 

Rik . . . Rik−m+1 {

(k−m−1)  q/ii1 ...ik−m−1

· d/(Rik−m x j )}

m=0

Proof. Since:

(0)

= p/i ,

(0)  q/i

= q/i ,

(0) j r/i

=0

(10.106)

Part 1: Control of the angular derivatives of the first derivatives of the x i

345

the lemma reduces for k = 0 to equation (10.25). Arguing by induction, let the formula for the angular derivatives of y j hold with k replaced by k − 1, that is, let: Rik−1 . . . Ri1 Ri y j =

(k−1)

p/ii1 ...ik−1 Tˆ j +

(k−1)  q/ii1 ...ik−1

· d/x j +

(k−1) j r/ii1 ...ik−1

(10.107)

We then apply Rik . By (10.24), (10.25) we have: Rik Tˆ j = −(1 − u + η0 t)−1 Rik x j + Rik y j = p/ik Tˆ j + (q/i − (1 − u + η0 t)−1 Rik ) · d/x j

(10.108)

k

Also, Ri k (

(k−1)  q/ii1 ...ik−1

(k−1)  q/ii1 ...ik−1 x j (10.109) (k−1)  j (k−1)  j [Rik , q/ii1 ...ik−1 ]x + q/ii1 ...ik−1 (Rik x ) (L / Rik (k−1) q/ii−1...ik−1 ) · d/x j + (k−1) q/ii1 ...ik−1 · d/(Rik x j )

· d/x j ) = Rik = =

Substituting (10.108) and (10.109) in the formula which results from (10.107) by applying Rik , we obtain the formula for Rik . . . Ri1 Ri y j given in the statement of the lemma, with the following recursion relations: (k) (k)  q/ii1 ...ik

=L / R ik (k) j r/ii1 ...ik

p/ii1 ...ik = (Rik + p/ik )

(k−1)  q/ii1 ...ik−1

= Ri k

+

(k−1)

(k−1) j r/ii1 ...ik−1

(k−1)

p/ii1 ...ik−1

(10.110)

p/ii1 ...ik−1 (q/ik − (1 − u + η0 t)−1 Rik ) (10.111)

+

(k−1)  q/ii1 ...ik−1

· d/(Rik x j )

(10.112)

The formulas for the coefficients (k) p/ii1 ...ik , (k) q/ii1 ...ik , (k)r/ii1 ...ik , given in the statement of the lemma then follow by applying Proposition 8.2 to these recursions, taking account of (10.106). j

Now, it is evident from equation (10.27) and the formulas for the coefficients j and (k)r/ii1 ...ik given in the statement of Lemma 10.4, that (k) q/ii1 ...ik contains

(k) q /ii1 ...ik

the angular derivatives of the Ri of order up to k and (k)r/ii1 ...ik contains the angular derivatives of the Ri of order up to k − 1, where by the angular derivatives of the Ri of order up to l we mean the St,u -tangential vectorfields L / R im . . . L / Ri1 Ri for m = 0, . . . , l. We shall thus derive the below expressions for these vectorfields. The basic formula from which these expressions shall be derived is the formula for j

L / Ri R j = [Ri , R j ] given by the following lemma. Let wi be the St,u -tangential vectorfields: wi = (wi )b · h/−1

(10.113)

(wi )b = i j m y j h mn d/x n

(10.114)

where (wi )b are the St,u 1-forms: 

(rectangular coordinates). Let us then define the St,u -tangential vectorfields q/˜ i by: q/˜ i = q/i − wi

(10.115)

346

Chapter 10. Control of the Angular Derivatives

Lemma 10.5 We have: [Ri , R j ] = −i j k Rk + λi q/˜ j − λ j q/˜ i Proof. Let (Ri )m be the components of the vectorfields Ri in rectangular coordinates: Ri = (Ri )m According to (10.35):

∂ ∂xm

◦

(Ri )m = ( Ri )m − λi Tˆ m Thus, we have: [Ri , R j ] = {Ri ((R j )m ) − R j ((Ri )m )}

∂ ∂xm

◦ ◦ ∂ = {Ri (( R j )m − λ j Tˆ m ) − R j (( Ri )m − λi Tˆ m )} m ∂x

(10.116)

Now (see (10.33)), ◦

Ri (( R j )m = Ri ( j km x k ) =  j km (Ri )k ◦

= − j mk (( Ri )k − λi Tˆ k ) = − j mk ilk x l +  j mk λi Tˆ k Since,  j mk ilk + mik  j lk + i j k mlk = 0 we then obtain: ◦

◦

◦

Ri (( R j )m − R j (( Ri )m = −i j k ( Rk )m + (ikm λ j −  j km λi )Tˆ k

(10.117)

Substituting (10.117) in (10.116) yields: ◦

[Ri , R j ] = −i j k Rk −λi v j + λ j vi

(10.118)

∂ −(Ri (λ j ) − R j (λi ))Tˆ − (λ j Ri (Tˆ m ) − λi R j (Tˆ m )) m ∂x where the vi are the vectorfields: vi = ikm Tˆ k

∂ ∂xm

(10.119)

By (10.24), (10.25), Ri (Tˆ m )

∂ = −(1 − u + η0 t)−1 Ri + p/i Tˆ + q/i ∂xm

(10.120)

Part 1: Control of the angular derivatives of the first derivatives of the x i

347

Substituting (10.120) in (10.118), the latter becomes: ◦

[Ri , R j ] = −i j k Rk −(Ri (λ j ) + p/i λ j − R j (λi ) − p/ j λi )Tˆ +λi (q/j − v j ) − λ j (q/i − vi ) where:

vi = vi + (1 − u + η0 t)−1 Ri

(10.121)

(10.122)

Now, in view of the fact that the vectorfield [Ri , R j ] is St,u -tangential, we may apply the projection operator  to the right-hand side of (10.121). This annihilates the second term, ◦

and, since  Ri = Ri , we obtain: [Ri , R j ] = −i j k Rk + λi (q/j − v j ) − λ j (q/i − vi )

(10.123)

In view of the definition (10.115), the lemma is then proved once we show that:

We have, in view of (10.122),

vi = wi

(10.124)

vi = v˜i

(10.125)

where:

◦

v˜i = vi + (1 − u + η0 t)−1 Ri

(10.126)

In view of the definition (10.119), (10.24) and (10.33), the vectorfields v˜i are given by, simply: ∂ (10.127) v˜i = ikm y k m ∂x We then have, in terms of the frame (X A : A = 1, 2), v˜i = h(v˜i , X B )(h/−1 ) B A X A

(10.128)

h(v˜i , X B ) = h mn (v˜i )m X nB = ikm y k h mn d/ B x n

(10.129)

and: for, X nB = d/ B x n . Comparing (10.129) with (10.114) we see that: h(v˜i , X B ) = (wi )b (X B )

(10.130)

Substituting (10.130) in (10.128) we conclude in view of (10.113) that: v˜i = wi

(10.131)

therefore by (10.125), the conclusion (10.124) follows. This completes the proof of the lemma.

348

Chapter 10. Control of the Angular Derivatives

Let us define the St,u -tangential vectorfields: n i = n i − wi

(10.132)

Then by (10.27) and (10.115) the St,u -tangential vectorfields q/˜ i are expressed as: q/˜ i = m  · Ri + n i

(10.133)

Thus, defining the T11 -type St,u tensorfields µi by: µi = λi m 

(10.134)

and the St,u -tangential vectorfields νi j = −ν j i by: νi j = λi n j − λ j n i

(10.135)

the formula of Lemma 10.5 takes the form: [Ri , R j ] = −i j k Rk + µi · R j − µ j · Ri + νi j

(10.136)

Lemma 10.6 For every non-negative integer k we have: / R i1 R j = L / R ik . . . L The coefficients

(k) α m j ;i1 ...ik

(k) m α j ;i1 ...ik

(k) m β j ;i1 ...ik

· Rm +

(k)

γ j :i1 ...ik

are constants, given by:

(k) m α j ;i1 ...ik

= (−1)k ik nk m ik−1 nk−1 nk . . . i1 n1 n2 δ nj 1

The coefficients (k) β j ;i1 ...ik = ( tensorfields given by: (k)

Rm +

(k) β m i1 ...ik

β j ;i1 ...ik =

k−1 

: m = 1, 2, 3) are triplets of T11 -type St,u

Aik . . . Aik−n+1

(k−n)

ρ j ;i1 ...ik

n=0

Here, the Ai are operators acting on triplets z = (z m : m = 1, 2, 3) of T11 -type St,u tensorfields, defined by:     m m n m n (Ai z) = L / Ri z − inm z + z · µi − z · µn δim n

and the (k) ρ j ;i1 ...ik = ( fields, given by: (k) m ρ j ;i1 ...ik

(k) ρ m j ;i1 ...ik

=

n

: m = 1, 2, 3) are the triplets of T11 -type St,u tensor-

(k−1) m α j ;i1 ...ik−1 µik

−

  n

 (k−1) n α j ;i1 ...ik µn

δimk

Part 1: Control of the angular derivatives of the first derivatives of the x i (k) γ

Finally, the coefficients (k)

γ j ;i1 ...ik =

j ;i1 ...ik

k−1 

349

are St,u -tangential vectorfields, given by:

(k−1−n) m α j ;i1 ...ik−1−n L / R ik

...L / Rik−n+1 νik−n m

n=0

+

k−1 



L / R ik . . . L / Rik−n+1

(k−1−n) m β j ;i1 ...ik−1−n

· νik−n m



n=0

Proof. With:

(0) m αj

(0) m βj

= δ mj ,

= 0,

(0)

γj = 0

(10.137)

the lemma is trivial for k = 0. Moreover, we have: (1) m α j ;i1

and, since

= −i1 n1 m δ nj 1 = −i1 j m

(1) m ρ j ;i1

(10.138)

= δ mj µi1 − µ j δim1 ,

(10.139)

also: (1) m β j ;i1

=

(1) m ρ j ;i1

(1)

= δ mj µi1 − µ j δim1 ,

γ j ;i1 = δ mj νi1 m = νi1 j

(10.140)

Thus, for k = 1 the lemma reduces to the formula (10.136). Arguing by induction, let the formula for the angular derivatives of R j hold with k replaced by k − 1, that is, let: / R i1 R j = L / Rik−1 . . . L

(k−1) m α j ;i1 ...ik−1 Rm

(k−1) m β j ;i1 ...ik−1

+

Applying L / Rik to this we obtain, since the coefficients L / R ik . . . L / R i1 R j =

(k−1) m α j ;i1 ...ik−1 [Rik ,

 + L / R ik +L / R ik

Rm ] 

(k−1) m β j ;i1 ...ik−1

(k−1)

(k−1) α m j ;i1 ...ik−1

· Rm +

(k−1)

· Rm +

γ j ;i1 ...ik−1

are constants,

(k−1) m β j ;i1 ...ik−1

· [Rik , Rm ]

γ j ;i1 ...ik−1

(10.141)

Substituting for [Rik , Rm ] from formula (10.136), that is: [Rik , Rm ] = −ik mn Rn + µik · Rm − µm · Rik + νik m then yields: (k−1) m α j ;i1 ...ik−1 Rn (k−1) m α j ;i1 ...ik−1 (µik · Rm

/ Ri1 R j = −ik mn L / R ik . . . L +

 + L / R ik

(k−1) m β j ;i1 ...ik−1

+ (k−1) β m j ;i1 ...ik−1 · (µik +L / Rik (k−1) γ j ;i1 ...ik−1



− µm · R i k ) +

· Rm − ik mn

(k−1) m α j ;i1 ...ik−1 νik m

(k−1) m β j ;i1 ...ik−1

· R m − µm · R i k ) +

· Rn

(k−1) m β j ;i1 ...ik−1

· νik m

(10.142)

350

Chapter 10. Control of the Angular Derivatives

This is seen to be a formula for the kth order angular derivatives of R j of the form given by the lemma, if we set: (k) m α j ;i1 ...ik (k) m β j ;i1 ...ik

(k−1) n α j ;i1 ...ik−1 (k−1) m β j ;i1 ...ik−1 − ik nm (k−1) β nj;i1 ...ik−1

= −ik nm =L / R ik +

(k−1) m β j ;i1 ...ik−1

· µi k −  + (k−1) α mj;i1 ...ik−1 µik − (k)

γ j ;i1 ...ik = L / R ik +

(k−1)

(10.143)

 · µn δimk  (k−1) n α j ;i1 ...ik−1 µn δimk

(10.144)

(k−1) m β j ;i1 ...ik−1

(10.145)



(k−1) n β j ;i1 ...ik−1

γ j ;i1 ...ik−1

(k−1) m α j ;i1 ...ik−1 νik m

+

· νik m

(k) γ These constitute recursion relations for the coefficients (k) α mj;i1 ...ik , (k) β m j ;i1 ...ik , j ;i1 ...ik , to be considered in succession. First, the recursion (10.143), together with the first condition of (10.137), yields the expression for (k) α mj;i1 ...ik given in the statement of the lemma.

Next, in terms of the definitions of (k) ρ j ;i1 ...ik and the operators Ai given in the statement of the lemma, the recursion (10.144) takes the form, simply: (k)

β j ;i1 ...ik = Aik

(k−1)

β j ;i1 ...ik−1 +

(k)

ρ j ;i1 ...ik

(10.146)

To this recursion we may directly apply Proposition 8.2, taking the space X to be the space of triplets of T11 -type St,u tensorfields. In view of the second condition of (10.137), the expression for (k) β j ;i1 ...ik given in the statement of the lemma follows. Finally, to the recursion (10.145) we apply Proposition 8.2, taking the space X to be the space of St,u -tangential vectorfields. In view of the third condition of (10.137), the expression for (k) γ j ;i1 ...ik given in the statement of the lemma follows. This completes the proof of Lemma 10.6. To analyze the operators Ai , we define the multiplication operators Mi . These are 1 3-dimensional matrices, the entries (Mi )m n : m, n = 1, 2, 3 of which are T1 -type of St,u tensorfields given by: m m (Mi )m (10.147) n = −inm I + δn µi − δi µn Here I is the T11 -type of St,u tensorfield which is the identity as a transformation in the tangent space of St,u at each point. These matrices act on triplets z = (z m : m = 1, 2, 3) of T11 -type of St,u tensorfields on the right:  (z · Mi )m = z n · (Mi )m (10.148) n n

so that with matrix multiplication defined according to:  (Mi · M j )m (Mi )kn · (M j )m n = k k

(10.149)

Part 1: Control of the angular derivatives of the first derivatives of the x i

351

we have: (z · Mi ) · M j = z · (Mi · M j )

(10.150)

The operators Ai are defined in terms of the matrices Mi by: Ai z = L / Ri z + z · Mi

(10.151)

We may also define the action of the operators Ai on matrices K , such as the Mi themselves, by: Ai K = L / Ri K + K · Mi (10.152) where matrix multiplication is defined according to (10.149). Note that if (with some abuse of notation) we also denote by I the identity matrix, that is the matrix whose entries are: m (I )m (10.153) n = δn I then according to (10.152) we have: Ai I = Mi

(10.154)

From the definitions (10.151), (10.152), we readily deduce the following formula, for any non-negative integer k:  (L / R )s1 z · (M)s2 (10.155) A ik . . . A i1 z = partitions

Here the sum is over all ordered partitions {s1 , s2 } of the set {1, . . . , k} into two ordered subsets s1 , s2 . Also, we denote: (M)s2 = Ain p . . . Ain1 I

: if s2 = {n 1 , . . . , n p }, p = |s2 |

(10.156)

Introducing the matrices Ii and Ni by: (Ii )m n = −inm I

(10.157)

m m (Ni )m n = δ n µi − δ i µn

(10.158)

we may write the definition (10.147) in the form: Mi = Ii + Ni

(10.159)

Let us introduce for any positive integer k and multi-index (i 1 , . . . , i k ) the matrices (k) I (k) B i1 ...ik and i1 ...ik by: (k) Ii1 ...ik = Ii1 · · · Iik (10.160) and:

(k)

Bi1 ...ik = Aik . . . Ai1 I −

(k)

Ii1 ...ik

(10.161)

352

Chapter 10. Control of the Angular Derivatives

We also set given by:

(0) I

= I,

(0) B

= 0. It is then readily seen that the matrices (k)

(

Ii1 ...ik )m n =

(k) m αn;i1 ...ik I

(k) I

i1 ...ik

are

(10.162)

The matrices (k) Bi1 ...ik contain terms of degree 1 up to k in the matrices Ni , and the terms of degree q contain up to k − q angular derivatives. Thus the (k) Bi1 ...ik contain the angular derivatives of the Ni of order up to k − 1. Also, (10.156) takes the form: (M)s2 =

( p)

Iin1 ...in p +

( p)

Bin1 ...in p

: if s2 = {n 1 , . . . , n p }, p = |s2 | In regard to the coefficients (10.143) twice to obtain: (k) m α j ;i1 ...ik

(k) α m j ;i1 ...ik ,

(10.163)

for k ≥ 2 we may apply the relation

= ik lm ik−1 nl

(k−2) n α j ;i1 ...ik−2

Since ik lm ik−1 nl = δmik−1 δnik − δmn δik−1 ik this is: (k) m α j ;i1 ...ik

= δmik−1

(k−2) ik α j ;i1 ...ik−2

− δik ik−1

(k−2) m α j ;i1 ...ik−2

(10.164)

|

(k−2) m α j ;i1 ...ik−2 |

(10.165)

It follows from (10.164) that for k ≥ 2: max |

m, j ;i1 ...ik

(k) m α j ;i1 ...ik |

=

max

m, j ;i1 ...ik−2

Now, by (10.138) and the first of (10.137) we have: max |

m, j ;i1

(1) m α j ;i1 |

= max | m, j

(0) m αj |

=1

(10.166)

It then follows that for each non-negative integer k we have: max |

m, j ;i1 ...ik

(k) m α j ;i1 ...ik |

=1

(10.167)

Ii1 ...ik )m n|=1

(10.168)

which implies that for each non-negative integer k: max |(

n,m;i1 ...ik

(k)

Given a non-negative integer l let us denote by E / lQ the bootstrap assumption that there is a constant C independent of s such that for all t ∈ [0, s]: Q

E /l

: max max Ril . . . Ri1 Qψα  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 i1 ...il

α

t

Part 1: Control of the angular derivatives of the first derivatives of the x i

353

Q We then denote by E / [l] the bootstrap assumption: Q

E / [l]

Q

Q

: E / 0 and . . . and E /l

Given a positive integer l let us denote by X / l the bootstrap assumption that there is a constant C independent of s such that for all t ∈ [0, s]: X / l : max L / Ril . . . L / Ri1 χ L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] i1 ...il

t

Let us also denote by X / 0 the bootstrap assumption:     η0 h /  ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]  X / 0 : χ − 1 − u + η0 t  This coincides with the assumption F2 of Chapter 6. We then denote by X / [l] the bootstrap assumption: X / [l] : X / 0 and . . . and X /l We are now ready to proceed with the estimates of the angular derivatives of the functions y j . Proposition 10.1 Let hypothesis H0 and the estimate (10.30) hold. Let also the bootstrap Q / [l−1] and X / [l−1] hold, for some positive integer l. Then if δ0 is suitably assumptions E / [l] , E small (depending on l) we have: max Rik . . . Ri1 y j  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

j ;i1 ...ik

t

for all k = 0, . . . , l. Proof. First, applying the starting point of the argument of Lemma 10.1 to σ by writing Ril . . . Ri1 σ = −2ψ µ Ril . . . Ri1 ψµ  −2(g −1 )µν ((R)s1 ψµ )((R)s2 ψν )

(10.169)

partitions

where the sum is over all partitions of {1, . . . , l} into two subsets s1 , s2 , each of which is ordered and contains at least one element, we see that assumption E / [l] implies: max Rik . . . Ri1 σ  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1

i1 ...ik

t

: for all k = 1, . . . , l

(10.170)

Also, noting that Qσ = −2ψ µ Qψµ and writing Rik . . . Ri1 Qσ = −2



((R)s1 ψ µ )((R)s2 Qψµ )

partitions

(10.171)

354

Chapter 10. Control of the Angular Derivatives

where the sum is over all ordered partitions {s1 , s2 } of {1, . . . , k} into two subsets, we Q / [l] imply: readily conclude that the bootstrap assumptions E / [l−1] and E max Rik . . . Ri1 Qσ  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1

i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.172)

The proof of the proposition is by induction. The statement of the proposition reduces for l = 0 to the estimate (10.30), which is here assumed. Let then the proposition hold with l replaced by l − 1. We then have: max Rik . . . Ri1 y j  L ∞ ( ε0 ) ≤ Cl−1 δ0 (1 + t)−1 [1 + log(1 + t)]

j ;i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.173)

Thus, the assumptions of Lemma 10.2 are satisfied with l replaced by l − 1. This lemma then yields: max Rik . . . Ri1 λi  L ∞ ( ε0 ) ≤ Cl−1 δ0 [1 + log(1 + t)]

i;i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.174)

and: max 

j ;i1 ...ik

(k) j δi1 ...ik  L ∞ ( ε0 ) t

≤ Cl−1 δ0 [1 + log(1 + t)] : for all k = 0, . . . , l

(10.175)

(0) δ j

(note that, by definition, = 0). From expression (10.39) and the estimates (10.53) and (10.175) we conclude that, if δ0 is suitably small (depending on l), max Rik . . . Ri1 x j  L ∞ ( ε0 ) ≤ C(1 + η0 t)

j ;i1 ...ik

t

: for all k = 0, . . . , l

(10.176)

From expression (10.24) we have: Rik . . . Ri1 Tˆ j = −(1 − u + η0 t)−1 Rik . . . Ri1 x j + Rik . . . Ri1 y j

(10.177)

By the estimates (10.173) and (10.176) it then follows that, if δ0 is suitably small (depending on l) we have: max Rik . . . Ri1 Tˆ j  L ∞ ( ε0 ) ≤ C

j ;i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.178)

Consider next the function ψTˆ . Recalling that ψTˆ = ψ j Tˆ j , we have:  Rik . . . Ri1 ψTˆ = ((R)s1 ψ j )((R)s2 Tˆ j ) partitions

|s1 | + |s2 | = k

(10.179)

Part 1: Control of the angular derivatives of the first derivatives of the x i

355

where the sum is over all ordered partitions {s1 , s2 } of {1, . . . , k} into two subsets. By the estimate (10.178) and assumption E / [l−1] it then follows that: max Rik . . . Ri1 ψTˆ  L ∞ ( ε0 ) ≤ C[l−1] δ0 (1 + t)−1

i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.180)

We turn to the St,u 1-form ψ. Recalling that ψ = ψ j d/x j , we have: / R i1 ψ = L / R ik . . . L



((R)s1 ψ j )d/((R)s2 x j )

partitions

|s1 | + |s2 | = k

(10.181)

where the sum is again over all ordered partitions {s1 , s2 } of {1, . . . , k} into two subsets. By hypothesis H0 and (10.176) we have, if k ≤ l − 1, d/((R)s2 x j ) L ∞ ( ε0 ) ≤ C t

(10.182)

/ [l−1] this yields, through expression since |s2 |+1 ≤ k +1 ≤ l. Together with assumption E (10.181), / R ik . . . L / Ri1 ψ L ∞ ( ε0 ) ≤ C[l−1] δ0 (1 + t)−1 max L

i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.183)

We also have the symmetric 2-covariant St,u tensorfield ω /, defined by (3.235) (or (4.89)). We can write this definition more simply as: ω / = (d/ψ j ) ⊗ (d/x j )

(10.184)

We then have: L / R ik . . . L / R i1 ω /=



(d/(R)s1 ψ j ) ⊗ (d/(R)s2 x j )

partitions

|s1 | + |s2 | = k

(10.185)

By hypothesis H0 and assumption E / [l−1] we have, if k ≤ l − 1, d/(R)s1 ψ j  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 t

(10.186)

since |s1 | + 1 ≤ k + 1 ≤ l. Together with (10.182) this yields , through expression (10.185), the estimate: / R ik . . . L / R i1 ω / L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 max L

i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.187)

356

Chapter 10. Control of the Angular Derivatives ε

Next, we estimate, in L ∞ (t 0 ), the angular derivatives of the functions bi , defined by (10.19), of order up to l − 1. We write, in regard to the first term in (10.19)      dH s1 d H ψ ˆ Ri σ = ((R)s2 ψTˆ )((R)s3 Ri σ ) (R) Ri k . . . Ri 1 dσ T dσ partitions

|s1 | + |s2 | + |s3 | = k ≤ l − 1

(10.188)

where the sum is over all ordered partitions {s1 , s2 , s3 } of {1, . . . , k} into three subsets. ε Since |s3 | + 1 ≤ k + 1 ≤ l, the last factor in the sum is bounded in L ∞ (t 0 ) by (10.170), ε while since |s2 | ≤ k ≤ l − 1, the second factor is bounded in L ∞ (t 0 ) by (10.180). As ε the first factor is also bounded in L ∞ (t 0 ) by a constant C, we obtain: 0  0 0 0 2 −2 0 Ri . . . Ri d H ψ ˆ Ri σ 0 1 0 ∞ ε0 ≤ Cl δ0 (1 + t) 0 k T dσ L (t ) : for all k = 0, . . . , l − 1

(10.189)

We write, in regard to the second term in (10.19),  ((R)s1 H )((R)s2 Tˆ j )((R)s3 Ri ψ j ) Rik . . . Ri1 (H Tˆ j Ri ψ j ) = partitions

|s1 | + |s2 | + |s3 | = k ≤ l − 1

(10.190) ε

Here, since |s3 | + 1 ≤ k + 1 ≤ l the last factor in the sum is bounded in L ∞ (t 0 ) by ε assumption E / [l] , while since |s2 | ≤ k ≤ l − 1 the second factor is bounded in L ∞ (t 0 ) ε by (10.178). As the first factor is also bounded in L ∞ (t 0 ) by a constant C, we obtain: Rik . . . Ri1 (H Tˆ j Ri ψ j ) L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 t

: for all k = 0, . . . , l − 1

(10.191)

Combining finally the estimates (10.189) and (10.191) we obtain: max Rik . . . Ri1 bi  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1

i;i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.192)

Next, we have the functions p/i , defined by (10.18). Writing:  ((R)s1 ψTˆ )((R)s2 bi ) Rik . . . Ri1 p/i = partitions

|s1 | + |s2 | = k ≤ l − 1

(10.193)

we estimate, using (10.180) and (10.192), max Rik . . . Ri1 p/i  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−2

i;i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.194)

Part 1: Control of the angular derivatives of the first derivatives of the x i

357

We now turn to the St,u -tangential vectorfields q/i , given by (10.27), with the T11 type St,u tensorfield m  given by (10.28), (10.29), and the St,u -tangential vectorfields n i by (10.23). Here we must first estimate the angular derivatives of the symmetric 2covariant St,u tensorfield k/ . This is given by equation (3.236), which we may write more simply as:  ψ0 1 H ψ0 1 dH  ψ⊗ ψ(Lσ ) − ( ψ ⊗ d/σ + d/σ ⊗ ψ) − ω / (10.195) k/ = 2α dσ 1 + ρH α 1 + ρH Consider first the last term in (10.195). We have:       1 H ψ0 1 H ψ0 L / R ik . . . L ω / = / R i1 /) (R)s1 ((L / R )s 2 ω α 1 + ρH α 1 + ρH partitions

|s1 | + |s2 | = k ≤ l − 1

(10.196)

Here, and in the following, if sm = {n 1 , . . . , n p },

p = |sm |

we denote by (L / R )sm the operator: (L / R )s m = L / R in p . . . L / R in Since

1

(10.197)

1 H ψ0 α 1 + ρH

is a smooth function of the ψα , and |s1 | ≤ k ≤ l − 1, the first factor in the sum in (10.196) ε is bounded by assumption E / [l−1] in L ∞ (t 0 ) by a constant C. On the other hand, since ε also |s2 | ≤ k ≤ l − 1 the second factor is bounded in L ∞ (t 0 ) by (10.187), we obtain: 0 0  0 0 1 H ψ0 −2 0 / R ik . . . L ω / 0 / R i1 max 0L 0 ∞ ε0 ≤ Cl δ0 (1 + t) i1 ...ik α 1 + ρH L (t ) : for all k = 0, . . . , l − 1

(10.198)

In regard to the first term in (10.195), we write:   1 d H ψ0 L / Rik ...L ( ψ⊗ d/σ + d/σ ⊗ ψ) / R i1 2α dσ 1 + ρ H       1 d H ψ0 s1 ((L / R )s2 ψ) ⊗ (d/(R)s3 σ ) + (d/(R)s3 σ ) ⊗ ((L = / R )s2 ψ) (R) 2α dσ 1 + ρ H partitions

|s1 | + |s2 | + |s3 | = k ≤l − 1 Again, since

(10.199) 1 d H ψ0 2α dσ 1 + ρ H

358

Chapter 10. Control of the Angular Derivatives

is a smooth function of the ψα , and |s1 | ≤ k ≤ l − 1, the first factor in the sum in ε (10.196) is bounded by assumption E / [l−1] in L ∞ (t 0 ) by a constant C. Moreover, since ε s 2 |s2 | ≤ k ≤ l − 1, the factor (L / R ) ψ is bounded in L ∞ (t 0 ) by (10.183). Since also ε s ∞ 3 |s3 | ≤ k ≤ l − 1, the factor d/(R) σ is bounded in L (t 0 ) by hypothesis H0 and (10.170): (10.200) d/(R)s3 σ  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 t

It follows that: 0 0  0 0 1 d H ψ0 2 −3 0 / R ik . . . L ( ψ ⊗ d/σ + d/σ ⊗ ψ) 0 / R i1 max L 0 ∞ ε0 ≤ Cl δ0 (1 + t) i1 ...ik 0 2α dσ 1 + ρ H L (t ) : for all k = 0, . . . , l − 1

(10.201)

Finally, also in regard to the first term in (10.195), we write:   1 d H ψ0 L / R ik . . . L ψ⊗ ψ(Lσ ) / R i1 2α dσ 1 + ρ H     1 d H ψ0 = ((L / R )s2 ψ) ⊗ ((L (R)s1 / R )s3 ψ)((R)s4 (Lσ )) 2α dσ 1 + ρ H partitions

|s1 | + |s2 | + |s3 | + |s4 | + k ≤ l − 1

(10.202) ε

Again, by assumption E / [l−1] the first factor in the above sum is bounded in L ∞ (t 0 ) by a constant C, the second and third factors are bounded by (10.183), and, since L = (1 + t)−1 Q, the fourth factor is bounded by (10.171). It follows that: 0 0  0 0 1 d H ψ0 3 −4 0 / R ik . . . L max 0L / R i1 ψ⊗ ψ(Lσ ) 0 0 ∞ ε0 ≤ Cl δ0 (1 + t) i1 ...ik 2α dσ 1 + ρ H L (t ) : for all k = 0, . . . , l − 1

(10.203)

Collecting the above results, (10.198), (10.201) and (10.203) we conclude that: max L / R ik . . . L / Ri1 k/  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2

i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.204)

Next we must consider the Lie derivatives of the induced acoustical metric h/ on the surfaces St,u with respect to the rotation vectorfields Ri . We have: L / Ri h/ = the symmetric 2-covariant St,u tensorfields write more simply as: (Ri )

(Ri )

π /

(Ri ) π / being

(10.205) given by (6.40), which we may

π / = 2λi (α −1 χ − k/ ) +H i j m ( ψ ⊗ d/x j + d/x j ⊗ ψ)ψm +H ( ψ ⊗ d/ψm + d/ψm ⊗ ψ)i j m x j dH ( ψ⊗ ψ)(Ri σ ) + dσ

(10.206)

Part 1: Control of the angular derivatives of the first derivatives of the x i

359

We shall estimate the angular derivatives of the (Ri ) π / of order up to l − 1. Consider the first term in (10.206). We have:  L / R ik . . . L / Ri1 (λi (α −1 χ − k/ )) = ((R)s1 λi )((L / R )s2 (α −1 χ − k/ )) partitions

|s1 | + |s2 | = k ≤ l − 1

(10.207)

ε

Since |s1 | ≤ k ≤ l − 1, the first factor is bounded in L ∞ (t 0 ) by (10.174). To estimate the second factor we write:  (L / R )s2 (α −1 χ) = ((R)s2,1 α −1 )((L / R )s2,2 χ) partitions

|s2,1 | + |s2,2 | = |s2 | ≤ k ≤ l − 1

(10.208)

where the sum is over all ordered partitions {s2,1 , s2,2 } of the set s2 into two ordered subsets. Since |s2,1 | ≤ l − 1, the first factor in the sum in (10.208) is bounded by assumption E / [l−1] by a constant C, while, since also |s2,2 | ≤ l − 1, by assumption X / [l−1] we have: (L / R )s2,2 χ L ∞ ( ε0 ) ≤ C(1 + t)−1

(10.209)

(L / R )s2 (α −1 χ) L ∞ ( ε0 ) ≤ C(1 + t)−1

(10.210)

t

It follows that:

t

(provided that δ0 is suitably small, depending on l). Taking also into account the estimate (10.204), we then conclude, through expression (10.207), that: max L / R ik . . . L / Ri1 (λi (α −1 χ − k/ )) L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

i;i1 ...ik

t

: for all k = 0, . . . , l − 1 (10.211) Consider the second term in (10.206). We have: L / R ik . . . L / Ri1 (H ψm ( ψ ⊗ d/x j + d/x j ⊗ ψ)) =  ((R)s1 H )((R)s2 ψm )(((L / R )s3 ψ) ⊗ (d/(R)s4 x j ) + (d/(R)s4 x j ) ⊗ ((L / R )s3 ψ)) partitions

|s1 | + |s2 | + |s3 | + |s4 | = k ≤ l − 1

(10.212)

where the sum is over all ordered partitions of the set {1, . . . , k} into four ordered subsets. Since |s1 |, |s2 | ≤ l − 1, by assumption E / [l−1] the first factor in the sum in (10.212) is ε bounded in L ∞ (t 0 ) by a constant C while the second factor is bounded by C[l−1] δ0 (1 + ε t)−1 . Since also |s3 |, |s4 | ≤ l − 1, the third factor is bounded in L ∞ (t 0 ) according to (10.182) and (10.183). We conclude that: max L / R ik . . . L / Ri1 (H i j m ( ψ ⊗ d/x j + d/x j ⊗ ψ)ψm ) L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−2

i;i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.213)

360

Chapter 10. Control of the Angular Derivatives

Consider the third term in (10.206). We have: / Ri1 (H x j ( ψ ⊗ d/ψm + d/ψm ⊗ ψ)) = L / R ik . . . L  ((R)s1 H )((R)s2 x j )(((L / R )s3 ψ) ⊗ (d/(R)s4 ψm ) + (d/(R)s4 ψm ) ⊗ ((L / R )s3 ψ)) partitions

|s1 | + |s2 | + |s3 | + |s4 | = k ≤ l − 1

(10.214)

Since |s1 |, |s3 | ≤ l − 1, the first and third factors in the sum in (10.214) are bounded ε in L ∞ (t 0 ) as above, while since |s2 |, |s4 | ≤ l − 1 the second factor is bounded in ε0 ε ∞ L (t ) by (10.176) and the fourth factor is bounded in L ∞ (t 0 ) according to (10.186). We conclude that: / R ik . . . L / Ri1 (H i j m x j ( ψ ⊗ d/ψm + d/ψm ⊗ ψ)) L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−2 max L

i;i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.215)

Consider finally the fourth term in (10.206). We have:   dH L / R ik . . . L ( ψ⊗ ψ)(Ri σ ) = / R i1 dσ    s1 d H ((L / R )s2 ( ψ⊗ ψ))((R)s3 Ri σ ) (R) dσ partitions

|s1 | + |s2 | + |s3 | = k ≤ l − 1

(10.216) ε

Again, the first factor in the sum (10.216) is bounded in L ∞ (t 0 ) by a constant C, while, since |s2 | ≤ l − 1, (10.183) implies: (L / R )s2 ( ψ⊗ ψ) L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−2 t

(10.217)

ε

Since also |s3 | ≤ l − 1, the last factor is bounded in L ∞ (t 0 ) by (10.170). We conclude that: 0  0 0 0 dH 3 −3 0 ( ψ ⊗ ψ )(R max 0 . . . L / σ ) L / R R i ik i1 0 0 ∞ ε0 ≤ Cl δ0 (1 + t) dσ i;i1 ...ik L (t ) : for all k = 0, . . . , l − 1

(10.218)

Putting together the results (10.211), (10.213), (10.215), (10.218), yields the desired estimate: max L / R ik . . . L / R i1

i;i1 ...ik

(Ri )

π / L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

: for all k = 0, . . . , l − 1 (10.219) We must also estimate the Lie derivatives of the reciprocal St,u metric /h−1 with respect to the rotation vectorfields Ri . We have: /−1 = −h/−1 · L / Ri h

(Ri )

π / · h/−1

(10.220)

Part 1: Control of the angular derivatives of the first derivatives of the x i

361

In general, for k ≥ 1: L / R ik . . . L / R i1 h /−1 =

k  j =1

(−1) j



(h/−1 · (L / R )s1 h/) · · · (h/−1 (L / R )s j h/) · h/−1 (10.221)

partitions

Here, the inner sum is over all partitions of the set {1, . . . , k} into j subsets s1 , . . . , s j , each consisting of at least one element. Each sm , m = 1, . . . , j , is ordered, and the order of the subsets in the partition {s1 , . . . , s j } matters (so that a permutation of the subsets in a given partition yields a partition which is considered distinct from the original). Since 1 ≤ |sm | ≤ k for each m = 1, . . . , j and each j = 1, . . . , k, in view of (10.205) and by virtue of the estimate (10.219) we then obtain: max L / R ik . . . L / R i1 h /−1  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

i1 ...ik

t

: for all k = 1, . . . , l

(10.222)

We turn to the coefficients m  and n i in (10.27). The T11 -type St,u tensorfield m  is given by (10.28) in terms of the 2-covariant St,u tensorfield m b , given by (10.29). Let us first consider L / R ik . . . L / Ri1 m b : for k = 0, . . . , l − 1 In reference to the second term in (10.29), by assumption X / 0 we have: 0 0 0 0 η0 h / 0χ − 0 ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] 0 (1 − u + η0 t) 0 L ∞ (tε0 ) Note that:

2  2     / 1  2η0 η0 h 2  = |χ|  χ − ˆ + trχ −   (1 − u + η0 t) 2 1 − u + η0 t 

For k ≥ 1 we have, by (10.205),   η0 h / L / Rik ...L =L / Rik ...L / R i1 χ − / Ri1 χ − η0 (1 − u + η0 t)−1 L / Rik ...L / R i2 1 − u + η0 t

(10.223)

(10.224)

(Ri1 )

π /

(10.225) thus, by assumption X / k and the estimate (10.219) with l replaced by l − 1, we have, for 1 ≤ k ≤ l − 1, 0  0 0 0 / η0 h 0 0L / . . . L / ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] χ − R R i1 0 ik 1 − u + η0 t 0 L ∞ (tε0 ) (10.226) We conclude that by assumption X / [l−1] and (10.219) with l replaced by l − 1, 0  0 0 0 / η0 h 0 max 0 . . . L / ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] χ − L / R R ik i1 0 i1 ...ik 1 − u + η0 t 0 L ∞ (tε0 ) : for all k = 0, . . . , l − 1

(10.227)

362

Chapter 10. Control of the Angular Derivatives

In reference to the first term in (10.29) we have, by assumption E / [l−1] (see (8.156)): max Rik . . . Ri1 (α −1 − η0−1 ) L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1

i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.228)

The angular derivatives of the last term in (10.29) of order up to l − 1 are bounded in ε L ∞ (t 0 ) using (10.180), (10.183), (10.170) and hypothesis H0, by: Cl δ03 (1 + t)−4 Using also the estimate (10.204) in regard to the third term in (10.29), we then obtain: max L / R ik . . . L / Ri1 m b  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.229)

By (10.28) we can express: L / R ik . . . L / R i1 m  =



((L / R )s1 m b ) · ((L / R )s2 h/−1 )

partitions

|s1 | + |s2 | = k ≤ l − 1

(10.230)

Since |s1 |, |s2 | ≤ l − 1, we can then estimate, using (10.222) (with l replaced by l − 1) and (10.229): max L / R ik . . . L / Ri1 m   L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.231)

The St,u -tangential vectorfields n i are given by (10.23). Writing:  L / R ik . . . L / Ri1 ( ψ · h /−1 ) = ((L / R )s1 ψ) · ((L / R )s2 h/−1 ) partitions

|s1 | + |s2 | ≤= k ≤ l − 1

(10.232)

we can estimate, using (10.183) and (10.222) (with l replaced by l − 1), since |s1 |, |s2 | ≤ l − 1, max L / R ik . . . L / Ri1 ( ψ · h /−1 ) L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1

i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.233)

Using (10.233) and (10.192) we then obtain the estimate: max L / R ik . . . L / Ri1 n i  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−2

i;i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.234)

Part 1: Control of the angular derivatives of the first derivatives of the x i

363

Next, we estimate the angular derivatives, of order up to l − 1, of the St,u -tangential vectorfields wi given by (10.113), with the St,u 1-forms (wi )b given by (10.114). We have:  / Ri1 (wi )b = i j m ((R)s1 h mn )((R)s2 y j )d/((R)s3 x n ) L / R ik . . . L partitions

|s1 | + |s2 | + |s3 | = k ≤ l − 1

(10.235)

Since h mn = δmn + H ψm ψn ε

and |s1 | ≤ l −1, the first factor in the sum (10.235) is bounded in L ∞ (t 0 ) by assumption E / [l−1] by a constant C, while, since also |s2 | ≤ l − 1, the second factor is bounded in ε L ∞ (t 0 ) by: C[l−1] δ0 (1 + t)−1 [1 + log(1 + t)] Since |s3 | ≤ l − 1 as well, the third factor is bounded as in (10.182). We conclude that: / R ik . . . L / Ri1 (wi )b  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] max L

i;i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.236)

Using also (10.222) (with l replaced by l − 1) we then obtain the estimate: / R ik . . . L / Ri1 wi  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] max L

i;i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.237)

From the estimates (10.234) and (10.237) it follows that (see (10.132)): / R ik . . . L / Ri1 n i  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] max L

i;i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.238)

Next, we consider the T11 type St,u tensorfields µi and the St,u -tangential vectorfields νi j which enter the formula (10.136) for [Ri , R j ] and are given by (10.134) and (10.135) respectively. Using the estimates (10.174), (10.231) and (10.238) we obtain, in a straightforward manner: max L / R ik . . . L / Ri1 µi  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−2 [1 + log(1 + t)]2

i;i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.239)

and: max L / R ik . . . L / Ri1 νi j  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−1 [1 + log(1 + t)]2

i, j ;i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.240)

364

Chapter 10. Control of the Angular Derivatives

(k) γ We turn to estimate the coefficients (k) β m j ;i1 ...ik of the expansion of j ;i1 ...ik and m (k) L / R ik . . . L / Ri1 R j given by Lemma 10.6. The coefficients α j :i1 ...ik , which are constants, have been shown to satisfy the bound (10.167). The triplets (k) β j ;i1 ...ik = ( (k) βim1 ...ik : m = 1, 2, 3) are given by Lemma 10.7 in terms of the operators Ai and the triplets (k) ρ (k) ρ m m : m = 1, 2, 3) be a triplet of j ;i1 ...ik = ( j ;i1 ...ik : m = 1, 2, 3). Let z = (z T11 -type St,u tensorfields. Let us define the following pointwise norm of z:

|z| = max |z m |

(10.241)

m

Here |z m | is the pointwise norm of z m as a linear transformation in the tangent space of ε St,u at each point. We then define the L ∞ norm of z on t 0 by: z L ∞ ( ε0 ) = sup |z|

(10.242)

ε

t

t 0

Let K = (K nm : n, m = 1, 2, 3) be a matrix of T11 -type St,u tensorfields. We then define the pointwise norm of K by: (10.243) |K | = max |K nm | n,m

and the

L∞

norm of K on

ε t 0

by: K  L ∞ ( ε0 ) = sup |K |

(10.244)

ε

t

t 0

The estimate (10.239) implies, in view of the expression (10.158) for the matrices Ni , / R ik . . . L / Ri1 Ni  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−2 [1 + log(1 + t)]2 max L

i;i1 ...ik

t

: for all k = 0, . . . , l − 1 In view of the structure of the matrices (10.163) this in turn implies: max 

i1 ...ik

(k)

(k) B

i1 ...ik

(10.245)

discussed in the paragraph preceding

Bi1 ...ik  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−2 [1 + log(1 + t)]2 t

: for all k = 1, . . . , l

(10.246)

Let us then consider expression (10.155) for an arbitrary triplet z of T11 -type St,u tensorfields. By expression (10.163), (10.168) and the estimate (10.246), we have: (M)s2  L ∞ ( ε0 ) ≤ C t

: for |s2 | ≤ l

(10.247)

(provided that δ0 is suitably small depending on l). It then follows from expression (10.155) that: max Aik . . . Ai1 z L ∞ ( ε0 ) ≤ Ck

i1 ...ik

k 

t

m=0

max L / R im . . . L / Ri1 z L ∞ ( ε0 )

i1 ...im

: for all k = 0, . . . , l

t

(10.248)

Part 1: Control of the angular derivatives of the first derivatives of the x i (k) ρ

Now, from the expression for the triplets (10.167) and (10.239) we obtain: (m)

/ Rim+n . . . L / Rim+1 max L

i1 ...im+n

j ;i1 ...ik

365

of Lemma 10.6 and the bounds

ρ j ;i1 ...im  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−2 [1 + log(1 + t)]2 t

: for all m and all n ≤ l − 1 Applying (10.248) to the triplet Aim+n . . . Aim+1

(m)

(m) ρ

(10.249)

yields:

j ;i1 ...im

n 

ρ j ;i1 ...im  L ∞ ( ε0 ) ≤ Cn t

L / Rim+ p . . . L / Rim+1

(m)

ρ j ;i1 ...im  L ∞ ( ε0 ) t

p=0

(10.250) By virtue of the estimates (10.249) we then obtain: Aim+n . . . Aim+1

(m)

ρ j ;i1 ...im  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−2 [1 + log(1 + t)]2 t

: for all m and all n ≤ l − 1 It then follows from the expression for the triplets these satisfy the following estimate: max 

(k)

j ;i1 ...ik

(k) β

(10.251) j ;i1 ...ik

given by Lemma 10.6 that

β j ;i1 ...ik  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−2 [1 + log(1 + t)]2 t

: for all k = 1, . . . , l

(10.252)

In a similar manner we obtain, more generally, using (10.249), the estimates: max

j ;i1 ...im+n

L / Rim+n . . . L / Rim+1

(m)

β j ;i1 ...im  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−2 [1 + log(1 + t)]2 t

: for all m and all n such that m + n ≤ l

(10.253)

Consider finally the expression for the coefficients (k) γ j ;i1 ...ik given in the statement of Lemma 10.6. By (10.240) and the bounds (10.167) the first sum in this expression is ε bounded in L ∞ (t 0 ) by: Cl δ02 (1 + t)−1 [1 + log(1 + t)]2

: for all k = 1, . . . , l

while using the estimates (10.253) and (10.240) we can bound the second sum in ε L ∞ (t 0 ) by: Cl δ04 (1 + t)−3 [1 + log(1 + t)]4 We conclude that the coefficients max 

j ;i1 ...ik

(k)

(k) γ

j ;i1 ...ik

: for all k = 1, . . . , l

satisfy the estimate:

γ j ;i1 ...ik  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−1 [1 + log(1 + t)]2 t

: for all k = 1, . . . , l

(10.254)

366

Chapter 10. Control of the Angular Derivatives

The expression for L / R ik . . . L / Ri1 R j given by Lemma 10.6 and the estimates (10.167), (10.252) and (10.254) for the coefficients in this expression, together with the elementary bound (6.121), that is: max Ri  L ∞ ( ε0 ) ≤ C(1 + η0 t)

(10.255)

t

i

imply, trivially: max L / R ik . . . L / Ri1 R j  L ∞ ( ε0 ) ≤ C(1 + η0 t)

j ;i1 ...ik

t

: for all k = 0, . . . , l

(10.256)

provided that δ0 is suitably small. We turn to the St,u -tangential vectorfields q/i , given by (10.27). We have: 

/ Ri1 (m  · Ri ) = L / R ik . . . L

((L / R )s1 m  ) · ((L / R )s 2 R i )

partitions

|s1 | + |s2 | = k ≤ l − 1

(10.257) ε

Since |s1 |, |s2 | ≤ l −1, the first factor in the above sum is bounded in L ∞ (t 0 ), according to (10.231), while the second factor is by (10.256) (replacing l by l − 1 here suffices) ε bounded in L ∞ (t 0 ) bounded by C(1 + η0 t). It follows that: / R ik . . . L / Ri1 (m  · Ri ) L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] max L

i;i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.258)

Combining this with the estimate (10.234) we then obtain: max L / R ik . . . L / Ri1 q/i  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

i;i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.259)

We finally consider the coefficients (k) p/ii1 ...ik , (k) q/ii1 ...ik , and (k)r/ii1 ...ik of the expression for Rik . . . Ri1 Ri y j given by Lemma 10.4. First, from the expression for the coefficients (k) p/ii1 ...ik given by Lemma 10.4 and the estimate (10.194) it readily follows that: j

max 

i;i1 ...ik

(k)

p/ii1 ...ik  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−2 t

: for all k = 0, . . . , l − 1

(10.260)

and, more generally: / Rim+n . . . L / Rim+1 max L

i;i1 ...im+n

(m)

p/ii1 ...im  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−2 t

: for all m and all n such that m + n ≤ l − 1

(10.261)

Part 1: Control of the angular derivatives of the first derivatives of the x i

367

Next, we consider the expression for the coefficients (k) q/ii1 ...ik given by Lemma ε 10.4. The first term in this expression is estimated in L ∞ (t 0 ) by (10.259). In reference to the second term (the sum) we write: / Rik−m+1 ( (k−m−1) p/ii1 ...ik−m−1 q/ik−m ) = L / R ik . . . L  ((R)s1 (k−m−1) p/ii1 ...ik−m−1 )((L / R )s2 q/ik−m ) partitions

|s1 | + |s2 | = m

(10.262)

where the sum is over all ordered partitions {s1 , s2 } of the set {k − m + 1, . . . , k} into two ordered subsets s1 , s2 . Here, k ≤ l − 1, thus, since |s1 | ≤ m, we have: |s1 | + k − m − 1 ≤ k − 1 ≤ l − 2, therefore the estimate (10.261) (even with l replaced by l − 1) gives: (R)s1

(k−m−1)

p/ii1 ...ik−m−1  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−2 t

Moreover, since also |s2 | ≤ m ≤ k − 1 ≤ l − 2 the estimate (10.259) (even with l replaced by l − 1) gives: (L / R )s2 q/ik−m  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

It follows that:

0 k−1 0 0 L / R ik . . . L / Rik−m+1 ( 0 0

(k−m−1)

m=0

0 0

0 p/ii1 ...ik−m−1 q/ik−m )0 0

ε

L ∞ (t 0 )

≤ Cl δ03 (1 + t)−3 [1 + log(1 + t)] : for all k ≤ l − 1

(10.263)

Again in reference to the second term in the expression for 10.4 we write:

(k) q /ii1 ...ik

given by Lemma

/ Rik−m+1 ( (k−m−1) p/ii1 ...ik−m−1 Rik−m ) L / R ik . . . L  ((R)s1 (k−m−1) p/ii1 ...ik−m−1 )((L / R )s2 Rik−m ) = partitions

|s1 | + |s2 | = m

(10.264) ε L ∞ (t 0 )

as above, while by (10.256) (even Here the first factor in the sum is bounded in ε with l replaced by l − 2) the second factor is bounded in L ∞ (t 0 ) by C(1 + η0 t). It follows that: 0 k−1 0 0 0 0 0 (k−m−1) L / R ik . . . L / Rik−m+1 ( p/ii1 ...ik−m−1 Rik−m )0 ≤ Cl δ02 (1 + t)−1 0 0 0 ε0 m=0

L ∞ (t )

: for all k ≤ l − 1 (10.265)

368

Chapter 10. Control of the Angular Derivatives

In view of the estimates (10.263) and (10.265), the second term in the expression for ε (k) q /ii1 ...ik given by Lemma 10.4 (the sum) is bounded in L ∞ (t 0 ) by Cl δ02 (1 + t)−2

: for all k ≤ l − 1

Recalling also the estimate (10.259) for the first term we conclude that: max 

i;i1 ...ik

(k)  q/ii1 ...ik  L ∞ ( ε0 ) t

≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] : for all k = 0, . . . , l − 1

(10.266)

In a similar way we obtain, more generally: max L / Rim+n . . . L / Rim+1

i;i1 ...im+n

(m)  q/ii1 ...im  L ∞ ( ε0 ) t

≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

: for all m and all n such that m + n ≤ l − 1 Finally, we consider the expression for the coefficients 10.4. We write:

(10.267) (k) r/ j ii1 ...ik

given by Lemma

Rik . . . Rik−m+1 { (k−m−1) q/ii1 ...ik−m−1 · d/(Rik−m x j )}  = ((L / R )s1 (k−m−1) q/ii1 ...ik−m−1 ) · d/((R)s2 Rik−m x j ) partitions

|s1 | + |s2 | = m

(10.268)

where again the sum is over all ordered partitions {s1 , s2 } of the set {k − m + 1, . . . , k}, m =≤ k−1, k ≤ l−2, into two ordered subsets s1 , s2 . Since |s1 |+k−m−1 ≤ k−1 ≤ l−2, ε the first factor in the above sum is bounded in L ∞ (t 0 ) by (10.267) (even with l replaced by l − 1), while also |s2 | + 2 ≤ m + 2 ≤ k + 1 ≤ l ε

the second factor is bounded in L ∞ (t 0 ) by a constant C by virtue of (10.176) and hypothesis H0 (as in (10.182)). We conclude: Rik . . . Rik−m+1 {

(k−m−1)  q/ii1 ...ik−m−1

· d/(Rik−m x j )} L ∞ ( ε0 ) t

≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] : for all m = 0, . . . , k − 1 and k ≤ l − 1 It follows that: max 

i;i1 ...ik

(k) j r/ii1 ...ik  L ∞ ( ε0 ) t

≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] : for all k = 0, . . . , l − 1

(10.269)

The estimates (10.260), (10.266) and (10.269), taken together, imply, through the expression for Rik . . . Ri1 Ri y j of Lemma 10.4, max Rik . . . Ri1 Ri y j  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

j ;i,i1 ...ik

t

: for all k = 0, . . . , l − 1 This completes the inductive step and the proof of Proposition 10.1.

(10.270)

Part 1: Control of the angular derivatives of the first derivatives of the x i

369

The foregoing proposition has a number of corollaries which we shall presently state. First, in conjunction with Lemma 10.2, we have: Corollary 10.1.a Under the assumptions of Proposition 10.1 we have: max Rik . . . Ri1 λi  L ∞ ( ε0 ) ≤ Cl δ0 [1 + log(1 + t)]

i;i1 ...ik

t

: for all k = 0, . . . , l and: max 

j ;ii1 ...ik

(k+1) j δii1 ...ik  L ∞ ( ε0 ) t

≤ Cl δ0 [1 + log(1 + t)]

: for all k = 0, . . . , l which, if δ0 is suitably small (depending on l), implies: max Rik . . . Ri1 x j  L ∞ ( ε0 ) ≤ C(1 + η0 t)

j ;i1 ...ik

t

: for all k = 0, . . . , l + 1 Next, from the proof of Proposition 10.1 we have: Corollary 10.1.b Under the assumptions of Proposition 10.1 we have: max Rik . . . Ri1 Tˆ j  L ∞ ( ε0 ) ≤ C

j ;i1 ...ik

t

: for all k = 0, . . . , l and: max Rik . . . Ri1 ψTˆ  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1

i1 ...ik

t

: for all k = 0, . . . , l Moreover, max L / R ik . . . L / Ri1 ψ L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1

i1 ...ik

t

: for all k = 0, . . . , l and: max L / R ik . . . L / R i1 ω / L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2

i1 ...ik

t

: for all k = 0, . . . , l − 1 Corollary 10.1.c Under the assumptions of Proposition 10.1 we have: / R ik . . . L / Ri1 k/  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 max L

i1 ...ik

: for all k = 0, . . . , l − 1

t

370

Chapter 10. Control of the Angular Derivatives

Corollary 10.1.d Under the assumptions of Proposition 10.1 we have: / R ik . . . L / R i1 max L

(Ri )

i;i1 ...ik

π / L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

: for all k = 0, . . . , l − 1 Corollary 10.1.e Under the assumptions of Proposition 10.1 we have, if δ0 is suitably small (depending on l), / R ik . . . L / Ri1 R j  L ∞ ( ε0 ) ≤ C(1 + η0 t) max L

j ;i1 ...ik

t

: for all k = 0, . . . , l More precisely, the coefficients of the expression for L / R ik . . . L / Ri1 R j of Lemma 10.6 satisfy: max | (k) α mj;i1 ...ik | = 1 : for all k m, j ;i1 ...ik

max 

m, j ;i1 ...ik

(k) m β j ;i1 ...ik  L ∞ ( ε0 ) t

≤ Cl δ02 (1 + t)−2 [1 + log(1 + t)]2

: for all k = 1, . . . , l (

(0) m βj

= 0)

max 

(k)

j ;i1 ...ik

γ j ;i1 ...ik  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−1 [1 + log(1 + t)]2 t

: for all k = 1, . . . , l (

(0)

γ j = 0)

Finally, we have: Corollary 10.1.f Under the assumptions of Proposition 10.1, the coefficients of the expression for Rik . . . Ri1 Ri y j of Lemma 10.4 satisfy: max 

i;i1 ...ik

(k)

p/ii1 ...ik  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−2 t

: for all k = 0, . . . , l − 1 max 

i:i1 ...ik

(k)  q/ii1 ...ik  L ∞ ( ε0 ) t

≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

: for all k = 0, . . . , l − 1 and: max 

i;i1 ...ik

(k) j r/ii1 ...ik  L ∞ ( ε0 ) t

: for all k = 0, . . . , l − 1

≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

Part 1: Control of the angular derivatives of the first derivatives of the x i

371

We now proceed with the L 2 estimates. We define: 0 0 0 0 η0 h/ 0 0 A0 = 0χ − 1 − u + η0 t 0 L 2 (tε0 ) and for k = 0:

/ R ik . . . L / Ri1 χ L 2 ( ε0 ) Ak = max L i1 ...ik

t

and: A[l] =

l 

Ak

k=0

Also, we define: WkQ

= max Rik . . . Ri1 Qψα  α;i1 ...ik

ε L 2 (t 0 )

,

Q W[l]

=

l 

WkQ

k=0

Proposition 10.2 Let the hypothesis H0 and the estimate (10.30) hold. Let l be a positive Q / [l∗ −1] and X / [l∗ −1] hold. Then if δ0 is integer and let the bootstrap assumptions E / [l∗ ] , E suitably small (depending on l) we have:

 Q Yk ≤ Cl Y0 + (1 + t)A[l−1] + W[l−1] + δ02 (1 + t)−2 W[l−1] for all k = 1, . . . , l. Proof. The proof of the proposition is by induction. The proposition extends trivially to the case l = 0. Let then the proposition hold with l replaced by 1, . . . , l − 1 and consider the case l. The assumptions of Proposition 10.2 coincide with those of Proposition 10.1 with l replaced by l∗ . Therefore the conclusion of Proposition 10.1 holds with l replaced by l∗ . Also, all the corollaries of Proposition 10.1, that is Corollaries 10.1.a,b,c,d,e,f, hold with l replaced by l∗ . Now, by virtue of Proposition 10.1 with l replaced by l∗ , the hypotheses of Lemma 10.3 are satisfied. The lemma in question then yields: max Rik . . . Ri1 λi  L 2 ( ε0 ) ≤ Cl {(1 + η0 t)Y[l] + δ0 W[l] }

i;i1 ...ik

t

: for all k = 0, . . . , l

(10.271)

and: max 

j ;ii1 ...ik

(k+1) j δii1 ...ik  L 2 ( ε0 ) t

: for all k = 0, . . . , l

≤ Cl {(1 + η0 t)Y[l] + δ0 W[l] } (10.272)

On the other hand, the hypotheses of Lemma 10.3 with l replaced by l − 1 are a fortiori satisfied. Therefore the conclusions (10.271) and (10.272) also hold when l is replaced by

372

Chapter 10. Control of the Angular Derivatives

l − 1, that is: max Rik . . . Ri1 λi  L 2 ( ε0 ) ≤ Cl−1 {(1 + η0 t)Y[l−1] + δ0 W[l−1] }

i;i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.273)

and: max 

j ;ii1 ...ik

(k+1) j δii1 ...ik  L 2 ( ε0 ) t

≤ Cl−1 {(1 + η0 t)Y[l−1] + δ0 W[l−1] }

: for all k = 0, . . . , l − 1

(10.274)

By the inductive hypothesis: l−1 

 Q Yk ≤ Cl−1 Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2]

(10.275)

k=1

Since Y[l−1] = Y0 +

l−1 

Yk ,

k=1

substituting in (10.273) and (10.274) we obtain: max Rik . . . Ri1 λi  L 2 ( ε0 ) ≤ t 

Q Cl−1 (1 + t) Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2]

i;i1 ...ik

: for all k = 0, . . . , l − 1

(10.276)

and: max 

j ;ii1 ...ik

(k+1) j δii1 ...ik  L 2 ( ε0 ) t

≤



Q Cl−1 (1 + t) Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2]

: for all k = 0, . . . , l − 1

(10.277)

Consider now Rik . . . Ri1 ψTˆ for k = 0, . . . , l−1, as expressed by (10.179). Consider a given term in the sum in (10.179). Now, we cannot have simultaneously |s1 | > l∗ and |s2 | > l∗ − 1. For then |s1 | ≥ l∗ + 1 and |s2 | ≥ l∗ , hence |s1 | + |s2 | ≥ 2l∗ + 1 ≥ l, contradicting the fact that |s1 | + |s2 | = k ≤ l − 1. We thus have two cases to consider: Case 1: |s2 | ≤ l∗ − 1

and:

Case 2: |s1 | ≤ l∗

Part 1: Control of the angular derivatives of the first derivatives of the x i

373

ε

ε

In Case 1, we place the first factor in L 2 (t 0 ) and the second factor in L ∞ (t 0 ), using the first statement of Corollary 10.1.b with l replaced by l∗ (actually l∗ −1 suffices). We then obtain: ((R)s1 ψ j )((R)s2 Tˆ j ) L 2 ( ε0 ) ≤ (R)s1 ψ j  L 2 ( ε0 ) (R)s2 Tˆ j  L ∞ ( ε0 ) t

t

t

≤ CW[l−1]

(10.278)

In Case 2, we express the second factor as in (10.87). To estimate the contribution of the first term on the right in (10.87) to the L 2 norm of the product ((R)s1 ψ j )((R)s2 Tˆ j ) ε

ε

ε

on t 0 , we place this term in L ∞ (t 0 ) using (10.53) and the factor (R)s1 ψ j in L 2 (t 0 ). We then obtain that the contribution in question is bounded by: CW[l∗ ]

(10.279) ε

To estimate the contribution of the second term in (10.87), we place this term in L 2 (t 0 ) ε using the inductive hypothesis (10.275), and the factor (R)s1 ψ j in L ∞ (t 0 ) using the assumption E [l∗ ] . We then obtain that the contribution in question is bounded by:

 Q (10.280) Cl δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] Finally, to estimate the contribution of the third term in (10.87), we place this term in ε L 2 (t 0 ) using (10.277) (replacing l by l − 1 here suffices), and the factor (R)s1 ψ j again ε in L ∞ (t 0 ). This yields an estimate of the form (10.280) (with l replaced by l − 1). Combining the above results we conclude that in Case 2: ((R)s1 ψ j )((R)s2 Tˆ j ) L 2 ( ε0 )

(10.281)

 Q ≤ CW[l∗ −1] + Cl δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] t

Combining finally the results of the two cases, (10.278) and (10.281) we conclude that: max Rik . . . Ri1 ψTˆ  L 2 ( ε0 ) t

 Q ≤ Cl W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2]

i1 ...ik

: for all k = 0, . . . , l − 1

(10.282)

/ Ri1 ψ, for k = 0, . . . , l − 1, as expressed by (10.181). Consider next L / R ik . . . L Consider a given term in the sum in (10.181). Here the two cases to be considered are: Case 1: |s2 | ≤ l∗

and:

Case 2: |s1 | ≤ l∗ − 1 ε

ε

In Case 1 we place the first factor in L 2 (t 0 ) and the second factor in L ∞ (t 0 ) using hypothesis H0 and the last statement of Corollary 10.1.a with l replaced by l∗ . We then obtain: ((R)s1 ψ j )d/((R)s2 x j ) L 2 ( ε0 ) ≤ (R)s1 ψ j  L 2 ( ε0 ) d/((R)s2 x j ) L ∞ ( ε0 ) t

t

≤ CW[l−1]

t

(10.283)

374

Chapter 10. Control of the Angular Derivatives

In Case 2 we first note that by virtue of hypothesis H0 we have, pointwise, |d/(R)s2 x j | ≤ C(1 + t)−1

3 

|Rq (R)s2 x j |

(10.284)

q=1

We then express Rq (R)s2 x j as in (10.86). That is, if s2 = {n 1 , . . . , n p }, we write: Rq (R)s2 x j =

( p+1) j x in ...in q p 1

p = |s2 |, −

( p+1) j δin ...in q p 1

(10.285)

Substituting in (10.284), yields: |d/(R)s2 x j | ≤ C(1 + t)−1

3  q=1

|

j ( p+1) j x in ...in q | + | ( p+1) δin ...in q | p p 1 1

 (10.286)

We then estimate: |(R)s1 ψ j ||

( p+1) j x in ...in q | L 2 ( ε0 ) p t 1

≤ (R)s1 ψ j  L 2 ( ε0 )  t

( p+1) j x in ...in q  L ∞ ( ε0 ) p t 1

≤ C(1 + t)W[l∗ −1]

(10.287)

and, using (10.277), noting that p + 1 = |s2 | + 1 ≤ k + 1,

k ≤ l − 1,

as well as assumption E / [l∗ −1] : |(R)s1 ψ j || ( p+1) δin ...in q | L 2 ( ε0 ) ≤ (R)s1 ψ j  L ∞ ( ε0 )  ( p+1) δin ...in q  L 2 ( ε0 ) p p t t t 1 1

 Q 2 −2 (10.288) ≤ Cl δ0 Y0 + (1 + t)A[l−2] + W[l−1] + δ0 (1 + t) W[l−2] j

j

Combining (10.287) and (10.288) we obtain, in view of (10.286), that in Case 2: ((R)s1 ψ j )d/((R)s2 x j ) L 2 ( ε0 ) (10.289) t

 Q ≤ CW[l∗ −1] + Cl δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] Combining finally the results of the two cases, (10.283) and (10.289) we conclude that: max L / R ik . . . L / Ri1 ψ L 2 ( ε0 ) ≤ t

 Q Cl W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2]

i1 ...ik

: for all k = 0, . . . , l − 1

(10.290)

Part 1: Control of the angular derivatives of the first derivatives of the x i

375

Consider next L / R ik . . . L / R i1 ω /, for k = 0, . . . , l − 1, as expressed by (10.185). Consider a given term in the sum in (10.185). Again the two cases to be considered are: Case 1: |s2 | ≤ l∗

and:

Case 2: |s1 | ≤ l∗ − 1

In Case 1 we proceed as in (10.283), obtaining: (d/(R)s1 ψ j ) ⊗ (d/(R)s2 x j ) L 2 ( ε0 ) ≤ d/((R)s1 ψ j ) L 2 ( ε0 ) d/((R)s2 x j ) L ∞ ( ε0 ) t

t

t

≤ CW[l]

(10.291)

In Case 2 we bound d/(R)s2 x j pointwise as in (10.286), and we estimate, as in (10.287), |d/((R)s1 ψ j ||

( p+1) j x in ...in q | L 2 ( ε0 ) p t 1

≤ d/((R)s1 ψ j ) L 2 ( ε0 )  t

( p+1) j x in ...in q  L ∞ ( ε0 ) p t 1

≤ CW[l∗ ]

(10.292)

and, as in (10.288), using assumption E / [l∗ ] : |d/((R)s1 ψ j ||

( p+1) j δin ...in q | L 2 ( ε0 ) p t 1

≤ d/((R)s1 ψ j ) L ∞ ( ε0 ) 

( p+1) j δin ...in q  L 2 ( ε0 ) p t 1

 Q ≤ Cl δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] t

(10.293)

Combining (10.292) and (10.293) we obtain, in view of (10.286), that in Case 2: d/((R)s1 ψ j ) ⊗ d/((R)s2 x j ) L 2 ( ε0 )

(10.294)

 Q ≤ C(1 + t)−1 W[l∗ ] + Cl δ0 (1 + t)−2 Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] t

Combining finally the results of the two cases, (10.291) and (10.294) we conclude that: / R ik . . . L / R i1 ω / L 2 ( ε0 ) max L t

 Q −1 ≤ Cl (1 + t) W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2]

i1 ...ik

: for all k = 0, . . . , l − 1

(10.295) ε

Next, we estimate, in L 2 (t 0 ), the angular derivatives of the functions bi , defined by (10.19), of order up to l −1. In regard to the first term in (10.19) we have the expression (10.188). Here the two cases to be considered are: Case 1: |s1 | + |s2 | ≤ l∗

and:

Case 2: |s3 | ≤ l∗ − 1

In Case 1 we estimate, using assumption E / [l∗ ] and the second statement of Corollary 10.1.b with l replaced by l∗ , 0 0  0 0 0 (R)s1 d H ((R)s2 ψ ˆ )((R)s3 Ri σ )0 0 0 2 ε0 T dσ L (t ) 0 0 0 s dH 0 s2 s3 1 0 ≤0 0(R) dσ 0 ∞ ε0 (R) ψTˆ  L ∞ (tε0 ) (R) Ri σ  L 2 (tε0 ) ≤

L (t ) −1 Cl δ0 (1 + t) W[l]

(10.296)

376

Chapter 10. Control of the Angular Derivatives

For, by Lemma 10.1 we have, since |s3 | + 1 ≤ l, (R)s3 Ri σ  L 2 ( ε0 ) ≤ Cl W[l] t

(10.297)

On the other hand in Case 2 we have, by assumption E / [l∗ ] , (R)s3 Ri σ  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 t

(10.298)

Also, since at most one of |s1 |, |s2 | may exceed l∗ we can estimate: 0 0  0 0 s2 0 (R)s1 d H ((R)s2 ψ ˆ )0 (10.299) 0 0 2 ε0 ≤ C(R) ψTˆ  L 2 (tε0 ) T dσ L (t )

 Q ≤ Cl W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2] (by (10.282)) if |s1 | ≤ l∗ , and: 0 0 0 0  0 0 s dH 0 0 −1 0 1 0 (R)s1 d H ((R)s2 ψ ˆ )0 0 ≤ Cl δ0 (1 + t) 0(R) T 0 0 ε dσ dσ 0 L 2 (tε0 ) L 2 (t 0 ) ≤ Cl δ0 (1 + t)−1 W[l−1]

(10.300)

(by Corollary 10.1.b with l replaced by l∗ and Lemma 10.1) if |s2 | ≤ l∗ . Thus, in general, 0 0  0 0 0 (R)s1 d H ((R)s2 ψ ˆ )0 (10.301) 0 2 ε0 0 T dσ L (t )

 Q ≤ Cl W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2] In view of (10.298) it follows that in Case 2: 0 0  0 0 0 (R)s1 d H ((R)s2 ψ ˆ )((R)s3 Ri σ )0 (10.302) 0 0 2 ε0 T dσ L (t )

 Q ≤ Cl δ0 (1 + t)−1 W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2] Combining the results of the two cases, (10.296) and (10.302), we conclude through expression (10.188) that: 0 0  0 0 0 Ri . . . Ri d H ψ ˆ Ri σ 0 1 0 k 0 2 ε0 T dσ L (t )

 Q ≤ Cl δ0 (1 + t)−1 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2] : for all k = 0, . . . , l − 1

(10.303)

In regard to the second term in (10.19) we have the expression (10.189). Again the two cases to be considered are:

Part 1: Control of the angular derivatives of the first derivatives of the x i

Case 1: |s1 | + |s2 | ≤ l∗

377

Case 2: |s3 | ≤ l∗ − 1

and:

In Case 1 we estimate, using assumption E / [l∗ ] and the first statement of Corollary 10.1.b with l replaced by l∗ , as |s1 |, |s2 | ≤ l∗ , ((R)s1 H )((R)s2 Tˆ j )((R)s3 Ri ψ j ) L 2 ( ε0 ) t

≤ (R)s1 H  L ∞( ε0 ) (R)s2 Tˆ j  L ∞ ( ε0 ) (R)s3 Ri ψ j  L 2 ( ε0 ) t

t

t

≤ CW[l]

(10.304)

In Case 2 we have, by assumption E / [l∗ ] , (R)s3 Ri ψ j  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 t

and at most one of |s1 |,|s2 | may exceed l∗ . If |s1 | ≤ l∗ we express (R)s2 Tˆ j as in (10.87). To estimate the contribution of the first term in (10.87) to the L 2 norm of the product ((R)s1 H )((R)s2 Tˆ j )((R)s3 Ri ψ j ) ε

ε

ε

on t 0 , we place this term in L ∞ (t 0 ) using (10.53), the factor (R)s1 H also in L ∞ (t 0 ) ε and the factor (R)s3 Ri ψ j in L 2 (t 0 ). We then obtain that the contribution in question is bounded by: (10.305) CW[l∗ ] ε

To estimate the contribution of the second term in (10.87) we place this term in L 2 (t 0 ) ε using the inductive hypothesis (10.275), and the other two factors in L ∞ (t 0 ). We then obtain that the contribution in question is bounded by:

 Q Cl δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] (10.306) Finally, to estimate the contribution of the third term in (10.87) we place this term in ε L 2 (t 0 ) using (10.277) (replacing l by l − 1 here suffices), and the other two factors ε ∞ in L (t 0 ). This yields an estimate of the form (10.306) (with l replaced by l − 1). Combining the above results we conclude that if |s1 | ≤ l∗ : ((R)s1 H )((R)s2 Tˆ j )((R)s3 Ri ψ j ) L 2 ( ε0 ) (10.307) t

 Q ≤ Cl W[l∗ ] + Cl δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] ε

On the other hand, if |s2 | ≤ l∗ we place the factor (R)s1 H in L 2 (t 0 ) using Lemma 10.1 ε (in this case we may assume that s1 is non-empty), and the other two factors in L ∞ (t 0 ) (using the first statement of Corollary 10.1.b with l replaced by l∗ ), obtaining: ((R)s1 H )((R)s2 Tˆ j )((R)s3 Ri ψ j ) L 2 ( ε0 ) t

≤ Cl δ0 (1 + t)−1 W[l−1]

(10.308)

378

Chapter 10. Control of the Angular Derivatives

Combining the results (10.307) and (10.308) we conclude that in Case 2: (10.309) ((R)s1 H )((R)s2 Tˆ j )((R)s3 Ri ψ j ) L 2 ( ε0 ) t

 Q ≤ CW[l∗ ] + Cl δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] Combining the results of the two cases, (10.304) and (10.309), we conclude that: Rik . . . Ri1 (H Tˆ j Ri ψ j  L 2 ( ε0 ) t

 Q −1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2] ≤ Cl W[l] + δ0 (1 + t) : for all k = 0, . . . , l − 1

(10.310)

Combining finally the results (10.303) and (10.310) for the two terms in (10.19) we obtain: max Rik . . . Ri1 bi  L 2 ( ε0 ) t

 Q ≤ Cl W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2]

i;i1 ...ik

: for all k = 0, . . . , l − 1

(10.311)

We turn to the functions p/i . Here we have expression (10.193). Considering the two cases |s1 | ≤ l∗ and |s2 | ≤ l∗ − 1, the L ∞ estimates given by the second statement of Corollary 10.1.b and (10.192) with l replaced by l∗ , and the L 2 estimates (10.282) and (10.311), we obtain: max Rik . . . Ri1 p/i  L 2 ( ε0 ) t

 Q −1 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2] ≤ Cl δ0 (1 + t)

i;i1 ...ik

: for all k = 0, . . . , l − 1

(10.312)

We proceed to the symmetric 2-covariant St,u tensorfield k/ , given by (10.195). We first consider Rik . . . Ri1 Qσ , for k = 0, . . . , l − 1, as expressed by (10.171). The two cases to be considered are |s1 | ≤ l∗ and |s2 | ≤ l∗ − 1. In the first case we place the first ε ε factor in L ∞ (t 0 ) using assumption E / [l∗ ] and the second factor in L 2 (t 0 ), while in the ε second case we place the first factor in L 2 (t 0 ) (we can assume that in this case s1 is ε non-empty) and the second factor in L ∞ (t 0 ) using assumption E / [lQ∗ −1] . In this way we obtain the estimate:

 Q + δ0 (1 + t)−1 W[l−1] max Rik . . . Ri1 Qσ  L 2 ( ε0 ) ≤ Cl W[l−1] i1 ...ik

: for all k = 0, . . . , l − 1

t

(10.313)

Let us then consider expression (10.195). We shall presently outline a general procedure. Let us first write out the first term in (10.195) as a sum of two terms. We have

Part 1: Control of the angular derivatives of the first derivatives of the x i

379

then three terms to consider. In each term there is only one factor of order 1, the other factors being of order 0. The factor of order 1 is Lσ in the first term, d/σ in the second / Ri1 k/ . We have the expressions (10.202), term, and ω / in the third. Consider then L / R ik . . . L (10.199), and (10.196), for the three corresponding terms from expression (10.195). In each of these expressions a certain subset s M of the set {1, . . . , k} is associated to the factor of order 1. Thus in (10.202) s M = s4 , in (10.199) s M = s3 , and in (10.196) s M = s2 . In estimating each of these three expressions we consider the following two cases:  Case 1: m = M |sm | ≤ l∗ and: Case 2: |s M | ≤ l∗ − 1 Note that we can assume that s M is non-empty in Case 1. In Case 1 we place the factor which is associated to the subset s M and is an angular ε derivative of the factor of order 1 in the corresponding term in (10.195) in L 2 (t 0 ), and ε0 ∞ we place all the other factors in L (t ) using either directly the assumptions of the present proposition (and their immediate consequences (10.170) and (10.171)), or Propoε sition 10.1 and its corollaries with l replaced by l∗ . These control in L ∞ (t 0 ) the angular derivatives, up to the order l∗ , of all relevant quantities of order 0. In Case 2 we place ε the factor associated to the subset s M in L ∞ (t 0 ) using either directly the assumptions of the present proposition, or Proposition 10.1 and its corollaries with l replaced by l∗ . We then place the factor associated to that subset sm , m = M, which has the maximal ε ε cardinality in L 2 (t 0 ), and all the other factors in L ∞ (t 0 ) using again either directly the assumptions of the present proposition, or Proposition 10.1 and its corollaries with l replaced by l∗ , noting that at most one of the sm , m = M can have cardinality exceeding l∗ , for otherwise:  |sm | ≥ 2(l∗ + 1) ≥ l + 1 m = M

contradicting the fact that 

|sm | = k − |s M | ≤ k ≤ l − 1

m = M

In this way we obtain, in regard to (10.202) and in view of (10.313) the estimate: 0 0  0 0 1 d H ψ0 0 max 0L / R i1 ψ⊗ ψ(Lσ ) 0 / R ik . . . L 0 2 ε0 i1 ...ik 2α dσ 1 + ρ H L (t )

1 2 Q ≤ Cl δ02 (1 + t)−3 W[l−1] + W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] : for all k = 0, . . . , l − 1

(10.314)

in regard to (10.199) the estimate: 0 0  0 0 1 d H ψ0 0 ( ψ ⊗ d/σ + d/σ ⊗ ψ) 0 max 0L / R i1 / R ik . . . L 0 2 ε0 i1 ...ik 2α dσ 1 + ρ H L (t )

 Q −2 −1 W[l] + δ0 (1 + t) Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2] ≤ Cl δ0 (1 + t) : for all k = 0, . . . , l − 1

(10.315)

380

Chapter 10. Control of the Angular Derivatives

and in regard to (10.196) and in view of (10.295) the estimate: 0 0  0 0 1 H ψ0 0 / R ik . . . L ω / 0 max L / R i1 0 2 ε0 i1 ...ik 0 α 1 + ρH L (t )

 Q −1 −1 W[l] + δ0 (1 + t) Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2] ≤ Cl (1 + t) : for all k = 0, . . . , l − 1

(10.316)

Combining finally the results (10.314), (10.315), (10.316), yields: max L / R ik . . . L / Ri1 k/  L 2 ( ε0 ) t

1 2 Q ≤ Cl (1 + t)−1 W[l] + δ02 (1 + t)−2 W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2]

i1 ...ik

: for all k = 0, . . . , l − 1

(10.317)

We turn to the symmetric 2-covariant St,u tensorfields (Ri ) π / given by (10.206). Here we have four terms each of which, except the second, is of the form considered in the above general procedure, that is, it contains only one factor of order 1, which in the first term is α −1 χ − k/ , in the third d/ψm and in the fourth Ri σ . In the second term we may take the factor d/x j in the role of the factor of order 1 in applying the general procedure, although it is of order 0 according to our convention. Applying in this way the above general procedure we obtain the following estimates. In regard to the first term, expression (10.207), and in view of the estimates (10.276) and (10.317), we obtain: / R ik . . . L / Ri1 (λi (α −1 χ − k/ )) L 2 ( ε0 ) ≤ max L t

 Q Cl δ0 (1 + t)−1 [1 + log(1 + t)] (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1]  Q +Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2]

i;i1 ...ik

: for all k = 0, . . . , l − 1

(10.318)

In regard to the second term, we have expression (10.212). Here we apply a variant of the above outlined procedure. Here s M = s4 so the corresponding factor is d/(R)s4 x j . Case 2 is treated according to the general procedure, however in Case 1 we bound d/(R)s4 x j pointwise as in (10.286), so, if s4 = {n 1 , . . . , n p }, p = |s4 |, we write: |d/(R)s4 x j | ≤ C(1 + t)−1

3  q=1

|

j ( p+1) j x in ...in q | + | ( p+1) δin ...in q | p p 1 1

 (10.319)

To estimate the contribution of the first term in parenthesis in (10.319) we deviate from ε the general procedure by placing this term in L ∞ (t 0 ). By (10.53): 

( p+1) j x i1 ...in q  L ∞ ( ε0 ) p t

≤ C(1 + t)

(10.320)

Part 1: Control of the angular derivatives of the first derivatives of the x i

381 ε

We then place one of the other factors, say (R)s2 ψm , in L 2 (t 0 ), and the remaining ε factors in L ∞ (t 0 ). We then obtain that the contribution in question is bounded by: Cl δ0 W[l∗ −1] To estimate the contribution of the second term in parenthesis in (10.319) we follow the ε general procedure by placing this term in L 2 (t 0 ) using (10.277), noting that p + 1 = |s4 | + 1 ≤ k + 1,

k ≤ l − 1,

ε L ∞ (t 0 ). We then obtain that the contribution in question

and we place all other factors in is bounded by:

 Q Cl δ02 (1 + t)−1 Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] Combining the above results through (10.319) (there is an overall factor of C(1 + t)−1 ), and then combining with the result for Case 2, yields:

max L / R ik . . . L / Ri1 (H i j m ( ψ ⊗ d/x j + d/x j ⊗ ψ)ψm ) L 2 ( ε0 ) t

 Q ≤ Cl δ0 (1 + t)−1 W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2]

i;i1 ...ik

: for all k = 0, . . . , l − 1

(10.321)

In regard to the third term, we have expression (10.214). Here s M = s4 . In treating the subcase of Case 2 where the sm , m = M, which has the maximal cardinality is s2 , we express (R)s2 x j as in (10.86). We then estimate the contribution of the first term in (10.86) ε by placing this term in L ∞ (t 0 ) using (10.53) and one of the other factors, say d/(R)s4 ψm , ε0 ε 2 in L (t ), and all remaining factors in L ∞ (t 0 ). We estimate the contribution of the ε second term in (10.86) by following the general procedure, placing this term in L 2 (t 0 ) ε0 ∞ using (10.277) (replacing l by l − 1 here suffices) and all other factors in L (t ). Combining the two results and then combining with the results for the other subcases and with the result for Case 1, then yields: max L / R ik . . . L / Ri1 (H i j m x j ( ψ ⊗ d/ψm + d/ψm ⊗ ψ)) L 2 ( ε0 ) t

 Q −1 −1 W[l] + δ0 (1 + t) Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2] ≤ Cl δ0 (1 + t)

i;i1 ...ik

: for all k = 0, . . . , l − 1

(10.322)

In regard to the fourth term in (10.206), we have expression (10.216). This is treated according to the general procedure in a straightforward manner. We obtain: 0 0  0 0 dH 0 ( ψ⊗ ψ)(Ri σ ) 0 max 0L / R i1 / R ik . . . L 0 2 ε0 i;i1 ...ik dσ L (t )

 Q 2 −2 −1 W[l] + δ0 (1 + t) Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2] ≤ Cl δ0 (1 + t) : for all k = 0, . . . , l − 1

(10.323)

382

Chapter 10. Control of the Angular Derivatives

Putting together the results (10.318), (10.321), (10.322), (10.323), yields the desired estimate: max L / R ik . . . L / Ri1 (Ri ) π / L 2 ( ε0 ) t

 Q ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1]  Q +Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2]

i;i1 ...ik

: for all k = 0, . . . , l − 1

(10.324) ε

Next we estimate in L 2 (t 0 ) the Lie derivatives of the reciprocal St,u metric h/−1 with respect to the rotation vectorfields Ri , of order up to l. In reference to expression (10.221), consider a term of the outer sum. Such a term corresponds to a given value of j ∈ {1, . . . , k}, where k = 1, . . . , l. Since j 

|sm | = k ≤ l

(10.325)

m=1

at most one of the |sm | may exceed l∗ . [Otherwise, if |sm  | and |sm  | both exceed l∗ then j 

|sm | ≥ |sm  | + |sm  | ≥ 2(l∗ + 1) ≥ l + 1

m=1

contradicting (10.325)]. We place the factor corresponding to the subset s M with the maxε ε imal cardinality in L 2 (t 0 ), and the other factors in L ∞ (t 0 ) using Corollary 10.1.d with l replaced by l∗ . Since there are j − 1 such other factors we obtain that the j th term in the ε outer sum in (10.221) is bounded in L 2 (t 0 ) by: k− j   j −1  Cl δ0 (1 + t)−1 [1 + log(1 + t)] max L / R in . . . L / R i1 n=0

(Ri )

i;i1 ...in

π / L 2 ( ε0 ) t

For, since each |sm | ≥ 1, we have: j − 1 + |s M | ≤

j 

|sm | = k

m=1

In the case of the first term in the outer sum in (10.221), namely the term corresponding ε to j = 1, we can be more precise. It is simply bounded in L 2 (t 0 ) by: L / R ik . . . L / R i2

(Ri1 )

π / L 2 ( ε0 ) t

We conclude that if δ0 is suitably small (depending on l) we have: L / R ik . . . L / R i1 h /−1  L 2 ( ε0 ) ≤ C

k−1 

t

n=0

max L / R in . . . L / R i1

i;i1 ...ik

: for all k = 1, . . . , l

(Ri )

π / L 2 ( ε0 ) t

(10.326)

Part 1: Control of the angular derivatives of the first derivatives of the x i

383

Substituting the estimate (10.324) we then obtain: L / R ik . . . L / R i1 h /−1  L 2 ( ε0 ) t

 Q −1 ≤ Cl δ0 (1 + t) [1 + log(1 + t)] (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1]  Q +Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] : for all k = 1, . . . , l

(10.327)

We turn to the coefficients m  and n i in (10.27). We first consider the angular derivatives of order up to l − 1 of the 2-covariant St,u tensorfield m b given by (10.29). Here we have four terms each of which, except the first, contains exactly one factor of order 1, which in the second term is η0 h/ , χ− 1 − u + η0 t in the third term is simply the term itself k/ , and in the fourth d/σ . In the first term we may take the factor h / in the role of the factor of order 1, although it is of order 0 according to our convention. Applying in a straightforward manner the general procedure outlined above we obtain the following estimates. In regard to the first term in (10.29), by (10.228) with l replaced by l∗ , (10.222) with l replaced by l∗ − 1, Lemma 10.1 applied to α −1 , and the estimate (10.327) with l replaced by l − 1, we obtain: max L / R ik . . . L / Ri1 {(α −1 − η0−1 )h/} L 2 ( ε0 ) t

 Q −1 ≤ Cl W[l−1] + δ0 (1 + t) Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−3] i1 ...ik

: for all k = 0, . . . , l − 1

(10.328)

In regard to the second term in (10.29) we obtain, using (10.327) with l replaced by l − 1 (and (10.226) with l∗ in the role of l), 0    0 0 0 η0 h/ −1 0 0 / R ik . . . L max 0L / R i1 α χ− 0 2 ε0 i1 ...ik 1 − u + η0 t L (t )

 Q −1 2 ≤ Cl A[l−1] + (1 + t) Y0 + W[l−1] + δ0 (1 + t)−2 W[l−3] : for all k = 0, . . . , l − 1

(10.329)

In regard to the fourth term we obtain: 0 0  0 0 dH 0 ψTˆ ψ ⊗ d/σ 0 / R i1 max 0L / R ik . . . L 0 2 ε0 i1 ...ik dσ L (t )

 Q 2 −3 −1 W[l] + δ0 (1 + t) Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2] ≤ Cl δ0 (1 + t) : for all k = 0, . . . , l − 1

(10.330)

384

Chapter 10. Control of the Angular Derivatives

Combining finally the results (10.328), (10.329), (10.330), with the result (10.317) for the third term, yields:

 Q / R ik . . . L / Ri1 m b  L 2 ( ε0 ) ≤ Cl (1 + t)−1 Y0 + (1 + t)A[l−1] + W[l] + δ02 W[l−1] max L i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.331)

We then consider the angular derivatives of order up to l − 1 of the T11 -type St,u tensorfield m  . These are expressed by (10.230), which derives from (10.28). In (10.28) the factor m b is of order 1 while the factor h/−1 is of order 0. Applying the general procedure outlined in the preceding to (10.230), using the L ∞ estimates (10.222) and (10.225) with l replaced by l∗ as well as the L 2 estimates (10.327) (with l replaced by l − 1) and (10.331), we conclude that:

 Q max L / R ik . . . L / Ri1 m   L 2 ( ε0 ) ≤ Cl (1 + t)−1 Y0 + (1 + t)A[l−1] + W[l] + δ02 W[l−1] i1 ...ik

t

: for all k = 0, . . . , l − 1

(10.332)

Next we consider the St,u -tangential vectorfields n i , given by (10.23). We first consider the angular derivatives of order up to l − 1 of ψ · h/−1 , expressed by (10.232). Taking the factor ψ in the role of the factor of order 1 in applying the general procedure (although it is in fact of order 0), and using the L ∞ estimates (10.222) and that of the third statement of Corollary 10.1.b with l replaced by l∗ (actually l∗ − 1 suffices) as well as the L 2 estimates (10.327) (with l replaced by l − 1) and (10.290), we obtain: / R ik . . . L / Ri1 ( ψ · h /−1 ) L 2 ( ε0 ) max L t

 Q ≤ Cl W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2] i1 ...ik

: for all k = 0, . . . , l − 1

(10.333)

On the other hand, revisiting the argument leading to the estimate (10.233) in the proof of Proposition 10.1, taking now into account the third statement of Corollary 10.1.b as well as (10.222), we can improve the L ∞ estimate (10.233) to: / R ik . . . L / Ri1 ( ψ · h/−1 ) L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 max L

i1 ...ik

t

: for all k = 0, . . . , l

(10.334)

under the assumptions of Proposition 10.1. Therefore under the assumptions of the present proposition the same estimate holds with l replaced by l∗ . Using this L ∞ estimate together with the L ∞ estimate (10.192) with l replaced by l∗ , as well as the L 2 estimates (10.333) and (10.311), in applying the general procedure to (10.23), we conclude that: / R ik . . . L / Ri1 n i  L 2 ( ε0 ) max L t

 Q −1 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2] ≤ Cl δ0 (1 + t)

i;i1 ...ik

: for all k = 0, . . . , l − 1

(10.335)

Part 1: Control of the angular derivatives of the first derivatives of the x i

385

Next we consider the St,u -tangential vectorfields wi given by (10.113), with the St,u 1-forms (wi )b given by (10.114). In (10.114) all three factors are of order 0, but we may take the factor d/x n in the role of the factor of order 1. Thus, in applying the general procedure to the expression (10.235) for the angular derivatives of order up to l − 1 of the (wi )b , we set s M = s3 . In treating Case 1 we follow the variant of the general procedure as in the argument leading to (10.321). That is, we bound d/(R)s3 x n pointwise as in (10.319) and we estimate the contribution of the first term in parenthesis by placing ε ε this term in L ∞ (t 0 ) using (10.320), placing the factor (R)s2 y j in L 2 (t 0 ) using the inductive hypothesis (10.275), noting that |s2 | ≤ l − 1, and placing the remaining factor ε / [l∗ ] . We then obtain that the contribution in (R)s1 h mn in L ∞ (t 0 ) using assumption E question is bounded by: 

Q Cl (1 + t) Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] To estimate the contribution of the second term in parentheses in the pointwise bound ε for d/(R)s3 x n , we follow the general procedure by placing this term in L 2 (t 0 ) using ε0 ∞ (10.277), and we place all other factors in L (t ) using Proposition 10.1 with l replaced / [l∗ ] . We then obtain that the contribution in question is bounded by: by l∗ and assumption E 

Q Cl δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] Combining the above results, noting that there is an overall factor of C(1 + t)−1 in (10.319), and then combining with the result for Case 2, yields: / R ik . . . L / Ri1 (wi )b  L 2 ( ε0 ) max L t

 Q ≤ Cl Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] i;i1 ...ik

; for all k = 0, . . . , l − 1

(10.336)

Using also the L 2 estimate (10.327) with l replaced by l − 1, as well as the L ∞ estimates (10.236) and (10.222) with l replaced by l∗ , we then obtain: max L / R ik . . . L / Ri1 wi  L 2 ( ε0 ) t

 Q ≤ Cl Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] i;i1 ...ik

; for all k = 0, . . . , l − 1

(10.337)

From the estimates (10.335) and (10.337) it follows that (see (10.132)): / R ik . . . L / Ri1 n i  L 2 ( ε0 ) max L t

 Q ≤ Cl δ0 (1 + t)−1 W[l] + Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] i;i1 ...ik

: for all k = 0, . . . , l − 1

(10.338)

386

Chapter 10. Control of the Angular Derivatives

We turn to the T11 -type St,u tensorfields µi and the St,u -tangential vectorfields νi j given by (10.134) and (10.135) respectively. Using the L 2 estimates (10.276), (10.332) and (10.338), and the L ∞ estimates given by the first statement of Corollary 10.1.a, (10.231) and (10.238), with l∗ in the role of l, we obtain, applying the general procedure outlined in the preceding in a straightforward manner, max L / R ik . . . L / Ri1 µi  L 2 ( ε0 )

i;i1 ...ik

t



Q ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] : for all k = 0, . . . , l − 1

(10.339)

and: / R ik . . . L / Ri1 νi j  L 2 ( ε0 ) max L

i, j ;i1 ...i j

t

≤ Cl δ0 [1 + log(1 + t)] δ0 (1 + t)−1 W[l] + Y0 + (1 + t)A[l−2]  Q +W[l−1] + δ02 (1 + t)−2 W[l−2] : for all k = 0, . . . , l − 1

(10.340)

The estimate (10.339) implies, in view of the expression (10.158) for the matrices Ni , max L / R ik . . . L / Ri1 Ni  L 2 ( ε0 )

i;i1 ...ik

t



Q ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] : for all k = 0, . . . , l − 1

(10.341)

In view of the structure of the matrices (k) Bi1 ...ik discussed in the paragraph preceding (10.163), this L 2 estimate together with the L ∞ estimate (10.245) with l∗ in the role of l yields: max 

i1 ...ik

(k)

Bi1 ...ik  L 2 ( ε0 ) t



Q ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] : for all k = 1, . . . , l

(10.342)

This is so because, in a term of degree q in (k) Bi1 ...ik there is at most one factor involving more than l∗ −1 angular derivatives (otherwise the term would contain at least 2l∗ ≥ l −1 angular derivatives contradicting the fact that it may contain at most k −q ≤ k −2 ≤ l −2 angular derivatives). Let us now consider expression (10.155) for an arbitrary triplet z of T11 -type St,u tensorfields and k = 0, . . . , l − 1. We have two cases to consider in regard to the terms in the sum in (10.155).

Part 1: Control of the angular derivatives of the first derivatives of the x i

Case 1: |s2 | ≤ l∗ In Case 1 the estimate:

Case 2: |s1 | ≤ l∗ − 1

and:

L∞

387

estimate (10.247) holds with l∗ in the role of l, hence we can

/ R )s1 z L 2 ( ε0 ) (M)s2  L ∞ ( ε0 ) (L / R )s1 z · (M)s2  L 2 ( ε0 ) ≤ (L t

t

≤C

l−1  m=0

t

max L / R im . . . L / Ri1 z L 2 ( ε0 )

i1 ...im

(10.343)

t

In Case 2 we express (M)s2 as in (10.163). We estimate the contribution of the first term in (10.163) by placing this term in L ∞ using (10.168): ( p)

(L / R )s 1 z ·

Iin1 ...in p  L 2 ( ε0 ) ≤ (L / R )s1 z L 2 ( ε0 )  t

( p)

t

≤C

l−1  m=0

Iin1 ...in p  L ∞ ( ε0 ) t

max L / R im . . . L / Ri1 z L 2 ( ε0 )

i1 ...im

(10.344)

t

We estimate the contribution of the second term in (10.163) by placing this term in L 2 using (10.342) with l replaced by l − 1, and placing the factor (L / R )s1 z in L ∞ : (L / R )s 1 z ·

( p)

Bin1 ...in p  L 2 ( ε0 ) ≤ (L / R )s1 z L ∞ ( ε0 )  t

t

( p)

Bin1 ...in p  L 2 ( ε0 ) t

≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] max max L / R im . . . L / Ri1 z L ∞ ( ε0 ) t m≤l∗ −1 i1 ...im

 Q 2 −2 · Y0 + (1 + t)A[l−2] + W[l−1] + δ0 (1 + t) W[l−2] (10.345) Putting together the above results we conclude that, for any triplet z of tensorfields:

T11 -type

of St,u

max Aik . . . Ai1 z L 2 ( ε0 )

i1 ...ik

≤ Cl

t

l−1  m=0

max L / R im . . . L / Ri1 z L 2 ( ε0 )

i1 ...im

t

+ Cl δ0 (1 + t)−1 [1 + log(1 + t)] max max L / R im . . . L / Ri1 z L ∞ ( ε0 ) t m≤l∗ −1 i1 ...im

 Q · Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] : for all k = 0, . . . , l − 1 From the expression for the triplets and (10.339) we obtain: / Rim+n . . . L / Rim+1 max L

(k) ρ

(10.346) j ;i1 ...ik

of Lemma 10.6 and the bounds (10.167)

(m)

ρ j ;i1 ...im  L ∞ ( ε0 ) t 

Q −1 ≤ Cl δ0 (1 + t) [1 + log(1 + t)] Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1]

i1 ...im+n

: for all m and all n ≤ l − 1

(10.347)

388

Chapter 10. Control of the Angular Derivatives

We then apply (10.346) to the triplet (m) ρ j ;i1 ...im using the L 2 bound (10.347) as well as the L ∞ bound (10.249) with l∗ in the role of l. This yields: Aim+n . . . Aim+1

(m)

ρ j ;i1 ...im  L 2 ( ε0 ) t 

Q −1 ≤ Cl δ0 (1 + t) [1 + log(1 + t)] Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] : for all m and all n ≤ l − 1

(10.348)

It then follows from the expression for the triplets these satisfy the following estimate: max 

j ;i1 ...ik

(k)

(k) β

j ;i1 ...ik

given by Lemma 10.6 that

β j ;i1 ...ik  L 2 ( ε0 ) t



Q ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] : for all k = 1, . . . , l

(10.349)

In a similar manner we obtain, more generally, the estimates: L / Rim+n . . . L / Rim+1

(m)

β j ;i1 ...im  L 2 ( ε0 ) t 

Q ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] max

j ;i1 ...im+n

: for all m and all n such that m + n ≤ l

(10.350)

We turn to consider the expression for the coefficients (k) γ j ;i1 ...ik given in the statement of Lemma 10.7. By (10.340) and the bounds (10.167) the first sum in this expression ε is bounded in L 2 (t 0 ) by:

Cl δ0 [1 + log(1 + t)] δ0 (1 + t)−1 W[l] + Y0 + (1 + t)A[l−2]  Q +W[l−1] + δ02 (1 + t)−2 W[l−2] : for all k = 1, . . . , l Consider an arbitrary term in the second sum:   / Rik−n+1 (k−1−n) β m · ν L / R ik . . . L i m k−n j ;i1 ...ik−1−n We can write this in the form:   (L / R )s 1

(k−1−n) m β j ;i1 ...ik−1−n

(10.351)

: n = 0, . . . , k − 1

(10.352)

   · (L / R )s2 νik−n m

partitions

|s1 | + |s2 | = n ≤ k − 1; k ≤ l

(10.353)

where the sum is over all ordered partitions {s1 , s2 } of the set {k − n + 1, . . . , k} into two ordered subsets s1 , s2 . We have two cases to consider.

Part 1: Control of the angular derivatives of the first derivatives of the x i

Case 1: |s2 | ≤ l∗ − 1

and:

389

Case 2: |s2 | ≥ l∗ ε

In Case 1 we can place the factor (L / R )s2 νik−n m in L ∞ (t 0 ) using the estimate / R )s1 (k−1−n) β m (10.240) with l∗ in the role of l. We then place the factor (L j ;i1 ...ik−1−n in ε

L 2 (t 0 ) using the estimate (10.350), which applies since

|s1 | + k − 1 − n ≤ k − 1 ≤ l − 1, as |s1 | ≤ n In fact the estimate (10.350) with l replaced by l − 1 suffices. This gives a bound in ε L 2 (t 0 ) by:

 Q Cl δ03 (1 + t)−2 [1 + log(1 + t)]3 Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] In Case 2, we have: |s1 | = n − |s2 | ≤ n − l∗ , hence: |s1 | + k − 1 − n ≤ k − 1 − l∗ ≤ l − 1 − l∗ ≤ l∗ Thus we can apply the estimate (10.253) with l∗ in the role of l to place the factor ε0 ε ∞ (L / R )s1 (k−1−n) β m / R )s2 νik−n m in L 2 (t 0 ) j ;i1 ...ik−1−n in L (t ). We then place the factor (L using the estimate (10.340), which applies since |s2 | ≤ n ≤ k − 1 ≤ l − 1. This gives a ε bound in L 2 (t 0 ) by:

Cl δ03 (1 + t)−2 [1 + log(1 + t)]3 δ0 (1 + t)−1 W[l] + Y0 + (1 + t)A[l−2]  Q +W[l−1] + δ02 (1 + t)−2 W[l−2] Combining the results for the two cases we conclude that the second sum in the expression ε for the coefficients (k) γ j ;i1 ...ik given by Lemma 10.6 is bounded in L 2 (t 0 ) by:

Cl δ03 (1 + t)−2 [1 + log(1 + t)]3 δ0 (1 + t)−1 W[l] + Y0 + (1 + t)A[l−2]  Q +W[l−1] + δ02 (1 + t)−2 W[l−2] : for all k = 1, . . . , l

(10.354)

Finally putting together the results (10.351), (10.354), we conclude that the coefficients (k) γ j ;i1 ...ik satisfy the estimates: max 

(k)

γ j ;i1 ...ik  L 2 ( ε0 ) t

≤ Cl δ0 [1 + log(1 + t)] δ0 (1 + t)−1 W[l] + Y0 + (1 + t)A[l−2]  Q +W[l−1] + δ02 (1 + t)−2 W[l−2] j ;i1 ...ik

: for all k = 1, . . . , l

(10.355)

We turn to the St,u -tangential vectorfields q/i , given by (10.27). We consider the angular derivatives of order up to l − 1. For the first term in (10.27) these are expressed by (10.257). Here we have two cases to consider.

390

Chapter 10. Control of the Angular Derivatives

Case 1: |s2 | ≤ l∗

and:

Case 2: |s1 | ≤ l∗ − 1

In Case 1 we appeal to the first statement of Corollary 10.1.e with l∗ in the role of l and to (10.332) to estimate: / R )s2 Ri ) L 2 ( ε0 ) ≤ (L / R )s1 m   L 2 ( ε0 ) (L / R )s2 Ri  L ∞ ( ε0 ) ((L / R )s1 m  ) · ((L t t t

 Q 2 −2 (10.356) ≤ Cl Y0 + (1 + t)A[l−1] + W[l] + δ0 (1 + t) W[l−1] In Case 2 we appeal to Lemma 10.6 to express, if s2 = {n 1 , . . . , n p }, p = |s2 | ≤ l − 1, (L / R )s 2 R i =

( p) m α j ;in ...in p 1

Rm +

( p) m β j ;in ...in p 1

· Rm +

( p)

γ j ;in1 ...in p

(10.357)

/ R )s 1 m  ) · To estimate the contribution of the first term in (10.357) to the L 2 norm of ((L ε0 ε0 s ∞ 2 ((L / R ) Ri ) on t , we place this term in L (t ) using (10.167), and the factor ε (L / R )s1 m  in L 2 (t 0 ) using (10.332). We then obtain that the contribution in question is bounded by:

 Q (10.358) Cl Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] To estimate the contribution of the second and third terms in (10.357) we place these ε terms in L 2 (t 0 ) using (10.349) and (10.355) respectively (replacing l by l − 1 here ε suffices), and we place the factor (L / R )s1 m  in L ∞ (t 0 ) using (10.231) with l∗ in the role of l. We then obtain that the contribution of the second term is bounded by:

 Q Cl δ02 (1 + t)−2 [1 + log(1 + t)]2 Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] (10.359) while the contribution of the third term is bounded by:

Cl δ02 (1 + t)−2 [1 + log(1 + t)]2 δ0 (1 + t)−1 W[l−1] + Y0 + (1 + t)A[l−3]  Q (10.360) +W[l−2] + δ02 (1 + t)−2 W[l−3] Combining the results (10.358), (10.359), (10.360), we conclude that in Case 2: ((L / R )s1 m  ) · ((L / R )s2 Ri ) L 2 ( ε0 ) t

 Q ≤ Cl Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2]

(10.361)

Combining this in turn with the result (10.356) for Case 1, we conclude that: max L / R ik . . . L / Ri1 (m  · Ri ) L 2 ( ε0 ) t

 Q ≤ Cl Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] i;i1 ...ik

: for all k = 0, . . . , l − 1

(10.362)

Part 1: Control of the angular derivatives of the first derivatives of the x i

391

We then combine the estimates (10.362) and (10.335) to obtain: max L / R ik . . . L / Ri1 q/i  L 2 ( ε0 ) t

 Q ≤ Cl Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] i;i1 ...ik

: for all k = 0, . . . , l − 1

(10.363)

We finally consider the coefficients (k) p/ii1 ...ik , (k) q/ii1 ...ik , and (k)r/ii1 ...ik of the expression for Rik . . . Ri1 Ri y j given by Lemma 10.4. First, from the expression for the coefficients (k) p/ii1 ...ik given by Lemma 10.4, using the L 2 estimate (10.312) together with the L ∞ estimate (10.194) with l∗ in the role of l, we readily deduce: j

max 

(k)

p/ii1 ...ik  L 2 ( ε0 ) t

 Q ≤ Cl δ0 (1 + t)−1 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2]

ii1 ...ik

: for all k = 0, . . . , l − 1

(10.364)

Moreover, max Rim+n . . . Rim+1 (m) p/ii1 ...im  L 2 ( ε0 ) t

 Q ≤ Cl δ0 (1 + t)−1 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2]

ii1 ...im+n

: for all m and all n such that m + n ≤ l − 1

(10.365)

Next, we consider the expression for the coefficients (k) q/ii1 ...ik given by Lemma ε 10.4. The first term in this expression is estimated in L 2 (t 0 ) by (10.363). In reference to the second term we have expression (10.262). Here, |s1 | + |s2 | = m ≤ k − 1,

k ≤l −1

We have two cases to consider. Case 1: |s2 | ≤ l∗ − 1

and:

Case 2: |s2 | ≥ l∗ ε

In Case 1 we can place the factor (L / R )s2 q/ik−m in L ∞ (t 0 ) using the estimate (10.259) with l∗ in the role of l. We then place the factor (R)s1 (k−m−1) p/ii1 ...ik−m−1 in ε L 2 (t 0 ) using the estimate (10.365), which applies since |s1 | + k − m − 1 ≤ k − 1 ≤ l − 2, as |s1 | ≤ m In fact the estimate (10.365) with l replaced by l − 1 suffices. This gives a bound in ε L 2 (t 0 ) by: Cl δ02 (1 + t)−2 [1 + log(1 + t)]

 Q × W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−3] + δ02 (1 + t)−2 W[l−3]

392

Chapter 10. Control of the Angular Derivatives

In Case 2 we have: |s1 | = m − |s2 | ≤ m − l∗ , hence: |s1 | + k − m − 1 ≤ k − 1 − l∗ ≤ l − 2 − l∗ ≤ l∗ − 1 Thus we can apply the estimate (10.261) with l∗ in the role of l to place the factor ε ε / R )s2 q/ik−m in L 2 (t 0 ) (R)s1 (k−m−1) p/ii1 ...ik−m−1 in L ∞ (t 0 ). We then place the factor (L using the estimate (10.363), which applies since |s2 | ≤ m ≤ k − 1 ≤ l − 2. In fact, ε replacing l by l − 1 suffices. This gives a bound in L 2 (t 0 ) by:

 Q Cl δ02 (1 + t)−2 Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] Combining the results for the two cases we conclude that: 0 k−1 0 0 0 0 0 (k−m−1)  L / R ik . . . L / Rik−m+1 ( p/ii1 ...ik−m−1 q/ik−m )0 0 0 0 ε m=0 L 2 (t 0 )

 Q ≤ Cl δ02 (1 + t)−2 Y0 + (1 + t)A[l−2] + [1 + log(1 + t)]W[l−1] + δ02 (1 + t)−2 W[l−2] : for all k ≤ l − 1

(10.366)

In reference to the second term in the expression for also have expression (10.264). Here, again |s1 | + |s2 | = m ≤ k − 1,

(k) q /ii1 ...ik

of Lemma 10.4, we

k ≤l −1

and we have two cases to consider. Case 1: |s2 | ≤ l∗ − 1

and:

Case 2: |s2 | ≥ l∗

In Case 1 we appeal to the first statement of Corollary 10.1.e with l∗ in the role of l (actually l∗ − 1 suffices) and to (10.365) (with l replaced by l − 1) to estimate: ((R)s1

(k−m−1)

p/ii1 ...ik−m−1 )((L / R )s2 Rik−m ) L 2 ( ε0 ) t

(10.367)

/ R )s2 Rik−m  L ∞ ( ε0 ) ≤ (L / R )s1 (k−m−1) p/ii1 ...ik−m−1  L 2 ( ε0 ) (L t t

 Q −1 2 Y0 + (1 + t)A[l−3] + δ0 (1 + t)−2 W[l−3] ≤ Cl δ0 W[l−1] + δ0 (1 + t) In Case 2 we express (L / R )s2 Rik−m as in (10.357). To estimate the contribution of the ε first term in (10.357) to the L 2 norm of ((R)s1 (k−m−1) p/ii1 ...ik−m−1 )((L / R )s2 Rik−m ) on t 0 , ε0 ∞ s (k−m−1) we place this term in L (t ) using (10.167), and the factor (R) 1 p/ii1 ...ik−m−1 in ε0 2 L (t ) using (10.365) with l replaced by l−1, noting that |s1 |+k−m−1 ≤ k−1 ≤ l−2. We then obtain that the contribution in question is bounded by:

 Q (10.368) Cl δ0 W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−3] + δ02 (1 + t)−2 W[l−3] To estimate the contribution of the second and third terms in (10.357) we place these terms ε in L 2 (t 0 ) using (10.349) and (10.355) respectively. Here, since |s2 | ≤ m ≤ k − 1 ≤

Part 1: Control of the angular derivatives of the first derivatives of the x i

l − 2 replacing l by l − 2 suffices. We then place the factor (R)s1 ε L ∞ (t 0 ) using (10.261) with l∗ in the role of l, noting that:

393 (k−m−1) p /ii1 ...ik−m−1

in

|s1 | + k − m − 1 = k − 1 − |s2 | ≤ l − 2 − l∗ ≤ l∗ − 1 We then obtain that the contribution of the second term is bounded by: 

Q Cl δ03 (1 + t)−2 [1 + log(1 + t)] Y0 + (1 + t)A[l−3] + W[l−2] + δ02 (1 + t)−2 W[l−3] (10.369) while the contribution of the third term is bounded by:

Cl δ03 (1 + t)−2 [1 + log(1 + t)] δ0 (1 + t)−1 W[l−2] + Y0 + (1 + t)A[l−4]  Q +W[l−3] + δ02 (1 + t)−2 W[l−4] (10.370) Combining the results (10.368), (10.369), (10.370), we conclude that in Case 2: ((R)s1 (k−m−1) p/ii1 ...ik−m−1 )((L / R )s2 Rik−m ) L 2 ( ε0 ) (10.371) t

 Q ≤ Cl δ0 W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−3] + W[l−2] + δ02 (1 + t)−2 W[l−3] Combining the results of the two cases, (10.367) and (10.371), we conclude that: 0 0 k−1 0 0 0 0 (k−m−1) L / R ik . . . L / Rik−m+1 ( p/ii1 ...ik−m−1 Rik−m )0 0 0 0 ε m=0 L 2 (t 0 )

 Q ≤ Cl δ0 W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−3] + δ02 (1 + t)−2 W[l−3] : for all k ≤ l − 1

(10.372)

The estimates (10.366) and (10.372) control the sum which constitutes the second term in the expression for q/ii1 ...ik of Lemma 10.4. Combining these estimates with the estimate for the first term provided by (10.363) we conclude that: max  (k) q/ii1 ...ik  L 2 ( ε0 ) t

 Q ≤ Cl Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] ii1 ...i

: for all k = 0, . . . , l − 1

(10.373)

In a similar way we obtain more generally: max L / Rim+n . . . L / Rim+1 (m) q/ii1 ...im  L 2 ( ε0 ) t

 Q ≤ Cl Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] ii1 ...im+n

: for all m and all n such that m + n ≤ l − 1

(10.374)

394

Chapter 10. Control of the Angular Derivatives

Finally, we consider the expression for the coefficients (k)r/ii1 ...ik given by Lemma 10.4. A given term in the sum is expressed by (10.268). Here, again j

|s1 | + |s2 | = m ≤ k − 1,

k ≤l −1

and as above we have two cases to consider. Case 1: |s2 | ≤ l∗ − 1

and:

Case 2: |s2 | ≥ l∗

In Case 1 we appeal to the last statement of Corollary 10.1.a with l∗ in the role of l, which together with hypothesis H0 implies: d/(R)s2 Rik−m x j  L ∞ ( ε0 ) ≤ C

(10.375)

t

Since also |s1 | + k − m − 1 ≤ k − 1 ≤ l − 2 we can estimate, using (10.374) (we may replace l by l − 1), ((L / R )s 1 ≤

(k−m−1)  q/ii1 ...ik−m−1 )

· d/((R)s2 Rik−m x j ) L 2 ( ε0 ) t

(10.376)

(L / R )s1 (k−m−1) q/ii1 ...ik−m−1  L 2 ( ε0 ) d/(R)s2 Rik−m x j  L ∞ ( ε0 ) t t

 Q ≤ Cl Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2]

In Case 2 we first note that if s2 = {n 1 , . . . , n p }, p = |s2 |, then as in (10.286) we have the pointwise bound: |d/(R)s2 Rik−m x j | ≤ C(1 + t)−1

3  | q=1

j ( p+2) j x ik−m in ...in q | + | ( p+2) δik−m in ...in q | p p 1 1



(10.377) To estimate the contribution of the first term in parentheses in (10.377) we place this term ε ε / R )s1 (k−m−1) q/ii1 ...ik−m−1 in L 2 (t 0 ) using in L ∞ (t 0 ) (see (10.320)) and the factor (L (10.374) (we may replace l by l − 1). We then obtain that the contribution in question is bounded by:

 Q Cl (1 + t) Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] To estimate the contribution of the second term in parenthesis in (10.377) we may appeal ε to (10.277) to place this term in L 2 (t 0 ), since p + 2 = |s2 | + 2 ≤ m + 2 ≤ k + 1, We then place the factor (L / R )s 1 the role of l, noting that:

(k−m−1) q /ii1 ...ik−m−1

k ≤l −1 ε

in L ∞ (t 0 ) using (10.267) with l∗ in

|s1 | + k − m − 1 = k − |s2 | − 1 ≤ k − l∗ − 1 ≤ l − l∗ − 2 ≤ l∗ − 1

Part 1: Control of the angular derivatives of the first derivatives of the x i

395

We then obtain that the contribution in question is bounded by: 

Q Cl δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] Combining the above results and noting that in (10.377) there is an overall factor C(1 + t)−1 we conclude that in Case 2: ((L / R )s 1

(k−m−1)  q/ii1 ...ik−m−1 ) · d/((R)s2 Rik−m x j ) L 2 ( ε0 ) t

 Q ≤ Cl Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] (10.378)

Combining the results of the two cases, (10.376) and (10.377) we conclude through exj pression (10.268) and the expression for (k)r/ii1 ...ik of Lemma 10.4 that: max  (k)r/ii1 ...ik  L 2 ( ε0 ) t

 Q ≤ Cl Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] j

ii1 ...i

: for all k = 0, . . . , l − 1

(10.379)

In conclusion, we have obtained the L 2 estimates (10.364), (10.373), (10.379), for the coefficients of the expansion of Rik . . . Ri1 Ri y j given by Lemma 10.4. Since these estimates hold for all k ≤ l − 1, in particular for k = l − 1, it follows that: max Ril−1 . . . Ri1 Ri y j  L 2 ( ε0 ) t

 Q ≤ Cl Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1]

Yl =

j ;ii1 ...il−1

(10.380)

This completes the inductive step and the proof of Proposition 10.2. The foregoing proposition has a number of corollaries analogous to those of Proposition 10.1. First, in conjunction with Lemma 10.3 we have: Corollary 10.2.a Under the assumptions of Proposition 10.2 we have: max Rik . . . Ri1 λi  L 2 ( ε0 ) t 

Q ≤ Cl (1 + t) Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] i;i1 ...ik

: for all k = 0, . . . , l and: max 

j ;ii1 ...ik

(k+1) j δii1 ...ik  L 2 ( ε0 ) t



Q ≤ Cl (1 + t) Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] : for all k = 0, . . . , l

396

Chapter 10. Control of the Angular Derivatives

Next, from the proof of Proposition 10.2 we have: Corollary 10.2.b Under the assumptions of Proposition 10.2 we have: max Rik . . . Ri1 ψTˆ  L 2 ( ε0 ) t

 Q ≤ Cl W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1]

i1 ...ik

: for all k = 0, . . . , l max L / R ik . . . L / Ri1 ψ L 2 ( ε0 ) t

 Q ≤ Cl W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1]

i1 ...ik

: for all k = 0, . . . , l and: max L / R ik . . . L / R i1 ω / L 2 ( ε0 ) t

 Q ≤ Cl (1 + t)−1 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2]

i1 ...ik

: for all k = 0, . . . , l − 1 Corollary 10.2.c Under the assumptions of Proposition 10.2 we have: / R ik . . . L / Ri1 k/  L 2 ( ε0 ) max L t

1 2 Q ≤ Cl (1 + t)−1 W[l] + δ02 (1 + t)−2 W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2]

i1 ...ik

: for all k = 0, . . . , l − 1 Corollary 10.2.d Under the assumptions of Proposition 10.2 we have: / R ik . . . L / Ri1 (Ri ) π / L 2 ( ε0 ) max L t

 Q −1 ≤ Cl δ0 (1 + t) [1 + log(1 + t)] (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1]  Q +Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] i;i1 ...ik

: for all k = 0, . . . , l − 1 Corollary 10.2.e Under the assumptions of Proposition 10.2, the coefficients of the ex/ Ri1 R j of Lemma 10.6 satisfy: pression for L / R ik . . . L max 

j ;i1 ...ik

(k)

β j ;i1 ...ik  L 2 ( ε0 )

≤ Cl δ0 (1 + t)

t

−1



Q [1 + log(1 + t)] Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1]

: for all k = 1, . . . , l

Part 1: Control of the angular derivatives of the first derivatives of the x i

max 

397

(k)

γ j ;i1 ...ik  L 2 ( ε0 ) t

≤ Cl δ0 [1 + log(1 + t)] δ0 (1 + t)−1 W[l] + Y0 + (1 + t)A[l−2]  Q +W[l−1] + δ02 (1 + t)−2 W[l−2] j ;i1 ...ik

: for all k = 1, . . . , l Finally we have: Corollary 10.2.f Under the assumptions of Proposition 10.2, the coefficients of the expansion of Rik . . . Ri1 Ri y j of Lemma 10.4 satisfy: max 

(k)

p/ii1 ...ik  L 2 ( ε0 ) t

 Q ≤ Cl δ0 (1 + t)−1 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2]

ii1 ...ik

: for all k = 0, . . . , l − 1 max  (k) q/ii1 ...ik  L 2 ( ε0 ) t

 Q ≤ Cl Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1]

ii1 ...i

: for all k = 0, . . . , l − 1 and: max  (k)r/ii1 ...ik  L 2 ( ε0 ) t

 Q ≤ Cl Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] j

ii1 ...i

: for all k = 0, . . . , l − 1 We shall now estimate the angular derivatives of the functions L i , the components of the vectorfield L in rectangular coordinates (recall that L 0 = 1). After this we shall proceed with the estimates for the St,u 1-form κ −1 ζ  and the St,u -tangential vectorfields (Ri ) Z . According to equations (6.47), (6.48): Lj =

η0 x j + zj 1 − u + η0 t

where:

(10.381)

H ψ0 ψ j (α − η0 )x j + (10.382) 1 − u + η0 t 1 + ρH As a corollary of Proposition 10.1 we then have, in connection with the first two statements of Corollary 10.1.b, the following. Let us recall the St,u 1-form ω / L defined by (4.90): (10.383) ω / L = L µ d/ψµ z j = −αy j +

398

Chapter 10. Control of the Angular Derivatives

as well as the functions ω L L and ω L Tˆ defined by (4.92) and (4.93) respectively: ω L L = L µ (Lψµ )

(10.384)

ˆi

(10.385)

ω L Tˆ = T (Lψi ) Corollary 10.1.g Under the assumptions of Proposition 10.1 we have: max Rik . . . Ri1 z j  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

j ;i1 ...ik

t

: for all k = 0, . . . , l, max Rik . . . Ri1 L j  L ∞ ( ε0 ) ≤ C

j ;i1 ...ik

t

: for all k = 0, . . . , l Moreover, max Rik . . . Ri1 ψ L  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1

i1 ...ik

t

: for all k = 1, . . . , l and: max L / R ik . . . L / R i1 ω / L  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2

i1 ...ik

t

: for all k = 0, . . . , l − 1 max L / R ik . . . L / Ri1 ω L L  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2

i1 ...ik

t

: for all k = 0, . . . , l − 1 / R ik . . . L / Ri1 ω L Tˆ  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 max L

i1 ...ik

t

: for all k = 0, . . . , l − 1 Also, as a corollary of Proposition 10.2 we have, in connection with the first statement of Corollary 10.2.b: Corollary 10.2.g Under the assumptions of Proposition 10.2 we have: max Rik . . . Ri1 z j  L 2 ( ε0 ) t

 Q ≤ Cl Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] j ;i1 ...ik

: for all k = 0, . . . , l

Part 1: Control of the angular derivatives of the first derivatives of the x i

399

Moreover, max Rik . . . Ri1 ψ L  L 2 ( ε0 ) t

 Q ≤ Cl W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1]

i1 ...ik

: for all k = 1, . . . , l and: max L / R ik . . . L / R i1 ω / L  L 2 ( ε0 ) t

 Q ≤ Cl (1 + t)−1 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2]

i1 ...ik

: for all k = 0, . . . , l − 1 max L / R ik . . . L / Ri1 ω L L  L 2 ( ε0 ) t

1 2 Q −1 W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + W[l−1] ≤ Cl (1 + t)

i1 ...ik

: for all k = 0, . . . , l − 1 max L / R ik . . . L / Ri1 ω L Tˆ  L 2 ( ε0 ) t

1 2 Q −1 W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + W[l−1] ≤ Cl (1 + t)

i1 ...ik

: for all k = 0, . . . , l − 1 We turn to the St,u 1-form κ −1 ζ  . This is given by equation (3.175): κ −1 ζ  =

 1 dH ψTˆ ψ(Lσ ) + (ψTˆ + α −1 ψ L )ψ L (d/σ ) + H (ψTˆ + α −1 ψ L )ω /L 2 dσ (10.386) ε

We first consider, under the assumptions of Proposition 10.1, L ∞ estimates on t 0 of the angular derivatives of κ −1 ζ  up to order l − 1. Using Corollaries 10.1.b, 10.1.g, and the estimates (10.170) and (10.172), we readily deduce: Corollary 10.1.h Under the assumptions of Proposition 10.1 we have: / R ik . . . L / Ri1 (κ −1 ζ  ) L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 max L

i1 ...ik

t

for all k = 0, . . . , l − 1 ε

We consider next, under the assumptions of Proposition 10.2, L 2 estimates on t 0 of the angular derivatives of κ −1 ζ  up to order l − 1. We follow the general procedure outlined in the paragraph after the estimate (10.313). In the term 1 dH ψ ˆ ψ(Lσ ) 2 dσ T

400

Chapter 10. Control of the Angular Derivatives

in (10.386), the factor of order 1 is Lσ . We appeal to Corollary 10.1.b with l replaced by l∗ and the estimate (10.313) to handle Case 1, the estimate (10.172) with l replaced by l∗ and Corollary 10.2.b with l reduced to l − 1 to handle Case 2. We then obtain that the ε angular derivatives of order up to l −1 of the term in question are bounded in L 2 (t 0 ) by:

1 2 Q Cl δ02 (1 + t)−3 W[l−1] + W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] In the term

1 dH (ψ ˆ + α −1 ψ L )ψ L (d/σ ) 2 dσ T in (10.386), the factor of order 1 is d/σ . We appeal to Corollaries 10.1.b and 10.1.g with l replaced by l∗ and Lemma 10.1 together with hypothesis H0 to handle Case 1, the estimate (10.170) with l replaced by l∗ and Corollaries 10.2.b and 10.2.g with l reduced to l − 1 to handle Case 2. We then obtain that the angular derivatives of order up to l − 1 ε of the term in question are bounded in L 2 (t 0 ) by:

 Q Cl (1 + t)−1 W[l] + δ02 (1 + t)−2 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2] In the last term in (10.386) the factor of order 1 is d/ψµ . We appeal to Corollaries 10.1.b and 10.1.g with l replaced by l∗ and to hypothesis H0 to handle Case 1, Corollaries 10.2.b and 10.2.g with l reduced to l − 1 to handle Case 2. In the subcase of Case 2 where the maximal number of angular derivatives falls on the factor L µ , we follow the variant of the general procedure. That is, we express through (10.381), if sm = {n 1 , . . . , n p }, p = |sm |, (see (10.86)): (R) L = sm

j

η0

( p) x j in 1 ...in p

1 − u + η0 t

−

η0

( p) δ j in 1 ...in p

1 − u + η0 t

+ (R)sm z j

(10.387)

and we estimate the contribution of the first term in (10.387) by placing this term in ε L ∞ (t 0 ) and the factor corresponding to the factor of order 1 (namely to d/ψµ ) in ε0 2 L (t ). To estimate the contribution of the other two terms in (10.387) we place these ε terms in L 2 (t 0 ) using Corollary 10.2.a with l reduced to l − 2 and Corollary 10.2.g with l reduced to l − 1. In this way we obtain that the angular derivatives of order up to l − 1 ε of the last term in (10.386) are bounded in L 2 (t 0 ) by:

 Q Cl (1 + t)−1 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2] Combining the above results we conclude: Corollary 10.2.h Under the assumptions of Proposition 10.2 we have: max L / R ik . . . L / Ri1 (κ −1 ζ  ) L 2 ( ε0 ) t

1 2 Q −1 2 W[l] + δ0 (1 + t)−2 W[l−1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] ≤ Cl (1 + t)

i1 ...ik

: for all k = 0, . . . , l − 1

Part 1: Control of the angular derivatives of the first derivatives of the x i

401

We turn to the St,u -tangential vectorfields (Ri ) Z . These are related to the St,u 1/ L by: forms (Ri ) π (Ri ) Z = (Ri ) π / L · h/−1 (10.388) According to equation (6.49):   / η0 h (Ri ) · Ri + i j m z j h mn d/x n + λi (κ −1 ζ  ) π /L = − χ − 1 − u + η0 t  1 dH i j m x j ψm ψ(Lσ ) + ψ L ψ(Ri σ ) − i j m x j ψm ψ L (d/σ ) + 2 dσ +H i j m x j ψ(Lψm ) (10.389) Using the corollaries of Proposition 10.1 we readily deduce: Corollary 10.1.i Under the assumptions of Proposition 10.1 we have: max L / R ik . . . L / R i1

i;i1 ...ik

(Ri )

π / L  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

: for all k = 0, . . . , l − 1 and: max L / R ik . . . L / R i1

i;i1 ...ik

(Ri )

Z  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

: for all k = 0, . . . , l − 1 Also, using the corollaries of Proposition 10.1 with l∗ in the role of l, as well as the corollaries of Proposition 10.2 we deduce: Corollary 10.2.i Under the assumptions of Proposition 10.2 we have: max L / R ik . . . L / Ri1 (Ri ) π / L  L 2 ( ε0 ) t

≤ Cl Y0 + (1 + t)A[l−1] + δ0 (1 + t)−1 [1 + log(1 + t)]W[l]  Q +W[l−1] + δ0 (1 + t)−1 W[l−1]

i;i1 ...ik

for all k = 0, . . . , l − 1 and: max L / R ik . . . L / Ri1 (Ri ) Z  L 2 ( ε0 ) t

≤ Cl Y0 + (1 + t)A[l−1] + δ0 (1 + t)−1 [1 + log(1 + t)]W[l]  Q +W[l−1] + δ0 (1 + t)−1 W[l−1]

i;i1 ...ik

for all k = 0, . . . , l − 1

402

Chapter 10. Control of the Angular Derivatives

Proof. Consider the St,u 1-forms (Ri ) π / L as expressed by (10.389). We apply the general procedure. In the first term in (10.389) the factor of order 1 is: χ−

η0 h/ 1 − u + η0 t

In Case 1 we appeal to Corollary 10.1.e with l∗ in the role of l and to Corollary 10.2.d with l reduced to l − 1. In Case 2 we express (L / R )s2 Ri as in (10.357) and we follow the ε variant of the general procedure in placing the term ( p) α mj;in ...in Rm in L ∞ (t 0 ) and the 1

p

ε

other factor, which corresponds to the factor of order 1, in L 2 (t 0 ). Using also Corollary 10.2.e with l reduced to l − 1 we obtain that the angular derivatives of order up to l − 1 ε of the first term in (10.389) are bounded in L 2 (t 0 ) by:

 Q Cl Y0 + (1 + t)A[l−1] + W[l−1] + δ02 (1 + t)−2 W[l−2] In the second term in (10.389) all factors are of order 0. We may take the factor d/x n in the role of the factor of order 1. In Case 1 we apply the variant of the general procedure as in the argument leading to the estimate (10.321), using a pointwise bound of the form (10.319). Using, in Case 1, Corollary 10.1.g with l∗ in the role of l and Corollaries 10.2.g and 10.2.a with l reduced to l − 1, and, in Case 2, Corollary 10.1.a with l∗ in the role of l and Corollary 10.2.g with l reduced to l − 1, we obtain that the angular derivatives of ε order up to l − 1 of the second term in (10.389) are bounded in L 2 (t 0 ) by:

 Q Cl Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] In the third term in (10.389) the factor of order 1 is κ −1 ζ  . Applying Corollary 10.1.a with l∗ in the role of l and Corollary 10.2.h, in Case 1, and Corollary 10.1.h with l∗ in the role of l and Corollary 10.2.a with l reduced to l − 1, in Case 2, we obtain that the angular ε derivatives of order up to l − 1 of the third term in (10.389) are bounded in L 2 (t 0 ) by:

  Q Cl δ0 (1 + t)−1 [1 + log(1 + t)] W[l] + δ02 (1 + t)−2 W[l−1] + Y0 + (1 + t)A[l−2] The angular derivatives of order up to l −1 of the remaining terms in (10.389) are straightε forward to estimate in L 2 (t 0 ), under the assumptions of Proposition 10.2, by the same methods. Combining the results yields the first statement of the present corollary. The second statement then follows through equation (10.388) from the first, combined with the estimate (10.327) with l reduced to l − 1 and with the first statement of Corollary 10.1.i with l∗ in the role of l and with the estimate (10.222) with l∗ in the role of l.

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

Part 2: Bounds for the quantities

403

(i 1 ...i l )

Q l and

(i 1 ...i l )

Pl

Our objective in the remainder of the present chapter is to obtain appropriate bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl , which, through (i1 ...il ) Bl , enter the final estimates of Chapter 8 for the 1-forms (i1 ...il ) xl . We begin with the investigation of the St,u 1-form i which appears in the Codazzi equation (8.104) and is given by (8.105). We have: i=

5 

(q)

(10.390)

i

q=1 ( P)∗

(1)

Here i is the principal part of i , the first term of β as given by (4.75): (1)

(1,1)

(1,2)

i= i + i

where:

(1,1)

i =

and:

(1,2)

i =

(10.391)



1 dH ψL D / σ ) ψ / 2 σ · ( ψ · h/−1 ) − (
2 dσ

(10.392)

 1 dH ψ( ψ · h/−1 ) · d/ Lσ − | ψ|2 d/ Lσ 2 dσ

(10.393)

(2)

In (10.390) i is the slowest decaying part of i :  (2) η0 1 dH −1 −1  −1 2 [ ψ( ψ · h/ ) · d/σ − | ψ| d/σ ] α (κ ζ ) − i= 1 − u + η0 t 2 dσ

(10.394)

(3)

and i is the leading acoustical part of i :      (3) 2η0 η0 h / −1 −1 = −α − trχ − χ − · / h I · (κ −1 ζ  ) i 1 − u + η0 t 1 − u + η0 t      1 dH η0 h/ − χ− ψ( ψ · h /−1 ) · · h/−1 · d/σ 2 dσ 1 − u + η0 t    η0 h/ · h/−1 · d/σ χ− (10.395) −| ψ|2 1 − u + η0 t (4)

(5)

(4)

( P)∗

Finally, in (10.390), i and i are lower order terms; i is the second term of β as given by (4.78): (4)

i =

1 d H −1 −ψ L [( ψ · h/−1 ) · k/ − ψtrk/ ] α 2 dσ

 + ψ( ψ · h /−1 ) · (κ −1 ζ  ) − | ψ|2 (κ −1 ζ  ) (Lσ )

(10.396)

404

Chapter 10. Control of the Angular Derivatives

and (see (4.86)):

(N)∗ ( A)∗ 1 (B)∗ 1 (C)∗ i = β = −F β − F2 β − F1 β 2 2 where (see (4.97)–(4.99)): (5)

( A)∗

β = (ω / L − (trω /)ω /L /·h /−1 ) · ω

(10.397)

(10.398)

(B)∗

β = (( ψ(Lσ ) − ψ L d/σ ) · h/−1 ) · ( ψ ⊗ d/σ − d/σ ⊗ ψ)

(10.399)

(C)∗

β = (Lσ )[ω /) ψ] / · ( ψ · h/−1 ) − (trω −1 /)ψ L ] +(d/σ )[ω / L · ( ψ · h/ ) − (trω +(d/σ · h /−1 ) · [ω /ψ L − 2 ψ ⊗ ω /L + ω / L ⊗ ψ]

(10.400)

In the above formulas all factors of h /−1 are written out explicitly, so that only pure contractions appear, which are denoted by a dot. The only exception is | ψ|2 which when written out explicitly reads: (10.401) | ψ|2 = ψ · h/−1 · ψ In (10.395) we have denoted by I the identity transformation in the tangent space to St,u at each point, as a T11 -type St,u tensorfield. In the next two lemmas we estimate the angular derivatives of the non-principal part of i . Lemma 10.7 Let hypothesis H0 and the estimate (10.30) hold. Let also the bootstrap Q / [l] and X / [l] hold, for some non-negative integer l. Then we have: assumptions E / [l+1] , E 0 0  0 0 5  (q) 0 0 0 / R ik . . . L max L / R i1  ≤Cl δ0 (1 + t)−3 : for all k = 0, . . . , l i 0 0 i1 ...ik 0 0 0 ε0 q=2 ∞ L (t )

provided that δ0 is suitably small (depending on l). Proof. The assumptions of the present lemma coincide with those of Proposition 10.1 with l + 1 in the role of l. Thus, Proposition 10.1 and all its corollaries hold with l + 1 in the role of l. (5)

Consider first i . By Corollaries 10.1.b and 10.1.g, with l replaced by l +1, together with the estimate (10.222) we obtain: ( A)∗

max L / R ik . . . L / Ri1 β  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−4

i1 ...ik

t

: for all k = 0, . . . , l

(10.402)

(B)∗

max L / R ik . . . L / Ri1 β  L ∞ ( ε0 ) ≤ Cl δ03 (1 + t)−5

i1 ...ik

t

: for all k = 0, . . . , l

(10.403)

(C)∗

/ R ik . . . L / Ri1 β  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−4 max L

i1 ...ik

t

: for all k = 0, . . . , l

(10.404)

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

405

Combining the above results yields: (5)

/ R ik . . . L / Ri1 i  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−4 max L t

i1 ...ik

: for all k = 0, . . . , l

(10.405)

Next, using Corollaries 10.1.c and 10.1.h, with l replaced by l + 1, together with the estimate (10.222) and Corollary 10.1.g, we obtain: (4)

/ R ik . . . L / Ri1 i  L ∞ ( ε0 ) ≤ Cl δ03 (1 + t)−5 max L t

i1 ...ik

: for all k = 0, . . . , l

(10.406)

(3)

We turn to i . By Corollary 10.1.d and the estimate (10.222): 0   0 0 0 η0 h/ −1 0 0 / R ik . . . L · h/ χ− max 0L / R i1 0 ∞ ε0 i1 ...ik 1 − u + η0 t L (t ) ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] : for all k = 0, . . . , l

(10.407)

Using also Corollary 10.1.h with l replaced by l +1 (and Corollary 10.1.b) we then obtain: (3)

/ R ik . . . L / Ri1 i  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)] max L t

i1 ...ik

: for all k = 0, . . . , l

(10.408)

Finally, using again Corollary 10.1.h with l replaced by l + 1 (and Corollary 10.1.b) we obtain: (2)

max L / R ik . . . L / Ri1 i  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−3 t

i1 ...ik

; for all k = 0, . . . , l

(10.409)

Combining the above results (10.405), (10.406), (10.408), (10.409), the lemma follows. Lemma 10.8 Let the hypothesis H0 and the estimate (10.30) hold. Let l be a positive Q integer and let the bootstrap assumptions E / [l∗ +1] , E / [l∗ ] and X / [l∗ ] hold. Then we have: 0 0  0 0 5  (q) 0 0 0   / R ik . . . L / R i1 max L i 0 0 i1 ...ik 0 0 0 2 ε0 q=2 L (t )

1 2 Q −2 2 ≤ Cl (1 + t) + δ0 (1 + t)−1 Y0 + (1 + t)A[l] W[l+1] + δ0 W[l] : for all k = 0, . . . , l provided that δ0 is suitably small (depending on l).

406

Chapter 10. Control of the Angular Derivatives

Proof. The assumptions of the present lemma coincide with those of Proposition 10.1 with l∗ + 1 in the role of l. Thus, Proposition 10.1 and all its corollaries hold with l∗ + 1 in the role of l. Moreover, the assumptions of the present lemma coincide with those of Proposition 10.2 with l∗ + 1 in the role of l∗ . Since (l + 1)∗ ≤ l∗ + 1, it follows that the assumptions of the present lemma imply those of Proposition 10.2 with l + 1 in the role of l. Thus, Proposition 10.2 and all its corollaries hold with l + 1 in the role of l. (5)

We begin again with i . Using the L 2 estimate of the third statement of Corollary 10.2.b with l replaced by l + 1, the L 2 estimate (10.327), which may be simplified to:

 Q L / R ik . . . L / R i1 h /−1  L 2 ( ε0 ) ≤ Cl Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] t

: for all k = 1, . . . , l

(10.410)

as well as the L ∞ estimates of the fourth statement of Corollary 10.1.b with l replaced by l∗ + 1 and (10.222) with l replaced by l∗ , we obtain: max L / R ik . . . L / Ri1 (ω /·h /−1 ) L 2 ( ε0 ) t

 Q ≤ Cl (1 + t)−1 W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1]

i1 ...ik

: for all k = 0, . . . , l

(10.411) ( A)∗

/ L , the first term of β (see (10.398)). Here the two factors are Consider now (ω /·h /−1 ) · ω on equal footing, being both of order 1. We have: / Ri1 ((ω /·h /−1 ) · ω /L ) = L / R ik . . . L



((L / R )s1 (ω / · h/−1 )) · ((L / R )s 2 ω /L )

partitions

|s1 | + |s2 | = k ≤ l

(10.412)

We have two cases to consider: Case 1: |s1 | ≤ l∗

and:

Case 2: s2 | ≤ l∗

(The negation of both cases is: |s1 | ≥ l∗ + 1 and |s2 | ≥ l∗ + 1, hence |s1 | + |s2 | ≥ 2l∗ +2 ≥ l +1, contradicting (10.412)). In Case 1 we apply the L ∞ estimates of the fourth statement of Corollary 10.1.b with l replaced by l∗ + 1 and (10.222) with l replaced by l∗ , which yield: max L / R ik . . . L / Ri1 (ω / · h/−1 ) L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2

i1 ...ik

t

: for all k = 0, . . . , l∗

(10.413)

together with the third statement of Corollary 10.2.g with l replaced by l + 1. In Case 2 we apply the L ∞ estimate of the fourth statement of Corollary 10.1.g with l replaced by l∗ + 1 together with the L 2 estimate (10.411). We then obtain, combining the results of

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

407

the two cases, max L / R ik . . . L / Ri1 ((ω /·h /−1 ) · ω / L ) L 2 ( ε0 ) t

 Q ≤ Cl δ0 (1 + t)−3 W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1]

i1 ...ik

: for all k = 0, . . . , l

(10.414)

( A)∗

The second term in β (see (10.398)) is similar. We conclude that; ( A)∗

/ R ik . . . L / Ri1 β  L 2 ( ε0 ) max L t i1 ...ik

 Q ≤ Cl δ0 (1 + t)−3 W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] : for all k = 0, . . . , l (B)∗

(10.415) ( A)∗

The structure of β is similar to that of β , consisting of two factors of order 1, namely /−1 and ( ψ ⊗ d/σ − d/σ ⊗ ψ). By (10.170) and (10.172) with l replaced ( ψ(Lσ ) − ψ L d/σ ) · h by l∗ + 1, together with the third statements of Corollaries 10.1.b and 10.1.g and (10.222) with l replaced by l∗ we have the L ∞ estimate: / R ik . . . L / Ri1 (( ψ(Lσ ) − ψ L d/σ ) · h/−1 ) L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 max L

i1 ...ik

t

: for all k = 0, . . . , l∗

(10.416)

On the other hand, by Lemma 10.1 and the estimate (10.313) with l replaced by l + 1, together with the second statements of Corollaries 10.2.b and 10.2.g and (10.410) we have the L 2 estimate: / R ik . . . L / Ri1 (( ψ(Lσ ) − ψ L d/σ ) · h/−1 ) L 2 ( ε0 ) max L t

1 2 Q ≤ Cl (1 + t)−1 W[l+1] + δ0 (1 + t)−1 W[l] + δ02 (1 + t)−2 Y0 + (1 + t)A[l−1]

i1 ...ik

: for all k = 0, . . . , l

(10.417)

Also, by (10.170) with l replaced by l∗ + 1 and the third statement of Corollary 10.1.b with l replaced by l∗ we have the L ∞ estimate: / R ik . . . L / Ri1 ( ψ ⊗ d/σ − d/σ ⊗ ψ) L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−3 max L

i1 ...ik

t

: for all k = 0, . . . , l∗

(10.418)

while by Lemma 10.1 with l replaced by l + 1 and the second statement of Corollary 10.2.b we have the L 2 estimate: / R ik . . . L / Ri1 ( ψ ⊗ d/σ − d/σ ⊗ ψ) L 2 ( ε0 ) max L t

 Q −2 −1 W[l+1] + δ0 (1 + t) Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] ≤ Cl δ0 (1 + t)

i1 ...ik

: for all k = 0, . . . , l

(10.419)

408

Chapter 10. Control of the Angular Derivatives ( A)∗

Applying the same method as in the L 2 estimate of the angular derivatives of β , using the above L ∞ estimates (10.416), (10.418), and L 2 estimates (10.417), (10.419), we obtain: (B)∗

max L / R ik . . . L / Ri1 β  L 2 ( ε0 ) t i1 ...ik

1 2 Q ≤ Cl δ02 (1 + t)−4 W[l+1] + δ0 (1 + t)−1 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] : for all k = 0, . . . , l

(10.420) (C)∗

The L 2 estimate of the angular derivatives of β proceeds in a similar manner. We obtain: (C)∗

/ R ik . . . L / Ri1 β  L 2 ( ε0 ) max L t i1 ...ik

1 2 Q −3 W[l+1] + δ0 (1 + t)−1 W[l] ≤ Cl δ0 (1 + t) + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] : for all k = 0, . . . , l

(10.421)

Combining finally the results (10.415), (10.420), (10.421), yields: (5)

/ R ik . . . L / Ri1 i  L 2 ( ε0 ) max L t i1 ...ik

1 2 Q ≤ Cl δ0 (1 + t)−3 W[l+1] + δ0 (1 + t)−1 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] : for all k = 0, . . . , l

(10.422)

(4)

We turn to i , given by (10.396). Here in each term we again have two factors of order 1, one of which is here Lσ and the other is k/ or κ −1 ζ  . By Corollary 10.2.c with l ε replaced by l + 1 the terms in which one of the factors is k/ are bounded in L 2 (t 0 ) with their angular derivatives up to order l by:

1 2 Q Cl δ02 (1 + t)−4 W[l+1] + W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] while by Corollary 10.2.h with l replaced by l + 1 the terms in which one of the factors ε is κ −1 ζ  are bounded in L 2 (t 0 ) with their angular derivatives up to order l by:

1 2 Q Cl δ03 (1 + t)−5 W[l+1] + W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] Combining, we conclude that: (4)

/ R ik . . . L / Ri1 i  L 2 ( ε0 ) max L t i1 ...ik

1 2 Q ≤ Cl δ02 (1 + t)−4 W[l+1] + W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] : for all k = 0, . . . , l

(10.423)

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

409

(3)

We turn to i , given by (10.395). Using the L 2 estimate (10.410) and the L ∞ estimate (10.222) with l replaced by l∗ we obtain: 0  0  0 0 η0 h/ −1 0 0 / R ik . . . L max L / R i1 χ− · h/ 0 2 ε0 i1 ...ik 0 1 − u + η0 t L (t )

 Q −2 ≤ Cl A[l] + δ0 (1 + t) [1 + log(1 + t)] Y0 + W[l] + δ02 (1 + t)−2 W[l−1] : for all k = 0, . . . , l

(10.424)

Using (10.424) together with Corollary 10.2.h with l replaced by l + 1 we obtain that the ε angular derivatives of order up to l of the first term in (10.395) are bounded in L 2 (t 0 ) by:

Cl δ0 (1 + t)−2 A[l] + (1 + t)−1 [1 + log(1 + t)]  Q × W[l+1] + δ0 (1 + t)−1 Y0 + δ02 (1 + t)−2 W[l] Using (10.424) (together with the second statement of Corollary 10.2.b) we also obtain that the angular derivatives of order l of the second term in (10.395) are bounded in ε L 2 (t 0 ) by:

Cl δ03 (1 + t)−4 A[l] + (1 + t)−1 [1 + log(1 + t)]  Q × W[l+1] + δ0 (1 + t)−1 Y0 + δ03 (1 + t)−3 W[l−1] Combining, we conclude that: (3)

/ R ik . . . L / Ri1 i  L 2 ( ε0 ) max L t i1 ...ik

1 −2 ≤ Cl δ0 (1 + t) A[l] + (1 + t)−1 [1 + log(1 + t)] W[l+1]  Q +δ0 (1 + t)−1 Y0 + δ02 (1 + t)−2 W[l−1] : for all k = 0, . . . , l

(10.425)

(2)

Finally, we have i , given by (10.394). Here we have two terms each of which contains a single factor of order 1. Using Corollary 10.2.h with l replaced by l + 1 we obtain: (2)

/ R ik . . . L / Ri1 i  L 2 ( ε0 ) max L t i1 ...ik

1 2 Q ≤ Cl (1 + t)−2 W[l+1] + δ02 (1 + t)−2 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] : for all k = 0, . . . , l

(10.426)

Combining finally the results (10.422), (10.423), (10.425), (10.426), yields the conclusion of the lemma.

410

Chapter 10. Control of the Angular Derivatives (1)

(1,1)

We now turn to investigate the principal part i , given by (10.391)–(10.393). In (1,1)

/ 2 σ and
/ σ = tr(D / 2 σ · h/−1 ). Thus, to estimate L / R ik . . . L / Ri1 i , we i we have D 2 must estimate L / R ik . . . L / R i1 D / σ . This shall be done by expressing the latter in terms of D / 2 (Rik . . . Ri1 σ ) and the commutators: (i1 ...ik )

ck = L / R ik . . . L / R i1 D / 2σ − D / 2 (Rik . . . Ri1 σ )

(10.427)

/1 the Lie In the following, given an St,u -tangential vectorfield X, we denote by (X ) π derivative of the connection induced on St,u with respect to X. This is a T21 -type St,u tensorfield, given in an arbitrary local frame field by: (X ) C π /1,AB

=

1 (D /A 2

(X )

π /B D + D /B

(X )

(X )

π / AD − D /D

π / AB )(h/−1 ) DC

(10.428)

(see equations (9.165), (9.166)). Being symmetric in the lower indices, (X ) π /1 at a point p ∈ St,u can be considered to be a linear map of T p∗ St,u into S2 (T p St,u ), the space of symmetric bilinear forms in T p St,u . In the proof of Lemma 9.5 we have derived the commutation formula (9.168), which holds on an arbitrary Riemannian manifold (M, g). In /), the formula in question reads: the case that (M, g) = (St,u , h /2 f ) − D / 2 (X f ) = − L / X (D

(X )

π /1 · d/ f

(10.429)

for any function f and any St,u -tangential vectorfield X. Lemma 10.9 The commutators (i1 ...ik )

ck = −

k−1 

(i1 ...ik ) c

k

are given by:

L / R ik . . . L / Rik−m+1 (

(Rik−m )

π /1 · d/(Rik−m−1 . . . Ri1 σ ))

m=0

Proof. Consider the definition (10.427) with k replaced by k − 1: L / Rik−1 . . . L / R i1 D /2σ − D / 2 (Rik−1 . . . Ri1 σ ) =

(i1 ...ik−1 )

ck−1

Let us apply L / Rik to this. We obtain: L / R ik . . . L / R i1 D /2σ − L / R ik D / 2 (Rik−1 . . . Ri1 σ ) = L / R ik

(i1 ...ik−1 )

ck−1

(10.430)

We then apply the commutation formula (10.429) taking f = Rik−1 . . . Ri1 σ and X = Rik to obtain: L / R ik D / 2 (Rik−1 . . . Ri1 σ ) − D / 2 (Rik . . . Ri1 σ ) = −

(Rik )

π /1 · d/(Rik−1 . . . Ri1 σ )

(10.431)

Adding (10.430) and (10.431) yields, in view of the definition (10.427), the recursion relation: (i1 ...ik )

ck = L / R ik

(i1 ...ik−1 )

ck−1 −

(Rik )

π /1 · d/(Rik−1 . . . Ri1 σ )

(10.432)

The lemma then results by applying Proposition 8.2 to this recursion, taking into account the fact that c0 = 0.

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

411

In view of the above lemma, to estimate the commutators (i1 ...ik ) ck we must esti/1 of order up to k − 1. Let us define the operator mate the angular derivatives of the (R j ) π ˇ/ which takes symmetric 2-covariant St,u -tensorfields ϑ into 3-covariant St,u -tensorfields D ˇ/ϑ, symmetric in the first two indices, given in an arbitrary local frame field by: D ˇ/ϑ) ABC = (D

1 (D / A ϑ BC + D / B ϑ AC − D / C ϑ AB ) 2

(10.433)

ˇ/, (10.428) takes the form, simply: In terms of the operator D (X )

π /1 =

(X )

(π /1 )b · h/−1 ,

(X )

ˇ/ (π /1 )b = D

(X )

π /

(10.434)

Following the derivation, in the proof of Lemma 9.5, of the commutation formula (9.167) for 1-forms α on an arbitrary Riemannian manifold (M, g), we deduce in the case of a symmetric 2-covariant St,u tensorfield ϑ and an St,u -tangential vectorfield X, in view of the fact that (X ) π /1 represents the Lie derivative with respect to X of the induced connection on St,u , the following commutation formula: (L / X (D /ϑ)) ABC = (D /(L /ϑ)) ABC − It follows that:

(X ) D π /1,AB ϑ DC

ˇ/ϑ − D ˇ/L L /X D /X ϑ = −

(X )

−

(X ) D π /1,AC ϑ B D

(10.435)

π /1 · ϑ

(10.436)

Using this commutation formula we derive by an argument similar to that of Lemma 10.9, the following lemma. Lemma 10.10 Let ϑ be an arbitrary symmetric 2-covariant St,u tensorfield. The following commutation formula holds for any non-negative integer k: ˇ/ϑ − D ˇ/L / R ik . . . L / R i1 D / R i1 ϑ L / R ik . . . L =−

k−1 

L / R ik . . . L / Rik−m+1 (

(Rik−m )

π /1 · L / Rik−m−1 . . . L / Ri1 ϑ)

m=0 (R j ) π /,

Applying the above lemma to the case ϑ = L / R ik . . . L / R i1

(R j )

ˇ/L (π /1 )b = D / R ik . . . L / R i1

(R j )

we obtain, in view of (10.434),

π /+

( j ;i1 ...ik )

dk

(10.437)

where: ( j ;i1 ...ik )

dk = −

k−1 

L / R ik . . . L / Rik−m+1 (

(Rik−m )

π /1 · L / Rik−m−1 . . . L / R i1

(R j )

π /)

(10.438)

m=0

Note that in (10.438) we have the angular derivatives of the (Rn ) π /1 , n = 1, 2, 3, of order up to k − 1. Let us recall the hypotheses H1 and H2 of Chapter 7, in addition to the hypothesis H0, which has already been We shall presently use the following hypothesis instead of hypothesis H2.

412

Chapter 10. Control of the Angular Derivatives

H2 There is a constant C (independent of s) such that for any 2-covariant symmetric St,u tensorfield ϑ we have, pointwise:     |L / R j ϑ|2 + |ϑ|2 |D /ϑ|2 ≤ C(1 + t)−2   j

It shall later be shown that, modulo the other assumptions, H2 implies H2 . Let us note that assumptions H0 and H1 imply that: There is a constant C (independent of s) such that for any function f we have, pointwise:  |R j2 R j1 f |2 (10.439) |D / 2 f |2 ≤ C(1 + t)−4 j1 , j2

For, as already noted in Chapter 7, applying hypothesis H1 to the St,u 1-form d/ f we obtain:  |D / 2 f |2 ≤ C(1 + t)−2 |d/ R j1 f |2 j1

Applying then hypothesis H0 to the function R j1 f yields (10.439). Lemma 10.11 Let the hypotheses H0, H1, H2 , and the estimate (10.30) hold. Let also Q / [l] and X / [l] hold, for some positive integer l. Then if the bootstrap assumptions E / [l+1] , E δ0 is suitably small (depending on l) we have: / R ik . . . L / R i1 max L

(R j )

j ;i1 ...ik

π /1  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

: for all k = 0, . . . , l − 1 Proof. The assumptions of the present lemma imply those of Proposition 10.1 with l + 1 in the role of l. Thus Proposition 10.1 and all its corollaries hold with l + 1 in the role of l. In particular, for l = 1, Corollary 10.1.d holds with l = 2, that is: max 

(R j )

π / L ∞ ( ε0 ) , max L / Ri t

j

j ;i

(R j )

π / L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] t

(10.440)

Hypothesis H2 applied to ϑ = ˇ/ max  D

(R j )

(R j ) π / then

implies:

π / L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]

(10.441)

π /1  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]

(10.442)

t

j

hence by (10.434) also: max  j

(R j )

t

Thus the lemma holds for l = 1.

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

413

We proceed by induction. Let the lemma hold with l replaced by l − 1 and consider the sum in (10.438), for k up to l − 1. The term corresponding to m is: / Rik−m+1 ( (Rik−m ) π /1 · L / Rik−m−1 . . . L / Ri1 (R j ) π /) L / R ik . . . L  s1 (Rik−m ) s2 = ((L /R ) π /1 ) · ((L /R ) L / Rik−m+1 . . . L / R i1

(R j )

π /)

partitions

|s1 | + |s2 | = m ≤ k − 1

(10.443)

where we are considering all ordered partitions {s1 , s2 } of the set {k − m + 1, . . . , k} into two ordered subsets s1 ,s2 . Since |s1 | ≤ m ≤ k − 1 ≤ l − 2 the first factor on the right is ε bounded in L ∞ (t 0 ) by the inductive hypothesis by: Cl δ0 (1 + t)−2 [1 + log(1 + t)] Also, since |s2 | ≤ m, we have |s2 | + k − m − 1 ≤ k − 1 ≤ l − 2 ε

thus the second factor is bounded in L ∞ (t 0 ) by Corollary 10.1.d (even with l reduced to l − 1) by: Cl δ0 (1 + t)−1 [1 + log(1 + t)] It follows that: max 

( j ;i1 ...ik )

j ;i1 ...ik

dk  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−3 [1 + log(1 + t)]2 t

: for all k = 0, . . . , l − 1

(10.444)

On the other hand, by hypothesis H2 applied to ϑ = L / R ik . . . L / R i1 10.1.d with l replaced by l + 1: ˇ/L / R ik . . . L / R i1 max  D

(R j )

j ;i1 ...ik

(R j ) π / and

Corollary

π / L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

: for all k = 0, . . . , l − 1

(10.445)

The estimates (10.445) and (10.444) yield, in view of (10.437), max L / R ik . . . L / R i1

(R j )

j ;i1 ...ik

(π /1 )b  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

: for all k = 0, . . . , l − 1

(10.446)

Together with the estimate (10.222) (even with l reduced to l − 1) this yields: / R ik . . . L / R i1 max L

j ;i1 ...ik

(R j )

π /1  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

: for all k = 0, . . . , l − 1 completing the inductive step and the proof of the lemma.

(10.447)

414

Chapter 10. Control of the Angular Derivatives

Lemma 10.12 Let the hypotheses H0, H1, H2 , and the estimate (10.30) hold. Let l be Q / [l∗ ] and X / [l∗ ] hold. Then if a positive integer and let the bootstrap assumptions E / [l∗ +1] , E δ0 is suitably small (depending on l) we have: max L / R ik . . . L / R i1

j ;i1 ...ik

(R j )

π /1  L 2 ( ε0 ) t

Q ≤ Cl (1 + t)−1 Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l]



: for all k = 0, . . . , l − 1 Proof. The assumptions of the present lemma imply those of Proposition 10.1 with l∗ +1 in the role of l, as well as those of Proposition 10.2 with l + 1 in the role of l. Thus, Proposition 10.1 and all its corollaries hold with l∗ + 1 in the role of l, and Proposition 10.2 and all its corollaries hold with l + 1 in the role of l. In particular, Corollary 10.2.d holds with l + 1 in the role of l, so, simplifying, we have: max L / R ik . . . L / Ri1 (R j ) π / L 2 ( ε0 ) t

 Q ≤ Cl Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l]

j ;i1 ...ik

: for all k = 0, . . . , l

(10.448)

Moreover, the assumptions of the present lemma coincide with those of Lemma 10.11 with l∗ in the role of l. Hence the conclusion of Lemma 10.11 holds with l∗ in the role of l: max L / R ik . . . L / R i1

(R j )

j ;i1 ...ik

π /1  L ∞ ( ε0 ) ≤ Cl∗ δ0 (1 + t)−2 [1 + log(1 + t)] t

: for all k = 0, . . . , l∗ − 1

(10.449)

In the case l = 1 (10.448) reads: max  j

(R j )

π / L 2 ( ε0 ) , max L / Ri (R j ) π / L 2 ( ε0 ) t t j ;i

 Q ≤ C Y0 + (1 + t)A[1] + W[2] + δ02 (1 + t)−2 W[1]

Hypothesis H2 applied to ϑ = ˇ/ max  D

(R j )

j

π / L 2 ( ε0 ) t

(10.450)

(R j ) π / then

implies:

 Q ≤ C(1 + t)−1 Y0 + (1 + t)A[1] + W[2] + δ02 (1 + t)−2 W[1] (10.451)

hence by (10.434) also: max  j

(R j )

 Q π /1  L 2 ( ε0 ) ≤ C(1 + t)−1 Y0 + (1 + t)A[1] + W[2] + δ02 (1 + t)−2 W[1] t

(10.452) Thus the lemma holds for l = 1.

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

415

We proceed by induction. Let the lemma hold with l replaced by l − 1 and consider the sum in (10.438), for k up to l − 1. The term corresponding to m is given by (10.443). We have the following two cases to consider. Case 1: |s1 | ≤ l∗ − 1

and:

Case 2: |s1 | ≥ l∗ ε

In Case 1 the first factor on the right in (10.443) is bounded in L ∞ (t 0 ) according to (10.449) by: Cl δ0 (1 + t)−2 [1 + log(1 + t)] while, since, as |s1 | ≤ m, we have |s2 | + k − m − 1 ≤ k − 1 ≤ l − 2, the second factor is bounded by Corollary 10.2.d with l reduced to l − 1 by:

 Q Cl Y0 + (1 + t)A[l−2] + W[l−1] + δ02 (1 + t)−2 W[l−2] In Case 2, since |s1 | ≤ m ≤ k − 1 ≤ l − 2, the first factor on the right in (10.443) is ε bounded in L 2 (t 0 ) by the inductive hypothesis by:

 Q Cl (1 + t)−1 Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] while, since |s2 | + k − m − 1 = k − |s1 | − 1 ≤ k − l∗ − 1 ≤ l − l∗ − 2 ≤ l∗ − 1, ε

the second factor is bounded in L ∞ (t 0 ) according to Corollary 10.1.d with l∗ in the role of l by: Cl δ0 (1 + t)−1 [1 + log(1 + t)] Combining the results of the two cases we conclude that: max 

( j ;i1 ...ik )

j ;i1 ...ik

dk  L 2 ( ε0 ) t

 Q ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] : for all k = 0, . . . , l − 1 / R ik . . . L / R i1 On the other hand, by hypothesis H2 applied to ϑ = L

(10.453) (R j ) π / and

(10.448):

ˇ/L max  D / R ik . . . L / Ri1 (R j ) π / L ∞ ( ε0 ) t

 Q −1 ≤ Cl (1 + t) Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l]

j ;i1 ...ik

: for all k = 0, . . . , l − 1

(10.454)

The estimates (10.454) and (10.453) yield, in view of (10.437), max L / R ik . . . L / Ri1 (R j ) (π /1 )b  L ∞ ( ε0 ) t

 Q ≤ Cl (1 + t)−1 Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l] j ;i1 ...ik

: for all k = 0, . . . , l − 1

(10.455)

416

Chapter 10. Control of the Angular Derivatives

Together with the L 2 estimate (10.410) (even with l reduced to l − 1) as well as the L ∞ estimates (10.446) and (10.222) with l∗ in the role of l, this yields: / R ik . . . L / Ri1 (R j ) π /1  L ∞ ( ε0 ) max L t

 Q ≤ Cl (1 + t)−1 Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l] j ;i1 ...ik

: for all k = 0, . . . , l − 1

(10.456)

completing the inductive step and the proof of the lemma. We are now ready to estimate the commutators (i1 ...ik ) ck given by Lemma 10.9, for k = 0, . . . , l. The mth term in the sum is expressed as: / Rik−m+1 ( (Rik−m ) π /1 · d/(Rik−m−1 . . . Ri1 σ )) L / R ik . . . L  ((L / R )s1 (Rik−m ) π /1 ) · d/((R)s2 Rik−m−1 . . . Ri1 σ ) = partitions

|s1 | + |s2 | = m ≤ k − 1

(10.457)

where we are again considering all ordered partitions {s1 , s2 } of the set {k − m + 1, . . . , k} into two ordered subsets s1 ,s2 . We consider first L ∞ estimates under the assumptions of Lemma 10.11. Since ε |s1 | ≤ m ≤ k − 1 ≤ l − 1, the first factor on the right in (10.457) is bounded in L ∞ (t 0 ) according to Lemma 10.11 by: Cl δ0 (1 + t)−2 [1 + log(1 + t)] while, since |s2 | ≤ m, hence |s2 | + k − m − 1 ≤ k − 1 ≤ l − 1, the second factor is ε bounded in L ∞ (t 0 ) by virtue of hypothesis H0 and assumption E / [l] by: Cl δ0 (1 + t)−2 It follows that under the assumptions of Lemma 10.1 we have: max 

(i1 ...ik )

i1 ...ik

ck  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)] t

: for all k = 0, . . . , l

(10.458)

We consider next L 2 estimates under the assumptions of Lemma 10.12. As we remarked above, the assumptions of Lemma 10.12 coincide with those of Lemma 10.11 with l∗ in the role of l, hence the conclusion (10.449) holds. We have two cases to consider in regard to the sum in (10.457). Case 1: |s1 | ≤ l∗ − 1

and:

Case 2: |s1 | ≥ l∗ ε

In Case 1 the first factor on the right in (10.457) is bounded in L ∞ (t 0 ) according to (10.449) by: Cl δ0 (1 + t)−2 [1 + log(1 + t)]

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

417

while in Case 2, since |s2 | + k − m − 1 ≤ k − 1 − |s1 | ≤ l − 1 − l∗ ≤ l∗ , ε

/ [l∗ +1] and hypothesis H0 by: the second factor is bounded in L ∞ (t 0 ) by assumption E Cl δ(1 + t)−2 On the other hand, since, in any case |s1 | ≤ m ≤ k − 1 ≤ l − 1, the first factor on the ε right in (10.457) is bounded in L 2 (t 0 ) according to Lemma 10.12, while, since also in any case |s2 | ≤ m, hence |s2 | + k − m − 1 ≤ k − 1 ≤ l − 1, the second factor is bounded ε in L 2 (t 0 ) by Lemma 10.1 and hypothesis H0 by: Cl (1 + t)−1 W[l] ε

Placing in L 2 (t 0 ) the second factor in Case 1 and the first factor in Case 2 and combining the results then yields, through the expression of Lemma 10.9 and (10.457), the estimate: max 

(i1 ...ik )

ck  L 2 ( ε0 ) ≤ t

 Q −3 Y0 + (1 + t)A[l] + [1 + log(1 + t)]W[l+1] + δ02 (1 + t)−2 W[l] Cl δ0 (1 + t)

i1 ...ik

: for all k = 0, . . . , l

(10.459)

under the assumptions of Lemma 10.12. We are now ready to estimate the angular derivatives, of order up to l, of princi(1)

(1,1)

pal part i of i . We begin with the L ∞ estimates. In reference to i (see (10.392)), ε to estimate the angular derivatives, of order up to l, of D / 2 σ in L ∞ (t 0 ) we must augment the assumptions of Lemma 10.11 replacing the assumption E / [l+1] by the stronger assumption E / [l+2] . By virtue of the pointwise estimate (10.439) applied to the function f = Rik . . . Ri1 σ we then obtain the L ∞ estimate: max D / 2 (Rik . . . Ri1 σ ) L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−3

i1 ...ik

t

: for all k = 0, . . . , l

(10.460)

Combining this with the estimate (10.458) we obtain, through (10.427), max L / R ik . . . L / R i1 D / 2 σ  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−3

i1 ...ik

t

: for all k = 0, . . . , l

(10.461)

Using this estimate we readily deduce: max L / R ik . . . L / R i1

i1 ...ik

(1,1)

i  L ∞ (tε0 ) ≤ Cl δ02 (1 + t)−4

: for all k = 0, . . . , l

(10.462)

418

Chapter 10. Control of the Angular Derivatives (1,2)

In reference to i (see (10.393)), to estimate the angular derivatives, of order up to l, ε of d/ Lσ in L ∞ (t 0 ) we must augment the assumptions of Lemma 10.11 replacing the Q Q / [l+1] . Using hypothesis H0 we then obtain: assumption E / [l] by the stronger assumption E max d/(Rik . . . Ri1 Qσ  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2

i1 ...ik

t

: for all k = 0, . . . , l

(10.463)

Using this estimate we readily deduce: max L / R ik . . . L / R i1

i1 ...ik

(1,2)

i  L ∞ (tε0 ) ≤ Cl δ03 (1 + t)−5

: for all k = 0, . . . , l

(10.464)

Combining the estimates (10.462) and (10.464) with the estimate of Lemma 10.7 we obtain the following lemma. Lemma 10.13 Let the hypotheses H0, H1, H2 , and the estimate (10.30) hold. Let also Q / [l+1] and X / [l] hold, for some positive integer l. Then the bootstrap assumptions E / [l+2] , E if δ0 is suitably small (depending on l) we have: / R ik . . . L / Ri1 i  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−3 max L

i1 ...ik

t

: for all k = 0, . . . , l We proceed to the L 2 estimates. By virtue of the pointwise estimate (10.439) applied to the function f = Rik . . . Ri1 σ we have the L 2 estimate: / 2 (Rik . . . Ri1 )σ  L 2 ( ε0 ) ≤ C(1 + t)−2 max D

i1 ...ik

t

≤ Cl (1 + t)

−2

max

j1 , j2 ;i1 ...ik

R j2 R j1 Rik . . . Ri1 σ  L 2 ( ε0 ) t

W[l+2]

: for all k = 0, . . . , l

(10.465)

where we have used Lemma 10.1 in the case G = σ with l + 2 in the role of l. This requires the bootstrap assumption E / [(l+2)∗ ] , which, since (l + 2)∗ = l∗ + 1, coincides with the assumption E / [l∗ +1] of Lemma 10.12. Combining the estimate (10.465) with the estimate (10.459) we obtain, through (10.427), / R ik . . . L / R i1 D / 2 σ  L 2 ( ε0 ) max L t

 Q ≤ Cl (1 + t)−2 W[l+2] + δ0 (1 + t)−1 Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W[l]

i1 ...ik

: for all k = 0, . . . , l

(10.466)

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

419

Using this estimate we readily deduce: (1,1)

max L / R ik . . . L / Ri1 i  L 2 ( ε0 ) t i1 ...ik

 Q ≤ Cl δ0 (1 + t)−3 W[l+2] + δ0 (1 + t)−1 Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W[l] : for all k = 0, . . . , l

(10.467)

By hypothesis H0 we have: max d/(Rik . . . Ri1 Qσ ) L 2 ( ε0 ) ≤ C(1 + t)−1 max R j Rik . . . Ri1 Qσ  L 2 ( ε0 ) t t i1 ...ik ; j

 Q ≤ Cl (1 + t)−1 W[l+1] + δ0 (1 + t)−1 W[l+1]

i1 ...ik

: for all k = 0, . . . , l

(10.468)

where we have used the estimate (10.313) with l + 2 in the role of l. This requires the Q bootstrap assumptions E / [(l+2)∗ ] and E / [(l+2)∗ −1] , which, since (l + 2)∗ = l∗ + 1, coincide

with the assumptions E / [l∗ +1] and E / [lQ∗ ] of Lemma 10.12, respectively. Using the estimate (10.468) we readily deduce: (1,2)

i  L 2 (tε0 )

1 2 Q ≤ Cl δ02 (1 + t)−4 W[l+1] + W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1]

max L / R ik . . . L / R i1

i1 ...ik

: for all k = 0, . . . , l

(10.469)

Combining the estimates (10.467) and (10.469) with the estimate of Lemma 10.8 we obtain the following lemma. Lemma 10.14 Let the hypotheses H0, H1, H2 , and the estimate (10.30) hold. Let l be a positive integer and let the bootstrap assumptions E / [l∗ +1] , E / [lQ∗ ] and X / [l∗ ] hold. Then if δ0 is suitably small (depending on l) we have: / R ik . . . L / Ri1 i  L 2 ( ε0 ) max L t

≤ Cl (1 + t)−2 W[l+1] + δ0 (1 + t)−1

i1 ...ik

Q × W[l+2] + Y0 + (1 + t)A[l] + δ0 (1 + t)−1 W[l+1]



: for all k = 0, . . . , l ε

We proceed to estimate, in L 2 (t 0 ), the St,u 1-form tion 8.4,     1 (Ri ) 1 (Ri ) (i1 ...il ) l il = L / Ril + tr π / ... L / Ri1 + tr 1 π / i 2 2   l−1   1 (Ri ) 1 L / Ril + tr l π + / ... L / Ril−k+1 + tr 2 2 k=0

(i1 ...il ) i

l

given by Proposi-

(10.470) (Ril−k+1 )

 π /

(i1 ...il−k )

ql−k

420

Chapter 10. Control of the Angular Derivatives

under the assumptions of Lemma 10.14. Note that the assumptions of Lemma 10.14 coincide with those of Lemma 10.12 and include those of Lemma 10.13 with l∗ − 1 in the role of l. In particular we have the L 2 estimate (10.448) as well as the L ∞ estimate of Corollary 10.1.d with l∗ + 1 in the role of l, that is: (R j )

max L / R ik . . . L / R i1

j ;i1 ...ik

π / L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

: for all k = 0, . . . , l∗

(10.471)

Let us begin with the first term in (10.470). We have:     1 (Ri ) 1 (Ri ) l 1 L / Ril + tr π / ... L / Ri1 + tr π / i 2 2 =L / Ril . . . L / R i1 i +

l 



l 

j =1 k1 <···
L / Ril · · ·

L / Ri

(10.472)



L / Ri

  1 (Rik j ) 1 tr tr π / ··· 2 2 kj

k1



(Rik ) 1

 π / ...L / R i1 i

/ Rik are missing and in their place we where, in the double sum, the operators L / R ik , . . . , L (Ri

)

j

1

(R

)

have the multiplication operators 12 tr k j π /, . . . , 12 tr ik1 π /. The first term in (10.472) is ε 0 2 the principal term, estimated in L (t ) by Lemma 10.14. Consider then the term in the outer sum in the second term in (10.472) corresponding to j = 1, . . . , l: l  k1 <···
 L / Ril · · ·

L / Ri



L / Ri

  1 (Rik j ) 1 tr tr π / ··· 2 2 kj

k1



(Rik ) 1

 π / ...L / R i1 i

(10.473)

Here there are l − j angular derivatives in all. There are two cases to be considered. Case 1: At most l∗ angular derivatives fall on one of the factors 1 tr 2

(Rik ) 1

π /,

...,

1 tr 2

(Rik ) j

π /

and:

Case 2: One of these factors receives more than l∗ angular derivatives. Then all the other factors receive at most l∗ angular derivatives. (Otherwise, there would be 2(l∗ +1) ≥ l +1 or more angular derivatives.) ε

In Case 1 we can apply (10.471) to estimate in L ∞ (t 0 ) the j factors corresponding (Rik )

(R

)

j π to the angular derivatives of the factors 12 tr /, . . . , 12 tr ik1 π /, and, since there are l − j angular derivatives in all, we can apply Lemma 10.14 with l reduced to l − j to ε estimate in L 2 (t 0 ) the factor corresponding to the angular derivatives of i . This gives an ε0 2 L (t ) bound by:  j Cl δ0 (1 + t)−1 [1 + log(1 + t)] (1 + t)−2

 Q · W[l− j +1] + δ0 (1 + t)−1 W[l− j +2] + Y0 + (1 + t)A[l− j ] + δ0 (1 + t)−1 W[l− j +1]

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

421

In Case 2 the number of angular derivatives falling on i is at most l∗ −1. (Otherwise, (R ) there would be at least l∗ + 1 angular derivatives falling on the factors 12 tr ik1 π /, . . . , 1 (Rik j ) π /, 2 tr

as well as l∗ or more angular derivatives falling on i , so there would have to be at least 2l∗ + 1 ≥ l angular derivatives in all.) We can then apply (10.448) with l (R

ε

)

(Ri

)

/, . . . , 12 tr k j π /, reduced to l − j to estimate in L 2 (t 0 ) the factor, among the 12 tr ik1 π ε receiving more than l∗ angular derivatives, (10.471) to estimate in L ∞ (t 0 ) the other ε such factors, and Lemma 10.13 with l∗ − 1 in the role of l to estimate in L ∞ (t 0 ) the ε factor corresponding to i . This gives an L 2 (t 0 ) bound by:  j −1  Cl δ0 (1 + t)−1 [1 + log(1 + t)] · δ0 (1 + t)−3 · 

Q Y0 + (1 + t)A[l− j ] + W[l− j +1] + δ02 (1 + t)−2 W[l− j ] Combining the results of the two cases we conclude that (10.473) is bounded in ε L 2 (t 0 ) by:   j −1 Cl δ0 (1 + t)−1 [1 + log(1 + t)] · δ0 (1 + t)−3 · 

Q [1 + log(1 + t)] W[l− j +1] + δ0 (1 + t)−1 W[l− j +2] + δ02 (1 + t)−2 W[l− j +1]  +Y0 + (1 + t)A[l− j ] (10.474) The double sum in (10.472) is then similarly bounded. Combining this result with the bound for the first term in (10.472) given by Lemma 10.14 we conclude that under the assumptions of Lemma 10.14 we have: 0    0 0 0 1 (Ri ) 1 (Ri ) 0 / ... L / Ri1 + tr 1 π / i0 (10.475) max 0 L / Ril + tr l π 0 2 ε0 i1 ...il 2 2 L (t )

 Q ≤ Cl (1 + t)−2 W[l+1] + δ0 (1 + t)−1 W[l+2] + Y0 + (1 + t)A[l] + δ0 (1 + t)−1 W[l+1]

Next we consider the second term in (10.470). Here we have the St,u 1-forms j , j ≥ 1, given in the statement of Proposition 8.4. Now, in terms of the opˇ/ (see (10.433)) the second term in the expression for (i1 ...i j ) q j in the statement erator D of Proposition 8.4 is simply: (i1 ...i j ) q

ˇ/ (D

(i1 ...i j −1 )

χˆ j −1 ) · (h/−1 ·

(R j ) ˆ

π / · h/−1 )

Also, the third term in the same expression is: (i1 ...i j −1 )

χˆ j −1 · (div /

(R j ) ˆ

π / · h/−1 )

422

Chapter 10. Control of the Angular Derivatives

Let us note from equation (10.428) that, in an arbitrary local frame field on St,u , (X ) C π /1,AB (h /−1 ) AB

1 −1 AB (h / ) (D /A 2 

π /B D + D / B (X ) π / AD − D /D  1 = (div / (X ) π /) D − d/ D tr (X ) π / (h/−1 ) DC 2 /ˆ D · (h/−1 ) DC = div / (X ) π =

(X )

(X )

π / AB )(h/−1 ) DC

Thus, defining, for any St,u -tangential vectorfield X, the St,u -tangential vectorfield tr (X ) π /1 , with components (X ) π /C /−1 ) AB , we have simply: 1,AB (h div /

(X ) ˆ

π / · h/−1 = tr

(X )

π /1

(10.476)

Hence, the third term in the expression for (i1 ...i j ) q j in the statement of Proposition 8.4 is simply: (R ) (i1 ...i j −1 ) χˆ j −1 · tr i j π /1 We conclude that the (i1 ...i j )

qj =

(i1 ...i j ) q

j

are given by:

1 (R j ) ˇ/ tr π /d/(Ri j −1 . . . Ri1 trχ) + ( D 4 (R ) + (i1 ...i j −1 ) χˆ j −1 · tr i j π /1

(i1 ...i j −1 )

χˆ j −1 ) · (h/−1 ·

(R j ) ˆ

π / · h/−1 ) (10.477)

To estimate the second term in (10.470), we must estimate the kth angular derivatives of (i1 ...i j ) q j for all j ≥ 1 and all k such that j + k ≤ l. First, we can show, in a straightforward manner, that under the assumptions of Lemma 10.11 we have: / Ri j +k . . . L / Ri j +1 max L

i1 ...i j +k

(i1 ...i j )

χˆ j  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

: for all j and all k such that j + k ≤ l

(10.478)

and, under the assumptions of Lemma 10.12 we have: / Ri j +k . . . L / Ri j +1 (i1 ...i j ) χˆ j  L 2 ( ε0 ) max L t

 Q ≤ Cl A[l] + (1 + t)−1 Y0 + W[l] + δ02 (1 + t)−2 W[l−1] i1 ...i j +k

: for all j and all k such that j + k ≤ l Next we must estimate, in regard to the second term in (10.477), ˇ/ L / Ri j +k . . . L / Ri j +1 D

(i1 ...i j −1 )

χˆ j −1

for all j ≥ 1 and all k such that j + k ≤ l. In the case k = 0 this reduces to: ˇ/ D

(i1 ...ik )

χˆ j −1

(10.479)

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

423 ε

ε

which can be estimated using hypothesis H2 in L ∞ (t 0 ) and in L 2 (t 0 ) by (10.478) and (10.479) respectively, with j, k replaced by j − 1, 1. This gives: ˇ/ D

(i1 ...i j −1 )

χˆ j −1  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)[1 + log(1 + t)] t

: for all j = 1, . . . , l

(10.480)

under the assumptions of Lemma 10.11, and: ˇ/ D

(i1 ...i j −1 )

χˆ j −1  L 2 ( ε0 ) t

 Q −1 A[l] + (1 + t)−1 Y0 + W[l] + δ02 (1 + t)−2 W[l−1] ≤ Cl (1 + t) : for all j = 1, . . . , l

(10.481)

under the assumptions of Lemma 10.12. In the case k ≥ 1 we apply Lemma 10.10, taking ϑ = ˇ/ L / Ri j +k . . . L / Ri j +1 D −

k−1 

(i1 ...i j −1 )

ˇ/L χˆ j −1 = D / Ri j +k . . . L / Ri j +1

L / Ri j +k . . . L / Ri j +k−m+1 (

(Ri j +k−m )

(i1 ...i j −1 ) χˆ (i1 ...i j −1 )

j −1 ,

to obtain:

χˆ j −1

π /1 · L / Ri j +k−m−1 . . . L / Ri j +1

(i1 ...i j −1 )

(10.482) χˆ j −1 )

m=0 ε

The first term on the right in (10.482) is estimated using hypothesis H2 , in L ∞ (t 0 ) using (10.478) with j, k replaced by j − 1, k + 1: ˇ/L D / Ri j +k . . . L / Ri j +1

(i1 ...i j −1 )

χˆ j −1  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] t

: for all j ≥ 1 and all k such that j + k ≤ l

(10.483)

ε

under the assumptions of Lemma 10.11, and in L 2 (t 0 ) using (10.479) with j, k replaced by j − 1, k + 1: ˇ/L D / Ri j +k . . . L / Ri j +1 (i1 ...i j −1 ) χˆ j −1 L 2 ( ε0 ) t

 Q −1 −1 ≤ Cl (1 + t) A[l] + (1 + t) Y0 + W[l] + δ02 (1 + t)−2 W[l−1] : for all j ≥ 1 and all k such that j + k ≤ l

(10.484)

under the assumptions of Lemma 10.12. ε To estimate the second term on the right in (10.482) in L ∞ (t 0 ) we appeal to Lemma 10.11. Since k ≤ l − j ≤ l − 1 and at most m ≤ k − 1 angular derivatives fall on (Ri j +k−m ) π /1 , Lemma 10.11 with l reduced to l − 1 suffices. Also, as at most k − 1 angular derivatives fall on (i1 ...i j −1 ) χˆ j −1, so the estimate (10.478) with l reduced to l − 1 suffices. ε We then obtain that the second term in (10.482) is bounded in L ∞ (t 0 ) by: Cl δ02 (1 + t)−4 [1 + log(1 + t)]2

(10.485)

424

Chapter 10. Control of the Angular Derivatives ε

To estimate the second term on the right in (10.482) in L 2 (t 0 ) we appeal to Lemma 10.12, as well as to Lemma 10.11 and (10.478) with l∗ in the role of l. This gives an ε L 2 (t 0 ) bound for the term in question by: 

Q Cl δ0 (1 + t)−3 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] (10.486) Combining the two L ∞ estimates (10.483) and (10.485) we conclude that under the assumptions of Lemma 10.11: ˇ/ / Ri j +1 D L / Ri j +k . . . L

(i1 ...i j −1 )

χˆ j −1  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] t

: for all j ≥ 1 and all k such that j + k ≤ l

(10.487)

Also, combining the two L 2 estimates (10.484) and (10.486) we conclude that under the assumptions of Lemma 10.12: ˇ/ (i1 ...i j −1 ) χˆ j −1  2 ε0 L / Ri j +k . . . L / Ri j +1 D L (t )

 Q −2 Y0 + (1 + t)A[l] + W[l] + δ02 (1 + t)−2 W[l−1] ≤ Cl (1 + t) : for all j ≥ 1 and all k such that j + k ≤ l

(10.488)

We now consider the second term on the right in (10.477). Under the assumptions of Lemma 10.11, using the estimate (10.487) and Corollary 10.1.d we obtain, in a straightforward manner, ˇ/ L / Ri j +k . . . L / Ri j +1 (( D

(i1 ...i j )

χˆ j −1 ) · (h/−1 ·

(Ri j ) ˆ

π / · h/−1 )) L ∞ ( ε0 ) t

≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)]2 : for all j ≥ 1 and all k such that j + k ≤ l

(10.489)

Also, under the assumptions of Lemma 10.12, using the estimate (10.488) together with Corollary 10.1.d with l∗ + 1 in the role of l, and (10.487) with l∗ in the role of l together with Corollary 10.2.d, we obtain in a straightforward manner: (R ) ˆ χˆ j −1 ) · (h/−1 · i j π / · h/−1 )) L 2 ( ε0 ) t 

Q −3 ≤ Cl δ0 (1 + t) [1 + log(1 + t)] Y0 + (1 + t)A[l] + W[l] + δ02 (1 + t)−2 W[l−1]

ˇ/ L / Ri j +k . . . L / Ri j +1 (( D

(i1 ...i j )

: for all j ≥ 1 and all k such that j + k ≤ l

(10.490)

Similar estimates hold in connection with the first term on the right in (10.477). We turn to the third term on the right in (10.477). Under the assumptions of Lemma 10.11, using the estimate (10.478) (l reduced to l − 1 here suffices) as well as Lemma 10.11 itself, we readily obtain: / Ri j +1 ( L / Ri j +k . . . L

(i1 ...i j −1 )

χˆ j −1 · tr

(Ri j )

π /1 ) L ∞ ( ε0 ) t

≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)]2 : for all j ≥ 1 and all k such that j + k ≤ l

(10.491)

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

425

Also, under the assumptions of Lemma 10.12, using the estimate (10.479) (l reduced to l − 1 here suffices) together with Lemma 10.11 with l∗ in the role of l, and (10.478) with l∗ in the role of l together with Lemma 10.12 itself, we readily obtain: L / Ri j +k . . . L / Ri j +1 (

(R )

(i1 ...i j −1 )

χˆ j −1 · tr i j π /1 ) L 2 ( ε0 ) t 

Q −3 ≤ Cl δ0 (1 + t) [1 + log(1 + t)] Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l] : for all j ≥ 1 and all k such that j + k ≤ l

(10.492)

In view of the above results we conclude that under the assumptions of Lemma 10.11 we have: / Ri j +k . . . L / Ri j +1 max L

(i1 ...i j )

i1 ...i j +k

q j  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)]2 t

: for all j ≥ 1 and all k such that j + k ≤ l

(10.493)

and under the assumptions of Lemma 10.12 we have: / Ri j +k . . . L / Ri j +1 max L

i1 ...i j +k

(i1 ...i j )

q j  L 2 ( ε0 ) t



Q ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l] : for all j ≥ 1 and all k such that j + k ≤ l

(10.494)

ε L 2 (t 0 )

the second term in (10.470) (the sum). We are now ready to estimate in Following the same argument as that leading from Lemmas 10.13 and 10.14 to the estimate (10.475), we deduce, under the assumptions of Lemma 10.12, using the estimate (10.493) with l∗ in the role of l (as noted earlier, the assumptions of Lemma 10.12 coincide with those of Lemma 10.11 with l∗ in the role of l) as well as the estimate (10.494), the following L 2 estimate for the second term in (10.470): 0 l−1  0    0 0 1 (Ri ) 1 (Ri 0 0 ) L / Ril + tr l π max 0 / ... L / Ril−k+1 + tr l−k+1 π / (i1 ...il−k ) ql−k 0 0 i1 ...il 0 2 2 ε k=0 L 2 (t 0 ) 

Q ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l] (10.495) Combining this estimate with the previously obtained estimate (10.475) for the first term in (10.470) yields the following. Proposition 10.3 Under the assumptions of Lemma 10.14 we have: max 

(i1 ...il )

i l  L 2 ( ε0 ) t

 Q −2 W[l+1] + δ0 (1 + t)−1 W[l+2] + δ0 (1 + t)−1 W[l+1] ≤ Cl (1 + t) 2 1 +δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l] i1 ...il

provided that δ0 is suitably small (depending on l).

426

Chapter 10. Control of the Angular Derivatives

We turn to the St,u 1-form (i1 ...il ) g˙l , given in terms of the St,u 1-form (i1 ...il ) gl by (8.411). In view of the expression for (i1 ...il ) gl given in the statement of Proposition 8.3, we have: (i1 ...il )

(i1 ...il )

g˙ l = L / Ril . . . L / R i1 g 0 − +

l−1 

L / Ril . . . L / Ril−k+1

wl

(i1 ...il−k )

yl−k

(10.496)

k=0

where the St,u 1-form is the St,u 1-form: (i1 ...il )

(i1 ...i j ) y

wl =

j

l−1 

is given in the statement of Proposition 8.3, and

L / Ril . . . L / Ril−k+1 L /

(Ri ) l−k Z

(i1 ...il−k−1 )

(i1 ...il ) w l

xl−k−1

k=0

−

l−1 

L /

i

(Ri ) l−k Z



. . il ) (i1 . l−k

xl−1

(10.497)

k=0

We begin with the first term in (10.496). The St,u 1-form g0 is given by equation (8.58). To estimate the angular derivatives of g0 of order l we must estimate the angular derivatives of the function gˇ of order l + 1. We thus first consider the function g. ˇ This (1)

(i)

function is defined by (8.44) through the functions g and g : i = 2, 3, 4. The function (1)

(1)

g is defined by (8.41) through the function g , itself defined by (8.28). The functions

(2)

(3)

(4)

g and g are defined by (8.29) and (8.30) respectively, while the function g is defined by (8.21). The formulas just mentioned, through which the function gˇ is defined, involve (N)

the functions Lµ, e, ˜ n, ν L , τ L , a, b, and tr α . We shall now express these in terms of quantities which we can directly estimate. First, Lµ is given by (8.52): Lµ = m + µe

(10.498)

with m and e given by (3.97) and (3.98): m=

1 dH (ψ L )2 (T σ ) 2 dσ

1 dH (Lσ ) − Fψ0 ω L L ψ L (ψ L + 2αψTˆ ) dσ 2α 2 Equation (8.38) expresses e˜ in terms of e. Substituting (10.500) we obtain: e=−

e˜ = −

1 1 dH dH (Lσ ) + ψ L ψ · d/σ − Fψ0 ω L L ψ L (2ψ L + 3αψTˆ ) 2 dσ 2 dσ 2α

(10.499) (10.500)

(10.501)

Equation (8.35) expresses n: n = −µ−1 ψ L (T σ ) − α −2 (ψ L + αψTˆ )(Lσ )+ ψ · d/σ

(10.502)

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

427

Equation (8.36) expresses ν L : ν L = −µ−1 ω L L (T σ ) − α −2 (ω L L + αω L Tˆ )(Lσ ) + ω / L · d/σ

(10.503)

while: τ L = Lσ Finally, a and b are given by (8.32) and (8.33) respectively. Substituting L = α −2 µL + 2T, we obtain:

a = −2µ−1 (Lσ )(T σ ) − α −2 (Lσ )2 + |d/σ |2

b = −2µ−1 (Lψ α )(T ψα ) − α −2 (Lψ α )(Lψα ) + d/ψ α · d/ψα α

(ψ = (g

(10.504) (10.505)

−1 αβ

) ψβ )

Substituting the above expressions in the formulas defining the function gˇ we find: gˇ = gˇ 0 + µgˇ 1

(10.506)

where the functions gˇ 0 and gˇ 1 do not involve µ, and are given by: gˇ 0 = c0,1 (T σ )(Lσ ) + c0,2 (T σ )( ψ · d/σ ) +c0,3 (T σ )ω L L + c0,4 (T ψ α )(Lψα )

(10.507)

and: gˇ 1 = c1,1 (Lσ )2 + c1,2 (Lσ )( ψ · d/σ ) + c1,3 ( ψ · d/σ )2 + c1,4 |d/σ |2 +c1,5 (Lσ )ω L L + c1,6 (Lσ )ω L Tˆ + c1,7 ( ψ · d/σ )ω L L + c1,8 ω / L · d/σ (N)

+c1,9 (Lψ α )(Lψα ) + c1,10d/ψ α · d/ψα + tr α

The coefficients in (10.507) are given by:    2 σ 1 2 d H −1 d d H c0,1 = ψ L + − 4 (1 − σ H ) dσ dσ dσ 2     σ dH 2 9 −2 + α (2ψ L + 3αψTˆ ) + 4 (1 − σ H ) dσ 2  dH c0,2 = ψ L3 dσ   1 dH − Fψ0 ψ L c0,3 = 2ψ L 1−σH dσ dH c0,4 = −2ψ L2 dσ

(10.508)

(10.509) (10.510) (10.511) (10.512)

428

Chapter 10. Control of the Angular Derivatives

and the coefficients in (10.508) are given by:   σ dH 1 d c1,1 = − 2 dσ 1 − σ H dσ  ψL d d H d2 H 1 + 2 (ψ L + αψTˆ ) 2 − ψ L −1 2 dσ dσ α dσ     ψL 5 dH 2 σ − 2 (2ψ L + 3αψTˆ ) (10.513) +(ψ L + αψTˆ ) 2 (1 − σ H ) α dσ      σ ψL d2 H 3 dH 2 + 2 (3ψ L + 4αψTˆ ) (10.514) c1,2 = ψ L − 2 + − dσ 2 (1 − σ H ) α dσ  2 2 dH c1,3 = −ψ L (10.515) dσ   σ dH dH d 1 c1,4 = ψ L2 −1 (10.516) − 2 dσ dσ 1 − σ H dσ    2ψ L 1 dH 1 σ − 2 (ψ L + αψTˆ ) + 2 (2ψ L + αψTˆ ) c1,5 = Fψ0 dσ 1−σH α α (1 − σ H ) (10.517) ψL 1 dH c1,6 = (10.518) α (1 − σ H ) dσ   1 dH c1,7 = 2Fψ0 ψ L − (10.519) 1 − σ H dσ ψL d H (10.520) c1,8 = − 1 − σ H dσ ψ2 d H c1,9 = − 2L (10.521) α dσ dH (10.522) c1,10 = ψ L2 dσ (N)

Moreover, tr α is given by (8.82)–(8.85). Note that each term in gˇ 0 and gˇ 1 has two factors of order 1, the coefficients being of order 0 and bounded in L ∞ by a constant. Given a non-negative integer l let us denote by E / lT the bootstrap assumption that there is a constant C independent of s such that for all t ∈ [0, s]: E / lT

: max max Ril . . . Ri1 T ψα  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 i1 ...il

α

t

T the bootstrap assumption: We then denote by E / [l] T E / [l]

: E / 0T and . . . and E / lT

We shall presently show that under the assumptions of Proposition 10.1 with l∗ + 1 in / [lQ∗ +1] and E / [lT∗ +1] , we can the role of l, augmented by the bootstrap assumptions E / [l∗ +2] , E

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

429 ε

estimate the angular derivatives of order up to l + 1 of the functions gˇ 0 and gˇ 1 in L 2 (t 0 ). First, since ω L L = L µ (Lψµ ), ω L Tˆ = Tˆ i (Lψi ), Q

assumption E / [l∗ +1] together with Proposition 10.1 and the 2nd statement of Corollary 10.1.g with l∗ + 1 in the role of l to estimate the first factors, yields: max Rik . . . Ri1 ω L L  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2

: ∀k ≤ l∗ + 1

(10.523)

max Rik . . . Ri1 ω L Tˆ  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2

: ∀k ≤ l∗ + 1

(10.524)

i1 ...ik

t

i1 ...ik

t

Also, in estimating the kth order angular derivatives of ω L L or ω L Tˆ for k ≤ l + 1 in ε L 2 (t 0 ), it is not possible that more than l∗ + 1 angular derivatives fall on the factor L µ and at the same time more than l∗ angular derivatives fall on the factor Lψµ , or that more than l∗ + 1 angular derivatives fall on the factor Tˆ i and at the same time more than l∗ angular derivatives fall on the factor Lψi . It follows that, using Proposition 10.2 and the 1st statement of Corollary 10.2.g with l + 1 in the role of l to estimate the first factors, max Rik . . . Ri1 ω L L  L 2 ( ε0 ) t

1 2 Q ≤ Cl (1 + t)−1 W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l] + W[l+1]

i1 ...ik

: ∀k ≤ l + 1

(10.525)

max Rik . . . Ri1 ω L Tˆ  L 2 ( ε0 ) t

1 2 Q ≤ Cl (1 + t)−1 W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l] + W[l+1]

i1 ...ik

: ∀k ≤ l + 1

(10.526)

Recalling that (see (10.383))

ω / L = L µ d/ψµ ,

assumption E / [l∗ +2] together with the 2nd statement of Corollary 10.1.g with l∗ + 1 in the role of l to estimate the first factor, yields: / R ik . . . L / R i1 ω / L  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 max L

i1 ...ik

t

: ∀k ≤ l∗ + 1

(10.527) ε

Also, in estimating the kth angular derivatives of ω / L for k ≤ l + 1 in L 2 (t 0 ), it is not possible that more than l∗ + 1 angular derivatives fall on the factor L µ and at the same time more than l∗ angular derivatives fall on the factor d/ψµ . It follows that, using the 1st statement of Corollary 10.2.g with l + 1 in the role of l to estimate the first factor, / R ik . . . L / R i1 ω / L  L 2 ( ε0 ) max L t

 Q ≤ Cl (1 + t)−1 W[l+2] + δ0 (1 + t)−1 Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W[l] i1 ...ik

: ∀k ≤ l + 1

(10.528)

430

Chapter 10. Control of the Angular Derivatives

Recalling that (see (10.184)) ω / = (d/ψi ) ⊗ (d/x i ), assumption E / [l∗ +2] together with the 3rd statement of Corollary 10.1.a with l∗ + 1 in the role of l to estimate the second factor, yields: max L / R ik . . . L / R i1 ω / L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 : ∀k ≤ l∗ + 1

i1 ...ik

(10.529)

t

ε

Also, in estimating the kth angular derivatives of ω / for k ≤ l + 1 in L 2 (t 0 ), it is not possible that more than l∗ angular derivatives fall on the factor d/ψi and at the same time more than l∗ + 1 angular derivatives fall on the factor d/x i . It follows that, using the 2nd statement of Corollary 10.2.a with l + 1 in the role of l to estimate the second factor, / R ik . . . L / R i1 ω / L 2 ( ε0 ) max L t

 Q ≤ Cl (1 + t)−1 W[l+2] + δ0 (1 + t)−1 Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W[l] i1 ...ik

: ∀k ≤ l + 1

(10.530)

Consider now expression (10.507) for the function gˇ 0 . Applying Rik . . . Ri1 with k ≤ l + 1, one of the order 1 factors receives at most l∗ + 1 angular derivatives and ε Q is thus bounded in L ∞ (t 0 ) by assumptions E / [lT∗ +1] , E / [l∗ +1] , E / [l∗ +2] , and the estimate ε

Q

T (10.523). The other order 1 factor is then bounded in L 2 (t 0 ) by W[l+1] , W[l+1] , W[l+2] , or by the estimate (10.525). Also, if one of the order 1 factors receives more than l∗ + 1 angular derivatives, then the coefficients receive at most l∗ angular derivatives and are ε thus bounded in L ∞ (t 0 ) by Proposition 10.1 and its corollaries with l∗ in the role of l. Otherwise, both order 1 factors receive at most l∗ +1 angular derivatives and are placed in ε ε L ∞ (t 0 ), in which case the angular derivatives of the coefficients are placed in L 2 (t 0 ) using Proposition 10.2 and its corollaries with l + 1 in the role of l. This argument yields the conclusion:

max Rik . . . Ri1 gˇ 0  L 2 ( ε0 ) t

Q T ≤ Cl δ0 (1 + t)−2 W[l+1] + W[l+1] + δ0 (1 + t)−1 W[l+2] 1 2 +δ0 (1 + t)−1 Y0 + (1 + t)A[l] i1 ...ik

: ∀k ≤ l + 1

(10.531) (N)

Consider the now expression (10.508) for the difference gˇ 1 − tr α . Applying Rik . . . Ri1 with k ≤ l +1, one of the order 1 factors receives at most l∗ +1 angular derivaε Q tives and is thus bounded in L ∞ (t 0 ) by assumptions E / [l∗ +1] , E / [l∗ +2] , and the estimates ε (10.523), (10.524) and (10.527). The other order 1 factor is then bounded in L 2 (t 0 ) by Q W[l+1] , W[l+2] , or by the estimates (10.525), (10.526) and (10.528). Also, if one of the order 1 factors receives more than l∗ + 1 angular derivatives, then the coefficients receive

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

431 ε

at most l∗ angular derivatives and are thus bounded in L ∞ (t 0 ) by Proposition 10.1 and its corollaries with l∗ in the role of l. Otherwise, both order 1 factors receive at most l∗ +1 ε angular derivatives and are placed in L ∞ (t 0 ), in which case the angular derivatives of ε0 2 the coefficients are placed in L (t ) using Proposition 10.2 and its corollaries with l + 1 in the role of l. This argument yields the conclusion: (N)

max Rik . . . Ri1 (gˇ 1 − tr α ) L 2 ( ε0 ) t i1 ...ik

1 2 Q ≤ Cl δ0 (1 + t)−3 W[l+1] + W[l+2] + δ0 (1 + t)−1 Y0 + (1 + t)A[l] : ∀k ≤ l + 1

(10.532) (N)

A similar argument applies to the function tr α , given by (8.82)–(8.85). Here we appeal to the L ∞ estimates (10.523), (10.527) and (10.529) and to the L 2 estimates (10.525), / [l∗ +2] , and to Proposition (10.528) and (10.530). We also appeal to assumptions E / [lQ∗ +1] , E 10.1 and its corollaries with l∗ in the role of l and to Proposition 10.2 and its corollaries with l + 1 in the role of l. We obtain, in this way, (N)

max Rik . . . Ri1 tr α  L 2 ( ε0 ) t i1 ...ik

1 2 Q ≤ Cl δ0 (1 + t)−3 W[l+1] + W[l+2] + δ0 (1 + t)−1 Y0 + (1 + t)A[l] : ∀k ≤ l + 1

(10.533)

Combining the estimates (10.532) and (10.533) then yields: max Rik . . . Ri1 gˇ 1  L 2 ( ε0 ) t

1 2 Q ≤ Cl δ0 (1 + t)−3 W[l+1] + W[l+2] + δ0 (1 + t)−1 Y0 + (1 + t)A[l] i1 ...ik

: ∀k ≤ l + 1

(10.534)

Moreover, under the same assumptions we readily obtain: (N)

max Rik . . . Ri1 (gˇ 1 − tr α ) L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−4

i1 ...ik

t

(N)

max Rik . . . Ri1 tr α  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−4

i1 ...ik

t

: ∀k ≤ l∗ + 1

(10.535)

: ∀k ≤ l∗ + 1

(10.536)

: ∀k ≤ l∗ + 1

(10.537)

and, combining, max Rik . . . Ri1 gˇ 1  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−4

i1 ...ik

t

We now introduce bootstrap assumptions in regard to the angular derivatives of the function µ. Given a positive integer l let us denote by M / l the bootstrap assumption that there is a constant C independent of s such that for all t ∈ [0, s]: M /

l

: max Ril . . . Ri1 µ L ∞ ( ε0 ) ≤ Cδ0 [1 + log(1 + t)] i1 ...il

t

432

Chapter 10. Control of the Angular Derivatives

Let us also denote by M /

the bootstrap assumption:

0

M /

0

: |µ − η0 | ≤ Cδ0 [1 + log(1 + t)]

We then denote by M / [l] the bootstrap assumption : M / [l] : M / 0 and . . . and M /

l

We also introduce the quantities: B0 = µ − η0  L 2 ( ε0 ) t

and for k = 0: Bk = max Rik . . . Ri1 µ L 2 ( ε0 ) i1 ...ik

t

and:

B[l] =

l 

Bk

k=0

In view of the estimates (10.531), (10.534) and (10.537) we readily obtain, through (10.506), the following lemma. Lemma 10.15 Let hypothesis H0 and the estimate (10.30) hold. Let also the bootstrap / [lQ∗ +1] , E / [l∗ +2] , as well as X / [l∗ ] and M / [l∗ ] hold, for some positive assumptions E / [lT∗ +1] , E integer l. We then have: ˇ L 2 ( ε0 ) max d/ Ril . . . Ri1 g t i1 ...il

Q T ≤ Cl δ0 (1 + t)−3 W[l+1] + W[l+1] + (1 + t)−1 [1 + log(1 + t)]W[l+2]  2 1 +δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l] + δ0 (1 + t)−2 B[l+1] provided that δ0 is suitably small (depending on l). We now return to equation (8.58) for the St,u 1-form g0, to estimate the remaining terms. The function eˇ is given by (8.59): eˇ = e − e

(10.538)

σ dH 1 (Lσ ) 4 (1 − σ H ) dσ

(10.539)

where the function e is given by (8.53): e = e − e˜ −

Substituting for the functions e and e˜ from (10.500) and (10.501) we obtain:   1 dH 1 ψL σ e= (ψ L + αψTˆ ) − (Lσ ) − ψ L ( ψ · d/σ ) 2 dσ 2 (1 − σ H ) α2

(10.540)

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

433

and: eˇ = −



1 dH 2 dσ

 σ 1 ψL (2ψ + 3αψ ) − ( ψ · d / σ ) − Fψ0 ω L L (Lσ ) − ψ L L Tˆ 2 (1 − σ H ) α2 (10.541)

From (10.500), (10.540) and (10.541) we readily deduce, under the assumptions of Propo/ [lQ∗ +1] and E / [l∗ +2] , the sition 10.1 with l∗ + 1 in the role of l augmented by assumptions E estimates: ˇ L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 max Rik . . . Ri1 (e, e, e)

i1 ...ik

: ∀k ≤ l∗ + 1

t

(10.542)

and: max Rik . . . Ri1 (e, e, e) ˇ L 2 ( ε0 ) t

1 2 Q ≤ Cl (1 + t)−1 W[l+1] + δ0 (1 + t)−1 W[l+2] + Y0 + (1 + t)A[l] i1 ...ik

: ∀k ≤ l + 1

(10.543)

It follows from the estimates (10.542), (10.543) that, in regard to the second term in (8.58), / Ril . . . L / Ri1 (µtrχd/e) ˇ L 2 ( ε0 ) max L t i1 ...il

  Q ≤ Cl (1 + t)−3 [1 + log(1 + t)] W[l+1] + δ0 (1 + t)−1 W[l+2] + Y0 + (1 + t)A[l]  (10.544) +δ0 (1 + t)−1 B[l] under the assumptions of Lemma 10.15 (except for the assumption E / [lT∗ +1] , which is not needed here). We turn to the function f 0 , which appears in the 3rd term in (8.58). This function ( P)

is given by (8.56). Here the principal term, of order 2, is tr α , given by (8.57). As we ε shall need to estimate up to l∗ angular derivatives of f 0 in L ∞ (t 0 ), we must introduce QQ ε the bootstrap assumption E / [l∗ ] to estimate in L ∞ (t 0 ) up to l∗ angular derivatives of the factor L(Lσ ) in the third term in parenthesis in (8.57). Here, given a positive integer l we QQ denote by E / l the bootstrap assumption that there is a constant C independent of s such that for all t ∈ [0, s]: QQ

E /l

: max max Ril . . . Ri1 Q Qψα  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 i1 ...il

α

t

QQ

and we denote by E / [l] the bootstrap assumption : QQ E / [l]

: E / 0Q Q and . . . and E / lQ Q

434

Chapter 10. Control of the Angular Derivatives ( P)

ε

We shall estimate up to l∗ angular derivatives of tr α in L ∞ (t 0 ) under the assump/ [lQ∗Q] . From tions of Lemma 10.13 with l∗ in the role of l, augmented by the assumption E (10.461) and (10.463) together with Corollaries 10.1.b and 10.1.g with l∗ in the role of l, the angular derivatives of order up to l∗ of the first three terms in parenthesis in (8.57) are ε bounded in L ∞ (t 0 ) by: Cl δ0 (1 + t)−3 On the other hand, in view also of Corollaries 10.1.c and 10.1.h with l∗ + 1 in the role of l as well as the estimate (10.542) (l∗ + 1 reduced to l∗ suffices), the angular derivatives of ε order up to l∗ of the last term in parenthesis in (8.57) are bounded in L ∞ (t 0 ) by: Cl δ02 (1 + t)−4 It follows that: ( P)

max Rik . . . Ri1 tr α  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−3

i1 ...ik

t

: ∀k ≤ l∗

(10.545)

( P)

Next we shall estimate up to l angular derivatives of tr α under the assumptions of QQ Lemma 10.14 augmented by the assumption E / [l∗ −1] . Using the estimates (10.466) and (10.468) we deduce that the angular derivatives of order up to l of the first three terms in ε parenthesis in (8.57) are bounded in L 2 (t 0 ) by:

Q QQ Cl (1 + t)−2 W[l+2] + δ0 (1 + t)−1 W[l+1] + δ02 (1 + t)−2 W[l] 1 2 +δ0 (1 + t)−1 Y0 + (1 + t)A[l] Using Corollaries 10.2.c and 10.2.h with l + 1 in the role of l, as well as the estimate (10.543) with l + 1 reduced to l, we deduce that the angular derivatives of order up to l of ε the last term in parenthesis in (8.57) are bounded in L 2 (t 0 ) by:

1 2 Q Cl δ0 (1 + t)−3 W[l+1] + W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] Combining, we conclude that: ( P)

max Rik . . . Ri1 tr α  L 2 ( ε0 ) t

Q QQ ≤ Cl (1 + t)−2 W[l+2] + δ0 (1 + t)−1 W[l+1] + δ02 (1 + t)−2 W[l] 1 2 +δ0 (1 + t)−1 Y0 + (1 + t)A[l] i1 ...ik

: ∀k ≤ l

(10.546)

Returning to the formula (8.56) for f 0 , the angular derivatives of order up to l∗ of ε the first term on the right are bounded in L ∞ (t 0 ) by: Cl δ0 (1 + t)−3

(10.547)

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

435

by virtue of the estimate (10.542) with l∗ +1 reduced to l∗ together with assumption X / [l∗ ] . ε Also, the angular derivatives of order up to l of this term are bounded in L 2 (t 0 ) by:

1 2 Q Cl (1 + t)−2 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l] + W[l+1] (10.548) by virtue of the estimate (10.543) with l + 1 reduced to l. Moreover, under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l we readily deduce that the angular derivatives of order up to l∗ of the second term on the right in (8.56) are bounded in ε L ∞ (t 0 ) by: Cl δ02 (1 + t)−4 (10.549) ε

while the angular derivatives of order up to l of this term are bounded in L 2 (t 0 ) by:

 Q Cl δ0 (1 + t)−3 W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W[l−1] (10.550) Combining the above estimates (10.547)–(10.550) with the estimates (10.545), (10.546) for the third term in (8.56) and the estimates (10.533) with l + 1 reduced to l and (10.536) with l∗ + 1 reduced to l∗ for the last term in (8.56), yields the following lemma. Lemma 10.16 Let the hypotheses H0, H1, H2 , and the estimate (10.30) hold. Let also Q QQ / [l∗ +1] , E / [l∗ ] and X / [l∗ ] hold, for some positive integer the bootstrap assumptions E / [l∗ +2] , E l. Then the function f 0 satisfies the estimates: max Rik . . . Ri1 f 0  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−3

i1 ...ik

t

: ∀k ≤ l∗

max Rik . . . Ri1 f 0  L 2 ( ε0 ) t

Q QQ ≤ Cl (1 + t)−2 W[l+2] + δ0 (1 + t)−1 W[l+1] + δ02 (1 + t)−2 W[l] 1 2 Q +W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l] i1 ...ik

: ∀k ≤ l provided that δ0 is suitably small (depending on l). Let us consider now the formula (8.58) for the St,u 1-form g0. The angular derivatives of order up to l of the first term on the right are controlled by Lemma 10.15 and / [l∗ ] together with the estimates (10.542), hypothesis H0. Using assumptions M / [l∗ ] and X (10.543), we readily deduce that the angular derivatives of order l of the second term on ε the right are bounded in L 2 (t 0 ) by:

  Q Cl (1 + t)−3 [1 + log(1 + t)] W[l+1] + δ0 (1 + t)−1 W[l+2] + Y0 + (1 + t)A[l]  +δ0 (1 + t)−1 B[l] (10.551)

436

Chapter 10. Control of the Angular Derivatives

Finally, using assumption M / [l∗ +1] together with Lemma 10.16 we deduce that the angular ε derivatives of order l of the third term on the right in (8.58) are bounded in L 2 (t 0 ) by:

Q QQ Cl δ0 (1 + t)−3 [1 + log(1 + t)] W[l+2] + δ0 (1 + t)−1 W[l+1] + δ02 (1 + t)−2 W[l]    Q +W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l] + (1 + t)−1 B[l+1] (10.552) Combining the above estimates yields the following lemma. Lemma 10.17 Let the hypotheses H0, H1, H2 and the estimate (10.30) hold. Let also Q QQ / [l∗ +1] , E / [l∗ ] and E / [lT∗ +1] , as well as X / [l∗ ] and M / [l∗ +1] the bootstrap assumptions E / [l∗ +2] , E hold, for some positive integer l. Then if δ0 is suitably small (depending on l) we have: / Ril . . . L / Ri1 g0  L 2 ( ε0 ) ≤ Cl (1 + t)−3 · max L t i1 ...il

  Q QQ T · [1 + log(1 + t)] W[l+1] + δ0 W[l+2] + δ03 (1 + t)−2 W[l] + δ0 W[l+1]    +δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l] + δ0 (1 + t)−1 B[l+1] We turn to the third term in the formula (10.496) for l−1 

L / Ril . . . L / Ril−k+1

(i1 ...il−k )

(i1 ...il ) g˙ , l

namely the sum:

yl−k

(10.553)

k=0

where the St,u 1-form (i1 ...i j ) y j , j = 1, . . . , l is given in the statement of Proposition 8.3. Substituting for Lµ from (10.498), and recalling definition (8.67), we have: (i1 ...i j )

y j = (Ri j µ) (i1 ...i j −1 ) a j −1  + µRi j (trχ − e) − Ri j m +

(Ri j )

 Z µ − e Ri j µ · d/(Ri j −1 . . . Ri1 trχ)

1 (Ri j trχ)d/(Ri j −1 . . . Ri1 fˇ) 2 : j = 1, . . . , l The St,u 1-form

(i1 ...i j −1 ) a

j −1

(i1 ...i j −1 )

where

(i1 ...i j −1 ) b (i1 ...i j −1 )

j −1

(10.554)

is given by formulas (8.79)–(8.81):

a j −1 = d/(Ri j −1 . . . Ri1 f0 ) +

(i1 ...i j −1 )

b j −1

(10.555)

is the St,u 1-form:

b j −1 =

j −2 

d/(Ri j −1 . . . Ri j −m

(Ri j −m−1 )

Z Ri j −m−2 . . . Ri1 trχ)

(10.556)

m=0

+

j −1 

j −1 

m=1 k1 <···
for j ≥ 2; b0 = 0.

(Rikm . . . Rik1 trχ)(d/ Ri j −1

ikm ...ik1 

...

Ri1 trχ)

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

437

As discussed in Chapter 8, the principal term of (i1 ...i j ) y j is contained in the factor j −1 of the first term on the right in (10.554) and the principal term of the latter is the term d/(Ri j −1 . . . Ri1 f 0 ), the first term on the right in (10.555), which has already been ε estimated by Lemma 10.16. To estimate the sum (10.553) in L 2 (t 0 ), we must estimate: (i1 ...i j −1 ) a

max L / Ri j +k . . . L / Ri j +1

(i1 ...i j )

i1 ...i j +k

y j  L 2 ( ε0 )

: for j + k = l

t

(10.557)

To estimate this requires estimating: max L / Ri j −1+k . . . L / Ri j

(i1 ...i j −1 )

a j −1 L 2 ( ε0 )

: for j + k ≤ l

(10.558)

a j −1  L ∞ ( ε0 )

: for j + k ≤ l∗

(10.559)

i1 ...i j −1+k

t

as well as: max L / Ri j −1+k . . . L / Ri j

(i1 ...i j −1 )

i1 ...i j −1+k

t

Since the first term on the right in (10.555) is already controlled by Lemma 10.16, estimating (10.558) and (10.559) reduces to estimating: max L / Ri j −1+k . . . L / Ri j

(i1 ...i j −1 )

i1 ...i j −1+k

b j −1 L 2 ( ε0 )

: for j + k ≤ l

(10.560)

b j −1  L ∞ ( ε0 )

: for j + k ≤ l∗

(10.561)

t

and: max L / Ri j −1+k . . . L / Ri j

(i1 ...i j −1 )

i1 ...i j −1+k

t

respectively. Consider then a given term in the first sum on the right in (10.556), the term: d/(Ri j −1 . . . Ri j −m

(Ri j −m−1 )

Z Ri j −m−2 . . . Ri1 trχ)

corresponding to a given m ∈ {0, . . . , j −2}. Applying L / Ri j −1+k . . . L / Ri j to this we obtain: / Ri j d/(Ri j −1 . . . Ri j −m L / Ri j −1+k . . . L

(Ri j −m−1 )

= d/(Ri j −1+k . . . Ri j Ri j −1 . . . Ri j −m =L / Ri j −1+k . . . L / Ri j −m d/(

(Ri j −m−1 )

Z Ri j −m−2 . . . Ri1 trχ)

(Ri j −m−1 )

Z Ri j −m−2 . . . Ri1 trχ)

Z Ri j −m−2 . . . Ri1 trχ)

=L / Ri j −1+k . . . L / Ri j −m L / (Ri j −m−1 ) (d/ Ri j −m−2 . . . Ri1 trχ) Z

/ Ri j −1+k . . . L / Ri j −m (d/ Ri j −m−2 . . . Ri1 trχ) =L / (Ri j −m−1 ) L Z

/ Ri j −1+k . . . L / Ri j −m ](d/ Ri j −m−2 . . . Ri1 trχ) − [L / (Ri j −m−1 ) , L Z

(10.562)

The first term on the right in the last of the above equalities is: L / (Ri j −m−1 ) d/(Ri j −1+k . . . Ri j −m Ri j −m−2 . . . Ri1 trχ) = L / X d/ρ Z

(10.563)

438

Chapter 10. Control of the Angular Derivatives

where we have taken X to be the St,u -tangential vectorfield: X=

(Ri j −m−1 )

Z

(10.564)

and ρ to be the function: ρ = Ri j −1+k . . . Ri j −m Ri j −m−2 . . . Ri1 trχ

(10.565)

We can estimate (10.563) pointwise by Lemma 8.6, taking in that lemma the function φ to be zero, so that the St,u 1-form ξ is simply: ξ = d/ρ

ηi = d/(Ri ρ)

and:

: i = 1, 2, 3

Lemma 8.6 then reduces to the pointwise estimate:    |L / X d/ρ| ≤ C(1 + t)−1 3|X| max |d/(Ri ρ)| + max |L / Ri X| |d/ρ| i

(10.566)

i

ε

Taking L ∞ norms on t 0 of both sides of (10.566) yields:  (10.567) L / X d/ρ L ∞ ( ε0 ) ≤ C(1 + t)−1 3X L ∞ ( ε0 ) max d/ Ri ρ L ∞ ( ε0 ) t t t i   + max L / Ri X L ∞ ( ε0 ) d/ρ L ∞ ( ε0 ) t

i

Also, since max |d/(Ri ρ)| ≤ i

we have:  max |d/(Ri ρ)| L 2 ( ε0 ) ≤

|d/(Ri ρ)|,

i



t

i



t

d/(Ri ρ) L 2 ( ε0 ) ≤ 3 max d/(Ri ρ) L 2 ( ε0 ) t

i

t

i

ε

hence, taking L 2 norms on t 0 of both sides of (10.566) we obtain:  (10.568) L / X d/ρ L 2 ( ε0 ) ≤ C(1 + t)−1 9X L ∞ ( ε0 ) max d/(Ri ρ) L 2 ( ε0 ) t t t i   + max L / Ri X L ∞ ( ε0 ) d/ρ L 2 ( ε0 ) i

t

t

Now, by Corollary 10.1.i with l∗ + 1 in the role of l we have: / Ri X L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] X L ∞ ( ε0 ) , max L t

i

t

(10.569)

provided that l∗ ≥ 1. Under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l, in particular the bootstrap assumption X / [l∗ ] , we have, in view of the fact that ρ is an angular derivative of trχ of order j + k − 2, d/ρ L ∞ ( ε0 ) , max d/(Ri ρ) L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] t

i

t

(10.570)

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

439

if j + k ≤ l∗ . It then follows from (10.567) that: L / X d/ρ L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−5 [1 + log(1 + t)]2

(10.571)

t

if j + k ≤ l∗ , hence, in regard to the first term on the right in the last of the equalities (10.562) (see (10.563)) we have the L ∞ estimate: L / (Ri j −m−1 ) L / Ri j −1+k . . . L / Ri j −m d/(Ri j −m−2 . . . Ri1 trχ) L ∞ ( ε0 ) Z

t

Cl δ02 (1 +

≤ t) : if j + k ≤ l∗

−5

[1 + log(1 + t)]

2

(10.572)

Moreover, under the same assumptions we have: d/ρ L 2 ( ε0 ) , max d/(Ri ρ) L 2 ( ε0 ) (10.573) t t i

 Q ≤ Cl (1 + t)−1 A[l] + (1 + t)−1 Y0 + W[l] + δ02 (1 + t)−2 W[l−1] if j + k ≤ l. It then follows from (10.568) that: L / X d/ρ L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] ·

t  Q · A[l] + (1 + t)−1 Y0 + W[l] + δ02 (1 + t)−2 W[l−1]

(10.574)

if j + k ≤ l, hence, in regard to the same term we have the L 2 estimate: L /

(Ri ) j −m−1

Z

L / Ri j −1+k . . . L / Ri j −m d/(Ri j −m−2 . . . Ri1 trχ) L 2 ( ε0 ) t

−3

≤ Cl δ0 (1 + t) [1 + log(1 + t)] · 

Q · A[l] + (1 + t)−1 Y0 + W[l] + δ02 (1 + t)−2 W[l−1] : if j + k ≤ l

(10.575)

We turn to the second term on the right in the last of the equalities (10.562) (the commutator). This is expressed by Lemma 8.5, taking X to be the St,u -tangential vectorfield of (10.564) and ξ to be the St,u 1-form:

where φ is the function:

ξ = d/φ

(10.576)

φ = Ri j −m−2 . . . Ri1 trχ

(10.577)

Denoting, as in (8.91), (in 1 ...inr ;i j −m−1 )

Z =L / R in r . . . L / R in

(i ...i )

(Ri j −m−1 ) 1

the St,u -tangential vectorfields k1 k j Y and St,u 1-forms Lemma 8.5, are in the present case given by: (in 1 ...inr )

Y =

(in 1 ...inr ;i j −m−1 )

Z,

(i1 ...im )

Z

(i1 ...im ) ξ m

(10.578) in the statement of

ξm = d/(Rim . . . Ri1 φ)

(10.579)

440

Chapter 10. Control of the Angular Derivatives

Thus by Lemma 8.5 the second term on the right in the last of the equalities (10.562) takes the form: / Ri j −1+k . . . L / Ri j −m ]d/φ [L / (Ri j −m−1 ) , L =−

Z k+m 

j −1+k 

r=1 n 1 <···
(10.580)

L / (in1 ...inr ;i j −m−1 ) Z d/ Ri j −1+k

inr ...in 1 

...

Ri j −m φ

To estimate a given term in the double sum we appeal to the pointwise estimate (10.566), taking now: X =

(in 1 ...inr ;i j −m−1 )

Z

(10.581)

ρ = Ri j −1+k

inr ...in 1 

Ri j −m φ

= Ri j −1+k

inr ...in 1 

Ri j −m Ri j −m−2 . . . Ri1 trχ

... ...

(R

(10.582)

)

i j −m−1 Z of order r , while ρ is an angular Note that X is an angular derivative of derivative of trχ of order k + j − 2 − r . Consider first L ∞ estimates for j + k ≤ l∗ . Since m ∈ {0, . . . , j − 2} we then have:

r ≤ k + m ≤ k + j − 2 ≤ l∗ − 2 Thus Corollary 10.1.i with l∗ in the role of l allows us to estimate: X L ∞ ( ε0 ) , max L / Ri X L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(10.583)

t

i

Moreover, since k + j − 2 − r ≤ k + j ≤ k + j − 3 ≤ l∗ − 3 / [l∗ −1] ) allow us the assumptions of Proposition 10.1 with l∗ in the role of l (in particular X to estimate: d/ρ L ∞ ( ε0 ) , max d/(Ri ρ) L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] t

t

i

(10.584)

It then follows from (10.567) that: L / X d/ρ L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−5 [1 + log(1 + t)]2

(10.585)

t

if j + k ≤ l∗ , hence, in regard to the second term on the right in the last of the equalities (10.562) (see (10.563)) we have the L ∞ estimate: [L /

) (Ri j −m−1

Z

,L / Ri j −1+k . . . L / Ri j −m ]d/(Ri j −m−2 . . . Ri1 trχ) L ∞ ( ε0 )

≤ Cl δ02 (1 + t)−5 [1 + log(1 + t)]2 : if j + k ≤ l∗

t

(10.586)

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

441

Consider next L 2 estimates for j + k ≤ l. If r ≤ l∗ − 1, then, under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l, Corollary 10.1.i with l∗ + 1 in the role of l holds, hence the estimates (10.583) hold as well. Moreover, since, in any case, k + j − 2 − r ≤ l − 2 − r ≤ l − 3, we have: (10.587) d/ρ L 2 ( ε0 ) , max d/(Ri ρ) L 2 ( ε0 ) t t i

 Q ≤ Cl (1 + t)−1 A[l−1] + (1 + t)−1 Y0 + W[l−1] + δ02 (1 + t)−2 W[l−2] for j + k ≤ l. It then follows from (10.568) that: L / X d/ρ L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] · (10.588)

t  Q · A[l−1] + (1 + t)−1 Y0 + W[l−1] + δ02 (1 + t)−2 W[l−2] if j + k ≤ l. If r = l∗ , then we still have the the first of the estimates (10.583): X L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

(10.589)

t

/ Ri X. but we no longer have the second, that is, we no longer have an L ∞ estimate for L Instead, we place the factor maxi |L / Ri X| in the second term in parenthesis on the righthand side in (10.566) in L 2 in taking L 2 norms of both sides of (10.566), using the fact that: / Ri X| L 2 ( ε0 ) ≤ 3 max L / Ri X L 2 ( ε0 ) ,  max |L t

i

and we place the factor |d/ρ| in L / X d/ρ L 2 ( ε0 ) t

t

i

L∞,

obtaining:  ≤ C(1 + t)−1 9X L ∞ ( ε0 ) max d/(Ri ρ) L 2 ( ε0 ) (10.590) t t i   +3 max L / Ri X L 2 ( ε0 ) d/ρ L ∞ ( ε0 ) i

t

t

Since in the case r = l∗ we have: k + j − 2 − r ≤ l − 2 − l∗ ≤ l∗ − 1, we can estimate, under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l (in particular X / [l∗ ] ): d/ρ L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] t

Moreover, since, in any case, r ≤ k + m ≤ k + j − 2 ≤ l − 2,

(10.591)

442

Chapter 10. Control of the Angular Derivatives

we have, by Corollary 10.2.i, X L 2 ( ε0 ) , max L / Ri X L 2 ( ε0 ) t t i

≤ Cl Y0 + (1 + t)A[l−1] + δ0 (1 + t)−1 [1 + log(1 + t)]W[l] + W[l−1]  Q +δ0 (1 + t)−1 W[l−1] (10.592) Using (10.589), the second of (10.587), the second of (10.592), and (10.591), we obtain from (10.590): L / X d/ρ L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] · t

1 2 · A[l−1] + (1 + t)−1 Y0 + W[l−1] + δ0 (1 + t)−2 [1 + log(1 + t)]W[l]  Q +δ0 (1 + t)−2 W[l−1] (10.593) if j +k ≤ l. Finally, if r ≥ l∗ +1, also in the first term in parenthesis on the right-hand side in (10.566) we place the factor |X| in L 2 in taking L 2 norms of both sides of (10.566), and the factor maxi |d/(Ri ρ)| in L ∞ , obtaining:  (10.594) L / X d/ρ L 2 ( ε0 ) ≤ C(1 + t)−1 3X L 2 ( ε0 ) max d/(Ri ρ) L ∞ ( ε0 ) t t t i   / Ri X L 2 ( ε0 ) d/ρ L ∞ ( ε0 ) +3 max L i

t

t

Since in the case r ≥ l∗ + 1 we have: k + j − 2 − r ≤ l − 3 − l∗ ≤ l∗ − 2, we can estimate, under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l (in particular X / [l∗ ] ): d/ρ L ∞ ( ε0 ) , max d/(Ri ρ) L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] t

i

t

(10.595)

Together with the estimates (10.592), these imply, through (10.594), L / X d/ρ L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] · t

1 2 · A[l−1] + (1 + t)−1 Y0 + W[l−1] + δ0 (1 + t)−2 [1 + log(1 + t)]W[l]  Q (10.596) +δ0 (1 + t)−2 W[l−1] if j + k ≤ l. Combining the results (10.588), (10.593), (10.596), of the three cases r ≤ l∗ − 1, r = l∗ , r ≥ l∗ + 1, respectively, yields: L / X d/ρ L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] · t

1 2 · A[l−1] + (1 + t)−1 Y0 + W[l−1] + δ0 (1 + t)−2 [1 + log(1 + t)]W[l]  Q (10.597) +δ0 (1 + t)−2 W[l−1]

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

443

if j + k ≤ l, hence, in regard to the second term on the right in the last of the equalities (10.562) (see (10.563)) we have the L 2 estimate: / Ri j −1+k . . . L / Ri j −m ]d/(Ri j −m−2 . . . Ri1 trχ) L 2 ( ε0 ) [L / (Ri j −m−1 ) , L Z

t

≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)]

1 2 · A[l−1] + (1 + t)−1 Y0 + W[l−1] + δ0 (1 + t)−2 [1 + log(1 + t)]W[l]  Q +δ0 (1 + t)−2 W[l−1] : if j + k ≤ l

(10.598)

Combining the L ∞ estimates (10.572) and (10.586) we obtain, through (10.562), the following L ∞ estimate in regard to the first sum on the right in (10.556): 0  0 j −2  0 0L  / Ri j d/(Ri j −1 . . . Ri j −m 0 / Ri j −1+k . . . L 0 m=0

0 0 0 (Ri j −m−1 )  Z Ri j −m−2 . . . Ri1 trχ) 0 0 0

≤ Cl δ02 (1 + t)−5 [1 + log(1 + t)]2 : if j + k ≤ l∗

ε

L ∞ (t 0 )

(10.599)

Also, combining the L 2 estimates (10.575) and (10.598) we obtain the following L 2 estimate in regard to the same sum: 0  0 0 0 j −2  0 0 (Ri j −m−1 ) 0L 0   . . . L / d / (R . . . R Z R . . . R trχ) / Ri j i j −1 i j −m i j −m−2 i1 0 Ri j −1+k 0 0 0 2 ε0 m=0 L (t ) 

Q ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] A[l] + (1 + t)−1 Y0 + W[l] + δ0 (1 + t)−1 W[l−1] : if j + k ≤ l

(10.600)

Consider next a given term in the second sum on the right in (10.556), the term: (Rikm . . . Rik1 trχ)(d/ Ri j −1

ikm ...ik1 

...

Ri1 trχ)

corresponding to a given m ∈ {1, . . . , j − 1}. The first factor is an angular derivative of trχ of order m, while the second factor is / Ri j , that is k angular an angular derivative of trχ of order j − m. Applying L / Ri j −1+k . . . L derivatives, at most k fall on each factor. So, the first factor contributes at most m + k ≤ j + k − 1 angular derivatives of trχ and the second factor contributes at most j − m + k ≤ j + k − 1 angular derivatives of trχ as well. Thus, if j + k ≤ l∗ , we obtain, under the assumptions of Proposition 10.1 with only l∗ in the role of l, the following L ∞ estimate

444

Chapter 10. Control of the Angular Derivatives

in regard to the second sum in (10.556): 0 0  0 0 j −1 j −1   0 0 ikm ...ik1  0L   / Ri j (Rikm ... Rik1 trχ)(d/ Ri j −1 . . . Ri1 trχ) 0 0 0 / Ri j −1+k ...L 0 0 m=1 k1 <···
ε

L ∞ (t 0 )

≤ Cl δ02 (1 + t)−5 [1 + log(1 + t)]2 : if j + k ≤l∗

(10.601)

Also, let j + k ≤ l. If at most l∗ − m angular derivatives fall on the first factor we can place the first factor in L ∞ using the assumptions of Proposition 10.1 with l∗ + 1 in the role of l. Otherwise, l∗ + 1 − m or more angular derivatives fall on the first factor, so at most k + m − l∗ − 1 fall on the second and, since ( j − m) + (k + m − l∗ − 1) ≤ l − l∗ − 1 ≤ l∗ , the second factor can then be placed in L ∞ using the same assumptions, and a similar L 2 bound results. We thus deduce, under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l, the following L 2 estimate in regard to the second sum in (10.556): 0 0  0 0 j −1 j −1   0 0 i ...i  km k1 0L   / Ri j (Rikm ... Rik1 trχ)(d/ Ri j −1 . . . Ri1 trχ) 0 0 0 / Ri j −1+k ...L 0 2 ε0 0 m=1 k1 <···
−3

≤ Cl δ0 (1 + t) [1 + log(1 + t)]· 

Q · A[l−1] + (1 + t)−1 Y0 + W[l−1] + δ02 (1 + t)−2 W[l−2] : if j + k ≤l

(10.602)

Combining the L ∞ estimates for the two terms on the right in (10.556), (10.599) and (10.601), yields: max L / Ri j −1+k . . . L / Ri j

(i1 ...i j −1 )

i1 ...i j −1+k

b j −1  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−5 [1 + log(1 + t)]2 t

: ∀ j + k ≤ l∗

(10.603)

under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l. Also, combining the L 2 estimates for the two terms on the right in (10.556), (10.600) and (10.602), yields: / Ri j −1+k . . . L / Ri j max L

i1 ...i j −1+k

(i1 ...i j −1 )

b j −1  L 2 ( ε0 ) t

−4

≤ Cl δ0 (1 + t) [1 + log(1 + t)] · 

Q · (1 + t)A[l] + Y0 + W[l] + δ0 (1 + t)−1 W[l−1] : ∀j + k ≤ l

(10.604)

under the same assumptions. The above estimates together with Lemma 10.16 then yield the following lemma.

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

445

Lemma 10.18 Let the hypotheses H0, H1, H2 , and the estimate (10.30) hold. Let also Q QQ / [l∗ +1] , E / [l∗ ] and X / [l∗ ] hold, for some positive integer the bootstrap assumptions E / [l∗ +2] , E l. Then, provided that δ0 is suitably small (depending on l), the St,u 1-form (i1 ...i j −1 ) a j −1 satisfies the following estimates: max L / Ri j −1+k . . . L / Ri j

(i1 ...i j −1 )

i1 ...i j −1+k

: ∀ j + k ≤ l∗ max L / Ri j −1+k . . . L / Ri j

i1 ...i j −1+k

a j −1  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−4 t

(i1 ...i j −1 )

a j −1  L 2 ( ε0 ) t

≤ Cl (1 + t)−3 W[l+2] + δ0 (1 + t)−1 W[l+1] + δ02 (1 + t)−2 W[l] 1 2 Q +W[l] + δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l] Q

QQ

: ∀j + k ≤ l We proceed to estimate (10.557). We are to apply L / Ri j +k . . . L / Ri j +1 to (i1 ...i j ) y j , given by equation (10.554) (note that j ≥ 1). In regard to the first term on the right in (10.554), using Lemma 10.18 and the bootstrap assumption M / [l∗ +1] we readily obtain: / Ri j +1 ((Ri j µ) L / Ri j +k . . . L

(i1 ...i j −1 )

a j −1 ) L 2 ( ε0 ) t

≤ Cl δ0 (1 + t)−4 [1 + log(1 + t)] ·

Q QQ · W[l+2] + δ0 (1 + t)−1 W[l+1] + δ02 (1 + t)−2 W[l] 2 1 Q +W[l] + δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l] + Cl δ0 (1 + t)−4 B[l] : for j + k = l

(10.605)

We consider next the second term on the right in (10.554). Here we must consider separately the contribution of the term µRi j trχ in the first factor. For, in estimating the kth order angular derivatives of the product (µRi j trχ)(d/ Ri j −1 . . . Ri1 trχ), where j + k = l, it is possible in the case that l is odd, so that l = 2l∗ + 1, that l∗ angular derivatives fall on the first factor and at the same time l∗ + 1 − j angular derivatives fall on the second. In this case, both factors will contribute angular derivatives of trχ of order l∗ + 1. Since one of these factors must be placed in L ∞ , we are forced to appeal to the stronger bootstrap assumption X / [l∗ +1] to handle this case. On the other hand, in the case that l is even no such difficulty arises and we only require the bootstrap assumption X / [l∗ ] . Noting that, in general,  l : if l is even (l + 1)∗ = ∗ (10.606) l∗ + 1 : if l is odd and: l∗ + (l + 1)∗ = l

(10.607)

446

Chapter 10. Control of the Angular Derivatives

we deduce, under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l aug/ [l∗ ] the estimate: mented by the bootstrap assumptions X / [(l+1)∗ ] and M L / Ri j +k . . . L / Ri j +1 ((µRi j trχ)(d/ Ri j −1 . . . Ri1 trχ)) L 2 ( ε0 ) ≤ t

Cl δ0 (1 + t)−4 [1 + log(1 + t)]2 · 

Q · (1 + t)A[l] + Y0 + δ0 (1 + t)−1 B[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] : for j + k = l

(10.608)

Consider next the contribution of the term Ri j m. The function m is given by (10.499). Using the assumptions of Proposition 10.1 with l∗ in the role of l augmented by the assumption E / [lT∗ ] we readily deduce: max Rik . . . Ri1 m L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1

i1 ...ik

t

: ∀k ≤ l∗

(10.609)

and, using the 2nd statement of Corollary 10.2.g, max Rik . . . Ri1 m L 2 ( ε0 ) t

T ≤ Cl W[l] + δ0 (1 + t)−1 W[l] i1 ...ik

+δ02 (1 + t)−2 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] Q

: ∀k ≤ l

 (10.610)

To estimate then the kth angular derivatives of the product (Ri j m)(d/ Ri j −1 . . . Ri1 trχ), where j + k = l, if at most l∗ − 1 angular derivatives fall on the first factor we place the first factor in L ∞ using the estimate (10.609). Otherwise, l∗ or more angular derivatives fall on the first factor, so at most k − l∗ fall on the second. Since k − l∗ + j = l − l∗ = (l + 1)∗ we can then place the second factor in L ∞ using assumption X / [(l+1)∗ ] . In view of the fact that the first factor contributes angular derivatives of m of order at most k + 1 ≤ l, so the estimate (10.610) applies, and the second factor contributes at most k + j = l angular derivatives of trχ, we obtain: / Ri j +1 ((Ri j m)(d/ Ri j −1 . . . Ri1 trχ)) L 2 ( ε0 ) L / Ri j +k . . . L t

 Q −3 T ≤ Cl δ0 (1 + t) (1 + t)A[l] + W[l] + Y0 + W[l] + δ02 (1 + t)−2 W[l−1] : for j + k = l

(10.611)

under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l augmented by the / [(l+1)∗ ] . assumptions E / [lT∗ ] and X

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

447

In a similar manner we can estimate the contribution of the term µRi j e in the first factor of the second term on the right in (10.554). Using the estimates (10.542) with l∗ +1 reduced to l∗ and (10.543) with l + 1 reduced to l we obtain: / Ri j +1 ((µRi j e)(d/ Ri j −1 . . . Ri1 trχ)) L 2 ( ε0 ) L / Ri j +k . . . L t

≤ Cl δ0 (1 + t)−4 [1 + log(1 + t)] ·

· (1 + t)A[l] + Y0 + δ0 (1 + t)−1 B[l−1]    Q +[1 + log(1 + t)] W[l] + δ0 (1 + t)−1 W[l+1] + W[l] : for j + k = l

(10.612)

under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l augmented by the assumptions X / [(l+1)∗ ] and M / [l∗ ] . (R )

ij Z µ − e Ri j µ in the first factor can The contribution of the remaining terms be estimated in a straightforward manner under the assumptions of Proposition 10.1 with / [l∗ +1] . We obtain: l∗ + 1 in the role of l, augmented by the assumption M 0  0 0 0 (R ) / Ri j +k . . . L / Ri j +1 ( i j Z µ − e Ri j µ)d/(Ri j −1 . . . Ri1 trχ) 0 2 ε0 0L

L (t )

≤

Cl δ02 (1 +

−4

2

t) [1 + log(1 + t)] · 

Q · (1 + t)A[l] + Y0 + (1 + t)−1 B[l] + W[l] + W[l−1]

: for j + k = l

(10.613)

We turn to the last term on the right in (10.554). Let us recall equation (8.54): fˇ = −2(m + µe)

(10.614)

From the L ∞ estimates (10.542) with l∗ + 1 reduced to l∗ and (10.609), together with assumption M / [l∗ ] , we obtain: max Rik . . . Ri1 fˇ L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1

i1 ...ik

t

: ∀k ≤ l∗

(10.615)

From the L 2 estimates (10.543) with l + 1 reduced to l and (10.610), together with assumption M / [l∗ ] , we obtain: max Rik . . . Ri1 fˇ L 2 ( ε0 ) t

  T ≤ Cl W[l] + δ0 (1 + t)−1 W[l] + (1 + t)−1 B[l]

i1 ...ik

  Q +(1 + t)−1 [1 + log(1 + t)] W[l] + δ0 (1 + t)−1 W[l+1] + Y0 + (1 + t)A[l−1] : ∀k ≤ l

(10.616)

448

Chapter 10. Control of the Angular Derivatives

Then to estimate k angular derivatives of the product (Ri j trχ)(d/ Ri j −1 . . . Ri1 fˇ), where j + k = l, if at most l∗ − j angular derivatives fall on the second factor we place the second factor in L ∞ using the estimate (10.615). Otherwise l∗ − j +1 or more angular derivatives fall on the second factor so at most k + j − l∗ − 1 = l − l∗ − 1 = (l + 1)∗ − 1 fall on the first factor. We can then place the first factor in L ∞ using assumption X / [(l+1)∗ ] . In view of the fact that the first factor contributes angular derivatives of trχ of order at most k + 1 ≤ l, while the second factor contributes at most k + j = l angular derivatives of fˇ so the estimate (10.616) applies, we obtain: L / Ri j +k . . . L / Ri j +1 ((Ri j trχ)(d/ Ri j −1 . . . Ri1 fˇ)) L 2 ( ε0 ) t  ≤ Cl δ0 (1 + t)−3 Y0 + (1 + t)A[l] + W[l]

  Q T +[1 + log(1 + t)] W[l] + (1 + t)−1 [1 + log(1 + t)] W[l] + δ0 (1 + t)−1 W[l+1]  +δ0 (1 + t)−2 B[l] : for j + k = l

(10.617)

From the estimates (10.605), (10.608), (10.611)–(10.613), (10.617), we conclude that: max L / Ri j +k . . . L / Ri j +1 (i1 ...i j ) y j  L 2 ( ε0 ) t

≤ Cl δ0 (1 + t)−3 (1 + t)−1 [1 + log(1 + t)]   Q QQ × W[l+2] + δ0 (1 + t)−1 W[l+1] + δ02 (1 + t)−2 W[l]   Q T +[1 + log(1 + t)] W[l] + (1 + t)−1 [1 + log(1 + t)]W[l] + W[l]  +Y0 + (1 + t)A[l] + (1 + t)−1 B[l]

i1 ...i j +k

: for j + k = l

(10.618)

This yields the following lemma. Lemma 10.19 Let the hypotheses H0, H1, H2 , and the estimate (10.30) hold. Let Q QQ / [l∗ +1] , E / [l∗ ] , E / [lT∗ ] , X / [(l+1)∗ ] and M / [l∗ +1] hold, also the bootstrap assumptions E / [l∗ +2] , E for some positive integer l. Then, provided that δ0 is suitably small (depending on l),

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

449

we have:

0 l−1 0 0 0 0 0 (i1 ...il−k ) L / Ril . . . L / Ril−k+1 yl−k 0 max 0 0 i1 ...il 0 ε k=0 L 2 (t 0 )

 Q ≤ Cl δ0 (1 + t)−3 (1 + t)−1 [1 + log(1 + t)] W[l+2] + δ0 (1 + t)−1 W[l+1]  QQ +δ02 (1 + t)−2 W[l]   Q T +[1 + log(1 + t)] W[l] + (1 + t)−1 [1 + log(1 + t)]W[l] + W[l]  +Y0 + (1 + t)A[l] + (1 + t)−1 B[l]

To estimate the St,u 1-form (i1 ...il ) g˙ l , given by (10.496), it remains for us to consider the St,u 1-form (i1 ...il ) wl , defined by (10.497). From formulas (8.87)–(8.90) of Chapter 8 we can express the first sum on the right in (10.497) as: l−1 

L / Ril . . . L / Ril−k+1 L /

) (Ri l−k Z

(i1 ...il−k−1 )

xl−k−1

k=0

=

l−1 

L / (Ril−k ) L / Ril . . . L / Ril−k+1

(i1 ...il−k−1 )

Z

xl−k−1

(10.619)

k=0

+

k l−1  

l 

k=1 j =1 m 1 <···<m j =l−k+1

L / (im 1 ...im j ;il−k ) L / Ril

im j ...im 1 

...

Z

L / Ril−k+1

(i1 ...il−k−1 )

xl−k−1

Moreover, in regard to the first term on the right in (10.619) we have formula (8.93): L / Ril . . . L / Ril−k+1 +

k 

(i1 ...il−k−1 )

xl−k−1 =

l 

j =1 m 1 <···<m j =l−k+1

i



(i1 . l−k . . il )

xl−1

(Rim j . . . Rim 1 µ)d/(Ril

(10.620)

im j ...im 1 il−k 

...

Ri1 trχ)

Now, the contribution of the first term on the right in (10.620) to the first sum on the right in (10.619) cancels the second sum on the right in (10.497). We conclude that the St,u 1-form (i1 ...il ) wl is given by: (i1 ...il )

=

wl

l−1  k 

l 

L /

k=1 j =1 m 1 <···<m j =l−k+1

+

l−1  k 

l 

k=1 j =1 m 1 <···<m j =l−k+1

(Ri ) l−k Z

 (Rim j . . . Rim 1 µ)d/(Ril

L / (im 1 ...im j ;il−k ) L / Ril Z

im j ...im 1 

...

im j ...im 1 il−k 

...

L / Ril−k+1

(10.621)  Ri1 trχ)

(i1 ...il−k−1 )

xl−k−1

450

Chapter 10. Control of the Angular Derivatives

A given term in each of the two triple sums corresponds to a given k ∈ {1, . . . , l − 1}, a given j ∈ {1, . . . , k} and a given ordered subset {m 1 , . . . , m j } of the set {l − k + 1, . . . , l} with j elements. A term in the first triple sum is the sum of two terms:   im j ...im 1 il−k  (Ril−k ) Z Rim j . . . Rim 1 µ d/(Ril ... Ri1 trχ) +(Rim j . . . Rim 1 µ)L / (Ril−k ) d/(Ril

im j ...im 1 il−k 

...

Z

Ri1 trχ)

(10.622)

ε

The first of these terms is readily estimated in L 2 (t 0 ), under the hypotheses of Proposi/ [l∗ ] and M / [l∗ +1] , by: tion 10.1 with l∗ in the role of l augmented by the assumptions X (10.623) Cl δ02 (1 + t)−4 [1 + log(1 + t)]2 · 

Q · (1 + t)A[l−1] + (1 + t)−1 B[l] + Y0 + W[l−1] + δ02 (1 + t)−2 W[l−2] To estimate the second of the two terms we use the inequalities (10.567) and (10.568) taking: X=

(Ril−k )

Z,

ρ = Ril

im j ...im 1 il−k 

...

Ri1 trχ

(10.624)

We have the following two cases. Case 1: j ≤ l∗ + 1

Case 2: j ≥ l∗ + 2

and:

In Case 1 we appeal to assumption M / [l∗ +1] to place the factor Rim j . . . Rim 1 µ in L ∞ and we use the estimate (10.568) to place the second factor in L 2 , obtaining: L / X d/ρ L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−4 [1 + log(1 + t)] · t 

Q · (1 + t)A[l] + Y0 + W[l] + δ02 (1 + t)−2 W[l−1] under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l. In Case 2 the first ε factor is bounded in L 2 (t 0 ) by B[l−1] and we use the estimate (10.567) to place the second factor in L ∞ , obtaining: L / X d/ρ L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−5 [1 + log(1 + t)]2 t

Combining the results of the two cases we conclude that the second of the two terms ε (10.622) is bounded in L 2 (t 0 ) by: (10.625) Cl δ02 (1 + t)−4 [1 + log(1 + t)]2 · 

Q · (1 + t)A[l] + (1 + t)−1 B[l−1] + Y0 + W[l] + δ02 (1 + t)−2 W[l−1] Combining the results (10.623), (10.625), for the two terms (10.622), we conclude that ε the first triple sum in (10.621) is bounded in L 2 (t 0 ) by: (10.626) Cl δ02 (1 + t)−4 [1 + log(1 + t)]2 · 

Q · (1 + t)A[l] + (1 + t)−1 B[l] + Y0 + W[l] + δ02 (1 + t)−2 W[l−1]

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

451

under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l augmented by the assumption M / [l∗ +1] . We turn to the second triple sum in (10.621). Consider a given term, corresponding to a given triplet k, j, {m 1, . . . , m j }. We can apply the formula (10.620) to express: L / Ril +

im j ...im 1 

...

L / Ril−k+1

k− j 

(i1 ...il−k−1 )

xl−k−1 =



(i1

il−k im 1 ...im j 

. . .

il )

(Rin p . . . Rin1 µ)d/(Ril

xl− j −1

(10.627)

in p ...in 1 ;im j ...im 1 ;il−k 

...

Ri1 trχ)

p=1 {n 1 ,...,n p }⊂{l−k+1,...,l}\{m 1 ,...,m j }

Here, the inner sum in the second term is over all ordered subsets {n 1 , . . . , n p } of the set {l − k + 1, . . . , l} \ {m 1 , . . . , m j }. In the last factor, the superscript i n p . . . i n1 ; i m j . . . i m 1 ; i l−k  indicates that the indices i l−k , i m 1 . . . i m j , as well as i n1 . . . i n p , are missing. Using (10.627) we can write the term of the second triple sum in (10.621) in question as the sum: L / (im 1 ...im j ;il−k )

(i1

il−k im 1 ...im j 

. . .

il )

Z

xl− j −1 + Ak, j,{m 1 ,...,m j } + Bk, j,{m 1 ,...,m j }

(10.628)

where: Ak, j,{m 1 ,...,m j } =

k− j 



(10.629)

p=1 {n 1 ,...,n p }⊂{l−k+1,...,l}\{m 1 ,...,m j }

(

(im 1 ...im j ;il−k )

Bk, j,{m 1 ,...,m j } =

Z Rin p . . . Rin1 µ)d/(Ril

k− j 

in p ...in 1 ;im j ...im 1 ;il−k 

...



Ri1 trχ) (10.630)

p=1 {n 1 ,...,n p }⊂{l−k+1,...,l}\{m 1 ,...,m j }

(Rin p . . . Rin1 µ)L / (im 1 ...im j ;il−k ) d/(Ril Z

in p ...in 1 ;im j ...im 1 ;il−k 

...

Ri1 trχ)

We consider first an arbitrary term of the double sum Ak, j,{m 1 ,...,m j } . Note that the number of angular derivatives in the second factor is l − p − j , which is at most l − 2, since both p, j ≥ 1. We distinguish the three cases: Case 1: p ≤ l∗ − 1 and j ≤ l∗ Case 2: p ≤ l∗ − 1 and j ≥ l∗ + 1 Case 3: p ≥ l∗ and j ≤ l∗ (The alternative p ≥ l∗ and j ≥ l∗ + 1 is impossible, for we would then have p + j ≥ l, contradicting the fact that p + j ≤ k ≤ l − 1.) In Case 1 we use assumption M / [l∗ ] and

452

Chapter 10. Control of the Angular Derivatives

Corollary 10.1.i with l∗ + 1 in the role of l to place the first factor in L ∞ . In Case 2 we have p + j ≥ l∗ + 2, so the number of angular derivatives in the second factor is at most l∗ −1, and the second factor is bounded in L ∞ by virtue of assumption X / [l∗ −1] . Moreover, since p ≤ l∗ − 1 and j ≤ k ≤ l − 1, by assumption M / [l∗ ] and Corollary 10.2.i, the first factor is bounded in L 2 according to: 

(im 1 ...im j ;il−k )

Z Rin p . . . Rin1 µ L 2 ( ε0 ) t

≤

(im 1 ...im j ;il−k )

Z  L 2 ( ε0 ) d/ Rin p . . . Rin1 µ L ∞ ( ε0 ) t t

 Q −1 ≤ Cl δ0 (1 + t) [1 + log(1 + t)] Y0 + (1 + t)A[l−1] + W[l] + δ0 (1 + t)−1 W[l−1] In Case 3 we have p + j ≥ l∗ + 1, so the number of angular derivatives in the second factor is at most l∗ , and the second factor is bounded in L ∞ by virtue of assumption X / [l∗ ] . Moreover, since p ≤ k − 1 ≤ l − 2 and j ≤ l∗ , by Corollary 10.1.i with l∗ + 1 in the role of l, the first factor is bounded in L 2 according to: 

(im 1 ...im j ;il−k )

≤

Z Rin p . . . Rin1 µ L 2 ( ε0 ) t

(im 1 ...im j ;il−k )

Z

ε L ∞ (t 0 )

d/ Rin p . . . Rin1 µ L 2 ( ε0 ) t

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]B[l−1] Combining the results of the three cases we conclude that: Ak, j,{m 1 ,...,m j }  L 2 ( ε0 ) t

≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)]2 · (10.631) 

Q · Y0 + (1 + t)A[l−1] + (1 + t)−1 B[l−1] + W[l] + δ0 (1 + t)−1 W[l−1] under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l augmented by assumption M / [l∗ ] . We consider next an arbitrary term of the double sum Bk, j,{m 1 ,...,m j } . Setting: X=

(im 1 ...im j ;il−k )

Z,

ρ = Ril

in p ...in 1 ;im j ...im 1 ;il−k 

...

Ri1 trχ

(10.632)

the second factor is simply L / X d/ρ. Here we distinguish four cases: Case 1: j ≤ l∗ − 1 and p ≤ l∗ Case 2: j ≤ l∗ − 1 and p ≥ l∗ + 1 Case 3: j = l∗ and p ≤ l∗ Case 4: j ≥ l∗ + 1 and p ≤ l∗ (the alternative j ≥ l∗ and p ≥ l∗ + 1 is impossible, for we would then have p + j ≥ l, contradicting the fact that p + j ≤ k ≤ l − 1).

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

453

In Cases 1, 3 and 4 the first factor is bounded in L ∞ by assumption M / [l∗ ] , while in Case 2 it is bounded in L 2 by B[l−2] (for, p ≤ k − 1 ≤ l − 2). In regard to the second factor, in Cases 1 and 2 we have: X L ∞ ( ε0 ) , max L / Ri X L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

t

i

(10.633)

by Corollary 10.1.i with l∗ + 1 in the role of l. In Case 3 we have: (10.634) X L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

 Q / Ri X L 2 ( ε0 ) ≤ Cl Y0 + (1 + t)A[l] + W[l+1] + δ0 (1 + t)−1 W[l] max L t

i

by Corollary 10.1.i with l∗ + 1 in the role of l and Corollary 10.2.i with l + 1 in the role of l. In Case 4 we have: X L 2 ( ε0 ) , max L / Ri X L 2 ( ε0 ) t t i

 Q ≤ Cl Y0 + (1 + t)A[l] + W[l+1] + δ0 (1 + t)−1 W[l]

(10.635)

by Corollary 10.2.i with l + 1 in the role of l. Also, in Case 1 we have: d/ρ L 2 ( ε0 ) , max d/ Ri ρ L 2 ( ε0 ) (10.636) t t i

 Q ≤ Cl (1 + t)−2 (1 + t)A[l−1] + Y0 + W[l−1] + δ02 (1 + t)−2 W[l−2] In Case 2 we have: d/ρ L ∞ ( ε0 ) , max d/ Ri ρ L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] t

t

i

(10.637)

by assumption X / [l∗ ] . In Case 3 we have, by assumption X / [l∗ ] , d/ρ L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] t

max d/ Ri ρ L 2 ( ε0 ) (10.638) t i

 Q ≤ Cl (1 + t)−2 (1 + t)A[l−1] + Y0 + W[l−1] + δ02 (1 + t)−2 W[l−2] Finally, in Case 4 the estimates (10.637) hold. In Case 1, we place the second factor in L 2 using inequality (10.568) and the estimates (10.633) and (10.636), which yields: L / X d/ρ L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−4 [1 + log(1 + t)] · 

t Q (10.639) · (1 + t)A[l−1] + Y0 + W[l−1] + δ02 (1 + t)−2 W[l−2] In Case 2, we place the second factor in L ∞ using inequality (10.567) and the estimates (10.633) and (10.637), which yields: L / X d/ρ L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−5 [1 + log(1 + t)]2 t

(10.640)

454

Chapter 10. Control of the Angular Derivatives

In Case 3, we place the second factor in L 2 using inequality (10.590) and the estimates (10.634) and (10.638), which yields: L / X d/ρ L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−4 [1 + log(1 + t)] · 

t Q · (1 + t)Y0 + A[l] + W[l+1] + δ0 (1 + t)−1 W[l]

(10.641)

Finally, in Case 4, we place the second factor in L 2 using inequality (10.594) and the estimates (10.635) and (10.637), obtaining: L / X d/ρ L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−4 [1 + log(1 + t)] · 

t Q · (1 + t)Y0 + A[l] + W[l+1] + δ0 (1 + t)−1 W[l]

(10.642)

Combining the results of the four cases we conclude that: Bk, j,{m 1 ,...,m j }  L 2 ( ε0 ) t

Cl δ02 (1 + t)−4 [1 +

log(1 + t)]2 · (10.643) 

Q −1 −1 · Y0 + (1 + t)A[l] + (1 + t) B[l−2] + W[l+1] + δ0 (1 + t) W[l]

≤

under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l augmented by assumption M / [l∗ ] . Finally, we consider the first of the terms (10.628). Here we must make full use of Lemma 8.6. We set: X=

(im 1 ...im j ;il−k )

Z,

ξ=

(i1

il−k im 1 ...im j 

. . .

il )

xl− j −1

(10.644)

Ri1 fˇ

(10.645)

so that: φ = Ril

im j ...im 1 il−k 

...

Ri1 trχ,

ρ = Ril

im j ...im 1 il−k 

...

Then the term in question is simply L / X ξ . Note that X is an angular derivative of (Ril−k ) Z of order j , which, since j ≤ k ≤ l − 1, is at most l − 1. Note also that φ and ρ are angular derivatives, of trχ and fˇ respectively, of order l − j − 1, which, since j ≥ 1, is at most l − 2. By Corollary 10.2.i with l increased to l + 1 we have:

 Q X L 2 ( ε0 ) , max L / Ri X L 2 ( ε0 ) ≤ Cl Y0 + (1 + t)A[l] + W[l+1] + δ0 (1 + t)−1 W[l] t

i

t

(10.646) Also, we have: d/φ L 2 ( ε0 ) , max d/(Ri φ) L 2 ( ε0 ) t t i

 Q ≤ Cl (1 + t)−2 (1 + t)A[l] + Y0 + W[l] + δ02 (1 + t)−2 W[l−1]

(10.647)

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

455

and by the estimate (10.616): d/ρ L 2 ( ε0 ) , max d/(Ri ρ) L 2 ( ε0 ) t t i

  T ≤ Cl (1 + t)−1 W[l] + δ0 (1 + t)−1 W[l+1] + (1 + t)−1 B[l] (10.648)   Q +(1 + t)−1 [1 + log(1 + t)] W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] According to Lemma 8.6 we have, pointwise:    |L / X ξ | ≤ C(1 + t)−1 |X| max |ηi | + 2 max |d/(Ri ρ)| + (1 + t)|d/µ||d/φ| i i     (10.649) / Ri X  (µ|d/φ| + |d/ρ|) + max L i

where: ηi = µd/(Ri φ) + d/(Ri ρ) =

(i1

il−k im 1 ...im j 

. . .

il i)

xl− j

(10.650)

In view of the estimates (10.647) and (10.648) we have: max ηi  L 2 ( ε0 ) t i

  Q T ≤ Cl (1 + t)−1 W[l] + (1 + t)−1 [1 + log(1 + t)] W[l] + W[l+1] + Y0 + (1 + t)A[l]  (10.651) +δ0 (1 + t)−2 B[l] We distinguish the following three cases. Case 1: j ≤ l∗ − 1, Case 2: j = l∗

and Case 3: j ≥ l∗ + 1

In Case 1 we have, by Corollary 10.1.i with l∗ + 1 in the role of l: X L ∞ ( ε0 ) , max L / Ri X L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(10.652)

t

i

We use the L ∞ estimates (10.652) together with the L 2 estimates (10.647), (10.648), (10.651), and the following consequence of the pointwise estimate (10.649): L / X ξ  L 2 ( ε0 ) ≤ C(1 + t)−1 · t   · X L ∞ ( ε0 ) 3 max ηi  L 2 ( ε0 ) + 6 max d/(Ri ρ) L 2 ( ε0 ) t t t i i  +(1 + t)d/µ L ∞ ( ε0 ) d/φ L 2 ( ε0 ) (10.653) t t    / Ri X L ∞ ( ε0 ) µ L ∞ ( ε0 ) d/φ L 2 ( ε0 ) + d/ρ L 2 ( ε0 ) + max L i

t

t

t

t

456

Chapter 10. Control of the Angular Derivatives

to obtain: L / X ξ  L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] · t  

Q T · W[l] + (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l] + W[l] + W[l+1]  +δ0 (1 + t)−2 B[l] (10.654) In Case 2 we have the first of the L ∞ estimates (10.652), but not the second. We use instead the second of the L 2 estimates (10.646) together with the L ∞ estimates: d/φ L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)]

(10.655)

d/ρ L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2

(10.656)

t

t

Since l − j − 1 = l − l∗ − 1 = (l + 1)∗ − 1, the estimate (10.655) holds if we appeal to the bootstrap assumption X / [(l+1)∗ ] . The estimate (10.656) is (10.615) with l∗ increased to l∗ + 1. It follows from the estimates (10.542) together with assumption M / [l∗ +1] and the estimate (10.609) with l∗ increased to l∗ + 1. The last follows readily under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l augmented by assumption E / [lT∗ +1] . Using the following consequence of the pointwise estimate (10.649): L / X ξ  L 2 ( ε0 ) ≤ C(1 + t)−1 · t   · X L ∞ ( ε0 ) 3 max ηi  L 2 ( ε0 ) + 6 max d/(Ri ρ) L 2 ( ε0 ) t t t i i  (10.657) +(1 + t)d/µ L ∞ ( ε0 ) d/φ L 2 ( ε0 ) t t    + 3 max L / Ri X L 2 ( ε0 ) µ L ∞ ( ε0 ) d/φ L ∞ ( ε0 ) + d/ρ L ∞ ( ε0 ) t

i

t

t

t

we then obtain: L / X ξ  L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−3 · t 

 Q T + (1 + t)−1 [1 + log(1 + t)]W[l] + δ0 (1 + t)−2 B[l] · [1 + log(1 + t)] W[l]  +W[l+1] + Y0 + (1 + t)A[l] (10.658) In Case 3, we have l − j − 1 = l − l∗ − 2 = (l + 1)∗ − 2, hence the L ∞ estimates d/φ L ∞ ( ε0 ) , max d/(Ri φ) L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] t

t

i

d/ρ L ∞ ( ε0 ) , max d/(Ri ρ) L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 t

i

t

(10.659) (10.660)

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

457

now hold, under the same assumptions as those under which the estimates (10.655) and (10.656), respectively, were established in Case 2. Using these together with the L 2 estimates (10.646) and the following consequence of the pointwise estimate (10.649): L / X ξ  L 2 ( ε0 ) ≤ C(1 + t)−1 · t   · X L 2 ( ε0 ) max ηi  L ∞ ( ε0 ) + 2 max d/(Ri ρ) L ∞ ( ε0 ) t t t i i  (10.661) +(1 + t)d/µ L ∞ ( ε0 ) d/φ L ∞ ( ε0 ) t t    ε ε ε ε + 3 max L / Ri X L 2 ( 0 ) µ L ∞ ( 0 ) d/φ L ∞ ( 0 ) + d/ρ L ∞ ( 0 ) t

i

t

t

t

we obtain:

 Q L / X ξ  L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−3 Y0 + (1 + t)A[l] + W[l+1] + +δ0 (1 + t)−1 W[l] t

(10.662) Combining the results of the three cases (10.654), (10.658), (10.662), we conclude that: L / X ξ  L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−3 · t 

 Q T + (1 + t)−1 [1 + log(1 + t)]W[l] + δ0 (1 + t)−2 B[l] · [1 + log(1 + t)] W[l]  +W[l+1] + Y0 + (1 + t)A[l] (10.663) under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l, augmented by the assumptions E / [l∗ +2] , E / [lQ∗ +1] (used in deriving (10.542)), E / [lT∗ +1] , X / [(l+1)∗ ] and M / [l∗ +1] . Combining the results (10.663), (10.631), (10.643), for the three terms of (10.628), ε we conclude that the second triple sum in (10.621) is bounded in L 2 (t 0 ) by:

  Q T + (1 + t)−1 [1 + log(1 + t)]W[l] + δ0 (1 + t)−2 B[l] Cl δ0 (1 + t)−3 [1 + log(1 + t)] W[l]  +W[l+1] + Y0 + (1 + t)A[l] (10.664) under the assumptions just stated. This estimate together with the estimate (10.626) for the first triple sum in (10.621) yields the following lemma. Lemma 10.20 Let the hypothesis H0 and the estimate (10.30) hold. Let also the bootQ strap assumptions E / [l∗ +2] , E / [l∗ +1] , E / [lT∗ +1] , X / [(l+1)∗ ] and M / [l∗ +1] hold, for some positive integer l. Then, provided that δ0 is suitably small (depending on l), we have: max  i1 ...il

(i1 ...il )

wl  L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−3 · t

T · Y0 + (1 + t)A[l] + W[l+1] + [1 + log(1 + t)]W[l]   Q +(1 + t)−1 [1 + log(1 + t)]2 W[l] + δ0 (1 + t)−1 B[l]

458

Chapter 10. Control of the Angular Derivatives

Recalling expression (10.496) we obtain, in view of Lemmas 10.17, 10.20 and 10.19, the following proposition. Proposition 10.4 Let the hypotheses H0, H1, H2 and the estimate (10.30) hold. Let / [lQ∗ +1] , E / [lQ∗Q] and E / [lT∗ +1] , as well as X / [(l+1)∗ ] and also the bootstrap assumptions E / [l∗ +2] , E M / [l∗ +1] hold, for some positive integer l. Then if δ0 is suitably small (depending on l) we have: g˙l  L 2 ( ε0 ) ≤ Cl (1 + t)−3 · t  

Q QQ T · [1 + log(1 + t)] W[l+1] + δ0 W[l+1] + W[l+2] + δ02 (1 + t)−2 W[l]   +δ0 Y0 + (1 + t)A[l] + (1 + t)−1 B[l+1]

max  i1 ...il

(i1 ...il )

Let us now consider the additional terms 2µD / χˆ ·

(i1 ...il )

χˆl + µd/

(i1 ...il ) ˙

hl

(10.665)

in equation (8.413) of Chapter 8. These terms constitute the difference (i1 ...il ) g¨l − (i1 ...il ) g˙ l . ˙ given by equation (8.410) (see also In fact, let us consider first the function (i1 ...il ) h, (8.68)): (i1 ...il ) ˙ (10.666) h = (i1 ...il ) h l − 2χˆ · (i1 ...il ) χˆl where the function

(i1 ...il ) h

l

is given by equation (8.68): (i1 ...il )

h l = Ril . . . Ri1 (|χ| ˆ 2)

(10.667)

These expressions involve implicitly the St,u metric h/. To estimate appropriately the function (i1 ...il ) h˙ we must re-express it so that its dependence on the metric h/ becomes clear. Given any St,u symmetric 2-covariant tensorfield s, we associate to it a T11 -type St,u tensorfield s $ by: s $ = h/−1 · s = s · h/−1 (10.668) In terms of components in an arbitrary local frame field:

We then have:

(s $ ) BA = (h /−1 ) AC sC B = s BC (h/−1 )C A

(10.669)

trs $ = (s $ ) AA = (h/−1 ) AC sC A = trs

(10.670)

The trace in the first member in (10.670), in contrast to the trace in the last member, commutes with Lie derivatives. That is, if X is an arbitrary St,u -tangential vectorfield we have: /X s$) (10.671) X (trs $ ) = tr(L Given any two T11 -type St,u tensorfields α and β we denote, as usual, by α · β their composition as linear transformations in the tangent space to St,u at each point. We also denote (α)2 = α · α. We then have: tr(s $ )2 = (s $ ) BA (s $ ) BA = (h/−1 ) AC (h/−1 ) B D sC B s D A = |s|2

(10.672)

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

459

In particular, the definition (10.667) takes the form: (i1 ...il )

h l = Ril . . . Ri1 (tr(χˆ $ )2 )

(10.673)

Noting that the Lie derivative of the composition satisfies the Leibniz rule, that is, if α and β are T11 -type St,u tensorfields and X is an arbitrary St,u -tangential vectorfield, we have: L / X (α · β) = (L / X α) · β + α · (L / X β) we obtain the following expression for the function (i1 ...il ) h l :  (i1 ...il ) hl = 2 tr((L / R )s1 χˆ $ · (L / R )s2 χˆ $ )

(10.674)

unordered partitions

Here the sum is over unordered partitions {s1 , s2 } of the set {1, . . . , l} into two ordered subsets s1 and s2 , that is, the partitions {s1 , s2 } and {s2 , s1 } are considered identical. Separating off the term where s1 = ∅, s2 = {1, . . . , l} we re-write this as: (i1 ...il )

h l = 2tr(χˆ $ · L / Ril . . . L / Ri1 χˆ $ )  +2 tr((L / R )s1 χˆ $ · (L / R )s2 χˆ $ )

(10.675)

non−empty unorderedpartitions

Now, (10.666) takes the form: (i1 ...il ) ˙

hl =

(i1 ...il )

h l − 2tr(χˆ $ ·

(i1 ...il ) $ χˆl )

(10.676)

Substituting from (10.675) we then obtain: (i1 ...il ) ˙

h l = 2tr(χˆ $ · +2

(i1 ...il )



εl ) tr((L / R )s1 χˆ $ · (L / R )s2 χˆ $ )

(10.677)

non−empty unorderedpartitions

where

(i1 ...il ) ε l

is the trace-free T11 -type St,u tensorfield: (i1 ...il )

εl = L / Ril . . . L / Ri1 χˆ $ −

(i1 ...il ) $ χˆl

(10.678)

Let now ϑ be a trace-free symmetric 2-covariant St,u tensorfield and consider the corresponding trace-free T11 -type St,u tensorfield ϑ $ . The following commutation formula then holds for any St,u -tangential vectorfield X: /ˆX ϑ)$ = − L / X ϑ $ − (L

(X ) $

π / ˆ·ϑ $

(10.679)

where, if α and β are trace-free T11 -type St,u tensorfields, we denote by αˆ·β their tracefree product: 1 αˆ·β = α · β − tr(α · β)I (10.680) 2

460

Chapter 10. Control of the Angular Derivatives

Note that the trace-free product satisfies the Leibniz rule: L / X (αˆ·β) = (L / X α)ˆ·β + αˆ·(L / X β)

(10.681)

Using the commutation formula (10.679) we deduce the following recursion relation for the (i1 ...il ) εl : (i1 ...il )

(i1 ...il−1 )

εl = L / Ril

(Ril ) $ (i1 ...il−1 ) $ π / ˆ· χˆl−1

εl−1 −

(10.682)

Applying Proposition 8.2 to this recursion then yields the formula: (i1 ...il )

εl = −

l−1 

L / Ril . . . L / Ril−k+1 (

(Ril−k ) $ (i1 ...il−k−1 ) $ π / ˆ· χˆl−k−1 )

(10.683)

k=0

From this formula, using Corollary 10.1.d with l∗ + 1 in the role of l, Corollary 10.2.d with l + 1 in the role of l, the estimate (10.478) with l∗ in the role of l and the estimate (10.479), we obtain the estimates: max 

(i1 ...il )

i1 ...il

εl  L 2 ( ε0 ) , max L /R j

(i1 ...il )

i1 ...il ; j

t

εl  L 2 ( ε0 ) t

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] · 

Q · Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l]

(10.684)

under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l. It follows that the ε contribution of the first term on the right in (10.677) to R j (i1 ...il ) h˙ l is bounded in L 2 (t 0 ) by:

 Q Cl δ02 (1 + t)−4 [1 + log(1 + t)]2 Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l] Under the same assumptions augmented by assumption X / [(l+1)∗ ] , the contribution of the ε second term in (10.677) (the sum) to R j (i1 ...il ) h˙ l is bounded in L 2 (t 0 ) by:

 Q δ0 (1 + t)−3 [1 + log(1 + t)] Y0 + (1 + t)A[l] + W[l] + δ02 (1 + t)−2 W[l−1] Combining and noting that by hypothesis H0: d/

(i1 ...il ) ˙

h l  L 2 ( ε0 ) ≤ C(1 + t)−1 max R j t

j

(i1 ...il ) ˙

h L 2 ( ε0 ) t

we obtain the following lemma. Lemma 10.21 Let hypothesis H0 and the estimate (10.30) hold. Let also the bootstrap Q / [l∗ ] and X / [(l+1)∗ ] hold for some positive integer l. Then we have: assumptions E / [l∗ +1] , E max µd/ i1 ...il

(i1 ...il ) ˙

h l  L 2 ( ε0 ) t

−4

≤ Cl δ0 (1 + t) [1 + log(1 + t)]2 · 

Q · Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l] provided that δ0 is suitably small (depending on l).

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

461 ε

Finally, the first of the terms (10.665) is readily estimated in L 2 (t 0 ) using hypothesis H2 to place the first factor in L ∞ , together with the estimate (10.479). We obtain: max µD / χˆ ·

(i1 ...il )

i1 ...il

χˆl  L 2 ( ε0 ) t

−4

≤ Cl δ0 (1 + t) [1 + log(1 + t)]2 · 

Q · Y0 + (1 + t)A[l] + W[l] + δ02 (1 + t)−2 W[l−1]

(10.685)

Proposition 10.4 together with Lemma 10.21 and the estimate (10.685) gives us an appropriate bound for  (i1 ...il ) g¨l  L 2 ( ε0 ) . Of the other quantities appearing in the t

quantity (i1 ...il ) Q l (t), defined by equation (8.424) of Chapter 8, and which itself appears integrated with respect to t in the quantity (i1 ...il ) B(t), defined by equation (8.427), the quantity  (i1 ...il ) i l  L 2 ( ε0 ) has already been bounded by Proposition 10.3 and the

quantity 

(i1 ...il ) χ ˆ

t

by the estimate (10.479) with j = l, k = 0. To control the (i ...i ) l 1 quantity Q l (t), it remains for us to bound the quantities d/ (i1 ...il ) fˇl  L 2 ( ε0 ) and 

(i1 ...il ) z

ε L 2 (t 0 )

t

l  L 2 ( ε0 ) . t

Now, from the estimate (10.616) and hypothesis H0 we have: max d/

(i1 ...il−1 )

fˇl−1  L 2 ( ε0 )

(10.686) t

  T ≤ Cl (1 + t)−1 W[l] + δ0 (1 + t)−1 W[l] + (1 + t)−1 B[l]   Q +(1 + t)−1 [1 + log(1 + t)] W[l] + δ0 (1 + t)−1 W[l+1] + Y0 + (1 + t)A[l−1]

i1 ...il−1

We need a similar estimate with l increased to l + 1: (10.687) fˇl  L 2 ( ε0 ) t

  T ≤ Cl (1 + t)−1 W[l+1] + δ0 (1 + t)−1 W[l+1] + (1 + t)−1 B[l+1]   Q +(1 + t)−1 [1 + log(1 + t)] W[l+1] + δ0 (1 + t)−1 W[l+2] + Y0 + (1 + t)A[l]

max d/

(i1 ...il )

i1 ...il

In fact, this follows from the estimate (10.610) with l increased to l + 1: max Rik . . . Ri1 m L 2 ( ε0 ) t

T ≤ Cl W[l+1] + δ0 (1 + t)−1 W[l+1] i1 ...ik

+δ02 (1 + t)−2 Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W[l] Q

: ∀k ≤ l + 1

 (10.688)

together with the estimates (10.543) and assumption M / [l∗ +1] . The estimate (10.688) is itself readily deduced under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l augmented by assumption E / [lT∗ ] .

462

Chapter 10. Control of the Angular Derivatives

The non-negative function readily obtain:

(i1 ...il ) z

l

is given by equation (8.416) of Chapter 8. We

max  (i1 ...il ) z l  L 2 ( ε0 ) t i1 ...il  ≤ l max  (Ri ) Z  L ∞ ( ε0 ) t i   · 3 max d/ (i1 ...il ) fˇl  L 2 ( ε0 ) + (1 + t)d/µ L ∞ ( ε0 ) max d/ Ril−1 . . . Ri1 trχ L 2 ( ε0 ) i1 ...il

+ max L /R j j ;i  · max 

t

(Ri )

t

i1 ...il−1

t

Z  L ∞ ( ε0 ) t

(i1 ...il−1 )

i1 ...il−1

xl−1  L 2 ( ε0 ) + max d/ t

(i1 ...il−1 )

i1 ...il−1

fˇl−1  L 2 ( ε0 )

 (10.689)

t

This implies, by Corollary 10.1.i with l = 2 and assumption M / [1] , max 

(i1 ...il )

i1 ...il

z l  L 2 ( ε0 )

(10.690)

t

≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]·  · max d/ (i1 ...il ) fˇl  L 2 ( ε0 ) + δ0 [1 + log(1 + t)] max d/ Ril−1 . . . Ri1 trχ L 2 ( ε0 ) t t i1 ...il i1 ...il−1 + max  (i1 ...il−1 ) xl−1  L 2 ( ε0 ) + max d/ (i1 ...il−1 ) fˇl−1  L 2 ( ε0 ) i1 ...il−1

t

i1 ...il−1

t

In view of the estimates (10.687), (10.647), (10.651), (10.686), we then conclude that: max  i1 ...il

(i1 ...il )

z l  L 2 ( ε0 ) t

(10.691)

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]·  

Q T + (1 + t)−1 [1 + log(1 + t)] W[l+1] + δ0 (1 + t)−1 W[l+2] · W[l+1]    +(1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l] + δ0 (1 + t)−2 B[l+1] The results which we have so far obtained allow us to bound the non-negative quantity (i1 ...il ) Q l , defined by (8.424) of Chapter 8. In view of the comparison inequalities (8.358), putting together the estimate (10.479) with j = l, k = 0, the estimate (10.687), Proposition 10.3, the estimate (10.691), Proposition 10.4, Lemma 10.21 and the estimate (10.685), yields the following proposition. Proposition 10.5 Let the hypotheses H0, H1, H2 and H2 and the estimate (10.30) hold. / [lQ∗ +1] , E / [lQ∗Q] and E / [lT∗ +1] , as well as X / [(l+1)∗ ] Let also the bootstrap assumptions E / [l∗ +2] , E

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

463

and M / [l∗ +1] hold, for some positive integer l. Then we have: (i1 ...il )

max i1 ...il

Q l ≤ Cl (1 + t)−4 ·

  Q QQ T · [1 + log(1 + t)] W[l+1] + δ0 W[l+1] + W[l+2] + δ02 (1 + t)−2 W[l]   +δ0 Y0 + (1 + t)A[l] + (1 + t)−1 B[l+1]

provided that δ0 is suitably small (depending on l). tity

We now consider the principal term in the defining expression (8.427) for the quannamely the term:  

(i1 ...il ) B , l

C(1 + t)−2

Here, the quantities

(i1 ...il )

(i1 ...il )

(0)

P l,a (t) + (1 + t)−1/2

(0)

P l,a ,

(i1 ...il )

(i1 ...il )

(1)

P l,a (t) µ−a m (t)

(1)

P l,a are defined by equations (8.371), (8.372) (0)

(1)

(i1 ...il )

fˇl (t)| L 2 ([0,ε0]×S 2 )

(i1 ...il )

of Chapter 8 in terms of the quantities P l , (i1 ...il ) P l , whose sum (see (8.370)) (i ...i ) bounds the quantity 1 l Pl itself defined by equation (8.369): (i1 ...il )

Now, d/

(i1 ...il )

Pl (t) = (1 + t)2 |d/

(10.692)

fˇl  L 2 ( ε0 ) has already been estimated in (10.687) above. However, a more t

precise estimate is needed here, because of the fact that the quantity (i1 ...il ) Pl , in contrast to (i1 ...il ) Q l , does not appear integrated with respect to t in the expression (8.427) for (i1 ...il ) B . l By the comparison inequalities (8.358) and hypothesis H0 we have:   (i1 ...il ) Pl ≤ C R j (i1 ...il ) fˇl  L 2 ( ε0 ) = C R j Ril . . . Ri1 fˇ L 2 ( ε0 ) (10.693) t

t

j

j

What is needed is a more precise L 2 estimate of the principal part of R j Ril . . . Ri1 fˇ. Let us recall that the function fˇ is given by equation (10.614) in terms of the functions m and e, themselves given (10.499) and (10.540) respectively. Noting that T σ = −2ψ α T ψα , we express: where the

coefficients m αT

Lσ = −2ψ α Lψα , m = m αT (T ψα )

(10.694)

are the functions: m αT = −

and:

d/σ = −2ψ α d/ψα ,

dH (ψ L )2 ψ α dσ α

e = eαL (Lψα ) + /e · (d/ψα )

(10.695)

(10.696)

464

Chapter 10. Control of the Angular Derivatives

where the coefficients eαL are the functions: eαL = −

dH dσ



 ψL σ 1 ψα (ψ + αψ ) − L Tˆ 2 (1 − σ H ) α2

(10.697)

α

and the coefficients /e are the St,u -tangential vectorfields: α

/e = Note that:

dH ψ L ψ$ ψ α ; dσ

|m 0T − k 3 | ≤ Cδ0 (1 + t)−1 ,

ψ$ = ψ · h/−1 |m iT | ≤ Cδ0 (1 + t)−1

(10.698)

(10.699)

where  is the constant defined by equation (8.223) of Chapter 8, while: α

|eαL | ≤ C,

|e/ | ≤ Cδ0 (1 + t)−1

(10.700)

We now express R j Ril . . . Ri1 m in the form: R j Ril . . . Ri1 m = m αT (R j Ril . . . Ri1 T ψα )  ((R)s1 m αT )((R)s2 T ψα ) +

(10.701)

partitions, s1 =∅

where we have separated the principal term, and the sum is over ordered partitions {s1 , s2 } of the set 1, . . . , l + 1, setting i l+1 = j , into two ordered subsets s1 and s2 , with s1 nonempty. Similarly, we express R j Ril . . . Ri1 e, in the form, separating the principal terms: α

R j Ril . . . Ri1 e = eαL (R j Ril . . . Ri1 Lψα ) + /e · (d/ R j Ril . . . Ri1 ψα )  + ((R)s1 eαL )((R)s2 Lψα ) partitions, s1 =∅



+

α

((L / R )s1 /e ) · (d/(R)s2 ψα )

(10.702)

partitions, s1 =∅

Writing also: R j Ril . . . Ri1 fˇ = −2R j Ril . . . Ri1 m − 2µR j Ril . . . Ri1 e  ((R)s1 µ)((R)s2 e) −2 partitions, s1 =∅

and substituting from (10.701) and (10.702), yields the decomposition: R j Ril . . . Ri1 fˇ = −2m αT (R j Ril . . . Ri1 T ψα )

α

−2µeαL (R j Ril . . . Ri1 Lψα ) − 2µe/ · (d/ R j Ril . . . Ri1 ψα ) +

(i1 ...il j )

nl+1

(10.703)

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

where

(i1 ...il j ) n

465

is the lower order part, of order l + 1,  (i1 ...il j ) nl+1 = −2 ((R)s1 µ)((R)s2 e)

l+1

partitions, s1 =∅



−2

((R)s1 m αT )((R)s2 T ψα )

partitions, s1 =∅



−2

µ((R)s1 eαL )((R)s2 Lψα )

partitions, s1 =∅



−2

α

µ((L / R )s1 /e ) · (d/(R)s2 ψα )

(10.704)

partitions, s1 =∅

From (10.703), taking into account (10.699), (10.700), we have:   R j Ril . . . Ri1 T ψα  L 2 ( ε0 ) R j Ril . . . Ri1 fˇ L 2 ( ε0 ) ≤ 2k 3 || + Cδ0 (1 + t)−1 t

+C

 α

µR j Ril . . . Ri1 Lψα  L 2 ( ε0 ) t

+Cδ0 (1 + t)−1

 α

+

t

α

(i1 ...il j )

µd/ R j Ril . . . Ri1 ψα  L 2 ( ε0 ) t

nl+1  L 2 ( ε0 ) t

(10.705)

where C is a constant which is independent of l. Under the assumptions of Proposition 10.5 we readily deduce: max 

(i1 ...il j )

nl+1  L 2 ( ε0 ) t

Q T ≤ Cl δ0 (1 + t)−1 W[l] + [1 + log(1 + t)]W[l] + W[l+1]    +δ0 (1 + t)−1 Y0 + (1 + t)A[l] + (1 + t)−1 B[l+1] (10.706) i1 ...il j

Now, the third term on the right in (10.705) is bounded by: ) −1 Cδ0 (1 + t) E0 [R j Ril . . . Ri1 ψα ]

(10.707)

α

Here we recall from Chapter 5 that, for any variation ψ, E0 [ψ] and E1 [ψ] denote the energies associated to the variation ψ and the vectorfields K 0 and K 1 respectively. These are ε functions of t, E0 [ψ](t) and E1 [ψ](t) being the energies corresponding to t 0 . Our objective in the following is, in regard to the first two terms on the right in (10.705), to express R j Ril . . . Ri1 T ψα and R j Ril . . . Ri1 Lψα as T R j Ril . . . Ri1 ψα and L R j Ril . . . Ri1 ψα , respectively, plus lower order terms. Then, using the fact, from Chapter 5, that for any variation ψ we have:

T ψ L 2 ( ε0 ) ≤ C E0 [ψ] t  √  µ(Lψ + νψ) L 2 ( ε0 ) ≤ C(1 + t)−1 E1 [ψ] (10.708) t

466

Chapter 10. Control of the Angular Derivatives

while: µν R j Ril . . . Ri1 ψα  L 2 ( ε0 ) ≤ C(1 + t)−1 [1 + log(1 + t)]W[l+1] t

we shall be able to bound the first term on the right in (10.705) by:   ) C k 3 || + δ0 (1 + t)−1 E0 [R j Ril . . . Ri1 ψα ]

(10.709)

(10.710)

α

and the second term on the right in (10.705) by: ) −1 1/2 C(1+t) [1+log(1+t)] E1 [R j Ril . . . Ri1 ψα ]+C(1+t)−1 [1+log(1+t)]W[l+1] α

(10.711) up to a bound for the lower order terms of a form similar to (10.706). We begin with the following lemma, which is analogous to Lemma 8.2. Lemma 10.22 Let Y be an arbitrary St,u -tangential vectorfield on the spacetime domain Wε∗0 . We have: [T, Y ] = (Y )  where (Y )  is an St,u -tangential vectorfield, associated to Y , and defined by the condition that for any vector V ∈ T Wε∗0 : h(

(Y )

, V ) =

(Y )

π(T, V )

In terms of the (L, T, X 1 , X 2 ) frame, (Y )

=

(Y )

A X A,

(Y )

A =

(Y )

πT B (h/−1 ) AB

Proof. Let s be the 1-parameter group generated by T and #s the 1-parameter group generated by Y . Then #s maps each surface St,u onto itself, while s maps St,u onto St,u+s . Hence #−s ◦ −s ◦ #s ◦ s (10.712) maps each St,u onto itself. Thus the orbit by (10.712) of any given point q ∈ St,u lies in St,u . But this orbit must coincide to O(s 2 ) as s → 0 with the integral curve of the commutator [T, Y ] through q, parametrized by s 2 . It follows that [T, Y ] = (Y )  is a vectorfield which is tangential to the surfaces St,u , consequently we can expand: (Y )

=

(Y )

B X B

Taking the h-inner product with X A we obtain: / AB h

(Y )

 B = h(

(Y )

, X A )

= h([T, Y ], X A ) = h(DT Y, X A ) − h(DY T, X A ) = (Y ) πT A − h(D X A Y, T ) − h(DY T, X A )

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

467

Now, since h(T, X A ) = 0 we have: −h(DY T, X A ) = h(T, DY X A ) hence, substituting, / AB h

(Y )

B = =

(Y ) (Y )

πT A + h(T, [Y, X A ]) πT A

for, the vectorfield [Y, X A ] is St,u -tangential, hence h-orthogonal to T , both Y , X A being St,u -tangential vectorfields. The lemma is thus proved. We can express, in rectangular coordinates:  ∂

[T, Ri ] = T (Ri ) j − Ri T j ∂x j Since T = κ Tˆ , we have:

(10.713)

Ri T j = (Ri κ)Tˆ j + κ Ri Tˆ j

Substituting for Ri Tˆ j from equation (10.17) we obtain: Ri T j = (Ri κ + κ p/i )Tˆ j + κq/i · d/x j

(10.714)

ε

Projecting to St,u on t 0 then yields:   ∂  Ri Tˆ j j = κq/i ∂x

(10.715)

On the other hand, from (10.33), (10.36), (Ri ) j = im j x m − λi Tˆ j hence:

T (Ri ) j = im j T m − (T λi )Tˆ j − λi T (Tˆ j )

(10.716)

According to (3.190)–(3.191) of Chapter 3: T (Tˆ j ) = pT Tˆ j + qT j

(10.717)

where pT is a function given by (3.197) and qT is an St,u -tangential vectorfield given by (3.201): (10.718) qT = (qT )b · h/−1 where (qT )b is the St,u 1-form: (qT )b = −d/κ −

  1 dH ψTˆ 2 ψ(T σ ) − κψTˆ (d/σ ) − H ψTˆ i (T ψi ) 2 dσ

(10.719)

468

Chapter 10. Control of the Angular Derivatives

Substituting (10.717) in (10.716) we obtain: T (Ri ) j

∂ = κvi − (T λi + λi pT )Tˆ − λi qT ∂x j

(10.720) ε

where the vi are the vectorfields defined by (10.119). Projecting to St,u on t 0 then yields:   ∂ (10.721)  T (Ri ) j j = κvi − λi qT ∂x Since the left-hand side of (10.713) is an St,u -tangential vectorfield, projecting both sides to St,u , we obtain, in view of (10.715) and (10.721), [T, Ri ] = κvi − λi qT − κq/i

(10.722)

Defining the vectorfields vi as in (10.122), equation (10.124) holds, hence: vi = wi − (1 − u + η0 t)−1 Ri Therefore with q/i defined as in (10.26) and

 q/˜ i

(10.723)

defined as in (10.115), we obtain:

 [T, Ri ] = −κ q/˜ i − λi qT

We have thus proved the following lemma. Lemma 10.23 We have: [T, Ri ] =

(Ri )

  = −κ q/˜ i − λi qT

Note that since by Lemma 10.22 we have: (Ri )

π /T =

(Ri )

 · h/

(10.724)

the formula of Lemma 10.23 gives a more convenient expression for the St,u 1-form (Ri ) π /T than the formulas (6.36) or (6.44) of Chapter 6. We shall presently obtain estimates for the (Ri )  under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l augmented by assumptions E / [lT∗ ] and M / [l∗ +1] . First from (10.718), (10.719) we deduce: max L / R ik . . . L / Ri1 qT  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

i1 ...ik

t

: ∀k ≤ l∗

(10.725)

and (using also (10.410)): / R ik . . . L / Ri1 qT  L 2 ( ε0 ) max L t

T ≤ Cl (1 + t)−1 B[l+1] + (1 + t)−1 [1 + log(1 + t)]W[l+1] + δ0 (1 + t)−1 W[l]

i1 ...ik

+δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] Q

: ∀k ≤ l



(10.726)

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

469

By the estimates (10.259) and (10.237) with l∗ + 1 in the role of l we have: 

max L / R ik . . . L / Ri1 q/˜ i  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

i;i1 ...ik

t

: ∀k ≤ l∗

(10.727)

and, by the estimates (10.363) and (10.337) with l + 1 in the role of l:  / R ik . . . L / Ri1 q/˜ i  L 2 ( ε0 ) max L t i;i1 ...ik

 Q ≤ Cl Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l]

: ∀k ≤ l

(10.728)

The above estimates (10.725)–(10.728) together with Corollary 10.1.a with l∗ in the role of l and Corollary 10.2.a yield the following lemma. Lemma 10.24 Under the assumptions of Proposition 10.1 with l∗ + 1 in the role of l augmented by assumptions E / [lT∗ ] and M / [l∗ +1] , we have the estimates: / R ik . . . L / R i1 max L

(Ri )

i;i1 ...ik

 L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]2 t

: ∀k ≤ l∗ and: / R ik . . . L / R i1 max L

(Ri )

 L 2 ( ε0 ) t

≤ Cl [1 + log(1 + t)] Y0 + (1 + t)A[l] + δ0 (1 + t)−1 B[l+1]  Q T +W[l+1] + δ02 (1 + t)−1 W[l] + δ02 (1 + t)−2 W[l]

i;i1 ...ik

: ∀k ≤ l provided that δ0 is suitably small (depending on l). Consider now the commutator (i1 ...il )

T Ril . . . Ri1 − Ril . . . Ri1 T =

AlT

(10.729)

ε

a differential operator acting on functions defined on t 0 . Using Lemma 10.23 we derive the following recursion relation: (i1 ...il )

AlT = Ril

(i1 ...il )

T Al−1 +

(Ril )

Ril−1 . . . Ri1

(10.730)

Ril−k−1 . . . Ri1

(10.731)

Applying Proposition 8.2 to this recursion yields: (i1 ...il )

AlT =

l−1  k=0

Ril . . . Ril−k+1

(Ril−k )

470

Chapter 10. Control of the Angular Derivatives

We then apply (10.729), (10.731) to the functions ψα , replacing l by l + 1 and setting i l+1 = j , to obtain: Ril+1 Ril . . . Ri1 T ψα = T Ril+1 Ril . . . Ri1 ψα −

l 

Ril+1 . . . Ril−k+2

(10.732) (Ril−k+1 )

Ril−k . . . Ri1 ψα

k=0

We shall now use the following lemma, which is analogous to Lemma 8.5 and proved in a similar way by induction. Lemma 10.25 For any St,u -tangential vectorfield X and any function φ and positive integer l we have: [X, Ril . . . Ri1 ]φ = −

l 

l 

(ik1 ...ik j )

Y Ril

ik j ...ik1 

...

Ri 1 φ

j =1 k1 <···
where

(ik1 ...ik j )

Y is the St,u -tangential vectorfield: (ik1 ...ik j )

Y =L / R ik . . . L / R ik X j

1

Consider then an arbitrary term of the sum on the right in (10.732), corresponding to some k ∈ {0, . . . , l}. Setting X=

(Ril−k+1 )

,

φ = Ril−k . . . Ri1 ψα

(10.733)

we write this term in the form, using Lemma 10.25, Ril+1 . . . Ril−k+2 Xφ = X · d/(Ril+1 . . . Ril−k+2 φ)  + (L / R )s1 X · d/(R)s2 φ

(10.734)

partitions, s1 =∅

where the sum is over all ordered partitions {s1 , s2 } of the set {l − k + 2, . . . , l + 1} into two ordered subsets s1 , s2 , s1 = ∅. The first term on the right in (10.734) is the product of X with an angular derivative of ψα of order l + 1. Placing the factor X in L ∞ using the first part of Lemma 10.24, we ε obtain a bound for this term in L 2 (t 0 ) by: Cl δ0 (1 + t)−2 [1 + log(1 + t)]2 W[l+1]

(10.735)

A term in the sum in (10.734) is the product of an angular derivative of X of order |s1 | with an angular derivative of ψα of order |s2 | + l − k + 1, and we have |s1 | + |s2 | = k. Thus, the sum in (10.734) contains angular derivatives of X of order at most k ≤ l and angular derivatives of ψα of order at most l (for, |s1 | ≥ 1).

Part 2: Bounds for the quantities (i1 ...il ) Q l and (i1 ...il ) Pl

471

We have the following two cases. Case 1: |s1 | ≤ l∗

and:

Case 2: |s1 | ≥ l∗ + 1

In Case 1 we use the first part of Lemma 10.24 to place the first factor in L ∞ , obtaining an L 2 bound by: Cl δ0 (1 + t)−2 [1 + log(1 + t)]2 W[l]

(10.736)

In Case 2 we have |s2 | ≤ k − l∗ − 1, hence: |s2 | + l − k + 1 ≤ l − l∗ ≤ l∗ + 1 and the bootstrap assumption E / [l∗ +1] allows us to place the second factor in L ∞ . Using also the second part of Lemma 10.24 in placing the first factor in L 2 , we obtain an L 2 bound by:

Cl δ0 (1 + t)−2 [1 + log(1 + t)] Y0 + (1 + t)A[l] + δ0 (1 + t)−1 B[l+1]  Q T +W[l+1] + δ02 (1 + t)−1 W[l] + δ02 (1 + t)−2 W[l] (10.737) Combining the above results (10.735)–(10.737) we conclude that: Ril+1 . . . Ril−k+2 Xφ L 2 ( ε0 )

[1 + log(1 + t)] Y0 + (1 + t)A[l] + δ0 (1 + t)−1 B[l+1]  Q T +[1 + log(1 + t)]W[l+1] + δ02 (1 + t)−1 W[l] + δ02 (1 + t)−2 W[l] t

≤ Cl δ0 (1 + t)

−2

: ∀k ∈ {0, . . . , l}

(10.738)

The sum on the right in (10.732) is then similarly bounded: 0 l 0 0 0 0 0 (Ril−k+1 ) Ril+1 . . . Ril−k+2 Ril−k . . . Ri1 ψα 0 0 0 0 ε k=0 L 2 (t 0 )

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] Y0 + (1 + t)A[l] + δ0 (1 + t)−1 B[l+1]  Q T +[1 + log(1 + t)]W[l+1] + δ02 (1 + t)−1 W[l] + δ02 (1 + t)−2 W[l] : ∀k ∈ {0, . . . , l}

(10.739)

A commutation relation similar to (10.732) with for L in the role of T , that is:

(Ri ) Z

in the role of

(Ri ) 

holds

Ril+1 Ril . . . Ri1 Lψα = L Ril+1 Ril . . . Ri1 ψα −

l  k=0

Ril+1 . . . Ril−k+2

(Ril−k+1 )

Z Ril−k . . . Ri1 ψα (10.740)

472

Chapter 10. Control of the Angular Derivatives

Using Corollary 10.1.i with l∗ + 1 in the role of l and Corollary 10.2.i with l + 1 in the role of l, in place of the estimates of Lemma 10.24, we deduce that: 0 0 l 0 0  0 0 (Ril−k+1 ) Ril+1 . . . Ril−k+2 Z Ril−k . . . Ri1 ψα 0 0µ 0 0 ε k=0 L 2 (t 0 )  ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] Y0 + (1 + t)A[l]  Q +[1 + log(1 + t)]W[l+1] + δ0 (1 + t)−1 W[l] : ∀k ∈ {0, . . . , l}

(10.741)

Finally, from (10.693), (10.705), expressions (10.732), (10.740) and the bounds (10.739), (10.741) and (10.706), we deduce, in view of (10.707)–(10.711), the following proposition. Proposition 10.6 Under the assumptions of Proposition 10.5, we have: (i1 ...il )

where: (i1 ...il )

Pl ≤

(0) P l=

(i1 ...il )

Ck 3 ||

(0) Pl

)

+

(i1 ...il )

(1) Pl

E0 [R j Ril . . . Ri1 ψα ]

j,α

and: (i1 ...il )

(1) P l=

Cδ0 (1 + t)−1

)

E0 [R j Ril . . . Ri1 ψα ]

j,α

+ C(1 + t)

−1

[1 + log(1 + t)]

1/2

)

E1 [R j Ril . . . Ri1 ψα ]

j,α

Q T [1 + log(1 + t)]W[l+1] + Cl δ0 (1 + t)−1 W[l] + [1 + log(1 + t)]W[l] 1   2 +(1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l] + B[l+1]

+ C(1 + t)

−1

where C is a constant which is independent of l.

Chapter 11

Control of the Spatial Derivatives of the First Derivatives of the x i . Assumptions and Estimates in Regard to µ Part 1: Control of the spatial derivatives of the first derivatives of the x i . The first part of the present chapter is concerned with the derivation of estimates for the spatial derivatives of the Tˆ x i = Tˆ i , of which at least one is a T -derivative. By spatial derivatives we mean the derivatives with respect to T and the rotation fields R j : j = 1, 2, 3. Combining with the estimates of Chapter 10 we then obtain estimates for all spatial derivatives of the first derivatives of the x i with respect to all five commutation fields. The derivation of the estimates of the present chapter is based on bootstrap assumptions in regard to µ, in addition to the bootstrap assumptions of Chapter 10 in regard to χ. The estimates are then used to obtain estimates for the spatial derivatives of the deformation tensors of the commutation fields. We begin by defining certain norms. In the following ξ is an arbitrary St,u tensorfield ε defined on t 0 . We first define L ∞ and L 2 norms which consider only angular derivatives. Given a non-negative integer l we define: ξ ∞,[l], ε0 = max max L / R in . . . L / Ri1 ξ  L ∞ ( ε0 ) t

ξ 2,[l], ε0 =

0≤n≤l i1 ...in l 

t

n=0

t

max L / R in . . . L / Ri1 ξ  L 2 ( ε0 )

i1 ...in

t

(11.1) (11.2)

We then define L ∞ and L 2 norms which take into consideration all spatial derivatives.

474

Chapter 11. Control of the Spatial Derivatives

Given a pair of non-negative integers k, l with k ≤ l, we define: /T )m ξ ∞,[l−m], ε0 ξ ∞,[k,l], ε0 = max (L t

ξ 2,[k,l], ε0 =

k 

t

(11.3)

t

0≤m≤k

(L /T )m ξ 2,[l−m], ε0

(11.4)

t

m=0

Also, we shall denote: ξ ∞,{l}, ε0 = ξ ∞,[l,l], ε0 , t

ξ 2,{l},t ε0 = ξ 2,[l,l], ε0

t

t

(11.5)

According to the above definitions ξ ∞,[k,l], ε0 is the maximum of the quantities: t

/ R in . . . L / Ri1 (L /T )m ξ  L ∞ ( ε0 ) max L

(11.6)

{(m, n) : 0 ≤ n ≤ l − m, 0 ≤ m ≤ k}

(11.7)

i1 ...in

t

over the trapezoid: in the 2-dimensional integral lattice, while ξ 2,[k,l], ε0 is the sum of the quantities: t

/ R in . . . L / Ri1 (L /T )m ξ  L 2 ( ε0 ) max L

i1 ...in

(11.8)

t

over the same trapezoid. Also, ξ ∞,{l}, ε0 is the maximum of the quantities (11.6) over t the triangle: {(m, n) : m, n ≥ 0, m + n ≤ l} (11.9) while ξ 2,{l}, ε0 is the sum of the quantities (11.8) over the same triangle. t We now introduce bootstrap assumptions which extend the bootstrap assumptions E / [l] of Chapter 10 to include all spatial derivatives. Given non-negative integers m, n not both zero, we denote by Em,n the bootstrap assumption that there is a constant C independent of s such that for all t ∈ [0, s]: Em,n

:

max Rin . . . Ri1 (T )m ψα  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1

α;i1 ...in

t

We also denote by E0,0 the bootstrap assumption: E0,0

: ψ0 − k L ∞ ( ε0 ) , max ψi  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 t

i

t

We then denote by E{l} the conjunction of the assumptions Em,n corresponding to the triangle (11.9). The constant C then depends on l only and shall be denoted by Cl to emphasize this dependence. Note that the assumptions E0,n coincide with the assumptions / [l] . E / n of Chapter 10, and the assumptions E{l} contain the assumptions E Given non-negative integers m, n not both zero, we denote by Wm,n the quantity: Wm,n = max Rin . . . Ri1 (T )m ψα  L 2 ( ε0 ) α;i1 ...in

t

(11.10)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

475

We also denote by W0,0 the quantity:  ε ε W0,0 = max ψ0 − k L 2 ( 0 ) , max ψi  L 2 ( 0 ) t

t

i

(11.11)

We then denote by W{l} the sum of the quantities Wm,n corresponding to the triangle (11.9). Note that the quantities W0,n coincide with the quantities Wn of Chapter 10, and the quantities W{l} dominate the quantities W[l] . Moreover, we have:  max ψ0 − k2,{l}, ε0 , max ψi 2,{l}, ε0 ≤ W{l} (11.12) t

t

i

The following lemma extends Lemma 10.1 to all spatial derivatives. Lemma 11.1 Let G be a smooth function of the (ψα : α = 0, 1, 2, 3) defined in a neighborhood of (k, 0, 0, 0), and let G 0 be the constant: G 0 = G(k, 0, 0, 0) Suppose that the bootstrap assumption E{l∗ } holds for some positive integer l. Then, if δ0 is suitably small (depending on l), we have: G − G 0 2,{l}, ε0 ≤ CW{l} t

where C is a constant which is independent of l. Proof. Consider the quantities: max Rin . . . Ri1 (T )m (G − G 0 ) L 2 ( ε0 )

i1 ...in

t

(11.13)

For m = n = 0, we express G − G 0 in the form: G − G 0 = G 0 (ψ0 − k) + G i ψi where: G

µ



1

= 0

∂G ((1 − λ)k + λψ0 , λψ1 , λψ2 , λψ3 )dλ ∂ψµ

(11.14)

(11.15)

The functions G µ being also smooth functions of the (ψα : α = 0, 1, 2, 3), assumption E0,0 implies: max G µ  L ∞ ( ε0 ) ≤ C (11.16) µ

t

hence by expression (11.14): G − G 0  L 2 ( ε0 ) ≤ CW0,0 t

For m = 0, n ≥ 1, the quantities (11.13) reduce to the quantities max Rin . . . Ri1 G L 2 ( ε0 )

i1 ...in

t

(11.17)

476

Chapter 11. Control of the Spatial Derivatives

to which Lemma 10.1 applies: max Rin . . . Ri1 G L 2 ( ε0 ) ≤ CW0,n + Cl δ0 (1 + t)−1

i1 ...in

t

: for n ≥ 1

n−1 

W0,n

n  =1

(11.18)

Summing over n ∈ {1, . . . , l} and adding (11.17) we obtain: G − G 0 2,[l], ε0 ≤ C

l 

t

W0,n + Cl δ0 (1 + t)−1

n=0

l−1 

W0,n

(11.19)

n=0

which if δ0 is suitably small, depending on l, implies: G − G 0 2,[l], ε0 ≤ C

l 

t

W0,n

(11.20)

n=0

for a constant C which is independent of l. Consider next the case m ≥ 1. We express: (T )m G =

m  k=1

∂k G ∂ψµ1 . . . ∂ψµk

(m,k) I µ1 ...µk

(11.21)

(m,k)

Here, for each k = 1, . . . , m, I µ1 ...µk is defined by considering all partitions of the unordered set {1, . . . , m} into k subsets each consisting of at least one element. Since the coefficient ∂k G ∂ψµ1 . . . ∂ψµk (m,k)

of I µ1 ...µk in (11.21) is symmetric under arbitrary permutations of the ψµ1 , . . . , ψµk , the partition being itself unordered. We have: (m,k) I µ1 ...µk =



Nm,k (m 1 , . . . , m k )((T )m 1 ψµ1 ) . . . ((T )m k ψµk )

(11.22)

m 1 +···+m k =m

where the m 1 , . . . , m k , the cardinalities of the subsets, are positive integers, and the symmetric function Nm,k (m 1 , . . . , m k ) is the number of such partitions where there is one subset of cardinality m 1 , etc., one subset of cardinality m k . If the m 1 , . . . , m k are all distinct then Nm,k (m 1 , . . . , m k ) is the multinomial coefficient. In general, if q, 1 ≤ q ≤ k, of the m 1 , . . . , m k are distinct, then if pi is the multiplicity of the i th value, we must divide by the number of permutations of pi elements. Thus Nm,k (m 1 , . . . , m k ) is in general given by: 1 m! Nm,k (m 1 , . . . , m k ) = . (11.23) m 1 ! . . . m k ! p1 ! . . . pq !

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

477

For k = 1 we have m 1 = m, (m,1) I µ1 =

Nm,1 (m) = 1,

(T )m ψµ1

(11.24)

while for k = m we have m 1 = · · · = m k = 1, q = 1, p1 = m, (m,m) I µ1 ...µk =

Nm,m (1, . . . , 1) = 1,

(T ψµ1 ) . . . (T ψµk )

(11.25)

We apply Rin . . . Ri1 to (11.21), for 0 ≤ n ≤ l − m. We then obtain: Rin . . . Ri1 (T ) G = m

m 



k=1

partitions |s1 |+|s2 |=n

 (R)s1

∂kG ∂ψµ1 . . . ∂ψµk

 (R)s2

(m,k) I µ1 ...µk

(11.26)

and: (R)s2

(m,k) I µ1 ...µk



=

Nm,k (m 1 , . . . , m k )

(11.27)

m 1 +···+m k =m



×

((R)s2,1 (T )m 1 ψµ1 ) . . . ((R)s2,k (T )m k ψµk )

partitions

|s2,1 |+···+|s2,k |=|s2 |

We separate the terms in (11.26) corresponding to k = 1 or s1 = ∅, re-writing (11.26) in the form, in view of (11.24), Rin . . . Ri1 (T )m G   ∂G = Rin . . . Ri1 (T )m ψµ1 ∂ψµ1  m   (m,k) ∂k G Rin . . . Ri1 I µ1 ...µk + ∂ψµ1 . . . ∂ψµk k=2    ∂G (R)s2 (T )m ψµ1 + (R)s1 ∂ψµ1

+

partitions, |s1 | =0 |s1 |+|s2 |=n m  

k=2 partitions, |s1 | =0 |s1 |+|s2 |=n

 (R)s1

∂k G ∂ψµ1 . . . ∂ψµk

 (R)s2

(m,k) I µ1 ...µk

(11.28)

ε

Now, the first term on the right in (11.28) is bounded in L 2 (t 0 ) by: CWm,n

(11.29)

478

Chapter 11. Control of the Spatial Derivatives

where C is a constant independent of m, n. Consider next a term in the first sum on the right in (11.28), corresponding to some k ∈ {2, . . . , m}. The first factor, ∂k G , ∂ψµ1 . . . ∂ψµk ε

is bounded in L ∞ (t 0 ) by a constant Ck , while the second factor, Ri n . . . Ri 1

(m,k) I µ1 ...µk ,

is given by (11.27) with s2 = {1, . . . , n}. Now, at most one of the |s2, j | + m j , j = 1, . . . , k, may exceed l∗ , for otherwise we would have:  n+m = (|s2, j | + m j ) ≥ 2l∗ + 2 ≥ l + 1, j

contradicting the fact that n + m ≤ l. Moreover, since the m j are positive integers and k ≥ 2, we have: max m j ≤ m − 1

and since also

j

max |s2, j | ≤ n j

we have: max(|s2, j | + m j ) ≤ n + m − 1 ≤ l − 1 j

ε

Thus, placing the factor (R)s2, j (T )m j ψµ j with the maximal |s2, j | + m j in L 2 (t 0 ) and ε all the other factors in L ∞ (t 0 ) using assumption E{l∗ } , we obtain, since there is at least one such other factor, Rin . . . Ri1

(m,k) I µ1 ...µk

 L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−1 W{l−1}

(11.30)

t

ε

We then conclude that the first sum on the right in (11.28) is bounded in L 2 (t 0 ) by: Cl δ0 (1 + t)−1 W{l−1}

(11.31)

Consider next a term in the second sum on the right in (11.28), corresponding to some non-empty subset s1 of {1, . . . , n}. We distinguish the two cases: Case 1: |s1 | ≤ l∗

and:

Case 2: |s1 | ≥ l∗ + 1

In Case 1 we may apply the argument of Lemma 10.1 to the function ∂G ∂ψµ1

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

479

taking L ∞ instead of L 2 norms, using assumption E{l∗ } , to conclude that: 0  0 0 s 0 0(R) 1 ∂ G 0 ≤ Cl δ0 (1 + t)−1 0 ∂ψµ1 0 L ∞ ( ε0 )

(11.32)

t

We then place the second factor in L 2 . Here: |s2 | + m ≤ n − 1 + m ≤ l − 1 hence:

(R)s2 (T )m ψµ1  L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−1 W{l−1}

(11.33)

t

In Case 2 we have: |s2 | + m ≤ n + m − l∗ − 1 ≤ l − l∗ − 1 ≤ l∗ ε

Assumption E{l∗ } then implies that the second factor is bounded in L ∞ (t 0 ) by: Cl δ0 (1 + t)−1 We then place the first factor in L 2 applying Lemma 10.1 to the function ∂G ∂ψµ1 obtaining, since |s1 | ≤ n ≤ l − m ≤ l − 1, 0  0 0 s 0 0(R) 1 ∂ G 0 0 0 2 ε0 ≤ CW[l−1] ∂ψ µ1

(11.34)

L (t )

Combining the results of the two cases we conclude that the second sum on the right in (11.28) is bounded in L 2 by: Cl δ0 (1 + t)−1 W{l−1} (11.35) Consider finally a term in the third sum on the right in (11.28) (the double sum), corresponding to some k ∈ {2, . . . , m} and some non-empty subset s1 of {1, . . . , n}. Again we distinguish the two cases: Case 1: |s1 | ≤ l∗

and:

Case 2: |s1 | ≥ l∗ + 1

In Case 1 we may apply the argument of Lemma 10.1 to the function ∂k G ∂ψµ1 . . . ∂ψµk taking L ∞ instead of L 2 norms, using assumption E{l∗ } , to conclude that: 0  0 0 s 0 ∂kG 0(R) 1 0 ≤ Cl δ0 (1 + t)−1 0 ∂ψµ1 . . . ∂ψµk 0 L ∞ ( ε0 ) t

(11.36)

480

Chapter 11. Control of the Spatial Derivatives

We then place the second factor in L 2 , following the argument leading to the estimate (11.30). Here max |s2, j | ≤ |s2 | ≤ n − 1, j

hence: max(|s2, j | + m j ) ≤ n − 1 + m − 1 ≤ l − 2 j

and we obtain: (R)s2

(m,k) I µ1 ...µk

 L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−1 W{l−2} t

(11.37)

In Case 2 we have: max |s2, j | ≤ |s2 | ≤ n − l∗ − 1 j

hence: max(|s2, j | + m j ) ≤ n − l∗ − 1 + m − 1 ≤ l − l∗ − 2 ≤ l∗ − 1 j

ε

and by assumption E{l∗ } the second factor is bounded in L ∞ (t 0 ) by: Cl δ0 (1 + t)−1 We then place the first factor in L 2 applying Lemma 10.1 to the function ∂k G ∂ψµ1 . . . ∂ψµk obtaining, since |s1 | ≤ n ≤ l − m ≤ l − 2, as the third sum is present only for m ≥ 2, 0  0 0 s 0 ∂kG 0(R) 1 0 ≤ CW[l−2] 0 ∂ψµ1 . . . ∂ψµk 0 L 2 ( ε0 )

(11.38)

t

Combining the results of the two cases we conclude that the second sum on the right in (11.28) is bounded in L 2 by: Cl δ0 (1 + t)−1 W{l−2} (11.39) Combining finally the results (11.29), (11.31), (11.35), (11.39), for the four terms on the right in (11.28) we conclude that, for m ≥ 1: max Rin . . . Ri1 (T )m G L 2 ( ε0 ) ≤ CWm,n

i1 ...in

t

(11.40)

for some constant C independent of m, n, provided that δ0 is suitably small (depending on l). Summing (11.40) over n ∈ {0, . . . , l − m}, then summing over m ∈ {1, . . . , l}, and adding (11.20), we obtain: G − G 0 2,{l}, ε0 ≤ CW{l} t

and the proof of the lemma is complete.

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

481

We continue with a product lemma for the norms (11.3) and (11.4), which shall be applied repeatedly in the sequel. Lemma 11.2 ε a. Let ξ1 , . . . , ξ N be arbitrary St,u tensorfields, defined on t 0 and let us denote by ξ1 · · · ξ N an arbitrary tensor product with contractions. Let also k, l be non-negative integers, k ≤ l. We then have:   N 3   ξ j  ε  ε ξ1 · · · ξ N 2,[k,l], ε0 ≤ Cl ∞,[k,l∗ ], 0 ξi 2,[k,l], 0 t

t

j =i

i=1

t

ε

b. Let ξ1 , . . . , ξ N be as above, and ϑ another arbitrary St,u tensorfield defined on t 0 . Let k, l be positive integers, k ≤ l. We then have: ξ1 · · · ξ N · ϑ2,[k−1,l−1], ε0 t

≤ Cl ϑ∞,[k−1,l∗ −1], ε0

N 

t

+ Cl

N 3

 

i=1

3 j =i

 ξ j ∞,[k−1,l∗ ], ε0  ξi 2,[k−1,l−1], ε0 t

t



ξi ∞,[k−1,l∗ ], ε0 t

ϑ2,[k−1,l−1], ε0 t

i=1

Part a shall be applied to St,u tensorfields ξ1 , . . . , ξ N of order at most 1, and part b shall be applied when the ξ1 , . . . , ξ N are of order at most 1, but the St,u tensorfield ϑ is of order 2. 

Proof. a. Let us apply (L /T )k , k  ∈ {0, . . . , k}, to ξ1 · · · ξ N . Then each factor receives at  most k T -derivatives, and in taking the  2,[l−k  ], ε0 - norm of the result, each factor t receives at most l − k  angular derivatives. Thus, each factor receives at most l spatial derivatives. Moreover, at most one factor receives more than l∗ spatial derivatives, for otherwise there would be at least 2(l∗ + 1) ≥ l + 1 spatial derivatives contrary to the fact that there are l spatial derivatives in all. It follows that:   N 3    ξ j  ε  ε (L /T )k (ξ1 · · · ξ N )2,[l−k  ], ε0 ≤ Cl ∞,[k  ,l∗ ], 0 ξi 2,[k  ,l], 0 (11.41) t

i=1

j =i

t

t

Noting that the right-hand side of (11.41) is non-decreasing in k  , we may replace k  by k on the right. Summing then over k  ∈ {0, . . . , k} yields part a. b. Here we distinguish the two cases: Case 1: ϑ receives at most l∗ − 1 spatial derivatives Case 2: ϑ receives l∗ or more spatial derivatives

and:

482

Chapter 11. Control of the Spatial Derivatives

Case 1 is handled as in part a and leads to the first term on the right in the estimate of part b. It then only remains to be shown that in Case 2 each ξi receives at most l∗ spatial derivatives. But this is indeed the case. For, if one of the ξi were to receive l∗ + 1 or more spatial derivatives, then, together with the spatial derivatives received by ϑ, there would be at least 2l∗ +1 ≥ l spatial derivatives, whereas in fact there are only l −1 spatial derivatives in all. As in Chapter 10 a primary role was played by the estimates, given by Propositions 10.1 and 10.2, for the angular derivatives of the rectangular components Tˆ i , or equivalently of the functions y i , a primary role shall be played in the present chapter by the estimates for the spatial derivatives of the Tˆ i of which at least one is a T -derivative. From Chapter 10, equation (10.717) we can express T Tˆ i in the form: T Tˆ i = pT Tˆ i + qT · d/x i

(11.42)

where pT is the function given by equation (3.197) of Chapter 3 and qT is the St,u tangential vectorfield (11.43) qT = (qT )b · h/−1 (see (10.718)), (qT )b being the St,u 1-form given by equation (10.719) of Chapter 10 (or by (3.201)). Setting: (11.44) ωT Tˆ = Tˆ i (T ψi ) we have:

1 dH (ψ ˆ )2 (T σ ) − H ψTˆ ωT Tˆ 2 dσ T   1 dH ψTˆ 2 ψ(T σ ) − κψTˆ (d/σ ) − H ψωT Tˆ (qT )b = −d/κ − 2 dσ pT = −

(11.45) (11.46)

The higher order T -derivatives of the Tˆ i shall be expressed recursively through the following lemma. Lemma 11.3 For any non-negative integer m we have: (T )m+1 Tˆ i = pT ,m Tˆ i + qT ,m · d/x i +

m−1 

r Tn ,m · d/(T )n Tˆ i

n=0

Here the pT ,m are functions and the qT ,m are St,u -tangential vectorfields determined by the recursion relations: pT ,m = T pT ,m−1 + pT pT ,m−1 + d/κ · qT ,m−1 qT ,m = L /T qT ,m−1 + qT pT ,m−1 and the initial conditions: pT ,0 = pT ,

qT ,0 = qT

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

483

Also, the r Tn ,m are St,u -tangential vectorfields determined by the recursion relations: r T0 ,m = L /T r T0 ,m−1 + κqT ,m−1 r Tn ,m = L /T r Tn ,m−1 + r Tn−1 ,m−1 r Tm−1 ,m

=

: for n ∈ {1, . . . , m − 2}

r Tm−2 ,m−1

and the initial condition: r T0 ,0 = 0 Proof. For m = 0 the sum on the right is absent and the formula reduces to (11.42). Let then the formula hold with m replaced by m − 1, that is, let: (T )m Tˆ i = pT ,m−1 Tˆ i + qT ,m−1 · d/x i +

m−2 

r Tn ,m−1 · d/(T )n Tˆ i

(11.47)

n=0

for some function pT ,m−1 and some St,u -tangential vectorfields qT ,m−1 and r Tn ,m−1 : n = 0, . . . , m − 2. Let us apply T to this. Then on the left we have (T )m+1 Tˆ i , and on the right, by (11.42): T ( pT ,m−1 Tˆ i ) = (T pT ,m−1 )Tˆ i + pT ,m−1 (T Tˆ i ) = (T pT ,m−1 + pT pT ,m−1 )Tˆ i + (qT pT ,m−1 ) · d/x i

(11.48)

/T qT ,m−1 ) · d/x i + qT ,m−1 · d/(κ Tˆ i ) T (qT ,m−1 · d/x i ) = (L

(11.49)

L /T d/x i = d/(T x i ) = d/T i = d/(κ Tˆ i )

(11.50)

and: for, Moreover, since

/T r Tn ,m−1 ) · d/(T )n Tˆ i + r Tn ,m−1 · d/(T )n+1 Tˆ i T (r Tn ,m−1 · d/(T )n Tˆ i ) = (L

(11.51)

on the right we also have: m−2  n=0

n T (rm−1 · d/(T )n Tˆ i ) = (L /T r T0 ,m−1 ) · d/Tˆ i +

m−2 

(L /T r Tn ,m−1 + r Tn−1 /(T )n Tˆ i ,m−1 ) · d

n=1

+ r Tm−2 /(T )m−1 Tˆ i ,m−1 · d

(11.52)

Adding (11.48), (11.49) and (11.52), the formula for (T )m+1 Tˆ i of the lemma results, if we define the function pT ,m and the St,u -tangential vectorfields qT ,m and r Tn ,m : n = 0, . . . , m − 1 according to the recursion relations of the lemma. Let us now investigate the recursions of the above lemma. First we have the recursion satisfied by:   pT ,m qT ,m

484

Chapter 11. Control of the Spatial Derivatives

Here we are considering columns



φ X



where φ is a function and X is a St,u -tangential vectorfield. We define the operator A acting on such columns by: (11.53) A=L /T + B where, naturally,

 L /T

φ X



 =

Tφ L /T X



and B is the multiplication operator:         φ φ pT φ + d/κ · X pT d/κ · B = = qT 0 qT φ X X In terms of the operator A the recursion in question takes the form, simply:     pT ,m pT ,m−1 =A qT ,m qT ,m−1 and the solution satisfying the initial condition     pT ,0 pT = qT ,0 qT 

is:

pT ,m qT ,m



 =A

m

pT qT

(11.54)

(11.55)

(11.56)  (11.57)

We turn to the 2-dimensional recursion satisfied by the St,u -tangential vectorfields r Tn ,m . The first recursion relation defines an ordinary recursion satisfied by the r T0 ,m : r T0 ,m = L /T r T0 ,m−1 + κqT ,m−1

(11.58)

To this we can apply Proposition 8.2 to obtain, in view of the initial condition r T0 ,0 = 0

(11.59)

the expression: r T0 ,m =

m−1 

(L /T ) j (κqT ,m−1− j )

(11.60)

j =0

Next, we consider the third recursion relation. This defines an ordinary recursion satisfied by the r Tm−1 ,m : m−2 r Tm−1 (11.61) ,m = r T ,m−1 It follows that: 0 r Tm−1 ,m = r T ,1 = κqT

the last by (11.61) for m = 1.

(11.62)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

485

Finally, we consider the second recursion relation, which is genuinely 2-dimensional. Setting k = m − n, this relation takes the form: m−1−k /T r Tm−k r Tm−k ,m = L ,m−1 + r T ,m−1

: for k ∈ {2, . . . , m − 1}

(11.63)

Given any positive integer k let us denote by Nk the set of positive integers: Nk = {m ≥ k}

(11.64)

On Nk we define the function x k with values in the space X of St,u -tangential vectorfields by: (11.65) x k (m) = r Tm−k ,m Let Lk be the linear map taking X -valued functions f defined on Nk−1 into X -valued functions defined on Nk by: /T ( f (m − 1)) (Lk f )(m) = L

: ∀m ∈ Nk

(11.66)

: ∀m ∈ Nk+1 , k ≥ 2

(11.67)

Then (11.63) takes the form: x k (m) − x k (m − 1) = (Lk x k−1 )(m) Setting: yk = Lk x k−1

(11.68)

and considering yk as given, we can view (11.67) as a difference equation in m for fixed k: x k (m) − x k (m − 1) = yk (m)

: ∀m ∈ Nk+1 , k ≥ 2

(11.69)

Let us sum (11.69) over m ∈ {k + 1, . . . , j }. On the left we then have the telescoping sum: j  (x k (m) − x k (m − 1)) = x k ( j ) − x k (k) m=k+1

We thus obtain, interchanging the labels m and j , x k (m) = x k (k) +

m 

yk ( j )

: ∀m ∈ Nk , k ≥ 2

(11.70)

j =k+1

Now, from (11.65) with m = k, x k (k) = ck = r T0 ,k

(11.71)

is given by (11.60), while from (11.68) and (11.66) the sum on the right is: m  j =k+1

yk ( j ) =

m 

(Lk x k−1 )( j ) =

j =k+1

m  j =k+1

L /T (x k−1 ( j − 1))

(11.72)

486

Chapter 11. Control of the Spatial Derivatives

Let us define a linear operator Mk acting on X -valued functions f defined on Nk by: (Mk f )(m) =

m−1 

(L /T f )( j )

(11.73)

j =k

In terms of the operator Mk and the constants ck , (11.70) becomes the following recursion in k for the X -valued functions x k : x k = ck + Mk x k−1

: k≥2

(11.74)

Here, we consider the ck as constant functions defined on Nk . Now, by (11.65) and (11.62) for k = 1 we have: : ∀m ∈ N1 (11.75) x 1 (m) = r Tm−1 ,m = κqT hence, by (11.71) for k = 1,

x 1 = c1

(11.76)

Thus (11.74) holds also for k = 1 if we set x 0 = 0: x k = ck + Mk x k−1

: k≥1

x0 = 0

(11.77)

To this recursion Proposition 8.2 applies, yielding: xk =

k−1 

Mk . . . Mk−i+1 ck−i

(11.78)

i=0

Now, in the case f is a constant function c (11.73) gives: (Mk c)(m) = (m − k)L /T c

: ∀m ∈ Nk

(11.79)

Let us define, for each non-negative integer l, the function Nl on the set of non-negative integers, with values in the same set, recursively by: N0 (n) = 1 : for all non-negative integers n (11.80) n  Nl−1 (m) : for all non-negative integers n and all positive integers l Nl (n) = m=1

In particular, we have: Nl (0) = 0

: for all positive integers l

(11.81)

In fact, it is readily seen that: Nl (n) =

n . . . (n + l − 1) l!

: for all non-negative integers n and all positive integers l (11.82)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

487

We can then show, by induction on j , that for all non-negative integers j we have: (Mk−i+ j . . . Mk−i+1 ck−i )(m) = N j (m − k + i − j )(L /T ) j ck−i

(11.83)

For j = 0 this is evident from the first of (11.80). The completion of the inductive step relies on the second of (11.80). Setting j = i in (11.83) we obtain: (Mk . . . Mk−i+1 ck−i )(m) = Ni (m − k)(L /T )i ck−i

: ∀m ∈ Nk

(11.84)

: ∀m ∈ Nk , ∀k ≥ 1

(11.85)

Substituting in (11.78) then yields: x k (m) =

k−1 

Ni (m − k)(L /T )i ck−i

i=0

We summarize the above results in the following lemma. Lemma 11.4 Let A be the operator  A=L /T + B acting on columns

φ X



where φ is a function and X a St,u -tangential vectorfield, B being the multiplication operator:   pT d/κ B= qT 0 Then the functions pT ,m and St,u -tangential vectorfields qT ,m are given by:     pT pT ,m = Am qT ,m qT Moreover, the St,u -tangential vectorfields r Tn ,m−1 , n ∈ {0, . . . , m − 1}, are given by: r Tn ,m

=

m−1−n 

Ni (n)

m−1−n 

(L /T ) j (κqT ,m−1−n− j )

j =i

i=0

where Ni (n) are the non-negative integers:  1 : if i = 0 Ni (n) = n...(n+i−1) : if i ≥ 1 i! The function pT and the St,u 1-form (qT )b can be directly estimated (see (11.45), (11.46)), however to estimate the St,u -tangential vectorfield qT , which enters all the above formulas, we must estimate the T -derivatives of h/−1 . Now, /−1 = −h /−1 · L /T h

(T )

π / · h/−1 ,

(T )

π /=L /T h/

(11.86)

488

Chapter 11. Control of the Spatial Derivatives

Thus estimating (L /T )m+1 h /−1 reduces to estimating (L /T )m (T ) π /. Now, from equations (10.5), (10.6) of Chapter 10 and (3.46) of Chapter 3 we have: (T )

π / = 2κθ = −2α −1 κχ + 2κk/

Let us define: χ = χ − and:

η0 h/ 1 − u + η0 t

(T ) 

π / = −2α −1 κχ  + 2κk/

We then have:

(T )

π /=

where λ is the function:

π / + λ(1 − u + η0 t)−1 h/

(11.87)

(11.88)

(11.89)

(T ) 

(11.90)

λ = −2η0 (α −1 κ)

(11.91)

/ . We shall be able to estimate in a more direct manner the T -derivatives of (T ) π (T ) From these estimates to recover estimates for the T -derivatives of π /, we shall use the / defined below. Their definition auxiliary symmetric 2-covariant St,u tensorfields (m;T ) π requires the polynomials pm (x), which we shall presently define. For each non-negative integer m we define the polynomial pm (x) in the real variable x, of degree m, by:  p0 (x) = 1 (11.92) pm (x) = (m + x) pm−1 (x) : for m ≥ 1 Thus, we have:

 pm (x) =

The identity:

1 : for m = 0 (m + x) . . . (1 + x)

 m! 1 + x

m−1  n=0

pn (x) (n + 1)!

: for m ≥ 1

(11.93)

 = pm (x)

(11.94)

follows from (11.92), and conversely. We can substitute for x any real-valued function, in particular the function λ. We then define, for any non-negative integer m: (m;T )

Note that:

π / = (L / T )m

(T )

π / − λpm (λ)(1 − u + η0 t)−m−1 h/

(0;T )

π /=

(T ) 

π /

(11.95) (11.96)

Note also that if in (11.91) we were to substitute α = η0 (the value in the constant state) and κ = 1, we would obtain λ = −2. We thus define: λ = λ + 2

(11.97)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

489

Lemma 11.5 Let l be a positive integer and m a non-negative integer, m ≤ l. Suppose that: /∞,[l−1], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] max  (Ri ) π t

i

and:

λ ∞,[m,l], ε0 ≤ Cl δ0 [1 + log(1 + t)] t

Then, if δ0 is suitably small (depending on l) we have: (k;T )

max 

π /∞,[l−k], ε0 ≤ Cl  t

0≤k≤m

(T ) 

π / ∞,[m,l], ε0 + Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

Proof. We apply (L /T )k to (11.90), for k ∈ {0, . . . , m}, to obtain: (L / T )k

(T )

π / = (L / T )k +

(T ) 



π /

k1 +k2 +k3 =k

Noting that:

(11.98) k! ((T )k1 (1 − u + η0 t)−1 )((T )k2 λ)((L /T )k3 h/) k1 !k2!k3 !

(T )k1 (1 − u + η0 t)−1 = k1 !(1 − u + η0 t)−k1 −1

and substituting in (11.95) with m replaced by k, we obtain: (k;T )

π / = (L / T )k +

(T ) 



π /

k1 +k2 +k3 =k

k! (1 − u + η0 t)−k1 −1 ((T )k2 λ)((L /T )k3 h/) k2 !k3 !

−λpk (λ)(1 − u + η0 t)−k−1 h/

(11.99)

In the sum on the right we express:  (L /T )k3 h /=

/h : if k3 = 0 (L /T )k3 −1 (T ) π / : if k3 ≥ 1

(11.100)

and, in the case k3 ≥ 1 we express, using (11.95) with k3 − 1 in the role of m, (L /T )k3 −1

(T )

π / = λpk3 −1 (λ)(1 − u + η0 t)−k3 h/ +

(k3 −1;T )

π /

(11.101)

Substituting in (11.99) and separating the term corresponding to k2 = 0, noting that by virtue of (11.94):   k! 1 : if k3 = 0 (1 − u + η0 t)−k1 −1 λh/ λpk3 −1 (λ)(1 − u + η0 t)−k3 : if k3 ≥ 1 k3 ! k1 +k3 =k   k   pk3 −1 (λ)  = k! 1 + λ (1 − u + η0 t)−k−1 λh/  k3 !  k3 =1

= λpk (λ)(1 − u + η0 t)−k−1 h/

(11.102)

490

Chapter 11. Control of the Spatial Derivatives

so this cancels with the last term on the right in (11.99), we obtain: (k;T )

π / = (L /T )k (T ) π /  k! (1 − u + η0 t)−k1 −1 λ + k3 !

(k3 −1;T )

π /

k1 +k3 =k k3 ≥1

+



k1 +k2 =k k2 ≥1

k! (1 − u + η0 t)−k1 −1 ((T )k2 λ)h/ k2 !



+

k1 +k2 +k3 =k k2 ,k3 ≥1



+

k1 +k2 +k3 =k k2 ,k3 ≥1

k! (1 − u + η0 t)−k1 −k3 −1 ((T )k2 λ)λpk3 −1 (λ)h/ k2 !k3 ! k! (1 − u + η0 t)−k1 −1 ((T )k2 λ) k2 !k3 !

(k3 −1;T )

π /

(11.103)

ε

We apply L / R in . . . L / Ri1 to (11.103), take the L ∞ norm on t 0 , the maximum over i 1 . . . i n , and the maximum over n ∈ {0, . . . , l − k}, obtaining the  ∞,[l−k], ε0 norm. The first t term on the right then contributes: (L / T )k

(T ) 

π / ∞,[l−k], ε0

(11.104)

t

We decompose the first sum on the right in (11.103) according to:

Setting:

λ = −2 + λ

(11.105)

k  = k3 − 1

(11.106)

the contribution of the term −2 is: 2

k−1  k  =0

(k 

k!  (1 − u + η0 t)k −k  + 1)!

(k  ;T )

π /∞,[l−k], ε0

(11.107)

t

On the other hand, since by the second assumption of the lemma we have: λ

(k  ;T )

(k  ;T )

π /∞,[l−k], ε0 ≤ Cl−k λ ∞,[l−k], ε0  t

t

π /∞,[l−k], ε0 t

≤ Cl δ0 [1 + log(1 + t)]

(m  ;T )

π /∞,[l−k], ε0 (11.108) t

the contribution of the term λ is bounded by: Cl δ0 [1 + log(1 + t)]

k−1  k  =0

(k 

k!  (1 − u + η0 t)k −k  + 1)!

(m  ;T )

π /∞,[l−k], ε0 t

(11.109)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

491

Consider now the contribution of the last sum in (11.103). Since here k2 ≥ 1, we have, by the second assumption of the lemma: (T )k2 λ∞,[l−k], ε0 = (T )k2 λ ∞,[l−k], ε0 t

(11.110)

t



≤ λ ∞,[k−1,l−1], ε0 ≤ Cl δ0 [1 + log(1 + t)] t

since as k3 ≥ 1, we have k2 ≤ k − 1. It follows that the contribution of the last sum in (11.103) is bounded by: Cl δ0 [1 + log(1 + t)] ×

 k−2 k−2−k  

k  =0

j =0

(11.111)

k!  (1 − u + η0 t)k −k+1+ j  ( j + 1)!(k  + 1)!

(k  ;T )

π /∞,[l−k], ε0 t

where we have set: k2 = 1 + j

(11.112)

Next, we consider the second sum in (11.103). This sum is: k−1  j =0

k! (1 − u + η0 t) j −k ((T ) j +1 λ )h/ ( j + 1)!

(11.113)

Now, we have:

 L / R in . . . L / Ri1 ((T ) j +1 λ )h/ =



((R)s1 (T ) j +1 λ )((L / R )s2 h/)

partitions

|s1 |+|s2 |=n

and if s2 = ∅, s2 = {n 1 , . . . , n p }, p = |s2 |, then: (L / R )s 2 h /=L / R in p . . . L / R in

(Rin ) 1

2

π /

Hence, the first assumption of the lemma implies: ((T ) j +1 λ )h /∞,[l−k], ε0 ≤ C(T ) j +1 λ ∞,[l−k], ε0 t

t

(11.114)

for k ≥ 1, provided that δ0 is suitably small (depending on l). Appealing also to the second assumption of the lemma we then obtain: /∞,[l−k], ε0 ≤ Cλ ∞,[k,l], ε0 ≤ Cl δ0 [1 + log(1 + t)] ((T ) j +1 λ )h t

t

: for j ∈ {0, . . . , k − 1}

(11.115)

It follows that the contribution of (11.113), the second sum in (11.103), is bounded by: Cl δ0 (1 + t)−1 [1 + log(1 + t)]

(11.116)

492

Chapter 11. Control of the Spatial Derivatives

Finally, we have the third sum in (11.103). We express (see (11.106)): λpk  (λ) = −2 pk  (−2) + qk  +1 (λ )

(11.117)

where qk  +1 (λ ) is a polynomial of degree k  + 1 in λ with vanishing constant term. The constant term −2 pk  (−2) vanishes for k  ≥ 2 (see (11.93)). By (11.115) (here j ∈ {0, . . . , k − 2}) the contribution of the constant term is bounded by: Cl δ0 (1 + t)−2 [1 + log(1 + t)]

(11.118)

(λ )

while the contribution of the polynomial qk  +1 is bounded by:    +1 k−2 k    Cl δ0 [1 + log(1 + t)] (δ0 [1 + log(1 + t)])i (1 + t)−2−k    k =0

(11.119)

i=1

Since      +1 k +1  k    (δ0 [1 + log(1 + t)])i (1 + t)−k ≤  δ0i  [1 + log(1 + t)]k +1 (1 + t)−k   i=1

i=1

≤

δ0 [1 + log(1 + t)] 1 − δ0

(11.120)

(as [1 + log(1 + t)](1 + t)−1 ≤ 1), (11.119) is in turn bounded by: Cl δ02 (1 + t)−2 [1 + log(1 + t)]2

(11.121)

provided that δ0 is suitably small. Combining the results (11.104), (11.107) and (11.109), (11.111), (11.116), (11.118) and (11.121), for the contributions of the five terms on the right in (11.103), we conclude that, simplifying, (k;T )



(T ) 

π /∞,[l−k], ε0 ≤ (L / T )k

π / ∞,[l−k], ε0

t

t

+Cl (1 + t)

−1

[1 + log(1 + t)]

k−1 



(k  ;T )

π /∞,[l−k], ε0 t

k  =0

+Cl δ0 (1 + t)−1 [1 + log(1 + t)]

(11.122)

Since: k−1  k  =0



(k  ;T )

π /∞,[l−k], ε0 ≤ t

k−1 



(k  ;T )

π /∞,[l−1−k  ], ε0 ≤ t

k  =0

k−1  k  =0



(k  ;T )

π /∞,[l−k  ], ε0 t

(11.123)

(11.122) reduces to a recursive inequality of the form: x k ≤ ck + a

k−1  k  =0

xk

: for k = 0, . . . , m

(11.124)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

493

where x k , ck and a are the non-negative quantities: xk = 

(k;T )

π /∞,[l−k], ε0 t

(T ) 

π / ∞,[l−k], ε0 + Cl δ0 (1 + t)−1 [1 + log(1 + t)]

ck = (L / T )k

t

a = Cl (1 + t)−1 [1 + log(1 + t)]

(11.125)

It follows that: x k ≤ ck + a

k−1 



(1 + a)k−1−k ck 

: for k = 0, . . . , m

(11.126)

k  =0

hence:  max x k ≤ max ck 1 + a

0≤k≤m

0≤k≤m

m−1 

 (1 + a)

m−1−k 

= (1 + a)m max ck 0≤k≤m

k  =0

(11.127)

Substituting the definitions (11.125) and recalling that (1 + t)−1 [1 + log(1 + t)] ≤ 1, we then obtain:

 /∞,[l−k], ε0 ≤ Cl  (T ) π / ∞,[m,l], ε0 + Cl δ0 (1 + t)−1 [1 + log(1 + t)] max  (k;T ) π t

0≤k≤m

t

and the lemma is proved. Lemma 11.6 Let l be a positive integer and m a non-negative integer, m ≤ l. Suppose that: /∞,[l∗ −1], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] max  (Ri ) π t

i



λ ∞,[m,l∗ ], ε0 ≤ Cl δ0 [1 + log(1 + t)] t

and: 

(T ) 

π / ∞,[m−1,l∗ −1], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

Then, if δ0 is suitably small (depending on l) we have: m  k=0



(k;T )

π /2,[l−k], ε0 ≤ Cl  t

(T ) 

π / 2,[m,l], ε0 t

 + Cl (1 + t)−1 λ 2,[m,l], ε0 + δ0 [1 + log(1 + t)] max  t

i

(Ri )

π /2,[l−2], ε0 t

Proof. The first two assumptions of the present lemma coincide with those of Lemma 11.5 with l∗ in the role of l. Thus, Lemma 11.5 holds with l∗ in the role of l. Together with the third assumption of the present lemma this implies that: max

0≤k≤m−1



(k;T )

π /∞,[l∗ −1−k], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(11.128)

494

Chapter 11. Control of the Spatial Derivatives

We again apply L / R in . . . L / Ri1 to (11.103) with k ∈ {0, . . . , m}, but we now take the ε0 norm on t , the maximum over i 1 . . . i n , and then the sum over n ∈ {0, . . . , l − k}, obtaining the  2,[l−k], ε0 norm. The first term on the right in (11.103) then contributes: L2

t

(L / T )k

(T ) 

π / 2,[l−k], ε0

(11.129)

t

We decompose again the first sum on the right in (11.103) according to (11.105). The contribution of the term −2 is now (see (11.106)): 2

k−1  k  =0

k!  (1 − u + η0 t)k −k  (k  + 1)!

≤ Cl (1 + t)−1

k−1 



(k  ;T )

(k  ;T )

π /2,[l−k], ε0 t

π /2,[l−1−k  ], ε0

(11.130)

t

k  =0

To bound the contribution of the term λ , we first estimate:

  /2,[l−k], ε0 ≤ Cl λ ∞,[l∗ ], ε0  (k ;T ) π /2,[l−k], ε0 λ (k ;T ) π t

t



+λ 

t

ε 2,[l−k],t 0

(k  ;T )



π /∞,[l∗ −k], ε0

 (11.131)

t

Now, by (11.128): max

0≤k  ≤k−1



(k  ;T )

π /∞,[l∗ −k], ε0 ≤ t

max

0≤k  ≤k−1



(k  ;T )

≤ Cl δ0 (1 + t)

π /∞,[l∗ −1−k  ], ε0 t

−1

[1 + log(1 + t)]

(11.132)

Thus, appealing also to the second assumption of the lemma we obtain: λ

(k  ;T )

π /2,[l−k], ε0 t

≤ Cl δ0 [1 + log(1 + t)] 

(k  ;T )

(11.133) 

π /2,[l−1−k  ], ε0 + (1 + t)−1 λ 2,[l−k], ε0 t

t

λ

It follows that the contribution of the term is bounded by:  k−1   −1 (k  ;T ) −1  Cl δ0 (1 + t) [1 + log(1 + t)]  π /2,[l−1−k  ], ε0 + (1 + t) λ 2,[l−k], ε0 k  =0

t

t

(11.134) Consider next the contribution of the last sum in (11.103). This contribution is (see (11.106), (11.112)): k−2 k−2−k   k  =0

j =0



k!  (1 − u + η0 t)k −k+1+ j ((T ) j +1 λ ) ( j + 1)!(k  + 1)!

(k  ;T )

π /2,[l−k], ε0 t

(11.135)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

To estimate ((T ) j +1 λ )

(k  ;T ) π /2,[l−k], ε0 t

L / R in . . . L / Ri1 ((T ) j +1 λ )

(k  ;T )

495

we write:

 π / =



((R)s1 (T ) j +1λ )((L / R )s 2

(k  ;T )

π /)

partitions

Here n ≤ l − k, j ≤ k − 2 − k  , k  ≤ k − 2. Consider a given term in the sum over partitions. If |s2 | ≤ l∗ − k  − 1 we estimate the term in L 2 by: λ 2,[k−1,l−1], ε0 

(k  ;T )

t

π /∞,[l∗ −1−k  ], ε0 t

Otherwise, |s2 | ≥ l∗ − k  , hence |s1 | = n − |s2 | ≤ l − k − l∗ + k  ≤ l∗ − k + 1 + k  and j + 1 + |s1 | ≤ k − 1 − k  + |s1 | ≤ l∗ . We then estimate the term in L 2 by: λ ∞,[k−1,l∗ ], ε0  t

(k  ;T )

π /2,[l−k], ε0 t

Combining we obtain: 

((T ) j +1 λ ) (k ;T ) π /2,[l−k], ε0 (11.136) t

 /∞,[l∗ −1−k  ], ε0 ≤ Cl λ 2,[k−1,l−1], ε0  (k ;T ) π t t   ;T )  (k + λ ∞,[k−1,l∗ ], ε0  π /2,[l−k], ε0 t t 

(k  ;T ) ≤ Cl δ0 [1 + log(1 + t)]  π /2,[l−2−k  ], ε0 + (1 + t)−1 λ 2,[k−1,l−1], ε0 t

t

by the second assumption of the lemma and (11.128). We then conclude that (11.135), the contribution of the last sum in (11.103) is bounded by: Cl δ0 (1 + t)−1 [1 + log(1 + t)]  k−2   (k  ;T ) −1  ×  π /2,[l−2−k  ], ε0 + (1 + t) λ 2,[k−1,l−1], ε0 t

k  =0

(11.137)

t

Next, we consider the second sum in (11.103), given by (11.113). To estimate (T ) j +1 λ h/2,[l−k], ε0 t

we refer to the expressions following (11.113). If |s2 | ≤ l∗ we appeal to the first assumption of the lemma to place the factor (L / R )s2 h/ in L ∞ . Otherwise, |s1 | ≤ l∗ − k, thus since j + 1 ≤ k, j + 1 + |s1 | ≤ l∗ , we may appeal to the second assumption of the lemma to place the factor (R)s1 (T ) j +1λ in L ∞ . In this way we obtain: /2,[l−k], ε0 ≤ Cλ 2,[k,l], ε0 +Cl δ0 [1+log(1+t)] max  (T ) j +1 λ h t

t

i

(Ri )

π /2,[l−1−k], ε0 t

(11.138)

496

Chapter 11. Control of the Spatial Derivatives

It follows that the contribution of (11.113), the second sum in (11.103), is bounded by, noting that here k ≥ 1:  /2,[l−2], ε0 (11.139) Cl (1 + t)−1 λ 2,[k,l], ε0 + δ0 [1 + log(1 + t)] max  (Ri ) π t

t

i

Finally, we have the third sum in (11.103). In reference to the expression (11.117), the contribution of the constant term is, by (11.138) bounded by, noting that here k ≥ 2:  Cl (1 + t)−2 λ 2,[k,l], ε0 + δ0 [1 + log(1 + t)] max  (Ri ) π (11.140) /2,[l−3], ε0 t

t

i

To estimate the contribution of the polynomial qk  +1 (λ ), we must estimate: ((T ) j +1 λ )qk  +1 (λ )h/2,[l−k], ε0 t

We write:

 / Ri1 ((T ) j +1 λ )qk  +1 (λ )h/ L / R in . . . L    (L / R )s1 ((T ) j +1 λ )h/ ((R)s2 qk  +1 (λ )) =

(11.141)

partitions

Consider an arbitrary term in the above sum. We distinguish the two cases: Case 1: |s2 | ≤ l∗

Case 2: |s2 | ≥ l∗ + 1

and:

In Case 1 we can estimate (R)s2 qk  +1 (λ ) L ∞ ( ε0 ) ≤ Cl

 +1 k

(δ0 [1 + log(1 + t)])i

t

(11.142)

i=1

by the second assumption of the lemma, hence the corresponding term in the sum in (11.141) is bounded in L 2 by:   k +1 Cl  (δ0 [1 + log(1 + t)])i  ((T ) j +1 λ )h/2,[l−k], ε0 t

i=1



 +1 k

≤ Cl 



(δ0 [1 + log(1 + t)])i  λ 2,[k,l], ε0 t

i=1

+δ0 [1 + log(1 + t)] max  i

(Ri )

π /2,[l−1−k], ε0 t

(11.143)

In Case 2 we have |s1 | ≤ l∗ − k, hence from (11.115) with l replaced by l∗ we have: /∞,[l∗ −k], ε0 ≤ Cl δ0 [1 + log(1 + t)] ((T ) j +1 λ )h t

(11.144)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

497

On the other hand we may apply Lemma 11.2a with k, l replaced by 0, l − k to deduce:   k +1 (δ0 [1 + log(1 + t)])i−1  λ 2,[l−k], ε0 (11.145) qk  +1 (λ )2,[l−k], ε0 ≤ Cl  t

t

i=1

Thus, in Case 2 the corresponding term in the sum in (11.141) is bounded in L 2 by:   k +1 (δ0 [1 + log(1 + t)])i  λ 2,[l−k], ε0 (11.146) Cl  t

i=1

Combining the results for the two cases we obtain: ((T ) j +1 λ )qk  +1 (λ )h/2,[l−k], ε0 t   k +1

≤ Cl  (δ0 [1 + log(1 + t)])i  λ 2,[k,l], ε0 t

i=1

+δ0 [1 + log(1 + t)] max  i

(Ri )

π /

(11.147)

ε 2,[l−1−k],t 0

It follows that the contribution of the polynomial qk  +1 (λ ) is bounded by:   k−2 k +1

   Cl (δ0 [1 + log(1 + t)])i  (1 + t)−2−k λ 2,[k,l], ε0 t

k  =0

i=1

+δ0 [1 + log(1 + t)] max  (Ri ) π /2,[l−1−k], ε0 t i

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] λ 2,[k,l], ε0 t +δ0 [1 + log(1 + t)] max  (Ri ) π /2,[l−3], ε0 t

i



(11.148)

by (11.120), noting that here k ≥ 2. Combining the results (11.129), (11.130) and (11.134), (11.137), (11.139), (11.140) and (11.148), for the contributions of the five terms on the right in (11.103), we conclude that, simplifying, 

(k;T )

π /2,[l−k], ε0 ≤ (L / T )k t

(T ) 

π / 2,[l−k], ε0 t

+ Cl (1 + t)−1 [1 + log(1 + t)] + Cl (1 + t)

−1



k−1  k  =0



(k  ;T )

π /2,[l−1−k  ], ε0



λ 2,[k,l], ε0 + δ0 [1 + log(1 + t)] max  t

(11.149)

t

i

(Ri )

π /2,[l−2], ε0 t

498

Chapter 11. Control of the Spatial Derivatives 



Replacing  (k ;T ) π /2,[l−1−k  ], ε0 by  (k ;T ) π /2,[l−k  ], ε0 in the sum on the right, reduces t t (11.149) to a recursive inequality of the form (11.124), where now: xk = 

(k;T )

π /2,[l−k], ε0 t

ck = (L / T )k

(T ) 

π / 2,[l−k], ε0 t  −1  λ 2,[k,l], ε0 + δ0 [1 + log(1 + t)] max  +Cl (1 + t) t

a = Cl (1 + t)

−1

(Ri )

π /2,[l−2], ε0 t

i

[1 + log(1 + t)]

(11.150)

The inequality (11.126) follows for each k ∈ {0, . . . , m}, which implies: m 

x k ≤ (1 + a)m

k=0

m 

ck

(11.151)

k=0

Substituting the definitions (11.150) and recalling that (1 + t)−1 [1 + log(1 + t)] ≤ 1, we then obtain: m 



k=0

(k;T )

π /2,[l−k], ε0 ≤ Cl 

(T ) 

π / 2,[m,l], ε0

t

t

 −1 λ 2,[m,l], ε0 + δ0 [1 + log(1 + t)] max  + Cl (1 + t) t

(Ri )

π /2,[l−2], ε0 t

i

and the lemma is proved. of

We use Lemmas 11.5 and 11.6 to derive L ∞ and L 2 estimates for the T -derivatives terms of estimates for the T -derivatives of (T ) π / .

(T ) π / in

Lemma 11.7 Under the assumptions of Lemma 11.5 we have: 

(T )

π / + 2(1 − u + η0 t)−1 h/∞,[l], ε0 ≤ Cl  t

(T ) 

π / ∞,[0,l], ε0 t

+ Cl δ0 (1 + t)−1 [1 + log(1 + t)] L /T

(T )

π / − 2(1 − u + η0 t)−2 h /∞,[l−1], ε0 ≤ Cl  t

(T ) 

π / ∞,[1,l], ε0 t

+ Cl δ0 (1 + t)

−1

[1 + log(1 + t)]

and: / T )k max (L

(T )

2≤k≤m

π /∞,[l−k], ε0 ≤ Cl  t

(T ) 

π / ∞,[m,l], ε0

+ Cl δ0 (1 + t)

t

−1

[1 + log(1 + t)]

Proof. We consider formula (11.95) with m replaced by k ∈ {0, . . . , m}: (L / T )k

(T )

π /=

(k;T )

π / + λpk (λ)(1 − u + η0 t)−k−1 h/

(11.152)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

499

We express λpk (λ) as in (11.117) and we bring the contribution of the constant term −2 pk (−2) on the left. Since from (11.93) we have:  p (−2) = 1    0 p1 (−2) = −1    pk (−2) = 0 : for k ≥ 2

(11.153)

we obtain:

L /T

(T )

π / + 2(1 − u + η0 t)−1 h /=

(0;T )

π / + q1 (λ )(1 − u + η0 t)−1 h/

(T )

π / − 2(1 − u + η0 t)−2 h /=

(1;T )

π / + q2 (λ )(1 − u + η0 t)−2 h/

(L / T )k π /=

(k;T )

π / + qk+1 (λ )(1 − u + η0 t)−k−1 h/

(11.154)

: for k ≥ 2 We first take the  ∞,[l], ε0 -norm of each side of the first of (11.154). Since: t

/∞,[l], ε0 ≤ Cλ ∞,[l], ε0 q1 (λ )h t

t

+Cl λ ∞,[l−1], ε0 max  t

(R j )

π /∞,[l−1], ε0 t

j

≤ Cl δ0 [1 + log(1 + t)]

(11.155)

(provided that δ0 is suitably small), the first statement of the lemma results by substituting the estimate for  (0;T ) π /∞,[l], ε0 of Lemma 11.5 with m = 0. t

Next, we take the  ∞,[l−1], ε0 -norm of each side of the second of (11.154). t Since: q2 (λ )h /∞,[l−1], ε0 t

 2  i  λ ∞,[l−1], ε0 ≤C

(11.156)

t

i=1

 2  i   λ ∞,[l−2], ε0 max  + Cl t

i=1

≤ Cl

j

(R j )

π /∞,[l−2], ε0 t

2  (δ0 [1 + log(1 + t)])i i=1

≤ 2Cl δ0 [1 + log(1 + t)](1 + t) by (11.120) with k  = 1 (provided that δ0 is suitably small), the second statement of the lemma results by substituting the estimate for  (1;T ) π /∞,[l−1], ε0 given by Lemma 11.5 t with m = 1.

500

Chapter 11. Control of the Spatial Derivatives

Finally, we take the  ∞,[l−k], ε0 -norm of each side of the third equation of t (11.154), and then take the maximum over k ∈ {2, . . . , m}. Here:  k+1  i    qk+1 (λ )h λ ∞,[l−k], ε0 /∞,[l−k], ε0 ≤ C (11.157) t

t

i=1

 k+1 

+ Cl



λ ∞,[l−1−k], ε0

i

t

 max 

i=1

≤ Cl

(R j )

π /∞,[l−1−k], ε0 t

j

k+1  (δ0 [1 + log(1 + t)])i ≤ 2Cl δ0 [1 + log(1 + t)](1 + t)k i=1

by (11.120) with k  = k (provided that δ0 is suitably small). In view of (11.157), the last statement of the lemma results after substituting the /∞,[l−k], ε0 given by Lemma 11.5. estimate for max2≤k≤m  (k;T ) π t

Lemma 11.8 Let the assumptions of Lemma 11.6 hold. Let us define: T[0,l] = 

(T )

T[1,l] = 

(T )

π / + 2(1 − u + η0 t)−1 h/2,[l], ε0 t

π / + 2(1 − u + η0 t)

+ L /T

(T )

−1

h/2,[l], ε0 t

π / − 2(1 − u + η0 t)

−2

h/2,[l−1], ε0 t

and for m ≥ 2: T[m,l] = 

(T )

π / + 2(1 − u + η0 t)−1 h/2,[l], ε0 t

+ L /T +

m 

(T )

π / − 2(1 − u + η0 t)−2 h/2,[l−1], ε0 t

(L / T )k

(T )

π /2,[l−k], ε0 t

k=2

We then have: T[m,l] ≤ Cl 

(T ) 

π / 2,[m,l], ε0  t +Cl (1 + t)−1 λ 2,[m,l], ε0 + δ0 [1 + log(1 + t)] max  t

(Ri )

i

π /

ε 2,[l−1],t 0

Proof. In reference to the formulas (11.154) we estimate qk+1 (λ )h /2,[l−k], ε0 t

We write, for n ≤ l − k,   L / R in . . . L / Ri1 qk+1 (λ )h / =

: for k ∈ {0, . . . , m}

   (R)s1 (qk+1 (λ )) ((L / R )s2 h/) partitions

(11.158)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

501

Consider an arbitrary term in the above sum. If |s1 | ≤ l∗ , the estimate (11.142) with k  = k holds and, if also s2 = ∅, the corresponding term in the sum in (11.158) is bounded in L 2 by: k+1   i Cl (δ0 [1 + log(1 + t)]) max  (R j ) π /2,[l−1−k], ε0 (11.159) t

j

i=1

Otherwise, |s1 | ≥ l∗ + 1 or s2 = ∅, in any case |s2 | ≤ l − k − |s1 | ≤ l∗ − k, hence by the first assumption of Lemma 11.6: (L / R )s2 h/ L ∞ ( ε0 ) ≤ C t

k

Using also the estimate (11.145) with = k, the corresponding term in the sum in (11.158) is then bounded in L 2 by: k+1   i−1 Cl (δ0 [1 + log(1 + t)]) (11.160) λ 2,[l−k], ε0 t

i=1

Combining the results for the two cases we obtain: qk+1 (λ )h /2,[l−k], ε0 t k+1   i−1 · ≤ Cl (δ0 [1 + log(1 + t)])

(11.161)

i=1

 · λ 2,[l−k], ε0 + δ0 [1 + log(1 + t)] max  t

j

Since: k+1 

(δ0 [1 + log(1 + t)])

i−1

(R j )

k+1 

≤

i=1

π /2,[l−1−k], ε0 t

 δ0i−1

[1 + log(1 + t)]k ≤ 2(1 + t)k

(11.162)

i=1

(provided that δ0 is suitably small), it follows that: qk+1 (λ )h /2,[l−k], ε0 t  k  ≤ Cl (1 + t) λ 2,[l−k], ε0 + δ0 [1 + log(1 + t)] max  t

j

(R j )

(11.163) π /2,[l−1−k], ε0 t

Taking then the  2,[l−k], ε0 -norms, for k = 0, k = 1, and k = 2, . . . , m, of each side t of the three formulas (11.154) respectively, substituting (11.163), summing appropriately, and substituting the estimate of Lemma 11.6 for m 



k=0

yields the result of the present lemma.

(k;T )

π /2,[l−k], ε0 t

502

Chapter 11. Control of the Spatial Derivatives

In establishing the primary propositions of the present chapter, which concern estimates for the spatial derivatives of the Tˆ i of which at least one is a T -derivative, we shall have to estimate, for given St,u 1-forms ξ , the commutators: (i1 ...in )

cm,n [ξ ] = L / R in . . . L / Ri1 (L / T )m D /ξ − D /L / R in . . . L / Ri1 (L / T )m ξ

(11.164)

The following lemma gives an expression for these commutators. Lemma 11.9 The commutators (i1 ...in )

(i1 ...in ) c

m,n [ξ ]

are given by:

cm,n [ξ ] = L / R in . . . L / Ri1 cm,0 [ξ ] −

n−1  j =0

L / R in . . . L / Rin− j +1

(Rin− j )

π /1 · L / Rin− j −1 . . . L / Ri1 (L / T )m ξ



where the cm,0 [ξ ] are given by: cm,0 [ξ ] = −

m−1 

(L / T )i

(T )

π /1 · (L /T )m−i−1 ξ



i=0

Here, (T ) π /1 is the Lie derivative with respect to T of the induced connection on St,u (see (9.178), (9.179)). Proof. Consider first the case n = 0. Then the definition (11.164) reduces to: / T )m D /ξ − d/(L / T )m ξ cm,0 [ξ ] = (L

(11.165)

Let us replace m by m − 1 in (11.165) and apply L /T to the resulting equation. We obtain: (L / T )m D /ξ − L /T D /(L /T )m−1 ξ = L /T cm−1,0 [ξ ]

(11.166)

Now, according to the commutation formula (9.180) of Chapter 9 we have, for any St,u 1-form α: /α) − D /(L /T α) = − (T ) π /1 · α (11.167) L /T (D Taking α = (L /T )m−1 ξ we obtain: /(L /T )m−1 ξ − D /(L / T )m ξ = − L /T D

(T )

π /1 · (L /T )m−1 ξ

(11.168)

Adding (11.166) and (11.168) yields, in view of the definition (11.165), cm,0 [ξ ] = L /T cm−1,0 [ξ ] −

(T )

π /1 · (L /T )m−1 ξ

(11.169)

Applying Proposition 8.2 to this recursion, noting that c0,0 [ξ ] = 0, we then obtain: cm,0 [ξ ] = −

m−1 

(L / T )i

i=0

(T )

π /1 · (L /T )m−i−1 ξ

 (11.170)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

503

Next, we consider the case n ≥ 1. Let us replace n by n − 1 in definition (11.164) and apply L / Rin to the resulting equation. We obtain: L / R in . . . L / Ri1 (L / T )m D /ξ − L / R in D /L / Rin−1 . . . L / Ri1 (L / T )m ξ = L / R in

(i1 ...in−1 )

cm,n−1 [ξ ] (11.171) We now appeal to the commutation formula (9.167) of Chapter 9, which holds on an arbitrary Riemannian manifold (M, g). In the case that (M, g) = (St,u , h/), the formula in question reads: /α) − D /(L / X α) = − (X ) π /1 · α (11.172) L / X (D for any St,u 1-form α and any St,u -tangential vectorfield X. Taking X = Rin and α = L / Rin−1 . . . L / Ri1 (L /T )m ξ we obtain: L / R in D /L / Rin−1 ...L / Ri1 (L / T )m ξ − D /L / Rin ...L / Ri1 (L / T )m ξ = −

(Rin )

π /1 · L / Rin−1 ...L / Ri1 (L / T )m ξ (11.173) Adding (11.171) and (11.173) yields, in view of the definition (11.164), (i1 ...in )

(i1 ...in−1 )

cm,n [ξ ] = L / R in

(Rin )

cm,n−1 [ξ ] −

π /1 · L / Rin−1 . . . L / Ri1 (L /T )m ξ (11.174)

Applying then Proposition 8.2 to this recursion we obtain: (i1 ...in )

cm,n [ξ ] = L / R in . . . L / Ri1 cm,0 [ξ ] −

n−1  j =0

(11.175)

L / R in . . . L / Rin− j +1

(Rin− j )

π /1 · L / Rin− j −1 . . . L / Ri1 (L / T )m ξ



and the lemma is proved. ˇ/ defined by /1 . In terms of the operator D We must now obtain estimates for (T ) π (10.433) of Chapter 10, we can express (see (9.179)): (T )

π /1 =

(T )

(π /1 )b · h/−1 ,

(T )

ˇ/ (π /1 )b = D

(T )

π /

(11.176)

(recall the analogous formulas (10.434)). We shall need the following analogue of Lemma 10.10. Lemma 11.10 Let ϑ be an arbitrary symmetric 2-covariant St,u tensorfield. The following commutation formula holds for any non-negative integer m: ˇ/ϑ − D ˇ/(L / T )m ϑ = − (L / T )m D

m−1 

(L / T )k (

(T )

π /1 · (L /T )m−k−1 ϑ)

k=0

The proof of the above lemma is straightforward. We proceed to: Lemma 11.11 Let l be a positive integer and m a non-negative integer m ≤ l − 1. Let also hypothesis H2 hold. Suppose that: λ ∞,[m,l], ε0 ≤ Cl δ0 [1 + log(1 + t)], t

 max  i

(T ) 

π / ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)], t

(Ri )

π /∞,[l−1], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

504

Chapter 11. Control of the Spatial Derivatives

and, if l ≥ 2, max 

(Ri )

π /1 ∞,[l−2], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

i

Then, if δ0 is suitably small (depending on l) we have: 

(T )

π /1 ∞,[m,l−1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

Proof. The assumptions of the present lemma include those of Lemma 11.7. Therefore under the present assumptions the conclusions of Lemma 11.7 hold. Substituting in these the bound for / ∞,[m,l], ε0  (T ) π t

given by the second assumption of the present lemma yields:  L /T

(T )

(T )

π / + 2(1 − u + η0 t)−1 h/∞,[l], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

π / − 2(1 − u + η0 t)−2 h/∞,[l−1], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

/ T )k max (L

(T )

π /∞,[l−k], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

2≤k≤m

(11.177) / is covariantly constant on each St,u , the second Now, since 2(1 − u + η0 t)−1 h equation of (11.176) can be written in the form: (T )

ˇ/( (π /1 )b = D

(T )

π / + 2(1 − u + η0 t)−1 h/)

(11.178)

Applying L / R in . . . L / Ri1 , for n ∈ {0, . . . , l − 1}, to this and using Lemma 10.10, taking in that lemma / + 2(1 − u + η0 t)−1 h/, ϑ = (T ) π we obtain: L / R in . . . L / R i1 −

n−1 

(T )

ˇ/L (π /1 )b = D / R in . . . L / R i1 (

L / R in . . . L / Rn− j +1

(Rin− j )

j =0

(T )

π / + 2(1 − u + η0 t)−1 h/)

π /1 · L / Rin− j −1 . . . L / R i1 (

(T )

(11.179)

π / + 2(1 − u + η0 t)−1 h/)



By hypothesis H2 and the first of the estimates (11.177), ˇ/L D / R in . . . L / R i1 ( ≤ C(1 + t)

−1

(T )

π / + 2(1 − u + η0 t)−1 h/) L ∞ ( ε0 ) t



(T )

π / + 2(1 − u + η0 t)

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

−1

h/∞,[l], ε0 t

(11.180)

Consider next a term j ∈ {0, . . . , n − 1} in the sum in (11.179). There are j angular (R ) / Rin− j +1 outside the parenthesis. Thus, in− j π /1 receives at most j ≤ derivatives L / R in . . . L

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

505 ε

n − 1 ≤ l − 2 angular derivatives, hence the corresponding factor is bounded in L ∞ (t 0 ) / + 2(1 − u + η0 t)−1 h/ receives at most by the last assumption of the lemma, while (T ) π ε n − 1 ≤ l − 2 angular derivatives, hence is bounded in L ∞ (t 0 ) by the first of the estimates (11.177) (even with l reduced to l − 2). It follows that the sum in (11.179) is ε bounded in L ∞ (t 0 ) by: Cl δ02 (1 + t)−3 [1 + log(1 + t)]2

(11.181)

Combining the results (11.180) and (11.181) for the two terms on the right in (11.179), and taking the maximum over i 1 . . . i n and then the maximum over n ∈ {0, . . . , l − 1}, we obtain: 

(T )

(π /1 )b ∞,[l−1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

(11.182)

t

Using then also the next-to-the-last assumption of the lemma we conclude that: 

(T )

π /1 ∞,[l−1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

Next, we apply Lemma 11.10 taking ϑ = equation of (11.176), for k ≥ 1, (L / T )k

(T )

(T ) π /

ˇ/(L (π /1 )b = D / T )k

(11.183)

to obtain, in view of the second

(T )

π / + dk

(11.184)

where: dk = −

k−1 

(L /T )k−1− p (

(T )

π /1 · (L /T ) p

(T )

π /)

p=0

=−

k−1 

Nq,k−1 ((L /T )k−1−q

(T )

π /1 ) · ((L / T )q

(T )

π /)

(11.185)

q=0

Here, Nq,k−1 , q ∈ {0, . . . , k − 1}, are the positive integers:  (k − 1 − q + r )! 1 (k − 1 − q)! r! q

Nq,k−1 =

(11.186)

r=0

For k = 1, we have N0,0 = 1 and (11.185) reduces to: d1 = −

(T )

π /1 ·

(T )

π /

/1 )b = 2(1 − u + η0 t)−1 (T ) (π (T ) (T ) − π /1 · ( π / + 2(1 − u + η0 t)−1 h/)

(11.187)

Moreover, since −2(1 − u + η0 t)−2 h/ is covariantly constant on each St,u , in the case k = 1, (11.185) may be written in the form: L /T

(T )

ˇ/(L (π /1 )b = D /T

(T )

π / − 2(1 − u + η0 t)−2 h/) + d1

(11.188)

506

Chapter 11. Control of the Spatial Derivatives

Applying L / R in . . . L / Ri1 , for n ∈ {0, . . . , l − 2}, to this and using Lemma 10.10, setting in that lemma / − 2(1 − u + η0 t)−2 h/, ϑ=L /T (T ) π we obtain: L / R in . . . L / R i1 L /T −

n−1  j =0

(T )

ˇ/L (π /1 )b = D / R in . . . L / Ri1 (L /T

L / R in . . . L / Rin− j +1

(Rin− j )

(T )

π / − 2(1 − u + η0 t)−2 h/)

π /1 · L / Rin− j −1 . . . L / Ri1 (L /T

(T )

π / − 2(1 − u + η0 t)−2 h/)

/ Ri1 d1 +L / R in . . . L



(11.189)

By hypothesis H2 and the second of the estimates (11.177), ˇ/L D / R in . . . L / Ri1 (L /T ≤ C(1 + t)

−1

(T )

L /T

π / − 2(1 − u + η0 t)−2 h/) L ∞ ( ε0 ) t

(T )

π / − 2(1 − u + η0 t)

−2

h/∞,[l−1], ε0

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

t

(11.190)

Consider next a term j ∈ {0, . . . , n − 1} in the sum in (11.189). There are j angular (R ) /1 receives at most j ≤ n − 1 ≤ l − 3 derivatives outside the parenthesis. Thus, in− j π ε0 ∞ angular derivatives and is bounded in L (t ) by the last assumption of the lemma / − 2(1 − u + η0 t)−2 h/ receives at most (even with l reduced to l − 1), while L /T (T ) π ε n − 1 ≤ l − 3 angular derivatives and is bounded in L ∞ (t 0 ) by the second of the estimates (11.177) (even with l reduced to l − 2). It follows that the sum in (11.189) is ε bounded in L ∞ (t 0 ) by: Cl δ02 (1 + t)−3 [1 + log(1 + t)]2

(11.191)

ε

Finally, the last term in (11.189) is bounded in L ∞ (t 0 ) by: d1 ∞,[l−2], ε0 t

To estimate this we consider the expression (11.187). The contribution of the first term on the right is by (11.182) (even with l reduced to l − 1) bounded by: Cl δ0 (1 + t)−3 [1 + log(1 + t)] while the contribution of the second term is by (11.183) (even with l reduced to l − 1) and the first of the estimates (11.177) bounded by: Cl δ02 (1 + t)−3 [1 + log(1 + t)]2 We thus obtain: d1 ∞,[l−2], ε0 ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)]2 t

(11.192)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

507

The results (11.190), (11.191) together with the estimate (11.192) imply: L /T

(T )

(π /1 )b ∞,[l−2], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

(11.193)

t

Using then also the first of the estimates (11.177) as well as the next-to-the-last assumption of the lemma we conclude that: L /T

(T )

π /1 ∞,[l−2], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

(11.194)

t

According to (11.183) and (11.194), we have shown that: (T )

(L / T )k

π /1 ∞,[l−1−k], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

: for k = 0, 1 (11.195)

t

We proceed by induction, to show that: (L / T )k

(T )

π /1 ∞,[l−1−k], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

: ∀k ≤ m (11.196)

t

which is evidently equivalent to the conclusion of the lemma. Given then k ∈ {2, . . . , m}, let: 

(T )

π /1 ∞,[l−1−k  ], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

: hold for all k  ≤ k − 1 (11.197) What needs to be shown is then the same for k  = k. Going back to the expression (11.185) we separate the terms corresponding to q = 0, 1 in the sum, writing: (L / T )k

t

(L / T )0

(T )

π / = −2(1 − u + η0 t)−1 h/ + (

(L / T )1

(T )

π / = 2(1 − u + η0 t)−2 h/ + (L /T

(T )

π / + 2(1 − u + η0 t)−1 h/),

(T )

π / − 2(1 − u + η0 t)−2 h/)

We then obtain: dk = 2(1 − u + η0 t)−1 N0,k−1 ((L /T )k−1

(T )

π /1 )b k−2 (T ) −2(1 − u + η0 t) N1,k−1 ((L /T ) π /1 )b k−1 (T ) (T ) −N0,k−1 ((L /T ) π /1 ) · ( π / + 2(1 − u + η0 t)−1 h/) k−2 (T ) /T ) π /1 ) · (L /T (T ) π / − 2(1 − u + η0 t)−2 h/) −N1,k−1 ((L −2

−

k−1 

Nq,k−1 ((L /T )k−1−q

(T )

π /1 ) · ((L / T )q

(T )

π /)

(11.198)

q=2

Here k ≥ 2. We now apply L / R in . . . L / Ri1 , for n ∈ {0, . . . , l − 1 − k} to (11.184). Using Lemma 10.10, setting in that lemma ϑ = (L / T )k

(T )

π /,

we obtain: L / R in . . . L / Ri1 (L / T )k −

n−1  j =0

(T )

ˇ/L (π /1 )b = D / R in . . . L / Ri1 (L / T )k

L / R in . . . L / Rin− j +1

(Rin− j )

(T )

π /

π /1 · L / Rin− j −1 . . . L / Ri1 (L / T )k

/ Ri1 dk +L / R in . . . L

(T )

 π / (11.199)

508

Chapter 11. Control of the Spatial Derivatives

By hypothesis H2 and the third of the estimates (11.177), ˇ/L / Ri1 (L / T )k D / R in . . . L ≤ C(1 + t)

−1

(T )

t

(L /T )

≤ Cl δ0 (1 + t)

−2

π / L ∞ ( ε0 )

k (T )

π /∞,[l−k], ε0 t

[1 + log(1 + t)]

(11.200)

Consider a term j ∈ {0, . . . , n − 1} in the sum in (11.199). Again, there are j angular (R ) derivatives outside the parenthesis. Thus, in− j π /1 receives at most j ≤ n −1 ≤ l −2−k ε0 ∞ angular derivatives and is bounded in L (t ) by the last assumption of the lemma (even with l reduced to l − k), while (L /T )k (T ) π / receives at most n − 1 ≤ l − 2 − k angular ε ∞ derivatives and is bounded in L (t 0 ) by the third of the estimates (11.177) (even with ε l reduced to l − 2). It follows that the sum in (11.199) is bounded in L ∞ (t 0 ) by: Cl δ02 (1 + t)−3 [1 + log(1 + t)]2

(11.201)

ε

Finally, the last term in (11.199) is bounded in L ∞ (t 0 ) by: dk ∞,[l−1−k], ε0 t

To estimate this we consider expression (11.198). The contributions of the first two terms on the right are bounded using the inductive hypothesis (11.197) by:

and:

Cl δ0 (1 + t)−3 [1 + log(1 + t)]

(11.202)

Cl δ0 (1 + t)−4 [1 + log(1 + t)]

(11.203)

respectively. The contributions of the next two terms are both bounded using the inductive hypothesis (11.197) and the first two of the estimates (11.177) by: Cl δ02 (1 + t)−3 [1 + log(1 + t)]2

(11.204)

Lastly, we have the sum in (11.198). This is present only for k ≥ 3. Consider a term /1 and (L /T )q (T ) π / q ∈ {2, . . . , k − 1} in this sum. Since each of the factors (L /T )k−1−q (T ) π receives at most l − 1 − k angular derivatives, the contribution of the sum is bounded by: / T )k Cl max (L k  ≤k−3

≤



(T )

π /1 ∞,[l−4−k  ], ε0 · t

Cl δ02 (1 + t)−3 [1 + log(1 +

t)]

max (L / T )q

2≤q≤k−1

(T )

π /∞,[l−1−k], ε0

2

t

(11.205)

by virtue of the inductive hypothesis (11.197) and the third of the estimates (11.177). Combining the results (11.202)–(11.205) we obtain: dk ∞,[l−1−k], ε0 ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)]2 t

(11.206)

The results (11.200), (11.201) together with the estimate (11.206) imply: (L / T )k

(T )

(π /1 )b ∞,[l−1−k], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.207)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

509

Using then also the estimates (11.177) as well as the next-to-the-last assumption of the lemma we conclude that: (T )

(L / T )k

π /1 ∞,[l−1−k], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.208)

This completes the inductive step and the proof of the lemma. To proceed with L 2 estimates for lemma shall also be used in the sequel.

(T ) π /1 ,

we shall need the following lemma. This

Lemma 11.12 Let k, l be positive integers k ≤ l and let: λ ∞,[k−1,l∗ −1], ε0 ≤ Cl δ0 [1 + log(1 + t)] t

max 

(Ri )



π /∞,[l∗ −1], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

i

(T ) 

π / ∞,[k−1,l∗ −1], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

Then, provided that δ0 is suitably small (depending on l), we have, for any St,u tensorfield ξ : ξ · h /2,[k−1,l−1], ε0 ,ξ · h /−1 2,[k−1,l−1], ε0 t t   ≤ Cl ξ 2,[k−1,l−1], ε0 + ξ ∞,[k−1,l∗ −1], ε0 max  t

t

(Ri )

i

 π /2,[l−2], ε0 + T[k−2,l−2] t

where the T[k,l] are the quantities defined in the statement of Lemma 11.8 (the term T[k−2,l−2] is present only for k ≥ 2). Proof. The assumptions of the present lemma include those of Lemma 11.7 with (k − 1, l∗ − 1) in the role of (m, l). Therefore under the present assumptions the conclusions of Lemma 11.7 hold with (m, l) replaced by (k − 1, l∗ − 1). Substituting in these the bound for  (T ) π / ∞,[k−1,l∗ −1], ε0 t

given by the third assumption of the present lemma yields: 

(T )

π / + 2(1 − u + η0 t)−1 h /∞,[l∗ −1], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

L /T

(T )

π / − 2(1 − u + η0 t)−2 h /∞,[l∗ −2], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

t t

/ T )i max (L

2≤i≤k−1

(T )

π /∞,[l∗ −1−i], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(11.209) We shall estimate: /)2,[l− j ], ε0 (L /T ) j −1 (ξ · h t

: for j = 1, . . . , k

(11.210)

510

Chapter 11. Control of the Spatial Derivatives

We begin with the case j = 1. In this case using the second assumption of the lemma we readily obtain:  ξ · h /2,[l−1], ε0 ≤ Cl ξ 2,[l−1], ε0 + ξ ∞,[l∗ −1], ε0 max  t

t

t

(Ri )

i

π /2,[l−2], ε0 t

(11.211) For j ≥ 2 we express: /) = ξ j −1 · h / (L /T ) j −1 (ξ · h + ( j − 1)((L /T ) j −2 ξ ) · ( (T ) π / + 2(1 − u + η0 t)−1 h/) ( j − 1)( j − 2) ) ((L /T ) j −3 ξ ) · (L + /(T / − 2(1 − u + η0 t)−2 h/) T π 2 j −2  ( j − 1)! + ((L /T ) j −2−i ξ ) · ((L /T )i (T ) π /) (11.212) (i + 1)!( j − 2 − i )! i=2

Here: ξ j −1 = (L /T ) j −1 ξ − 2( j − 1)(1 − u + η0 t)−1 (L /T ) j −2 ξ + ( j − 1)( j − 2)(1 − u + η0 t)−2 (L /T ) j −3ξ

(11.213)

Only the first two terms in (11.212) and (11.213) are present for j = 2. Also, the sum in (11.212) is present only for j ≥ 4. We are to bound the  2,[l− j ], ε0 norm of (11.212). In regard to the first term, t using the second assumption of the lemma we deduce: ξ j −1 · h /2,[l− j ], ε0 t  ≤ Cl ξ j −1 2,[l− j ], ε0 + ξ j −1 ||∞,[l∗ − j ], ε0 max  t

t

(Ri )

(11.214) π /2,[l−1− j ], ε0 t

i

Now, from (11.213) we readily obtain: ξ j −1 ∞,[l∗ − j ], ε0 ≤ Cl ξ ∞,[ j −1,l∗ −1], ε0

(11.215)

ξ j −1 2,[l− j ], ε0 ≤ Cl ξ 2,[ j −1,l−1], ε0

(11.216)

t

t

t

t

Substituting in (11.214) then yields: ξ j −1 · h /2,[l− j ], ε0 t  ≤ Cl ξ 2,[ j −1,l−1], ε0 + ξ ||∞,[ j −1,l∗ −1], ε0 max  t

t

i

(Ri )

(11.217) π /2,[l−1− j ], ε0 t

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

511

The contribution of the second term in (11.212) is bounded by:

/ + 2(1 − u + η0 t)−1 h/∞,[l∗ −1], ε0 Cl ξ 2,[ j −2,l−2], ε0  (T ) π t

t

+ξ ∞,[ j −2,l∗ −1], ε0 

(T )

t

π / + 2(1 − u + η0 t)

−1

h/2,[l− j ], ε0 t

or, substituting the first of the estimates (11.209),

Cl δ0 (1 + t)−1 [1 + log(1 + t)]ξ 2,[ j −2,l−2], ε0 t

(T )

+ξ ∞,[ j −2,l∗ −1], ε0  t

π / + 2(1 − u + η0 t)

−1



h/2,[l− j ], ε0

 (11.218)

t

Similarly, the contribution of the third term in (11.212) is bounded by:

Cl ξ 2,[ j −3,l−3], ε0 L /T (T ) π / − 2(1 − u + η0 t)−2 h/∞,[l∗ −2], ε0 t t  −2 (T ) +ξ ∞,[ j −3,l∗ −1], ε0 L /T π / − 2(1 − u + η0 t) h/2,[l− j ], ε0 t

t

or, substituting the second of the estimates (11.209),

Cl δ0 (1 + t)−1 [1 + log(1 + t)]ξ 2,[ j −3,l−3], ε0 t

/T +ξ ∞,[ j −3,l∗ −1], ε0 L t

(T )

π / − 2(1 − u + η0 t)−2 h/2,[l− j ], ε0



(11.219)

t

Finally, the contribution of the sum in (11.212) is bounded by:  Cl ξ 2,[ j −4,l−4], ε0 max (L /T )i (T ) π /∞,[l∗ −1−i], ε0 t

2≤i≤ j −2

t

+ ξ 

ε ∞,[ j −4,l−4],t 0

j −2 

(L /T )

i (T )

π /

ε 2,[l− j ],t 0

i=2

or, substituting the third of the estimates (11.209),  Cl δ0 (1 + t)−1 [1 + log(1 + t)]ξ 2,[ j −4,l−4], ε0 t

+ ξ ∞,[ j −4,l−4], ε0

j −2 

t

(L / T )i

(T )

π /2,[l− j ], ε0 t

(11.220)

i=2

Combining the estimates (11.217)–(11.220) for the contributions of the four terms on the right in (11.212) and recalling the definition of the quantities T[k,l] (see Lemma 11.8) we conclude that: /)2,[l− j ], ε0 (L /T ) j −1 (ξ · h t  ≤ Cl ξ 2,[ j −1,l−1], ε0 + ξ ∞,[ j −1,l∗ −1], ε0 t t   (Ri ) ε × max  π /2,[l−1− j ], 0 + T[ j −2,l−2] i

t

(11.221)

512

Chapter 11. Control of the Spatial Derivatives

Summing then over j ∈ {2, . . . , k} and adding the estimate (11.211) yields the estimate of the lemma for ξ · /h2,[k−1,l−1], ε0 . The estimate for ξ · h/−1 2,[k−1,l−1], ε0 follows t t in a similar manner. We proceed with L 2 estimates for

(T ) π /1 .

Lemma 11.13 Let l be a positive integer and m a non-negative integer m ≤ l − 1. Let also hypothesis H2 hold. Suppose that: λ ∞,[m,l∗ ], ε0 ≤ Cl δ0 [1 + log(1 + t)], t

 max 

(T ) 

π / ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)], t

(Ri )

i

π /∞,[l∗ −1], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

and (if l∗ ≥ 2): max 

(Ri )

i

π /1 ∞,[l∗ −2], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

Then,if δ0 is suitably small (depending on l) we have: 

(T )

π /1 2,[m,l−1], ε0  t ≤ Cl (1 + t)−1 T[m,l] + δ0 (1 + t)−1 [1 + log(1 + t)] ·  /2,[l−2], ε0 + (1 + t) max  · max  (Ri ) π t

i

(Ri )

i

 π /1 2,[l−2], ε0 t

Proof. The assumptions of the present lemma include those of Lemma 11.7 with (m, l∗ ) in the role of (m, l). Therefore under the present assumptions the conclusions of Lemma 11.7 hold with (m, l) replaced by (m, l∗ ). Substituting in these the bound for 

(T ) 

π / ∞,[m,l∗ ], ε0 t

given by the second assumption of the present lemma yields:  L /T

(T )

(T )

π / + 2(1 − u + η0 t)−1 h/∞,[l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

π / − 2(1 − u + η0 t)−2 h /∞,[l∗ −1], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

max (L /T )

k (T )

π /∞,[l∗ −k], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

2≤k≤m

(11.222) Also, the assumptions of the present lemma coincide with those of Lemma 11.11 with (m, l∗ ) in the role of (m, l). Therefore under the present assumptions the conclusion of Lemma 11.11 holds with (m, l) replaced by (m, l∗ ), that is we have: 

(T )

π /1 ∞,[m,l∗ −1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.223)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

513

We follow the proof of Lemma 11.11. We consider again expression (11.179) for n ∈ {0, . . . , l − 1}. By hypothesis H2 , ˇ/L D / R in . . . L / R i1 (

(T )

π / + 2(1 − u + η0 t)−1 h/) L 2 ( ε0 ) t

≤ C(1 + t)−1 

(T )

π / + 2(1 − u + η0 t)−1 h/2,[l], ε0 t

≤ C(1 + t)−1 T[0,l]

(11.224) (R

)

/1 receives Consider next a term j ∈ {0, . . . , n − 1} in the sum in (11.179). Again, in− j π at most j ≤ n − 1 ≤ l − 2 angular derivatives and (T ) π / + 2(1 − u + η0 t)−1 h/ receives at most n − 1 ≤ l − 2 angular derivatives, hence the corresponding factors are bounded in ε /1 2,[l−2], ε0 and T[0,l−2] respectively. If at most l∗ − 2 angular L 2 (t 0 ) by maxi  (Ri ) π (R

)

t

ε

/1 the corresponding factor is bounded in L ∞ (t 0 ) by the last derivatives fall on in− j π assumption of the lemma. Otherwise, at most n − l∗ ≤ l − 1 − l∗ ≤ l∗ angular derivatives ε fall on (T ) π / + 2(1 − u + η0 t)−1 h /, so the corresponding factor is bounded in L ∞ (t 0 ) by the first of the estimates (11.222). In this way we obtain an L 2 bound for the sum in (11.179) by:   −2 (Ri ) ε Cl δ0 (1 + t) [1 + log(1 + t)] T[0,l−2] + (1 + t) max  π /1 2,[l−2], 0 (11.225) t

i

Combining the results (11.224) and (11.225) for the two terms on the right in (11.179), and taking the maximum over i 1 . . . i n and then summing over n ∈ {0, . . . , l − 1}, we obtain: (T )

(π /1 )b 2,[l−1], ε0  t −1 ≤ Cl (1 + t) T[0,l] + δ0 [1 + log(1 + t)] max 



i

(Ri )



(11.226)

π /1 2,[l−2], ε0 t

Applying then Lemma 11.12 with k = 1, taking ξ = (T ) (π /1 )b , we conclude that, by (11.223) with m = 0 and (11.226),

/1 2,[l−1], ε0 ≤ Cl (1 + t)−1 T[0,l] + δ0 (1 + t)−1 [1 + log(1 + t)]·  (T ) π t   /2,[l−2], ε0 + (1 + t) max  (Ri ) π /1 2,[l−2], ε0 (11.227) · max  (Ri ) π t

i

t

i

Next, we consider again expression (11.189) for n ∈ {0, . . . , l − 2}. By hypothesis H2 , ˇ/L D / R in . . . L / Ri1 (L /T

(T )

≤ C(1 + t)−1 L /T ≤ C(1 + t)−1 T[1,l]

π / − 2(1 − u + η0 t)−2 h/) L 2 ( ε0 ) t

(T )

π / − 2(1 − u + η0 t)−2 h/2,[l−1], ε0 t

(11.228)

514

Chapter 11. Control of the Spatial Derivatives (R

)

Consider a term j ∈ {0, . . . , n − 1} in the sum in (11.189). Again, in− j π /1 receives at / − 2(1 − u + η0 t)−2 h/ receives at most j ≤ n − 1 ≤ l − 3 angular derivatives and L /T (T ) π most n − 1 ≤ l − 3 angular derivatives, hence the corresponding factors are bounded in ε L 2 (t 0 ) by maxi  (Ri ) π /1 2,[l−3], ε0 and T[1,l−2] respectively. If at most l∗ − 2 angular (R

t

)

ε

/1 , the corresponding factor is bounded in L ∞ (t 0 ) by the last derivatives fall on in− j π assumption of the lemma. Otherwise, at most n − l∗ ≤ l − 2 − l∗ ≤ l∗ − 1 angular / − 2(1 − u + η0 t)−2 h/, so the corresponding factor is bounded derivatives fall on L /T (T ) π ε in L ∞ (t 0 ) by the second of the estimates (11.222). In this way we obtain an L 2 bound for the sum in (11.189) by:   (11.229) /1 2,[l−3], ε0 Cl δ0 (1 + t)−2 [1 + log(1 + t)] T[1,l−2] + (1 + t) max  (Ri ) π t

i

ε

The last term in (11.189) is bounded in L 2 (t 0 ) by: d1 2,[l−2], ε0 t

To estimate this we consider the expression (11.187). The contribution of the first term on the right is by (11.226) with l reduced to l − 1 bounded by:  −2 (Ri ) ε Cl (1 + t) π /1 2,[l−3], 0 T[0,l−1] + δ0 [1 + log(1 + t)] max  (11.230) t

i

To estimate the contribution of the second term we apply Lemma 11.2.b with (1, l − 1) /1 . Using also (11.223) with in the role of (k, l), taking the factor of order 2 ϑ to be (T ) π m = 0 and the first inequality of (11.222) (and (11.227) with l reduced to l − 1) we then deduce that the contribution of the second term is bounded by:  −2 Cl δ0 (1 + t) [1 + log(1 + t)] T[0,l−1] + δ0 (1 + t)−1 [1 + log(1 + t)]·   (Ri ) (Ri ) ε ε · max  (11.231) π /2,[l−3], 0 + (1 + t) max  π /1 2,[l−3], 0 t

i

t

i

Combining the results (11.230) and (11.231) yields: d1 2,[l−2], ε0

(11.232)

t

≤ Cl (1 + t)−2 [1 + log(1 + t)] T[0,l−1] + δ0 (1 + t)−1 [1 + log(1 + t)]·   (Ri ) (R ) i · max  π /2,[l−3], ε0 + (1 + t) max  π /1 2,[l−3], ε0 t

i

t

i

The results (11.228), (11.229) together with the estimate (11.232) imply: (T )

(π /1 )b 2,[l−2], ε0 t

−1 ≤ Cl (1 + t) T[1,l] + δ0 (1 + t)−1 [1 + log(1 + t)]·  /2,[l−3], ε0 + (1 + t) max  · max  (Ri ) π

L /T

i

t

i

(11.233)

(Ri )

 π /1 2,[l−3], ε0 t

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

515

Combining this with the previous estimate (11.227) we obtain: 

(T )

(π /1 )b 2,[1,l−1], ε0 t

−1 ≤ Cl (1 + t) T[1,l] + δ0 (1 + t)−1 [1 + log(1 + t)]·  /2,[l−2], ε0 + (1 + t) max  · max  (Ri ) π t

i

(Ri )

 π /1 2,[l−2], ε0 t

i

Applying then Lemma 11.12 with k = 2, taking ξ = (11.223) with m = 1 and (11.234), 

(11.234)

(T ) (π /1 )b ,

we conclude that, by

(T )

π /1 2,[1,l−1], ε0

t −1 T[1,l] + δ0 (1 + t)−1 [1 + log(1 + t)]· ≤ Cl (1 + t)  · max  (Ri ) π /2,[l−2], ε0 + (1 + t) max  t

i

(11.235)

(Ri )

 π /1 2,[l−2], ε0 t

i

We proceed by induction. Given k ∈ {2, . . . , m}, let: 

(T )

π /1 2,[k−1,l−1], ε0 t

−1 T[k−1,l] + δ0 (1 + t)−1 [1 + log(1 + t)]· ≤ Cl (1 + t)  · max  (Ri ) π /2,[l−2], ε0 + (1 + t) max  t

i

(11.236)

(Ri )

 π /1 2,[l−2], ε0 t

i

hold and consider expression (11.199) for n ∈ {0, . . . , l − 1 − k}. By hypothesis H2 , ˇ/L / Ri1 (L / T )k D / R in . . . L

(T )

π / L 2 ( ε0 ) ≤ C(1 + t)−1 (L / T )k t

(T )

π /2,[l−k], ε0 t

≤ C(1 + t)−1 T[k,l]

(11.237) (R

)

in− j π /1 receives Consider a term j ∈ {0, . . . , n − 1} in the sum in (11.199). Again, k (T ) at most j ≤ n − 1 ≤ l − 2 − k angular derivatives and (L /T ) π / receives at most n − 1 ≤ l − 2 − k angular derivatives, hence the corresponding factors are bounded in ε /1 2,[l−2−k], ε0 and T[k,l−2] respectively. If at most l∗ − 2 angular L 2 (t 0 ) by maxi  (Ri ) π

(R

)

t

ε

/1 , the corresponding factor is bounded in L ∞ (t 0 ) by the last derivatives fall on in− j π assumption of the lemma. Otherwise, at most n − l∗ ≤ l − 1 − k − l∗ ≤ l∗ − k angular ε /, so the corresponding factor is bounded in L ∞ (t 0 ) by derivatives fall on (L /T )k (T ) π 2 the third of the estimates (11.222). In this way we obtain an L bound for the sum in (11.199) by:   /1 2,[l−2−k], ε0 (11.238) Cl δ0 (1 + t)−2 [1 + log(1 + t)] T[k,l−2] + (1 + t) max  (Ri ) π i

ε

Finally, the last term in (11.199) is bounded in L 2 (t 0 ) by: dk 2,[l−1−k], ε0 t

t

516

Chapter 11. Control of the Spatial Derivatives

To estimate this we consider expression (11.198). The contributions of the first two terms are both bounded using the inductive hypothesis (11.236) by:

Cl (1 + t)−2 T[k−1,l] + δ0 (1 + t)−1 [1 + log(1 + t)]· (11.239)   · max  (Ri ) π /2,[l−2], ε0 + (1 + t) max  (Ri ) π /1 2,[l−2], ε0 t

i

t

i

The contributions of the next two terms are both bounded using the inductive hypothesis (11.236) and the first two estimates of (11.222) by:

Cl δ0 (1 + t)−2 [1 + log(1 + t)] T[k−1,l] + δ0 (1 + t)−1 [1 + log(1 + t)]· (11.240)   /2,[l−2], ε0 + (1 + t) max  (Ri ) π /1 2,[l−2], ε0 · max  (Ri ) π t

i

t

i

Lastly, we have the sum in (11.198), which is present only for k ≥ 3. Consider a term q ∈ {2, . . . , k − 1} in this sum. Since each of (L /T )k−1−q (T ) π /1 and (L /T )q (T ) π / receives at most l − 1 − k angular derivatives, the corresponding factors are bounded in L 2 (tε ) /1 2,[k−3,l−4], ε0 and T[k−1,l−2] respectively. If at most l∗ − 1 angular derivatives by  (T ) π t

ε

/1 , the corresponding factor is bounded in L ∞ (t 0 ) by (11.223). fall on (L /T )k−1−q (T ) π /T )q (T ) π / and the Otherwise, at most l − 1 − k − l∗ ≤ l∗ − k angular derivatives fall on (L ε0 ∞ corresponding factor is bounded in L (t ) by the last of the estimates (11.222). In this way we obtain an L 2 bound for the sum in (11.198) by, using the inductive hypothesis (11.236),

Cl δ0 (1 + t)−2 [1 + log(1 + t)] T[k−1,l] + δ0 (1 + t)−1 [1 + log(1 + t)]· (11.241)   /2,[l−2], ε0 + (1 + t) max  (Ri ) π /1 2,[l−2], ε0 · max  (Ri ) π t

i

t

i

Combining the results (11.239)–(11.241) we obtain: dk 2,[l−1−k], ε0

(11.242)

≤ Cl (1 + t)−2 [1 + log(1 + t)] T[k−1,l] + δ0 (1 + t)−1 [1 + log(1 + t)]·   · max  (Ri ) π /2,[l−2], ε0 + (1 + t) max  (Ri ) π /1 2,[l−2], ε0 t

i

t

t

i

The results (11.237), (11.238) together with the estimate (11.242) imply: (L / T )k

(T )

(π /1 )b 2,[l−1−k], ε0 t

−1 ≤ Cl (1 + t) T[k,l] + δ0 (1 + t)−1 [1 + log(1 + t)]·  /2,[l−2], ε0 + (1 + t) max  · max  (Ri ) π i

t

i

(11.243)

(Ri )

 π /1 2,[l−2], ε0 t

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

517

Now, the inductive hypothesis (11.236) implies through Lemma 11.12 applied to /: ξ = (T ) π 

(T )

(π /1 )b 2,[k−1,l−1], ε0 t

−1 T[k−1,l] + δ0 (1 + t)−1 [1 + log(1 + t)]· ≤ Cl (1 + t)  · max  (Ri ) π /2,[l−2], ε0 + (1 + t) max  t

i

(11.244)

(Ri )

 π /1 2,[l−2], ε0 t

i

Adding (11.243) to (11.244) yields: 

(T )

(π /1 )b 2,[k,l−1], ε0 t

−1 T[k,l] + δ0 (1 + t)−1 [1 + log(1 + t)]· ≤ Cl (1 + t)  /2,[l−2], ε0 + (1 + t) max  · max  (Ri ) π t

i

(Ri )

π /1 2,[l−2], ε0 t

i

Applying then Lemma 11.12 with k + 1 in the role of k, taking ξ = 

(11.245) 

(T ) (π /1 )b ,

(T )

π /1 2,[k,l−1], ε0

t −1 T[k,l] + δ0 (1 + t)−1 [1 + log(1 + t)]· ≤ Cl (1 + t)  /2,[l−2], ε0 + (1 + t) max  · max  (Ri ) π t

i

i

yields: (11.246)

(Ri )

 π /1 2,[l−2], ε0 t

This completes the inductive step and the proof of the lemma. We now apply Lemmas 11.11 and 11.13 to obtain estimates for the commutators m,n [ξ ], defined by (11.164)

(i1 ...in ) c

Lemma 11.14 Let k, l be positive integers, k ≤ l. Let also hypothesis H2 hold. Suppose that: λ ∞,[k−1,l], ε0 ≤ Cl δ0 [1 + log(1 + t)], t



(T ) 

π / ∞,[k−1,l], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)], t

max 

(Ri )

π /∞,[l−1], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

i

and: max 

(Ri )

i

π /1 ∞,[l−1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

Then, for any St,u 1-form ξ we have: max

max 

m≤k,n≤l−m i1 ...in

(i1 ...in )

cm,n [ξ ] L ∞ ( ε0 ) ≤ Cl δ0 (1 +t)−2[1 +log(1 +t)]ξ ∞,[k,l−1], ε0 t

provided that δ0 is suitably small (depending on l).

t

518

Chapter 11. Control of the Spatial Derivatives

Proof. The assumptions of the present lemma include those of Lemma 11.11 with k − 1 in the role of m. Therefore the conclusion of Lemma 11.11 holds with m replaced by k − 1, that is, we have: 

(T )

π /1 ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

Consider the expression for

(i1 ...in ) c

m,n [ξ ]

(11.247)

given by Lemma 11.9. Here,

{(m, n) : n ≤ l − m, m ≤ k} / Ri1 cm,0 [ξ ]. Now Consider the first term in this expression, namely the term L / R in . . . L cm,0 [ξ ] is also expressed by Lemma 11.9 as a sum over i ∈ {0, . . . , m − 1}. Consider an arbitrary term in this sum, corresponding to a given i ∈ {0, . . . , m−1}. It can be expressed as a sum of terms each of which has two factors, the first factor being a T -derivative of (T ) π /1 of order at most i ≤ m − 1 ≤ k − 1 and the second factor being a T -derivative of ξ of order at most m − 1 ≤ k − 1. To L / R in . . . L / Ri1 cm,0 [ξ ] the term in question contributes a sum of terms with two factors, the first of which is a spatial derivative of (T ) π /1 of order at most n + m − 1 ≤ l − 1 and the second is a spatial derivative of ξ of order at most n + m − 1 ≤ l − 1. It follows that: / Ri1 cm,0 [ξ ] L ∞ ( ε0 ) ≤ Cl  L / R in . . . L t

(T )

π /1 ∞,[k−1,l−1], ε0 ξ ∞,[k−1,l−1], ε0 t

t

(11.248) Substituting the estimate (11.247) we then obtain: L / R in . . . L / Ri1 cm,0 [ξ ] L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]ξ ∞,[k−1,l−1], ε0 t

t

(11.249) Consider next the second term in the expression for (i1 ...in ) cm,n [ξ ] of Lemma 11.9, namely the sum over j ∈ {0, . . . , n − 1}. Consider an arbitrary term in this sum, corresponding to a given j ∈ {0, . . . , n − 1}. It can be expressed as a sum of terms with (Rin− j ) two factors, the first of which is an angular derivative of π /1 of order at most j ≤ n − 1 ≤ l − 1 and the second is an angular derivative of (L /T )m ξ of order at most n − 1, thus a spatial derivative of ξ of order at most m + n − 1 ≤ l − 1. It follows that the ε second term in the expression for (i1 ...in ) cm,n [ξ ] is bounded in L ∞ (t 0 ) by: Cl max  i

(Ri )

π /1 ∞,[l−1], ε0 ξ ∞,[k,l−1], ε0 t

t

or, substituting from the last assumption of the lemma, by: Cl δ0 (1 + t)−2 [1 + log(1 + t)]ξ ∞,[k,l−1], ε0 t

(11.250)

In view of the results (11.249) and (11.250) for the two terms in the expression for (i1 ...in ) c m,n [ξ ], the lemma follows.

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

519

Lemma 11.15 Let k, l be positive integers, k ≤ l. Let also hypothesis H2 hold. Suppose that: λ ∞,[k−1,l∗ ], ε0 ≤ Cl δ0 [1 + log(1 + t)], t



(T ) 

π / ∞,[k−1,l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)], t

max 

(Ri )

π /∞,[l∗ −1], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

i

and: max 

(Ri )

i

π /1 ∞,[l∗ −1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

Then, for any St,u 1-form ξ we have: max

max 

(i1 ...in )

m≤k,n≤l−m i1 ...in

≤ Cl δ0 (1 + t)

cm,n [ξ ] L 2 ( ε0 ) t

−2

[1 + log(1 + t)]ξ 2,[k,l−1], ε0 t  + Cl (1 + t)−1 ξ ∞,[k,l∗ ], ε0 T[k−1,l] + (1 + t) max  t

(Ri )

i

+δ0 (1 + t)−1 [1 + log(1 + t)] max 

(Ri )

i

π /1 2,[l−1], ε0 t

π /2,[l−2], ε0 t

provided that δ0 is suitably small (depending on l). Proof. The assumptions of the present lemma include those of Lemma 11.11 with (k − 1, l∗ ) in the role of (m, l). Therefore the conclusion of Lemma 11.11 holds with (m, l) replaced by (k − 1, l∗ ), that is, we have: 

(T )

π /1 ∞,[k−1,l∗ −1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

(11.251)

t

Also, the assumptions of the present lemma include those of Lemma 11.13 with k − 1 in the role of m. Therefore the conclusion of Lemma 11.13 holds with m replaced by k − 1, that is, we have: (T )

π /1 2,[k−1,l−1], ε0 t

−1 ≤ Cl (1 + t) T[k−1,l] + δ0 (1 + t)−1 [1 + log(1 + t)]·  /2,[l−2], ε0 + (1 + t) max  · max  (Ri ) π



i

t

i

(11.252)

(Ri )

 π /1 2,[l−2], ε0 t

Following the proof of Lemma 11.14, we consider the contribution to / Ri1 cm,0 [ξ ] L / R in . . . L

of the term i ∈ {0, . . . , m − 1}

in the sum expressing cm,0 [ξ ]. As discussed above the term is expressed as a sum of terms /1 of order at most l −1, of with two factors, the first of which is a spatial derivative of (T ) π

520

Chapter 11. Control of the Spatial Derivatives

which at most k − 1 are T -derivatives, and the second is a spatial derivative of ξ of order at most l − 1, of which at most k − 1 are T -derivatives. Thus the first factor is bounded ε in L 2 (t 0 ) by  (T ) π /1 2,[k−1,l−1], ε0 , t

ε

which is bounded by (11.252), and the second factor is bounded in L 2 (t 0 ) by ξ 2,[k−1,l−1], ε0 t

Now the total number of spatial derivatives falling on (T ) π /1 and ξ is n + m − 1 ≤ l − 1. ε If at most l∗ − 1 fall on (T ) π /, then the corresponding factor is bounded in L ∞ (t 0 ) by (11.251). Otherwise, at most l − 1 − l∗ ≤ l∗ spatial derivatives fall on ξ in which case the ε corresponding factor is bounded in L ∞ (t 0 ) by ξ ∞,[k−1,l∗ ], ε0 t

It follows that: L / R in . . . L / Ri1 cm,0 [ξ ] L 2 ( ε0 )

(11.253)

t

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]ξ 2,[k−1,l−1], ε0 t

−1 + Cl (1 + t) ξ ∞,[k−1,l∗ ], ε0 T[k−1,l] + δ0 (1 + t)−1 [1 + log(1 + t)]· t   (Ri ) π /2,[l−2], ε0 + (1 + t) max  (Ri ) π /1 2,[l−2], ε0 · max  t

i

t

i

Consider next the contribution of the term j ∈ {0, . . . , n − 1} in the sum which is the second term in the expression for (i1 ...in ) cm,n [ξ ] of Lemma 11.9. As discussed above the term is expressed as a sum of terms with two factors, the first of which is an angular (R ) derivative of in− j π /1 of order at most l − 1, and the second is a spatial derivative of ξ of order at most l − 1, of which at most k are T -derivatives. Thus the first factor is bounded ε in L 2 (t 0 ) by: max  (Ri ) π /1 2,[l−1], ε0 , t

i

and the second factor is bounded in

ε L 2 (t 0 )

by:

ξ 2,[k,l−1], ε0 t

(R

)

Now the total number of angular derivatives falling on in− j π /1 and (L /T )m ξ is n − 1 ≤ (Rin− j ) l − 1. If at most l∗ − 1 fall on π /1 , then the corresponding factor is bounded in ε L ∞ (t 0 ) by the last assumption of the lemma. Otherwise, at most n − 1 − l∗ angular derivatives fall on (L /T )m ξ , resulting in a spatial derivative of ξ of order at most m + n − ε 1 − l∗ ≤ l − 1 − l∗ ≤ l∗ , thus the corresponding factor is bounded in L ∞ (t 0 ) by ξ ∞,[k,l∗ ], ε0 t

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

521

(i1 ...in ) c

It follows that the second term in the expression for ε L 2 (t 0 ) by:

m,n [ξ ]

is bounded in

Cl δ0 (1 + t)−2 [1 + log(1 + t)]ξ 2,[k,l−1], ε0 t

+ Cl ξ ∞,[k,l∗ ], ε0 max  t

i

(Ri )

π /1 2,[l−1], ε0

(11.254)

t

In view of the results (11.253) and (11.254) for the two terms in the expression for (i1 ...in ) c m,n [ξ ], the lemma follows. Q Given non-negative integers m, n, we denote by Em,n the bootstrap assumption that there is a constant C independent of s such that for all t ∈ [0, s]: Q Em,n

:

max Rin . . . Ri1 (T )m Qψα  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1

α;i1 ...in

t

Q Q We then denote by E{l} the conjunction of the assumptions Em,n corresponding to the triangle (11.9). The constant C then depends on l only and shall be denoted by Cl to Q emphasize this dependence. Note that the assumptions E0,n coincide with the assumptions Q

Q

Q

E / n of Chapter 10, and the assumptions E{l} contain the assumptions E / [l] . Given non-negative integers m, l, m ≤ l, we denote by M[m,l] the bootstrap assumption that there is a constant Cl independent of s such that for all t ∈ [0, s]: M[m,l]

: µ − η0 ∞,[m,l], ε0 ≤ Cl δ0 [1 + log(1 + t)] t

In view of the fact that µ = ακ, the bootstrap assumption M[m,l] is equivalent, modulo the bootstrap assumption E{l} , to the assumption that there is a constant Cl independent of s such that for all t ∈ [0, s]: κ − 1∞,[m,l], ε0 ≤ Cl δ0 [1 + log(1 + t)] t

(11.255)

Proposition 11.1 Let hypotheses H0, H1, H2 and the estimate (10.30) of Chapter 10 Q / [l] , hold for some non-negative hold. Let also the bootstrap assumptions E{l+1} , E{l} , and X integer l. Moreover, let the bootstrap assumption M[m,l+1] hold for some non-negative integer m ≤ l. Then if δ0 is suitably small (depending on l) we have: max (T )k+1 Tˆ i ∞,[l−k], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

i

: for all k = 0, . . . , m Proof. The assumptions of the present proposition include those of Proposition 10.1 with l + 1 in the role of l. Therefore under the present assumptions Proposition 10.1 and all its corollaries follow with l + 1 in the role of l. In particular, we have (Corollaries 10.1.b and 10.1.g with l replaced by l + 1): ψ L − k∞,[l+1], ε0 ≤ Cl δ0 (1 + t)−1

(11.256)

ψTˆ ∞,[l+1], ε0 ≤ Cl δ0 (1 + t)−1

(11.257)

 ψ∞,[l+1], ε0 ≤ Cl δ0 (1 + t)−1

(11.258)

t t

t

522

Chapter 11. Control of the Spatial Derivatives

and: ω L L ∞,[l], ε0 ≤ Cl δ0 (1 + t)−2

(11.259)

ω L Tˆ ∞,[l], ε0 ≤ Cl δ0 (1 + t)−2

(11.260)

ω / L ∞,[l], ε0 ≤ Cl δ0 (1 + t)−2

(11.261)

ω /∞,[l], ε0 ≤ Cl δ0 (1 + t)−2

(11.262)

t t t

t

Moreover, from (11.44) and the definition: ω /Tˆ = Tˆ i d/ψi we readily obtain: ω /Tˆ ∞,[l], ε0 ≤ Cl δ0 (1 + t)−2

(11.263)

ωT Tˆ ∞,[l], ε0 ≤ Cl δ0 (1 + t)−1

(11.264)

t t

Now, the estimates (11.257) and (11.264) imply, through the definition (11.45),  pT ∞,[l], ε0 ≤ Cl δ02 (1 + t)−2 t

(11.265)

while the estimates (11.257), (11.258), (11.264), together with the estimate (11.255) with (0, l + 1) in the role of (m, l) (which is equivalent to assumption M[0,l+1] modulo assumption E{l+1} ) imply, through the definition (11.46), (qT )b ∞,[l], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(11.266)

(provided that δ0 is suitably small depending on l). The estimate (11.266) together with Corollary 10.1.d in turn implies: qT ∞,[l], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(11.267)

In view of formula (11.42) and the estimates (11.265) and (11.267) the statement of the proposition for k = 0 follows. We proceed by induction on k for fixed m and l. Let k ∈ {1, . . . , m} and let: 

max (T )k +1 Tˆ i ∞,[l−k  ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

i

: hold for all

k

= 0, . . . , k − 1

(11.268)

We first establish, using the inductive hypothesis (11.268) L ∞ estimates for the T -derivatives of up to the kth order of the ψ and ω components. We begin with ψTˆ . Since ψTˆ = Tˆ i ψi ,

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

523

we have, for k  ∈ {1, . . . , k}: k

 (T ) ψTˆ = Tˆ i (T )k ψi +

 −1 k

j =0

k!  ((T ) j +1 Tˆ i )((T )k −1− j ψi )  ( j + 1)!(k − 1 − j )! (11.269)

We take the  ∞,[l+1−k  ], ε0 norm of this. The contribution of the first term on the right t is by Proposition 10.1 and assumption E{l+1} bounded by: Cl δ0 (1 + t)−1 while the contribution of the sum is by the inductive hypothesis (11.268) bounded by: Cl δ02 (1 + t)−2 [1 + log(1 + t)] Combining we conclude that: 

(T )k ψTˆ ∞,[l+1−k  ], ε0 ≤ Cl δ0 (1 + t)−1 t

: for all k  ∈ {1, . . . , k}

(11.270)

which, together with the estimate (11.257) yields: ψTˆ ∞,[k,l+1], ε0 ≤ Cl δ0 (1 + t)−1 t

Since

ωT Tˆ = Tˆ i (T ψi ),

ω L Tˆ = Tˆ i (Lψi ),

(11.271)

ω /Tˆ = Tˆ i (d/ψi ),

these are analogous to ψTˆ with T ψi , Lψi , d/ψi , in the role of ψi , respectively. We thus obtain, in an analogous manner, taking the  ∞,[l−k  ],t norm for k  ∈ {1, . . . , k} and combining with (11.264), (11.260), (11.263), the estimates: ωT Tˆ ∞,[k,l], ε0 ≤ Cl δ0 (1 + t)−1 t

ω L Tˆ ∞,[k,l], ε0 ≤ Cl δ0 (1 + t)

−2

t

ω /Tˆ ∞,[k,l], ε0 ≤ Cl δ0 (1 + t)−2 t

(11.272) (11.273) (11.274)

Next, we consider the expression (2.66) for L i : H ψ0 ψi L i = −α Tˆ i − 1 + ρH

(11.275)

From the inductive hypothesis (11.268) together with Proposition 10.1 and assumption E{l+1} it readily follows that: 

max (T )k +1 L i ∞,[l−k  ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

i

: for all

k

= 0, . . . , k − 1

(11.276)

524

Chapter 11. Control of the Spatial Derivatives

Since ψ L = ψ0 + L i ψi , we have, for k  ∈ {1, . . . , k}, 





(T )k ψ L = (T )k ψ0 + L i (T )k ψi +

 −1 k

k!  ((T ) j +1 L i )((T )k −1− j ψi ) ( j + 1)!(k  − 1 − j )! j =0 (11.277)

We take the  ∞,[l+1−k  ], ε0 norm of this. The contribution of the first term on the right t is by assumption E{l+1} bounded by: Cl δ0 (1 + t)−1 The contribution of the second term is by Proposition 10.1 bounded by: Cl δ0 (1 + t)−1 while the contribution of the sum is by (11.276) bounded by: Cl δ02 (1 + t)−2 [1 + log(1 + t)] Combining we conclude that: 

(T )k ψ L ∞,[l+1−k  ], ε0 ≤ Cl δ0 (1 + t)−1 t

: for all k  ∈ {1, . . . , k}

(11.278)

which, together with the estimate (11.256) yields: ψ L − k∞,[k,l+1], ε0 ≤ Cl δ0 (1 + t)−1 t

(11.279)

Since ω L L = L µ (Lψµ ) = Lψ0 + L i (Lψi ),

ω / L = L µ (d/ψµ ) = d/ψ0 + L i (d/ψi ),

these are analogous to ψ L − k with Lψ0 , d/ψ0 , in the role of ψ0 − k, and Lψi , d/ψi , in the role of ψi , respectively. We thus obtain, in an analogous manner, taking the  ∞,[l−k  ],t norm for k  ∈ {1, . . . , k} and combining with (11.259), (11.261), the estimates: ω L L ∞,[k,l], ε0 ≤ Cl δ0 (1 + t)−2

(11.280)

ω / L ∞,[k,l], ε0 ≤ Cl δ0 (1 + t)−2

(11.281)

t t

Finally, since ψ = (d/x i )ψi ,

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

525

for k  ∈ {1, . . . , k} we have: k

k

(L /T ) ψ = (d/x )(T ) ψi + i

 −1 k

j =0

k!  (d/(T ) j +1 x i )(T )k −1− j ψi ( j + 1)!(k  − 1 − j )! (11.282)

We take the  ∞,[l+1−k  ], ε0 norm of this. By Corollary 10.1.a and assumption E{l+1} t the contribution of the first term on the right is bounded by: Cl δ0 (1 + t)−1

(11.283)

To estimate the contribution of the sum we note that: (T ) j +1 x i = (T ) j T i = (T ) j (κ Tˆ i ) = ((T ) j κ)Tˆ i +

j −1  j  =0

( j

(11.284)

j!   ((T ) j −1− j κ)((T ) j +1 Tˆ i )  + 1)!( j − 1 − j )!

Here, j ∈ {0, . . . , k  − 1}, k  ∈ {1, . . . , k}. By (11.255) with l replaced by l + 1 (which is equivalent, modulo assumption E{l+1} , to assumption M[m,l+1] ) and Proposition 10.1: ((T ) j κ)Tˆ i ∞,[l+2−k  ], ε0 ≤ Cl [1 + log(1 + t)] t

while by (11.255) and the inductive hypothesis (11.268): 



((T ) j −1− j κ)((T ) j +1 Tˆ i )∞,[l+2−k  ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]2 t

: for all j  ∈ {0, . . . , j − 1}

It then follows through (11.284) and hypothesis H0 that: d/(T ) j +1 x i ∞,[l+1−k  ], ε0 t

≤ C(T ) j +1 x i ∞,[l+2−k  ], ε0 ≤ Cl (1 + t)−1 [1 + log(1 + t)] t

: for all j ∈ {0, . . . , k  − 1}

(11.285)

(provided that δ0 is suitably small depending on l). In view of the estimate (11.285) we conclude, by assumption E{l+1} , that the contribution of the sum in (11.282) is bounded by: (11.286) Cl δ0 (1 + t)−2 [1 + log(1 + t)] Combining the results (11.283) and (11.286) for the contributions of the two terms on the right in (11.282) we conclude that: 

(L /T )k ψ∞,[l+1−k  ], ε0 ≤ Cl δ0 (1 + t)−1 t

(11.287)

Together with the estimate (11.258) this yields:  ψ∞,[k,l+1], ε0 ≤ Cl (1 + t)−1 t

(11.288)

526

Chapter 11. Control of the Spatial Derivatives

Since ω / = (d/x i ) ⊗ (d/ψi ), this is analogous to ψ with d/ψi in the role of ψi . We thus obtain, in an analogous manner, taking the  ∞,[l−k  ],t norm for k  ∈ {1, . . . , k} and combining with (11.262), the estimate: ω /∞,[k,l], ε0 ≤ Cl δ0 (1 + t)−2 (11.289) t

The estimates (11.289) and (11.288) imply through expression (10.195) the following estimate for k/ : k/ ∞,[k,l], ε0 ≤ Cl δ0 (1 + t)−2 (11.290) t

Also, the estimates (11.281) and (11.271), (11.279), (11.288) imply through expression (10.386) the following estimate for κ −1 ζ  : κ −1 ζ  ∞,[k,l], ε0 ≤ Cl δ0 (1 + t)−2 t

(11.291)

Moreover, recalling from Chapter 3, equations (3.233) and (3.168), that: 1 dH ψ⊗ ψ(T σ ) 2 dσ 1 dH ψ L ψ(T σ ) ζ = ζ − 2 dσ

κk/ = κk/ +

(11.292) (11.293)

the estimates (11.290), (11.291) and (11.255) with m = k imply (using also (11.279), (11.288)): κk/∞,[k,l], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

(11.294)

ζ ∞,[k,l], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

(11.295)

t

t

We turn to the function pT and the St,u 1-form (qT )b given by (11.45) and (11.46) respectively. Using the estimates (11.272) and (11.271) we obtain:  pT ∞,[k,l], ε0 ≤ Cl δ02 (1 + t)−2 t

(11.296)

Also, using the estimates (11.272) and (11.271), (11.288), as well as the estimate (11.255) with (k, l + 1) in the role of (m, l) (which is equivalent to assumption M[k,l+1] modulo assumption E{l+1} ) we obtain: (qT )b ∞,[k,l], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(11.297)

To complete the inductive step we need to obtain a similar estimate for the St,u -tangential vectorfield qT , given through (qT )b by (11.43). This requires estimating the T -derivatives of h /−1 up to order k, which amounts to estimating the T -derivatives of (T ) π / up to order k − 1. The required estimates would follow from Lemma 11.7 if we can show that: 

(T ) 

π / ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(11.298)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

527

Because the assumptions of Lemma 11.7, which are those of Lemma 11.5, hold by virtue of Corollary 10.1.d and the estimate (11.255). Now, from the definition (11.89) and the estimate (11.294) we see that the estimate (11.298) would in turn follow if we can show that: (11.299) χ  ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−1 t

A stronger estimate is in fact provided by the following lemma, to the proof of which we shall presently turn. Lemma 11.16 Let the assumptions of Proposition 11.1 hold. Let also the inductive hypothesis (11.268) hold, for some k ∈ {1, . . . , m}. Then, provided that δ0 is suitably small (depending on l), the following estimate holds: χ  ∞,[k,l], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

Proof. As remarked above, under the present assumptions Proposition 10.1 and all its corollaries hold with l + 1 in the role of l. From the definition (11.88) and Corollary 10.1.d we obtain: χ  ∞,[l], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.300)

We proceed, by induction on k  , to show that: 

(L /T )k χ  ∞,[l−k  ], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

: for all k  ∈ {1, . . . , k}

(11.301)

which, together with (11.300), yields the conclusion of the lemma. In fact we shall show that (11.268) together with the estimate: χ  ∞,[k−1,l], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.302)

implies: (L /T )k χ  ∞,[l−k], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.303)

Replacing then k by k  ∈ {1, . . . , k} in this statement yields the inductive step associated to (11.301). From the definition (11.88) we have: /T χ − L /T χ  = L

η0 h/ / η0 (T ) π − 1 − u + η0 t (1 − u + η0 t)2

(11.304)

Now, L /T χ is given by formula (4.109) of Chapter 4. Substituting in (11.304) this formula, and also substituting for (T ) π / in terms of χ  from (11.89)–(11.90), we obtain the  following formula for L /T χ : ( P) (N) 1 ˜ )− γ − γ /ζ + D /ζ / 2 µ + (D L /T χ  =  + D 2

(11.305)

528

Chapter 11. Control of the Spatial Derivatives

Here:  =

7 

n

(11.306)

n=1

and: 1 = 1,1 + 1,2 1,1 = (1 − u + η0 t)−2 (α −1 κ − η0−1 )η02 h/ 1,2 = −α −1 κχ  · χ $

(11.307) −1

2 = −2(1 − u + η0 t) η0 (κk/)   1 3 = α −1 (κ −1 ζ  ) ⊗ η + η ⊗ (κ −1 ζ  ) 2 4 = 4,1 + 4,2 + 4,3 1 dH (T σ )ψ L (ψ L + αψTˆ )(1 − u + η0 t)−1 η0 h/ 4,1 = − α −2 2 dσ dH 1 (T σ )ψ L (ψ L + αψTˆ )χ  4,2 = − α −2 2 dσ  1 dH 1 (T σ ) ψ ⊗ (χ $ · ψ) + (χ $ · ψ)⊗ ψ 4,3 = − α −2 2 dσ 2 5 = 5,1 + 5,2

(11.308) (11.309)

(11.310)

5,1 = −α −1 κe (1 − u + η0 t)−1 η0 h/ 5,2 = −α −1 κe χ 

(11.311)

6 = 6,1 + 6,2 6,1 = (1 − u + η0 t)−1 η0 κk/ 1 6,2 = κ(k/ · χ $ + χ $ · k/ ) 2   1 dH (ψ L − αψTˆ ) ψ ⊗ (χ $ · d/σ ) + (χ $ · d/σ )⊗ ψ 7 = − α −1 κ 4 dσ

(11.312) (11.313)

In the above formulas the dependence on h/ is explicit (the · denotes only a contraction). In particular, χ $ and χ $ are defined according to (10.668) of Chapter 10 and by (11.88) we have: η0 I + χ $ (11.314) χ$ = 1 − u + η0 t where I denotes the identity transformation in the tangent space to St,u at each point. In ˜ denotes the transpose of D (11.305) D /ζ /ζ as a bilinear form in the tangent space to St,u at each point. We are to apply (L /T )k−1 to (11.305) and then take the  ∞,[l−k], ε0 norm, obt taining on the left (L /T )k χ  ∞,[l−k], ε0 t

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

529

The contribution of any given term on the right-hand side, will then be bounded by the  ∞,[k−1,l−1], ε0 norm of that term. t

Now, from the definition (11.89), (11.302) together with (11.294) and (11.255) imply 

(T ) 

π / ∞,[k−1,l], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]2

(11.315)

t

By Lemma 11.7 with k − 1 in the role of m we then obtain:  L /T

(T )

(T )

π / + 2(1 − u + η0 t)−1 h/∞,[l], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

π / − 2(1 − u + η0 t)−1 h/∞,[l−1], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

max

2≤k  ≤k−1

(L / T )k



(T )

π /∞,[l−k  ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(11.316) These estimates, even with (k, l) reduced to (k − 1, l − 2), together with Corollary 10.1.d, even with l reduced to l − 1, imply that for any St,u tensorfield ξ we have: /−1 ∞,[k−1,l−1], ε0 ≤ Cl ξ ∞,[k−1,l−1], ε0 ξ · h /∞,[k−1,l−1], ε0 , ξ · h t

t

t

(11.317)

provided that δ0 is suitably small (depending on l). This is an L ∞ analogue of Lemma 11.12, proved along the same lines in a straightforward manner. Applying (11.317) to χ  , we obtain, by (11.302), χ $ ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.318)

We may also apply (11.317) to obtain, in view of the estimate (11.255), 1,1 ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.319)

Moreover, by (11.302) and (11.318), 1,2 ∞,[k−1,l−1], ε0 ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)]3 t

Hence:

1 ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.320)

(11.321)

Next, by the estimate (11.294): 2 ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] t

(11.322)

Next, since (3.65): η = ζ + d/µ by the estimate (11.295) and assumption M[k,l+1] we have: η∞,[k,l], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(11.323)

530

Chapter 11. Control of the Spatial Derivatives

Together with the estimate (11.291) this yields: 3 ∞,[k−1,l−1], ε0 ≤ Cl δ02 (1 + t)−3 [1 + log(1 + t)] t

(11.324)

Next, we may apply (11.317) to obtain, using the estimates (11.279) and (11.271), 4,1 ∞,[k−1,l−1] ε0 ≤ Cl δ0 (1 + t)−2 t

(11.325)

Moreover, using the estimate (11.318) we obtain: 4,2 ∞,[k−1,l−1], ε0 ≤ Cl δ02 (1 + t)−3 [1 + log(1 + t)] t

(11.326)

To estimate 4,3 , we first consider χ $ · ψ. From (11.314) we have: χ $ · ψ = (1 − u + η0 t)−1 η0 ψ + χ $ · ψ

(11.327)

The estimates (11.288) and (11.318) then yield: χ $ · ψ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−2 t

(11.328)

Using this estimate we obtain: 4,3 ∞,[k−1,l−1], ε0 ≤ Cl δ03 (1 + t)−4 t

(11.329)

Combining the results (11.325), (11.326) and (11.329) we obtain: 4 ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−2 t

(11.330)

The function e which appears in the term 5 is given by equation (3.101) of Chapter 3:

dH 1 (ψ ˆ )2 (Lσ ) + Fψ0 ω L Tˆ 2 T dσ The estimates (11.271), (11.273) imply: e =

e ∞,[k,l], ε0 ≤ Cl δ0 (1 + t)−2 t

(11.331)

(11.332)

We may then apply (11.317) to obtain, using (11.332) (and (11.255)), 5,1 ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.333)

Moreover, using also (11.318) we obtain: 5,2 ∞,[k−1,l−1], ε0 ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)]2 t

Hence:

5 ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.334)

(11.335)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

531

Next, by the estimates (11.290) and (11.255): 6,1 ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)]

(11.336)

t

Moreover, using also the estimate (11.318), 6,2 ∞,[k−1,l−1], ε0 ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)]2

(11.337)

t

Hence:

6 ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)]

(11.338)

t

To estimate 7 , we first consider χ $ · d/σ . From (11.314) we have: χ $ · d/σ = (1 − u + η0 t)−1 η0 d/σ + χ $ · d/σ

(11.339)

The estimate (11.318) then yields: χ $ · d/σ ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−3

(11.340)

t

Using this estimate together with (11.271), (11.279), (11.288) and (11.255) we obtain: 7 ∞,[k−1,l−1], ε0 ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)]

(11.341)

t

Combining finally the results (11.321), (11.322), (11.324), (11.330), (11.335), (11.338), (11.341), yields:  ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

(11.342)

t

(N)

We turn to the last term on the right in (11.305), namely the term γ . This is given by (4.88) and (4.103)–(4.105) of Chapter 4. The last three may be written more simply in the form:  1  κ ω /L ⊗ ω /Tˆ + ω /Tˆ ⊗ ω / L − 2ω L Tˆ ω / 2 (B) 1 γ = (κψTˆ (d/σ ) − (T σ ) ψ) ⊗ (ψ L (d/σ ) − (Lσ ) ψ) 2  + (ψ L (d/σ ) − (Lσ ) ψ) ⊗ (κψTˆ (d/σ ) − (T σ ) ψ) (C) 1  γ = κ (d/σ ) ⊗ (ψ L ω /Tˆ + ψTˆ ω / L − 2ω L Tˆ ψ) 2  + (ψ L ω /Tˆ + ψTˆ ω / L − 2ω L Tˆ ψ) ⊗ (d/σ )   1 /Tˆ + ω + κ(Lσ ) ψ ⊗ ω /Tˆ ⊗ ψ − 2ψTˆ ω / 2 1 /L + ω + (T σ ) ( ψ ⊗ ω / L ⊗ ψ − 2ψ L ω /) 2 ( A)

γ =

(11.343)

(11.344)

(11.345)

532

Chapter 11. Control of the Spatial Derivatives

Using the estimates (11.274), (11.281), (11.273), (11.289), and (11.271), (11.279), (11.288), as well as the estimate (11.255), we obtain: ( A)

 γ ∞,[k−1,l−1], ε0 ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)] t

(B)

 γ ∞,[k−1,l−1], ε0 ≤ Cl δ03 (1 + t)−4

(11.346) (11.347)

t

(C)

 γ ∞,[k−1,l−1], ε0 ≤ Cl δ02 (1 + t)−3

(11.348)

t

It follows that: (N)

 γ ∞,[k−1,l−1], ε0 ≤ Cl δ02 (1 + t)−3

(11.349)

t

( P)

We turn to the next-to-the-last term on the right in (11.305), namely the term γ . This is given by equation (4.82) of Chapter 4: ( P)

( P P)

γ = γ

( P P)

Here γ

+

1 dH κl 2 dσ

(11.350)

is the principal part (of order 2), given by: ( P P)

( P P) 1

γ = γ

( P P) 2

+ γ

( P P) 3

+ γ

( P P) 4

+ γ

(11.351)

where: ( P P) γ 1 ( P P) γ 2 ( P P) γ 3 ( P P) γ 4

1 dH ψ L ( ψ ⊗ d/(T σ ) + d/(T σ )⊗ ψ) 4 dσ 1 dH κψTˆ ( ψ ⊗ d/(Lσ ) + d/(Lσ )⊗ ψ) = 4 dσ 1 dH ( ψ⊗ ψ)L(T σ ) =− 2 dσ 1 dH κψTˆ ψ L (D =− / 2σ ) 2 dσ =

(11.352) (11.353) (11.354) (11.355)

In (11.350), l is a lower order term (of order 1), given by equation (4.84) of Chapter 4, or, equivalently, l = l1 + l2 l1 = l1,1 + l1,2 + l1,3

  1 l1,1 = α −1 (Lσ ) ψ ⊗ (ψTˆ − α −1 ψ L )(κ −1 ζ  ) − ψ L α −1 d/α 2    + (ψTˆ − α −1 ψ L )(κ −1 ζ  ) − ψ L α −1 d/α ⊗ ψ

(11.356)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

533

l1,2 = −α −1 (Lσ )( ψ⊗ ψ)e l1,3 = α −1 (Lσ )ψ L ψTˆ k/

(11.357)

l2 = l2,1 + l2,2   1 l2,1 = − ψ L ψ ⊗ (k/ · (d/σ )$ ) + (k/ · (d/σ )$ )⊗ ψ 2 l2,2 = −( ψ⊗ ψ)(κ −1 ζ  ) · (d/σ )$

(11.358)

The dependence on h / in the above formulas is explicit and we have denoted: (d/σ )$ = (d/σ ) · h/−1

(11.359)

We shall first estimate the lower order term l. Assumption E{l+1} implies: d/α∞,{l}, ε0 ≤ Cl δ0 (1 + t)−2 t

(11.360)

Together with the estimate (11.291) and the estimates (11.271), (11.279), (11.288), this yields: (11.361) l1,1 ∞,[k−1,l−1], ε0 ≤ Cl δ03 (1 + t)−5 t

Moreover, by the estimate (11.332): l1,2 ∞,[k−1,l−1], ε0 ≤ Cl δ04 (1 + t)−6 t

(11.362)

and, by the estimate (11.290): l1,3 ∞,[k−1,l−1], ε0 ≤ Cl δ03 (1 + t)−5

(11.363)

l1 ∞,[k−1,l−1], ε0 ≤ Cl δ03 (1 + t)−5

(11.364)

t

Hence:

t

To estimate l2 we first consider (d/σ )$ . Applying (11.317) we obtain, in view of assumption E{l+1} , (11.365) (d/σ )$ ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−2 t

Using this estimate together with the estimates (11.290) and (11.291) yields: l2,1 ∞,[k−1,l−1], ε0 ≤ Cl δ03 (1 + t)−5

(11.366)

l2,2 ∞,[k−1,l−1], ε0 ≤ Cl δ04 (1 + t)−6

(11.367)

l2 ∞,[k−1,l−1], ε0 ≤ Cl δ03 (1 + t)−5

(11.368)

t

and:

t

Hence:

t

Combining finally the results (11.364), (11.368), yields: l∞,[k−1,l−1], ε0 ≤ Cl δ03 (1 + t)−5 t

(11.369)

534

Chapter 11. Control of the Spatial Derivatives ( P P)

We turn to the principal part γ . By assumption E{l+1} and hypothesis H0: d/(T σ )∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−2 t

(11.370)

In view also of the estimates (11.279) and (11.288), it follows that: ( P P) 1

 γ

∞,[k−1,l−1], ε0 ≤ Cl δ02 (1 + t)−3 t

(11.371)

Q and hypothesis H0: By assumption E{l}

d/(Lσ )∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−3 t

(11.372)

In view also of the estimates (11.255), (11.271) and (11.288), it follows that: ( P P) 2

 γ

( P P) 3

To estimate γ

∞,[k−1,l−1], ε0 ≤ Cl δ03 (1 + t)−5 [1 + log(1 + t)] t

(11.373)

we write: L(T σ ) = T (Lσ ) +  · d/σ

(11.374)

and  is given by equation (3.71) of Chapter 3:  = −(ζ + η)$ = −(ζ + η) · h/−1

(11.375)

Applying (11.317) we obtain, in view of the estimates (11.295) and (11.323), ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(11.376)

It then follows that:  · d/σ ∞,[k−1,l−1], ε0 ≤ Cl δ02 (1 + t)−3 [1 + log(1 + t)] t

(11.377)

Q

Moreover, by assumption E{l} : T (Lσ )∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−2 t

(11.378)

In view of the estimates (11.378) and (11.377) we conclude from (11.374) that: L(T σ )∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−2 t

(11.379)

In view also of the estimate (11.288), it follows that: ( P P) 3

 γ

∞,[k−1,l−1], ε0 ≤ Cl δ03 (1 + t)−4 t

(11.380)

Part 1: Control of the spatial derivatives of the first derivatives of the x i . ( P P) 4.

We now consider γ

535

In this case we must estimate D / 2 σ ∞,[k−1,l−1], ε0 t

Setting ξ = d/σ in (11.164) we obtain: L / R in . . . L / Ri1 (L /T ) j D / 2σ = D / 2 Rin . . . Ri1 (T ) j σ +

(i1 ...in )

c j,n [d/σ ]

(11.381)

Here we take j ∈ {0, . . . , k − 1} and n ∈ {0, . . . , l − 1 − j }. Now the assumptions of Proposition 11.1 include the assumptions of Lemma 10.11. Therefore under the present assumptions the conclusion of Lemma 10.11 holds, that is, we have: max 

(Ri )

π /1 ∞,[l−1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

(11.382)

t

i

Recalling also the estimate (11.315), we see that the assumptions of Lemma 11.14 are all verified. They are a fortiori verified if in that lemma (k, l) is replaced by (k − 1, l − 1). Therefore, for any St,u 1-form ξ we have: (i1 ...in )

max 

max

j ≤k−1,n≤l−1− j i1 ...in

≤ Cl δ0 (1 + t)

c j,n [ξ ] L ∞ ( ε0 ) t

−2

[1 + log(1 + t)]ξ ∞,[k−1,l−2], ε0

(11.383)

t

Taking in (11.383) ξ = d/σ we obtain (using assumption E{l−1} ): max 

max

(i1 ...in )

j ≤k−1,n≤l−1− j i1 ...in

c j,n [d/σ ] L ∞ ( ε0 ) t

≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)]

(11.384)

On the other hand, by hypotheses H1 and H0, and assumption E{l+1} , max D / 2 Rin . . . Ri1 (T ) j σ  L ∞ ( ε0 )

max

j ≤k−1,n≤l−1− j i1 ...in

t

≤ C(1 + t)

−2

max (T ) (σ − k )∞,[l+1− j ], ε0 j

2

j ≤k−1

t

= C(1 + t)−2 σ − k 2 ∞,[k−1,l+1], ε0 t

≤ Cl δ0 (1 + t)−3

(11.385)

In view of the estimates (11.384) and (11.385) we conclude from (11.381) that: D / 2 σ ∞,[k−1,l−1] ε0 = t

max L / R in . . . L / Ri1 (L /T ) j D / 2 σ  L ∞ ( ε0 )

max

j ≤k−1,n≤l−1− j i1 ...in

≤ Cl δ0 (1 + t)

−3

t

(11.386)

In view also of the estimates (11.255), (11.271) and (11.279), it follows that: ( P P) 4

 γ

∞,[k−1,l−1], ε0 ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)] t

(11.387)

536

Chapter 11. Control of the Spatial Derivatives

Combining the results (11.371), (11.373), (11.380), (11.387), we conclude that: ( P P)

 γ

∞,[k−1,l−1], ε0 ≤ Cl δ02 (1 + t)−3

(11.388)

t

Taking also into account the estimate (11.369) we then obtain: ( P)

 γ ∞,[k−1,l−1], ε0 ≤ Cl δ02 (1 + t)−3

(11.389)

t

We turn to the second term on the right in (11.305), namely the term D / 2 µ. Setting ξ = d/µ in (11.164) we obtain: L / R in . . . L / Ri1 (L /T ) j D / 2µ = D / 2 Rin . . . Ri1 (T ) j µ +

(i1 ...in )

c j,n [d/µ]

(11.390)

Again, j ∈ {0, . . . , k − 1} and n ∈ {0, . . . , l − 1 − j }. Taking in (11.383) ξ = d/µ we obtain (using assumption M[k−1,l−1] ): max 

max

(i1 ...in )

j ≤k−1,n≤l−1− j i1 ...in

≤

c j,n [d/µ] L ∞ ( ε0 ) t

Cl δ02 (1 + t)−3 [1 +

log(1 + t)]

2

(11.391)

On the other hand, by hypotheses H1 and H0, and assumption M[k−1,l+1] , max D / 2 Rin . . . Ri1 (T ) j µ L ∞ ( ε0 )

max

j ≤k−1,n≤l−1− j i1 ...in

t

≤ C(1 + t)

−2

= C(1 + t)

max (T ) (µ − η0 )∞,[l+1− j ], ε0 j

j ≤k−1 −2

t

µ − η0 ∞,[k−1,l+1], ε0 t

≤ Cl δ0 (1 + t)

−2

[1 + log(1 + t)]

(11.392)

In view of the estimates (11.391) and (11.392) we conclude from (11.390) that: D / 2 µ∞,[k−1,l−1] ε0 = t

max

max L / R in . . . L / Ri1 (L /T ) j D / 2 µ L ∞ ( ε0 )

j ≤k−1,n≤l−1− j i1 ...in

≤ Cl δ0 (1 + t)

−2

t

[1 + log(1 + t)]

(11.393)

Finally, we have the third term on the right in (11.305), namely the term (1/2)(D /ζ + ˜ ). Taking in (11.383) ξ = ζ and using the estimate (11.295) (even with (k, l) reduced D /ζ to (k − 1, l − 2)) we obtain: max 

max

j ≤k−1,n≤l−1− j i1 ...in

≤

(i1 ...in )

c j,n [ζ ] L ∞( ε0 )

Cl δ02 (1 + t)−4 [1 + log(1 +

t

t)]2

(11.394)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

537

On the other hand, by hypothesis H1 and the estimate (11.295) (with k reduced to k − 1): max D /L / R in . . . L / Ri1 (L /T ) j ζ  L ∞ ( ε0 )

max

j ≤k−1,n≤l−1− j i1 ...in

t

≤ C(1 + t)

−1

max (L /T ) ζ ∞,[l− j ], ε0 j

j ≤k−1

t

= C(1 + t)−1 ζ ∞,[k−1,l], ε0 t

≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)]

(11.395)

In view of the estimates (11.394) and (11.395) we conclude through (11.164) with ζ in the role of ξ : D /ζ ∞,[k−1,l−1], ε0 ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] t

(11.396)

˜ , by virtue of the fact that Lie derivatives commute with The same estimate holds for D /ζ transposition. Combining the results (11.342), (11.393), (11.396), (11.389), (11.349), for the five terms on the right in (11.305) we arrive at the estimate (11.303). This completes the inductive step and the proof of Lemma 11.16. We now resume the proof of Proposition 11.1. Lemma 11.16, together with the estimates (11.294) and (11.255) imply (11.315) with k − 1 increased to k. The estimates (11.316) with k −1 increased to k follow. Then (11.317) holds with (k −1, l −1) increased to (k, l). Applying this to ξ = (qT )b , we obtain, in view of the estimate (11.297), qT ∞,[k,l], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t] t

(11.397)

Using the estimates (11.296) and (11.397), as well as (11.255), we deduce, from the first statement of Lemma 11.4, the estimates:  pT , j ∞,[k− j,l− j ], ε0 ≤ Cl δ02 (1 + t)−2 [1 + log(1 + t)]2 t

: for all j ∈ {0, . . . , k}

(11.398)

qT , j ∞,[k− j,l− j ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

: for all j ∈ {0, . . . , k}

(11.399)

We write the second statement of Lemma 11.4 in the form (setting m = k, j = k − 1 − n − j  ): r Tn ,k =

k−1−n  i=0

Ni (n)

k−1−n−i  j  =0



(L /T )k−1−n− j (κqT , j  )

(11.400)

538

Chapter 11. Control of the Spatial Derivatives

Here n ∈ {0, . . . , k − 1}. Taking the  ∞,[l−k+n], ε0 norm of this we obtain, using t (11.399) (and (11.255)), r Tn ,k ∞,[l−k], ε0 t

≤

k−1−n 

Ni (n)

k−1−n−i  j  =0

i=0

≤ Cl

max

j  ∈{0,...,k−1−n}

κqT , j  ∞,[k−1−n− j  ,l−1− j  ], ε0 t

κqT , j  ∞,[k−1− j  ,l−1− j  ], ε0 t

≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]2 : for all n ∈ {0, . . . , k − 1}

(11.401)

Now, according to Lemma 11.3 (setting m = k): (T )k+1 Tˆ i = pT ,k Tˆ i + qT ,k · d/x i +

k−1 

r Tn ,k · d/(T )n Tˆ i

(11.402)

n=0

We take the  ∞,[l−k], ε0 norm of this. By (11.398) and (11.399) with j = k and Propot sition 10.1 (first statement of Corollary 10.1.b and last statement of Corollary 10.1.a): max  pT ,k Tˆ i ∞,[l−k], ε0 ≤ Cl δ02 (1 + t)−2 [1 + log(1 + t)]2 i

t

max qT ,k · d/x i ∞,[l−k], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] i

t

(11.403) (11.404)

Also, by (11.401), the inductive hypothesis (11.268), and Proposition 10.1 (first statement of Corollary 10.1.b): 0k−1 0 0 0 0 0 max 0 r Tn ,k · d/(T )n Tˆ i 0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]2 (11.405) 0 0 i ε0 n=0

∞,[l−k],t

Combining the results (11.403), (11.404), (11.405), we conclude that: max (T )k+1 Tˆ i ∞,[l−k], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] i

t

(11.406)

This completes the inductive step and the proof of Proposition 11.1. The foregoing proposition has the following corollaries. Corollary 11.2.a Under the assumptions of Proposition 11.1 the coefficients of the expression for (T )k+1 Tˆ i , k = 0, . . . , m, of Lemma 11.3 satisfy:  pT ,k ∞,[m−k,l−k], ε0 ≤ Cl δ02 (1 + t)−2 [1 + log(1 + t)]2 t

: for all k ∈ {0, . . . , m} qT ,k ∞,[m−k,l−k], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

: for all k ∈ {0, . . . , m}

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

539

and: r Tn ,k ∞,[l−k+n], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]2 t

: for all k ∈ {0, . . . , m}, n ∈ {0, . . . , k − 1} Also, the component functions L i satisfy the estimate: max (T )k+1 L i ∞,[l−k], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

i

: for all k = 0, . . . , m Corollary 11.1.b Under the assumptions of Proposition 11.1 the following estimates hold: ψTˆ ∞,[m+1,l+1], ε0 ≤ Cl δ0 (1 + t)−1 t

ψ L − k∞,[m+1,l+1], ε0 ≤ Cl δ0 (1 + t)−1 t

 ψ∞,[m+1,l+1], ε0 ≤ Cl δ0 (1 + t)−1 t

and: ωT Tˆ ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−1 t

ω L Tˆ ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 t

ω L L ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 t

ω /Tˆ ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 t

ω / L ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 t

ω /∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 t

Also, we have: k/ ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 t

κ −1 ζ  ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 t

κk/∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

ζ ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

The next corollary is obtained from Lemma 11.16 by substituting Proposition 11.1 itself in place of the inductive hypothesis (11.268), and then substituting the result in Lemma 11.7 and Lemma 11.11. Corollary 11.1.c Under the assumptions of Proposition 11.1 we have: χ  ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

540

Chapter 11. Control of the Spatial Derivatives

Moreover, we have:  L /T

(T )

(T )

π / + 2(1 − u + η0 t)−1 h/∞,[l], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

−2

h/∞,[l−1], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

(T )

π /∞,[l−k], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

π / − 2(1 − u + η0 t) / T )k max (L

2≤k≤m

t

t

and: 

(T )

π / + 2(1 − u + η0 t)−1 h /∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

Also: ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

and: 

Q Wm,n

(T )

π /1 ∞,[m,l−1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

We proceed with the L 2 estimates. Given non-negative integers m, n, we denote by the quantity: Q = max Rin . . . Ri1 (T )m Qψα  L 2 ( ε0 ) Wm,n α;i1 ...in

t

(11.407)

Q Q the sum of the quantities Wm,n corresponding to the triangle We then denote by W{l} Q (11.9). Note that the quantities W0,n coincide with the quantities WnQ of Chapter 10, and Q Q the quantities W{l} dominate the quantities W[l] . Moreover, we have:

max Lψα 2,{l}, ε0 ≤ (1 + t)−1 W{l} Q

α

t

(11.408)

Given non-negative integers m, l, m ≤ l we denote: B[m,l] = µ − η0 2,[m,l], ε0 t

(11.409)

Note that the quantities B[0,l] coincide with the quantities B[l] of Chapter 10. Since µ = ακ, it follows from Lemma 11.1 that under the bootstrap assumptions E{l∗ } and M[m,l∗ ] , we have:   κ − 12,[m,l], ε0 ≤ Cl B[m,l] + [1 + log(1 + t)]W{l} (11.410) t

Proposition 11.2 Let the hypotheses H0, H1, H2 and the estimate (10.30) of Chap/ [l∗ ] , hold for some ter 10 hold. Let also the bootstrap assumptions E{l∗ +1} , E{lQ∗ } , and X

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

541

non-negative integer l. Moreover, let the bootstrap assumption M[m,l∗ +1] hold for some non-negative integer m ≤ l (where by definition M[m,l∗ +1] coincides with M{l∗ +1} = M[l∗ +1,l∗ +1] for m ≥ l∗ + 1). Then if δ0 is suitably small (depending on l) we have: m  k=0

max (T )k+1 Tˆ i 2,[l−k], ε0 t

i

2 1 ≤ Cl (1 + t)−1 B[m,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] Q + [1 + log(1 + t)] W{l+1} + δ0 (1 + t)−2 [1 + log(1 + t)]2 W{l−1}



Proof. The assumptions of the present proposition coincide with those of Proposition 11.1 with (m, l∗ ) in the role of (m, l). Therefore under the present assumptions Proposition 11.1 and its corollaries hold with (m, l∗ ) in the role of (m, l). (If m ≥ l∗ then Proposition 11.1 and its corollaries hold with (l∗ , l∗ ) in the role of (m, l)). Moreover, the present assumptions include those of Proposition 10.1 with l∗ + 1 in the role of l, as well as those of Proposition 10.2 with l + 1 in the role of l. Therefore under the present assumptions Proposition 10.1 and all its corollaries follow with l∗ + 1 in the role of l, and Proposition 10.2 and all its corollaries follow with l + 1 in the role of l. In particular, we have (Corollaries 10.2.b and 10.2.g with l replaced by l + 1): ψ L − k2,[l+1], ε0 t

 Q ≤ Cl W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W[l]

(11.411)

ψTˆ 2,[l+1], ε0

t  Q ≤ Cl W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W[l]

(11.412)

 ψ2,[l+1], ε0

t  Q ≤ Cl W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W[l]

(11.413)

and: ω L L 2,[l], ε0

(11.414)

ω L Tˆ 2,[l], ε0

(11.415)

1 2 Q ≤ Cl (1 + t)−1 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + W[l] t

1 2 Q ≤ Cl (1 + t)−1 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + W[l] t

ω / L 2,[l], ε0

(11.416)

 Q ≤ Cl (1 + t)−1 W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] t

542

Chapter 11. Control of the Spatial Derivatives

ω /2,[l], ε0

(11.417)

 Q ≤ Cl (1 + t)−1 W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] t

Moreover, we readily obtain: ω /Tˆ 2,[l], ε0

(11.418)

 Q ≤ Cl (1 + t)−1 W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] t

ωT Tˆ 2,[l], ε0 t

 Q ≤ Cl W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] Let us define: Uk,l =

k  k  =0

 max (T )k +1 Tˆ i 2,[l−k  ], ε0

i

t

(11.419)

(11.420)

The estimates (11.412) and (11.419) yield, through the definition (11.45), (11.421)  pT 2,[l], ε0 ≤ Cl δ0 (1 + t)−1 · t

 Q W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] while the estimates (11.412), (11.413), (11.419), yield, through the definition (11.46), (qT )b 2,[l], ε0 ≤ C(1 + t)−1 κ − 12,[l+1], ε0 (11.422) t t

 Q + Cl δ0 (1 + t)−1 W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] The estimate (11.422) together with Corollary 10.2.d as well as the L ∞ estimates (11.266) and Corollary 10.1.d, both with l∗ in the role of l, in turn yields:

(11.423) qT 2,[l], ε0 ≤ Cl (1 + t)−1 κ − 12,[l+1], ε0 t t  Q +δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] + W[l+1] + δ02 (1 + t)−2 W[l−1] By virtue of the estimates (11.421) and (11.423), we obtain from formula (11.42), in view also of Proposition 10.2 and Corollary 10.2.a,

U0,l ≤ Cl (1 + t)−1 κ − 12,[l+1], ε0 (11.424) t  Q +δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] + W[l+1] + δ02 (1 + t)−2 W[l−1] We shall prove the proposition by deriving a recursive, in k, inequality, for the quantities Uk,l , for fixed l. We first establish L 2 estimates for the T -derivatives of up

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

543

to the kth order of the ψ and ω components, in terms of Uk−1,l . We begin with ψTˆ . For k  ∈ {1, . . . , k} we have the formula (11.269). We now take the  2,[l+1−k  ], ε0 norm t of this. The contribution of the first term on the right is by Proposition 10.2 bounded by:

 Q Cl W{l+1} + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] while, using Proposition 11.1 with l∗ in the role of l, we find that the contribution of the sum is bounded by:   Cl δ0 (1 + t)−1 [1 + log(1 + t)]W{l} + Uk−1,l Combining we conclude that: 

(T )k ψTˆ 2,[l+1−k  ], ε0 (11.425) t

 Q ≤ Cl W{l+1} + δ0 (1 + t)−1 Uk−1,l + Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] : for all k  ∈ {1, . . . , k} which, together with the estimate (11.412) yields: ψTˆ 2,[k,l+1], ε0 (11.426) t

 Q ≤ Cl W{l+1} + δ0 (1 + t)−1 Uk−1,l + Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W{l} In a similar manner we obtain the estimates: ωT Tˆ 2,[k,l], ε0 (11.427) t

 Q ≤ Cl W{l+1} + δ0 (1 + t)−1 Uk−1,l−1 + Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W{l−1} ω L Tˆ 2,[k,l], ε0

(11.428)

1 2 Q ≤ Cl (1 + t)−1 W{l} + δ0 (1 + t)−1 Uk−1,l−1 + Y0 + (1 + t)A[l−1] + W{l} t

ω /Tˆ 2,[k,l], ε0 ≤ Cl (1 + t)−1 W{l+1} t

+ δ0 (1 + t)−1 Uk−1,l−1 + Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W{l−1} Q

(11.429) 

Next, from the expression (11.275), using Proposition 10.2 and Lemma 11.1 we deduce: k−1  k  =0

 max (T )k +1 L i 2,[l−k  ], ε0 ≤ Cl Uk−1,l + W{l+1} i

(11.430)

t

+ δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] Q

We consider ψ L . For k  ∈ {1, . . . , k} we have formula (11.277).



544

Chapter 11. Control of the Spatial Derivatives

We take the  2,[l+1−k  ], ε0 norm of this. The contribution of the first term on the t right is bounded by: W{l+1} The contribution of the second term is by Proposition 10.2 (Corollaries 10.2.a and 10.2.g) bounded by:

 Q Cl W{l+1} + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] while, using (11.430) we find that the contribution of the sum is bounded by:   Cl δ0 (1 + t)−1 [1 + log(1 + t)]W{l} + Uk−1,l Combining we conclude that:

 (T )k ψ L 2,[l+1−k  ], ε0 ≤ Cl W{l+1} + t

δ0 (1 + t)

−1

Q Uk−1,l + Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1]



: for all k  ∈ {1, . . . , k}

(11.431)

which, together with the estimate (11.411) yields:

ψ L − k2,[k,l+1], ε0 ≤ Cl W{l+1} + t

δ0 (1 + t)−1 Uk−1,l + Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W{l} Q



(11.432)

In a similar manner, using (11.430) with l reduced to l − 1 we obtain the estimates:

Q (11.433) ω L L 2,[k,l], ε0 ≤ Cl (1 + t)−1 W{l} + t  1 2 δ0 (1 + t)−1 Uk−1,l−1 + Y0 + (1 + t)A[l−1] + W{l}

ω / L 2,[k,l], ε0 ≤ Cl (1 + t)−1 W{l+1} + t

δ0 (1 + t)−1 Uk−1,l−1 + Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W{l−1} Q

(11.434) 

Next, we consider ψ. For k  ∈ {1, . . . , k} we have formula (11.282). We take the  2,[l+1−k  ], ε0 norm of this. By Corollary 10.2.a the contribution of t the first term on the right is bounded by:

 Q (11.435) Cl W{l+1} + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] To estimate the contribution of a term 

(d/(T ) j +1 x i )(T )k −1− j ψi

: j ∈ {0, . . . , k  − 1}

(11.436)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

545

in the sum in (11.282), we consider expression (11.284) for (T ) j +1 x i . The contribution of the first term on the right in (11.284) to (11.436) is:      d/(((T ) j κ)Tˆ i ) (T )k −1− j ψi = (d/Tˆ i )((T ) j κ)(T )k −1− j ψi + Tˆ i (d/(T ) j κ)(T )k −1− j ψi (11.437) Consider the first term on the right in (11.437). To estimate its  2,[l+1−k  ], ε0 norm we t write:

  / Ri1 (d/Tˆ i )((T ) j κ)(T )k −1− j ψi L / R in . . . L

   = (d/(R)s1 Tˆ i )(R)s2 ((T ) j κ)(T )k −1− j ψi (11.438) partitions

Here n ≤ l + 1 − k  . If |s1 | ≤ l∗ we use Proposition 10.1 with l∗ + 1 in the role of l to bound: d/(R)s1 Tˆ i  L ∞ ( ε0 ) ≤ C(1 + t)−1 t

ε

obtaining an L 2 (t 0 ) bound for the corresponding term in the sum in (11.438) by: 

C(1 + t)−1 ((T ) j κ)(T )k −1− j ψi 2,[l+1−k  ], ε0 t

 −1 [1 + log(1 + t)]W{l} + δ0 (1 + t)−1 κ − 12,[ j,l], ε0 ≤ Cl (1 + t) t

Otherwise, |s1 | ≥ l∗ + 1, therefore |s2 | = n − |s1 | ≤ l∗ + 1 − k  and we can bound:

   (R)s2 ((T ) j κ)(T )k −1− j ψi  L ∞ ( ε0 ) ≤ ((T ) j κ)(T )k −1− j ψi ∞,[l∗ +1−k  ], ε0 t

t

≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] by assumptions M[ j,l∗ ] and E{l∗ } . In this case, we express the factor d/(R)s1 Tˆ i as in (10.87) of Chapter 10. That is, if s1 = {n 1 , . . . , n p }, p = |s1 |, we write: d/(R)s1 Tˆ i = −(1 − u + η0 t)−1 d/

( p) i x in ...in p 1

+d/(R)s1 y i + (1 − u + η0 t)−1 d/

( p) i δin ...in p 1 ε

0 ∞ We estimate the contribution of the first

term on the right byplacing this term in L (t ) ε 0 s j k −1− j 2 ψi in L (t ). This yields an using (10.53), and the factor (R) 2 ((T ) κ)(T )

ε

L 2 (t 0 ) bound similar to that of the case |s1 | ≤ l∗ above. On the other hand, we estimate the contributions of the second and third terms on the right by placing these terms in ε 10.2.a (note that L 2 (t 0 ) using Proposition 10.2 with l + 1 in the role of l and Corollary

 

|s1 |+1 ≤ n +1 ≤ l +2 −k  ≤ l +1), and placing the factor (R)s2 ((T ) j κ)(T )k −1− j ψi ε

in L ∞ (t 0 ), using the estimate above. We then obtain that the contributions in question are bounded by: 

Q Cl δ0 (1 + t)−2 [1 + log(1 + t)] Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l]

546

Chapter 11. Control of the Spatial Derivatives

Combining the above results we conclude that: 

(d/Tˆ i )((T ) j κ)(T )k −1− j ψi 2,[l+1−k  ], ε0 ≤ Cl δ0 (1 + t)−2 κ − 12,[ j,l], ε0 t t

−1 + Cl (1 + t) [1 + log(1 + t)] W{l+1} + (11.439)  Q δ0 (1 + t)−1 Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W[l] In a similar manner, using also assumption M[ j,l∗ +1] , we obtain, in regard to the second term on the right in (11.437), 

Tˆ i (d/(T ) j κ)(T )k −1− j ψi 2,[l+1−k  ], ε0 ≤ Cl δ0 (1 + t)−2 κ − 12,[ j,l+1], ε0 t t

−1 + Cl (1 + t) [1 + log(1 + t)] W{l} + (11.440)  Q δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] Combining (11.439), (11.440), yields:     d/(((T ) j κ)Tˆ i ) (T )k −1− j ψi 2,[l+1−k  ], ε0 ≤ Cl δ0 (1 + t)−2 κ − 12,[ j,l+1], ε0 t t

−1 + Cl (1 + t) [1 + log(1 + t)] W{l+1} + (11.441)  Q δ0 (1 + t)−1 Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W[l] This is the bound for the contribution of the first term on the right in (11.284) to the  2,[l+1−k  ], ε0 norm of (11.436). The sum on the right in (11.284) is present only for t j ≥ 1 (hence k ≥ k  ≥ 2). By virtue of Proposition 11.1 with l∗ in the role of l and assumption M[ j −1,l∗ ] we have: 0 0 0 j −1 0 0 0 j! j −1− j  j  +1 ˆ i 0 0 ((T ) κ)((T ) ) T 0 0   0  ( j + 1)!( j − 1 − j )! 0 ε j =0

2,[l+2−k  ],t 0

≤ Cl [1 + log(1 + t)] j −1     (T ) j +1 Tˆ i 2,[l+2−k  ], ε0 + δ0 (1 + t)−1 (T ) j −1− j (κ − 1)2,[l+2−k  ], ε0 · t

j  =0

t

Now, since l − j  ≥ l + 1 − j ≥ l + 2 − k  , we have: j −1 

(T )

j  =0

j  +1

Tˆ i 2,[l+2−k  ], ε0 ≤ t

j −1  j  =0

 (T ) j +1 Tˆ i 2,[l− j  ], ε0 = U j −1,l t

Also, since j − 1 − j  + l + 2 − k  ≤ l + j + 1 − k  ≤ l, we have: j −1  j  =0



(T ) j −1− j (κ − 1)2,[l+2−k  ], ε0 ≤ κ − 12,[ j −1,l], ε0 t

t

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

547

Therefore, substituting we obtain: 0 0 0 0 0 0 j −1 j !   j −1− j j +1 ˆ i 0 0 ((T ) κ)((T ) T )0 0   0 0 j  =0 ( j + 1)!( j − 1 − j )! ε 2,[l+2−k  ],t 0

 ≤ Cl [1 + log(1 + t)] U j −1,l + δ0 (1 + t)−1 κ − 12,[ j −1,l], ε0

(11.442)

t

Proposition 11.1 with l∗ in the role of l and assumption M[ j −1,l∗ ] moreover yield: 0 0 0 0 j −1 0 0 j !   j −1− j j +1 ˆ i 0 0 ((T ) T )0 κ)((T ) 0   0 0 j  =0 ( j + 1)!( j − 1 − j )!

ε

∞,[l∗ +2−k  ],t 0

≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]2

(11.443)

In estimating the contribution of the sum on the right in (11.284) to the  2,[l+1−k  ], ε0 t



norm of (11.436), if the number of angular derivatives falling on the factor (T )k −1− j ψi is at most (l∗ +1)−(k −1− j ) we appeal to assumption E{l∗ +1} to place the resulting factor in ε ε L ∞ (t 0 ), and we use the estimate (11.442) to place in L 2 (t 0 ) the factor resulting from the contribution of the sum in (11.284) to the factor d/(T ) j +1 x i . Otherwise, (l∗ + 2) −  (k  − 1 − j ) or more angular derivatives fall on the factor (T )k −1− j ψi , therefore at most (l + 1 − k  ) − (l∗ + 2) + (k  − 1 − j ) ≤ l∗ − 1 − j ≤ l∗ − k  angular derivatives fall on the factor corresponding to d/(T ) j +1 x i , so we can appeal to ε (11.443) to place that factor in L ∞ (t 0 ) (l∗ reduced to l∗ − 1 in (11.443) would suffice), ε the other factor being bounded in L 2 (t 0 ) by W{l− j } . In this way we obtain that the contribution of the sum on the right in (11.284) to the  2,[l+1−k  ], ε0 norm of (11.436) t is bounded by:

Cl δ0 (1 + t)−2 [1 + log(1 + t)] U j −1,l + δ0 (1 + t)−1 κ − 12,[ j −1,l], ε0 t  +[1 + log(1 + t)]W{l− j } (11.444) Combining the results (11.441) and (11.444) for the contributions of the two terms on the right in (11.284) we conclude that: 

(d/(T ) j +1 x i )(T )k −1− j ψi 2,[l+1−k  ], ε0 t

−1 ≤ Cl (1 + t) [1 + log(1 + t)]W{l+1} + δ0 (1 + t)−2 κ − 12,[ j,l+1], ε0 t

+δ0 (1 + t)

−2

[1 + log(1 + t)] U j −1,l + Y0 + (1 + t)A[l] +

: ∀ j ∈ {0, . . . , k  − 1}

Q δ02 (1 + t)−2 W[l]



(11.445)

548

Chapter 11. Control of the Spatial Derivatives

It follows that the  2,[l+1−k  ], ε0 norm of the sum in (11.282) is bounded by: t

Cl (1 + t)−1 [1 + log(1 + t)]W{l+1} + δ0 (1 + t)−2 κ − 12,[k−1,l+1], ε0 t

+δ0 (1 + t)

−2

[1 + log(1 + t)] Uk−2,l + Y0 + (1 +

Q t)A[l] + δ02 (1 + t)−2 W[l]



(11.446) Combining then the results (11.435) and (11.446) for the two terms on the right in (11.282) we conclude that:

 (L /T )k ψ2,[l+1−k  ], ε0 ≤ Cl W{l+1} + δ0 (1 + t)−2 κ − 12,[k−1,l+1], ε0 t

t

+ δ0 (1 + t) +δ0 (1 + t)

−2

[1 + log(1 + t)]Uk−2,l

−1

Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W[l] Q



: for all k  ∈ {1, . . . , k}

(11.447)

which, combined with the estimate (11.413) yields:

 ψ2,[k,l+1], ε0 ≤ Cl W{l+1} + δ0 (1 + t)−2 κ − 12,[k−1,l+1], ε0 t

t

+ δ0 (1 + t) +δ0 (1 + t)

−2

−1

[1 + log(1 + t)]Uk−2,l Q Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W{l}

 (11.448)

In a similar manner we obtain:

ω /2,[k,l], ε0 ≤ Cl W{l+1} + δ0 (1 + t)−2 κ − 12,[k−1,l], ε0 t

t

+ δ0 (1 + t)

−2

[1 + log(1 + t)]Uk−2,l−1

+δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W{l−1} Q

 (11.449)

In the following we shall use the estimates (11.426), (11.432), (11.448), for the components of ψ, with l +1 reduced to l. We shall use the estimates (11.427), (11.428), (11.433), (11.434) and (11.449), for the components of ω, as they stand. The estimate (11.448) with l + 1 reduced to l together with the estimate (11.449) and the corresponding L ∞ estimates with l∗ in the role of l from Corollary 11.1.b, imply through expression (10.195), using Lemma 11.2.a, the following estimate for k/ :

k/ 2,[k,l], ε0 ≤ Cl (1 + t)−1 δ0 (1 + t)−2 [1 + log(1 + t)]Uk−2,l−1 t 1 2 + δ0 (1 + t)−2 κ − 12,[k−1,l], ε0 + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] t  Q +W{l+1} + δ02 (1 + t)−2 W{l} (11.450) Also, the estimates (11.426), (11.432), (11.448), with l + 1 reduced to l, together with the estimate (11.434) and the corresponding L ∞ estimates with l∗ in the role of l from

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

549

Corollary 11.1.b, imply through expression (10.386), using Lemma 11.2.a, the following estimate for κ −1 ζ  :

κ −1 ζ  2,[k,l], ε0 ≤ Cl (1 + t)−1 δ0 (1 + t)−1 Uk−1,l−1 t 1 2 3 + δ0 (1 + t)−4 κ − 12,[k−1,l], ε0 + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] t  Q (11.451) +W{l+1} + δ02 (1 + t)−2 W{l} Moreover, in view of expressions (11.292), (11.293), the estimates (11.450), (11.451) imply (using also (11.432), (11.448), with l + 1 reduced to l):

κk/2,[k,l], ε0 ≤ Cl (1 + t)−1 δ0 (1 + t)−2 [1 + log(1 + t)]2 Uk−2,l−1 t

+ δ0 (1 + t)−1 κ − 12,[k,l], ε0 t 1 2 −1 + δ0 (1 + t) [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q +[1 + log(1 + t)] W{l+1} + δ02 (1 + t)−2 W{l}

(11.452)

ζ 2,[k,l], ε0 ≤ Cl (1 + t)−1 δ0 (1 + t)−1 [1 + log(1 + t)]Uk−1,l−1 t

+ δ0 (1 + t)−1 κ − 12,[k,l], ε0 t 1 2 + δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q (11.453) +[1 + log(1 + t)] W{l+1} + δ02 (1 + t)−2 W{l} We turn to pT and (qT )b given by (11.45) and (11.46). The estimate (11.426), with l +1 reduced to l, together with the estimate (11.427) and the corresponding L ∞ estimates with l∗ in the role of l from Corollary 11.1.b, yield, using Lemma 11.2.a,

 pT 2,[k,l], ε0 ≤ Cl δ0 (1 + t)−1 δ0 (1 + t)−1 Uk−1,l−1 t 1 2 + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1]  Q (11.454) +W{l+1} + δ03 (1 + t)−3 W{l−1} Also, the estimates (11.426) and (11.448) with l + 1 reduced to l, together with the estimate (11.427) and the corresponding L ∞ estimates with l∗ in the role of l from Corollary 11.1.b, yield, using Lemma 11.2.a,

(qT )b 2,[k,l], ε0 ≤ Cl δ02 (1 + t)−2 Uk−1,l−1 t

+ (1 + t)−1 κ − 12,[k,l+1], ε0 t 1 2 + δ02 (1 + t)−2 Y0 + (1 + t)A[l−1] +δ0 (1 + t)−1 W{l+1} + δ03 (1 + t)−3 W{l−1} Q

 (11.455)

550

Chapter 11. Control of the Spatial Derivatives

To derive a recursive inequality for the quantities U[k,l] , we need to derive a similar estimate for the St,u -tangential vectorfield qT in terms of U[k−1,l−1] . This requires an appropriate estimate for the quantity T[k−1,l−1] defined in the statement of Lemma 11.8. Now, the assumptions of Lemma 11.8, which are those of Lemma 11.6, all follow under the assumptions of the present proposition, by Corollary 10.1.d, with l∗ in the role of l, assumption M[m,l∗ ] , and Corollary 11.1.c with l∗ in the role of l. Lemma / 2,[k,l], ε0 , λ 2,[k,l], ε0 and 11.8 gives a bound for the quantity T[k,l] in terms of  (T ) π t

t

maxi  (Ri ) π /2,[l−1], ε0 . From (11.89) we obtain, using the estimate (11.452) and Lemma t 11.2.a and Corollary 11.1.c with l∗ in the role of l, 

(T ) 

π / 2,[k,l], ε0 ≤ C[1 + log(1 + t)]χ  2,[k,l], ε0 t

t −1 + Cl (1 + t) [1 + log(1 + t)] δ0 (1 + t)−2 [1 + log(1 + t)]Uk−2,l−1  + δ0 (1 + t)−1 κ − 12,[k,l], ε0 + Y0 + (1 + t)A[l−1] t  Q 2 (11.456) +W{l+1} + δ0 (1 + t)−2 W{l}

Also, by Lemma 11.1 and Lemma 11.2.a,

 λ 2,[k,l], ε0 ≤ Cl κ − 12,[k,l], ε0 + [1 + log(1 + t)]W{l} t

t

(11.457)

Also, from Corollary 10.2.d we have:

 Q /2,[l−1], ε0 ≤ Cl Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] max  (Ri ) π t

i

(11.458) Substituting in the conclusion of Lemma 11.8 the estimates (11.456)–(11.458) yields the following bound for the quantity T[k,l] :

T[k,l] ≤ Cl [1 + log(1 + t)]χ  2,[k,l], ε0 t

+ δ0 (1 + t)

−3

[1 + log(1 + t)] Uk−2,l−1 2

2 1 κ − 12,[k,l], ε0 + δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] t  Q −1 +(1 + t) [1 + log(1 + t)] W{l+1} + δ02 (1 + t)−2 W{l} (11.459)

+ (1 + t)

−1

The above bound is in terms of χ  2,[k,l], ε0 . The appropriate bound of this quantity in t turn in terms of the quantity Uk−2,l−1 is provided by the following lemma, to the proof of which we shall presently turn. Lemma 11.17 Let the assumptions of Proposition 11.2 hold. Then, provided that δ0 is suitably small (depending on l), the following estimate holds, for each k ∈ {0, . . . , m}:

χ  2,[k,l], ε0 ≤ Cl (1 + t)−1 δ0 (1 + t)−2 [1 + log(1 + t)]Uk−2,l−1 t   + (1 + t)−1 κ − 12,[k−1,l+1], ε0 + Y0 + (1 + t)A[l] t  Q −1 +W{l+1} + (1 + t) [1 + log(1 + t)]W{l}

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

551

Proof. From the definition (11.88) and Corollary 10.2.d we obtain:

 Q χ  2,[l], ε0 ≤ Cl (1 + t)−1 Y0 + (1 + t)A[l] + W[l] + δ02 (1 + t)−2 W[l−1] (11.460) t

Therefore the lemma holds for k = 0. To prove the lemma for k ∈ {1, . . . , m} we shall derive a recursive, in k, inequality for the quantities χ  2,[k,l], ε0 , for fixed l. t

/T )k−1 to (11.305) and We thus consider equation (11.305) for L /T χ  . We apply (L take the  2,[l−k], ε0 norm. We then obtain on the left t

(L /T )k χ  2,[l−k], ε0 = χ  2,[k,l], ε0 − χ  2,[k−1,l], ε0 t

t

t

and the contribution of any given term on the right-hand side is bounded by the  2,[k−1,l−1], ε0 norm of that term. t Now, the assumptions of Proposition 11.2 imply the assumptions of Lemma 11.12. Substituting in the conclusion of that lemma the bound for maxi  (Ri ) π /2,[l−2], ε0 from t (11.458) with l reduced to l − 1 as well as the bound for T[k−2,l−2] from (11.459) with (k, l) reduced to (k − 2, l − 2), yields the following conclusion: For any St,u tensorfield ξ we have: ξ · h /2,[k−1,l−1], ε0 , ξ · h /−1 2,[k−1,l−1], ε0 t

t

≤ Cl ξ 2,[k−1,l−1], ε0

t ε + Cl ξ ∞,[k−1,l∗ −1], 0 [1 + log(1 + t)]χ  2,[k−2,l−2], ε0 t

t

−3

+ δ0 (1 + t) [1 + log(1 + t)] Uk−4,l−3 + (1 + t)−1 κ − 12,[k−2,l−2], ε0 + Y0 + (1 + t)A[l−2] t  Q 2 (11.461) +W{l−1} + δ0 (1 + t)−2 W{l−2} 2

Applying (11.461) to χ  we obtain, in view of Corollary 11.1.c with l∗ in the role of l, χ $ 2,[k−1,l−1], ε0 ≤ Cl χ  2,[k−1,l−1], ε0 t

t −2 + Cl δ0 (1 + t) [1 + log(1 + t)] δ0 (1 + t)−3 [1 + log(1 + t)]2 Uk−4,l−3   + (1 + t)−1 κ − 12,[k−2,l−2], ε0 + Y0 + (1 + t)A[l−2] t  Q 2 (11.462) +W{l−1} + δ0 (1 + t)−2 W{l−2} We may also apply (11.461) to obtain (see (11.307)): 1,1 2,[k−1,l−1], ε0 ≤ Cl (1 + t)−2 · t

δ0 [1 + log(1 + t)]2 χ 

 + δ0 (1 + t)−3 [1 + log(1 + t)]Uk−4,l−3 2 1 + κ − 12,[k−1,l−1], ε0 + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−2] t  Q +[1 + log(1 + t)] W{l−1} + δ03 (1 + t)−2 W{l−2} (11.463) ε 2,[k−2,l−2],t 0

552

Chapter 11. Control of the Spatial Derivatives

Moreover, by (11.462), 1,2 2,[k−1,l−1], ε0 t

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]2 ·

χ  2,[k−1,l−1], ε0 + δ02 (1 + t)−5 [1 + log(1 + t)]3 Uk−4,l−3 t   −2 + δ0 (1 + t) κ − 12,[k−1,l−1], ε0 + [1 + log(1 + t)] Y0 + (1 + t)A[l−2] t  Q −2 (11.464) +δ0 (1 + t) [1 + log(1 + t)] W{l−1} + δ02 (1 + t)−2 W{l−2} Hence: 1 2,[k−1,l−1], ε0 t

−2 ≤ Cl δ0 (1 + t) [1 + log(1 + t)]2 χ  2,[k−1,l−1], ε0 t

+

δ02 (1 +

t)

−5

[1 + log(1 + t)] Uk−4,l−3 3

  + (1 + t)−2 κ − 12,[k−1,l−1], ε0 + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−2] t  Q −2 +(1 + t) [1 + log(1 + t)] W{l−1} + δ03 (1 + t)−2 W{l−2} (11.465) Next, by the estimate (11.452) with (k, l) reduced to (k − 1, l − 1) (see (11.308)): 2 2,[k−1,l−1], ε0 t

2 −4 ≤ Cl δ0 (1 + t) [1 + log(1 + t)]2 Uk−3,l−2

  + δ0 (1 + t)−3 κ − 12,[k−1,l−1], ε0 + [1 + log(1 + t)] Y0 + (1 + t)A[l−2] t  Q −2 (11.466) +(1 + t) [1 + log(1 + t)] W{l} + δ02 (1 + t)−2 W{l−1}

Next, by the estimate (11.453),

η2,[k,l], ε0 ≤ Cl (1 + t)−1 δ0 (1 + t)−1 [1 + log(1 + t)]Uk−1,l−1 t

+ κ − 12,[k,l+1], ε0 t

1 2 + δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q (11.467) +[1 + log(1 + t)] W{l+1} + δ02 (1 + t)−2 W{l} Using this estimate together with the estimate (11.451), with (k, l) reduced to (k−1, l−1), as well as the corresponding L ∞ estimates with l∗ in the role of l from Corollary 11.1.b,

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

553

we obtain, using Lemma 11.2.a (see (11.309)): 3 2,[k−1,l−1], ε0 t

2 −3 ≤ Cl δ0 (1 + t) [1 + log(1 + t)]Uk−2,l−2

  + δ0 (1 + t)−3 κ − 12,[k−1,l], ε0 + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−2] t  Q (11.468) +δ0 (1 + t)−2 [1 + log(1 + t)] W{l} + δ02 (1 + t)−2 W{l−1}

To estimate 4 (see (11.310)), we first consider χ $ · ψ, expressed by (11.327). The estimate (11.448) with (k, l + 1) reduced to (k − 1, l − 1) together with the estimate (11.462) and the corresponding L ∞ estimates with l∗ in the role of l from Corollaries 11.1.b and 11.1.c yield, using Lemma 11.2.a, χ $ · ψ2,[k−1,l−1], ε0

t −1 ≤ Cl (1 + t) δ0 χ  2,[k−1,l−1], ε0 + δ0 (1 + t)−2 [1 + log(1 + t)]Uk−3,l−2 t   −1 −1 (1 + t) κ − 12,[k−2,l−1], ε0 + Y0 + (1 + t)A[l−2] + δ0 (1 + t) t  Q 3 −3 (11.469) +W{l−1} + δ0 (1 + t) W{l−2} Using this estimate we deduce: 4 2,[k−1,l−1], ε0

t −1 δ0 χ  2,[k−1,l−1], ε0 + δ02 (1 + t)−2 Uk−2,l−2 ≤ Cl (1 + t) t

+ δ0 (1 + t)−2 κ − 12,[k−2,l−1], ε0 t 2 1 + δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l−2]  Q + W{l} + δ03 (1 + t)−3 [1 + log(1 + t)]W{l−2}

(11.470)

To estimate 5 (see (11.311)), we first consider the function e , given by (11.331). The estimate (11.426) with (k, l + 1) reduced to (k − 1, l − 1) together with the estimate (11.428) with (k, l) reduced to (k − 1, l − 1) and the corresponding L ∞ estimates with l∗ in the role of l from Corollary 11.1.b yield, using Lemma 11.2.a,

e 2,[k−1,l−1], ε0 ≤ Cl (1 + t)−1 δ0 (1 + t)−1 Uk−2,l−2 t 1 2 + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2]  Q +W{l−1} + δ0 (1 + t)−1 W{l−1}

(11.471)

554

Chapter 11. Control of the Spatial Derivatives

Using this estimate we deduce: 5 2,[k−1,l−1], ε0

t −2 ≤ Cl (1 + t) δ0 [1 + log(1 + t)]χ  2,[k−1,l−1], ε0 t

+ δ0 (1 + t)

−1

+ δ0 (1 + t)

−1

[1 + log(1 + t)]Uk−2,l−2

  κ − 12,[k−1,l−1], ε0 + [1 + log(1 + t)] Y0 + (1 + t)A[l−2] t  Q (11.472) +[1 + log(1 + t)] W{l−1} + δ0 (1 + t)−1 W{l−1}

Next, using the estimate (11.450) with (k, l) reduced to (k−1, l−1) and the estimate (11.462) we deduce (see (11.312)): 6 2,[k−1,l−1], ε0

t −2 ≤ Cl (1 + t) δ0 [1 + log(1 + t)]χ  2,[k−1,l−1], ε0 t

+ δ0 (1 + t)−2 [1 + log(1 + t)]Uk−3,l−2

  + δ0 (1 + t)−1 κ − 12,[k−1,l−1], ε0 + [1 + log(1 + t)] Y0 + (1 + t)A[l−2] t  Q 2 (11.473) +[1 + log(1 + t)] W{l} + δ0 (1 + t)−2 W{l−1} To estimate 7 (see (11.313)), we first consider χ $ · d/σ , expressed by (11.339). The estimate (11.462) and the corresponding L ∞ estimate with l∗ in the role of l from Corollary 11.1.c yield, using Lemma 11.2.a, χ $ · d/σ 2,[k−1,l−1], ε0 t

≤ Cl (1 + t)−2 δ0 χ  2,[k−1,l−1], ε0 t

+ δ03 (1 + t)−5 [1 + log(1 + t)]3 Uk−4,l−3 + δ02 (1 + t)−3 [1 + log(1 + t)]κ − 12,[k−2,l−2], ε0 t 2 1 + δ02 (1 + t)−2 [1 + log(1 + t)] Y0 + (1 + t)A[l−2]  Q +W{l} + δ04 (1 + t)−4 [1 + log(1 + t)]W{l−2}

(11.474)

Using this estimate we deduce: 7 2,[k−1,l−1], ε0

t ≤ Cl δ0 (1 + t)−3 δ0 [1 + log(1 + t)]χ  2,[k−1,l−1], ε0 t

+ δ0 (1 + t)−2 [1 + log(1 + t)]2 Uk−2,l−2 + δ0 (1 + t)−1 κ − 12,[k−1,l−1], ε0 t 2 1 + δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l−2]  Q +[1 + log(1 + t)] W{l} + δ03 (1 + t)−3 W{l−1}

(11.475)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

555

Combining finally the results (11.465), (11.466), (11.468), (11.470), (11.472), (11.473), (11.475), yields:  2,[k−1,l−1], ε0

t −1 δ0 χ  2,[k−1,l−1], ε0 ≤ Cl (1 + t) t

−2

+ δ0 (1 + t) [1 + log(1 + t)]Uk−2,l−2 + (1 + t)−1 κ − 12,[k−1,l], ε0 t 2 1 + δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l−2]  Q +W{l} + (1 + t)−1 [1 + log(1 + t)]W{l−1}

(11.476)

(N)

We turn to the last term on the right in (11.305), the term γ , given by (4.88) of Chapter 4 and (11.343)–(11.345). Applying Lemma 11.2.a, using the estimates (11.426), (11.432), (11.448), with (k, l + 1) reduced to (k − 1, l − 1), together with the estimates (11.428), (11.429), (11.434), (11.449), with (k, l) reduced to (k − 1, l − 1) and the corresponding L ∞ estimates with l∗ in the role of l from Corollary 11.1.b, we find: (N)

 γ 2,[k−1,l−1], ε0

t ≤ Cl δ0 (1 + t)−2 δ0 (1 + t)−2 [1 + log(1 + t)]Uk−2,l−2 + δ0 (1 + t)−2 κ − 12,[k−1,l−1], ε0 t 1 2 + δ0 (1 + t)−1 Y0 + (1 + t)A[l−2] +W{l} + (1 + t)−1 [1 + log(1 + t)]W{l−1} Q

 (11.477)

( P)

We turn to the term γ , given by (11.350). We first consider the lower term l given by (11.356)–(11.358). We note here that by Lemma 11.1 (and hypothesis H0) : d/α2,{l−1}, ε0 ≤ Cl W{l}

(11.478)

t

Also, by (11.461) applied to ξ = d/σ : (d/σ )$ 2,[k−1,l−1], ε0

t −1 ≤ Cl (1 + t) δ0 (1 + t)−1 [1 + log(1 + t)]χ  2,[k−2,l−2], ε0 t

+ δ02 (1 + t)−4 [1 + log(1 + t)]2 Uk−4,l−3

  + δ0 (1 + t)−1 (1 + t)−1 κ − 12,[k−2,l−2], ε0 + Y) + (1 + t)A[l−2] t  Q 3 −3 (11.479) +W{l} + δ0 (1 + t) W{l−2}

556

Chapter 11. Control of the Spatial Derivatives

Using the estimates (11.478), (11.479), and (11.471), as well as the estimates (11.426), (11.432), (11.448), with (k, l + 1) reduced to (k − 1, l − 1), and the estimates (11.450), (11.451), with (k, l) reduced to (k − 1, l − 1), and the corresponding L ∞ estimates with l∗ in the role of l from Corollary 11.1.b, we find: l2,[k−1,l−1], ε0 t

≤

Cl δ02 (1 + t)−4

δ0 (1 + t)−1 [1 + log(1 + t)]χ  2,[k−2,l−2], ε0 t

+ δ0 (1 + t) + δ0 (1 + t)

−1

−1

Uk−2,l−2

  κ − 12,[k−2,l−1], ε0 + Y0 + (1 + t)A[l−2] t  Q (11.480) +W{l} + W{l−1} (1 + t)

−1

( P P)

We turn to the principal part γ (see (11.351)). Consider first (11.352). By hypothesis H0 and assumption E{l∗ +1} we have: d/(T σ )2,[k−1,l−1], ε0 = t

k−1  j =0

( P P) γ 1,

given by

d/(T ) j +1 σ 2,[l−1− j ], ε0

≤ C(1 + t)−1

t

k−1  j =0

(T ) j +1σ 2,[l− j ], ε0 t

= C(1 + t)−1 T σ 2,[k−1,l], ε0 t

≤ Cl (1 + t)−1 W{l+1}

(11.481)

We then apply Lemma 11.2.b taking ϑ = d/(T σ ). Using the estimates (11.432), (11.448) with (k, l + 1) reduced to (k − 1, l − 1) and the corresponding L ∞ estimates with l∗ in the role of l from Corollary 11.1.b, together with the estimate (11.481) and assumption E{l∗ +1} , which implies the estimate (11.370) with l∗ in the role of l, we obtain: ( P P) 1

 γ

2,[k−1,l−1], ε0

t −2 δ0 (1 + t)−2 [1 + log(1 + t)]Uk−2,l−2 ≤ Cl δ0 (1 + t)

  + δ0 (1 + t)−1 (1 + t)−1 κ − 12,[k−2,l−1], ε0 + Y0 + (1 + t)A[l−2] t  Q 3 −3 +W{l+1} + δ0 (1 + t) W{l−2} (11.482) ( P P) 2,

Consider next γ E{l∗ +1} we have:

Q

given by (11.353). By hypothesis H0 and assumptions E{l∗ } and

d/(Lσ )2,[k−1,l−1], ε0 = t

k−1  j =0

d/(T ) j (Lσ )2,[l−1− j ], ε0 t

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

≤ C(1 + t)−1

k−1  j =0

557

(T ) j (Lσ )2,[l− j ], ε0 = C(1 + t)−1 Lσ 2,[k−1,l], ε0 t

t

 Q ≤ Cl (1 + t)−2 W{l} + δ0 (1 + t)−1 W{l}

(11.483)

We then apply Lemma 11.2.b taking ϑ = d/(Lσ ). Using the estimates (11.426), (11.448) with (k, l + 1) reduced to (k − 1, l − 1) and the corresponding L ∞ estimates with l∗ in the role of l from Corollary 11.1.b, together with the estimate (11.483) and assumption E{lQ∗ } , which implies the estimate (11.372) with l∗ in the role of l, we obtain: ( P P) 2

 γ

2,[k−1,l−1], ε0

t 2 −4 δ0 (1 + t)−1 [1 + log(1 + t)]Uk−2,l−2 ≤ Cl δ0 (1 + t)

  + δ0 (1 + t)−1 κ − 12,[k−1,l−1], ε0 + [1 + log(1 + t)] Y0 + (1 + t)A[l−2] t  Q +[1 + log(1 + t)] W{l} + W{l} (11.484) ( P P)

To estimate γ 3 (see (11.354)) we consider the expression (11.374) for L(T σ ),  being given by (11.375). Applying (11.461) and using the estimates (11.453) and (11.467) with (k, l) reduced to (k − 1, l − 1) as well as the corresponding L ∞ estimates (11.295) and (11.323) with l∗ in the role of l, we obtain:

2,[k−1,l−1], ε0 ≤ Cl (1 + t)−1 δ0 [1 + log(1 + t)]2 χ  2,[k−2,l−2], ε0 t

t

+ δ0 (1 + t)−1 [1 + log(1 + t)]Uk−2,l−2 + κ − 12,[k−1,l], ε0 t 1 2 + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−2] +[1 + log(1 + t)] W{l} + δ02 (1 + t)−2 W{l−1} Q

 (11.485)

In view of the fact that: T (Lσ )2,[k−1,l−1], ε0 = (1 + t)−1 T (Qσ )2,[k−1,l−1], ε0 t

t

≤ (1 + t)−1 Qσ 2,[k,l], ε0 t

 Q −1 ≤ Cl (1 + t) W{l} + δ0 (1 + t)−1 W{l}

(11.486)

we obtain through expression (11.374), using Lemma 11.2.a and the estimate (11.485) to handle the second term, L(T σ )2,[k−1,l−1], ε0 t

−1 2 δ0 (1 + t)−2 [1 + log(1 + t)]2 χ  2,[k−2,l−2], ε0 ≤ Cl (1 + t) t

558

Chapter 11. Control of the Spatial Derivatives

+ δ02 (1 + t)−3 [1 + log(1 + t)]Uk−2,l−2 + δ0 (1 + t)−2 κ − 12,[k−1,l], ε0 t 1 2 + δ02 (1 + t)−2 [1 + log(1 + t)] Y0 + (1 + t)A[l−2]  Q +W{l} + δ0 (1 + t)−1 [1 + log(1 + t)]W{l}

(11.487)

We then apply Lemma 11.2.b taking ϑ = L(T σ ). Using the estimate (11.448) with (k, l + 1) reduced to (k − 1, l − 1) and the corresponding L ∞ estimate with l∗ in the role of l from Corollary 11.1.b, together with the estimate (11.487) and assumption E{lQ∗ } , which implies the estimate (11.379) with l∗ in the role of l, we obtain: ( P P) 3

 γ

2,[k−1,l−1], ε0

t 2 −3 ≤ Cl δ0 (1 + t) δ02 (1 + t)−2 [1 + log(1 + t)]2 χ  2,[k−2,l−2], ε0 t

+ δ0 (1 + t)

−2

[1 + log(1 + t)]Uk−2,l−2   (1 + t) κ − 12,[k−1,l], ε0 + Y0 + (1 + t)A[l−2] + δ0 (1 + t) t  Q (11.488) +W{l} + W{l} −1

( P P) 4.

We turn to γ

−1

In this case we must estimate D / 2 σ 2,[k−1,l−1], ε0 t

We consider again the expression (11.381) taking j ∈ {0, . . . , k − 1} and n ∈ {0, . . . , l − 1 − j }. Now the assumptions of Proposition 11.2 include the assumptions of Lemma 10.11 with l∗ in the role of l. Therefore under the present assumptions we have: max 

(Ri )

i

π /1 ∞,[l∗ −1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.489)

Moreover, the first statement Corollary 11.1.c with l∗ in the role of l implies (11.315) with l∗ in the role of l. Thus, the assumptions of Lemma 11.15 are all verified. They are a fortiori verified if in that lemma (k, l) is replaced by (k − 1, l − 1). Substituting the bound for T[k−2,l−1] from (11.459) with (k, l) reduced to (k − 2, l − 1), as well as the bound for maxi  (Ri ) π /1 2,[l−2], ε0 from Lemma 10.12 with l reduced to l − 1, noting that the t present assumptions include those of Lemma 10.12, yields the following conclusion: For any St,u 1-form ξ we have: max

max 

j ≤k−1,n≤l−1− j i1 ...in

≤ Cl δ0 (1 + t) + Cl (1 + t)

−2

−1

(i1 ...in )

c j,n [ξ ] L 2 ( ε0 ) t

[1 + log(1 + t)]ξ 2,[k−1,l−2], ε0 t

ξ ∞,[k−1,l∗ ], ε0 · t

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

559

[1 + log(1 + t)]χ  2,[k−2,l−1], ε0 t

+ δ0 (1 + t)−3 [1 + log(1 + t)]2 Uk−4,l−2 + (1 + t)−1 κ − 12,[k−2,l−1], ε0 t

Q +Y0 + (1 + t)A[l−1] + W{l} + δ02 (1 + t)−2 W{l−1}

 (11.490)

Taking in (11.490) ξ = d/σ we obtain, using assumption E{l∗ +1} , max  (i1 ...in ) c j,n [d/σ ] L 2 ( ε0 ) t

≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)]χ  2,[k−2,l−1], ε0 max

j ≤k−1,n≤l−1− j i1 ...in

t

+ δ0 (1 + t)−3 [1 + log(1 + t)]2 Uk−4,l−2 + (1 + t)−1 κ − 12,[k−2,l−1], ε0

(11.491) 

t

Q +Y0 + (1 + t)A[l−1] + [1 + log(1 + t)]W{l} + δ02 (1 + t)−2 W{l−1}

On the other hand, by hypotheses H1 and H0: max

max D / 2 Rin . . . Ri1 (T ) j σ  L 2 ( ε0 )

j ≤k−1,n≤l−1− j i1 ...in

≤ C(1 + t)−2

t

k−1  j =0

(T ) j (σ − k 2 )2,[l+1− j ], ε0 t

= C(1 + t)−2 σ − k 2 2,[k−1,l+1], ε0 t

≤ Cl (1 + t)

−2

W{l+1}

(11.492)

In view of the estimates (11.491) and (11.492) we conclude from (11.381) that: D / 2 σ 2,[k−1,l−1], ε0 t

=

k−1 l−1−  j j =0 n=0

max L / R in . . . L / Ri1 (L /T ) j D / 2 σ  L 2 ( ε0 )

i1 ...in

t

≤ Cl (1 + t)−2 W{l+1}

+ Cl δ0 (1 + t)−3 [1 + log(1 + t)]χ  2,[k−2,l−1], ε0 t

+ δ0 (1 + t)

−3

[1 + log(1 + t)] Uk−4,l−2 2

+ (1 + t)−1 κ − 12,[k−2,l−1], ε0 t

Q +Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W{l−1} ( P P)

 (11.493)

/ 2 σ . Using the estimates To estimate γ 4 we then apply Lemma 11.2.b taking ϑ = D (11.426), (11.432), with (k, l + 1) reduced to (k − 1, l − 1) and the corresponding L ∞

560

Chapter 11. Control of the Spatial Derivatives

estimates with l∗ in the role of l from Corollary 11.1.b, together with the estimate (11.493) and assumption E{l∗ +1} , which implies the estimate (11.386) with l∗ in the role of l, we obtain: ( P P) 4

 γ

2,[k−1,l−1], ε0

t −3 ≤ Cl δ0 (1 + t) δ0 (1 + t)−1 [1 + log(1 + t)]2 χ  2,[k−2,l−1], ε0 t

−1

+ δ0 (1 + t) [1 + log(1 + t)]Uk−2,l−2 + δ0 (1 + t)−1 κ − 12,[k−1,l−1], ε0 t 2 1 + δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q +[1 + log(1 + t)] W{l+1} + δ03 (1 + t)−3 W{l−1}

(11.494)

Combining the results (11.482), (11.484), (11.488), (11.494), and taking also into account the estimate (11.480), we conclude that: ( P)

 γ 2,[k−1,l−1], ε0

t −2 δ0 (1 + t)−2 [1 + log(1 + t)]2 χ  2,[k−2,l−1], ε0 ≤ Cl δ0 (1 + t) t

+ δ0 (1 + t)

−2

+ δ0 (1 + t)

−2

[1 + log(1 + t)]Uk−2,l−2

κ − 12,[k−1,l], ε0 t 1 2 + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1]  Q +W{l+1} + δ0 (1 + t)−1 W{l}

(11.495)

We turn to the second term on the right in (11.305), the term D / 2 µ. We consider again the expression (11.390) taking j ∈ {0, . . . , k − 1} and n ∈ {0, . . . , l − 1 − j }. Taking in (11.490) ξ = d/µ we obtain, using assumption M[k−1,l∗ +1] , max 

(i1 ...in )

c j,n [d/µ] L 2 ( ε0 ) t

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] [1 + log(1 + t)]χ  2,[k−2,l−1], ε0 max

j ≤k−1,n≤l−1− j i1 ...in

t

−3

+ δ0 (1 + t) [1 + log(1 + t)] Uk−4,l−2 + (1 + t)−1 κ − 12,[k−1,l−1], ε0 2

t

+Y0 + (1 + t)A[l−1] + W{l} + δ02 (1 + t)−2 W{l−1} Q

On the other hand, by hypotheses H1 and H0: max

max D / 2 Rin . . . Ri1 (T ) j µ L 2 ( ε0 )

j ≤k−1,n≤l−1− j i1 ...in

≤ C(1 + t)−2

t

k−1  j =0

(T ) j (µ − η0 )2,[l+1− j ], ε0 t

 (11.496)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

561

= C(1 + t)−2 µ − η0 2,[k−1,l+1], ε0 (11.497) t

 ≤ Cl (1 + t)−2 κ − 12,[k−1,l+1], ε0 + [1 + log(1 + t)]W{l+1} t

In view of the estimates (11.496) and (11.497) we conclude from (11.390) that: D / 2 µ2,[k−1,l−1], ε0 t

=

k−1 l−1−  j j =0 n=0

max L / R in . . . L / Ri1 (L /T ) j D / 2 µ L 2 ( ε0 )

i1 ...in

t

 ≤ Cl (1 + t)−2 κ − 12,[k−1,l+1], ε0 + [1 + log(1 + t)]W{l+1} t

−2 + Cl δ0 (1 + t) [1 + log(1 + t)] [1 + log(1 + t)]χ  2,[k−2,l−1], ε0 t

+ δ0 (1 + t)−3 [1 + log(1 + t)]2 Uk−4,l−2 Q +Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W{l−1}

 (11.498)

˜ ). Finally, we have the third term on the right in (11.305), the term (1/2)(D /ζ + D /ζ Taking in (11.490) ξ = ζ and using the estimate (11.453) with (k, l) reduced to (k − 1, l − 2) as well as the estimate (11.295) with (k − 1, l∗ ) in the role of (k, l), we obtain: max

max 

(i1 ...in )

c j,n [ζ ] L 2 ( ε0 ) t

≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] [1 + log(1 + t)]χ  2,[k−2,l−1], ε0

j ≤k−1,n≤l−1− j i1 ...in

t

−1

+ δ0 (1 + t) [1 + log(1 + t)]Uk−2,l−2 + (1 + t)−1 κ − 12,[k−1,l−1], ε0 t

(11.499) 

+Y0 + (1 + t)A[l−1] + [1 + log(1 + t)] W{l} + δ02 (1 + t)−2 W{l−1} Q

On the other hand, by hypotheses H1 and the estimate (11.453) with k reduced to k − 1: max

max D /L / R in . . . L / Ri1 (L /T ) j ζ  L 2 ( ε0 )

j ≤k−1,n≤l−1− j i1 ...in

≤ C(1 + t)−1

t

k−1  j =0

(L /T ) j ζ 2,[l− j ], ε0 t

= C(1 + t)−1 ζ 2,[k−1,l], ε0 t

−2 −1 ≤ Cl (1 + t) δ0 (1 + t) [1 + log(1 + t)]Uk−2,l−1 + δ0 (1 + t)−1 κ − 12,[k−1,l], ε0 t 2 1 + δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q +[1 + log(1 + t)] W{l+1} + δ02 (1 + t)−2 W{l}

(11.500)

562

Chapter 11. Control of the Spatial Derivatives

In view of the estimates (11.499) and (11.500) we conclude through (11.164) with ζ in the role of ξ that: D /ζ 2,[k−1,l−1], ε0 t

−2 ≤ Cl (1 + t) δ0 (1 + t)−1 [1 + log(1 + t)]2 χ  2,[k−2,l−1], ε0 t

+ δ0 (1 + t)−1 [1 + log(1 + t)]Uk−2,l−1 + δ0 (1 + t)−1 κ − 12,[k−1,l], ε0 t 2 1 −1 + δ0 (1 + t) [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q +[1 + log(1 + t)] W{l+1} + δ02 (1 + t)−2 W{l}

(11.501)

˜ , by virtue of the fact that Lie derivatives commute with The same estimates hold for D /ζ transposition. Combining the results (11.476), (11.498), (11.501), (11.495), (11.477), for the five terms on the right in (11.305) we arrive at the following inequality: χ  2,[k,l], ε0 − χ  2,[k−1,l], ε0 = (L /T )k χ  2,[l−k], ε0 t

t

(11.502)

t

≤ Cl δ0 (1 + t)−1 χ  2,[k−1,l−1], ε0 + ak,l t

where:

ak,l = Cl (1 + t)−1 δ0 (1 + t)−2 [1 + log(1 + t)]Uk−2,l−1 + (1 + t)−1 κ − 12,[k−1,l+1], ε0 t 2 1 + δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q +W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l}

The inequality (11.502) implies the ordinary recursive inequality:   χ  2,[k,l], ε0 ≤ 1 + Cl δ0 (1 + t)−1 χ  2,[k−1,l], ε0 + ak,l t

t

(11.503)

(11.504)

which yields: k   k k− j  ||χ  2,[k,l], ε0 ≤ 1 + Cl δ0 (1 + t)−1 χ  2,[0,l], ε0 + a j,l 1 + Cl δ0 (1 + t)−1 t

t

j =1

(11.505) Since the ak,l are non-decreasing in k, this in turn implies, provided that δ0 is suitably small (depending on l):

 (11.506) χ  2,[k,l], ε0 ≤ Cl χ  2,[l], ε0 + ak,l t

t

Substituting finally in (11.506) the estimate (11.460) and the definition (11.503), we obtain the conclusion of the lemma.

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

563

We now resume the proof of Proposition 11.2. Substituting in (11.459) the result of Lemma 11.17, yields the following estimate for the quantity T[k,l] :

T[k,l] ≤ Cl (1 + t)−1 δ0 (1 + t)−2 [1 + log(1 + t)]2 Uk−2,l−1 (11.507) 2 1 +κ − 12,[k,l+1], ε0 + [1 + log(1 + t)] Y0 + (1 + t)A[l] t  Q +[1 + log(1 + t)] W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} Consider now Lemma 11.12 with (k +1, l +1) in the role of (k, l). Since (l +1)∗ ≤ l∗ +1, the assumptions hold if: λ ∞,[k,l∗ ], ε0 ≤ Cl δ0 [1 + log(1 + t)] t

max 

(Ri )

t

i



π /∞,[l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

(T ) 

π / ∞,[k,l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(11.508)

The first inequality of (11.508) follows from assumptions M[k,l∗ ] and E{l∗ } . The second is Corollary 10.1.d with l∗ + 1 in the role of l, and the third follows from Corollary 11.1.c with l∗ in the role of l. Thus, the assumptions (11.508) all follow from the assumptions of Proposition 11.2. Therefore, under the present assumptions the conclusion of Lemma 11.12 holds with (k +1, l +1) in the role of (k, l). This implies that for any St,u tensorfield ξ we have: ξ · h /2,[k,l], ε0 , ξ · h /−1 2,[k,l], ε0 t  t  ε ε ≤ Cl ξ 2,[k,l], 0 + ξ ∞,[k,l∗ ], 0 max  t

t

i

(Ri )

(11.509)  π /2,[l−1], ε0 + T[k−1,l−1] t

Substituting from Corollary 10.2.d and the bound (11.507) (with (k, l) replaced by (k − 1, l − 1)) this becomes: /−1 2,[k,l], ε0 ≤ Cl ξ 2,[k,l], ε0 ξ · h /2,[k,l], ε0 , ξ · h t

t

t

(11.510)

+ Cl ξ ∞,[k,l∗ ], ε0 · t

· δ0 (1 + t)−3 [1 + log(1 + t)]2 Uk−3,l−2   + (1 + t)−1 κ − 12,[k−1,l], ε0 + Y0 + (1 + t)A[l−1] t  Q −2 +W{l} + (1 + t) [1 + log(1 + t)]2 W{l−1} (this is to be compared with (11.461)). Applying this to ξ = (qT )b , we obtain, by virtue of the estimate (11.455) and the estimate (11.397) with l∗ in the role of l,

(11.511) qT 2,[k,l], ε0 ≤ Cl (1 + t)−1 δ02 (1 + t)−1 Uk−1,l−1 + κ − 12,[k,l+1], ε0 t t 2 1 +δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q + δ0 [1 + log(1 + t)] W{l+1} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1}

564

Chapter 11. Control of the Spatial Derivatives

Consider now the first two recursion relations of Lemma 11.3, replacing m by j ∈ {0, . . . , k}. Taking the  2,[k− j,l− j ], ε0 norm we obtain: t

 pT , j 2,[k− j,l− j ], ε0 t

≤  pT , j −12,[k+1− j,l+1− j ], ε0 t

+ Cl  pT ∞,[k− j,l∗ ], ε0  pT , j −12,[k− j,l− j ], ε0

 + pT 2,[k− j,l− j ], ε0  pT , j −1∞,[k− j,l∗ − j ], ε0 t t

+ Cl d/κ∞,[k− j,l∗ ], ε0 qT , j −12,[k− j,l− j ], ε0 t t  +d/κ2,[k− j,l− j ], ε0 qT , j −1∞,[k− j,l∗ − j ], ε0 t

t

t

t

(11.512)

qT , j 2,[k− j,l− j ], ε0 t

≤ qT , j −1 2,[k+1− j,l+1− j ], ε0 t

ε + Cl qT ∞,[k− j,l∗ ], 0  pT , j −12,[k− j,l− j ], ε0 t

t

+qT 2,[k− j,l− j, ε0  pT , j −1∞,[k− j,l∗ − j ], ε0 t

t

 (11.513)

Setting, for fixed (k, l) and with j ∈ {0, . . . , k}, x j =  pT , j 2,[k− j,l− j ], ε0 , t

y j = qT , j 

ε 2,[k− j,l− j ],t 0

(11.514)

and: a =  pT 2,[k,l], ε0 = x 0 t

b = qT 2,[k,l], ε0 = y0 t

c = d/κ2,[k,l], ε0 t

(11.515)

(11.512)–(11.513) imply by assumptions M[k,l∗ +1] and E{l∗ +1} and Corollary 11.1.a with l∗ in the role of l, x j ≤ x j −1 + ε2 (x j −1 + a) + ε(y j −1 + c) y j ≤ y j −1 + ε(x j −1 + εb) where:

ε = Cl δ0 (1 + t)−1 [1 + log(1 + t)]

The inequalities (11.516) may be written in the form:     xj x j −1 ≤ (I + E) +a yj y j −1

(11.516) (11.517)

(11.518)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

565

where E is the matrix with non-negative entries and a the column with non-negative entries, given by:  2    2 ε a + εc ε ε E= (11.519) a= ε 0 ε2 b (I is the identity matrix). It follows that: 

xj yj



 ≤ (I + E) j

x0 y0



  j −1  + (I + E)i  a

(11.520)

i=0

Now, if ε is suitably small we have: |E| ≤ Cε (matrix norm) and (11.520) implies: x j ≤ x 0 + C j ε(x 0 + y0 ) + C j (ε2 a + εc + ε2 b) y j ≤ y0 + C j ε(x 0 + y0 ) + C j (ε2 a + εc + ε2 b)

(11.521)

In view of the definitions (11.514) and (11.515), substituting for a and b the bounds (11.454) and (11.511) respectively, and noting that: c ≤ C(1 + t)−1 κ − 12,[k,l+1], ε0 t

yields the estimates:  pT , j 2,[k− j,l− j ], ε0 t

≤ Cl δ0 (1 + t)

−1

δ0 (1 + t)−1 Uk−1,l−1

+ (1 + t)−1 [1 + log(1 + t)]κ − 12,[k,l+1], ε0 t 1 2 + δ0 (1 + t)−1 [1 + log(1 + t)]2 Y0 + (1 + t)A[l−1]  Q +W{l+1} + δ0 (1 + t)−3 W{l−1} : for all j ∈ {0, . . . , k}

(11.522)

qT , j 2,[k− j,l− j ], ε0 t

−1 δ02 (1 + t)−1 Uk−1,l−1 + κ − 12,[k,l+1], ε0 ≤ Cl (1 + t) t 2 1 + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] +δ0 [1 + log(1 + t)] W{l+1} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1} Q

: for all j ∈ {0, . . . , k}



(11.523)

566

Chapter 11. Control of the Spatial Derivatives

Next, we consider formula (11.400). Taking the  2,[l−k+n], ε0 norm we obtain, t using (11.523), r Tn ,k 2,[l−k+n], ε0 t

≤

k−1−n 

Ni (n)

k−1−n−i  j  =0

i=0

≤ Cl

max

j  ∈{0,...,k−1−n}

κqT , j  2,[k−1−n− j  ,l−1− j  ], ε0 t

κqT , j  2,[k−1− j  ,l−1− j  ], ε0 t

≤ Cl (1 + t)−1 [1 + log(1 + t)] ·

δ02 (1 + t)−1 Uk−2,l−2 + κ − 12,[k−1,l], ε0 t 2 1 + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−2] Q +δ0 [1 + log(1 + t)] W{l} + (1 + t)−2 [1 + log(1 + t)]2 W{l−2}

: for all n ∈ {0, . . . , k − 1}

 (11.524)

Finally, we take the  2,[l−k], ε0 norm of formula (11.402). We then obtain on the left: t

(T )

k+1

Tˆ i 2,[l−k], ε0 = Uk,l − Uk−1,l t

(see (11.420)). By (11.522) and (11.523) with j = k and Proposition 10.2 together with Proposition 10.1 and Corollary 11.1.a with l∗ in the role of l: max  pT ,k Tˆ i 2,[l−k], ε0 t i

−1 δ0 (1 + t)−1 Uk−1,l−1 ≤ Cl δ0 (1 + t) + (1 + t)−1 [1 + log(1 + t)]κ − 12,[k,l+1], ε0 t 1 2 + δ0 (1 + t)−1 [1 + log(1 + t)]2 Y0 + (1 + t)A[l−1]  Q +W{l+1} + δ0 (1 + t)−3 [1 + log(1 + t)]2 W{l−1}

(11.525)

max qT ,k · d/x i 2,[l−k], ε0 t i

≤ Cl (1 + t)−1 δ02 (1 + t)−1 Uk−1,l−1 + κ − 12,[k,l+1], ε0 t 2 1 + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]

(11.526) 

+δ0 [1 + log(1 + t)] W{l+1} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1} Q

By (11.524) with n = 0 and Proposition 10.2 (note that k ≥ 1) together with Proposition 10.1 with l∗ + 1 in the role of l and Corollary 11.1.a with l∗ in the role of l: r T0 ,k · d/Tˆ i 2,[l−k], ε0 t

≤ Cl (1 + t)−2 [1 + log(1 + t)]·

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

567

δ02 (1 + t)−1 Uk−2,l−2 + κ − 12,[k−1,l], ε0 t 2 1 + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] Q +δ0 [1 + log(1 + t)] W{l} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1}

(11.527) 

Also, by (11.524) together with Proposition 11.1 and Corollary 11.1.a with l∗ in the role of l: 0 k−1 0 0 0 0 n n ˆ i0 max 0 r T ,k · d/(T ) T 0 0 i 0 ε n=1 2,[l−k],t 0  max max r Tn ,k ∞,[l∗ −k+n], ε0 Uk−2,l−1 ≤ Cl (1 + t)−1 +

k−1 

n∈{1,...,k−1}

r Tn ,k ∞,[l−k], ε0 t

n=1

t

i



max

n∈{1,...,k−1}

n



ˆi

(T ) T ∞,[l∗ +1−n], ε0 t

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]2 Uk−2,l−1

+ Cl δ0 (1 + t)−3 [1 + log(1 + t)]2 κ − 12,[k−1,l], ε0 t 2 1 + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−2] +δ0 [1 + log(1 + t)] W{l} + (1 + t)−2 [1 +

Q log(1 + t)]2 W{l−2}

(11.528) 

Combining (11.527) and (11.528) we obtain: 0 k−1 0 0 0 0 0 r Tn ,k · d/(T )n Tˆ i 0 max 0 0 i 0 ε0 n=0

2,[l−k],t

−2

≤ Cl (1 + t) [1 + log(1 + t)] ·

δ0 [1 + log(1 + t)]Uk−2,l−1 + κ − 12,[k−1,l], ε0 t 2 1 + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] Q +δ0 [1 + log(1 + t)] W{l} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1}

(11.529) 

In view of the results (11.525), (11.526), (11.529), for the three terms on the right in (11.402) we arrive at the following recursive inequality: Uk,l − Uk−1,l ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]2 Uk−1,l−1 + bk,l where:

bk,l = Cl (1 + t)−1 κ − 12,[k,l+1], ε0 t 1 + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] +W{l+1} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1} Q

(11.530)

 (11.531)

568

Chapter 11. Control of the Spatial Derivatives

The inequality (11.530) implies the ordinary recursive inequality:   Uk,l ≤ 1 + Cl δ0 (1 + t)−2 [1 + log(1 + t)]2 Uk−1,l + bk,l

(11.532)

which yields: k  Uk,l ≤ 1 + Cl δ0 (1 + t)−2 [1 + log(1 + t)]2 U0,l +

k  k− j  1 + Cl δ0 (1 + t)−2 [1 + log(1 + t)]2 b j,l

(11.533)

j =1

Since the bk,l are non-decreasing in k, this in turn implies:   Uk,l ≤ Cl U0,l + bk,l

(11.534)

Substituting finally in (11.534) the estimate (11.424) and the definition (11.531) we obtain:

Uk,l ≤ Cl (1 + t)−1 κ − 12,[k,l+1], ε0 t 1 + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q +W{l+1} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1} (11.535) Since we have:

  κ − 12,[k,l+1], ε0 ≤ Cl B[k,l+1] + [1 + log(1 + t)]W{l+1}

(11.536)

t

the estimate (11.535) for k = 0, . . . , m, yields the proposition. The foregoing proposition has the following corollaries. Corollary 11.1.a Under the assumptions of Proposition 11.2 the coefficients of the expression for (T )k+1 Tˆ i , k = 0, . . . , m, of Lemma 11.3 satisfy:  pT ,k 2,[m−k,l−k], ε0

1 (1 + t)−1 [1 + log(1 + t)] B[m,l+1] 2  +δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q +W{l+1} + δ0 (1 + t)−3 W{l−1} t

≤ Cl δ0 (1 + t)

−1

: for all k ∈ {0, . . . , m} qT ,k 2,[m−k,l−k], ε0 t    ≤ Cl (1 + t)−1 B[m,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] +[1 + log(1 + t)] W{l+1} + δ0 (1 + t)−2 [1 + log(1 + t)]2 W{l−1} Q

: for all k ∈ {0, . . . , m}



Part 1: Control of the spatial derivatives of the first derivatives of the x i .

569

and: r Tn ,k 2,[l−k+n], ε0 t

≤ Cl (1 + t)−1 [1 + log(1 + t)] ·    B[m,l] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−2] +[1 + log(1 + t)] W{l} + δ0 (1 + t)−2 [1 + log(1 + t)]2 W{l−2} Q



: for all k ∈ {0, . . . , m}, n ∈ {0, . . . , k − 1} Also, the component functions L i satisfy the estimate: m  k=0

max (T )k+1 L i 2,[l−k], ε0 t

i

1 2  ≤ Cl (1 + t)−1 B[m,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q +W{l+1} + δ0 (1 + t)−3 [1 + log(1 + t)]3 W{l−1} Corollary 11.2.b Under the assumptions of Proposition 11.2 the following estimates hold:  ψTˆ 2,[m+1,l+1], ε0 ≤ Cl W{l+1} + t  Q δ0 (1 + t)−1 (1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l] + δ0 (1 + t)−2 W{l}  ψ L − k2,[m+1,l+1], ε0 ≤ Cl W{l+1} + t

δ0 (1 + t)

−1

Q (1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l] + δ0 (1 + t)−2 W{l}

  ψ2,[m+1,l+1], ε0 ≤ Cl W{l+1} + t

δ0 (1 + t)

−1

Q (1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l] + δ0 (1 + t)−2 W{l}





and:  ωT Tˆ 2,[m,l], ε0 ≤ Cl W{l+1} + t

δ0 (1 + t)

−1

(1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1] + δ0 (1 + t)−2 W{l−1} Q

Q ω L Tˆ 2,[m,l], ε0 ≤ Cl (1 + t)−1 W{l} + t

δ0 (1 + t)

−1

(1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1] + W{l}





570

Chapter 11. Control of the Spatial Derivatives

Q ω L L 2,[m,l], ε0 ≤ Cl (1 + t)−1 W{l} + t

δ0 (1 + t)

−1

(1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1] + W{l}



 ω /Tˆ 2,[m,l], ε0 ≤ Cl (1 + t)−1 W{l+1} + t

δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1] + δ0 (1 + t)−2 W{l−1} Q

 ω / L 2,[m,l], ε0 ≤ Cl (1 + t)−1 W{l+1} + t

δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1] + δ0 (1 + t)−2 W{l−1} Q

 ω /2,[m,l], ε0 ≤ Cl (1 + t)−1 W{l+1} + t

Q δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1] + δ0 (1 + t)−2 W{l−1}







Also, we have:

Q k/ 2,[m,l], ε0 ≤ Cl (1 + t)−1 W{l+1} + δ02 (1 + t)−2 W{l} t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1]



Q κ −1 ζ  2,[m,l], ε0 ≤ Cl (1 + t)−1 W{l+1} + δ02 (1 + t)−2 W{l} t  +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1]

 Q κk/2,[m,l], ε0 ≤ Cl (1 + t)−1 [1 + log(1 + t)] W{l+1} + δ02 (1 + t)−2 W{l} t 1 2  +δ0 (1 + t)−1 B[m,l] + [1 + log(1 + t)] Y0 + (1 + t)A[l−1]

 Q ζ 2,[m,l], ε0 ≤ Cl (1 + t)−1 [1 + log(1 + t)] W{l+1} + δ02 (1 + t)−2 W{l} t 1 2  +δ0 (1 + t)−1 B[m,l] + [1 + log(1 + t)] Y0 + (1 + t)A[l−1] The next corollary is obtained from Lemma 11.17 by substituting Proposition 11.2 and then substituting the result in the estimates (11.507), (11.485), and in Lemma 11.13. Corollary 11.2.c Under the assumptions of Proposition 11.2 we have:

χ  2,[m,l], ε0 ≤ Cl (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] t  Q +W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l}

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

571

Moreover we have:

 1 T[m,l] ≤ Cl (1 + t)−1 B[m,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l]  Q +W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l}

and: 

(T )

π / + 2(1 − u + η0 t)−1 h/2,[m,l], ε0 ≤ Cl T[m,l] t

Also:

   2,[m,l], ε0 ≤ Cl (1 + t)−1 B[m,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] t  Q +[1 + log(1 + t)] W{l+1} + δ0 (1 + t)−2 [1 + log(1 + t)]2 W{l}

and: 

(T )

 1 π /1 2,[m,l−1], ε0 ≤ Cl (1 + t)−2 B[m,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] t  Q +W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l}

We now revisit Lemma 11.12. Under the assumptions of Proposition 11.12 the following bounds hold: λ ∞,[m,l∗ ], ε0 ≤ Cl δ0 [1 + log(1 + t)] t

max  

(Ri )

i (T ) 

π /∞,[l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

π / ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(11.537)

These bounds imply the assumptions of Lemma 11.12 with (m, l) in the role of (k − 1, l − 1). In fact, revisiting the proof of Lemma 11.12 with (m, l) in the role of (k − 1, l − 1) we see that the bounds (11.537) complemented with ξ ∞,[m,l∗ −1], ε0 suffice and we t obtain: ξ · h /2,[m,l], ε0 , ξ · h /−1 2,[m,l], ε0 t t   ≤ Cl ξ 2,[m,l], ε0 + ξ ∞,[m,l∗ −1], ε0 max  t

t

i

(Ri )

(11.538)  π /2,[l−1], ε0 + T[m−1,l−1] t

Substituting for maxi  (Ri ) π /2,[l−1], ε0 from Corollary 10.1.d and for T[m−1,l−1] from t Corollary 11.2.c then yields the following. Corollary 11.2.d Under the assumptions of Proposition 11.2, for any St,u tensorfield ξ we have: ξ · h /2,[m,l], ε0 , ξ · h /−1 2,[m,l], ε0 t t

ε ≤ Cl ξ 2,[m,l], 0 t

Q + ξ ∞,[m,l∗ −1], ε0 W{l} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1} t  +(1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1]

572

Chapter 11. Control of the Spatial Derivatives

To derive the estimates of the later part of the present chapter, we shall make use of the following two lemmas. Lemma 11.18 Under the assumptions of Proposition 11.2, for an arbitrary St,u tensorfield ξ we have:

ξ · d/x i 2,[m,l], ε0 ≤ Cl ξ 2,[m,l], ε0 t

t

+ ξ ∞,[m,l∗ −1], ε0 (1 + t)

−1

B[m−1,l] + Y0 + (1 + t)A[l−1]  Q +W{l} + δ0 (1 + t)−2 W{l−1} t

Note that in the case that ξ is an St,u -tangential vectorfield and we have a contraction, ξ · d/x i = ξ i are simply the components of the vectorfield ξ in rectangular coordinates. Proof. Consider first: ξ · d/x i 2,[l], ε0 t

We have: / Ri1 (ξ · d/x i ) = L / R in . . . L





 (L / R )s1 ξ · d/(R)s2 x i

(11.539)

partitions |s1 |+|s2 |=n≤l

We write, as in (10.86) of Chapter 10, if s2 = {m 1 , . . . , m p }, p = |s2 |, (R)s2 x i =

( p) i x im ...im p 1

−

( p) i δim ...im p 1

(11.540)

Thus a term in the sum in (11.539) is of the form: (L / R )s1 ξ · d/

( p) i x im ...im p 1

Since, from Chapter 10, d/

− (L / R )s1 ξ · d/

( p) i x im ...im p  L ∞ ( ε0 ) t 1

( p) i δim ...im p 1

≤C

(11.541)

(11.542)

ε

the first term in (11.541) is bounded in L 2 (t 0 ) by: Cξ 2,[l], ε0 t

If p = |s2 | ≤ l∗ + 1 we can bound, using Corollary 10.1.a with l∗ + 1 in the role of l, d/

( p) i δim ...im p  L ∞ ( ε0 ) t 1

≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

Otherwise |s1 | = n − |s2 | ≤ n − l∗ − 2 ≤ l − l∗ − 2 ≤ l∗ − 1, hence we have: (L / R )s1 ξ  L ∞ ( ε0 ) ≤ ξ ∞,[l∗ −1], ε0 t

t

(11.543)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

573

and by Corollary 10.2.a (note that p ≤ n ≤ l) we can bound:

 Q d/ ( p) δiim ...im p  L 2 ( ε0 ) ≤ Cl Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] t 1 (11.544) It follows from the above that:

ξ · d/x i 2,[l], ε0 ≤ Cl ξ 2,[l], ε0 (11.545) t t  Q +ξ ∞,[l∗ −1], ε0 Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] t

Consider next: L /T (ξ · d/x i 2,[l−1], ε0 t

We have: L /T (ξ · d/x i ) = (L /T ξ ) · d/x i + ξ · d/T i

(11.546)

To the first term on the right in (11.546) we can apply the above, taking L /T ξ in the role of ξ . Since in the analogue of (11.539) n ≤ l − 1, in the case that |s2 | ≤ l∗ + 1 does not hold we now have |s1 | ≤ l∗ − 2. We thus obtain:

/T ξ 2,[l−1], ε0 (L /T ξ ) · d/x i 2,[l−1], ε0 ≤ Cl L (11.547) t t  Q +L /T ξ ∞,[l∗ −2], ε0 Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] t

To bound the contribution of the second term on the right in (11.546) we shall use the following, which requires only the assumptions of Proposition 10.2: For any function f we have:

 f Tˆ i 2,[l], ε0 ≤ Cl  f 2,[l], ε0 (11.548) t t  Q + f ∞,[l∗ ], ε0 Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] t

To demonstrate this we write: Rin . . . Ri1 ( f Tˆ i ) =



((R)s1 f )((R)s2 Tˆ i )

(11.549)

partitions |s1 |+|s2 |=n≤l

If |s2 | ≤ l∗ we have (R)s2 Tˆ i  L ∞ ( ε0 ) ≤ C, by Proposition 10.1 with l∗ in the role of l, t

hence we can bound the corresponding term in the sum in (11.549) in L 2 by: C f 2,[l], ε0 t

Otherwise |s1 | ≤ l∗ . Writing then as in (10.82) of Chapter 10, if s2 = {m 1 , . . . , m p }, p = s2 , (R)s2 Tˆ i = −(1 − u + η0 t)−1

( p) i x im ...im p 1

+ (1 − u + η0 t)−1

( p) i δim ...im p 1

+ (R)s2 y i (11.550)

574

Chapter 11. Control of the Spatial Derivatives

we estimate the contribution of the first term by placing it in L ∞ , obtaining an L 2 bound for the corresponding term in the sum in (11.549) similar to that of the case |s2 | ≤ l∗ . We estimate the contributions of the other two terms by placing them in L 2 using Proposition 10.2, Corollary 10.2.a, obtaining an L 2 bound for the corresponding term in the sum in (11.549) by:  Q Cl  f ∞,[l∗ ], ε0 Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] t

Combining the results for the two cases then yields (11.548). We write the first term in (11.546) as: ξ · d/(κ Tˆ i ) = ξ · d/Tˆ i + ξ · d/((κ − 1)Tˆ i )

(11.551)

By an argument similar to that leading to (11.548), appealing to Proposition 10.1 with l∗ + 1 in the role of l, we deduce:

ξ · d/Tˆ i 2,[l−1], ε0 ≤ Cl (1 + t)−1 ξ 2,[l−1], ε0 (11.552) t t  Q + ξ ∞,[l∗ −1], ε0 Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] t

On the other hand applying (11.548) to the function f = κ − 1 and appealing also to Proposition 10.1 with l∗ + 1 in the role of l and to assumption M[0,l∗ +1] we obtain: ξ · d/((κ − 1)Tˆ i )2,[l−1], ε0 t

 −1 δ0 [1 + log(1 + t)]ξ 2,[l−1], ε0 + ξ ∞,[l∗ −1], ε0 (κ − 1)Tˆ i 2,[l], ε0 ≤ Cl (1 + t) t t t

−1 δ0 [1 + log(1 + t)]ξ 2,[l−1], ε0 ≤ Cl (1 + t) t 1 + ξ ∞,[l∗ −1], ε0 B[0,l] + [1 + log(1 + t)]W[l] t   Q (11.553) + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] Combining (11.552) and (11.553) we obtain: ξ · d/T i 2,[l−1], ε0 (11.554)

t 1 ≤ Cl (1 + t)−1 [1 + log(1 + t)]ξ 2,[l−1], ε0 + ξ ∞,[l∗ −1], ε0 B[0,l] t t   Q 2 −2 +[1 + log(1 + t)] Y0 + (1 + t)A[l−1] + W[l] + δ0 (1 + t) W[l−1] Combining then the estimates (11.547), (11.554), for the contributions of the two terms on the right in (11.546) we conclude that: L /T (ξ · d/x i )2,[l−1], ε0 t

≤ Cl ξ 2,[1,l], ε0 + ξ ∞,[1,l∗ −1], ε0 (1 + t)−1 B[0,l] t

(11.555)

t

+Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] Q



Part 1: Control of the spatial derivatives of the first derivatives of the x i .

575

Consider finally: (L /T )k (ξ · d/x i )2,[l−k], ε0 t

: for 2 ≤ k ≤ m

We have: /T )k ξ ) · d/x i + k((L /T )k−1 ξ ) · d/T i (L /T )k (ξ · d/x i ) = ((L +

k 

k! ((L /T )k− j ξ ) · d/(T ) j −1 T i j !(k − j )!

j =2

(11.556)

To the first term on the right in (11.556) we can apply the argument leading to the estimate (11.545), taking (L /T )k ξ in the role of ξ . Since in the analogue of (11.539) n ≤ l − k, in the case that |s2 | ≤ l∗ + 1 does not hold we now have |s1 | ≤ l∗ − 1 − k. We thus obtain:

/T )k ξ 2,[l−k], ε0 ((L /T )k ξ ) · d/x i 2,[l−k], ε0 ≤ Cl (L t

(11.557)  Q Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] t

+(L /T )k ξ ∞,[l∗ −1−k], ε0 t

To the second term on the right in (11.556) we can apply the argument leading to the estimate (11.554), with (L /T )k−1 ξ in the role of ξ . Taking into account the fact that we are now taking the  2,[l−k], ε0 norm instead of the  2,[l−1], ε0 norm, we obtain: t

t

((L /T )k−1 ξ ) · d/T i 2,[l−1], ε0 t

−1 ≤ Cl (1 + t) [1 + log(1 + t)](L /T )k−1 ξ 2,[l−k], ε0

(11.558)

t

+ (L /T )

k−1

ξ

ε ∞,[l∗ −k],t 0

  Q × B[0,l] + [1 + log(1 + t)] Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1]

Consider finally a term: ((L /T )k− j ξ ) · d/(T ) j −1 T i

:2≤ j ≤k

in the sum in (11.556). Using Proposition 10.1 with l∗ + 1 in the role of l, Proposition 11.1 with l∗ in the role of l and assumption M[k−1,l∗ +1] we obtain: ((L /T )k− j ξ ) · d/(T ) j −1 T i 2,[l−k], ε0 ≤ t

−1 δ0 [1 + log(1 + t)](L Cl (1 + t) /T )k− j ξ 2,[l−k], ε0 t

+(L /T ) : for 2 ≤ j ≤ k

k− j

ξ

ε ∞,[l∗ −1−k+ j ],t 0

(T )

j −1

T i 2,[l+1−k], ε0



t

(11.559)

576

Chapter 11. Control of the Spatial Derivatives

We have: (T ) j −1 T i = ((T ) j −1κ)Tˆ i +

j −1  p=1

( j − 1)! ((T ) j −1− p κ)((T ) p Tˆ i ) p!( j − 1 − p)!

(11.560)

To the first term on the right in (11.560) we may apply (11.548), taking f = (T ) j −1 κ. Here we are to take the  2,[l+1−k], ε0 norm instead of the  2,[l], ε0 norm, therefore t t in the case that |s2 | ≤ l∗ does not hold in the argument leading to the proof of (11.548), we now have |s1 | ≤ l∗ + 1 − k. Since:   (T ) j −1 κ2,[l+1−k], ε0 ≤ κ − 12,[k−1,l], ε0 ≤ C B[k−1,l] + [1 + log(1 + t)]W{l} t

t

and, by assumption M[k−1,l∗ ] : (T ) j −1 κ∞,[l∗ +1−k], ε0 ≤ κ − 1∞,[k−1,l∗ ], ε0 ≤ Cl δ0 [1 + log(1 + t)] t

t

we then obtain (noting that k ≥ 2):  ((T ) j −1 κ)Tˆ i 2,[l+1−k], ε0 ≤ Cl B[k−1,l] + [1 + log(1 + t)]W{l} t

+δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−2] + δ02 (1 + t)−2 W[l−2] Q

: for 2 ≤ j ≤ k



(11.561)

On the other hand by Proposition 11.2, even with l reduced to l −1, together with assumption M[k−2,l∗ ] and Proposition 11.1, even with l∗ − 1 in the role of l, we can estimate: 0 0 0 0 0 j −1 0 ( j − 1)! j −1− p p ˆi 0 0 ((T ) κ)((T ) T )0 0 p!( j − 1 − p)! 0 p=1 0 ε 2,[l+1−k],t 0   j −1    ≤ Cl [1 + log(1 + t)] δ0 (1 + t)−1 κ − 12,[ j −2,l−1], ε0 + (T ) p Tˆ i 2,[l+1−k], ε0 t t   p=1    ≤ Cl (1 + t)−1 [1 + log(1 + t)] B[k−2,l] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−2]   Q + [1 + log(1 + t)] W{l} + δ0 (1 + t)−2 [1 + log(1 + t)]2 W{l−2} : for 2 ≤ j ≤ k

(11.562)

Combining the results (11.561) and (11.562) yields: (T ) j −1 T i 2,[l+1−k], ε0 ≤ t  Cl B[k−1,l] + [1 + log(1 + t)]W{l} +δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−2] + (1 + t)−2 W{l−2} Q

: for 2 ≤ j ≤ k

 (11.563)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

Substituting this in (11.559) we obtain: 0 0 0 0 k 0 0 k! k− j j −1 i 0 0 ((L /T ) ξ ) · d/(T ) T 0 0 j !(k − j )! 0 0 j =2

577

(11.564)

ε 2,[l−k],t 0

≤ Cl (1 + t)−1 δ0 [1 + log(1 + t)]ξ 2,[k−2,l−2], ε0 t 1 + ξ ∞,[k−2,l∗ −1], ε0 B[k−1,l] + [1 + log(1 + t)]W{l} t   Q +δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−2] + (1 + t)−2 W{l−2}

Combining the estimates (11.557), (11.558), (11.564), for the three terms on the right in (11.556) we conclude that:

(L /T )k (ξ · d/x i )2,[l−k], ε0 ≤ Cl ξ 2,[k,l], ε0 t

t

+ ξ ∞,[k,l∗ −1], ε0 (1 + t)

−1

B[k−1,l] + Y0 + (1 + t)A[l−1]  Q +W{l} + δ0 (1 + t)−2 W{l−1} t

: for 2 ≤ k ≤ m

(11.565)

Combining finally the estimates (11.545), (11.555), (11.565), yields the lemma. Lemma 11.19 Under the assumptions of Proposition 11.2, for an arbitrary St,u tensorfield ξ we have:

ξ · D / 2 x i 2,[m,l], ε0 ≤ Cl (1 + t)−1 ξ 2,[m,l], ε0 t t

−1 + ξ ∞,[m,l∗ ], ε0 (1 + t) B[m−1,l+1] + Y0 + (1 + t)A[l] t  Q +W{l+1} + (1 + t)−2 [1 + log(1 + t)]2 W{l} Proof. Consider first: ξ · D / 2 x i 2,[l], ε0 t

We have: / Ri1 (ξ · D /2x i ) = L / R in . . . L





 (L / R )s1 ξ · ((L / R )s 2 D /2x i )

(11.566)

partitions |s1 |+|s2 |=n≤l

To treat the factor (L / R )s 2 D / 2 x i , consider: / R i1 D /2x i L / R in . . . L We are afterwards going to replace the index set {1, . . . , n} by its subset s2 . By Lemma 11.9 with m = 0 we have: / R i1 D /2x i = D / 2 (Rin . . . Ri1 x i ) + L / R in . . . L

(i1 ...in )

c0,n [d/x i ]

(11.567)

578

Chapter 11. Control of the Spatial Derivatives

(i1 ...in )

c0,n [d/x i ] = −

n−1  j =0

L / R in . . . L / Rin− j +1

(Rin− j )

π /1 · d/ Rin− j −1 . . . Ri1 x i

 (11.568)

Now, by hypothesis H1 we have, pointwise: |D / 2 (Rin . . . Ri1 x i )| ≤ C(1 + t)−1 max |d/(R j Rin . . . Ri1 x i | j  ≤ C(1 + t)−1 max |d/ (n+1) x ii1 ...in j | + max |d/ j

j

(11.569) (n+1) i δi1 ...in j |

(see (11.540)). Applying Proposition 10.1, Corollary 10.1.a, with l∗ + 1 in the role of l, we then obtain: max max D / 2 (Rin . . . Ri1 x i ) L ∞ ( ε0 ) ≤ C(1 + t)−1 n≤l∗ i1 ...in

(11.570)

t

provided that δ0 is suitably small (depending on l). Using also Lemma 10.11 with l∗ in the role of l we deduce: max max 

(i1 ...in )

n≤l∗ i1 ...in

c0,n [d/x i ] L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

(11.571)

t

Combining, we conclude through (11.567) that: D / 2 x i ∞,[l∗ ], ε0 ≤ C(1 + t)−1

(11.572)

t

provided that δ0 is suitably small (depending on l). It follows that if in the sum on the right in (11.566) we have |s2 | ≤ l∗ , then the corresponding term is bounded in L 2 by: C(1 + t)−1 ξ 2,[l], ε0

(11.573)

t

Otherwise, |s2 | ≥ l∗ + 1, hence |s1 | ≤ l∗ . Consider, in this case, the contribution of the / R )s 2 D / 2 x i . In regard to this term we have (11.569) with term D / 2 (R)s2 x i in the factor (L i {1, . . . , n} replaced by s2 . To estimate the contribution of C(1 + t)−1 max j |d/ (|s2 |+1) x ... j| ∞ s 2 2 1 we place it in L and the factor (L / R ) ξ in L , obtaining an L bound as in (11.573). i 2 To estimate the contribution of C(1 + t)−1 max j |d/ (|s2 |+1) δ... j | we place it in L using Proposition 10.2, Corollary 10.2.a, with l + 1 in the role of l and the factor (L / R )s1 ξ in ∞ 2 L , obtaining an L bound by:  Q (11.574) Cl (1 + t)−1 ξ ∞,[l∗ ], ε0 Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l] t

On the other hand, using also Lemma 10.12 we deduce the following L 2 estimate for the commutator (11.568): max max  n≤l i1 ...in

(i1 ...in )

c0,n [d/x i ] L 2 ( ε0 )

(11.575)

t

≤ Cl (1 + t)−1 Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l] Q



Part 1: Control of the spatial derivatives of the first derivatives of the x i .

579

It follows that the contribution of the commutator term in (L / R )s 2 D / 2 x i to the correspond2 ing term in the sum in (11.566) is bounded in L as in (11.574). Combining then the results for the two cases, we conclude that:

ξ · D / 2 x i 2,[l], ε0 ≤ Cl (1 + t)−1 ξ 2,[l], ε0 + (11.576) t t  Q ξ ∞,[l∗ ], ε0 Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l] t

Consider next: / 2 x i )2,[l−1], ε0 L /T (ξ · D t

We have: L /T (ξ · D / 2 x i ) = (L /T ξ ) · D /2x i + ξ · L /T D /2x i

(11.577)

To the first term on the right in (11.577) we can apply the above, taking L /T ξ in the role of ξ . Since in the analogue of (11.566) n ≤ l − 1, in the case that |s2 | ≥ l∗ + 1 we now have |s1 | ≤ l∗ − 1. We thus obtain:

/ 2 x i 2,[l−1], ε0 ≤ Cl (1 + t)−1 L (11.578) /T ξ 2,[l−1], ε0 + (L /T ξ ) · D t t  Q L /T ξ ∞,[l∗ −1], ε0 Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l] t

To treat the second term on the right in (11.577) we consider: / R i1 L /T D /2x i L / R in . . . L

: for n ≤ l − 1

By Lemma 11.9 with m = 1 we have: L / R in . . . L / R i1 L /T D /2x i = D / 2 (Rin . . . Ri1 T i ) +

(i1 ...in )

c1,n [d/x i ]

(11.579)

where: c1,0 [d/x i ] = − (i1 ...in )

(T )

π /1 · d/x i

i

(11.580) i

c1,n [d/x ] = L / R in . . . L / Ri1 c1,0 [d/x ] −

n−1  j =0

(11.581)

L / R in . . . L / Rin− j +1

(Rin− j )

π /1 · d/ Rin− j −1 . . . Ri1 T i



Now, by hypotheses H1 and H0 we have, pointwise: |D / 2 (Rin . . . Ri1 T i )| ≤ C(1 + t)−1 max |d/(R j Rin . . . Ri1 T i | j

≤ C(1 + t)

−2

max |R j2 R j1 Rin . . . Ri1 T i | j1 , j2

(11.582)

or, writing T i = κ Tˆ i = Tˆ i + (κ − 1)Tˆ i , |D / 2 (Rin . . . Ri1 T i )| (11.583)  −2 i i ˆ ˆ ≤ C(1 + t) max |R j2 R j1 Rin . . . Ri1 T | + max |R j2 R j1 Rin . . . Ri1 ((κ − 1)T )| j1 , j2

j1 , j2

580

Chapter 11. Control of the Spatial Derivatives

Appealing to Proposition 10.1 with l∗ + 1 in the role of l and to assumption M[0,l∗ +1] we then obtain: / 2 (Rin . . . Ri1 T i ) L ∞ ( ε0 ) ≤ C(1 + t)−2 [1 + log(1 + t)] max max D

n≤l∗ −1 i1 ...in

t

(11.584)

provided that δ0 is suitably small (depending on l). Also, by Corollary 11.1.c with l∗ in the role of l, / R in . . . L / Ri1 c[1,0] [d/x i ] L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] (11.585) max max L

n≤l∗ −1 i1 ...in

t

and by Lemma 10.11 with l∗ in the role of l the sum on the right in (11.581) is bounded ε in L ∞ (t 0 ) by: Cl δ0 (1 + t)−3 [1 + log(1 + t)]2 Combining we obtain: max max 

n≤l∗ −1 i1 ...in

(i1 ...in )

c1,n [d/x i ] L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.586)

Combining then the estimates (11.584) and (11.586) yields, through (11.579), L /T D / 2 x i ∞,[l∗ −1], ε0 ≤ C(1 + t)−2 [1 + log(1 + t)]

(11.587)

t

provided that δ0 is suitably small (depending on l). Writing:  / Ri1 (ξ · L /T D /2x i ) = ((L / R )s1 ξ ) · ((L / R )s 2 L /T D /2x i ) L / R in . . . L

(11.588)

partitions |s1 |+|s2 |=n≤l−1

the estimate (11.587) applies in the case that |s2 | ≤ l∗ − 1. We then obtain an L 2 bound for the corresponding term in the sum in (11.588) by: C(1 + t)−2 [1 + log(1 + t)]ξ 2,[l−1], ε0

(11.589)

t

Otherwise, |s2 | ≥ l∗ , hence |s1 | ≤ n − l∗ ≤ l∗ . Consider, in this case, the contribution of / R )s 2 L /T D / 2 x i . In regard to this term we have (11.583) the term D / 2 (R)s2 T i in the factor (L with {1, . . . , n} replaced by s2 . To estimate the contribution of C(1 + t)−2 max |R j2 R j1 (R)s2 Tˆ i | j1 , j2

we express, as in (11.550), R j2 R j1 (R)s2 Tˆ i = −(1 − u + η0 t)−1 +(1 − u +

(|s2 |+2) i x ... j1 j2 −1 (|s2 |+2) i η0 t) δ... j1 j2

(11.590) + R j2 R j1 (R) y

s2 i

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

581

To estimate the partial contribution of the first term on the right in (11.590) we place it in / R )s1 ξ in L 2 obtaining an L 2 bound by: L ∞ and the factor (L C(1 + t)−2 ξ 2,[l−1], ε0

(11.591)

t

To estimate the contribution of the other two terms on the right in (11.590) we place them in L 2 using Proposition 10.2 and Corollary 10.2.a with l + 1 in the role of l, noting that |s2 | + 2 ≤ n + 2 ≤ l + 1, and the factor (L / R )s1 ξ in L ∞ obtaining an L 2 bound by: Q Cl (1 + t)−2 ξ ∞,[l∗ ], ε0 Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l] t

 (11.592)

On the other hand, the contribution of C(1 + t)−2 max |R j2 R j1 (R)s2 ((κ − 1)Tˆ i )| j1 , j2

is bounded by: (11.593) C(1 + t)−2 ξ ∞,[l∗ ], ε0 (κ − 1)Tˆ i 2,[l+1], ε0 t t 1 ≤ Cl (1 + t)−2 ξ ∞,[l∗ ], ε0 B[0,l+1] + [1 + log(1 + t)]W[l+1] t   Q +δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W[l] using (11.548) with l replaced by l + 1, applied to f = κ − 1. Combining the results (11.591)–(11.593) we obtain that, in the case |s2 | ≥ l∗ , the contribution of the term D / 2 (R)s2 T i in the factor (L / R )s 2 L /T D / 2 x i to the corresponding term in the sum in (11.588) 2 is bounded in L by: C(1 + t)−2 ξ 2,[l−1], ε0

 ξ ∞,[l∗ ], ε0 B[0,l+1] t

+Cl (1 + t)

−2

t

(11.594) 

Q +[1 + log(1 + t)] Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l]

To estimate the contribution of the commutator, given by (11.580), (11.581) with {1, . . . , n} replaced by s2 , we first apply the estimate (11.545) with (T ) π /1 in the role of ξ and l reduced to l − 1, to obtain, using Corollary 11.2.c as well as Corollary 11.1.c with l∗ in the role of l,  c1,0 [d/x i ]2,[l−1], ε0 ≤ Cl (1 + t)−2 B[0,l+1] + [1 + log(1 + t)] · .

(11.595)  Q Y0 + (1 + t)A[l] + W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l}

t

To estimate the sum on the right in (11.581) we follow the approach leading to the estimate (11.554), using Lemma 10.12 with l reduced to l − 1 as well as Lemma 10.11 with

582

Chapter 11. Control of the Spatial Derivatives

l∗ − 1 in the role of l. Recalling that here n ≤ l − 1, we obtain that the sum in question is bounded in L 2 by:

Cl (1 + t)−2 [1 + log(1 + t)] δ0 (1 + t)−1 B[0,l−1]  Q (11.596) +Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] Combining the results (11.595) and (11.596) we conclude that: max max 

n≤l−1 i1 ...in

(i1 ...in )

c1,n [d/x i ] L 2 ( ε0 ) t

1  ≤ Cl (1 + t)−2 B[0,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] +W{l+1} + (1 + t)

−1

[1 +

Q log(1 + t)]W{l}

(11.597) 

It follows that, in the case |s2 | ≥ l∗ , the contribution of the commutator term in the factor (L / R )s 2 L /T D / 2 x i to the corresponding term in the sum in (11.588) is bounded in L 2 by:  (11.598) Cl (1 + t)−2 ξ ∞,[l∗ ], ε0 B[0,l+1] + [1 + log(1 + t)]· t  Q · Y0 + (1 + t)A[l] + W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} Combining then the results (11.594) and (11.598), for the case |s2 | ≥ l∗ , with the earlier result (11.589), for the case |s2 | ≤ l∗ − 1, yields: ξ · L /T D / 2 x i 2,[l−1], ε0 (11.599) t

≤ Cl (1 + t)−2 [1 + log(1 + t)]ξ 2,[l−1], ε0 t 1  + ξ ∞,[l∗ ], ε0 B[0,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] t  Q +W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} This is the estimate for the second term on the right in (11.577). Combining, in turn, this with the estimate (11.578) for the first term, we conclude that:

L /T (ξ · D / 2 x i )2,[l−1], ε0 ≤ Cl (1 + t)−1 ξ 2,[1,l], ε0 (11.600) t

t

+ ξ ∞,[1,l∗ ], ε0 (1 + t) t

−1

B[0,l+1] + Y0 + (1 + t)A[l]

Q +W{l+1} + (1 + t)−2 [1 + log(1 + t)]2 W{l}



Consider finally: / 2 x i )2,[l−k], ε0 (L /T )k (ξ · D t

: for 2 ≤ k ≤ m

We have: / 2 x i ) = ((L / T )k ξ ) · D / 2 x i + k((L /T )k−1 ξ ) · L /T D /2x i (L /T )k (ξ · D +

k  j =2

k! ((L /T )k− j ξ ) · (L /T ) j D /2x i j !(k − j )!

(11.601)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

583

To the first term on the right in (11.601) we can apply the argument leading to the estimate (11.576), taking (L /T )k ξ in the role of ξ . Since in the analogue of (11.566) n ≤ l − k, in the case that |s2 | ≥ l∗ + 1 we now have |s1 | ≤ l∗ − k. We thus obtain:

((L / T )k ξ ) · D / 2 x i 2,[l−k], ε0 ≤ Cl (1 + t)−1 (L (11.602) /T )k ξ 2,[l−k], ε0 t t  Q +(L /T )k ξ ∞,[l∗ −k], ε0 Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l] t

To the second term on the right in (11.601) we can apply the argument leading to the estimate (11.599), with (L /T )k−1 ξ in the role of ξ . Taking into account the fact that we are now taking the  2,[l−k], ε0 norm instead of the  2,[l−1], ε0 norm, we obtain: t

(L /T )

k−1

t

2 i

ξ ·L /T D / x 2,[l−k], ε0 (11.603) t

≤ Cl (1 + t)−2 [1 + log(1 + t)](L /T )k−1 ξ 2,[l−k], ε0 t 1  k−1 ε + (L /T ) ξ ∞,[l∗ +1−k], 0 B[0,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] t  Q +W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l}

Consider finally a term: ((L /T )k− j ξ ) · (L /T ) j D /2x i

:2≤ j ≤k

in the sum in (11.601). To estimate the  2,[l−k], ε0 norm of this we apply L / R in . . . L / R i1 t for n ≤ l − k, obtaining:  ((L / R )s1 (L /T )k− j ξ ) · ((L / R )s2 (L /T ) j D /2x i ) (11.604) partitions

|s1 |+|s2 |=n≤l−k

To treat the second factor we consider: L / R in . . . L / Ri1 (L /T ) j D /2x i

: for 2 ≤ j ≤ k

We are afterwards going to replace the index set {1, . . . , n} by its subset s2 . By Lemma 11.9 we have: / Ri1 (L /T ) j D /2x i = D / 2 (Rin . . . Ri1 (T ) j −1 T i ) + L / R in . . . L

(i1 ...in )

c j,n [d/x i ]

(11.605)

where: c j,0 [d/x i ] = −

j −1 

(L / T )q

(T )

π /1 · d/(T ) j −1−q x i



q=0

=−

j −1  q=0

Nq, j −1 (L /T ) j −1−q

(T )

π /1 · d/(T )q x i

(11.606)

584

Chapter 11. Control of the Spatial Derivatives

= −(L /T ) j −1 +

j −1 

(T )

π /1 · d/x i − j (L /T ) j −2

Nq, j −1 (L /T ) j −1−q

(T )

(T )

π /1 · d/T i

π /1 · d/(T )q−1 T i

q=2 (i1 ...in )

c j,n [d/x ] = L / R in . . . L / Rin c j,0 [d/x i ] i

−

n−1 

(11.607)

L / R in . . . L / Rin− p+1

(Rin− p )

π /1 · d/ Rin− p−1 . . . Ri1 (T ) j −1 T i



p=0

In (11.606) the coefficients Nq, j −1 are the positive integers defined by (11.186). We first take n ≤ l∗ − j in (11.605) to derive an estimate for: / 2 x i ∞,[l∗ − j ], ε0 (L /T ) j D t

:2≤ j ≤k

Consider the first term on the right in (11.605). By hypotheses H1 and H0: max max D / 2 (Rin . . . Ri1 (T ) j −1 T i  L ∞ ( ε0 ) ≤ C(1 + t)−2 (T ) j −1 T i 2,[l∗ +2− j ], ε0

n≤l∗ − j i1 ...in

t

t

≤ Cl δ0 (1 + t)

−2

[1 + log(1 + t)] (11.608)

using Propositions 10.1 and 11.1 with l∗ in the role of l and assumption M[k−1,l∗ +1] . Consider next the first term on the right in (11.607). From (11.606) we deduce, using Corollary 11.1.c with l∗ in the role of l, together with Propositions 10.1 and 11.1, even with l∗ − 1 in the role of l and l∗ − 2 in the role of l respectively, as well as assumption M[k−2,l∗ −1] : c j,0 [d/x i ]∞,[l∗ − j ], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.609)

Consider then the sum on the right in (11.607). Using Lemma 10.11 even with l∗ − 2 in the role of l, together with Propositions 10.1 and 11.1, even with l∗ −2 in the role of l, and ε assumption M[k−1,l∗ −1] , we deduce that the sum in question is bounded in L ∞ (t 0 ) by: Cl δ02 (1 + t)−3 [1 + log(1 + t)]2

(11.610)

Combining the last two results yields: max max 

n≤l∗ − j i1 ...in

(i1 ...in )

c j,n [d/x i ] L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.611)

Combining this, in turn, with (11.608), we conclude, through (11.605) that: (L /T ) j D / 2 x i ∞,[l∗ − j ], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

:2≤ j ≤k

Next, we take n ≤ l − j in (11.605) to derive an estimate for: (L /T ) j D / 2 x i 2,[l− j ], ε0 t

:2≤ j ≤k

(11.612)

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

585 ε

By hypotheses H1 and H0 the first term on the right in (11.605) is bounded in L 2 (t 0 ) by: C(1 + t)−2 (T ) j −1 T i 2,[l+2− j ], ε0 t  −2 B[ j −1,l+1] + [1 + log(1 + t)]W{l+1} ≤ Cl (1 + t) +δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] + (1 + t)−2 W{l−1} Q



: for 2 ≤ j ≤ k

(11.613)

where we have applied (11.563) with k set equal to j and l increased to l + 1 (this relies only on the assumptions of Proposition 11.2). Consider next the first term on the right in (11.606). We may apply to this term the /1 · d/x i in the role of ξ . In view of the fact that we estimate (11.545) taking (L /T ) j −1 (T ) π ε are considering the  2,[l− j ], 0 norm instead of the  2,[l], ε0 norm, we may replace t t ξ ∞,[l∗ −1], ε0 on the right by ξ 2,[l∗ −1− j ], ε0 , obtaining: t

t

j −1 (T )

(L /T ) π /1 · d/x 2,[l− j ], ε0 t

j −1 (T ) /T ) π /1 2,[l− j ], ε0 + (L /T ) j −1 ≤ Cl (L i

(11.614) (T )

π /1 ∞,[l∗ −1− j ], ε0 · t  Q 2 −2 Y0 + (1 + t)A[l−1] + W[l] + δ0 (1 + t) W[l−1] t

Since (L /T ) j −1 (L /T )

j −1 (T )

(T )

π /1 2,[l− j ], ε0 ≤ 

(T )

π /1 ∞,[l∗ −1− j ], ε0 ≤ 

(T )

t

π /1 2,[ j −1,l−1], ε0 , t

t

π /1 ∞,[ j −1,l∗ −2], ε0 , t

substituting from Corollary 11.1,c, with l∗ − 1 in the role of l, and Corollary 11.2.c, we then obtain: (L /T ) j −1

(T )

π /1 · d/x i 2,[l− j ], ε0 t  1 −2 B[ j −1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] ≤ Cl (1 + t)

(11.615)

+W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} Q



Consider next the second term on the right in (11.606). We may apply to this term the estimate (11.554) taking (L /T ) j −2 (T ) π /1 · d/x i in the role of ξ . In view of the fact that we are considering the  2,[l− j ], ε0 norm instead of the  2,[l−1], ε0 norm, we may t t replace ξ ∞,[l∗ −1], ε0 on the right by ξ 2,[l∗ − j ], ε0 , obtaining: t

(L /T )

j −2 (T )

t

π /1 · d/T 2,[l− j ], ε0 t

/T ) j −2 (T ) π ≤ Cl (1 + t)−1 [1 + log(1 + t)](L /1 2,[l− j ], ε0 (11.616) t 1  + (L /T ) j −2 (T ) π /1 ∞,[l∗ − j ], ε0 B[0,l] + [1 + log(1 + t)] Y0 + (1 + t)A[l−1] t  Q +W[l] + δ02 (1 + t)−2 W[l−1] i

586

Chapter 11. Control of the Spatial Derivatives

Since (L /T ) j −2 (L /T ) j −2

(T )

(T )

π /1 2,[l− j ], ε0 ≤ 

(T )

π /1 2,[ j −2,l−2], ε0 ,

π /1 ∞,[l∗ − j ], ε0 ≤ 

(T )

π /1 ∞,[ j −2,l∗ −2], ε0 ,

t

t

t

t

substituting from Corollary 11.1,c, with l∗ − 1 in the role of l, and Corollary 11.2.c, with l reduced to l − 1, we then obtain: (L /T ) j −2

(T )

π /1 · d/T i 2,[l− j ], ε0 (11.617) t  1 −3 ≤ Cl (1 + t) [1 + log(1 + t)] B[ j −2,l] + [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q +W{l} + (1 + t)−1 [1 + log(1 + t)]W{l−1} Finally, the  2,[l− j ], ε0 norm of the third term on the right in (11.606) (the sum) is t bounded by: Cl (1 + t)−1

j −1 

(L /T ) j −1−q

(T )

π /1 2,[l− j ], ε0 (T )q−1 T i ∞,[l∗ +1−q], ε0 t

t

q=2

 +(L /T ) j −1−q (T ) π /1 ∞,[l∗ − j +q], ε0 (T )q−1 T i 2,[l+1− j ], ε0 t t  /1 2,[ j −3,l−3], ε0 max (T )q−1 T i ∞,[l∗ +1−q], ε0 ≤ Cl (1 + t)−1  (T ) π t

+

(T )

π /1 ∞,[ j −3,l∗ −1], ε0

j −1 

t

2≤q≤ j −1

t



(T )q−1 T i 2,[l+1− j ], ε0 t

(11.618)

q=2

Using then Corollary 11.2.c with l reduced to l −2 and Corollary 11.1.c with l∗ in the role of l, together with the estimate (11.563) with (q, j ) in the role of ( j, k), and the estimate: (T )q−1 T i ∞,[l∗ +1−q], ε0 ≤ Cl δ0 [1 + log(1 + t)] t

: for 2 ≤ q ≤ j

(11.619)

which follows from Propositions 10.1 and 11.1 with l∗ − 1 in the role of l together with assumption M[ j −1,l∗ ] , we obtain that the  2,[l− j ], ε0 norm of the third term on the t right in (11.606) (the sum) is bounded by:

Cl δ0 (1 + t)−3 [1 + log(1 + t)] B[ j −1,l] + [1 + log(1 + t)] Y0 + (1 + t)A[l−2]  Q (11.620) + W{l} + (1 + t)−1 [1 + log(1 + t)]W{l−2} Combining the results (11.615), (11.617), (11.620), for the three terms on the right in (11.606), we conclude that, for 2 ≤ j ≤ k:

c j,0 [d/x i ]2,[l− j ], ε0 ≤ Cl (1 + t)−2 B[ j −1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] t  Q (11.621) + W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} ε

This is the bound in L 2 (t 0 ) for the first term on the right in (11.607), for n ≤ l − j .

Part 1: Control of the spatial derivatives of the first derivatives of the x i .

587

Consider finally the sum on the right in (11.607). For n ≤ l − j this sum is bounded ε in L 2 (t 0 ) by:  /1 2,[l−1− j ], ε0 (T ) j −1 T i ∞,[l∗ +1− j ], ε0 Cl (1 + t)−1 max  (Ri ) π t t i + max  (Ri ) π /1 ∞,[l∗ −1], ε0 (T ) j −1 T i 2,[l− j ], ε0 (11.622) t

i

t

Using then Lemma 10.12 with l reduced to l − j and Lemma 10.11 with l∗ in the role of l, together with the estimate (11.563) with l reduced to l − 1 and k set equal to j and the ε estimate (11.619) for q = j , we obtain that the sum in question is bounded in L 2 (t 0 ) by:

Cl δ0 (1 + t)−2 [1 + log(1 + t)] (1 + t)−1 B[ j −1,l−1] + Y0 + (1 + t)A[l−2]  Q +W{l−1} + δ0 (1 + t)−2 W{l−2} (11.623) Combining this with (11.621) yields the following estimate for the commutator (11.607): max max 

(i1 ...in )

n≤l− j i1 ...in

c j,n [d/x i ] L 2 ( ε0 ) t

 1 ≤ Cl (1 + t)−2 B[ j −1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] +W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} Q

: for 2 ≤ j ≤ k

 (11.624)

Combining (11.624), in turn, with the estimate (11.613) we conclude through (11.605) that: / 2 x i 2,[l− j ], ε0 (L /T ) j D t  1 ≤ Cl (1 + t)−2 B[ j −1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] +W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} Q

: for 2 ≤ j ≤ k

 (11.625)

Going back to (11.604), if |s2 | ≤ l∗ − j we can place the second factor in L ∞ appealing to the estimate (11.612). Otherwise, we have |s1 | ≤ l∗ − k + j and we place the first factor in L ∞ , noting that, in this case: /T )k− j ξ  L ∞ ( ε0 ) ≤ ξ ∞,[k−2,l∗ ], ε0 (L / R )s1 (L t

t

We thus obtain, using also the estimate (11.625): (L /T )k− j ξ · (L /T ) j D / 2 x i 2,l−k, ε0 t

≤ Cl ξ 2,[k−2,l−2], ε0 (L /T ) j D / 2 x i ∞,[l∗ − j ], ε0 t t  j 2 i +ξ ∞,[k−2,l∗ ], ε0 (L /T ) D / x 2,[l− j ], ε0 t

t

588

Chapter 11. Control of the Spatial Derivatives

≤ Cl (1 + t)−2 δ0 [1 + log(1 + t)]ξ 2,[k−2,l−2], ε0 t 1 + ξ ∞,[k−2,l∗ ], ε0 B[k−1,l+1] + [1 + log(1 + t)]· t   Q Y0 + (1 + t)A[l] + W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} : for 2 ≤ j ≤ k

(11.626)

The sum in (11.601) then satisfies a similar estimate. Combining this with the estimates (11.602) and (11.603) for the first two terms on the right in (11.601) we conclude that:

(L /T )k (ξ · D / 2 x i )2,[l−k], ε0 ≤ Cl (1 + t)−1 ξ 2,[k,l], ε0 t

+ ξ ∞,[k,l∗ ], ε0 (1 + t) t

t

−1

B[k−1,l+1] + Y0 + (1 + t)A[l]

+W{l+1} + (1 + t)−2 [1 + log(1 + t)]2 W{l} Q

: for all 2 ≤ k ≤ m

 (11.627)

Combining finally the estimates (11.576), (11.600) and (11.627) yields the lemma.

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

Part 2: Bounds for the quantities

(i 1 ...i l )

589

Q m,l and

(i 1 ...i l )

 Pm,l

Our objective in the remainder of the present chapter is to obtain appropriate bounds  , which, through (i1 ...il ) B  , enter the final for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l m,l  . The quantities (i1 ...il ) Q  (t) are estimates of Chapter 9 for the functions (i1 ...il ) x m,l m,l defined by equation (9.284) of Chapter 9, the principal term in that formula being the last term:  (t) L 2 ([0,ε0 ]×S 2 )  (i1 ...il ) g˙ m,l ε

 We must thus obtain estimates for the functions (i1 ...il ) g˙ m,l in L 2 (t 0 ). Now, the func  are defined in terms of the functions (i1 ...il ) gm,l by equations (9.270), tions (i1 ...il ) g˙ m,l  (i ...i ) l (9.271). The functions 1 gm,l are in turn defined in the statements of Propositions 9.1  and 9.2. The first term in the resulting expression for (i1 ...il ) gm,l is the term:

Ril . . . Ri1 (T )m gˇ  where gˇ  is the function defined by equation (9.68) of Chapter 9. Note that for m = l = 0 we have:  = gˇ  (11.628) g0,0 (see (9.98)), while according to (9.270) with l = 0:   g˙ 0,0 = g0,0 − ξ · x0

(11.629)

where ξ is the St,u 1-form defined by equation (9.83) of Chapter 9. To obtain estimates  it is more appropriate to consider in place of gˇ  a function gˇ˙  for the functions (i1 ...il ) g˙ m,l defined so that we have:  = gˇ˙  (11.630) g˙ 0,0 in analogy with (11.628). Recalling from Chapter 8 that: x 0 = µd/trχ + d/ fˇ, in view of (11.629) the appropriate definition is: gˇ˙  = gˇ  − ξ $ · (µd/trχ + d/ fˇ)

(11.631)

Here we have introduced the St,u -tangential vectorfield ξ $ corresponding to the St,u 1form ξ : ξ $ = ξ · h/−1 (11.632) so that the dependence on the induced acoustical metric h/ is made explicit and the formulas contain only contractions. According to the statement concluding with (9.82) in Chapter 9, the function gˇ˙  does not contain a principal acoustical part, that is an acoustical part of order 2. Our objective in the following shall be to estimate: gˇ˙  2,[m,l], ε0 t

590

Chapter 11. Control of the Spatial Derivatives

Substituting in (11.631) for gˇ  from equation (9.68) of Chapter 9 we obtain: gˇ˙  = −ξ $ · d/ fˇ − 2µ(d/µ)$ · (i − d/e) 1 d log  1 (Lσ ) f 0 − (µtrχ + 2m + 2µe)µf 1 + 2 dσ 2 +n˙ 0 + µ(n˙ 1 + n˙ 2 )

(11.633)

Here, the function n˙ 0 is defined by a formula analogous to (9.51) for n 0 , but with the function τ¨ , to be presently defined, in the role of τ˙  , that is:   1  2 dH n˙ 0 = − L ψ L (LT σ ) 2 dσ    1 dH  + ψ L2 ν(LT σ ) − 2ζ · d/T σ + τ¨ (11.634) 2 dσ The function τ¨ is defined by a formula analogous to (9.39) for τ˙  but with the function v˙  = v  + κ 2 (d/σ )$ · d/trχ in the role of v  , that is:

τ¨ = −1 (v˙  + T τ˜ +

(T )

δ τ˜ )

(11.635)

(11.636)

According to the statement concluding with (9.35) in Chapter 9, the function v˙  does not contain principal acoustical terms, and according to the statement concluding with (9.52) neither does τ¨ nor n˙ 0 . In (11.633), the function n˙ 1 is defined by: n˙ 1 = n 1 − ψ L

dH (T σ ) ψ$ · d/trχ dσ

(11.637)

the function n 1 being given by (9.53). According to the statement concluding with (9.80), the function n˙ 1 does not contain principal acoustical terms. Finally, in (11.633), the function n˙ 2 is defined by a formula analogous to (9.65) for n 2 , that is: n˙ 2 = µn˙ 2 +

  dH 1 L µψ L ψ Lˆ / ψµ
/ σ + L(µFψ0 L µ )
2 dσ

(11.638)

The function n˙ 2 which is here in the role of the function n 2 given by (9.60), is defined by: $  dH dH 1 −1 (Lσ ) ψ + 2Fψ0 ω d/σ · d/trχ n˙ 2 = n 2 + α ψTˆ / L + ψ L ψ Lˆ dσ 2 dσ

(11.639)

According to the statement concluding with (9.81) in Chapter 9, the function n˙ 2 does not contain principal acoustical terms and neither does n˙ 2 . Now, the function v  is defined by equation (9.34) of Chapter 9 in terms of the function v: /µ v  = v − (T σ )


 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

We thus have:

/ µ + κ 2 (d/σ )$ · d/trχ v˙  = v − (T σ )


591

(11.640)

The function v is given by equations (9.18)–(9.24) of Chapter 9. We have: v = v1 + v2 + v3

(11.641)

where v1 , v2 and v3 are given by equations (9.19), (9.20) and (9.21)–(9.24) respectively. Let us define: v˙1 = −1 v1 v˙2 = −1 v2 − (T σ )
/ µ + κ 2 (d/σ )$ · d/trχ −1 v˙3 =  v3

(11.642)

v˙ = v˙1 + v˙2 + v˙3

(11.643)

and: Then according to the statement concluding with (9.33) in Chapter 9, the functions v˙1 , v˙2 and v˙3 contain no principal acoustical parts, and we have: v˙  = v˙

(11.644)

Our first objective shall be to estimate: v ˙ 2,[m,l], ε0 t

We first derive explicit expressions for the functions v˙1 , v˙2 , v˙3 , in terms of the acoustical quantities, the derivatives of σ and the components of ψ and ω. Substituting in (9.19) for the components of (T ) π˜ from table 6.12 of Chapter 6 we obtain: v˙1 = (ν − α −1 κν)(L Lσ + ν Lσ ) + 2(T (α −1 κ))(L)2 σ − · d/ Lσ − α −1 κ · d/ Lσ



/ 2 σ )$ −(T µ + µT log )
/ σ + µtr (χˆ − α −1 χˆ )$ · (D

(11.645)

In analogy with (9.25) of Chapter 9 we have: L v˙2 = v˙2L Lσ + v˙2 Lσ + v/˙ 2 · d/σ

(11.646)

/µ v˙2L = −1 v2L + (1/2)α −1 κ


(11.647)

/µ v˙2 = −1 v2 − (1/2)
−1 2 v/˙ 2 =  v/2 + κ (d/trχ)$

(11.648)

where:

L

L

(11.649)

592

Chapter 11. Control of the Spatial Derivatives

According to the statements concluding with (9.29), (9.30)and (9.32) of Chapter 9, the L functions v˙2L , v˙2 and the St,u -tangential vectorfield v/˙ 2 do not contain principal acoustical terms. To obtain an explicit expression for v˙2L we substitute in (9.26) for the components of (T ) π ˜ from table 6.12 of Chapter 6. Substituting also for Lµ from the propagation equation (9.1) of Chapter 9 and for Ltrχ from the propagation equation (8.55) of Chapter 8 and noting that: T trχ = T trχ  + 2η0 (1 − u + η0 t)−2 ,    /T χ  − tr T trχ  = tr L

and that:

(T ) $

π / · χ $

 (11.650)

while, by (11.305): ( P) (N)   tr L /T χ  = tr +
/ µ + div / ζ − tr γ −tr γ

(11.651)

L L + v˙2,N v˙2L = v˙2,P

(11.652)

we find: L is the principal part, given by: where v˙2,P L v˙2,P = α −2 T m + α −1 κ(T e − 2T L log α)

+(T )2 log  + (1/2)α −1 κ T L log  +T (κtrk/) + (1/2)α −1 κ L(κtrk/) ( P)

+α −1 κtr γ −(1/2)α −2 κ 2 f 0

(11.653)

L is the lower order part, given by: while v˙2,N L v˙2,N + (1 − u + η0 t)−2 = (e − 2L log α + L log  − trχ)T (α −2 µ)

+ (1/2) · d/(α −2 µ) + mT (α −2 ) − (1/2)α −4 µtrχ {m + µ(e − 2L log α)}

 + (1/2)α −4 µ2 |χ  |2 + 2η0 (1 − u + η0 t)−1 trχ  + (1/2)(−α −1 κtrχ + κtrk/ + T log )(2T log  + α −1 κ L log )  (N) + α −1 κ tr γ +tr( (T ) π /$ · χ $ ) − tr − (1/4)λ (3 − λ )(1 − u + η0 t)−2 L

(11.654)

To obtain an explicit expression for v˙2 we substitute in (9.27) for the components of (T ) π ˜ from table 6.12 of Chapter 6. Substituting again for Lµ from the propagation equation (9.1) of Chapter 9 and for Ltrχ from the propagation equation (8.55) of Chapter 8 we find: L L L (11.655) v˙2 = v˙2,P + v˙2,N

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

593

L

where v˙2,P is the principal part, given by: L

v˙2,P = (1/2)T L log  + (1/2)L(κtrk/) −(1/2)α −1 κ f 0 + div / ζ

(11.656)

L

while v˙2,N is the lower order part, given by: 

L v˙2,N − η0 (1 − u + η0 t)−2 = (1/2)α −1 κ |χ  |2 + 2η0 (1 − u + η0 t)−1 trχ  −(1/2)α −2 trχ {m + µ(e − 2L log α)} +(1/2)(−α −1 κtrχ + κtrk/ + T log )(L log ) (11.657) −(1/2)η0 (1 − u + η0 t)−2 λ In regard to the St,u -tangential vectorfield v/˙ 2 , it is more convenient to work with the corresponding St,u 1-form (v/˙ 2 )b : (v/˙ 2 )b = v/˙ 2 · h/

(11.658)

Substituting in (9.28) for the components of (T ) π˜ from table 6.12 and taking into account the propagation equation (9.1), the Codazzi equation (8.104) of Chapter 8, and also the fact that L /L h / = 2χ (see equation (10.4) of Chapter 10), while L /T h/ = (T ) π /, we find: (v/˙ 2 )b = (v/˙ 2,P )b + (v/˙ 2,N )b

(11.659)

where (v/˙ 2,P )b is the principal part, given by: / L ζ + d/m + µd/e) + 2L /T ζ (v/˙ 2,P )b = α −1 κ (2L / (κ k/ˆ ) −µd/T log  − 2κ 2i + 2µdiv

(11.660)

while (v/˙ 2,N )b is the lower order part, given by: (v/˙ 2,N )b = 2(κk/) ·  − (T µ + µT log )d/ log 

 + (1/2)α −2 (m + µ(3e − 2L log α)) + α −1 κ L log  d/µ 

+ α −2 (m + µ(e − 2L log α)) + 2α −1 κ L log  + 2T log  ζ $

− 2χˆ $ · d/(κ 2 ) + 2(κ k/ˆ )$ · d/µ + 2µ(α −1 χˆ $ + κ k/ˆ ) · d/ log 

(11.661)

In analogy with (9.21) of Chapter 9 we have: L v˙3 = v˙3L Lσ + v˙3 Lσ + v/˙ 3 · d/σ

v˙3L = −1 v3L ,

v˙3 = −1 v3 , L

L

v/˙ = −1 v/3

(11.662)

594

Chapter 11. Control of the Spatial Derivatives

Substituting in (9.22)–(9.24) for the components of

(T ) π ˜

from table 6.12 we find:

v˙3L − 2(1 − u + η0 t)−2 = (1/2)λ (−4 + λ )(1 − u + η0 t)−2 

+ (1/2)(−2 + λ )(1 − u + η0 t) −1



−1

(−2α

−1

−1

(11.663) 

κtrχ + 3κtrk/ + T log )



+ (1/2)(−α κtrχ + 2κtrk/)(−α κtrχ + κtrk/ + T log ) + (2η0 (1 − u + η0 t)−1 + trχ  )T (α −1 κ) + (1/2) · d/(α −1 κ)

 L v˙3 = −(1/2)(L log ) −2α −1 κη0 (1 − u + η0 t)−1 − α −1 κtrχ  + κtrk/ + T log  (11.664)

v/˙ 3 = − (−2 + λ )(1 − u + η0 t)−1 − α −1 κtrχ  + 2κtrk/ +T log  + (1/2) (m + µ(e − 2L log α))} 

(11.665) L

We shall begin our estimates with v˙2 . We consider first the principal parts v˙2L , v˙2 , (v/˙ 2 )b , given by (11.653), (11.656), (11.660), respectively. The first two contain the term / L (κk/) L(κtrk/) and the last contains the term L / L ζ . We thus need to obtain estimates for L and L / L ζ , which require estimates for L / L k/ and L / L (κ −1 ζ  ). These in turn require estimates for LψTˆ , Lψ L , L / L ψ, as well as for L /L ω /, L /L ω / L , to which we presently turn. We have: (11.666) LψTˆ = ω L Tˆ + ψi L(Tˆ i ) According to equation (3.159) of Chapter 3: L(Tˆ i ) = p L Tˆ i + q L · d/x i where p L is the function given by (3.164):  1 dH p L = −ψTˆ ψ ˆ (Lσ ) + H ω L Tˆ 2 dσ T

(11.667)

(11.668)

while q L is the St,u -tangential vectorfield given by (3.169): q L = −(κ −1 ζ  )$ −

1 dH ψ ˆ ( ψ$ (Lσ ) − ψ L (d/σ )$ ) − H ψ$ ω L Tˆ 2 dσ T

(11.669)

Substituting (11.667) in (11.666) we obtain: LψTˆ = ω L Tˆ + p L ψTˆ + q L · ψ

(11.670)

Lψ L = ω L L + ψi L(L i )

(11.671)

We have: According to equation (3.144) of Chapter 3: L(L i ) = a L Tˆ i + b L · d/x i

(11.672)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

where a L is the function given by (3.153):  1 dH −1 ψ L (Lσ ) + H ω L L aL = α ψL 2 dσ

595

(11.673)

while b L is the St,u -tangential vectorfield given by (3.156): bL = −

1 dH ψ L (2 ψ$ (Lσ ) − ψ L (d/σ )$ ) − H ψ$ ω L L 2 dσ

(11.674)

Substituting (11.672) in (11.671) we obtain: Lψ L = ω L L + a L ψTˆ + b L · ψ

(11.675)

/ L , we have: In view of the fact that Lψi d/x i = ω L / L ψ = ω / L + ψi d/ L i

(11.676)

or, substituting for L i from equation (10.381) of Chapter 10, L / L ψ = ω / L + η0 (1 − u + η0 t)−1 ψ + ψi d/z i

(11.677)

Next, in view of the representation ω / L = L µ d/ψµ we have: L /L ω / L = L µ d/ Lψµ + L(L i )d/ψi

(11.678)

Substituting from (11.672) we obtain: L /L ω / L = L µ d/ Lψµ + a L ω /Tˆ + b L · ω /

(11.679)

Moreover, since ω / = d/ψi ⊗ d/x i , we have: L /L ω / = d/ Lψi ⊗ d/x i + d/ψi ⊗ d/ L i

(11.680)

or, substituting for L i from (10.381), / = d/ Lψi ⊗ d/x i + η0 (1 − u + η0 t)−1 ω / + d/ψi ⊗ d/z i L /L ω

(11.681)

Using the above formulas we deduce, as further corollaries of Propositions 11.1 and 11.2 the following. Corollary 11.1.e Under the assumptions of Proposition 11.1 we have:  p L ∞,[m,l], ε0 ≤ Cl δ02 (1 + t)−3 t

q L ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 t

a L ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 t

b L ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 t

596

Chapter 11. Control of the Spatial Derivatives

and: LψTˆ ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 t

Lψ L ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 t

L / L ψ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 t

and: L /L ω / L ∞,[m,l−1], ε0 ≤ Cl δ0 (1 + t)−3 t

L /L ω /∞,[m,l−1], ε0 ≤ Cl δ0 (1 + t)−3 t

Corollary 11.2.e Under the assumptions of Proposition 11.2 we have:

Q  p L 2,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 W{l} t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1] + W{l}

Q q L 2,[m,l], ε0 ≤ Cl (1 + t)−1 W{l+1} + δ0 (1 + t)−1 W{l} t  +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1]

Q a L 2,[m,l], ε0 ≤ Cl (1 + t)−1 W{l} t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1] + W{l}

Q b L 2,[m,l], ε0 ≤ Cl (1 + t)−1 W{l+1} + δ0 (1 + t)−1 W{l} t  +δ02 (1 + t)−2 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1]





and:

Q + δ0 (1 + t)−1 W{l+1} LψTˆ 2,[m,l], ε0 ≤ Cl (1 + t)−1 W{l} t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1]

Q Lψ L 2,[m,l], ε0 ≤ Cl (1 + t)−1 W{l} + δ0 (1 + t)−1 W{l+1} t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1]  L / L ψ2,[m,l], ε0 ≤ Cl (1 + t)−1 W{l+1}





t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] + δ0 (1 + t)−2 W{l} Q



 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

and:

597

Q / L 2,[m,l], ε0 ≤ Cl (1 + t)−2 W{l+1} + δ0 (1 + t)−1 W{l+1} L /L ω t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1]

Q L /L ω /2,[m,l], ε0 ≤ Cl (1 + t)−2 W{l+1} + W{l+1} t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]





To obtain the bound for L /L ω /2,[m,l], ε0 we have appealed to Lemma 11.18, taking Q

t

ξ = d/ Lψi to estimate, using assumption E{l∗ } :

Q d/ Lψi ⊗ d/x i 2,[m,l], ε0 ≤ Cl (1 + t)−2 W{l+1} t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1] + W{l}

(11.682) 

We now consider L / L k/ and L / L (κ −1 ζ  ). Now k/ and κ −1 ζ  are given by formulas (10.195) and (10.386) of Chapter 10 respectively. We can apply L / L to these formulas and use the estimates of the above corollaries to estimate L / L k/ and L / L (κ −1 ζ  ). However, both formulas contain, in the first term, the factor Lσ . In applying L to this factor, the factor QQ (L)2 σ results. Thus, to obtain the L ∞ estimates we must introduce the assumption E{l−1} to be presently specified. QQ Given non-negative integers m, n, we denote by Em,n the bootstrap assumption that there is a constant C independent of s such that for all t ∈ [0, s]: QQ Em,n

max Rin . . . Ri1 (T )m (Q)2 ψα  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1

:

α;i1 ...in

t

QQ

QQ

We then denote by E{l} the conjunction of the assumptions Em,n corresponding to the triangle (11.9). The constant C then depends on l only and, as usual, shall be denoted by QQ Cl to emphasize this dependence. Note that the assumptions E0,n coincide with the asQQ QQ sumptions E / nQ Q of Chapter 10, and the assumptions E{l} contain the assumptions E / [l] . QQ defined as follows. Given non-negative inteWe also introduce the quantities W{l} QQ

gers m, n, we denote by Wm,n the quantity: QQ Wm,n = max Rin . . . Ri1 (T )m (Q)2 ψα  L 2 ( ε0 ) α;i1 ...in

t

(11.683)

QQ QQ We then denote by W{l} the sum of the quantities Wm,n corresponding to the triangle

QQ (11.9). Note that the quantities W0,n coincide with the quantities WnQ Q of Chapter 10, QQ

QQ

and the quantities W{l} dominate the quantities W[l] . Moreover, we have:   QQ Q max (L)2 ψα 2,{l}, ε0 ≤ (1 + t)−2 W{l} + W{l} α

t

(11.684)

598

Chapter 11. Control of the Spatial Derivatives

Corollary 11.1.f Under the assumptions of Proposition 11.1 augmented by the assumpQQ tion E{l−1} , we have: L / L k/ ∞,[m,l−1], ε0 ≤ Cl δ0 (1 + t)−3 t

L / L (κ −1 ζ  )∞,[m,l−1], ε0 ≤ Cl δ0 (1 + t)−3 t

and: L / L (κk/)∞,[m,l−1], ε0 ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] t

L / L ζ ∞,[m,l−1], ε0 ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] t

Corollary 11.2.f Under the assumptions of Proposition 11.2 augmented by the assumption E{lQ∗Q−1} , we have:

Q QQ L / L k/ 2,[m,l], ε0 ≤ Cl (1 + t)−2 W{l+1} + δ02 (1 + t)−2 W{l} t

+W{l+1} + δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

Q QQ L / L (κ −1 ζ  )2,[m,l], ε0 ≤ Cl (1 + t)−2 W{l+1} + δ02 (1 + t)−2 W{l}



t

+δ0 (1 + t)

−1

W{l+1} + δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]



and: L / L (κk/)2,[m,l], ε0

t  Q QQ −2 [1 + log(1 + t)] W{l+1} + δ02 (1 + t)−2 W{l} + W{l+1} ≤ Cl (1 + t)  1 2 +δ0 (1 + t)−1 B[m,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] L / L ζ 2,[m,l], ε0 t

 Q QQ −2 [1 + log(1 + t)] W{l+1} + δ02 (1 + t)−2 W{l} ≤ Cl (1 + t)  1 2 +δ0 W{l+1} + δ0 (1 + t)−1 B[m,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] The estimates for L / L (κk/), L / L ζ in the above corollaries are obtained from the es/ L (κ −1 ζ ) through the relations (11.292), (11.293). Here we substitute timates for L / L k/ , L for Lκ from the propagation equation: Lκ = α −1 (m + µe ),

e = e − L log α

(11.685)

which follows from the propagation equation for µ, (9.1) of Chapter 9, and coincides with equation (3.99) of Chapter 3.

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

599

We now turn to estimate v˙2L , given by (11.653), in  2,[m,l], ε0 norm. In regard to t the first term on the right in (11.653), we obtain, using Corollary 11.2.b,  m2,[m+1,l+1], ε0 ≤ Cl W{l+2} (11.686) t  Q +δ02 (1 + t)−2 (1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l] + δ0 (1 + t)−2 W{l} To estimate the second and fifth terms on the right in (11.653) in this norm, we remark that the estimates of Corollary 11.2.b for the components of ω as well as for k/ and κ −1 ζ also hold, under the same assumptions, with (m, l) increased to (m + 1, l + 1). Moreover, under the additional assumption M[m+1,l∗ +1] , the estimates for κk/ and ζ hold with (m, l) increased to (m + 1, l + 1) as well. We thus obtain from (11.331), under the assumptions of Proposition 11.2, in regard to the second term on the right in (11.653),

Q (11.687) e 2,[m+1,l+1], ε0 ≤ Cl (1 + t)−1 W{l+1} t  +δ0 (1 + t)−1 (1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l] + W{l+1} Moreover, recalling that α is a function of the ψµ , so that: L log α =

∂α Lψµ ∂ψµ

(11.688)

we obtain, using Lemma 10.1 with l increased to l + 1 together with assumptions E{l∗ +1} , Q E{l∗ } ,   Q L log α2,[m+1,l+1], ε0 ≤ Cl (1 + t)−1 W{l+1} + δ0 (1 + t)−1 W{l+1} t

It follows that e satisfies an estimate similar to (11.687):

Q e2,[m+1,l+1], ε0 ≤ Cl (1 + t)−1 W{l+1} t

+δ0 (1 + t)−1 (1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l] + W{l+1}

(11.689)

(11.690) 

The third and fourth terms on the right in (11.653) are likewise readily estimated using Lemma 10.1 with l increased to l + 1. In regard to the fifth term on the right in (11.653) we have: L /T (κk/)2,[m,l], ε0 ≤ κk/2,[m+1,l+1], ε0 (11.691) t

t

and, under the assumptions of Proposition 11.2 augmented by the assumption M[m+1,l∗ +1] , we have the estimate of Corollary 11.2.b with (m, l) increased to (m + 1, l + 1). Writing then /$ · (κk/)$ (11.692) T (κtrk/) = tr (L /T (κk/))$ − (T ) π

600

Chapter 11. Control of the Spatial Derivatives

to estimate the first term on the right, the principal term, we apply Corollary 11.2.d, with L /T (κk/) in the role of ξ , taking also into account the fact that under the augmented assumptions Corollary 11.1.b with (m + 1, l∗ ) in the role of l holds, hence we have the L ∞ estimate: L /T (κk/)∞,[m,l∗ −1], ε0 ≤ κk/∞,[m+1,l∗ ], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

t

(11.693) The second term on the right in (11.692), a lower order term, is readily estimated using Corollaries 11.2.b, 11.2.c and 11.2.d. In this way we obtain the following estimate for the fifth term on the right in (11.653): T (κtrk/)2,[m,l], ε0 (11.694)

t  Q ≤ Cl (1 + t)−1 [1 + log(1 + t)] W{l+2} + δ0 (1 + t)−2 W{l+1} 1 2  +δ0 (1 + t)−1 B[m+1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] In regard to the sixth term on the right in (11.653) we similarly express: L(κtrk/) = tr (L / L (κk/))$ − 2χ $ · (κk/)$

(11.695)

To estimate the first term on the right, the principal term, we apply Corollary 11.2.d, with L / L (κk/) in the role of ξ , and use the L 2 estimate for this of Corollary 11.2.f as well as the L ∞ estimate from Corollary 11.1.f with l∗ in the role of l. The second term on the right in (11.695) is readily estimated using Corollaries 11.2.b, 11.2.c and 11.2.d. In this way we obtain: L(κtrk/)2,[m,l], ε0 (11.696)

t  Q QQ ≤ Cl (1 + t)−2 [1 + log(1 + t)] W{l+1} + δ02 (1 + t)−2 W{l} + W{l+1} 1 2  +δ0 (1 + t)−1 B[m,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] In regard to the seventh term on the right we have the estimate (11.495). It is readily seen that under the assumptions of Proposition 11.2 this estimate holds with (k − 1, l − 1) replaced by (m, l). Substituting in the resulting estimate for Um−1,l−1 from Proposition 11.2 and for χ  2,[m−1,l], ε0 from Corollary 11.2.c, we obtain: t

Q  γ 2,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 W{l+2} + δ0 (1 + t)−1 W{l+1} t  +δ0 (1 + t)−1 (1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l] ( P)

(11.697)

Also, under the assumptions of Proposition 11.2 we have the L ∞ estimate (11.389) with (k − 1, l − 1) replaced by (m, l∗ − 1): ( P)

 γ ∞,[m,l∗ −1], ε0 ≤ Cl δ02 (1 + t)−2 t

(11.698)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

601

We can then apply Corollary 11.2.d to obtain:

( P) Q tr γ 2,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 W{l+2} + δ0 (1 + t)−1 W{l+1} t  +δ0 (1 + t)−1 (1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l]

(11.699)

We turn to the eighth (last) term on the right in (11.653). The function f 0 is given by equation (8.56) of Chapter 8, which we may write in the form: f 0 = 2η0 (1 − u + η0 t)−1 e + trχ  e  dH η0 (1 − u + η0 t)−1 ψ+ ψ · χ $ · (d/σ )$ +ψ L dσ ( P)

(N)

−tr α −tr α

(11.700)

The angular derivatives of this function have already been estimated by Lemma 10.16. ( P)

(N)

In regard to the last two terms on the right in (11.700), namely the terms tr α , tr α , ( P) (N)

we shall estimate first the symmetric 2-covariant St,u tensorfields α , α , themselves in

( P)

 2,[m,l], ε0 norm and then appeal to Corollary 11.2.d to estimate their traces. Now α t is given by equation (4.72) of Chapter 4, which we may write in the form: ( P)

( P P)

α = α +

1 d H −1 α ψ L ( ψ ⊗ (κ −1 ζ  ) + (κ −1 ζ  )⊗ ψ) 2 dσ  +α −1 (ψ L )2 k/ + ( ψ⊗ ψ)e (Lσ )

(11.701)

( P P)

with α , the principal part, given by: ( P P)

α

=

1 dH [ψ L ( ψ ⊗ (d/ Lσ ) + (d/ Lσ )⊗ ψ) 2 dσ  −(ψ L )2 (D / 2 σ ) − ( ψ⊗ ψ)(L)2 σ

(N)

(11.702)

( A) (B) (C)

Also, α is given by equation (4.85) of Chapter 4, with α , α , α , given by equations (4.94), (4.95), (4.96), respectively, which we may write in the form: ( A)

α =ω /L ⊗ ω /L − ωL L ω /

(B)

α = [ψ L (d/σ )− ψ(Lσ )] ⊗ [ψ L (d/σ )− ψ(Lσ )]

(11.703) (11.704)

(C)

α = d/σ ⊗ (ψ L ω / L − ψω L L ) + (ψ L ω / L − ψω L L ) ⊗ d/σ /− ψ ⊗ ω /L − ω / L ⊗ ψ) −(Lσ )(2ψ L ω

(11.705)

In regard to D / 2 σ , under the assumptions of Proposition 11.2, the estimate (11.493) holds with (k − 1, l − 1) replaced by (m, l). Substituting in the resulting estimate for Um−3,l−1

602

Chapter 11. Control of the Spatial Derivatives

from Proposition 11.2 and for χ  2,[m−1,l], ε0 from Corollary 11.2.c, we obtain: t

D / 2 σ 2,[m,l], ε0 ≤ Cl (1 + t)−2 W{l+2} + δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] t  Q (11.706) +Y0 + (1 + t)A[l] + (1 + t)−2 [1 + log(1 + t)]2 W{l} We also have the estimate (11.483) with (k − 1, l − 1) replaced by (m, l):   Q d/ Lσ 2,[m,l], ε0 ≤ Cl (1 + t)−2 W{l+1} + δ0 (1 + t)−1 W{l+1} t

and:

  QQ Q (L)2 σ 2,[m,l], ε0 ≤ Cl (1 + t)−2 W{l} + W{l} + δ0 (1 + t)−1 W{l} t

(11.707)

(11.708)

Using also the estimates of Corollary 11.2.b for ψ L , ψ, we then obtain: ( P P)

 α 2,[m,l], ε0

t Q QQ −2 ≤ Cl (1 + t) + δ02 (1 + t)−2 W{l} W{l+2} + δ0 (1 + t)−1 W{l+1} +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

 (11.709)

The remaining terms in (11.701) are lower order terms readily estimated using the estimates of Corollary 11.2.b. We thus obtain: ( P)

 α 2,[m,l], ε0 t

Q QQ −2 W{l+2} + δ0 (1 + t)−1 W{l+1} + δ02 (1 + t)−2 W{l} ≤ Cl (1 + t) +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

 (11.710)

Also, from the estimates of Corollary 11.1.b with l∗ in the role of l we readily deduce: ( P)

 α ∞,[m,l∗ −1], ε0 ≤ Cl δ0 (1 + t)−3

(11.711)

t

Applying then Corollary 11.2.d yields: ( P)

tr α 2,[m,l], ε0

t Q QQ −2 ≤ Cl (1 + t) + δ02 (1 + t)−2 W{l} W{l+2} + δ0 (1 + t)−1 W{l+1} +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] ( A) (B) (C)

 (11.712)

Consider next α , α , α . These are readily estimated by applying Lemma 11.2.a and using the estimates of Corollary 11.1.b with l∗ in the role of l together with those of

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

603

Corollary 11.2.b. We obtain:

( A) Q  α 2,[m,l], ε0 ≤ Cl δ0 (1 + t)−3 W{l+1} + W{l} t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1]

(B) Q  α 2,[m,l], ε0 ≤ Cl δ0 (1 + t)−3 W{l+1} + δ0 (1 + t)−1 W{l} t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1]

(C) Q  α 2,[m,l], ε0 ≤ Cl δ0 (1 + t)−3 W{l+1} + W{l} t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1] Combining the above we conclude that:

(N) Q  α 2,[m,l], ε0 ≤ Cl δ0 (1 + t)−3 W{l+1} + W{l} t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1]

 (11.713)

 (11.714)

 (11.715)

 (11.716)

From Corollary 11.1.b with l∗ in the role of l we also have: (N)

 α ∞,[m,l∗ −1], ε0 ≤ Cl δ02 (1 + t)−4

(11.717)

t

Applying then Corollary 11.2.d yields:

(N) Q tr α 2,[m,l], ε0 ≤ Cl δ0 (1 + t)−3 W{l+1} + W{l} t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1]

 (11.718)

The remaining terms in (11.700) are readily estimated by applying Lemma 11.2.a and Corollary 11.2.d and using the estimates of Corollaries 11.1.b and 11.1.c with l∗ in the role of l together with those of Corollaries 11.2.b and 11.2.c. In this way we obtain:  f 0 2,[m,l], ε0

Q QQ ≤ Cl (1 + t)−2 W{l+2} + W{l+1} + δ02 (1 + t)−2 W{l} t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

 (11.719)

The estimates (11.686), (11.689) and (11.690), (11.694), (11.696), (11.699), (11.719), in regard to the terms on the right in (11.653), yield: L 2,[m,l], ε0 v˙2,P t

Q QQ ≤ Cl W{l+2} + (1 + t)−1 [1 + log(1 + t)]W{l+1} + δ02 (1 + t)−4 [1 + log(1 + t)]2 W{l}  1 2 +δ0 (1 + t)−2 B[m+1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] (11.720)

604

Chapter 11. Control of the Spatial Derivatives L

We turn to v˙2,P , given by (11.658). Here the first term on the right is straightforward to estimate, while the second and third terms have already been estimated by (11.696) and (11.719) respectively, while the last term is estimated using the estimate of Corollary 11.2.b for ζ with (k − 1, l − 1) replaced by (m, l), together with Corollary 11.2.d. We thus obtain: L

v˙2,P 2,[m,l], ε0 t

  Q QQ −1 W{l+1} ≤ Cl (1 + t) + (1 + t)−1 [1 + log(1 + t)] W{l+2} + δ02 (1 + t)−2 W{l} 1 2  (11.721) +δ0 (1 + t)−2 B[m,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] We proceed to (v/˙ 2,P )b , given by (11.660). Here the first term on the right is estimated using Corollary 11.2.f and the estimates (11.686) and (11.690), the second term is estimated using the estimate of Corollary 11.2.b for ζ with (k − 1, l − 1) replaced by (m, l), while the third term is straightforward to estimate. We turn to the fourth term on the right in (11.660). The St,u 1-form i is given by formulas (10.390)–(10.400) of Chapter 10. Using the estimates (11.706) and (11.707) we obtain:

(1,1) Q  i 2,[m,l], ε0 ≤ Cl δ0 (1 + t)−3 W{l+2} + δ0 (1 + t)−3 [1 + log(1 + t)]2 W{l} t  (11.722) +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

(1,2) Q  i 2,[m,l], ε0 ≤ Cl δ02 (1 + t)−4 W{l+1} + W{l+1} t +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1]

 (11.723)

The remaining terms in i are readily estimated by applying Lemma 11.2.a and Corollary 11.2.d and using the estimates of Corollaries 11.1.b and 11.1.c with l∗ in the role of l together with those of Corollaries 11.2.b and 11.2.c. In this way we obtain: i 2,[m,l], ε0

Q ≤ Cl (1 + t)−2 W{l+2} + δ0 (1 + t)−2 [1 + log(1 + t)]W{l+1} t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

 (11.724)

which generalizes the estimate of Lemma 10.14. We turn to the fifth (last) term on the right in (11.660). Here we shall use the following. Let ϑ be an arbitrary symmetric 2-covariant St,u tensorfield. Denoting by ϑˆ the trace-free part of ϑ, we have: ˇ/ϑ) div / ϑˆ = tr(h/−1 · D (11.725) We shall apply this to the symmetric 2-covariant St,u tensorfield κk/ to estimate div / (κ k/ˆ ). Applying L / R in . . . L / Ri1 to the formula of Lemma 11.10 and adding to the result the formula of Lemma 10.10, with k replaced by n, we obtain: ˇ/ϑ − D ˇ/L L / Ri . . . L / Ri (L / T )m D / Ri (L /T )m ϑ = (i1 ...in ) cm,n [ϑ] (11.726) / Ri . . . L n

1

n

1

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

605

where: (i1 ...in )

cm,n [ϑ] = −

m−1 

L / R in . . . L / Ri1 (L / T )k



(T )

π /1 · (L /T )m−k−1 ϑ

 (11.727)

k=0

−

n−1  j =0

L / R in . . . L / Rin− j +1



(Rin− j )

π /1 · L / Rin− j −1 . . . L / Ri1 (L / T )m ϑ



Using Corollary 11.1.c and Lemma 10.11 we deduce in a straightforward manner the following corollary of Proposition 11.1. Corollary 11.1.g Under the assumptions of Proposition 11.1, for any symmetric 2covariant St,u tensorfield ϑ we have: max

max 

(i1 ...in )

k≤m,n≤l−k i1 ...in

ck,n [ϑ] L ∞ ( ε0 ) ≤ Cl δ0 (1+t)−2 [1+log(1+t)]ϑ∞,[m,l−1], ε0 t

t

Also, for any St,u 1-form ξ we similarly have: max

max 

(i1 ...in )

k≤m,n≤l−k i1 ...in

ck,n [ξ ] L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]ξ ∞,[m,l−1], ε0 t

t

The statement for St,u 1-forms is the conclusion of Lemma 11.14. Using Corollary 11.2.c and Lemma 10.12, as well as Corollary 11.1.c and Lemma 10.11 with l∗ in the role of l, we deduce in a straightforward manner the following corollary of Proposition 11.2. Corollary 11.2.g Under the assumptions of Proposition 11.2, for any symmetric 2covariant St,u tensorfield ϑ we have: max  (i1 ...in ) ck,n [ϑ] L 2 ( ε0 ) t

−1 −1 δ0 (1 + t) [1 + log(1 + t)]ϑ2,[m,l−1], ε0 ≤ Cl (1 + t) max

k≤m,n≤l−k i1 ...in

t

−2

Q

[1 + log(1 + t)]2 W{l}  +(1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

+ ϑ∞,[m,l∗ ], ε0 W{l+1} + (1 + t) t

Also, for any St,u 1-form ξ we similarly have: max  (i1 ...in ) ck,n [ξ ] L 2 ( ε0 ) t

≤ Cl (1 + t)−1 δ0 (1 + t)−1 [1 + log(1 + t)]ξ 2,[m,l−1], ε0 max

k≤m,n≤l−k i1 ...in

t

−2

Q [1 + log(1 + t)]2 W{l}  +(1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

+ ξ ∞,[m,l∗ ], ε0 W{l+1} + (1 + t) t

606

Chapter 11. Control of the Spatial Derivatives

The statement for St,u 1-forms is obtained by substituting in the conclusion of /1 2,[l−1], ε0 their bounds from Corollary 11.2.c Lemma 11.15 for T[m−1,l] and maxi (Ri ) π t and Lemma 10.12 respectively. We apply the first statement of Corollary 11.2.g to κk/ to obtain, using the estimates for this of Corollary 11.1.b with l∗ in the role of l and of Corollary 11.2.b, max

max  (i1 ...in ) ck,n [κk/] L 2( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] · t 

 Q [1 + log(1 + t)] W{l+1} + (1 + t)−2 [1 + log(1 + t)]W{l}  (11.728) +(1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

k≤m,n≤l−k i1 ...in

On the other hand, by hypothesis H2 : ˇ/L D / R in . . . L / Ri1 (L /T )k (κk/) L 2( ε0 ) ≤ C(1 + t)−1 (L /T )k (κk/)2,[l+1−k], ε0 t

t

≤ C(1 + t)−1 κk/2,[m,l+1], ε0 t

: for all k ≤ m and all n ≤ l − k

(11.729)

Thus, by the estimate for κk/ of Corollary 11.2.b with (k − 1, l − 1) replaced by (m, l) we obtain through (11.726) with ϑ = κk/: −2 ˇ/(κk/) ε D · (11.730) 2,[m,l],t 0 ≤ Cl (1 + t)

  Q [1 + log(1 + t)] W{l+2} + δ0 (1 + t)−2 W{l+1} 1 2  +δ0 (1 + t)−1 B[m,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l]

From formula (11.725) and Corollary 11.2.d we then obtain: div / (κ k/ˆ )2,[m,l], ε0 ≤ Cl (1 + t)−2 · (11.731) t

  Q [1 + log(1 + t)] W{l+2} + δ0 (1 + t)−2 W{l+1} 1 2  +δ0 (1 + t)−1 B[m,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] Combining the above results for the five terms on the right in (11.660) we conclude that: (v/˙ 2,P )b 2,[m,l], ε0 ≤ Cl (1 + t)−1 t

  Q QQ · [1 + log(1 + t)] W{l+2} + (1 + t)−1 [1 + log(1 + t)] W{l+1} + δ02 (1 + t)−2 W{l} 1 2  +δ0 (1 + t)−1 B[m+1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] (11.732) L , v˙ L , (v We now turn to the lower order parts v˙2,N /˙ 2,N )b , given by (11.654), 2,N (11.657), (11.661), respectively. These are estimated in a straightforward manner by applying Lemma 11.2.a and Corollary 11.2.d and using the estimates of Corollaries 11.1.b

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

607

and 11.1.c with l∗ in the role of l together with those of Corollaries 11.2.b and 11.2.c. In (N)

L , we appeal, more directly, to the case of the terms tr and tr γ , which occur in v˙2,N the estimates (11.476) and (11.477) with (k − 1, l − 1) replaced by (m, l). Substituting in these for χ  2,[m,l], ε0 from Corollary 11.2.c and for Um−1,l−1 from Proposition 11.2, t in fact yields:

Q  2,[m,l], ε0 ≤ Cl (1 + t)−1 W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} t 1 2  (11.733) +(1 + t)−1 B[m,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l]

(N) Q  γ 2,[m,l], ε0 ≤ Cl δ0 (1 + t)−2 W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} t  (11.734) +δ0 (1 + t)−1 (1 + t)−1 B[m,l] + Y0 + (1 + t)A[l−1] In this way, we obtain: L + (1 − u + η0 t)−2 2,[m,l], ε0 v˙2,N

t  Q −1 ≤ Cl (1 + t) [1 + log(1 + t)] W{l+1} + W{l}   +B[m+1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l]

(11.735)

v˙2,N − η0 (1 − u + η0 t)−2 2,[m,l], ε0 t

Q −1 −1 ≤ Cl (1 + t) W{l+1} + (1 + t) [1 + log(1 + t)]W{l} 1 2  +(1 + t)−1 B[m,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l]

(11.736)

L

(v/˙ 2,N )b 2,[m,l], ε0 t

  Q −1 [1 + log(1 + t)] W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} ≤ Cl δ0 (1 + t)   +(1 + t)−1 B[m+1,l+1] + [1 + log(1 + t)]2 Y0 + (1 + t)A[l] (11.737) Combining the estimates (11.18) and (11.735) yields: v˙2L + (1 − u + η0 t)−2 2,[m,l], ε0 t

Q QQ ≤ Cl W{l+2} + (1 + t)−1 [1 + log(1 + t)]W{l+1} + δ02 (1 + t)−4 [1 + log(1 + t)]2 W{l} 1 2  (11.738) +(1 + t)−1 B[m+1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] v˙2 − η0 (1 − u + η0 t)−2 2,[m,l], ε0 t

Q QQ −1 W{l+2} + W{l+1} + δ02 (1 + t)−3 [1 + log(1 + t)]W{l} ≤ Cl (1 + t)  1 2 +(1 + t)−1 B[m,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] L

(11.739)

608

Chapter 11. Control of the Spatial Derivatives

(v/˙ 2 )b 2,[m,l], ε0 t  1 −1 [1 + log(1 + t)] W{l+2} ≤ Cl (1 + t)   Q QQ +(1 + t)−1 [1 + log(1 + t)] W{l+1} + δ02 (1 + t)−2 W{l}   +δ0 (1 + t)−1 B[m+1,l+1] + [1 + log(1 + t)]2 Y0 + (1 + t)A[l]

(11.740)

We also deduce, under the same assumptions, v˙2L + (1 − u + η0 t)−2 ∞,[m,l∗ −1], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

(11.741)

v˙2 − η0 (1 − u + η0 t)−2 ∞,[m,l∗ −1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

(11.742)

(v/˙ 2 )b ∞,[m,l∗ −1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

(11.743)

t

L

t

t

The above results lead immediately to the conclusion that under the assumptions of Proposition 11.2 augmented by the assumptions E{l∗ +2} , E{lQ∗ +1} , E{lQ∗Q} and M[m+1,l∗ +1] , we have: v˙2 2,[m,l], ε0

(11.744)

  Q QQ ≤ Cl (1 + t)−2 [1 + log(1 + t)] W{l+2} + δ0 W{l+1} + δ03 (1 + t)−3 W{l}  1 2 +δ0 (1 + t)−1 B[m+1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] t

We turn to v˙1 , given by (11.645). To estimate the first term on the right in (11.63), recalling that L = 2T + α −1 κ we write:

so that

[L, L] = 2 + (L(α −1 κ))L,

L Lσ = L Lσ + 2 · d/σ + (L(α −1 κ))Lσ

and we use the following estimate for the principal term:

Q L Lσ 2,[m,l], ε0 ≤ Cl (1 + t)−1 W{l+1} + δ0 (1 + t)−1 W{l+1} t  QQ +(1 + t)−1 [1 + log(1 + t)]W{l} + δ0 (1 + t)−2 B[m,l]

(11.745)

To estimate the second term on the right in (11.645), we use the estimate (11.708). To estimate the third term we use the following estimate:  d/ Lσ 2,[m,l], ε0 ≤ Cl (1 + t)−1 W{l+2} t  Q +(1 + t)−1 [1 + log(1 + t)]W{l+1} + δ0 (1 + t)−2 B[m,l+1] (11.746) To estimate the fourth term we use the estimate (11.707). Finally, to estimate the last two terms on the right in (11.645) we use the estimate (11.706) and Corollary 11.2.d.

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

609

Applying then Lemma 11.2.a and using the estimates just mentioned together with those of Corollaries 11.2.b and 11.2.c, and those of Corollaries 11.1.b and 11.1.c with l∗ in the role of l, we obtain: v˙1 2,[m,l], ε0

(11.747)   Q QQ ≤ Cl (1 + t)−2 [1 + log(1 + t)] W{l+2} + W{l+1} + W{l} 1 2  +δ0 (1 + t)−2 B[m+1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] t

under the assumptions of Proposition 11.2 augmented by assumptions E{l∗ +2} , E{lQ∗ +1} , E{lQ∗Q} and M[m+1,l∗ +1] .

L Finally, we consider v˙3 , given by (11.662), with v˙3L , v˙3 , v/˙ 3 , given by (11.663), (11.664), (11.665), respectively. These, being of lower order, are estimated in a straightforward manner by applying Lemma 11.2.a and Corollary 11.2.d and using the estimates of Corollaries 11.1.b and 11.1.c with l∗ in the role of l together with those of Corollaries 11.2.b and 11.2.c. Under the assumptions of Proposition 11.2, we deduce:

v˙3L − 2(1 − u + η0 t)−2 2,[m,l], ε0 t

  Q −1 ≤ Cl (1 + t) [1 + log(1 + t)] W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l}   (11.748) +B[m+1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] L

v˙3 2,[m,l], ε0

Q + δ0 W{l+1} [1 + log(1 + t)]W{l} 1 2  +δ0 (1 + t)−1 B[m,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] t

≤ Cl (1 + t)

−2

v/˙ 3 2,[m,l], ε0 ≤ Cl (1 + t)−1 [1 + log(1 + t)]· t

Q W{l+1} + δ0 (1 + t)−1 [1 + log(1 + t)]W{l}  1 2 +(1 + t)−1 B[m,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l]

(11.749)

(11.750)

Moreover, under the same assumptions we obtain the L ∞ estimates: v˙3L − 2(1 − u + η0 t)−2 ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(11.751)

v˙3 ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)]

(11.752)

v/˙ 3 ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]2

(11.753)

L

t

t

610

Chapter 11. Control of the Spatial Derivatives

It then follows that: v˙3 2,[m,l], ε0

(11.754)

  Q ≤ Cl (1 + t)−2 [1 + log(1 + t)] W{l} + δ0 (1 + t)−1 [1 + log(1 + t)]W{l+1} 1 2  +δ0 (1 + t)−1 B[m+1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] t

Combining the results (11.747), (11.744), (11.754), yields the following lemma. Lemma 11.20 Under the assumptions of Proposition 11.2 augmented by the assumpQ QQ tions E{l∗ +2} , E{l∗ +1} , E{l∗ } and M[m+1,l∗ +1] we have the L 2 estimate: v ˙ 2,[m,l], ε0

  Q QQ ≤ Cl (1 + t)−2 [1 + log(1 + t)] W{l+2} + W{l+1} + W{l} 1 2  +δ0 (1 + t)−1 B[m+1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] t

as well as the L ∞ estimate: v ˙ ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] t

We now consider the function τ¨ , defined by (11.636). Setting: τ˙ = −1 τ˜

(11.755)

equation (11.636) becomes, in view of (11.644), τ¨ = v˙ + T τ˙ + (T log  +

(T )

δ)τ˙

(11.756)

Now, the function τ˜ is given by equation (9.38) of Chapter 9. In terms of τ˙ this is:   τ˙ = −2 −(Lψ α )(Lψα ) + µ(d/ψ α )$ · d/ψα (11.757) To estimate (d/ψ α )$ 2,[m+1,l+1], ε0 , we would like to apply Corollary 11.2.d with t (m, l) replaced by (m+1, l+1). To do so, however, would seem to require the assumptions of Proposition 11.2 with (m+1, l +1) in the role of (m, l). To avoid this, we appeal instead to Lemma 11.12 with (k, l) replaced by (m + 2, l + 2). Then l∗ is replaced by l∗ + 1 and we must verify the assumptions: λ ∞,[m+1,l∗ ], ε0 ≤ Cl δ0 [1 + log(1 + t)] t

max 

(Ri )

i



π /∞,[l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(T ) 

π / ∞,[m+1,l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(11.758)

The first of these follows if we augment the assumptions of Proposition 11.2 by the assumption M[m+1,l∗ ] . The second follows by Corollary 10.1.d with l∗ +1 in the role of l. To

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

611

verify the third assumption we note that if in Lemma 11.16 we replace the inductive hypothesis (11.268) by Proposition 11.1 itself, then the conclusion holds also for k = m + 1. Therefore under the assumptions of Proposition 11.2 we have: χ  ∞,[m+1,l∗ ], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.759)

It then follows from definition (11.89) that if the assumptions of Proposition 11.2 are augmented by assumption M[m+1,l∗ ] , then we have: 

(T ) 

π / ∞,[m+1,l∗ ], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]2 t

(11.760)

which implies the third inequality of (11.758). Consequently, Lemma 11.12 holds with (m, l) replaced by (m + 2, l + 2), under the assumptions of Proposition 11.2 augmented by assumption M[m+1,l∗ ] . /2,[l], ε0 of Corollary Substituting in the conclusion the estimate for maxi  (Ri ) π t 10.2.d with l + 1 in the role of l as well as the estimate for T[m,l] of Corollary 11.2.c, we then obtain the conclusion of Corollary 11.2.d with (m, l) replaced by (m + 1, l + 1). Applying this to ξ = d/ψ α we obtain:

(d/ψ α )$ 2,[m+1,l+1], ε0 ≤ Cl (1 + t)−1 W{l+2} + δ0 (1 + t)−1 · (11.761) t  Q (1 + t)−2 [1 + log(1 + t)]2 W{l} + (1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l] under the assumptions of Proposition 11.2 augmented by assumption M[m+1,l∗ ] . Using this estimate we readily deduce, applying Lemma 11.2.a with (m + 1, l + 1) in the role of (k, l), τ˙ 2,[m+1,l+1], ε0

(11.762)

Q ≤ Cl δ0 (1 + t)−2 W{l+2} + W{l+1} 1 2  +δ0 (1 + t)−2 B[m+1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] t

Q

under the assumptions of Proposition 11.2 augmented by the assumptions E{l∗ +2} , E{l∗ +1} and M[m+1,l∗ +1] . Moreover, under the same assumptions, τ˙ ∞,[m+1,l∗ +1], ε0 ≤ Cl δ02 (1 + t)−3 t

(11.763)

Combining the estimates (11.762) (11.763) with those of Lemma 11.20, we conclude through (11.756) that under the assumptions of Lemma 11.20 we have: τ¨ 2,[m,l], ε0

(11.764)

  Q QQ ≤ Cl (1 + t)−2 [1 + log(1 + t)] W{l+2} + W{l+1} + W{l} 1 2  +δ0 (1 + t)−1 B[m+1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] t

612

Chapter 11. Control of the Spatial Derivatives

as well as:

τ¨ 2,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] t

(11.765)

Using finally the estimates (11.764) and (11.765), we deduce through (11.634), in a straightforward manner, the following estimate for the function n˙ 0 : n˙ 0 2,[m,l], ε0

(11.766)

  Q QQ ≤ Cl (1 + t)−2 [1 + log(1 + t)] W{l+2} + W{l+1} + W{l}  1 2 +δ0 (1 + t)−1 B[m+1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] t

We proceed to estimate the functions n˙ 1 and n˙2  , given by (11.637) and (11.638)– (11.639), respectively, the functions n 1 and n 2 being themselves given by equations (9.53) and (9.60) of Chapter 9. Here, the main step shall be to estimate:
/ Tˆ i + α −1 (d/trχ)$ · d/x i

(11.767)


/ L i − (d/trχ)$ · d/x i

(11.768)

A suitable estimate for:

will then follow through expression (11.275). According to the statements concluding in (9.75), (9.76), respectively, the expressions (11.767), (11.768) do not contain principal acoustical terms. More precisely, we consider: / Tˆ i + α −1 (d/trχ)$ · d/x i + η0 (1 − u + η0 t)−1 α −1
/ xi Ui =


(11.769)

Then by the second of the equations (9.74) of Chapter 9, taking also account of the Codazzi equation (8.104) of Chapter 8, we have: U i = a Tˆ i + b1 · d/x i + b2 · D /2x i

(11.770)

a = div / p/ + | p/|2

(11.771)

where a is the function: b1 is the St,u -tangential vectorfield: b1 = div / q/ + p/$ · q/ + α −1 χ $ · p/$ + α −1 i $ + α −2 χ $ · (d/α)$

(11.772)

and b2 is the symmetric 2-contravariant St,u tensorfield: b2 =

 1  $ q/ + q/˜$ − α −1 χ $$ 2

(11.773)

In (11.771)–(11.773) p/ is the St,u 1-form given by equation (3.226) of Chapter 3: p/ = −

1 dH (ψ ˆ )2 d/σ − H ψTˆ ω /Tˆ 2 dσ T

(11.774)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

613

and q/ is the T11 -type St,u tensorfield: q/ = q/ + α −1 χ $

(11.775)

where q/ is the T11 -type St,u tensorfield given by equations (3.230), (3.232), of Chapter 3. We have: 1 dH ψ ˆ (d/σ ⊗ ψ$ − ψ ⊗ (d/σ )$ ) − H ω q/ = k/$ − /Tˆ ⊗ ψ$ (11.776) 2 dσ T Note that in (11.773), q/$ is the 2-contravariant St,u tensorfield with components, in an arbitrary local frame field for St,u , (q/$ ) AB = (h/−1 ) AC (q/ )CB , q/˜$ is its transpose:

(q/˜$ ) AB = (q/$ ) B A

while χ $$ is the symmetric 2-contravariant St,u tensorfield: (χ $$ ) AB = (h/−1 ) AC (h/−1 ) B D χC D Also, in (11.170), div / q/ is the St,u -tangential vectorfield with components: (div / q/ ) A = D / B q/A /−1 ) BC (D /q/ )CA B B = (h We shall estimate U i 2,[m,l], ε0 t

under the assumptions of Proposition 11.2 augmented by the assumptions E{l∗ +2} , E{lQ∗ +1} . Consider first the function a. Using the estimate for ω /Tˆ Corollary 11.2.b with l replaced by l + 1 together with Corollary 11.1.b with l∗ in the role of l we obtain:

(11.777)  p/2,[m,l+1], ε0 ≤ Cl δ0 (1 + t)−2 W{l+2} + δ0 (1 + t)−1 · t  Q (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] + δ0 (1 + t)−2 W{l} Also, using the estimate for ω /Tˆ of Corollary 11.1.b with l replaced by l∗ + 1, which holds under the augmented assumptions, we obtain:  p∞,[m,l∗ +1], ε0 ≤ Cl δ02 (1 + t)−3 t

(11.778)

/ p/) we first estimate D / p/ using the estimates (11.777), To estimate div / p/ = tr(h /−1 · D (11.778) together with the second statement of Corollary 11.2.g and then apply Corollary 11.2.d. We obtain: div / p/2,[m,l], ε0

W{l+2} + δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1]  Q +Y0 + (1 + t)A[l] + (1 + t)−2 [1 + log(1 + t)]2 W{l} t

≤ Cl δ0 (1 + t)

−3

(11.779)

614

Chapter 11. Control of the Spatial Derivatives

The estimates (11.777)–(11.779) imply: a2,[m,l], ε0

≤ Cl δ0 (1 + t)−3 W{l+2} + δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1]  Q +Y0 + (1 + t)A[l] + (1 + t)−2 [1 + log(1 + t)]2 W{l} t

(11.780)

Using the estimate (11.778) we also obtain: a∞,[m,l∗ ], ε0 ≤ Cl δ02 (1 + t)−4

(11.781)

t

Consider next the St,u -tangential vectorfield b1 , given by (11.772). Here the main step is to estimate div / q/ , the intrinsic divergence on St,u of the T11 -type St,u tensorfield q/ . To do this we consider the corresponding 2-covariant St,u tensorfield q/b :

We then have:

q/b = q/ · h/

(11.782)

div / q/ = (div / q/b )$

(11.783)

From (11.776): q/b = k/ −

1 dH ψ ˆ (d/σ ⊗ ψ− ψ ⊗ d/σ ) − H ω /Tˆ ⊗ ψ 2 dσ T

(11.784)

Using the fact that if ξ and η are two arbitrary St,u 1-forms we have: div / (ξ ⊗ η) = (div / ξ )η + ξ $ · D /η we deduce:

(11.785)

 1 dH / σ − ψ$ · D ψTˆ ψ
/ 2 σ + (d/σ )$ · D / ψ − (div / ψ)d/σ 2 dσ   $ /ˆ · D / ψ − H ψdiv / ω /Tˆ + ω T    dH 1 ψ(d/σ )$ − (d/σ ) ψ$ · d/ ψTˆ − ψ(d/ H )$ · ω /Tˆ (11.786) − 2 dσ

div / q/b = div / k/ −

To estimate div / k/ , the first term on the right in (11.786), we write: div / k/ = div / k/ˆ + (1/2)d/trk/ To estimate div / k/ˆ we apply the approach which we followed to derive the estimate (11.731). The term d/trk/ is straightforward to estimate. We then obtain, combining, div / k/ 2,[m,l], ε0 t

Q −2 W{l+2} + δ0 (1 + t)−2 W{l+1} ≤ Cl (1 + t) +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

 (11.787)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

615

The second term on the right in (11.786) is estimated using (11.706) together with the estimate: D / ψ2,[m,l], ε0 t

Q −1 W{l+1} + δ0 (1 + t)−3 [1 + log(1 + t)]2 W{l} ≤ Cl (1 + t) +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

 (11.788)

obtained by using hypothesis H1 together with the second statement of Corollary 11.2.g. The third term on the right in (11.786) is estimated using the estimate: D /ω /Tˆ 2,[m,l], ε0 t

Q −2 ≤ Cl (1 + t) W{l+2} + δ0 (1 + t)−3 [1 + log(1 + t)]2 W{l} +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

 (11.789)

obtained by using hypothesis H1, the estimate for ω /Tˆ of Corollary 11.2.b with l replaced by l + 1, together with the second statement of Corollary 11.2.g. The remaining terms on the right in (11.786) are readily estimated, being of lower order. In this way, we deduce the estimate: div / q/b 2,[m,l], ε0 t

Q −2 W{l+2} + δ0 (1 + t)−2 W{l+1} ≤ Cl (1 + t) +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

 (11.790)

under the assumptions of Proposition 11.2 augmented by the assumptions E{l∗ +2} , E{lQ∗ +1} . Under the same assumptions we also deduce the L ∞ estimate: div / q/b ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−3

(11.791)

t

In view of (11.783), applying Corollary 11.2.d it then follows that: div / q/ 2,[m,l], ε0 t

Q −2 ≤ Cl (1 + t) W{l+2} + δ0 (1 + t)−2 W{l+1} +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

 (11.792)

We also readily obtain the estimates: q/ 2,[m,l], ε0

Q ≤ Cl (1 + t)−1 W{l+1} + δ0 (1 + t)−2 W{l} t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1] q/ ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−2 t

 (11.793) (11.794)

616

Chapter 11. Control of the Spatial Derivatives

which require only the assumptions of Proposition 11.2. The estimates (11.777), (11.778), (11.793), (11.794) allow us to estimate the 2nd and 3rd terms on the right in (11.772). The 4th term on the right in (11.772) has already been estimated by (11.724) while the 5th term is straightforward to estimate. Combining, we obtain: b1 2,[m,l], ε0

Q ≤ Cl (1 + t)−2 W{l+2} + δ0 (1 + t)−2 [1 + log(1 + t)]W{l+1}  +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] t

(11.795)

We also obtain, using (11.791), the L ∞ estimate: b1 ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−3 t

(11.796)

Finally, using the estimate (11.793) and Corollary 11.2.c together with Corollary 11.2.d we obtain: b2 2,[m,l], ε0

Q ≤ Cl (1 + t)−1 W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l}  +(1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] t

(11.797)

We also obtain, using the estimate (11.794) and Corollary 11.1.c with l∗ in the role of l, the L ∞ estimate: b2 ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.798)

We are now ready to estimate U i as given by expression (11.770). To estimate the first term on the right however, we must extend the estimate (11.548) to all spatial derivatives. For 1 ≤ k ≤ m we write: (T )k ( f Tˆ i ) = ((T )k f ))Tˆ i +

k  j =1

k! ((T )k− j f )(T ) j Tˆ i j !(k − j )!

(11.799)

We are to estimate the  2,[l−k], ε0 norm of this equality. To the first term on the right t

in (11.799) we apply (11.548) taking (T )k f in the role of f . Since here we are considering the  2,[l−k], ε0 norm instead of the  2,[l], ε0 norm, we may replace the t t  ∞,[l∗ ], ε0 norm on the right in (11.548) by the  ∞,[l∗ −k], ε0 norm. We then obt t tain:

(11.800) ((T )k f )Tˆ i 2,[l−k], ε0 ≤ Cl (T )k f 2,[l−k], ε0 t t  Q +(T )k f ∞,[l∗ −k], ε0 Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] t

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

617

We handle the sum in (11.799) using Proposition 11.1 with l∗ in the role of l and Proposition 11.2 with l reduced to l − 1. We obtain: 0 0 0 0 0 k 0 k! k− j j ˆ i0 0 ((T ) f )(T ) T 0 (11.801) 0 j !(k − j )! 0 0 ε j =1

2,[l−k],t 0

≤ Cl (1 + t)−1 δ0 [1 + log(1 + t)] f 2,[k−1,l−1], ε0 t  1  ε +  f ∞,[k−1,l∗ ], 0 B[k−1,l] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−2] t   Q +[1 + log(1 + t)] W{l} + δ0 (1 + t)−2 [1 + log(1 + t)]2 W{l−2} Combining (11.800) and (11.801) we obtain, for 1 ≤ k ≤ m:

(T )k ( f Tˆ i )2,[l−k], ε0 ≤ Cl  f 2,[k,l], ε0 t

+f

t

(1 + t)

−1

B[k−1,l] + Y0 + (1 + t)A[l−1]  Q +W{l} + δ0 (1 + t)−2 W{l−1}

ε ∞,[k,l∗ ],t 0

(11.802)

Summing over k ∈ {1, . . . , m} and adding the estimate (11.548) yields the following result, which extends the estimate (11.548) to all spatial derivatives: Under the assumptions of Proposition 11.2, for any function f we have:

 f Tˆ i 2,[m,l], ε0 ≤ Cl  f 2,[m,l], ε0 t

t

+  f ∞,[m,l∗ ], ε0 (1 + t)

−1

B[m−1,l] + Y0 + (1 + t)A[l−1]  Q (11.803) +W{l} + δ0 (1 + t)−2 W{l−1} t

Applying this with the function a in the role of f and using the estimates (11.780) and (11.781) we obtain: a Tˆ i 2,[m,l], ε0 ≤ Cl δ0 (1 + t)−3 · t

Q W{l+2} + δ0 (1 + t)−3 [1 + log(1 + t)]2 W{l} +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

 (11.804)

Next, we apply Lemma 11.18 with the St,u -tangential vectorfield b1 in the role of ξ . Using the estimates (11.795), (11.796), we obtain: b1 · d/x i 2,[m,l], ε0 ≤ Cl (1 + t)−2 · t

Q W{l+2} + δ0 (1 + t)−2 [1 + log(1 + t)]W{l+1} +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

 (11.805)

618

Chapter 11. Control of the Spatial Derivatives

Finally, we apply Lemma 11.19 with the symmetric 2-contravariant St,u tensorfield b2 in the role of ξ . Using the estimates (11.797), (11.798), we obtain: b2 · D / 2 x i 2,[m,l], ε0 ≤ Cl (1 + t)−2 · t

Q W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} +(1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

 (11.806)

Combining then (11.804)–(11.806) we conclude that: U i 2,[m,l], ε0 ≤ Cl (1 + t)−2 ·

t Q W{l+2} + (1 + t)−1 [1 + log(1 + t)]W{l+1} +(1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

 (11.807)

Let us now define: Vi =
/ L i − (d/trχ)$ · d/x i − η0 (1 − u + η0 t)−1
/ xi In view of expression (11.275) we have: V i = −αU i − 2(d/α)$ · d/Tˆ i − (
/ α)Tˆ i −
/



H ψ0ψi 1 + ρH

(11.808)

 (11.809)

To estimate the last two terms in (11.809) we apply the following: Let G be a smooth function of the (ψα : α = 0, 1, 2, 3) defined in a neighborhood of (k, 0, 0, 0), as in Lemma 11.1. Then, under the assumptions of Proposition 11.2 we have: D / 2 G2,[m,l], ε0 ≤ Cl (1 + t)−2 · .

t Q W{l+2} + δ0 (1 + t)−1 (1 + t)−2 [1 + log(1 + t)]2 W{l}  (11.810) +(1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] This follows by applying Lemma 11.9 to the St,u 1-form d/G. By hypotheses H1 and H0 and Lemma 11.1 with l +2 in the role of l (assumption E{l∗ +1} ), the principal term satisfies the estimate: max

max D / 2 Rin . . . Ri1 (T )k G L 2 ( ε0 ) ≤ C(1 + t)−2 W{l+2}

k≤m,n≤l−k i1 ...in

t

On the other hand by the second statement of Corollary 11.2.g and Lemma 11.1, the commutator satisfies the estimate: max

max 

k≤m,n≤l−k i1 ...in

(i1 ...in )

ck,n [d/G] L 2 ( ε0 ) t

Q ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)]W{l+1} + (1 + t)−2 [1 + log(1 + t)]2 W{l}  +(1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

Combining then yields (11.810).

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

619

Using (11.810) to estimate the last two terms on the right in (11.809) we obtain an estimate for V i similar to the estimate (11.807). In conclusion, we have proved the following lemma. Lemma 11.21 Under the assumptions of Proposition 11.2 augmented by the assumptions E{l∗ +2} , E{lQ∗ +1} , we have: U i 2,[m,l], ε0 , V i 2,[m,l], ε0 ≤ Cl (1 + t)−2 · t

t Q −1 W{l+2} + (1 + t) [1 + log(1 + t)]W{l+1} +(1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] and:



U i ∞,[m,l∗ ], ε0 , V i ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] t

t

Using the above lemma we shall prove the following. Lemma 11.22 Under the assumptions of Proposition 11.2 augmented by the assumptions E{l∗ +2} , E{lQ∗ +1} , we have the L 2 estimates: 
/ ψTˆ + α −1 ψ$ · d/trχ2,[m,l], ε0 t

Q −2 W{l+2} + δ0 (1 + t)−2 [1 + log(1 + t)]W{l+1} ≤ Cl (1 + t)  +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] 
/ ψ L − ψ$ · d/trχ2,[m,l], ε0

t Q −2 W{l+2} + δ0 (1 + t)−2 [1 + log(1 + t)]W{l+1} ≤ Cl (1 + t)  +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] 
/ ψ Lˆ + α −2 ψ$ · d/trχ2,[m,l], ε0 t

Q −2 W{l+2} + δ0 (1 + t)−2 [1 + log(1 + t)]W{l+1} ≤ Cl (1 + t)  +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] and: $

/ Li − ω / L · d/trχ2,[m,l], ε0 (Lψi )
t

Q −3 ≤ Cl (1 + t) δ0 (1 + t)−1 W{l+2} + W{l+1} +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]



620

Chapter 11. Control of the Spatial Derivatives

Moreover, under the same assumptions we have the L ∞ estimates: 
/ ψTˆ + α −1 ψ$ · d/trχ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−3 t


/ ψ L − ψ$ · d/trχ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−3 t


/ ψ Lˆ + α −2 ψ$ · d/trχ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−3 t

and: $

/ Li − ω / L · d/trχ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−4 (Lψi )
t

Proof. In view of the definition (11.769) we have: / xi
/ ψTˆ + α −1 ψ$ · d/trχ = ψi U i − η0 (1 − u + η0 t)−1 α −1 ψi
+Tˆ i
/ ψi + 2(d/Tˆ i )$ · d/ψi

(11.811)

We take the  2,[m,l], ε0 norm. To estimate the contribution of the first term on the right t in (11.811), we apply Lemma 11.21. To estimate the contribution of the second term, we apply Lemma 11.19 with the symmetric 2-contravariant St,u tensorfield ψi h/−1 in the role of the St,u tensorfield ξ . We have: /−1 ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 ψi h t

(11.812)

and, by Corollary 11.2.d:

Q ψi h /−1 2,[m,l], ε0 ≤ Cl W{l} + δ0 (1 + t)−1 (1 + t)−2 [1 + log(1 + t)]2 W{l−1} t  (11.813) +(1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1]

Applying Lemma 11.19 we then obtain: ψi
/ x i 2,[m,l], ε0 ≤ Cl (1 + t)−1 · t

Q W{l+1} + δ0 (1 + t)−1 (1 + t)−2 [1 + log(1 + t)]2 W{l}  +(1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

(11.814)

We also obtain, using (11.572), (11.587), (11.612), ψi
/ x i ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−2 t

(11.815)

To estimate the contribution of the third term on the right in (11.811) we apply the estimate (11.803) with
/ ψi in the role of the function f . Using the second statement of Corollary 11.2.g applied to the St,u 1-form d/ψµ we deduce: D / 2 ψµ 2,[m,l], ε0 ≤ Cl (1 + t)−2 · t

Q W{l+2} + δ0 (1 + t)−1 (1 + t)−2 [1 + log(1 + t)]2 W{l}  (11.816) +(1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

621

Also, using the second statement of Corollary 11.1.g we deduce: D / 2 ψµ ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−3 t

(11.817)

Applying Corollary 11.2.d then yields similar estimates for
/ ψµ . Finally, the contribution of the last term in (11.811), a lower order term, is readily estimated. The L 2 estimate of the lemma for
/ ψTˆ + α −1 ψ $ · d/trχ then follows. The L ∞ estimate is straightforward. In view of definition (11.808) we have: / xi
/ ψ L − ψ$ · d/trχ = ψi V i + η0 (1 − u + η0 t)−1 ψi
+
/ ψ0 + L i
/ ψi + 2(d/ L i )$ · d/ψi

(11.818)

The first two terms on the right in (11.818) are similar to the first two terms on the right in (11.811). The fourth term on the right in (11.818) is similar to the third term on the right in (11.811), while the third term on the right in (11.818) has already been estimated by (11.816). The last term is again a lower order term, readily estimated. The L 2 estimate of the lemma for
/ ψ L − ψ$ · d/trχ then follows. The L ∞ estimate is straightforward. The estimates for
/ ψ Lˆ + α −2 ψ$ · d/trχ similarly follow. Finally, in view of (11.808) we have: $

(Lψi )
/ Li − ω / L · d/trχ = (Lψi )V i + η0 (1 − u + η0 t)−1 (Lψi )
/ xi

(11.819)

Taking the  2,[m,l], ε0 norm, to estimate the contribution of the first term on the right t in (11.819), we apply Lemma 11.21 . To estimate the contribution of the second term, we /−1 in the role of ξ . We have: apply Lemma 11.19 with (Lψi )h /−1 ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−2 (Lψi )h t

(11.820)

and, by Corollary 11.2.d:

1 Q (Lψi )h /−1 2,[m,l], ε0 ≤ Cl (1 + t)−1 W{l} + δ0 (1 + t)−1 W{l} t  +(1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1]

(11.821)

Applying Lemma 11.19 we then obtain:

1 Q (Lψi )
/ x i 2,[m,l], ε0 ≤ Cl (1 + t)−2 W{l} + δ0 (1 + t)−1 W{l+1} t  +(1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

(11.822)

$

The L 2 estimate of the lemma for (Lψi )
/ Li − ω / L · d/trχ then follows. The L ∞ estimate is again straightforward. We now consider the function n˙ 1 given by equation (11.637). In this equation, the function n 1 is given by equations (9.47), (9.43) of Chapter 9:    $  1 2 dH 1  $ 2 dH 2 dH (d/ log ) · d/T σ + d/ ψ L / ψL n1 = − ψL · d/T σ +
(11.823) 2 dσ dσ 2 dσ

622

Chapter 11. Control of the Spatial Derivatives

Substituting in (11.637) we obtain the following expression for n˙ 1 :   dH (T σ )
/ ψ L − ψ$ · d/trχ dσ $    dH dH 1 d/ log  + (d/T σ ) · d/ ψ L2 − ψ L2 dσ 2 dσ      dH 1 2 dH dH ψL
+ 2ψ L (d/ψ L )$ · d/ + |d/ψ L |2 + (T σ ) / 2 dσ dσ dσ

n˙ 1 = ψ L

(11.824)

Taking the  2,[m,l], ε0 norm, the contribution of the first term on the right in (11.824) t is estimated using Lemma 11.22. The contribution of the second term is estimated by applying Lemma 11.2.a and using Corollary 11.2.b and Corollary 11.2.d. Finally to estimate the contribution of the third term we apply the estimate (11.810) taking G = d H /dσ . In this way we deduce:

Q n˙ 1 2,[m,l], ε0 ≤ Cl δ0 (1 + t)−3 W{l+2} + δ0 (1 + t)−2 [1 + log(1 + t)]W{l+1} t  +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] (11.825) We also obtain: n˙ 1 2,[m,l∗ ], ε0 ≤ Cl δ02 (1 + t)−4

(11.826)

t

Next we consider the function n˙ 2 given by equation (11.639). In this equation the function n 2 is given by equations (9.60)–(9.62) of Chapter 9. Substituting in (11.639) we obtain the following expression for n˙ 2 : n˙ 2 = −

  1 dH / ψ L − ψ$ · d/trχ (Lσ ) ψ Lˆ
2 dσ  

 / ψ Lˆ + α −2 ψ$ · d/trχ + 2(d/ψ L )$ · d/ψ Lˆ +ψ L
      $ dH 1 dH ψ L ψ Lˆ
/ · d/(ψ L ψ Lˆ ) −(Lσ ) + d/ 2 dσ dσ   dH · (d/ Lσ )$ −d/ ψ L ψ Lˆ dσ  dH η0 (1 − u + η0 t)−1
−ψ L ψ Lˆ / σ + χ $$ · D / 2 σ + (d/σ )$ · i  dσ  $

−Fψ0 (Lψi )
/ Li − ω / L · d/trχ

$ / L · i + (d/ Lψi )$ · d/ L i −2Fψ0 ω

+η0 (1 − u + η0 t)−1 (L µ
/ ψµ ) + χ $$ · (L µ D / 2 ψµ ) −2(d/(Fψ0 ))$ · d/ω L L − ω L L
/ (Fψ0 )

 (11.827)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

623

Taking the  2,[m,l], ε0 norm, the contribution of the first term on the right in (11.827) is t estimated using Lemma 11.22. To estimate the contribution of the second term we apply the estimate (11.810) taking G = d H /dσ . The contribution of the third term is readily estimated. To estimate the contribution of the fourth term on the right in (11.827) we use the estimates (11.706), (11.724), Corollary 11.2.c and Corollary 11.2.d. The contribution of the fifth term is estimated using Lemma 11.22. Consider next the sixth term on the right in (11.827). To estimate the partial contri$ bution of ω / L · i , we use the estimate (11.724) and Corollary 11.2.d. To estimate the partial contribution of (d/ Lψi )$ · d/ L i we use the following: Under the assumptions of Proposition 11.2, for any St,u tangential tensorfield ξ we have:

ξ · d/ L i 2,[m,l], ε0 ≤ Cl (1 + t)−1 ξ 2,[m,l], ε0 t

t

+ ξ ∞,[m,l∗ ], ε0 (1 + t)

−1

B[m−1,l+1] + Y0 + (1 + t)A[l]  Q +W{l+1} + δ0 (1 + t)−2 W{l} (11.828) t

To demonstrate this we first estimate: ξ · d/ L i 2,[l], ε0 t

We write: / Ri1 (ξ · d/ L i ) = L / R in . . . L



((L / R )s1 ξ )(d/(R)s2 L i )

(11.829)

partitions |s1 |+|s2 |=n≤l

Here we distinguish the case where |s2 | ≤ l∗ (Case 1), from the complementary case where |s2 | ≥ l∗ + 1 so that |s1 | ≤ l∗ (Case 2). In Case 1 we appeal to Corollary 10.1.g with l∗ + 1 in the role of l and hypothesis H0 to estimate the corresponding term in the sum in (11.829) by: C(1 + t)−1 ξ 2,[l], ε0 t

In Case 2 we express L i as in equation (10.381) of Chapter 10, so that if s2 = {n 1 ,...,n p }, p = |s2 |, we have: (R)s2 L i = η0 (1 − u + η0 t)−1

( p) i x in ...in p 1

+η0 (1 − u + η0 t)−1

( p) i δin ...in p 1

+ (R)s2 z i

(11.830)

To estimate the partial contribution of the first term on the right in (11.830) we place it in L ∞ , obtaining an L 2 bound for the corresponding term in the sum in (11.829) as in Case 1. To estimate the partial contributions of the second and third terms on the right in (11.830) we place them in L 2 using Corollary 10.2.a and Corollary 10.2.g with l replaced by l + 1, respectively. We then obtain an L 2 bound for the corresponding term in the sum in (11.829) by:  Q Cl (1 + t)−1 ξ ∞,[l∗ ], ε0 Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l] t

624

Chapter 11. Control of the Spatial Derivatives

Combining the results of the two cases we conclude that:

ξ · d/ L i 2,[l], ε0 ≤ Cl (1 + t)−1 ξ 2,[l], ε0 t

t

+ ξ ∞,[l∗ ], ε0 Y0 + (1 + t)A[l] + W[l+1] + δ02 (1 + t)−2 W[l] Q



t

(11.831)

Next, we estimate: (L /T )k (ξ · d/ L i )2,[l−k], ε0 t

: for 1 ≤ k ≤ m

We write: /T )k ξ ) · d/ L i + (L /T )k (ξ · d/ L i ) = ((L

k  j =1

k! k− j ((L / ξ ) · d/(T ) j L i j !(k − j )! T

(11.832)

To estimate the contribution of the first term on the right in (11.832) we apply the estimate (11.831) taking (L /T )k ξ in the role of ξ . Since here we are taking the  2,[l−k], ε0 norm t instead of the  2,[l], ε0 norm, we may replace the  ∞,[l∗ ], ε0 norm on the right in t t (11.831) by the  ∞,[l∗ −k], ε0 norm. We then obtain: t

/T )k ξ 2,[l−k], ε0 ((L /T )k ξ ) · d/ L i 2,[l−k], ε0 ≤ Cl (1 + t)−1 (L t t  Q 2 + ξ ∞,[k,l∗ ], ε0 Y0 + (1 + t)A[l] + W[l+1] + δ0 (1 + t)−2 W[l] (11.833) t

To estimate the contribution of the sum in (11.832) we apply Corollary 11.2.a and Corollary 11.1.a with l∗ in the role of l. We then obtain: 0 0 0 0 k 0 0 k! k− j j i0 0 ((L /T ξ ) · d/(T ) L 0 0 0 0 j =1 j !(k − j )! ε0 2,[l−k],t

≤ Cl δ0 (1 + t)

−2

[1 + log(1 + t)]ξ 2,[k−1,l−1], ε0 t

1 −1 −1 B[k−1,l+1] + Cl (1 + t) ξ ∞,[k−1,l∗ ], ε0 (1 + t) t 2  +δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q +W{l+1} + δ0 (1 + t)−3 [1 + log(1 + t)]3 W{l−1}

(11.834)

Combining (11.833), (11.834), we obtain:

(L /T )k (ξ · d/ L i )2,[l−k], ε0 ≤ Cl (1 + t)−1 ξ 2,[k,l], ε0 t

t

+ ξ ∞,[k,l∗ ], ε0 (1 + t)−1 B[k−1,l+1] + Y0 + (1 + t)A[l] t  Q +W{l+1} + δ0 (1 + t)−2 W{l} : for 1 ≤ k ≤ m Combining then (11.831) and (11.835), yields the estimate (11.828).

(11.835)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

625

We apply (11.828) taking ξ = (d/ Lψi )$ to estimate the partial contribution of (d/ Lψi )$ · d/ L i in the sixth term on the right in (11.827). To estimate the partial contribution of the last two terms in parenthesis in the sixth term, we must estimate L µ D / 2 ψµ . We write: / 2 ψµ = D / 2 ψ0 + L i D / 2 ψi Lµ D For D / 2 ψµ we have the estimates (11.816), (11.817). To estimate L i D / 2 ψi we apply an i i analogue of the estimate (11.803) with L replacing Tˆ and the symmetric 2-covariant St,u tensorfield D / 2 ψi in the role of the function f . We obtain: / 2 ψµ 2,[m,l], ε0 L µ D

t Q −2 W{l+2} + δ0 (1 + t)−3 [1 + log(1 + t)]2 W{l} ≤ Cl (1 + t) +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

 (11.836)

To estimate the contribution of the seventh term on the right in (11.827) we apply the estimate for ω L L of Corollary 11.2.b with l replaced by l + 1. Finally, to estimate the contribution of the eighth term on the right in (11.827) we use the estimate (11.810) applied to G = Fψ0 . Combining the results for the eight terms on the right in (11.827) we conclude that:

Q n˙ 2 2,[m,l], ε0 ≤ Cl (1 + t)−3 W{l+2} + W{l+1} t  (11.837) +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] We also deduce:

n˙ 2 ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−4 t

(11.838)

function n˙ 2

given in terms of n˙ 2 by (11.638). Here we must estimate We turn to the the additional terms on the right. In regard to the second term on the right in (11.638) we have:     dH dH = (m + µe)ψ L ψ Lˆ + µψ L ψ Lˆ L L µψ L ψ Lˆ dσ dσ

 dH (11.839) +µ (Lψ L )ψ Lˆ + ψ L (Lψ Lˆ ) dσ Using the estimates (11.686), (11.690) with (m + 1, l + 1) reduced to (m, l) together with Corollary 11.2.e (and Corollary 11.1.e with l∗ in the role of l) we obtain: 0  0 0 0 0 L µψ L ψ ˆ d H 0 0 L dσ 0 ε 2,[m,l],t 0

Q ≤ Cl W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} 1 2  . (11.840) +δ0 (1 + t)−2 B[m,l] + [1 + log(1 + t)] Y0 + (1 + t)A[l−1] The contribution of the second term on the right in (11.638) is then estimated using (11.839) together with the estimate (11.706).

626

Chapter 11. Control of the Spatial Derivatives

In regard to the third term on the right in (11.638) we have: L(µFψ0 ) = (m + µe)Fψ0 + µL(Fψ0 )

(11.841)

and, from (11.672): L(µFψ0 L i ) = (L(µFψ0 ))L i + µFψ0 (a L Tˆ i + b L · d/x i )

(11.842)

We first obtain the estimates: L(µFψ0 )∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−1

(11.843)

t

and: L(µFψ0 )2,[m,l], ε0 t

Q ≤ Cl W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} 1 2  +δ0 (1 + t)−2 B[m,l] + [1 + log(1 + t)] Y0 + (1 + t)A[l−1]

(11.844)

To estimate then the first term on the right in (11.842) we apply the analogue of the estimate (11.803) with L i replacing Tˆ i and L(µFψ0 ) in the role of the function f . To estimate a L Tˆ i in the second term on the right in (11.842) we apply the estimate (11.803) itself and use the estimates for the function a L of Corollary 11.2.e and Corollary 11.1.e with l∗ in the role of l. Finally to estimate b L · d/x i in the second term on the right in (11.842) we apply Lemma 11.18 and use the estimates for b L of Corollary 11.2.e and Corollary 11.1.e with l∗ in the role of l. We obtain in this way the estimates: L(µFψ0 L i )2,[m,l], ε0 t

Q ≤ Cl W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} +δ0 (1 + t)−1 (1 + t)−1 B[m,l] + Y0 + (1 + t)A[l−1] and:



L(µFψ0 L i )∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 t

(11.845)

(11.846)

The contribution of the third term on the right in (11.638) is then estimated by applying Lemma 11.2.a and using the estimates (11.845), (11.846) together with the estimates (11.816), (11.817). Combining finally with the previous estimates (11.837), (11.838) for n˙ 2 , we conclude that:

  Q n˙ 2 2,[m,l], ε0 ≤ Cl (1 + t)−3 [1 + log(1 + t)] W{l+2} + W{l+1} t  1 2 +δ0 (1 + t)−1 B[m,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] (11.847) and:

n˙ 2 ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−4 [1 + log(1 + t)] t

(11.848)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

627

We now turn to the remaining terms in the expression (11.633) for gˇ˙  . We first consider the functions f 0 and f1 , given by equations (9.49) and (9.64) of Chapter 9 respectively: 1 2 dH ψ (LT σ ) 2 L dσ dH 1 (
/ σ ) − Fψ0 L µ (
/ ψµ ) f 1 = − ψ L ψ Lˆ 2 dσ

f 0 =

(11.849) (11.850)

To estimate f 0 we first estimate LT σ . Writing LT σ = 2(T )2 σ + α −1 κ T Lσ + α −1 κ · d/σ and using the estimates for  of Corollary 11.1.c with l∗ in the role of l and of Corollary 11.2.c we obtain: (11.851) L T σ ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 t

and:

Q L T σ 2,[m,l], ε0 ≤ Cl W{l+2} + (1 + t)−1 [1 + log(1 + t)]W{l+1} (11.852) t   +δ0 (1 + t)−2 B[m,l+1] + δ0 (1 + t)−1 [1 + log(1 + t)]2 Y0 + (1 + t)A[l−1]

Using these estimates we deduce:  f0 ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−1

(11.853)

Q  f0 2,[m,l], ε0 ≤ Cl W{l+2} + (1 + t)−1 [1 + log(1 + t)]W{l+1} t 1   + δ0 (1 + t)−2 B[m,l+1] + δ0 Y0 + (1 + t)A[l−1]

(11.854)

t

and:

Using the estimates (11.706) and (11.836) we deduce:  f1 2,[m,l], ε0

Q ≤ Cl (1 + t)−2 W{l+2} + δ0 (1 + t)−3 [1 + log(1 + t)]2 W{l}  +δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] t

We also obtain:

 f1 ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−3 t

(11.855)

(11.856)

With the help of the estimates (11.853)–(11.856) we can estimate, in  2,[m,l], ε0 t norm, the third and fourth terms on the right in (11.633) in a straightforward manner. The second term on the right in (11.633) is likewise estimated using the estimates (11.690) (with m + 1 reduced to m) and (11.724). Finally, we have −ξ $ · d/ fˇ

628

Chapter 11. Control of the Spatial Derivatives

the first term on the right in (11.633). Here ξ is the St,u 1-form given by equation (9.83) of Chapter 9: /L ξ = −d/µ − 2µFψ0 ω

 dH [ψ L (T σ ) − κψTˆ (Lσ )] ψ − κψ L (α −1 ψ L + ψTˆ )(d/σ ) (11.857) + dσ and fˇ is the function defined by equation (8.54) of Chapter 8: fˇ = −2(m + µe)

(11.858)

which plays a leading role in that chapter. Now e is a function with similar properties to e, satisfying an estimate similar to (11.690). In view of (11.686) and (11.690) (with m + 1 reduced to m) we obtain: d/ fˇ2,[m,l], ε0

(11.859)

Q ≤ Cl (1 + t)−1 W{l+2} + (1 + t)−1 [1 + log(1 + t)]W{l+1} 1 2  +δ0 (1 + t)−2 B[m,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] t

We also obtain:

d/ fˇ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−2 t

(11.860)

Next, using Corollary 11.1.b with l∗ in the role of l and Corollary 11.2.b we deduce: ξ ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

and:

(11.861)

  Q ξ 2,[m,l], ε0 ≤ Cl (1 + t)−1 [1 + log(1 + t)] W{l+1} + δ02 (1 + t)−2 W{l} t   (11.862) +B[m,l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1]

Applying then Corollary 11.2.d yields: ξ $ 2,[m,l], ε0 ≤ Cl (1 + t)−1 ·

t   Q [1 + log(1 + t)] W{l+1} + δ0 (1 + t)−2 [1 + log(1 + t)]2 W{l}   +B[m,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] (11.863) We also obtain: ξ $ ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(11.864)

The first term on the right in (11.633) is then readily estimated by applying Lemma 11.2.a and using the estimates (11.859), (11.860), (11.863), (11.864).

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

629

Combining finally the above results with the earlier results (11.766), (11.825) and (11.826), (11.847) and (11.848), yields, through expression (11.633), the following proposition. Proposition 11.3 Let the hypotheses H0, H1, H2 and the estimate (10.30) of Chapter / [l∗ ] ,, hold for 10 hold. Let also the bootstrap assumptions E{l∗ +2} , E{lQ∗ +1} , E{lQ∗Q} and X some non-negative integer l. Moreover, let the bootstrap assumption M[m+1,l∗ +1] hold for some non-negative integer m ≤ l. Then if δ0 is suitably small (depending on l) we have: gˇ˙  2,[m,l], ε0

  Q QQ [1 + log(1 + t)] W{l+2} + W{l+1} + W{l} 1 2  +δ0 (1 + t)−1 B[m+1,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] t

≤ Cl (1 + t)

−2

ε

 , defined by (9.270)– We proceed to estimate in L 2 (t 0 ) the functions (i1 ...in ) g˙ m,n (9.271) of Chapter 9, for n = l − m. Note that in Chapter 9 l stands for the number of angular derivatives, whereas in the present Chapter l stands for the total number of spatial derivatives, the number of T derivatives in both chapters being denoted by m. Thus, the number of angular derivatives, presently denoted by n, corresponds to l − m in Chapter  are defined in the statements 9. In the formulas (9.270)–(9.271) the functions (i1 ...in ) gm,n  of Propositions 9.1 and 9.2, in terms of the function gˇ . We shall presently re-express the  in terms of the function gˇ˙  , which has been estimated by Proposition functions (i1 ...in ) g˙ m,n 11.3. Let us define the functions: (i1 ...in )

 wm,n =

m−1 

 Rin . . . Ri1 (T )k x m−k−1,0

k=0

− m (i1 ...in )

 wm,n =

n−1 

(i1 ...in )  x m−1,n

Rin . . . Rin−k+1

(11.865)

(Rin−k )

Z

(i1 ...in−k−1 )  x m,n−k−1

k=0

−

n−1 

(Rin−k )

Z

i



(i1 .n−k . . in )  x m,n−1

(11.866)

k=0  , (i1 ...in ) w  According to the discussion in Chapter 9, the functions (i1 ...in ) wm,n m,n do not contain terms of the top order l + 2. From Propositions 9.1 and 9.2 we then have: (i1 ...in )  gm,n

− m

(i1 ...in )  x m−1,n

m 

= Rin . . . Ri1 (T ) gˇ + +

m−1  k=0

(i1 ...in )

−

n−1 

(Rin−k )

Z

k=0   wm,n + (i1 ...in ) wm,n n−1 

 Rin . . . Ri1 (T )k ym−k,0 +

i

Rin . . . Rin−k+1

k=0



(i1 .n−k . . in )  x m,n−1

(i1 ...in−k )  ym,n−k

(11.867)

630

Chapter 11. Control of the Spatial Derivatives

We now substitute for gˇ  in terms of gˇ˙  from (11.631). Consider first the case m = 0. We have: (11.868) Rin . . . Ri1 gˇ  = Rin . . . Ri1 gˇ˙  + ξ $ · (i1 ...in ) x n + (i1 ...in ) u n where

(i1 ...in ) u

are the functions defined by:   (i1 ...in ) un = ξ $ · ((R)s1 µ)d/(R)s2 trχ + (L / R )s1 ξ $ · (L / R )s 2 x 0 n

partitions s1 =∅

(11.869)

partitions s1 =∅

Here, in both sums we are considering all ordered partitions {s1 , s2 } of the set {1, . . . , n} into two ordered subsets s1 , s2 , with s1 non-empty. Note that the functions (i1 ...in ) u n do not contain terms of order n + 2. Substituting (11.868) in (11.867) with m = 0 and  = 0, comparing with (9.270) we obtain, noting that (i1 ...in ) w0,n (i1 ...in )  g˙ 0,n

= Rin . . . Ri1 gˇ˙  + +

n−1 

(i1 ...in )

Rin . . . Rin−k+1

un +

(i1 ...in )

 w0,n

(i1 ...in−k )  y0,n−k

(11.870)

k=0

For m ≥ 1 we first consider (see (11.631)): T gˇ  = T gˇ˙  + ξ $ · (µd/T trχ + d/T fˇ) +ξ $ · (T µ)d/trχ + (L /T ξ $ ) · x 0

(11.871)

Recalling from Chapter 9 that:  = µ
/ µ − fˇ x 0,0

we can write (11.871) in the form:  T gˇ  = T gˇ˙  + ξ $ · d/x 0,0 + u 0,0

(11.872)

/T ξ $ ) · x 0 u 0,0 = ξ $ · x˙0 + (L

(11.873)

Here, u 0,0 is the function:

where x˙0 is the St,u 1-form: x˙0 = µd/(T trχ  −
/ µ) + (T µ)d/trχ  − (d/µ)
/ µ + d/(T fˇ + fˇ ) Note, from (11.650)–(11.651) that:   ( P) (N) / ζ − tr γ −tr γ / µ = −tr (T ) π /$ · χ $ + tr + div T trχ  −


(11.874)

(11.875)

is of order 2 but contains no acoustical part of order 2. Thus u 0,0 is of order 3, but contains no acoustical part of order 3. Applying (T )m−1 to (11.872) and recalling from Chapter 9 that:  = µRin . . . Ri1 (T )m
/ µ − Rin . . . Ri1 (T )m fˇ x m,n

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

we obtain:

631

 + u m−1,0 (T )m gˇ  = (T )m gˇ˙  + ξ $ · d/x m−1,0

(11.876)

u m−1,0 = (T )m−1 u 0,0 + u m−1,0

(11.877)

(m − 1)! ξ $ · d/(((T )k µ)(T )m−1−k
/ µ) k!(m − 1 − k)!     + (L /T )k ξ $ · d/(T )m−1−k x 0,0

(11.878)

where: and: u m−1,0 =

m−1  k=1

Note that the function u m−1,0 is of order m + 1. Applying Rin . . . Ri1 to (11.876) we then obtain: Rin . . . Ri1 (T )m gˇ  = Rin . . . Ri1 (T )m gˇ˙  + ξ $ · d/ Here

(i1 ...in ) u  m−1,n

+

(i1 ...in )  u m−1,n

(11.879)

are the functions defined by:

(i1 ...in )  u m−1,n

where

(i1 ...in )  x m−1,n

(i1 ...in ) u  m−1,n

= Rin . . . Ri1 (T )m−1 u 0,0 +

(i1 ...in )  u m−1,n

(11.880)

are the functions defined, in terms of the u m−1,0 , by:

(i1 ...in )  u m−1,n

= Rin . . . Ri1 u m−1,0    +ξ $ · d/ ((R)s1 µ)(R)s2 (T )m−1
/µ partitions s1 =∅



+

 (L / R )s1 ξ $ · d/(R)s2 x m−1,0

(11.881)

partitions s1 =∅

Here again, in both sums we are considering all ordered partitions {s1 , s2 } of the set {1, . . . , n} into two ordered subsets s1 , s2 , with s1 non-empty. Note that the functions (i1 ...in ) u  m−1,n do not contain terms of order m + n + 2. Substituting (11.879) in (11.867) and comparing with (9.271) we obtain, for m ≥ 1: (i1 ...in )  g˙ m,n

= Rin . . . Ri1 (T )m gˇ˙  + (i1 ...in ) u m−1,n m−1 

+

Ri n . . .

k=0

+

(11.882) (i1 ...in )

 wm,n

 Ri1 (T )k ym−k,0

+

(i1 ...in )

+

n−1 

 wm,n

Rin . . . Rin−k+1

(i1 ...in−k )  ym,n−k

k=0

The first term on the right in (11.870) and (11.882) is, for n = l − m, estimated in ε ε L 2 (t 0 ) by Proposition 11.3. We proceed to estimate, in L 2 (t 0 ), the remaining terms on the right in (11.870) and (11.882). We begin by deriving a bound for u 0,0 2,[m−1,l−1], ε0 t

632

Chapter 11. Control of the Spatial Derivatives

Consider first (11.875). Here, the terms on the right have already been estimated in esL ((11.653)), the sum timating v˙2 . In particular, the 4th term on the right occurs in v˙2,P L of the 1st, 2nd and 5th terms occurs in v˙2,N ((11.654)), while the 3rd term occurs in L

v˙2,P ((11.656)). Combining the estimates, we obtain, under the assumptions of Proposition 11.3: / µ2,[m,l], ε0 T trχ  −
t

Q ≤ Cl (1 + t)−1 W{l+2} + (1 + t)−1 [1 + log(1 + t)]W{l+1} 1 2  +(1 + t)−1 B[m,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l]

(11.883)

Under the same assumptions we also obtain the L ∞ estimate: T trχ  −
/ µ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

(11.884)

t

To estimate D / 2 µ, we consider expression (11.390). In regard to the principal term on the right, we have, by hypotheses H1 and H0: max

max D / 2 Rin . . . Ri1 (T ) j µ L 2 ( ε0 ≤ C(1 + t)−2 B[m−1,l+1]

j ≤m−1,n≤l−1− j i1 ...in

t

In regard to the commutator term, we apply the second statement of Corollary 11.2.g (with (m, l) reduced to (m − 1, l − 1)) to the St,u 1-for d/µ to obtain: max 

(i1 ...in )

c j,n [d/µ] L 2 ( ε0 ) t

Q ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] W{l} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1}  +(1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1] max

j ≤m−1,n≤l−1− j i1 ...in

It follows that:

 D / 2 µ2,[m−1,l−1], ε0 ≤ Cl (1 + t)−2 B[m−1,l+1] t 1 + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] Q +W{l} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1}

 (11.885)

We also obtain, using assumption M[m−1,l∗ +1] and the second statement of Corollary 11.1.g (with l∗ − 1 in the role of l): D / 2 µ∞,[m−1,l∗ −1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.886)

Applying Corollary 11.2.d (with (m, l) reduced to (m − 1, l − 1)) then yields:  
/ µ2,[m−1,l−1], ε0 ≤ Cl (1 + t)−2 B[m−1,l+1] t 1 + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q + W{l} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1} (11.887)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

633

We also obtain: 
/ µ∞,[m−1,l∗ −1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.888)

The function fˇ is defined by (9.67) of Chapter 9: fˇ = f 0 + µ2 f 1 The estimates (11.853)–(11.856) imply:

Q  fˇ 2,[m,l], ε0 ≤ Cl W{l+2} + (1 + t)−1 [1 + log(1 + t)]W{l+1} t 1 2 + δ0 (1 + t)−2 B[m,l+1] + Y0 + (1 + t)A[l] and:

 fˇ ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 t

(11.889)

(11.890)

(11.891)

Using the estimates (11.883)–(11.884), (11.887)–(11.888), (11.859)–(11.860), (11.890)– (11.891), we deduce through (11.874) the following estimates for x˙0 : x˙0 2,[m−1,l−1], ε0

t Q −1 ≤ Cl (1 + t) W{l+2} + (1 + t)−1 [1 + log(1 + t)]W{l+1}

(11.892)

+(1 + t)−1 [1 + log(1 + t)] (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] x˙0 ∞,[m−1,l∗ −1], ε0 ≤ Cl δ0 (1 + t)−2 t



(11.893)

On the other hand, Corollary 11.2.c (with m reduced to m − 1) and (11.859) (with (m, l) reduced to (m − 1, l − 1)) imply the following L 2 estimate for x 0 : x 0 2,[m−1,l−1], ε0

t Q −1 W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} ≤ Cl (1 + t)

(11.894)

+(1 + t)−1 [1 + log(1 + t)] (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]



We also obtain, using assumption X / [l∗ ] , the L ∞ estimate: x 0 ∞,[m−1,l∗ −1], ε0 ≤ Cl δ0 (1 + t)−2 t

(11.895)

However, the estimate (11.895) is insufficient to estimate the product (L /T ξ $ ) · x 0 in ε  2,[m−1,l−1], 0 norm. For, without strengthening assumption M[m,l∗ +1] the L ∞ est timate (11.864) cannot be improved, and that, together with (11.894), allows us to handle the case where the factor L /T ξ $ receives at most l∗ − 1 spatial derivatives. In the complementary case, the factor x 0 receives at most l −1 −l∗ = (l +1)∗ −1 spatial derivatives. To

634

Chapter 11. Control of the Spatial Derivatives

handle this case, we must strengthen assumption X / [l∗ ] , to X / [(l+1)∗ ] , as in Lemma 10.19. This assumption, together with (11.860), implies the L ∞ estimate: x 0 ∞,[m−1,(l+1)∗ −1], ε0 ≤ Cl δ0 (1 + t)−2 t

(11.896)

which, together with (11.863) allows us to handle the complementary case in question. It follows that: (L /T ξ $ ) · x 0 2,[m−1,l−1], ε0 (11.897) t

  Q ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l}   +(1 + t)−1 B[m,l+1] + [1 + log(1 + t)]2 Y0 + (1 + t)A[l] Using the estimates (11.892)–(11.893) together with (11.863)–(11.864) we also obtain, applying Lemma 11.2.b with x˙0 in the role of ϑ, ξ $ · x˙ 0 2,[m−1,l−1], ε0 (11.898)

t   Q ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] W{l+2} + (1 + t)−1 [1 + log(1 + t)]W{l+1}   +(1 + t)−1 B[m,l+1] + [1 + log(1 + t)]2 Y0 + (1 + t)A[l] Combining finally (11.887), (11.898), we conclude that:

u 0,0 2,[m−1,l−1], ε0 ≤ Cl δ0 (1 + t)−2 (1 + t)−1 B[m,l+1] + [1 + log(1 + t)]· t   Q (11.899) · W{l+2} + (1 + t)−1 [1 + log(1 + t)] W{l+1} + Y0 + (1 + t)A[l] Next, we derive a bound for u m−1,0 2,[l−m], ε0 t

Consider the contribution to this of a term k ∈ {1, . . . , m − 1} in the sum in (11.878). By the estimates (11.863)–(11.864) and (11.887)–(11.888), we obtain, in regard to the first term in parenthesis, ξ $ · d/(((T )k µ)(T )m−1−k
/ µ)2,[l−m], ε0 ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)]2 · t    B[m−1,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]   Q +[1 + log(1 + t)] W{l} + δ0 (1 + t)−2 [1 + log(1 + t)]2 W{l−1} : for k ∈ {1, . . . , m − 1}

(11.900)

In regard to the second term in parenthesis, we may appeal to the estimate (11.864) in the case that the factor (L /T )k ξ $ receives at most l∗ − k angular derivatives. In the  receives at most l − m − l − 1 + k = complementary case, the factor d/(T )m−1−k x 0,0 ∗

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

635

 receives a total of at most (l + 1) − 1 (l + 1)∗ − 1 − m + k angular derivatives, that is, x 0,0 ∗ spatial derivatives of which at most m − 2 are T -derivatives. It follows that the assumption M[m−2,(l+1)∗ +1] is needed to handle the complementary case. We introduce, in fact, the assumption M[m,(l+1)∗ +1] . This assumption, together with (11.891) implies (compare with (11.888)):  ∞,[m,(l+1)∗ −1], ε0 ≤ Cl δ0 (1 + t)−1 (11.901) x 0,0 t

which, together with (11.863) suffices to handle the complementary case. We thus obtain, in regard to the second term in parenthesis in (11.878),  ((L /T )k ξ $ ) · d/(T )m−1−k x 0,0 2,[l−m], ε0 ≤ Cl δ0 (1 + t)−2 · t

 1 2 (1 + t)−1 B[m−1,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]   Q +[1 + log(1 + t)] W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l}

: for k ∈ {1, . . . , m − 1}

(11.902)

Combining the estimates (11.900) and (11.902) we conclude, through (11.878), that: u m−1,0 2,[l−m], ε0 t

  Q −2 [1 + log(1 + t)] W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} ≤ Cl δ0 (1 + t) 1 2  (11.903) + (1 + t)−1 B[m−1,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] ε

The above estimate bounds in L 2 (t 0 ) the first term on the right in (11.881) for n = l − m. Consider next the second term on the right in (11.881). Using the estimates (11.863)–(11.864), (11.887)–(11.888) we deduce that for n = l − m this is bounded in ε L 2 (t 0 ) by:  Cl δ02 (1 + t)−4 [1 + log(1 + t)]2 B[m−1,l+1] + δ0 [1 + log(1 + t)]·  Q (11.904) · Y0 + (1 + t)A[l−1] + W{l} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1} Consider finally the third term on the right in (11.881), for n = l − m. In the case that |s1 | ≤ l∗ we place the factor (L / R )s1 ξ $ in L ∞ appealing to (11.864), and use the estimate

1  x m−1,0 2,[l−m], ε0 ≤ Cl (1 + t)−2 [1 + log(1 + t)] B[m−1,l+1] t  2 + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q (11.905) + W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} which follows from (11.887), (11.890). In the complementary case we have |s2 | = l − m − |s1 | ≤ l − m − l∗ − 1 = (l + 1)∗ − 1 − m

636

Chapter 11. Control of the Spatial Derivatives

 It follows that to place the factor d/(R)s2 x m−1,0 in L ∞ assumption M[m−1,(l+1)∗ +1] is required. We are in fact assuming M[m,(l+1)∗ +1] , which, together with (11.891) implies (compare with (11.901)):  x m−1,0 ∞,[(l+1)∗ −m], ε0 ≤ Cl δ0 (1 + t)−1 t

(11.906)

This, together with (11.863) suffices to handle the complementary case. In this way we ε obtain that for n = l − m the third term on the right in (11.881) is bounded in L 2 (t 0 ) by:

  Q Cl δ0 (1 + t)−2 [1 + log(1 + t)] W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} (11.907)   +(1 + t)−1 B[m−1,l+1] + δ0 (1 + t)−2 [1 + log(1 + t)]3 Y0 + (1 + t)A[l−1] Combining the estimates (11.903), (11.904), (11.907), we conclude, through (11.881), that: max 

(i1 ...il−m )  u m−1,l−m  L 2 ( ε0 ) t

(11.908)   Q ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} 1 2  +(1 + t)−1 B[m−1,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]

i1 ...il−m

Combining finally the estimates (11.899) and (11.908) we conclude, through (11.880), that for m ≥ 1: max 

i1 ...il−m

(i1 ...il−m )  u m−1,l−m  L 2 ( ε0 ) t

(11.909) 

 Q ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] W{l+2} + (1 + t)−1 [1 + log(1 + t)]W{l+1} 1 2  +(1 + t)−1 B[m,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]

under the assumptions of Proposition 11.3, augmented by the assumptions X / [(l+1)∗ ] and M[m,(l+1)∗ +1] . ε For m = 0 we must estimate, in L 2 (t 0 ), the functions (i1 ...il ) u l given by (11.869) with n = l. The contribution of the first term on the right in (11.869) is readily seen to be bounded by:

 (11.910) Cl δ02 (1 + t)−3 [1 + log(1 + t)]2 (1 + t)−1 B[0,l] + (1 + t)A[l] In regard to the second term on the right in (11.869), in the case that |s1 | ≤ l∗ we appeal to (11.864) with m = 0 to place the factor (L / R )s1 ξ $ in L ∞ . In the complementary case / R )s2 x 0 in we have |s2 | = l − |s1 | ≤ l − l∗ − 1 = (l + 1)∗ − 1, thus, to place the factor (L ∞ L , we must again appeal to assumption X / [(l+1)∗ ] . Using also  x 0 2,[l−1], ε0 ≤ Cl (1 + t)−1 [1 + log(1 + t)]A[l] t

Q +W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l}   + δ0 (1 + t)−2 B[0,l] + [1 + log(1 + t)]Y0

(11.911)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

637

which follows from (11.859) with (m, l) replaced by (0, l − 1), as well as the estimate (11.863) with m = 0, we obtain that the contribution of the second term in (11.869) is bounded by:

  Q Cl δ0 (1 + t)−2 [1 + log(1 + t)] W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l}   +(1 + t)−1 B[0,l+1] + [1 + log(1 + t)]2 Y0 + (1 + t)A[l] (11.912) Combining then the bounds (11.910), (11.912), we obtain: max 

(i1 ...il )

u l  L 2 ( ε0 ) (11.913) t

  Q ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l}   +(1 + t)−1 B[0,l+1] + [1 + log(1 + t)]2 Y0 + (1 + t)A[l] i1 ...il

under the assumptions of Proposition 11.3, augmented by assumption X / [(l+1)∗ ] . We turn to the functions set n = l − m. We have:

(i1 ...in ) w  m,n

given by (11.865). Here m ≥ 1 and we are to

   = x m−1,0 + Sk,m−1 (T )k x m−k−1,0

(11.914)

where:  = Sk,m−1

k  j =1

k! ((T ) j µ)(T )m− j −1
/µ j !(k − j )!

(11.915)

and:  Rin . . . Ri1 x m−1,0 =

(i1 ...in )  x m−1,n

+

(i1 ...in )

Sm−1,n

(11.916)

((R)s1 µ)((R)s2 (T )m−1
/ µ)

(11.917)

where: (i1 ...in )

Sm−1,n =



partitions |s1 |+|s2 |=n,s1 =∅

By (11.914),     = [(T )k , ]x m−k−1,0 + x m−1,0 + Sk,m−1 (T )k x m−k−1,0

(11.918)

and by (11.916),  Rin . . . Ri1 x m−1,0  = [Rin . . . Ri1 , ]x m−1,0 +

(i1 ...in )  x m−1,n

+

(i1 ...in )

Sm−1,n

(11.919)

638

Chapter 11. Control of the Spatial Derivatives

In view of (11.918)–(11.919), (11.865) takes the form: (i1 ...in )

  wm,n = m[Rin . . . Ri1 , ]x m−1,0 m−1 

+

   Rin . . . Ri1 [(T )k , ]x m−k−1,0

k=1 m−1 

+

 Rin . . . Ri1 Sk,m−1

k=1

+m

(i1 ...in )

Sm−1,n

(11.920)

To express the commutators in (11.920), we shall apply the following lemma. Lemma 11.23 Let A1 , . . . , An and B be linear operators acting on some space X. We then have: [ An . . . A1 , B] =

n−1 

An . . . An−m+1 [ An−m , B] An−m−1 . . . A1

m=0

Proof. Setting, for each m = 1, . . . , n, Cm = [ Am . . . A1 , B] we derive the following recursion relation: Cm = Am Cm−1 + [ Am , B] Am−1 . . . A1 This holds also for m = 1 if we set C0 = 0. The lemma then follows by applying Proposition 8.2 to this recursion. By Lemma 11.23, for any function φ we have: [Rin . . . Ri1 , ]φ =

n−1 

  / Rin−q  · d/ Rin−q−1 . . . Ri1 φ Rin . . . Rin−q+1 L

(11.921)

q=0

and: [(T )k , ]φ =

k−1 

  (T ) p L /T  · d/(T )k− p−1 φ

p=0

=

k−1 

 Nr,k−1 (L /T )r+1  · d/(T )k−r−1 φ

(11.922)

r=0  are the positive integers: where the coefficients Nr,k−1  Nr,k−1 =

k−1  p=r

p! r !( p − r )!

 Note that Nr,k−1 = Nk−1−r,k−1 (see (11.186)).

(11.923)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

639

Consider the first term on the right in (11.920). In regard to this term we have  . Consider a term q ∈ {0, . . . , n − 1} in the sum in (11.921). (11.921) with φ = x m−1,0 There are q angular derivatives outside the parenthesis. If at most l∗ −1 angular derivatives fall on L / Rin−q , we place the corresponding factor in L ∞ using Corollary 11.1.c with l∗ in  , the role of l. Otherwise, at most q − l∗ angular derivatives fall on d/ Rin−q−1 . . . Ri1 x m−1,0  resulting in a factor which is an angular derivative of x m−1,0 of order at most n − l∗ = (l + 1)∗ − m. We then place this factor in L ∞ using the estimate (11.906) (which relies on assumption M[m,(l+1)∗ +1] ). We thus obtain, with n = l − m, m ≥ 1,  [Rin . . . Ri1 , ]x m−1,0  L 2 ( ε0 ) (11.924) t

  ≤ Cl δ0 (1 + t)−2 2,[n], ε0 + [1 + log(1 + t)]x m−1,0 2,[n], ε0 t t

  Q −2 −1 [1 + log(1 + t)] W{l+1} + (1 + t) [1 + log(1 + t)]W{l} ≤ Cl δ0 (1 + t) 1 2  +(1 + t)−1 B[m−1,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]

by Corollary 11.2.c and the estimate (11.905). Consider next a term k ∈ {1, . . . , m − 1} in the sum, which is the second term on the right in (11.920). Here, m ≥ 2 and we are again to set n = l − m. To bound the term ε in question in L 2 (t 0 ), we must bound:  [(T )k , ]x m−k−1,0 2,[l−m], ε0

(11.925)

t

 . Let us consider the contribution In regard to this we have (11.922) with φ = x m−k−1,0  (L /T )r+1  · d/(T )k−r−1 x m−k−1,0 2,[l−m], ε0 t

(11.926)

of a term r ∈ {0, . . . , k − 1} in the sum on the right in (11.922) to (11.925). If at most l∗ − r − 1 angular derivatives fall on (L /T )r+1  we place the corresponding factor in L ∞ using Corollary 11.1.c with l∗ in the role of l. Otherwise, at most l − m − l∗ + r =  (l + 1)∗ − m + r angular derivatives fall on d/(T )k−r−1 x m−k−1,0 , resulting in a factor which is bounded in L ∞ by:  C(1 + t)−1 x m−k−1,0 ∞,[k−r−1,(l+1)∗ −m+k], ε0 ≤ Cl δ0 (1 + t)−2 t

(11.927)

by assumption M[m,(l+1)∗ +1] and the estimate (11.891) (compare with (11.901)). Noting also that, from (11.914),    = x m−2−r,0 + Sk−r−1,m−r−2 , (T )k−r−1 x m−k−1,0

we then obtain that (11.926) is bounded by:

Cl δ0 (1 + t)−2 2,[m−1,l−1], ε0 (11.928) t     +[1 + log(1 + t)] x m−2−r,0 2,[l+1−m], ε0 + Sk−r−1,m−r−2 2,[l+1−m], ε0 t

t

640

Chapter 11. Control of the Spatial Derivatives

Using Corollary 11.2.c, (11.905), and the estimate:  2,[l+1−m], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] · Sk,m−1 t  1 B[m−1,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] Q +W{l} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1}

: for k ∈ {1, . . . , m − 1}

 (11.929)

which follows from (11.887)–(11.888), we then obtain that (11.928) is bounded by:

  Q Cl δ0 (1 + t)−2 [1 + log(1 + t)] W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} 1 2  (11.930) +(1 + t)−1 B[m−1,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] It follows that (11.925), hence also the second term on the right in (11.920), is similarly bounded. ε The third term on the right in (11.920) is bounded in L 2 (t 0 ) by (n = l − m): m−1 

  · d/ Sk,m−1 2,[l−m], ε0 t

k=1

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] δ0 (1 + t)−1 [1 + log(1 + t)]2,[l−m], ε0 t   ε + max Sk,m−1 2,[l+1−m], 0 t

1≤k≤m−1 2 Cl δ0 (1 + t)−4 [1 +

2

log(1 + t)] · ≤

  Q [1 + log(1 + t)] W{l} + δ0 (1 + t)−2 [1 + log(1 + t)]2 W{l−1}   + B[m−1,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]

(11.931)

by Corollary 11.2.c and the estimate (11.929). ε Finally, the fourth term on the right in (11.920) is bounded in L 2 (t 0 ) by (n = l − m): Cl δ0 (1 + t)−1 [1 + log(1 + t)]d/

(i1 ...il−m )

Sm−1,l−m  L 2 ( ε0 ) t

(11.932)

Using the estimates (11.887)–(11.888) we readily deduce: d/

(i1 ...il−m )

Sm−1,l−m  L 2 ( ε0 )

(11.933) 

/ µ2,[m−1,l−1], ε0 + B[0,l] ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] (1 + t)2 
t  −3 ≤ Cl δ0 (1 + t) [1 + log(1 + t)] B[m−1,l+1] + δ0 [1 + log(1 + t)]·  Q Y0 + (1 + t)A[l−1] + W{l} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1} t

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

641

hence (11.932) is bounded by:  Cl δ02 (1 + t)−4 [1 + log(1 + t)]2 B[m−1,l+1] + δ0 [1 + log(1 + t)]· Q Y0 + (1 + t)A[l−1] + W{l} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1}

(11.934) 

Combining then the results (11.924), (11.930), (11.931), (11.934), for the four terms on the right in (11.920) we conclude that:  wm,l−m  L 2 ( ε0 (11.935) t

  Q ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} 1 2  +(1 + t)−1 B[m−1,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]

max 

(i1 ...il−m )

i1 ...il−m

under the assumptions of Proposition 11.3, augmented by assumption M[m,(l+1)∗ +1] . (i1 ...in ) w  m,n

We turn to the functions n = l − m. We have: Rin . . . Rin−k+1

(i1 ...in−k−1 )  x m,n−k−1

=

given by (11.866). Here again we are to set i



(i1 .n−k . . in )  x m,n−1

+

i



(i1 .n−k . . in )  Sm,n−1

(11.936)

where: i



(i1 .n−k . . in )  Sm,n−1



=



((R)s1 µ)(R)s2 (T )m
/ µ, s2 = s2



{1, . . . , n − k − 1}

partitions |s1 |+|s2 |=k, s1 =∅

(11.937)

and we are considering all ordered partitions {s1 , s2 } of the set {n − k + 1, . . . , n} into two ordered subsets s1 , s2 , with s1 non-empty. By (11.936), Rin . . . Rin−k+1

(Rin−k )

= [Rin . . . Rin−k+1 , +

(Rin−k )

Z

(i1 ...in−k−1 )  x m,n−k−1 (Rin−k )  Z ] (i1 ...in−k−1 ) x m,n−k−1

Z

i



(i1 .n−k . . in )  x m,n−1

+

(Rin−k )

Z

i



(i1 .n−k . . in )  Sm,n−1

(11.938)

In view of (11.938), (11.866) takes the form: (i1 ...in )

 wm,n =

n−1 

[Rin . . . Rin−k+1 ,

(Rin−k )

Z]

(i1 ...in−k−1 )  x m,n−k−1

k=0

+

n−1  k=0

(Rin−k )

Z

i



(i1 .n−k . . in )  Sm,n−1

(11.939)

642

Chapter 11. Control of the Spatial Derivatives

Consider a term k ∈ {0, . . . , n − 1} in the first sum on the right in (11.939). By Lemma 11.23 the term in question is given by: [Rin . . . Rin−k+1 , =

k−1 

(Rin−k )

Rin . . . Rin− j +1

j =0

with φ =

Z ]φ

 L / Rin− j

(i1 ...in−k−1 ) x  m,n−k−1 .

(Rin−k )

  Z · d/ Rin− j −1 . . . Rin−k+1 φ

(11.940)

There are j angular derivatives outside the parenthesis in

the sum on the right in (11.940). If at most l∗ −1 angular derivatives fall on L / Rin− j (Rin−k ) Z we place the corresponding factor in L ∞ using Corollary 10.1.i with l∗ + 1 in the role of l. Otherwise, at most j − l∗ angular derivatives fall on d/ Rin− j −1 . . . Rin−k+1 φ resulting in  of order at most k − l∗ . We a factor which is an angular derivative of (i1 ...in−k−1 ) x m,n−k−1 ∞ then place this factor in L using the fact that by assumption M[m,(l+1)∗ +1] and (11.891) we have: 

(i1 ...in−k−1 )  x m,n−k−1 ∞,[k−l∗ ], ε0 t

≤ Cl δ0 (1 + t)−1

: for n = l − m

(11.941)

(compare with (11.901)). It follows that the first sum on the right in (11.939) is for n = ε l − m bounded in L 2 (t 0 ) by:  −2  Cl δ0 (1 + t) [1 + log(1 + t)] max  (i1 ...in−k−1 ) x m,n−k−1 2,[k], ε0 t 0≤k≤n−1 + max  (Ri ) Z 2,[l−1], ε0 (11.942) t

i

Now, the estimates (11.887)–(11.888), (11.890), imply: max 

0≤k≤n−1

(i1 ...in−k−1 )  x m,n−k−1 2,[k], ε0 t

Q ≤ Cl W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l+1} 1 + (1 + t)−2 [1 + log(1 + t)] B[m,l+1] 2  +δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] : for n = l − m

(11.943)

By (11.943) and Corollary 10.2.i, (11.942) is bounded by:

  Q Cl δ0 (1 + t)−2 [1 + log(1 + t)] W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l}  (11.944) + (1 + t)−2 [1 + log(1 + t)]2 B[m,l+1] + Y0 + (1 + t)A[l−1] Finally, the second sum on the right in (11.939) is by Corollary 10.1.i bounded in ε L 2 (t 0 ) by: Cl δ0 (1 + t)−1 [1 + log(1 + t)]

n−1  k=0

d/

i



(i1 .n−k . . in )  Sm,n−1  L 2 ( ε0 ) t

(11.945)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

643

Using the estimates (11.887)–(11.888) we readily deduce, for 0 ≤ k ≤ n − 1, n = l − m, d/

i



(i1 .n−k . . in )  Sm,n−1  L 2 ( ε0 ) t

(11.946)

 ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] (1 + t)2 
/ µ2,[m,l−1], ε0 + B[0,l] t  −3 ≤ Cl δ0 (1 + t) [1 + log(1 + t)] B[m,l+1] + δ0 [1 + log(1 + t)]·  Q Y0 + (1 + t)A[l−1] + W{l} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1}

hence (11.945) is bounded by:

 Cl δ02 (1 + t)−4 [1 + log(1 + t)]2 B[m,l+1] + δ0 [1 + log(1 + t)]· · Y0 + (1 + t)A[l−1] + W{l} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1} Q

 (11.947)

Combining then the results (11.944), (11.947), for the two terms on the right in (11.939) we conclude that:  max  (i1 ...il−m ) wm,l−m  L 2 ( ε0 ) (11.948) t

  Q ≤ Cl δ0 [1 + log(1 + t)] W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l}  +(1 + t)−2 [1 + log(1 + t)]2 B[m,l+1] + Y0 + (1 + t)A[l−1]

i1 ...il−m

We turn to the sums: m−1 

n−1 

 Rin . . . Ri1 (T )k ym−k,0 ,

k=0

Rin . . . Rin−k+1

(i1 ...in−k )  ym,n−k

(11.949)

k=0

which appear in (11.882). The first of these vanishes for m = 0, thus only the second appears in (11.870), which corresponds to the case m = 0. Setting n = l − m, we are ε to obtain L 2 (t 0 ) estimates for each of these sums. The first of the sums (11.949) is ε 2 bounded in L (t 0 ) by (changing m − k to k): m 

 yk,0 2,[m−k,l−k], ε0 t

(11.950)

k=1 ε

while the second is bounded in L 2 (t 0 ) by (changing n − k to k): l−m  k=1

max 

i1 ...ik

(i1 ...ik )  ym,k 2,[l−m−k], ε0 t

(11.951)

 are given in the statement of Proposition 9.1: The functions yk,0   = −(T µ)ak−1,0 yk,0

+(T Lµ − µT trχ − µ)(T )k−1
/µ  $$ 1  + (T trχ) fˇk−1,0 − 2µ L /)χˆ ·µ /ˆ 2,k−1,0 /ˆT χˆ − (tr (T ) π 2

(11.952)

644

Chapter 11. Control of the Spatial Derivatives

 is given by equation (9.117) of Chapter 9: Here, k ≥ 1. The function a0,0  = e
/ µ − 2(d/µ)$ · div / χˆ +
/ m + µ
/ e + 2(d/µ)$ · d/e a0,0

(11.953)

  and for each k ≥ 2 the function ak−1,0 is expressed in terms of the function a0,0 by Lemma 9.4:    = (T )k−1 a0,0 + bk−1,0 (11.954) ak−1,0

where:  = bk−1,0

k−2 k−1   (T ) j (T )k−2− j
/µ+ (T )k−1− j cj j =0

(11.955)

j =1

and, for each j ∈ {1, . . . , k − 1},  /µ − 2 L cj = −(T trχ)(T ) j −1
/ˆT χˆ − (tr

(T )

π /)χˆ

$$

·µ /ˆ 2, j −1,0

(11.956)

 According to the discussion in Chapter 9, the function yk,0 is of order k + 2 and its   . Moreover, principal part is contained in ak−1,0 , more precisely, in the term (T )k−1 a0,0  the function a0,0 is of order 3 but contains no acoustical part of order 3. The function  bk−1,0 is of order k + 1, while the function cp is of order p + 1, so the second sum in (11.955) is only of order k. We begin our estimates by deriving a bound for  a0,0 2,[m−1,l−1], ε0 t

We consider expression (11.953). The contributions of the first and fourth terms on the right are estimated using (11.690) (with m + 1 reduced to m − 1) and (11.887). The contribution of the third term is estimated using (11.686) (with m + 1 reduced to m − 1). To estimate the contribution of the second term we take account of the Codazzi equation ((8.104)) to express this term as −(d/µ)$ ·(d/trχ +2i ) and use Corollary 11.2.c and (11.724). Finally, the contribution of the fifth term on the right in (11.953), a lower order term, is readily estimated. We obtain:  a0,0 2,[m−1,l−1], ε0

t Q ≤ Cl (1 + t)−2 W{l+2} + (1 + t)−1 [1 + log(1 + t)]W{l+1}

(11.957)

+δ0 (1 + t)−1 [1 + log(1 + t)] (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]



using only the assumptions of Proposition 11.3. Next, we estimate:  bk−1,0 2,[m−k,l−k], ε0 t

: for 1 ≤ k ≤ m

(11.958)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

645

The contribution to (11.958) of a term j ∈ {0, . . . , k − 2} in the first sum on the right in (11.955) is bounded by: / µ2,[m−k+ j,l−k+ j ], ε0  · d/(T )k−2− j


t −2 ≤ Cl δ0 (1 + t) [1 + log(1 + t)] 
/ µ2,[m−2,l−1], ε0 t  −1 +(1 + t) 2,[m−2,l−2], ε0 t

≤ Cl δ0 (1 + t)−4 [1 + log(1 + t)] ·    B[m−2,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] +[1 + log(1 + t)] W{l} + δ0 (1 + t)−2 [1 + log(1 + t)]2 W{l−1} Q

: for 1 ≤ k ≤ m, 0 ≤ j ≤ k − 2

 (11.959)

by (11.887) (with m reduced to m − 1) and Corollary 11.2.c (with (m, l) reduced to (m − 2, l − 2)). The contribution to (11.958) of the second sum on the right in (11.955) is bounded by: k−1  cj 2,[m−1− j,l−1− j ], ε0 (11.960) j =1

t

Rewriting (11.956) in the form: cj = −2η0 (1 − u + η0 t)−2 (T ) j −1
/ µ − (T trχ  )(T ) j −1
/µ $$  /)χˆ  ·µ /ˆ 2, j −1,0 −2 L /ˆT χˆ  − (tr (T ) π

(11.961)

we deduce, using (11.887) and Corollary 11.2.c with (m, l) reduced to (m − 1, l − 1), cj 2,[m−1− j,l−1− j ], ε0 ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] · t

  (1 + t)−1 B[m−2,l] + δ0 Y0 + (1 + t)A[l−1]   Q +δ0 W{l} + (1 + t)−1 [1 + log(1 + t)]W{l−1} : for 1 ≤ k ≤ m, 1 ≤ j ≤ k − 1

(11.962)

Then (11.960) is similarly bounded. Combining the results (11.959) and (11.962) we then obtain:  bk−1,0 2,[m−k,l−k], ε0 ≤ Cl (1 + t)−3 [1 + log(1 + t)] · t

  −1 (1 + t) B[m−2,l+1] + δ0 Y0 + (1 + t)A[l−1]   Q +δ0 W{l} + (1 + t)−1 [1 + log(1 + t)]W{l−1}

: for 1 ≤ k ≤ m

(11.963)

646

Chapter 11. Control of the Spatial Derivatives

In view of the estimates (11.957) and (11.963) we conclude, through (11.954), that:  ak−1,0 2,[m−k,l−k], ε0 t

Q −2 W{l+2} + (1 + t)−1 [1 + log(1 + t)]W{l+1} ≤ Cl (1 + t)

  +(1 + t)−1 [1 + log(1 + t)] (1 + t)−1 B[m−1,l+1] + δ0 Y0 + (1 + t)A[l]

: for 1 ≤ k ≤ m

(11.964)

using only the assumptions of Proposition 11.3. Under the same assumptions we also obtain the L ∞ estimate:  ak−1,0 ∞,[m−k,l∗ −k], ε0 ≤ Cl δ0 (1 + t)−3 t

: for 1 ≤ k ≤ m

(11.965)

We now consider expression (11.952). The contribution of the first term on the right in (11.952) to (11.950) is bounded using the estimates (11.964)–(11.965). The second ˙ )k−1
/ µ, where φ˙ is the function: term on the right in (11.952) is φ(T φ˙ = T m + µT e + eT µ − 2η0 (1 − u + η0 t)−2 µ − µT trχ  − µ

(11.966)

(taking account of the propagation equation for µ). Using the estimates (11.686), (11.690), and Corollary 11.2.c we deduce:

Q −1 ˙ ε0 ≤ Cl W{l+1} + (1 + t) φ [1 + log(1 + t)]W{l} (11.967) 2,[m−1,l−1],t  +(1 + t)−1 [1 + log(1 + t)] (1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l] We also obtain:

−1 ˙ ε φ ∞,[m−1,l∗ −1], 0 ≤ Cl δ0 (1 + t) t

(11.968)

The contribution of the second term on the right in (11.952) to (11.950) is then bounded using (11.967) and (11.968), together with (11.887) and (11.888) with l∗ replaced by (l + 1)∗ , which follows from assumption M[m,(l+1)∗ +1] . The contribution of the third term on the right in (11.952) to (11.950) is bounded using (11.890). Finally, the contribution of the fourth term on the right in (11.952) to (11.950) is bounded using Corollary 11.2.c and the estimate (11.885). Combining the results then yields: m 

 yk,0 2,[m−k,l−k], ε0 t

k=1

  Q ≤ Cl (1 + t)−2 [1 + log(1 + t)] W{l+2} + (1 + t)−1 [1 + log(1 + t)]W{l+1}  +δ0 (1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l] (11.969)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

Consider finally (11.951). The functions Proposition 9.2: (i1 ...ik )  ym,k

= −(Rik µ)

647

(i1 ...ik ) y  m,k

are given in the statement of

(i1 ...ik−1 )  am,k−1

+(Rik Lµ − µRik trχ − (Rik ) Z µ)(Rik−1 . . . Ri1 (T )m
/ µ) 1  + (Rik trχ) (i1 ...ik−1 ) fˇm,k−1 2  $$ −2µ L /)χˆ · (i1 ...ik−1 ) µ (11.970) /ˆ 2,m,k−1 / Rˆ ik χˆ − (tr (Rik ) π Here, 1 ≤ k ≤ l − m. The functions (i1 ...ik−1 )  am,k−1

(i1 ...ik−1 ) a  m,k−1

are given by Lemma 9.4:

 = Rik−1 . . . Ri1 (T )m a0,0 +

(i1 ...ik−1 )  bm,k−1

(11.971)

where: (i1 ...ik−1 )  bm,k−1

 = Rik−1 . . . Ri1 bm,0

+

k−2 

Rik−1 . . . Rik− j

(Rik− j −1 )

Rik−1 . . . Ri j +1

(i1 ...i j )  cm, j

Z Rik− j −2 . . . Ri1 (T )m
/µ

j =0

+

k−1 

(11.972)

j =1

and, for each j ∈ {1, . . . , k − 1}, (i1 ...i j )  cm, j

= −(Ri j trχ)(Ri j −1 . . . Ri1 (T )m
/ µ) $$  (R ) /)χˆ · (i1 ...i j −1 ) µ −2 L / Rˆ i j χˆ − (tr i j π /ˆ 2,m, j −1

(11.973)

Now, to obtain a bound for (11.951) we must estimate: max 

i1 ...ik−1

(i1 ...ik−1 )  am,k−1 2,[l−m−k], ε0 t

: for 1 ≤ k ≤ l − m

(11.974)

The contribution to (11.974) of the first term on the right in (11.971) is already estimated by (11.957), increasing m − 1 to m. To estimate the contribution of the second term we must estimate: max 

i1 ...ik−1

(i1 ...ik−1 )  bm,k−1 2,[l−m−k], ε0 t

: for 1 ≤ k ≤ l − m

(11.975)

The contribution to (11.975) of the first term on the right in (11.972) is already estimated by (11.963), setting k = m and then increasing m−1 to m. We must also estimate however the contribution to (11.975) of the two sums on the right in (11.972).

648

Chapter 11. Control of the Spatial Derivatives

Using Corollary 10.1.i with l∗ + 1 in the role of l and the estimate (11.888) with m − 1 increased to m we deduce that the contribution to (11.975) of the first sum on the right in (11.972) is bounded by:  / µ2,[m,l−1], ε0 + (1 + t)−1 max  (Ri ) Z 2,[l−2], ε0 Cl δ0 (1 + t)−2 [1 + log(1 + t)] 
t t i

−3 −1 ≤ Cl δ0 (1 + t) [1 + log(1 + t)] (1 + t) B[m,l+1] + Y0 + (1 + t)A[l−1]  Q +W{l} + δ0 (1 + t)−1 W{l−1} (11.976) where we have appealed to the estimate (11.887) with m − 1 increased to m and to Corollary 10.2.i. On the other hand, the contribution to (11.975) of the second sum on the right in (11.972) is bounded by: k−1  j =1

max 

i1 ...i j

(i1 ...i j )  cm, j 2,[l−1−m− j ], ε0 t



≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] D / 2 µ2,[m,l−2], ε0 + χ  2,[l−1], ε0 t t

−3 −1 ≤ Cl δ0 (1 + t) [1 + log(1 + t)] (1 + t) B[m,l] + Y0 + (1 + t)A[l−1]  Q +W{l} + δ0 (1 + t)−2 W{l−1} (11.977)

from (11.973), using the estimate (11.885) with (m − 1, l − 1) replaced by (m, l − 2) and Corollary 10.2.d. Combining the above results yields: max 

i1 ...ik−1

(i1 ...ik−1 )  bm,k−1 2,[l−m−k], ε0 t

  ≤ Cl (1 + t)−3 [1 + log(1 + t)] (1 + t)−1 B[m,l+1] + δ0 Y0 + (1 + t)A[l−1]   Q +δ0 W{l} + (1 + t)−1 [1 + log(1 + t)]W{l−1} : for 1 ≤ k ≤ l − m

(11.978)

Combining then (11.978) with (11.957) with m − 1 increased to m we conclude that: max 

i1 ...ik−1

(i1 ...ik−1 )  am,k−1 2,[l−m−k], ε0 t

Q ≤ Cl (1 + t)−2 W{l+2} + (1 + t)−1 [1 + log(1 + t)]W{l+1}

  +(1 + t)−1 [1 + log(1 + t)] (1 + t)−1 Bm,l+1] + δ0 Y0 + (1 + t)A[l]

: for 1 ≤ k ≤ l − m

(11.979)

We also obtain the L ∞ estimate: max 

i1 ...ik−1

(i1 ...ik−1 )  am,k−1 ∞,[l∗ −m−k], ε0 t

≤ Cl δ0 (1 + t)−3

(11.980)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

649

We now consider expression (11.970). The contribution of the first term on the right in (11.970) to (11.951) is bounded using the estimates (11.979)–(11.980). The second / µ), where φik is the function: term on the right in (11.970) is φik (Rik−1 . . . Ri1 (T )m
φik = Rik m + µRik e + e Rik µ − µRik trχ  −

(Rik )

Zµ

(11.981)

(taking account of the propagation equation for µ). Using estimates (11.686), (11.690), and Corollary 10.2.i we deduce:

Q φik 2,[l−1], ε0 ≤ Cl W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} t  (11.982) + (1 + t)−1 [1 + log(1 + t)] δ0 (1 + t)−1 B[0,l] + Y0 + (1 + t)A[l] We also obtain:

φik ∞,[l∗ −1], ε0 ≤ Cl δ0 (1 + t)−1 t

(11.983)

The contribution of the second term on the right in (11.970) to (11.951) is then bounded using (11.982) and (11.983), together with (11.887) and (11.888) with l∗ replaced by (l + 1)∗ , which follows from assumption M[m,(l+1)∗ +1] . The contribution of the third term on the right in (11.970) to (11.951) is bounded using (11.890). Finally, the contribution of the fourth term on the right in (11.970) to (11.951) is bounded using Corollary 10.2.d and the estimate (11.885). Combining the results then yields: l−m  k=1

max 

i1 ...ik

(i1 ...ik )  ym,k 2,[l−m−k], ε0 t

  Q ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] W{l+2} + (1 + t)−1 [1 + log(1 + t)]W{l+1}  +(1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l] (11.984) We now return to expressions (11.870) and (11.882). Proposition 11.3 together with the estimates (11.913), (11.909), (11.935), (11.948), (11.969), (11.984), yields the following proposition. Proposition 11.4 Under the assumptions of Proposition 11.3, augmented by the assumptions X / [(l+1)∗ ] and M[m,(l+1)∗ +1] we have: max 

(i1 ...il−m )  g˙ m,l−m  L 2 ( ε0 ) t

≤

  Q QQ Cl (1 + t)−2 [1 + log(1 + t)] W{l+2} + W{l+1} + W{l}  +δ0 (1 + t)−1 B[m+1,l+1] + Y0 + (1 + t)A[l]

i1 ...il−m

We proceed to bound the quantities (i1 ...il−m ) Q m,l−m defined by equation (9.284) of Chapter 9. All except the second and third terms on the right in that equation have already

650

Chapter 11. Control of the Spatial Derivatives

been estimated. The terms in question involve the functions (i1 ...in ) dm,n and the symmetric trace-free 2-covariant tensorfields (i1 ...in ) em,n defined in the statements of Propositions 9.3 and 9.4 respectively. With the functions (i1 ...in ) µm,n defined by equation (9.159) of Chapter 9: (i1 ...in ) µm,n = Rin . . . Ri1 (T )m µ (11.985) we have: (i1 ...in )

dm,n = Rin . . . Ri1 dm,0 +

n−1 

Rin . . . Rin−k+1

(11.986) 

(Rin−k ) $$

(i1 ...in−k−1 )

π / ·D /2

µm,n−k−1



k=0

+

n−1 

 Rin . . . Rin−k+1 tr

(Rin−k )

(i1 ...in−k−1 )

π /1 · d/

µm,n−k−1



k=0

where: m−1 

dm,0 =

(T )

k



(T ) $$



2

π / ·D / µm−k−1,0 +

k=0

m−1 

 (T )k tr

(T )

π /1 · d/µm−k−1,0

k=0



(11.987)

and: (i1 ...in )

em,n = L /ˆR in . . . L /ˆR i1 em,0 + +

1 2

n−1 

(11.988)

/ˆR in−k+1 L /ˆR in . . . L



(Rin−k ) ˆ

π /
/

(i1 ...in−k−1 )



µm,n−k−1

k=0

n−1 

L /ˆR in . . . L /ˆR in−k+1



(Rin−k ) ˆ

π /1 ·

(i1 ...in−k−1 )

µm,n−k−1



k=0

where: em,0 =

m−1 1   ˆ k  L /T 2

 m−1   k  π /
/ µm−k−1,0 + L /ˆT

(T ) ˆ

k=0

(T ) ˆ

π /1 · d/µm−k−1,0

k=0



(11.989)

We are to estimate: max 

i1 ...il−m

(i1 ...il−m )

dm,l−m  L 2 ( ε0 ) , t

(i1 ...il−m )

max 

i1 ...il−m

em,l−m  L 2 ( ε0 ) t

(11.990)

To estimate the first of (11.990) we must first estimate: dm,0 2,[l−m], ε0 t

(11.991)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

651

Now, the contribution to (11.991) of the first and second sums on the right in (11.987) is bounded by: m−1 



(T ) $$

π / ·D / 2 µm−k−1,0 2,[k,l−m+k], ε0 t

k=0

and m−1 

tr

(T )

π /1 · d/µm−k−1,0 2,[k,l−m+k], ε0 t

(11.992)

k=0

respectively. To estimate the first of (11.992), we apply the formula (11.164) taking ξ = d/µm−k−1,0 to obtain: L / R in . . . L / Ri1 (L /T ) j D / 2 µm−k−1,0 =D / 2 Rin . . . Ri1 (T ) j µm−k−1,0 +

(i1 ...in )

c j,n [d/µm−k−1,0 ]

(11.993)

Consider first the L ∞ estimate for this, taking j ≤ k, n ≤ l∗ − m + k − j , 0 ≤ k ≤ m − 1. Applying hypotheses H1, H0 and assumption M[m−1,l∗ +1] to estimate the principal term, and the second statement of Corollary 11.1.g to estimate the commutator term, we obtain: max

0≤k≤m−1

D / 2 µm−k−1,0 ∞,[k,l∗ −m+k], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

(11.994)

Consider next the L 2 estimate for (11.993), taking j ≤ k, n ≤ l−m+k− j , 0 ≤ k ≤ m−1. Applying hypotheses H1, H0 to estimate the principal term, and the second statement of Corollary 11.2.g to estimate the commutator term, we obtain: D / 2 µm−k−1,0 2,[k,l−m+k], ε0 t  1 −2 B[m−1,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] ≤ Cl (1 + t)  Q +W{l} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1} (11.995) max

0≤k≤m−1

Using the estimates (11.994)–(11.995) together with the estimate for (T ) π / + 2(1 − u + η0 t)−1 h / of Corollary 11.1.c with l∗ in the role of l and that of Corollary 11.2.c (as well as Corollary 11.2.d), we then deduce that the first of (11.992) is bounded by:

Cl (1 + t)−2 [1 + log(1 + t)] (1 + t)−1 B[m−1,l+1] 1 + δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q (11.996) +W{l} + (1 + t)−1 [1 + log(1 + t)]W{l−1} The second of (11.994) is readily estimated using the estimate for (T ) π /1 of Corollary 11.1.c with l∗ in the role of l and that of Corollary 11.2.c (as well as Corollary 11.2.d).

652

Chapter 11. Control of the Spatial Derivatives

We deduce a bound by:

Cl δ0 (1 + t)−2 [1 + log(1 + t)] (1 + t)−1 B[m−1,l+1] 1 + (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l]  Q +W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l}

(11.997)

Combining the bounds (11.996), (11.997), we conclude that:

dm,0 2,[l−m], ε0 ≤ Cl (1 + t)−2 [1 + log(1 + t)] (1 + t)−1 B[m−1,l+1] t 1 + δ0 (1 + t)−1 [1 + log(1 + t)] Y0 + (1 + t)A[l]  Q (11.998) +W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} ε

Next we must estimate, in L 2 (t 0 ), the two sums on the right in (11.986), for n = l − m. To estimate the first sum we apply formula (11.164) to ξ = d/

(i1 ...il−m−k−1 )

µm,l−m−k−1

to obtain: L / R in . . . L / R i1 D /2

(i1 ...il−m−k−1 )

=D / Ri n . . . Ri 1 2

µm,l−m−k−1

(i1 ...il−m−k−1 )

µm,l−m−k−1 +

(11.999) (i1 ...in )

c0,n [d/

(i1 ...il−m−k−1 )

µm,l−m−k−1 ]

Consider first the L ∞ estimate for this, taking n ≤ k − l∗ − 1, 0 ≤ k ≤ l − m − 1. Applying hypotheses H1, H0 and assumption M[m,l∗ +1] to estimate the principal term, and the second statement of Corollary 11.1.g with (m, l) replaced by (0, k − l∗ − 1) to estimate the commutator term, we obtain: max

max

0≤k≤l−m−1 i1 ...il−m−k−1

D /2

(i1 ...il−m−k−1 )

µm,l−m−k−1 ∞,[k−l∗ −1], ε0 t

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

(11.1000)

Consider next the L 2 estimate for (11.999), taking n ≤ k, 0 ≤ k ≤ l − m − 1. Applying hypotheses H1, H0 to estimate the principal term, and the second statement of Corollary 11.2.g with (m, l) replaced by (0, k) to estimate the commutator term, we obtain: max

max

0≤k≤l−m−1 i1 ...il−m−k−1

≤ Cl (1 + t)

−2

D /2

(i1 ...il−m−k−1 )

µm,l−m−k−1 2,[k], ε0 t

 1 B[m,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q +W{l} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1} (11.1001)

Using the estimates (11.1000)–(11.1001) together with Corollary 10.1.d with l∗ + 1 in the role of l and with Corollary 10.2.d, we deduce that the first sum on the right in (11.986)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

653

ε

is, for n = l − m, bounded in L 2 (t 0 ) by:

Cl δ0 (1 + t)−2 [1 + log(1 + t)] (1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l−1]  Q (11.1002) +W{l} + δ0 (1 + t)−2 W{l−1} The second sum on the right in (11.986) is readily estimated using Lemma 10.11 with l∗ ε in the role of l and Lemma 10.12. We deduce a bound in L 2 (t 0 ) by:

Cl δ0 (1 + t)−2 [1 + log(1 + t)] (1 + t)−1 B[m,l] + Y0 + (1 + t)A[l]  Q (11.1003) +W{l+1} + δ0 (1 + t)−2 W{l} Combining finally the estimates (11.998), (11.1002), (11.1003), for the three terms on the right in (11.986), we conclude that: max 

(i1 ...il−m )

dm,l−m  L 2 ( ε0 ) t

  ≤ Cl (1 + t)−2 [1 + log(1 + t)] (1 + t)−1 B[m,l+1] + δ0 Y0 + (1 + t)A[l]   Q +δ0 W{l+1} + (1 + t)−2 [1 + log(1 + t)]2 W{l} (11.1004) i1 ...il−m

The estimate for max 

i1 ...il−m

(i1 ...il−m )

em,l−m  L 2 ( ε0 ) t

is similar. The above estimates yield bounds for the second and third terms on the right in equation (9.284). The other terms on the right in that equation having been already bounded by the estimates previously obtained, the last by Proposition 11.4, we arrive at the conclusion expressed by the following proposition. Proposition 11.5 Under the assumptions of Proposition 11.3, augmented by the assumptions X / [(l+1)∗ ] and M[m,(l+1)∗ +1] , we have: Q m,l−m (t)

  Q QQ + W{l} ≤ Cl (1 + t)−3 [1 + log(1 + t)] W{l+2} + W{l+1} max

(i1 ...il−m )

i1 ...il−m

+δ0 (1 + t)−1 B[m+1,l+1] + Y0 + (1 + t)A[l]

tity



We now consider the principal term in the defining expression (9.287) for the quannamely the term:

(i1 ...in ) B  , m,n

 C(1 + t)

−1

(i1 ...in )

(0) 

P

m,n,a

(t) + (1 + t)

−1/2 (i1 ...in )

(1) 

P

 m,n,a

(t) µ−a m (t)

654

Chapter 11. Control of the Spatial Derivatives

Here, the quantities

(i1 ...in )

(0) 

P

(i1 ...in )

m,n,a ,

(1) 

P

m,n,a

are defined by equations (9.243), (0)

(1)

(9.244) of Chapter 9, in terms of the quantities (i1 ...in ) P  m,n , (i1 ...in ) P  m,n , whose sum  itself defined by equation (9.241): (see (9.242)) bounds the quantity (i1 ...in ) Pm,n (i1 ...in )

 Pm,n (t) = (1 + t)

(i1 ...in )

 (t) L 2 ([0,ε0]×S 2 ) fˇm,n

(11.1005)

 (t), for n = l − m. Now, since (see We shall presently derive a bound for (i1 ...in ) Pm,n   = Rin . . . Ri1 (T )m fˇ , the quantity  (i1 ...il−m ) fˇm,l−m  L 2 ( ε0 ) has al(9.94)) (i1 ...in ) fˇm,n t ready been estimated in (11.890) above. However, a more precise estimate is needed  , in contrast to (i1 ...in ) Q  , does here, because of the fact that the quantity (i1 ...in ) Pm,n m,n  . By the not appear integrated with respect to t in the expression (9.287) for (i1 ...in ) Bm,n comparison inequalities (8.358) we have: (i1 ...il−m )

 Pm,l−m ≤ C

(i1 ...il−m ) ˇ f m,l−m  L 2 ( ε0 ) t

= CRil−m . . . Ri1 (T )m fˇ  L 2 ( ε0 ) t

(11.1006)

What is needed is a more precise estimate of the principal part of Ril−m . . . Ri1 (T )m fˇ . Recall that the function fˇ is given by equation (11.889) in terms of the functions f 0 , f1 , themselves defined by equations (11.849), (11.850), respectively. Consider first the contribution of the term f 0 in fˇ to Ril−m . . . Ri1 (T )m fˇ . Since: T σ = −2ψ α T ψα , LT σ = −2ψ α (2(T )2 ψα + α −2 µLT ψα ) − 2(Lψ α )(T ψα ) defining, as in (10.695), m αT = − we have:

dH (ψ L )2 ψ α dσ

  + f 0,N f 0 = f 0,P

(11.1007) (11.1008)

 where f 0,P is the principal part of f 0 :  f 0,P = m αT (2(T )2 ψα + α −2 µLT ψα )

(11.1009)

 and f 0,N is the lower order part of f 0 :  f 0,N = −ψ L2

dH (Lψ α )(T ψα ) dσ

(11.1010)

We then obtain the decomposition: Ril−m . . . Ri1 (T )m f 0 = 2m αT Ril−m . . . Ri1 (T )m+2 ψα +α −2 µm αT Ril−m . . . Ri1 (T )m LT ψα +

(i1 ...il−m )  n m,l−m

(11.1011)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

where

(i1 ...il−m ) n  m,l−m ,

(i1 ...il−m )  n m,l−m

655

of order l + 1, is given by:

= 2Ril−m . . . Ri1

 m  k=1



+2

m! ((T )k m αT )((T )m−k+2 ψα ) k!(m − k)!



((R)s1 m αT )((R)s2 (T )m+2 ψα )

partitions

|s1 |+|s2 |=l−m,s1 =∅



+Ril−m . . . Ri1 

+

m  k=1

m! ((T )k (α −2 µm αT ))((T )m−k LT ψα ) k!(m − k)!



((R)s1 (α −2 µm αT ))((R)s2 (T )m LT ψα )

partitions

|s1 |+|s2 |=l−m,s1 = =∅  +Ril−m . . . Ri1 (T )m f 0,N

(11.1012)

From (11.1011), taking into account (10.699) of Chapter 10, we have: Ril−m . . . Ri1 (T )m f 0  L 2 ( ε0 ) t   3 −1 ≤ 2k || + Cδ0 (1 + t) Ril−m . . . Ri1 (T )m+2 ψα  L 2 ( ε0 ) +C +



t

α

µRil−m . . . Ri1 (T )m LT ψα  L 2 ( ε0 )

α (i1 ...il−m )  n m,l−m  L 2 ( ε0 ) t

t

(11.1013)

where C is a constant which is independent of l. Moreover, using the estimate for ψ L of Corollary 11.2.c with (m + 1, l + 1) reduced to (m, l) as well as the estimate for  of Corollary 11.2.c with (m, l) reduced to (m − 1, l − 1) ( appears in expressing LT ψα as T Lψα + ψα ), we readily deduce: max 

i1 ...il−m

(i1 ...il−m )  n m,l−m  L 2 ( ε0 ) t

Q ≤ Cl δ0 (1 + t)−1 W{l+1} + [1 + log(1 + t)]W{l} 1  2 +(1 + t)−1 B[m,l] + δ0 Y0 + (1 + t)A[l−1]

(11.1014)

Consider next the contribution of the term µ2 f 1 in fˇ to Ril−m . . . Ri1 (T )m fˇ . Since: d/σ = −2ψ α d/ψα , / ψα − 2(d/ψ α )$ · (d/ψα )
/ σ = −2ψ α
setting: m˜ α = ψ L ψ Lˆ

dH α ψ − Fψ0 L α dσ

(11.1015)

656

Chapter 11. Control of the Spatial Derivatives

we have:

  f 1 = f 1,P + f 1,N

(11.1016)

 where f 1,P is the principal part of f 1 :  = m˜ α
/ ψα f 1,P

(11.1017)

 is the lower order part of f 1 : and f 1,N  = ψ L ψ Lˆ f 1,N

dH (d/ψ α )$ · (d/ψα ) dσ

(11.1018)

We then obtain the decomposition: Ril−m . . . Ri1 (T )m (µ2 f1 ) = µ2 m˜ α Ril−m . . . Ri1 (T )m
/ ψα + where

(i1 ...il−m ) n  m,l−m ,

(i1 ...il−m )  n m,l−m

(i1 ...il−m )  n m,l−m

(11.1019)

of order l + 1, is given by:

= Ril−m . . . Ri1 +

 m 



k=1

m! ((T )k (µ2 m˜ α ))((T )m−k
/ ψα ) k!(m − k)!



((R)s1 (µ2 m˜ α ))((R)s2 (T )m
/ ψα )

partitions

|s1 |+|s2 |=l−m,s1 =∅  +Ril−m . . . Ri1 (T )m (µ2 f 1,N )

(11.1020)

In view of the decomposition (11.1019) we have: Ril−m . . . Ri1 (T )m (µ2 f 1 ) L 2 ( ε0 ) t  µRil−m . . . Ri1 (T )m
/ ψα  L 2 ( ε0 ) +  ≤ C[1 + log(1 + t)] α

t

(11.1021) (i1 ...il−m )  n m,l−m  L 2 ( ε0 ) t ε

where C is a constant which is independent of l. To estimate, in L 2 (t 0 ), the first term on the right in (11.1020) we use the following analogue of (11.801) with L i in the role of Tˆ i (and (m, k) replacing (k, j )): 0 m 0 0 0 m! 0 m−k k i0 ((T ) f )(T ) L 0 0 0 0 k!(m − k)! ε k=1 2,[l−m],t 0

≤ Cl (1 + t)−1 δ0 [1 + log(1 + t)] f 2,[m−1,l−1], ε0 t 1   +  f ∞,[m−1,l∗ ], ε0 B[m−1,l] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−2] t  Q +(1 + t)W{l} + δ0 (1 + t)−2 [1 + log(1 + t)]3 W{l−2} (11.1022)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

657

(see Corollary 11.2.a). Applying this in the case f =
/ ψi , yields, using (11.816) (with (m, l) reduced to (m − 1, l − 1)) and (11.817) (with m reduced to m − 1), 0 0 m 0 0 m! 0 0 ((T )m−k
/ ψi )(T )k L i 0 0 0 0 k!(m − k)! ε k=1 2,[l−m],t 0

  Q ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] W{l+1} + δ0 (1 + t)−3 [1 + log(1 + t)]2 W{l−1} 1 2  +(1 + t)−1 B[m−1,l] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] (11.1023) Using also Corollary 11.2.b we then obtain that the first term on the right in (11.1020) is ε bounded in L 2 (t 0 ) by: Cl (1 + t)−2 [1 + log(1 + t)] · (11.1024) 

 Q −3 2 [1 + log(1 + t)] W{l+1} + δ0 (1 + t) [1 + log(1 + t)] W{l−1} 1 2  +δ0 (1 + t)−1 B[m,l] + [1 + log(1 + t)] Y0 + (1 + t)A[l−1] ε

To estimate, in L 2 (t 0 ), the second term on the right in (11.1020) we use the following estimate, for an arbitrary function f , related to the estimate (11.548), and proved in a similar manner using Corollary 10.2.g together with Corollary 10.1.g with l∗ in the role of l: 0 0 0 0 0 0  0 0 s1 s2 i 0 0 ((R) f )(R) L 0 0 0 0 0|s |+|spartitions 0 ε 1 2 |=n≤l,s2 =∅ L 2 (t 0 )

≤ Cl  f 2,[l−1], ε0 +  f ∞,[l∗ ], ε0 t t  Q (11.1025) × Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] We apply this in the case f = (T )m
/ ψi , replacing on the left l by l − m. Then on the right  f 2,[l−1], ε0 ,  f ∞,[l∗ ], ε0 are replaced by  f 2,[l−m−1], ε0 ,  f ∞,[l∗ −m], ε0 , t t t t respectively. Interchanging also the roles of the subsets s1 and s2 we then obtain: 0 0 0 0 0 0  0 0 s1 i s2 m 0 0 ((R) L )((R) (T )
/ ψ ) i 0 0 0 0 partitions 0 0|s |+|s |≤l−m,s =∅ ε 1 2 1 L 2 (t 0 )  1 ≤ Cl / ψi ∞,[m,l∗ ], ε0 Y0 + (1 + t)A[l−1] 
/ ψi 2,[m,l−1], ε0 + 
t

t

i

+W[l] + δ02 (1 + t)−2 W[l−1] Q



658

Chapter 11. Control of the Spatial Derivatives

Q ≤ Cl (1 + t)−2 W{l+1} + δ0 (1 + t)−1 (1 + t)−2 [1 + log(1 + t)]2 W{l−1}  (11.1026) +(1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1] by (11.816) with l replaced by l − 1 and (11.817). Using also Corollary 11.2.b we obtain ε that the second term on the right in (11.1020) is bounded in L 2 (t 0 ) by: Cl δ0 (1 + t)−3 [1 + log(1 + t)]B[0,l]

Q + Cl (1 + t)−2 [1 + log(1 + t)]2 W{l+1} + δ0 (1 + t)−1 (1 + t)−2 [1 + log(1 + t)]2 W{l−1}  +(1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1] (11.1027) ε

Finally, the third term on the right in (11.1020) is readily bounded in L 2 (t 0 ) by: Cl δ02 (1 + t)−4 [1 + log(1 + t)]B[m,l]

+ Cl δ0 (1 + t)−3 [1 + log(1 + t)]2 W{l+1} + δ0 (1 + t)−1 · · (1 + t)−2 [1 + log(1 + t)]W{l−1} + Y0 + (1 + t)A[l−1] Q

 (11.1028)

Combining the results (11.1024), (11.1027), (11.1028), yields: max  (i1 ...il−m ) n m,l−m  L 2 ( ε0 ) ≤ Cl (1 + t)−2 [1 + log(1 + t)]· t 

 Q · [1 + log(1 + t)] W{l+1} + δ0 (1 + t)−3 [1 + log(1 + t)]2 W{l−1} 1 2  +δ0 (1 + t)−1 B[m,l] + 1 + log(1 + t)] Y0 + (1 + t)A[l−1]

i1 ...il−m

(11.1029)

We now return to (11.1013). Our objective in the following is, in regard to the first two terms on the right in (11.1013), to express Ril−m . . . Ri1 (T )m+2 ψα

and

Ril−m . . . Ri1 (T )m LT ψα

T Ril−m . . . Ri1 (T )m+1 ψα

and

L Ril−m . . . Ri1 (T )m+1 ψα ,

as respectively, plus lower order terms. Then using the fact that for any variation ψ (10.708) of Chapter 10 holds, we shall be able to bound the first two terms on the right in (11.1013) by:  )  3 −1 C k || + δ0 (1 + t) E0 [Ril−m . . . Ri1 (T )m+1 ψα ] α

+C(1 + t)−1 [1 + log(1 + t)]1/2

)

 α

+ C(1 + t)

−1

E1 [Ril−m . . . Ri1 (T )m+1 ψα ]

[1 + log(1 + t)]W{l+1}

up to a bound for the lower order terms of a form similar to (11.1014).

(11.1030)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

659

In view of the formulas (10.729), (10.731), of Chapter 10, that is; T Ri n . . . Ri 1 − Ri n . . . Ri 1 T =

n−1 

Rin . . . Rin−k+1

(Rin−k )

Rin−k−1 . . . Ri1

(11.1031)

k=0

we have: Ril−m . . . Ri1 (T )m+2 ψα = T Ril−m . . . Ri1 (T )m+1 ψα −

l−m−1 

Ril−m . . . Ril−m−k+1

(Ril−m−k )

(11.1032) Ril−m−k−1 . . . Ri1 (T )m+1 ψα

k=0 ε

To estimate in L 2 (t 0 ) the sum on the right, we observe that it is given by: l−m−1 



k=0

partitions |s1 |+|s2 |=k

(L / R )s 1

(Ril−m−k )



 · d/(R)s2 (T )m+1 ψα

(11.1033)

Here, in the inner sum, we are considering all ordered partitions {s1 , s2 } of the set {l − m − k + 1, . . . , l − m} into two ordered subsets s1 , s2 . Also, s2 = s2



{1, . . . , l − m − k − 1}

ε

To estimate in L 2 (t 0 ) a term in the inner sum in (11.1033) , we use Lemma 10.24. If ε |s1 | ≤ l∗ , we place the first factor in L ∞ (t 0 ) using the first statement of that lemma. Otherwise, |s1 | ≥ l∗ + 1, hence |s2 | = k − |s1 | ≤ k − l∗ − 1 and: m + 2 + |s2 | = |s2 | + l − k + 1 ≤ l − l∗ ≤ l∗ + 1 and we place the second factor in L ∞ using assumption E{l∗ +1} . Using also the second ε statement of Lemma 10.24 we obtain in this way that (11.1033) is bounded in L 2 (t 0 ) by:  Cl δ0 (1 + t)−2 [1 + log(1 + t)]2 W{l+1} + max 

(Ri )

i

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] ·

Q [1 + log(1 + t)]W{l+1} + δ02 (1 + t)−2 W{l−1}  +δ0 (1 + t)−1 B[0,l] + Y0 + (1 + t)A[l−1]

2,[l−1], ε0 t

(11.1034)

660

Chapter 11. Control of the Spatial Derivatives

It then follows through (11.1032) that:  Ril−m . . . Ri1 (T )m+2 ψα  L 2 ( ε0 ) t

α

≤



T Ril−m . . . Ri1 (T )m+1 ψα  L 2 ( ε0 ) t

α

+ Cl δ0 (1 + t)−2 [1 + log(1 + t)] ·

Q [1 + log(1 + t)]W{l+1} + δ02 (1 + t)−2 W{l−1}  +δ0 (1 + t)−1 B[0,l] + Y0 + (1 + t)A[l−1]

(11.1035)

As observed in Chapter 10, a commutation formula similar to (11.1031) holds with in the role of (Ri )  holds for L in the role of T , that is:

(Ri ) Z

L Ri n . . . Ri 1 − Ri n . . . Ri 1 L =

n−1 

Rin . . . Rin−k+1

(Rin−k )

Z Rin−k−1 . . . Ri1

(11.1036)

k=0

It follows that, taking n = l − m, Ril−m . . . Ri1 L(T )m+1 ψα = L Ril−m . . . .Ri1 (T )m+1 ψα −

l−m−1 

Ril−m . . . Ril−m−k+1

(Ril−m−k )

(11.1037)

Z Ril−m−k−1 . . . Ri1 (T )m+1 ψα

k=0

By Lemma 11.23 we have: [(T )m , L] = −

m−1 

(T )k (T )m−k−1

(11.1038)

k=0

hence: Ril−m . . . Ri1 (T )m LT ψα = Ril−m . . . Ri1 L(T )m+1 ψα −

m−1 

(11.1039)

Ril−m . . . Ri1 (T )k (T )m−k ψα

k=0 ε

Now, µ times the sum on the right in (11.1037) is bounded in L 2 (t 0 ) by: C[1 + log(1 + t)]

l−m−1  k=0



(Ril−m−k )

Z · d/ Ril−m−k−1 . . . Ri1 (T )m+1 ψα 2,[k], ε0 t

(11.1040)

Using Corollary 10.1.i with l∗ + 1 in the role of l and Corollary 10.2.i we obtain that (11.1040) is bounded by:

Q Cl δ0 (1 + t)−2 [1 + log(1 + t)] [1 + log(1 + t)]W{l+1} + δ0 (1 + t)−1 W{l−1}  +Y0 + (1 + t)A[l−1] (11.1041)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

661 ε

Also, µ times the sum on the right in (11.1039) is bounded in L 2 (t 0 ) by: C[1 + log(1 + t)]

m−1 

 · d/(T )m−k ψα 2,[k,l−m+k], ε0 t

(11.1042)

k=0

Using Corollary 11.1.c with l∗ in the role of l and Corollary 11.2.c with (m, l) reduced to (m − 1, l − 1) we obtain that (11.1042) is bounded by: Cl δ0 (1 + t)−2 [1 + log(1 + t)]· 

 Q · [1 + log(1 + t)] W{l+1} + δ0 (1 + t)−3 [1 + log(1 + t)]2 W{l−1}  1 2 + (1 + t)−1 B[m−1,l] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−2]

(11.1043)

In view of the estimates (11.1041), (11.1043), it follows through (11.1037), (11.1039) that:  µRil−m . . . Ri1 (T )m LT ψα  L 2 ( ε0 ) t

α

≤

 α

µL Ril−m . . . Ri1 (T )m+1 ψα  L 2 ( ε0 ) t

+ Cl δ0 (1 + t)−2 [1 + log(1 + t)] ·

Q [1 + log(1 + t)]W{l+1} + δ0 (1 + t)−1 W{l−1}  +(1 + t)−1 B[m−1,l] + Y0 + (1 + t)A[l−1]

(11.1044)

In view of (11.1030), inequalities (11.1035), (11.1044), together with the estimate (11.1014), imply through (11.1013) that: Ril−m . . . Ri1 (T )m f0  L 2 ( ε0 ) t   ) ≤ C k 3 || + δ0 (1 + t)−1 E0 [Ril−m . . . Ri1 (T )m+1 ψα ] α

+ C(1 + t)−1 [1 + log(1 + t)]1/2

) α

E1 [Ril−m . . . Ri1 (T )m+1 ψα ]

Q + C(1 + t)−1 [1 + log(1 + t)]W{l+1} + Cl δ0 (1 + t)−1 [1 + log(1 + t)]W{l} 1 2  (11.1045) +(1 + t)−1 B[m,l] + [1 + log(1 + t)] Y0 + (1 + t)A[l−1] where C is a constant which is independent of l. / ψα We finally turn to (11.1021). Our objective here is to express Ril−m . . . Ri1 (T )m
as
/ (Ril−m . . . Ri1 (T )m ψα ) plus lower order terms. Since:
/ ψα = tr(D / 2 ψα )$ ,

(D / 2 ψα )$ = h/−1 · D / 2 ψα

662

Chapter 11. Control of the Spatial Derivatives

we have: Ril−m . . . Ri1 (T )m
/ ψα = Ril−m . . . Ri1 (T )m tr(D / 2 ψα )$  $ / Ril−m . . . L = tr L / Ri1 (L / T )m D / 2 ψα   m    m! + tr L / Ril−m . . . L (L /T )k (h/−1 ) · (L / R i1 /T )m−k D / 2 ψα k!(m − k)! k=1            s1 −1 s2 m 2 / R ) (L /T ) D / ψα (L / R ) (h/ ) · (L + tr     partitions   |s1 |+|s2 |=l−m,s1 =∅ (11.1046) Applying formula (11.164) to the St,u 1-form d/ψα we obtain: L / Ril−m . . . L / Ri1 (L / T )m D / 2 ψα =D / 2 (Ril−m . . . Ri1 (T )m ψα ) +

(i1 ...il−m )

cm,l−m [d/ψα ]

(11.1047)

Hence the first term on the right in (11.1046) is given by:  $ / Ri1 (L / T )m D / 2 ψα tr L / Ril−m . . . L =
/ (Ril−m . . . Ri1 (T )m ψα ) + tr



(i1 ...il−m )

cm,l−m [d/ψα ]

$

(11.1048)

Taking account of the fact that: L /T (h/−1 ) = −

(T ) $$

π /

and using Corollary 11.1.c with l∗ in the role of l and the estimate (11.817), together with Corollary 11.2.c and the estimate (11.816) with (m, l) replaced by (m − 1, l − 1), we ε deduce that µ times the first sum on the right in (11.1046) is bounded in L 2 (t 0 ) by:

Cl [1 + log(1 + t)] (1 + t)−1 [1 + log(1 + t)]D / 2 ψα 2,[m−1,l−1], ε0 t  −3 (T ) −1 +δ0 (1 + t)  π / + (1 − u + η0 t) h/2,[m−1,l−1], ε0 t

≤ Cl (1 + t)−3 [1 + log(1 + t)]·

Q [1 + log(1 + t)]W{l+1} + δ0 (1 + t)−2 [1 + log(1 + t)]2 W{l−1}  1 2 +δ0 (1 + t)−1 B[m−1,l] + [1 + log(1 + t)] Y0 + (1 + t)A[l−1] Similarly, taking account of the fact that L / Ri (h/−1 ) = −

(Ri ) $$

π /

(11.1049)

 Part 2: Bounds for the quantities (i1 ...il ) Q m,l and (i1 ...il ) Pm,l

663

and using Corollary 10.1.d with l∗ in the role of l and the estimate (11.817), together with Corollary 10.2.d and the estimate (11.816) with l replaced by l − 1, we deduce that µ ε times the second sum on the right in (11.1046) is bounded in L 2 (t 0 ) by:

Cl [1 + log(1 + t)] δ0 (1 + t)−1 [1 + log(1 + t)]D / 2 ψα 2,[m,l−1], ε0 t +δ0 (1 + t)−3 max  (Ri ) π /2,[l−1], ε0 t i

Q −3 ≤ Cl δ0 (1 + t) [1 + log(1 + t)] [1 + log(1 + t)]W{l+1} + δ0 (1 + t)−2 W{l−1}  (11.1050) +δ0 (1 + t)−2 [1 + log(1 + t)]B[m−1,l] + Y0 + (1 + t)A[l−1] ε

Finally, to estimate in L 2 (t 0 ) µ times the second term on the right in (11.1048), we apply the second statement of Corollary 11.2.g to the St,u 1-form d/ψα . This yields a bound by: Cl δ0 (1 + t)−3 [1 + log(1 + t)] · 

 Q [1 + log(1 + t)] W{l+1} + (1 + t)−2 [1 + log(1 + t)]W{l}  +(1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] (11.1051) In view of the estimates (11.1049)–(11.1051), it follows through (11.1046), (11.1048) that:  α

µRil−m . . . Ri1 (T )m
/ ψα  L 2 ( ε0 ) t

≤

 α

µ
/ (Ril−m . . . Ri1 (T )m ψα ) L 2 ( ε0 ) t

+ Cl (1 + t)−3 [1 + log(1 + t)] · 

 Q [1 + log(1 + t)] W{l+1} + δ0 (1 + t)−2 [1 + log(1 + t)]W{l}  (11.1052) +δ0 (1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l] Moreover, by hypothesis H1 the first term on the right in (11.1052) is bounded by:  α

µ
/ (Ril−m . . . Ri1 (T )m ψα ) L 2 ( ε0 )

(11.1053)

t

≤ C(1 + t)−1

 α, j

µd/(R j Ril−m . . . Ri1 (T )m ψα ) L 2 (σ ε0 )

≤ C(1 + t)−2 [1 + log(1 + t)]1/2

t

) α, j

E1 [R j Ril−m . . . Ri1 (T )m ψα ]

664

Chapter 11. Control of the Spatial Derivatives

The bounds (11.1052)–(11.1053) together with the estimate (11.1029) imply through (11.1021) that: Ril−m . . . Ri1 (T )m (µ2 f 1 ) L 2 ( ε0 ) t ) −2 3/2 ≤ C(1 + t) [1 + log(1 + t)] E1 [R j Ril−m . . . Ri1 (T )m ψα ] α, j −2

+ Cl (1 + t) [1 + log(1 + t)]· 

 Q · [1 + log(1 + t)] W{l+1} + δ0 (1 + t)−3 [1 + log(1 + t)]2 W{l}  1 2 + δ0 (1 + t)−1 B[m,l+1] + [1 + log(1 + t)] Y0 + (1 + t)A[l] (11.1054) The bounds (11.1045), (11.1054), yield finally the following proposition. Proposition 11.6 Under the assumptions of Proposition 11.5 we have: (i1 ...il−m )

where: (i1 ...il−m )

 Pm,l−m ≤

(0) 

P

m,l−m =

(i1 ...il−m )

Ck || 3

(0) 

P

m,l−m

)

+

(i1 ...il−m )

(1) 

P

m,l−m

E0 [Ril−m . . . Ri1 (T )m+1 ψα ]

α

and: (i1 ...il−m )

(1) 

P

m,l−m =

Cδ0 (1 + t)

−1

) 

E0 [Ril−m . . . Ri1 (T )m+1 ψα ]

α

+ C(1 + t)−1 [1 + log(1 + t)]1/2

)

 α

+ C(1 + t)−2 [1 + log(1 + t)]3/2

) α, j

E1 [Ril−m . . . Ri1 (T )m+1 ψα ] E1 [R j Ril−m . . . Ri1 (T )m ψα ]

  + C(1 + t)−1 [1 + log(1 + t)] + Cl (1 + t)−2 [1 + log(1 + t)]2 W{l+1}

1 Q + Cl δ0 (1 + t)−1 [1 + log(1 + t)]W{l} + (1 + t)−1 B[m,l+1] 2  +[1 + log(1 + t)] Y0 + (1 + t)A[l] where C is a constant which is independent of l.

Chapter 12

Recovery of the Acoustical Assumptions. Estimates for Up to the Next to the Top Order Angular Derivatives of χ and Spatial Derivatives of µ Q

In the first part of the present chapter we shall show that the assumptions E{1} and E{0} on ε Wεs0 , together with certain primitive assumptions on the initial conditions on 00 , imply i i pointwise estimates for κ, the functions λi , y , y , in particular the bound (10.30) for the y i , assumed in all the propositions of Chapters 10 and 11, as well as sharp upper and lower bounds for r . Moreover, we shall show that the same assumptions imply hypothesis H0 on Wεs0 . In the second part of the present chapter we shall show that the assumptions E{l+2} , initial conditions H1, H2 and H2 . Moreover, we shall show that, under additional appropriate assumptions on the initial conditions, the acoustical assumptions M[m,l+1] , for m = 0, . . . , l + 1, also follow.

Q QQ , E{l} , on Wεs0 , together with appropriate assumptions on the E{l+1} ε0 on 0 , imply the acoustical assumptions X / [l] , as well as hypotheses

Finally, in the third part of the present chapter we shall derive estimates for the Q quantities A[l] and B[m+1,l+1] , for m = 0, . . . , l, under the assumptions E{l∗ +2} , E{l∗ +1} , ε

QQ

E{l∗ } , on Wεs0 , together with appropriate assumptions on the initial conditions on 00 . We begin with the following elementary proposition.

Q Proposition 12.1 Let assumptions E{1} and E{0} hold on Wεs0 , and let the initial data satisfy:

κ − 1 L ∞ ( ε0 ) ≤ Cδ0 0

666

Chapter 12. Recovery of the Acoustical Assumptions

Then, provided that δ0 is suitably small, for all t ∈ [0, s] we have: κ − 1 L ∞ ( ε0 ) ≤ Cδ0 [1 + log(1 + t)] t

where C is a constant which is independent of s. Proof. Let us recall the propagation equation for κ, equation (3.99) of Chapter 3: Lκ = α −1 m + κe

(12.1)

where (see (3.97)): 1 dH (ψ L )2 (T σ ) 2 dσ

(12.2)

dH 1 (ψTˆ )2 (Lσ ) + Fψ0 ω L Tˆ 2 dσ

(12.3)

σ − k 2  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1

(12.4)

α −1 − η0−1  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 , hence α −1 ≤ C on Wεs0

(12.5)

m= and (see (3.98)): e = Now, assumption E{0} implies:

t

and:

t

provided that δ0 is suitably small. Moreover, recalling from Chapter 6 the elementary bounds: (12.6) |Tˆ i | ≤ 1, |L i | ≤ 1 (see (6.52), (6.54)) assumption E{0} also implies: ψTˆ  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 ,

ψ L − k L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1

t

t

(12.7)

Q

Also, assumption E{0} implies: ω L Tˆ  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2

(12.8)

t

Since by assumptions E{1} , E{0} we also have: T σ  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 , t

Lσ  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 t

(12.9)

it follows that these assumptions imply the pointwise bounds: α −1 |m| ≤ Cδ0 (1 + t)−1 ,

|e | ≤ Cδ0 (1 + t)−2

Writing the propagation equation (12.1) in the form: L(κ − 1) = e (κ − 1) + α −1 m + e

: on Wεs0

(12.10)

Chapter 12. Recovery of the Acoustical Assumptions

667

and noting that for any function f : L| f | ≤ |L f | we obtain, substituting the bounds (12.10), the following ordinary differential inequality for |κ − 1| along the integral curves of L: L|κ − 1| ≤ Cδ0 (1 + t)−2 |κ − 1| + Cδ0 (1 + t)−1

(12.11)

ε

Integrating this from 00 and taking into account the assumption on the initial conditions, the proposition follows. ◦

Let us recall from Chapter 6 the functions λi : ◦

◦

λi =  Ri , Tˆ 

(12.12)

(see (6.122)), where  ,  denotes the Euclidean inner product on t . Lemma 12.1 Let assumptions E{1} , E{0} , hold on Wεs0 , and let the initial data satisfy: ◦

max  λi  L ∞ ( ε0 ) ≤ Cδ0 i

0

s

Moreover, let hypothesis H0 hold on Wε00 for some s0 ∈ (0, s]. Then, provided that δ0 is suitably small, we have, for all t ∈ [0, s0 ]: ◦

max  λi  L ∞ ( ε0 ) ≤ Cδ0 [1 + log(1 + t)] i

t

◦

Proof. We recall from Chapter 6 that the functions λi satisfy along the integral curves of L the equation: ◦ ◦ ◦ H ψ0 i j m ψ j Tˆ m (12.13) L λi = p L λi + Ri , q L  − 1 + ρH (see (6.129)). Here, p L is the function defined by equation (3.164) of Chapter 3:  1 dH ψTˆ (Lσ ) + H ω L Tˆ p L = −ψTˆ (12.14) 2 dσ and q L is the St,u -tangential vectorfield given by equation (3.169) of Chapter 3. With (q L )b = q L · h/ we have: (q L )b = −κ −1 ζ  −

1 dH ψ ˆ ( ψ(Lσ ) − ψ L (d/σ )) − H ψω L Tˆ 2 dσ T

(12.15)

668

Chapter 12. Recovery of the Acoustical Assumptions

Q The assumptions E{0} and E{0} imply (12.7), (12.8) and the second of (12.9), hence also:

 p L  L ∞ ( ε0 ) ≤ Cδ02 (1 + t)−3

: for all t ∈ [0, s]

t

(12.16)

Let us recall the fact that κ −1 ζ  is given by (see (3.175)):  1 dH ψTˆ ψ(Lσ ) + (ψTˆ + α −1 ψ L )ψ L (d/σ ) κ −1 ζ  = 2 dσ +H (ψTˆ + α −1 ψ L )ω /L

(12.17)

Now, ψ = ψi d/x i satisfies, from Chapter 6, the inequality:  | ψ|2 ≤ ρ, where ρ = 3i=1 (ψi )2

(12.18)

(this is the inequality immediately preceding (6.60)). It follows that assumption E{0} implies: (12.19)  ψ L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 t

Q similarly Since ω / L = (Lψi )d/x i is like ψ but with Lψi in the role of ψi , assumption E{0} implies: (12.20) ω / L  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 t

Now, by virtue of hypothesis H0, we have, pointwise: |d/σ | ≤ C(1 + t)−1 max |Ri σ |

(12.21)

max Ri σ  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1

(12.22)

i

and by assumption E{1} : t

i

It follows that the assumptions of the lemma imply: max d/σ  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 t

i

: for all t ∈ [0, s0 ]

(12.23)

The estimates (12.7), (12.19), the second of (12.9), and (12.23), (12.20), yield, through expression (12.17), κ −1 ζ   L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 t

: for all t ∈ [0, s0 ]

(12.24)

The remaining terms on the right in (12.15) are bounded pointwise using (12.7), (12.19), the second of (12.9), and (12.23), (12.8). Noting that the magnitude of any St,u -tangential vectorfield is equal, pointwise, to the magnitude of the corresponding St,u 1-form, so that: |q L | = |(q L )b | we then obtain: q L  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 t

: for all t ∈ [0, s0 ]

(12.25)

Chapter 12. Recovery of the Acoustical Assumptions

669

Recalling from Chapter 6 that the induced acoustical metric h on t dominates the Euclidean metric, we have: q L  ≤ |q L | √ where, as in Chapter 6, we denote by V  = V, V  the Euclidean magnitude of a vector V which is tangential to t . Thus, (12.25) implies: q L  ≤ Cδ0 (1 + t)−2

s

: on Wε00

(12.26)

In view of the fact that by the elementary bound: ε

r ≤ 1 + η0 t : in t 0

(12.27)

of Chapter 6 (see (6.109)), we have: ◦

 Ri 2 = r 2 − (x i )2 ≤ r 2 ≤ (1 + η0 t)2

ε

: in t 0

(12.28)

it then follows that the second term on the right in (12.13) is bounded pointwise by: ◦

◦

| Ri , q L | ≤  Ri q L  ≤ (1 + η0 t)q L  ≤ Cδ0 (1 + t)−2 : on Wε00 s

(12.29)

The third term on the right in (12.13) is by assumption E{0} and the first of (12.6) bounded pointwise by: Cδ0 (1 + t)−1 : on Wεs0 (12.30) s

In view of the bounds (12.16), (12.29), (12.30), equation (12.13) yields on Wε00 the following ordinary differential inequality along the integral curves of L: ◦

◦

L| λi | ≤ Cδ02 (1 + t)−3 | λi | + Cδ0 (1 + t)−1

(12.31)

ε

Integrating this from 00 and taking into account the assumption on the initial conditions, the lemma follows. According to equation (6.123) of Chapter 6, the functions ◦

λi = h( Ri , Tˆ )

(12.32)

◦

(see (6.115)) are given, in terms of the functions λi by: ◦

◦

λi =λi +H Rim ψm ψTˆ

(12.33)

In view of (12.28) and the first of (12.7), it follows that under the assumptions of Lemma 12.1 we have: max λi  L ∞ ( ε0 ) ≤ Cδ0 [1 + log(1 + t)] i

t

: for all t ∈ [0, s0 ]

(12.34)

670

Chapter 12. Recovery of the Acoustical Assumptions

The result of Proposition 12.1 coincides with the estimate (6.81) of Chapter 6. As was shown in Chapter 6, the estimate (6.81) together with the first of the elementary bounds (12.6) implies the lower bound (6.113) for r : r ≥ 1 − u + η0 t − Cδ0 u[1 + log(1 + t)]

(12.35)

which, if δ0 is suitably small, implies the lower bound (6.114): r ≥ C −1 (1 + η0 t)

(12.36)

We conclude that under the assumptions of Proposition 12.1 the estimates (12.35), (12.36) hold on Wεs0 . Let us recall from Chapter 6 the Euclidean projection operator  to the Euclidean coordinate spheres (equation (6.151)):  ij = δ ij − r −2 x i x j

(12.37)

(rectangular coordinates). According to equation (6.152) we have:  Tˆ 2 = r −2



◦

(λi )2

(12.38)

i

Lemma 12.1 together with the lower bound (12.36) then implies:  Tˆ  ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

s

: in Wε00

(12.39)

Let us recall from Chapter 6 the Euclidean outward unit normal N to the Euclidean coordinates spheres (equation (6.154)): N=

xi ∂ r ∂xi

(12.40)

(rectangular coordinates). According to equation (6.159) we have: 1 − N, Tˆ 2 = H (ψTˆ )2 + T 2

(12.41)

The estimate (12.39) together with the first of (12.7) implies, by the argument presented in Chapter 6, the inequalities (6.161): 0 ≤ 1 + N, Tˆ  ≤ Cδ02 (1 + t)−2 [1 + log(1 + t)]2

(12.42)

These inequalities together with the result of Proposition 12.1 yield the estimate (6.165): |r − 1 − η0 t + u| ≤ Cδ0 u[1 + log(1 + t)] which implies the estimate (6.166):   1  1  −  ≤ Cδ0 u(1 + t)−2 [1 + log(1 + t)] r 1 − u + η0 t 

(12.43)

(12.44)

Chapter 12. Recovery of the Acoustical Assumptions

671

Let us recall from Chapter 10 the vectorfield: y  = Tˆ + N =  Tˆ + (1 + N, Tˆ )N

(12.45)

The estimates (12.39) and (12.42) imply: y   ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(12.46)

which, together with (12.44) and the elementary bound (12.27), implies the estimate (10.30) of Chapter 10 for the functions y i : max y i  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] t

i

(12.47)

We conclude that under the assumptions of Proposition 12.1 and Lemma 12.1, the estis mates (12.43) and (12.46) hold on Wε00 and the estimate (12.47) holds for all t ∈ [0, s0 ]. We now turn to the following fundamental proposition. Proposition 12.2 Consider a surface St,u for which the following hold: sup y   < 1

inf r > 0

St,u

St,u

Then for every St,u 1-form ξ we have:   2

−1 

|ξ | ≤ 1 − sup y  2

inf r

St,u

St,u

−2  3

(ξ(Ri ))2

i=1

pointwise on St,u . ◦

Proof. Since Ri =  Ri , we have, in terms of components in rectangular coordinates with respect to the background Euclidean metric on t : ◦

(Ri )a = ab ( Ri )b = ab ikb x k

(12.48)

hence: 3 3   (Ri )a (Ric ) = ab ikb x k cd ild x l i=1

i=1

= (δkl δbd − δkd δbl )x k x l ab cd = (r 2 δbd − x b x d )ab cd

(12.49)

where we have used the fact that: 3  i=1

ikb ild = δkl δbd − δkd δbl

(12.50)

672

Chapter 12. Recovery of the Acoustical Assumptions

It follows that: 3 

(ξ(Ri )) = 2

i=1

=

 3 

 (Ri )

i=1 (r 2 δbd

a

(Ric )

ξa ξc

− x b x d )(ab ξa )(cd ξc )

(12.51)

Since ξ is an St,u 1-form, we have: ab ξa = ξb We thus obtain: 3 

(ξ(Ri ))2 = (r 2 δbd − x b x d )ξb ξd

i=1

= r 2 (ξ 2 − (ξ(N))2 )

(12.52)

recalling from (12.40) that x a = r N a . Here ξ  is the magnitude of ξ with respect to the background Euclidean metric on t : " # 3 # ξ  = $ (ξa )2 (12.53) a=1

In view of (12.45) and the fact that ξ(Tˆ ) = 0, we have:

hence:

ξ(N) = ξ(y  ) = y a ξa

(12.54)

|ξ(N)| = |ξ(y  )| ≤ y  ξ 

(12.55)





and:

 2

ξ  − (ξ(N)) ≥ 1 − sup y  2

2

ξ 2

(12.56)

St,u

Substituting in (12.52) then yields:   2  3  2  2 1 − sup y  ξ 2 (ξ(Ri )) ≥ inf r St,u

i=1

(12.57)

St,u ε

Recall now from Chapter 6 that the induced acoustical metric h on t 0 dominates the Euclidean metric. It follows that: |ξ |2 =

3 

(h

a,b=1

−1 ab

) ξa ξb ≤

3  (ξa )2 = ξ 2 a=1

In view of (12.57), (12.58), the proposition follows.

(12.58)

Chapter 12. Recovery of the Acoustical Assumptions

673

As a corollary of Proposition 12.2 we have the following. Corollary 12.2.a Suppose that there are positive constants ε1 and ε2 < 1 such that: r ≥ ε1 η0 (1 + t) and sup y  2 ≤ 1 − ε2 inf ε ε

t 0

t 0

ε

Then hypothesis H0 holds on t 0 . In fact, for any function f , differentiable on the St,u , ε we have, pointwise on t 0 :  |d/ f |2 ≤ C(1 + t)−2 (Ri f )2 i

where C is the constant: C=

1 ε2 ε12 η02

Proof. This follows immediately by applying Proposition 12.2 to the St,u 1-form ξ = d/ f . Since sup u = ε0 ≤ ε t 0

1 2

and η0 < 1

fixing any positive constant ε1 ≤

1 , 2

the lower bound (12.35) implies that: inf r > ε1 η0 (1 + t) ε

t 0

: for all t ∈ [0, s]

(12.59)

provided that δ0 is suitably small. Also, fixing any positive constant ε2 < 1, the estimate (12.46) implies that: sup y   < 1 − ε2 ε

: for all t ∈ [0, s0 ]

(12.60)

t 0

provided that δ0 is suitably small. Let now s 0 be the maximal value of s0 ∈ [0, s] such that hypothesis H0 holds on s Wε00 . We shall show that in fact s 0 = s. For, suppose that s 0 < s. Since we can take s s0 = s 0 in Lemma 12.1, the estimate (12.46) holds on Wε00 , hence also inequality (12.60) holds for all t ∈ [0, s 0 ], in particular at t = s 0 . However, (12.60) being a strict inequality, it must, by continuity, hold in a suitably small interval [s 0 , s∗ ) ⊂ [s 0 , s]. This contradicts the maximality of s 0 . We conclude that hypothesis H0 holds on Wεs0 . From Lemma 12.1 with s0 = s and the estimates which follow we then obtain the following proposition. Q Proposition 12.3 Let assumptions E{1} , E{0} , hold on Wεs0 , and let the initial data satisfy: ◦

κ − 1 L ∞ ( ε0 ) ≤ Cδ0 and max  λi  L ∞ ( ε0 ) ≤ Cδ0 0

i

0

674

Chapter 12. Recovery of the Acoustical Assumptions

Then if δ0 is suitably small, hypothesis H0 holds on Wεs0 . Moreover, we have, on Wεs0 : 1 + η0 t − u − Cδ0 u[1 + log(1 + t)] ≤ r ≤ 1 + η0 t + min {0, −u + Cδ0 u[1 + log(1 + t)]} y   ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] and, for all t ∈ [0, s], the estimates: κ − 1 L ∞ ( ε0 ) ≤ Cδ0 [1 + log(1 + t)] t

◦

max  λi  L ∞ ( ε0 ) , max λi  L ∞ ( ε0 ) ≤ Cδ0 [1 + log(1 + t)] t

i

and:

t

i

max y i  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] t

i

In the sequel we shall also make use of the following proposition, which is analogous to Proposition 12.2. Proposition 12.4 Consider a surface St,u for which the following hold: inf r > 0

St,u

sup y   < 1 St,u

Then for every symmetric 2-covariant St,u tensorfield ϑ we have:   2

|ϑ| ≤ 1 − sup y  2

St,u

−2  inf r

−4  3

St,u

(ϑ(Ri , R j ))2

i, j =1

pointwise on St,u . Proof. We have: 3 

  3 3   (ϑ(Ri , R j ))2 = (Ri )a (Ri )c  (R j )b (R j )d  ϑab ϑcd 

i, j =1

i=1

(12.61)

j =1

We substitute from (12.49) for the first two factors on the right. In fact, noting that, from (12.40), (12.45), x b = r N b = r (y b − Tˆ b ), and Tˆ b ab = 0, we have: x b ab = r y b ab

(12.62)

3  (Ri )a (Ri )c = r 2 (δbd − y b y d )ab cd

(12.63)

(12.49) takes the form:

i=1

Chapter 12. Recovery of the Acoustical Assumptions

675

Substituting then from (12.63) for the first two factors on the right in (12.61), we obtain: 3 









(ϑ(Ri , R j ))2 = r 4 (δa  c − y a y c )(δb d  − y b y d )ϑa  b ϑc d 

i, j =1

  = r 4 ϑ2 − 2i y  ϑ2 + (ϑ(y  , y  ))2

(12.64)

where i y  ϑ is the St,u 1-form with rectangular components: 

(i y  ϑ)b = y a ϑa  b

(12.65)

Consider the expression: ϑ2 − 2i y  ϑ2 + (ϑ(y  , y  ))2 in the right-hand side of (12.64), at a given point. It is invariant under the orthogonal group. By a suitable proper orthogonal transformation of the rectangular coordinates we can arrange so that y 2 = y 3 = 0 at the given point. Then this expression becomes: 2     1 − (y 1 )2 (ϑ11 )2 + (ϑ22 )2 + (ϑ33 )2 + 2 1 − (y 1 )2 (ϑ12 )2 + (ϑ13 )2 + 2(ϑ23 )2 2    (ϑ11 )2 + (ϑ22 )2 + (ϑ33 )2 + 2(ϑ12 )2 + 2(ϑ13 )2 + (ϑ23 )2 ) ≥ 1 − (y 1 )2 2  = 1 − y  2 ϑ2 We conclude that:

 2 ϑ2 − 2i y  ϑ2 + (ϑ(y  , y  ))2 ≥ 1 − y  2 ϑ2

(12.66)

Thus, (12.64) implies: 3 

 2 (ϑ(Ri , R j ))2 ≥ r 4 1 − y  2 ϑ2

(12.67)

i, j =1 ε

Taking into account the fact that, since the induced acoustical metric h on t 0 dominates the Euclidean metric, we have: |ϑ|2 =

3 

(h

−1 ac

a,b,c,d=1

) (h

−1 bd

) ϑab ϑcd ≤

3 

(ϑab )2 = ϑ2

(12.68)

a,b=1

the proposition then follows. We now turn to the propagation equation (4.106) of Chapter 4 for the symmetric 2-covariant St,u tensorfield χ: ( P) (N) dH 1 ( ψ ⊗ (χ $ · d/σ ) + (χ $ · d/σ )⊗ ψ))− α − α L / L χ = χ · χ $ + eχ + ψ L 2 dσ (12.69)

676

Chapter 12. Recovery of the Acoustical Assumptions ( P)

Here, α is given by (4.72):

( P)

( P P)

( P N)

α = α + α

(12.70)

( P P)

where α is the principal part:  ( P P) 1 dH α = ψ L ( ψ ⊗ d/(Lσ ) + d/(Lσ )⊗ ψ) − ψ L2 D / 2 σ − ( ψ⊗ ψ)(L)2 σ 2 dσ

(12.71)

( P N)

and α is the lower order part:  ( P N) 1 d H −1 α = α ψ L ( ψ ⊗ (κ −1 ζ  ) + (κ −1 ζ  )⊗ ψ) + α −1 ψ L2 k/ + ( ψ⊗ ψ)e (Lσ ) 2 dσ (12.72) (N)

Also, in (12.69), α is the lower order term given by (4.85) and (4.94)–(4.96): (N)

( A) 1 (B) 1 (C) α = −F α − F2 α − F1 α 2 2

(12.73)

where: ( A)

α =ω /L ⊗ ω /L − ωL L ω /

(12.74)

(B)

α = (ψ L (d/σ ) − (Lσ ) ψ) ⊗ (ψ L (d/σ ) − (Lσ ) ψ)

(12.75)

(C)

α = (d/σ ) ⊗ (ψ L ω / L − ω L L ψ) + (ψ L ω / L − ω L L ψ) ⊗ (d/σ ) −(Lσ )(2ψ L ω /− ψ ⊗ ω /L − ω / L ⊗ ψ)

(12.76)

Taking account of the fact that L / L h/ = 2χ the propagation equation (12.69) is seen to be equivalent to the following propagation equation for the symmetric 2-covariant St,u tensorfield: η0 h/ 1 − u + η0 t  1  χ · a + χ  ˜· a + b L / L χ  = χ  · χ $ + 2 1 where a is the T1 type St,u tensorfield: χ = χ −

a = eI + ψ L

dH (d/σ )$ ⊗ ψ dσ

(12.77)

(12.78)

and b is the symmetric 2-covariant St,u tensorfield: b=

( P) (N) 1 η0 dH η0 eh/ + ψL ( ψ ⊗ d/σ + d/σ ⊗ ψ) − α − α 1 − u + η0 t 2 1 − u + η0 t dσ (12.79)

Chapter 12. Recovery of the Acoustical Assumptions

677

Q QQ Lemma 12.2 Let assumptions E{2} , E{1} , E{0} , hold on Wεs0 , and let the initial data satisfy: ◦

κ − 1 L ∞ ( ε0 ) ≤ Cδ0 ,

max  λi  L ∞ ( ε0 ) ≤ Cδ0 i

0

and:

0

χ   L ∞ ( ε0 ) ≤ Cδ0 0

s Wε00

Moreover, let hypothesis H1 hold on for some s0 ∈ (0, s]. Then, provided that δ0 is suitably small, we have, for all t ∈ [0, s0 ]: χ   L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] t

Proof. The assumptions of the lemma contain those of Proposition 12.3. Therefore, under the present assumptions the conclusions of Proposition 12.3 hold. We shall first estimate the coefficients a and b in the propagation equation (12.77). The function e is given by equation (3.98): e=− Now

1 dH (Lσ ) − Fψ0 ω L L ψ L (ψ L + 2αψTˆ ) 2α 2 dσ

(12.80)

ω L L = L µ (Lψµ ) = Lψ0 + L i (Lψi )

is like ψ L − k = L µ ψµ − k = ψ0 − k + L i ψi , but with Lψ0 , Lψi , in the role of ψ0 − k, ψi , respectively. Thus, as assumption E{0} implies the second of the estimates (12.7), Q assumption E{0} implies: ω L L  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2

(12.81)

t

The estimates (12.7), the second of (12.9), together with (12.81), imply: e L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2

(12.82)

t

This estimate, together with the estimate (12.23), which now holds for all t ∈ [0, s], since by Proposition 12.3 hypothesis H0 holds on Wεs0 , and the estimates (12.19) and the second of (12.7), yields: a L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 t

: for all t ∈ [0, s]

(12.83)

Using the same estimates we deduce that the first term on the right in (12.79) is bounded ε in L ∞ (t 0 ) by: Cδ0 (1 + t)−3 : for all t ∈ [0, s] (12.84) ε

while the second term on the right in (12.79) is bounded in L ∞ (t 0 ) by: Cδ02 (1 + t)−4

: for all t ∈ [0, s]

(12.85)

678

Chapter 12. Recovery of the Acoustical Assumptions (N)

Next we consider the term α . According to inequality (6.62) of Chapter 6: |ω /|2 ≤

3 

|d/ψi |2

(12.86)

i=1

By hypothesis H0 and assumption E{1} this implies: ω / L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 t

: for all t ∈ [0, s]

(12.87)

In view of the estimates (12.20), (12.81), (12.87), the second of (12.9), (12.23) (which holds for all t ∈ [0, s]), the second of (12.7), (12.19), we deduce from expressions (12.74)–(12.76) the estimates: ( A)

 α  L ∞ ( ε0 ) ≤ Cδ02 (1 + t)−4 , t

(B)

 α  L ∞ ( ε0 ) ≤ Cδ02 (1 + t)−4 , t

(C)

 α  L ∞ ( ε0 ) ≤ Cδ02 (1 + t)−4 ,

(12.88)

t

for all t ∈ [0, s]. It follows that: (N)

 α  L ∞ ( ε0 ) ≤ Cδ02 (1 + t)−4 t

( P)

: for all t ∈ [0, s]

(12.89)

( P N)

We turn to the term α . We consider first the lower order part α , given by (12.72). Let us recall the fact that k/ is given by (see (10.195))  ψ0 1 H ψ0 1 dH  ψ⊗ ψ(Lσ ) − ( ψ ⊗ d/σ + d/σ ⊗ ψ) − ω / (12.90) k/ = 2α dσ 1 + ρH α 1 + ρH Using the estimate (12.87) together with the second of (12.9), (12.23) (which holds for all t ∈ [0, s]), and (12.19), we obtain: k/  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 t

: for all t ∈ [0, s].

(12.91)

This, together with the estimate (12.24), which now holds for all t ∈ [0, s] (since by Proposition 12.3 hypothesis H0 holds on Wεs0 ), the estimate (12.82), as well as the second of (12.9), the second of (12.7), and (12.19), then yield: ( P N)

 α

 L ∞ ( ε0 ) ≤ Cδ02 (1 + t)−4 t

: for all t ∈ [0, s]

(12.92)

( P P)

QQ Q Finally, we consider the principal part α , given by (12.71). By assumptions E{0} , E{0} , E{0} , we have:

(L)2 σ  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−3 t

: for all t ∈ [0, s]

(12.93)

Chapter 12. Recovery of the Acoustical Assumptions

679

By virtue of hypothesis H0, we have, pointwise: |d/ Lσ | ≤ C(1 + t)−1 max |Ri Lσ |

: on Wεs0

(12.94)

: for all t ∈ [0, s]

(12.95)

i

Q hence, by assumptions E{1} , E{1} , we have:

d/ Lσ  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−3 t

To estimate D / 2 σ we must appeal to hypothesis H1 as well as hypothesis H0. First, by hypothesis H1 applied to the St,u 1-form d/σ , we have, pointwise: / Ri d/σ | |D / 2 σ | ≤ C(1 + t)−1 max |L i

s

: on Wε00

Then, since L / Ri d/σ = d/ Ri σ , we have, by hypothesis H0, |L / Ri d/σ | ≤ C(1 + t)−1 max |R j Ri σ | j

: on Wεs0

Thus, combining, we obtain, pointwise: |D / 2 σ | ≤ C(1 + t)−2 max |R j Ri σ | i, j

s

: on Wε00

(12.96)

hence, by assumption E{2} : D / 2 σ  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−3 t

: for all t ∈ [0, s0 ]

(12.97)

The estimates (12.93), (12.95), (12.97), together with the previous estimates, the second of (12.7) and (12.19), imply: ( P P)

 α  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−3 t

: for all t ∈ [0, s0 ]

(12.98)

The estimates (12.84), (12.85), (12.89), (12.92) and (12.98), yield: b L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−3 t

: for all t ∈ [0, s0 ]

(12.99)

Let now ϑ, ϑ  , be a pair of symmetric 2-covariant St,u tensorfields. Their pointwise inner product, with respect to the induced acoustical metric h/, is expressed relative to an arbitrary local frame field by: (ϑ, ϑ  ) = (h/−1 ) AC (h/−1 ) B D ϑ AB ϑC D

(12.100)

In terms of the corresponding T11 type tensorfields ϑ $ , ϑ $ , ϑ$ = h /−1 · ϑ = ϑ · h/−1 , we have:

ϑ $ = h/−1 · ϑ  = ϑ  · h/−1 ,

(ϑ, ϑ $ ) = tr(ϑ $ · ϑ $ )

(12.101)

680

Chapter 12. Recovery of the Acoustical Assumptions

Thus:

/ L ϑ $ ) · ϑ $ ) + tr(ϑ $ · (L / L ϑ $ )) L(ϑ, ϑ  ) = tr((L

Since

(12.102)

/−1 ) = −2χ $$ = −2h/−1 · χ · h/−1 L / L (h

we have:

L / L ϑ $ = (L / L ϑ)$ − 2χ $ · ϑ,

L / L ϑ $ = (L / L ϑ  )$ − 2χ $ · ϑ 

It follows that: / L ϑ) + (ϑ, L /L ϑ ) L(ϑ, ϑ  ) = (ϑ  , L −2tr(ϑ $ · χ $ · ϑ $ ) − 2tr(ϑ $ · χ $ · ϑ $ )

(12.103)

Taking in particular ϑ  = ϑ we obtain: |ϑ|L|ϑ| =

1 L(ϑ, ϑ) = (ϑ, L / L ϑ) − 2tr(ϑ $ · χ $ · ϑ $ ) 2

Writing χ$ = we have: tr(ϑ $ · χ $ · ϑ $ ) =

(12.104)

η0 I + χ $ 1 − u + η0 t η0 |ϑ|2 + tr(ϑ $ · χ $ · ϑ $ ) 1 − u + η0 t

and (12.104) takes the form: |ϑ|L|ϑ| = (ϑ, L / L ϑ) −

2η0 |ϑ|2 + 2tr(ϑ $ · χ  · ϑ $ ) 1 − u + η0 t

(12.105)

Now, the following inequality holds: |tr(ϑ $ · χ $ · ϑ $ )| ≤ |χ  ||ϑ|2

(12.106)

To see this, we work in an orthonormal (relative to h/) frame (E 1 , E 2 ) of eigenvectors of χ  at a point. Let λ1 , λ2 be the corresponding eigenvalues. We then have: tr(ϑ $ · χ $ · ϑ $ ) = λ1 (ϑ11 )2 + (λ1 + λ2 )(ϑ12 )2 + λ2 (ϑ22 )2 Since |ϑ|2 = (ϑ11 )2 + 2(ϑ12 )2 + (ϑ22 )2 it follows that:

|tr(ϑ $ · χ  · ϑ $ )| ≤ max{|λ1 |, |λ2 |}|ϑ|2

In view of the fact that: max{|λ1 |, |λ2 |} ≤ the inequality (12.106) then follows.



(λ1 )2 + (λ2 )2 = |χ  |

Chapter 12. Recovery of the Acoustical Assumptions

681

Taking into account the inequality (12.106) as well as the fact that |(ϑ, L / L ϑ)| ≤ |ϑ||L / L ϑ| we deduce from (12.105):     / L ϑ| L (1 − u + η0 t)2 |ϑ| ≤ (1 − u + η0 t)2 2|χ  ||ϑ| + |L

(12.107)

We now apply this to the case ϑ = χ  . In view of the facts that:

|χ  · χ $ | = |(χ $ )2 | = (λ1 )4 + (λ2 )4 ≤ (λ1 )2 + (λ2 )2 = |χ  |2 , |χ  · a| = |χ  ˜· a| ≤ |a||χ  | we then obtain the inequality:     L (1 − u + η0 t)2 |χ  | ≤ (1 − u + η0 t)2 (3|χ  | + |a|)|χ  | + |b|

(12.108)

For the sake of clarity, in the following argument we shall be specific about the numerical constants. Let P(t) be the property: P(t) :

χ   L ∞ ( ε0 ) ≤ C0 δ0 (1 + t  )−2 [1 + log(1 + t  )] : for all t  ∈ [0, t] t

Choosing C0 suitably large, we have, by the assumptions of the lemma on the initial conditions, χ   L ∞ ( ε0 ) < C0 δ0 (12.109) 0

It follows by continuity that P(t) is true for sufficiently small positive t. Let then t0 be the least upper bound of the set of values of t ∈ [0, s0 ] for which P(t) holds. Then by t continuity P(t0 ) is true. Hence, in Wε00 , we have, in view of the estimate (12.83): 3|χ  | + |a| ≤ (3C0 + C)δ0 (1 + t)−2 [1 + log(1 + t)]

t

: in Wε00

(12.110)

Substituting in (12.108) and taking into account the estimate (12.99) we obtain the following inequality along the integral curves of L:     L (1 − u + η0 t)2 |χ  | ≤ (3C0 + C)δ0 (1 + t)−2 [1 + log(1 + t)] (1 − u + η0 t)2 |χ  | +Cδ0 (1 + t)−1

(12.111)

Consider a given integral curve of L. Setting, along the given integral curve: x(t) = (1 − u + η0 t)2 |χ  |

(12.112)

the inequality (12.111) takes the form: dx ≤ fx +g dt

(12.113)

682

Chapter 12. Recovery of the Acoustical Assumptions

where: f (t) = (3C0 + C)δ0 (1 + t)−2 [1 + log(1 + t)],

g(t) = Cδ0 (1 + t)−1

(12.114)

Integrating from t = 0 yields: x(t) ≤ e

!t 0

f (t  )dt 





t

x(0) +

e−

! t 0

f (t  )dt 

g(t  )dt 

(12.115)

0

Since

x(0) ≤ χ   L ∞ ( ε0 )

(12.116)

0

while x(t) ≥

 η 2 0

2

(1 + t)2 |χ  |

(as u ≤ ε0 ≤ 1/2 and η0 < 1)

taking into account the facts that: t ∞ f (t  )dt  ≤ f (t  )dt  = 2(3C0 + C)δ0 , 0

0



t

(12.117)

g(t  )dt  = Cδ0 log(1 + t)

0

(12.118) we deduce from (12.115):  η 2 

0 ε (1 + t)2 |χ  | ≤ e2(3C0+C)δ0 χ   L ∞ ( ε0 ) + Cδ0 log(1 + t) : on t 0 (12.119) 0 2 ε

Taking the supremum on t 0 then yields: χ   L ∞ ( ε0 ) ≤ t



2 η0

2

 e2(3C0+C)δ0 χ   L ∞ ( ε0 ) + Cδ0 log(1 + t) (1+t)−2 (12.120) 0

This holds for all t ∈ [0, t0 ]. Let us now fix C0 large enough so that: C 0 δ0 ≥

8 η02

χ   L ∞ ( ε0 ) and C0 > 0

8 η02

C

(12.121)

(the first is possible by the assumptions of the lemma on the initial conditions). Then, provided that δ0 satisfies the smallness condition: δ0 ≤

log 2 2(3C0 + C)

(12.122)

the inequality (12.120) implies: χ   L ∞ ( ε0 ) < C0 δ0 (1 + t)−2 [1 + log(1 + t)] t

: for all t ∈ [0, t0 ]

(12.123)

It follows by continuity that P(t) is true for some t > t0 . This contradicts the definition of t0 , unless t0 = s0 . Consequently, the property P(s0 ) is true, in fact the strict inequality (12.123) holds for all t ∈ [0, s0 ], which implies the conclusion of the lemma.

Chapter 12. Recovery of the Acoustical Assumptions

683

Let us now investigate hypothesis H1. We consider L / Ri ξ for an arbitrary St,u 1-form ξ . The components of this in rectangular coordinates are given by: / Ri ξ )a + ξm (D / R i )m (L / Ri ξ )a = (D a, Hence, we have:



|L / R i ξ |2 =



i

i

+2

(D / Ri ξ )a = (Ri )m (D /ξ )ma

(12.124)

|D / R i ξ |2  −1 (h )ab (D / Ri ξ )a ξn (D / Ri )nb i

 −1 + (h )ab ξm ξn (D / R i )m / Ri )nb a (D

(12.125)

i

Consider the first term on the right in (12.125). We have:  −1  |D / R i ξ |2 = (h )ab (D / Ri ξ )a (D / R i ξ )b i

i



= (h

−1 ab

)

  m n /ξ )ma (D (Ri ) (Ri ) (D /ξ )nb

(12.126)

i

Substituting from (12.63) and noting that: n m /ξ )ma (D /ξ )nb = (D /ξ )ca (D /ξ )db c d (D

we obtain:



|D / Ri ξ |2 = r 2 (h

−1 ab

) (δcd − y c y d )(D /ξ )ca (D /ξ )db

(12.127)

i

Consider the symmetric 2-covariant St,u tensorfield ϑ with rectangular components: ϑcd = (h ◦

◦

−1 ab

) (D /ξ )ca (D /ξ )db

◦

(12.128) ε

Let µ1 , µ2 , µ3 , be the eigenvalues of ϑ, considered as a tensorfield on t 0 , relative to the background Euclidean metric. Since ϑ is positive semi-definite, we have: ◦

µi ≥ 0 : i = 1, 2, 3 It follows that:

◦

max µi ≤ i

where



◦

◦

µi =tr ϑ

(12.129) (12.130)

i ◦

tr ϑ = δcd ϑcd

(12.131)

is the trace of ϑ with respect to the background Euclidean metric. The inequality (12.130) implies: ◦

y c y d ϑcd ≤ y  2 tr ϑ

(12.132)

684

Chapter 12. Recovery of the Acoustical Assumptions

hence, in reference to (12.127), (h

◦  ) (δcd − y c y d )(D /ξ )ca (D /ξ )db = (δcd − y c y d )ϑcd ≥ 1 − y  2 tr ϑ (12.133)

−1 ab

Recalling from Chapter 6 the fact that (see (6.58)): δcd = (h we have:

−1 cd

)

H ψc ψd 1 + ρH

+

◦

tr ϑ = δcd ϑcd ≥ (h

(12.134)

−1 cd

) ϑcd = trϑ.

(12.135)

Substituting in (12.133), assuming that y   ≤ 1, and noting that: trϑ = (h

−1 cd

) (h

−1 ab

) (D /ξ )ca (D /ξ )db = |D /ξ |2

(12.136)

(12.127) is seen to imply: 

|D / Ri ξ |2 ≥ r 2 (1 − y  2 )|D /ξ |2 .

(12.137)

i

Consider next the third term on the right in (12.125). Here, we must first obtain a suitable expression for D / Ri . Let (X A : A = 1, 2) be an arbitrary local frame field for St,u . We have (see Chapter 6): ◦

◦

/(D h / X A Ri , X B ) = h(D X A Ri , X B ) and D X A Ri = (D X A ) Ri +(D X A Ri ) hence: ◦

◦

/(D h / X A Ri , X B ) = h((D X A ) Ri , X B ) + h(D X A Ri , X B )

(12.138)

Now, according to the last of equations (6.25): ◦

h((D X A ) Ri , X B ) = −λi θ AB

(12.139)

while according to equation (6.34): ◦

◦

h(D X A Ri , X B ) = h mn X lA ilm X nB + H ψ B ( Ri )m (d/ A ψm ) (12.140) ◦ ◦ 1 dH 1 dH {(d/ A σ ) ψ B − (d/ B σ ) ψ A } (( Ri )m ψm ) ( Ri σ ) ψ A ψ B + + 2 dσ 2 dσ ◦

Expressing the factor Ri σ in the third term on the right in (12.140) as: ◦

Ri σ = Ri σ + κ −1 λi T σ

Chapter 12. Recovery of the Acoustical Assumptions

685

adding (12.139), (12.140), and recalling that (see (6.38)): −θ AB +

1 dH  ψ A ψ B T σ = −θ AB = α −1 χ AB − k/AB 2κ dσ

we obtain:

◦

/(D h / X A Ri , X B ) = ((νi )b ) AB + ((τi )b ) AB where:

(12.141)

◦

((νi )b ) AB = h mn X lA ilm X nB

(12.142) ◦

are the components in the frame (X A : A = 1, 2) of the 2-covariant St,u tensorfields νi , whose rectangular components are given by: ◦





(νi )ln = ll nn h mn il  m

(12.143)

((τi )b ) AB are the components in the frame (X A : A = 1, 2) of the 2-covariant St,u tensorfield: ◦

(12.144) (τi )b = λi (α −1 χ − k/ ) + H ( Ri )m (d/ψm )⊗ ψ  ◦ 1 dH + (Ri σ ) ψ⊗ ψ + ((d/σ )⊗ ψ− ψ ⊗ (d/σ ))(( Ri )m ψm ) 2 dσ ◦

◦

Introducing the T11 type St,u tensorfields νi , τi , corresponding to (νi )b , (τi )b , respectively: ◦

◦

νi = (νi )b · h/−1

τi = (τi )b · h/−1

(12.145)

we conclude from (12.141) that D / Ri is given by: ◦

D / Ri =νi +τi

(12.146)

◦

Now, by (12.143), the rectangular components of νi are given by: ◦

◦

(νi )lk = (νi )ln (h Since

−1 nk

)







= ll nn il  m h mn (h

nn h mn (h

−1 nk

)

= km

−1 nk

)

(12.147)

this reduces to, simply: ◦



(νi )lk = km ll il  m

(12.148)

and the rectangular components of D / Ri are given by: 

(D / Ri )lk = km ll il  m + (τi )lk

(12.149)

686

Chapter 12. Recovery of the Acoustical Assumptions

Using the expression (12.149) we obtain, in view of (12.50):  n a  b (D / R i )m / Ri )nb = m a (D m  n   b (δa  b δm  n  − δa  n  δb m  ) i

+

  n b a (τi )m (τi )nb m a n  b ib n  + m  a ia  m  i

i

 n + (τi )m a (τi )b

(12.150)

i

Consider then the third term on the right in (12.125). Substituting (12.150) we find:  −1   −1 (h )ab ξm ξn (D / R i )m / Ri )nb = δa  b (h )ab aa bb ξ 2 − |ξ |2 (12.151) a (D i

+2

 −1  (h )ab (τi · ξ )a (vi )b + |τi · ξ |2 i

i

where the vi are the St,u 1-forms with rectangular components: 

(vi )b = bb ib n ξn Now, since (h

−1 ab





) aa bb = (h

−1 a  b

)

(12.152)

  − Tˆ a Tˆ b

(12.153)

and, from equations (6.58) of Chapter 6, δa  b = h a  b − H ψa  ψb we obtain: δa  b (h

−1 ab



(12.154)



) aa bb = 2 − H | ψ|2

(12.155)

therefore, in view of (12.58) the sum of the first two terms on the right in (12.151) is not less than: (12.156) (1 − H | ψ|2 )ξ 2 The third term on the right in (12.151) is bounded in absolute value by: ) )   2 2 |τi · ξ ||vi | ≤ 2 |τi ||ξ ||vi | ≤ 2|ξ | |τi | |vi |2 i

i

i

i

Using (12.50) we obtain:  −1    |vi |2 = (h )ab aa bb ia  m  ib n ξm  ξn i

i

= (h

−1 ab





) aa bb (δa  b δm  n − δa  n δb m  )ξm  ξn

= δa  b (h

−1 ab





) aa bb ξ 2 − |ξ |2

(12.157)

Chapter 12. Recovery of the Acoustical Assumptions

hence, by (12.155),



687

|vi |2 = (2 − H | ψ|2 )ξ 2 − |ξ |2

(12.158)

i

Now, by (12.134), ξ 2 = δab ξa ξb = (h

−1 ab

) ξa ξb +

H H (ψa ξa )(ψb ξb ) ≤ |ξ |2 + ρξ 2 1 + ρH 1 + ρH

hence: ξ 2 ≤ (1 + ρ H )|ξ |2

(12.159)

In view of (12.159), equation (12.158) implies: ) |vi |2 ≤ (1 + ρ H )|ξ |

(12.160)

i

By (12.157) and (12.160), the third term on the right in (12.151) is bounded in absolute value by: ) 2(1 + ρ H ) |τi |2 |ξ |2 (12.161) i

Since the fourth term on the right in (12.151) is non-negative, we conclude from (12.156), (12.161), that the third term on the right in (12.125) is not less than:   ) 2 1 − H | ψ| − 2(1 + ρ H ) |τi |2  |ξ |2 (12.162) i

provided that H |ψ|2 ≤ 1. Finally, we consider the second term on the right in (12.125):  −1  −1 (h )ab (D / Ri ξ )a ξn (D / Ri )nb = 2 (h )ab (Ri )m (D /ξ )ma ξn (D / Ri )nb 2 i

(12.163)

i

From (12.48) and (12.149) we obtain, using (12.50),   k n b n (Ri )m (D / Ri )nb = m m  ikm  x (n  b ib n  + (τi )b ) i

(12.164)

i 

n b m k = (δkb δm  n − δkn δm  b )m m  n  b ib n  + m  x

=

n b b m n n m k m n  n  (b x ) − b (n  x ) + m  x





ikm  (τi )nb

i

ikm  (τi )nb

i

Consider the first term on the right in (12.164). We have, in view of (12.154), 











bb x b = r bb N b = r δb c bb N c = r (h b c − H ψb ψc )bb N c

688

Chapter 12. Recovery of the Acoustical Assumptions

 By (12.45) and the fact that h b c bb Tˆ c = 0, we have: 



h b c bb N c = h b c bb y c Hence, substituting, we obtain: 





bb x b = r bb (h b c y c − H ψ N ψb )

(12.165)

ψ N = N i ψi

(12.166)

where: Introducing the St,u -tangential vectorfield: /y =  · y  

(12.167)



and noting that bb h b c = bc h b b , (12.165) takes the form: 





bb x b = r (h b b y/b − H ψ N ψb )

(12.168)

Consider next the second term on the right in (12.164). Here we have: 



nn x n = r nn N n = r nn y n or, simply:





nn x n = r y/n

(12.169)

In view of (12.168), (12.169), equation (12.164) reduces to:      n k (Ri )m (D / Ri )nb = r m /b − H ψ N ψb ) − m /n + m ikm  (τi )nb b y n  n  (h b b y m N i

i

(12.170) Hence, from (12.163), the second term on the right in (12.125) is given by:    −1  ab /ξ )(ξ · y/ ) + (h ) (ηi )a (τi · ξ )b 2r ζn ξn − tr(D

(12.171)

i

where ζ is the St,u 1-form with rectangular components: ζn = (D /ξ )n a (y/a − H ψ N ( ψ$ )a )

(12.172)

and the ηi are the St,u 1-forms with rectangular components: (ηi )a = N k ikm  (D /ξ )m  a Now, by (12.134), for any pair of St,u 1-forms ζ , ξ we have: ζn ξn = δm  n ζm  ξn = (h

−1 m  n 

)

ζm  ξn +

H ψm  ψn ζm  ξn 1 + ρH

(12.173)

Chapter 12. Recovery of the Acoustical Assumptions

689

hence, denoting: ψ · ζ = ψm  ζm  , we have: |ζn ξn | ≤ |ζ ||ξ | +

ψ · ξ = ψm  ξm  H |ψ · ζ ||ψ · ξ | 1 + ρH

By (12.159): (ψ · ζ )2 ≤ ρζ 2 ≤ (1 + ρ H )|ζ |2, (ψ · ξ )2 ≤ ρξ 2 ≤ (1 + ρ H )|ξ |2, hence: |ψ · ζ ||ψ · ξ | ≤ (1 + ρ H )|ζ ||ξ |, and substituting above we obtain, for any pair of St,u 1-forms ζ , ξ : |ζn ξn | ≤ (1 + ρ H )|ζ ||ξ |

(12.174)

In the present case, where ζ is given by (12.172), recalling that (see (12.18)): √ |ψ N |, | ψ| ≤ ρ we have:

|ζ | ≤ |D /ξ |(|y/ | + ρ H )

(12.175)

hence by (12.174) the first term in parenthesis in (12.171) is bounded in absolute value by: (12.176) (1 + ρ H )|D /ξ ||ξ |(|y/| + ρ H ) The second term in parenthesis in (12.171) is bounded in absolute value by: √ 2|D /ξ ||ξ ||y/ |

(12.177)

where we have used the fact that for any 2-covariant St,u tensorfield M we have: 1 (trM)2 ≤ |M|2 2 Finally, the third term in parenthesis in (12.171) is bounded in absolute value by: ) )  2 |ηi ||τi ||ξ | ≤ |ξ | |τi | |ηi |2 (12.178) i

i

i

Moreover, using (12.50) we obtain:  −1  |ηi |2 = (h )ab N k N l ikm  iln (D /ξ )m  a (D /ξ )n b i

i

= (h

−1 ab

) N k N l (δkl δm  n − δkn δlm  )(D /ξ )m  a (D /ξ )n b

= (h

−1 ab

≤ (h

−1 ab





) ((D /ξ )n a (D /ξ )n b − N m N n (D /ξ )m  a (D /ξ )n b ) ) (D /ξ )n a (D /ξ )n b ≤ (1 + ρ H )|D /ξ |2

(12.179)

690

Chapter 12. Recovery of the Acoustical Assumptions

The last inequality is proved as follows. By (12.134) we have: (h

−1 ab

) (D /ξ )n a (D /ξ )n b = |D /ξ |2 +

H |ψ · D /ξ |2 1 + ρH

where ψ · D /ξ is the St,u 1-form with rectangular components: /ξ )n a (ψ · D /ξ )a = ψn (D By (12.128), (12.130): |ψ · D /ξ |2 = (h

◦

−1 ab

) ψm  ψn (D /ξ )m  a (D /ξ )n b = ψm  ψn ϑm  n ≤ ρ tr ϑ

Now, by (12.134), (12.135), denoting ψm  ψn ϑm  n = ψ · ϑ · ψ, ◦

tr ϑ = trϑ +

H ρH ◦ tr ϑ ψ · ϑ · ψ ≤ trϑ + 1 + ρH 1 + ρH

hence:

◦

tr ϑ ≤ (1 + ρ H )trϑ

(12.180)

In view of (12.136) we then conclude that: /ξ |2 |ψ · D /ξ |2 ≤ ρ(1 + ρ H )|D Substituting this above yields: (h

−1 ab

) (D /ξ )n a (D /ξ )n b ≤ (1 + ρ H )|D /ξ |2

(12.181)

which is the last inequality in (12.179). Combining the above results we conclude that the second term on the right in (12.125) is bounded in absolute value by:   )   √

|τi |2 |D /ξ ||ξ | (12.182) 2r (1 + ρ H )(ρ H + |y/ |) + 2|y/ | + 1 + ρ H   i

Combining the results (12.137), (12.182), (12.162), for the three terms on the righthand side of (12.125), yields the following proposition. Proposition 12.5 Consider a surface St,u for which the following hold: sup y   ≤ 1

sup H | ψ|2 ≤ 1

St,u

St,u

Then for every St,u 1-form ξ we have: 



|L / Ri ξ |2 ≥ r 2 (1 − y  2 )|D /ξ |2 + 1 − H | ψ|2 − 2(1 + ρ H )

i

− 2r pointwise on St,u .

  

)

 |τi |2  |ξ |2

i

√

(1 + ρ H )(ρ H + |y/ |) + 2|y/ | + 1 + ρ H

) i

|τi |2

  

|D /ξ ||ξ |

Chapter 12. Recovery of the Acoustical Assumptions

691

As a corollary of Proposition 12.5 we have the following. Corollary 12.5.a Suppose that: sup |ψ0 − k|, sup ε t 0

ε t 0

sup y   ≤ Cδ0 ε

and

t 0

√ ρ ≤ Cδ0

sup

)

ε

t 0

|τi |2 ≤ Cδ0

i

for some fixed positive constant C. Then if δ0 is suitably small, for any St,u 1-form ξ , ε differentiable on the St,u , we have, pointwise on t 0 : 

|L / R i ξ |2 ≥

i

 1 2 r |D /ξ |2 + |ξ |2 2

If also: inf r ≥ ε1 η0 (1 + t) ε

t 0

ε

(see Corollary 12.2.a) then hypothesis H1 holds on t 0 . In fact, for any St,u 1-form ξ , ε differentiable on the St,u , we have, pointwise on t 0 :  |L / R i ξ |2 |D /ξ |2 ≤ C(1 + t)−2 i

where C is the constant: C=

2 ε12 η02

Proof. The assumptions imply: 1 y  ≤ , 2 

ρ H + 2(1 + ρ H )

)

|τi |2 ≤

i

1 4

and: (1 + ρ H )(ρ H +





1 + ρ H y ) +





2(1 + ρ H )y  +



1 + ρH

) i

|τi |2 ≤

1 4

if δ0 is suitably small. In view of (12.18), the first two imply that the coefficients of /ξ |2 , |ξ |2 in the inequality of Proposition 12.5 are not less than 3/4, while since r 2 |D

|y/ | ≤ |y  | ≤ 1 + ρ H y   ε

(for any vector V tangent to t 0 we have: |V |2 ≤ (1 + ρ H )V 2 ), the coefficient of −2r |D /ξ ||ξ | in the same inequality is not greater than 1/4. The first statement of the

692

Chapter 12. Recovery of the Acoustical Assumptions

corollary then follows (for, 2r |D /ξ ||ξ | is not greater than r 2 |D /ξ |2 + |ξ |2 ), and the second statement is an immediate consequence. Let us now return to Lemma 12.2 and let s 0 be the maximal value of s0 ∈ [0, s] s such that hypothesis H1 holds on Wε00 . We shall show that in fact s 0 = s. For, suppose that s 0 < s. Now the first assumption of Corollary 12.5.a holds for all t ∈ [0, s] by virtue of assumption E{0} , while the second and fourth likewise hold for all t ∈ [0, s] by virtue of Proposition 12.3. Also, from the definition (12.144), the estimate for λi of Proposition 12.3, the estimate (12.91), hypothesis H0, assumption E{1} , and the bound (12.28), we obtain: max sup |(τi )b − λi α −1 χ| ≤ Cδ02 (1 + t)−2 [1 + log(1 + t)] i

ε

t 0

: for all t ∈ [0, s]

(12.183)

ε

Thus, if for some t ∈ [0, s], χ  satisfies on t 0 the estimate: χ   L ∞ ( ε0 ) ≤ C0 (1 + t)−1

(12.184)

t

for some fixed positive constant C0 , then we have: max τi  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] t

i

(12.185)

ε

so on t 0 the third assumption of Corollary 12.5.a holds as well, hence hypothesis H1 ε holds on t 0 . On the other hand, taking s0 = s 0 in Lemma 12.2, yields the conclusion that the estimate (12.186) χ   L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] t

holds for all t ∈ [0, s 0 ], in particular at t = s 0 . But, if δ0 is suitably small this implies that: 1 χ   L ∞ ( ε0 ) ≤ C0 (1 + t)−1 : at t = s 0 t 2 therefore by continuity (12.184) must hold in a suitably small interval [s 0 , s∗ ] ⊂ [s 0 , s]. But then all assumptions of Corollary 12.5.a hold for all t ∈ [0, s∗ ], hence hypothesis H1 holds on Wεs0∗ , contradicting the maximality of s 0 . We conclude that hypothesis H1 holds on Wεs0 . From Lemma 12.2 with s0 = s we then obtain the following proposition. Q QQ Proposition 12.6 Let assumptions E{2} , E{1} , E{0} , hold on Wεs0 , and let the initial data satisfy: ◦

κ − 1 L ∞ ( ε0 ) ≤ Cδ0 ,

max  λi  L ∞ ( ε0 ) ≤ Cδ0 i

0

and:

0

χ   L ∞ ( ε0 ) ≤ Cδ0 0

Then, if δ0 is suitably small, hypothesis H1 holds on Wεs0 . Moreover, we have, for all t ∈ [0, s]: χ   L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] t

Chapter 12. Recovery of the Acoustical Assumptions

693

In view of the fact that the assumptions of Proposition 12.6 contain those of Proposition 12.3, it follows that under the assumptions of Proposition 12.6, the assumptions of Proposition 10.1 hold for l = 1. Therefore the conclusions of Proposition 10.1 and all its corollaries hold for l = 1. / Ri ϑ for an arbitrary Let us now investigate hypotheses H2 and H2 . We consider L symmetric 2-covariant St,u tensorfield ϑ. The components of this in rectangular coordinates are given by: (L / Ri ϑ)ab = (D / Ri ϑ)ab + ϑmb (D / R i )m / R i )m a + ϑam (D b, m /ϑ)mab (D / Ri ϑ)ab = (Ri ) (D Hence, we have:   |L / Ri ϑ|2 = |D / Ri ϑ|2 i

+2

 (h

i −1 ac

) (h

(12.187)

(12.188)

−1 bd

) (ϑmb (D / R i )m / R i )m / Ri ϑ)cd a + ϑam (D b )(D

i

 −1 −1 + (h )ac (h )bd (ϑmb (D / R i )m / R i )m / Ri )nc + ϑcn (D / Ri )nd ) a + ϑam (D b )(ϑnd (D i

Consider the first term on the right in (12.188). We have:   −1 −1 |D / Ri ϑ|2 = (h )ac (h )bd (D / Ri ϑ)ab (D / Ri ϑ)cd i

i

= (h

−1 ac

) (h

 

−1 bd

)

(12.189)

 (Ri ) (Ri ) m

n

/ϑ)ncd (D /ϑ)mab (D

i

Substituting from (12.63) and noting that: n m /ϑ)mab (D /ϑ)ncd = (D /ϑ)kab (D /ϑ)lcd k l (D

we obtain: 

|D / Ri ϑ|2 = r 2 (h

−1 ac

) (h

−1 bd

) (δkl − y k y l )(D /ϑ)kab (D /ϑ)lcd

(12.190)

i

Consider the symmetric 2-covariant St,u tensorfield ϕ with rectangular components: ϕkl = (h

−1 ac

) (h

−1 bd

) (D /ϑ)kab (D /ϑ)lcd

(12.191)

Since ϕ is positive semi-definite, we have, as in (12.132), (12.135), ◦

y k y l ϕkl ≤ y  2 tr ϕ and

◦

tr ϕ ≥ trϕ

(12.192)

Assuming that y   ≤ 1 and noting that: trϕ = (h

−1 kl

) (h

−1 ac

) (h

−1 bd

) (D /ϑ)kab (D /ϑ)lcd = |D /ϑ|2

(12.193)

694

Chapter 12. Recovery of the Acoustical Assumptions

(12.190) is then seen to imply:  |D / Ri ϑ|2 ≥ r 2 (1 − y  2 )|D /ϑ|2

(12.194)

i

Consider next the third term on the right in (12.188). This is the sum of two terms: 2III1 + 2III2 

where: III1 = (h

−1 ac

) (h

−1 bd

) ϑmb ϑnd

 (D / R i )m / Ri )nc a (D i

III2 = (h

−1 ac

) (h

−1 bd

) ϑam ϑnd

 

 (12.195) 

(D / R i )m / Ri )nc b (D

(12.196)

i

Substituting (12.150) in (12.195) we find: III1 = (h

−1 ac

−1









) (h )bd ϑm  b ϑn d (δa  c aa cc δm  n − na m c )   −1 −1 (h )ac (h )bd (wi )ab (τi · ϑ)cd + |τi · ϑ|2 +2 i

(12.197)

i

where the wi are the 2-covariant St,u tensorfields with rectangular components: 

(wi )ab = aa ia  m  ϑm  b

(12.198)

and: (τi · ϑ)ab = (τi )m a ϑmb Substituting from (12.155) in the first term on the right in (12.197) and in view of (12.68), we obtain that this term is (2 − H | ψ|2 )δm  n (h

−1 bd

) ϑm  b ϑn d − |ϑ|2 ≥ (1 − H | ψ|2 )|ϑ|2

(12.199)

provided that H | ψ|2 ≤ 1. The second term on the right in (12.197) is bounded in absolute value by: ) )  2 2 |wi ||τi · ϑ| ≤ 2|ϑ| |wi | |τi |2 (12.200) i

i

i

Using (12.50) we obtain:  −1    −1 |wi |2 = (h )ac (h )bd aa cc ia  m  ic n ϑm  b ϑn d i

i

= (h = (h

−1 ac

) (h

−1 bd

−1 bd





) aa cc (δa  c δm  n − δa  n δc m  )ϑm  b ϑn d

) δa  c (h

−1 ac





) aa cc δm  n ϑm  b ϑn d − |ϑ|2

Chapter 12. Recovery of the Acoustical Assumptions

695

hence, by (12.155),  −1 |wi |2 = (2 − H | ψ|2 )δm  n (h )bd ϑm  b ϑn d − |ϑ|2

(12.201)

i

Let M be the symmetric 2-covariant St,u tensorfield with rectangular components: Mm  n = (h We have: δm  n (h

−1 bd

) ϑm  b ϑn  d ◦

−1 bd

) ϑm  b ϑn d =tr M,

(12.202)

|ϑ|2 = trM

(12.203)

Now M is positive semi-definite, therefore the analogue of (12.130) holds for M. It follows that (see (12.132)): ◦

ψ · M · ψ = ψm  ψn Mm  n ≤ ρ tr M

(12.204)

It then follows by (12.134) that: ◦

tr M = trM + hence:

ρH ◦ H ψ · M · ψ ≤ trM + tr M 1 + ρH 1 + ρH ◦

tr M ≤ (1 + ρ H )trM

(12.205)

In view of (12.203) and (12.205), (12.201) implies: ) |wi |2 ≤ (1 + ρ H )|ϑ|

(12.206)

i

By (12.200) and (12.206), the second term on the right in (12.197) is bounded in absolute value by: ) |τi |2 |ϑ|2 (12.207) 2(1 + ρ H ) i

We conclude from (12.199), (12.207), that:   )  III1 ≥ 1 − H | ψ|2 − 2(1 + ρ H ) |τi |2  |ϑ|2 + |τi · ϑ|2 i

(12.208)

i

provided that H | ψ|2 ≤ 1. Substituting (12.150) in (12.196) we find: III2 = (h

−1 ac

−1









) (h )bd ϑam  ϑn d (δb c δm  n bb cc − nb m (12.209) c )  −1  −1 −1 −1 +2 (h )ac (h )bd (τi ˜· ϑ)ab (wi )cd + (h )ac (h )bd (τi ˜· ϑ)ab (τi · ϑ)cd i

i

696

Chapter 12. Recovery of the Acoustical Assumptions

where:

(τi ˜· ϑ)ab = (τi · ϑ)ba

Since (h

−1 ac



) cc = aa  (h

−1 a  c

)

,

(h

−1 bd



) bb = dd  (h

−1 d  b

)

we have, in view of (12.154), δb c (h

−1 ac

) (h

−1 bd





) bb cc = δb c (h

−1 a  c

)

(h

−1 d  b

)

= (h b c − H ψb ψc )(h = (h

−1 a  d 

)

aa  dd 

−1 a  c

)

aa  dd  − H (h

(h

−1 d  b

−1 d  b

)

)

aa  dd 

ψb (h

−1 a  c

)

ψc aa  dd 

Consequently, the first term on the right in (12.209) is given by: δm  n ((h

−1 a  d 

)

◦

ϑm  a ϑn d  − H ξm  ξn ) − (trϑ)2 =tr M − H ξ 2 − (trϑ)2

where ξ is the St,u 1-form with rectangular components: ξm  = ψb (h

−1 b d 

)

ϑd  m 

(12.210)

and M is as in (12.202). Since ◦

tr M ≥ trM = |ϑ|2 and (see (12.159)): ξ 2 ≤ (1 + ρ H )|ξ |2 ≤ (1 + ρ H )ρ|ϑ|2 ,

(12.211)

it follows that the first term on the right in (12.209) is not less than:

Since:

(1 − ρ H (1 + ρ H ))|ϑ|2 − (trϑ)2

(12.212)

|τi ˜· ϑ| = |τi · ϑ|

(12.213)

the second term on the right in (12.209) is bounded in absolute value in the same way as the second term on the right in (12.197) (see (12.207)). Finally, again by (12.213), the third term on the right in (12.209) is bounded in absolute value by:  |τi · ϑ|2 (12.214) i

We conclude that:  III2 ≥ 1 − ρ H (1 + ρ H ) − 2(1 + ρ H )

) i

 |τi |2  |ϑ|2 − (trϑ)2 −



|τi · ϑ|2

i

(12.215)

Chapter 12. Recovery of the Acoustical Assumptions

697

Combining (12.208) and (12.215) yields:  III1 + III2 ≥ 2 − H | ψ|2 − ρ H (1 + ρ H ) − 4(1 + ρ H )

)

 |τi |2  |ϑ|2 − (trϑ)2

i

(12.216) Consider finally the second term on the right in (12.188). This is: 4II where: II = (h

−1 ac

) (h

−1 bd

) ϑmb

 

 k (D /ϑ)kcd (D / i R)m a (Ri )

(12.217)

i

Substituting from (12.170) we obtain:  −1 −1 /ϑ)d II = r (h )bd ϑm  b ιm  d − (h )bd (ϑ · y/ )b (trD +

 −1 −1 (h )ac (h )bd (τi · ϑ)ab (i )cd

(12.218)

i

Here ι is the 2-covariant St,u tensorfield with rectangular components: /ϑm  cd (y/c − H ψ N ( ψ$ )c ) ιm  d = (D

(12.219)

and the i are the symmetric 2-covariant St,u tensorfields with rectangular components: /ϑ)k  cd (i )cd = N l ilk  (D

(12.220)

Also, in (12.218), (trD /ϑ)d = (h

−1 ac

) (D /ϑ)acd ,

(ϑ · y/ )b = ϑbm y/m

Now, by (12.134) we have:     −1    −1 bd     −1 bd (h )m n + (h ) ϑm b ιm d  = (h ) ≤ |ϑ||ι| +

   H ψm  ψn ϑm  b ιn d  1 + ρH

H |ψ · ϑ||ψ · ι| 1 + ρH

(12.221)

where ψ · ϑ, ψ · ι, are the St,u 1-forms with rectangular components: (ψ · ϑ)a = ψm  ϑm  a ,

(ψ · ι)a = ψm  ιm  a

By (12.202)–(12.205): |ψ · ϑ|2 = (h

−1 ab

) ψm  ϑm  a ψn ϑn b = ψm  ψn Mm  n ◦

= ψ · M · ψ ≤ ρ tr M ≤ ρ(1 + ρ H )trM = ρ(1 + ρ H )|ϑ|2

(12.222)

698

Chapter 12. Recovery of the Acoustical Assumptions

Defining also the symmetric 2-covariant St,u tensorfield K with rectangular components: K m  n = (h we have: |ψ · ι|2 = (h

−1 ab

) ιm  a ιm  b

(12.223)

−1 ab

) ψm  ιm  a ψn ιn b = ψm  ψn K m  n

hence, by virtue of the fact that K is positive semi-definite: ◦

|ψ · ι|2 = ψ · K · ψ ≤ ρ tr K ≤ ρ(1 + ρ H )trK = ρ(1 + ρ H )|ι|2

(12.224)

From (12.222), (12.224) we obtain: |ψ · ϑ||ψ · ι| ≤ ρ(1 + ρ H )|ϑ||ι|

(12.225)

Substituting this in (12.221) we obtain:     −1 bd (h ) ϑm  b ιm  d  ≤ (1 + ρ H )|ϑ||ι| Since also:

(12.226)

|ι| ≤ |D /ϑ|(|y/ | + ρ H )

(12.227)

we conclude that the first term in parenthesis in (12.218) is bounded in absolute value by: (1 + ρ H )|ϑ||D /ϑ|(|y/| + ρ H )

(12.228)

Next, since: 1 |trD /ϑ|2 ≤ |D /ϑ|2 2 the second term in parenthesis in (12.218) is bounded in absolute value by: √  2|y/ ||ϑ||D /ϑ|

(12.229)

Finally the third term in parenthesis in (12.218) is bounded in absolute value by: ) )  2 |ϑ||τi ||i | ≤ |ϑ| |τi | |i |2 (12.230) i

i

i

Moreover, using (12.50) we obtain:  −1  −1 |i |2 = (h )ac (h )bd N k N l ikm  iln (D /ϑ)m  ab (D /ϑ)n cd i

i

= (h = (h ≤ (h

−1 ac

−1 bd

−1 ac

−1 bd

−1 ac

−1 bd

) (h ) (h ) (h

) N k N l (δkl δm  n − δkn δlm  )(D /ϑ)m  ab (D /ϑ)n cd 



) ((D /ϑ)m  ab (D /ϑ)m  cd − N m N n (D /ϑ)m  ab (D /ϑ)n cd ) ) (D /ϑ)m  ab (D /ϑ)m  cd ≤ (1 + ρ H )|D /ϑ|2

The last inequality is proved as follows. By (12.134) we have: (h

−1 ac

) (h

−1 bd

) (D /ϑ)m  ab (D /ϑ)m  cd = |D /ϑ|2 +

H |ψ · D /ϑ|2 1 + ρH

(12.231)

Chapter 12. Recovery of the Acoustical Assumptions

699

where ψ · D /ϑ is the symmetric 2-covariant St,u tensorfield with rectangular components: (ψ · D /ϑ)ab = ψm  (D /ϑ)m  ab Recalling the definition (12.191), |ψ · D /ϑ|2 = (h

−1 ac

) (h

◦

−1 bd

) ψm  ψn (D /ϑ)m  ab (D /ϑ)n cd = ψm  ψn ϕm  n ≤ ρ tr ϕ

by the analogue of (12.130) for ϕ, which holds by virtue of the fact that ϕ is positive semi-definite. By (12.134), (12.135), denoting ψm  ψn ϕm  n = ψ · ϕ · ψ, ◦

tr ϕ = trϕ +

ρH ◦ H ψ · ϕ · ψ ≤ trϕ + tr ϕ 1 + ρH 1 + ρH

hence:

◦

tr ϕ ≤ (1 + ρ H )trϕ

(12.232)

In view of (12.193) we then conclude that: |ψ · D /ϑ|2 ≤ ρ(1 + ρ H )|D /ϑ|2 Substituting this above yields: (h

−1 ac

) (h

−1 bd

) (D /ϑ)m  ab (D /ϑ)m  cd ≤ (1 + ρ H )|D /ϑ|2

(12.233)

which is the last inequality in (12.231). Combining the above results we conclude that:   )  

√ |II| ≤ r (1 + ρ H )(ρ H + |y/ |) + 2|y/ | + 1 + ρ H |τi |2 |D /ϑ||ϑ| (12.234)   i

Combining the results (12.194), (12.234), (12.216), for the three terms on the right in (12.188), yields the following proposition. Proposition 12.7 Consider a surface St,u for which the following hold: sup y   ≤ 1

sup H | ψ|2 ≤ 1

St,u

St,u

Then for every symmetric 2-covariant St,u tensorfield ϑ we have:  |L / Ri ϑ|2 ≥ r 2 (1 − y  2 )|D /ϑ|2   i ) + 2 2 − H | ψ|2 − ρ H (1 + ρ H ) − 4(1 + ρ H ) |τi |2  |ϑ|2 − 2(trϑ)2 i

 ) 

√ − 4r (1 + ρ H )(ρ H + |y/ |) + 2|y/ | + 1 + ρ H |τi |2 |D /ϑ||ϑ|    

i

pointwise on St,u .

700

Chapter 12. Recovery of the Acoustical Assumptions

As a corollary of Proposition 12.7 we have the following. Corollary 12.7.a Suppose that: sup |ψ0 − k|, sup ε t 0

ε t 0



sup y  ≤ Cδ0 ε

and

t 0

√ ρ ≤ Cδ0

sup

)

ε

t 0

|τi |2 ≤ Cδ0

i

for some fixed positive constant C. Then if δ0 is suitably small, for any symmetric 2ε covariant St,u tensorfield ϑ, differentiable on the St,u , we have, pointwise on t 0 :  i

1 |L / Ri ϑ|2 ≥ r 2 |D /ϑ|2 + 2|ϑ|2 − 2(trϑ)2 2

If also: inf r ≥ ε1 η0 (1 + t) ε

t 0

ε

(see Corollary 12.2.a) then hypotheses H2 and H2 hold on t 0 . In fact, for any symmetric ε 2-covariant St,u tensorfield ϑ, differentiable on the St,u , we have, pointwise on t 0 :    2 −2 2 2 |D /ϑ| ≤ C(1 + t) |L / Ri ϑ| + 2|ϑ| i

where C is the constant: C=

2 ε12 η02

and if the tensorfield ϑ is trace-free, we have the stronger inequality:  |L / Ri ϑ|2 |D /ϑ|2 ≤ C(1 + t)−2 i

Proof. The assumptions imply: 1 y  ≤ , 2 

ρ H (2 + ρ H ) + 4(1 + ρ H )

)

|τi |2 ≤

i

and: (1 + ρ H )(ρ H +



1 + ρ H y  ) +



2(1 + ρ H )y   +



1 + ρH

1 2

) i

|τi |2 ≤

1 4

if δ0 is suitably small. In view of (12.18), the first two imply that the coefficients of r 2 |D /ϑ|2 and |ϑ|2 in the inequality of Proposition 12.7 are not less than 3/4 and 3 respectively, while the coefficient of −2r |D /ϑ||θ | in the same inequality is not greater than

Chapter 12. Recovery of the Acoustical Assumptions

701

1/2. Since 2r |D /ϑ||ϑ| is not greater than (1/2)r 2|D /ϑ|2 + 2|ϑ|2 , the first statement of the corollary then follows. The trace-free case of the second statement is an immediate consequence, while the general case follows noting that (trϑ)2 ≤ 2|ϑ|2 . We remark that the assumptions of Corollary 12.7.a coincide with the assumptions of Corollary 12.5.a. Having shown that the assumptions of Corollary 12.5.a follow from those of Proposition 12.6, we conclude that under the assumptions of Proposition 12.6 hypotheses H2 and H2 hold on Wεs0 . In the sequel we shall make use of the following proposition, which is based on Propositions 12.2 and 12.4. Proposition 12.8 Let the assumptions of Proposition 12.6 hold and let X be an arbitrary St,u -tangential vectorfield defined on Wεs0 and differentiable on the St,u . Then, for any St,u 1-form ξ we have, pointwise:    −1 |L / X ξ | ≤ C(1 + t) / Ri ξ | + |ξ | + |ξ | max |L / Ri X| |X| max |L i

i

Also, for any symmetric 2-covariant St,u tensorfield ϑ we have, pointwise:    −1 |X| max |L |L / X ϑ| ≤ C(1 + t) / Ri ϑ| + |ϑ| + |ϑ| max |L / Ri X| i

i

Proof. Consider first the case of a St,u 1-form ξ . By Proposition 12.2 it holds, pointwise, that:  |(L / X ξ )(Ri )| (12.235) |L / X ξ | ≤ C(1 + t)−1 i

Now, we have: (L / X ξ )(Ri ) = X (ξ(Ri )) − ξ([X, Ri ]) = X · d/(ξ(Ri )) + ξ(L / Ri X)

(12.236)

hence: |(L / X ξ )(Ri )| ≤ |X||d/(ξ(Ri ))| + |ξ ||L / Ri X|

(12.237)

By hypothesis H0: |d/(ξ(Ri ))| ≤ C(1 + t)−1



|R j (ξ(Ri ))|

(12.238)

R j (ξ(Ri )) = (L / R j ξ )(Ri ) + ξ([R j , Ri ])

(12.239)

/ R j ξ ||Ri | + |ξ ||[R j , Ri ]| |R j (ξ(Ri ))| ≤ |L

(12.240)

j

and we have: hence: Now, by Corollary 10.1.e with l = 1: |Ri |, |[R j , Ri ]| ≤ C(1 + η0 t)

(12.241)

702

Chapter 12. Recovery of the Acoustical Assumptions

It follows that:

  / R j ξ | + |ξ | |R j (ξ(Ri ))| ≤ C(1 + t) |L

(12.242)

Substituting this in (12.238) we obtain:   |d/(ξ(Ri ))| ≤ C max |L / R j ξ | + |ξ |

(12.243)

j

Substituting (12.243) in turn in (12.237) we obtain:    |(L / X ξ )(Ri )| ≤ C |X| max |L / R j ξ | + |ξ | + |ξ ||L / Ri X| j

(12.244)

Substituting finally (12.244) in (12.235) yields the statement of the proposition in regard to St,u 1-forms. Consider next the case of a symmetric 2-covariant St,u tensorfield ϑ. By Proposition 12.4 it holds, pointwise, that:  |(L / X ϑ)(Ri , R j )| (12.245) |L / X ϑ| ≤ C(1 + t)−2 i, j

Now, we have: (L / X ϑ)(Ri , R j ) = X (ϑ(Ri , R j )) − ϑ([X, Ri ], R j ) − ϑ(Ri , [X, R j ]) = X · d/(ϑ(Ri , R j )) + ϑ(L / Ri X, R j ) + ϑ(Ri , L / R j X)

(12.246)

hence:   / Ri X| + |Ri ||L / R j X| |(L / X ϑ)(Ri , R j )| ≤ |X||d/(ϑ(Ri , R j ))| + |ϑ| |R j ||L / Rk X| (12.247) ≤ |X||d/(ϑ(Ri , R j ))| + C(1 + t)|ϑ| max |L k

in view of the first of the bounds (12.241). By hypothesis H0:  |d/(ϑ(Ri , R j ))| ≤ C(1 + t)−1 |Rk (ϑ(Ri , R j ))|

(12.248)

k

and we have: / Rk ϑ)(Ri , R j ) + ϑ([Rk , Ri ], R j ) + ϑ(Ri , [Rk , R j ]) Rk (ϑ(Ri , R j )) = (L

(12.249)

hence, in view of the bounds (12.241):   |Rk (ϑ(Ri , R j ))| ≤ C(1 + t)2 |L / Rk ϑ| + |ϑ|

(12.250)

Substituting this in (12.248) we obtain:   / Rk ϑ| + |ϑ| |d/(ϑ(Ri , R j ))| ≤ C(1 + t) max |L k

(12.251)

Chapter 12. Recovery of the Acoustical Assumptions

703

Substituting (12.251) in turn in (12.247) we obtain:    |(L / X ϑ)(Ri , R j )| ≤ C |X| max |L / Rk ϑ| + |ϑ| + |ϑ| max |L / Rk X| k

k

(12.252)

Substituting finally (12.252) in (12.245) yields the statement of the proposition in regard to symmetric 2-covariant St,u tensorfields. In the sequel we shall also make use of the following commutation lemma, which is analogous to Lemma 8.3. Lemma 12.3 Let Y be an arbitrary St,u -tangential vectorfield and ϑ an arbitrary 2covariant St,u tensorfield, defined on Wεs0 . Then we have: /Y ϑ − L /Y L /L ϑ = L / (Y ) Z ϑ L /L L Proof. Following the proof of Lemma 8.3, we may restrict ourselves to a given Cu . In defining L / L ϑ, L /Y ϑ we are considering the extension of ϑ to T Cu by the condition ϑ(L, W ) = ϑ(W, L) = 0 for any W ∈ T Cu . Thus, with W an arbitrary vectorfield tangential to Cu we have: (L L ϑ)(L, W ) = L(ϑ(L, W )) − ϑ([L, L], W ) − ϑ(L, [L, W ]) = 0 ([L, W ] is tangential to Cu ). Similarly, (L L ϑ)(W, L) = 0. It follows that: L /L ϑ = LL ϑ

(12.253)

However, with X an arbitrary St,u -tangential vectorfield, we have, by Lemma 8.2: (LY ϑ)(L, X) = Y (ϑ(L, X)) − ϑ([Y, L], X) − ϑ(L, [Y, X]) = ϑ([L, Y ], X) = ϑ( (Y ) Z , X) ([Y, X] is St,u -tangential). Similarly, (LY ϑ)(X, L) = ϑ(X,

(Y ) Z ).

Also,

(LY ϑ)(L, L) = Y (ϑ(L, L)) − ϑ([Y, L], L) − ϑ(L, [Y, L]) = 0 /Y ϑ is defined by restricting LY ϑ to T St,u and then ([Y, L] is tangential to Cu ). Since L extending to T Cu by the condition (L /Y ϑ)(L, W ) = (L /Y ϑ)(W, L) = 0 : for any W ∈ T Cu , it follows that, on the manifold Cu , L /Y ϑ = LY ϑ − (ϑ · where ϑ ·

(Y ) Z

and

(Y ) Z

(ϑ · (

(Y )

(Y )

(Y )

Z ) ⊗ dt − dt ⊗ (

(Y )

Z · ϑ)

(12.254)

: for all X ∈ T St,u

(12.255)

· ϑ are the St,u 1-forms defined by: Z )(X) = ϑ(X,

Z · ϑ)(X) = ϑ(

(Y )

(Y )

Z ),

Z , X)

704

Chapter 12. Recovery of the Acoustical Assumptions

This is so because (12.254) holds when evaluated on a pair (X 1 , X 2 ) of St,u -tangential vectorfields, by virtue of the definition of L /Y ϑ and the fact that X 1 t = X 2 t = 0. When evaluated on (L, X), where X is any St,u tangential vectorfield, the left-hand side vanishes, and the right-hand side likewise vanishes by virtue of the definition of (Y ) Z · ϑ and the fact that Xt = 0 while Lt = 1. Similarly, when evaluated on (X, L), where X is any St,u tangential vectorfield, the left-hand side vanishes, and the right-hand side likewise vanishes by virtue of the definition of ϑ · (Y ) Z . Finally, when evaluated on (L, L) both sides vanish, recalling that (LY ϑ)(L, L) = 0. Consider now the evaluation of L /L L /Y ϑ − L /Y L /L ϑ on the St,u frame vectorfields X A . This evaluation is: (L L (L /Y ϑ) − LY (L / L ϑ))(X A , X B )

(12.256)

Substituting from (12.253), (12.254) this becomes: (L L LY ϑ − L L (ϑ · (Y ) Z ) ⊗ dt − dt ⊗ L L ( = (L L LY ϑ − LY L L ϑ)(X A , X B )

(Y )

Z · ϑ) − LY L L ϑ)(X A , X B ) (12.257)

for, L L dt = d(Lt) = 0 and dt (X A ) = dt (X B ) = 0. Therefore (12.256) reduces to: (L L LY ϑ − LY L L ϑ)(X A , X B ) = (L[L ,Y ] ϑ)(X A , X B ) = (L

(Y ) Z

ϑ)(X A , X B )

(12.258)

The lemma thus follows. Let us now define, given a positive integer l and a multi-index (i 1 . . . i l ), the symmetric 2-covariant St,u tensorfield: (i1 ...il )  χl

=L / Ril . . . L / R i1 χ 

(12.259)

We shall derive from the propagation equation (12.77), using the above lemma, a propagation equation for (i1 ...il ) χ  . We first apply Lemma 11.23 taking X in that lemma to be the space of symmetric 2-covariant St,u tensorfields, A1 , . . . , An to be the operators L / R i1 , . . . , L / Rin and B to be the operator L / L to obtain: / R i1 , L / L ]χ  = [L / Ril . . . L

l−1 

L / Ril . . . L / Ril−k+1 [L / Ril−k , L / L ]L / Ril−k−1 . . . L / R i1 χ 

(12.260)

k=0

/ Ril−k−1 . . . L / Ri1 χ  , we obtain: By Lemma 12.3 applied to the case Y = Ril−k , ϑ = L [L / Ril−k , L / L ]L / Ril−k−1 . . . L / Ri1 χ  = −L / (Ril−k ) L / Ril−k−1 . . . L / R i1 χ  Z

(12.261)

Chapter 12. Recovery of the Acoustical Assumptions

705

Substituting in (12.260) then yields: L /L

(i1 ...il )  χl

=

l−1 

L / Ril . . . L / Ril−k+1 L / (Ril−k ) L / Ril−k−1 . . . L / R i1 χ  Z

k=0

+L / Ril . . . L / Ri1 (L /L χ )

(12.262)

We shall express the last term using the propagation equation (12.77). First, writing: χ  · χ $ = χ  · h/−1 · χ 

(12.263)

we express: / Ri1 (χ  · χ $ ) = L / Ril . . . L where the remainder (i1 ...il )

(i1 ...il )  χl

(i1 ...il ) $ χl

· χ $ + χ  ·

(i1 ...il )

+

rl

(12.264)

(i1 ...il ) r

l is given by:  (L / R )s1 χ  · (L / R )s2 h/−1 · (L / R )s 3 χ 

rl =

(12.265)

partitions

|s1 |+|s2 |+|s3 |=l;|s1 |,|s3 | =l

where the sum is over all ordered partitions {s1 , s2 , s3 } of the set {1, . . . , l} into three ordered subsets s1 , s2 , s3 , such that |s1 |, |s3 | = l. Next, we have: / Ri1 (χ  · a) = L / Ril . . . L where

(i1 ...il ) s l

(i1 ...il )  χl

·a+

(i1 ...il )

sl

(12.266)

is the 2-covariant St,u tensorfield:  (i1 ...il ) sl = (L / R )s1 χ  · (L / R )s 2 a

(12.267)

partitions |s1 |+|s2 |=l,|s1 | =l

We denote by

(i1 ...il ) s˜ l

the transpose of

L / Ril . . . L / Ri1 (L /L χ ) =

1 2

(i1 ...il ) s . l

(i1 ...il )  χl

· a +

From equation (12.77) we then obtain: (i1 ...il )  ˜  χl · a



+

(i1 ...il )

bl

(12.268)

where a  is the T11 type St,u tensorfield: a  = a + 2χ $ and

(i1 ...il ) b l

(12.269)

is the symmetric 2-covariant St,u tensorfield:

(i1 ...il )

bl = L / Ril . . . L / R i1 b +

1 2

(i1 ...il )

sl +

(i1 ...il )

 s˜l +

(i1 ...il )

rl

(12.270)

Equation (12.262) with the last term given by (12.268) constitutes the propagation equation for (i1 ...il ) χl .

706

Chapter 12. Recovery of the Acoustical Assumptions

Given a non-negative integer l let us denote by X / l the statement that there is a constant C independent of s such that for all t ∈ [0, s]: X / l : max L / Ril . . . L / Ri1 χ   L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] i1 ...il

t

We then denote by X / [l] the statement: X / [l]

: X / 0 and . . . and X / l

The following lemma relates X / [l−1] to X / [l−1] in the context of Proposition 10.1. Lemma 12.4 Let hypothesis H0 and the estimate (10.30) of Chapter 10 hold. Let also Q / [l−1] and X / [l−1] hold, for some positive integer l. Then the bootstrap assumptions E / [l] , E the assumption X / [l−1] also holds. / 0 , the lemma is trivial for l = 1. We proceed by Proof. Since X / 0 coincides with X induction. Let then the lemma hold with l replaced by l − 1, for some l ≥ 2. That is, let Q / [l−1] , E / [l−1] , hypothesis H0 and the estimate (10.30), imply X / [l−2] . X / [l−2] together with E But then the assumptions of Proposition 10.1 with l replaced by l − 1 hold. Therefore the conclusion of Proposition 10.1 and those of all its corollaries hold with l replaced by l − 1. In particular, the conclusion of Corollary 10.1.d holds with l replaced by l − 1, that is: max 

(Ri )

π /∞,[l−2], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] : for all t ∈ [0, s] (12.271) t

i

Now for any positive integer n we have: / R i1 χ = L / R in . . . L / R i1 χ  + L / R in . . . L

η0 L /R . . . L / R i2 1 − u + η0 t in

(Ri1 )

π /

(12.272)

hence: max L / R in . . . L / Ri1 χ L ∞ ( ε0 )

i1 ...in

(12.273)

t

≤ max L / R in . . . L / Ri1 χ   L ∞ ( ε0 ) + C(1 + t)−1 max  i1 ...in

t

j

(R j )

π /∞,[n−1], ε0 t

Therefore, taking n = l − 1, X / [l−1] together with (12.271) imply X / [l−1] . This completes the inductive step and the proof of the lemma. Proposition 12.9 Let the assumptions of Proposition 12.6 hold. Let, in addition, the Q QQ assumptions E{l+2} , E{l+1} , E{l} , hold on Wεs0 for some non-negative integer l, and let the initial data satisfy: χ  ∞,[l], ε0 ≤ Cl δ0 0

Then there is a constant Cl independent of s such that for all t ∈ [0, s] we have: χ  ∞,[l], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] t

that is,

X / [l]

holds on

Wεs0 .

Chapter 12. Recovery of the Acoustical Assumptions

707

Proof. For l = 0 the proposition reduces to Proposition 12.6. We proceed by induction. Let then the proposition hold with l replaced by l − 1, for some l ≥ 1. Then X / [l−1] holds s s on Wε0 , hence by Lemma 12.4, X / [l−1] holds on Wε0 as well. Thus the assumptions of Proposition 10.1 all hold. Therefore the conclusion of Proposition 10.1 and those of all its corollaries hold. By Proposition 12.6 and the estimate (12.83) we have: a   L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] t

: for all t ∈ [0, s]

(12.274)

We shall presently estimate (i1 ...il ) bl (see (12.270)) using the corollaries of Proposition 10.1 together with the assumptions of the present proposition for l. First, by Corollaries 10.1.b, 10.1.g: ψTˆ ∞,[l], ε0 , ψ L − k∞,[l], ε0 ,  ψ∞,[l], ε0 ≤ Cl δ0 (1 + t)−1 t

t

t

: for all t ∈ [0, s]

(12.275)

Next, since ω L L is like ψ L − k but with Lψ0 , Lψi , in the role of ψ0 − k, ψi , respectively, ω / L is like ψ but with Lψi in the role of ψi , and ω / is like ψ but with d/ψi in the role of ψi , Q by virtue of assumptions E{l} , E{l+1} we obtain: ω L L ∞,[l], ε0 , ω / L ∞,[l], ε0 , ω /∞,[l], ε0 ≤ Cl δ0 (1 + t)−2 t

t

t

: for all t ∈ [0, s]

(12.276)

The same assumptions imply: Lσ ∞,[l], ε0 , d/σ ∞,[l], ε0 ≤ Cl δ0 (1 + t)−2 t

t

: for all t ∈ [0, s]

(12.277)

The estimates (12.275)–(12.277) yield, through (12.80), (12.17), (12.90): e∞,[l], ε0 , κ −1 ζ  ∞,[l], ε0 , k/ ∞,[l], ε0 ≤ Cl δ0 (1 + t)−2 t

t

t

: for all t ∈ [0, s]

(12.278)

The estimates (12.275)–(12.277) also yield, through (12.73)–(12.76): (N)

 α ∞,[l], ε0 ≤ Cl δ02 (1 + t)−4 : for all t ∈ [0, s] t

(12.279)

Moreover, using the estimates (12.278) we obtain, through (12.72): ( P N)

 α

∞,[l], ε0 ≤ Cl δ02 (1 + t)−4 : for all t ∈ [0, s] t

(12.280)

( P P)

We turn to α , given by (12.71). Here we must estimate (L)2 σ , d/ Lσ , as well as QQ Q 2 , E{l} imply: D / σ in  ∞,[l], ε0 norm. First, the assumptions E{l} t

(L)2 σ ∞,[l], ε0 ≤ Cl δ0 (1 + t)−3 : for all t ∈ [0, s] t

(12.281)

708

Chapter 12. Recovery of the Acoustical Assumptions

Q Also, the assumptions E{l+1} , E{l+1} , together with hypothesis H0 imply:

d/ Lσ ∞,[l], ε0 ≤ Cl δ0 (1 + t)−3 : for all t ∈ [0, s] t

(12.282)

In the case of D / 2 σ we have, by Lemma 10.9 (see (10.427)): L / R ik . . . L / R i1 D /2σ = D / 2 Ri k . . . Ri 1 σ +

(i1 ...ik )

ck

(12.283)

where: (i1 ...ik )

ck = −

k−1 

L / R ik . . . L / Rik−m+1 (

(Rik−m )

π /1 · d/(Rik−m−1 . . . Ri1 σ ))

(12.284)

m=0

Here, k ≤ l. In the term corresponding to m ∈ {0, . . . , k − 1} in the sum in (12.284), (Rik−m ) π /1 receives at most m angular derivatives. Now, since the assumptions of the present proposition include those of Proposition 12.6, by the remark following the proof of Corollary 12.7.a, under the present assumptions hypotheses H2 and H2 (as well as H0 and H1) hold on Wεs0 . Taking also into account the inductive hypothesis, which as we have already noted, implies that X / [l−1] holds on Wεs0 , the assumptions of Lemma 10.11 all hold with l − 1 in the role of l. Therefore we have: max  i

(Ri )

π /1 ∞,[l−2], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] : for all t ∈ [0, s] (12.285) t

It follows that, for k ≤ l − 1: max max 

(i1 ...ik )

k≤l−1 i1 ...ik

ck  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)] : for all t ∈ [0, s] t

(12.286) For k = l, on the other hand, we write: (i1 ...il )

cl = −L / Ril . . . L / R i2 ( −

l−2 

(Ri1 )

π /1 · d/σ )

L / Ril . . . L / Ril−m+1 (

(Ril−m )

π /1 · d/(Ril−m−1 . . . Ri1 σ )) (12.287)

m=0

The sum on the right in (12.287) is bounded as in (12.286). Moreover, expanding the first term on the right in (12.287): / R i2 ( L / Ril . . . L

(Ri1 )

π /1 · d/σ ) = (L / Ril . . . L / Ri2 (Ri1 ) π /1 ) · d/σ  (L / R )s1 (Ri1 ) π /1 · d/(R)s2 σ +

(12.288)

partitions |s1 |+|s2 |=l−1,s2 =∅

the sum in (12.288) is likewise bounded as in (12.286). Consequently: 

(i1 ...il )

cl + (L / Ril . . . L / R i2

(Ri1 )

π /1 ) · d/σ  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)]

: for all t ∈ [0, s]

t

(12.289)

Chapter 12. Recovery of the Acoustical Assumptions (Ri1 )

We shall now estimate L / Ril . . . L / R i2 (Ri1 )

709

π /1 pointwise. Since

π /1 =

(Ri1 )

(π /1 )b · h/−1

we have, by Lemma 10.11 with l reduced to l − 1 and Corollary 10.1.d, L / Ril . . . L / R i2

/ R i2 To estimate L / Ril . . . L

lemma ϑ = (10.434)):

(Ri1 )

(Ri1 )

π /1 − (L / Ril . . . L / R i2

Cl δ02 (1 +

≤ (Ri1 )

(Ri1 )

t)

−3

(π /1 )b ) · h/−1  L ∞ ( ε0 ) t

[1 + log(1 + t)]

2

: for all t ∈ [0, s]

(12.290)

(π /1 )b pointwise we apply Lemma 10.10 taking in that

π / and replacing {1, . . . , k} by {2, . . . , l}. In view of the fact that (see (Ri1 )

ˇ/ (π /1 )b = D

(Ri1 )

π /

we then obtain: (Ri1 )

(π /1 )b

ˇ/L / R i2 =D / Ril . . . L

(Ri1 )

L / Ril . . . L / R i2

(12.291) π /−

l−2 

L / Ril . . . L / Ril−m+1 (

(Ril−m )

π /1 · L / Ril−m−1 . . . L / R i2

(Ri1 )

π /)

m=0

In the sum on the right in (12.291) we have angular derivatives of (Ril−m ) π /1 of order at most m ≤ l − 2, so Lemma 10.11 with l reduced to l − 1 applies, and angular derivatives of (Ri1 ) π / of order at most l − 2, so Corollary 10.1.d applies. We thus obtain that the sum ε on the right in (12.291) is bounded in L ∞ (t 0 ) by: Cl δ02 (1 + t)−3 [1 + log(1 + t)]2

: for all t ∈ [0, s]

(12.292)

On the other hand, by hypothesis H2 the first term on the right in (12.291) is bounded pointwise by: /R j L / Ril . . . L / Ri2 (Ri1 ) π /| (12.293) C(1 + t)−1 max |L j

Now, according to equation (10.206) of Chapter 10 we have: (Ri )

π / − 2λi α −1 χ  = 2λi (α −1 η0 (1 − u + η0 t)−1 h/ − k/ ) +H i j m ( ψ ⊗ d/x j + d/x j ⊗ ψ)ψm +H ( ψ ⊗ d/ψm + d/ψm ⊗ ψ)i j m x j dH ( ψ⊗ ψ)(Ri σ ) + dσ

(12.294)

By Corollary 10.1.a: max λi ∞,[l], ε0 ≤ Cl δ0 [1 + log(1 + t)] t

i

: for all t ∈ [0, s]

(12.295)

and: max d/x i ∞,[l], ε0 ≤ C i

t

: for all t ∈ [0, s]

(12.296)

710

Chapter 12. Recovery of the Acoustical Assumptions

These, together with the estimates (12.275) and the third of (12.278), imply: 

(Ri )

π /−2λi α −1 χ  ∞,[l], ε0 ≤ Cl δ0 (1+t)−1 [1+log(1+t)] : for all t ∈ [0, s] (12.297) t

Moreover, we have, pointwise on Wεs0 : |L /R j L / Ril . . . L / Ri2 (λi α −1 χ  )| ≤ |λi α −1 ||L /R j L / Ril . . . L / R i2 χ  | +Cl λi α −1 ∞,[l], ε0 χ  ∞,[l−1], ε0 t

t

≤ Cδ0 [1 + log(1 + t)]|L /R j L / Ril . . . L / R i2 χ  | +Cl δ02 (1 + t)−2 [1 + log(1 + t)]2

(12.298)

by (12.295) and X / [l−1] (the inductive hypothesis). It follows that: max |L /R j L / Ril . . . L / R i2 j

(Ri1 )

π /| ≤ Cδ0 [1 + log(1 + t)] max |L /R j L / Ril . . . L / R i2 χ  | j

−1

+Cl δ0 (1 + t) [1 + log(1 + t)] : pointwise on Wεs0

(12.299)

Going back to (12.291), the bounds (12.292) and (12.299) imply: |L / Ril . . . L / R i2

(Ri1 )

(π /1 )b | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] max |L /R j L / Ril . . . L / R i2 χ  | j

−2

+Cl δ0 (1 + t) [1 + log(1 + t)] : pointwise on Wεs0

(12.300)

hence, by (12.290), also: |L / Ril . . . L / R i2

(Ri1 )

π /1 | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] max |L /R j L / Ril . . . L / R i2 χ  | j

+Cl δ0 (1 + t)

−2

[1 + log(1 + t)]

: pointwise on Wεs0

(12.301)

In view of (12.289) we then conclude that: |

(i1 ...il )

cl | ≤ Cδ02 (1 + t)−3 [1 + log(1 + t)] max |L /R j L / Ril . . . L / R i2 χ  | j

+Cl δ02 (1 +

t)

: pointwise on

−4

[1 + log(1 + t)]

Wεs0

(12.302)

Using hypotheses H1 and H0 and assumption E{l+2} we then obtain through (12.283) for k = l: |L / Ril . . . L / R i1 D / 2 σ | ≤ Cδ02 (1 + t)−3 [1 + log(1 + t)] max |L /R j L / Ril . . . L / R i2 χ  | j

−3

+Cl δ0 (1 + t) : pointwise on Wεs0

(12.303)

Chapter 12. Recovery of the Acoustical Assumptions

711

We also obtain through (12.283) for k ≤ l − 1, using assumption E{l+1} and the bound (12.286): D / 2 σ ∞,[l−1], ε0 ≤ Cl δ0 (1 + t)−3 t

: for all t ∈ [0, s]

(12.304)

( P P)

Returning to the expression (12.71) for α , (12.275) and the estimates (12.281), (12.282), imply: 0 0 0( P P) 1 d H 2 2 0 0 α + ψ D / σ0 ≤ Cl δ02 (1 + t)−4 : for all t ∈ [0, s] (12.305) 0 0 ε 2 dσ L ∞,[l],t 0 On the other hand, by the estimates (12.303), (12.304) and (12.275):     1 d H 2 2  L ψ D / R i1 / σ   / Ril . . . L 2 dσ L ≤ Cδ02 (1 + t)−3 [1 + log(1 + t)] max |L /R j L / Ril . . . L / Ri2 χ  | + Cl δ0 (1 + t)−3 j

: pointwise on

Wεs0

(12.306)

Combining the results (12.279), (12.280), (12.305), (12.306), we conclude through (12.79) (using also (12.275), (12.277) and the first of (12.278)) that: / Ri1 b| ≤ Cδ02 (1 + t)−3 [1 + log(1 + t)] max |L /R j L / Ril . . . L / R i2 χ  | |L / Ril . . . L j

−3

+Cl δ0 (1 + t) : pointwise on Wεs0

(12.307)

The first of the estimates (12.278), together with (12.275), (12.277) and Corollary 10.1.d, implies: a∞,[l], ε0 ≤ Cl δ0 (1 + t)−2 t

: for all t ∈ [0, s]

(12.308)

This together with the inductive hypothesis X / [l−1] yields, through (12.267): 

(i1 ...il )

sl  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)] t

: for all t ∈ [0, s]

(12.309)

Also, X / [l−1] together with Corollary 10.1.d yields, through (12.265): 

(i1 ...il )

rl  L ∞ ( ε0 ) ≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)]2 t

: for all t ∈ [0, s]

(12.310)

Combining the bounds (12.307), (12.309) and (12.310), we obtain, through (12.270): |

(i1 ...il )

bl | ≤ Cδ02 (1 + t)−3 [1 + log(1 + t)] max |L /R j L / Ril . . . L / R i2 χ  | j

−3

+Cl δ0 (1 + t) : pointwise on Wεs0

(12.311)

712

Chapter 12. Recovery of the Acoustical Assumptions

The estimates (12.274) and (12.311), imply, through (12.268): / Ri1 (L / L χ  )| ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] max | |L / Ril . . . L

( j1 ... jl ) 

χ|

j1 ... jl

+Cl δ0 (1 + t)−3 : pointwise on Wεs0

(12.312)

What remains to be estimated is the sum on the right in (12.262). Consider a given term in this sum, corresponding to k ∈ {0, . . . , l − 1}. Let us recall at this point Lemma 8.5. Now this lemma holds with ξ being an St,u tensorfield of any type. We may thus / Ril−k−1 . . . L / Ri1 χ  in apply Lemma 8.5 with the symmetric 2-covariant St,u tensorfield L the role of ξ (and the index set {l − k + 1, . . . , l} in the role of the index set {1, . . . , l}) to express the term in question in the form: L / Ril . . . L / Ril−k+1 L / +

k 

(Ri ) l−k Z

L / Ril−k−1 . . . L / R i1 χ  = L / (Ril−k )

l 

L /

p=1 m 1 <···<m p =l−k+1

(im 1 ...im p ;il−k )

(i1

i

Z



(i1 . l−k . . il )  χl−1

im 1 ...im p il−k 

. . .

il )  χl−1− p

(12.313)

(Ril−k )

Z

(12.314)

Z

where (see (8.91)): (im 1 ...im p ;il−k )

Z =L / R im p . . . L / R im

1

To estimate the terms on the right in (12.313), we apply the second statement of Proposition 12.8. In the case of the first term on the right in (12.313), we then obtain:     i  . . il )   (i1 . l−k L χl−1   / (Ril−k ) Z    i  i  (Ril−k ) −1 (i1 . l−k . . il j )  (i1 . l−k . . il )  ≤ C(1 + t) | Z | max | χl | + | χl−1 | j il−k   | max | ( j ;il−k ) Z | (12.315) +| (i1 . . . il ) χl−1 j

Here we must distinguish the case l = 1 from the complementary case l ≥ 2. In the case l ≥ 2, Corollary 10.1.i together with the inductive hypothesis X / [l−1] allows us to conclude that the right-hand side of (12.315) is bounded pointwise in Wεs0 by: Cl δ0 (1 + t)−2 [1 + log(1 + t)] max | j1 ... jl

( j1 ... jl )  χl | + Cl δ02 (1 + t)−4 [1 + log(1 + t)]2

(12.316)

In the case l = 1, Corollary 10.1.i does not suffice to estimate ( j ;il−k ) Z . Recalling that (Ri ) Z = (Ri ) π /L · h /−1 we refer instead to equation (10.389) of Chapter 10 to express: (Ri )

π / L + χ  · Ri = i j m z j h mn d/x n + λi (κ −1 ζ  )  1 dH i j m x j ψm ψ(Lσ ) + ψ L ψ(Ri σ ) − i j m x j ψm ψ L (d/σ ) + 2 dσ +H i j m x j ψ(Lψm ) (12.317)

Chapter 12. Recovery of the Acoustical Assumptions

713

By Corollary 10.1.g with l = 1 together with the estimates (12.295), (12.296), (12.275), (12.277) and the second of (12.278), all with l = 1, imply: 

(Ri )

π / L +χ  · Ri ∞,[1], ε0 ≤ Cδ0 (1+t)−1 [1+log(1+t)] t

: for all t ∈ [0, s] (12.318)

Moreover, we have, pointwise on Wεs0 : |L / R j (χ  · Ri )| ≤ |L / R j χ  ||Ri | + |χ  ||L / R j Ri | ≤ C(1 + t)|L / R j χ  | + Cδ0 (1 + t)−1 [1 + log(1 + t)]

(12.319)

by Corollary 10.1.e with l = 1, and X / 0 . Combining (12.319) with (12.318) yields: (Ri )

|L /R j

π / L | ≤ C(1 + t)|L / R j χ  | + Cδ0 (1 + t)−1 [1 + log(1 + t)] : pointwise on Wεs0

(12.320)

hence by Corollary 10.1.d with l = 1 also: (Ri )

|L /R j

Z | ≤ C(1 + t)|L / R j χ  | + Cδ0 (1 + t)−1 [1 + log(1 + t)] : pointwise on Wεs0

(12.321)

Using this to estimate the last term on the right in (12.315), we conclude that also in the case l = 1 a pointwise estimate for the right-hand side of (12.315) of the form (12.316) holds. Consider next a term in the double sum in (12.313), corresponding to an ordered subset {m 1 , . . . , m p } ⊂ {l − k + 1, . . . , l}, p ∈ {1, . . . , k}. Note that the double sum is present only for k ≥ 1, hence l ≥ 2. Applying the second statement of Proposition 12.8 we obtain, in regard to the term in question,    im ...im i   (i1 1 . . p. l−k il )  −1  L χ · / (im 1 ...im p ;il−k ) l−1− p  ≤ C(1 + t)  Z    im 1 ...im p il−k  im 1 ...im p il−k  . . . il j )  . . . il )  χl− p | + | (i1 χl−1− p | | (im 1 ...im p ;il−k ) Z | max | (i1 j im 1 ...im p il−k  . . . il )  χl−1− p | max | (im 1 ...im p j ;il−k ) Z | (12.322) +| (i1 j

Since p ≥ 1, we have, by the inductive hypothesis X / [l−1] : 

(i1

max 

im 1 ...im p il−k 

. . .

(i1

il )  χl−1− p  L ∞ ( ε0 ) t

im 1 ...im p il−k 

. . .

j

il j )  χl− p  L ∞ ( ε0 ) t

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)], ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] : for all t ∈ [0, s]

(12.323)

Also, since p ≤ k ≤ l − 1, we have, by Corollary 10.1.i: 

(im 1 ...im p ;il−k )

Z  L ∞ ( ε0 ) ≤ Cl δ0 (1 +t)−1 [1 +log(1 +t)] t

: for all t ∈ [0, s] (12.324)

714

Chapter 12. Recovery of the Acoustical Assumptions

Moreover, Corollary 10.1.i suffices to estimate k = l − 1: max 

(im 1 ...im p j ;il−k )

(im 1 ...im p j ;il−k )

Z except in the case p =

Z  L ∞ ( ε0 ) ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

j

: for all t ∈ [0, s], unless p = k = l − 1

(12.325)

On the other hand, using Corollary 10.1.g together with the estimates (12.295), (12.296), (12.275), (12.277) and the second of (12.278) we obtain, through (12.317): 

(Ri )

π / L +χ  · Ri ∞,[l], ε0 ≤ Cl δ0 (1+t)−1 [1+log(1+t)] t

: for all t ∈ [0, s] (12.326)

Moreover, we have, pointwise on Wεs0 : 

/ R j1 (χ  · Ri )| ≤ |L / R jl . . . L / R j1 χ  ||Ri | + |L / R jl . . . L

|(L / R )s1 χ  ||(L / R )s 2 R i |

partitions

|s1 |+|s2 |=l,s2 =∅

≤ C(1 + t)|

( j1 ... jl )  χl |

+ Cl δ0 (1 + t)−1 [1 + log(1 + t)]

(12.327)

by Corollary 10.1.e and X / [l−1] . Combining (12.327) with (12.326) yields: / R j1 |L / R jl . . . L

(Ri )

π / L | ≤ C(1 + t)| ( j1... jl ) χl | + Cl δ0 (1 + t)−1 [1 + log(1 + t)] (12.328) : pointwise on Wεs0

hence by Corollary 10.1.d also: |

( j1 ... jl ;i)

Z | ≤ C(1 + t)| : pointwise

( j1 ... jl )  χl | + Cl δ0 (1 + s on Wε0

t)−1 [1 + log(1 + t)] (12.329)

Substituting the above estimates (12.323)–(12.325), (12.329), in (12.322), we conclude that: For p = k = l − 1:   i  i ...i   (i1 m 1 . m. p. l−k il )  L χl−1− p   / (im 1 ...im p ;il−k ) Z ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] max |

( j1 ... jl )  χl |

j1 ... jl

+ Cl δ02 (1 + t)−4 [1 + log(1 + t)]2

: pointwise on Wεs0 Otherwise:   L  / (im 1 ...im p ;il−k ) Z

(i1

(12.330)

im 1 ...im p il−k 

. . .

: pointwise on Wεs0

 

il )  χl−1− p 

≤ Cl δ02 (1 + t)−4 [1 + log(1 + t)]2 (12.331)

We thus obtain for the double sum in (12.313) a pointwise bound similar to (12.316). Combining, we conclude that the sum on the right in (12.262) is bounded pointwise in a

Chapter 12. Recovery of the Acoustical Assumptions

similar manner:  l−1   L / Ril . . . L / Ril−k+1 L /  

(Ri ) l−k Z

715

  ...L / R i1 χ   

L / Ril−k−1

k=0

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] max | j1 ... jl

( j1 ... jl )  χl |

+ Cl δ02 (1 + t)−4 [1 + log(1 + t)]2

: pointwise on Wεs0

(12.332)

The estimates (12.312) and (12.332) together imply, through (12.262): |L /L

(i1 ...il )  χl |

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] max | j1 ... jl

+Cl δ0 (1 + t)−3

( j1 ... jl )  χl |

: pointwise on Wεs0

(12.333)

Applying then (12.107), taking ϑ = (i1 ...il ) χl , and using X / 0 , we obtain the following inequality along the integral curves of L:   (12.334) L (1 − u + η0 t)2 | (i1 ...il ) χl |   ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] (1 − u + η0 t)2 max | ( j1 ... jl ) χl | + Cl δ0 (1 + t)−1 j1 ... jl

Consider a given integral curve of L. Setting, along the given integral curve: (i1 ...il )

xl (t) = (1 − u + η0 t)2 |

(i1 ...il )  χl |

(12.335)

xl + gl

(12.336)

the inequality (12.334) takes the form: d

(i1 ...il ) x

l

≤ fl max

( j1 ... jl )

j1 ... jl

dt where:

fl (t) = Cl δ0 (1 + t)−2 [1 + log(1 + t)], Integrating from t = 0 yields: (i1 ...il )

xl (t) ≤

(i1 ...il )

xl (0) +

t

fl (t  ) max

gl (t) = Cl δ0 (1 + t)−1

( j1 ... jl )

j1 ... jl

0

 xl (t  ) + gl (t  ) dt 

(12.337)

(12.338)

Taking the maximum over i 1 . . . i l , on both sides, and setting: x l (t) = max i1 ...il

(i1 ...il )

xl (t)

we deduce the linear integral inequality: t   x l (t) ≤ x l (0) + fl (t  )x l (t  ) + gl (t  ) dt  0

(12.339)

(12.340)

716

Chapter 12. Recovery of the Acoustical Assumptions

which implies: x l (t) ≤ e

!t 0

fl (t  )dt 





t

x l (0) +

e−

! t

fl (t  )dt 

0

g(t  )dt 

(12.341)

0

Since x l (0) ≤ max  i1 ...il

(i1 ...il )  χl  L ∞ ( ε0 ) 0

≤ C l δ0

by the assumption of the proposition on the initial conditions, while x l (t) ≥

 η 2 0

2

(1 + t)2 max | i1 ...il

(i1 ...il )  χl |

taking into account the facts that: t ∞ fl (t  )dt  ≤ fl (t  )dt  = 2Cl , 0

(as u ≤ ε0 ≤ 1/2 and η0 < 1)



0

t

gl (t  )dt  = Cl δ0 log(1 + t)

0 ε

and taking the supremum on t 0 , we conclude that: max  i1 ...il

(i1 ...il )  χl  L ∞ ( ε0 ) t

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

: for all t ∈ [0, s] (12.342)

provided that δ0 is suitably small (depending on l). This, together with the inductive hypothesis X / [l−1] , yields X / [l] , completing the inductive step and the proof of the proposition. From Lemma 12.4 with l +1 in the role of l we conclude that under the assumptions of Proposition 12.9, X / [l] holds on Wεs0 . Thus the assumptions of Proposition 10.1 all hold with l + 1 in the role of l. Therefore the conclusion of Proposition 10.1 and those of all its corollaries hold with l + 1 in the role of l. Moreover, under the same assumptions Lemma 10.11 holds. We now proceed to show that, under additional assumptions on the initial conditions, the M[m,l+1] , for m = 0, . . . , l + 1, also follow. Proposition 12.10 Let the assumptions of Proposition 12.9 hold for some non-negative integer l. Let, in addition the initial data satisfy: µ − η0 ∞,[m,l+1], ε0 ≤ Cl δ0 0

for some m ∈ {0, . . . , l + 1}. Then there is a constant Cl independent of s such that for all t ∈ [0, s] we have: µ − η0 ∞,[m,l+1], ε0 ≤ Cl δ0 [1 + log(1 + t)] t

that is, M[m,l+1] holds on

Wεs0 .

Proof. We consider first the case m = 0. Since M[0,0] follows directly from Proposition 12.1, we may apply induction on l. Assuming then M[0,l] we are to establish M[0,l+1] under the assumptions of the proposition for m = 0.

Chapter 12. Recovery of the Acoustical Assumptions

717

We consider the fundamental propagation equation (equation (9.1) of Chapter 9): Lµ = eµ + m

(12.343)

We apply Ril+1 . . . Ri1 to this equation. Taking X in Lemma 11.23 to be the space of differentiable functions on Wεs0 , A1 , . . . , An to be the operators Ri1 , . . . , Rin and B to be the operator L, acting on X, we obtain: [Ril+1 . . . Ri1 , L] = −

l 

Ril+1 . . . Ril+2−k

(Ril+1−k )

Z Ril−k . . . Ri1

(12.344)

k=0

Hence, defining:

(i1 ...il+1 )

µ0,l+1 = Ril+1 . . . Ri1 µ

(12.345)

we deduce the following propagation equation for the functions L

(i1 ...il+1 )

µ0,l+1 =

l 

Ril+1 . . . Ril+2−k (

(Ril+1−k )

Z · d/

(i1 ...il ) µ

(i1 ...il−k )

0,l+1 :

µ0,l−k )

k=0

+e where the remainder

(i1 ...il+1 )

(i1 ...il+1 ) r  0,l+1

(i1 ...il+1 )  r0,l+1

µ0,l+1 + Ril+1 . . . Ri1 m +

(i1 ...il+1 )  r0,l+1

(12.346)

is given by: 

=

((R)s1 e)((R)s2 µ)

(12.347)

partitions |s1 |+|s2 |=l+1,s1 =∅

Now, by Corollaries 10.1.b, 10.1.g with l + 1 in the role of l we have: ψTˆ ∞,[l+1], ε0 , ψ L − k∞,[l+1], ε0 ≤ Cl δ0 (1 + t)−1 t

t

: for all t ∈ [0, s]

(12.348)

Q

Also, by assumptions E{l+2} , E{l+1} : T σ ∞,[l+1], ε0 ≤ Cl δ0 (1 + t)−1 t

Lσ ∞,[l+1], ε0 ≤ Cl δ0 (1 + t)−2 t

: for all t ∈ [0, s]

(12.349)

Moreover, since ω L L is like ψ L − k but with Lψ0 , Lψi in the role of ψ0 − k, ψi , reQ spectively, Corollary 10.1.g with l + 1 in the role of l together with assumption E{l+1} imply: (12.350) ω L L ∞,[l+1], ε0 ≤ Cl δ0 (1 + t)−2 : for all t ∈ [0, s] t

The estimates (12.348), the second of (12.349), and (12.350), yield, through (12.80): e∞,[l+1], ε0 ≤ Cl δ0 (1 + t)−2 t

: for all t ∈ [0, s]

(12.351)

718

Chapter 12. Recovery of the Acoustical Assumptions

Also, recalling that the function m is given by (3.97): m=

1 dH (ψ L )2 (T σ ) 2 dσ

(12.352)

the second of the estimates (12.348) together with the first of (12.349) yield: m∞,[l+1], ε0 ≤ Cl δ0 (1 + t)−1 t

: for all t ∈ [0, s]

(12.353)

The estimate (12.351) together with the inductive hypothesis M[0,l] imply: 

(i1 ...il+1 )  r0,l+1  L ∞ ( ε0 ) t

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

: for all t ∈ [0, s] (12.354)

Consider finally a term in the sum on the right in (12.346), corresponding to k ∈ {0, . . . , l}. We express the term in question as: Ril+1 . . . Ril+2−k (

(Ril+1−k )



+

Z · d/

(i1 ...il−k )

((L /R )

µ0,l−k ) =

s1 (Ril+1−k )

(Ril+1−k )

Z ) · d/(R)

Z · d/

s2 (i1 ...il−k )

i



(i1 l+1−k . . . il+1 )

µ0,l−k

µ0,l

(12.355)

partitions |s1 |+|s2 |=k,|s2 | =k

Consider first the sum on the right in (12.355). By Corollary 10.1.i with l + 1 in the role of l: max  i

(Ri )

Z ∞,[l], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

: for all t ∈ [0, s] (12.356)

ε

This allows us to bound in L ∞ (t 0 ) the first factor in the summand since |s1 | ≤ k ≤ l. Moreover, by H0 and the inductive hypothesis M[0,l] , the second factor in the summand ε is bounded in L ∞ (t 0 ) by: Cl δ0 (1 + t)−1 [1 + log(1 + t)] since 1 + |s2 | + l − k ≤ l. It follows that the sum on the right in (12.355) is bounded in ε L ∞ (t 0 ) by: Cl δ02 (1 + t)−2 [1 + log(1 + t)]2 (12.357) On the other hand by (12.356) and H0 the first term on the right in (12.355) is bounded pointwise in Wεs0 by: Cδ0 (1 + t)−2 [1 + log(1 + t)] max |

i



(i1 l+1−k . . . il+1 j )

j

µ0,l+1 |

(12.358)

Combining the results (12.357) and (12.358) we conclude that the sum on the right in (12.346) is bounded pointwise in Wεs0 by: Cl δ0 (1 + t)−2 [1 + log(1 + t)] max | j1 ... jl+1

+Cl δ02 (1 + t)−2 [1 + log(1 + t)]2

( j1 ... jl+1 )

µ0,l+1 | (12.359)

Chapter 12. Recovery of the Acoustical Assumptions

719

The estimates (12.351), (12.353), (12.354), and (12.359), yield through (12.346) the following inequality along the integral curves of L: L(|

(i1 ...il+1 )

µ0,l+1 |) ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] max |

( j1 ... jl+1 )

j1 ... jl+1

+Cl δ0 (1 + t)−1

µ0,l+1 | (12.360)

Considering a given integral curve of L and setting, along this curve, (i1 ...il+1 )  x 0,l+1 (t)

=|

(i1 ...il+1 )

µ0,l+1 |

(12.361)

the inequality (12.360) takes the form: d

(i1 ...il+1 ) x  0,l+1

dt

≤ fl max

j1 ... jl+1

( j1 ... jl+1 )  x 0,l+1

+ gl

(12.362)

where fl (t) = Cl δ0 (1 + t)−2 [1 + log(1 + t)],

gl (t) = Cl δ0 (1 + t)−1

(12.363)

This is identical in form to (12.336)–(12.337). Setting then: x 0,l+1 (t) = max

i1 ...il+1

(i1 ...il+1 )  x 0,l+1 (t)

(12.364)

and noting that by the assumption of the proposition on the initial conditions for m = 0: x 0,l+1 (0) ≤ Cl δ0 an analysis analogous to that applied to (12.336) yields: max 

i1 ...il+1

(i1 ...il+1 )

µ0,l+1  L ∞ ( ε0 ) ≤ Cl δ0 [1 + log(1 + t)] t

: for all t ∈ [0, s]

(12.365)

provided that δ0 is suitably small (depending on l). This, together with the inductive hypothesis M[0,l] , yields M[0,l+1] , completing the inductive step. This proves the proposition in the case m = 0. To prove the proposition for m ∈ {1, . . . , l + 1} we apply induction on m. Replacing for convenience m by m + 1, we assume M[m,l+1] for some m ∈ {0, . . . , l}, and we are to establish M[m+1,l+1] under the assumption: µ − η0 ∞,[m+1,l+1], ε0 ≤ Cl δ0

(12.366)

0

on the initial conditions. Since we are assuming M[m,l+1] , the assumptions of Proposition 11.1 are all satisfied, therefore the conclusions of Proposition 11.1 and of all its corollaries hold.

720

Chapter 12. Recovery of the Acoustical Assumptions

We now first apply (T )m+1 to Lµ to obtain, using Lemma 11.23, L(T )m+1 µ = (T )m+1 Lµ +

m  (T )k (T )m−k µ

(12.367)

k=0

We then apply Rin . . . Ri1 to this equation, to obtain, using again Lemma 11.23, L Rin . . . Ri1 (T )m+1 µ = Rin . . . Ri1 (T )m+1 Lµ m  Rin . . . Ri1 (T )k (T )m−k µ +

(12.368)

k=0

+

n−1 

Rin . . . Rin−k+1

(Rin−k )

Z Rin−k−1 . . . Ri1 (T )m+1 µ

k=0

Let us define, as in Chapter 9, the functions: (i1 ...in )

µm+1,n = Rin . . . Ri1 (T )m+1 µ

(12.369)

Applying Rin . . . Ri1 (T )m+1 to the propagation equation (12.343) gives: Rin . . . Ri1 (T )m+1 Lµ = e

(i1 ...in )

µm+1,n + Rin . . . Ri1 (T )m+1 m +

(i1 ...in )  rm+1,n

(12.370) where the remainder (i1 ...in )  rm+1,n

(i1 ...in ) r  m+1,n

is given by:



=

((R)s1 e)((R)s2 (T )m+1 µ)

(12.371)

partitions |s1 |+|s2 |=n,s1 =∅

+Rin . . . Ri1

m+1  k=1

(m + 1)! ((T )k e)((T )m+1−k µ) k!(m + 1 − k)!



Substituting (12.370) in (12.368) we then obtain the following propagation equation for the functions (i1 ...in ) µm+1,n : L

(i1 ...in )

µm+1,n = e +

(i1 ...in ) m 

µm+1,n + Rin . . . Ri1 (T )m+1 m +

(i1 ...in )  rm+1,n

Rin . . . Ri1 (T )k (T )m−k µ

(12.372)

k=0

+

n−1 

Rin . . . Rin−k+1

(Rin−k )

Z Rin−k−1 . . . Ri1 (T )m+1 µ

k=0

Now, by Corollary 11.1.b we have: ψTˆ ∞,[m+1,l+1], ε0 , ψ L − k∞,[m+1,l+1], ε0 ≤ Cl δ0 (1 + t)−1 t

: for all t ∈ [0, s]

t

(12.373)

Chapter 12. Recovery of the Acoustical Assumptions

721

Q Also, by assumptions E{l+2} , E{l+1} :

T σ ∞,[m+1,l+1], ε0 ≤ Cl δ0 (1 + t)−1 t

Lσ ∞,[m+1,l+1], ε0 ≤ Cl δ0 (1 + t)−2 t

: for all t ∈ [0, s]

(12.374)

Moreover, since ω L L is like ψ L − k but with Lψ0 , Lψi in the role of ψ0 − k, ψi , respectively, Corollary 10.1.g with l + 1 in the role of l and Corollary 11.1.a together with Q assumption E{l+1} imply: ω L L ∞,[m+1,l+1], ε0 ≤ Cl δ0 (1 + t)−2

: for all t ∈ [0, s]

t

(12.375)

The estimates (12.373), the second of (12.374), and (12.375), yield, through (12.80): e∞,[m+1,l+1], ε0 ≤ Cl δ0 (1 + t)−2 t

: for all t ∈ [0, s]

(12.376)

and the second of the estimates (12.373) together with the first of (12.374) yield: m∞,[m+1,l+1], ε0 ≤ Cl δ0 (1 + t)−1 t

: for all t ∈ [0, s]

(12.377)

Also, the estimate (12.376) together with the inductive hypothesis M[m,l+1] imply: 

(i1 ...in )  rm+1,n  L ∞ ( ε0 ) t

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] : for all t ∈ [0, s] : for n ≤ l − m

(12.378)

provided that either n = 0, so that the first sum on the right in (12.371) vanishes, or Mm+1,m+n holds. Moreover, by Corollary 11.1.c: ∞,[m,l], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

: for all t ∈ [0, s]

(12.379)

To establish M[m+1,l+1] , we establish M[m+1,m+1+n] for n = 0, . . . , l − m by induction on n. We begin with the case n = 0, that is with M[m+1,m+1] . In the case n = 0 the second sum on the right in (12.372) is absent and the equation reduces to:  + Lµm+1,0 = eµm+1,0 + (T )m+1 m + rm+1,0

m  (T )k (T )m−k µ

(12.380)

k=0 ε

The sum on the right in (12.380) is bounded in L ∞ (t 0 ) by: m 

 · d/(T )m−k µ∞,[k,k], ε0 ≤ Cm (1 + t)−1 ∞,[m,m], ε0 µ − η0 ∞,[m,m+1], ε0 t

t

t

k=0

≤ Cm δ02 (1 + t)−2 [1 + log(1 + t)]2

(12.381)

722

Chapter 12. Recovery of the Acoustical Assumptions

by (12.379) with l = m, and M[m,m+1] . Appealing also to (12.82), (12.377) with l = m, and (12.378) with n = 0, we deduce from (12.380) the following inequality along the integral curves of L: L(|µm+1,0 |) ≤ Cδ0 (1 + t)−2 |µm+1,0 | + Cm δ0 (1 + t)−1

(12.382)

from which, together with the assumption µ − η0 ∞,[m+1,m+1], ε0 ≤ Cm δ0 0

on the initial conditions, the estimate: µm+1,0  L ∞ ( ε0 ) ≤ Cm δ0 [1 + log(1 + t)]

: for all t ∈ [0, s]

t

(12.383)

readily follows. This estimate, together with M[m,m+1] yields M[m+1,m+1] . Next, let M[m+1,m+n] hold for some n ∈ {1, . . . , l − m}. We shall show that then M[m+1,m+1+n] holds as well. Consider the first sum on the right in (12.372). Since n ≤ ε l − m this sum is bounded in L ∞ (t 0 ) by: m 

 · d/(T )m−k µ∞,[k,l+k−m], ε0 ≤ Cl (1 + t)−1 ∞,[m,l], ε0 µ − η0 ∞,[m,l+1], ε0 t

t

t

k=0

≤ Cl δ02 (1 + t)−2 [1 + log(1 + t)]2

(12.384)

by (12.379) and M[m,l+1] . Consider next the second sum on the right in (12.372). Consider a term in this sum corresponding to k ∈ {0, . . . , n − 1}. We express the term in question as: Rin . . . Rin−k+1 ( +

(Rin−k )



Z · d/ Rin−k−1 . . . Ri1 (T )m+1 µ) = ((L / R )s 1

(Rin−k )

(Rin−k )

Z · d/

i



(i1 .n−k . . in )

s2

Z ) · d/(R) (T )m+1 µ

µm+1,n−1 (12.385)

partitions |s1 |+|s2 |=k,s1 =∅

where the sum on the right is over all ordered partitions {s1 , s2 } of the set {n−k+1, . . . , n} into two ordered subsets s1 , s2 , with s1 non-empty. Also:  s2 = s2 {1, . . . , n − k − 1} Since |s1 | ≤ k ≤ n − 1 ≤ l − m − 1, the first factor in the summand is bounded in ε L ∞ (t 0 ) by (12.356). Also, since |s2 | ≤ k − 1, we have: |s2 | = |s2 | + n − k − 1 ≤ n − 2 and 1 + |s2 | + m + 1 ≤ n + m ε

therefore the second factor is bounded in L ∞ (t 0 ) by H0 and the inductive hypothesis M[m+1,m+n] . We thus obtain that the sum on the right in (12.385) is bounded in ε L ∞ (t 0 ) by: (12.386) Cl δ02 (1 + t)−2 [1 + log(1 + t)]2

Chapter 12. Recovery of the Acoustical Assumptions

723

On the other hand, by (12.356) and H0 the first term on the right in (12.385) is bounded pointwise in Wεs0 by: Cδ0 (1 + t)−2 [1 + log(1 + t)] max |

i



(i1 .n−k . . in j )

j

µm+1,n |

(12.387)

Combining the results (12.386) and (12.387) we conclude that the second sum on the right in (12.372) is bounded pointwise in Wεs0 by: Cl δ0 (1 + t)−2 [1 + log(1 + t)] max |

( j1 ... jn )

j1 ... jn

µm+1,n |

+Cl δ02 (1 + t)−2 [1 + log(1 + t)]2

(12.388)

The estimates (12.376)–(12.378), (12.384), (12.388), yield through (12.372) the following inequality along the integral curves of L: L(|

(i1 ...in )

µm+1,n |) ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)] max |

( j1 ... jn )

j1 ... jn

µm+1,n |

+Cl δ0 (1 + t)−1

(12.389)

Considering a given integral curve of L and setting, along this curve, (i1 ...in )  x m+1,n (t)

=|

(i1 ...in )

µm+1,n |

(12.390)

the inequality (12.389) takes the form: d

(i1 ...in ) x  m+1,n

≤ fl max

j1 ... jn

dt

( j1 ... jn )  x m+1,n

+ gl

(12.391)

where fl (t) = Cl δ0 (1 + t)−2 [1 + log(1 + t)],

gl (t) = Cl δ0 (1 + t)−1

(12.392)

This is identical in form to (12.336)–(12.337). Setting then: x m+1,n (t) = max i1 ...in

(i1 ...in )  x m+1,n (t)

(12.393)

and noting that by the assumptions of the proposition on the initial conditions: x m+1,n (0) ≤ Cl δ0 an analysis analogous to that applied to (12.336) yields: max 

i1 ...in

(i1 ...in )

µm+1,n  L ∞ ( ε0 ) ≤ Cl δ0 [1 + log(1 + t)] t

: for all t ∈ [0, s]

(12.394)

provided that δ0 is suitably small (depending on l). This, together with the inductive hypothesis M[m+1,m+n] , yields M[m+1,m+1+n] , completing the inductive step and the proof of the proposition.

724

Chapter 12. Recovery of the Acoustical Assumptions

Let us now define, for any non-negative integer k, the quantities: Ak = max L / R ik . . . L / Ri1 χ   L 2 ( ε0 ) i1 ...ik

(12.395)

t

and, for any non-negative integer l, the quantities: A[l] =

l 

Ak

(12.396)

k=0

The following lemma relates the quantities A[l] to the quantities A[l] which we have used so far, in the context of Proposition 10.2. It is the L 2 analogue of Lemma 12.4. Lemma 12.5 Let hypothesis H0 and the estimate (10.30) of Chapter 10 hold. Let l be a positive integer and let the assumptions E / [l∗ ] , E / [lQ∗ −1] as well as X / [l∗ −1] hold. Then we have:  Q A[l] ≤ A[l] + Cl A[l−1] + Cl (1 + t)−1 Y0 + W[l] + δ02 (1 + t)−2 W[l−1] Proof. From (12.272) we obtain, for k ≥ 1: Ak ≤ Ak +

η0 max max L / R jk−1 . . . L / R j1 1 − u + η0 t i j1 ... jk−1

(Ri )

π / L 2 ( ε0 )

(12.397)

t

Summing over k ∈ {1, . . . , l} and adding A0 = A0

(12.398)

we deduce, for l ≥ 1: A[l] ≤ A[l] +

3η0 max  1 − u + η0 t i

(Ri )

π /2,[l−1], ε0

(12.399)

t

/ [l∗ −1] , Now, by Lemma 12.4 with l∗ in the role of l, the assumptions of the lemma imply X therefore all the assumptions of Proposition 10.2 hold. Then by Corollary 10.2.d we have:  Q max  (Ri ) π /2,[l−1], ε0 ≤ Cl Y0 + (1 + t)A[l−1] + W[l] + δ02 (1 + t)−2 W[l−1] t

i

(12.400) Substituting this in (12.399) yields the recursive inequality: A[l] ≤ A[l] + Cl A[l−1] + Cl (1 + t)−1 Y0 + W[l] + δ02 (1 + t)−2 W[l−1] Q



: for l ≥ 1 A[0] = A[0] The lemma then readily follows.

(12.401)

Chapter 12. Recovery of the Acoustical Assumptions

725

Proposition 12.11 Let the assumptions of Proposition 12.6 hold. Let l be a non-negative Q QQ integer and let the assumptions E{l∗ +2} , E{l∗ +1} , E{l∗ } , hold on Wεs0 . Moreover let the initial data satisfy: χ  ∞,[l∗ ], ε0 ≤ Cl δ0 0

Then X / [l] holds on Wεs0 and there is a constant Cl independent of s such that for all t ∈ [0, s] we have:

A[l] (t) ≤ Cl (1 + t)−1 A[l] (0) + δ0 Y0 (0) t   Q QQ + (1 + t  )−1 W[l+2] (t  ) + W[l+1] (t  ) + δ02 (1 + t  )−2 W[l] (t  ) dt  0

Proof. The assumptions of the present proposition include those of Proposition 12.9 with l∗ in the role of l, therefore X / [l∗ ] holds on Wεs0 . Consequently, by Lemma 12.4 with / [l∗ ] holds on Wεs0 . Thus, under the assumptions of the present l∗ + 1 in the role of l, also X proposition the assumptions of Proposition 10.1 all hold with l∗ + 1 in the role of l, hence also the assumptions of Proposition 10.2 hold with l + 1 in the role of l. Therefore the conclusions of Proposition 10.1 and of all its corollaries hold with l∗ + 1 in the role of l, and the conclusions of Proposition 10.2 and of all its corollaries hold with l + 1 in the role of l. ε We shall presently estimate (i1 ...il ) bl (see (12.270)) in L 2 (t 0 ). First, by Corollaries 10.1.b, 10.1.g with l∗ in the role of l: ψTˆ ∞,[l∗ ], ε0 , ψ L − k∞,[l∗ ], ε0 ,  ψ∞,[l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 t

t

t

: for all t ∈ [0, s]

(12.402)

Q Moreover, by virtue of the assumptions E{l∗ } , E{l∗ +1} , the estimates (12.276) and (12.277) hold with l∗ in the role of l:

ω L L ∞,[l∗ ], ε0 , ω / L ∞,[l∗ ], ε0 , ω /∞,[l∗ ], ε0 ≤ Cl δ0 (1 + t)−2 t

t

t

: for all t ∈ [0, s]

(12.403)

Lσ ∞,[l∗ ], ε0 , d/σ ∞,[l∗ ], ε0 ≤ Cl δ0 (1 + t)−2 t

t

: for all t ∈ [0, s]

(12.404)

The estimates (12.402)–(12.404) imply the estimates (12.278) with l∗ in the role of l: e∞,[l∗ ], ε0 , κ −1 ζ  ∞,[l∗ ], ε0 , k/ ∞,[l∗ ], ε0 ≤ Cl δ0 (1 + t)−2 t

t

: for all t ∈ [0, s]

t

(12.405)

Next, by Corollaries 10.2.b, 10.2.g: ψTˆ 2,[l], ε0 , ψ L − k2,[l], ε0 ,  ψ2,[l], ε0 t t

t  Q −1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] ≤ C L W[l] + δ0 (1 + t) : for all t ∈ [0, s]

(12.406)

726

Chapter 12. Recovery of the Acoustical Assumptions

Also, by Corollary 10.2.b with l + 1 in the role of l: ω /2,[l], ε0

 Q ≤ Cl (1 + t)−1 W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] t

: for all t ∈ [0, s]

(12.407)

and by Corollary 10.2.g with l + 1 in the role of l: ω / L 2,[l], ε0

 Q ≤ Cl (1 + t)−1 W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] t

: for all t ∈ [0, s]

(12.408)

and:

1 2 Q ω L L 2,[l], ε0 ≤ Cl (1 + t)−1 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + W[l] t

: for all t ∈ [0, s]

(12.409)

Moreover, we have: Lσ 2,[l], ε0 ≤ Cl (1 + t)−1 W[l] + δ0 (1 + t)−1 W[l] Q



t

d/σ2,[l], ε0 ≤ Cl (1 + t)−1 W[l+1] t

: for all t ∈ [0, s]

(12.410)

Also, by Corollary 10.2.c with l + 1 in the role of l: k/ 2,[l], ε0

t

1 2 Q ≤ Cl (1 + t)−1 W[l+1] + δ02 (1 + t)−2 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1]

: for all t ∈ [0, s]

(12.411)

and by Corollary 10.2.h with l + 1 in the role of l: κ −1 ζ  2,[l], ε0 t

≤ Cl (1 + t)

−1

1 2 Q W[l+1] + δ02 (1 + t)−2 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1]

: for all t ∈ [0, s]

(12.412)

Moreover, using (12.402)–(12.404) together with (12.406), (12.409), (12.410), we deduce from (12.80):

1 2 Q e2,[l], ε0 ≤ Cl (1 + t)−1 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + W[l] t

: for all t ∈ [0, s]

(12.413)

Chapter 12. Recovery of the Acoustical Assumptions

727

Next, using (12.402)–(12.404) together with (12.406)–(12.410) we deduce from (12.74)– (12.76):

1 2 (N) Q  α 2,[l], ε0 ≤ Cl δ0 (1 + t)−3 W[l+1] + W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] t

: for all t ∈ [0, s]

(12.414)

Also, using (12.402)–(12.404) together with (12.406), (12.410)–(12.412), we deduce from (12.72): ( P N)

 α

1 2 Q 2,[l], ε0 ≤ Cl δ0 (1 + t)−3 W[l+1] + W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] t

: for all t ∈ [0, s]

(12.415)

( P P)

We turn to α (see (12.71)). By (12.281), (12.282), (12.303), (12.304), with l∗ in the role of l and X / [l∗ ] : (L)2 σ ∞,[l∗ ], ε0 ≤ Cl δ0 (1 + t)−3 t

d/ Lσ ∞,[l∗ ], ε0 ≤ Cl δ0 (1 + t)−3 t

D / 2 σ ∞,[l∗ ], ε0 ≤ Cl δ0 (1 + t)−3 t

: for all t ∈ [0, s] We also have the L 2 estimates: (L)2 σ 2,[l], ε0 ≤ Cl (1 + t)−2 W[l] + W[l] + δ0 (1 + t)−1 W[l] t  Q d/ Lσ 2,[l], ε0 ≤ Cl (1 + t)−2 W[l+1] + δ0 (1 + t)−1 W[l+1] QQ

Q

(12.416) 

t

: for all t ∈ [0, s]

(12.417)

Using the above (as well as (12.402), (12.406)) we deduce: 0 0

0( P P) 1 d H 2 2 0 QQ Q −3 0 α + 0 δ0 (1 + t)−1 W[l] ψ D / σ ≤ C δ (1 + t) + W[l+1] l 0 L 0 0 ε) 2 dσ 2,[l],t 1 2 +W[l+1] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] : for all t ∈ [0, s]

(12.418)

and: 0 0 01 dH 2 2 0 0 / σ0 ≤ CD / 2 σ 2,[l], ε0 + 0 0 2 dσ ψ L D ε t 2,[l],t 0

 Q Cl δ0 (1 + t)−3 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + δ02 (1 + t)−2 W[l−1] : for all t ∈ [0, s]

(12.419)

728

Chapter 12. Recovery of the Acoustical Assumptions

provided that δ0 is suitably small (depending on l). Now D / 2 σ 2,[l], ε0 has already been t estimated in Chapter 10, by (10.466):  D / 2 σ 2,[l], ε0 ≤ Cl (1 + t)−2 W[l+2] t  Q +δ0 (1 + t)−1 Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W[l] (12.420) This estimate relies on Lemma 10.12, and uses only the assumptions of that lemma, which / [l∗ +1] , E / [lQ∗ ] , and X / [l∗ ] , all of which have been are H0, H1, H2 , the estimate (10.30), E shown to follow under the assumptions of the present proposition. Substituting the estimate (12.420) we obtain: 0 0 01 dH 2 2 0 0 / σ0 0 2 dσ ψ L D 0 ε 2,[l],t 0

 Q ≤ Cl (1 + t)−2 W[l+2] + δ0 (1 + t)−1 Y0 + (1 + t)A[l] + δ02 (1 + t)−2 W[l] : for all t ∈ [0, s]

(12.421)

which, combined with (12.418), yields:

( P P) Q QQ + δ02 (1 + t)−2 W[l]  α 2,[l], ε0 ≤ Cl (1 + t)−2 W[l+2] + δ0 (1 + t)−1 W[l+1] t 1 2 +δ0 (1 + t)−1 Y0 + (1 + t)A[l] : for all t ∈ [0, s]

(12.422)

We proceed to estimate b∞,[l], ε0 . The contributions of the last two terms on the t right in the expression (12.79) for b are bounded by the estimates (12.414), (12.415) and (12.422), while the contributions of the remaining terms are readily bounded using (12.402), (12.404), (12.405) and Corollary 10.1.d with l∗ in the role of l, together with (12.406), (12.410), (12.413) and Corollary 10.2.d. Substituting also the bound for A[l] in terms of A[l] of Lemma 12.5, we obtain:

Q QQ b2,[l], ε0 ≤ Cl (1 + t)−2 W[l+2] + W[l+1] + δ02 (1 + t)−2 W[l] t 1 2 +δ0 (1 + t)−1 Y0 + (1 + t)A[l] : for all t ∈ [0, s]

(12.423)

We also estimate a∞,[l], ε0 . Using again (12.402), (12.404) and Corollary 10.1.d with t l∗ in the role of l, together with (12.406), (12.410), Corollary 10.2.d, and (12.413), we deduce through expression (12.78):

1 2 Q a2,[l], ε0 ≤ Cl (1 + t)−1 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + W[l+1] t

: for all t ∈ [0, s]

(12.424)

Chapter 12. Recovery of the Acoustical Assumptions

729

Moreover, we have the L ∞ estimate (12.308) with l∗ in the role of l: a∞,[l∗ ], ε0 ≤ Cl δ0 (1 + t)−2 t

(i1 ...il ) s , l

Consider now 

(i1 ...il )

sl  L 2 ( ε0 ) t

≤ Cl δ0 (1 + t)

−3

: for all t ∈ [0, s]

(12.425)

given by (12.267). Using (12.424), (12.425) and X / [l∗ ] we obtain:

   Q [1 + log(1 + t)] W[l] + δ0 (1 + t)−1 Y0 + W[l+1] + (1 + t)A[l]

: for all t ∈ [0, s]

(12.426)

Consider next (i1 ...il )rl , given by (12.265). Using Corollary 10.1.d with l∗ in the role of l, Corollary 10.2.d and X / [l∗ ] we obtain: 

(i1 ...il )

rl  L 2 ( ε0 ) t

≤ Cl δ0 (1 + t)

−2

[1 + log(1 + t)]A[l−1]

+ Cl δ02 (1 + t)−4 [1 + log(1 + t)]2 Y0 + W[l] + δ02 (1 + t)−2 W[l−1] Q

: for all t ∈ [0, s]

 (12.427)

Combining finally the bounds (12.423), (12.426), (12.427), we obtain, through expression (12.270): (i1 ...il )



bl  L 2 ( ε0 ) t

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]A[l]

 Q QQ + Cl (1 + t)−2 W[l+2] + W[l+1] + δ02 (1 + t)−2 W[l] + δ0 (1 + t)−1 Y0 : for all t ∈ [0, s]

(12.428) ε

Next we must estimate in L 2 (t 0 ) the sum on the right in (12.262). A term k ∈ {0, . . . , l − 1} in this sum is expressed in the form (12.313). For the first term on the right in (12.313) we have the pointwise estimate (12.315). This implies the L 2 estimate: 0 0 0 0 i  . . il )  0 (i1 . l−k 0L χl−1 0 ≤ C(1 + t)−1 · 0 / (Ril−k )  

ε

Z

L 2 ( 0 )

t  (Ril−k ) Z  L ∞ ( ε0 ) 3 max  t

+

i

j

i



(i1 . l−k . . il j )  χl  L 2 ( ε0 ) t



+

i



(i1 . l−k . . il )  χl−1  L 2 ( ε0 ) t

(i1 . l−k . . il )  χl−1  L 2 ( ε0 ) max  ( j ;il−k ) Z  L ∞ ( ε0 ) t t j





(12.429)

If l ≥ 2, so that l∗ ≥ 1, we apply the above estimate as it stands, appealing to Corollary 10.1.i with l∗ + 1 in the role of l to bound: 

(Ril−k )

Z  L ∞ ( ε0 ) , max  t

j

( j ;il−k )

Z  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] t

(12.430)

730

Chapter 12. Recovery of the Acoustical Assumptions

We then obtain: 0 0 0L 0 / (Ril−k )

i

0 0



(i1 . l−k . . il )  0 χl−1 0 Z

ε L 2 (t 0 )

≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]A[l]

: for l ≥ 2

(12.431)

On the other hand, if l = 1, so that l∗ = 0, we estimate the last term in parenthesis in (12.315) in L 2 by: 3χ   L ∞ ( ε0 ) max  t

( j ;i1 )

j

Z  L 2 ( ε0 )

(12.432)

t

Now, by Corollary 10.2.i with l + 1 in the role of l, we have, substituting for A[l] in terms of A[l] from Lemma 12.5, max  i

(Ri )

 Q Z 2,[l], ε0 ≤ Cl Y0 + (1 + t)A[l] + W[l+1] + δ0 (1 + t)−1 W[l] (12.433) t

By Proposition 12.6 and the estimate (12.433) with l = 1, (12.432) is bounded by:

 Q Cδ0 (1 + t)−2 [1 + log(1 + t)] Y0 + (1 + t)A[1] + W[2] + δ0 (1 + t)−1 W[1] (12.434) Thus, in the case l = 1, in place of the bound (12.431) we obtain: 0 0 0L / (Ri ) Z χ  0 L 2 ( ε0 ) ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]A[1] t

+ Cδ0 (1 + t)−3 [1 + log(1 + t)] Y0 + W[2] + δ0 (1 + t)−1 W[1] Q

: for l = 1



(12.435)

Consider next a term in the double sum in (12.313), corresponding to an ordered subset {m 1 , . . . , m p } ⊂ {l − k + 1, . . . , l}, p ∈ {1, . . . , k}. For the term in question, we have the pointwise estimate (12.322). We distinguish three cases: Case 1: p ≤ l∗ − 1, Case 2: p = l∗ ,

Case 3: p ≥ l∗ + 1

In Case 1, Corollary 10.1.i with l∗ + 1 in the role of l implies: 

(im 1 ...im p ;il−k )

≤ Cl δ0 (1 + t)

Z  L ∞ ( ε0 ) , max  t

−1

j

[1 + log(1 + t)]

(im 1 ...im p j ;il−k )

Z  L ∞ ( ε0 ) t

(12.436)

Chapter 12. Recovery of the Acoustical Assumptions

731

The pointwise estimate (12.322) then implies (noting that p ≥ 1): 0 0  im ...im i 0 0 (i1 1 . . p. l−k il )  0 0L / χ ≤ C(1 + t)−1 · (im 1 ...im p ;il−k ) l−1− p 0 0 Z  

 (im 1 ...im p ;il−k ) ε Z  L ∞ ( 0 ) 3 max  t

(im 1 ...im p j ;il−k )

j

(i1

im 1 ...im p il−k 

. . .

il j )  χl− p  L 2 ( ε0 ) t

j

+ + max 

ε

L 2 (t 0 )

Z

(i1

im 1 ...im p il−k 

. . .

ε L ∞ (t 0 )



(i1

il )  χl−1− p  L 2 ( ε0 ) t



im 1 ...im p il−k 

. . .

il )  χl−1− p  L 2 ( ε0 ) t

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]A[l−1]



(12.437)

In Case 2, Corollary 10.1.i with l∗ + 1 in the role of l implies the first of (12.436), but not the second. We appeal instead to Corollary 10.2.i with l + 1 in the role of l, that is to (12.433), to estimate, noting that p ≤ k ≤ l − 1,

 Q max  (im 1 ...im p j ;il−k ) Z  L 2 ( ε0 ) ≤ Cl Y0 + (1 + t)A[l] + W[l+1] + δ0 (1 + t)−1 W[l] t

j

(12.438) / [l∗ ] to estimate: Since in Case 2 we have l − 1 − p = l − 1 − l∗ ≤ l∗ , we may appeal to X 

(i1

im 1 ...im p il−k 

. . .

il )  χl−1− p  L ∞ ( ε0 ) t

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

(12.439)

In view of (12.438), (12.439), the pointwise estimate (12.322) implies, in Case 2: 0 0 i ...i i  0 0 (i1 m 1 . m. p. l−k il )  0 0L / χ (12.440) l−1− p 0 0 (im 1 ...im p ;il−k ) Z  −1 ≤ C(1 + t) 

ε

L 2 (t 0 )

 (im 1 ...im p ;il−k ) Z  L ∞ ( ε0 ) 3 max  t

+ +3

(i1

(i1

im 1 ...im p il−k 

im 1 ...im p il−k 

. . .

j

. . .

(i1

im 1 ...im p il−k 

. . .

il j )  χl− p  L 2 ( ε0 ) t

il )  χl−1− p  L 2 ( ε0 ) t



il )  χl−1− p  L ∞ ( ε0 ) max  (im 1 ...im p j ;il−k ) Z  L 2 ( ε0 ) t t j



≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] Y0 + (1 + t)A[l] + W[l+1] + δ0 (1 + t)−1 W[l] Q



In Case 3 apply the estimate (12.438) and we also appeal to (12.433) with l reduced to l − 1 to estimate:

 Q  (im 1 ...im p ;il−k ) Z  L 2 ( ε0 ) ≤ Cl Y0 + (1 + t)A[l−1] + W[l] + δ0 (1 + t)−1 W[l−1] t (12.441) Since in Case 3 we have l − p ≤ l − l∗ − 1 ≤ l∗ , we may appeal to X / [l∗ ] to estimate: max  j

(i1

im 1 ...im p il−k 

. . .

il j )  χl− p  L ∞ ( ε0 ) t

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

(12.442)

732

Chapter 12. Recovery of the Acoustical Assumptions

as well as (12.439). In view of (12.441), (12.442), the pointwise estimate (12.322) implies, in Case 3: 0 0 im ...im i  0 0 (i1 1 . . p. l−k il )  0L χl−1− p 0 (12.443) 0 0 / (im 1 ...im p ;il−k ) Z  ≤ C(1 + t)−1 

ε

L 2 (t 0 )

 (im 1 ...im p ;il−k ) Z  L 2 ( ε0 ) max  t

+3

(i1

im 1 ...im p il−k 

. . .

im 1 ...im p il−k 

j

(i1

+

(i1

im 1 ...im p il−k 

. . .

. . .

il j )  χl− p  L ∞ ( ε0 ) t

il )  χl−1− p  L ∞ ( ε0 ) t



il )  χl−1− p  L ∞ ( ε0 ) max  (im 1 ...im p j ;il−k ) Z  L 2 ( ε0 ) t t j



≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] Y0 + (1 + t)A[l] + W[l+1] + δ0 (1 + t)−1 W[l] Q



Combining the results (12.437), (12.440), (12.443), for the three cases, we conclude that ε the double sum in (12.313) is bounded in L 2 (t 0 ) by:  Q Cl δ0 (1 +t)−3 [1 +log(1 +t)] Y0 + (1 + t)A[l] + W[l+1] + δ0 (1 + t)−1 W[l] (12.444) Combining finally this result with the earlier results (12.431), (12.435), for the first term on the right in (12.313), we conclude that the sum on the right in (12.262) is bounded in ε L 2 (t 0 ), in general, by: 0 0 l−1 0 0 0 0 L / Ril . . . L / Ril−k+1 L / (Ril−k ) L / Ril−k−1 . . . L / R i1 χ 0 0 Z 0 0 ε k=0 L 2 (t 0 )  Q ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] Y0 + (1 + t)A[l] + W[l+1] + δ0 (1 + t)−1 W[l] : for all t ∈ [0, s]

(12.445)

In view of the estimates (12.428) and (12.445), we obtain through (12.262), (12.268), recalling also the estimate (12.274): 2 1 L / L (i1 ...il ) χl  L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−3 [1 + log(1 + t)] Y0 + (1 + t)A[l] t  Q QQ +Cl (1 + t)−2 W[l+2] + W[l+1] + δ02 (1 + t)−2 W[l] : for all t ∈ [0, s]

(12.446)

Let us define the non-negative functions: (i1 ...il ) (i1 ...il )

(i1 ...il )  χl | 2  (i1 ...il )  + η0 t) (2|χ || χl |

φl = (1 − u + η0 t)2 | ρl = (1 − u

(i1 ...il ) χ  l

Then by (12.107) applied to ϑ = L

(i1 ...il )

(12.447) + |L / L (i1 ...il ) χl |)

(12.448)

we have:

φl ≤

(i1 ...il )

ρl

(12.449)

Chapter 12. Recovery of the Acoustical Assumptions

733

To (i1 ...il ) φl , (i1 ...il ) ρl , we associate the functions depending on the parameters t, u, by: (i1 ...il )

(i1 ...il )

φl (t, u) =

(i1 ...il )

φl ◦ "t,u ,

(i1 ...il ) φ (t, u), (i1 ...il ) ρ (t, u), l l

ρl (t, u) = s (i1 ...il ) ρl ◦ "t,u

on S 2 ,

(12.450)

where "t,u is the diffeomorphism of S 2 onto St,u defined in Chapter 8. In terms of (i1 ...il ) φ (t, u), (i1 ...il ) ρ (t, u), (12.449) reads: l l ∂ ∂t

(i1 ...il )

(i1 ...il )

φl (t, u) ≤

ρl (t, u)

(12.451)

Integrating this with respect to t from t = 0, we obtain: (i1 ...il )

φl (t, u) ≤

(i1 ...il )



t

φl (0, u) +

(i1 ...il )

ρl (t  , u)dt 

(12.452)

0

Taking the L 2 norm of both sides on [0, ε0 ] × S 2 yields: 

(i1 ...il )

φl (t) L 2 ([0,ε0]×S 2 ) ≤ 

(i1 ...il )

φl (0) L 2 ([0,ε0 ]×S 2 ) +

t



(i1 ...il )

0

ρl (t  ) L 2 ([0,ε0 ]×S 2 ) (12.453)

By the comparison inequalities (8.356) of Chapter 8 we then obtain: (1 + t)−1  (i1 ...il ) φl  L 2 ( ε0 ) t  t ≤ C  (i1 ...il ) φl  L 2 ( ε0 ) + (1 + t  )−1  0

(i1 ...il )

ρl  L 2 ( ε0 ) dt 



t

0

(12.454)

Replacing l by k ∈ {0, . . . , l}, taking on both sides the maximum over i 1 , . . . , i k and then summing over k ∈ {0, . . . , l} yields: (1 + t)−1 ≤C

l 

 l  k=0

k=0

max 

(i1 ...ik )

i1 ...il

max 

(12.455)

t

(i1 ...ik )

i1 ...ik

φk  L 2 ( ε0 )

φk  L 2 ( ε0 ) + 0

t

 −1

(1 + t )

0

l  k=0

 max 

(i1 ...ik )

i1 ...ik

ρk  L 2 ( ε0 ) dt



t

Now, from (12.447) we have: l  k=0 l  k=0

max 

(i1 ...ik )

max 

(i1 ...ik )

i1 ...il

i1 ...il

φk  L 2 ( ε0 ) ≥ C −1 (1 + t)2 A[l] (t) t

φk  L 2 ( ε0 ) ≤ A[0] (0) 0

(12.456)

734

Chapter 12. Recovery of the Acoustical Assumptions

and from (12.448), X / [0] , and the estimate (12.446): l 

max 

(i1 ...ik )

i1 ...ik

k=0

ρk  L 2 ( ε0 ) ≤ t

/ L (i1 ...il ) χl  L 2 ( ε0 ) Cδ0 [1 + log(1 + t)]A[l] (t) + C(1 + t)2 L t 1 2 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] Y0 (t) + (1 + t)A[l] (t)  Q QQ (t) + δ02 (1 + t)−2 W[l] (t) (12.457) + Cl W[l+2] (t) + W[l+1] Substituting (12.456) and (12.457) in (12.455), we obtain the following linear integral inequality for the quantity (1 + t)A[l] (t): (1 + t)A[l] (t)



(12.458) t

≤ CA[l] (0) + Cl δ0 (1 + t  )−2 [1 + log(1 + t  )] · (1 + t  )A[l] (t  )dt  0 t + Cl δ0 (1 + t  )−2 [1 + log(1 + t  )]Y0 (t  )dt  0 t  Q QQ + Cl (1 + t  )−1 W[l+2] (t  ) + W[l+1] (t  ) + δ02 (1 + t)−2 W[l] (t  ) dt  0

This implies:

t (1 + t)A[l] (t) ≤ CA[l] (0) + Cl δ0 (1 + t  )−2 [1 + log(1 + t  )]Y0 (t  )dt  (12.459) 0 t  Q QQ Cl (1 + t  )−1 W[l+2] (t  ) + W[l+1] (t  ) + δ02 (1 + t)−2 W[l] (t  ) + 0

provided that δ0 is suitably small (depending on l). To prove the proposition, what remains to be done is to obtain a suitable estimate for Y0 (t) in terms of Y0 (0). To obtain this estimate we derive a propagation equation for the functions y i . According to the definition (10.24) of the functions y i we have: Tˆ i = −

xi + yi 1 − u + η0 t

(12.460)

hence:

η0 x i Li + + Ly i 1 − u + η0 t (1 − u + η0 t)2 Also from (10.381), (10.382) we have: L Tˆ i = −

(12.461)

Li =

η0 x i + wi − αy i 1 − u + η0 t

(12.462)

wi =

H ψ0ψi (α − η0 )x i + 1 − u + η0 t 1 + ρH

(12.463)

where:

Chapter 12. Recovery of the Acoustical Assumptions

735

Substituting (12.462) in (12.461) we obtain: L Tˆ i =

(αy i − wi ) + Ly i (1 − u + η0 t)

(12.464)

On the other hand, according to (11.667) we have: L(Tˆ i ) = p L Tˆ i + q L · d/x i

(12.465)

Comparing (12.464) and (12.465) we conclude that the y i satisfy the propagation equation: η0 y i = w˜ i (12.466) Ly i + 1 − u + η0 t where:

wi − (α − η0 )y i + p L Tˆ i + q L · d/x i 1 − u + η0 t

w˜ i =

(12.467)

Now, using the estimate (12.47) we obtain that the first term on the right in (12.467) is ε bounded in L 2 (t 0 ) by: (12.468) C(1 + t)−1 W{0} (t) while from expressions (11.668), (11.669), we deduce that:   Q  p L  L 2 ( ε0 ) ≤ Cδ0 (1 + t)−2 W{1} (t) + W{0} (t) t   Q q L  L 2 ( ε0 ) ≤ C(1 + t)−1 W{1} (t) + W{0} (t) t

(12.469) (12.470)

hence combining we obtain:   Q max w˜ i  L 2 ( ε0 ) ≤ C(1 + t)−1 W{1} (t) + W{0} (t) t

i

(12.471)

We now integrate the propagation equation (12.466) along the integral curves of L. We then obtain, in acoustical coordinates: t (1−u +η0t)y i (t, u, ϑ) = (1−u)y i (0, u, ϑ)+ (1−u +η0 t  )w˜ i (t  , u, ϑ)dt  (12.472) 0

Taking the L 2 norm on [0, ε0 ] × S 2 we then obtain: (1 + t)y i (t) L 2 ([0,ε0 ]×S 2 )  t ≤ C y i (0) L 2 ([0,ε0]×S 2 ) + (1 + t  )w˜ i  L 2 ([0,ε0]×S 2 ) dt 

(12.473)

0

which is equivalent to: y  L 2 ( ε0 ) i

t

 t i i  ≤ C y  L 2 ( ε0 ) + w˜  L 2 ( ε0 ) dt 0

0

t

(12.474)

736

Chapter 12. Recovery of the Acoustical Assumptions

Taking the maximum over i = 1, 2, 3 and substituting the bound (12.471) then yields:  t   Q Y0 (t) ≤ C Y0 (0) + (1 + t  )−1 W{1} (t  ) + W{0} (t  ) dt  (12.475) 0

which implies: t (1 + t  )−2 [1 + log(1 + t  )]Y0 (t  )dt  (12.476) 0  t   Q  ≤ C Y0 (0) + (1 + t  )−2 [1 + log(1 + t)]2 W{1} (t  ) + W{0} (t ) dt  0

Substituting this on the right in the estimate (12.459), the integral on the right in (12.476) is absorbed into the last integral on the right in (12.459) and the statement of the proposition results. Proposition 12.12 Let the assumptions of Proposition 12.11 hold for some non-negative integer l. Let m ∈ {0, . . . , l + 1}, and let the initial data satisfy, in addition: µ − η0 ∞,[m,l∗ +1], ε0 ≤ Cl δ0 0

Then M[m,l∗ +1] holds on Wεs0 and there is a constant Cl independent of s such that for all t ∈ [0, s] we have: (1 + t)−1 B[m,l+1] (t)  t    (1 + t  )−1 W{l+2} (t  )+ ≤ Cl B[m,l+1] (0) + δ0 Y0 (0) + A[l] (0) +  −1

(1 + t )



[1 + log(1 + t )]

0   Q QQ W{l+1} (t  ) + δ03 (1 + t  )−2 W{l} (t  ) dt 

Proof. The assumptions of the present proposition include those of Proposition 12.10 with l∗ in the role of l, therefore M[m,l∗ +1] holds on Wεs0 . Thus, under the assumptions of the present proposition the assumptions of Proposition 11.1 all hold with l∗ in the role of l, hence also the assumptions of Proposition 11.2 hold. Therefore the conclusions of Proposition 11.1 and of all its corollaries hold with l∗ in the role of l, and the conclusions of Proposition 11.2 and of all its corollaries hold as well. We consider the propagation equation (12.372), replacing again m + 1 by m, so that ε m ∈ {0, . . . , l + 1} as in the statement of the proposition. We must estimate in L 2 (t 0 ) the terms on the right-hand side. First, according to the estimates (11.686), (11.690), of Chapter 11, with m + 1 reduced to m:  (12.477) m2,[m,l+1], ε0 ≤ Cl W{l+2} t  Q +δ02 (1 + t)−2 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] + δ0 (1 + t)−2 W{l}

Q e2,[m,l+1], ε0 ≤ Cl (1 + t)−1 W{l+1} t

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l] + W{l+1}



(12.478)

Chapter 12. Recovery of the Acoustical Assumptions

737

Also, from Corollary 11.1.b with l∗ in the role of l we obtain: e∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−2

(12.479)

t

 , given by (12.371) with m+1 replaced by m. Here n ≤ l+1−m. Consider next (i1 ...in )rm,n Consider the summand:

((R)s1 e)((R)s2 (T )m µ)

: |s1 | + |s2 | = n, s1 = ∅

of the first sum. If |s2 | ≤ l∗ + 1 − m we appeal to M[m,l∗ +1] to place the second factor in ε L ∞ (t 0 ). Otherwise, we have |s1 | ≤ n − (l∗ + 2 − m) ≤ l − l∗ − 1 ≤ l∗ and we appeal to ε (12.479) to place the first factor in L ∞ (t 0 ). Noting that |s2 | ≤ n − 1 ≤ l − m, we then ε0 2 obtain an L (t ) bound for the first sum by: Cl δ0 (1 + t)−2 B[m,l] + Cl [1 + log(1 + t)]e2,[l+1], ε0

(12.480)

t

Consider next the summand:   Rin . . . Ri1 ((T )k e)((T )m−k µ)

: k ∈ {1, . . . , m}

of the second sum, to which Rin . . . Ri1 is applied. If at most l∗ + 1 + k − m of the n angular derivatives fall on (T )m−k µ we appeal to M[m,l∗ +1] to place the corresponding ε factor in L ∞ (t 0 ). Otherwise, at most n − (l∗ + 2 + k − m) ≤ l − l∗ − 1 − k ≤ l∗ − k angular derivatives fall on (T )k e and we appeal to (12.479) to place the corresponding ε ε factor in L ∞ (t 0 ). Noting that k ≥ 1, we then obtain an L 2 (t 0 ) bound for the second term on the right in (12.371) (with m + 1 replaced by m) by: Cl δ0 (1 + t)−2 B[m−1,l] + Cl [1 + log(1 + t)]e2,[m,l+1], ε0 t

(12.481)

Combining (12.480), (12.481), and substituting the bound (12.478), we obtain:

 max max  (i1 ...in )rm,n  L 2 ( ε0 ) ≤ Cl (1 + t)−1 δ0 (1 + t)−1 B[m,l] (12.482) t n≤l+1−m i1 ...in   Q +[1 + log(1 + t)] W{l+1} + δ0 (1 + t)−1 Y0 + (1 + t)A[l] + W{l+1} ε

What remains to be estimated, in L 2 (t 0 ), are the two sums on the right in (12.372), ε with m + 1 replaced by m. The first of these sums is bounded in L 2 (t 0 ) by: m−1 

 · d/(T )m−1−k µ2,[k,l+k+1−m], ε0 t

(12.483)

k=0

Here in the summand each of the factors , d/(T )m−1−k µ, receives at most k T -derivatives, thus µ receives at most m − 1 T -derivatives, and there are l + k + 1 − m spatial derivatives. Now by Corollary 11.1.c with l∗ in the role of l: ∞,[m,l∗ ], ε0 ≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] t

(12.484)

738

Chapter 12. Recovery of the Acoustical Assumptions

If at most l∗ spatial derivatives fall on  we appeal to this estimate to place the corε responding factor in L ∞ (t 0 ). Otherwise, l∗ + 1 or more spatial derivatives fall on , hence at most (l + k + 1 − m) − (l∗ + 1) ≤ l∗ + k + 1 − m spatial derivatives fall on d/(T )m−1−k µ, for a total of at most l∗ + 1 spatial derivatives ε falling on µ, and we appeal to M[m−1,l∗ +1] to place the corresponding factor in L ∞ (t 0 ). In this way we deduce that (12.483) is bounded by: Cl δ0 (1 + t)−2 [1 + log(1 + t)]B[m−1,l+1] + Cl δ0 (1 + t)

−1

(12.485)

[1 + log(1 + t)]2,[m−1,l], ε0 t

−2

≤ Cl δ0 (1 + t) [1 + log(1 + t)] ·    B[m−1,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] +[1 + log(1 + t)] W{l+1} + δ0 (1 + t)−2 [1 + log(1 + t)]2 W{l} Q



by Corollary 11.2.c with m reduced to m − 1. Finally, we have the second sum on the right in (12.372) (with m + 1 replaced by m). Consider a term in this sum, corresponding to k ∈ {0, . . . , n − 1}: Rin . . . Rin−k+1 ( =

(Rin−k )



Z · d/ Rin−k−1 . . . Ri1 (T )m µ)

((L / R )s 1

(Rin−k )



Z ) · d/(R)s2 (T )m µ

(12.486)

partitions |s1 |+|s2 |=k

s2 = s2



{1, . . . , n − k − 1}

If |s1 | ≤ l∗ we appeal to Corollary 10.1.i with l∗ + 1 in the role of l to place the first factor ε in the summand in L ∞ (t 0 ). Otherwise, |s2 | ≤ k − l∗ − 1 and |s2 | = |s2 | + n − k − 1 ≤ n − l∗ − 2 ≤ l − l∗ − 1 − m ≤ l∗ − m hence 1 + |s2 | + m ≤ l∗ + 1 and we appeal to M[m,l∗ +1] to place the second factor in the ε summand in L ∞ (t 0 ). Since also |s1 | ≤ k ≤ n − 1 ≤ l − m and 1 + |s  | + m = |s2 | + n − k + m ≤ n + m ≤ l + 1 we obtain in this way that the second sum in (12.372) (with m + 1 replaced by m) is bounded in L 2 by: Cl δ0 (1 + t)−2 [1 + log(1 + t)]B[m,l+1] + Cl δ0 (1 + t)

−1

[1 + log(1 + t)] max  i

−1

(12.487) (Ri )

Z 2,[l−m], ε0 t

≤ Cl δ0 (1 + t) [1 + log(1 + t)] · 

Q (1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l] + W{l+1} + δ0 (1 + t)−1 W{l} by Corollary 10.2.i with l + 1 in the role of l.

Chapter 12. Recovery of the Acoustical Assumptions

739

In view of the estimates (12.477), (12.482), (12.485), (12.487), we obtain from (12.372) (with m + 1 replaced by m), recalling also the estimate (12.82), and substituting for A[l] in terms of A[l] from Lemma 12.5: L

(i1 ...in )

µm,n  L 2 ( ε0 )

(12.488)

t

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]B[m,l+1]

  Q + Cl W{l+2} + (1 + t)−1 [1 + log(1 + t)] W{l+1} + δ0 Y0 + (1 + t)A[l] : for n ≤ l + 1 − m Defining the non-negative functions: (i1 ...in )  φm,n (i1 ...in )  ρm,n

=|

(i1 ...in )

= |L

µm,n |

(i1 ...in )

(12.489)

µm,n |

(12.490)

we have, as in (12.449), (i1 ...in )  φm,n

L

≤

(i1 ...in )  ρm,n

(12.491)

Following the argument leading from (12.449) to (12.454) we then obtain:  (1 + t)−1  (i1 ...in ) φm,n  L 2 ( ε0 ) t  t (i1 ...in )  ε ≤C  φm,n  L 2 ( 0 ) + (1 + t  )−1  0

0

(i1 ...in )  ρm,n  L 2 ( ε0 ) dt  



(12.492)

t

Taking on both sides the maximum over i 1 . . . i n , replacing m by k ∈ {0, . . . , m} and then summing over n ∈ {0, . . . , l + 1 − k} and over k ∈ {0, . . . , m} yields: (1 + t)−1

m l+1−k   k=0 n=0

≤C

 m l+1−k  

max 

i1 ...in

k=0 n=0



+

max 

i1 ...in

t

(i1 ...in )  φk,n  L 2 ( ε0 ) t

(12.493)

(i1 ...in )  φk,n  L 2 ( ε0 ) 0

 −1

(1 + t )

0

m l+1−k   k=0 n=0

 max 

i1 ...in

(i1 ...in )  ρk,n  L 2 ( ε0 ) dt   t

Now, from (12.489) we have, for all t ∈ [0, s]: m l+1−k   k=0 n=0

max 

i1 ...in

(i1 ...in )  φk,n  L 2 ( ε0 ) t

= B[m,l+1] (t)

(12.494)

740

Chapter 12. Recovery of the Acoustical Assumptions

and from (12.490) and the estimate (12.488): m l+1−k  

max 

i1 ...ik

k=0 n=0

(i1 ...ik )

ρk  L 2 ( ε0 ) t

(12.495)

≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]B[m,l+1]

  Q + Cl W{l+2} + (1 + t)−1 [1 + log(1 + t)] W{l+1} + δ0 Y0 + (1 + t)A[l] Substituting (12.494) and (12.495) in (12.493), we obtain the following linear integral inequality for the quantity (1 + t)−1 B[m,l+1] (t): (1 + t)−1 B[m,l+1] (t) ≤ CB[m,l+1] (0) t + Cl δ0 (1 + t  )−2 [1 + log(1 + t  )] · (1 + t  )−1 B[m,l+1] (t  )dt  0 t + Cl δ0 (1 + t  )−2 [1 + log(1 + t  )]Y0 (t  )dt  0 t  Q + Cl (1 + t  )−1 W{l+2} (t  ) + (1 + t  )−1 [1 + log(1 + t  )]W{l+1} (t  ) dt  0 t + Cl δ0 (1 + t  )−1 [1 + log(1 + t  )]A[l] (t  )dt  (12.496) 0

This implies: t (1 + t)−1 B[m,l+1] (t) ≤ CB[m,l+1] (0) + Cl δ0 (1 + t  )−2 [1 + log(1 + t  )]Y0 (t  )dt  0 t  Q + Cl (1 + t  )−1 W{l+2} (t  ) + (1 + t  )−1 [1 + log(1 + t  )]W{l+1} (t  ) dt  0 t + Cl δ0 (1 + t  )−1 [1 + log(1 + t  )]A[l] (t  )dt  (12.497) 0

provided that δ0 is suitably small (depending on l). Substituting the estimate for A[l] (t) of Proposition 12.11 we find, in regard to the last integral on the right in (12.497): t (1 + t  )−1 [1 + log(1 + t  )]A[l] (t  )dt  (12.498) 0  t 1 ≤ Cl A[l] (0) + δ0 Y0 (0) + (1 + t  )−2 [1 + log(1 + t  )] W[l+2] (t  ) 0   Q QQ +W[l+1] (t  ) + δ02 (1 + t  )−2 W[l] (t  ) dt  substituting also the estimate (12.476) for the first integral on the right in (12.497), the proposition follows.

Chapter 13

The Error Estimates Involving the Top Order Spatial Derivatives of the Acoustical Entities. The Energy Estimates. Recovery of the Bootstrap Assumptions. Statement and Proof of the Main Theorem: Existence up to Shock Formation Part 1: Derivation of the properties C1, C2, C3 In the first part of the present chapter we shall establish the properties C1, C2, C3, of the function µ, introduced in Chapter 5, on which the proof of Theorem 5.1 relies. To establish these properties we shall make use of Proposition 8.6. The assumptions of this proposition are those of Proposition 8.5 together with the assumption I0 of Chapter 6, which coincides with the assumption of Proposition 12.1 on the initial data. The assumptions of Proposition 8.5, in turn, are the assumptions of Lemma 8.10 together with the bootstrap assumptions ELT 3, ELL 3. Finally, the assumptions of Lemma 8.10 are basic bootstrap assumptions A1, A2, A3, introduced in Chapter 5, the bootstrap assumptions E1, E2, F2, of Chapter 6, together with the additional bootstrap assumption E / 30 . Now, assumptions A1 and A2 follow from assumption E1, while assumption A3 follows from Proposition 12.1. Assumption F2 coincides with X / 0 which has been established by Proposition 12.6. Q Assumption E1 is E{0} while assumption E2 follows from E{1} , E{0} , and hypothesis H0, QQ which has been established by Proposition 12.3. Assumption ELL 3 follows from E{0} ,

742

Chapter 13. Error Estimates

Q Q E{0} , while assumption ELT 3 follows from E{1} , E{1} , and the estimate for  L ∞ ( ε0 ) t of Corollary 11.1.c with m = l = 0. The assumptions of Corollary 11.1.c are those of Proposition 11.1 and follow from Proposition 12.10. Finally, assumption E / 30 follows from E{2} and hypotheses H1 and H0, hypothesis H1 having been established by Proposition 12.6. We conclude from the above that the assumptions of Proposition 8.6 all follow from the assumptions of Proposition 12.10 with m = l = 0, namely the assumptions of Proposition 12.9 with l = 0, which coincide with those of Proposition 12.6, together with the following assumption on the initial data:

µ − η0 ∞,[0,1], ε0 ≤ Cδ0

(13.1)

0

Proposition 13.1 Let the assumptions of Proposition 12.6 hold and let the initial data satisfy, in addition, the estimate (13.1). Then on Wεs0 we have: µ−1 (Lµ)+ ≤ (1 + t)−1 [1 + log(1 + t)]−1 + A(t) where A(t) is the non-negative function: A(t) = Cδ0 (1 + t)−2 [1 + log(1 + t)]

Since

s

A(t)dt ≤ Cδ0

0

C being a constant independent of s, we conclude that the property C1 holds on Wεs0 . Proof. We appeal to Proposition 8.6 to express, in acoustical coordinates:   1 ∂ µˆ s −1 µ (Lµ)+ = µˆ s ∂t +   Eˆ s (u, ϑ)(1 + t)−1 + Qˆ 1,s (t, u, ϑ) + = 1 + Eˆ s (u, ϑ) log(1 + t) + Qˆ 0,s (t, u, ϑ)

(13.2)

Consider a given (u, ϑ) ∈ [0, ε0 ] × S 2 . There are three cases to consider. Case 1: Eˆ s (u, ϑ) = 0,

Case 2: Eˆ s (u, ϑ) > 0, Case 3: Eˆ s (u, ϑ) < 0

In Case 1, (13.2) reduces to:   ( Qˆ 1,s (t, u, ϑ))+ 1 ∂ µˆ s = µˆ s ∂t + 1 + Qˆ 0,s (t, u, ϑ)

(13.3)

Now, from Proposition 8.6 we have: [1 + log(1 + t)] (1 + t) [1 + log(1 + t)] | Qˆ 1,s (t, u, ϑ)| ≤ Cδ0 (1 + t)2 | Qˆ 0,s (t, u, ϑ)| ≤ Cδ0

(13.4) (13.5)

Part 1: Derivation of the properties C1, C2, C3

which imply: 1 µˆ s



∂ µˆ s ∂t

743

 +

≤ Cδ0

[1 + log(1 + t)] (1 + t)2

(13.6)

In Case 2, we have:   Eˆ s (u, ϑ)(1 + t)−1 + Qˆ 1,s (t, u, ϑ) ≤ Eˆ s (u, ϑ)(1 + t)−1 + | Qˆ 1,s (t, u, ϑ)| +

hence, substituting in (13.2),   1 ∂ µˆ s Eˆ s (u, ϑ)(1 + t)−1 + | Qˆ 1,s (t, u, ϑ)| ≤ µˆ s ∂t + 1 + Eˆ s (u, ϑ) log(1 + t) + Qˆ 0,s (t, u, ϑ) Eˆ s (u, ϑ)(1 + t)−1 = 1 + Eˆ s (u, ϑ) log(1 + t)   1 Eˆ s (u, ϑ) 1 + − (1 + t) 1 + Eˆ s (u, ϑ) log(1 + t) + Qˆ 0,s (t, u, ϑ) 1 + Eˆ s (u, ϑ) log(1 + t) +

| Qˆ 1,s (t, u, ϑ)| 1 + Eˆ s (u, ϑ) log(1 + t) + Qˆ 0,s (t, u, ϑ)

Since

| Eˆ s (u, ϑ)| ≤ Cδ0 ≤ 1

(13.7)

(13.8)

(assuming that δ0 is suitably small), the first term on the right in (13.7) is bounded by: 1 (1 + t)[1 + log(1 + t)]

(13.9)

The factor in parenthesis in the second term on the right in (13.7) is bounded in absolute value by: | Qˆ 0,s (t, u, ϑ)|   1 + Eˆ s (u, ϑ) log(1 + t) + Qˆ 0,s (t, u, ϑ) 1 + Eˆ s (u, ϑ) log(1 + t) ≤

| Qˆ 0,s (t, u, ϑ)|   1 + Eˆ s (u, ϑ) log(1 + t) 1 − | Qˆ 0,s (t, u, ϑ)| 1

therefore the second term on the right in (13.7) is bounded in absolute value by: 1 | Qˆ 0,s (t, u, ϑ)| Cδ0  ≤ (1 + t)[1 + log(1 + t)] 1 − | Qˆ 0,s (t, u, ϑ)| (1 + t)2

(13.10)

by (13.4), taking δ0 suitably small. Also, by (13.5) (and (13.4)) the third term on the right in (13.7) is bounded in absolute value by: Cδ0

[1 + log(1 + t)] (1 + t)2

(13.11)

744

Chapter 13. Error Estimates

Combining the results (13.9), (13.10), (13.11), for the three terms on the right in (13.7), we conclude that in Case 2:   1 [1 + log(1 + t)] 1 ∂ µˆ s ≤ (13.12) + Cδ0 µˆ s ∂t + (1 + t)[1 + log(1 + t)] (1 + t)2 Finally, we consider Case 3. In this case we define: t1 (u, ϑ) = e

−

1 2 Eˆ s (u,ϑ)

−1

(13.13)

There are two subcases to consider. Subcase 3a: t ≤ t1 (u, ϑ),

Subcase 3b: t > t1 (u, ϑ)

In Subcase 3a we have: log(1 + t) ≤ −

1 ˆ 2 E s (u, ϑ)

hence:

1 2 In view of the fact that by (13.4), taking δ0 suitably small: 1 + Eˆ s (u, ϑ) log(1 + t) ≥

(13.14)

1 : for all t ∈ [0, s] | Qˆ 0,s (t, u, ϑ)| ≤ 4

(13.15)

it follows that: µˆ s (t, u, ϑ) = 1 + Eˆ s (u, ϑ) log(1 + t) + Qˆ 0,s (t, u, ϑ) ≥ hence, from (13.2): 1 µˆ s



∂ µˆ s ∂t



 +

≤4

Eˆ s (u, ϑ) + Qˆ 1,s (t, u, ϑ) (1 + t)

≤ 4| Qˆ 1,s (t, u, ϑ)| [1 + log(1 + t)] ≤ Cδ0 (1 + t)2

1 4

(13.16)

 +

(13.17)

by (13.5). In Subcase 3b we have, by (13.5):

  Eˆ s (u, ϑ) ˆ [1 + log(1 + t)] 1 ∂ µˆ s ˆ (t, u, ϑ) = + Q 1,s (t, u, ϑ) ≤ E s (u, ϑ) + Cδ0 ∂t (1 + t) (1 + t) (1 + t) (13.18) Let us set (13.19) τ = log(1 + t) so that t = eτ − 1

The mapping t → τ is an orientation preserving diffeomorphism of [0, ∞) onto itself. Also, [1 + log(1 + t)] (13.20) = e−τ (1 + τ ) = f (τ ) (1 + t)

Part 1: Derivation of the properties C1, C2, C3

745

is a decreasing function of τ . Thus t > t1 (u, ϑ) corresponds to τ > τ1 (u, ϑ), where: τ1 (u, ϑ) = −

1

 f (τ ) < f (τ1 (u, ϑ)) = 1 −

and to:

(13.21)

2 Eˆ s (u, ϑ) 

1 2 Eˆ s (u, ϑ)

1

e 2 Eˆ s (u,ϑ)

(13.22)

It then follows from (13.18) that in Subcase 3b:    1 1 ∂ µˆ s 1 ˆ s (u,ϑ) 2 E ˆ e (t, u, ϑ) < E s (u, ϑ) + Cδ0 1 − ∂t (1 + t) 2 Eˆ s (u, ϑ) Eˆ s (u, ϑ) = (13.23) (1 − 2Cδ0 g(x)) (1 + t) where we have set: x = −2 Eˆ s (u, ϑ) > 0 and g(x) =

  1 1 −1 1+ e x x x

(13.24)

Now, 0 < x ≤ Cδ0 , the function g(x) is bounded on (0, 1], and we have: lim g(x) = 0

x→0+

(13.25)

It follows that: 1 − 2Cδ0 g(x) > 0 if δ0 is suitably small, hence from (13.23): ∂ µˆ s (t, u, ϑ) < 0 ∂t Therefore in Subcase 3b:

1 µˆ s



∂ µˆ s ∂t

(13.26)

 +

=0

(13.27)

In view of the results (13.17), (13.27), for the two subcases we conclude that in Case 3:   1 ∂ µˆ s [1 + log(1 + t)] ≤ Cδ0 (13.28) µˆ s ∂t + (1 + t)2 Combining the results (13.6), (13.12), (13.28), we conclude that in Wεs0 :   1 ∂ µˆ s 1 + A(t) ≤ µˆ s ∂t + (1 + t)[1 + log(1 + t)] where: A(t) = Cδ0 and the proposition is proved.

[1 + log(1 + t)] (1 + t)2

(13.29)

(13.30)

746

Chapter 13. Error Estimates

Proposition 13.2 Let the assumptions of Proposition 12.10 hold for m = 2, l = 1. Then on Wεs0 we have: µ−1 (T µ)+ ≤ Bs (t) Here Bs (t) is the non-negative function:

(1 + τ ) Bs (t) = C δ0 √ + Cδ0 (1 + τ ) σ −τ where τ = log(1 + t), σ = log(1 + s). We have: s

(1 + t)−2 [1 + log(1 + t)]4 Bs (t)dt ≤ C δ0 0

where C is a constant independent of s. Moreover, the property C2 holds on Wεs0 . Proof. We shall again make use of Proposition 8.6. Let us denote Eˆ s,m =

min

(u,ϑ)∈[0,ε0 ]×S 2

Eˆ s (u, ϑ)

(13.31)

There are two cases, according as to whether Eˆ s,m ≥ 0, or Eˆ s,m < 0. The case that Eˆ s,m ≥ 0 is much easier and shall be treated below. In the case that Eˆ s,m < 0, we set: Eˆ s,m = −δ1 , δ1 > 0

(13.32)

 be the open subset of [0, ε ] × S 2 defined by: Let Vs− 0  δ1  Vs− = (u, ϑ) ∈ [0, ε0 ] × S 2 : Eˆ s (u, ϑ) < − 2

(13.33)

ε

 0 We denote by Us,t − the corresponding open subset of t , namely the set of points on ε0  . Now, from Proposition t with acoustical coordinates (t, u, ϑ), such that (u, ϑ) ∈ Vs− 8.6 we have: (13.34) µˆ s (s, u, ϑ) = 1 + Eˆ s (u, ϑ) log(1 + s)

hence, with σ = log(1 + s): 0≤

min

(u,ϑ)∈[0,ε0]×S 2

µˆ s (s, u, ϑ) = 1 − δ1 σ

(13.35)

ε

 Consider then a point p ∈ t 0 \ Us,t −. The acoustical coordinates of p are (t, u, ϑ) with 2  (u, ϑ) ∈ [0, ε0 ] × S \ Vs− . With τ = log(1 + t) we then have, by Proposition 8.6 and (13.35), since τ ≤ σ :

µˆ s (t, u, ϑ) = 1 + Eˆ s (u, ϑ) log(1 + t) + Qˆ 0,s (t, u, ϑ) 1 ≥ 1 − δ1 τ − | Qˆ 0,s (t, u, ϑ)| 2 1 1 ≥ − | Qˆ 0,s (t, u, ϑ)| ≥ 2 4

(13.36)

Part 1: Derivation of the properties C1, C2, C3

747

by (13.15). Since µ( p) = µ[1],s (u, ϑ)µˆ s (t, u, ϑ)

(13.37)

η0 2

(13.38)

and by Proposition 8.6: inf

(u,ϑ)∈[0,ε0

]×S 2

µ[1],s (u, ϑ) ≥

it follows that:

η0 (13.39) 8 ε In the case that Eˆ s,m ≥ 0 a similar conclusion holds for all p ∈ t 0 , for, by Proposition 8.6 and (13.4) we have in this case: µ( p) ≥

µˆ s (t, u, ϑ) ≥ 1 + Qˆ 0,s (t, u, ϑ) ≥ 1 − | Qˆ 0,s (t, u, ϑ)| ≥

3 4

(13.40)

hence by (13.37), (13.38): 3η0 (13.41) 8 Now, according to Proposition 12.10 with m = 2, l = 1, M[2,2] holds on Wεs0 . In particular, we have: (13.42) sup |T µ| ≤ Cδ0 [1 + log(1 + t)] : for all t ∈ [0, s] µ( p) ≥

ε

t 0

In view of (13.39), (13.41), and (13.42), it follows that:   µ−1 (T µ)+ ( p) ≤ Cδ0 [1 + log(1 + t)]

(13.43)

ε ε  for any p ∈ t 0 , t ∈ [0, s], in the case that Eˆ s,m ≥ 0, for any p ∈ t 0 \ Us,t −, t ∈ [0, s], ˆ in the case that E s,m < 0.  It remains for us to consider the case where Eˆ s,m < 0 and p ∈ Us,t −. Let us set:

τ1 =

1 , t1 = eτ1 − 1 2δ1

(13.44)

There are two subcases to consider, according as to whether t ≤ t1 or t > t1 . In the subcase t ≤ t1 , which corresponds to τ ≤ τ1 , we have, by Proposition 8.6 and (13.35): µˆ s (t, u, ϑ) = 1 + Eˆ s (u, ϑ) log(1 + t) + Qˆ 0,s (t, u, ϑ) ≥ 1 − δ1 τ − | Qˆ 0,s (t, u, ϑ)| 1 1 − | Qˆ 0,s (t, u, ϑ)| ≥ (13.45) 2 4 by (13.15), as in (13.36). By (13.37), (13.38), the lower bound (13.39) follows, which, together with (13.42) yields:   µ−1 (T µ)+ ( p) ≤ Cδ0 [1 + log(1 + t)] (13.46) ≥

 also for any p ∈ Us,t −, in the subcase t ≤ t1 .

748

Chapter 13. Error Estimates

   Finally, it remains for us to estimate µ−1 (T µ)+ ( p) in the subcase p ∈ Us,t −, t > t1 . If (T µ)( p) ≤ 0, then (T µ)+ ( p) = 0 and the estimate is trivial. Otherwise (T µ)( p) > ε  0, and we consider the integral curve of T , on t 0 , through p. The intersection of Us,t − with this integral curve is an open subset of the integral curve, corresponding to an open subset J of the parameter interval (0, ε0 ] (the integral curves of T are parametrized by u). Thus J is a countable union of open intervals. The point 0 is not a boundary point of J , while the point ε0 may be contained in J as a boundary point of the rightmost interval. Let the point p correspond to the parameter value u ∗ . Then u ∗ ∈ J . Consider the interval I, the component of J to which u ∗ belongs. Now I is of the form I = (a, b), a > 0, b < ε0 , if I is not the rightmost interval, while either I = (a, ε0 ] or I = (a, ε0 ), a > 0, if I is the rightmost interval. Consider the function µ along the given integral curve as a function of u. Then T µ is the function dµ/du, and (T )2 µ the function d 2 µ/du 2 along the same integral curve. Consider the subinterval (a, u ∗ ] of I. Then either (Case 1) dµ/du > 0 throughout (a, u ∗ ], or (Case 2) there is a rightmost value of u in (a, u ∗ ] where dµ/du = 0, and, denoting this rightmost value by u, we have dµ/du > 0 on (u, u ∗ ]. In Case 1 we have: (13.47) µ(u ∗ ) > µ(a) Now a ∈ / J , therefore the point q at parameter value a along the integral curve in question ε  belongs to t 0 \ Us,t −. Consequently the lower bound (13.39) holds at q: µ(q) ≥

η0 8

(13.48)

Since according to (13.47) µ( p) > µ(a), this implies: µ(q) >

η0 8

(13.49)

which together with (13.42) yields:   µ−1 (T µ)+ ( p) ≤ Cδ0 [1 + log(1 + t)]

(13.50)

In Case 2 the mean value of d 2 µ/du 2 on [u, u ∗ ] is positive. Denoting then by: γ (t) = sup(T )2 µ ε

(13.51)

t 0

we have γ (t) > 0. Consider then some u ∈ [u, u ∗ ]. We have: u∗ 2 d µ  dµ dµ (u ∗ ) − (u) = (u )du  2 du du du u that is:

(T µ)(u ∗ ) − (T µ)(u) =

u∗

((T )2 µ)(u  )du 

u

≤ γ (t)(u ∗ − u)

(13.52)

Part 1: Derivation of the properties C1, C2, C3

749

Therefore: (T µ)(u) ≥ (T µ)(u ∗ ) − γ (t)(u ∗ − u) 1 ≥ (T µ)(u ∗ ) 2 : for all u ∈ [u 1 , u ∗ ] where: u1 = u∗ −

(T µ)(u ∗ ) 2γ (t)

(13.53)

(13.54)

This shows in particular that u < u 1 , for otherwise (13.53) would apply in the case u = u, yielding (T µ)(u) ≥ (1/2)(T µ)(u ∗ ) > 0, contradicting the fact that (T µ)(u) = 0. Now, since T µ > 0 throughout (u, u ∗ ], in particular on (u, u 1 ], we have: µ(u 1 ) > µ(u)

(13.55)

Moreover, by (13.53), (13.54), we have: u∗ (T µ)(u)du µ(u ∗ ) − µ(u 1 ) = u1

≥

1 ((T µ)(u ∗ ))2 (T µ)(u ∗ )(u ∗ − u 1 ) = 2 4γ (t)

(13.56)

Also, denoting (as in Chapters 8 and 9): µm (t) = min µ ε

(13.57)

µ(u) ≥ µm (t)

(13.58)

t 0

we have: Combining the inequalities (13.58), (13.55) and (13.56), we conclude that (recalling that the parameter value u ∗ corresponds to the point p): µ( p) − µm (t) ≥

((T µ)( p))2 4γ (t)

(13.59)

We thus obtain:   (T µ)( p) µ−1 (T µ)+ ( p) = µ( p) (T µ)( p) ≤ ((T µ)( p))2 + µm (t) 4γ (t )

= 2 γ (t) f ε (x)

(13.60)

µm (t) > 0

(13.61)

where, with: x=

(T µ)( p) > 0, √ 2 γ (t)

ε=

750

Chapter 13. Error Estimates

f ε (x) is the function:

x (13.62) x 2 + ε2 Now, the function fε (x) achieves its maximum at x = ε and the maximum value is 1/2ε. Thus: 1 (13.63) f ε (x) ≤ 2ε hence from (13.60): √   γ (t) −1 µ (T µ)+ ( p) ≤ √ (13.64) µm (t) f ε (x) =

Consider now µm (t). From Proposition 8.6 we have (see (13.37), (13.38)): η0 µˆ s (t, u, ϑ) 2   η0 1 + Eˆ s (u, ϑ) log(1 + t) + Qˆ 0,s (t, u, ϑ) = 2  η0 1 − δ1 τ − | Qˆ 0,s | ≥ 2

µ(t, u, ϑ) ≥

(13.65)

Since δ1 σ ≤ 1 this implies: µm (t) ≥

 η0 δ1 (σ − τ ) − | Qˆ 0,s | 2

(13.66)

Now, from Proposition 8.6 we have: | Qˆ 0,s (t, u, ϑ)| ≤ Cδ0 b(t, s)

(13.67)

where b(t, s) is defined by (8.286) of Chapter 8: b(t, s) =

[1 + log(1 + t)] (s − t) (1 + t) (1 + s)

(13.68)

According to (8.311) of Chapter 8: 0 ≤ b(t, s) ≤

(1 + τ ) (σ − τ ) eτ

Substituting (13.67), (13.69), in (13.66) we obtain:   η0 δ0 (1 + τ ) µm (t) ≥ δ1 1 − C (σ − τ ) 2 δ1 e τ As we are considering the subcase τ > τ1 , we have:   1 + 2δ11 (1 + τ ) (1 + τ1 ) < = 1 eτ eτ1 e 2δ1

(13.69)

(13.70)

Part 1: Derivation of the properties C1, C2, C3

The factor

751

  1 − 1 1+ e 2δ1 2δ1

1 δ1

is bounded and tends to zero as δ1 → 0. Therefore, if δ0 is suitably small we have: 1−C

δ0 (1 + τ ) 1 ≥ τ δ1 e 2

Since also 2δ1 ≥ 1/τ , it follows from (13.70) that: η0 (σ − τ ) 8 (1 + τ )

(13.71)

γ (t) ≤ Cδ0 (1 + τ )

(13.72)

µm (t) ≥ On the other hand, by M2,0 we have:

In view of the bounds (13.71), (13.72), we conclude from (13.64) that:  

(1 + τ ) µ−1 (T µ)+ ( p) ≤ C δ0 √ σ −τ

(13.73)

This is the bound in Case 2. Combining this with the earlier bound (13.50) for Case 1, we  conclude that for any p ∈ Us,t −, t ∈ (t1 , s], we have:  

(1 + τ ) µ−1 (T µ)+ ( p) ≤ C δ0 √ + Cδ0 (1 + τ ) σ −τ

(13.74)

 Combining this with the bound (13.46) for the case that p ∈ Us,t − , t ∈ [0, t1 ], and the ε0  ˆ bound (13.43) for the case that p ∈ t \ Us,t − and the case that E s,m ≥ 0, we conclude that on Wεs0 we have, in general:

where:

We have:



s

µ−1 (T µ)+ ≤ Bs (t)

(13.75)

(1 + τ ) + Cδ0 (1 + τ ) Bs (t) = C δ0 √ σ −τ

(13.76)

(1 + t)−2 [1 + log(1 + t)]4 Bs (t)dt =

0



σ

(1 + τ )4 Bs (t)e−τ dτ

0

and:

σ 2

0

(1 + τ )5 −τ e dτ ≤ √ σ −τ



σ 2

(1 + τ )5 −τ e dτ √ τ

∞

(1 + τ )5 −τ e dτ = C(independent of s) √ τ

0

≤ 0

752

Chapter 13. Error Estimates

while:

σ σ 2

It follows that:

σ (1 + τ )5 −τ dτ 5 − σ2 √ √ e dτ ≤ (1 + σ ) e σ σ −τ σ −τ 2 √ 5 − σ2 = 2σ (1 + σ ) e ≤ C(independent of s)

s

(1 + t)−2 [1 + log(1 + t)]4 Bs (t)dt ≤ C δ0

0

where C is a constant independent of s. Consider now µ−1 (Lµ + Lµ)+ . Since L = α −2 µL + 2T , we have: µ−1 (Lµ + Lµ)+ ≤ α −2 |Lµ| + µ−1 (Lµ)+ + 2µ−1 (T µ)+

(13.77)

The first term on the right in (13.77) is bounded by Cδ0 (1 + t)−1 . The second term on the right is bounded by Proposition 13.1, however a different bound for this term is more appropriate here. Going back to the proof of Proposition 13.1, we treat Case 2 in a slightly different manner, namely by estimating the first term on the right in (13.7) by: Cδ0 (1 + t)−1 (using simply (13.8)). We then obtain in Case 2 the bound   Cδ0 1 ∂ µˆ s ≤ µˆ s ∂t + (1 + t)

(13.78)

in place of the bound (13.12). Combining this with the bounds (13.6), (13.28), for the other two cases, we obtain: µ−1 (Lµ)+ ≤ Cδ0 (1 + t)−1

(13.79)

Together with the bound (13.75) this yields, through (13.77): µ−1 (Lµ + Lµ)+ ≤ Bs (t) where: Noting that:

: on Wεs0

Bs (t) = 2Bs (t) + Cδ0 (1 + t)−1

s 0

(1 + t)−2 [1 + log(1 + t)]4 Bs (t)dt ≤ C δ0

(13.80) (13.81)

(13.82)

where C is a constant independent of s, we conclude that property C2 holds on Wεs0 , and the proof of Proposition 13.2 is complete. Proposition 13.3 Let the assumptions of Proposition 13.1 hold and let U be the region: U = {x ∈ Wε∗0 : µ < η0 /4}

Part 1: Derivation of the properties C1, C2, C3

753

Then there is a positive constant C independent of s such that in U



Wεs0 we have:

Lµ ≤ −C −1 (1 + t)−1 [1 + log(1 + t)]−1 that is, property C3 holds on Wεs0 .  ε Proof. Consider a point p ∈ U t 0 , t ∈ [0, s], and let its acoustical coordinates be (t, u, ϑ). First we show that Eˆ s (u, ϑ) < 0. For, otherwise we have, from Proposition 8.6: µˆ s (t, u, ϑ) ≥ 1 + Qˆ 0,s (t, u, ϑ) ≥ 1 − Cδ0 ≥

1 2

if δ0 is suitably small, hence: µ(t, u, ϑ) = µ[1],s (u, ϑ)µˆ s (t, u, ϑ) ≥

η0 4

in contradiction with the fact that p ∈ U. Next we show that log(1 + t) = τ > τ2 , where: τ2 =

1 , 4δ

: δ = − Eˆ s (u, ϑ) > 0

(13.83)

For, otherwise we have, from Proposition 8.6: 3 1 µˆ s (t, u, ϑ) = 1 − δτ + Qˆ 0,s (t, u, ϑ) ≥ − Cδ0 ≥ 4 2 if δ0 is suitably small, hence again µ(t, u, ϑ) ≥ η0 /4 in contradiction with the fact that p ∈ U. Now, from Proposition 8.6: Lµ = µ[1],s and:

∂ µˆ s ∂t

δ ∂ µˆ s (t, u, ϑ) = − + Qˆ 1,s (t, u, ϑ) ∂t (1 + t)

Moreover, the bound (13.5) holds. It follows that:  δ Cδ0 (1 + τ ) ∂ µˆ s (t, u, ϑ) ≤ − 1− ∂t (1 + t) δ eτ Now τ > τ2 implies:   1 (1 + τ2 ) 1 (1 + τ ) < = 1 + e− 4δ eτ eτ2 4δ Since the function

  1 1 1 1+ e− 4δ δ 4δ

(13.84)

(13.85)

(13.86)

754

Chapter 13. Error Estimates

is bounded and tends to zero as δ → 0, it follows that: 1 Cδ0 (1 + τ ) ≤ τ δ e 2 provided that δ0 is suitably small. Hence, from (13.86): 1 δ ∂ µˆ s (t, u, ϑ) ≤ − ∂t 2 (1 + t)

(13.87)

Since log(1 + t) = τ > τ2 = 1/4δ, this implies: 1 1 ∂ µˆ s (t, u, ϑ) ≤ − ∂t 8 (1 + t)[1 + log(1 + t)]

(13.88)

Then by (13.84) and the fact that, by Proposition 8.6, µ[1],s ≥ η0 /2, it follows that: Lµ ≤ −C −1 (1 + t)−1 [1 + log(1 + t)]−1 with C = 16/η0

(13.89)

and the proposition is proved. To complete the first part of the present chapter we shall define a function ω, as needed in Chapter 5. Proposition 13.4 Let the assumptions of Proposition 12.6 hold and let the initial data satisfy, in addition, the estimate (13.1). Let us define: ω = 2η0 (1 + t) Then the function ω has the properties D1, D2, D3, D4, and D5, required in Chapter 5. Proof. Property D1 is obvious. We have: Lω = 2η0 , T ω = 0, d/ω = 0

(13.90)

By the first condition of (13.90):   η0 ω η0 (1 + t) = 2η0 1 − Lω − (1 − u + η0 t) (1 − u + η0 t) (1 − u − η0 ) = 2η0 (1 − u + η0 t) hence:

    η0 ω  Lω − ≤ C  (1 − u + η0 t)  (1 + t)

(13.91)

(13.92)

On the other hand, we have, by equation (5.10) of Chapter 5: 1 (trχ + L log ) 2 η0 1 = + (trχ  + L log ) (1 − u + η0 t) 2

ν=

(13.93)

Part 1: Derivation of the properties C1, C2, C3

hence, by Proposition 12.6:   ν − 

  η0  ≤ Cδ0 [1 + log(1 + t)] (1 − u + η0 t)  (1 + t)2

755

(13.94)

Multiplying (13.94) by ω and adding the result to (13.92) yields: |Lω − νω| ≤ C(1 + t)−1 [1 + log(1 + t)]

(13.95)

which implies property D2. By the first two conditions of (13.90): Lω = 2η0 α −1 κ

(13.96)

|Lω| ≤ C[1 + log(1 + t)]

(13.97)

hence, by Proposition 12.1:

which implies property D3. Property D4 is obvious from the third condition of (13.90). It remains for us to verify property D5. To compute µ
h˜ ω we appeal to the expression (3.133) of Chapter 3 for 
h˜ f for an arbitrary function f . Taking f = ω, in view of the third condition of (13.90) this expression reduces to: −µ
h˜ ω = L(Lω) + ν Lω + ν Lω

(13.98)

L(Lω) = 2η0 L(α −1 κ)

(13.99)

Now, from (13.96):

Also, from (13.96) and the first condition of (13.90): ν Lω + ν Lω = 2η0 (να −1 κ + ν) = η0 (α −1 κtrχ + trχ + α −1 κ L log  + L log ) = 2η0 (κtrk/ + α −1 κ L log  + T log )

(13.100)

by equations (5.10), (5.11) of Chapter 5, and the fact that (see Chapter 3): α −1 κχ + χ = 2κk/ We thus obtain:



−µ
h˜ ω = 2η0 L(α −1 κ) + κtrk/ + α −1 κ L log  + T log 

Using Corollary 11.1.b with m = l = 0 then yields:   sup µ|
h˜ ω| ≤ Cδ0 (1 + t)−1 ε

t 0

which implies property D5. The proof of the proposition is therefore complete.

(13.101)

(13.102)

756

Chapter 13. Error Estimates

The remaining assumptions on which Theorem 5.1 relies, assumptions B, readily follow from assumptions X / [1] and M[1,2] , established by Propositions 12.9 with l = 1 and 12.10 with m = l = 1, respectively. Now Propositions 10.5, 10.6, 11.5, 11.6, require the maximal order assumptions on the acoustical quantities, namely the assumptions X / [(l+1)∗ ] and M{(l+1)∗ +1} (recall that by definition M[m,l] = M[l,l] = M{l} if m ≥ l). These assumptions are established by Propositions 12.9 and 12.10 on the basis of the bootstrap Q QQ assumptions E{(l+1)∗ +2} , E{(l+1)∗ +1} , E{(l+1)∗ +1} . We recall from Chapter 11 the definiQ

QQ

tion of the bootstrap assumptions E{k} , E{k} , E{k} . We now make these assumptions more precise by setting the constant appearing on the right equal to 1. We define, more generp 5 67 8 Q... Q ally, the bootstrap assumptions E{q} , for p = 0, . . . , q as follows. First, we say that the bootstrap assumption E0,0 holds on Wεs0 if for all t ∈ [0, s]: : ψ0 − k L ∞ ( ε0 ) , max ψi  L ∞ ( ε0 ) ≤ δ0 (1 + t)−1

E0,0

t

i

t

For non-negative integers p, m, n not all zero we say that the bootstrap assumption p

5 67 8 Q... Q Em,n holds on Wεs0 if for all t ∈ [0, s]: p

5 67 8 Q... Q Em,n

: p

max Rin . . . Ri1 (T )m (Q) p ψα  L ∞ ( ε0 ) ≤ δ0 (1 + t)−1

α;i1 ...in

t

p

5 67 8 5 67 8 Q... Q Q... Q We then define E{q} to be the conjunction of the assumptions Em,n corresponding to the triangle: {(m, n) : m, n ≥ 0, m + n ≤ q} Finally we define the bootstrap assumption E{{k}} to be the conjunction of the assumptions: p 5 67 8 Q... Q : p = 0, . . . , k E{k− p} From now on by bootstrap assumption we mean the assumption E{{(l+1)∗ +2}} . We have shown in the preceding that all the assumptions which we have so far used follow from the bootstrap assumption together with appropriate assumptions on the initial conditions. p 5 67 8 Q... Q The assumptions E{q} , for p = 3, . . . , (l + 1)∗ + 2, q = (l + 1)∗ + 2 − p, contained in the bootstrap assumption, suffice to control in an appropriate manner the derivatives with respect to Q of the (Y ) π˜ of the commutation fields Y .

Part 2: The error estimates of the acoustical entities

757

Part 2: The error estimates involving the top order spatial derivatives of the acoustical entities The second part of the present chapter is concerned with the error estimates involving the top order spatial derivatives of the acoustical entities χ and µ. We remark that, in view of the fact that L / L χ and Lµ are expressed by the basic propagation equations ((12.69) and (12.343)) in terms of χ and µ respectively, the derivatives of the acoustical entities χ and µ which are actually of the top order, are in fact the top order spatial derivatives of these entities. This is so since derivatives of χ and µ containing differentiations with respect to L are expressed in terms of lower order spatial derivatives of these entities. In the following, given a positive integer n, we denote by (α;I1 ...In−1 ) ψn the nth order variation (see Chapters 6 and 7): (α;I1 ...In−1 )

where

(α) ψ 1

ψn = Y In−1 . . . Y I1

(α)

ψ1

is the 1st order variation:  ψ0 − k : for α = 0 (α) ψ1 = ψα : for α = 1, 2, 3

(13.103)

(13.104)

Here the capital indices I1 , . . . , In−1 range over the set {1, 2, 3, 4, 5} and we have, as in Chapters 6 and 7: Y1 = T, Y2 = Q, Yi+2 = Ri : i = 1, 2, 3

(13.105)

For n ≥ 2 we may express (13.103) more simply as: (α;I1 ...In−1 )

ψn = Y In−1 . . . Y I1 ψα

: for n ≥ 2

(13.106)

We shall require the multi-indices (I1 . . . In−1 ) to be of the following form. First, there should be a string of 2’s (which may be empty). Then, there should be a string of 1’s (which may also be empty). Finally, there should be a multi-index from the set {3, 4, 5}. If the string of 2’s has length p, the string of 1’s has length m, and the multi-index from the set {3, 4, 5} has length n, then we have: p

m

5 67 85 67 8 (I1 . . . I p+m+n ) = (2 . . . 21 . . . 1 i 1 + 2 . . . i n + 2)

(13.107)

where (i 1 . . . i n ) is a multi-index from the set {1, 2, 3}. Thus, if p + m + n = 0 we have, from (13.106): (α;I1 ...I p+m+n )

ψ p+m+n+1 = Rin . . . Ri1 (T )m (Q) p ψα

(13.108)

We shall also use in the following, in place of (13.103), the simpler notation: (I1 ...In−1 )

ψn = Y In−1 . . . Y I1 ψ1

(13.109)

758

Chapter 13. Error Estimates

whenever we do not wish to be specific about which of the 1st order variations (13.104) we are considering. To the nth order variation (I1 ...In−1 ) ψn is associated the nth order source function (I1 ...In−1 ) ρ by: n (13.110)
h˜ (I1 ...In−1 ) ψn = (I1 ...In−1 ) ρn (I1 ...In−1 ) ρ˜ n

The re-scaled source function (7.17) of Chapter 7:

(I1 ...In−1 )

is defined through

ρ˜n = 2 µ

(I1 ...In−1 )

(I1 ...In−1 ) ρ n

by equation

ρn

(13.111)

Then according to (7.18) of Chapter 7, the re-scaled sources satisfy for n ≥ 2 the following recursion formula: (I1 ...In−1 )

ρ˜n = (Y In−1 +

(Y In−1 )

δ)

(I1 ...In−2 )

ρ˜n−1 +

(Y In−1 ;I1 ...In−2 )

σn−1

(13.112)

and we have ((7.22)): ρ˜1 = 0

(13.113)

In (13.112) the function (Y In−1 ) δ is defined by equation (7.21) and depends only on the commutation field Y In−1 , while the function (Y In−1 ;I1 ...In−2 ) σn−1 is defined by equations (7.19), (7.20), and depends on the commutation field Y In−1 as well as on the n − 1st order variation (I1 ...In−2 ) ψn−1 . We shall apply Proposition 8.2 to the recursion (13.112), taking the space X in that proposition to be the space of functions on Wε∗0 . We replace n by n + 1 in (13.112) so that we have: (I1 ...In )

ρ˜n+1 = (Y In +

(Y In )

δ)

(I1 ...In−1 )

ρ˜n +

(Y In ;I1 ...In−1 )

σn

(13.114)

for n = 1, 2, . . . . Then x n in Proposition 8.2 corresponds to (I1 ...In−1 ) ρ˜n+1 , so that by (13.113) we have x 0 = 0. Also, yn corresponds to (Y In ;I1 ...In−1 ) σn , and An corresponds to the operator Y In + (Y In ) δ. Applying then Proposition 8.2 yields: (I1 ...In )

ρ˜n+1 =

n−1 

(Y In +

(Y In )

δ) . . . (Y In−m+1 +

(Y In−m+1 )

δ)

(Yn−m ;I1 ...In−m−1 )

σn−m

m=0

(13.115) or, setting n = l + 1, m = k: (I1 ...Il+1 )

ρ˜l+2 =

l  (Y Il+1 +

(Y Il+1 )

δ) . . . (Y Il+2−k +

(Yl+2−k )

δ)

(Yl+1−k ;I1 ...Il−k )

σl+1−k

k=0

(13.116) We shall focus our attention on the terms involving the top order spatial derivatives of the acoustical entities, namely the l + 1st order spatial derivatives of χ and the l + 2nd order spatial derivatives of µ. These appear only in the top order, l +1st, spatial derivatives ˜ Now, according to equation (7.36) of Chapter 7 we have: of the (Y ) π. (Y )

σn−1 =

(Y )

σ1,n−1 +

(Y )

σ2,n−1 +

(Y )

σ3,n−1

(13.117)

Part 2: The error estimates of the acoustical entities

759

where (Y ) σ1,n−1 , (Y ) σ2,n−1 , (Y ) σ3,n−1 , are given by equations (7.43), (7.44), (7.45) and (7.46)–(7.48), respectively. Of these only (Y ) σ2,n−1 contains the spatial derivatives of (Y ) π, ˜ being a sum of terms which are products of 1st derivatives of (Y ) π˜ with 1st derivatives of ψn−1 . In contrast, (Y ) σ1,n−1 is a sum of terms which are products of components of (Y ) π˜ with 2nd derivatives of ψn−1 , and (Y ) σ3,n−1 is a sum of terms which are triple products of trχ, trχ, , d/(α −1 κ), or L(α −1 κ), which, like the (Y ) π, ˜ contain acoustical entities of order at most 1, with components of (Y ) π˜ and 1st derivatives of ψn−1 . ˜ occur only in the We conclude that the derivatives of top order, l + 1, of the (Y ) π, term k = l in the sum in (13.116): (Y Il+1 )

(Y Il+1 +

δ) . . . (Y I2 +

more precisely, in: Y Il+1 . . . Y I2

(Y I1 )

(Y I2 )

δ)

(Y I1 )

σ1

σ2,1

(13.118) (13.119)

Furthermore, since we are only concerned about top order spatial derivatives of the acoustical entities, we may replace, in (13.119), (Y I1 ) σ2,1 , given by (7.44) with n = 2 and Y = Y I1 , by its top order spatial derivative part: π /)Lψ1 − L /T (Y I1 ) Z˜ · d/ψ1 −(1/2)div / (Y I1 ) Z˜ Lψ1 − (1/2)div / (Y I1 ) Z˜ Lψ1 +(1/2)d/ (Y I1 ) π˜ L L · (d/ψ1 )$ + div / (µ (Y I1 ) π /ˆ˜ ) · (d/ψ1 )$

(1/2)T (tr

(Y I1 ) ˜

(13.120)

Here we have replaced (1/4)L(tr (Y I1 ) π / L (Y I1 ) Z˜ by their top order spa/˜ )Lψ1 and −(1/2)L tial derivative parts (1/2)T (tr (Y I1 ) π /T (Y I1 ) Z˜ · d/ψ1 , respectively. We remark /˜ )Lψ1 and −L finally that the only expressions of the form (13.119) which may contain top order spatial derivatives of the acoustical entities are those for which (I2 . . . Il+1 ) does not contain any 2’s, therefore either p = 0, in which case Y I1 is one of T , R j : j = 1, 2, 3, or p = 1, in which case Y I1 = Q. In the following, we denote by [ ] P.A. , the principal acoustical part, expressed in terms of χ  and µ, the difference χ − χ  = η0 (1 − u + η0 t)−1 h/ being of lower order. Consider first the case p = 1, Y I1 = Q. The components of (Q) π˜ are given by table 6.13 of Chapter 6. Noting that: [] P.A. = −(d/µ)$ , we have: (Q)

π˜ L L

(Q) (Q)

tr

Z˜ Z˜

  P.A.  P.A. P.A.

[ν] P.A. =

1  trχ 2

(13.121)

=0 =0 = 2(1 + t)(d/µ)$

 = 2(1 + t)trχ  π / P.A.  (Q) ˆ˜ π / = 2(1 + t)χˆ  (Q) ˜

P.A.

(13.122)

760

Chapter 13. Error Estimates

From (13.120) we then obtain:  2 1  (Q) σ2,1 = (1 + t) (T trχ  −
/ µ)Lψ1 + 2µ(div / χˆ  ) · (d/ψ1 )$ P.A. (13.123) P.A.

Now by equation (8.104) of Chapter 8: 1 2 2div / χˆ  − d/trχ  P.A. = 0

(13.124)

Also, by equation (11.875) of Chapter 11: 1 2 T trχ  −
/ µ P.A. = 0

(13.125)

In view of (13.124) and (13.125), (13.123) reduces to:  (Q) σ2,1 = (1 + t)µ(d/trχ  ) · (d/ψ1 )$

(13.126)

P.A.

Consider next the case p = 0, Y I1 = R j . By equations (6.173) and (6.10) of Chapter 6:

(R j )

hence:

π˜ L L = −2R j µ − 2µR j 

(R j )

π˜ L L

 P.A.

(13.127)

= −2R j µ

(13.128)

= −R j · χ 

(13.129)

By equation (10.389) of Chapter 10: (R j )

Since:

(R j )

Z˜ = 

it follows that:

(R j )

(R j )

We have:

(R j )

π /L

Z˜

 P.A.

(R j )

Z,

 P.A.

π / L = α −1 κ

Z=

(R j )

π / L · h/−1

= −R j · χ $ (R j )

(13.130)

(R j )

π /L + 2

π /T

and according to equation (10.724) of Chapter 10: (R j )

where

(R j ) 

π /T =

(R j )

 · h/

is given by Lemma 10.23. Since: (R j )

it follows that:

Z˜ =  (R j )

(R j )

Z,

 Z˜ =  α −1 κ

(R j )

Z=

(R j )

(R j )

Z +2

π / L · h/−1

(R j )



 (13.131)

Now, according to Lemma 10.23 we have: (R j )



 = −κ q/˜ j − λ j qT

(13.132)

Part 2: The error estimates of the acoustical entities

761

Here qT is given by equations (10.718), (10.719) of Chapter 10, from which we obtain: 1 2 [qT ] P.A. = −(d/κ)$ P.A. = −α −1 (d/µ)$ (13.133)  Also, in (13.132), q/˜ j is defined through q/j by equation (10.115):  q/˜ j = q/j − w j

Since w j , given by equations (10.113), (10.114), is of lower order, we have:    = q/j q/˜ j P.A.

P.A.

Furthermore, q/j is given by equation (10.27): q/j = m  · R j + n j and m  is given by equations (10.28), (10.29), from which we obtain: 1 2 m P.A. = −α −1 χ $ Also, n j is given by equation (10.23) through b j : n j = b j ψ$ Since b j , given by (10.19), does not contain principal acoustical terms, the same is true for n j . The above then yield:  1 2  = m  · R j P.A. = −α −1 R j · χ $ (13.134) q/˜ j P.A.

In view of (13.133), (13.134), we obtain from (13.132):    (R j )  = α −1 κ R j · χ $ + λ j (d/µ)$

(13.135)

Combining this with (13.130) we then obtain, through (13.131):    (R j ) ˜ = α −1 κ R j · χ $ + 2λ j (d/µ)$ Z

(13.136)

P.A.

P.A.

Next, from equation (6.10) of Chapter 6: (R j ) ˜

π /=

(R j )

π / + (R j )h/

Since the second term on the right is of lower order, we have:   (R j ) ˜ π / =  (R j ) π / P.A.

P.A.

(13.137)

762

Chapter 13. Error Estimates

Now, by equation (10.206) of Chapter 10:  (R j ) π /

= 2λ j α −1 χ 

(13.138)

= 2λ j α −1 χ 

(13.139)

= 2λ j α −1 trχ 

(13.140)

= 2λ j α −1 χˆ 

(13.141)

P.A.

 π /

Hence:

(R j ) ˜

In particular: tr

 π /

(R j ) ˜

 π /

and:

P.A.

(R j ) ˆ˜

P.A.

P.A.

Substituting (13.128), (13.130), (13.136), (13.140), (13.141), in (13.120) we obtain:  (R j ) σ2,1 = λ j α −1 (T trχ  )Lψ1 + (R j · L /T χ  ) · (d/ψ1 )$ P.A.

+(1/2)div / (R j · χ $ )Lψ1   / (R j · χ $ ) + 2λ j (
−(1/2)α −1 κdiv / µ) Lψ1 2 −(d/ R j µ) · (d/ψ1 )$ + 2λ j κdiv / χˆ  · (d/ψ1 )$ P.A. (13.142) Using (13.124), (13.125) and the fact that by equation (11.305) of Chapter 11:  / 2µ =0 (13.143) L /T χ  − D P.A.

this reduces to: (R j )

σ2,1

 P.A.

1 = div / (R j · χ $ )T ψ1 + λ j κd/trχ  · (d/ψ1 )$  +(R j · D / 2 µ − d/ R j µ) · (d/ψ1 )$ P.A.

(13.144)

Consider div / (R j · χ $ ). In terms of components in an arbitrary local frame field for St,u we have: div / (R j · χ $ ) = D / A (R Bj χ BA ) = R Bj (D / A χ BA ) + (D / A R Bj )χ BA and, with R j B = h / BC R Cj : (D / A R Bj )χ BA = (D / A R j B )χ AB =

1 1 (D /ARjB + D / B R j A )χ AB = 2 2

(R j )

π / AB χ AB

We conclude that: 1  div / (R j · χ $ ) = R j · div / χ  + tr 2

(R j ) $

π / · χ $



The second term is of lower order. In view of (13.124) it follows that: 1 2 1 2 div / (R j · χ $ ) P.A. = R j · div / χ  P.A. = R j trχ 

(13.145)

(13.146)

Part 2: The error estimates of the acoustical entities

763

Consider finally (R j · D / 2 µ − d/ R j µ) · (d/ψ1 )$ . Let X be the St,u -tangential vectorfield: X = (d/ψ1 )$ Then we have:

(13.147)

Rj · D / 2 µ · (d/ψ1 )$ = D / 2 µ · (X, R j )

and: D / 2 µ · (X, R j ) = X R j µ − d/µ · (D /X Rj) = X R j µ − d/µ · (D / R j X) + d/µ · [R j , X]

(13.148)

Therefore: (R j · D / 2 µ − d/ R j µ) · (d/ψ1 )$ = D / 2 µ · (X, R j ) − X R j µ / R j X) = d/µ · L / R j X − d/µ · (D

(13.149)

Now: L / R j X = (d/ R j ψ1 )$ − (d/ψ1 )$ · D / R j X = R j · (D / ψ1 ) 2

(R j ) $

π /

(13.150)

$

(13.151)

are both of order 2, but contain no acoustical parts of order 2. It follows that:  (R j · D / 2 µ − d/ R j µ) · (d/ψ1 )$ =0

(13.152)

In view of (13.146) and (13.152), (13.144) reduces finally to:  (R j ) σ2,1 = (R j trχ  )T ψ1 + κλ j d/trχ  · (d/ψ1 )$

(13.153)

P.A.

P.A.

Finally, we consider the case p = 0, Y I1 = T . The components of by table 6.12 of Chapter 6. According to this table, we have: (T )

Z˜ =

(T ) ˜

π /L · h /−1 = ,

(T )

Z˜ =

(T ) ˜

π / L · h/−1 = α −1 κ

(T ) π ˜

are given (13.154)

Taking also into account (13.121) and the fact that since χ = −α −1 κχ + 2κk/ we have:   1 1 trχ = − α −1 κtrχ  , [χ] ˆ P.A. = −α −1 κ χˆ  (13.155) [ν] P.A = 2 2 P.A we obtain from table 6.12: (T )

π˜ L L

(T ) (T )

tr

Z˜ Z˜

  P.A.  P.A. P.A.

= −2T µ = −(d/µ)$ = −α −1 κ(d/µ)$

 = −2α −1 κtrχ  π / P.A.  (T ) ˆ˜ π / = −2α −1 κ χˆ  (T ) ˜

P.A.

(13.156)

764

Chapter 13. Error Estimates

Substituting the above in (13.120) we obtain:  (T ) σ2,1 = −α −1 κ(T trχ  )Lψ1 + (d/T µ) · (d/ψ1 )$ P.A.

+(1/2)(
/ µ)Lψ1 + (1/2)α −1 κ(
/ µ)Lψ1  $ 2  −(d/T µ) · (d/ψ1 ) − 2κ div / χˆ · (d/ψ1 )$

P.A.

(13.157)

Note the crucial cancellation of the two terms containing the factor d/T µ, one (in the first line on the right) coming from the term −L /T (Y I1 ) Z˜ · d/ψ1 in (13.120) and the other (in the third line on the right) from the term (1/2)d/ (Y I1 ) π˜ L L · (d/ψ1 )$ in the same expression. Taking then into account (13.124) and (13.125), (13.157) reduces to:  (T σ2,1 = (
/ µ)T ψ1 − κ 2 d/trχ  · (d/ψ1 )$ (13.158) P.A.

We conclude from the above that the top order variations (α;I1 ...Il+1 ) ψl+2 to which the re-scaled sources (α;I1 ...Il+1 ) ρ˜l+2 containing the top order spatial derivatives of the acoustical entities are as follows. In the case Y I1 = Q, the variations (see (13.107), (13.108)): (α;21...1i1 +2...in +2)

ψl+2 = Rin . . . Ri1 (T )m Qψα

: m+n =l

(13.159)

where there are m 1’s in the superscript on the left. By (13.119) and (13.126) the principal acoustical part of the corresponding re-scaled sources are:  (α;21...1i1 +2...in +2) ρ˜l+2 P.A. 2 1 = (1 + t)µ(d/ Rin . . . Ri1 (T )m trχ  ) · (d/ψα )$ P.A.  (1 + t)µ(d/ Rin . . . Ri1 trχ  ) · (d/ψα )$ : if m = 0 = (1 + t)µ(d/ Rin . . . Ri1 (T )m−1
/ µ) · (d/ψα )$ : if m ≥ 1 :m +n =l

(13.160)

where again there are m 1’s in the superscript on the left, and in the case m ≥ 1 we have used (13.125). In the case Y I1 = R j , where, setting i 1 = j we have I1 . . . Il+1 = i 1 + 2 . . . i l+1 + 2, the variations (see (13.107), (13.108)): (α;i1 +2...il+1 +2)

ψl+2 = Ril+1 . . . Ri1 ψα

(13.161)

By (13.119) and (13.153) the principal acoustical part of the corresponding re-scaled sources is:  (α;i1 +2...il+1 +2) ρ˜l+2 = (Ril+1 . . . Ri1 trχ  )T ψα P.A.

+κλi1 (d/ Ril+1 . . . Ri2 trχ  ) · (d/ψα )$

(13.162)

Part 2: The error estimates of the acoustical entities

765

In the case Y I1 = T , the variations (see (13.107), (13.108)): (α;1...1i1 +2...in +2)

ψl+2 = Rin . . . Ri1 (T )m+1 ψα

: m+n =l

(13.163)

where there are m + 1 1’s in the superscript on the left. By (13.119) and (13.158) the principal acoustical part of the corresponding re-scaled sources is:  (α;1...1i1 +2...in +2) ρ˜l+2 P.A.

= (Rin . . . Ri1 (T )m
/ µ)T ψα  2 κ (d/ Rin . . . Ri1 trχ  ) · (d/ψα )$ : if m = 0 + κ 2 (d/ Rin . . . Ri1 (T )m−1
/ µ) · (d/ψα )$ : if m ≥ 1 :m+n =l (13.164) where again there are m + 1 1’s in the superscript on the left, and in the case m ≥ 1 we have used (13.125). We see from formulas (13.160), (13.162), (13.164), that the terms which result in the most difficult error integrals to estimate are the terms: (Ril+1 . . . Ri1 trχ  )T ψα

(13.165)

in (13.162), and (Rin . . . Ri1 (T )m
/ µ)T ψα

: m+n =l

(13.166)

in (13.164). These are the terms proportional to T ψα in (13.162) and (13.164). The other terms in these two expressions are proportional to (d/ ψ)$ . Besides containing at least one factor of κ, which makes the estimates much easier in the region of small µ, these terms have a decay factor of (1 + t)−2 [1 + log(1 + t)]2 relative to the leading terms (13.165) and (13.166). The terms in the expression (13.160) are also proportional to (d/ ψ)$ , and, besides containing the factor µ which makes the estimates much easier in the region of small µ, these terms have a decay factor of (1 + t)−1 [1 + log(1 + t)] relative to (13.165) (in the case m = 0) and (13.166) (in the case m ≥ 1). Consequently, we may restrict attention to the estimation of the contribution of the leading terms (13.165) and (13.166) to the error integrals. We recall from Chapter 7 that the error integrals corresponding to an nth order variation ψn are the integrals: ρ˜n (K 0 ψn )dt  du  dµh/ (13.167) − Wut

associated to the vectorfield K 0 , and ρ˜n (K 1 ψn + ωψn )dt  du  dµh/ −

(13.168)

Wut

(see (7.27), (7.28)). Since : K 0 ψn = Lψn + (η0−1 + α −1 κ)Lψn

(13.169)

766

Chapter 13. Error Estimates

while K 1 ψn + ωψn = (ω/ν)(Lψn + νψn )

(13.170)

(see equations (5.15), (5.16) of Chapter 5), the coefficient of the term Lψn in (13.169) has a decay factor of (1 + t)−2 [1 + log(1 + t)] relative to the coefficient of the same term in (13.170). Therefore it suffices to bound the error integral (13.168) and the contribution of the term Lψn in (13.169) to the error integral (13.167). This contribution is bounded in absolute value by: Wεt

|ρ˜n ||Lψn |dt  dudµh/

(13.171)

0

We are thus to estimate the contributions of the leading terms (13.165) and (13.166) to the corresponding integrals (13.171) and (13.168). We begin with the contribution of (13.165) to the corresponding integral (13.171). Recalling that this is associated to the variation (13.161), the contribution in question is: |Ril+1 . . . Ri1 trχ  ||T ψα ||L Ril+1 . . . Ri1 ψα |dt  dudµh/ (13.172) Wεt

0



≤C

t

  sup µ−1 |T ψα | µRil+1 . . . Ri1 trχ   L 2 ( ε0 ) L Ril+1 . . . Ri1 ψα  L 2 ( ε0 ) dt  t

0  ε0 t

t

Now, in view of the definitions (8.60), (8.67), (8.369), of Chapter 8, we have: µRil+1 . . . Ri1 trχ   L 2 ( ε0 ) ≤ C(1 + t)µd/ Ril . . . Ri1 trχ   L 2 ( ε0 ) t t

 2 (i1 ...il ) (i1 ...il ) ˇ fl (t)| L 2 ([0,ε0 ]×S 2 ) ≤ C(1 + t) | xl (t)| L 2 ([0,ε0]×S 2 ) + |d/

 = C (1 + t)2 | (i1 ...il ) xl (t)| L 2 ([0,ε0]×S 2 ) + (i1 ...il ) Pl (t) (13.173) Substituting the estimate (8.437) we then obtain: µRil+1 . . . Ri1 trχ   L 2 ( ε0 ) t  2 ≤ C(1 + t) (1 + t)−2

(i1 ...il )

Pl (t) +

+ Cl δ0 (1 + t)−1 [1 + log(1 + t)]2

(i1 ...il )



t

Bl (t)

(13.174)



(1 + t  )[1 + log(1 + t  )]Bl (t  )dt 

0

Here, the quantity (i1 ...il )

(i1 ...il ) B (t) l

Bl (t) = C(1 + t)

is defined by equation (8.427): 

−2

(i1 ...il )

(0)

P l,a (t) + (1 + t)

+C(1 + t)−3 [1 + log(1 + t)]2



t

(1 + t  )3

0 2 (i1 ...il )

+C(1 + t)−3 [1 + log(1 + t)] 

−1/2 (i1 ...il )

(i1 ...il )

(1)

P l,a (t) µ−a m (t)

Q l (t  )dt 

xl (0) L 2 ( ε0 ) 0



(13.175)

Part 2: The error estimates of the acoustical entities

Also (see (8.429))

767

(i1 ...il )

Bl (t) = max i1 ...il

(i1 ...il )

The quantities

(0)

P l,a ,

(i1 ...il )

Bl (t)

(1)

(i1 ...il )

P l,a are defined by equations (8.371), (8.372):

(0)

P l,a (t) = sup {µam (t  )

(i1 ...il )

t  ∈[0,t ]

(i1 ...il )

(1)

P l,a (t) = sup {(1 + t  )1/2 µam (t  )

(0) Pl

(i1 ...il )

t  ∈[0,t ]

(0)

(13.176)

(t  )}

(1) Pl

(13.177)

(t  )}

(13.178)

(1)

The quantities (i1 ...il ) P l , (i1 ...il ) P l are defined in the statement of Proposition 10.6 and their sum bounds the quantity (i1 ...il ) Pl (see also (8.370)): (i1 ...il )

Pl (t) ≤

(i1 ...il )

(0) Pl

(0)

(t) +

(i1 ...il )

(1) Pl

(t)

(13.179)

(1)

Note that the quantities (i1 ...il ) P l,a , (i1 ...il ) P l,a are non-decreasing. The leading contribution to the integral on the right in (13.172) comes from the first term on the right in (13.174) in which the first term on the right in (13.175) is substituted and the bound (see (8.373)):   (i1 ...il )

Pl (t) ≤ µ−a m (t)

is substituted for  C

(i1 ...il ) P (t), l

(i1 ...il )

(i1 ...il )

(0)

P l,a (t) + (1 + t)−1/2

(i1 ...il )

(1)

P l,a (t)

namely from:

(0)

P l,a (t) + (1 + t)

−1/2 (i1 ...il )

(1)



P l,a (t) µ−a m (t)

The actual borderline contribution is the contribution from: ) (0) (i1 ...il ) E0 [R j Ril . . . Ri1 ψα ](t) P l (t) = Ck 3 ||

(13.180)

(13.181)

j,α

To estimate this borderline contribution, we shall make use of the assumption J below. Let us recall the scaling operator S (see (1.93) of Chapter 1) and the rotation operators ◦

Ri , i = 1, 2, 3 (see (6.9) of Chapter 6), of the background Minkowskian metric: S = xα

∂ − 1, ∂xα

◦

Ri =

∂ 1 i j k  j k = i j k x j k : i = 1, 2, 3 2 ∂x

(13.182)

(rectangular coordinates). Let us also recall the original wave function φ. Then our assumption is that there is a positive constant C independent of s such that in Wεs0 ,

768

Chapter 13. Error Estimates

J:

◦

|Sφ|, |T Sφ| ≤ Cδ0 (1 + t)−1 , : i = 1, 2, 3

◦

| Ri φ|, |T Ri φ| ≤ Cδ0 (1 + t)−1 ,

We shall establish this assumption in the sequel, on the basis of the bootstrap assumption. We shall presently use this assumption to derive a pointwise estimate for the T ψα in terms of Lµ. Let us consider the t -tangential vectorfield: V =

3  ∂ (T ψi ) i ∂x

(13.183)

i=1

(rectangular coordinates). Recalling from Chapter 6 the Euclidean outward unit normal N to the Euclidean coordinate spheres: 3  xi ∂ N= r ∂xi

(13.184)

i=1

(rectangular coordinates) we decompose V into its components which are tangential and orthogonal, relative to the induced Euclidean metric on t , to the Euclidean coordinate spheres: V = V  + V ⊥,

V ⊥ = V, NN,

V 2 = V  2 + V ⊥ 2

(13.185)

where, as in Chapter 6,  ,  denotes the Euclidean pointwise inner product, and   the Euclidean pointwise magnitude. Consider the vector (outer) Euclidean product N × V . We have: (13.186) N × V 2 = V  2 The i th rectangular component of N × V is given by: r (N × V )i =

3 

 i j k x j T

j,k=1

=

3  j,k=1

∂φ ∂xk



   j ∂φ j ∂φ i j k T x − (T x ) k ∂xk ∂x

(13.187)

Noting that T x j = T j , the j th rectangular component of the t -tangential vectorfield T , we obtain, in view of the second equation of (13.182), ◦

r (N × V )i = T Ri φ −

3 

i j k T j ψk

(13.188)

j,k=1

By virtue of assumption J it then follows that: r N × V  ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] : in Wεs0

(13.189)

Part 2: The error estimates of the acoustical entities

769

where we have also taken into account the fact that (see (12.6) of Chapter 12): |T i | = κ|Tˆ i | ≤ κ ≤ C[1 + log(1 + t)] ε

In view of the lower bound (12.43) of Chapter 12 for r on t 0 , and (13.186), we then obtain: sup V   ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] : for all t ∈ [0, s] ε

(13.190)

t 0

By (13.183)–(13.185) and (13.190) it follows that:     xi sup T ψi − V, N ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] : for all t ∈ [0, s] ε r  0

(13.191)

t

Consider next: Sφ = x α

 ∂φ − φ = tψ0 + x i ψi − φ α ∂x 3

(13.192)

i=1

We have: T Sφ = t T ψ0 + Since

T xi

=

Ti

and T φ =

3 

x i T ψi +

i=1

3

i i=1 T ψi ,

T Sφ = t T ψ0 +

3 

ψi T x i − T φ

i=1

the last two terms cancel and we obtain:

3 

x i T ψi = t T ψ0 + r V, N

(13.193)

i=1

in view of the fact that by (13.183)–(13.184): V, N =

3  xi T ψi r

(13.194)

i=1

By virtue of assumption J it then follows that:     t sup V, N + T ψ0  ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] : for all t ∈ [0, s] ε r  0

(13.195)

t

Combining (13.191) and (13.195) we conclude that:     txi  sup T ψi + 2 T ψ0  ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] : for all t ∈ [0, s] ε r  0

(13.196)

t

Let us revisit at this point the beginning of the proof of Proposition 8.5. We have: Lµ = m + eµ

770

Chapter 13. Error Estimates

and, according to (8.222): m = m0 + m1,

m 0 = k 3 T ψ0

while according to the estimates (8.227) and (8.234): |m 1 | ≤ Cδ0 (1 + t)−2 ,

|e| ≤ Cδ0 (1 + t)−2

It follows that: sup |Lµ − k 3 T ψ0 | ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] : for all t ∈ [0, s] ε

(13.197)

t 0

In view of (13.196), the lower bound (12.43) for r , and (13.197), we have, pointwise in Wεs0 :   max µ−1 |T ψα | ≤ Cµ−1 |T ψ0 | + Cµ−1 δ0 (1 + t)−2 [1 + log(1 + t)] α  C −1 µ |Lµ| + Cµ−1 δ0 (1 + t)−2 [1 + log(1 + t)] ≤ 3 k || ε

Taking the supremum on t 0 we then obtain the desired estimate:      C −2 max sup µ−1 |T ψα | ≤ 3 δ (1 + t) [1 + log(1 + t)] sup µ−1 |Lµ| + Cµ−1 m 0 α ε k ||  ε0  0 t

t

(13.198) We shall appeal to this estimate in estimating the borderline contribution (13.181), through (13.177), (13.180) and (13.174), to the integral on the right in (13.172). On the other hand, in estimating all other contributions, through (13.174), to the same integral, we simply use the estimate:   −1 max sup µ−1 |T ψα | ≤ Cµ−1 (13.199) m δ0 (1 + t) α

ε

t 0

implied by the bootstrap assumption E{1} . Let us define, for non-negative real numbers a and p, the quantities:      (i1 ...il )  G0,l+2;a, p (t) = sup [1 + log(1 + t  )]−2 p µ2a E0 [R j Ril . . . Ri1 ψα ](t  ) m (t )  t  ∈[0,t ]  j,α

(13.200) These quantities are non-decreasing functions of t and we have: )    p E0 [R j Ril . . . Ri1 ψα ](t  ) ≤ µ−a (t )[1 + log(1 + t )] m

(i1 ...il ) G 0,l+2;a, p (t)

j,α

: for all t  ∈ [0, t]

(13.201)

Part 2: The error estimates of the acoustical entities

771

hence, in view of the definition (13.177), the borderline contribution (13.181) to (13.180) is bounded by:  p (i1 ...il ) G (t)[1 + log(1 + t)] (13.202) Ck 3 ||µ−a 0,l+2;a, p (t) m Also, in regard to the last factor in the integral on the right in (13.172) we have:  (13.203) L Ril+1 . . . Ri1 ψα  L 2 ( ε0 ) ≤ E0 [Ril+1 . . . Ri1 ψα ](t) t  p (i1 ...il ) G ≤ µ−a 0,l+2;a, p (t) m (t)[1 + log(1 + t)] Substituting the estimates (13.198), (13.202) and (13.203) in the integral on the right in (13.172) the factors k 3 || cancel and we obtain that the borderline contribution to the integral in question is bounded by: t   −1 −1  −2  C (13.204) sup µ |Lµ| + Cµm δ0 (1 + t ) [1 + log(1 + t )] · ε

0

t 0

  2p µ−2a m (t )[1 + log(1 + t )]

(i1 ...il )

G0,l+2;a, p (t  )dt 

Now, the partial contribution of the second term in the first factor in the integrant is actually not borderline. We shall show how to estimate contributions of this type afterwards, in connection with the estimate for the contribution of the term ) E1 [R j Ril . . . Ri1 ψα ] (13.205) C(1 + t)−1 [1 + log(1 + t)]1/2 j,α (1)

in the expression for (i1 ...il ) P l of Proposition 10.6, through (13.178), (13.180) and (13.174), to the integral on the right in (13.172). For the present we focus attention on the borderline integral: t     2 p (i1 ...il ) sup µ−1 |Lµ| µ−2a G0,l+2;a, p (t  )dt  (13.206) C m (t )[1 + log(1 + t )] 0  ε0 t

Since |Lµ| = max{−(Lµ)− , (Lµ)+ } ≤ −(Lµ)− + (Lµ)+ we have:

      sup µ−1 |Lµ| ≤ sup −µ−1 (Lµ)− + sup µ−1 (Lµ)+ ε

t 0

ε

t 0

ε

(13.207)

t 0

Substituting (13.207) in (13.206), let us consider first the contribution of the first term on the right in (13.207). According to definition (8.273) of Chapter 8:   (13.208) sup −µ−1 (Lµ)− = M(t) ε

t 0

772

Chapter 13. Error Estimates

therefore the contribution in question is: t   2p C M(t  )µ−2a m (t )[1 + log(1 + t )]

(i1 ...il )

G0,l+2;a, p (t  )dt 

0

= C[1 + log(1 + t)]2 p

(i1 ...il )

G0,l+2;a, p (t)I2a (t)

(13.209)

Here, I2a (t) is the integral of Lemma 8.11, and we have used the fact that [1+log(1+t)]2 p and (i1 ...il ) G0,l+2;a, p (t) are non-decreasing functions of t. Now, according to Lemma 8.11 we have: (13.210) I2a (t) ≤ C(2a)−1 µ−2a m (t) where C is a constant independent of a, provided that a ≥ 2 and δ0 is suitably small depending on a. We conclude that (13.209) is bounded by: C −2a µ (t)[1 + log(1 + t)]2 p 2a m

(i1 ...il )

G0,l+2;a, p (t)

(13.211)

Consider next the contribution of the second term on the right in (13.207). Proposition 13.1 implies:   sup µ−1 (Lµ)+ ≤ C(1 + t)−1 [1 + log(1 + t)]−1 (13.212) ε

t 0

therefore the contribution in question is bounded by: t   −1  2 p−1 (i1 ...il ) µ−2a G0,l+2;a, p (t  )dt  C m (t )(1 + t ) [1 + log(1 + t )]

(13.213)

0

Now, by Corollary 2 of Lemma 8.11:  −2a µ−2a m (t ) ≤ Cµm (t)

(13.214)

where C is a constant independent of a, provided that a ≥ 2 and δ0 is suitably small depending on a. Taking also into account the fact that (i1 ...il ) G0,l+2;a, p (t) is a nondecreasing function of t, it follows that (13.213) is bounded by: t (i1 ...il ) Cµ−2a (t) G (t) (1 + t  )−1 [1 + log(1 + t  )]2 p−1 dt  (13.215) 0,l+2;a, p m 0

Since

t

(1 + t  )−1 [1 + log(1 + t  )]2 p−1 dt  =

0

 1  [1 + log(1 + t)]2 p − 1 2p

this is bounded by: C −2a µ (t)[1 + log(1 + t)]2 p 2p m

(i1 ...il )

G0,l+2;a, p (t)

(13.216)

Part 2: The error estimates of the acoustical entities

773

We have thus estimated the borderline contributions to the integral on the right in (13.172). We proceed to estimate the remaining contributions. Let us define: E0,[n] (t) =

n 

E0,m (t)

(13.217)

m=1

where E0,n represents the sum of the energies associated to the vectorfield K 0 of all the nth order variations (13.104), (13.106) of the form specified in the paragraph just following (13.106). We then define: 

  G0,[n];a, p (t) = sup [1 + log(1 + t  )]−2 p µ2a (t )E (t ) (13.218) 0,[n] m t  ∈[0,t ]

Let us also define:  E1,[n] (t) =

n 

 E1,m (t)

(13.219)

m=1  represents the sum of the energies associated to the vectorfield K of all the nth where E1,n 1 order variations (13.104), (13.106) of the form specified in the paragraph just following (13.106). We then define, for q ≥ p: 

    G1,[n];a,q (t) = sup [1 + log(1 + t  )]−2q µ2a (13.220) m (t )E1,[n] (t ) t  ∈[0,t ]

We now consider the contribution of the term (13.205) in the estimate for (i1 ...il ) Pl of Proposition 10.6, through (13.178) and (13.174), to the integral on the right in (13.172). From the definitions (13.219), (13.220), we have: )   E1 [R j Ril . . . Ri1 ψα ](t  ) ≤ E1,[l+2] (t  ) j,α

   q  (t )[1 + log(1 + t )] G1,[l+2];a,q (t) ≤ µ−a m : for all t  ∈ [0, t]

(13.221)

hence, in view of the definition (13.178), the contribution of (13.205) to (13.180) is bounded by:   Cq (1 + t)−1/2 µ−a (13.222) m (t) G1,[l+2];a,q (t) where Cq is a constant depending on q: Cq = C sup {(1 + t)−1/2 [1 + log(1 + t)]q+1/2 } t ∈[0,∞]

Also, by (13.203) and the definitions (13.217), (13.218) we have:  p L Ril+1 . . . Ri1 ψα  L 2 ( ε0 ) ≤ µ−a G0,[l+2];a, p (t) m (t)[1 + log(1 + t)] t

(13.223)

774

Chapter 13. Error Estimates

Using also the estimate (13.199) we then conclude that the contribution of the term (13.205) through (13.174) to the integral on the right in (13.172) is bounded by: t    C q δ0 (1 + t  )−3/2 [1 + log(1 + t  )] p µ−2a−1 (t ) G0,[l+2];a, p (t  )G1,[l+2];a,q (t  )dt  m 0

(13.224) To estimate this we consider the two cases occurring in the proof of Lemma 8.11: Case 1) Eˆ s,m ≥ 0,

Case 2) Eˆ s,m < 0

In Case 1 we have the lower bound (8.281):

which implies:

µm (t) ≥ 1 − Cδ0 : for all t ∈ [0, s]

(13.225)

µ−2a−1 (t) ≤ C : for all t ∈ [0, s] m

(13.226)

provided that δ0 a is suitably small (see (8.283)). It follows that in Case 1, (13.224) is bounded by: t   (1 + t  )−3/2 [1 + log(1 + t  )] p G0,[l+2];a, p (t  )G1,[l+2];a,q (t  )dt  (13.227) C q δ0 0

In Case 2 we set, as in the proof of Lemma 8.11: Eˆ s,m = −δ1 , δ1 > 0

(13.228)

Following the proof of Lemma 8.11 with a replaced by 2a, we then set: 1

t1 = e 4aδ1 − 1

(13.229)

and we consider the two subcases: Subcase 2a) t  ≤ t1 ,

Subcase 2b) t  > t1

In Subcase 2a we have the lower bound (8.296) with a replaced by 2a, that is: µm (t  ) ≥ 1 −

1 a

(13.230)

Hence, noting that (1 − 1/a)−2a−1 is bounded for a ∈ [2, ∞), if t ≤ t1 (13.224) is bounded by: t   (1 + t  )−3/2 [1 + log(1 + t  )] p G0,[l+2];a, p (t  )G1,[l+2];a,q (t  )dt  (13.231) C q δ0 0

and if t > t1 we have: t1    (1 + t  )−3/2 [1 + log(1 + t  )] p µ−2a−1 (t ) G0,[l+2];a, p (t  )G1,[l+2];a,q (t  )dt  m 0 t1   ≤C (1 + t  )−3/2 [1 + log(1 + t  )] p G0,[l+2];a, p (t  )G1,[l+2];a,q (t  )dt  0

(13.232)

Part 2: The error estimates of the acoustical entities

775

In Subcase 2b we have the lower bound (8.326) with a replaced by 2a:   1 µm (t  ) ≥ 1 − (1 − δ1 τ  ), τ  = log(1 + t  ) a

(13.233)

 In view of the fact that the G0,[n];a, p (t), G1,[n];a,q (t) are non-decreasing functions of t it follows that: t    (1 + t  )−3/2 [1 + log(1 + t  )] p µ−2a−1 (t ) G0,[l+2];a, p (t  )G1,[l+2];a,q (t  )dt  m t1

  ≤ C G0,[l+2];a, p (t)G1,[l+2];a,q (t) · t (1 + t  )−3/2 [1 + log(1 + t  )] p (1 − δ1 τ  )−2a−1 dt 

(13.234)

t1

The integral in (13.234) is: t (1 + t  )−3/2 [1 + log(1 + t  )] p (1 − δ1 τ  )−2a−1 dt  t1 τ ≤ (1 + t1 )−1/2 [1 + log(1 + t1 )] p (1 − δ1 τ  )−2a−1 dτ  τ1

≤ (1 + t1 )−1/2 [1 + log(1 + t1 )] p

1 (1 − δ1 τ )−2a 2aδ1

≤ 2ϕ1+ p (4aδ1)(1 − δ1 τ )−2a

(13.235)

Here for any real number r we denote by ϕr (x) the following function on the positive real line:   1 1 r (13.236) ϕr (x) = e− 2x 1 + x The function ϕr (x) decreases to 0 exponentially as x → 0. Since δ1 ≤ Cδ0 , we have: ϕr (4aδ1) ≤ ϕr (Caδ0 )

(13.237)

provided that Caδ0 is suitably small. In view also of the lower bound (8.337) with a replaced by 2a: 1 (1 − δ1 τ )−2a µ−2a (13.238) m (t) ≥ C we conclude that the right-hand side of (13.234) is bounded by:   Cϕ1+ p (Caδ0 )µ−2a (13.239) m (t) G0,[l+2];a, p (t)G1,[l+2];a,q (t) Combining with (13.232) we obtain that if t > t1 (13.224) is bounded by:   −2a  (t) (13.240) Cq δ0 µm (t) ϕ1+ p (Caδ0 ) G0,[l+2];a, p (t)G1,[l+2];a,q t   −3/2  p     + (1 + t ) [1 + log(1 + t )] G0,[l+2];a, p (t )G1,[l+2];a,q (t )dt 0

776

Chapter 13. Error Estimates

Combining finally with the earlier results (13.231), for the subcase t ≤ t1 , and (13.227), for Case 1, we conclude that the contribution of the term (13.205) through (13.174) to the integral on the right in (13.172) is bounded by:    Cq δ0 µ−2a (t) ϕ (Caδ ) G0,[l+2];a, p (t)G1,[l+2];a,q (t) (13.241) 1+ p 0 m t   + (1 + t  )−3/2 [1 + log(1 + t  )] p G0,[l+2];a, p (t  )G1,[l+2];a,q (t  )dt  0

for

Let us now consider the contribution of the second term in the expression (13.175) to the first term on the right in (13.174), namely the contribution of: t −1 2 (1 + t  )3 (i1 ...il ) Q l (t  )dt  (13.242) C(1 + t) [1 + log(1 + t)]

(i1 ...il ) B (t) l

0

to the estimate for µRil+1 . . . Ri1

trχ  

L 2 (t 0 ) . ε

Here, we recall the estimate for maxi1 ...il (i1 ...il ) Q l of Proposition 10.5. The principal part of the right-hand side is:   Q QQ T + W[l+2] + δ02 (1 + t)−2 W[l] Cl (1 + t)−4 [1 + log(1 + t)] W[l+1] + δ0 W[l+1] (13.243) We shall estimate below the contribution of this principal part, through (13.242) and (13.174), to the integral on the right in (13.172). We have, for each non-negative integer n: )  d/ Rin . . . Ri1 ψα 2 2 ε0 Wn+1 ≤ C(1 + t) L (t )

i1 ...in ,α

−1/2

≤ Cµm

  E1,n+1

It follows that:

−1/2

W[l+2] ≤ Cl µm

(13.244) 

 E1,[l+2] + W0

(13.245)

and by inequality (5.72) of Chapter 5: √ W0 ≤ ε0 max sup ψα  L 2 (St,u ) α

≤ Cε0

u∈[0,ε0 ]

)

E0 [ψα ] = Cε0 E0,[1]

(13.246)

α

Similarly, for each non-negative integer n: )  T d/ Rin . . . Ri1 T ψα 2 2 Wn+1 ≤ C(1 + t) −1/2

≤ Cµm



i1 ...in ,α  E1,n+2

ε

L (t 0 )

(13.247)

Part 2: The error estimates of the acoustical entities

It follows that:

−1/2

T W[l+1] ≤ C l µm

777



 E1,[l+2] + W0T

(13.248)

and by inequality (5.72): √ ε0 max sup T ψα  L 2 (St,u )

W0T ≤

α

≤ Cε0

u∈[0,ε0 ]

)

E0 [T ψα ] ≤ Cε0 E0,[2]

(13.249)

α

Also, for each non-negative integer n: Q Wn+1 ≤ C(1 + t) −1/2

≤ Cµm

) 



d/ Rin . . . Ri1 Qψα 2 2

ε

L (t 0 )

i1 ...in ,α  E1,n+2

(13.250)

It follows that: −1/2

Q

W[l+1] ≤ Cl µm



 E1,[l+2] + W0

Q

(13.251)

and by inequality (5.72): √ ε0 max sup Qψα  L 2 (St,u )

W0Q ≤

α

≤ Cε0

u∈[0,ε0 ]

)

E0 [Qψα ] ≤ Cε0 E0,[2]

(13.252)

α

Similarly, for any non-negative integer n: QQ Wn+1 ≤ C(1 + t)

) 

d/ Rin . . . Ri1 (Q)2 ψα 2 2

−1/2

≤ Cµm

  E1,n+3

It follows that: −1/2

QQ

W[l] ≤ Cl µm

ε

L (t 0 )

i1 ...in ,α

(13.253)  QQ  E1,[l+2] + W0

(13.254)

and by inequality (5.72): W0Q Q ≤

√ ε0 max sup (Q)2 ψα  L 2 (St,u )

≤ Cε0

α

u∈[0,ε0 ]

) α

E0 [(Q)2 ψα ] ≤ Cε0 E0,[3]

(13.255)

778

Chapter 13. Error Estimates

In view of (13.245), (13.246), (13.258), (13.249), (13.251), (13.252), (13.254) and (13.255), (13.243) is bounded by:  

−1/2  E1,[l+2] + Cε0 E0,[3] Cl (1 + t)−4 [1 + log(1 + t)] µm

(13.256)

therefore the contribution of (13.243) to (13.242) is bounded by: Cl (1 + t)−1 [1 + log(1 + t)]2 (13.257) t 



−1/2  × (1 + t  )−1 [1 + log(1 + t  )] µm (t  ) E1,[l+2] (t  ) + Cε0 E0,[3] (t  ) dt  0

We shall estimate below the partial contribution  of the principal term in the last factor in −1/2   the above integrant, namely the term µm (t ) E1,[l+2] (t  ). From the definition (13.220) we have:   −1/2 −a−1/2    µm (t  ) E1,[l+2] (t  ) ≤ µm (t )[1 + log(1 + t  )]q G1,[l+2];a,q (t) (13.258) Therefore the contribution in question is bounded by:   Cl (1 + t)−1 [1 + log(1 + t)]2 Ja,q (t) G1,[l+2];a,q (t)

(13.259)

where:

t

Ja,q (t) =

−a−1/2

(1 + t  )−1 µm

(t  )[1 + log(1 + t  )]q+1 dt 

(13.260)

0

To estimate Ja,q (t), we consider again the two cases occurring in the proof of Lemma 8.11, as in the estimation of (13.224). In Case 1 we have the lower bound (13.225) hence also (13.226), which implies:

t

Ja,q (t) ≤ C

 −1

(1 + t )



[1 + log(1 + t )]

q+1





τ

dt = C

0

(1 + τ  )q+1 dτ 

0

C C (1 + τ )q+2 = [1 + log(1 + t)]q+2 ≤ (q + 2) (q + 2)

(13.261)

Similarly, in Subcase 2a, we obtain if t ≤ t1 : Ja,q (t) ≤

C [1 + log(1 + t)]q+2 (q + 2)

(13.262)

and if t > t1 : Ja,q (t1 ) ≤

C C [1 + log(1 + t)]q+2 < [1 + log(1 + t)]q+2 (q + 2) (q + 2)

(13.263)

Part 2: The error estimates of the acoustical entities

779

In Subcase 2b we have the lower bound (13.233) which implies (recalling that a ≥ 1): t Ja,q (t) − Ja,q (t1 ) ≤ C[1 + log(1 + t)]q+1 (1 − δ1 τ  )−a−1/2 (1 + t  )−1 dt  t1 τ = C(1 + τ )q+1 (1 − δ1 τ  )−a−1/2 dτ  τ1

(1 + τ )q+1

≤

C   (1 − δ1 τ )−a+1/2 δ1 a − 1 2

≤

C −a+1/2 (1 + τ )q+1 µm (t) aδ1

(13.264)

where in the last step we have used the upper bound (8.335) (with a replaced by 2a), which implies: 1 −a+1/2 µm (t) ≥ (1 − δ1 τ )−a+1/2 C (see (8.337)). Since τ ≥ τ1 = 1/4aδ1, (13.264) implies: −a+1/2

Ja,q (t) − Ja,q (t1 ) ≤ C[1 + log(1 + t)]q+2 µm

(t)

(13.265)

In view of (13.261), (13.262), (13.263) and (13.265), we conclude that, in general: −a+1/2

Ja,q (t) ≤ C[1 + log(1 + t)]q+2 µm hence (13.259) is bounded by: −a+1/2

Cl (1 + t)−1 [1 + log(1 + t)]q+4 µm

(t)

  (t) G1,[l+2];a,q (t)

(13.266)

(13.267)

This bounds the contribution under consideration, through (13.242), to the estimate for µRil+1 . . . Ri1 trχ   L 2 ( ε0 ) . In view of (13.223) and the estimate (13.199), the corret sponding contribution to the integral on the right in (13.172) is then bounded by: t  −2a−1/2   C l δ0 (1 + t  )−2 [1 + log(1 + t  )] p+q+4 µm (t ) G0,[l+2];a, p (t  )G1,[l+2];a,q (t  )dt  0

(13.268) This is estimated by following an argument similar to that used to estimate (13.224). We obtain in this way a bound by: 

−2a+1/2   Cl δ0 µm (t)[1 + log(1 + t)]2 p ϕ5+q− p (Caδ0 ) G0,[l+2];a, p (t)G1,[l+2];a,q (t) t   + (1 + t  )−2 [1 + log(1 + t  )]4+q− p G0,[l+2];a, p (t  )G1,[l+2];a,q (t  )dt  0

(13.269) where for any real number r we denote by ϕr (x) the following function on the positive real line:   1 r  − 1x ϕr (x) = e (13.270) 1+ x

780

Chapter 13. Error Estimates

Note that the function ϕr (x) decreases to 0 exponentially as x → 0, and that, since δ1 ≤ Cδ0 , ϕr (4aδ1) ≤ ϕr (Caδ0 ) provided that Caδ0 is suitably small. Let us finally consider the contribution of the last term in the expression (13.175) for (i1 ...il ) Bl (t), the initial data term, to the first term on the right in (13.174), namely the contribution of: C(1 + t)−1 [1 + log(1 + t)]2 

(i1 ...il )

xl (0) L 2 ( ε0 )

(13.271)

0

to the estimate for µRil+1 . . . Ri1 trχ   L 2 ( ε0 ) . In view of (13.223) and the estimate t (13.199), the corresponding contribution to the integral on the right in (13.172) is bounded by: t   (1 + t  )−2 [1 + log(1 + t  )] p+2 µ−a−1 (t ) G0,[l+2];a, p (t  )dt  Cδ0  (i1 ...il ) xl (0) L 2 ( ε0 ) m 0

0

(13.272) This is again estimated by following an argument similar to that used to estimate (13.224). We obtain a bound by: 

 (t) ϕ (Caδ ) G0,[l+2];a, p (t) C p δ0  (i1 ...il ) xl (0) L 2 ( ε0 ) µ−a 0 m 3+ p 0 t   −2  2+ p   + (1 + t ) [1 + log(1 + t )] (13.273) G0,[l+2];a, p (t )dt 0

In regard to the contribution of the last term on the right in (13.174), we remark that it is bounded by: Cl δ0 (1 + t)−1 [1 + log(1 + t)]4 sup {(1 + t  )2 µam (t  )Bl (t  )}µ−a m (t)

(13.274)

t  ∈[0,t ]

Thus, relative to the term (1 + t)2 (i1 ...il ) Bl there is here an extra factor of Cl δ0 (1 + t)−1 [1 + log(1 + t)]4 , consequently the corresponding contribution is absorbed in the estimates already made. We now consider the contribution of (13.166) to the corresponding integral (13.171). Recalling that this is associated to the variation (13.163), the contribution in question is: |Ril−m . . . Ri1 (T )m
/ µ||T ψα ||L Ril−m . . . Ri1 (T )m+1 ψα |dt  dudµh/ Wεt

0

≤C



t

  sup µ−1 |T ψα |

0  ε0 t

× µRil−m . . . Ri1 (T )m
/ µ L 2 ( ε0 ) L Ril−m . . . Ri1 (T )m+1 ψα  L 2 ( ε0 ) dt  t

: m = 0, . . . , l

t

(13.275)

Part 2: The error estimates of the acoustical entities

781

Now, in view of the definitions (9.84), (9.94), (9.241), of Chapter 9, we have: µRil−m . . . Ri1 (T )m
/ µ L 2 ( ε0 ) t 

(i1 ...il−m )   ≤ C(1 + t) | x m,l−m (t)| L 2 ([0,ε0 ]×S 2 ) + | (i1 ...il−m ) fˇm,l−m (t)| L 2 ([0,ε0 ]×S 2 ) 

  (t)| L 2 ([0,ε0 ]×S 2 ) + (i1 ...il−m ) Pm,l−m (t) (13.276) = C (1 + t)| (i1 ...il−m ) x m,l−m Substituting the estimate (9.299) we then obtain: µRil−m . . . Ri1 (T )m
/ µ L 2 ( ε0 ) t   −1 (i1 ...il−m )   ≤ C(1 + t) (1 + t) Pm,l−m (t) + (i1 ...il−m ) Bm,l−m (t)  t + Cl δ0 (1 + t)−2 [1 + log(1 + t)]2 (1 + t  )[1 + log(1 + t  )]Bl (t  )dt  0  m t      + [1 + log(1 + t )]Bk,l−k (t )dt k=0 0

: for m = 0, . . . , l Here, the quantities (i1 ...il−m )

(i1 ...il−m ) B  m,l−m (t)

 Bm,l−m (t) 

= C(1 + t)

−1

(i1 ...il−m )

(0) 

P

m,l−m,a

+ C(1 + t)−2 [1 + log(1 + t)]2 + C(1 +

(13.277)

are defined by equation (9.287):

(t) + (1 + t)

−1/2 (i1 ...il−m )

(1) 

P

 m,l−m,a

µ−a m (t)

t

(1 + t  )2 (i1 ...il−m ) Q m,l−m (t  )dt  0  t)−2 [1 + log(1 + t)]2  (i1 ...il−m ) x m,l−m (0) L 2 ( ε0 ) 0

(13.278)

Also (see (9.289))  Bm,l−m (t) = max

(i1 ...il−m )

(0) 

(1) 

i1 ...il−m

The quantities (9.244):

(i1 ...il−m )

(i1 ...il−m )

(i1 ...il−m )

(1) 

P

(0) 

P

P

m,l−m,a ,

m,l−m,a

m,l−m,a

(i1 ...il−m )

P

m,l−m,a

(t) = sup {µam (t  )

(t) = sup {(1 + t  )1/2 µam (t  )

(13.279)

are defined by equations (9.243),

(i1 ...il−m )

t  ∈[0,t ]

t  ∈[0,t ]

 Bm,l−m (t)

(0) 

P

m,l−m

(i1 ...il−m )

(1) 

P

(t  )}

m,l−m

(13.280)

(t  )}

(13.281)

782

Chapter 13. Error Estimates (0)

(1)

The quantities (i1 ...il−m ) P  m,l−m , (i1 ...il−m ) P  m,l−m are defined in the statement of  (see also (9.242)): Proposition 11.6 and their sum bounds the quantity (i1 ...il−m ) Pm,l−m (i1 ...il−m )

 Pm,l−m (t) ≤

(i1 ...il−m )

(0) 

P

m,l−m

(t) +

(0) 

(i1 ...il−m )

(1) 

P

m,l−m

(t)

(13.282)

(1) 

Note that the quantities (i1 ...il−m ) P m,l−m,a , (i1 ...il−m ) P m,l−m,a are non-decreasing. The leading contribution to the integral on the right in (13.275) comes from the first term on the right in (13.277) in which the first term on the right in (13.278) is substituted and the bound (see (9.245)): (i1 ...il−m )

≤

 Pm,l−m (t  ) 

 µ−a m (t )

is substituted for  C

(i1 ...il−m )

(i1 ...il−m )

(0) 

P

m,l−m,a

(i1 ...il−m ) P  m,l−m , (0) 

P

m,l−m,a

 −1/2 (i1 ...il−m )

(t) + (1 + t )

(1) 

P

 (t)

m,l−m,a

namely from:

(t) + (1 + t)

−1/2 (i1 ...il−m )

(1) 

P

 m,l−m,a

(t) µ−a m (t)

The actual borderline contribution is the contribution from: ) (0)  (i1 ...il−m )  P m,l−m = Ck 3 || E0 [Ril−m . . . Ri1 (T )m+1 ψα ]

(13.283)

(13.284)

α

We shall appeal to the estimate (13.198) in estimating the borderline contribution (13.284), through (13.280), (13.283) and (13.277), to the integral on the right in (13.275). On the other hand, in estimating all other contributions, through (13.277), to the same integral, we simply use the estimate (13.199). Let us define, for non-negative real numbers a and p, the quantities: (i1 ...il−m )

= sup

G0,m,l+2;a, p (t) 

t  ∈[0,t ]

 [1 + log(1 + t  )]−2 p µ2a m (t )



(13.285)  E0 [Ril−m . . . Ri1 (T )m+1 ψα ](t  )

α

These quantities are non-decreasing functions of t and we have: ) E0 [Ril−m . . . Ri1 (T )m+1 ψα ](t  ) α   p ≤ µ−a m (t )[1 + log(1 + t )]

: for all t  ∈ [0, t]



(i1 ...il−m ) G 0,m,l+2;a, p (t)

(13.286)

Part 2: The error estimates of the acoustical entities

783

hence, in view of the definition (13.280), the borderline contribution (13.284) to (13.283) is bounded by:  p (i1 ...il−m ) G (t)[1 + log(1 + t)] (13.287) Ck 3 ||µ−a 0,m,l+2;a, p (t) m Also, in regard to the last factor in the integral on the right in (13.275) we have:  (13.288) L Ril−m . . . Ri1 (T )m+1 ψα  L 2 ( ε0 ) ≤ E0 [Ril−m . . . Ri1 (T )m+1 ψα ](t) t  p (i1 ...il−m ) G ≤ µ−a 0,m,l+2;a, p (t) m (t)[1 + log(1 + t)] Substituting the estimates (13.198), (13.287) and (13.288) in the integral on the right in (13.275), the factors k 3 || cancel and we obtain that the borderline contribution to the integral in question is bounded by:   t     −2  C δ (1 + t ) [1 + log(1 + t )] sup µ−1 |Lµ| + Cµ−1 · (13.289) 0 m  0   ε0 t

  2p µ−2a m (t )[1 + log(1 + t )]

(i1 ...il−m )

G0,m,l+2;a, p (t  )dt 

Now, the partial contribution of the second term in the first factor in the integrant is actually not borderline. We shall show how to estimate contributions of this type afterwards, in connection with the estimate for the contribution of the terms )  −1 1/2 C(1 + t) [1 + log(1 + t)] E1 [Ril−m . . . Ri1 (T )m+1 ψα ] (13.290) α

+ C(1 + t)−2 [1 + log(1 + t)]3/2

)

E1 [R j Ril−m . . . Ri1 (T )m ψα ]

α, j (1)

in the expression for (i1 ...il−m ) P  m,l−m of Proposition 11.6, through (13.281), (13.283) and (13.277), to the integral on the right in (13.275). For the present we focus attention on the borderline integral: t     2 p (i1 ...il−m ) C sup µ−1 |Lµ| µ−2a G0,m,l+2;a, p (t  )dt  (13.291) m (t )[1 + log(1 + t )] 0  ε0 t

This integral is estimated in exactly the same way as the integral (13.206). We obtain that it is bounded by (see (13.211) and (13.216)):   1 1 2 p (i1 ...il−m ) C + µ−2a G0,m,l+2;a, p (t) (13.292) m (t)[1 + log(1 + t)] 2a 2p We proceed to estimate the remaining contributions to the integral on the right in (13.275). We first consider the contribution of the terms (13.290) in the estimate for

784

Chapter 13. Error Estimates

(i1 ...il−m ) P  m,l−m

of Proposition 11.6, through (13.281) and (13.277), to the integral on the right in (13.275). From the definitions (13.219), (13.220) we have: )  α

)

E1 [Ril−m . . . Ri1 (T )m+1 ψα ](t  ) ≤

E1 [R j Ril−m



 E1,[l+2] (t  )

   q  ≤ µ−a (t )[1 + log(1 + t )] G1,[l+2];a,q (t) m   . . . Ri1 (T )m ψα ](t  ) ≤ E1,[l+2] (t  )

α, j

   q  (t )[1 + log(1 + t )] G1,[l+2];a,q (t) ≤ µ−a m : for all t  ∈ [0, t]

(13.293)

hence, in view of the definition (13.281), the contribution of (13.290) to (13.283) is bounded by:   (13.294) Cq (1 + t)−1/2 µ−a m (t) G1,[l+2];a,q (t) where Cq is a constant depending on q. Also, by (13.288) and the definitions (13.217), (13.218) we have:  p L Ril−m . . . Ri1 (T )m+1 ψα  L 2 ( ε0 ) ≤ µ−a G0,[l+2];a, p (t) (13.295) m (t)[1 + log(1 + t)] t

Using also the estimate (13.199) we then conclude that the contribution of the terms (13.290) through (13.277) to the integral on the right in (13.275) is bounded by:

t

C q δ0 0

   (1 + t  )−3/2 [1 + log(1 + t  )] p µ−2a−1 (t ) G0,[l+2];a, p (t  )G1,[l+2];a,q (t  )dt  m

(13.296) This is identical in form to (13.224) and is therefore bounded by (13.241). Let us next consider the contribution of the second term in the expression (13.278)  for (i1 ...il−m ) Bm,l−m (t) to the first term on the right in (13.277), namely the contribution of t

C(1 + t)−1 [1 + log(1 + t)]2

(1 + t  )2

(i1 ...il−m )

0

Q m,l−m (t  )dt 

(13.297)

/ µ L 2 ( ε0 ) . Here, we recall the estimate for to the estimate for µRil−m . . . Ri1 (T )m
maxi1 ...il−m side is:

(i1 ...il−m ) Q  m,l−m

t

of Proposition 11.5. The principal part of the right-hand

  Q QQ Cl (1 + t)−3 [1 + log(1 + t)] W{l+2} + W{l+1} + W{l}

(13.298)

We shall estimate below the contribution of this principal part, through (13.297) and (13.277), to the integral on the right in (13.275).

Part 2: The error estimates of the acoustical entities

785

For each pair of non-negative integers m, n we have (see (11.10)): )  Wm,n+1 ≤ C(1 + t) d/ Rin . . . Ri1 ψα 2 2 ε0 L (t )

i1 ...in ,α

−1/2

≤ Cµm

  E1,n+m+1

(13.299)

and, for each non-negative integer m we have: ) Wm+1,0 ≤ (T )m+1 ψα 2 2

ε

L (t 0 )

α

≤ C E0,m+1

(13.300)

It follows that: W{l+2} =



Wm,n

n+m≤l+2

=



n+m≤l+1

Wm,n+1 +



Wm+1,0 + W0

m≤l+1

  

−1/2  ≤ C l µm E1,[l+2] + E0,[l+2] + W0

(13.301)

Similarly, we obtain:   

−1/2 Q  ≤ C l µm E1,[l+2] + E0,[l+2] + W0Q W{l+1}   

−1/2 QQ  W{l} ≤ C l µm E1,[l+2] + E0,[l+2] + W0Q Q

(13.302) (13.303)

Therefore, taking also into account (13.246), (13.252), (13.255), we conclude that (13.298) is bounded by:  

−1/2  (13.304) E1,[l+2] + E0,[l+2] Cl (1 + t)−3 [1 + log(1 + t)] µm (as we may assume that l ≥ 1). Consequently, the contribution of (13.298) to (13.297) is bounded by: Cl (1 + t)−1 [1 + log(1 + t)]2 t 



−1/2  × (1 + t  )−1 [1 + log(1 + t  )] µm (t  ) E1,[l+2] (t  ) + E0,[l+2] (t  ) (13.305) 0

By virtue of (13.258) and the fact that, from (13.218): 

  p E0,[l+2] (t  ) ≤ µ−a G0,[l+2];a, p (t) m (t )[1 + log(1 + t )]

(13.306)

786

Chapter 13. Error Estimates

(13.305) is in turn bounded by, recalling the definition (13.260): 

   Cl (1 + t)−1 [1 + log(1 + t)]2 Ja,q (t) G1,[l+2];a,q (t) + Ja−1/2, p (t) G0,[l+2];a, p (t) (13.307) Substituting the estimate (13.266) and the same with (a, q) replaced by (a − 1/2, q) we conclude that the contribution of (13.298), through (13.297) to the estimate for µRil−m . . . Ri1 (T )m
/ µ L 2 ( ε0 ) t

is bounded by:   (t) G1,[l+2];a,q (t)  +Cl (1 + t)−1 [1 + log(1 + t)] p+4 µ−a+1 (t) G0,[l+2];a, p (t) m −a+1/2

Cl (1 + t)−1 [1 + log(1 + t)]q+4 µm

(13.308)

In view of (13.295) and the estimate (13.199), the corresponding contribution to the integral on the right in (13.275) is then bounded by: t  −2a−1/2   (t ) G0,[l+2];a, p (t  )G1,[l+2];a;q (t  )dt  Cl δ0 (1 + t  )−2 [1 + log(1 + t  )] p+q+4 µm 0 t    + Cl δ0 (1 + t  )−2 [1 + log(1 + t  )]2 p+4 µ−2a (13.309) m (t )G0,[l+2];a, p (t )dt 0

The first of the terms in (13.309) coincides with (13.268) and is estimated by (13.269) while the second is estimated in a similar manner by:

Cl δ0 µ−2a+1 [1 + log(1 + t)]2 p ϕ5 (Caδ0 )G0,[l+2];a, p (t) m t  (13.310) + (1 + t  )−2 [1 + log(1 + t  )]4 G0,[l+2];a, p (t  )dt  0

Let us finally consider the contribution of the last term in the expression (13.278)  for (i1 ...il−m ) Bm,l−m (t), the initial data term, to the first term on the right in (13.277), namely the contribution of: C(1 + t)−1 [1 + log(1 + t)]2 

(i1 ...il−m )  x m,l−m (0) L 2 ( ε0 ) 0

(13.311)

/ µ L 2 ( ε0 ) . In view of (13.295) and the estimate to the estimate for µRil−m . . . Ri1 (T )m
t (13.199), the corresponding contribution to the integral on the right in (13.275) is bounded by: Cδ0 

(i1 ...il−m )  x m,l−m (0) L 2 ( ε0 ) 0



t

× 0

 (1 + t  )−2 [1 + log(1 + t  )] p+2 µ−a−1 (t  ) G0,[l+2];a, p (t  )dt  m

(13.312)

Part 2: The error estimates of the acoustical entities

787

This is similar to (13.272), and is bounded by: 

  C p δ0  (i1 ...il−m ) x m,l−m (0) L 2 ( ε0 ) µ−a (t) ϕ3+ (Caδ ) G0,[l+2];a, p (t) 0 m p 0 t  + (1 + t  )−2 [1 + log(1 + t  )]2+ p G0,[l+2];a, p (t  )dt  (13.313) 0

In regard to the contribution of the last term on the right in (13.277), we remark that it is bounded by:  Cl δ0 (1 + t)−1 [1 + log(1 + t)]4 +

m 

sup {(1 + t  )2 µam (t  )Bl (t  )}

t  ∈[0,t ]

 

sup {(1 + t )µ (t a

 k=0 t ∈[0,t ]



 )Bk,l−k }

µ−a m (t)

(13.314)

 Thus, relative to the terms (1 + t)2 (i1 ...il ) Bl and (1 + t) (i1 ...il−m ) Bm,l−m there is here an extra factor of Cl δ0 (1+t)−1 [1+log(1+t)]4 , consequently the corresponding contribution is absorbed in the estimates already made. We now consider the contribution of (13.165) to the corresponding integral (13.168). Recalling that this is associated to the variation (13.161), the contribution in question is: (ω/ν)(Ril+1 . . . Ri1 trχ  )(T ψα )((L + ν)Ril+1 . . . Ri1 ψα )dt  du  dµh/ (13.315) − Wut

One may think at first sight that this integral can be directly estimated using the flux F1 [Ril+1 . . . Ri1 ψα ] associated to the vectorfield K 1 and the variation Ril+1 . . . Ri1 ψα . Such an estimate would however involve the L 2 norm of Ril+1 . . . Ri1 trχ, a top order acoustical entity, on the spacetime region Wut , and our estimate (13.174) for this entity would only allow us to bound its L 2 norm on Wut by a quantity growing with t like √ −a−1/2 tµm . Instead, we proceed as follows. First, since /h˜ = h/,

dµh/˜ = dµh/

the integral (13.315) is: (ω/ν)(T ψα )(Ril+1 . . . Ri1 trχ  )((L + ν)Ril+1 . . . Ri1 ψα )dt  du  dµh/˜ −

(13.316)

Wut

Let us consider, for an arbitrary function f , the integral: (L f + 2ν f )dt  du  dµh/˜ Wut



Let: F(t, u) =

St,u

f dµh/˜

(13.317)

788

Chapter 13. Error Estimates

Then we have:

∂F = ∂t

St,u

(L f + 2ν f )dµh/˜

(13.318)

Therefore: Wut





∂F     (t , u )dt du  0 0 ∂t u   F(t, u  ) − F(0, u  ) du  = 0 = f du  dµh/˜ − f du  dµh/˜

(L f + 2ν f )dt  du  dµh/˜ =

u

t

tu

0u

(13.319)

Going back to the integral (13.316), we write the integrant in the form: − (ω/ν)(T ψα )(Ril+1 . . . Ri1 trχ  )((L + ν)Ril+1 . . . Ri1 ψα )   = −(L + 2ν) (ω/ν)(T ψα )(Ril+1 . . . Ri1 trχ  )(Ril+1 . . . Ri1 ψα )   + (Ril+1 . . . Ri1 ψα )(L + ν) (ω/ν)(T ψα )(Ril+1 . . . Ri1 trχ  ) By (13.319) with the function (ω/ν)(T ψα )(Ril+1 . . . Ri1 trχ  )(Ril+1 . . . Ri1 ψα ) in the role of the function f , we then conclude that the integral (13.316) is equal to: (ω/ν)(T ψα )(Ril+1 . . . Ri1 trχ  )(Ril+1 . . . Ri1 ψα )du  dµh/˜ (13.320) −

+ +

tu 0u

Wut

(ω/ν)(T ψα )(Ril+1 . . . Ri1 trχ  )(Ril+1 . . . Ri1 ψα )du  dµh/˜   (Ril+1 . . . Ri1 ψα )(L + ν) (ω/ν)(T ψα )(Ril+1 . . . Ri1 trχ  ) dt  du  dµh/˜

We first consider the hypersurface integral: − (ω/ν)(T ψα )(Ril+1 . . . Ri1 trχ  )(Ril+1 . . . Ri1 ψα )du  dµh/˜ tu

(13.321)

(The other hypersurface integral is in any case expressible in terms of the initial data.) Let f, g be arbitrary functions defined on St,u and X an arbitrary vectorfield defined on and tangential to St,u . We have: f (Xg)dµh/˜ = X ( f g)dµh/˜ − g(X f )dµh/˜ St,u

and:

St,u



St,u

X ( f g)dµh/˜ =

St,u

div /˜ ( f g X)dµh/˜ −

St,u

St,u

(div /˜ X) f gdµh/

Part 2: The error estimates of the acoustical entities



Since

we obtain:

1 div /˜ ( f g X)dµh/˜ = 0, while div /˜ X = tr 2 St,u 



St,u

789

f (Xg)dµh/˜ = −

St,u

1 g(X f ) + tr 2

(X ) ˜

(X ) ˜

π /,



π / f g dµh/˜

(13.322)

Applying (13.322), taking: X = Ril+1 , g = Ril . . . Ri1 trχ  ,

f = (ω/ν)(T ψα )(Ril+1 . . . Ri1 ψα )

we conclude that (13.321) is equal to the sum: H0 + H1 + H2

(13.323)

of three hypersurface integrals: H0 = (ω/ν)(T ψα )(Ril . . . Ri1 trχ  )(Ril+1 Ril+1 . . . Ri1 ψα )du  dµh/˜

(13.324)

(ω/ν)(Ril+1 T ψα )(Ril . . . Ri1 trχ  )(Ril+1 . . . Ri1 ψα )du  dµh/˜

(13.325)



tu



tu

H1 = H2 =

tu

(T ψα )(Ril . . . Ri1 trχ  )(Ril+1 . . . Ri1 ψα ) 

1 × Ril+1 (ω/ν) + tr 2

(Ril+1 ) ˜



π /(ω/ν) du  dµh/˜

(13.326)

Now, by Proposition 12.6 and Corollary 10.1.d with l = 1, recalling that ω is constant on each St,u , 1 (ν/ω)|Ril+1 (ω/ν)| + |tr 2

(Ril+1 ) ˜

π /| ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]

(13.327)

It follows that the integral H2 has an extra decay factor of δ0 (1 + t)−1 [1 + log(1 + t)] relative to the integral H1. We shall therefore confine attention in the following to the integrals H0 and H1. We first consider the integral H0. We have: |H0| ≤ C ε (1 + t)3 |T ψα ||Ril . . . Ri1 trχ  ||d/ Ril+1 . . . Ri1 ψα |dudµh/ (13.328) t 0

To proceed we must obtain a sharp estimate for Ril . . . Ri1 trχ  . An estimate for this in ε L 2 (t 0 ) follows directly from the estimate for χ  2,[l], ε0 of Proposition 12.11. Tot gether with Corollary 11.2.d with m = 0, in which the bound for A[l−1] in terms of A[l−1] of Lemma 12.5 is substituted, the estimate of Proposition 12.11 yields:

χ $ 2,[l], ε0 ≤ Cl A[l] + δ0 (1 + t)−2 [1 + log(1 + t)]· t  Q (13.329) W{l} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1} + Y0

790

Chapter 13. Error Estimates

Since trχ  is actually the pure trace trχ $ , a similar estimate holds for trχ  2,[l], ε0 . t However, a more precise estimate for Ril . . . Ri1 trχ  is needed here. To obtain the required estimate we derive a propagation equation for Ril . . . Ri1 trχ  . Since     Ltrχ  = tr L / L χ  − 2tr χ · χ $   2η0 trχ  − 2|χ  |2 = tr L /L χ  − 1 − u + η0 t taking the trace of the propagation equation (12.77) of Chapter 12 we obtain: Ltrχ  + where:

2η0 trχ  = ρ0 1 − u + η0 t

(13.330)

ρ0 = trb + tr(χ  · a) − |χ  |2

(13.331)

Applying Ril . . . Ri1 to this equation and using Lemma 11.23 then yields the propagation equation: L Ril . . . Ri1 trχ  +

2η0 Ri . . . Ri−1 trχ  = 1 − u + η0 t l

(i1 ...il )

ρl

(13.332)

where (i1 ...il )

ρl = Ril . . . Ri1 ρ0 +

l−1 

Ril . . . Ril−k+1

(Ril−k )

Z Ril−k−1 . . . Ri1 trχ 

(13.333)

k=0

The principal part of the function ρ0 is contained in the term trb. Now b is given by ( P)

(12.79), from which we see that the principal part of trb is contained in the term −tr α , ( P P)

( P P)

more precisely in −tr α , where α is given by (12.79). This term must be treated in a more precise manner in the present context. Noting that: d/σ = −2ψ α d/ψα
/ σ = −2ψ α
/ ψα − 2(d/ψ α )$ · d/ψα

Lσ = −2ψ α Lψα = −2(1 + t)−1 ψ α Qψα d/ Lσ = −2(1 + t)−1 ψ α d/ Qψα − 2(1 + t)−1 (d/ψα )(Qψα )

(L)2 σ = −2(1 + t)−1 ψ α L(Qψα ) − 2(1 + t)−2 (Qψ α − ψ α )(Qψα ) we can express: ( P P)

Ril . . . Ri1 tr α

α

= −m αT Ril . . . Ri1
/ ψα − 2(1 + t)−1/e · d/ Ril . . . Ri1 Qψα −(1 + t)−1 n αL Ril . . . Ri1 L Qψα +

(i1 ...il )

n˜ l

(13.334)

α

Here, m αT , /e are the coefficients defined by (10.695), (10.698) of Chapter 10, while: n αL = −

dH | ψ|2 ψ α dσ

(13.335)

Part 2: The error estimates of the acoustical entities

In (13.334)

(i1 ...il ) n˜

791

is a lower order term (of order l + 1), given by:  (i1 ...il ) n˜ l = − ((R)s1 m αT )(R)s2
/ ψα l

partitions, s1 =∅

−2(1 + t)−1



α

((L / R )s1 /e ) · d/(R)s2 Qψα

partitions, s1 =∅

−(1 + t)



−2

((R)s1 n αL ) · (R)s2 (Q)2 ψα

partitions, s1 =∅

+Ril . . . Ri1 n˜ 0

(13.336)

where: n˜ 0 =

dH 2 ψ L (d/ψ α )$ · d/ψα − 2(1 + t)−1 ψ L ( ψ$ · d/ψ α )Qψα dσ  +(1 + t)−2 | ψ|2 (Qψ α − ψ α )Qψα

(13.337)

In the three sums in (13.336) we are considering all ordered partitions {s1 , s2 } of the set {1, . . . , l} into two ordered subsets s1 , s2 , such that s1 = ∅, thus |s1 | + |s2 | = l, |s1 | = 0. Under the assumptions of Proposition 10.5 we readily deduce the estimate:

Q QQ  (i1 ...il ) n˜ l  L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−3 W{l+1} + W{l} + δ0 (1 + t)−1 W{l−1} t 1 2 +δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] (13.338) Let us recall that according to (10.699), (10.700): |m 0T − k 3 | ≤ Cδ0 (1 + t)−1 ,

|m iT | ≤ Cδ0 (1 + t)−1

α

Moreover, we have:

(13.339)

|e/ | ≤ Cδ0 (1 + t)−1

(13.340)

|n αL | ≤ Cδ02 (1 + t)−2

(13.341)

Let us define the functions: (i1 ...il )

(0) ρl

= k 3 
/ Ril . . . Ri−1 ψ0

(i1 ...il )

(1) ρl

= (m 0T − k 3 )
/ Ril . . . Ri1 ψ0 + m iT
/ Ril . . . Ri1 ψi + 2(1 + t)

−1 α

(13.342)

/e · d/ Ril . . .

Ri1 Qψα + (1 + t)−1 n αL (L

(13.343) + ν)(Ril . . . Ri1 Qψα )

We can then re-express (13.334) in the form: ( P P)

Ril . . . Ri1 tr α

(0) (1) ρ l − (i1 ...il ) ρ l −m αT (i1 ...il ) cl;α − (1 + t)−1 n αL (i1 ...il ) dl;α +(1 + t)−1 n αL ν Ril . . . Ri1 Qψα + (i1 ...il ) n˜ l

=−

(i1 ...il )

(13.344)

792

Here

Chapter 13. Error Estimates (i1 ...il ) c , (i1 ...il ) d l;α l;α (i1 ...il ) (i1 ...il )

are the commutators:

cl;α = Ril . . . Ri1
/ ψα −
/ Ril . . . Ri1 ψα

(13.345)

dl;α = Ril . . . Ri1 L Qψα − L Ril . . . Ri1 Qψα

(13.346)

By (11.1046), (11.1048) of Chapter 11 with m = 0:  $ (i1 ...il ) cl;α = tr (i1 ...il ) c0,l [d/ψα ]      + tr ((L / R )s1 (h/−1 )) · (L / R )s 2 D / 2 ψα  

(13.347)

partitions,s1 =∅

Following the procedure leading to the estimates (11.1050), (11.1051), we obtain:   (i1 ...il ) cl;α  L 2 ( ε0 ) ≤ Cl δ0 (1 + t)−3 · t

α

  Q [1 + log(1 + t)] W{l+1} + (1 + t)−2 [1 + log(1 + t)]W{l}  (13.348) +Y0 + (1 + t)A[l]

By (10.740) of Chapter 10 with l + 1 replaced by l and ψα by Qψα : (i1 ...il )

dl;α = −

l−1 

Ril . . . Ril−k+1

(Ril−k )

Z Ril−k−1 . . . Ri1 Qψα

(13.349)

k=0

and in analogy with estimate (10.741) we obtain:   (i1 ...il ) dl;α  L 2 ( ε0 ) α

(13.350)

t

 Q ≤ Cl δ0 (1 + t)−2 Y0 + (1 + t)A[l−1] + [1 + log(1 + t)]W[l] + W[l]

We also note that by (13.341): n αL ν Ril . . . Ri1 Qψα  L 2 ( ε0 ) ≤ Cδ02 (1 + t)−3 W[l] Q

(13.351)

t

The remaining terms in the expression (12.79) for b have all been estimated in the proof of Proposition 12.11. Similar estimates hold for the traces of these terms. In particular, by the estimate (12.413) for the function e, the trace of the first term on the right in (12.179) is bounded in  2,[l], ε0 norm by: t

1 2 Q Cl (1 + t)−2 W[l] + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + W[l]

(13.352) (0)

This term has the same decay factor as the leading principal term (i1 ...il ) ρ l . The remaining terms in the expression (13.331) for ρ0 are estimated in  2,[l], ε0 norm, noting that t

Part 2: The error estimates of the acoustical entities

793

      tr χ  · a is the pure trace tr χ $ · a while |χ  |2 is the pure trace tr χ $ · χ $ , using the estimate (13.329) for χ $ and the estimate (12.424) for a. Finally, the sum on the right ε in (13.333) is estimated in L 2 (t 0 ) using Corollary 10.2.i and the estimate (13.329), as well as Corollary 10.1.i with l replaced by l∗ + 1 and property X / [l∗ ] . Combining the above results, substituting for A[l] in terms of A[l] from Lemma 12.5, and defining by: (i1 ...il )

ρl =

(i1 ...il )

(0) ρl

+

(i1 ...il )

(1) ρl

+

(i1 ...il )

(2) ρl

(i1 ...il )

(2) ρl

(13.353)

we obtain: (i1 ...il )

(2) ρl

Q  L 2 ( ε0 ) ≤ Cl (1 + t)−2 W{l} (13.354) t

   QQ + δ0 (1 + t)−3 [1 + log(1 + t)] W{l+1} + Y0 + (1 + t)A[l] + δ0 (1 + t)−1 W{l−1}



On the other hand, by hypothesis H1 we have, pointwise: ) (0) | (i1 ...il ) ρ l | ≤ C(1 + t)−1 k 3 || |d/ R j Ril . . . Ri1 ψα |2

(13.355)

j,α

and: |

(1) (i1 ...il ) ρ l

| ≤ Cδ0 (1 + t)−2

+

) α



 ) 

|d/ R j Ril . . . Ri1 ψα |2

(13.356)

j,α

  |(L + ν)Ril . . . Ri1 Qψα |2 + |d/ Ril . . . Ri−1 Qψα |2 

where, to obtain (13.356) we have used the bounds (13.339)–(13.341). These pointwise inequalities imply: ) (0) −1/2 E1 [R j Ril . . . Ri1 ψα ](t)  (i1 ...il ) ρ l  L 2 ( ε0 ) ≤ Ck 3 ||(1 + t)−2 µm (t) t

j,α

(13.357)

and: 

(i1 ...il )

(1) ρl

−1/2

 L 2 ( ε0 ) ≤ Cδ0 (1 + t)−3 µm (t) · (13.358) t   ) )   E1 [R j Ril . . . Ri1 ψα ](t) + E1 [Ril . . . Ri1 Qψα ](t)   α j,α

while: µ1/2

(i1 ...il )

(0) ρl

 L 2 ( ε0 ) ≤ Ck 3 ||(1 + t)−2

)

t

j,α

E1 [R j Ril . . . Ri1 ψα ](t)

(13.359)

794

Chapter 13. Error Estimates

and: µ1/2

(i1 ...il )

(1) ρl

 ) 

 L 2 ( ε0 ) ≤ Cδ0 (1 + t)−3 · t

E1 [R j Ril

. . . Ri1 ψα ](t) +

) α

j,α

E1 [Ril . . . Ri1 Qψα ](t)

(13.360)   

We express the propagation equation (13.332) in acoustical coordinates (t, u, ϑ), so ε that L becomes ∂/∂t, and integrate it along each integral curve of L from 00 to obtain: (1 − u + η0 t)2 (Ril . . . Ri1 trχ  )(t, u, ϑ) = (1 − u)2 (Ril . . . Ri1 trχ  )(0, u, ϑ) +

(13.361)



t

(1 − u + η0 t  )2

(i1 ...il )

ρl (t  , u, ϑ)dt 

0

It follows that: |Ril . . . Ri1 trχ  |(t, u, ϑ)  ≤ C(1 + t)

−2



|Ril . . . Ri1 trχ |(0, u, ϑ) +

2 

(i1 ...il )

(k) Al

 (t, u, ϑ)

(13.362)

k=0

where: (i1 ...il )



(k)

t

A (t, u, ϑ) =

(1 + t  )2 |

(i1 ...il )

(k) ρl

|(t  , u, ϑ)dt 

: k = 0, 1, 2 (13.363)

0

Note that by the comparison inequalities (8.355), (8.356) of Chapter 8 we have: t (k) (k)  (i1 ...ik ) A l (t) L 2 ([0,ε0 ]×S 2 ) ≤ (1 + t  )2  (i1 ...il ) ρ l (t  ) L 2 ([0,ε0]×S 2 ) dt  0 t (k) ≤C (1 + t  ) (i1 ...il ) ρ l  L 2 ( ε0 ) dt  (13.364) 0

t

We now substitute the pointwise estimate (13.362) in (13.328). The borderline con(0)

tribution is the contribution from (i1 ...il ) A l . This contribution is the borderline integral: (0) C ε (1 + t)|T ψα | (i1 ...il ) A l |d/ Ril+1 . . . Ri1 ψα |dudµh/ (13.365) t 0

Recalling from Chapter 8 the partition of [0, ε0] × S 2 into the subsets Vs− , Vs+ , defined by (8.360), (8.361), respectively, we shall consider separately the integrals over the corε − + responding regions Us,t , Us,t of t 0 :   ε − (13.366) = (t, u, ϑ) ∈ t 0 : (u, ϑ) ∈ Vs− Us,t   ε0 + Us,t = (t, u, ϑ) ∈ t : (u, ϑ) ∈ Vs+ (13.367)

Part 2: The error estimates of the acoustical entities

795

− We consider first the integral over Us,t . We have:

− Us,t

(i1 ...il )

(1 + t)|T ψα |

(0) Al

≤ (1 + t) sup |T ψα |

|d/ Ril+1 . . . Ri1 ψα |dudµh/

(i1 ...il )

− Us,t

−1/2

≤ C(1 + t)µm

(0) Al

(t) sup |T ψα |

(13.368)

 L 2 (U − ) d/ Ril+1 . . . Ri1 ψα  L 2 ( ε0 ) s,t

(i1 ...il )

− Us,t

t

(0) Al

 (t) L 2 (Vs− ) E1 [Ril+1 . . . Ri1 ψα ](t)

by the comparison inequality (8.355). Now by the estimates (13.196), (13.197), we have, ε pointwise on t 0 : max |T ψα | ≤ α

 C −2 (1 + t) [1 + log(1 + t)] |Lµ| + Cδ 0 k 3 ||

(13.369)

− In estimating the integral over Us,t we can assume that Vs− = ∅, thus, recalling the definitions (8.275) and (8.284) of Chapter 8,

min

(u,ϑ)∈Vs−

Eˆ s (u, ϑ) = Eˆ s,m = −δ1 , δ1 > 0

(13.370)

Now, from Proposition 8.6 we have:  (Lµ)(t, u, ϑ) = µ[1],s (u, ϑ) It follows that:

 sup |Lµ| ≤ C − Us,t

Eˆ s (u, ϑ) + Qˆ 1,s (t, u, ϑ) (1 + t)

Cδ0 [1 + log(1 + t)] δ1 + (1 + t) (1 + t)2

 (13.371)

(13.372)

hence, substituting in (13.369), max sup |T ψα | ≤ α

− Us,t

C 3 k ||



Cδ0 [1 + log(1 + t)] δ1 + (1 + t) (1 + t)2

(13.373)

On the other hand, from (13.364) for k = 0 and the estimate (13.357), 

(i1 ...il )

(0) Al

(t) L 2 ([0,ε0]×S 2 ) (13.374) t ) −1/2 E1 [R j Ril . . . Ri1 ψα ](t  )dt  ≤ Ck 3 || (1 + t  )−1 µm (t  ) 0

j,α

796

Chapter 13. Error Estimates

Substituting (13.373) and (13.374) in (13.368), the factors k 3 || cancel and we obtain that (13.368) is bounded by:  Cδ0 [1 + log(1 + t)]   −1/2 E1 [Ril+1 . . . Ri1 ψα ](t) Cµm (t) δ1 + (1 + t) t ) −1/2 · (1 + t  )−1 µm (t  ) E1 [R j Ril . . . Ri1 ψα ](t  )dt  (13.375) 0

j,α

We now define (see (13.200), (13.220)):   (i1 ...il )     G1,l+2;a,q (t) = sup [1 + log(1 + t  )]−2q µ2a (t ) E [R R . . . R ψ ](t ) j il i1 α m 1 t  ∈[0,t ]

j,α

(13.376) Then (13.375) is bounded by: 

Cδ0 [1 + log(1 + t)] C[1 + log(1 + t)] δ1 + (1 + t) t  dt −a−1/2   · (i1 ...il ) G1,l+2;a,q µm (t ) (t) · (1 + t  ) 0 2q

−a−1/2 µm (t)



(13.377)

To estimate the integral in (13.377), we follow the argument of Lemma 8.11. Since we are assuming that Vs− = ∅, we only have Case 2 to consider. Setting again, as in (8.292), 1

t1 = e 2aδ1 − 1, we have the two subcases of Case 2 as in the proof of Lemma 8.11. In Subcase 2a the −a−1/2  lower bound (8.296) holds, hence µm (t ) ≤ C and t t dt  dt  −a−1/2  ≤ C = C log(1 + t) (13.378) µm (t )  (1 + t  ) 0 0 (1 + t ) therefore: t  Cδ0 dt  −a−1/2  (1 + t) δ1 + µm (t ) [1 + log(1 + t)] (1 + t  ) 0   C [1 + log(1 + t)]2 ≤ C δ1 log(1 + t) + Cδ0 (1 + t) a for, in Subcase 2a δ1 log(1 + t) ≤ δ1 log(1 + t1 ) = while

1 2a

1 [1 + log(1 + t)]2 ≤ C and Cδ0 ≤ (1 + t) a

(13.379)

Part 2: The error estimates of the acoustical entities

797

In Subcase 2b we first estimate the contribution of t1 dt  C −a−1/2  ≤ C log(1 + t1 ) = µm (t )  (1 + t ) 2aδ1 0 (by (13.378)) to the left-hand side of (13.379). This contribution is then bounded by:   C C [1 + log(1 + t)] [1 + log(1 + t1 )] ≤ δ1 + Cδ0 δ1 + Cδ0 (1 + t) 2aδ1 (1 + t1 ) 2aδ1    1   C C 1 C Cδ0 − 1 + Cϕ2 (2aδ1 ) ≤ (13.380) ≤ 1+ e 2aδ1 ≤ 1+ 2a δ1 2aδ1 2a a (see (13.270)). Next, we estimate the contribution of: t dt  −a−1/2  µm (t ) (1 + t  ) t1 By the lower bound (8.326), t −a−1/2  µm (t ) t1

dt  ≤C (1 + t  )



τ

(1 − δ1 τ τ1 −a+1/2 µm (t)

(13.381)

 −a−1/2

)

dτ 

C C (1 − δ1 τ )−a+1/2 ≤ δ1 (a − 1/2) δ1 (a − 1/2)

≤

(13.382)

where the last step is by virtue of the upper bound (8.335). It follows that the contribution of (13.381) to the left-hand side of (13.379) is bounded by:  −a+1/2 [1 + log(1 + t)] C µm (t) δ1 + Cδ0 (1 + t) δ1 (a − 1/2)  −a+1/2 Cµm (t) Cδ0 [1 + log(1 + t1 )] ≤ 1+ (a − 1/2) δ1 (1 + t1 ) −a+1/2

≤

−a+1/2

 2Cµm (t)  (t) Cµm 1 + Cϕ2 (2aδ1) ≤ (a − 1/2) (a − 1/2)

(13.383)

Combining the results (13.379), (13.383) of the two subcases we conclude that, in general:  t −a+1/2 dt  [1 + log(1 + t)] µm (t) −a−1/2  ≤ C (13.384) µm (t ) δ1 + Cδ0 ) (1 + t) (1 + t (a − 1/2) 0 hence (13.377), therefore also (13.375) and the integral on the left in (13.368), is bounded by: C  µ−2a (t)[1 + log(1 + t)]2q (i1 ...il ) G1,l+2;a,q (t) (13.385) (a − 1/2) m + We now consider the integral over Us,t : (0) (1 + t)|T ψα | (i1 ...il ) A l |d/ Ril+1 . . . Ri1 ψα |dudµh/ +

Us,t

(13.386)

798

Chapter 13. Error Estimates

+ Here we want to make use of the fact that in Us,t µ is bounded from below by a positive constant. We thus estimate (13.386) by:

  (1 + t) sup µ−1 |T ψα | µ1/2

(i1 ...il )

+ Us,t

(0) Al

  ≤ C(1 + t) sup µ−1 |T ψα | (µ1/2

 L 2 (U + ) µ1/2 d/ Ril+1 . . . Ri1 ψα  L 2 ( ε0 )

(i1 ...il )

+ Us,t

s,t

t

 (0) A l )(t) L 2 (Vs+ ) E1 [Ril+1

. . . Ri1 ψα ](t) (13.387)

+ In view of the fact that in Us,t we have µ ≥ C −1 , the pointwise estimate (13.369) implies:      C max sup µ−1 |T ψα | ≤ 3 sup µ−1 |Lµ| + Cδ0 (1 + t)−2 [1 + log(1 + t)] α k || U + U+ s,t

s,t

(13.388) Let us define:

Eˆ s,M =

max

(u,ϑ)∈[0,ε0]×S 2

Setting:

Eˆ s (u, ϑ)

δ2 = Eˆ s,M

(13.389)

(13.390)

we can assume in estimating (13.388) that δ2 > 0. From Proposition 8.6 we have: µ−1 Lµ =

Eˆ s (u, ϑ)(1 + t)−1 + Qˆ 1,s (t, u, ϑ) 1 + Eˆ s (u, ϑ) log(1 + t) + Qˆ 0,s (t, u, ϑ)

(13.391)

+ : In view of the fact that Eˆ s (u, ϑ) ≥ 0 in Vs+ , (13.391) implies that in Us,t

µ−1 |Lµ| ≤ ≤ ≤

Eˆ s (u, ϑ)(1 + t)−1 + | Qˆ 1,s (t, u, ϑ)| 1 + Eˆ s (u, ϑ) log(1 + t) − | Qˆ 0,s (t, u, ϑ)| Eˆ s (u, ϑ)(1 + t)−1 + Cδ0 (1 + t)−2 [1 + log(1 + t)] 1 + Eˆ s (u, ϑ) log(1 + t) − Cδ0 (1 + t)−1 [1 + log(1 + t)] Eˆ s (u, ϑ)(1 + t)−1 1 + Eˆ s (u, ϑ) log(1 + t) − Cδ0 (1 + t)−1 [1 + log(1 + t)] [1 + log(1 + t)] +C  δ0 (13.392) (1 + t)2

With η = Eˆ s (u, ϑ) log(1 + t) ≥ 0, we write

ε = Cδ0 (1 + t)−1 [1 + log(1 + t)] > 0,

1 ε 1 1 = + ≤ + Cε, 1+η−ε 1 + η (1 + η − ε)(1 + η) 1+η

Part 2: The error estimates of the acoustical entities

then since

799

Eˆ s (u, ϑ)(1 + t)−1 ε ≤ C  δ02 (1 + t)−2 [1 + log(1 + t)]

+ (13.392) implies that in Us,t :

µ−1 |Lµ| ≤

Eˆ s (u, ϑ)(1 + t)−1 + Cδ0 (1 + t)−2 [1 + log(1 + t)] ˆ 1 + E s (u, ϑ) log(1 + t)

(13.393)

The first term on the right in (13.393) is, with the above notation, η 1 (1 + t) log(1 + t) 1 + η

(13.394)

+ This being an increasing function of η, it achieves its maximum in Us,t where η achieves its maximum value (13.395) η M = δ2 log(1 + t) + in Vs+ . The supremum of (13.394) in Us,t is therefore:

ηM δ2 1 1 = (1 + t) log(1 + t) 1 + η M (1 + t) [1 + δ2 log(1 + t)]

(13.396)

It follows that: sup µ−1 |Lµ| ≤ + Us,t

1 [1 + log(1 + t)] δ2 + Cδ0 (1 + t) [1 + δ2 log(1 + t)] (1 + t)2

(13.397)

Substituting this in (13.388) and the result in (13.387), we conclude that the integral (13.386) is bounded by:  C [1 + log(1 + t)] δ2 + Cδ0 (1 + t) k 3 || 1 + δ2 log(1 + t)  (0) (13.398) (µ1/2 (i1 ...il ) A l (t) L 2 (Vs+ ) E1 [Ril+1 . . . Ri1 ψα ](t) Now from (13.363) for k = 0 we have:  1/2 t (0) 1/2 (i1 ...il )  2 µ(t, u, ϑ) (1+t ) |(µ1/2 (µ A l )(t, u, ϑ) = µ(t  , u, ϑ) 0

(i1 ...il )

(0) ρ l )(t  , u, ϑ)|dt 

(13.399)

It follows that: (0)

(µ1/2 (i1 ...il ) A l )(t) L 2 (Vs+ )  1/2  t µ(t, u, ϑ)  2 sup ≤ (1 + t ) (µ1/2  0 (u,ϑ)∈Vs+ µ(t , u, ϑ)  1/2  t µ(t, u, ϑ)  ≤C (1 + t ) sup µ1/2  0 (u,ϑ)∈Vs+ µ(t , u, ϑ)

(13.400) (i1 ...il )

(0) ρ l )(t  ) L 2 (Vs+ ) dt 

(i1 ...il )

(0) ρl

 L 2 ( ε0 ) dt  t

800

Chapter 13. Error Estimates

where in the last step we have used the comparison inequalities (8.355), (8.356) of Chapter 8. From Proposition 8.6 we have, for (u, ϑ) ∈ Vs+ , µ(t, u, ϑ) 1 + Eˆ s (u, ϑ) log(1 + t) + Qˆ 0,s (t, u, ϑ) = (13.401)  µ(t , u, ϑ) 1 + Eˆ s (u, ϑ) log(1 + t  ) + Qˆ 0,s (t  , u, ϑ) 1 + Eˆ s (u, ϑ) log(1 + t) + Cδ0 (1 + t)−1 [1 + log(1 + t)] ≤ 1 + Eˆ s (u, ϑ) log(1 + t  ) − Cδ0 (1 + t  )−1 [1 + log(1 + t  )] Now, since Cδ0

[1 + log(1 + t  )] 1 ≤ Cδ0 ≤ ,  (1 + t ) 2

the denominator in the fraction on the right in (13.401) is ≥

 1 1 + Eˆ s (u, ϑ) log(1 + t  ) ≥ 1 + Eˆ s (u, ϑ) log(1 + t  ) 2 2

Similarly, since Cδ0

1 [1 + log(1 + t)] ≤ Cδ0 ≤ , (1 + t) 2

the numerator in the fraction on the right in (13.401) is ≤

 3 3 + Eˆ s (u, ϑ) log(1 + t) ≤ 1 + Eˆ s (u, ϑ) log(1 + t) 2 2

Therefore, for (u, ϑ) ∈ Vs+ it holds that µ(t, u, ϑ) 1 + Eˆ s (u, ϑ) log(1 + t) ≤3  µ(t , u, ϑ) 1 + Eˆ s (u, ϑ) log(1 + t  )

(13.402)

Now, for a > b > 0, x ≥ 0, the function f (x) =

1 + ax 1 + bx

is increasing in x. Hence, taking a = log(1 + t), b = log(1 + t  ), x = Eˆ s (u, ϑ), the ratio on the right in (13.402) achieves its maximum in Vs+ where x achieves its maximum δ2 in Vs+ . Therefore: sup

(u,ϑ)∈Vs+

1 + δ2 log(1 + t) µ(t, u, ϑ) ≤3  µ(t , u, ϑ) 1 + δ2 log(1 + t  )

(13.403)

Part 2: The error estimates of the acoustical entities

801

Substituting this and the estimate (13.359) in (13.400) we obtain: (µ1/2

(i1 ...il )

≤ Ck 3 ||

(0) A l )(t) L 2 (Vs+ )

t) 0



(13.404) )

1 + δ2 log(1 + t) 1 + δ2 log(1 + t  ) )

j,α

E1 [R j Ril ... Ri1 ψα ](t  )

dt  (1 + t  )

1 + δ2 log(1 + t) dt   q −a  [1 + log(1 + t )] µ (t ) m 1 + δ2 log(1 + t  ) (1 + t  ) 0 ) t t  1 + δ2 log(1 + t) 3 −a  q dt  (i ...i ) l 1 [1 + log(1 + t ≤ Ck ||µm (t) G1,l+2;a,q )] 1 + δ2 log(1 + t  ) (1 + t  ) 0 0

≤ Ck 3 ||

(i1 ...il ) G  1,l+2;a,q

t

where in the last step we have used the fact that according to Corollary 2 of Lemma 8.11, −a  µ−a m (t ) ≤ Cµm (t). Substituting the estimate (13.404) in (13.398) and noting that   q (i1 ...il ) G  E1 [Ril+1 . . . Ri1 ψα ](t) ≤ µ−a m (t)[1 + log(1 + t)] 1,l+2;a,q (t), the factors k 3 || cancel and we obtain that (13.398) hence also (13.387) and the integral (13.386) is bounded by: q (i1 ...il )  G1,l+2;a,q (t) (13.405) Cµ−2a m (t)[1 + log(1 + t)]   δ2 [1 + log(1 + t)]3/2 Iq;δ2 (t) ·

+ Cδ0 (1 + t) 1 + δ2 log(1 + t)



where: Iq;δ2 (t) =

t 0

[1 + log(1 + t  )]q dt 

1 + δ2 log(1 + t  ) (1 + t  )

Setting x = 1 = log(1 + t  ), the integral Iq;δ2 (t) takes the form: 1+log(1+t ) xq Iq;δ2 (t) = √ dx 1 + δ2 (x − 1) 1

(13.406)

(13.407)

Now, since δ2 ≤ 1 we have: 1 + δ2 (x − 1) ≥ δ2 + δ2 (x − 1) = δ2 x hence: 1 Iq;δ2 (t) ≤ √ δ2



1+log(1+t ) 0

1 [1 + log(1 + t)]q+1/2 x q−1/2 d x = √ (q + 1/2) δ2

Also, trivially, since the denominator of the integrant in (13.407) is ≥ 1, 1+log(1+t ) [1 + log(1 + t)]q+1 Iq;δ2 (t) ≤ xqdx = (q + 1) 0

(13.408)

(13.409)

802

Chapter 13. Error Estimates

We use the bound (13.408) in estimating the product: δ2 [1 + log(1 + t)]q+1/2 √ Iq;δ2 (t) ≤

δ2 (q + 1/2) 1 + δ2 log(1 + t) 1 + δ2 log(1 + t) √

δ2 1 + log(1 + t) [1 + log(1 + t)]q [1 + log(1 + t)]q =

≤ (13.410) (q + 1/2) (q + 1/2) 1 + δ2 log(1 + t) δ2



where in the last step we have used the fact that δ2 ≤ 1. We use the bound (13.409) in estimating the product: [1 + log(1 + t)]3/2 [1 + log(1 + t)]3/2 [1 + log(1 + t)]q+1 Iq;δ2 (t) ≤ Cδ0 (1 + t) (1 + t) (q + 1) C  δ0 [1 + log(1 + t)]5/2 [1 + log(1 + t)]q = Cδ0 ≤ [1 + log(1 + t)]q (13.411) (1 + t) (q + 1) (q + 1)

Cδ0

We conclude that (13.405) hence also the integral (13.386) is bounded by: C µ−2a (t)[1 + log(1 + t)]2q (q + 1/2) m

(i1 ...il )  G1,l+2;a,q (t)

(13.412)

We have thus estimated the borderline integral (13.365). We proceed to estimate the remaining contributions to the hypersurface integral H0, bounded by (13.328). These are (1)

the contributions from (i1 ...il ) A l and tions we simply appeal to the bound:

(i1 ...il )

(2) Al

in (13.362). To estimate these contribu-

max sup |T ψα | ≤ Cδ0 (1 + t)−1 α

ε

(13.413)

t 0

ε

and estimate directly the integrals over the entire t 0 . These contributions are then bounded by: (k) C ε (1 + t)|T ψα | (i1 ...il ) A l |d/ Ril+1 . . . Ri1 ψα |dudµh/ t 0



≤ Cδ0 ≤ Cδ0 

ε t 0

(k) Al

(i1 ...il )

(i1 ...il )

|d/ Ril+1 . . . Ri1 ψα |dudµh/

(k) Al

 L 2 ( ε0 ) d/ Ril+1 . . . Ri1 ψα  L 2 ( ε0 ) t t  (k) −1/2 (i1 ...il )  ≤ Cδ0 µm (t) A l (t) L 2 ([0,ε0]×S 2 ) E1,[l+2] (t) : for k = 1, 2

(13.414)

by the comparison inequalities (8.355), (8.356) of Chapter 8. Consider first the contribution of estimate (13.358): 

(i1 ...il )

(1) ρl

(i1 ...il )

−1/2

 L 2 ( ε0 ) ≤ Cµm t

(1) Al.

From (13.364) and the fact that by the

  (t)δ0 (1 + t)−3 E1,[l+2] (t)

(13.415)

Part 2: The error estimates of the acoustical entities

we obtain: 

(i1 ...il )



(1) Al

t

(t) L 2 ([0,ε0 ]×S 2 ) ≤ Cδ0

803

−1/2

µm

  (t  )(1 + t  )−2 E1,[l+2] (t  )dt 

0   ≤ Cδ0 G1,[l+2];a,q (t)Ja,q (t)

where:  (t) = Ja,q



t

−a−1/2

µm

(t  )(1 + t  )−2 [1 + log(1 + t  )]q dt 

(13.416)

(13.417)

0

To estimate this integral we consider again the two cases occurring in the proof of Lemma 8.11. In Case 1 we have the lower bound (13.226), which implies: t  (t) ≤ C (1 + t  )−2 [1 + log(1 + t  )]q dt  ≤ Cq (13.418) Ja,q 0

Similarly, in Subcase 2a:

 (t1 ) ≤ Cq Ja,q

(13.419)

In Subcase 2b we have the lower bound (13.233), which, with

 Cq (t1 ) = sup (1 + t  )−1 [1 + log(1 + t  )]q t  ≥t1

implies:   (t) − Ja,q (t1 ) ≤ C · Cq (t1 ) Ja,q

≤C· We have:



t

(1 + t  )−1 (1 − δ1 τ  )−a−1/2dt 

t1

Cq (t1 ) −a+1/2 µm (t) aδ1

  Cq (t1 ) = sup e−τ (1 + τ  )q f (x) = e−x x q

(13.421)

τ1 = log(1 + t1 )

τ  ≥τ1

Now, the function:

(13.420)

on [1, ∞)

is decreasing in x for x ≥ q. Hence if τ1 = 1/4aδ1 ≥ q, which is the case if aqδ0 is suitably small, then:   1 q − 1 (13.422) Cq (t1 ) = e−τ1 (1 + τ1 )q = e 4aδ1 1 + 4aδ1 Therefore, from (13.421),   Ja,q (t) − Ja,q (t1 ) ≤

  C 1 q − 4aδ1 −a+1/2 1 µm e (t) 1+ 4aδ1 4aδ1 −a+1/2

 ≤ Cϕq+1 (4aδ1)µm

(t)

−a+1/2

 ≤ Cϕq+1 (Caδ0 )µm

(t)

(13.423)

804

Chapter 13. Error Estimates

We conclude that, in general: −a+1/2

  Ja,q (t) ≤ Cq + Cϕq+1 (Caδ0 )µm

(t)

(13.424)

Substituting in (13.416) and the result in (13.414) for k = 1, then yields: (1) C ε (1 + t)|T ψα | (i1 ...il ) A l |d/ Ril+1 . . . Ri1 ψα |dudµh/ t 0

(13.425)

 −a−1/2   q ≤ Cδ02 G1,[l+2];a,q (t) Cq µm (t) + Cϕq+1 (Caδ0 )µ−2a m (t) [1 + log(1 + t)]

Next, we consider the contribution of for 

(i1 ...il )

(2) ρl

(i1 ...il )

(2) Al.

We have (13.364) for k = 2 and

 L 2 ( ε0 ) we have the estimate (13.354). Here we shall only consider the t

Q contribution of the leading term on the right in (13.354), namely the term Cl (1+t)−2 W{l} .

The contribution of this term to the bound (13.364) for 

t

Cl 0

(i1 ...il )

(2) Al

(t) L 2 ([0,ε0]×S 2 ) is:

(1 + t  )−1 W{l} (t  )dt  Q

(13.426)

Now by the inequality (5.72) of Chapter 5 we have:

Q  W{l} (t ) ≤ Cε0 E0,[l+2] (t  )

   p ≤ Cε0 µ−a (t )[1 + log(1 + t )] G0,[l+2];a, p m

(13.427)

hence (13.426) is bounded by: Cl ε0 [1 + log(1 + t)]

p





t

G0,[l+2];a, p (t) 0

  (1 + t  )−1 µ−a m (t )dt

(13.428)

The last integral coincides with the integral Ja−1/2,−1(t) (see definition (13.260)), therefore by the bound (13.266) it is bounded by: C[1 + log(1 + t)]µ−a+1 m Consequently (13.428), hence also (13.426), is bounded by:  Cl ε0 [1 + log(1 + t)] p+1 µ−a+1 (t) G0,[l+2];a, p (t) m

(13.429)

We conclude that the contribution to the integral on the left in (13.414) for k = 2 is bounded by:  −2a+1/2  C l ε 0 δ 0 µm (t)[1 + log(1 + t)] p+q+1 G0,[l+2];a, p (t)G1,[l+2];a,q (t) (13.430)

Part 2: The error estimates of the acoustical entities

805

Finally, we consider the contribution of the hypersurface integral H1, given by (13.325). We have: |H1| ≤ C ε (1 + t)2 |Ril+1 T ψα ||Ril . . . Ri1 trχ  ||Ril+1 . . . Ri1 ψα |dudµh/ (13.431) t 0

By virtue of the bound max sup |Ril+1 T ψα | ≤ Cδ0 (1 + t)−1 α

we have:

|H1| ≤ Cδ0

ε

t 0

(13.432)

ε

t 0

(1 + t)|Ril . . . Ri1 trχ  ||Ril+1 . . . Ri1 ψα |dudµh/

≤ Cδ0 (1 + t)Ril . . . Ri1 trχ   L 2 ( ε0 ) Ril+1 . . . Ri1 ψα  L 2 ( ε0 ) t

t



2

≤ Cδ0 (1 + t) (Ril . . . Ri1 trχ )(t) L 2 ([0,ε0 ]×S 2 ) W{l+1} (t)

(13.433)

Now by (13.362), (13.364): (1 + t)2 (Ril . . . Ri1 trχ  )(t) L 2 ([0,ε0]×S 2 )   2  (k)  (i ...i ) l 1 C Ril . . . Ri1 trχ  L 2 ( ε0 ) +  A l (t) L 2 ([0,ε0 ]×S 2 ) 0

(13.434)

k=0

 

≤ C Ril . . . Ri1 trχ  L 2 ( ε0 ) + 0

2 

t



(1 + t )

(i1 ...il )

(k) ρl

k=0 0

  L 2 ( ε0 ) dt



t

Here we need only consider the leading principal contribution, namely that of   L 2 ( ε0 ) . This contribution to (13.434) is bounded by (see (13.357)):

(i1 ...il )

(0) ρl

t



t

Ck 3 ||

−1/2

(1 + t  )−1 µm

(t  )

)

0

E1 [R j Ril . . . Ri1 ψα ](t  )dt 

(13.435)

j,α

t  −a−1/2   ≤ Ck || G1,[l+2];a,q (t) (1 + t  )−1 µm (t )[1 + log(1 + t  )]q dt  3

0

The last integral coincides with the integral Ja,q−1 (see definition (13.260)), therefore by the bound (13.266) it is bounded by: −a+1/2

C[1 + log(1 + t]q+1 µm

(t)

Consequently the leading principal contribution to (13.434) is bounded by:  −a+1/2  (t)[1 + log(1 + t)]q+1 G1,[l+2];a,q (t) Ck 3 ||µm

(13.436)

806

Chapter 13. Error Estimates

On the other hand, as in (13.427) we have:

W{l+1} ≤ Cε0 E0,[l+2] (t)

 p ≤ Cε0 µ−a G0,[l+2];a, p (t) m (t)[1 + log(1 + t)]

(13.437)

Substituting the bounds (13.436) (through (13.434)) and (13.437) in (13.433), we conclude that the leading principal contribution to H1 is bounded by:  −2a+1/2  Cε0 δ0 µm (t)[1 + log(1 + t)] p+q+1 G0,[l+2];a, p (t)G1,[l+2];a,q (t) (13.438) the same in form as the bound (13.430). We turn to the spacetime integral in (13.320) namely:   (Ril+1 . . . Ri1 ψα )(L + ν) (ω/ν)(T ψα )(Ril+1 . . . Ri1 trχ  ) dt  du  dµh/˜

(13.439)

Wut

Here we shall make use of the fact that (L + ν)T ψα decays faster than (1 + t)−2 . To establish this fact we consider the wave equation satisfied by the ψα , as in the proof of Lemma 8.10. In fact, in analogy with equations (8.192), (8.193) of Chapter 8 we have, for each α = 0, 1, 2, 3: (L + ν)Lψα = ρα (13.440) where: ρα = µ
/ ψα − ν Lψα − 2ζ · d/ψα + µ

d log  d/σ · d/ψα dσ

(13.441)

Under the same assumptions as those of Lemma 8.10 an estimate similar to (8.194) holds for each α = 0, 1, 2, 3, that is, we have: max |ρα | ≤ Cδ0 (1 + t)−3 [1 + log(1 + t)] α

(13.442)

Since 2T ψα = Lψα − α −1 κ Lψα , (13.440) implies:

where:

(L + ν)T ψα = τα

(13.443)

2τα = ρα − (L + ν)(α −1 κ Lψα )

(13.444)

Now, we have: (L + ν)(α −1 κ Lψα ) = α −1 κ(L)2 ψα + ((L + ν)(α −1 κ))Lψα QQ Q From the propagation equation (3.99) for κ and the assumptions E{0} , E{0} we deduce:

max |(L + ν)(α −1 κ Lψα )| ≤ Cδ0 (1 + t)−3 [1 + log(1 + t)] α

(13.445)

Part 2: The error estimates of the acoustical entities

807

Combining (13.442) and (13.445) we obtain: max |τα | ≤ Cδ0 (1 + t)−3 [1 + log(1 + t)] α

(13.446)

This estimate actually relies only on the assumptions of Proposition 12.6. Similarly, under the assumptions of Proposition 12.9 with l = 1 together with those of Proposition 12.10 with m = l = 0, we deduce: max τα ∞,[1], ε0 ≤ Cδ0 (1 + t)−3 [1 + log(1 + t)] α

t

(13.447)

Next, we consider the factor ω/ν. Setting: ν =

1 (trχ  + L log ) 2

(13.448)

we have:

η0 + ν (1 − u + η0 t) Recalling that ω = 2η0 (1 + t) we then obtain: ν=

(13.449)

(L − 2ν)(ω/ν) = γ ν −2 ω where: η0 (1 − u − η0 ) γ = + (1 + t)(1 − u + η0 t)2



(13.450)

4η0 1 − 1+t 1 − u + η0 t



− Lν 

(13.451)

Proposition 12.9 with l = 1 implies: γ ∞,[1], ε0 ≤ Cδ0 (1 + t)−3 [1 + log(1 + t)] t

(13.452)

By (13.443) and (13.450) we have, in regard to the integrant in (13.439),   (L + ν) (ω/ν)(T ψα )(Ril+1 . . . Ri1 trχ  )   (13.453) = (ω/ν) (T ψα )(L + 2ν)(Ril+1 . . . Ri1 trχ  ) + τ˜α (Ril+1 . . . Ri1 trχ  ) where:

τ˜α = τα + γ ν −1 T ψα

Note that by (13.447) and (13.452) (and the estimate for 12.9 with l = 1) we have:

(13.454) ν

resulting from Proposition

max τ˜α ∞,[1], ε0 ≤ Cδ0 (1 + t)−3 [1 + log(1 + t)] α

t

(13.455)

Substituting the expression (13.455) in (13.439), we are thus to estimate the spacetime integral: (Ril+1 . . . Ri1 ψα )(ω/ν) (13.456) Wut

  × (T ψα )(L + 2ν)(Ril+1 . . . Ri1 trχ  ) + τ˜α (Ril+1 . . . Ri1 trχ  ) dt  du  dµh/˜

808

Chapter 13. Error Estimates

Writing: (L + 2ν)Ril+1 . . . Ri1 trχ  = Ril+1 (L + 2ν)Ril . . . Ri1 trχ  +

(Ril+1 )

(13.457)



Z Ril . . . Ri1 trχ − 2(Ril+1 ν)(Ril . . . Ri1 trχ  )

we integrate by parts on each St,u using (13.322), first taking X = Ril+1 to obtain: (ω/ν)(T ψα )(Ril+1 . . . Ri1 ψα )Ril+1 ((L + 2ν)Ril . . . Ri1 trχ  )dµh/˜ St,u



(ω/ν)(T ψα )(Ril+1 Ril+1 . . . Ri1 ψα )(L + 2ν)Ril . . . Ri1 trχ  dµh/˜

=−

St,u



St,u

−

(ω/ν)(Ril+1 T ψα )(Ril+1 . . . Ri1 ψα )(L + 2ν)Ril . . . Ri1 trχ  dµh/˜ 

1 − (T ψα ) Ril+1 (ω/ν) + (ω/ν)tr 2 St,u

π /

(Ril+1 ) ˜

× (Ril+1 . . . Ri1 ψα )(L + 2ν)Ril . . . Ri1 trχ  dµh/˜ and then taking X = (Ril+1 ) Z to obtain: (ω/ν)(T ψα )(Ril+1 . . . Ri1 ψα ) St,u



=−

(

(Ril+1 )

St,u

−

(ω/ν)(

St,u

− St,u

(T ψα )

(Ril+1 )

(13.458)

Z (Ril . . . Ri1 trχ  )dµh/˜

Z · d/ Ril+1 . . . Ri1 ψα )(ω/ν)(T ψα )(Ril . . . Ri1 trχ  )dµh/˜ (Ril+1 )

Z · d/T ψα )(Ril+1 . . . Ri1 ψα )(Ril . . . Ri1 trχ  )dµh/˜

(Ril+1 )

Z · d/(ω/ν) + (ω/ν)div /˜

(Ril+1 )

 Z

× (Ril+1 . . . Ri1 ψα )(Ril . . . Ri1 trχ  )dµh/˜

(13.459)

Also, taking again X = Ril+1 , (Ril+1 . . . Ri1 ψα )(ω/ν)τ˜α Ril+1 . . . Ri1 trχ  dµh/˜ St,u



=−

St,u



St,u

− − St,u

(ω/ν)τ˜α (Ril+1 Ril+1 . . . Ri1 ψα )(Ril . . . Ri1 trχ  )dµh/˜ (ω/ν)(Ril+1 τ˜α )(Ril+1 . . . Ri1 ψα )(Ril . . . Ri1 trχ  )dµh/˜ 

τ˜α (Ril+1

1 . . . Ri1 ψα ) Ril+1 (ω/ν) + (ω/ν)tr 2

(13.460) π / (Ril . . . Ri1 trχ  )dµh/˜

(Ril+1 ) ˜

In view of (13.458)–(13.460) the spacetime integral (13.456) becomes: −V0 − V1 − V2

(13.461)

Part 2: The error estimates of the acoustical entities

809

where: V0 = V0,0 + V0,1

V0,0 =

Wut

(ω/ν)(T ψα )(Ril+1 Ril+1 . . . Ri1 ψα )(L + 2ν)Ril . . . Ri1 trχ  dt  du  dµh/˜ (13.463)

V0,1 =

(13.462)

Wut

+ (T ψα )

(Ril+1 )

(13.464)



Z · d/ Ril+1 . . . Ri1 ψα Ril . . . Ri1 trχ  dt  du  dµh/˜

V1 = V1,0 + V1,1

V1,0 =

(ω/ν) τ˜α (Ril+1 Ril+1 . . . Ri1 ψα )

Wut

(13.465)

(ω/ν)(Ril+1 T ψα )(Ril+1 . . . Ri1 ψα )(L + 2ν)Ril . . . Ri1 trχ  dt  du  dµh/˜ (13.466)

V1,1 =

Wut

(ω/ν) Ril+1 τ˜α +

(Ril+1 )

Z · d/T ψα

 (13.467)

· (Ril . . . Ri1 ψα )(Ril . . . Ri1 trχ  )dµh/˜

and: V2 = V2,0 + V2,1 V2,0

(13.468)



1 (Ri ) ˜ l+1 = (T ψα ) Ril+1 (ω/ν) + tr π / 2 Wut

(13.469)

· (Ril+1 . . . Ri1 ψα )(L + 2ν)Ril . . . Ri1 trχ  dt  du  dµh/˜

V2,1 =

Wut

(Ril+1 . . . Ri1 ψα )(Ril . . . Ri1 trχ  ) · 

 1 τ˜α Ril+1 (ω/ν) + (ω/ν)tr 2

+(T ψα )

(Ril+1 )

 π /

(Ril+1 ) ˜

Z · d/(ω/ν) + (ω/ν)div /˜

(13.470) (Ril+1 )

 Z

dt  du  dµh/˜

In the following we shall focus attention on the two leading integrals V0,0 and V1,0 . In view of the estimates (13.327), (13.455) and the estimate max  j

(R j )

Z ∞,[1], ε0 ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] t

(13.471)

of Corollary 10.1.i with l = 2, the other integrals contain decay factors of at least δ0 (1 + t)−1 [1 + log(1 + t)] relative to the leading integrals. We first consider the integral V0,0 . We have: |V0,0 | ≤ C (1 + t  )3 |T ψα ||d/ Ril+1 . . . Ri1 ψα ||(L + 2ν)Ril . . . Ri1 trχ  |dt  dudµh/ Wεt

0

(13.472)

810

Chapter 13. Error Estimates

From the propagation equation (13.332) we have: (L + 2ν)Ril . . . Ri1 trχ  = where:

(i1 ...il )

(i1 ...il )

ρ˜l =

(i1 ...il )

ρ˜l

(13.473)

ρl + 2ν  Ril . . . Ri1 trχ 

(13.474)

We write, as in (13.353), (i1 ...il )

(i1 ...il )

ρ˜l =

where: (i1 ...il )

(2)

ρ˜ l =

In view of the bound (13.329), estimate (13.354) for

(0) ρl

(i1 ...il )

(i1 ...il )

(2) (i1 ...il ) ρ . l

+ (2) ρl

(i1 ...il )

(1) ρl

+

(i1 ...il )

(2)

ρ˜ l

(13.475)

+2ν  Ril . . . Ri1 trχ 

(13.476)

(2)

ρ˜ l satisfies an estimate identical in form to the (0)

The borderline contribution to (13.472) is the contribution from (i1 ...il ) ρ l . This is the borderline integral: (0) C (1 + t  )3 |T ψα ||d/ Ril+1 . . . Ri1 ψα || (i1 ...il ) ρ l |dt  dudµh/ Wεt

0

t

≤C

  (1 + t  )3 sup µ−1 |T ψα |

(13.477)

ε

0

t 0

× µ1/2 d/ Ril+1 . . . Ri1 ψα  L 2 ( ε0 ) µ1/2 t

(i1 ...il )



(0) ρl

 L 2 ( ε0 ) dt  

t

Here we substitute the estimates (13.198) for sup ε0 µ−1 |T ψα | and the estimate (13.359) for

µ1/2 (i1 ...il )

(0) ρl

t

 L 2 ( ε0 ) . The factors t

k 3 ||

then cancel. Now the partial

contribution of the second term on the right in (13.198) is actually not borderline. We shall show how to estimate contributions of this type afterwards, in connection with the (1)

estimate for the contribution of (i1 ...il ) ρ l . For the present we focus attention on the actual borderline integral, which represents the partial contribution of the first term on the right in (13.198): t )   −1   C sup µ |Lµ| E1 [Ril+1 . . . Ri1 ψα ](t ) E1 [R j Ril . . . Ri1 ψα ](t  )dt  0  ε0 t



≤C ≤C

j,α t

  sup µ−1 |Lµ| E1 [R j Ril . . . Ri1 ψα ](t  )dt 

0  ε0 t t

    2q sup µ−1 |Lµ| µ−2a m (t )[1 + log(1 + t ]

0  ε0 t

(13.478)

j,α (i1 ...il )  G1,l+2;a,q (t  )dt 

Part 2: The error estimates of the acoustical entities

811

This is formally identical to the borderline integral (13.206), with q replacing p and (i1 ...il ) G  (i1 ...il ) G 0,l+2;a, p . Thus, from the estimates (13.211) and 1,l+2;a,q replacing (13.216) we conclude that (13.478) is bounded by:   1 1 2q (i1 ...il )  + C µ−2a G1,l+2;a,q (t) (13.479) m (t)[1 + log(1 + t)] 2a 2q (1)

We proceed to consider the contribution of (i1 ...il ) ρ l to (13.472). To estimate this and all remaining contributions we simply appeal to the estimate (13.199) for T ψα . Using the estimate (13.360) we then obtain that the contribution in question is bounded by: (1) C (1 + t  )3 |T ψα ||d/ Ril+1 . . . Ri1 ψα || (i1 ...il ) ρ l |dt  dudµh/ Wεt

0



t

≤ Cδ0 ≤ Cδ02 ≤

Cδ02

t 0 t 0

(i1 ...il )

t

0



 1/2 (1 + t  )2 µ−1 d/ Ril+1 . . . Ri1 ψα  L 2 ( ε0 ) µ1/2 m (t )µ

(1) ρl

 L 2 ( ε0 ) dt  t

    (1 + t  )−2 µ−1 m (t )E1,[l+2] (t )dt  (1 + t  )−2 [1 + log(1 + t  )]2q µ−1−2a (t  )G1,[l+2];a,q (t  )dt  m

(13.480)

The last integral is estimated in a similar way as the integral (13.224). We find that (13.480) is bounded by: 2q Cδ02 µ−2a m (t)[1 + log(1 + t)]  t    −2    × ϕ1 (Caδ0 )G1,[l+2];a,q (t) + (1 + t ) G1,[l+2];a, p (t )dt

(13.481)

0

We consider next the contribution to (13.472) of the leading term Cl (1 + t)−2 W{l} Q

in the estimate for 

(i1 ...il )

(2)

ρ˜ l  L 2 ( ε0 ) (see (13.354)). This contribution is bounded by: t



t

−1/2

µm (t  )µ1/2 d/ Ril+1 . . . Ri1 ψα  L 2 ( ε0 ) W{l} (t  )dt  t t −2a−1/2  ≤ C l ε 0 δ0 (1 + t  )−1 µm (t )[1 + log(1 + t  )] p+q 0   × G1,[l+2];a;q (t  )G0,[l+2];a, p (t  )dt    = Cl ε0 δ0 J2a, p+q−1 (t) G1,[l+2];a,q (t)G0,[l+2];a, p (t)

C l δ0

Q

0

(13.482)

where Ja,q (t) is the integral (13.260). According to (13.266) we have: −2a+1/2

J2a, p+q−1(t) ≤ C[1 + log(1 + t)] p+q+1 µm

(t)

(13.483)

812

Chapter 13. Error Estimates

hence (13.482) is bounded by: −2a+1/2

C l ε 0 δ 0 µm

  (t)[1 + log(1 + t)] p+q+1 G1,[l+2];a,q (t)G0,[l+2];a, p (t)

(13.484)

Finally, we consider the integral V1,0 , given by (13.466). By the bound (13.432) we have:

|V1,0| ≤ Cδ0

Wεt t

≤ Cδ0 0

(1 + t  )|Ril+1 . . . Ri1 ψα ||

(i1 ...il )

ρ˜l |dt  dudµh/

0

(1 + t  )W{l+1} (t  )

(i1 ...il )

ρ˜l  L 2 ( ε0 ) dt  t

Here we need only consider the leading principal contribution, namely that of   L 2 ( ε0 ) . By (13.357) and (13.437) this contribution is bounded by:

(13.485) (i1 ...il )

(0)

ρ˜ l

t



t

Cε0 δ0 0

−2a−1/2

(1 + t  )−1 µm

  (t  )[1 + log(1 + t  ] p+q G0,[l+2];a, p (t  )G1,[l+2];a,q (t  )dt  (13.486)

This is identical in form to (13.482), which has already been bounded by (13.484). This completes the estimates for the spacetime integrals V0 , V1 , V2 . We have thus completed the estimate of the contribution of the leading term (13.165) to the corresponding error integral (13.168), associated to K 1 . We now consider the contribution of (13.166) to the corresponding integral (13.168). Recalling that this is associated to the variation (13.163), the contribution in question is: − (ω/ν)(T ψα )(Ril−m . . . Ri1 (T )m
/ µ)(L + ν)(Ril−m . . . Ri1 (T )m+1 ψα )dt  du  dµh/˜ Wut

(13.487)

Again, one may think at first sight that this integral can be directly estimated using the flux F1 [Ril−m . . . Ri1 (T )m+1 ψα ] associated to the vectorfield K 1 and the variation Ril−m . . . Ri1 (T )m+1 ψα . Such an estimate would however involve the L 2 norm of / µ, a top order acoustical entity, on the spacetime region Wut , and our Ril−m . . . Ri1 (T )m
estimate (13.277) for this entity would only allow us to bound its L 2 norm on Wut by a √ −a−1/2 quantity growing with t like tµm . Instead, we proceed in a similar way as in our treatment of the integral (13.315). That is, we write the integrant in (13.487) in the form: −(ω/ν)(T ψα )(Ril−m . . . Ri1 (T )m
/ µ)((L + ν)Ril−m . . . Ri1 (T )m+1 ψα ) 

/ µ)(Ril−m . . . Ri1 (T )m+1 ψα ) = −(L + 2ν) (ω/ν)(T ψα )(Ril−m . . . Ri1 (T )m
  / µ) + (Ril−m . . . Ri1 (T )m+1 ψα )(L + ν) (ω/ν)(T ψα )(Ril−m . . . Ri1 (T )m


Part 2: The error estimates of the acoustical entities

813

By (13.319) with the function (ω/ν)(T ψα )(Ril−m . . . Ri1 (T )m
/ µ)(Ril−m . . . Ri1 (T )m+1 ψα ) in the role of the function f , we then conclude that the integral (13.487) is equal to: (ω/ν)(T ψα )(Ril−m . . . Ri1 (T )m
/ µ)(Ril−m . . . Ri1 (T )m+1 ψα )du  dµh/˜ −

+ +

tu 0u

Wut

(ω/ν)(T ψα )(Ril−m . . . Ri1 (T )m
/ µ)(Ril−m . . . Ri1 (T )m+1 ψα )du  dµh/˜ (Ril−m . . . Ri1 (T )m+1 ψα )(L + ν)   × (ω/ν)(T ψα )(Ril−m . . . Ri1 (T )m
/ µ) dt  du  dµh/˜

We first consider the hypersurface integral: − (ω/ν)(T ψα )(Ril−m . . . Ri1 (T )m
/ µ)(Ril−m . . . Ri1 (T )m+1 ψα )du  dµh/˜ tu

(13.488)

(13.489)

(The other hypersurface integral is in any case expressible in terms of the initial data). We define the St,u -tangential vectorfields: (i1 ...il−m )

Ym,l−m = (d/ Ril−m . . . Ri1 (T )m µ)$

(13.490)

Then with (i1 ...il−m )

rm,l−m = Ril−m . . . Ri1 (T )m
/µ −
/ Ril−m . . . Ri1 (T )m µ

(13.491)

noting that, since h /˜ = h /, for any St,u -tangential vectorfield X: div /˜ X = div / X + X · d/ log 

(13.492)

we have: Ril−m . . . Ri1 (T )m
/ µ = div /˜

(i1 ...il−m )

Ym,l−m +

(i1 ...il−m )

r˜m,l−m

(13.493)

Ym,l−m · d/ log 

(13.494)

where: (i1 ...il−m )

r˜m,l−m =

(i1 ...il−m )

rm,l−m −

(i1 ...il−m )

For (i1 ...il−m )rm,l−m formulas analogous to (11.1046), (11.1048) of Chapter 11 hold with the function µ in the role of the function ψα , that is, we have:  $ (i1 ...il−m ) rm,l−m = tr (i1 ...il−m ) cm,l−m [d/µ]  m  m! ((L /T )k (h/−1 )) · (L + tr L / Ril−m . . . L / R i1 /T )m−k D / 2µ k!(m − k)! k=1   s1 −1 s2 s2 2 + tr ((L / R ) (h/ )) · (L / R ) (L /T ) D / µ (13.495) partitions |s1 |+|s2 |=l−m,s1 =∅

814

Chapter 13. Error Estimates ε

We shall presently estimate, in L 2 (t 0 , (i1 ...il−m )rm,l−m and (i1 ...il−m )r˜m,l−m . First, taking k = m − 1 in the estimates (11.994) and (11.995) and then replacing m − 1 by m, we obtain, since µ0,0 = µ (see (11.985)): D / 2 µ∞,[m,l∗ −1], ε0 ≤ Cl δ0 (1 + t)−2 [1 + log(1 + t)]

(13.496)

t

and:

D / 2 µ2,[m,l−1], ε0 ≤ Cl (1 + t)−2 B[m,l+1] t

+δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] Q +W{l} + (1 + t)−2 [1 + log(1 + t)]2 W{l−1}

 (13.497)

Using then, as in (11.1049), Corollary 11.1.c with l∗ in the role of l and the bound (13.496) together with Corollary 11.2.c and the bound (13.497) with m − 1 in the role of m, we ε deduce that the second term on the right in (13.495) (the first sum) is bounded in L 2 (t 0 ) by:

Cl (1 + t)−1 [1 + log(1 + t)]D / 2 µ2,[m−1,l−1], ε0 t  +δ0 (1 + t)−2 [1 + log(1 + t)] (T ) π / + (1 − u + η0 t)−1 h/2,[m−1,l−1], ε0 t  1 ≤ Cl (1 + t)−3 [1 + log(1 + t)] B[m−1,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q (13.498) +W{l} + (1 + t)−1 [1 + log(1 + t)]W{l−1} Also, using, as in (11.1050), Corollary 10.1.d with l∗ in the role of l and the estimate (13.496) together with Corollary 10.2.d and the estimate (13.497), we deduce that the ε third term on the right in (13.495) (the second sum) is bounded in L 2 (t 0 ) by:

/ 2 µ2,[m,l−1], ε0 Cl δ0 (1 + t)−1 [1 + log(1 + t)]D t +δ0 (1 + t)−2 [1 + log(1 + t)] max  (R j ) π /2,[l−1], ε0 t j

−2 −1 ≤ Cl δ0 (1 + t) [1 + log(1 + t)] (1 + t) B[m,l+1]  Q (13.499) +Y0 + (1 + t)A[l−1] + W{l} + δ0 (1 + t)−2 W{l−1} ε

Finally, to estimate in L 2 (t 0 ) the first term on the right in (13.495) we apply the second statement of Corollary 11.2.g to the St,u 1-form d/µ. This yields a bound by:

Cl δ0 (1 + t)−2 [1 + log(1 + t)] (1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l]  Q (13.500) +W{l+1} + (1 + t)−2 [1 + log(1 + t)]2 W{l}

Part 2: The error estimates of the acoustical entities

815

Combining (13.498)–(13.500) and substituting for A[l] in terms of A[l] from Lemma 12.5 we conclude that:

 (i1 ...il−m )rm,l−m  L 2 ( ε0 ) ≤ Cl (1 + t)−2 [1 + log(1 + t)] (1 + t)−1 B[m,l+1] t  Q  (13.501) + δ0 Y0 + (1 + t)A[l] + W{l+1} + (1 + t)−2 [1 + log(1 + t)]2 W{l} Moreover, since 

(i1 ...il−m )

Ym,l−m  L 2 ( ε0 ) ≤ C(1 + t)−1 B[m,l+1] t

while

d/ log  L ∞ ( ε0 ) ≤ Cδ0 (1 + t)−2 t

(13.502)

(13.503)

(i1 ...il−m ) r˜ m,l−m

satisfies a similar estimate as (i1 ...il−m )rm,l−m . Substituting now (13.493) in (13.489) the hypersurface integral becomes: (ω/ν)(T ψα )div /˜ (i1 ...il−m ) Ym,l−m (Ril−m . . . Ri1 (T )m+1 ψα )du  dµh/˜ − tu



−

tu

(ω/ν)(T ψα )

(i1 ...il−m )

r˜m,l−m (Ril−m . . . Ri1 (T )m+1 ψα )du  dµh/˜

(13.504)

Now, if f is an arbitrary function defined on St,u and X an arbitrary vectorfield defined on and tangential to St,u , we have: ˜ f div / Xdµh/˜ = − X · d/ f dµh/˜ (13.505) St,u

St,u

Applying (13.505) to the first of the integrals in (13.504), taking X = (i1 ...il−m ) Ym,l−m and f = (ω/ν)(T ψα )(Ril−m . . . Ri1 (T )m+1 ψα ), we obtain that (13.504) is equal to the sum of three hypersurface integrals: H0 + H1 + H2 where: H0

=

H1 = H2



=

tu

tu

tu

(ω/ν)(T ψα )

(i1 ...il−m )

(ω/ν)(d/T ψα ) ·

(i1 ...il−m )

(T ψα ) d/(ω/ν) · −(ω/ν)

Ym,l−m · d/(Ril−m . . . Ri1 (T )m+1 ψα )du  dµh/˜ Ym,l−m (Ril−m . . . Ri1 (T )m+1 ψα )du  dµh/˜

(i1 ...il−m )

(i1 ...il−m )

(13.506)

(13.507)

(13.508)

Ym,l−m

 r˜m,l−m (Ril−m . . . Ri1 (T )m+1 ψα )du  dµh/˜

(13.509)

816

Chapter 13. Error Estimates

Now, by (13.501) and (13.502) (see also (13.327)), Ym,l−m · d/(ω/ν) − (i1 ...il−m )r˜m,l−m  L 2 ( ε0 ) t

1 −2 −1 ≤ Cl (1 + t) [1 + log(1 + t)] (1 + t) B[m,l+1] + δ0 Y0 + (1 + t)A[l]  Q (13.510) +W{l+1} + (1 + t)−2 [1 + log(1 + t)]2 W{l}

(ν/ω)

(i1 ...il−m )

Comparing on one hand (13.509) with (13.508) and on the other (13.510) with (13.502), we see that the integral H2 has an extra decay factor of (1 +t)−1 [1 +log(1 +t)] relative to the integral H1 . (We have taken account of the fact that d/T ψα decays faster than T ψα by a factor of (1 + t)−1 .) We shall therefore confine attention in the following to the integrals H0 and H1 . We first consider the integral H0 . We have, recalling the definition (13.490),  |H0| ≤ C ε (1 + t)2 |T ψα ||d/ Ril−m ... Ri1 (T )m µ||d/ Ril−m ... Ri1 (T )m+1 ψα |dudµh/ t 0

≤C

ε

t 0

   (1 + t)|T ψα |  |R j Ril−m ... Ri1 (T )m µ| |d/ Ril−m ... Ri1 (T )m+1 ψα |dudµh/ j

(13.511) Let us set j = i l−m+1 . To proceed we must obtain a sharper estimate for Ril−m+1 . . . Ri1 (T )m µ than that of Proposition 12.11. We shall presently derive the required estimate. Writing, as in (12.345), (12.369) of Chapter 12: (i1 ...il−m+1 )

µm,l−m+1 = Ril−m+1 . . . Ri1 (T )m µ (here m = 0, . . . , l)

(13.512)

we have the propagation equations (12.346), for m = 0, and (12.372), for m ≥ 1, which we can write in combined form as: L

(i1 ...il−m+1 )

µm,l−m+1 =

(i1 ...il−m+1 )  ρm,l−m+1

(13.513)

where: (i1 ...il−m+1 )  ρm,l−m+1

=e

(i1 ...il−m+1 )

µm,l−m+1 + Ril−m+1 . . . Ri1 (T )m m

+µRil−m+1 . . . Ri1 (T ) e + m

+ +

m−1  k=0 l−m 

(13.514)

(i1 ...il−m+1 )  rm,l−m+1

Ril−m+1 . . . Ri1 (T )k (T )m−1−k µ Ril−m+1 . . . Ril−m−k+2

(Ril−m−k+1 )

Z Ril−m−k . . . Ri1 (T )m µ

k=0

Here we have defined the functions: (i1 ...il−m+1 )  rm,l−m+1

=

(i1 ...ilm +1 )  rm,l−m+1

− (Ril−m+1 . . . Ri1 (T )m e)µ

(13.515)

Part 2: The error estimates of the acoustical entities

817

  where the (i1 ...il−m+1 )rm,l−m+1 are given by (13.369). The functions (i1 ...il−m+1 )rm,l−m+1 are of order l + 1 and contain spatial derivatives of µ of order at most l of which at most m are T -derivatives. Explicitly, for m = 0:  (i1 ...il+1 )  r0,l+1 = ((R)s1 e)((R)s2 µ) (13.516) partitions

|s1 |+|s2 |=l+1,|s1 |,|s2 | =0

and for m ≥ 1: (i1 ...il−m+1 )  rm,l−m+1



=

((R)s1 e)((R)s2 (T )m µ)

(13.517)

partitions

|s1 |+|s2 |=l−m+1,|s1 | =0



+

((R)s1 (T )m e)((R)s2 µ)

partitions

|s1 |+|s2 |=l−m+1,|s2 | =0

+Ril−m+1 . . . Ri1

m−1  k=1

m! ((T )k e)((T )m−k µ) k!(m − k)!



As in (10.694) of Chapter 10 we write: m = m αT (T ψα )

(13.518)

and we express: Ril−m+1 . . . Ri1 (T )m m = m αT Ril−m+1 . . . Ri1 (T )m+1 ψα +

(i1 ...il−m+1 )

(0) n˜  m,l−m+1

(13.519) where: (i1 ...il−m+1 )

(0) n˜  m,l−m+1



=

((R)s1 m αT )((R)s2 (T )m+1 ψα )

(13.520)

partitions

|s1 |+|s2 |=l−m+1,|s1 | =0



+Ril−m+1 . . . Ri1

m  k=1

m! ((T )k m αT )((T )m−k+1 ψα ) k!(m − k)!



Also, in analogy with (10.696) we write: e = eαL (Lψα )

(13.521)

where the coefficients eαL are the functions (see equation (3.98) of Chapter 3): eαL =

1 dH ψ L (ψ L + 2αψTˆ )ψ α − Fψ0 L α α 2 dσ

(13.522)

818

Chapter 13. Error Estimates

and we have: |eαL | ≤ C

(13.523)

We then express: Ril−m+1 . . . Ri1 (T )m e = eαL Ril−m+1 . . . Ri1 (T )m Lψα +

(i1 ...il−m+1 )

(1) n˜  m,l−m+1

(13.524)

where: (i1 ...il−m+1 )

(1) n˜  m,l−m+1



=

((R)s1 eαL )((R)s2 (T )m Lψα )

(13.525)

partitions

|s1 |+|s2 |=l−m+1,|s1 | =0



+Ril−m+1 . . . Ri1

m  k=1

m! ((T )k eαL )((T )m−k Lψα ) k!(m − k)!



We define: (i1 ...il−m+1 )

(0) ρ  m,l−m+1 =

k 3 Ril−m+1 . . . Ri1 (T )m+1 ψ0

(13.526)

and: (i1 ...il−m+1 )

(1) ρ  m,l−m+1

= (m 0T − k 3 )Ril−m+1 . . . Ri1 (T )m+1 ψ0 +m iT Ril−m+1 . . . Ri1 (T )m+1 ψi +(1 + t)−1 µeαL Ril−m+1 . . . Ri1 (T )m Qψα

(13.527)

We then have: Ril−m+1 . . . Ri1 (T )m m + µRil−m+1 . . . Ri1 (T )m e =

(0) (i1 ...il−m+1 )  ρ m,l−m+1

+

(13.528)

(1) (i1 ...il−m+1 )  ρ m,l−m+1

+

(i1 ...il−m+1 )  n˜ m,l−m+1

where: (i1 ...il−m+1 )  n˜ m,l−m+1

=

(i1 ...il−m+1 )

(0) n˜  m,l−m+1

+µ

(i1 ...il−m+1 )

(1) n˜  m,l−m+1

(13.529)

and (13.514) takes the form: (i1 ...il−m+1 )  ρm,l−m+1

=

(0) (i1 ...il−m+1 )  ρ m,l−m+1

(13.530) +

(1) (i1 ...il−m+1 )  ρ m,l−m+1

+

(2) (i1 ...il−m+1 )  ρ m,l−m+1

Part 2: The error estimates of the acoustical entities

819

where: (i1 ...il−m+1 )

(2) ρ  m,l−m+1

=e

(i1 ...il−m+1 )

µm,l−m+1 (i1 ...il−m+1 )  + n˜ m,l−m+1 m−1 

(13.531) (i1 ...il−m+1 )  rm,l−m+1

+

Ril−m+1 . . . Ri1 (T )k (T )m−1−k µ

+

k=0

+

l−m 

Ril−m+1 . . . Ril−m−k+2

(Ril−m−k+1 )

Z Ril−m−k . . . Ri1 (T )m µ

k=0 (2)

ε

We must now obtain an L 2 (t 0 ) estimate for (i1 ...il−m+1 ) ρ  m,l−m+1 analogous to ε (13.354). The first term on the right in (13.531) is bounded in L 2 (t 0 ) by: Cδ0 (1 + t)−1 B[m,l+1]

(13.532)

From (13.530) we readily deduce:  To estimate

(i1 ...il−m+1 )

(i1 ...il−m+1 )

(0) n˜  m,l−m+1

(1) n˜  m,l−m+1

 L 2 ( ε0 ) ≤ Cδ0 (1 + t)−1 W{l+1}

(13.533)

t

we write (see (13.522)): α eαL = eα L − Fψ0 L

(13.534)

The contribution of the term eα L through (13.525) to  readily seen to be bounded by:

Q Cl δ0 (1 + t)−2 W{l+1} + W{l}

(i1 ...il−m+1 )

(1) n˜  m,l−m+1

+δ0 (1 + t)−1 (1 + t)−1 B[m−1,l+1] + Y0 + (1 + t)A[l]

 L 2 ( ε0 ) is t

 (13.535)

To estimate the contribution of the term −Fψ0 L α through the first sum in (13.525), we write as in (10.381) of Chapter 10: Lj =

η0 x j + zj 1 − u + η0 t

and use the expression (10.94): (R)s1 x j = Rin p . . . Rin1 x j =

( p) j x in ...in p 1 ε

−

( p) j δin ...in , p 1

s1 = {n 1 , . . . , n p }, p = |s1 | ≤ l + 1, and the L ∞ (t 0 ) bound (10.53) for well as the bound of Corollary 10.2.a for

( p) δ j in 1 ...in p

( p) x j in 1 ...in p

as

and the bound of Corollary 10.2.g

820

Chapter 13. Error Estimates

with l replaced by l + 1 for z j . We obtain in this way that the contribution in question is bounded by:

  Q + δ0 (1 + t)−1 Y0 + (1 + t)A[l] (13.536) Cl δ0 (1 + t)−2 W{l+1} + W{l} To estimate the contribution of the term −Fψ0 L α through the second sum in (13.525), we simply apply Corollary 11.2.a with m reduced to m − 1. This yields a bound for the contribution in question by:

1 Q + (1 + t)−1 B[m−1,l+1] Cl δ0 (1 + t)−2 W{l+1} + W{l} 2  (13.537) +δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1] Combining (13.535)–(13.537) we conclude that: 

(i1 ...il−m+1 )

(1) n˜  m,l−m+1

 L 2 ( ε0 ) t

≤ Cl δ0 (1 + t)−2 (13.538)

1 2  Q × W{l+1} + W{l} + (1 + t)−1 B[m−1,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l] Combining in turn (13.533) and (13.538) we conclude that: 

(i1 ...il−m+1 )  n˜ m,l−m+1  L 2 ( ε0 ) t

Q ≤ Cδ0 W{l+1} + (1 + t)−1 [1 + log(1 + t)]W{l} (13.539) 1  2 +(1 + t)−2 [1 + log(1 + t)] B[m−1,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l]

We turn to the third term on the right in (13.531). Using the estimate (11.690) of Chapter 11 with (m + 1, l + 1) reduced to (m, l) we obtain, through (13.516), (13.517), 

(i1 ...il−m+1 )  rm,l−m+1  L 2 ( ε0 ) t

  Q ≤ Cδ0 [1 + log(1 + t)](1 + t)−1 W{l} + δ0 (1 + t)−1 Y0 + (1 + t)A[l−1] + W{l}  +(1 + t)−2 B[m,l] (13.540) ε

Finally, the two sums on the right in (13.531) have already been estimated in L 2 (t 0 ) in Chapter 12, the first by (see (12.485)):    Cl δ0 (1 + t)−2 [1 + log(1 + t)] B[m−1,l+1] + δ0 [1 + log(1 + t)] Y0 + (1 + t)A[l−1]  Q +[1 + log(1 + t)] W{l+1} + δ0 (1 + t)−2 [1 + log(1 + t)]2 W{l} (13.541) and the second by (see (12.487)): Cl δ0 (1 + t)−1 [1 + log(1 + t)]

 Q × (1 + t)−1 B[m,l+1] + Y0 + A[l] + W{l+1} + δ0 (1 + t)−1 W{l}

(13.542)

Part 2: The error estimates of the acoustical entities

821

Combining the results (13.532), (13.539), (13.540), (13.541), (13.542), for the five terms on the right in (13.531) we conclude that: 



(i1 ...il−m j )

(2) ρ  m,l−m+1

 L 2 ( ε0 ) t

j

≤ Cl δ0 (1 + t)−1 [1 + log(1 + t)] 

Q × W{l+1} + W{l} + (1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l]

(13.543)

Moreover, from (13.526), (13.527), we have, in analogy with (13.357)–(13.360), 



(i1 ...il−m j )

(0) ρ  m,l−m+1

−1/2

 L 2 ( ε0 ) ≤ Ck 3 ||µm

)

t

α

j

E1 [Ril−m . . . Ri1 (T )m+1 ψα ](t) (13.544)





(i1 ...il−m j )

(1) ρ  m,l−m+1

 L 2 ( ε0 )

j

≤

−1/2 Cδ0 (1 + t)−1 µm

(13.545)

t

) α

E1 [Ril−m . . . Ri1 (T )m+1 ψα ](t)

+ Cδ0 (1 + t)−1 [1 + log(1 + t)]1/2

) α

E1 [Ril−m . . . Ri1 (T )m Qψα ](t)

and: 

µ

1/2 (i1 ...il−m j )

(0) ρ  m,l−m+1

3

 L 2 ( ε0 ) ≤ Ck ||

)

t

j

α

E1 [Ril−m . . . Ri1 (T )m+1 ψα ](t) (13.546)



µ1/2

(i1 ...il−m j )

(1) ρ  m,l−m+1

j

≤ Cδ0 (1 + t)

−1

) α

+ Cδ0 (1 + t)

−1

 L 2 ( ε0 )

(13.547)

t

E1 [Ril−m . . . Ri1 (T )m+1 ψα ](t)

[1 + log(1 + t)]

) α

E1 [Ril−m . . . Ri1 (T )m Qψα ](t)

Comparing (13.543) to (13.544) we see that, in contrast to (13.354) compared to (13.357), the right-hand side of (13.543) contains no leading terms.

822

Chapter 13. Error Estimates

We now integrate the propagation equation (13.513) along each integral curve of L ε from 00 to obtain:  | (i1 ...il−m j ) µm,l−m+1 (t, u, ϑ)| (13.548) j

≤C



(i1 ...il−m j )

|

µm,l−m+1 (0, u, ϑ)| +

j

2 

(i1 ...il−m )



(k) 

A m,l−m (t, u, ϑ)

k=0

where: (i1 ...il−m )

(k) 

A m,l−m (t, u, ϑ) =

t 0

(i1 ...il−m j )

|

(k) ρ  m,l−m+1

|(t  , u, ϑ)dt 

j

: k = 0, 1, 2

(13.549)

By the comparison inequalities (8.355), (8.356) of Chapter 8 we have: 

(i1 ...il−m )

(k) 

A m,l−m (t) L 2 ([0,ε0 ]×S 2 ) t (k) ≤  (i1 ...il−m j ) ρ  m,l−m+1 (t  ) L 2 ([0,ε0]×S 2 ) dt  0



j t

≤

(1 + t  )−1



0



(i1 ...il−m j )

(k) ρ  m,l−m+1

 L 2 ( ε0 ) dt  t

j

(13.550)

Remark that (13.543)–(13.550) are analogous to (13.354), (13.357)–(13.364). We now substitute the pointwise estimate (13.548) in (13.511), noting that:   |R j Ril−m . . . Ri (T )m µ| = | (i1 ...il−m j ) µm,l−m+1 | j

j

The borderline contribution is the contribution from is the borderline integral: C

ε t 0

(1 + t)|T ψα |

(i1 ...il−m )

(0) 

(i1 ...il−m )

(0) A

m,l−m . This contribution

A m,l−m |d/ Ril−m . . . Ri1 (T )m+1 ψα |dudµh/

(13.551)

This is entirely analogous to the integral (13.365) and it is estimated in exactly the same manner. Defining (see (13.376)): (i1 ...il−m )  G1,m,l+2;a,q (t)



= sup

t  ∈[0,t ]

 [1 + log(1 + t  )]−2q µ2a m (t )

 α

(13.552)  E1 [Ril−m . . . Ri1 (T )m+1 ψα ](t  )

Part 2: The error estimates of the acoustical entities

823

we obtain (see (13.385), (13.412)) that the borderline integral (13.551) is bounded by:   1 1 2q (i1 ...il−m )  + µ−2a G1,m,l+2;a,q (t) (13.553) C m (t)[1 + log(1 + t)] a − 1/2 q + 1/2 The remaining contributions to (13.511) are estimated in exactly the same manner as the remaining contributions to (13.328). In view of the bound (13.413) these contributions are bounded by: C

ε t 0

(1 + t)|T ψα |

≤ Cδ0 ≤ Cδ0  ≤

ε t 0

(i1 ...il−m )

A m,l−m |d/ Ril−m . . . Ri1 (T )m+1 ψα |dudµh/

(k) 

(i1 ...il−m )

(i1 ...il−m )

(k) 

A m,l−m |d/ Ril−m ...Ri1 (T )m+1 ψα |dudµh/

(k) 

A m,l−m  L 2 ( ε0 ) d/ Ril−m . . . Ri1 (T )m+1 ψα  L 2 ( ε0 ) t

−1/2 Cδ0 µm (t) (i1 ...il−m )

(k) 

A m,l−m

t

  (t) L 2 ([0,ε0 ]×S 2 ) E1,[l+2] (t)

: for k = 1, 2

(13.554)

which is analogous to (13.414). Here we need only consider the contribution of (i1 ...il−m )

(1) A

(2)

because (i1 ...il−m ) A m,l−m besides being of lower order does not now contain any leading terms, as already remarked above. From (13.550) and the fact that by the estimate (13.545): 



m,l−n ,

(i1 ...il j )

(1) ρ  m,l−m+1

−1/2

 L 2 ( ε0 ) ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)]1/2 µm t

  (t) E1,[l+2] (t)

j

(13.555) we obtain: 

(i1 ...il )

(1) 

A m,l−m (t) L 2 ([0,ε0 ]×S 2 ) t  −1/2  ≤ Cδ0 µm (t  )(1 + t  )−2 [1 + log(1 + t  )]1/2 E1,[l+2] (t  )dt  0    ≤ Cδ0 Ja,q+1/2 (t) G1,[l+2];a,q (t) (13.556)

 (t) is the integral (13.417). Substituting for J  where Ja,q a,q+1/2 (t) the estimate (13.424) with q replaced by q + 1/2 and substituting the result in (13.554) then yields:

C

ε

t 0

(1 + t)|T ψα |

(i1 ...il−m )

(1) 

A m,l−m |d/ Ril−m . . . Ri1 (T )m+1 ψα |dudµh/

(13.557)

 −a−1/2   q ≤ Cδ02 G1,[l+2];a,q (t) Cq µm (t) + Cϕq+3/2 (Caδ0 )µ−2a m (t) [1 + log(1 + t)]

824

Chapter 13. Error Estimates

Finally, we consider the hypersurface integral H1 , given by (13.508). We have: |H1 | ≤ C

ε t 0

(1 + t)|d/T ψα |



|R j Ril−m . . . Ri1 (T )m µ||Ril−m . . . Ri1 (T )m+1 ψα |dudµh/

j

(13.558) By virtue of the bound max sup |d/T ψα | ≤ Cδ0 (1 + t)−2 α

(13.559)

ε

t 0

(following from the bound (13.432)) we have: |H1 | ≤ Cδ0

ε t 0

≤ Cδ0 (1 + t)−1 ≤ Cδ0



(1 + t)−1 

|R j Ril−m . . . Ri1 (T )m µ||Ril−m . . . Ri1 (T )m+1 ψα |dudµh/

j

Ril−m . . . Ri1 (T )m µ L 2 ( ε0 ) Ril−m . . . Ri1 (T )m+1 ψα  L 2 ( ε0 ) t

t

j



(R j Ril−m . . . Ri1 (T )m µ)(t) L 2 ([0,ε0 ]×S 2 ) W{l+1} (t)

(13.560)

j

which is analogous to (13.433). Now from (13.548), (13.550): 

(R j Ril−m . . . Ri1 (T )m µ)(t) L 2 ([0,ε0]×S 2 )

j

≤C

+

  

R j Ril−m . . . Ri1 (T )m µ L 2 ( ε0 ) 0

j

2 



(i1 ...il−m )



(k) 

A m,l−m (t) L 2 ([0,ε0]×S 2 )

k=0

≤C

+

  

R j Ril−m . . . Ri1 (T )m µ L 2 ( ε0 ) 0

j

2 

t

(1 + t  )−1

k=0 0





(i1 ...il−m

(k) j) 

ρ

m,l−m+1

 L 2 ( ε0 ) dt  t

j

  

(13.561)

which is analogous to (13.434). Here we need only consider the leading principal contribution, namely that of  j



(i1 ...il−m j )

(0) ρ  m,l−m+1

 L 2 ( ε0 ) . t

Part 2: The error estimates of the acoustical entities

825

This contribution to (13.561) is bounded by (see (13.544)): ) t 3  −1 −1/2  Ck || (1 + t ) µm (t ) E1 [Ril−m . . . Ri1 (T )m+1 ψα ](t  )dt  0

α

t  −a−1/2   ≤ Ck 3 || G1,[l+2];a,q (t) (1 + t  )−1 µm (t )[1 + log(1 + t  )]q dt  0  3  = Ck ||Ja,q−1 (t) G1,[l+2];a,q (t) (13.562) Substituting for Ja,q−1 (t) the bound (13.266) with q replaced by q − 1 and the resulting bound through (13.561) to (13.560), and substituting also the bound (13.437) for W{l+1} , we obtain that the leading principal contribution to H1 is bounded by:  −2a+1/2  Cε0 δ0 µm (t)[1 + log(1 + t)] p+q+1 G0,[l+2];a, p (t)G1,[l+2];a,q (t) (13.563) the same in form as the bound (13.438). We turn to the spacetime integral in (13.488), namely:   (Ril−m . . . Ri1 (T )m+1 ψα )(L + ν) (ω/ν)(T ψα )(Ril−m . . . Ri1 (T )m
/ µ) dt  du  dµh/˜ Wut

(13.564) As in (13.453) we have, in regard to the integrant in (13.564),   / µ) (13.565) (L + ν) (ω/ν)(T ψα )(Ril−m . . . Ri1 (T )m
  m m = (ω/ν) (T ψα )(L + 2ν)(Ril−m . . . Ri1 (T )
/ µ) + τ˜α (Ril−m . . . Ri1 (T )
/ µ) / µ. We substitute on the right-hand side the expression (13.493) for Ril−m . . . Ri1 (T )m
Here we apply the following. Let X be an arbitrary St,u -tangential vectorfield. We then have: (L + 2ν)div /˜ X = div /˜ (L / L X + 2ν X)

(13.566)

To establish this we work in acoustical coordinates (t, u, ϑ A : A = 1, 2). We then have: ∂ ∂ϑ A

∂ 1 ∂XA ∂

div / X= √ ( deth /X A) = + X A A log deth/ A A ∂ϑ ∂ϑ deth / ∂ϑ X = XA

(13.567)

and: ∂ div / X ∂t



∂2 X A ∂2 ∂XA ∂ = + log deth/ + X A log deth/ A A A ∂t∂ϑ ∂t ∂ϑ ∂t∂ϑ

Ldiv / X=

(13.568)

826

Chapter 13. Error Estimates

Now: L / L X = [L, X] =

∂XA ∂ ∂t ∂ϑ A

(13.569)

and:

∂ log deth/ = trχ ∂t Therefore comparing with (13.567) we obtain:

(13.570)

Ldiv / X = div / (L / L X) + X · d/trχ

(13.571)

In view of (13.492) and the definition of ν, this is equivalent to: L div /˜ X = div /˜ (L / L X) + 2X · d/ν

(13.572)

which is in turn equivalent to (13.566). We appeal to (13.566) taking X = (i1 ...il−m ) Ym,l−m after substituting the expression (13.493) in (13.565). Substituting then the result in (13.564) and integrating by parts on each St,u we obtain: (ω/ν)(T ψα )(Ril−m . . . Ri1 (T )m+1 ψα )(L + 2ν)div /˜ (i1 ...il−m ) Ym,l−m dt  du  dµh/˜ Wut



=−

Wut



Wut

− −

Wut

(ω/ν)((L / L + 2ν)

(i1 ...il−m )

(ω/ν)((L / L + 2ν)

(i1 ...il−m )

Ym,l−m ) · d/ Ril−m . . . Ri1 (T )m+1 ψα dt  du  dµh/˜

Ym,l−m ) · d/T ψα dt  du  dµh/˜

(T ψα )(Ril−m . . . Ri1 (T )m+1 ψα ) × ((L / L + 2ν)

(i1 ...il−m )

Ym,l−m ) · d/(ω/ν)dt  du  dµh/˜

and we are left with: (ω/ν)(T ψα )(Ril−m . . . Ri1 (T )m+1 ψα )(L + 2ν) Wut

(i1 ...il−m )

(13.573)

r˜m,l−m dt  du  dµh/˜ (13.574)

We also integrate by parts on each St,u : (ω/ν)τ˜α (Ril−m . . . Ri1 (T )m+1 ψα )div /˜ Wut



=−

Wut



Wut

− −

Wut

(ω/ν)τ˜α

(i1 ...il−m )

(i1 ...il−m )

Ym,l−m dt  du  dµh/˜

Ym,l−m · d/(Ril−m . . . Ri1 (T )m+1 ψα )dt  du  dµh/˜

(ω/ν)(Ril−m . . . Ri1 (T )m+1 ψα ) τ˜α (Ril−m . . . Ri1 (T )m+1 ψα )

(i1 ...il−m )

(i1 ...il−m )

Ym,l−m · d/τ˜α dt  du  dµh/˜

Ym,l−m · d/(ω/ν)dt  du  dµh/˜

(13.575)

Part 2: The error estimates of the acoustical entities

and we are left with: (ω/ν)τ˜α (Ril−m . . . Ri1 (T )m+1 ψα )

827

(i1 ...il−m )

Wut

r˜m,l−m dt  du  dµh/˜

(13.576)

In view of (13.573)–(13.576) the spacetime integral (13.564) becomes:

where:  V0,0 =



 V0,1

 = V1,0

 V1,1

(ω/ν)(T ψα )((L / L + 2ν)

Wut

(13.577)

  V0 = V0,0 + V0,1

(13.578)

(i1 ...il−m )

Ym,l−m ) · d/ Ril−m . . . Ri1 (T )m+1 ψα dt  du  dµh/˜ (13.579)



=

V0 − V1 − V2

Wut

(ω/ν)τ˜α

(i1 ...il−m )

Ym,l−m · d/ Ril−m . . . Ri1 (T )

m+1





ψα dt du dµh/˜

  V1 = V1,0 + V1,1

Wut

(ω/ν)(Ril−m . . . Ri1 (T )m+1 ψα )((L / L + 2ν)

(13.581) (i1 ...il−m )

Ym,l−m ) · d/T ψα dt  du  dµh/˜ (13.582)



=

Wut

(ω/ν)(Ril−m . . . Ri1 (T )

and:

m+1

ψα )

(i1 ...il−m )





Ym,l−m · d/τ˜α dt du dµh/˜

  + V2,1 V2 = V2,0  V2,0 =

(13.583)

(13.584)

Wut

(T ψα )(Ril−m . . . Ri1 (T )m+1 ψα )

((L / L + 2ν)

(i1 ...il−m )

Ym,l−m ) · d/(ω/ν)  +(ω/ν)(L + 2ν) (i1 ...il−m )r˜m,l−m dt  du  dµh/˜

 V2,1 =

(13.580)

(13.585)

(Ril−m . . . Ri1 (T )m+1 ψα )τ˜α Wut (i1 ...il−m ) Ym,l−m · (d/(ω/ν) + (ω/ν) (i1 ...il−m )r˜m,l−m )dt  du  dµh/˜

(13.586)

 and V  . In In the following we shall focus attention on the two leading integrals V0,0 1,0 view of the estimates (13.455), (13.327) and (13.501) the other integrals contain decay factors of at least (1 + t)−1 [1 + log(1 + t)] relative to the two leading integrals.  . We have: We first consider the integral V0,0  | |V0,0

≤C

(13.587)  2

Wεt

(1 + t ) |T ψα ||d/ Ril−m . . . Ri1 (T ) 0

m+1

ψα ||(L / L + 2ν)

(i1 ...il−m )

Yl−m |dt  dudµh/

828

Chapter 13. Error Estimates

Since according to (13.490): d/ Ril−m . . . Ri1 (T )m µ = h/ ·

(i1 ...il−m )

Ym,l−m

(13.588)

/ = 2χ, we have: and L /L h d/ L Ril−m . . . Ri1 (T )m µ = L / L d/ Ril−m . . . Ri1 (T )m µ = 2χ ·

(i1 ...il−m )

Ym,l−m + h/ · L /L

(i1 ...il−m )

Ym,l−m

(13.589)

  1 1 / + χˆ = ν − L log  h/ + χˆ  χ = trχh 2 2

Since

defining: 1 χ˜  = χˆ  − (L log )h/ 2

(13.590)

we then obtain: / · (L h / L + 2ν)

(i1 ...il−m )

Ym,l−m = d/ L Ril−m . . . Ri1 (T )m µ − 2χ˜  ·

(i1 ...il−m )

Ym,l−m (13.591)

Substituting for L Ril−m . . . Ri1 (T )m µ from (13.513) with l + 1 replaced by l then yields: h/ · (L / L + 2ν)

(i1 ...il−m )

Ym,l−m = d/

(i1 ...il−m )  ρm,l−m

$ − 2χ˜  · d/ Ril−m . . . Ri1 (T )m µ

(13.592) which implies: |(L / L + 2ν)

(i1 ...il−m )

Ym,l−m | ≤

2 

|d/

(i1 ...il−m )

(k) 

ρ

m,l−m

| + 2|χ˜  ||d/ Ril−m . . . Ri1 (T )m µ|

k=0

(13.593) Now, from (13.526) with l + 1 replaced by l: d/

(i1 ...il−m )

(0) ρ  m,l−m =

k 3 d/ Ril−m . . . Ri1 (T )m+1 ψ0

hence: µ1/2 d/

(i1 ...il−m )

(0) ρ  m,l−m

 L 2 ( ε0 ) ≤ Ck 2 ||(1+t)−1 t

)

 α

(13.594)

E1 [Ril−m . . . Ri1 (T )m+1 ψα ](t) (13.595)

Also, from (13.527) with l + 1 replaced by l: d/

(i1 ...il−m )

(1) ρ  m,l−m = (m 0T

− k 3 )d/ Ril−m ... Ri1 (T )m+1 ψ0 + m iT d/ Ril−m ... Ri1 (T )m+1 ψi

+ (1 + t)−1 µeαL d/ Ril−m ... Ri1 (T )m Qψα + (d/m αT )(Ril−m ... Ri1 (T )m+1 ψα ) + (1 + t)−1 (d/(µeαL ))(Ril−m ... Ri1 (T )m Qψα )

(13.596)

Part 2: The error estimates of the acoustical entities

829

hence, in view of the facts that: sup |d/m αT | ≤ Cδ0 (1 + t)−2 ,

sup |d/eαL | ≤ Cδ0 (1 + t)−2

ε t 0

(13.597)

ε

t 0

we obtain: µ1/2 d/

(i1 ...il−m )

(1) ρ  m,l−m

   L 2 ( ε0 ) ≤ Cδ0 (1 + t)−2 E1,[l+2] (t)

(13.598)

t

Also, by virtue of (13.543) we have: |d/

(2) ρ  m,l−m

| + 2|χ˜  ||d/ Ril−m . . . Ri1 (T )m µ| L 2 ( ε0 ) t

Q −2 ≤ Cl δ0 (1 + t) [1 + log(1 + t)] W{l+1} + W{l} (i1 ...il−m )

(13.599)

+(1 + t)−1 B[m,l+1] + Y0 + (1 + t)A[l]

 (0)

The borderline contribution to (13.587) is the contribution from d/ (i1 ...il−m ) ρ  m,l−m . This is the borderline contribution : (0) (1 + t  )2 |T ψα ||d/ Ril−m . . . Ri1 (T )m+1 ψα ||d/ (i1 ...il−m ) ρ  m,l−m |dt  dudµh/ C Wεt

0



t

≤C 0

  (1 + t  )2 sup µ−1 |T ψα | µ1/2 d/ Ril−m . . . Ri1 (T )m+1 ψα  L 2 ( ε0 ) t

ε

t 0

· µ1/2 d/

(i1 ...il−m )

(0) ρ  m,l−m

 L 2 ( ε0 ) dt  t

(13.600)

  Here we substitute the estimates (13.198) for sup ε0 µ−1 |T ψα | and the estimate (13.595) for µ1/2 d/

t

(0) (i1 ...il−m ) ρ  m,l−m

 L 2 ( ε0 ) . The factors k 3 || then cancel. Now, the t

partial contribution of the second term on the right in (13.198) is in fact not borderline. The actual borderline integral, which represents the partial contribution of the first term on the right in (13.198), is: t   sup µ−1 |Lµ| E1 [Ril−m . . . Ri1 (T )m+1 ψα ](t  ) C 0  ε0 t

·  ≤C

1 1 + 2a 2q



) α

E1 [Ril−m . . . Ri1 (T )m+1 ψα ](t  )dt 

2q µ−2a m (t)[1 + log(1 + t)]

(compare with (13.478), (13.479)).

(i1 ...il−m )  G1,m,l+2;a,q (t)

(13.601)

830

Chapter 13. Error Estimates (1)

We proceed to consider the contribution of d/ (i1 ...il−m ) ρ  m,l−m to (13.587). Here   we simply appeal to the estimate (13.199) for sup ε0 µ−1 |T ψα | . Substituting also the estimate (13.598) for µ1/2 d/

(1) (i1 ...il−m ) ρ  m,l−m

t

 L 2 ( ε0 ) , we obtain that the contribution

in question is bounded by: (1 + t  )2 |T ψα ||d/ Ril−m . . . Ri1 (T )m+1 ψα ||d/ C Wεt

0



t

≤ Cδ0 0

(i1 ...il−m )

(1) ρ  m,l−m

|dt  dudµh/

 1/2 (1 + t  )µ−1 d/ Ril−m . . . Ri1 (T )m+1 ψα  L 2 ( ε0 ) m (t )µ t

· µ1/2 d/ ≤

t

(1) (i1 ...il−m )  ρ m,l−m

 L 2 ( ε0 ) dt  t

2q Cδ02 µ−2a m (t)[1 + log(1 + t)]

 t    −2    · ϕ1 (Caδ0 )G1,[l+2];a,q (t) + (1 + t ) G1,[l+2];a,q (t )dt (13.602) 0

(compare with (13.480)).  , given by (13.582). By virtue of the bound Finally, we consider the integral V1,0 (13.559) we have:  |V1,0 | ≤ Cδ0 |Ril−m . . . Ri1 (T )m+1 ψα ||(L / L + 2ν) (i1 ...ilm ) Ym,l−m |dt  dudµh/

Wεt t

≤ Cδ0 0

0

W{l+1} (t  )(L / L + 2ν)

(i1 ...il−m )

Ym,l−m  L 2 ( ε0 ) dt  dudµh/ t

(13.603)

Here we need only consider the leading principal contribution, namely that of d/

(i1 ...il−m )

(0) ρ  m,l−m .

d/

By (13.595),

(i1 ...il−m )

(0) ρ  m,l−m

−1/2

 L 2 ( ε0 ) ≤ Cµm t

  (1 + t)−1 E1,[l+2] (t),

hence taking also into account the estimate (13.437) the contribution in question is bounded by: t −2a−1/2  Cε0 δ0 (1 + t  )−1 µm (t )[1 + log(1 + t  )] p+q (13.604) 0   · G0,[l+2];a, p (t  )G1,[l+2];a,q (t  )dt  This is identical in form to (13.482), which has already been bounded by (13.484). This completes the estimates for the spacetime integrals V0 , V1 , V2 . We have thus completed the estimate of the contribution of the leading term 166 to the corresponding error integral (13.168), associated to K 1 . The error estimates involving the top order spatial derivatives of the acoustical entities have now been completed.

Part 3: The energy estimates

831

Part 3: The energy estimates We now proceed to the third part of the present chapter, the derivation of the energy estimates. We first consider the energy estimates of top order l + 2. Of these the most delicate are those corresponding to the variations (13.161) and (13.163), to which the borderline error integrals are associated. For each of the variations, we first consider the integral identity associated to the vectorfield K 1 and then the integral identity associated to the vectorfield K 0 (see Chapter 5). For each variation ψ, of any order, there is an additional borderline integral associated to K 1 , contributed by the term Q 1,3 [ψ] in the fundamental energy estimate associated to ψ (see (5.100), (5.164)–(5.170)). This is the integral L(t, ε0 )[ψ] defined by (5.242) of Chapter 5:

t

L(t, ε0 )[ψ] = 0

(1 + t  )−1 [1 + log(1 + t  )]−1 E1 [ψ](t  )dt 

(13.605)

Consider now the integral identities corresponding to the variations (13.161) and the vectorfield K 1 . In each of these we have the borderline hypersurface integral (13.365) bounded by the sum of (13.385) and (13.412):   1 1 2q (i1 ...il )  + C µ−2a G1,l+2;a,q (t) (13.606) m (t)[1 + log(1 + t)] a − 1/2 q + 1/2 We also have the borderline spacetime integral (13.477) bounded by (13.479):   1 1 2q (i1 ...il )  C + µ−2a G1,l+2;a,q (t) (13.607) m (t)[1 + log(1 + t)] 2a 2q We also have the remaining integrals, bounded, in the case of (13.425) by:  Cq δ02 [1 + log(1 + t)]2q µ−2a m (t)G1,[l+2];a,q

(13.608)

and in the case of (13.430), by:

 2q  Cl δ0 µ−2a G (t)[1 + log(1 + t)] (t) + G (t) 0,[l+2];a, p m 1,[l+2];a,q

(13.609)

provided that: q ≥ p+1

(13.610)

and (13.438) is the same as (13.430). In the case of (13.481) we have a bound by: 2q  Cδ02 µ−2a m (t)[1 + log(1 + t)] G1,[l+2];a, p (t)

(13.611)

and in the case of (13.484) a bound by:

 2q  Cl δ0 µ−2a G1,[l+2];a,q (t) + G0,[l+2];a, p (t) m (t)[1 + log(1 + t)]

(13.612)

832

Chapter 13. Error Estimates

provided that (13.610) holds. Finally, (13.486) is the same as (13.484). Combining we see that all the remaining integrals associated to the variations (13.161) and to K 1 containing the top order spatial derivatives of the acoustical entities are bounded by:  2q  Cq,l δ0 µ−2a G1,[l+2];a,q (t) + G0,[l+2];a, p (t) m (t)[1 + log(1 + t)]

   · δ0 Y0 (0) + A[l] (0) + B{l+1} (0) + G1,[l+2];a,q (t) + G0,[l+2];a, p (t)

(13.613)

Consider now the error integrals associated to the same variations, or in fact any of the top order variations, and to K 1 , or to K 0 , which contain the lower order spatial derivatives of the acoustical entities and are contributed by the remaining terms in the sum in (13.116). Consider an arbitrary term in this sum, corresponding to some k ∈ {0, . . . , l}: (Y Il+1 +

(Y Il+1 )

δ) . . . (Y Il+2−k +

(Yl+2−k )

δ)

(Yl+1−k ;I1 ...Il−k )

σl+1−k

(13.614)

Here there is a total of k derivatives with respect to the commutation fields acting on (Y ) σ (Y ) σ l+1−k . In view of the fact that l+1−k has the structure described in the paragraph following (13.117), in considering the partial contribution of each term in (Y ) σ1,l+1−k , if the factor which is a component of (Y ) π˜ receives more than (l + 1)∗ derivatives with respect to the commutation fields, then the factor which is a 2nd derivative of ψl+1−k receives at most k − (l + 1)∗ − 1 derivatives of the commutation fields, thus corresponds to a derivative of the ψα of order at most: k − (l + 1)∗ + 1 + l − k = l∗ + 1 ε

therefore this factor is bounded in L ∞ (t 0 ) by the bootstrap assumption. Also, in considering the partial contribution of each term in (Y ) σ2,l+1−k , if the factor which is a 1st derivative of (Y ) π˜ receives more than (l + 1)∗ − 1 derivatives with respect to the commutation fields, then the factor which is a 1st derivative of ψl+1−k receives at most k−(l +1)∗ derivatives with respect to the commutation fields, thus corresponds to a derivative of the ψα of order at most: k − (l + 1)∗ + 1 + l − k = l∗ + 1 ε

therefore this factor is again bounded in L ∞ (t 0 ) by the bootstrap assumption. Similar considerations apply to (Y ) σ3,l+1−k . We conclude that for all terms in the sum in (13.116) of which one factor is a derivative of the (Y ) π˜ of order more than (l +1)∗ , the other factor ε is then a derivative of the ψα of order at most l∗ + 1 and is thus bounded in L ∞ (t 0 ) by the bootstrap assumption. Of these terms we have already estimated the contribution of those containing the top order spatial derivatives of the acoustical entities. The contributions of the remaining terms are then bounded using Propositions 12.11 and 12.12. We obtain a bound for these contributions to the error integrals associated to K 1 and to any

Part 3: The energy estimates

833

of the variations, up to the top order, by: q Cl δ0 µ−a m (t)[1 + log(1 + t)] ·

   δ0 Y0 (0) + A[l] (0) + B{l+1} (0) + G1,[l+2];a,q (t) + G0,[l+2];a, p (t)  ε0 1/2 t   · F1,[l+2] (u )du 0



ε0

+Cl δ0

1/2  t F1,[l+2] (u  )du  + Cl δ0 K [l+2] (t, ε0 )

0



ε0 0

t F1,[l+2] (u  )du 

(13.615) 1/2

the last two terms in (13.615) contributed by the terms in (13.116) containing the top order derivatives of the acoustical entities of which however one is a derivative with respect to L (hence are expressible in terms of the top order derivatives of the ψα ). Here, we have defined: n  t t F1,[n] (u) = F1,m (u) (13.616) m=1 t (u) represents the sum of the fluxes associated to the vectorfield K of all where F1,n 1 the nth order variations (13.104), (13.106) of the form specified in the paragraph just following (13.106). On the other hand, all the other terms in the sum (13.116) contain derivatives of the (Y ) π ˜ of order at most (l + 1)∗ , thus spatial derivatives of χ  of order at most (l + 1)∗ and ε spatial derivatives of µ of order at most (l + 1)∗ + 1, which are bounded in L ∞ (t 0 ) by virtue of Propositions 12.9 and 12.10 and the bootstrap assumption. The contributions of these terms to the error integrals associated to K 0 or to K 1 and to any of the variations, up to the top order, are bounded exactly as in Chapter 7. For the error integrals associated to K 1 we obtain for Lemma 7.6 the bound:  ε0 1/2 ε0  1/2 t   t   K [l+2] (t, ε0 ) Cl F1,[l+2] (u )du + Cl F1,[l+2] (u )du 0

0



ε0

+ Cl 0

t F1,[l+2] (u  )du 

 0

t

1/2

·

(13.617)

  1/2  (1 + t  )−2 [1 + log(1 + t  )]2 E1,[l+2] (t  ) + ε02 E0,[l+2] (t  ) dt 

Here, we have defined: K [n] (t, u) =

n 

K m (t, u)

(13.618)

m=1

where K n (t, u) represents the sum of the integrals  −1 −1 ων µ (Lµ)− |d/ψ|2 dµh K [ψ](t, u) = − t Wu 2

(13.619)

(see (5.169) of Chapter 5) of all the nth order variations (13.104), (13.106) of the form specified in the paragraph just following (13.106).

834

Chapter 13. Error Estimates

Finally, for each variation ψ we have the error integrals:

7  Wεt k=1 0

Q 0,k [ψ]dµh ,

8  Wεt k=1 0

Q 1,k [ψ]dµh

(13.620)

of the fundamental energy estimate associated to ψ and to K 0 and K 1 respectively (see (5.88)–(5.94) and (5.98)–(5.104) of Chapter 5). According to the estimate (5.245) of Chapter 5 we have:

8  Wεt 0

1 3 Q 1,k [ψ]dµh ≤ − K [ψ](t, ε0 ) + C M[ψ](t, ε0 ) + L[ψ](t, ε0 ) 2 2 k=1 t ˜  )E1 [ψ](t  )dt  + (13.621) A(t 0

Here, M[ψ](t, ε0 ) is given by (5.241):

4

M[ψ](t, ε0 ) = E 0 [ψ](t)[1 + log(1 + t)] + 0

ε0

F1t [ψ](u  )du 

(13.622)

As in Chapter 5 the term −(1/2)K [ψ](t, ε0) is to be brought to the left-hand side of the integral inequality associated to ψ and K 1 . Let us define, for any variation ψ, in analogy with (13.218), (13.220),

   G0;a, p [ψ](t) = sup [1 + log(1 + t  )]−2 p µ2a (t )E [ψ](t ) (13.623) 0 m t  ∈[0,t ]

 G1;a,q [ψ](t) = sup

t  ∈[0,t ]

    [1 + log(1 + t  )]−2q µ2a (t )E [ψ](t ) m 1

 Note that G0;a, p [ψ](t), G1;a,q [ψ](t), are non-decreasing functions of t. Recalling from Chapter 8 the definition of µm (see (8.271), (8.272)):  µm (t) µm (t) = min , 1 , µm (t) = min µ= min µ(t, u, ϑ) ε η0 (u,ϑ)∈[0,ε0]×S 2 t 0

(13.624)

(13.625)

ε

we define µm,u by replacing [0, ε0] × S 2 ] by [0, u] × S 2 , or t 0 by tu :  µm,u (t) µm,u (t) = min , 1 , µm,u (t) = min µ= min µ(t, u  , ϑ) η0 tu (u  ,ϑ)∈[0,u]×S 2 (13.626) Note that µm,u (t) is a non-increasing function of u at each t. We then define:

 t  t  H1;a,q [ψ](u) = sup [1 + log(1 + t  )]−2q µ2a (t )F [ψ](u) (13.627) m,u 1 t  ∈[0,t ]

 V1;a,q [ψ](t, u) =

0

u

t H1;a,q [ψ](u  )du 

(13.628)

Part 3: The energy estimates

835

t  Note that H1;a,q [ψ](u) is a non-decreasing function of t at each u while V1;a,q [ψ](t, u) is a non-decreasing function of u at each t as well as a non-decreasing function of t at each u. We have, for each u ∈ [0, ε0]: u u t   2q t   F1 [ψ](u )du ≤ µ−2a m,u  (t)[1 + log(1 + t)] H1;a,q [ψ](u )du 0

0

2q  ≤ µ−2a m,u (t)[1 + log(1 + t)] V1;a,q [ψ](t, u)

We also define: t H1,[n];a,q (u) = sup

t  ∈[0,t ]



 t  (t )F (u) [1 + log(1 + t  )]−2q µ2a m,u 1,[n]

 V1,[n];a,q (t, u) =

0

u

t H1,[n];a,q (u  )du 

(13.629)

(13.630) (13.631)

t  (u) is a non-decreasing function of t at each u, V1;[n],a,q (t, u) is a nonThen H1;[n]a,q decreasing function of u at each t as well as a non-decreasing function of t at each u, and for each u ∈ [0, ε0]: u u t   2q t   F1,[n] (u )du ≤ µ−2a m,u  (t)[1 + log(1 + t)] H1,[n];a,q (u )du 0

0

2q  ≤ µ−2a m,u (t)[1 + log(1 + t)] V1,[n];a,q (t, u)

(13.632)

Going back to (13.622), we have, in regard to the first term on the right, E 0 [ψ](t) = sup E0 [ψ](t  ) t  ∈[0,t ]

≤ sup

t  ∈[0,t ]



  2p  µ−2a m (t )[1 + log(1 + t )] G0;a, p [ψ](t )

≤ C[1 + log(1 + t)]2 p µ−2a m (t)G0;a, p [ψ](t)

(13.633)

(by Corollary 2 of Lemma 8.11). At this point we set: q = p+2

(13.634)

which is consistent with (13.610). Then (13.633) implies that the first term on the right in (13.622) is bounded by: 2q Cµ−2a m (t)[1 + log(1 + t)] G0;a, p [ψ](t)

(13.635)

The second term on the right in (13.622) being bounded according to (13.629) with u = ε0 , we conclude that: 

2q  M[ψ](t, ε0 ) ≤ µ−2a [ψ](t, ε0 ) CG0;a, p [ψ](t) + V1;a,q m (t)[1 + log(1 + t)]

 2q  CG ≤ µ−2a (t)[1 + log(1 + t)] (t) + V (t, ε ) 0 0,[l+2];a, p m 1,[l+2];a,q (13.636) for all variations ψ of the form (13.106) of order not exceeding l + 2.

836

Chapter 13. Error Estimates

In regard to the third term on the right in (13.621), which involves the borderline integral (13.605), we have, by Corollary 2 of Lemma 8.11, t  (t)G [ψ](t) (1 + t  )−1 [1 + log(1 + t  )]2q−1 dt  L[ψ](t, ε0 ) ≤ Cµ−2a m 1;a,q 0

C −2a  ≤ µ (t)[1 + log(1 + t)]2q G1;a,q [ψ](t) 2q m

(13.637)

Finally, in regard to the last term on the right in (13.621) we have: t t     −2a 2q   ˜ ˜  )G  A(t )E1 [ψ](t )dt ≤ Cµm (t)[1 + log(1 + t)] A(t 1;a,q [ψ](t )dt 0 0 t −2a 2q   ˜  )G  ≤ Cµm (t)[1 + log(1 + t)] A(t 1,[l+2];a,q (t )dt 0

(13.638) for all variations ψ of the form (13.106) of order not exceeding l + 2. We also recall from Chapter 5 that in the integral identity (5.66) (with u = ε0 ) corresponding to the variation ψ and to K 1 we have the hypersurface integrals bounded according to (5.252):      2 2  ε (1/2)(Lω + νω)ψ − ε (1/2)(Lω + νω)ψ    t 0  0 0

≤ CE 0 [ψ](t)[1 + log(1 + t)]4 2q ≤ Cµ−2a m (t)[1 + log(1 + t)] G0;a, p [ψ](t)

by (13.633), (13.634). We turn to the bound (13.617). Let us define:

   (t )K (t , u) I[n];a,q (t, u) = sup [1 + log(1 + t  )]−2q µ2a [n] m,u t  ∈[0,t ]

(13.639)

(13.640)

This is a non-decreasing function of t at each u. Then, in view of (13.632) with u = ε0 , we can bound (13.617) by: 2q Cl µ−2a m (t)[1 + log(1 + t)] ·  t    (1 + t  )−2 [1 + log(1 + t  )]2 G0,[l+2];a, p (t  ) + G1;[l+2];a,q (t  ) dt  0    +V1,[l+2];a,q (t, ε0 ) + I[l+2];a,q (t, ε0 )1/2 V1,[l+2];a,q (t, ε0 )1/2 (13.641)

and the last term in parenthesis is bounded by: δ 1  I[l+2];a,q (t, ε0 ) + V1,[l+2];a,q (t, ε0 ) 2 2δ for any positive constant δ (we shall choose δ below).

(13.642)

Part 3: The energy estimates

837

Finally, (13.615) is bounded by:

 2 2q Y0 (0) + A[l] (0) + B{l+1} (0)) Cl δ0 µ−2a (13.643) m (t)[1 + log(1 + t)]   1   +δ(G1,[l+2];a,q (t) + G0,[l+2];a, p (t)) + δ I[l+2];a,q (t, ε0 ) + 1 + V1,[l+2];a,q (t, ε0 ) δ We now consider the integral identity corresponding to the vectorfield K 1 ((5.66) with u = ε0 ) and to the variations (13.161) R j Ril . . . Ri1 ψα , j = 1, 2, 3, α = 0, 1, 2, 3, for a given multi-index (i 1 . . . i l ). Summing over j and α, we then obtain from (13.606), (13.607), (13.637), in regard to the borderline integrals, and from (13.613), (13.636), (13.638), (13.639), (13.641), (13.642), in regard to the remaining contributions, the following:   E1 [R j Ril . . . Ri1 ψα ](t) + F1t [R j Ril . . . Ri1 ψα ](ε0 ) j,α

j,α

1 −2q K [R j Ril . . . Ri1 ψα ](t, ε0 ) µ2a + m (t)[1 + log(1 + t)] 2 j,α   1 (i1 ...il )  1  + ≤C G1,[l+2];a,q (t) + Cq,l δ0 G1,[l+2];a,q (t) a q t  + Cq,l R[l+2];a,q (t, ε0 ) + C A(t  )G1,[l+2];a,q (t  )dt  (13.644) 0

Here,   1  (t, ε0 ) R[l+2];a,q (t, ε0 ) = D[l+2] + G0,[l+2];a, p (t) + 1 + V1,[l+2];a,q δ +δ I[l+2];a,q (t, ε0 ) (13.645) and D[l+2] stands for the quantity: 2  D[l+2] = E0,[l+2] (0) + P[l+2] P[l+2] = Y0 (0) + A[l] (0) + B{l+1} (0) +

 i1 ...il



(i1 ...il )

xl (0) L 2 ( ε0 ) + 0

(13.646) l 





m=0 i1 ...il−m

(i1 ...il−m )  x m,l−m (0) L 2 ( ε0 ) 0

ε

The quantity D[l+2] refers to the initial data on 00 . Also, ˜ + Cl (1 + t)−2 [1 + log(1 + t)]2 A(t) = A(t)

(13.647)

It is crucial that the constant C in front of the first term on the right in (13.644) is independent of a, q or l.

838

Chapter 13. Error Estimates

Now all the estimates we have made so far depend only on the bootstrap assumption and on assumption J, which refer to Wεs0 , on the assumptions of Propositions 12.6, 12.9, ε 12.10 on the initial data on 00 , and on the fact that ε0 ≤ 1/2. These assumptions hold a fortiori if ε0 is replaced by any u ∈ (0, ε0 ]; therefore all our estimates hold with ε0 replaced by any u ∈ (0, ε0 ]. With this replacement, µm (t) is replaced by µm,u (t) (see (13.626)) and, for any variation ψ E0 [ψ](t) and E1 [ψ](t) are replaced by E0u [ψ](t) and  E1u [ψ](t) respectively. Thus G0;a, p [ψ](t), G1;a,q [ψ](t) are replaced by: u G0;a, p [ψ](t) = sup

  u  [1 + log(1 + t  )]−2 p µ2a (t )E [ψ](t ) m,u 0

(13.648)

u G1;a,q [ψ](t) = sup



 u  (t )E [ψ](t ) [1 + log(1 + t  )]−2q µ2a m,u 1

(13.649)

t  ∈[0,t ] t  ∈[0,t ]

respectively (see (13.623), (13.624)), and (i1 ...il ) G0,l+2;a, p (t), (i1 ...il ) G  (i1 ...il−m ) G  1,l+2;a, p (t), 1,m,l+2;a,q (t), are replaced by:

(i1 ...il−m ) G 0,m,l+2;a, p (t),

(i1 ...il ) u G0,l+2;a, p (t)

= sup

 

t  ∈[0,t ] 

 [1 + log(1 + t  )]−2 p µ2a m,u (t )



E0u [R j Ril . . . Ri1 ψα ](t  )

j,α

  

(i1 ...il−m ) u G0,m,l+2;a, p (t)



= sup

t  ∈[0,t ]

 [1 + log(1 + t  )]−2 p µ2a m,u (t )

 α

(13.651)  E0u [Ril−m . . . Ri1 (T )m+1 ψα ](t  )

(i1 ...il ) u G1,l+2;a,q (t)

= sup

 

t  ∈[0,t ] 

 [1 + log(1 + t  )]−2q µ2a m,u (t )



E1u [R j Ril . . . Ri1 ψα ](t  )

j,α

(i1 ...il−m ) u G1,m,l+2;a,q (t)



= sup

t  ∈[0,t ]

 [1 + log(1 + t  )]−2q µ2a m,u (t )

 α

(13.650)

 

(13.652)

 (13.653) 

E1u [Ril−m . . . Ri1 (T )m+1 ψα ](t  )

 , are respectively (see (13.200), (13.285), (13.376), (13.552)). Also, G0,[n];a, p , G1,[n];a,q replaced by: 

u  u  (13.654) [1 + log(1 + t  )]−2 p µ2a G0,[n];a, m,u (t )E0,[n] (t ) p (t) = sup t  ∈[0,t ]

u (t) = sup G1,[n];a,q

t  ∈[0,t ]

 u  [1 + log(1 + t  )]−2q µ2a m,u (t )E1,[n] (t )

 (13.655)

Part 3: The energy estimates

839

u respectively (see (13.218), (13.220)). (Moreover, D[l+2] is replaced by D[l+2] ). Thus, in place of the inequality (13.644) we obtain, for all u ∈ [0, ε0],and all t ∈ [0, s]:   E1u [R j Ril . . . Ri1 ψα ](t) + F1t [R j Ril . . . Ri1 ψα ](u) j,α

j,α

1 −2q + K [R j Ril . . . Ri1 ψα ](t, u) µ2a m,u (t)[1 + log(1 + t)] 2 j,α   1 (i1 ...il ) u 1 u + ≤C G1,[l+2];a,q (t) + Cq,l δ0 G1,[l+2];a,q (t) a q t u A(t  )G1,[l+2];a,q (t  )dt  (13.656) + Cq,l R[l+2];a,q (t, u) + C 0

and we have:

  1 u u  V1,[l+2];a,q + G0,[l+2];a, (t) + 1 + (t, u) R[l+2];a,q (t, u) = D[l+2] p δ (13.657) + δ I[l+2];a,q (t, u)

u where the quantity D[l+2] refers to the initial on 0u :

 2 u u u D[l+2] = E0,[l+2] (0) + P[l+2] u u P[l+2] = Y0u (0) + Au [l] (0) + B{l+1} (0)

+





(i1 ...il )

xl (0) L 2 ( u ) + 0

i1 ...il

(13.658) l  



m=0 i1 ...il−m

(i1 ...il−m )  x m,l−m (0) L 2 ( u ) 0 ε

u 2 0 the quantities Y0u (0), Au [l] (0), B{l+1} (0), being defined by replacing the L norms over 0 by L 2 norms over 0u in the definitions of the corresponding quantities Y0 (0), A[l] (0), B{l+1} (0). Keeping only the term  E1u [R j Ril . . . Ri1 ψα ](t) j,α

on the left in (13.656), we have: −2q µ2a m,u (t)[1 + log(1 + t)]

 ≤C

1 1 + a q





E1u [R j Ril . . . Ri1 ψα ](t)

j,α (i1 ...il ) u G1,[l+2];a,q (t) +



t

+ Cq,l R[l+2];a,q (t, u) + C 0

u Cq,l δ0 G1,[l+2];a,q (t)

u A(t  )G1,[l+2];a,q (t  )dt 

(13.659)

840

Chapter 13. Error Estimates

The same holds with t replaced by any t  ∈ [0, t]. Now the right-hand side of (13.659) is a non-decreasing function of t at each u. The inequality corresponding to t  thus holds a fortiori if we again replace t  by t on the right-hand side. Taking then the supremum over all t  ∈ [0, t] on the left-hand side we obtain, in view of the definition (13.652),   1 (i1 ...il ) u 1 (i1 ...il ) u u + G1,[l+2];a,q (t) ≤ C G1,[l+2];a,q (t) + Cq,l δ0 G1,[l+2];a,q (t) a q t u + Cq,l R[l+2];a,q (t, u) + C A(t  )G1,[l+2];a,q (t  )dt  (13.660) 0

If a and q are chosen suitably large so that:   1 1 1 + ≤ C a q 2

(13.661)

then (13.660) will imply: 1 2

(i1 ...il ) u G1,[l+2];a,q (t)

u ≤ Cq,l δ0 G1,[l+2];a,q (t)

(13.662)

+Cq,l R[l+2];a,q (t, u) + C 0

t

u A(t  )G1,[l+2];a,q (t  )dt 

The actual choice of a and q shall be specified below. Consider now the integral identities corresponding to the variations (13.163) and the vectorfield K 1 . In each of these we have the borderline hypersurface integral (13.551) bounded by (13.553):   1 1 2q (i1 ...il−m )  + µ−2a G1,m,l+2;a,q (t) (13.663) C m (t)[1 + log(1 + t)] a − 1/2 q + 1/2 We also have the borderline spacetime integral (13.600) bounded by (13.601):   1 1 2q (i1 ...il−m )  + µ−2a G1,m,l+2;a,q (t) (13.664) C m (t)[1 + log(1 + t)] 2a 2q We also have the remaining integrals, bounded, in the case of (13.557) by (13.608), in the case of (13.563) by (13.609) (recalling (13.634)), in the case of (13.602) by (13.611), and in the case of (13.604) by (13.612). Combining we see that all the remaining integrals associated to the variations (13.163) and to K 1 containing the top order spatial derivatives of the acoustical entities are bounded by (13.613). On the other hand, by the discussion following (13.613) and concluding with (13.617), the error integrals associated to the variations (13.163) and to K 1 contributed by all other terms in (13.116) are bounded by the sum of (13.615) and (13.617). Finally, for the variations in question we have the integrals (13.620), the second of which, associated to K 1 , is bounded according to (13.621) and (13.636)–(13.638) with (13.163) in the role of ψ. Moreover, we have (13.639) with (13.163) in the role of ψ. Taking also into account (13.641)–(13.643), the integral identity

Part 3: The energy estimates

841

corresponding to K 1 ((5.66) with u = ε0 ) and to the variations Ril−m . . . Ri1 (T )m+1 ψα , α = 0, 1, 2, 3, for a given multi-index (i 1 . . . i l−m ), summed over α, yields:   E1 [Ril−m . . . Ri1 (T )m+1 ψα ](t) + F1t [Ril−m . . . Ri1 (T )m+1 ψα ](ε0 ) α

α

1 m+1 −2q K [Ril−m . . . Ri1 (T ) ψα ](t, ε0 ) µ2a + m (t)[1 + log(1 + t)] 2 j,α   1 (i1 ...il−m )  1  + ≤C G1,m,[l+2];a,q (t) + Cq,l δ0 G1,[l+2];a,q (t) a q t  + Cq,l R[l+2];a,q (t, ε0 ) + C A(t  )G1,[l+2];a,q (t  )dt  (13.665) 0

It is again crucial that the constant C in front of the first term on the right in (13.665) is independent of a, q or l. By the discussion following (13.647) and concluding with (13.656), we can replace ε0 by any u ∈ (0, ε0 ] in (13.665) obtaining, for all u ∈ [0, ε0 ] and all t ∈ [0, s]:    E1u [Ril−m . . . Ri1 (T )m+1 ψα ](t) + F1t [Ril−m . . . Ri1 (T )m+1 ψα ](u) α

α

+  ≤C

1 2

1 1 + a q



K [Ril−m . . . Ri1 (T )

m+1



−2q ψα ](t, u) µ2a m,u (t)[1 + log(1 + t)]

α (i1 ...il−m ) u G1,m,[l+2];a,q (t)



t

+ Cq,l R[l+2];a,q (t, u) + C 0

Keeping only the term  α

u + Cq,l δ0 G1,[l+2];a,q (t) u A(t  )G1,[l+2];a,q (t  )dt 

(13.666)

E1u [Ril−m . . . Ri1 (T )m+1 ψα ](t)

on the left in (13.666) we have: −2q µ2a m,u (t)[1 + log(1 + t)]

 ≤C

1 1 + a q



 α

E1u [Ril−m . . . Ri1 (T )m+1 ψα ](t)

(i1 ...il−m ) u G1,m,[l+2];a,q (t)



+ Cq,l R[l+2];a,q (t, u) + C 0

t

u + Cq,l δ0 G1,[l+2];a,q (t)

u A(t  )G1,[l+2];a,q (t  )dt 

(13.667)

The same holds with t replaced by any t  ∈ [0, t]. Now the right-hand side of (13.667) is a non-decreasing function of t at each u. The inequality corresponding to t  thus holds a

842

Chapter 13. Error Estimates

fortiori if we again replace t  by t on the right-hand side. Taking then the supremum over all t  ∈ [0, t] on the left-hand side we obtain, in view of the definition (13.653), (i1 ...il−m ) u G1,m,[l+2];a,q (t)



≤C

1 1 + a q



(i1 ...il−m ) u G1,m,[l+2];a,q (t)



+ Cq,l R[l+2];a,q (t, u) + C

t

0

u + Cq,l δ0 G1,[l+2];a,q (t)

u A(t  )G1,[l+2];a,q (t  )dt 

(13.668)

If a and q are chosen suitably large so that (13.661) holds, then (13.668) will imply: 1 2

(i1 ...il−m ) u G1,m,[l+2];a,q (t)

(13.669)

u ≤ Cq,l δ0 G1,[l+2];a,q (t) + Cq,l R[l+2];a,q (t, u) + C



t 0

u A(t  )G1,[l+2];a,q (t  )dt 

Consider finally the integral identities corresponding to any of the other variations ψ of order up to l + 2, of the form specified in the paragraph following (13.106), and to the vectorfield K 1 . In each of these we have only one borderline integral, namely the integral (13.605) (with 0 replaced by u), bounded by (13.637) (with 0 replaced by u). As all the remaining integrals in these identities are bounded by: t u u (t) + Cq,l R[l+2];a,q (t, u) + C A(t  )G1,[l+2];a,q (t  )dt  Cq,l δ0 G1,[l+2];a,q 0

we obtain, for each such variation ψ, the inequality:  1 u t −2q E1 [ψ](t) + F1 [ψ](u) + K [ψ](t, u) µ2a m,u (t)[1 + log(1 + t)] 2 C u u G ≤ [ψ](t) + Cq,l δ0 G1,[l+2];a,q (t) 2q 1;a,q t u A(t  )G1,[l+2];a,q (t  )dt  (13.670) + Cq,l R[l+2];a,q (t, u) + C 0

Keeping only the term

E1u [ψ](t)

on the left in (13.670), and noting that by virtue of the condition (13.661) the coefficient C/2q of the first term on the right is not greater than 1/2, we have: −2q u µ2a E1 [ψ](t) m,u (t)[1 + log(1 + t)] 1 u u [ψ](t) + Cq,l δ0 G1,[l+2];a,q (t) ≤ G1;a,q 2 t u + Cq,l R[l+2];a,q (t, u) + C A(t  )G1,[l+2];a,q (t  )dt  0

(13.671)

Part 3: The energy estimates

843

The same holds with t replaced by any t  ∈ [0, t]. The right-hand side of (13.671) being a non-decreasing function of t at each u, the inequality corresponding to t  holds a fortiori if we again replace t  by t on the right. Taking then the supremum over all t  ∈ [0, t] on the left we obtain, in view of the definition (13.649), 1 u u [ψ](t) ≤ Cq,l δ0 G1,[l+2];a,q (t) G 2 1;a,q



t

+ Cq,l R[l+2];a,q (t, u) + C 0

u A(t  )G1,[l+2];a,q (t  )dt 

(13.672)

u Now, according to the definition of E1,[l+2] (t) we have,

u E1,[l+2] (t) =

   i1 ...il

+



E1u [R j Ril . . . Ri1 ψα ](t)

j,α

 

l   m=0 i1 ...il−m

+



α

   

E1u [Ril−m

. . . Ri1 (T )

m+1

ψα ](t)

E1u [ψ](t)

(13.673)

ψ

where the last sum is over all the other variations ψ of order up to l + 2, as specified above. Noting that for any set of N non-negative functions x 1 (t), . . . , x N (t) we have: sup

N 

t  ∈[0,t ] n=1

x n (t  ) ≤

N 

sup x n (t  )

 n=1 t ∈[0,t ]

(13.674)

it follows, in view of the definitions (13.649), (13.652), (13.653), (13.655), that: u (t) ≤ G1,[l+2];a,q



(i1 ...il ) u G1,l+2;a,q (t)

+

i1 ...il

+



l  

(i1 ...il−m ) u G1,m,l+2;a,q (t)

m=0 i1 ...il−m u G1;a,q [ψ](t)

(13.675)

ψ

Thus, summing inequalities (13.662) over i 1 . . . i l , summing inequalities (13.669) over i 1 . . . i l−m and over m = 0, . . . , l, and summing inequalities (13.672) over all the other variations ψ of order up to l + 2, as specified above, and then adding the resulting three inequalities, we obtain: 1 u u (t) ≤ Cq,l δ0 G1,[l+2];a,q (t) G 2 1,[l+2];a,q

(13.676)

+Cq,l R[l+2];a,q (t, u) + Cl 0

t

u A(t  )G1,[l+2];a,q (t  )dt 

844

Chapter 13. Error Estimates

for new constants Cq,l , Cl . Requiring then δ0 to satisfy the smallness condition: 1 4

Cq,l δ0 ≤

(13.677)

(13.676) implies: u G1,[l+2];a,q (t)



t

≤ Cq,l R[l+2];a,q (t, u) + Cl 0

u A(t  )G1,[l+2];a,q (t  )dt 

(13.678)

for new constants Cq,l , Cl . This is a linear integral inequality, with respect to t, for the u function G1,[l+2];a,q (t). By (13.647), (5.243) and Proposition 13.1, we have:

s

A(t)dt ≤ Cl : independent of s

(13.679)

0

In view of the fact that R[l+2];a,q (t, u) is a non-decreasing function of t at each u, (13.678) then implies: u (t) ≤ Cq,l R[l+2]:a,q (t, u) (13.680) G1,[l+2];a,q (for a new constant Cq,l ), hence also: t u A(t  )G1,[l+2];a,q (t  )dt  ≤ Cq,l R[l+2];a,q (t, u)

(13.681)

0

(for a new constant Cq,l ). Substituting the estimates (13.680), (13.681), in the right-hand sides of (13.656), (13.666), and (13.670), omitting in each case the first term in parenthesis on the left, we obtain:      1 F1t [R j Ril . . . Ri1 ψα ](u) + K [R j Ril . . . Ri1 ψα ](t, u)   2 j,α

j,α

log(1 + t)]−2q ≤ Cq,l R[l+2];a,q (t, u)

· µ2a m,u (t)[1 + 

 α

F1t [Ril−m

. . . Ri1 (T )

m+1

(13.682)

1 ψα ](u) + K [Ril−m . . . Ri1 (T )m+1 ψα ](t, u) 2 α



−2q · µ2a m,u (t)[1 + log(1 + t)]

≤ Cq,l R[l+2];a,q (t, u) 

F1t [ψ](u) +

1 −2q K [ψ](t, u) µ2a m,u (t)[1 + log(1 + t)] 2 ≤ Cq,l R[l+2];a,q (t, u)

(13.683)

(13.684)

Part 3: The energy estimates

845

t Summing then as in (13.673), yields, in view of the definitions of F1,[l+2] (u), K [l+2] (t, u),  1 t −2q (u) + K [l+2] (t, u) µ2a F1,[l+2] m,u (t)[1 + log(1 + t)] 2 ≤ Cq,l R[l+2];a,q (t, u) (13.685)

(for a new constant Cq,l ). Now the definition (13.657) of R[l+2];a,q (t, u) reads: R[l+2];a,q (t, u) = δ I[l+2];a,q (t, u) + N[l+2];a,q (t, u) where: N[l+2];a,q (t, u) =

u D[l+2]

u + G0,[l+2];a, p (t)

  1  V1,[l+2];a,q + 1+ (t, u) δ

(13.686)

(13.687)

Keeping only the term (1/2)K [l+2] (t, u) on the left in (13.685) we have: 1 2a µ (t)[1 + log(1 + t)]−2q K [l+2] (t, u) ≤ Cq,l δ I[l+2];a,q (t, u) + Cq,l N[l+2];a,q (t, u) 2 m,u (13.688) The same holds with t replaced by any t  ∈ [0, t]. The right-hand side of (13.688) being a non-decreasing function of t at each u, the inequality corresponding to t  holds a fortiori if we again replace t  by t on the right. Taking then the supremum over all t  ∈ [0, t] on the left we obtain, in view of the definition (13.640), 1 I[l+2];a,q (t, u) ≤ Cq,l δ I[l+2];a,q (t, u) + Cq,l N[l+2];a,q (t, u) 2 We now choose δ according to: Cq,l δ =

1 4

(13.689)

(13.690)

Then (13.689) implies: I[l+2];a,q (t, u) ≤ Cq,l N[l+2];a,q (t, u)

(13.691)

Substituting the estimate (13.691) in (13.686) and the result in (13.685), and keeping t (u) on the left, we obtain: only the term F1,[l+2]   −2q t  µ2a F1,[l+2] (u) ≤ Cq,l Q [l+2];a, p (t, u) + V1,[l+2];a,q (t, u) m,u (t)[1 + log(1 + t)] (13.692) (for a new constant Cq,l ) where: u u Q [l+2];a, p (t, u) = D[l+2] + G0,[l+2];a, p (t)

(13.693)

Note that Q [l+2];a, p (t, u) is a non-decreasing function of t at each u. Now, the inequality (13.692) holds with t replaced by any t  ∈ [0, t]. The right-hand side being a nondecreasing function of t at each u, the inequality corresponding to t  holds a fortiori if we

846

Chapter 13. Error Estimates

again replace t  by t on the right. Taking then the supremum over all t  ∈ [0, t] on the left we obtain, in view of the definition (13.630):   t  (u) ≤ Cq,l Q [l+2];a, p (t, u) + V1,[l+2];a,q (t, u) (13.694) H1,[l+2];a,q  Recalling the definition (13.631) of V1,[l+2];a,q (t, u), this is a linear integral inequality, t with respect to u, for the function H1,[l+2];a,q (u): u t t H1,[l+2];a,q (u) ≤ Cq,l Q [l+2];a, p (t, u) + Cq,l H1,[l+2];a,q (u  )du  (13.695) 0

Defining the functions: 

u

u G 0,[n];a, p (t) = sup G0,[n];a, p (t) u  ∈[0,u]

(13.696)

which are non-decreasing functions of u at each t as well as non-decreasing functions of t at each u, we may replace Q [l+2];a, p (t, u) on the right in (13.695) by: u

u Q [l+2];a, p (t, u) = D[l+2] + G 0,[l+2];a, p (t)

(13.697)

which is also a non-decreasing function of u at each t as well as a non-decreasing function of t at each u. Recalling that [0, ε0] is a bounded interval, (13.695) then implies: t H1,[l+2];a,q (u) ≤ Cq,l Q [l+2];a, p (t, u)

(13.698)

(for a new constant Cq,l ), hence also:  (t, u) ≤ Cq,l ε0 Q [l+2];a, p (t, u) V1,[l+2];a,q

(13.699)

Substituting this in (13.687) and the result in (13.691) we obtain: I[l+2];a,q (t, u) ≤ Cq,l Q [l+2];a, p (t, u)

(13.700)

(for a new constant Cq,l ). Substituting finally this in (13.686) and the result in (13.680) yields: u (t) ≤ Cq,l Q [l+2];a, p (t, u) (13.701) G1,[l+2];a,q (for a new constant Cq,l ). Consider now the integral identities corresponding to the variations (13.161) and to the vectorfield K 0 . In each of these we have the borderline integral (13.206), bounded by the sum of (13.211) and (13.216):   1 1 2 p (i1 ...il ) + µ−2a G0,l+2;a, p (t) (13.702) C m (t)[1 + log(1 + t)] 2a 2p We also have the remaining integrals, bounded in the case of (13.224) by (13.241), hence by:

 2p  G0,[l+2];a, p (t) + G1,[l+2];a,q Cq δ0 µ−2a (t) (13.703) m (t)[1 + log(1 + t)]

Part 3: The energy estimates

847

and in the case of (13.268) by (13.269), hence by (13.703) as well. Also, the integral (13.272), bounded by (13.273), hence by:  2p C p δ0 µ−2a m (t)[1 + log(1 + t)] P[l+2] G0,[l+2];a, p (t)

(13.704)

Combining we see that all the remaining integrals associated to the variations (13.161) and to K 0 containing the top order spatial derivatives of the acoustical entities are bounded by:  2p  C p,l δ0 µ−2a G0,[l+2];a, p (t) + G1,[l+2];a,q (t) · m (t)[1 + log(1 + t)]

   P[l+2] + G0,[l+2];a, p (t) + G1,[l+2];a,q (t) (13.705) (recall (13.634)). By the discussion following (13.613), the contributions of the terms in the sum (13.116) of which one factor is a derivative of the (Y ) π˜ of order more than (l + 1)∗ , but which do not contain top order spatial derivatives of the acoustical entities, are bounded using Propositions 12.11 and 12.12 and the bootstrap assumption. We obtain a bound for these contributions to the error integrals associated to K 0 and to any of the variations, up to the top order, by:  

p  (t) + G0,[l+2];a, p (t) P[l+2] + G1,[l+2];a,q Cl δ0 µ−a m (t)[1 + log(1 + t)]  t 1/2 ε0 t × (1 + t  )−3/2 E0,[l+2] (t  )dt  + F0,[l+2] (u  )du  0 ε0 0 t   + C l δ0 F0,[l+2] (u )du 0

 + C l δ0

ε0

0

t F 1,[l+2] (u  )du 



× 

0

t

1/2

(1 + t  )−3/2 E0,[l+2] (t  )dt  +



ε0 0

1/2

+ Cl δ0 K [l+2] (t, ε0 )  t  −3/2   × (1 + t ) E0,[l+2] (t )dt + 0

ε0 0

t F0,[l+2] (u  )du 

t F0,[l+2] (u  )du 

1/2

1/2 (13.706)

the last three terms in (13.706) contributed by the terms in (13.116) containing the top order derivatives of the acoustical entities of which however one is a derivative with respect to L (hence are expressible in terms of the top order derivatives of the ψα ). Here, we have defined: n  t t (u) = F0,m (u) (13.707) F0,[n] m=1

848

Chapter 13. Error Estimates

t where F0,n (u) represents the sum of the fluxes associated to the vectorfield K 0 of all the nth order variations (13.104), (13.106) of the form specified in the paragraph just following (13.106). Also:

 t t  F 1,[n] (u) = sup [1 + log(1 + t  )]−4 F1,[n] (u) (13.708) t  ∈[0,t ]

K [n] (t, u) = sup

t  ∈[0,t ]

 [1 + log(1 + t  )]−4 K [n] (t  , u)

(13.709)

On the other hand, all the other terms in the sum (13.116) contain derivatives of the of order at most (l + 1)∗ , thus spatial derivatives of χ  of order at most (l + 1)∗ and ε spatial derivatives of µ of order at most (l + 1)∗ + 1, which are bounded in L ∞ (t 0 ) by virtue of Propositions 12.9 and 12.10 and the bootstrap assumption. The contributions of these terms to the error integrals associated to K 0 or to K 1 and to any of the variations, up to the top order, are bounded exactly as in Chapter 7. For the error integrals associated to K 0 we obtain from Lemma 7.6 the bound: t ε0 t F0,[l+2] (u  )du  (13.710) Cl (1 + t  )−3/2 E0,[l+2] (t  )dt  + Cl (Y ) π ˜

0



ε0

+ Cl 0

t F 1,[l+2] (u  )du 



+ Cl K [l+2] (t,ε0 )

1/2

0

1/2 



t

 −3/2

(1 + t )





ε0

E0,[l+2] (t )dt +

0 t



(1 + t  )−3/2 E0,[l+2] (t  )dt  +

0



ε0 0

0

t F0,[l+2] (u  )du 

t F0,[l+2] (u  )du 

1/2

1/2

Note that by the definitions (13.630), (13.640), and the condition (13.634), we have, for all t  ∈ [0, t]: 

t   2 p t (u) ≤ µ−2a [1 + log(1 + t  )]−4 F1,[n] m,u (t )[1 + log(1 + t )] H1,[n];a,q (u) 

[1 + log(1 + t )]

−4



K [n] (t , u) ≤

  2p µ−2a m,u (t )[1 + log(1 + t )] I[l+2];a,q (t, u)

(13.711) (13.712)

hence, taking the supremum over t  ∈ [0, t], we obtain, in view of Corollary 2 of Lemma 8.11, t

2 p t F 1,[n] (u) ≤ Cµ−2a m,u (t)[1 + log(1 + t)] H1,[n];a,q (u)

K [n] (t, u) ≤

Cµ−2a m,u (t)[1 +

2p

log(1 + t)] I[l+2];a,q (t, u)

(13.713) (13.714)

In view of the definition (13.631) and the fact that µm,u (t) is a non-increasing function of t at each u we then have: u t 2p  F 1,[n] (u  )du  ≤ µ−2a (13.715) m,u [1 + log(1 + t)] V1,[n];a,q (t, u) 0

We also define: t H0,[n];a, p (u) = sup

V0,[n];a, p (t, u) =

t  ∈[0,t ] u 0

  t [1 + log(1 + t  )]−2 p µ2a m,u (t )F0,[n] (u)

t   H0,[n];a, p (u )du

(13.716) (13.717)

Part 3: The energy estimates

849

t Then H0,[n];a, p (u) is a non-decreasing function of t at each u, V0,[n];a, p (t, u) is a nondecreasing function of u at each t as well as a non-decreasing function of t at each u, and for each u ∈ [0, ε0]: u u   2p t   F0,[n] (u )du ≤ µ−2a m,u  (t)[1 + log(1 + t)] H0,[n];a, p (u )du 0

0

2p ≤ µ−2a m,u (t)[1 + log(1 + t)] V0,[n];a, p (t, u)

(13.718)

In view of (13.218), (13.714), (13.715), (13.718), we can bound (13.706) by: 2p · Cl δ0 µ−2a m (t)[1 + log(1 + t)]

 2   P[l+2] + δ (G1,[l+2];a,q (t) + G0,[l+2];a, p (t))     t 1  −3/2   (1 + t ) G0,[l+2];a, p (t )dt + V0,[l+2];a, p (t, ε0 ) + 1+  δ 0    +δ  V1,[l+2];a,q (t, ε0 ) + I[l+2];a,q (t, ε0 ) (13.719)

and we can bound (13.710) by: 2p · Cl µ−2a m (t)[1 + log(1 + t)]     t 1  −3/2   1+  (1 + t ) G0,[l+2];a, p (t )dt + V0,[l+2];a, p (t, ε0 ) δ 0    +δ  V1,[l+2];a,q (13.720) (t, ε0 ) + I[l+2];a,q

for any positive constant δ  (we shall choose δ  below). Finally, to each variation ψ and to K 0 is associated the first of the error integrals (13.620). According to the estimate (5.267) of Chapter 5 we have:

7  Wεt k=1 0 t

+C

0



t

Q 0 [ψ]dµh ≤ 0



(1 + t  )−2 [1 + log(1 + t  )]4 Bs (t  )E 1 [ψ](t  )dt  

(1 + t  )−1 [1 + log(1 + t  )]−2 (E 1 [ψ](t  ) + E 0 [ψ](t  ))dt 

+ C V0 [ψ](t,ε0 )  1/2 t  −1  −2   + C V0 [ψ](t,ε0 ) + (1 + t ) [1 + log(1 + t )] E 0 [ψ](t )dt (V1 [ψ](t,ε0 ))1/2 0 t + C (1 + t  )−2 [1 + log(1 + t  )]2 V1 [ψ](t  ,ε0 )dt  0



t

+ C K [ψ](t,ε0 )1/2

(1 + t  )−1 [1 + log(1 + t  )]−2 E 0 [ψ](t  )dt 

1/2

0

+ C K [ψ](t,ε0 )1/2 (V0 [ψ](t,ε0 ))1/2

(13.721)

850

Chapter 13. Error Estimates

It follows that: 7  Q 0 [ψ]dµh ≤ Wεt k=1 0

t

0

 B˜ s (t  )E 1 [ψ](t  )dt 

   t 1  −1  −2   +C 1+  (1 + t ) [1 + log(1 + t )] E 0 [ψ](t )dt + V0 [ψ](t, ε0 ) δ 0 + δ  V1 [ψ](t, ε0 ) t +C (1 + t  )−2 [1 + log(1 + t  )]2 V1 [ψ](t  , ε0 )dt  0

+ δ  K [ψ](t, ε0 )

(13.722)

where (see (5.273)): B˜ s (t) = (1 + t)−2 [1 + log(1 + t)]4 Bs (t) + C(1 + t)−1 [1 + log(1 + t)]−2 Since



E 1 [ψ](t) = sup

t  ∈[0,t ]

≤ sup

t  ∈[0,t ]

[1 + log(1 + t  )]−4 E1 [ψ](t  )

(13.723)



  2p   µ−2a m (t )[1 + log(1 + t )] G1;a,q [ψ](t )



2p  ≤ Cµ−2a m (t)[1 + log(1 + t)] G1;a,q [ψ](t)

(13.724)

the first term on the right in (13.722) is bounded by: t −2a 2p  Cµm (t)[1 + log(1 + t)] [ψ](t  )dt  B˜ s (t  )G1;a,q

(13.725)

0

Similarly, in regard to the second term on the right in (13.722), we have, in view of (13.633), t (1 + t  )−1 [1 + log(1 + t  )]−2 E 0 [ψ](t  )dt  (13.726) 0 t 2p (t)[1 + log(1 + t)] (1 + t  )−1 [1 + log(1 + t  )]−2 G0;a, p [ψ](t  )dt  ≤ Cµ−2a m 0

Moreover, defining: t H0;a, p [ψ](u) = sup

t  ∈[0,t ]



 t [1 + log(1 + t  )]−2 p µ2a m,u (t )F0 [ψ](u)



V0;a, p [ψ](t, u) = 0

u



t   H0;a, p [ψ](u )du

we have, in analogy with (13.629) and (13.718): t 2p V0 [ψ](t, u) = F0t (u  )du  ≤ µ−2a m,u (t)[1 + log(1 + t)] V0;a, p [ψ](t, u) 0

(13.727)

(13.728)

(13.729)

Part 3: The energy estimates

851

In regard to the third and fourth terms on the right in (13.722), recalling from Chapter 5 that: u V1 [ψ](t, u) =

t

0

F 1 [ψ](u  )du 

(13.730)

(see (5.248)) we have, using an inequality analogous to (13.713), 2p  V1 [ψ](t, u) ≤ Cµ−2a m,u (t)[1 + log(1 + t)] V1;a,q [ψ](t, u)

(13.731)

hence also: t (1 + t  )−2 [1 + log(1 + t  )]2 V1 [ψ](t  , ε0 )dt  (13.732) 0 t 2p  ≤ Cµ−2a (1 + t  )−2 [1 + log(1 + t  )]2 V1;a,q [ψ](t  , ε0 )dt  m (t)[1 + log(1 + t)] 0

Finally, in analogy with (13.714) we have, in regard to the last term on the right in (13.722), 2p K [ψ](t, ε0 ) ≤ Cµ−2a m (t)[1 + log(1 + t)] Ia,q [ψ](t, ε0 )

where: Ia,q [ψ](t, u) = sup

t  ∈[0,t ]

   [1 + log(1 + t  )]−2q µ2a m,u (t )K [ψ](t , u)

(13.733)

(13.734)

Substituting the above in (13.722) we obtain:

7  Wεt k=1 0



t

Q 0 [ψ]dµh ≤ C 0

 [ψ](t  )dt  B˜ s (t  )G1;a,q

   t 1  −1  −2   +C 1+  (1 + t ) [1 + log(1 + t )] G0;a, p [ψ](t )dt + V0;a, p [ψ](t, ε0 ) δ 0 t   + Cδ  V1;a,q [ψ](t, ε0 ) + C (1 + t  )−2 [1 + log(1 + t  )]2 V1;a,q [ψ](t  , ε0 )dt  0

+ Cδ  Ia,q [ψ](t, ε0 )

(13.735)

Consequently, for any variation ψ of order up to l + 2, of the form specified in the paragraph following (13.106) we have:

7  Wεt k=1 0



t

Q 0 [ψ]dµh ≤ C 0

 B˜ s (t  )G1,[l+2];a,q (t  )dt 

 t  1  −1  −2   +C 1+  (1 + t ) [1 + log(1 + t )] G0,[l+2];a, p (t )dt + V0,[l+2];a, p (t,ε0 ) δ 0 t    + Cδ V1,[l+2];a,q (t,ε0 ) + C (1 + t  )−2 [1 + log(1 + t  )]2 V1,[l+2];a,q (t  ,ε0 )dt  + Cδ  I[l+2],a,q (t,ε0 )

0

(13.736)

852

Chapter 13. Error Estimates

We now consider the integral identity corresponding to the vectorfield K 0 ((5.67) with u = ε0 ) and to the variations (13.161) R j Ril . . . Ri1 ψα , j = 1, 2, 3, α = 0, 1, 2, 3, for a given multi-index (i 1 . . . i l ). Summing over j and α, we then obtain from (13.702), in regard to the borderline integral, and from (13.705), (13.720), (13.736), (13.719), in regard to the remaining contributions, the following: 

E0 [R j Ril ... Ri1 ψα ](t) +

j,α





−2 p F0t [R j Ril ... Ri1 ψα ](ε0 ) µ2a m (t)[1 + log(1 + t)]

j,α



1 1 (i1 ...il ) + G0,[l+2];a, p (t) + C p,l D[l+2] a p 

 + C p,l δ0 G0,[l+2];a, p (t) + G1,[l+2];a,q (t) t  +C (t  )dt  B˜ s (t  )G1,[l+2];a,q 0 t  + C (1 + t  )−2 [1 + log(1 + t  )]2 V1,[l+2];a,q (t  ,ε0 )dt  0   t 1  −1  −2   + Cl 1 +  (1 + t ) [1 + log(1 + t )] G0,[l+2];a, p (t )dt + V0,[l+2];a, p (t,ε0 ) δ 0

  (t,ε0 ) + I[l+2];a,q (t,ε0 ) (13.737) + Cl δ  V1,[l+2];a,q

≤C

We now substitute on the right the estimates (13.699)–(13.701) to obtain:   

j,α

E0 [R j Ril ... Ri1 ψα ](t) +



 j,α



 

F0t [R j Ril ... Ri1 ψα ](ε0) µ2a (t)[1 + log(1 + t)]−2 p  m

1 1 (i1 ...il ) + G0,[l+2];a, p (t) + C p,l (δ0 + δ  )G 0,[l+2];a, p (t) (13.738) a p   t 1 + C p,l 1 +  B˜ s (t  )G 0,[l+2];a, p (t  )dt  + V0,[l+2];a, p (t,ε0 ) D[l+2] + δ 0

≤C

Here, we have used the fact that by Proposition 13.2:

s

B˜ s (t)dt ≤ C : independent of s

(13.739)

0

It is crucial that the constant C in front of the first term on the right in (13.738) is independent of a, p or l. By the discussion following (13.647) and concluding with (13.656), inequality (13.738) holds with ε0 replaced by any u ∈ (0, ε0 ], that is, we have, for all u ∈ [0, ε0 ]

Part 3: The energy estimates

853

and all t ∈ [0, s]:   −2 p E0u [R j Ril ... Ri1 ψα ](t) + F0t [R j Ril ... Ri1 ψα ](u) µ2a m,u (t)[1 + log(1 + t)] j,α



j,α



1 1 (i1 ...il ) u u + G0,[l+2];a, p (t) + C p,l (δ0 + δ  )G 0,[l+2];a, p (t) (13.740) a p   t 1 u u D[l+2] + + C p,l 1 +  B˜ s (t  )G 0,[l+2];a, p (t  )dt  + V0,[l+2];a, p (t,u) δ 0

≤C

Keeping only the term 

E0u [R j Ril . . . Ri1 ψα ]

j,α

on the left in (13.740), we have: −2 p µ2a m,u (t)[1 + log(1 + t)]



E0u [R j Ril . . . Ri1 ψα ](t)

(13.741)

j,α

 1 (i1 ...il ) u 1 u + ≤C G0,[l+2];a, p (t) + C p,l (δ0 + δ  )G 0,[l+2];a, p (t) a p   t 1 u  u   ˜ + C p,l 1 +  D[l+2] + Bs (t )G 0,[l+2];a, p (t )dt + V0,[l+2];a, p (t, u) δ 0 

The same holds with t replaced by any t  ∈ [0, t]. Now the right-hand side of (13.741) is a non-decreasing function of t at each u. The inequality corresponding to t  thus holds a fortiori if we again replace t  by t on the right-hand side. Taking then the supremum over all t  ∈ [0, t] on the left-hand side we obtain, in view of the definition (13.650), (i1 ...il ) u G0,[l+2];a, p (t)



(13.742)



1 1 (i1 ...il ) u u + G0,[l+2];a, p (t) + C p,l (δ0 + δ  )G 0,[l+2];a, p (t) a p   t 1 u  u   ˜ + C p,l 1 +  Bs (t )G 0,[l+2];a, p (t )dt + V0,[l+2];a, p (t, u) D[l+2] + δ 0

≤C

If a and p are chosen suitably large so that:   1 1 1 C + ≤ a p 2

(13.743)

then (13.742) will imply: 1 2

(i1 ...il ) u G0,[l+2];a, p (t)

u

≤ C p,l (δ0 + δ  )G 0,[l+2];a, p (t) (13.744)   t 1 u  u   ˜ + C p,l 1 +  D[l+2] + Bs (t )G 0,[l+2];a, p (t )dt + V0,[l+2];a, p (t, u) δ 0

854

Chapter 13. Error Estimates

Consider now the integral identities corresponding to the variations (13.163) and the vectorfield K 0 . In each of these we have the borderline integral (13.291), bounded by (13.292),   1 1 2 p (i1 ...il−m ) µ−2a G0,m,l+2;a, p (t) (13.745) C + m (t)[1 + log(1 + t)] 2a 2p We also have the remaining integrals, bounded in the case of (13.296) by (13.703) and in the case of (13.309) by the sum of (13.269) and (13.310), hence also by (13.703). We also have the integral (13.312) bounded by (13.313), hence by (13.704). Combining we see that all the remaining integrals associated to the variations (13.163) and to K 0 containing the top order spatial derivatives of the acoustical entities are bounded by (13.705). On the other hand, the error integrals associated to the variations (13.163) contributed by all other terms in (13.116) are bounded by the sum of (13.706) and (13.710), hence by the sum of (13.720) and (13.719). Finally for the variations in question we have, associated to K 0 , the first of the integrals (13.620), which is bounded by (13.736). Consequently, the integral identity corresponding to K 0 ((5.67) with u = ε0 ) and to the variations Ril−m . . . Ri1 (T )m+1 ψα , α = 0, 1, 2, 3, for a given multi-index (i 1 . . . i l−m ), summed over α, yields, after substituting the estimates (13.699)–(13.701):     m+1 t m+1 E0 [Ril−m . . . Ri1 (T ) ψα ](t) + F0 [Ril . . . Ri1 (T ) ψα ](ε0 ) (13.746) α

α

−2 p

+ log(1 + t)]  1 (i1 ...il−m ) 1 + ≤C G0,m,[l+2];a, p (t) + C p,l (δ0 + δ  )G 0,[l+2];a, p (t) a p   t 1 + C p,l 1 +  B˜ s (t  )G 0,[l+2];a, p (t  )dt  + V0,[l+2];a, p (t, ε0 ) D[l+2] + δ 0 

· µ2a m (t)[1

It is again crucial that the constant C in front of the first term on the right in (13.746) is independent of a, p or l. By the discussion following (13.647) and concluding with (13.656), we can replace ε0 by any u ∈ (0, ε0 ] in (13.746), obtaining, for all u ∈ [0, ε0 ] and all t ∈ [0, s]:     u m+1 t m+1 E0 [Ril−m . . . Ri1 (T ) ψα ](t) + F0 [Ril . . . Ri1 (T ) ψα ](u) (13.747) α

α

 ≤C

−2 p · µ2a m,u (t)[1 + log(1 + t)]

 1 (i1 ...il−m ) u 1 u + G0,m,[l+2];a, p (t) + C p,l (δ0 + δ  )G 0,[l+2];a, p (t) a p   t 1 u u + C p,l 1 +  B˜ s (t  )G 0,[l+2];a, p (t  )dt  + V0,[l+2];a, p (t, u) + D[l+2] δ 0

Keeping only the term

 α

E0u [Ril−m . . . Ri1 (T )m+1 ψα ]

Part 3: The energy estimates

855

on the left in (13.747), we have: −2 p µ2a m,u (t)[1 + log(1 + t)]

 α

E0u [Ril−m . . . Ri1 (T )m+1 ψα ](t)

(13.748)

 1 (i1 ...il−m ) u 1 u + G0,m,[l+2];a, p (t) + C p,l (δ0 + δ  )G 0,[l+2];a, p (t) a p   t 1 u u + C p,l 1 +  D[l+2] + B˜ s (t  )G 0,[l+2];a, p (t  )dt  + V0,[l+2];a, p (t, u) δ 0 

≤C

The same holds with t replaced by any t  ∈ [0, t]. Now the right-hand side of (13.748) is a non-decreasing function of t at each u. The inequality corresponding to t  thus holds a fortiori if we again replace t  by t on the right-hand side. Taking then the supremum over all t  ∈ [0, t] on the left-hand side we obtain, in view of the definition (13.651), (i1 ...il−m ) u G0,m,[l+2];a, p (t)



(13.749)



1 1 (i1 ...il−m ) u u + G0,m,[l+2];a, p (t) + C p,l (δ0 + δ  )G 0,[l+2];a, p (t) a p   t 1 u  u   ˜ + C p,l 1 +  Bs (t )G 0,[l+2];a, p (t )dt + V0,[l+2];a, p (t, u) D[l+2] + δ 0

≤C

If a and p are chosen suitably large so that (13.743) holds, then (13.742) will imply: 1 2

u

(i1 ...il−m ) u G0,m,[l+2];a, p (t)

≤ C p,l (δ0 + δ  )G 0,[l+2];a, p (t) (13.750)   t 1 u  u   ˜ + C p,l 1 +  Bs (t )G 0,[l+2];a, p (t )dt + V0,[l+2];a, p (t, u) D[l+2] + δ 0

Consider finally the integral identities corresponding to any of the other variations ψ of order up to l + 2, of the form specified in the paragraph following (13.106), and to the vectorfield K 0 . These identities contain no borderline integrals and all error integrals involved are bounded by: u

(13.751) C p,l (δ0 + δ  )G 0,[l+2];a, p (t)   t 1 u  u   ˜ + C p,l 1 +  Bs (t )G 0,[l+2];a, p (t )dt + V0,[l+2];a, p (t, u) D[l+2] + δ 0 We thus obtain, for each such variation ψ the inequality:  u  −2 p E0 [ψ](t) + F0t [ψ](u) µ2a m,u (t)[1 + log(1 + t)] ≤

u C p,l (δ0 + δ  )G 0,[l+2];a, p (t)

(13.752)

  t 1 u u + C p,l 1 +  + D[l+2] B˜ s (t  )G 0,[l+2];a, p (t  )dt  + V0,[l+2];a, p (t, u) δ 0

Keeping only the term E0u [ψ]

856

Chapter 13. Error Estimates

on the left in (13.753), and noting that the right-hand side is a non-decreasing function of t at each u, we may replace t by any t  ∈ [0, t] on the left while keeping t on the right, thus, taking the supremum over all t  ∈ [0, t] we obtain, in view of the definition (13.648), u

u  G0;a, (13.753) p [ψ](t) ≤ C p,l (δ0 + δ )G 0,[l+2];a, p (t)   t 1 u  u   ˜ + C p,l 1 +  D[l+2] + Bs (t )G 0,[l+2];a, p (t )dt + V0,[l+2];a, p (t, u) δ 0 u Now, according to the definition of E0,[l+2] (t) we have, in analogy with (13.673),

u E0,[l+2] (t) =

   i1 ...il

+



E0u [R j Ril . . . Ri1 ψα ](t)

j,α

l  



 α

m=0 i1 ...il−m

+



   

E0u [Ril−m

. . . Ri1 (T )

m+1

ψα ](t)

E0u [ψ](t)

(13.754)

ψ

where the last sum is over all the other variations ψ up to order l +2, as specified above. It follows, by virtue of (13.674), and in view of the definitions (13.648), (13.650), (13.651), (13.654), that: u G0,[l+2];a, p (t) ≤



(i1 ...il ) u G0,l+2;a, p (t)

+

i1 ...il

+



l  

(i1 ...il−m ) u G0,m,l+2;a, p (t)

m=0 i1 ...il−m u G0;a, p [ψ](t)

(13.755)

ψ

Thus, summing inequalities (13.744) over i 1 . . . i l , summing inequalities (13.750) over i 1 . . . i l−m and over m = 0, . . . , l, and summing inequalities (13.753) over all the other variations ψ of order up to l + 2, as specified above, and then adding the resulting three inequalities, we obtain: 1 u u G0,[l+2];a, p (t) ≤ C p,l (δ0 + δ  )G 0,[l+2];a, p (t) (13.756) 2   t 1 u u + C p,l 1 +  B˜ s (t  )G 0,[l+2];a, p (t  )dt  + V0,[l+2];a, p (t, u) + D[l+2] δ 0 for a new constant C p,l . Requiring then δ0 to satisfy the smallness condition: C p,l δ0 ≤

1 8

(13.757)

Part 3: The energy estimates

857

and then choosing δ  according to: C p,l δ  =

1 8

(13.758)

(13.756) implies: 1 u 1 u G (t) ≤ G 0,[l+2];a, p (t) 2 0,[l+2];a, p 4  u +C p,l D[l+2] +



t 0

(13.759)  u   ˜ Bs (t )G 0,[l+2];a, p (t )dt + V0,[l+2];a, p (t, u)

for a new constant C p,l . The same holds with u replaced by any u  ∈ [0, u]. Now the right-hand side of (13.759) is a non-decreasing function of u at each t. The inequality corresponding to u  thus holds a fortiori if we again replace u  by u on the right-hand side. Taking then the supremum over all u  ∈ [0, u] on the left-hand side we obtain, in view of the definition (13.696),  t 1 u u u G 0,[l+2];a, p (t) ≤ C p,l D[l+2] + B˜ s (t  )G 0,[l+2];a, p (t  )dt  + V0,[l+2];a, p (t, u) 4 0 (13.760) u

This is a linear integral inequality, with respect to t, for the function G 0,[l+2];a, p (t). In view of (13.739) and the fact that V0,[l+2];a, p (t, u) is a non-decreasing function of t at each u, (13.760) implies:

 u u G 0,[l+2];a, p (t) ≤ C p,l D[l+2] + V0,[l+2];a, p (t, u) (13.761) (for a new constant C p,l ), hence also: t 

u u + V0,[l+2];a, p (t, u) B˜ s (t  )G 0,[l+2];a, p (t  )dt  ≤ C p,l D[l+2]

(13.762)

0

(for a new constant C p,l ). Substituting the estimates (13.761), (13.762), in the right-hand sides of (13.740), (13.747), and (13.752), omitting in each case the first term in parenthesis on the left we obtain:  −2 p µ2a F0t [R j Ril . . . Ri1 ψα ](u) m,u (t)[1 + log(1 + t)] j,α 

u ≤ C p,l D[l+2] + V0,[l+2];a, p (t, u) (13.763)  −2 p µ2a F0t [Ril . . . Ri1 (T )m+1 ψα ](u) m,u (t)[1 + log(1 + t)]

 α u + V0,[l+2];a, p (t, u) (13.764) ≤ C p,l D[l+2] −2 p t µ2a F0 [ψ](u) m,u (t)[1 + log(1 + t)] 

u ≤ C p,l D[l+2] + V0,[l+2];a, p (t, u)

(13.765)

858

Chapter 13. Error Estimates

t Summing then as in (13.754), yields, in view of the definition of F0,[l+2] (u), 

−2 p t u µ2a (t)[1 + log(1 + t)] F (u) ≤ C + V (t, u) (13.766) D p,l 0,[l+2];a, p m,u 0,[l+2] [l+2]

(for a new constant C p,l ). Since the inequality holds with t replaced by any t  ∈ [0, t] and the right-hand side is a non-decreasing function of t at each u, the inequality corresponding to t  holds a fortiori if we again replace t  by t on the right. Taking then the supremum over all t  ∈ [0, t] on the left we obtain, in view of the definition (13.716):

 t u D (u) ≤ C + V (t, u) (13.767) H0,[l+2];a, p,l 0,[l+2];a, p [l+2] p Recalling the definition (13.717) of V0,[l+2];a, p (t, u), this is a linear integral inequality, t with respect to u, for the function H0,[l+2];a, p (u): u t u t   H0,[l+2];a, (13.768) H0,[l+2];a, p (u) ≤ C p,l D[l+2] + C p,l p (u )du 0

u Since [0, ε0 ] is a bounded interval while D[l+2]

is a non-decreasing function of u, (13.768)

implies: t u H0,[l+2];a, p ≤ C p,l D[l+2]

(13.769)

(for a new constant C p,l ), hence also: u V0,[l+2];a, p (t, u) ≤ C p,l ε0 D[l+2]

(13.770)

Substituting this in (13.761) we obtain: u

u G 0,[l+2];a, p (t) ≤ C p,l D[l+2]

(13.771)

Substituting finally (13.771) in (13.697) and the result in (13.698), (13.700), (13.701), yields: u t u G1,[l+2];a,q (t), H1,[l+2];a,q (u), I[l+2];a,q (t, u) ≤ C p,l D[l+2]

(13.772)

This completes the top order energy estimates. At this point we shall specify the choice of a and p (recall from (13.634) that q = p + 2). We take a to be of the form: a = [a] +

3 4

(13.773)

where [a] denotes the integral part of a. We then choose [a] to be the smallest positive integer so that with C the largest of the two constants in (13.661) and (13.743) we have: C 3 ≤ a 8

(13.774)

We then choose p so that with the same constant C we have: C 1 = p 8

(13.775)

Part 3: The energy estimates

859

Then the conditions (13.661) and (13.743) are both satisfied and thereafter a, p and q are fixed. We then fix l so that: (13.776) l∗ ≥ [a] + 4 The reason for this requirement shall become apparent in the sequel. Once this requirement is imposed, l is fixed as well, therefore we shall not denote the dependence of the various constants on a, p, q, or l. Actually, the constants shall now depend only on the Lagrangian function L(σ ) (see Chapter 1) through the function H (σ ) and on

the constant k associated to the surrounding constant state, in particular through η0 = 1 − k H (k 2). We shall still require δ0 to be suitably small in relation to these constants. We now consider again the estimates for the energies E0,[l+1] and E1,[l+1] , associated to K 0 and K 1 respectively. We shall presently obtain improved estimates for these next to the top order energies, using the estimates for the top order energies just obtained. We shall show that the next to the top order quantities satisfy estimates similar to (13.769), (13.771), (13.772), but with a replaced by b =a−1

(13.777)

p replaced by 0, and, accordingly, q replaced by 2. Let us first consider the error integrals associated to the variations of order l + 1 and to K 1 , involving the highest spatial derivatives of the acoustical entities. The leading contributions are: 1) The contribution of (13.165) with l + 1 replaced by l to the corresponding integral (13.168), namely the integral (13.315) with l + 1 replaced by l: (ω/ν)(Ril . . . Ri1 trχ  )(T ψα )((L + ν)Ril . . . Ri1 ψα )dt  du  dµh/˜ (13.778) − Wut

2) The contribution of (13.166) with l replaced by l − 1 to the corresponding integral (13.168), namely the integral (13.487) with l replaced by l − 1: (ω/ν)(T ψα )(Ril−1−m ... Ri1 (T )m
/µ)((L + ν)Ril−1−m ... Ri1 (T )m+1 ψα )dt  du  dµh/˜ − Wut

(13.779)

for m = 0, . . . , l − 1. By virtue of the bound (13.413) the integral (13.778) is bounded by: Cδ0 (1 + t  )|Ril . . . Ri1 trχ  ||(L + ν)Ril . . . Ri1 ψα |dt  dudµh/ Wut



≤ Cδ0

1/2  2

Wut



|Ril . . . Ri1 trχ | dt du dµh/



·



1/2  2

Wut

2





(1 + t ) |(L + ν)Ri1 . . . Ri1 ψα | dt du dµh/

(13.780)

860

Chapter 13. Error Estimates

Now the last factor is bounded by:  C

u 0

while the first factor is:



t 0

which is bounded in terms of:

t F1,[l+1] (u  )du 

1/2

Ril . . . Ri1 trχ  2L 2 ( u ) dt 

(13.781) 1/2 (13.782)

t



t 0



 Au [l] (t )

2

dt 

1/2 (13.783) ε

0 2 Here the quantity Au [l] (t) is defined by replacing the L norms over t in the definition  2 u of the quantity A[l] (t) by L norms over t . Proposition 12.11 applies with ε0 replaced by u, giving a bound for Au [l] (t) in terms of: t  Qu Q Qu  u C(1 + t)−1 (1 + t  )−1 W[l+2] (t  ) + W[l+1] (t  ) + δ02 (1 + t  )−2 W[l] (t ) dt 

0

(13.784) Qu Q Qu u where the quantities W[l+2] (t), W[l+1] (t), W[l] (t) are defined by replacing the L 2 ε

Q

QQ

norms over t 0 in the definition of the quantities W[l+2] (t), W[l+1] (t), W[l] (t) by L 2 norms over tu . Now we have: u W[l+2] (t) + W[l+1] (t) + δ02 (1 + t)−2 W[l] (t)  −1/2 u (t) ≤ Cµm,u E1,[l+2]  −a−1/2 u (t)[1 + log(1 + t)]q G1,[l+2];a,q (t) ≤ Cµm,u Qu

Q Qu

(13.785)

Therefore (13.784) is bounded by: t  −a−1/2  u C(1 + t)−1 G1,[l+2];a,q (t) (1 + t  )−1 µm,u (t )[1 + log(1 + t  )]q dt  0  u u ≤ C(1 + t)−1 G1,[l+2];a,q (t)Ja,q−1 (t)  −a+1/2 u ≤ C(1 + t)−1 [1 + log(1 + t)]q+1 µm,u (t) D[l+2] (13.786) u (t) is the integral (13.260) with ε replaced by u and we have used (13.266) Here Ja,q 0 with ε0 replaced by u as well as the bounds (13.772). It follows that the contribution of (13.784) to (13.783) is bounded by:  t 1/2    u (1 + t  )−2 [1 + log(1 + t  )]2q+2 µ−2a+1 (t )dt C D[l+2] m,u 0  u ≤ C D[l+2] µ−a+1 (13.787) m,u (t)

Part 3: The energy estimates

861

the last integral being estimated by following an argument similar to that used to estimate (13.224) (with ε0 replaced by u). In fact (13.782) is similarly bounded. Since by (13.632) with a replaced by b and q by 2 we have: u t 4  F1,[l+1] (u  )du  ≤ µ−2b (13.788) m,u (t)[1 + log(1 + t)] V1,[l+1];b,2 (t, u) 0

we conclude that (13.780) is bounded by:   2 u  Cδ0 µ−2b D[l+2] V1,[l+1];b,2 (t, u) m,u (t)[1 + log(1 + t)]

(13.789)

if b is defined according to (13.777). By virtue of the bound (13.413) the integral (13.779) is bounded by: (1 + t  )|Ril−1−m . . . Ri1 (T )m
/ µ||(L + ν)Ril−1−m . . . Ri1 (T )m+1 ψα |dt  du  dµh/ Cδ0 Wut



≤ Cδ0

1/2 Wut



·





|Ril−1−m . . . Ri1 (T )
/ µ| dt du dµh/ m

2

1/2  2

Wut

(1 + t ) |(L + ν)Ril−1−m . . . Ri1 (T )

m+1





ψα | dt du dµh/ 2

(13.790)

Now the last factor is bounded by (13.781) while the first factor is:  t 1/2 m  Ril−1−m . . . Ri1 (T )
/ µ L 2 ( u ) dt

(13.791)

which is bounded in terms of:  t 2 1/2  u (1 + t  )−4 B[m,l+1] (t  ) dt 

(13.792)

t

0

0

ε

u Here the quantity B[m,l+1] (t) is defined by replacing the L 2 norms over t 0 in the definition of the quantity B[m,l+1] (t) by L 2 norms over tu . Proposition 12.12 applies with ε0 u replaced by u, giving a bound for B[m,l+1] (t) in terms of: t

u C(1 + t) (1 + t  )−1 W{l+2} (t  ) + (1 + t  )−1 [1 + log(1 + t  )] 0  Qu Q Qu  (t  ) + δ03 (1 + t  )−2 W{l} (t ) dt  (13.793) · W{l+1}

Now we have:

 Qu Q Qu u (t  ) + (1 + t  )−1 [1 + log(1 + t  )] W{l+1} (t  ) + δ03 (1 + t  )−2 W{l} (t  ) W{l+2}  

 −1/2 u u (t) + E0,[l+2] (t) (13.794) ≤ C µm,u E1,[l+2] 

  −a−1/2 u u (t)[1 + log(1 + t)]q G1,[l+2];a,q (t) + G0,[l+2];a, ≤ Cµm,u p (t)

862

Chapter 13. Error Estimates

Therefore (13.793) is bounded by:

   u u C(1 + t) G1,[l+2];a,q (t) + G0,[l+2];a, p (t) t −a−1/2  × (1 + t  )−1 µm,u (t )[1 + log(1 + t  )]q dt  0

   u u u ≤ C(1 + t) G1,[l+2];a,q (t) + G0,[l+2];a, (t) Ja,q−1 (t) p  −a+1/2 u ≤ C(1 + t)[1 + log(1 + t)]q+1 µm,u (t) D[l+2]

(13.795)

where we have used the bounds (13.771), (13.772). It follows that the contribution of (13.793) to (13.792) is bounded by:  t 1/2   −2  2q+2 −2a+1   u (1 + t ) [1 + log(1 + t )] µm,u (t )dt C D[l+2] 0  u ≤ C D[l+2] µ−a+1 (13.796) m,u (t) In fact (13.791) is similarly bounded. In view of (13.788) we then conclude that (13.790) is bounded by:   2 u  (t)[1 + log(1 + t)] D (13.797) Cδ0 µ−2b m,u [l+2] V1,[l+1];b,2 (t, u) if b is defined according to (13.777). We proceed to consider the error integrals associated to the variations of order l + 1 and to K 0 , involving the highest spatial derivatives of the acoustical entities. The leading contributions are: 1) The contribution of (13.165) with l + 1 replaced by l to the corresponding integral (13.171), namely the integral on the left in (13.172) with l + 1 replaced by l (and ε0 by u): |Ril . . . Ri1 trχ  ||T ψα ||L Ril . . . Ri1 ψα |dt  du  dµh/˜ (13.798) Wut

2) The contribution of (13.166) with l replaced by l − 1 to the corresponding integral (13.171), namely the integral (13.275) with l + 1 replaced by l: |Ril−1−m . . . Ri1 (T )m
/ µ||T ψα ||L Ril−1−m . . . Ri1 (T )m+1 ψα |dt  du  dµh/˜ Wut

(13.799)

By virtue of the bound (13.413) the integral (13.798) is bounded by: (1 + t  )−1 |Ril . . . Ri1 trχ  ||L Ril . . . Ri1 ψα |dt  du  dµh/ Cδ0 Wut



t

≤ Cδ0 0

(1 + t  )−1 Ril . . . Ri1 trχ   L 2 ( u ) L Ril . . . Ri1 ψα  L 2 ( u ) dt  t

t

(13.800)

Part 3: The energy estimates

863

Substituting the bound (13.786) for the first factor in the integrant on the right and noting the definition (13.777) of b and the fact that:   u u (t) ≤ Cµ−b (t) G0,[l+1];b,0 (t) (13.801) L Ril . . . Ri1 ψα  L 2 (tu ) ≤ C E0,[l+1] m,u we obtain that (13.800) is bounded by: t   −2b−1/2  u u Cδ0 D[l+2] G0,[l+1];b,0 (t) (1 + t  )−2 [1 + log(1 + t  )]q+1 µm,u (t )dt  0   −2b+1/2 u u ≤ Cδ0 µm,u (t) D[l+2] G0,[l+1];b,0 (t) (13.802) the last integral being estimated by following an argument similar to that used to estimate (13.224) (with ε0 replaced by u). By virtue of the bound (13.413) the integral (13.799) is bounded by: (1 + t  )−1 |Ril−1−m . . . Ri1 (T )m
/ µ| |L Ril−1−m . . . Ri1 (T )m+1 ψα |dt  du  dµh/ Cδ0 Wut



t

≤ Cδ0 0

(1 + t  )−1 Ril−1−m . . . Ri1 (T )m
/ µ L 2 ( u ) t

× L Ril−1−m . . . Ri1 (T )

m+1

ψα  L 2 ( u ) dt t



(13.803)

Substituting the bound (13.795) (multiplied by C(1 + t)−2 ) for the first factor in the integrant on the right and noting again the definition (13.777) of b and the fact that:   u u (t) ≤ Cµ−b L Ril−1−m . . . Ri1 (T )m+1 ψα  L 2 ( u ) ≤ C E0,[l+1] m,u (t) G0,[l+1];b,0 (t) t

(13.804) we obtain that (13.803) is bounded by: t   −2b−1/2  u u Cδ0 D[l+2] G0,[l+1];b,0 (t) (1 + t  )−2 [1 + log(1 + t  )]q+1 µm,u (t )dt  0   −2b+1/2 u u ≤ Cδ0 µm,u (t) D[l+2] G0,[l+1];b,0 (t) (13.805) a bound of the same form as (13.802). Consider now the error integrals associated to the variations of order l + 1 and to K 1 , or to K 0 , which contain the lower order spatial derivatives of the acoustical entities and are contributed by the remaining terms in the sum (13.116) with l + 2 replaced by l + 1. By the discussion following (13.613) of these error integrals, those contributed by the terms in which one factor is a derivative of the (Y ) π˜ of order more than (l + 1)∗ are bounded, using Propositions 12.11 and 12.12 and the bootstrap assumption, by (13.615), hence by (13.643), with ([l + 2]; a, q) replaced by ([l + 1]; b, 2) (and ε0 by u), thus by:

4 u u u + δ(G1,[l+1];b,2 (t) + G0,[l+1];b,2 (t)) D[l+2] Cl δ0 µ−2b m,u (t)[1 + log(1 + t)]   1  (t, u) (13.806) +δ I[l+1];b,2 (t, u) + 1 + V1,[l+1];b,2 δ

864

Chapter 13. Error Estimates

in the case of the error integrals associated to K 1 , by (13.706), hence by (13.719), with ([l + 2]; a, p) replaced by ([l + 1]; b, 0) (and ε0 by u), thus by:

u u u (t) D[l+2] + δ  (G1,[l+1];b,2 (t) + G0,[l+1];b,0 (t)) Cl δ0 µ−2b m,u     t 1  −3/2 u   + 1+  (1 + t ) G0,[l+1];b,0 (t )dt + V0,[l+1];b,0(t, u) δ 0    (t, u) + I[l+1];b,2 (t, u) (13.807) +δ  V1,[l+1];b,0 in the case of the error integrals associated to K 0 . In (13.806), (13.807), the arbitrary positive constants δ, δ  shall be chosen appropriately below. In fact, (13.806) and (13.807) also bound the error integrals associated to K 1 and to K 0 respectively, and to any of the variations of order up to l, of the form specified in the paragraph following (13.106), contributed by the terms in 13.116 in which one factor is a derivative of (Y ) π˜ of order more than (l + 1)∗ . On the other hand, the error integrals associated to any of the variations of order up to l + 1 contributed by the terms in the sum (13.116) containing derivatives of the (Y ) π˜ of order at most (l + 1)∗ are bounded by (13.617) with [l + 2] replaced by [l + 1] (and ε0 by u): u t C F1,[l+1] (u  )du  0



u

+C 

0 u

+C 0



t F1,[l+1] (u  )du  t F1,[l+1] (u  )du 

1/2



K [l+1] (t, u)

1/2

1/2 (13.808)

  1/2 u u (1 + t  )−2 [1 + log(1 + t  )]2 E1,[l+1] (t  ) + u 2 E0,[l+1] (t  ) dt  0  4   ≤ Cµ−2b (t)[1 + log(1 + t)] + (V1,[l+1];b,2 (t, u))1/2 (I[l+1];b,2 (t, u))1/2 V1,[l+1];b,2 m,u t

×

 + (V1,[l+1];b,2 (t, u))1/2  t   1/2 u u × (1 + t  )−2 [1 + log(1 + t  )]2 G1,[l+1];b,2 (t  ) + G0,[l+1];b,0 (t  ) dt  0

in the case of the error integrals associated to K 1 , and by (13.710) with [l + 2] replaced by [l + 1] (and ε0 by u): t u C (1 + t  )−3/2 E0,[l+1] (t  )dt  (13.809) 0 u t F0,[l+1] (u  )du  +C 0

 +C 0

u

t F 1,[l+1] (u  )du 

1/2  0

t

u (1 + t  )−3/2 E0,[l+1] (t  )dt 



u

+ 0

t F0,[l+1] (u  )du 

1/2

Part 3: The energy estimates





1/2

865 t

u (1 + t  )−3/2 E0,[l+1] (t  )dt 



u

t F0,[l+1] (u  )du 

1/2

+ C K [l+1] (t, u) + 0 0  t u ≤ Cµ−2b (1 + t  )−3/2 G0,[l+1];b,0 (t  )dt  + V0,[l+1];b,0 (t, u) m,u (t) 0

 + (V1,[l+1];b,2 (t, u))1/2



t 0



t

+ (I[l+1];b,2 (t, u))1/2 0

(1 +

u t  )−3/2 G0,[l+1];b,0 (t  )dt 

1/2 + V0,[l+1];a,0(t, u)

u (1 + t  )−3/2 G0,[l+1];b,0 (t  )dt  + V0,[l+1];b,0(t, u)

1/2

in the case of the error integrals associated to K 0 . Finally, for each variation ψ of order up to l + 1, of the form specified in the paragraph following (13.106), we have the error integrals (13.620). Consider first the second of these integrals, which is associated to K 1 . Summing (13.621) over all such variations we obtain: 8   1 3 Q 1,k [ψ]dµh ≤ − K [l+1] (t, u) + C M[l+1] (t, u) + L [l+1] (t, u) 2 2 Wut k=1 ψ t   ˜  )E u + A(t (13.810) 1,[l+1] (t )dt 0

Here, M[l+1] (t, u) =



M[ψ](t, u),

L [l+1] (t, u) =

ψ



L[ψ](t, u)

(13.811)

ψ

From (13.622) and (13.633) with (a, p) replaced by (b, 0), noting the fact that there is a fixed number of variations under consideration, we obtain: u u t M[l+1] (t, u) ≤ C[1 + log(1 + t)]4 sup E0,[l+1] (t  ) + F1,[l+1] (u  )du  t  ∈[0,t ]

≤

4 µ−2b m,u (t)[1 + log(1 + t)]

0



u  (t) + V1,[l+1];b,2 (t, u) CG0;b,0

(13.812)

Also, from (13.605):

t

L [l+1] (t, u) = 0

u (1 + t  )−1 [1 + log(1 + t  )]−1 E1,[l+1] (t  )dt 

(13.813)

To estimate this borderline integral appropriately, we must make use of the following variant of Corollary 2 of Lemma 8.11: Variant of Corollary 2: Let b be a positive constant and k any constant greater than 1. Then if δ0 is suitably small, depending on an upper bound for b and a lower bound greater than 1 for k, we have, for all t ∈ [0, t], t ∈ [1, s], and u ∈ (0, ε0 ]:  −b µ−b m,u (t ) ≤ kµm,u (t)

866

Chapter 13. Error Estimates

To show this, we revisit the proof of Corollary 2. In the following, we shall take the constant a in the proof of that corollary to be a suitably large constant, having nothing to do with the choice of a which we made above in relation to the top order energy estimates. In Case 1 of this proof (8.345) holds (with ε0 replaced by u), hence:  µ−b m,u (t )

µ−b m,u (t)

≤ (1 − Cδ0 )−b ≤ k

(13.814)

provided that δ0 is suitably small, depending on an upper bound for b and a lower bound greater than 1 for k. In Subcase 2a the lower bound (8.296) holds (with ε0 replaced by u) for any given a, provided that aδ0 is suitably small, hence:   2 −b ≤ 1− ≤k a µ−b m,u (t)

 µ−b m,u (t )

(13.815)

if a is chosen suitably large, depending on an upper bound for b and a lower bound greater than 1 for k. In Subcase 2b the lower bound (8.326) as well as the upper bound 8.302 hold (with ε0 replaced by u) for any given a, provided that aδ0 is suitably small, hence:  µ−b m,u (t )

µ−b m,u (t)

 ≤

1− 1+

2 a 2 a

−b 

1 − δ1 τ 1 − δ1 τ 



b ≤

1− 1+

2 a 2 a

−b ≤ k

(13.816)

if again a is chosen suitably large, depending on an upper bound for b and a lower bound greater than 1 for k. We have thus established the required variant of Corollary 2. We now set:

9 k=

4 3

(13.817)

in the above variant of Corollary 2, and we proceed to estimate L [l+1] (t, u). We have, from (13.813) and the definition (13.220) with (a, q) replaced by (b, 2), t u   L [l+1] (t, u) ≤ (1 + t  )−1 [1 + log(1 + t  )]3 µ−2b m,u G1,[l+1];b,2 (t )dt 0 t 4 u ≤ µ−2b (t)G (t) (1 + t  )−1 [1 + log(1 + t  )]3 dt  1,[l+1];b,2 3 m,u 0 1 u ≤ µ−2b (t)[1 + log(1 + t)]4 G1,[l+1];b,2 (t) (13.818) 3 m,u Consider now the integral identity (see (5.66)) corresponding to the vectorfield K 1 and to the variations ψ of order up to l +1 of the form specified in the paragraph following (13.106). In each of these identities we also have the hypersurface integrals bounded according to (13.639) with (a, p) replaced by (b, 0) (and ε0 by u) by: 4 u Cµ−2b m,u (t)[1 + log(1 + t)] G0,[l+1];b,0 (t)

(13.819)

Part 3: The energy estimates

867

Summing over all such variations we then obtain, from (13.818), and from (13.789), (13.797), (13.806), (13.808), (13.810), (13.812), the following:  1 u t −4 (t) + F1,[l+1] (u) + K [l+1] (t, u) µ2b E1,[l+1] m,u (t)[1 + log(1 + t)] 2 1 u u ≤ G1,[l+1];b,2 (t) + Cδ0 δG1,[l+1];b,2 (t) 2 t  u + C R[l+1];b,2 (t, u) + C A(t  )G1,[l+1];b,2 (t  )dt  (13.820) 0

Here, δ is any positive constant, and:  R[l+1];b,2

=

u D[l+2]

u + G0,[l+1];b,0 (t)

  1  + 1+ (t, u) + δ I[l+1];b,2 (t, u) V1,[l+1];b,2 δ (13.821)

(we shall choose δ below). u Keeping only the term E1,[l+1] (t) in parenthesis on the left in (13.820) we have: 1 u −4 u u µ2b m,u (t)[1 + log(1 + t)] E1,[l+1] (t) ≤ G1,[l+1];b,2 (t) + Cδ0 δG1,[l+1];b,2 (t) 2 t  u + C R[l+1];b,2 (t, u) + C A(t  )G1,[l+1];b,2 (t  )dt  (13.822) 0

The same holds with t replaced by any t  ∈ [0, t]. The right-hand side of (13.822) being a non-decreasing function of t at each u, the inequality corresponding to t  holds a fortiori if we again replace t  by t on the right. Taking then the supremum over all t  ∈ [0, t] on the left we obtain, in view of the definition (13.655), t 1 u  u G (t) ≤ C R[l+1];b,2 (t, u) + C A(t  )G1,[l+1];b,2 (t  )dt  (13.823) 3 1,[l+1];b,2 0 provided that: 1 (13.824) 6 From this point we proceed as in the argument leading from (13.678) to (13.680), (13.681), to obtain: Cδ0 δ ≤



t 0

u  G1,[l+1];b,2 (t) ≤ C R[l+1];b,2 (t, u)

(13.825)

u  A(t  )G1,[l+1];b,2 (t  )dt  ≤ C R[l+1];b,2 (t, u)

(13.826)

Substituting these in the right-hand side of (13.820) and omitting the first term in parenthesis on the left, we obtain:  1 t −4  F1,[l+1] (u) + K [l+1] (t, u) µ2b ≤ C R[l+1];b,2 (t, u) (13.827) m,u (t)[1 + log(1 + t)] 2

868

Chapter 13. Error Estimates

for a new constant C. From this point we proceed exactly as in the argument leading from (13.685) to (13.698)–(13.701), choosing δ according to (see (13.690)): Cδ =

1 4

(13.828)

We obtain in this way the estimates: 

u t G1,[l+1];b,2 (t), H1,[l+1];b,2 (u), I[l+1];b,2 (t, u) ≤ C Q [l+1];b,2 (t, u) 

 V1,[l+1];b,2 (t, u) ≤ Cε0 Q [l+1];b,2 (t, u)

where:



u

u Q [l+1];b,2 (t, u) = D[l+2] + G 0,[l+1];b,0 (t)

(13.829) (13.830) (13.831)

Consider finally the integral identity (see (5.67)) corresponding to the vectorfield K 0 and to the variations ψ of order up to l + 1 of the form specified in the paragraph following (13.106). In each of these identities we have the first of the error integrals (13.620). Summing (13.735) over all such variations we obtain (there is a fixed number of variations under consideration):  ψ

7  Wut k=1



t

Q 0 [ψ]dµh ≤ C 0

u B˜ s (t  )G1,[l+1];b,2 (t  )dt 

  t  1  −1  −2 u   +C 1+  (1 + t ) [1 + log(1 + t )] G0,[l+1];b,0 (t )dt + V0,[l+1];b,0 (t, u) δ 0 t    + Cδ V1,[l+1];b,2(t, u) + C (1 + t  )−2 [1 + log(1 + t  )]2 V1,[l+1];b,2 (t, u)dt  + Cδ  I[l+1];b,2 (t, u)

0

(13.832)

Here, as in (13.807), δ  is an arbitrary positive constant which shall be chosen appropriately below. Summing over all such variations we then obtain, from (13.802), (13.805), (13.807), (13.809), (13.832), the following:

 u t −4 E0,[l+1] (t) + F0,[l+1] (t) µ2b m,u (t)[1 + log(1 + t)]     δ0  u u u ≤ Cδ0 δ G0,[l+1];b,0 (t) + G1,[l+1];b,2 + C 1 +  D[l+2] δ t u +C (t  )dt  B˜ s (t  )G1,[l+1];b,2 0   t 1 u +C 1+  (1 + t  )−1 [1 + log(1 + t  )]−2 G0,[l+1];b,0 (t  )dt  δ 0 t    + Cδ V1,[l+1];b,2 (t, u) + C (1 + t  )−2 [1 + log(1 + t  )]2 V1,[l+1];b,2 (t, u)dt  0   1 + C 1 +  V0,[l+1];b,0 (t, u) + Cδ  I[l+1];b,2 (t, u) (13.833) δ

Part 3: The energy estimates

869

We now substitute on the right the estimates (13.829), (13.830) to obtain:

 u t −4 E0,[l+1] (t) + F0,[l+1] (t) µ2b m,u (t)[1 + log(1 + t)]     t δ0 1 u  u u B˜ s (t  )G 0,[l+1];b,0(t  )dt  ≤ Cδ G 0,[l+1];b,0 (t) + C 1 +  D[l+2] + C 1 +  δ δ 0   1 + C 1 +  V0,[l+1];b,0(t, u) (13.834) δ u (t) in parenthesis on the left in (13.834), and noting Keeping only the term E0,[l+1] that the right-hand side is a non-decreasing function of t at each u, we may replace t by any t  ∈ [0, t] on the left while keeping t on the right, thus, taking the supremum over all t  ∈ [0, t] we obtain, in view of the definition (13.654),   δ0 u  u u G0,[l+1];b,0 (t) ≤ Cδ G 0,[l+1];b,0 (t) + C 1 +  D[l+2] δ   t 1 u B˜ s (t  )G 0,[l+1];b,0 (t  )dt  +C 1+  δ  0  1 + C 1 +  V0,[l+1];b,0 (t, u) (13.835) δ

We now choose δ  according to: Cδ  =

1 2

(13.836)

Then (13.835) implies: 1 u u G (t) ≤ CD[l+2] +C 2 0,[l+1];b,0



t 0

u B˜ s (t  )G 0,[l+1];b,0 (t  )dt  + C V0,[l+1];b,0 (t, u)

(13.837)

(for new constants C). The same holds with u replaced by any u  ∈ [0, u]. Now the right-hand side of (13.837) is a non-decreasing function of u at each t. The inequality corresponding to u  thus holds a fortiori if we again replace u  by u on the right-hand side. Taking then the supremum over all u  ∈ [0, u] on the left-hand side we obtain, in view of the definition (13.696), t 1 u u u B˜ s (t  )G 0,[l+1];b,0 (t  )dt  + C V0,[l+1];b,0 (t, u) G (t) ≤ CD[l+2] + C 2 0,[l+1];b,0 0 (13.838) From this point we proceed exactly as in the argument leading from (13.760) to the estimates (13.769)–(13.771). We obtain in this way the estimates: u

t u G 0,[l+1];b,0 (t), H0,[l+1];b,0 (u) ≤ CD[l+2]

(13.839)

Substituting finally the first term of (13.839) in (13.831) and the result in (13.829) yields: u t u G1,[l+1];b,2 (t), H1,[l+1];b,2 (u), I[l+1];b,2 (t, u) ≤ CD[l+2]

(13.840)

870

Chapter 13. Error Estimates

We have thus obtained improved energy estimates of the next to the top order, namely of order l + 1. We proceed in this way, taking at the nth step an = a − n and bn = an − 1 = a − (n + 1) = an+1

(13.841)

in the role of a and b respectively, the argument beginning in the paragraph containing (13.777) and concluding with the estimates (13.839), (13.840), being step 0. For n ≥ 1 we have 2 and 0 in the role of q and p respectively. The nth step is otherwise exactly the same as the 0th step above, as long as bn > 0, that is as long as n ≤ [a] − 1. In estimating the integrals Jaun ,1 in (13.786), (13.795), as well as the integrals

t 0

n +1  (1 + t  )−2 [1 + log(1 + t  )]6 µ−2a (t )dt  m,u

(13.842)

for n ≥ 1, we follow the argument leading from (13.260) to the estimate (13.266) in the case of Jaun ,1 , the argument leading from (13.224) to (13.240) in the case of (13.842). However, when referring to Lemma 8.11 in these arguments, we now mean Lemma 8.11 with a set equal to 4 in that lemma. Then in Case 1, (13.225) holds which implies, in analogy with (13.226): n −1 µ−2a (t) ≤ C : for all t ∈ [0, s] m,u

(13.843)

as the an have a fixed upper bound, namely a. Also, the subcases of Case 2 are defined according to: 1

t1 = e 8δ1 − 1

(13.844)

and in Subcase 2a we have the lower bound: µm,u (t  ) ≥

1 2

(13.845)

in place of (13.230), hence again: n −1  µ−2a (t ) ≤ C, m,u

(13.846)

while in Subcase 2b we have the lower bound: µm,u ≥

1 (1 − δ1 τ  ), τ  = log(1 + t  ) 2

(13.847)

in place of (13.233), as well as the upper bound (8.335): µm (t) ≤

3 (1 − δ1 τ ) 2

(13.848)

Part 3: The energy estimates

871

from the proof of Lemma 8.11 with a set equal to 4, which implies, in analogy with (13.238): 1 n (1 − δ1 τ )−2an τ = log(1 + t) µ−2a (13.849) m,u ≥ C noting again that the an have a fixed upper bound. We thus obtain in Subcase 2b, in place of (13.264), Jaun ,1 (t) − Jaun ,1 (t1 ) ≤ C(1 + τ )2 (1 − δ1 τ  )−an −1/2 dτ  τ1τ

≤

C (1 +   (1 − δ1 τ )−an +1/2 δ1 a n − 1 2

≤C

τ )2

[1 + log(1 + t)]3 −an +1/2   µm,u an − 12

(13.850)

Since for any n ≤ [a]: an −

3 1 1 1 1 ≥ a[a] − = − = 2 2 4 2 4

(13.851)

the denominator in (13.850) is bounded from below by a positive constant even for n = [a], hence combining with the results for Subcase 2a and for Case 1 we conclude that: −a +1/2

Jaun ,1 (t) ≤ C[1 + log(1 + t)]3 µm,un

: for all n = 1, . . . , [a]

(13.852)

Also, for n ≤ [a] − 1, so that an ≥ 7/4, we have in Subcase 2b, in regard to the integral (13.842), in place of (13.235),

t t1

(1 + t  )−2 [1 + log(1 + t  )]6 (1 − δ1 τ  )−2an +1 dt  τ ≤ (1 + t1 )−1 [1 + log(1 + t1 )]6 (1 − δ1 τ  )−2an +1 dτ  τ1

(1 − δ1 τ )−2an +2 2(an − 1) (1 − δ1 τ )−2(an −1) ≤ C(1 + t1 )−1 [1 + log(1 + t1 )]7 2(an − 1)

≤ (1 + t1 )−1

[1 + log(1 + t1 δ1

)]6

n −1) ≤ Cµ−2(a m,u

(13.853)

hence combining with the results for Subcase 2a and for Case 1 (where (13.842) is simply bounded by a constant) we conclude that the integral (13.842) is bounded by: n −1) : for all n = 1, . . . , [a] − 1 Cµ−2(a m,u

(13.854)

872

Chapter 13. Error Estimates

Moreover, in (13.802) and (13.805) we may replace the integrals on the left by the integral (13.842) and we obtain bounds at the nth step by:   u u n (t) D (13.855) Cδ0 µ−2b m,u [l+2] G0,[l+1−n];bn ,0 (t) Following then the argument of step 0, we obtain at the nth step, for n = 1, . . . , [a] − 1, the estimates: u

t u G 0,[l+1−n];bn ,0 (t), H0,[l+1−n];b (u) ≤ CD[l+2] n ,0 u t u G1,[l+1−n];b (t), H1,[l+1−n];b (u), I[l+1−n];bn ,2 (t, u) ≤ CD[l+2] n ,2 n ,2

: for all (t, u) ∈ [0, s] × [0, ε0 ]

(13.856)

Here the constants C may depend on n, but n is in any case bounded. We now make the last step n = [a]. In this case, we have an = a[a] = 3/4 and the estimate (13.852) reads: −1/4

u J3/4,1 (t) ≤ C[1 + log(1 + t)]3 µm,u

(13.857)

while in regard to the integral (13.842), we now have, from (13.853), in Subcase 2b, since −2a[a] + 1 = −1/2: t (1 + t  )−2 [1 + log(1 + t  )]6 (1 − δ1 τ  )−1/2 dt  t1 τ ≤ (1 + t1 )−1 [1 + log(1 + t1 )]6 (1 − δ1 τ  )−1/2 dτ  τ1

[1 + log(1 + t1 )]6 (1 − δ1 τ1 )1/2 ≤ (1 + t1 )−1 δ1 1/2 1/2 (7/8) ≤ 8(1 + t1 )−1 [1 + log(1 + t1 )]7 ≤C 1/2

(13.858)

hence combining with the results for Subcase 2a and for Case 1 (where (13.842) is in any case bounded by a constant) we conclude that the integral (13.842) is in the case n = [a] simply bounded by a constant. We can thus set: bn = b[a] = 0

(13.859)

in the last step, and proceed otherwise exactly as in the preceding steps. We thus arrive at the estimates: u

t u G 0,[l+1−[a]];0,0 (t), H0,[l+1−[a]];0,0 (u) ≤ CD[l+2] u t u G1,[l+1−[a]];0,2 (t), H1,[l+1−[a]];0,2 (u), I[l+1−[a]];0,2 (t, u) ≤ CD[l+2]

: for all (t, u) ∈ [0, s] × [0, ε0 ]

(13.860)

Part 3: The energy estimates

873

These are the desired estimates. For, according to the definitions (13.630), (13.640), (13.654), (13.655), (13.696), (13.716), we have:

 u u G 0,[l+1−[a]];0,0(t) = sup E0,[l+1−[a]] (t  ) (13.861) t  ∈[0,t ]

t t H0,[l+1−[a]];0,0 (u) = F0,[l+1−[a]] (u) 

u u G1,[l+1−[a]];0,2 = sup [1 + log(1 + t  )]−4 E1,[l+1−[a]] (t  )

(13.862) (13.863)

t  ∈[0,t ]

t (u) = sup H1,[l+1−[a]];0,2

t  ∈[0,t ]

I[l+1−[a]];0,2 (t, u) = sup

t  ∈[0,t ]



t (u) [1 + log(1 + t  )]−4 F1,[l+1−[a]]



[1 + log(1 + t  )]−4 K [l+1−[a]] (t  , u)

and the weights µm,u have been eliminated.

(13.864)  (13.865)

874

Chapter 13. Error Estimates

Part 4: Recovery of assumption J. Recovery of the bootstrap assumption. Proof of the main theorem We now establish assumption J on the basis of the bootstrap assumption. Recall that the estimate (13.198), on which the borderline estimates relied, was itself established using this assumption. To establish assumption J we consider the first order variations:  Sφ : for α = 0 (α) ˜ (13.866) ψ1 = ◦ Ri φ : for α = i = 1, 2, 3 To these Theorem 5.1 applies and we obtain, in particular: E0u [

(α)

ψ˜ 1 ](t) ≤ CE u [

(α)

ψ˜ 1 ](0) : for all (t, u) ∈ [0, s] × [0, ε0]

(13.867)

We also consider the higher order variations, corresponding to the first order variations (α) ψ ˜ 1: (α;I1 ...In−1 ) ˜ (13.868) ψn = Y In−1 . . . Y I1 (α) ψ˜ 1 (compare with (13.103)) and we require the multi-indices (I1 . . . In−1 ) to be of the form specified in the paragraph following (13.106) and concluding with (13.107). We consider u (t) the sum of the energies correall such variations of order up to 4. We denote by E˜0,[4] sponding to the vectorfield K 0 , the hypersurface tu , and to all such variations of order up to 4. Now the integral identities corresponding to the variations in question and to the vectorfields K 0 and K 1 contain derivatives of the (Y ) π˜ of order at most 3, which, in view ε of the condition (13.776), are bounded in L ∞ (t 0 ) by Propositions 12.9 and 12.10 and the bootstrap assumption. The error integrals involved are then all bounded exactly as in Chapter 7 and in accordance with the remark following Lemma 7.6 can all be absorbed in the error integrals of the fundamental energy estimate. Consequently, the conclusions of Theorem 5.1 hold for the variations in question and we obtain, in particular: u u (t) ≤ C D˜ [4] : for all (t, u) ∈ [0, s] × [0, ε0 ] E˜0,[4]

(13.869)

u u = E˜0,[4] (0) D˜ [4]

(13.870)

where we denote:

Applying inequality (5.72) of Chapter 5 we then conclude that: u S˜[4] (t, u) ≤ Cε0 D˜ [4] : for all (t, u) ∈ [0, s] × [0, ε0 ]

(13.871)

where we denote by S˜[4] the integral on St,u (with respect to dµh/ ) of the sum of the squares of all the variations under consideration. In particular, we have:    2 2 2 |R j1 Sφ| + |R j2 R j1 Sφ| dµh/ ≤ S˜[4] (t, u) |Sφ| + St,u

j1

j1 , j2

(13.872)

Part 4: Recovery of assumption J





◦

| Ri φ|2 + St,u

St,u



◦

|R j1 Ri φ|2 +



◦ |R j2 R j1 Ri φ|2 dµh/ ≤ S˜[4] (t, u)

j1 , j2

j1







875

: i = 1, 2, 3 (13.873)    |T Sφ|2 + |R j1 T Sφ|2 + |R j2 R j1 T Sφ|2 dµh/ ≤ S˜[4] (t, u) j1 , j2

j1

◦

|T Ri φ|2 +



St,u

◦

|R j1 T Ri φ|2 +



(13.874) ◦ |R j2 R j1 T Ri φ|2 dµh/ ≤ S˜[4] (t, u)

j1 , j2

j1

: i = 1, 2, 3

(13.875)

We are now in a position to apply the following Sobolev-type lemma. Lemma 13.1 Let f be a function on St,u with square-integrable derivatives up to 2nd order. We denote:       S[2] [ f ] = | f |2 + |R j1 f |2 + |R j2 R j1 f |2 dµh/  St,u  j1 , j2

j1

Then f ∈ L ∞ (St,u ) and there is a positive numerical constant C such that: sup | f | ≤ C(1 + η0 t)−1 (S[2] [ f ])1/2 St,u

Proof. Noting that by the comparison inequality (8.355) we have: C −1 (1 + η0 t)2 ≤ A(t, u) ≤ C(1 + η0 t)2

(13.876)

where A(t, u) is the area of St,u , hypothesis H0 in the form established by Corollary 12.2.a of Proposition 12.2 implies:       2 2 2 2 |f| + |d/ f | + |Ri f | dµh/ + A(t, u) |d/ Ri f | dµh/ ≤ CS[2] [ f ] St,u

St,u

i

i

(13.877) We now apply the isoperimetric Sobolev inequality of St,u (see [O]). If g is an arbitrary function which is integrable and with integrable derivative on St,u , then g is square-integrable on St,u and, denoting by g the mean value of g on St,u 1 g= gdµh/ , (13.878) A(t, u) St,u we have:

2



(g − g) dµh/ ≤ I (t, u) 2

St,u

St,u

|d/g|dµh/

(13.879)

876

Chapter 13. Error Estimates

where I (t, u) is the isoperimetric constant of St,u : I = sup U

min{Area(U ), Area(U c )} (Perimeter(∂U ))2

(13.880)

where the supremum is over all domains U with C 1 boundary ∂U in St,u , and U c = St,u \ U denotes the complement of U in St,u . Since: g L 2 (St,u ) = |g|A1/2 , |g| ≤ A−1  f  L 2 (St,u ) (13.881) it follows that: g L 2 (St,u ) ≤ where:

√

I  gW 1 (St,u )

(13.882)

1

gW 1 (St,u ) = d/g L 1 (St,u ) + A−1/2 g L 1 (St,u )

(13.883)

I  = max{I, 1}

(13.884)

1

and: We have:

 f W 1 (St,u ) = 2

1

|d/( f )|dµh/ + A 2

St,u

−1/2

St,u

| f |2 dµh/

(13.885)

and, since d/( f 2 ) = 2 f d/ f , 1/2 



|d/( f )|dµh/ ≤ 2 2

St,u

| f | dµh/

1/2

2

St,u

|d/ f | dµh/ 2

St,u

≤ C A−1/2 S[2] [ f ]

(13.886)

by (13.877). Since the second term on the right in (13.885) is similarly bounded by (13.877), we obtain: (13.887)  f 2 W 1 (St,u ) ≤ C A−1/2 S[2] [ f ] 1

We have: 0 0 0 0 0 0 |Ri f |2 0 0 0 0 i

and, since d/

 i

  2      2  −1/2 = |Ri f |  dµh/ + A |Ri f |2 dµh/ d/  St,u  St,u

W11 (St,u )

i

i

(13.888)

  |Ri f |2 = 2 i (Ri f )d/(Ri f ),

  2 1/2  1/2        2  2 2 |Ri f |  dµh/ ≤ 2 |Ri f | dµh/ |d/ Ri f | dµh/ d/  St,u  St,u St,u i

i

≤ CA

−1/2

S[2] [ f ]

i

(13.889)

Part 4: Recovery of assumption J

877

by (13.877). Since the second term on the right in (13.888) is similarly bounded by (13.877), we obtain: 0 0 0 0 0 20 |Ri f | 0 ≤ C A−1/2 S[2] [ f ] (13.890) 0 0 0 1 W1 (St,u )

i

 We now apply (13.882) first taking g = f 2 and then taking g = i |Ri f |2 to obtain, in view of (13.887), (13.890),  1/2  2        √ |Ri f |2 ≤ C I  A−1/2 S[2] [ f ] (13.891) | f |4 + dµh/    St,u  i

Hypothesis H0 in the form established by Corollary 12.2.a of Proposition 12.2 then implies: √  f 2W 4 (S ) ≤ C I  A−3/2S[2] [ f ] (13.892) 1

t,u

where, for an arbitrary function f on St,u we denote:  f W 4 (St,u ) = d/ f  L 4 (St,u ) + A−1/2  f  L 4 (St,u ) 1

(13.893)

From this point we adapt to the present situation an argument found in Gilbarg– Trudinger [G–T]. We rescale the metric h/ on St,u , setting: /hˆ = A−1 h / so that dµh/ˆ = A−1 dµh/

(13.894)

to a metric h /ˆ of unit area. Taking account of the fact that relative to the new metric we have: −1

/ˆ ) BC ∂ B f ∂C f = A(h/−1 ) BC ∂ B f ∂C f = A|d/ f |2h/ |d/ f |2ˆ = (h h/

we have:

 f W 4 (St,u ,h/) = A−1/4  f W 4 (St,u ,h/ˆ ) 1

1

hence (13.892) reads:  f 2

W14 (St,u ,h/ˆ )

≤C

√

I  A−1 S[2] [ f ]

Moreover, (13.882), (13.883) read relative to h/ˆ : √ g L 2 (St,u ,h/ˆ ) ≤ I  gW 1 (St,u ,h/ˆ ) 1

(13.895)

(13.896) (13.897)

(13.898)

where: gW 1 (St,u ,h/ˆ ) = d/g L 1 (St,u ,h/ˆ ) + g L 1 (St,u ,h/ˆ )

(13.899)

1 |f| f˜ = √  I  f W14 (St,u ,h/ˆ )

(13.900)

1

We now set:

878

Chapter 13. Error Estimates

Then f˜ ≥ 0 and taking g = f˜k , k > 1, in (13.898) we obtain: √  f˜k  L 2 (St,u ,h/ˆ ) ≤ I   f˜k W 1 (St,u ,h/ˆ )

(13.901)

1

Since d/( f˜k ) = k f˜k−1 d/ f˜ we have, by H¨older’s inequality: d/( f˜k ) L 1 (St,u ,h/ˆ ) ≤ k f˜k−1  L 4/3 (St,u ,h/ˆ ) d/ f˜ L 4 (St,u ,h/ˆ )

(13.902)

and:  f˜k  L 1 (St,u ,h/ˆ ) =  f˜k−1 f˜ L 1 (St,u ,h/ˆ ) ≤  f˜k−1  L 4/3 (St,u ,h/ˆ )  f˜ L 4 (St,u ,h/ˆ ) hence, adding,

 f˜k W 1 (St,u ,h/ˆ ) ≤ k f˜k−1  L 4/3 (St,u ,h/ˆ )  f˜W 4 (St,u ,h/ˆ ) 1

(13.903) (13.904)

1

Now, by virtue of the definition (13.900) we have: 1  f˜W 4 (St,u ,h/ˆ ) = √ 1 I

(13.905)

Substituting in (13.904) and the result in (13.901) yields:  f˜k  L 2 (St,u ,h/ˆ ) ≤ k f˜k−1  L 4/3 (St,u ,h/ˆ )

(13.906)

which is equivalent to:  f˜ L 2k (St,u ,h/ˆ ) ≤ k 1/ k  f˜

1−(1/ k) L (4/3)(k−1) (St,u ,h/ˆ )

This implies:

(13.907)

1−(1/ k)  f˜ L 2k (St,u ,h/ˆ ) ≤ k 1/ k  f˜ (4/3)k L

(13.908)

(St,u ,h/ˆ )

for, by virtue of the fact that St,u has unit area with respect to h/ˆ , the norm g L p (St,u ,h/ˆ ) for a given function g is a non-decreasing function of the exponent p. The ratio of the exponent on the left in (13.908) to the exponent on the right is 2/(4/3) = 3/2. We now set:  n 3 k= : n = 1, 2, 3, . . . (13.909) 2 For n = 1 the exponent on the right in (13.908) is 2, and taking g = f˜ in (13.898) we obtain, by (13.900):  f˜ L 2 (St,u ,h/ˆ ) ≤

√

I   f˜W 1 (St,u ,h/ˆ ) = 1

 f W 1 (St,u ,h/ˆ ) 1

 f W 4 (St,u ,h/ˆ )

≤1

(13.910)

1

It follows from (13.908)–(13.910), by induction on n, that for every n = 1, 2, 3, . . . :  f˜ L 2(3/2)n (St,u ,h/ˆ ) ≤

 nm=1 m(3/2)−m 3 2

(13.911)

Part 4: Recovery of assumption J

879

Taking the limit n → ∞ we then obtain: sup f˜ ≤ c

(13.912)

St,u

where c is the constant: −m  ∞  6 m=1 m(3/2) 3 3 c= = 2 2

In view of the definition (13.900), (13.912) is equivalent to: √ sup | f | ≤ c I   f W 4 (St,u ,h/ˆ ) 1

St,u

(13.913)

Substituting finally from (13.896) and (13.892) yields: sup | f | ≤ cC I 3/4 A−1/2(S[2] [ f ])1/2

(13.914)

St,u

To complete the proof of the lemma, what remains to be done is to obtain an upper bound for I (St,u ), the isoperimetric constant of St,u . Now the integral curves of T on a ε given t 0 define a diffeomorphism of St,0 onto each St,u , u ∈ [0, ε0 ]. A domain Uu ⊂ St,u with C 1 boundary ∂Uu is mapped by the inverse onto a domain U0 ⊂ St,0 with C 1 boundary ∂U0 . Consider the image Uu  of the domain U0 on each St,u  , u  ∈ [0, u], under the diffeomorphism. Then from the definition L /T h/ = 2κθ ε

of the second fundamental form θ of St,u relative to t 0 , we have: d  Area(U ) = κtrθ dµh/ u du  St,u  and:

d Perimeter(∂Uu  ) = du 

(13.915)

∂Uu 

κθ (V, V )ds

(13.916)

where V is the unit tangent vectorfield and ds the element of arc length of ∂Uu  . Now we have shown that the bootstrap assumption implies: κ|θ | ≤ C(1 + t)−1 [1 + log(1 + t)] ≤ C hence (since also |θ (V, V )| ≤ |θ |) from (13.915), (13.916) we obtain:    d     Area(U ) u  ≤ CArea(Uu  )  du     d     du  Perimeter(∂Uu  ) ≤ CPerimeter(∂Uu  )

(13.917)

(13.918) (13.919)

880

Chapter 13. Error Estimates

Therefore integrating with respect to u  on [0, u] yields: C −1 Area(U0 ) ≤ Area(Uu ) ≤ CArea(U0 ) C −1 Perimeter(∂U0 ) ≤ Perimeter(∂Uu ) ≤ CPerimeter(∂U0 ) : for all u ∈ [0, ε0]

(13.920)

It follows that: C −1 I (St,0 ) ≤ I (St,u ) ≤ C I (St,0 ) : for all u ∈ [0, ε0]

(13.921)

Now:

1 2π as St,0 is simply a round sphere in Euclidean space. We have thus obtained an upper bound for I (St,u ) by a numerical constant. In view also of (13.876), the lemma then follows from (13.914). I (St,0 ) =

Applying Lemma 13.1 noting from (13.872)–(13.876) that: ◦

S[2] [Sφ] ≤ S˜[4] (t, u)

S[2] [ Ri φ] ≤ S˜[4] (t, u) : i = 1, 2, 3

S[2] [T Sφ] ≤ S˜[4] (t, u)

S[2] [T Ri φ] ≤ S˜[4] (t, u) : i = 1, 2, 3

◦

(13.922)

then by Lemma 13.1 and (13.871) we have: ◦

u sup |Sφ| ≤ C(1 + t)−1 D˜ [4]

u sup | Ri φ| ≤ C(1 + t)−1 D˜ [4] : i = 1, 2, 3

u sup |T Sφ| ≤ C(1 + t)−1 D˜ [4]

u sup |T Ri φ| ≤ C(1 + t)−1 D˜ [4] : i = 1, 2, 3

ε

t 0

ε

t 0

ε

t 0

◦

ε

t 0

(13.923)

it follows that there is a constant C such that if: u ≤ C −1 δ0 D˜ [4]

(13.924)

then assumption J holds on Wεs0 . We have thus established assumption J. We are now ready to recover the bootstrap assumption and proceed to the proof of the main theorem of this monograph. Let us denote by S[n] (t, u) the integral on St,u (with respect to dµh/ ) of the sum of the squares of all the variations (13.104), (13.106), of order up to n, of the form specified in the paragraph following (13.106). By inequality (5.72) of Chapter 5 we have: u (t) : for all (t, u) ∈ [0, s] × [0, ε0] S[l+1−[a]] (t, u) ≤ Cε0 E0,[l+1−[a]]

(13.925)

Hence, in view of (13.861), by the first of the estimates (13.860): u : for all (t, u) ∈ [0, s] × [0, ε0] S[l+1−[a]] (t, u) ≤ Cε0 D[l+2]

(13.926)

Part 4: Recovery of assumption J

881

Now, for any variation ψ of the form specified in the paragraph following (13.106) of order up to l − 1 − [a] we have: S[2] [ψ] ≤ S[l+1−[a]]

(13.927)

For, ψ, R j1 ψ : j1 = 1, 2, 3, R j2 R j1 ψ : j1, j2 = 1, 2, 3, are themselves variations included in S[l+1−[a]] . Applying Lemma 13.1 to any such variation ψ of order up to l − 1 − [a] we obtain, in view of (13.926), (13.927): √  u : for all (t, u) ∈ [0, s] × [0, ε0 ] (13.928) sup |ψ| ≤ C(1 + t)−1 ε0 D[l+2] St,u

That is, we obtain (see (13.103), (13.104)): max max Rin . . . Ri1 (T )m (Q) p α

(α)

i1 ...in

√

ψ1  L ∞ ( ε0 ) ≤ C(1 + t)−1 ε0 D[l+2] t

: for all p + m + n ≤ l − 2 − [a] and all t ∈ [0, s]

(13.929)

ε0 . D[l+2] denotes D[l+2]

Comparing with the statement of assumption √  u in the E{{k}} we conclude that assumption E{{l−2−[a]}} holds on Wεs0 with C ε0 D[l+2] role of δ0 . Thus, if l − 2 − [a] ≥ (l + 1)∗ + 2

where we recall that

to the condition (13.776), we recover the which, since l = l∗ + (l + 1)∗ , is equivalent

√ bootstrap assumption E{{(l+1)∗ +2}} with C ε0 D[l+2] in the role of δ0 . Let now s∗ be the least upper bound of the set of values of s in the interval [0, t∗ε0 ] such that the bootstrap assumption holds on Wεs0 . We recall from Chapter 3 that t∗ε0 is defined by: (13.930) t∗ε0 = inf t∗ (u) u∈[0,ε0

where t∗ (u) is the greatest lower bound of the extent of the generators of Cu , in the parameter t, in the domain of the maximal solution. We note here that the Cu do not contain cut loci. This follows from the fact that the angle between the outward unit normal to the surface St,u with respect to the Euclidean metric on t and the outward unit normal N to the Euclidean coordinate spheres does not exceed a fixed constant times δ0 . This fact in turn readily follows from the estimate (12.42) of Chapter 12. In view of the bound on χ, the second fundamental form of the sections St,u relative to Cu , the Cu do not contain focal points (that is, points along a generator of Cu which are conjugate to S0,u ) either. The absence of cut loci or focal points implies that a bicharacteristic generator of Cu cannot leave the boundary of the domain of dependence , in the domain of the maximal solution , of the exterior of the surface S0,u in the initial hyperplace 0 . (For the notion of cut locus and of focal or conjugate points in Riemannian geometry, see [C–E]. For the corresponding notions in Lorentzian geometry see [P]). Thus, unless t∗ε0 = ∞, there is ε on t∗ε0 at least one point which belongs to the boundary of the domain of the maximal 0 solution and not to the domain itself. We shall presently show that in fact s∗ coincides with t∗ε0 , provided that the initial data satisfies the smallness condition: √

(13.931) C ε0 D[l+2] < δ0

882

Chapter 13. Error Estimates

For, otherwise, that is if s∗ < t∗ε0 , then by continuity the bootstrap assumption holds on Wεs0∗ as well, hence by the above, (13.929) holds with l−2−[a] replaced by (l+1)∗ +2 and s replaced by s∗ . By virtue of the smallness condition (13.931) and continuity however, the bootstrap assumption must also hold for some s > s∗ contradicting the definition of s∗ . We conclude that s∗ = t∗ε0 and (13.929) holds for all t ∈ [0, t∗ε0 ). We thus have uniform pointwise estimates for the ψα on Wε∗0 up to order (l + 1)∗ + 2. Propositions 12.9 and 12.10 then give uniform pointwise estimates for χ  up to order (l + 1)∗ and for µ up to order (l + 1)∗ + 1. It follows that the ψα , χ  and µ thus also α, κ, and ε the induced acoustical metric h /, extend smoothly in acoustical coordinates to t∗ε0 and 0

t∗ε

Wε0 0 . Also the rectangular components h µν of the acoustical spacetime metric, being functions of the ψα , likewise extend. Moreover, since the bootstrap assumption holds on Wε∗0 , all the estimates we have derived, in particular the energy estimates (13.860), (13.856), (13.769)–(13.772), and the L 2 acoustical estimates of Propositions 12.11, 12.12, as well as the top order acoustical estimates, hold for all t ∈ [0, t∗ε0 ). The estimates not containing the weights µm,u , such as the estimates (13.860), then extend to t = t∗ε0 as well. Now, we must have: µm (t∗ε0 ) = 0 (13.932) For otherwise the Jacobian determinant
of the transformation from acoustical to rectanε gular coordinates (see Chapter 3) has a positive minimum on t∗ε0 , therefore the inverse 0 ε transformation is regular and the ψα extend smoothly in rectangular coordinates to t 0 . ε0 However, once the ψα extend to functions of the rectangular coordinates on t∗ε which 0 belong to the Sobolev space H3, then the standard local existence theorem applies and we obtain an extension of the solution to a development containing an extension of all the characteristic hypersurfaces Cu , u ∈ [0, ε0 ], up to a value t1 of t for some t1 > t∗ , in contradiction with the definition of t∗ε0 . We conclude that there is at least one point ε ε on t∗0ε0 where µ vanishes. By the same argument, at each point x ∗ ∈ t∗ε0 which lies 0 on the boundary of the domain of the maximal solution, µ(x ∗ ) = 0. For, otherwise, µ has a positive minimum in a suitable neighborhood of x ∗ , hence the solution is locally extendible at x ∗ in contradiction with the fact that x ∗ lies on the boundary of the domain of the maximal solution. Now, from Proposition 8.6, taking t = s we have: µˆ s (s, u, ϑ) = 1 − Eˆ s (u, ϑ) log(1 + s)

(13.933)

and we recall that: µˆ s (s, u, ϑ) =

µ(s, u, ϑ) , µ[1],s (u, ϑ)

Eˆ s (u, ϑ) =

E s (u, ϑ) µ[1],s (u, ϑ)

(13.934)

and that µ[1],s (u, ϑ) is bounded from below by η0 /2. We also recall that according to (8.250):      E s (u, ϑ) − 1 k 3 Ps (u, ϑ) ≤ Cδ0 (1 + s)−1 [1 + log(1 + s)] (13.935)   2

Part 4: Recovery of assumption J

883

where, from Lemma 8.10: Ps (u, ϑ) = (1 + s)(Lψ0 )(s, u, ϑ)

(13.936)

Now by the bounds (13.929) we have: √

|Ps (u, ϑ)| ≤ C ε0 D[l+2]

(13.937)

Hence we obtain: √

|E s (u, ϑ)| ≤ Ck 3 || ε0 D[l+2] + Cδ0 (1 + s)−1 [1 + log(1 + s)] Since

(13.938)

log(1 + s)[1 + log(1 + s)] ≤ C, (1 + s)

this implies: √

log(1 + s)|E s (u, ϑ)| ≤ Ck 3 || ε0 D[l+2] log(1 + s) + Cδ0

(13.939)

Substituting in (13.933) then yields: √

µˆ s (s, u, ϑ) ≥ 1 − Cδ0 − Ck 3 || ε0 D[l+2] log(1 + s) It follows that: log(1 + t∗ε0 ) ≥

1

√ Ck 3 || ε



0

D[l+2]

(13.940)

(13.941)

(for a new constant C), for, otherwise, µ would be bounded from below by a positive ε constant on t∗ε0 contradicting the conclusions of the previous paragraph. 0 It remains for us to analyze the requirements on the initial data. We recall from Chapter 2 that u on 0 is defined by: u = 1−r

(13.942)

Thus, in rectangular coordinates we have on 0 : ∂i u = −N i ,

Ni =

xi r

(13.943)

where N is the Euclidean outward unit normal to the Euclidean coordinate spheres. According to (2.68) of Chapter 2, κ is defined by: κ −2 = (h −1

−1 i j

) ∂i u∂ j u

(13.944)

The components (h )i j of the reciprocal of the induced acoustical metric on the t being given by (see (6.58)): H ψi ψ j −1 (h )i j = δi j − (13.945) 1 + ρH

884

Chapter 13. Error Estimates

it follows that on 0 κ, is given by: κ −2 = 1 −

H (ψ N )2 1 + ρH

ψ N = N i ψi

(13.946)

According to (2.67) Tˆ is given by: −1 Tˆ i = κ(h )i j ∂ j u

(13.947)

It then follows from (13.943), (13.945), that on 0 the rectangular components of Tˆ are given by:   H ψi ψ N Tˆ i = −κ N i − (13.948) 1 + ρH Thus, the commutation field T is given on 0 in rectangular coordinates by:   H ψi ψ N ∂ T = −κ 2 N i − 1 + ρ H ∂xi Then by 3.b11 the vectorfield L is given on 0 in rectangular coordinates by:   H ψi ∂ ∂ i (ψ + ακ N − + ακψ ) L= 0 N ∂x0 1 + ρH ∂xi

(13.949)

(13.950)

The commutation field Q coincides on 0 with L. Consider next the orthogonal projection to St,u in t , relative to h. It is given in rectangular coordinates by: ab = δba − Tˆ a h bc Tˆ c

(13.951)

The level surfaces S0,u of u on 0 are Euclidean spheres centered at the origin. Now the ◦

vectorfields Ri are tangential to these spheres. It follows that the commutation fields Ri ◦

coincide on 0 with the Ri : ◦

◦

Ri =  Ri = Ri = i j k x j

∂ ∂xk

(13.952)

◦

Moreover, the functions λi defined by (12.12) are, in view of (13.948), given on 0 by: ◦

λi = r κ

H ψN i j k N j ψk 1 + ρH

(13.953)

To derive an expression for χ on 0 , we consider the formula (3.221) of Chapter 3, expressed in rectangular coordinates: d/a Tˆ i = p/a Tˆ i + q/ia

(13.954)

Part 4: Recovery of assumption J

885

Note that for any function f on t we have d/a f = ia ∂i f . According to (3.230) we have: 1 dH  ψ ˆ ( ψc d/a σ − ψa d/c σ ) − H ψc ω q/ba h bc = θac − /a Tˆ (13.955) 2 dσ T Note that ψa = ia ψi , ω / ˆ = Tˆ i d/a ψi . Since by (13.954): aT

q/ba = bi d/a Tˆ i

(13.956)

 1   1 h ac ci d/b Tˆ i + h bc ci d/a Tˆ i + H ψa ω /bTˆ + ψb ω /a Tˆ 2 2

(13.957)

we then obtain:  θab =

From (13.948) we obtain: ci d/a Tˆ i and we have:

 =

−κci d/a

H ψi ψ N N − 1 + ρH



i

d/a N i = a ∂ j N i = r −1 a  ij j

j

(13.958)

(13.959)

where  is the Euclidean projection operator to the Euclidean coordinate spheres:  ij = δi j − N i N j

: in rectangular coordinates

(13.960)

Note that by (13.948), (13.951), we have: H ψN i ψ j 1 + ρH j H ψN j N a  ij ψ j a  ij = ai + κ 2 1 + ρH  ij Tˆ j = κ

(13.961) (13.962)

Noting also that by (13.943), (13.947), (13.951), we have, in rectangular coordinates, h bc ci = h bi − κ 2 N b N i

(13.963)

we then obtain, from (13.959):   H ψN N a  ij ψ j h bc ci d/ N i = h bi ai + κ 2 1 + ρH H ψN = ab + H ψi ψb ai + κ 2 N a h bi  ij ψ j 1 + ρH

(13.964)

Substituting this in (13.958) and (13.957) then yields: ◦

 θab = κ θ ab + f ab

where

◦

θ ab = −r −1 ab

: in rectangular coordinates

(13.965)

(13.966)

886

Chapter 13. Error Estimates

is the second fundamental form of the spheres S0,u in 0 with respect to the inward normal −N and the Euclidean metric on 0 . Note that: ◦

L ◦ θ= 0

: i = 1, 2, 3

Ri

(13.967)

Also, f ab is the symmetric 2-covariant S0,u tensorfield, given in rectangular coordinates by:  κ H ψN (N a h bi + N b h ai ) ij ψ j f ab = − H ψi (ψa bi + ψb ai ) + κ 2 2 1 + ρH      H ψi ψ N H ψi ψ N κ c + h bc d/a + i h ac d/b 2 1 + ρH 1 + ρH  1  + H ψa ω (13.968) /bTˆ + ψb ω /a Tˆ 2 Finally from (3.232) of Chapter 3 we have:  χab = −α(θab − k/ab )

(13.969)

and k/ is given by (3.236): k/ab

 1 dH ψ0 ( ψa d/b σ − ψb d/a σ ) = ψa ψb (Lσ ) − 2α dσ (1 + ρ H ) 1 H ψ0 − ω /ab α (1 + ρ H )

(13.970)

where ω /ab = ib d/a ψi . Consider now the initial data for the nonlinear wave equation (1.57). The functions ψ0 , ψi : i = 1, 2, 3 are given in the exterior of the sphere of radius 1 − ε0 with center at the origin and satisfy ∂i ψ j = ∂ j ψi . Outside the unit sphere with center at the origin, the initial data coincide with those of a constant state ψ0 = k, ψi = 0 : i = 1, 2, 3. We assume that ψ0 − k, ψi : i = 1, 2, 3 are, in rectangular coordinates, functions belonging ε to the Sobolev space Hl+2 (00 ) with vanishing traces on the unit sphere. We set: D[l+2] =



∂i ψα 2

α,i

ε

Hl+1 (00 )

(13.971)

The nonlinear wave equation in the form (see (1.64)): (h −1 )µν ∂µ ψν = 0,

∂µ ψν = ∂ν ψµ

(13.972)

allows us to express ∂0 ψ0 and ∂0 ψi and in terms of the ∂i ψ0 and ∂i ψ j , for, ∂0 ψi = ∂i ψ0 and we have: (13.973) (h −1 )00 = −1 − F(ψ0 )2 ≤ −1

Part 4: Recovery of assumption J

887

We can then express recursively ∂0k ψα : α = 0, 1, 2, 3 for k = 1, . . . , l + 2 in terms of the data ∂ik . . . ∂i1 ψα : α = 0, 1, 2, 3; i 1 , . . . , i n = 1, 2, 3. Now the standard Sobolev inequalities yield: max ∂in . . . ∂i1 (ψ0 − k) L ∞ ( ε0 ) , max max ∂in . . . ∂i1 ψ j  L ∞ (0 ε0 ) 0 j i1 ...in √

≤ C ε0 D[l+2] : for n = 0, . . . , l − 1 (13.974)

i1 ...in

It then follows that also: max ∂in . . . ∂i1 ∂0k (ψ0 − k) L ∞ ( ε0 ) , max max ∂in . . . ∂i1 ∂0k ψ j  L ∞ (0 ε0 ) 0 j i1 ...in √

≤ C ε0 D[l+2] : for n, k ≥ 0, n + k ≤ l − 1 (13.975)

i1 ...in

Now from (13.946), (13.949), (13.950), and (13.952) κ and the rectangular components ε of the vectorfields T , L and Ri are on 00 smooth functions of the rectangular coordinates and the ψα . It then follows that: E0,[l+2] (0) ≤ C D[l+2]

(13.976)

provided that D[l+2] is suitably small. Also, from (13.651) and (13.974) with n = 0 we obtain: ◦

max  λi  L ∞ ( ε0 ) ≤ Cε0 D[l+2] i

(13.977)

0

From (13.948) and (13.974) we obtain: κ − 1∞,{l−1}, ε0 ≤ Cε0 D[l+2]

(13.978)

0

From (13.967) and (13.969), the components f ab and k/ab are smooth functions of the rectangular coordinates, the ψα and their first derivatives with respect to the rectangular coordinates. In view also of (13.966), we then obtain from (13.975), through (13.964), (13.968): √

χ  ∞,{l−2}, ε0 ≤ C ε0 D[l+2] (13.979) 0

The assumptions of Propositions 12.3, 12.6, 12.9, 12.10, on the initial conditions are then recovered provided that: √

ε0 D[l+2] ≤ C −1 δ0 (13.980) Moreover, using the same formulas together with (13.975) we deduce:

Y0 (0) ≤ C D[l+2]

B{l+1} (0) ≤ C D[l+2]

A[l] (0) ≤ C D[l+2]

(13.981) (13.982) (13.983)

888

Chapter 13. Error Estimates

and: 

(i1 ...il )



m=0 i1 ...il−m



(13.984)

≤ C D[l+2]

(13.985)

0

i1 ...il l  

xl (0) L 2 ( ε0 ) ≤ C D[l+2]

(i1 ...il−m )  x m,l−m (0) L 2 ( ε0 ) 0

Combining then yields (see (13.656)): D[l+2] ≤ C D[l+2]

(13.986)

and we can replace ε0 by any u ∈ (0, ε0 ] to obtain: u u D[l+2] ≤ C D[l+2] : for all u ∈ (0, ε0 ]

where: u = D[l+2]

 α,i

∂i ψα 2Hl+1 ( u ) 0

(13.987)

(13.988)

In conclusion we have proved the following theorem, which is the main theorem of this monograph. Theorem 13.1 Let ( p, s, u µ : µ = 0, 1, 2, 3) be initial data for the fluid equations (1.20) on 0 which correspond to the initial data of a constant state p = p0 , s = s0 , u 0 = 1, u i = 0 : i = 1, 2, 3 outside the unit sphere with center at the origin in 0 . Let also the initial data be irrotational and isentropic outside the the sphere of radius 1 − ε0 , 0 < ε0 ≤ 1/2 with center at the origin in 0 . Then we have initial data (φ, ∂0 φ) for the nonlinear wave equation√(1.57) outside the sphere of radius 1 − ε0 with center at the origin in 0 , where ∂µ φ = − σ u µ , √ σ is the enthalpy: √ ρ+p , σ = n √ and G(σ ) σ is n as a function of the enthalpy at s = s0 . The initial data (φ, ∂0 φ) coincide with those of a constant state φ = kt,

k=

ρ0 + p0 √ σ0 = n0 ε

outside the unit sphere with center at the origin in 0 . Consider the annular region 00 in 0 bounded by the two concentric spheres. Then there is a positive integer [a] such that the following hold. Let l be a positive integer such that l∗ ≥ [a] + 4

Part 4: Recovery of assumption J

889

and suppose that the functions ψ0 − k, ψi : i = 1, 2, 3 corresponding to the initial ε data on 00 are functions of the rectangular coordinates belonging to the Sobolev space ε0 Hl+2 (0 ) with vanishing traces on the unit sphere. Then setting D[l+2] =



∂i ψα 2

ε

Hl+1 (00 )

α,i

there is a suitably small positive constant δ 0 and a suitably large positive constant C, such that for any δ0 ∈ (0, δ0 ], if: √

C ε0 D[l+2] < δ0 the following conclusions hold: (i) Let u be the function 1 − r on 0 and Su,0 the spheres of radius 1 − u with center at the origin, the level surfaces of u in 0 . We consider, in the domain of the maximal solution corresponding to the given initial data, the family {Cu : u ∈ [0, ε0 ]} of outgoing characteristic hypersurfaces corresponding to the family {S0,u : u ∈ [0, ε0]}: Cu



0 = S0,u : ∀u ∈ [0, ε0 ]

with each bicharacteristic generator of each Cu extending in the domain of the maximal solution as long at it remains on the boundary of the domain of dependence of the exterior of S0,u in 0 . Then the bicharacteristic generators of each Cu have no future end points except on the boundary of the domain of the maximal solution . Let t∗ (u) be the least upper bound of the extent of the generators of Cu , in the parameter t, in the domain of the maximal solution, and let: t∗ε0 = inf t∗ (u) u∈[0,ε0 ]

We define for each (t, u) ∈ [0, t∗ε0 ) × [0, ε0 ] the closed surface: St,u = Cu



t

ε

Then either t∗ε0 = ∞ or there is on t∗ε0 at least one point which belongs to the boundary 0 of the domain of the maximal solution and not to the domain itself. (ii) We have the lower bound: log(1 + t∗ε0 ) ≥ where  is the constant: = In particular, if  = 0 we have t∗ε0 = ∞.

1

√ Ck 3 || ε dH (σ0 ) dσ

0



D[l+2]

890

Chapter 13. Error Estimates

(iii) For all t ∈ [0, t∗ε0 ) we have the L ∞ bounds: √

max Rin . . . Ri1 (T )m (Q) p (ψ0 − k) L ∞ ( ε0 ) ≤ C(1 + t)−1 ε0 D[l+2] t i1 ...in √

max Rin . . . Ri1 (T )m (Q) p ψ j  L ∞ ( ε0 ) ≤ C(1 + t)−1 ε0 D[l+2] j i1 ...in

t

: for all p + m + n ≤ l − 2 − [a] and with µ and χ the acoustical entities defined by the family {Cu : u ∈ [0, ε0 ]} and for each Cu : u ∈ [0, ε0 ] by the family of sections {St,u : t ∈ [0, t∗ε0 )}: (µ/η0 ) − 1∞,{l−3−[a]}, ε0 ≤ Cδ0 [1 + log(1 + t)] t



χ ∞,{l−4−[a]}, ε0 ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)] t

where χ = χ −

η0 h/ , 1 − u + η0 t

/ is the induced acoustical metric on St,u and η0 is the sound speed in the constant state. h (iv) The ψα , α, κ, µ, h /, and χ, extend continuously with their first l − 2 − [a] derivatives in the case of ψα , α, l −3−[a] derivatives in the case of κ, µ, h/, and l −4−[a] derivatives in the case of χ, in acoustical coordinates to t∗ ε0 . The rectangular components Tˆ µ and L µ (Tˆ 0 = 0, L 0 = 1) of the vectorfields T and L likewise extend continuously with their first l − 3 − [a] derivatives in acoustical coordinates to t∗ε0 , and so do the rectangular components h µν of the acoustical spacetime metric. The functions κ and µ so extended ε vanish at each point in t∗ε0 which lies on the boundary of the domain of the maximal so0 lution , being positive everywhere else. The Jacobian determinant
of the transformation from acoustical to rectangular coordinates also vanishes at the same points, being positive everywhere else. The first derivatives Tˆ i ∂i ψα of the ψα with respect to the rectangular coordinates blow up at these points, and so do the corresponding derivatives of the h µν . More precisely, what blows up is the component Tˆ i Tˆ j ∂i ψ j . Moreover, there is a positive constant C such that in the subdomain U of Wε∗0 where µ < η0 /4, which contains a spacetime neighborhood of each of the points in question, we have: Lµ ≤ −C −1 (1 + t)−1 [1 + log(1 + t)]−1 (v) The following energy estimates hold, for all u ∈ [0, ε0]: 

u u sup (t) ≤ C D[l+2] E0,[l+1−[a]] t ∈[0,t∗ε0 ]

t∗ε

sup

t ∈[0,t∗ε0 ]

u 0 F0,[l+1−[a]] (u) ≤ C D[l+2]

 u u [1 + log(1 + t)]−4 E1,[l+1−[a]] (t) ≤ C D[l+2]

Part 4: Recovery of assumption J

891

sup

 t u [1 + log(1 + t)]−4 F1,[l+1−[a]] (u) ≤ C D[l+2]

sup

 u [1 + log(1 + t)]−4 K [l+1−[a]] (t, u) ≤ C D[l+2]

t ∈[0,t∗ε0 ] t ∈[0,t∗ε0 ]

where u D[l+2] =

 α,i

∂i ψα 2Hl+1 ( u ) 0

and 0u is the annular region on 0 bounded by S0,u and the unit sphere S0,0 . Moreover, setting a = [a] +

3 4

we have, for each n = 0, . . . , [a] − 1, and for all u ∈ [0, ε0 ], the estimates: sup

t ∈[0,t∗ε0 )

sup

t ∈[0,t∗ε0 )

sup

t ∈[0,t∗ε0 )

 u u µ2(a−n−1) (t)E0,[l+1−n] (t) ≤ C D[l+2] m,u

 t u µ2(a−n−1) (t)F0,[l+1−n] (u) ≤ C D[l+2] m,u

 −4 u u µ2(a−n−1) (t)[1 + log(1 + t)] E (t) ≤ C D[l+2] m,u 1,[l+1−n]

sup

 −4 t u µ2(a−n−1) (t)[1 + log(1 + t)] F (u) ≤ C D[l+2] m,u 1,[l+1−n]

sup



u µ2(a−n−1) (t)[1 + log(1 + t)]−4 K [l+1−n] (t, u) ≤ C D[l+2] m,u

t ∈[0,t∗ε0 ) t ∈[0,t∗ε0 )

Furthermore, there is a positive real number p such that with q = p + 2 we have the top order energy estimates: sup

t ∈[0,t∗ε0 )

sup

t ∈[0,t∗ε0 )

sup

 −2 p u u µ2a (t)[1 + log(1 + t)] E (t) ≤ C D[l+2] m,u 0,[l+2]

 −2 p t u µ2a (t)[1 + log(1 + t)] F (u) ≤ C D[l+2] m,u 0,[l+2]

t ∈[0,t∗ε0 )

 −2q u u µ2a E1,[l+2] (t) ≤ C D[l+2] m,u (t)[1 + log(1 + t)]

sup



−2q t u µ2a F1,[l+2] (u) ≤ C D[l+2] m,u (t)[1 + log(1 + t)]

sup



−2q u µ2a K [l+2] (t, u) ≤ C D[l+2] m,u (t)[1 + log(1 + t)]

t ∈[0,t∗ε0 ) t ∈[0,t∗ε0 )

892

Chapter 13. Error Estimates

(vi) The following L 2 acoustical estimates hold, for all u ∈ [0, ε0 ] and all t ∈ [0, t∗ε0 ]:  −1 3 u Au D[l+2] [l−[a]−1] (t) ≤ C(1 + t) [1 + log(1 + t)]  u u B{l−[a]} (t) ≤ C(1 + t)[1 + log(1 + t)]3 D[l+2] Moreover, for each n = 1, . . . , [a] and for all u ∈ [0, ε0 ] and all t ∈ [0, t∗ε0 ):  −1 3 −a+n+1/2 u Au (t) D[l+2] [l−n] (t) ≤ C(1 + t) [1 + log(1 + t)] µm,u  −a+n+1/2 u u B{l−n+1} (t) ≤ C(1 + t)[1 + log(1 + t)]3 µm,u (t) D[l+2] and for n = 0 and for all u ∈ [0, ε0 ] and all t ∈ [0, t∗ε0 ):  u (t) D[l+2]  −a+1/2 u u B{l+1} (t) ≤ C(1 + t)[1 + log(1 + t)]q+1 µm,u (t) D[l+2] −a+1/2

−1 q+1 µm,u Au [l] (t) ≤ C(1 + t) [1 + log(1 + t)]

Finally, for all u ∈ [0, ε0 ] and all t ∈ [0, t∗ε0 ) we have the top order acoustical estimates:  2p u max µRil+1 . . . Ri1 trχ   L 2 (tu ) ≤ Cµ−2a D[l+2] m,u (t)[1 + log(1 + t)] i1 ...il+1

l  m=0

 2p u max µRil−m . . . Ri1
/ µ L 2 (tu ) ≤ Cµ−2a D[l+2] m,u (t)[1 + log(1 + t)]

i1 ...il−m

Chapter 14

Sufficient Conditions on the Initial Data for the Formation of a Shock in the Evolution In the present chapter we shall establish sharp sufficient conditions on the initial data for the formation of a shock in the evolution. In the first part of the chapter we shall investigate the problem in the isentropic irrotational context, while in the second part we shall remove the irrotational as well as the isentropic restrictions and shall prove a general theorem of shock formation in fluid mechanics. In the first part the setup is as in Theorem 13.1. Following this theorem we must now find sharp sufficient conditions on the initial data which will guarantee that the function ε µ becomes zero somewhere on t 0 at some finite t. That value of t shall then be t∗ε0 . To accomplish our aim we shall make use of what has been established in Chapter 8 by Lemma 8.10, Proposition 8.5 and Proposition 8.6. Taking t = s in Proposition 8.6 we obtain: (14.1) µ(s, u, ϑ) = µ[1],s (u, ϑ)µˆ s (s, u, ϑ) where µ[1],s (u, ϑ) satisfies: 3η0 η0 ≤ µ[1],s (u, ϑ) ≤ 2 2

(14.2)

µˆ s (s, u, ϑ) = 1 + Eˆ s (u, ϑ) log(1 + s)

(14.3)

while µˆ s (s, u, ϑ) is given by:

Here: Eˆ s (u, ϑ) =

E s (u, ϑ) µ1,[s] (u, ϑ)

and according to (8.250):      E s (u, ϑ) − 1 k 3 Ps (u, ϑ) ≤ Cδ0 (1 + s)−1 [1 + log(1 + s)]   2

(14.4)

(14.5)

894

Chapter 14. Sufficient Conditions on the Initial Data

where Ps (u, ϑ) is the function defined by Lemma 8.10: Ps (u, ϑ) = (1 + s)(Lψ0 )(s, u, ϑ)

(14.6)

In (14.5) and in all the following the positive constant δ0 shall be as in Theorem 13.1. In ε view of the above, a sharp sufficient condition for µ to become zero somewhere on s 0 at some finite s, is an upper bound for min(u,ϑ)∈[0,ε0]×S 2 Ps (u, ϑ) by a negative constant for all sufficiently large s in the case that  > 0, a lower bound for max(u,ϑ)∈[0,ε0]×S 2 Ps (u, ϑ) by a positive constant for all sufficiently large s in the case that  < 0. As was already shown in Theorem 5.1 no shocks can form in the case  = 0 as in this case we have t∗ε0 = ∞. Thus in the following we assume  = 0. The sharp sufficient condition just stated is however not a condition on the initial data. To find the corresponding condition on the initial data, we apply the following method. Let us revisit the wave equation for ψ0 in the form encountered in the proof of Lemma 8.10 (see equations (8.192), (8.193)):

where:

L(Lψ0 ) + ν Lψ0 + ν Lψ0 = ρ0

(14.7)

ρ0 = µ
/ ψ0 + µh /−1 (d/ log , d/ψ0 ) − 2h/−1 (ζ, d/ψ0 )

(14.8)

Consider the function: τ = (1 − u + η0 t)Lψ0 − (ψ0 − k)

(14.9)

Lτ = (1 − u + η0 t)L(Lψ0 ) − Lψ0 + η0 Lψ0

(14.10)

We have: Substituting for L(Lψ0 ) from (14.7) yields: Lτ = ω

(14.11)

where: ω = −[(1 − u + η0 t)ν − η0 ]Lψ0 − [(1 − u + η0 t)ν + 1]Lψ + (1 − u + η0 t)ρ0 (14.12) From conclusions (iii) of Theorem 13.1 we have, recalling that: 1 (trχ + L log ) , 2 1 1 ν + α −1 κν = α −1 κ L log  + L log  + κtrk/, 2 2 ν=

the bounds: |(1 − u + η0 t)ν − η0 | ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] |(1 − u + η0 t)ν + 1| ≤ Cδ0 [1 + log(1 + t)]

(14.13) (14.14)

Chapter 14. Sufficient Conditions on the Initial Data

895

Let now f be an arbitrary function defined on Wεs0 . Let us denote by f (t, u) the mean value of f on St,u with respect to the measure dµh/˜ : 1 ˜ f (t, u) = f dµh/˜ , Area(t, u) = dµh/˜ (14.15) ˜ Area(t, u) St,u St,u From (13.314)–(13.315) we have: ∂ f dµh/˜ = (L f + 2ν f )dµh/˜ ∂t St,u S t,u ∂ ˜ ˜ 2νdµh/˜ = 2ν Area(t, u) Area(t, u) = ∂t St,u Hence:

1 ∂f (t, u) = ˜ ∂t Area(t, u)

(14.17)

St,u

(L f + 2ν f − 2ν f )dµh/˜

Noting that for any pair of functions f ,g, on St,u we have: ( f g − f g)dµh/˜ = ( f − f )(g − g)dµh/˜ St,u

(14.16)

(14.18)

(14.19)

St,u

we can write (14.18) in the form: ∂f (t, u) = L f + 2(ν − ν)( f − f ) ∂t

(14.20)

We apply the formula (14.20) to the function τ obtaining, in view of (14.11), ∂τ (t, u) = ω + 2(ν − ν)(τ − τ ) ∂t We integrate this with respect to t on [0, s] to obtain: s  τ (s, u) = τ (0, u) + ω + 2(ν − ν)(τ − τ ) (t, u)dt

(14.21)

(14.22)

0

˜ u  ), and integrate with respect to Finally we replace u by u  ∈ [0, u], multiply by Area(0,  u on [0, u] to obtain: u ˜ τ (s, u  )Area(0, u  )du  (14.23) 0  s u u  ˜ = τ dµh/˜ du  + ω + 2(ν − ν)(τ − τ ) (t, u  )dt du  Area(0, u ) 0

S0,u

0

0

We shall estimate the second integral on the right in (14.23). Interchanging the order of integration this integral is: s  u     ˜ I (s, u) = (14.24) Area(0, u ) ω + 2(ν − ν)(τ − τ ) (t, u )du dt 0

0

896

Chapter 14. Sufficient Conditions on the Initial Data

˜ ˜ Since Area(0, u  ) ≤ Area(0, 0) = 4π, we have: s J (t, u)dt |I (s, u)| ≤ C

(14.25)

0

where J (t, u) is the integral:



u

J (t, u) =

2 1 |ω| + 2|ν − ν||τ − τ | (t, u  )du 

(14.26)

0

We consider first the contribution of the term |ω|, through (14.26), to the integral on the right in (14.25). Now ω is given by (14.12), and we consider first the third term on the right in (14.12). We have: (1 − u + η0 t) ρ0 dµh/˜ (14.27) (1 − u + η0 t)ρ0 (t, u) = ˜ Area(t, u) St,u Now, for an arbitrary function f on St,u , we have:    ∂f ∂ 1 −1 AB ˜ ˜ ˜ deth/ A (h/ )
/f =  ∂ϑ A ∂ϑ deth /˜  

∂f 1 ∂ −1 AB (h/ ) = −1
deth/ A /f = √ ∂ϑ  deth / ∂ϑ A

(14.28)

Thus, integrating by parts on St,u , we obtain: µ
/ ψ0 dµh/˜ = µ
/˜ ψ0 dµh/˜ St,u St,u  −1 −1 µh /˜ (d/, d/ψ0 ) + h/˜ (d/µ, d/ψ0 ) dµh/˜ =− S t,u  =− µh /−1 (d/ log , d/ψ0 ) + h/−1 (d/µ, d/ψ0 ) dµh/˜

(14.29)

St,u

Recalling that we then conclude that:

− (d/µ + 2ζ )$ = − (η + ζ )$ = , St,u

ρ0 dµh/

= St,u

 · d/ψ0 dµh/˜

(14.30)

Now from conclusions (iii) of Theorem 13.1 we have: || ≤ Cδ0 (1 + t)−1 [1 + log(1 + t)] Thus (14.30) implies, through (14.27), (1 − u

+ η0 t)|ρ0 (t, u)|

(1 − u + η0 t) ≤ sup || ˜ (Area(t, u))1/2 St,u ≤ Cδ0 (1 + t)

−1

(14.31)



1/2 2

St,u

|d/ψ0 | dµh/ 

1/2 2

[1 + log(1 + t)] St,u

|d/ψ0 | dµh/

(14.32)

Chapter 14. Sufficient Conditions on the Initial Data

897

Let us estimate the contribution of this to the integral on the right in (14.25). This contribution is then, in view of the fact that u ∈ [0, ε0 ], bounded by √ Cδ0 ε0



s 0

√ ≤ Cδ0 ε0

(1 + t)



s



s

 [1 + log(1 + t)]

1/2 tu

1/2



|d/ψ0 |

2

dt

2

tu

0

Since the integral

−1

|d/ψ0 | dt

(14.33)

(1 + t)−2 [1 + log(1 + t)]2 dt

0

is bounded by a constant independent of s. Now the integral on the right in (14.33) is: |d/ψ0 |2 dtdu  dµh/ (14.34) Wus

This integral has already been estimated in Chapter 7 by (7.100) for any variation. In particular for the first order variation (0)ψ1 = ψ0 − k we have:  u 2  s   |d/ψ0 | dtdu dµh/ ≤ C K [ψ0 − k](s, u) + F0 [ψ0 − k](u )du Wus

0

≤ CE0u [ψ0 − k](0)

(14.35)

the last step by virtue of Theorem 5.1. We thus conclude that (14.33) is bounded by: √  Cδ0 ε0 E0u [ψ0 − k](0) (14.36) We consider next the contributions of the first two terms on the right in (14.12), through |ω| to the integral on the right in (14.25). By (14.13) the contribution of the first term on the right in (14.12) to |ω| is bounded by: Cδ0 (1 + t)−1 [1 + log(1 + t)] ˜ Area(t, u)

St,u

|Lψ0 |dµh/˜

Cδ0 (1 + t)−1 [1 + log(1 + t)] ≤ ˜ (Area(t, u))1/2 ≤ Cδ0 (1 + t)

−2



1/2 |Lψ0 | dµh/˜ 2



St,u

[1 + log(1 + t)]

1/2 |Lψ0 | dµh/˜

(14.37)

hence the corresponding contribution to J (t, u) is bounded by:  √ Cδ0 ε0 (1 + t)−2 [1 + log(1 + t)] E0u [ψ0 − k](t)

(14.38)

2

St,u

898

Chapter 14. Sufficient Conditions on the Initial Data

and to the integral on the right in (14.25) by: 1/2  s √ Cδ0 ε0 (1 + t)−2 E0u [ψ0 − k](t)dt 0 √  u ≤ Cδ0 ε0 E0 [ψ0 − k](0)

(14.39)

by virtue of Theorem 5.1, noting again that the integral s (1 + t)−2 [1 + log(1 + t)]2 dt 0

is bounded by a constant independent of s. By (14.14) the contribution of the second term on the right in (14.12) to ω is bounded by: Cδ0 [1 + log(1 + t)] |Lψ0 |dµh/˜ ˜ Area(t, u) St,u  1/2 Cδ0 [1 + log(1 + t)] 2 ≤ |Lψ0 | dµh/˜ ˜ (Area(t, u))1/2 St,u 1/2  −1 2 ≤ Cδ0 (1 + t) [1 + log(1 + t)] |Lψ0 | dµh/˜ (14.40) St,u

The contribution of this to the integral on the right in (14.25) is then bounded by: 1/2  s √ −1 2 Cδ0 ε0 (1 + t) [1 + log(1 + t)] |Lψ0 | dt 0

√ ≤ Cδ0 ε0 √

≤ Cδ0 ε0

 

s

tu

0 u

tu

1/2 |Lψ0 |2 dt

F0s [ψ0



− k](u )du 0  √ ≤ Cδ0 ε0 E0u [ψ0 − k](0)



1/2

(14.41)

by virtue again of Theorem 5.1. Combining the above results (14.36), (14.39), (14.41), we conclude that the contribution of the term |ω| to the integral on the right in (14.25) is bounded by: √  (14.42) Cδ0 ε0 E0u [ψ0 − k](0) Finally, we consider the contribution of the term 2|ν − ν||τ − τ | to the integral on the right in (14.25). We have: 1 |ν − ν||τ − τ | = |ν − ν||τ − τ |dµh/˜ (14.43) ˜ Area(t, u) St,u

Chapter 14. Sufficient Conditions on the Initial Data

899

Now, by (14.13):         η0 η0  + ν −  |ν − ν| ≤ ν −   1 − u + η0 t 1 − u + η0 t  ≤ Cδ0 (1 + t)−2 [1 + log(1 + t)]

(14.44)

Therefore: Cδ0 (1 + t)−2 [1 + log(1 + t)] |ν − ν||τ − τ | ≤ ˜ Area(t, u) Cδ0 (1 + t)−2 [1 + log(1 + t)] ≤ ˜ (Area(t, u))1/2 We have: St,u

|τ − τ |2 dµh/˜ =

St,u

St,u

|τ − τ |dµh/˜



1/2 2

|τ − τ | dµh/˜

(14.45)

|ψ0 − k|2 dµh/

(14.46)

St,u

St,u

τ 2 dµh/˜ −

2

≤



τ dµh/˜ ≤ C(1 + t)

2

St,u

τ 2 dµh/˜



2

St,u

|Lψ0 | dµh/ + C

St,u

in view of the definition (14.9). The contribution of the first integral on the right in (14.46) to the right-hand side of (14.45) is bounded by: Cδ0 (1 + t)

−2

1/2

 [1 + log(1 + t)]

|Lψ0 |

2

(14.47)

St,u

This coincides with (14.37), hence the corresponding contribution to the integral on the right in (14.25) is bounded by (14.39). Finally, the contribution of the second integral on the right in (14.46) to the right-hand side of (14.45) is bounded by: Cδ0 (1 + t)−3 [1 + log(1 + t)]

1/2

 St,u

|ψ0 − k|2 dµh/

√  ≤ Cδ0 (1 + t)−3 [1 + log(1 + t)] ε0 E0u [ψ0 − k](t)

(14.48)

by inequality (5.72), hence the corresponding contribution to the integral on the right in (14.25) is also bounded by (14.39). Combining the above results we conclude that the contribution of the term 2|ν − ν||τ − τ | to the integral on the right in (14.25) is also bounded by (14.42). We have thus arrived at the following estimate for the integral I (s, u) defined by (14.24) √  (14.49) |I (s, u)| ≤ Cδ0 ε0 E0u [ψ0 − k](0)

900

Chapter 14. Sufficient Conditions on the Initial Data

It then follows from (14.23) that:    u  √    ˜ τ (s, u  )Area(0, u  )du  − τ dµh/˜ du   ≤ Cδ0 ε0 E0u [ψ0 − k](0)   0  u

(14.50)

0

Now from (14.6) and (14.9) we have: Ps (u, ϑ) =

  (η0 − 1 + u) τ 1 + τ + ψ0 − k η0 (1 − u + η0 s) η0

Let us denote: Q(u) =

0u

τ dµh/˜ du 

Consider the case  > 0. If the initial data satisfy: √  Q(u) ≤ −2Cδ0 ε0 E0u [ψ0 − k](0) < 0

for some u ∈ (0, ε0 ]

(14.51)

(14.52)

(14.53)

for some u ∈ (0, ε0 ] with the constant C as in (14.50), then by virtue of 14.50 we have: u 1 ˜ (14.54) τ (s, u  )Area(0, u  )du  ≤ Q(u) < 0 2 0 It follows that: min τ (s, u  ) ≤ ! u

u  ∈[0,ε0 ]

0

(1/2)Q(u) <0 ˜ Area(0, u  )du 

hence also: min

(u  ,ϑ)∈[0,u]×S 2

τ (s, u  , ϑ) ≤ ! u 0

(1/2)Q(u) ˜ Area(0, u  )du 

(14.55)

(14.56)

It then follows from (14.51), in view of the fact that by conclusions (iii) of Theorem 13.1 the second term on the right is bounded in absolute value by Cδ0 (1 + s)−1 , that: min

(u  ,ϑ)∈[0,u]×S 2

Ps (u, ϑ) ≤ ! u 0

(1/4)Q(u) <0 ˜ Area(0, u  )du 

(14.57)

for all s large enough so that: !u 1 + s ≥ −Cδ0

0

˜ Area(0, u  )du  (1/4)Q(u)

(14.58)

Thus by the discussion in the beginning of the chapter t∗ε0 is finite. In fact, (14.57) implies, in view of (14.5), that: (1/16)k 3Q(u) E s (u, ϑ) ≤ ! u <0 ˜ (u  ,ϑ)∈[0,u]×S 2 Area(0, u  )du  min

0

(14.59)

Chapter 14. Sufficient Conditions on the Initial Data

901

for all s large enough so that: !u ˜ Area(0, u  )du  1+s ≥ −Cδ0 0 1 + log(1 + s) (1/16)k 3Q(u)

(14.60)

where C is the constant on the right in (14.5). But then it follows from (14.3) in view of (14.2) that s is bounded from above according to: !u ˜ Area(0, u  )du  log(1 + s) ≤ − 0 (14.61) (1/24)k 3Q(u) Since the upper bound on s implied by (14.61) exceeds the lower bounds implied by (14.58) and (14.60), it follows that t∗ε0 satisfies the upper bound: !u ˜ Area(0, u  )du  (14.62) log(1 + t∗ε0 ) ≤ − 0 (1/24)k 3Q(u) In the case  < 0 a similar argument shows that if the initial data satisfy: √  Q(u) ≥ 2Cδ0 ε0 E0u [ψ0 − k](0) > 0 for some u ∈ (0, ε0 ]

(14.63)

for some u ∈ (0, ε0 ] with the constant C as in (14.50), then t∗ε0 is finite, in fact satisfies the upper bound: !u ˜ Area(0, u  )du  (14.64) log(1 + t∗ε0 ) ≤ 0 (1/24)k 3Q(u) Consider now the initial data integral Q(u) (see (14.52)). If we replace the measure dµh/˜ by dµh/ , then since the difference of  from unity, its ε value in the constant state, is bounded on 00 by Cδ0 , the change in the integral will be

u bounded by Cδ0 E0 [ψ0 − k](0). If we in turn replace the measure dµh/ by the measure dµg/ where g/ is the metric on S0,u induced by the Euclidean metric on 0 , namely the metric of the round

sphere of radius r = 1 − u, then the change in the integral will again be bounded by Cδ0 E0u [ψ0 − k](0), since the difference from the Euclidean metric of the ε induced acoustical metric h on 00 is also bounded by Cδ0 . Moreover we may replace ε L in the definition (14.9) of τ on 00 by what it would be if the constant state where to extend throughout 0 , namely by: ◦

L= We then obtain:

∂ 1 ∂ − Ni i η0 ∂ x 0 ∂x

◦

τ 0 = r L ψ0 − (ψ0 − k)

(14.65)

ε

: on 00

(14.66)

The difference of the resulting integral: 0u

τ 0d 3x

(14.67)

902

Chapter 14. Sufficient Conditions on the Initial Data

from Q(u) will again be bounded by Cδ0 E0u [ψ0 − k](0). We have: ◦

L ψ0 = ∂0 ψ0 − ψ N ,

ψ N = N i ψi

(14.68)

Consider the nonlinear wave equation (1.57) on 0 . We can express the reciprocal acoustical metric h −1 in rectangular coordinates in the form: ◦ −1

(h −1 )µν = (h ◦ −1

where h

)µν − δ µν

is the reciprocal acoustical metric in the constant state:   ◦ −1 1 µ (h )µν = (g −1 )µν − − 1 δ0 δ0ν 2 η0

(14.69)

(14.70)

and: 1 1 µ µ δ µν = (F − F(k 2 ))ψ µ ψ ν + F(δ0 − ψ µ )(δ0ν + ψ ν ) + F(δ0ν − ψ ν )(δ0 + ψ µ ) (14.71) 2 2 ε

ε

is bounded on 00 by Cδ0 . The nonlinear wave equation (1.57) then takes on 00 the form: −η0−2 ∂0 ψ0 + ∂i ψi = δ µν ∂µ ψν (14.72)  u (0). and we can bound the L 1 norm on 0u of r times the right-hand side by: δ0 E0,[1] Therefore defining  r (η0 ∂i ψi − N i ∂i ψ0 ) − (ψ0 − k) d 3 x (14.73) Q 0 (u) = 0u

we have:

 u |Q 0 (u) − Q(u)| ≤ C  δ0 E0,[1] (0)

Hence if in the case  > 0 we have:  √ u (0) < 0 Q 0 (u) ≤ −2(C  + C ε0 )δ0 E0,[1]

: for some u ∈ (0, ε0 ]

(14.74)

(14.75)

where C is the constant appearing in (14.53), then (14.53) holds and we have: Q(u) ≤

1 Q 0 (u) 2

hence the following holds, in place of (14.54): u 1 ˜ τ (s, u  )Area(0, u  )du  ≤ Q 0 (u) < 0 4 0

(14.76)

(14.77)

Chapter 14. Sufficient Conditions on the Initial Data

903

Similarly, if in the case  < 0 we have:  √ u Q 0 (u) ≥ 2(C  + C ε0 )δ0 E0,[1] (0) > 0

: for some u ∈ (0, ε0 ]

(14.78)

where C is the constant appearing in (14.63), then (14.63) holds and we have: Q(u) ≥ hence:



u

1 Q 0 (u) 2

˜ τ (s, u  )Area(0, u  )du  ≥

0

(14.79)

1 Q 0 (u) > 0 4

(14.80)

Consider now the integral (14.73). The first term in parenthesis can be integrated by parts to give: i 3 r (η0 ∂i ψi −N ∂i ψ0 )d x = r ((ψ0 −k)−η0 ψ N )dµg/ + (3(ψ0 −k)−η0 ψ N )d 3 x 0u

0u

S0,u

(14.81)

We have thus proved the following theorem. Theorem 14.1 Let (φ, ∂0 φ) be initial data for the nonlinear wave equation (1.57) outside the sphere of radius 1 − ε0 with center at the origin in 0 , fulfilling the conditions of Theorem 13.1. Let u be the function 1 − r on 0 and let: r ((ψ0 − k) − η0 ψ N )dµg/ + (2(ψ0 − k) − η0 ψ N )d 3 x Q 0 (u) = 0u

S0,u

Then there is a positive constant C such that for each δ0 ∈ (0, δ 0 ] as in Theorem 13.1 the following hold. Suppose that  = 0 and in the case  > 0 we have:  u Q 0 (u) ≤ −Cδ0 E0,[1] (0) < 0 : for some u ∈ (0, ε0 ] while in the case  < 0 we have:  u (0) > 0 Q 0 (u) ≥ Cδ0 E0,[1]

: for some u ∈ (0, ε0 ]

Then t∗ε0 is finite and in both cases we have the following upper bound on t∗ε : log(1 + t∗ε0 ) ≤

C u k 3 |||Q 0 (u)|

We now turn to the general problem of shock formation in fluid mechanics, removing the irrotational as well as the isentropic restrictions. We are still within the framework of the geometric construction of Chapter 2, as we assume that the initial data coincide

904

Chapter 14. Sufficient Conditions on the Initial Data

with those of a constant state outside the unit sphere S0,0 with center at the origin in 0 . We thus have: p = p0 s = s0 , u 0 = 1, u i = 0 : i = 1, 2, 3 : in the exterior of S0,0 in 0

(14.82)

Then by the results of Chapter 1 the portion of the fluid which on 0 lies outside S0,0 remains isentropic and irrotational as long as the solution remains smooth. We consider the initial value problem for the general equations of motion (1.20) for the time interval [0, η0−1 ]. If ( p− p0, s −s0 , u i : i = 1, 2, 3) belong at some t to the Sobolev space Hk (t ), we denote:  u i 2Hk (t ) (14.83) E [k] (t) = s − s0 2Hk (t ) +  p − p0 2Hk (t ) + +

k 

i

 (∂0 ) s2Hk−l (t ) l

+ (∂0 )

l

p2Hk−l (t )

+



 (∂0 )l u i 2Hk−l (t )

i

l=1

Now we can apply the standard theory of symmetric hyperbolic systems to show that   there is a suitably small positive constant δ 0 such that for all δ0 ∈ (0, δ 0 ], if the initial data ( p − p0, s − s0 , u i : i = 1, 2, 3) at t = 0 belong to the Sobolev space Hk (0 ) for some k ≥ 3 and 

D[k] < δ0 , where D[k] = s − s0 2Hk (0 ) +  p − p02Hk (0 ) + u i 2Hk (0 ) (14.84) i

then the system (1.20) has a solution in the fixed time interval [0, η0−1 ] such that ( p − p0 , s − s0 , u i : i = 1, 2, 3) belong at each t ∈ [0, η0−1 ] to the Sobolev space Hk (t ), and moreover there are constants C and C  such that: E [k] (t) ≤ C E [k] (0) ≤ C  D[k]

: for all t ∈ [0, η0−1 ]

(14.85)

Furthermore, defining the acoustical function u by imposing the final condition u = 2 − r : on η−1

(14.86)

0

and, as before, requiring that the level sets Cu of u be outgoing characteristic hypersurfaces, then, setting, for m = 0, . . . , k:  u E [m] (t) = s − s0 2Hm ( u ) +  p − p0 2Hm ( u ) + u i 2Hm ( u ) (14.87) t

+

m 

t

t

i

 (∂0 ) s2Hm−l ( u ) t l

+ (∂0 )

l

p2Hm−l ( u ) t

+

l=1



 (∂0 )l u i 2Hm−l ( u ) t

i

we have, for all m = 0, . . . , k: : for all t ∈ [0, η0−1 ] and all u ∈ [0, ε0]  = s − s0 2Hm ( u ) +  p − p0 2Hm ( u ) + u i 2Hm ( u )

u u u (t) ≤ C E [m] (0) ≤ C  D[m] E [m] u where D[m]

0

0

i

0

(14.88)

Chapter 14. Sufficient Conditions on the Initial Data

905

the solution on tu depending only on the initial data on 0u . This local existence theorem is in fact proved by using the energy estimates provided by the integral on t of the energy current (1.27) corresponding to the variations ( p, ˙ s˙ , u) ˙ of order up to k generated by the translations Tµ = ∂/∂ x µ : µ = 0, 1, 2, 3, that is, the variations of the form: (Tµm . . . Tµ1 p, Tµm . . . Tµ1 s, LTµm . . . LTµ1 u) : µ1 , . . . , µm = 0, 1, 2, 3; m = 1, . . . , k

(14.89)

Now (14.84), (14.85) imply through the standard Sobolev inequality on t that: s − s0  L ∞ (t ) ,  p − p0  L ∞ (t ) , max u i  L ∞ (t ) ≤ Cδ0 i

: for all t ∈ [0, η0−1 ]

(14.90)

and: max ∂α s L ∞ (t ) , max ∂α p L ∞ (t ) , max max ∂α u i  L ∞ (t ) ≤ Cδ0 α

α

α

i

: for all t ∈ [0, η0−1 ]

(14.91)

and, taking k ≥ 4, also: max ∂α ∂β s L ∞ (t ) , max ∂α ∂β p L ∞ (t ) , α,β

α,β

max max |∂α ∂β u i  L ∞ (t ) ≤ Cδ0 i

α,β

: for all t ∈ [0, η0−1 ]

(14.92)

where the equations (1.20) are used to express the derivatives with respect to x 0 in terms of the derivatives with respect to the x i : i = 1, 2, 3. The above imply that the rectangular components of the acoustical metric satisfy: ◦

max h µν − h µν  L ∞ (t ) ≤ Cδ0 µ,ν

where

: for all t ∈ [0, η0−1 ]

◦

h µν = gµν + (1 − η02 )δµ0 δν0

(14.93)

(14.94)

are the components of the acoustical metric in the constant state. Also: max ∂α h µν  L ∞ (t ) ≤ Cδ0

: for all t ∈ [0, η0−1 ]

(14.95)

max ∂α ∂β h µν  L ∞ (t ) ≤ Cδ0

: for all t ∈ [0, η0−1 ]

(14.96)

α,µ,ν

and:

α,β,µ,ν

It follows that the acoustical curvature components in rectangular coordinates satisfy: max Rµναβ  L ∞ (t ) ≤ Cδ0

µ,ν,α,β

: for all t ∈ [0, η0−1 ]

(14.97)

906

Chapter 14. Sufficient Conditions on the Initial Data

The estimates (14.90)–(14.96) imply through equations (3.3), (3.4), (3.9), of Chapter 3, that the rectangular components of the second fundamental form of the t relative to h satisfy: max ki j  L ∞ (t ) | ≤ Cδ0

: for all t ∈ [0, η0−1 ]

(14.98)

max ∂m ki j  L ∞ (t ) ≤ Cδ0

: for all t ∈ [0, η0−1 ]

(14.99)

i, j

and:

m,i, j

By (14.86) on η−1 we have: 0

∂i u = −N i ,

Ni =

xi r

(14.100)

in accordance with (13.943), hence from (2.68), (2.55), we obtain, by virtue of the estimates (14.90) at t = η0−1 , κ − 1 L ∞ ( ε0 ) ≤ Cδ0 (14.101) η−1 0

Also, the estimates (14.99) imply: k(Tˆ , Tˆ ) L ∞ ( ε0 ) ≤ Cδ0 t

: for all t ∈ [0, η0−1 ]

(14.102)

(the fact that the induced acoustical metric h on t dominates the Euclidean metric implies that maxi |Tˆ i | ≤ 1). Noting also that by the estimates (14.90)–(14.92) we have: α − η0  L ∞ (t ) , max ∂µ α L ∞ (t ) , max ∂µ ∂ν α L ∞ (t ) ≤ Cδ0 : for all t ∈ [0, η0−1 ] µ

µ,ν

(14.103) integrating the propagation equation (3.78) of Chapter 3, in the form L log κ = −Tˆ i ∂i α + αki j Tˆ i Tˆ j

(14.104)

and taking into account (14.101) yields:  log κ L ∞ ( ε0 ) , L log κ ε0 ≤ Cδ0 t

t

: for all t ∈ [0, η0−1 ]

(14.105)

: for all t ∈ [0, η0−1 ]

(14.106)

and (in view of (14.103)):  log µ L ∞ ( ε0 ) , L log µ ε0 ≤ Cδ0 t

t

Using the estimates (14.98), (14.103), and (14.105), we obtain from the formula (3.64) of Chapter 3 the following estimate for the 1-form ζ : ζ  L ∞ ( ε0 ) ≤ Cδ0 t

: for all t ∈ [0, η0−1 ]

(14.107)

Here the pointwise magnitude of an St,u 1-form such as ζ is defined to be its magnitude ε as a 1-form on t 0 with respect to the Euclidean metric. Thus, in rectangular coordinates:  (ζi )2 |ζ |2 = i

Chapter 14. Sufficient Conditions on the Initial Data

907

Next we express Tˆ i and L i as in (12.460), (12.462): xi + yi (1 − u + η0 t) η0 x i + wi − αy i Li = 1 − u + η0 t

Tˆ i = −

(14.108) (14.109)

where now, from the general expression (2.65): wi =

H σ u 0u i (α − η0 ) + (1 − u + η0 t) 1 + ρH

(14.110)

We then obtain as in Chapter 12 the following propagation equation for the y i : αy i = wi 1 − u + η0 t

(14.111)

wi + p L Tˆ i + q Li 1 − u + η0 t

(14.112)

Ly i + where: wi =

Here p L and q Li are defined by the formula (see (3.159)): L(Tˆ i ) = p L Tˆ i + q Li

(14.113)

where the q Li are the rectangular components of an St,u -tangential vectorfield. Now the formula (3.141) of Chapter 3 for the connection coefficients of the acoustical metric in rectangular coordinates holds only in the isentropic irrotational case, but we may appeal directly to the general expression (3.139). By virtue of the estimate (14.95) we then obtain: (14.114) max !αβν  L ∞ (t ) ≤ Cδ0 : for all t ∈ [0, η0−1 ] α,β,ν

From (3.160), (3.163) we have: p L = −!αβν L α Tˆ β Tˆ ν

(14.115)

while from (3.162), (3.165), (3.166) we have: q Li h i j = −κ −1 ζ j − !αβi L α Tˆ β ij

(14.116)

where  denotes as in all preceding chapters beginning with Chapter 6 the h-orthogonal projection on t to the local tangent plane to St,u : ij = δ ij − Tˆ i h j k Tˆ k and not as in Chapter 1 the g-orthogonal projection in spacetime to the local simultaneous space of the fluid. Using the estimates (14.105), (14.107), (14.114) (and the elementary

908

Chapter 14. Sufficient Conditions on the Initial Data

bound |L i | < 1 of Chapter 6 which holds in general as it expresses the fact that the sound cone is contained within the light cone), we obtain: : for all t ∈ [0, η0−1 ]

 p L  L ∞ ( ε0 ) , max q Li  L ∞ ( ε0 ) ≤ Cδ0 t

t

i

(14.117)

Also, directly from the estimates (14.90) we have: max wi  L ∞ ( ε0 ) ≤ Cδ0

: for all t ∈ [0, η0−1 ]

(14.118)

max wi  L ∞ ( ε0 ) ≤ Cδ0

: for all t ∈ [0, η0−1 ]

(14.119)

t

i

It follows that:

t

i

Now from (2.67) and (14.100), on η−1 we have: 0

x −1 −1 Tˆ i = −κ(h )i j N j = −κ(h )i j r

j

(14.120)

From (14.105) and the estimate (14.93) which implies: max h i j − δi j  L ∞ (t ) , max (h i, j

−1 i j

i, j

) − δi j  L ∞ (t ) ≤ Cδ0

: for all t ∈ [0, η0−1 ]

we obtain, comparing with (14.108) at t = η0−1 , recalling (14.86), max y i  L ∞ ( ε0

η−1 0

i

)

≤ Cδ0

(14.121)

(14.122)

Integrating then the propagation equation (14.111) and using (14.123) and (14.126) yields: (14.123) max y i  L ∞ ( ε0 ) ≤ Cδ0 : for all t ∈ [0, η0−1 ] t

i

Next we consider the propagation equation (3.38) of Chapter 3 for χ in the following form as a propagation equation for χ  = χ − η0 (1 − u + η0 t)−1 h/:   η0 h/ 2η0 χ  = (L log µ) + χ  + χ $ · χ  − α (14.124) L /L χ  + 1 − u + η0 t 1 − u + η0 t Here α denotes the curvature component which is a symmetric 2-covariant St,u tensorfield with rectangular components: αmn = Riαjβ im L α n L β j

ε0 , η0−1

To obtain an estimate for χ  on 

(14.125)

we consider equation (13.956): bi d/a Tˆ i = q/ba

(14.126)

Chapter 14. Sufficient Conditions on the Initial Data

909

Now from (3.224), (3.227), (3.228) we have: q/ba h bc = θac − !i j k ia Tˆ j kc

(14.127)

Recalling that θ = −α −1 χ + k/ we then obtain:   α −1 η0 ba −1  + (h )bc −α −1 χac bi d/a Tˆ i = − + k/ac − !i j k ia Tˆ j kc 1 − u + η0 t

(14.128)

On the other hand on η−1 we have, from (14.120): 0

−1 bi d/a Tˆ i = −κbi d/a ((h )i j N j )   −1 −1 = −κbi d/a N i + N j d/a (h )i j + ((h )i j − δi j )d/a N j

(14.129)

while at each t (see (13.959)): d/a N i = r −1 a  ij j

(14.130)

Now, on η−1 we have, by (14.120), 0

where:

ij −  ij = N j N i

(14.131)

N i = (1 − κ 2 )N i − κ 2 ((h)ik − δik )N k

(14.132)

d/a N i = r −1 (ia − a N j N i )

(14.133)

It follows that on η−1 : 0

j

Substituting in (14.129) we obtain that on η−1 : 0

j bi d/a Tˆ i = −κ r −1 (ba − bi N i a N j )

(14.134)  −1 −1 j +N j bi d/a (h )i j + r −1 ((h )i j − δi j )bi (a − ka N k N k )

We now compare (14.128) and (14.134), recalling that on η−1 u = 2 − r . In view of the 0 estimates (14.98), (14.101), (14.103), (14.114), (14.121) and the estimates: max ∂m h i j  L ∞ (t ) , max ∂m (h

m,i, j

m,i, j

−1 i j

)  L ∞ (t ) ≤ Cδ0

: for all t ∈ [0, η0−1 ] (14.135)

which follow from the estimates (14.95), as well as the bound: max N i  L ∞ ( ε0 ) ≤ Cδ0 i

(14.136)

0

ε0 : η0−1

which follows from (14.98) and (14.121), we conclude that on  χ   L ∞ ( ε0

η−1 0

)

≤ Cδ0

(14.137)

910

Chapter 14. Sufficient Conditions on the Initial Data

Here the pointwise magnitude of a 2-covariant St,u tensorfield such as χ  is defined to ε be its magnitude as a 2-covariant tensorfield on t 0 with respect to the Euclidean metric. Thus, in rectangular coordinates:  |χ  |2 = (χij )2 i, j

Integrating then the propagation equation (14.124) using the estimates (14.97), which imply: (14.138) max αmn  L ∞ (t ) ≤ Cδ0 : for all t ∈ [0, η0−1 ] m,n

as well as the estimate (14.106), and taking into account (14.137), we conclude that: χ   L ∞ ( ε0 ) ≤ Cδ0 t

: for all t ∈ [0, η0−1 ]

(14.139)

Now, according to equation (3.223) of Chapter 3 we have:

and by (3.223) and (3.225):

d/a Tˆ i = p/a Tˆ i + q/ia

(14.140)

p/a = −!i j k ia Tˆ j Tˆ k

(14.141)

The estimates (14.114) imply:  p/ L ∞ ( ε0 ) ≤ Cδ0 t

: for all t ∈ [0, η0−1 ]

(14.142)

Also, in view of the estimate (14.139) we obtain through (14.126), (14.128): : for all t ∈ [0, η0−1 ]

max q/i  L ∞ ( ε0 ) ≤ C t

i

It follows that:

: for all t ∈ [0, η0−1 ]

max d/Tˆ i  L ∞ ( ε0 ) ≤ C t

i

(14.143)

(14.144)

Going now back to the propagation equation (14.104), differentiating the equation tangentially to the St,u and using the estimates (14.99), (14.103), and (14.144), we deduce: L / L d/ log κ L ∞ ( ε0 ) ≤ Cδ0 t

: for all t ∈ [0, η0−1 ]

(14.145)

From (13.943), (2.68), we deduce, using the estimates (14.135): d/ log κ L ∞ ( ε0

η0−1

)

≤ Cδ0

(14.146)

Integrating then the inequality (14.145) and taking into account (14.146), we conclude that: (14.147) d/ log κ L ∞ ( ε0 ) ≤ Cδ0 : for all t ∈ [0, η0−1 ] t

Chapter 14. Sufficient Conditions on the Initial Data

911

It then follows, in view of the estimates (14.103), that also: d/ log µ L ∞ ( ε0 ) ≤ Cδ0 t

: for all t ∈ [0, η0−1 ]

(14.148)

Now the estimate on maxi u i  L ∞ (t ) of (14.90) implies that along each flow line we have: uµ dr (14.149) = 0 ∂µ r ≤ Cδ0 : for all t ∈ [0, η0−1 ] dt u Integrating and taking into account the fact that on 0 we have a constant state outside the unit sphere with center at the origin, it follows that on η−1 the flow is isentropic and 0 irrotational outside a sphere of radius r0 ≤ 1 + Cδ0

(14.150)

with center at the origin. Taking δ0 suitably small, this implies in particular that the flow is ε isentropic and irrotational on  0−1 . Translating in time by −η0−1 and rescaling by a factor η0

of 2 we obtain irrotational isentropic initial data outside the sphere of radius 1 − (1/2)ε0 with center at the origin in 0 which coincide with the initial data of the constant state outside the corresponding unit sphere. Moreover the bounds (14.85) and (14.88) imply that:  ∂i ψα 2 ε0 /2 < Cδ0 (14.151) D[k] = α,i

0

Thus taking k = l + 2 and δ0 = C −1 δ0 for a suitably large positive constant C, the assumptions of Theorem 13.1 are all satisfied with ε0 replaced by ε0 /2. Going now back to the general solution in the time interval [0, η0−1 ], along the integral curves of T on each t , t ∈ [0, η0−1 ], we have:   r dr i i i i i i ˆ = Tr = T N = κT N = κ − +y N (14.152) du 1 − u + η0 t where we have substituted for Tˆ i from (14.108). We can write this in the form:     d r 1 (κ − 1)r i i = − + κy N (14.153) du 1 − u + η0 t (1 − u + η0 t) 1 − u + η0 t By virtue of (14.105), (14.123), and the elementary bound r ≤ 1 + η0 t in the interior ε of C0 , the right-hand side is bounded in t 0 by Cδ0 for all t ∈ [0, η0−1 . It then follows, integrating from St,0 where r = 1 + η0 t, that:     r     1 − u + η t − 1 ≤ Cδ0 u 0 hence also: |r − (1 − u + η0 t)| ≤ Cδ0 u

: for all u ∈ [0, ε0] and all t ∈ [0, η0−1 ]

(14.154)

912

Chapter 14. Sufficient Conditions on the Initial Data

Thus, at each t ∈ [0, η0−1 ] and for each u ∈ [0, ε0 ], the surface St,u is contained in the annular region in t bounded by the spheres of radii 1 − u + η0 t − Cδ0 u and 1 − u + η0 t + Cδ0 u. Given now any first order variation ( p, ˙ s˙ , u) ˙ of the solution ( p, s, u) through solutions of the equations of motion (1.20), we define the 1-form ξ by:   (14.155) ξµ = −n −1 u µ p˙ + (ρ + p)u˙ µ ; u µ = gµν u ν , u˙ µ = gµν u˙ ν √ Note that if β is the 1-form defined by equation (1.44) of Chapter 1, then σ being the enthalpy per particle , in view of the basic thermodynamic relation √˙ σ = n −1 p˙ + θ s˙ where θ is the temperature, we have: ξµ = β˙µ + θ s˙ u µ

(14.156)

˙ We consider, in general, the functions: Therefore in the isentropic case ξ coincides with β. i = L µ ξµ , We have:

i = L µ ξµ

(14.157)

Li = −n −1 (L µ u µ )L p˙ − n −1 (ρ + p)L µ L u˙ µ + B0

(14.158)

where   B0 = n −2 (Ln) (L µ u µ ) p˙ + (ρ + p)L µ u˙ µ − n −1 (Lρ + L p)L µ u˙ µ   (14.159) −n −1 (L L µ ) u µ p˙ + (ρ + p)u˙ µ Note that by virtue of the L ∞ estimates which we have derived above we can estimate:  √ (14.160) B0  L 1 (tu ) ≤ Cδ0 ε0 E˙ u (t) : for all t ∈ [0, η0−1 ] and all u ∈ [0, ε0 ] where:

E˙ u (t) = ˙s 2L 2 ( u ) +  p ˙ 2L 2 ( u ) + t

t



u˙ i 2L 2 ( u )

i

t

(14.161)

In particular, in regard to the coefficient L L µ in the last term on the right in (14.160) we have: µ L L µ = (∇ L L)µ = (D L L)µ − !αβ L α L β and from table 3.117 of Chapter 3 we have h(D L L, X) = −2ζ(X) : for every vector X , therefore:

L L µ = −2(h −1 )µν (ζν + !αβν L α L β )

(14.162)

Chapter 14. Sufficient Conditions on the Initial Data

913

The estimates (14.107) and (14.114) then give: max L L µ  L ∞ ( ε0 ) ≤ Cδ0 µ

t

: for all t ∈ [0, η0−1 ]

(14.163)

Returning to (14.158), we now appeal to the equations of variation (1.23) to express the first two terms on the right. In the following we denote by P the g-orthogonal projection in spacetime to the local simultaneous space of the fluid (to avoid confusion with , the h-orthogonal projection on t to the local tangent plane to the St,u ). From equations (1.23) we have: ∂µ p˙ = Pµν ∂ν p˙ − u µ u ν ∂ν p˙ = −(ρ + p)(u ν ∂ν u˙ µ − η2 u µ ∂ν u˙ ν ) + B1,µ

(14.164)

where B1,µ = −[(ρ˙ + p)u ˙ ν + (ρ + p)u˙ ν ]∂ν u µ − P˙µν ∂ν p +[u˙ ν ∂ν p + q∂ ˙ ν u ν ]u µ

(14.165)

Note that by virtue of the L ∞ estimates (14.91) we can estimate:  √ max B1,µ  L 1 (tu ) ≤ Cδ0 ε0 E˙ u (t) : for all t ∈ [0, η0−1 ] and all u ∈ [0, ε0] µ

(14.166) Now, (14.164) gives: L p˙ = −(ρ + p)(L µ u ν ∂ν u˙ µ − η2 L µ u µ ∂ν u˙ ν ) + L µ B1,µ

(14.167)

Substituting in (14.158) we then obtain:  Li = n −1 (ρ + p) g(L, u)(L µ u ν − η2 g(L, u)(g −1 )µν ) − L µ L ν ∂ν u˙ µ + B2 (14.168) where:

B2 = B0 − n −1 g(L, u)L µ B1,µ

(14.169)

Now according to (1.36) we have: (g −1 )µν = (h −1 )µν + (η−2 − 1)u µ u ν

(14.170)

In view of the constraint (1.22) on u, ˙ that is:

we have:

u µ u˙ µ = 0

(14.171)

u µ ∂ν u˙ µ = −u˙ µ ∂ν u µ

(14.172)

914

Chapter 14. Sufficient Conditions on the Initial Data

hence: u µ u ν ∂ν u˙ µ = −u˙ µ u ν ∂ν u µ = (ρ + p)−1 u˙ µ Pµν ∂ν p = (ρ + p)−1 u˙ ν ∂ν p

(14.173)

where we have used the equations of motion (1.20) and the fact that by the constraint (14.171), Pµν u˙ µ = u˙ ν . In view of (14.170) and (14.173) we have: (g −1 )µν ∂ν u˙ µ = (h −1 )µν ∂ν u˙ µ + (η−2 − 1)(ρ + p)−1 u˙ ν ∂ν p

(14.174)

Substituting this on the right in (14.168) we obtain:  Li = n −1 (ρ + p) g(L, u)(L µ u ν − η2 g(L, u)(h −1 )µν ) − L µ L ν ∂ν u˙ µ + B3 (14.175) where:

B3 = B2 − n −1 (1 − η2 )g(L, u)g(L, u)u˙ µ ∂µ p

Finally, expressing

(h −1 )µν

(14.176)

in the form:

(h −1 )µν = −

1 (L µ L ν + L µ L ν ) + (h/−1 )µν 2µ

(14.177)

where (h /−1 )µν are the rectangular components of the symmetric 2-contravariant St,u tensorfield h /−1 : /h−1 = (h /−1 ) AB X A ⊗ X B

: expanded in the X 1 , X 2 frame

(14.178)

(see (3.116)), we obtain:  Li = n −1 (ρ + p) −η2 g(L, u)g(L, u)(h/−1 )µν + γ µν ∂ν u˙ µ + B3

(14.179)

where γ is a 2-contravariant tensorfield in spacetime with rectangular components:   η2 g(L, u) µ ν µν µ ν µ ν (L L + L L ) − L µ L ν (14.180) γ = g(L, u) L u + 2µ Note that B3 can be estimated in the same way as B0 , that is, we have:  √ B3  L 1 (tu ) ≤ Cδ0 ε0 E˙ u (t) : for all t ∈ [0, η0−1 ] and all u ∈ [0, ε0 ]

(14.181)

Let us expand the tensorfield γ in the null frame L, L, X 1 , X 2 of the acoustical metric. Expanding in the same frame the fluid velocity u we have: u = u L L + u L L + u/ A X A ,

u/ = u

(14.182)

Substituting in (14.180) we then obtain the following expansion for γ : γ = γ LL L ⊗ L + γ LL L ⊗ L + γ LL L ⊗ L + γ LAL ⊗ X A

(14.183)

Chapter 14. Sufficient Conditions on the Initial Data

915

where the coefficients are given by: γ L L = g(L, u)u L  γ

η2 g(L, u) = g(L, u) u + 2µ

LL



L

γ LL =

η2 g(L, u)g(L, u) −1 2µ

γ L A = g(L, u)u/ A

(14.184)

Now, we have: uL = −

1 h(L, u), 2µ

uL = −

1 h(L, u) 2µ

and since: h µν = gµν + (1 − η2 )u µ u ν

(14.185)

(see (1.37)) we also have: h(L, u) = η2 g(L, u),

h(L, u) = η2 g(L, u)

(14.186)

Therefore the first two coefficients of (14.184) become: γ LL = −

η2 (g(L, u))2 , 2µ

γ LL = 0

(14.187)

It is crucial that the coefficient γ L L vanishes. For by virtue of this the expression γ µν ∂ν u˙ µ on the right in (14.179) becomes: γ µν ∂ν u˙ µ = U µ L u˙ µ + g(L, u)L µ u/ · d/u˙ µ where U µ are the rectangular components of the vectorfield:   η2 (g(L, u))2 η2 g(L, u)g(L, u) U =− L+ −1 L 2µ 2µ

(14.188)

(14.189)

As there are no derivatives of u˙ µ with respect to L in (14.188), in multiplying by a weight function and integrating on Cus = {x ∈ Cu : x 0 ∈ [0, s]}, we may integrate by parts, thereby removing the derivatives of u˙ µ . In fact the contribution of the second term on the right in (14.188) may be integrated by parts on each St,u . There is one more adjustment to be made however since the vectorfield U does not vanish in the constant state. In fact, in the constant state u 0 = 1, u i = 0 : i = 1, 2, 3, κ = 1, µ = α = η0 , hence: g(L, u) = −L 0 = −1,

g(L, u) = −η0−1

: in the constant state

(14.190)

and we obtain: 1 U = − (η0−1 L + L) = g(L, u)u 2

: in the constant state

(14.191)

916

Chapter 14. Sufficient Conditions on the Initial Data

To overcome this difficulty we note that by virtue of the constraint (14.171) we have: u ν L u˙ ν = −u˙ ν Lu ν hence:

U µ L u˙ µ = U µ Pµν L u˙ ν + U µ u µ u˙ ν Lu ν

(14.192)

The contribution of the second term on the right in (14.192) can be absorbed in B3 . We thus obtain the following expression for Li (compare with (14.179)):  Li = n −1 (ρ + p) −η2 g(L,u)g(L,u)(h/−1 )µν ∂ν u˙ µ +U µ L u˙ µ + g(L,u)L µ u/ · d/u˙ µ + B4 (14.193) where: and:

U  = PU

(14.194)

B4 = B3 + n −1 (ρ + p)g(U, u)u˙ µ Lu µ

(14.195)

can be estimated in the same way as B2 , that is, we have:  √ B4  L 1 (tu ) ≤ Cδ0 ε0 E˙ u (t) : for all t ∈ [0, η0−1 ] and all u ∈ [0, ε0 ]

(14.196)

Now from (14.189), (14.194) we have:

hence:

g(U, u) = −g(L, u)

(14.197)

U  = U − g(L, u)u

(14.198)

which by (14.191) vanishes in the constant state. Consider now the integral of the right-hand side of (14.193) on St,u . The contribution to this integral of the third term in square brackets is, integrating by parts on St,u , n −1 (ρ + p)g(L, u)L µ u / · d/u˙ µ dµh/ = −N1 (t, u) − N2 (t, u) − N3 (t, u) (14.199) St,u

where: N1 (t, u) =

St,u



St,u

N2 (t, u) = N3 (t, u) = St,u

(L µ u˙ µ )u/ · d/(n −1 (ρ + p)g(L, u))dµh/ n −1 (ρ + p)g(L, u)u˙ µ u/ · d/ L µ dµh/ n −1 (ρ + p)g(L, u)(L µ u˙ µ )div / u/dµh/

(14.200)

Chapter 14. Sufficient Conditions on the Initial Data

and we can estimate: u u u     |N1 (t, u )|du , |N2 (t, u )|du , |N3 (t, u  )|du  0 0 0  √ ≤ Cδ0 ε0 E˙ u (t) : for all t ∈ [0, η0−1 ] and all u ∈ [0, ε0 ] Noting that for an arbitrary function f we have (see (13.314), (13.315)): ∂ f dµh/ = (L f + trχ f )dµh/ ∂t St,u St,u

917

(14.201)

(14.202)

the contribution of the second term in square brackets to the integral of the right-hand side of (14.193) on St,u is: ∂ n −1 (ρ + p)U µ L u˙ µ dµh/ = K (t, u) − L 1 (t, u) − L 2 (t, u) (14.203) ∂t St,u where:

K (t, u) =

St,u



St,u

L 1 (t, u) = L 2 (t, u) = St,u

n −1 (ρ + p)U µ u˙ µ dµh/ u˙ µ L(n −1 (ρ + p)U µ )dµh/ trχn −1 (ρ + p)U µ u˙ µ dµh/

and we can estimate: u u u |K (t, u  )|du  , |L 1 (t, u  )|du  , |L 2 (t, u  )|du  0 0 0  √ ≤ Cδ0 ε0 E˙ u (t) : for all t ∈ [0, η0−1 ] and all u ∈ [0, ε0 ]

(14.204)

(14.205)

We turn to the contribution of the first term in square brackets on the right-hand side of (14.193): (14.206) −η2 g(L, u)g(L, u)(h/−1 )µν ∂ν u˙ µ Considering u˙ µ to be the rectangular components of a 1-form in spacetime, which we denote by u˙ in the following, we write: λ u˙ λ ∂ν u˙ µ = Dν u˙ µ + !νµ

(14.207)

Substituting the expansion (14.178) for h/−1 we have: (h /−1 )µν Dν u˙ µ = (h/−1 ) AB X A · D X B u˙

(14.208)

The 1-form u˙ can be expanded as: u˙ = adt + bdu + v

(14.209)

918

Chapter 14. Sufficient Conditions on the Initial Data

where v is an St,u 1-form. Evaluating this on L and T yields: a = L · u˙ = L µ · u µ , and: vµ = νµ u˙ ν ,

νµ = δµν +

b = T · u˙ = T µ u˙ µ

(14.210)

1 h µλ (L ν L λ + L ν L λ ) 2µ

(14.211)

In view of the expansion (14.209) we have: D X B u˙ = a D X B dt + b D X B du + D X B v + (X B a)dt + (X B b)du

(14.212)

/ X B v, we then obtain: Noting that X A · dt = X A · du = 0, while X A · D X B v = X A · D X A · D X B u˙ = a X A · D X B dt + b X A · D X B du + X A · D /XBv

(14.213)

Now according to equations (2.40) and (2.10), (2.11), of Chapter 2: ∂µ t = α −2 h µν B ν ∂µ u = −µ−1 h µν L ν

(14.214)

It follows that: X A · D X B dt = −h(X A , D X B (α −2 B)) = −α −2 h(X A , D X B B) = −α −2 k/ AB

(14.215)

X A · D X B du = −h(X A , D X B (µ−1 L)) = −µ−1 h(X A , D X B L) = −µ−1 χ AB (14.216) Substituting in (14.213) and substituting also the coefficients a and b from (14.210) we then obtain: /XBv − X A · D X B u˙ = X A · D

1 1 (L µ u˙ µ )χ AB − (L µ u˙ µ )χ AB 2µ 2µ

(14.217)

and substituting in turn in (14.208) yields: (h /−1 )µν Dν u˙ µ = div / v−

1 1 (L µ u˙ µ )trχ − (L µ u˙ µ )trχ 2µ 2µ

(14.218)

The contribution of the first term on the right in (14.218), through (14.207) and (14.206) to the integral of (14.193) on St,u is, integrating by parts, − n −1 (ρ + p)η2 g(L, u)g(L, u)div / vdµh/ = M(t, u) (14.219) St,u

where: M(t, u) = St,u

v $ · d/(n −1 (ρ + p)η2 g(L, u)g(L, u))dµh/

(14.220)

Chapter 14. Sufficient Conditions on the Initial Data

919

with v $ the St,u -tangential vectorfield: v $ = v · h/−1 , and we can estimate: u  √ |M(t, u  )|du  ≤ Cδ0 ε0 E˙ u (t) 0

: for all t ∈ [0, η0−1 ] and all u ∈ [0, ε0] (14.221)

Moreover, the second term in (14.207) can be absorbed in B4 , defining: λ B5 = B4 − n −1 (ρ + p)η2 g(L, u)g(L, u)(h/−1 )µν !νµ u˙ λ

(14.222)

Then in view of the estimates (14.114), B5 can be estimated in the same way as B4 , that is, we have:  √ B5  L 1 (tu ) ≤ Cδ0 ε0 E˙ u (t) : for all t ∈ [0, η0−1 ] and all u ∈ [0, ε0 ] (14.223) We have thus considered all the contributions to (14.193), with the exception of the contribution, through (14.206) of the last two terms on the right in (14.218). The contribution which remains to be investigated is:  η2 (14.224) n −1 (ρ + p) g(L, u)g(L, u) (L µ u˙ µ )trχ + (L µ u˙ µ )trχ 2µ  η2 = − g(L, u)g(L, u) trχi + trχi + n −1 (g(L, u)trχ + g(L, u)trχ) p˙ 2µ where we have substituted for L µ u˙ µ and L µ u˙ µ in terms of i and i from the definitions (14.157). Note that since in the constant state we have: 2η0 2 , trχ = − : in the constant state (14.225) r r the coefficient of p˙ in (14.224) vanishes in the constant state. We now consider the variation ( p, ˙ s˙ , u) ˙ to be the variation generated by the time translations (see Chapter 1), so that: trχ =

u˙µ = ∂0 u µ

(14.226)

τ  = (1 − u + η0 t)i − n −1 0 ( p − p0 )

(14.227)

s˙ = ∂0 s, We then define:

p˙ = ∂0 p,

where i corresponds to this particular variation, and n 0 and p0 are the positive constants corresponding to the surrounding constant state. We then have:  η2  g(L, u)g(L, u) trχi + trχi Lτ = (1 − u + η0 t) Li + 2µ  −1 + n (g(L, u)trχ + g(L, u)trχ) p˙ + (1 − u + η0 t)R

(14.228)

920

Chapter 14. Sufficient Conditions on the Initial Data

where the remainder R is given by:   η0 η2 R= − g(L, u)g(L, u)trχ i 1 − u + η0 t 2µ η2 g(L, u)g(L, u)n −1 (g(L, u)trχ + g(L, u)trχ) p˙ 2µ   2 n −1 0 −1 η 2 g(L, u)(g(L, u)) − + n Lp 2µ 1 − u + η0 t

−

−

η2 g(L, u)g(L, u)trχn −1 (ni + g(L, u)L p) 2µ

(14.229)

Now, in view of (14.224) the first term on the right in (14.228) is precisely what we have already estimated, while the coefficients of the first three terms on the right in (14.229) vanish in the constant state. What therefore remains to be considered is, in regard to the last term, ni +g(L, u)L p. Recalling that the variation we are considering is that generated by time translations we have, from (14.157), i = −n −1 (g(L, u)∂0 p + (ρ + p)L µ ∂0 u µ ) Writing ∂0 p = L p − L i ∂i p, we then have: ni + g(L, u)L p = g(L, u)L i ∂i p − (ρ + p)L µ ∂0 u µ

(14.230)

In regard to the second term on the right in (14.230), we can write: ∂0 u µ = u ν ∂ν u µ − (u 0 − 1)∂0 u µ − u i ∂i u µ and we can substitute from the equations of motion (1.20) to obtain: −(ρ + p)L µ u ν ∂ν u µ = Pµν L µ ∂ν p = L ν ∂ν p + g(L, u)u ν ∂ν p = (1 + g(L, u)u 0 )∂0 p + g(L, u)u i ∂i p + L i ∂i p

(14.231)

Substituting then in (14.230) we obtain: ni + g(L, u)L p = (1 + g(L, u)u 0 )∂0 p + (1 + g(L, u))L i ∂i p + g(L, u)u i ∂i p (14.232) In view of (14.190) the coefficients of ∂0 p and ∂i p vanish in the constant state. It then follows that: √  u R L 1 (tu ) ≤ Cδ0 ε0 E [1] (t) : for all t ∈ [0, η0−1 ] and all u ∈ [0, ε0 ] (14.233) We now follow the proof of Theorem 14.1, considering the mean value of τ  on St,u . Here however, the mean value is meant with respect to the measure dµh/ . Thus for

Chapter 14. Sufficient Conditions on the Initial Data

921

an arbitrary function f we presently denote by f (t, u) the mean value of f on St,u with respect to the measure dµh/ . We then have, in place of (14.20), ∂f (t, u) = L f + (trχ − trχ)( f − f ) ∂t Thus, taking f = τ  we have:

(14.234)

∂τ  (t, u) = Lτ  + (trχ − trχ)(τ  − τ  ) ∂t

(14.235)

Integrating this with respect to t on [0, η0−1 we obtain: η−1  0 −1   τ (η0 , u) = τ (0, u) + Lτ  + (trχ − trχ)(τ  − τ  ) (t, u)dt

(14.236)

0

We then replace u by u  ∈ [0, u], multiply by Area(0, u  ), and integrate with respect to u  on [0, u] to obtain: u τ  (η0−1 , u  )Area(0, u  )du  (14.237) 0   u η−1  0 τ  dµh/ du  + Area(0, u  ) Lτ  + (trχ − trχ)(τ  − τ  ) (t, u  )dt du  = 0u

0

0

In estimating the contribution of the second term in square brackets in (14.193) to the second integral on the right in (14.237), we note that this term contributes to:  −1  u η0 Area(0, u  ) Lτ  (t, u  )dt du  0

0



u

= 0





η0−1

Area(0, u )



(Area(t, u ))

−1



0

the corresponding contribution to

 

St,u 

St,u



Lτ dµh/ dt du 

(14.238)

Lτ  dµh/

being given by (14.203). Thus, the contribution in question to (14.238) is:  −1    u η0 ∂ K Area(0, u  ) (Area(t, u  ))−1 − L 1 − L 2 (t, u  )du  dt ∂t 0 0 u u = Area(0, u  )(Area(η0−1 , u  ))−1 K (η0−1 , u  )du  − K (0, u  )du  0 0 u  − Area(0, u ) (14.239) 0    −1  η0  −1  ∂(Area(t, u )) −1   + Area(t, u) (L 1 + L 2 ) (t, u )du dt K (t, u ) × ∂t 0

922

Chapter 14. Sufficient Conditions on the Initial Data

In view of the estimates (14.205) we then obtain that this is bounded in absolute value by: ) √ √  u Cδ0 ε0 sup E˙ u (t) ≤ C  δ0 ε0 E [1] (0) (14.240) t ∈[0,η0−1 ]

u since we have E˙ u (t) ≤ C E [1] (t) and by (14.88) with m = 1:

sup t ∈[0,η0−1 ]

u u E [1] (t) ≤ C E [1] (0)

The contributions of all the other terms to the second integral on the right in (14.237) are readily estimated using the estimates which we have already derived. We then obtain that √  u  the integral in question is bounded in absolute value by Cδ0 ε0 E [1] (0). It then follows from (14.237) that:    u  √  u   τ  (η0−1 , u  )Area(0, u  )du  − τ  dµh/ du   ≤ Cδ0 ε0 E [1] (0) (14.241)   0  u 0

ε0 η0−1

Now, on 

the flow is isentropic and irrotational. Let us then compare τ  with τ as

defined previously (see (14.9)) in the irrotational isentropic case. From (14.156), (14.157), we have, recalling that in the irrotational isentropic case β˙µ = ∂µ φ˙ (see (1.55)), i = Lψ0 , Recalling also that ψ0 =

i = Lψ0

ε0 η0−1

: on 

(14.242)

√ 0 √ ε σ u (see (1.44)) and k = σ0 , we conclude that on  0−1 : η0

τ − τ =

√ 0 √ σ u − σ0 − n −1 0 ( p − p0 )

(14.243)

√ Now in the√isentropic case the basic thermodynamic relation d σ = n −1 d p + θ ds reduces to d σ = n −1 d p. Thus, setting: pλ = (1 − λ) p0 + λp : for λ ∈ [0, 1] we obtain:

√ √ σ = σ ( p, s0 ) = σ0 + n −1 0 ( p − p0 ) + F( p; p0 , s0 )( p − p0 ) where



1

F( p; p0, s0 ) = 0

 n −1 ( pλ, s0 ) − n −1 dλ 0

(14.244)

(14.245)

(14.246)

The estimates (14.90) imply: F L ∞ (

η−1 0

)

≤ Cδ0 ,

u 0 − 1 L ∞ (

η−1 0

)

≤ Cδ02

(14.247)

Chapter 14. Sufficient Conditions on the Initial Data

923

It then follows through (14.243) that: τ  − τ  L 1 ( u η

−1 )

 √  u √ u ≤ C ε0 E [1] (η0−1 ) ≤ C  ε0 E [1] (0)

(14.248)

0

This implies that:  u  −1      τ (η0 , u )Area(0, u )du −  0

u 0

 

τ (η0−1 , u  )Area(0, u  )du  

√  u ≤ C ε0 E [1] (0) (14.249)

the mean values on St,u being with respect to dµh/ for τ  , with respect to dµh/˜ for τ . From this point we proceed as in the proof of Theorem 14.1, except that after replacing u by u  ∈ [0, u] in (14.22), where the t interval is now [η0−1 , s], we multiply by Area(0, u  ) and integrate with respect to u  ∈ [0, u] to obtain, in place of (14.23), u τ (s, u  )Area(0, u  )du  0 u = τ (η0−1 , u  )Area(0, u  )du  0   u  s   + Area(0, u ) ω + 2(ν − ν)(τ − τ ) (t, u )dt du  (14.250) η0−1

0

Denoting by I  (s, u) the second integral on the right, we have: s |I (s, u)| ≤ C J (t, u)dt η0−1

(14.251)

where J is the integral (14.26). This is identical in form to (14.25) and the remainder of the argument proceeds exactly as in the proof of Theorem 14.1. Combining with (14.241) and (14.249) we are led to the conclusion that there is a constant C such that with Q  (u) = τ  dµh/ du  (14.252) 0u

if  > 0 and: or if  < 0 and:

√  u (0) < 0 : for some u ∈ (0, ε0 ] Q  (u) ≤ −Cδ0 ε0 E [1]  √ u (0) > 0 : for some u ∈ (0, ε ] Q  (u) ≥ Cδ0 ε0 E [1] (14.253) 0

then t∗ε0 is finite, in fact satisfies the upper bound: log(1 + t∗ε0 ) ≤ (recall that δ0 = C −1 δ0 ).

C u k 3 |||Q  (u)|

(14.254)

924

Chapter 14. Sufficient Conditions on the Initial Data

Following again the argument of the proof of Theorem 14.1, we compare Q  (u) with the integral τ 0 d 3 x (14.255) 0u

where: and: We show that:

τ 0 = ri 0 − n −1 0 ( p − p0 )

(14.256)

−1 i i 0 = n −1 0 (η0 ∂0 p − (ρ0 + p0 )N ∂0 u i )

(14.257)

         u (0) ≤ C  δ  D u τ 0 d 3 x  ≤ Cδ0 E [1]  Q (u) − 0 [1]   u

(14.258)

0

Next, we replace ∂0 p by −η02 (ρ0 + p0 )∂i u i and ∂0 u i by −(ρ0 + p0 )−1 ∂i p. In view of the  u equations of motion (1.20) the difference of the integrals is again bounded by C  δ0 D[1] . ◦ r (u)

Finally, we replace the integral over 0u by the integral over the annular region 0m bounded by the sphere of radius rm (u) = minS0,u r and the unit sphere, both centered at the origin. Since S0,u is contained in the annular region bounded by the spheres of radii rm (u) and r M (u) centered at the origin, where r M (u) = maxS0,u , while from (14.154), r M (u) − rm (u) ≤ Cuδ0 , it follows that the difference of the integral over 0u from the 9 ◦ ◦  rm (u) rm (u)  integral over 0 is bounded by C δ0 u D[1] , where: ◦

r D[1] (0) = s − s0 2

◦ H1 (0r )

+  p − p0 2

◦ H1 (0r )

+

 i

u i 2

◦

H1 (0r )

We are then left with the integral Q 0 (rm (u)), where:   −1 i −1 i 3 Q 0 (r ) = ◦ r −η0 n −1 0 (ρ0 + p0 )∂i u − n 0 N ∂ p − n 0 ( p − p0 ) d x 0r

(14.259)

(14.260)

◦

Integrating by parts and denoting by S r , the sphere of radius r with center at the origin, ◦

the inner boundary of the region 0r , we then obtain:   −1 N dS Q 0 (r ) = ◦ r n −1 0 ( p − p0 ) + η0 n 0 (ρ0 + p0 )u Sr   −1 N d3x + ◦ 2n −1 ( p − p ) + η n (ρ + p )u 0 0 0 0 0 0 0r

◦

(14.261)

where d S is the area element on the sphere Sr induced by the Euclidean metric on 0 .

Chapter 14. Sufficient Conditions on the Initial Data

925

We have thus arrived at the following general theorem. Theorem 14.2 Let ( p, s, u µ : µ = 0, 1, 2, 3) be initial data on 0 for the equations of motion (1.20) such that the data coincide with those of a constant state ( p0, s0 , u 0 = 1, u i = 0 : i = 1, 2, 3) outside the unit sphere with center at the origin. Let l and δ0 be as in Theorem 13.1, and let the initial data belong to the Sobolev space Hl+2 (0 ). Then there is a suitably large constant C such that for all δ0 ∈ (0, δ 0 ] the following hold, provided that:

D[l+2] < C −1 δ0 where: D[l+2] = s − s0 2Hl+2 (0 ) +  p − p0 2Hl+2 (0 ) +



u i 2Hl+2 (0 )

i

(i) With η0 the sound speed in the surrounding constant state, the solution exists in the time interval [0, η0−1 ], belongs at each t ∈ [0, η0−1 ] to the Sobolev space Hl+2 (t ) and, denoting by:  u i 2Hm (t ) E [m] (t) = s − s0 2Hm (t ) +  p − p0 2Hm (t ) + i m    k 2 k 2 + (∂0 )k u i 2Hm−k (t ) (∂0 ) s Hm−k (t ) + (∂0 ) p Hm−k (t ) + k=1

i

we have, for each m = 0, . . . , l + 2: sup t ∈[0,η0−1 ]

E [m] (t) ≤ C E [m] (0) ≤ C  D[m]

where: D[m] = s − s0 2Hm (0 ) +  p − p0 2Hm (0 ) +



u i 2Hm (0 )

i

The solution induces irrotational isentropic initial data in the annular region in η−1 0 bounded by the sphere of radius 3/2 and that of radius 2, both centered at the origin and Theorem 13.1 applies to describe the future evolution in the domain of dependence of this annular region. ◦

◦

(ii) Let us denote by 0r the annular region in 0 bounded by Sr , sphere of radius r and the unit sphere, both centered at the origin. Let us also denote: ◦  r D[1] = s − s0 2 ◦ +  p − p0 2 ◦ + u i 2 ◦ H1 (0r )

and: Q 0 (r ) =

◦

Sr



+

H1 (0r )

i

H1 (0r )

  −1 N r n −1 ( p − p ) + η n (ρ + p )u dS 0 0 0 0 0 0 ◦

0r

  −1 N 2n −1 d3x ( p − p ) + η n (ρ + p )u 0 0 0 0 0 0

926

Chapter 14. Sufficient Conditions on the Initial Data

Moreover, we denote by k and  the constants: k=

√

 σ0 ,

=

∂H ∂σ

 s=s0

√ corresponding to the surrounding constant state, where σ = n −1 (ρ + p) is the enthalpy per particle and H = (1 − η2 )/σ where η is the sound speed (η−2 = (∂ρ/∂ p)s ). Then there is a positive constant C such that: 9  ◦



√  r < 0 : for some r ∈ [2/3, 1) δ0 + 1 − r D[1] if  > 0 and: Q 0 (r ) ≤ −C δ0 9  ◦



√  r > 0 : for some r ∈ [2/3, 1) δ0 + 1 − r D[1] or if  < 0 and: Q 0 (r ) ≥ C δ0 then the maximal time of existence t∗ (r ) which corresponds to the domain of the maximal ◦

solution associated to the restriction of the initial data to the exterior of S r is finite, in fact satisfies the upper bound: log(1 + t∗ (r )) ≤

C  (1 − r ) k 3 |||Q 0 (r )|

Chapter 15

The Nature of the Singular Hypersurface. The Invariant Curves. The Trichotomy Theorem. The Structure of the Boundary of the Domain of the Maximal Solution In the present chapter we shall take ε0 to be a variable with range (0, 1/2], in agreement with the setup of Theorem 13.1. To emphasize that ε0 is now taken to be a variable, we denote it simply by ε. Let us denote as in Chapter 2 by t∗ (u) the greatest lower bound of the generators of Cu in the domain of the maximal solution , and by t∗ε the greatest lower bound of t∗ (u) for u ∈ [0, ε]. Evidently t∗ε is a non-increasing function of ε. Then Theorem 13.1 applies for each ε ∈ (0, ε0 ), where we now denote by ε0 the maximal real number in the interval (0, 1/2] such that the smallness condition: √

C ε0 D[l+2] (ε0 ) ≤ δ0 ,

D[l+2] (ε0 ) =

 α,i

∂i ψα 2

ε

Hl+1 (00 )

holds. For, given any such ε we can find a δ0 ∈ (0, δ 0 ] such that the smallness condition of Theorem 13.1 holds with ε0 in the role of ε, that is, we have: √

C ε D[l+2] (ε) < δ0 Now according to Theorem 13.1, for each ε ∈ (0, ε0 ), the solution, which exists in the classical sense in Wεt∗ε \ tε∗ε extends smoothly in acoustical coordinates to tε∗ε , but there is a non-empty set of points K ε ⊂ tε∗ε where µ vanishes, being positive on the complement tε∗ε . Thus the set K ε is also the set of minima of the function µ on tε∗ε . If q is a point of K ε , then either q is an interior minimum of µ on t∗ε , in which case

928

Chapter 15. The Nature of the Singular Hypersurface

we have:  µ(q) = 0,

     ∂µ ∂µ ∂ 2µ (q) ≥ 0 (q) = 0, (q) = 0 : A = 1, 2, ∂u ∂ϑ A ∂u 2

: at an interior zero

(15.1)

and we shall assume that the non-degeneracy condition   ∂ 2µ (q) > 0 : at an interior zero ∂u 2

(15.2)

holds, or q is a boundary minimum, necessarily on St∗ε ,ε (for on the other boundary of ε , namely on St ,0 , we have µ = η0 ), in which case we have: t∗ε ∗ε     ∂µ ∂µ µ(q) = 0, (q) ≤ 0, (q) = 0 : A = 1, 2 ∂u ∂ϑ A : at a boundary zero (15.3) and we shall assume the non-degeneracy condition   ∂µ (q) < 0 : at a boundary zero ∂u

(15.4)

holds. Let now u m (ε) be the minimal value of u ∈ (0, ε) for which there is an interior zero of µ on t∗ε . Then if we replace ε by ε ∈ (0, ε], as we decrease ε , it follows directly from the definitions that t∗ε stays constant up to the point where ε reaches the value u m (ε), that is, we have: t∗ε = t∗ε : for all ε ∈ [u m (ε), ε]

(15.5) 

Thereafter t∗ε increases, µ being everywhere positive on tε∗u m (ε) , and for a while K ε consists only of boundary zeros. This situation either persists for all ε ∈ (0, u m (ε)) or, as we decrease ε there is a first ε1 ∈ (0, u m (ε)) at which one or more interior zeros of µ appear on tε∗ε1 , from which point the preceding repeat with ε1 in the role of ε. 1 In the arguments of the present chapter, the following fact, contained in conclusion (iv) of Theorem 13.1, plays an important role. Namely, that for every ε ∈ (0, ε0 ) the following upper bound for Lµ holds in the region Uε ⊂ Wεt∗ε where µ < η0 /4: Lµ ≤ −C −1 (1 + t)−1 [1 + log(1 + t)]−1 : in Uε

(15.6)

The precise form of this bound is not important in the present chapter, only the fact that, for finite t, Lµ is bounded from above by a negative constant in the region where µ < η0 /4. Let us now denote:  Wεt∗ε (15.7) Vε0 = ε∈(0,ε0 )

Chapter 15. The Nature of the Singular Hypersurface

929

This is the domain covered by Theorem 13.1, applied in the above manner (we are disregarding here the exterior domain where the constant state holds). This domain is contained in the closure of Wε0 (see Chapter 2), the domain of the maximal solution, and the set: Jε0 =



K ε ⊂ Vε0

(15.8)

ε∈(0,ε0 )

of zeros of µ in Vε0 is part of the singular boundary of Wε0 . Later in this chapter we shall show how to apply Theorem 13.1 in a more general manner and also how to suitably extend this theorem so that a larger part, if not all, of the closure of Wε0 is covered. Our purpose in the following is to describe the acoustical spacetime structure in a neighborhood of a point of the singular boundary of Wε0 . Consider a value ε ∈ (0, ε0 ) for which there is an interior zero of µ on tε∗ε , that is, there is a u 0 ∈ (0, ε) and a zero of µ belonging to St∗ε ,u 0 . By the non-degeneracy condition (15.2) the minimum of ∂ 2 µ/∂u 2 over all zeros of µ on St∗ε ,u 0 is positive. It follows that there are u 1 ∈ [0, u 0 ), u 2 ∈ (u 0 , ε] such that in the annular region:  St∗ε ,u (15.9) u∈(u 1 ,u 2 )

tε∗ε

in there are no zeros of µ except on St∗ε ,u 0 itself. If u 0 = u m (ε), there are no zeros of µ on St∗ε ,u for any u ∈ [0, u 0 ) and we can take u 1 = 0. According to the discussion above, a boundary zero of µ on tε∗ε belongs to the surface St∗ε ,ε and by the non-degeneracy condition (15.4) there is a u 1 < ε such that there are no zeros of µ on St∗ε ,u for any u ∈ (u 1 , ε). Moreover, there is a u 0 > ε such that for  ε ∈ [ε, u 0 ] there is a corresponding boundary zero of µ on tε∗ε belonging to the surface St∗ε ,ε and no interior zeros, and there is a u 2 > u 0 such that for ε ∈ (u 0 , u 2 ] we have t∗ε = t∗u 0 and on t∗ε we have an interior zero of µ on St∗ε ,u 0 and no zeros of µ on St∗ε ,u , for any u ∈ [0, ε ), u = u 0 . In this case we consider the annular region  u∈(u 1 ,u 2 )

St∗u 2 ,u

(15.10)

in tu∗u2 . This is analogous to (15.9) with u 2 = ε. 2 Consider the manifold: Mε0 = [0, ∞) × [0, ε0 ) × S 2

(15.11)

of all possible (t, u, ϑ) values with u ∈ [0, ε0). Then the domain Vε0 in Minkowski spacetime corresponds to the domain Vε0 = {(t, u, ϑ) ∈ Mε0 : t ≤ t∗u }

(15.12)

in Mε0 and the annular region (15.9) corresponds to the annular region: {t∗ε } × (u 1 , u 2 ) × S 2

(15.13)

930

Chapter 15. The Nature of the Singular Hypersurface

in {t∗ε } × [0, ε) × S 2 . The mapping (t, u, ϑ) ∈ Vε0 → x(t, u, ϑ) ∈ Vε0 , x = (x α : α = 0, 1, 2, 3)

(15.14)

where the x α are rectangular coordinates in Minkowski spacetime, is a homeomorphism of Vε0 onto Vε0 , which is locally also a diffeomorphism except at the points of the subset Jε0 , the image of which is the subset Jε0 of Vε0 , where the function µ vanishes. On the domain (15.12) we have the acoustical metric h, expressed by (2.37) of Chapter 2 (see also (2.48)) in the form: h = −2µdtdu + α −2 µ2 du 2 + h/ AB (dϑ A +  A du)(dϑ B +  B du)

(15.15)

Recall that α(t, u, ϑ) is a positive function which is less than 1 and that at each (t, u), / AB (t, u, ϑ) are the components of a positive definite metric on S 2 . According to Theh orem 13.1 the above acoustical components of h are smooth functions of the acoustical coordinates on the domain Vε0 , including the boundary of this domain. Moreover, the properties of α and h / AB just mentioned hold on the boundary of Vε0 as well. However on the subset Jε0 of this boundary the function µ vanishes, hence, in view of the fact that

√ −deth = µ deth/ (15.16) the metric (15.15) degenerates on Jε0 . We now consider a smooth extension of the acoustical metric h to t > t∗ (u). The extension is to satisfy the following two conditions: (i) The function α remains positive and less than 1, and the metric h/ remains a positive definite metric on S 2 . (ii) The function µ is extended in such a way that ∂µ/∂t is bounded from above by a negative constant where µ < η0 /4. Then for each (u, ϑ) ∈ (0, ε0 ) × S 2 the following alternative holds: Either there is a first t∗ (u, ϑ) ≥ t∗ (u) where µ vanishes, or µ is positive for all t ≥ t∗ (u). Consider the subset D˜ ⊂ (0, ε0 ) × S 2 consisting of those points where the first alternative holds. Then by property (ii) of the extension D˜ is an open set. Also, noting that in the case of an interior zero of µ on tε∗ε belonging to St∗ε ,u 0 we have t∗ε = t∗ (u 0 ), there is a ϑ0 ∈ S 2 such ˜ In the case of a boundary zero on that µ(t∗ (u 0 ), u 0 , ϑ0 ) = 0 . Therefore (u 0 , ϑ0 ) ∈ D. tε∗ε , for each ε ∈ [ε, u 0 ] there is a ϑ∗ (ε ) ∈ S 2 such that µ(t∗ (ε ), ε , ϑ∗ (ε )) = 0 (as t∗ε = t∗ (ε )). Therefore in this case D˜ contains the curve

Consider:

{(ε , ϑ∗ (ε )) : ε ∈ [u 0 , ε]}

(15.17)

˜ H˜ = {(t∗ (u, ϑ), u, ϑ) : (u, ϑ) ∈ D}

(15.18)

This is the zero level set of the function µ over the domain D˜ ⊂ (0, ε0 ) × S 2 . Now µ is a smooth function which by virtue of property (ii) has no critical points on its zero level set. Therefore H˜ is a smooth graph.

Chapter 15. The Nature of the Singular Hypersurface

931

˜ Since H˜ is a graph over D˜ ⊂ (0, ε0 ) × S 2 , (u, ϑ) can be used as coordinates on H, ˜ and in these coordinates h ∗ , the metric induced on H, is given by, from (15.15): /∗ ) AB (dϑ A + ∗A du)(dϑ B + ∗B du) h ∗ = (h

(15.19)

(h /∗ ) AB (u, ϑ) = h/ AB (t∗ (u, ϑ), u, ϑ)

(15.20)

where: are the components of a positive definite metric on S 2 , and: ∗A (u, ϑ) =  A (t∗ (u, ϑ), u, ϑ)

(15.21)

Now the metric (15.19) is degenerate: deth ∗ = 0

(15.22)

We see that although the hypersurface H˜ is singular, being a hypersurface where the spacetime metric h degenerates, from the point of view of its intrinsic geometry it is just like a regular null hypersurface in a regular spacetime. At each point q ∈ H˜ there is a unique null line L q ⊂ Tq H˜ which we may consider to be the linear span of a non-zero null vector V (q). Thus we have a null vectorfield V on H˜ and we can express it in terms of the coordinates (u, ϑ) by: V = Vu

∂ ∂ +V /, V /= VA A ∂u ∂ϑ

(15.23)

˜ the vector V Note that at each q = (t∗ (u, ϑ), u, ϑ) ∈ H, /(q) is tangent to the surface S∗u of constant u through q, which projects to the domain {u} × Bu where Bu is the domain: ˜ ⊂ S2 Bu = {ϑ ∈ S 2 : (u, ϑ) ∈ D} Since at each q ∈ H˜ only the linear span L q of V (q) is determined, we may, without loss of generality, set V u = 1. Then the integral curves of V are parametrized by u. We have: 0 = h(V, V ) = h ∗ (V, V ) = h /∗ (V /, V /) + 2h/∗ (∗ , V /) + h/∗ (∗ , ∗ ) = h/∗ (V / + ∗ , V / + ∗ ) (15.24) Since h /∗ is a positive definite metric, this holds if and only if: V / = −∗

(15.25)

Therefore the vectorfield V is expressed in (u, ϑ) coordinates on H˜ by: V = Now, let X (q) = X t (q)

∂ − ∗ ∂u

∂ ∂ ∂ + X u (q) + XA A ∂t ∂u ∂ϑ

(15.26)

(15.27)

932

Chapter 15. The Nature of the Singular Hypersurface

˜ Then X (q) ∈ Tq H˜ if and only if: be an arbitrary vector in Tq Mε0 , q ∈ H. X (q)(t − t∗ (u, ϑ)) = 0

(15.28)

Substituting the expansion (15.27) this is seen to be equivalent to: X t (q) =

∂t∗ u ∂t∗ A X (q) + X (q) ∂u ∂ϑ A

(15.29)

In particular the vectorfield V as a vectorfield in spacetime along H˜ is expressed by:   ∂t∗ ∂ ∂t∗ ∂ ∂ V = − ∗A A + − ∗A A (15.30) ∂u ∂ϑ ∂t ∂u ∂ϑ ˜ there is a unique real number λ such that Given any vector X (q) ∈ Tq Mε0 , q ∈ H, ˜ In fact it is readily seen from the condition (15.29), recalling that X (q) − λL(q) ∈ Tq H. L = ∂/∂t, that: ∂t∗ u ∂t∗ A X (q) − X (q) λ = X t (q) − ∂u ∂ϑ A This defines a projection operator ∗ from Tq Mε0 to Tq H˜ by: ∗ X (q) = X (q)−λL(q). ˜ It then readily follows from the above that the We may call ∗ the L projection to H. vectorfield V is given by: (15.31) V = ∗ T We call the integral curves of V the invariant curves. The singular surface H˜ is ruled by these curves. The invariant curves as 1-dimensional submanifolds of H˜ are independent of the choice of acoustical function u, being the integral manifolds of the 1˜ on H. ˜ Only the parametrization of these curves, dimensional distribution {L q : q ∈ H} which is a matter of convenience, depends on the choice of an acoustical function. The invariant curves have zero arc length. Now, we may adapt the coordinates (u, ϑ) on H˜ so that the coordinate lines ϑ = const. coincide with the invariant curves. This choice is equivalent to the condition: ∗ = 0

(15.32)

We then call the corresponding acoustical coordinates canonical. We recall from Chapter 2 that we can set  = 0 along any hypersurface H with the property that each integral curve of L intersects H at a single point. The singular hypersurface H˜ does have this property, as we have seen, by virtue of condition (ii), and the condition (15.32) corresponds ˜ Henceforth we shall only use acoustical coordinates which are to setting  = 0 along H. canonical. In these coordinates the induced metric (15.19) takes the form, simply: h ∗ = (h/∗ ) AB dϑ A dϑ B By virtue of condition (15.32), we have, for t ≤ t∗ (u, ϑ): t∗ (u,ϑ)  A  ∂ A  (t, u, ϑ) = − (t  , u, ϑ)dt  ∂t t

(15.33)

Chapter 15. The Nature of the Singular Hypersurface

Since also:

µ(t, u, ϑ) = −

t∗ (u,ϑ)  ∂µ 

∂t

t

it follows that the functions:

933

(t  , u, ϑ)dt 

ˆ A = µ−1  A 

(15.34)

which a priori are defined only for t < t∗ (u, ϑ), are actually given by:   mean value on [t, t∗ (u, ϑ)] of (∂ A /∂t)( , u, ϑ) A ˆ  (t, u, ϑ) = mean value on [t, t∗ (u, ϑ)] of {(∂µ/∂t)( , u, ϑ)}

(15.35)

˜ hence extend smoothly to t = t∗ (u, ϑ), that is, to H. We now consider the character of the singular hypersurface H˜ from the extrinsic point of view, that is, from the point of view of how it is imbedded in the acoustical spacetime. To do this, we consider the reciprocal acoustical metric, a quadratic form in the cotangent space to the spacetime manifold at each regular point, given by (see (3.116)): h −1 = −(1/2µ)(L ⊗ L + L ⊗ L) + (h/−1 ) AB X A ⊗ X B Since

∂ L= , ∂t

L=α

−1

κ L + 2T = α

−2

(15.36)

  ∂ ∂ A ∂ ˆ − µ µ +2 ∂t ∂u ∂ϑ A

(15.36) takes in canonical acoustical coordinates the form:   ∂ ∂ ∂ ∂ ∂ ∂ µh −1 = − ⊗ + ⊗ (15.37) − µα −2 ⊗ ∂t ∂u ∂u ∂t ∂t ∂t   ∂ ∂ ∂ ∂ ∂ ∂ ˆA ˆA +µ + µ(h/−1 ) AB ⊗ +  ⊗ ⊗ A A A ∂t ∂ϑ ∂ϑ ∂t ∂ϑ ∂ϑ B ˜ h −1 blows up at H, ˜ µh −1 in We see that although, due to the degeneracy of h on H, ˜ The character of H˜ from the extrinsic point of view is then fact extends smoothly to H. ˜ We have: determined by the sign of the invariant µ(h −1 )αβ ∂α µ∂β µ on H. µ(h −1 )αβ ∂α µ∂β µ = −2

∂µ ∂µ ∂t  ∂u

(15.38)   2 ∂µ ∂µ A ∂µ ∂µ −2 ∂µ −1 AB ˆ + (h/ ) +µ −α +2  ∂t ∂t ∂ϑ A ∂ϑ A ∂ϑ B

On H˜ µ vanishes and this reduces to, simply: µ(h −1 )αβ ∂α µ∂β µ = −2

∂µ ∂µ ∂t ∂u

(15.39)

In view of condition (ii) we obtain that µ(h −1 )αβ ∂α µ∂β µ is > 0, = 0, < 0, at a point q ∈ H˜ according as to whether (∂µ/∂u)(q) is > 0, = 0, < 0. We conclude that H˜

934

Chapter 15. The Nature of the Singular Hypersurface

is space-like, null, or time-like, at q according as to whether (∂µ/∂u)(q) is < 0, = 0, > 0. Now the boundary of the domain of the maximal solution cannot be time-like at any point. Therefore the time-like part of H˜ cannot be part of the boundary of the domain of the maximal solution. It corresponds to the boundary of some artificial extension and for this reason will not be considered further. We shall therefore focus henceforth on the ˜ which we denote by H and its boundary ∂H, where H˜ is null. space-like part of H, Consider then the open subset D ⊂ D˜ defined by: D = {(u, ϑ) ∈ D˜ : (∂µ/∂u)(t∗ (u, ϑ), u, ϑ) < 0}

(15.40)

(canonical acoustical coordinates). The boundary of D in D˜ is given by: ∂D = {(u, ϑ) ∈ D˜ : (∂µ/∂u)(t∗ (u, ϑ), u, ϑ) = 0}

(15.41)

Then H is the graph (15.18) over D: H = {(t∗ (u, ϑ), u, ϑ) : (u, ϑ) ∈ D}

(15.42)

and the boundary ∂H of H in H˜ is the graph (15.18) over ∂D: ∂H = {(t∗ (u, ϑ), u, ϑ) : (u, ϑ) ∈ ∂D}

(15.43)

Assuming that the non-degeneracy condition: (u, ϑ) ∈ ∂D implies (∂ 2 µ/∂u 2 )(t∗ (u, ϑ), u, ϑ) = 0
holds, ∂D splits into the disjoint union ∂D = ∂− D ∂+ D where:

(15.44)

∂− D = {(u, ϑ) ∈ ∂D : (∂ 2 µ/∂u 2 )(t∗ (u, ϑ), u, ϑ) > 0} ∂+ D = {(u, ϑ) ∈ ∂D : (∂ 2 µ/∂u 2 )(t∗ (u, ϑ), u, ϑ) < 0}

(15.45)
The boundary ∂H of H in H˜ similarly splits into the disjoint union ∂H = ∂− H ∂+ H where ∂− H and ∂+ H are the graphs (15.18) over ∂− D and ∂+ D respectively: ∂− H = {(t∗ (u, ϑ), u, ϑ) : (u, ϑ) ∈ ∂− D} ∂+ H = {(t∗ (u, ϑ), u, ϑ) : (u, ϑ) ∈ ∂+ D}

(15.46)

If there is an interior zero of µ on tε∗ε corresponding to (u 0 , ϑ0 ), then (u 0 , ϑ0 ) belongs to ∂− D. If there is a boundary zero of µ on tε∗ε , then by the non-degeneracy condition (15.4), D contains the curve (15.17) except for the point (u 0 , ϑ∗ (u 0 )) which belongs to ∂− D. Now on H˜ we have µ = 0, that is, we have: µ(t∗ (u, ϑ), u, ϑ) = 0 : for all (u, ϑ) ∈ D˜ Differentiating this equation implicitly with respect to u we obtain:     ∂t∗ ∂µ/∂u (u, ϑ) = − (t∗ (u, ϑ), u, ϑ) ∂u ∂µ/∂t

(15.47)

(15.48)

Chapter 15. The Nature of the Singular Hypersurface

935

By virtue of property (ii) of the extension we then have:

and:

D = {(u, ϑ) ∈ D˜ : (∂t∗ /∂u)(u, ϑ) > 0}

(15.49)

∂D = {(u, ϑ) ∈ D˜ : (∂t∗ ∂u)(u, ϑ) = 0}

(15.50)

(canonical acoustical coordinates). Moreover, differentiating again (15.48) with respect to u and evaluating the result on ∂D, we obtain, in view of (15.41) and (15.50):     ∂ 2 µ/∂u 2 ∂ 2 t∗ (u, ϑ) = − (t∗ (u, ϑ), u, ϑ) : on ∂D (15.51) ∂µ/∂t ∂u 2 Comparing with (15.45) we conclude that: ∂− D = {(u, ϑ) ∈ ∂D : (∂ 2 t∗ /∂u 2 )(u, ϑ) > 0} ∂+ D = {(u, ϑ) ∈ ∂D : (∂ 2 t∗ /∂u 2 )(u, ϑ) < 0}

(15.52)

Thus if we consider a connected component of H and the corresponding components of ∂− H, ∂+ H, then the component of ∂− H is the past boundary of H, the component of ∂+ H, which may be empty, its future boundary, the function t∗ reaching a minimum, along each invariant curve, at ∂− H, a maximum at ∂+ H. ˜ In canonical acoustical coordinates Consider now the function f = ∂µ/∂u on H. ˜ on H we have:   ∂µ f (u, ϑ) = (15.53) (t∗ (u, ϑ), u, ϑ) ∂u Differentiating with respect to u, that is, along the invariant curves, and evaluating the result on ∂H, we obtain, in view of (15.50):     ∂ 2µ ∂f (15.54) (t∗ (u, ϑ), u, ϑ) : on ∂H (u, ϑ) = ∂u ∂u 2 ˜ Thus ∂H is the zero level set of a smooth function, namely f , on the smooth manifold H, and by (15.54) and the non-degeneracy condition (15.44) this is a non-critical level set. It follows that ∂H and its two components ∂− H and ∂+ H are smooth. Moreover, since ˜ transversally, they are both ∂− H and ∂+ H intersect the invariant curves, generators of H, space-like surfaces in the acoustical spacetime. Finally, we note that ∂+ H cannot be part of the boundary of the domain of the maximal solution. This is seen as follows. For any given component of ∂+ H, there is ˜ whose future boundary is the given a component of H˜ \ H, the time-like part of H, component of ∂+ H. This, or in fact any, component of H˜ \ H has a past boundary, for, limu→0 t∗ (u) = ∞, therefore t∗ (u, ϑ) must have intervals of increase along each invariant line as we approach u = 0. The past boundary of the component in question is then also the past boundary of the next outward (that is, in the direction of decreasing u along

936

Chapter 15. The Nature of the Singular Hypersurface

the invariant curves) component of H, a component of ∂− H. But then the domain of the maximal solution terminates at the incoming characteristic hypersurface C generated by the incoming null normals to this component of ∂− H. This cuts off the component of ∂+ H, as it lies to the future of C. [See Figure 1].

H C

C C

H

H

Figure 1 We summarize the above results in the following proposition. Proposition 15.1 Consider a smooth extension of the acoustical metric h to t > t∗ (u), satisfying the following two conditions: (i) The function α remains positive and less than 1, and the metric h/ remains a positive definite metric on S 2 . (ii) The function µ is extended in such a way that ∂µ/∂t is bounded from above by a negative constant where µ < η0 /4. Then there is an open subset D˜ ⊂ (0, ε0 ) × S 2 and a smooth graph ˜ H˜ = {(t∗ (u, ϑ), u, ϑ) : (u, ϑ) ∈ D} where µ vanishes, being positive below this graph. The singular hypersurface H˜ with its induced metric h ∗ is smooth and has the intrinsic geometry of a regular null hypersurface in a regular spacetime. It is ruled by invariant curves of vanishing arc length. The tangent line to an invariant curve at a point q is the linear span of the vector ∗ T (q), the L ˜ Canonical acoustical coordinates are defined by taking the ϑ = projection of T (q) to H. const. coordinate lines on H˜ to be the invariant curves. On the other hand, from the point of view of how H˜ is embedded in the acoustical spacetime, the extrinsic point of view, H˜ is at a point q space-like, null, or time-like, according as to whether, in canonical acoustical coordinates, ∂µ/∂u is < 0, = 0, > 0, at q, or equivalently, as to whether ∂t∗ /∂u is < 0, = 0, > 0, at q. The time-like part of H˜ cannot be part of the boundary of

Chapter 15. The Nature of the Singular Hypersurface

937

the domain of the maximal solution. Moreover, denoting by H the space-like part of H˜ and by ∂H its boundary, under the non-degeneracy condition: ∂ 2 µ/∂u 2 = 0 : on ∂H the boundary splits into the disjoint union of ∂− H and ∂+ H, where ∂− H and ∂+ H correspond to ∂ 2 µ/∂u 2 being > 0 and < 0 respectively, or equivalently ∂ 2 t∗ /∂u 2 being > 0 and < 0 respectively. Each of ∂− H, ∂+ H, is a smooth space-like surface in the acoustical spacetime. For each connected component of H the corresponding components of ∂− H and ∂+ H are respectively its past and future boundaries, the sets of past and future end points of its invariant curves. Finally, ∂+ H, which, in contrast to ∂− H, may be empty, cannot be part of the boundary of the domain of the maximal solution. We now turn to the investigation of the past null geodesic (or characteristic) cone of an arbitrary point q on the singular boundary of the domain of the maximal solution. domain of According to the above, such a point belongs either to H, or to ∂− H. The ε
the maximal solution or maximal development of the initial data on 00 0E , where 0E denotes the exterior of the unit sphere in 0 , where the constant state holds, has like any development of the initial data in question the property that for each point q in this domain and for each past directed curve issuing at q which is causal with respect to the acoustical metric (that is, its tangent vector at any point p belongs to the closure of the open past cone defined by the
acoustical metric h in the tangent space at p) terminates in the past at a point of 0 0E . In particular, each past directed null geodesic issuing ε
at q terminates in the past at a point of 00 0E . Suppose now that q is a zero of µ on tε∗ε for some ε ∈ (0, ε0 ). Then q ∈ Wεt∗ε and the union of Wεt∗ε with the domain E t∗ε0 in Minkowski spacetime exterior to the cone C0 and bounded by the hyperplanes 0 and t∗ε
, where the constant state holds, is a development of the restriction of the initial
data to 0ε 0E , therefore each past directed null geodesic issuing at q remains in Wεt∗ε E t∗ε
and terminates in the past at a point of 0ε 0E . Therefore for the investigation of the past null geodesic cone of a point q on the singular boundary of Vε0 the extension of the metric h to t > t∗ (u) is irrelevant. (We are identifying in this discussion H and V with their images in Minkowski spacetime). Now, the null geodesic flow of a Lorentzian manifold (M, g) is the Hamiltonian flow on T ∗ M generated by the Hamiltonian: H=

1 −1 µν (g ) (q) pµ pν 2

(15.55)

on the surface H = 0. Here (q µ : µ = 1, . . . , n), n = dimM, are local coordinates on M (called in the Hamiltonian context canonical coordinates), and expanding p ∈ Tq∗ M in the basis (dq 1 (q), . . . , dq n (q)): p = pµ dq µ (q) the coefficients ( pµ : µ = 1, . . . , n) of the expansion are linear coordinates on Tq M (called in the Hamiltonian context canonical momenta). We thus have local coordinates

938

Chapter 15. The Nature of the Singular Hypersurface

(q µ : µ = 1, . . . , n; pµ : µ = 1, . . . , n) on T ∗ M and the equations defining a Hamiltonian flow are the canonical equations: ∂H dq µ = , dτ ∂ pµ

∂H d pµ =− dτ ∂qµ

(15.56)

Each level set of the Hamiltonian H is invariant by the Hamiltonian flow. We may thus consider the restriction of the flow to any given level set of H , in particular to the surface H = 0. Let now: ˜ H (q, p) = (q) H(q, p) (15.57) ˜ where  is a positive function on M. Then the Hamiltonian flow of H on its zero level set is equivalent, up to reparametrization, to the Hamiltonian flow of H on its zero level set. More precisely, let τ → (q(τ ), p(τ )) be a solution of the canonical equations (15.56) on the surface H = 0. Then defining a new parameter τ˜ by: d τ˜ = (q(τ )) dτ

(15.58)

τ˜ → (q( ˜ τ˜ ), p( ˜ τ˜ )) = (q(τ ), p(τ )) is a solution of (15.56) with H replaced by H˜ and τ by τ˜ on the surface H˜ = 0. This is so because dq µ dτ ∂ H −1 d q˜ µ ∂ H˜ = =  = d τ˜ dτ d τ˜ ∂ pµ ∂ pµ d pµ dτ ∂H d p˜ µ ∂( H˜ ) −1 = = − µ −1 = −  d τ˜ dτ d τ˜ ∂q ∂q µ ∂ ∂ H˜ ∂ H˜ = − µ − H˜ −1 µ = − µ : on H˜ = 0 ∂q ∂q ∂q In the case that H is the Hamiltonian (15.55), then 1 H˜ = (g˜ −1 )µν (q) pµ pν , g˜ µν = gµν 2 and the above corresponds to the fact that null geodesics are invariant, up to reparametrization, under conformal transformations of the metric. In the case of our acoustical spacetime (Mε0 , h), the reciprocal metric h −1 is as ˜ however µh −1 , given in canonical acoustical coordinates by we have seen singular at H, ˜ Taking then advantage of the above remarks we take the (15.29), extends smoothly to H. Hamiltonian to be:  1 −2 2 1 −1 AB A ˆ (15.59) H = − pu pt + µ − α pt + pt  p/ A + (h/ ) p/ A p/ B 2 2 Here pt , pu , and p/ A : A = 1, 2 are the momenta conjugate to the coordinates t, u, ϑ A : A = 1, 2, respectively. Thus the p/ A are the components of the angular momentum p/. Given a solution τ → (q(τ ), p(τ )), q = (t, u, ϑ), p = ( pt , pu , p/), then defining s by: ds = µ(q(τ )) dτ

(15.60)

Chapter 15. The Nature of the Singular Hypersurface

939

and taking the inverse of the mapping τ → s(τ ), then s → (q(τ (s)), p(τ (s)) is an affinely parametrized null geodesic of the acoustical metric h, s being the affine parameter, and conversely. The canonical equations (15.56) take for the Hamiltonian (15.59) the following form:   ∂H dt ˆ A p/ A = = − pu + µ −α −2 pt +  dτ ∂ pt du ∂H = = − pt dτ ∂ pu   ∂H dϑ A ˆA (15.61) = µ (h/−1 ) AB p/ B + pt  = dτ ∂ p/ A  ∂H ∂µ d pt 1 −1 AB 1 −2 2 A ˆ =− =− − α pt + pt  p/ A + (h/ ) p/ A p/ B dτ ∂t ∂t 2 2   ˆA ∂ 1 ∂(h/−1 ) AB −3 ∂α 2 −µ α p + pt p/ A − p/ A p/ B ∂t t ∂t 2 ∂t  ∂H ∂µ 1 −2 2 d pu 1 −1 AB A ˆ − α pt + pt  p/ A + (h/ ) p/ A p/ B =− =− dτ ∂u ∂u 2 2   A −1 ) AB ˆ ∂(h / ∂α ∂  1 − µ α −3 p 2 + pt p/ A − p/ A p/ B ∂u t ∂u 2 ∂u  d p/ A ∂H 1 ∂µ 1 ˆ B p/ B + (h/−1 ) BC p/ B p/C = − A = − A − α −2 pt2 + pt  dτ ∂ϑ ∂ϑ 2 2   B ˆ 1 ∂(h/−1 ) BC ∂ −3 ∂α 2 −µ α p + pt p/ B − p/ B p/C ∂ϑ A t ∂ϑ A 2 ∂ϑ A

(15.62)


Given a point q0 ∈ H ∂− H, we shall study the solutions of the system (15.61)–(15.62), subject to the condition H = 0, which end at q0 = (t0 , u 0 , ϑ0 ), t0 = t∗ (u 0 , ϑ0 ), at τ = 0. The solutions are then studied for τ ≤ 0. These are the null geodesics of the acoustical metric ending at q0 , that is, the past null geodesic cone of q0 , with the orientation of each geodesic reversed so that they become future-directed ending at q0 instead of past directed issuing at q0 . Now the condition H = 0 (see (15.59)) defines at a regular point p ∈ Mε0 , where µ( p) > 0, a double cone in T p∗ Mε0 and we are considering the backward part of this cone. However at the singular point q0 , where µ(q0 ) = 0, the condition H = 0 reduces to: pu pt = 0

(15.63)

Thus the double cone degenerates to the two hyperplanes pt = 0 and pu = 0 and the backward part of the cone degenerates to the following three pieces:

940

Chapter 15. The Nature of the Singular Hypersurface

(i) The negative half-hyperplane: pt = 0, pu < 0. (ii) The negative half-hyperplace: pu = 0, pt = 0. (iii) The plane pt = pu = 0. We thus have a trichotomy of the past null geodesic cone of q0 . The null geodesics ending at q0 such that their momentum covector at q0 belongs to (i) are the outgoing null geodesics. These contain the generator of the characteristic hypersurface Cu 0 through q0 , parametrized by t, which is the following solution of (15.61)–(15.62) and the condition H = 0: (15.64) t = t0 + τ, u = u 0 , ϑ = ϑ0 ; pt = 0, pu = −1, p/ = 0 The null geodesics ending at q0 such that their momentum vector at q0 belongs to (ii) are the incoming null geodesics. Finally, the null geodesics ending at q0 such that their momentum vector at q0 belongs to (iii) we simply call the other null geodesics. To obtain an intuitive picture of how the null cone in T p∗ Mε0 degenerates as p, a regular point, approaches q0 , a point on the singular boundary, we consider the picture in 4-dimensional Euclidean space with the rectangular coordinates x 1 , x 2 , y, z, taking: /−1 ) AB p/ A p/ B , x 12 + x 22 = (h

y = pu , z = α −1 pt

(15.65)

ˆ = An adequate picture of the degeneration is obtained if we assume for simplicity that  0. The equation of the double cone H = 0 then becomes: 2yz = κ(x 12 + x 22 − z 2 )

(15.66)

Note that the double cone contains the y axis as well as the line: 1 x 1 = x 2 = 0, y = − κz 2

(15.67)

Consider the point P0 on the positive y axis at Euclidean distance 1 from the origin. The coordinates of P0 are then (0, 0, 1, 0). Consider also the point P1 on the line (15.67) at Euclidean distance 1 from the origin in the positive z direction. The coordinates of P1 are:   (0, 0, −(κ/2)/ (κ/2)2 + 1, 1/ (κ/2)2 + 1) Let H1 be the hyperplane: 1 z = −λ(y − 1), λ =

(κ/2)2 + 1 + (κ/2)

(15.68)

passing through the points P0 , P1 , and ruled by planes parallel to the (x 1 , x 2 ) plane. Then the intersection of the hyperplane H1 with the double cone is the spheroid on H1 which projects to the following spheroid on the (x 1 , x 2 , y) hyperplane: ) x 12 + x 22 1 λ (y − y0 )2 1 + κλ , b= , y0 = + = 1; a = 2 2 κ(2 + κλ) 2 + κλ 2 + κλ a b (15.69)

Chapter 15. The Nature of the Singular Hypersurface

941

with semimajor axis a, semiminor axis b, centered at y0 on the y axis. The spheroid on H1 is centered at the point (0, 0, y0 , z 0 ), z 0 = −λ(y0 − 1), and has semimajor axis a = √ a in a plane through this center parallel to the (x 1 , x 2 ) plane and semiminor axis b = 1 + λ2 b in a line through the center in H1 orthogonal to this plane. As the regular point p approaches the singular point q0√, κ → 0, hence λ → 1, y0 → 1/2, z 0 → √ 1/2, b → 1/ 2, but a  → ∞, in fact 2κa  → 1. The hyperplane H1 becomes the hyperplane z = −(y − 1), and the spheroid on H1, the intersection with the double cone, degenerates to the planes y = 1, z = 0, and y = 0, z = 1, parallel to the (x 1 , x 2 ) plane, at which the hyperplane z = −(y − 1) intersects the hyperplanes z = 0 and y = 0. This shows how the double cone degenerates to the two hyperplanes z = 0 and y = 0. We return to the study of the null geodesics ending at q0 . The value of p0 = (( pt )0 , ( pu )0 , p/0 ) determines in which of the three classes the null geodesic belongs. Note that p0 = 0, so that (q0 , p0 ) is a regular non-critical point of the Hamiltonian system (15.61)–(15.62). Now we can impose in each class a normalization condition by making use of the following remark. Let τ → (q(τ, p(τ )) be a solution of the canonical equations (15.56) associated to the Hamiltonian (15.55) and corresponding to the conditions q(0) = q0 , p(0) = p0, at τ = 0. Then for any positive constant a, τ → (q(aτ ), ap(aτ )) is the solution corresponding to the conditions q(0) = q0 , p(0) = ap0. Thus the new conditions give a null geodesic which is simply a reparametrization of the null geodesic arising from the original conditions. This is so because if q(τ ˜ ) = q(aτ ), p(τ ˜ ) = ap(aτ ), we have:   µ   µ dq ∂H d q˜ (q(aτ ), p(aτ )) (τ ) = a (aτ ) = a dτ dτ ∂ pµ   ∂H ˜ )) p˜ ν (τ ) = (q(τ ˜ ), p(τ ˜ )) = a(g −1)µν (q(aτ )) pν (aτ ) = (g −1 )µν (q(τ ∂ pµ       d p˜ µ 2 d pµ 2 ∂H (τ ) = a (aτ ) = −a (q(aτ ), p(aτ )) dτ dτ ∂q µ   1 ∂(g −1 )λν = − a2 (q(aτ )) pλ(aτ ) pν (aτ ) 2 ∂q µ     1 ∂(g −1 )λν ∂H (q(τ ˜ )) p˜ λ (τ ) p˜ ν (τ ) = − (q(τ ˜ ), p(τ ˜ )) =− 2 ∂q µ ∂q µ Given an outgoing null geodesic ending at q0 we can, by suitable choice of the scale factor a, set ( p0 )u = −1. Thus, the outgoing null geodesics ending at q0 correspond to the plane: ( p0 )t = 0, ( p0 )u = −1 : in Tq∗0 Mε0 (15.70) Given an incoming null geodesic ending at q0 we can, by suitable choice of the scale factor a, set ( pt )0 = −1. Thus, the incoming null geodesics ending at q0 correspond to the plane: ( p0 )u = 0, ( p0)t = −1 : in Tq∗0 Mε0 (15.71)

942

Chapter 15. The Nature of the Singular Hypersurface

Finally, given one of the other null geodesics ending at q0 we can, by suitable choice of the scale factor a, set |( p/)0| = (h/−1 ) AB (q0 )( p/0 ) A ( p/0 ) B = 1. Thus, the other null geodesics ending at q0 correspond to the circle: ( p0 )t = 0, ( p0 )u = 0, | p/0 | = 1 : in Tq∗0 Mε0

(15.72)

Consider the class of outgoing null geodesics ending at q0 . For this class equations (15.61) evaluated at τ = 0 give:    A   du dϑ dt (0) = 1, (0) = 0, (0) = 0 (15.73) dτ dτ dτ Also, equations (15.62) give:     d pt 1 ∂µ (0) = − (q0 )| p/0|2 dτ 2 ∂t     d pu 1 ∂µ (0) = − (q0 )| p/0|2 dτ 2 ∂t     1 ∂µ d p/ A (0) = − (q0 )| p/0|2 dτ 2 ∂ϑ A

(15.74)

Differentiating equations (15.61) with respect to τ and evaluating the result at τ = 0 using (15.74) and the fact that by (15.73):     dµ(q(τ )) ∂µ (q0 ) = (15.75) dτ ∂t τ =0 we obtain:





   ∂µ ∂µ ˆ A (q0 )( p/0 ) A (q0 )| p/0 |2 + (q0 ) ∂u ∂t     d 2u 1 ∂µ (0) = (q0 )| p/0 |2 2 ∂t dτ 2     d 2ϑ A ∂µ (0) = (q0 )(h/−1 ) AB (q0 )( p/0) B 2 dτ ∂t d 2t dτ 2

(0) =

1 2



(15.76)

In view of (15.73), (15.76) the tangent vector to an outgoing null geodesic ending at q0 has the following expansion as τ → 0:         ∂µ dt 1 ∂µ 2 A ˆ (τ ) = 1 + (q0 )| p/0 | + (q0 ) (q0 )( p/0 ) A τ + O(τ 2 ) dτ 2 ∂u ∂t     du 1 ∂µ (τ ) = (q0 )| p/0 |2 τ + O(τ 2 ) dτ 2 ∂t  A   dϑ ∂µ /−1 ) AB (q0 )( p/0 ) B τ + O(τ 2 ) (15.77) (τ ) = (q0 )(h dτ ∂t

Chapter 15. The Nature of the Singular Hypersurface

943

We conclude that in canonical acoustical coordinates the tangent vectors to all outgoing null geodesics ending at q0 tend as we approach q0 to the tangent vector of the generator of Cu 0 through q0 . Consider the class of incoming null geodesics ending at q0 . For this class equations (15.61) evaluated at τ = 0 give:    A   du dϑ dt (0) = 0, (0) = 1, (0) = 0 (15.78) dτ dτ dτ Also, equations (15.62) give:      d pt 1 ∂µ ˆ A (q0 )( p/0 ) A + | p/0|2 (0) = − (q0 ) −α −2 (q0 ) − 2 dτ 2 ∂t      1 ∂µ d pu ˆ A (q0 )( p/0 ) A + | p/0|2 (0) = − (q0 ) −α −2 (q0 ) − 2 dτ 2 ∂t      1 ∂µ d p/ A −2 ˆ B (q0 )( p/0 ) B + | p/0|2 ) −α (q ) − 2  (q (0) = − 0 0 dτ 2 ∂ϑ A

(15.79)

Differentiating equations (15.61) with respect to τ and evaluating the result at τ = 0 using (15.79) and the fact that by (15.78):     dµ(q(τ )) ∂µ = (15.80) (q0 ) dτ ∂u τ =0 we obtain:





    1 ∂µ (0) = (q0 ) α −2 (q0 ) + | p/0|2 2 ∂u      ∂µ d 2ϑ A −1 AB ˆ A (q0 ) (0) = (q ) (h / ) (q )( p / ) −  0 0 0 B ∂u dτ 2 d 2t dτ 2

(15.81)

The right-hand side of the first equation of (15.81) is negative when q0 ∈ H. However in the case that q0 ∈ ∂− H, we have:       d 2ϑ A dµ(q(τ )) d 2t (0) = 0, (0) = 0 (15.82) = 0, dτ dτ 2 dτ 2 τ =0 We must then proceed to the third derivatives. Using (15.82) we find in this case:     d 2 µ(q(τ )) ∂ 2µ (q0 ) = (15.83) dτ 2 ∂u 2 τ =0

and differentiating the second equation of (15.62) with respect to τ and evaluating the result at τ = 0 gives:      1 ∂ 2µ d 2 pu −2 A 2 ˆ (0) = (q (15.84) ) α (q ) + 2  (q )( p / ) − | p / | 0 0 0 0 A 0 2 ∂u 2 dτ 2

944

Chapter 15. The Nature of the Singular Hypersurface

Differentiating then the second and third equations of (15.61) with respect to τ a second time and using (15.83) and (15.84) we then obtain that in the case q0 ∈ ∂− H:       1 ∂ 2µ d 3t −2 2 (0) = (q ) α (q ) + | p / | 0 0 0 2 ∂u 2 dτ 3      d 3ϑ A ∂ 2µ −1 AB ˆ A (q0 ) ) (h / ) (q )( p / ) −  (15.85) (0) = (q 0 0 0 B dτ 3 ∂u 2 Note that the right-hand side of the first equation of (15.85) is positive. The tangent vector to an incoming null geodesic ending at q0 has the following expansion as τ → 0:   du (τ ) = 1 + O(τ ) dτ       dt 1 ∂µ (τ ) = (q0 ) α −2 (q0 ) + | p/0|2 τ + O(τ 2 ) dτ 2 ∂u    A  ∂µ dϑ ˆ A (q0 ) τ + O(τ 2 ) (τ ) = (q0 ) (h /−1 ) AB (q0 )( p/0 ) B −  (15.86) dτ ∂u by (15.78), (15.81), in the case that q0 ∈ H, and:   du (τ ) = 1 + O(τ ) dτ       1 ∂ 2µ dt (q0 ) α −2 (q0 ) + | p/0 |2 τ 2 + O(τ 3 ) (τ ) = 2 dτ 4 ∂u    A  1 ∂ 2µ dϑ −1 AB ˆ A (q0 ) τ 2 + O(τ 3 ) (q (τ ) = ) (h / ) (q )( p / ) −  0 0 0 B dτ 2 ∂u 2

(15.87)

by (15.78), (15.83), (15.85), in the case that q0 ∈ ∂− H. Consider finally the class of the other null geodesics ending at q0 . For this class equations (15.61) and (15.62) evaluated at τ = 0 give:      A dt du dϑ (0) = 0, (0) = 0, (0) = 0 (15.88) dτ dτ dτ 

   d pt 1 ∂µ (0) = − (q0 ) dτ 2 ∂t     d pu 1 ∂µ (0) = − (q0 ) dτ 2 ∂u   d p/ A 1 ∂µ (0) = − dτ 2 ∂ϑ A

(15.89)

Note that the right-hand side of the first equation of (15.89) is positive. Differentiating equation (15.61) with respect to τ and evaluating the result at τ = 0 using (15.89) and

Chapter 15. The Nature of the Singular Hypersurface

the fact that by (15.88):



dµ(q(τ )) dτ

945

 τ =0

=0

(15.90)

we obtain: 



 ∂µ (q0 ) ∂u     d 2u 1 ∂µ (0) = (q0 ) dτ 2 2 ∂t   d 2ϑ A (0) = 0 dτ 2 d 2t dτ 2

(0) =

1 2



(15.91)

Note that the right-hand side of the second equation of (15.91) is negative, while the right-hand side of the first equation of (15.91) is negative when q0 ∈ H and vanishes when q0 ∈ ∂− H. Differentiating the third equation of (15.61) a second time with respect to τ and evaluating the result at τ = 0 yields:     d 2 µ(q(τ )) d 3ϑ A (h/−1 ) AB (q0 )( p/0 ) B (15.92) (0) = dτ 3 dτ 2 τ =0

Now from (15.88) and (15.91) we obtain:   d 2 µ(q(τ )) dτ 2  τ =0           2 ∂µ ∂µ ∂µ d t d 2u d 2ϑ A (0) (0) (0) (q0 ) + (q0 ) + (q0 ) = ∂t ∂u ∂ϑ A dτ 2 dτ 2 dτ 2   ∂µ ∂µ = (15.93) (q0 ) ∂t ∂u Substituting in (15.92) then yields:     ∂µ ∂µ d 3ϑ A (0) = (q0 )(h/−1 ) AB (q0 )( p/0) B ∂t ∂u dτ 3

(15.94)

In the case that q0 ∈ ∂− H the right-hand side of the first equation of (15.91), of (15.93), and of (15.94) all vanish:         d 2t d 2ϑ A d 3ϑ A d 2 µ(q(τ )) = 0, (0) = 0, (0) = (0) = 0 dτ 2 dτ 2 dτ 2 dτ 3 τ =0 (15.95)

946

Chapter 15. The Nature of the Singular Hypersurface

Differentiating the first equation of (15.61) a second time with respect to τ and evaluating the result at τ = 0 then yields:     d 2 pu d 3t (0) = − (0) (15.96) dτ 3 dτ 2 Also, differentiating the second equation of (15.62) with respect to τ and evaluating the result at τ = 0 yields:     1 d(∂µ/∂u)(q(τ )) d 2 pu (0) = − (15.97) 2 dτ dτ 2 τ =0 However, from (15.88):           d(∂µ/∂u)(q(τ )) ∂ 2µ ∂ 2µ dt du (q0 ) (0) + (0) (q0 ) = 2 dτ ∂t∂u dτ dτ ∂u τ =0    A ∂ 2µ dϑ + (0) = 0 (15.98) (q ) 0 A ∂ϑ ∂u dτ hence:



d 2 pu dτ 2



 (0) = 0 and

d 3t dτ 3

 (0) = 0

(15.99)

We must then proceed to the fourth derivatives. Differentiating the first of (15.61) a third time with respect to τ and evaluating the result at τ = 0 yields:       d 4t d 3 pu d 3 µ(q(τ )) ˆ A (q0 )( p/0 ) A (0) = − (0) + (15.100)  dτ 4 dτ 3 dτ 3 τ =0

Also, differentiating the second equation of (15.62) a second time with respect to τ and evaluating the result at τ = 0 yields:     d 3 pu 1 d 2 (∂µ/∂u)(q(τ )) (0) = − (15.101) 2 dτ 3 dτ 2 τ =0

Now, in view of (15.88) and (15.91):   d 2 (∂µ/∂u)(q(τ )) dτ 2 τ =0             2 2 ∂ 2µ ∂ 2µ ∂ µ d t d 2u d 2ϑ A (0) + (q0 ) (0) + (0) (q0 ) (q0 ) = ∂t∂u ∂ϑ A ∂u dτ 2 ∂u 2 dτ 2 dτ 2       ∂ 2µ d 2u 1 ∂µ ∂ 2 µ = (15.102) (q0 ) (0) = (q0 ) ∂u 2 dτ 2 2 ∂t ∂u 2

Chapter 15. The Nature of the Singular Hypersurface

947

On the other hand, by (15.88), (15.91), (15.95), (15.99), and the fact that (∂µ/∂u)(q0) = 0,   d 3 µ(q(τ )) dτ 3 τ =0             ∂µ d 3t d 3u d 3ϑ A ∂µ ∂µ = (0) + (0) + (0) (q0 ) (q0 ) (q0 ) ∂t ∂u ∂ϑ A dτ 3 dτ 3 dτ 3         d 2t d(∂µ/∂u)(q(τ )) d 2u d(∂µ/∂t)(q(τ )) (0) + 2 (0) +2 2 2 dτ dτ τ =0 dτ τ =0 dτ     d(∂µ/∂ϑ A )(q(τ )) d 2ϑ A +2 (0) = 0 (15.103) dτ dτ 2 τ =0 Substituting then (15.102) in (15.101) and the result in (15.100), yields, in view of (15.103),     1 ∂µ ∂ 2 µ d 4t (15.104) (0) = (q0 ) dτ 4 4 ∂t ∂u 2 Differentiating the third equation of (15.61) a third time and evaluating the result at τ = 0 we obtain, taking into account (15.103),     d 4ϑ d 3 µ(q(τ )) (0) = (h/−1 ) AB (q0 )( p/0 ) B = 0 (15.105) dτ 4 dτ 3 τ =0

Finally, we differentiate the third equation of (15.61) a fourth time and evaluate the result at τ = 0 to obtain:     d 5ϑ A d 4 µ(q(τ )) (0) = (h/−1 ) AB (q0 )( p/0 ) B (15.106) dτ 5 dτ 4 τ =0

and we have:   d 4 µ(q(τ )) dτ 4 τ =0         ∂µ d 4t d 2 (∂µ/∂u)(q(τ )) d 2u = (0) + 3 (0) (q0 ) ∂t dτ 4 dτ 2 dτ 2 τ =0    ∂µ 2 ∂ 2 µ (q0 ) (15.107) = ∂t ∂u 2 by (15.102), (15.104), and the second of (15.91). Substituting then in (15.106) yields:      ∂µ 2 ∂ 2 µ d 5ϑ A (0) = (q0 )(h/−1 ) AB (q0 )( p/0 ) B (15.108) ∂t dτ 5 ∂u 2

948

Chapter 15. The Nature of the Singular Hypersurface

According to the above results the tangent vector to any of the other null geodesics ending at q0 has the following expansion as τ → 0:     dt 1 ∂µ (τ ) = (q0 )τ + O(τ 2 ) dτ 2 ∂u     du 1 ∂µ (τ ) = (q0 )τ + O(τ 2 ) dτ 2 ∂t    A 1 ∂µ ∂µ dϑ (τ ) = (q0 )(h/−1 ) AB (q0 ))( p/0 ) B τ 2 + O(τ 3 ) (15.109) dτ 2 ∂t ∂u by (15.91), (15.94), in the case q0 ∈ H, and:     dt 1 ∂µ ∂ 2 µ (q0 )τ 3 + O(τ 4 ) (τ ) = dτ 24 ∂t ∂u 2     du 1 ∂µ (τ ) = (q0 )τ + O(τ 2 ) dτ 2 ∂t     A dϑ 1 ∂µ 2 ∂ 2 µ (q0 )(h/−1 ) AB (q0 )( p/0 ) B τ 4 + O(τ 5 ) (τ ) = dτ 24 ∂t ∂u 2

(15.110)

in the case that q0 ∈ ∂− H. To obtain the picture in Minkowski spacetime, we now investigate the behavior of the tangent vector to a null geodesic ending at q0 , in each of the three classes, as t → t0 , in rectangular coordinates. The rectangular components L˜ µ of the tangent vector L˜  of a null geodesic ending at q0 , parametrized by τ , are given by: ∂ x µ dt ∂ x µ du ∂ x µ dϑ A dxµ = + + L˜ µ = dτ ∂τ dτ ∂u dτ ∂ϑ A dτ

(15.111)

If the null geodesic is parametrized by t instead of τ the tangent vector is: L =

L˜  dt/dτ

(15.112)

and its rectangular components are given by: L µ =

∂ x µ du ∂ x µ dϑ A ∂xµ + + ∂t ∂u dt ∂ϑ A dt

(15.113)

dϑ A dϑ A /dτ = dt dt/dτ

(15.114)

where: du du/dτ = , dt dt/dτ Recalling from Chapter 2 that: ∂xµ = Lµ, ∂t

∂x0 ∂x0 = = 0, ∂u ∂ϑ A

∂xi = T i +  A X iA , ∂u

∂xi = X iA ∂ϑ A

(15.115)

Chapter 15. The Nature of the Singular Hypersurface

949

we then obtain: L 0 = 1,

ˆ A X iA ) L i = L i + µ(α −1 Tˆ i + 

dϑ A du + X iA dt dt

(15.116)


Consider the class of the outgoing null geodesics ending at q0 ∈ H ∂− H. Then from (15.77) we have: du dϑ A = 0, lim =0 (15.117) lim t →t0 dt t →t0 dt Since limt →t0 µ = 0, it follows that for all outgoing null geodesics ending at q0 we have: lim L i = L i (q0 )

t →t0

(15.118)

We conclude that in Minkowski spacetime the tangent vectors to all outgoing null geodesics ending at q0 tend as we approach q0 to the tangent vector L(q0 ) of the generator of Cu 0 through q0 . The vector L(q0 ) is therefore an invariant vector independent of the choice of acoustical function u. From the point of view of Minkowski spacetime this vector is equal, up to a multiplicative constant, to the invariant vector V (q0 ), defined by (15.23), since the vector T (q0 ) vanishes from the point of view of Minkowski spacetime.
Consider the class of the incoming null geodesics ending at q0 ∈ H ∂− H. If q0 ∈ H, then taking into account the fact that by (15.80):   ∂µ (15.119) µ(q(τ )) = (q0 )τ + O(τ 2 ) ∂u we obtain from (15.86): lim µ

t →t0

1 du = , −2 dt (1/2)(α (q0 ) + | p/0 |2 )

lim

t →t0

ˆ A (q0 ) (h/−1 ) AB ( p/) B −  dϑ A = dt (1/2)(α −2 (q0 ) + | p/0 |2 ) (15.120)

If q0 ∈ ∂− H, then taking into account the fact that by (15.82), (15.83):   1 ∂ 2µ (q0 )τ 2 + O(τ 3 ) µ(q(τ )) = 2 ∂u 2

(15.121)

we obtain from (15.87): lim µ

t →t0

1 du = , −2 dt (1/2)(α (q0 ) + | p/0 |2 )

lim

t →t0

ˆ A (q0 ) (h/−1 ) AB ( p/) B −  dϑ A = dt (1/2)(α −2 (q0 ) + | p/0 |2 ) (15.122)

which coincides with (15.120). It follows that in both cases we have: lim L i = L i (q0 ) +

t →t0

(h/−1 ) AB (q0 )X iA (q0 )( p/0) B α −1 (q0 )Tˆ i (q0 ) + (1/2)(α −2 (q0 ) + | p/0 |2 ) (1/2)(α −2 (q0 ) + | p/0 |2 ) (15.123)

950

Chapter 15. The Nature of the Singular Hypersurface

Note that this limit depends on p/0 . For p/0 = 0 the limit vector is (L + 2α Tˆ )(q0 ) = ˆ 0 ), the conjugate to L(q0 ), the null vector in the orthogonal complement of α 2 (q0 ) L(q Tq0 St0 ,u 0 . As p/0 tends to ∞ the limit vector tends to L(q0 ).
Consider finally the class of the other null geodesics ending at q0 ∈ H ∂− H. If q0 ∈ H, then taking into account the fact that by (15.90), (15.93):   1 ∂µ ∂µ (15.124) µ(q(τ )) = (q0 )τ 2 + O(τ 3 ) 2 ∂t ∂u we obtain from (15.109): lim µ

t →t0

du = 0, dt

lim

t →t0

dϑ A =0 dt

(15.125)

If q0 ∈ ∂− H, then taking into account the fact that by (15.90), (15.95), (15.103), (15.107):    ∂µ 2 ∂ 2 µ 1 (q0 )τ 4 + O(τ 5 ) (15.126) µ(q(τ )) = 24 ∂t ∂u 2 we obtain from (15.110): lim µ

t →t0

du = 0, dt

lim

t →t0

dϑ A =0 dt

(15.127)

as well. It follows that in both cases we have simply: lim L i = L i (q0 )

(15.128)

t →t0

Thus, from the point of view of Minkowski spacetime, the tangent vectors to all the other null geodesics ending at q0 tend, as for all outgoing null geodesics ending at q0 , to the vector L(q0 ) as we approach q0 .
We now consider, for a fixed point q0 ∈ H ∂− H and each of the three classes of null geodesics ending at q0 , the mapping: (τ ; p/0) → (t, u, ϑ) = F(τ ; p/0),

F = (F t , F u , F /)

(15.129)

where for the outgoing and incoming null geodesics: p/0 ∈ Tϑ0 S 2 which may be identified with 2 , while for the other null geodesics p/0 takes values in the unit circle in Tϑ0 S 2 , which may be identified with S 1 ⊂ 2 . In each of the three cases the mapping (15.129) is smooth, the range of F t being restricted to t ≥ 0. Let us consider first the mapping: τ → F t (τ ; p/0)

(15.130)

for fixed p/0 . We shall presently show that this mapping is a diffeomorphism of [τ0 , 0] onto [0, t0 ] in the case of the outgoing null geodesics, a diffeomorphism of [τ0 , 0) onto [0, t0 ) in the case of the incoming null geodesics as well as in the case of the other null geodesics,

Chapter 15. The Nature of the Singular Hypersurface

951

the difference of the latter from the former being, in this respect, that (dt/dτ )(0) = 1 for the outgoing null geodesics (see (15.77)), while (dt/dτ )(0) = 0 for the incoming and the other null geodesics (see (15.86), (15.87), (15.109), (15.110)). Here τ0 = τ0 ( p/0). Now the expressions (15.77) and (15.86), (15.87), (15.109), (15.110), show that (dt/dτ )(τ ) > 0 for τ ∈ [τ1 , 0), τ1 suitably small. Thus either (dt/dτ )(τ ) > 0 for all τ ∈ [τ0 , 0), or there is a first τ∗ where (dt/dτ )(τ∗ ) = 0. Now in general, for a Hamiltonian of the form (15.55), the condition H = 0 is equivalent to the following condition on the tangent vector: 1 dq µ dq ν gµν =0 (15.131) 2 dτ dτ In the present case, for our Hamiltonian (15.59), the condition H = 0 is equivalent to the condition:  2  A  B  1 du 1 dt du dϑ dϑ ˆA ˆ B = 0 (15.132) + α −2 µ / AB + µ + µ + h − dτ dτ 2 dτ 2 dτ dτ It follows that at τ∗ the tangent vector vanishes: du dϑ A dt (τ∗ ) = (τ∗ ) = (τ∗ ) = 0 dτ dτ dτ hence by equations (15.61) the momentum covector vanishes as well: pt (τ∗ ) = pu (τ∗ ) = p/ A (τ∗ ) = 0 Then with: (t∗ , u ∗ , ϑ∗ ) = (t (τ∗ ), u(τ∗ ), ϑ(τ∗ )), ((t, u, ϑ); ( pt , pu , p/)) = ((t∗ , u ∗ , ϑ∗ ); (0, 0, 0)) is a solution of the Hamiltonian system (15.61)–(15.52) coinciding with our solution at τ = τ∗ . By the uniqueness theorem for the Hamiltonian system (15.61)–(15.52) our null geodesic must coincide with the above constant solution for all τ , in particular we must have: t (0) = t∗ in contradiction with the fact that t (0) = t0 > t∗ . This rules out the second alternative above and we conclude that (dt/dτ )(τ ) > 0 for all τ ∈ [τ0 , 0). It follows that the mapping (15.130) is a diffeomorphism of [τ0 , 0] onto [0, t0 ] in the case of the outgoing null geodesics, a diffeomorphism of [τ0 , 0) onto [0, t0 ) in the cases of the incoming and the other null geodesics. We can thus invert the mapping (15.130) obtaining a continuous mapping of [0, t0 ] onto [τ0 , 0] by: t → (F t )−1 (t; p/0 )

(15.133)

which is smooth on [0, t0 ] in the case of the outgoing null geodesics, on [0, t0 ) in the cases of the incoming and the other null geodesics. Consider then the mapping: (t; p/0 ) → (t, u, ϑ) = G(t; p/0 ) = F((F t )−1 (t; p/0 ); p/0)

(15.134)

952

Chapter 15. The Nature of the Singular Hypersurface

and for fixed t ∈ [0, t0 ] the mapping: p/0 → (t, u, ϑ) = G t ( p/0 ) where G t ( p/0) = G(t; p/0 )

(15.135)

where, as above, p/0 ∈ 2 for the outgoing and incoming null geodesics, while p/0 ∈ S 1 for the other null geodesics. This is the domain of the mapping (15.135), while its range is the hypersurface {t} × (−∞, ε0 ) × S 2 in the extension of the manifold Mε0 (see (15.11)) to:  M E = [0, ∞) × (−∞, ε0 ) × S 2 Meε0 = Mε0 where M E = [0, ∞) × (−∞, 0) × S 2 corresponds to the domain of the surrounding constant state. The hypersurface {t} × (−∞, ε0 ) × S 2 corresponds to the domain in the hyperplane t in Minkowski spacetime where u > ε0 . The mapping (15.135) is smooth for each t ∈ [0, t0 ), and for t = t0 , G t0 is the constant mapping: G t0 ( p/0 ) = G(t0 ; p/0) = q0 (15.136) Also, writing: G = (G t , G u , G /),

G t = (G tt , G ut , G /t )

(15.137)

we have: G tt = G t = 1 We have:



∂G /A ∂t



 (t; p/0) =

dϑ A dt

 (t) =

(15.138) (dϑ A /dτ )(τ ) (dt/dτ )(τ )

(15.139)

and:       ∂2G /A d 2ϑ A (d 2 ϑ A /dτ 2 )(dt/dτ ) − (d 2 t/dτ 2 )(dϑ A /dτ ) (t; p/0 ) = (t) = (τ ) ∂t 2 dt 2 (dt/dτ )3 (15.140) In the case of the outgoing null geodesics we obtain, from (15.73): 

∂G /A ∂t

 (t0 ; p/0) = 0

(15.141)

and, from (15.76): 

/A ∂2G ∂t 2



 (t0 ; p/0) =

 ∂µ (q0 )(h/−1 ) AB (q0 )( p/0) B ∂t

(15.142)

Chapter 15. The Nature of the Singular Hypersurface

953

Let us consider the matrix ∂ G / tA /∂( p/0) B . By (15.141), (15.142):         ∂ ∂2 ∂µ ∂G /A ∂G /A (t0 ; p/0) = 0, (t0 ; p/0 ) = (q0 )(h/−1 ) AB (q0 ) ∂t ∂( p/0) B ∂t ∂t 2 ∂( p/0 ) B (15.143) It follows that:     ∂G / tA 1 ∂µ (15.144) ( p/0 ) = (q0 )(h/−1 ) AB (q0 )((t − t0 )2 ) + O((t − t0 )3 ) ∂( p/0 ) B 2 ∂t and: / t /∂ p/0 ) = (det∂ G

1 4



∂µ ∂t

2

(q0 )deth/−1 (t − t0 )4 + O((t − t0 )5 )

(15.145)

Therefore in the case of the outgoing null geodesics the mapping (15.135) is of maximal rank at each p/0 ∈ 2 for t suitably close to t0 . By the implicit function theorem we conclude that in the case of the outgoing null geodesics the image of 2 by the mapping (15.135) is, for t suitably close to t0 , an immersed disk in the hypersurface {t} × (−∞, ε0 ) × S 2 ⊂ Meε0 , which corresponds to the hyperplane t in Minkowski spacetime. To show that this image is in fact an embedded disk we show that the mapping (15.135) is one-to-one for t suitably close to t0 . For | p/0 | ≤ B and any given B > 0, this follows from the fact that by (15.141), (15.142):   1 ∂µ (q0 )(h G / tA ( p/0 ) = /−1 ) AB (q0 )( p/0 ) B (t − t0 )2 + O((t − t0 )3 ) (15.146) 2 ∂t provided that t is suitably close to t0 depending on B. To handle the case that | p/0 | > B, we choose a different normalization of the momentum covector at q0 for the class of outgoing geodesics ending at q0 than that given by (15.70). We now choose the scale factor a so that: (15.147) | p/0 | = 1 Then pu = −1/| p/0|. This corresponds to the diffeomorphism:   p/0 1 f : p/0 → − , | p/0| | p/0 |

(15.148)

which maps the exterior of the disk of radius B in 2 onto (0, −1/B) × S 1 . We can then show that the mapping G t ◦ f −1 is one-to-one on (0, −1/B) × S 1 so that the image of (0, −1/B) × S 1 by this mapping, which is the image of the exterior of the disk of radius B by G t , is an embedded annulus in {t} × (−∞, ε0 ) × S 2 ⊂ Meε0 not intersecting the image of the disk of radius B by G t , provided that t is suitably close to t0 . It then follows that the image of 2 by G t is an embedded disk in {t} × (−∞, ε0 ) × S 2 for t suitably close to t0 . In the case of the other null geodesics, (F t )−1 is not differentiable at t = t0 , so we first investigate the mapping F (see (15.129)). We choose a basis ω1 , ω2 for Tϑ0 S 2

954

Chapter 15. The Nature of the Singular Hypersurface

which is orthonormal relative to h /−1 (q0 ) and express p/0 , which belongs to the unit circle 2 in Tϑ0 S in the form: (15.149) p/0 = ω1 cos ϕ + ω2 sin ϕ In the case that q0 ∈ H, we have, from (15.88), (15.91), (15.94):    t   ∂F 1 ∂µ ∂2 Ft t F (0; ϕ) = t0 , (0, ϕ) = (0; ϕ) = 0, (q0 ) ∂τ 2 ∂u ∂τ 2 and:  A

F / (0; ϕ) = 

/A ∂3 F ∂τ 3

ϑ0A ,



 (0; ϕ) =

∂F /A ∂τ



 (0; ϕ) =

/A ∂2 F ∂τ 2

(15.150)

 (0; ϕ) = 0

 ∂µ ∂µ (q0 )(h/−1 ) AB (q0 )( p/0 ) B (ϕ); ∂t ∂u

( p/0 ) A = (ω1 ) A cos ϕ + (ω2 ) A sin ϕ It follows that in the case that q0 ∈ H we have:   1 ∂µ F t (τ ; ϕ) = t0 + (q0 )τ 2 + O(τ 3 ) 4 ∂u   1 ∂µ ∂µ F / A (τ ; ϕ) = ϑ0A + (q0 )(h/−1 ) AB (q0 )( p/0 ) B (ϕ)τ 3 + O(τ 4 ) 6 ∂t ∂u

(15.151)

(15.152)

hence, since G / tA (ϕ) = F / A ((F t )−1 (t; ϕ); ϕ), 4 (∂µ/∂t)(q0 ) (h/−1 ) AB (q0 )( p/0 ) B (ϕ)(t0 − t)3/2 + O((t0 − t)2 ) √ 3 −(∂µ/∂u)(q0) (15.153) Moreover, (15.150) implies:    t     t ∂ ∂F ∂2 ∂ Ft ∂F (0; ϕ) = (0; ϕ) = (0; ϕ) = 0 (15.154) ∂ϕ ∂τ ∂ϕ ∂τ 2 ∂ϕ G / tA (ϕ) = ϑ0A −



hence:

∂ Ft ∂ϕ

 (τ ; ϕ) = O(τ 3 )

(15.155)

Also, (15.151) implies:   A   / ∂F /A ∂2 ∂ F (0; ϕ) = (0; ϕ) = (0; ϕ) = 0 (15.156) ∂ϕ ∂τ 2 ∂ϕ     A  ∂3 ∂ F ∂µ ∂µ / (0; ϕ) = (q0 )(h/−1 ) AB (q0 )(−(ω1 ) B sin ϕ + (ω2 ) B cos ϕ) ∂t ∂u ∂τ 3 ∂ϕ



∂F /A ∂ϕ





∂ ∂τ



Chapter 15. The Nature of the Singular Hypersurface

955

hence:  A   ∂F / 1 ∂µ ∂µ /−1 ) AB (q0 )(−(ω1 ) B sinϕ + (ω2 ) B cosϕ)τ 3 + O(τ 4 ) (τ ;ϕ) = (q0 )(h ∂ϕ 6 ∂t ∂u (15.157) Differentiating the equation F t ((F t )−1 (t; ϕ); ϕ) = t implicitly with respect to ϕ yields: ∂ F t /∂ϕ ∂(F t )−1 =− t ∂ϕ ∂ F /∂τ

(15.158)

Now (15.150) implies: 

∂ Ft ∂τ



1 (τ ; ϕ) = 2



 ∂µ (q0 )τ + O(τ 2 ) ∂u

(15.159)

Substituting (15.155) and (15.159) in (15.158) yields: ∂(F t )−1 = O(τ 2 ) ∂ϕ

(15.160)

Differentiating G / A (t; ϕ) = F / A ((F t )−1 (t; ϕ); ϕ) with respect to ϕ we obtain: ∂F / A ∂(F t )−1 ∂F /A ∂G / tA = + ∂ϕ ∂τ ∂ϕ ∂ϕ

(15.161)

Now (15.151) implies: 1 ∂F /A = ∂τ 2



 ∂µ ∂µ (q0 )(h/−1 ) AB (q0 )( p/0 ) B (ϕ) ∂t ∂u

(15.162)

In view of (15.160) and (15.162) the first term on the right in (15.161) is O(τ 4 ), while the second term is given by (15.157). We conclude that: ∂G / tA = a(h /−1 ) AB (q0 )(−(ω1 ) B sin ϕ + (ω2 ) B cos ϕ)τ 3 + O(τ 4 ) ∂ϕ where: a=

1 6



 ∂µ ∂µ (q0 ) ∂t ∂u

(15.163)

(15.164)

which gives: / AB (q0 ) h

/ tB ∂G / tA ∂ G ∂ϕ ∂ϕ

/−1 ) AB (−(ω1 ) A sin ϕ + (ω2 ) A cos ϕ)(−(ω1 ) A sin ϕ + (ω2 ) A cos ϕ)τ 6 + O(τ 7 ) = a 2 (h = a 2 τ 6 + O(τ 7 )

(15.165)

956

Chapter 15. The Nature of the Singular Hypersurface

The above hold in the case that q0 ∈ H. In the case q0 ∈ ∂− H similar calculations using (15.88), (15.95), (15.96), (15.104), (15.105), (15.108), give: G / tA (ϕ) = ϑ0A −

8 (−(∂µ/∂t)(q0))3/4 (t0 − t)5/4 + O((t0 − t)3/2 ) 5 ((1/6)(∂ 2µ/∂u 2 )(q0 ))1/4

∂G / tA = b(h /−1 ) AB (q0 )(−(ω1 ) B sin ϕ + (ω2 ) B cos ϕ)τ 5 + O(τ 6 ) ∂ϕ where: 1 b= 120 and: / AB (q0 ) h



∂µ ∂t

2

 ∂ 2µ (q0 ) ∂u 2

/ tB ∂G / tA ∂ G = b2τ 10 + O(τ 11 ) ∂ϕ ∂ϕ

(15.166)

(15.167)

(15.168)

(15.169)

The formulas (15.165) and (15.169) imply in the case of the other null geodesics ending at q0 that the mapping (15.135) is of maximal rank at each p/0 ∈ S 1 for t suitably close to t0 . By the implicit function theorem we conclude that in the case of the other null geodesics the image of S 1 by the mapping (15.135) is, for t suitably close to t0 , an immersed circle in the hypersurface {t} × (−∞, ε0 ) × S 2 ⊂ Meε0 , which corresponds to the hyperplane t in Minkowski spacetime. Moreover, we deduce from (15.153) and (15.166) that the mapping (15.135) is one-to-one for t suitably close to t0 . It then follows that the image of S 1 by the mapping (15.135) is actually an embedded circle. This circle is the boundary of the embedded disk corresponding to the outgoing null geodesics. Similar arguments show that in the case of the incoming null geodesics the image of 2 by the mapping (15.135) is an embedded disk for t suitably close to t0 . The boundary of this disk is the circle corresponding to the other null geodesics. We have thus proved the following theorem. [See Figure 2].
Theorem 15.1 Let q0 = (t0 , u 0 , ϑ0 ) be a point belonging to H ∂− H, the singular boundary of the domain of the maximal solution. Then the backward past null cone of q0 in Tq∗0 Mε0 degenerates to the three pieces: (i) The negative half-hyperplane: pt = 0, pu < 0. (ii) The negative half-hyperplace: pu = 0, pt = 0. (iii) The plane pt = pu = 0. There is a corresponding trichotomy of the past null geodesic cone of q0 into three classes of null geodesics ending at q0 . The outgoing null geodesics, the incoming null geodesics, and the other null geodesics, corresponding to (i), (ii), and (iii), respectively. Let all these null geodesics be parametrized by t. Then, in Minkowski spacetime, the following hold: The tangent vectors to all outgoing null geodesics ending at q0 tend as we approach q0 to the tangent vector L(q0 ) of the generator of Cu 0 through q0 . The vector L(q0 ) is thus an invariant null vector associated to the singular point q0 .

Chapter 15. The Nature of the Singular Hypersurface

957

q

out

in Figure 2

The tangent vector to an incoming null geodesic ending at q0 with angular momentum p/0 at q0 , tends as t → t0 to the following limit: L i (q0 ) +

(h/−1 ) AB (q0 )X iA (q0 )( p/0 ) B α −1 (q0 )Tˆ i (q0 ) + (1/2)(α −2 (q0 ) + | p/0 |2 ) (1/2)(α −2 (q0 ) + | p/0|2 )

ˆ 0 ), while as p/0 → ∞ tends to L(q0 ). which, for p/0 = 0 is equal to α 2 (q0 ) L(q The tangent vectors to all the other null geodesics ending at q0 likewise tend as t → t0 to L(q0 ). Consider for each of the pieces (i), (ii), (iii), above, and for each t ∈ [0, t0 ], the mapping: p0 → G t ( p0 ) where G t ( p0 ) is the point at parameter value t along the null geodesic ending at parameter value t0 at the point q0 with momentum covector p0 . Then for t = t0 and suitably close to t0 the following hold: The image of (i), which corresponds to the points of intersection of the outgoing null geodesics ending at q0 with the hyperplane t , is an embedded disk. The image of (ii), which corresponds to the points of intersection of the incoming null geodesics ending at q0 with the hyperplane t , is also an embedded disk.

958

Chapter 15. The Nature of the Singular Hypersurface

The image of (iii), which corresponds to the points of intersection of the other null geodesics ending at q0 with the hyperplane t , is an embedded circle, the common boundary of the two disks. Consider now
again the mapping (15.129) for the class of outgoing null geodesics ∈ H ∂− H. We now wish to make the dependence on the point q0 ∈ ending at q 0
H ∂− H explicit, so we denote F(τ ; p/0 ) by F(τ ; q0 ; p/0). What we wish to investigate presently is the change from one acoustical function u to another u  , such that the null geodesic generators of the level sets Cu  of u  , the characteristic hypersurfaces corresponding to u  , which have future end points on the singular boundary of the domain of the maximal solution, belong to the class of outgoing null geodesics ending at these points. We can view this change as generated by a coordinate transformation:


u 0 = u 0 (u 0 , ϑ0 ),

ϑ0 = ϑ0 (ϑ0 )

(15.170)

on H ∂− H, preserving the invariant curves. We also impose the condition that the transformation ϑ0 → ϑ0 is orientation preserving and that on each invariant curve the transformation u 0 → u 0 is also orientation preserving:  ∂u 0 ∂ϑ0 >0 (15.171) > 0, det ∂u 0 ∂ϑ0
If the coordinates of a point q0 ∈ H ∂− H in the unprimed system are: (t0 , u 0 , ϑ0 ), t0 = t∗ (u 0 , ϑ0 ), then the coordinates of the same point in the primed system are: (t0 , u 0 , ϑ0 ), t0 = t∗ (u 0 , ϑ0 ), where: t∗ (u 0 , ϑ0 ) = t∗ (u 0 , ϑ0 ) (15.172) hence t0 = t0 . We shall determine p/0 as a function of (u 0 , ϑ0 ), such that the hypersurfaces, given in parametric form by: t = M t (τ, u 0 , ϑ0 ) = F t (τ ; (u 0 , ϑ0 ); p/0(u 0 , ϑ0 )) u = M u (τ, u 0 , ϑ0 ) = F u (τ ; (u 0, ϑ0 ); p/0 (u 0 , ϑ0 ))

ϑ=M / (τ, u 0 , ϑ0 ) = F /(τ ; (u 0 , ϑ0 ); p/0(u 0 , ϑ0 ))

(15.173)

for constant values of u 0 , are null hypersurfaces with respect to the acoustical metric, namely the level sets Cu  of the new acoustical function u  . 0
In (15.173), (t∗ (u 0 , ϑ0 ), u 0 , ϑ0 ) are the coordinates of a point q0 ∈ H ∂− H in the primed system, hence we have: F t (0; (u 0, ϑ0 ); p/0 ) = t∗ (u 0 , ϑ0 ) F u (0; (u 0, ϑ0 ); p/0 ) = u 0 F /(0; (u 0 , ϑ0 ); p/0 ) = ϑ0

(15.174)

The hypersurfaces in question are generated by the outgoing null geodesics corresponding to constant values of (u 0 , ϑ0 ), and the hypersurfaces are null hypersurfaces if and only if

Chapter 15. The Nature of the Singular Hypersurface

959

these null geodesics are orthogonal, with respect to the acoustical metric, to the sections   Sτ,u  corresponding to constant values of τ . Let then X A be the tangent vectorfield to the 0

ϑ0A coordinate lines on these sections and, as before, L˜  be the tangent vectorfield to the null geodesics, as parametrized by τ . Then the orthogonality condition reads:

and we have:

h( L˜  , X A ) = 0 : A = 1, 2

(15.175)

d Ft ∂ d Fu ∂ dF /A ∂ L˜  = + + dτ ∂t dτ ∂u dτ ∂ϑ A

(15.176)

and: X A = X tA

∂ ∂ ∂ + X u + X B A A ∂t ∂u ∂ϑ B

(15.177)

where: ∂ Ft ∂ Ft + ∂( p/0 ) B ∂ϑ0A u ∂F ∂ Fu = + A ∂( p/0 ) B ∂ϑ0

X tA = X u A

X C A =

∂( p/0) B ∂ϑ0A ∂( p/0) B ∂ϑ0A

∂F /C ∂F /C ∂( p/0) B + ∂( p/0 ) B ∂ϑ0A ∂ϑ0A

(15.178)

Now, (15.173) defines a change of coordinates in the spacetime domain covered by the hypersurfaces in question, from (t, u, ϑ) coordinates to (τ, u 0 , ϑ0 ) coordinates, and in the new coordinates we have, simply: ∂ , L˜  = ∂τ It follows that:

X A =

∂ ∂ϑ0A

(15.179)

[ L˜  , X A ] = 0

Now by (15.60) the integral curves of the vectorfield µ−1 L˜  geodesics, hence: Let us then define:

D L˜  L˜  = µ−1 ( L˜  µ) L˜  ,

(15.180) are affinely parametrized null

h( L˜  , L˜  ) = 0

ι A = h( L˜  , X A )

We have, along the integral curves of L˜  : dι A = L˜  (h( L˜  , X A )) = h(D L˜  L˜  , X A ) + h( L˜  , D L˜  X A ) dτ and from (15.180): D L˜  X A = D X A L˜  , hence: h( L˜  , D L˜  X A ) =

1  X (h( L˜  , L˜  )) = 0 2 A

(15.181) (15.182)

960

Chapter 15. The Nature of the Singular Hypersurface

while by (15.181): h(D L˜  L˜  , X A ) = µ−1

dµ ιA dτ

It follows that:

d (µ−1 ι A ) = 0 dτ In view of the form (15.15) of the acoustical metric and (15.176) we have: d F u u d F t u d F u t XA − µ X A + α −2 µ2 X dτ dτ dτ A   B  dF / d F u  C ˆ C X u ˆB X A + µ +h / BC + µ A dτ dτ

(15.183)

ι A = −µ

At τ = 0 this reduces to:

  dF / B C X =0 (ι A )τ =0 = /hBC dτ τ =0

(15.184)

(15.185)

by (15.73). Moreover, we have:     t   /B d F u d F u t −1 C −1 d F X + X lim (µ ι A ) = − + h/ BC X A lim µ τ =0 τ →0 τ →0 dτ A dτ A τ =0 dτ (15.186) Now by (15.73):



d Ft dτ





 =0

(15.187)

  dF /A = (h/−1 ) AB (q0 )( p/0 ) B µ−1 τ →0 dτ

(15.188)

τ =0

= 1,

d Fu dτ

τ =0

and by (15.75), (15.76): lim

while by (15.174), (15.177), (15.178): (X u A )τ =0 =

∂u 0 , ∂ϑ0A

(X B A )τ =0 =

∂ϑ0B

∂ϑ0A

(15.189)

Substituting in (15.186) we obtain: lim (µ−1 ι A ) = −

τ →0

∂ϑ0B ∂u 0 + ( p / ) 0 B ∂ϑ0A ∂ϑ0A

(15.190)

Therefore the condition limτ →0 (µ−1 ι A ) = 0 is equivalent to the following definition of p/0 as a function of (u 0 , ϑ0 ): ∂u 0 ∂ϑ0B (15.191) ( p/0 ) A = ∂ϑ0B ∂ϑ0A

Chapter 15. The Nature of the Singular Hypersurface

961

Once p/0 is defined according to (15.192), equation (15.183) implies that ι A = 0 for all τ , that is, the orthogonality condition (15.175) is everywhere satisfied and the hypersurfaces of constant u 0 , given by (15.173) are outgoing characteristic hypersurfaces. In the following we denote (u 0 , ϑ0 ) simply by (u  , ϑ  ). At fixed q0 , p/0 we invert the mapping (15.130), to obtain the mapping (15.133). Let us denote: f (t  , u  , ϑ  ) = (F t )−1 (t  ; (u  , ϑ  ); p/0(u  , ϑ  ))

(15.192)

We then substitute in (15.173) to obtain the following transformation from the original acoustical coordinates (t, u, ϑ) to the new acoustical coordinates (t  , u  , ϑ  ): t = t u = N u (t  , u  , ϑ  ) = F u ( f (t  , u  , ϑ  ); (u  , ϑ  ); p/0(u  , ϑ  )) ϑ=N /(t  , u  , ϑ  ) = F /( f (t  , u  , ϑ  ); (u  , ϑ  ); p/0(u  , ϑ  ))

(15.193)

The acoustical metric has in the new acoustical coordinates a form similar to (15.15): h = −2µ dt  du  + α −2 µ2 du 2 + h/AB (dϑ A + A du  )(dϑ B + B du  )

(15.194)

where we have new coefficients α  , µ , A , and h/AB . We have, in arbitrary local coordinates (see (2.41)): α −2 = −(h −1 )µν ∂µ t  ∂ν t  = −(h −1 )µν ∂µ t∂ν t = α −2

(15.195)

so the functions α and α  coincide. The function µ is given by (see (2.10), (2.11), (2.13)): L  = µ Lˆ  ,

Lˆ µ = −(h −1 )µν ∂ν u 

(15.196)

and we have (in arbitrary local coordinates): 1 = −(h −1 )µν ∂µ t  ∂ν u  = −(h −1 )µν ∂µ t∂ν u  µ

(15.197)

Here L  , related to L˜  by (15.112), has the same integral curves as L˜  , but parametrized by t instead of τ , while the integral curves of Lˆ  are also the same but affinely parametrized. As we have shown above, the integral curves of the vectorfield µ−1 L˜  are likewise the same and affinely parametrized. Now two different affine parameters s and s  along the same geodesic, giving the same orientation, satisfy s  = as + b where a and b are constants along the geodesic, a > 0. It follows that there is a function a(u  , ϑ  ) > 0 such that: µ−1 L˜  = a Lˆ 

(15.198)

On the other hand, from the second condition of (15.196) we obtain, working in the original acoustical coordinates:     ∂u ∂ ∂ ∂u  ˆ + (15.199) lim (µ L ) = τ →0 ∂u τ =0 ∂t ∂t τ =0 ∂u

962

Chapter 15. The Nature of the Singular Hypersurface

while

∂ lim L˜  = ∂t

(15.200)

τ →0

Now, from (15.193) we obtain: ∂t ∂t ∂t = 1, = 0, =0 (15.201)   ∂t ∂u ∂ϑ A u u u u u ∂u ∂ F ∂ f ∂u ∂F ∂f ∂F ∂u ∂F ∂f ∂F = , = + , = +      A A ∂t ∂τ ∂t ∂u ∂τ ∂u ∂u ∂ϑ ∂τ ∂ϑ ∂ϑ A A A A A A A A ∂ϑ ∂F / ∂ f ∂ϑ ∂F / ∂f ∂F / ∂ϑ ∂F / ∂f ∂F /A = , = + , = + ∂t  ∂τ ∂t  ∂u  ∂τ ∂u  ∂u  ∂ϑ B ∂τ ∂ϑ B ∂ϑ B In the above, we are considering F t , F u and F / as functions of τ , u  , ϑ  , according to: F t (τ, u  , ϑ  ) = F t (τ ; (u  , ϑ  ); p/0 (u  , ϑ  )) F u (τ, u  , ϑ  ) = F u (τ ; (u  , ϑ  ); p/0(u  , ϑ  )) /(τ ; (u  , ϑ  ); p/0 (u  , ϑ  )) F /(τ, u  , ϑ  ) = F

(15.202)

At τ = 0, which corresponds to t  = t∗ (u  , ϑ  ), so that: f (t∗ (u  , ϑ  ), u  , ϑ  ) = 0

(15.203)

∂ Fu = 0, ∂τ

(15.204)

we have, from (15.73): ∂ Ft = 1, ∂τ

∂F /A =0 ∂τ

hence the second and third of (15.201) reduce at τ = 0 to: ∂u = 0, ∂t  ∂ϑ A = 0, ∂t 

∂u ∂ Fu ∂u ∂ Fu = , = ∂u  ∂u  ∂ϑ A ∂ϑ A A A A ∂ϑ ∂F / ∂ϑ ∂F /A = , =   B ∂u ∂u ∂ϑ ∂ϑ B

(15.205)

Now at τ = 0 we have, according to (15.174): F t (0, u  , ϑ  ) = t∗ (u  , ϑ  ) = t∗ (u 0 (u  , ϑ  ), ϑ0 (ϑ  ))

F u (0, u  , ϑ  ) = u 0 (u  , ϑ  ) F /(0, u  , ϑ  ) = ϑ0 (ϑ  )

(15.206)

(see also (15.170)). Hence, at τ = 0: ∂ Fu ∂u 0 ∂ Fu ∂u 0 = , = ∂u  ∂u  ∂ϑ A ∂ϑ A ∂ϑ0A ∂F /A ∂F /A = 0, = ∂u  ∂ϑ B ∂ϑ B

(15.207)

Chapter 15. The Nature of the Singular Hypersurface

963

Substituting in (15.205) we conclude that the Jacobian matrix of the transformation (15.193) is at τ = 0 given by:   1 0 0 0  0 ∂u 0 /∂u  ∂u 0 /∂ϑ 1 ∂u 0 /∂ϑ 2  ∂(t, u, ϑ)   (15.208) = 0 ∂ϑ01 /∂ϑ 1 ∂ϑ01 /∂ϑ 2  ∂(t  , u  , ϑ  )  0 0 0 ∂ϑ02 /∂ϑ 1 ∂ϑ02 /∂ϑ 2 Computing the reciprocal matrix we then find:   ∂u 1 = , ∂u τ =0 ∂u 0 /∂u 



∂u  ∂t

 τ =0

=0

(15.209)

Substituting in (15.199) and comparing with (15.198) and (15.200) we conclude that: a(u  , ϑ  ) =

∂u 0 >0 ∂u 

(15.210)

(see (15.171)) and from (15.195), (15.197) a µ = µ d F t /dτ

(15.211)

We see that the ratio µ /µ is bounded from above and below by positive constants. Also, since α  = α we have: κ µ = (15.212) κ µ Next we consider the new coefficients A and h/AB . We have (see (15.194)): ∂xµ ∂xν (15.213) ∂ϑ A ∂u    ∂ϑ B ∂ϑ C ∂ϑ B ∂u ∂u ∂ϑ B ˆ 2 ) ∂u ∂u + µ ˆB + = µ2 (α −2 + || + / h BC ∂ϑ A ∂u  ∂ϑ A ∂u  ∂ϑ A ∂u  ∂ϑ A ∂u 

/AB B = h Au  = h µν A = h

Here, From (15.208) we have:

ˆ 2=h ˆ A ˆ B, || / AB  

∂ϑ A ∂u 

ˆ A = h/ AB  ˆB 

 τ =0

=0

In
view of (15.213) this implies that  vanishes at τ = 0, that is, on the singular boundary H ∂− H, a reflection of the fact that the new acoustical coordinates (t  , u  , ϑ  ) are also canonical. Since, by (15.211), ∂µ /∂t  < 0 at τ = 0 it follows that ˆ  = µ−1   is regular at τ = 0.

(15.214)

964

Chapter 15. The Nature of the Singular Hypersurface

We have: ∂xµ ∂xν (15.215) ∂ϑ A ∂ϑ B  C  ∂u ∂u ∂ϑ C ∂u ∂ϑ ∂u ∂ϑ C ∂ϑ D ˆ 2) ˆC + µ  + = µ2 (α −2 + || + / h CD ∂ϑ A ∂ϑ B ∂ϑ B ∂ϑ A ∂ϑ A ∂ϑ B ∂ϑ A ∂ϑ B

/hAB = h µν

Let: Y  = Y A Let also:

∂ , ∂ϑ A

/h (Y  , Y  ) = h/AB Y A Y B

∂ϑ C ϑ D /h˜ AB = h /C D A , ∂ϑ ∂ϑ B

/h˜ (Y  , Y  ) = h/˜ AB Y A Y B

(15.216)

(15.217)

We then deduce from (15.215) the following lower bound for /h in terms of h/. For any vector Y  as in (15.216) it holds that /h (Y  , Y  ) ≥

/h˜ (Y  , Y  ) ˆ ) ˆ 1 + α 2 h/(,

(15.218)

Reversing the roles of the coordinates (t, u, ϑ) and (t  , u  , ϑ  ), we obtain in a similar manner a lower bound for h / in terms of h/ . Let: Y =YA Let also:

∂ , ∂ϑ A

h/(Y, Y ) = h/ AB Y A Y B

∂ϑ C ∂ϑ D  /h˜ AB = h /C D , ∂ϑ A ∂ϑ B

  /h˜ (Y, Y ) = h/˜ AB Y A Y B

(15.219)

(15.220)

Then for any vector Y as in (15.219) it holds that /(Y, Y ) ≥ h

/h (Y, Y ) ˆ ,  ˆ ) 1 + α 2 h/ (

(15.221)

We summarize the above results in the following proposition. Proposition 15.2 Let u  be any other acoustical function such that the null geodesic generators of the level sets Cu  of u  which have future end points on the singular boundary of the domain of the maximal solution belong to the class of outgoing null geodesics ending at these points. The change from the original acoustical function u to the new acoustical function u  can be viewed as generated by a coordinate transformation u 0 = u 0 (u 0 , ϑ0 ),

ϑ0 = ϑ0 (ϑ0 )


on H ∂− H preserving the invariant curves, so that both systems of acoustical coordinates are canonical. We require that the transformation ϑ0 → ϑ0 be orientation preserving

Chapter 15. The Nature of the Singular Hypersurface

965

and that on each invariant curve the transformation u 0 → u 0 be also orientation preserving. The characteristic hypersurfaces Cu  are given in parametric form in terms of the 0 original acoustical coordinates (t, u, ϑ) by: t = F t (τ, u 0 , ϑ0 ) u = F u (τ, u 0 , ϑ0 )

ϑ=F /(τ, u 0 , ϑ0 ) where:

F t (τ, u 0 , ϑ0 ) = F t (τ ; q0; p/0(q0 ))

F u (τ, u 0 , ϑ0 ) = F u (τ ; q0 ; p/0(q0 )) F /(τ, u 0 , ϑ0 ) = F /(τ : q0 ; p/0(q0 ))


Here q0 is the point on H ∂− H with coordinates (t0 , u 0 , ϑ0 ) in the original system and (t0 , u 0 , ϑ0 ) in the new system, where t0 = t∗ (u 0 , ϑ0 ) = t∗ (u 0 , ϑ0 ) = t0 Also, p/0 (q0 ) =

∂u 0 dϑ A ∂ϑ0A 0

is the angular momentum at q0 . In the above τ → (F t (τ ; q0 ; p/0), F u (τ ; q0 ; p/0), F /(τ ; q0 ; p/0 ))
is the outgoing null geodesic, parametrized by τ , ending at τ = 0 at q0 ∈ H ∂− H with angular momentum p/0 , described in the original system of acoustical coordinates. Denoting by τ = f (t  , u  , ϑ  ) the solution of F t (τ, u  , ϑ  ) = t  for fixed (u  , ϑ  ), the transformation between the original system of acoustical coordinates (t, u, ϑ) and the new system of acoustical coordinates (t  , u  , ϑ  ) is given by: t = t u = N u (t  , u  , ϑ  ) ϑ=N /(t  , u  , ϑ  ) where: N u (t  , u  , ϑ  ) = F u ( f (t  , u  , ϑ  ), u  , ϑ  ) /( f (t  , u  , ϑ  ), u  , ϑ  ) N /(t  , u  , ϑ  ) = F

966

Chapter 15. The Nature of the Singular Hypersurface

The acoustical metric has the same form in the new acoustical coordinates: ˆ A du)(dϑ B + µ ˆ B du) h = −2µdtdu + α −2 µ2 du 2 + h/ AB (dϑ A + µ ˆ A du  )(dϑ B + µ  ˆ B du  ) = −2µ dt  du  + α −2 µ2 du 2 + h/AB (dϑ A + µ  where α  = α, the ratio µ /µ is bounded above and below by positive constants, while the metrics h / and h / are uniformly equivalent in a neighborhood of the singular boundary.
We now revisit the mapping (15.135) for a fixed point q0 ∈ H ∂− H and for the class of outgoing null geodesics ending at q0 . Setting t = 0 we have the mapping of 2 into 0 by: p/0 → (0, u, ϑ) = G 0 ( p/0 ) /( f ( p/0 ); p/0)) G 0 ( p/0 ) = G(0, p/0) = (0, F u ( f ( p/0); p/0 ), F where τ = f ( p/0 ) is the solution of F t (τ ; p/0) = 0

(15.222)

Let u  be a function defined on 0 , which has a unique maximum p0 on 0 , and is smooth and without critical points on 0 \{ p0 }. We may consider u  as a function of the acoustical coordinates (0, u, ϑ) on 0 minus the origin, defined on (−∞, 1) × S 2 \ { p0 }. Consider then the following function on 2 : g( p/0 ) = u  (G( p/0))

(15.223)

Suppose that the least upper bound of this function on 2 is achieved at some p/0,M ∈ 2 . Such a maximum point corresponds by the mapping (15.222) to a point p M ∈ 0 , p M = p0 , where: / 0 ( p/0,M ) p M = (0, u M , ϑ M ), u M = G u0 ( p/0,M ), ϑ M = G and we have:

that is:

(15.224)

∂g = 0 : at p/0,M ∂( p/0 ) A

(15.225)

/ 0B ∂u  ∂ G u0 ∂u  ∂ G + = 0 : at p/0,M ∂u ∂( p/0 ) A ∂ϑ B ∂( p/0) A

(15.226)

   Consider the level surface S0,u  of u on 0 through the point p M . We shall show that L is  − h-orthogonal to S0,u  at p M . This is equivalent to the following. Denoting by C (q0 ) the − past null geodesic cone of the point q0 on the singular boundary, and by Cout (q0 ) the part − of C (q0 ) corresponding to the outgoing null geodesics ending at q0 , then the statement −  is that the surfaces Cout (q0 ) and S0,u  have the same tangent plane at p M :

   −  (q0 ) 0 = T p M S0,u T p M Cout 

(15.227)

To show this, we shall produce a basis for the left-hand side and show that this is also a basis for the right-hand side.

Chapter 15. The Nature of the Singular Hypersurface

967

Consider the vectorfields: YA =

∂ Fu ∂ ∂F /B ∂ Ft ∂ ∂ + + ∂( p/0 ) A ∂t ∂( p/0) A ∂u ∂( p/0 ) A ∂ϑ B

: A = 1, 2

(15.228)

− These are tangential to the sections of Cout (q0 ) which correspond to constant values of −  τ , hence are h-orthogonal to L˜ . For each point p ∈ Cout (q0 ), (Y 1 ( p), Y 2 ( p)) is a basis − for the tangent plane at p to the τ = const. section of Cout (q0 ) through p, provided that the vectors Y 1 ( p), Y 2 ( p) are linearly independent, that is, provided that the point p is not conjugate to q0 . We project the Y A to the t as follows. We find suitable functions λ A , A = 1, 2, such that the vectorfields:

Y A = Y A − λ A L 

(15.229)

are tangential to the t . The vectorfields Y A , being h-orthogonal to L  hence tangential  − − to Cout (q0 ), shall then be tangential to the sections Cout (q0 ) t . As L  is given by (see (15.176)): ∂ F u /∂τ ∂ ∂F / A /∂τ ∂ ∂ + + (15.230) L = ∂t ∂ F t /∂τ ∂u ∂ F t /∂τ ∂ϑ A we find: ∂ Ft λA = (15.231) ∂( p/0 ) A and:   ∂ F t A ∂ Fu ∂ Ft ∂ Ft ∂ Fu ∂ Y = − ∂τ ∂( p/0 ) A ∂τ ∂( p/0) A ∂τ ∂u   B t t B ∂F ∂F ∂ / ∂F / ∂F − (15.232) + ∂( p/0) A ∂τ ∂( p/0 ) A ∂τ ∂ϑ B  −   (q0 ) 0 , provided that p M is not Then (Y 1 ( p M ), Y 2 ( p M )) is a basis for T p M Cout conjugate to q0 . Now, we have, from (15.222): ∂ G u0 ∂ Fu ∂ f ∂ Fu = + ∂( p/0) A ∂τ ∂( p/0) A ∂( p/0 ) A ∂G / 0B ∂F /B ∂ f ∂F /B = + ∂( p/0) A ∂τ ∂( p/0) A ∂( p/0) A

(15.233)

Also, differentiating the equation F t ( f ( p/0 ); p/0) = 0 implicitly with respect to p/0 yields: ∂ F t /∂( p/0 ) A ∂f =− ∂( p/0) A ∂ F t /∂τ Substituting in (15.233) we then obtain:   ∂ G u0 1 ∂ Fu ∂ Ft ∂ Ft ∂ Fu = − + ∂( p/0) A ∂τ ∂( p/0) A ∂τ ∂( p/0) A ∂ F t /∂τ   ∂G / 0B /B ∂F /B ∂ F t ∂ Ft ∂ F 1 = − + t ∂( p/0) A ∂τ ∂( p/0 ) A ∂τ ∂( p/0) A ∂ F /∂τ

(15.234)

(15.235)

968

Chapter 15. The Nature of the Singular Hypersurface

Comparing with (15.232) we see that: Y A =

∂G / 0B ∂ G u0 ∂ ∂ + ∂( p/0) A ∂u ∂( p/0) A ∂ϑ B

(15.236)

hence (15.226) reads, simply: Y A ( p M )u  = 0

: A = 1, 2

(15.237)

 , provided that the vectors We conclude that (Y 1 ( p M ), Y 2 ( p M )) is a basis for T p M S0,u  1 2 Y ( p M ), Y ( p M ) are linearly independent, that is, provided that the point p M is not conjugate to q0 . We have thus demonstrated (15.227) under the condition that p M is not conjugate to q0 . However, the equality (15.227) holds even in the case that p M is conjugate to q0 , but in this case we must approach p M , which corresponds to τ = f ( p/0,M ) along the outgoing null geodesic ending at q0 with angular momentum p/0,M , by a sequence of non-conjugate points corresponding to τ = τn → f ( p/0,M ) along the same geodesic.  The equality (15.227), is equivalent to L  being h-orthogonal to S0,u  at p M . Thus,  extending u to an acoustical function so that its level sets are outgoing characteristic   , the generator of the outgoing characteristic hypersurfaces Cu  with Cu  0 = S0,u  hypersurface Cu  through p M coincides with the outgoing null geodesic ending at q0 with angular momentum p/0,M . Let us now consider again our main problem from a more general point of view. Our problem is the initial value problem for the equations of motion of a perfect fluid, with initial data on the hyperplane 0 which coincide with those of a constant state outside a compact subset P0 ⊂ 0 . We may take S0,0 to be any sphere in 0 containing P0 . The construction of Chapter 2 then gives an acoustical function u based on S0,0 . If the initial data is isentropic and irrotational, Theorem 13.1 can be applied after suitable translation and rescaling, giving a development of the initial data in a domain Vε0 as in (15.7) corresponding to the annular region on 0 bounded by S0,0 and the concentric sphere of radius 1 − ε0 that of S0,0 . Then by the local uniqueness theorem the union of all the developments each of which corresponds to a sphere S0,0 containing P0 is also a development of the initial data, contained in the maximal development, the union of the domains Vε0 being contained in the closure of the domain of the maximal solution. In the general case, where the initial data is neither irrotational nor isentropic, we first apply the first part of Theorem 14.2 to cover an initial time interval of duration equal to the radius of S0,0 divided by η0 . After this time interval a suitable isentropic irrotational annular region forms and Theorem 13.1 can again be applied. Consider now a point q0 belonging to the singular boundary of the domain of the maximal solution such that q0 ∈ Vε0 where Vε0  . is the domain associated to the acoustical function u  , corresponding to the sphere S0,0  = inf      Thus q0 ∈ tε for some ε ∈ (0, ε0 ), where t∗ε u ∈[0,ε] t∗ (u ) and t∗ (u ) is the least ∗ε  upper bound of the extent of the generators of Cu  in the domain of the maximal solution. Then for any given acoustical function u, corresponding to a given sphere S0,0 , the above argument (with the roles of u and u  reversed) shows that there is an outgoing characteris , tic hypersurface Cu associated to u and a generator of Cu reaching q0 , thus of extent t∗ε

Chapter 15. The Nature of the Singular Hypersurface

969

contained in Vε0 , hence in the union of the domains Vε0 . In view of the properties of the transformation from one acoustical function to another, expounded by Proposition 15.2, Theorem 13.1 covers the union of the domains Vε0 . However we can extend the domain covered by Theorem 13.1 still further in the following manner. We can replace the hyperplanes t , by the hyperplanes α;t where for any fixed α = (α1 , α2 , α3 ) ∈ 3 such that  α = α12 + α22 + α32 < 1/η0 and any t ∈ , α;t is the hyperplane in Minkowski spacetime given by: x0 = t +

3 

αi x i

i=1

These hyperplanes are space-like with respect to the acoustical metric h 0 associated to the surrounding constant state: h 0 = −η02 (d x 0 )2 +

3 

(d x i )2

i=1

In fact, the hyperplane α;t together with the metric h 0 , induced from h 0 , is isometric to Euclidean space. The hyperplane α;0 is then space-like relative to the actual acoustical metric h provided that the departure of the data induced on α;0 from those of the constant state is suitably small. We then consider as above a sphere S0,0 in 0 such that the initial data on 0 coincide with those of the constant state outside S0,0 , and proceed to construct the corresponding acoustical function u. The surfaces Sα;t,0 = C0 α;t are then round spheres. The first variations are defined exactly as before, for, these are adapted to the rest frame of the constant state. Of the commutation fields T, Q, Ri : i = 1, 2, 3, which define the higher order variations, Q is defined exaclty as before, T is now defined to be  tangential to the hyperplanes α;t , h-orthogonal to the surfaces Sα;t,u = Cu α;t and satisfying T u = 1, while the Ri : i = 1, 2, 3 are defined to be the h-orthogonal projections to the surfaces Sα;t,u , of the rotations associated to h 0 which leave the hyperplanes α;t and the spheres Sα;t,0 invariant. We note that the hyperplanes α;t remain space-like relative to the acoustical metric provided that the initial departure from the constant state is suitably small. We obtain in this way an extension of Theorem 13.1 and a domain Vα;ε0 defined in a similar manner as the domain Vε0 , but with the hyperplanes α;t in the role of the hyperplanes t . The domain covered by Theorem 13.1 is then maximally extended to be the union of all the domains Vα;ε0 , and the remarks of the previous paragraph apply to this maximal extension. We now consider a given component of H and its past boundary, the corresponding component of ∂− H. By Proposition 15.1 these are smooth 3-dimensional and 2dimensional submanifolds of Mε0 , respectively. We shall now investigate the image of the component of H and the image of the corresponding component of ∂− H in Minkowski spacetime. For clarity we distinguish these images from the sets themselves and we

970

Chapter 15. The Nature of the Singular Hypersurface

denote by H and ∂− H the image in Minkowski spacetime of H and ∂− H respectively. Thus we are considering a component of H and the corresponding component of ∂− H , the images in Minkowski spacetime of the given component of H and its past boundary, the corresponding component of ∂− H. Now by Proposition 15.1 ∂− H is a space-like surface in (Mε0 , h) intersecting the invariant curves transversally. Thus there is a smooth function u ∗ defined on a connected domain B ⊂ S 2 , such that the component of ∂− H is given in canonical acoustical coordinates by: (15.238) {(t∗ (u ∗ (ϑ), ϑ), u ∗ (ϑ), ϑ) : ϑ ∈ B} and the corresponding component of H is given by: {(t∗ (u, ϑ), u, ϑ) : u ∗ (ϑ) > u > u(ϑ), ϑ ∈ B}

(15.239)

where a given invariant curve, parametrized by u, either extends to any u > 0, in which case u(ϑ) = 0, or ends at u = u(ϑ), the point of intersection with the incoming characteristic hypersurface C generated by the incoming null normals to the next outward component of ∂− H. Consider then the component of ∂− H , which is the image of (15.238) in Minkowski spacetime. This is the image in Minkowski spacetime of the domain B ⊂ S 2 under the mapping: ϑ → x = q(ϑ), where q µ (ϑ) = x µ (t∗ (u ∗ (ϑ), ϑ), u ∗ (ϑ), ϑ) : µ = 0, 1, 2, 3 (15.240) This mapping is one-to-one. We shall presently show that the mapping is of maximal rank. Consider the 2-dimensional matrix m with entries: µ

m AB = h µν X A

∂q ν ∂ϑ B

(15.241)

Now, we have (see (15.115)): ∂x0 ∂xi = 1, = Li ∂t ∂t ∂x0 ∂x0 ∂xi ˆ A X iA ), = = µ(α −1 Tˆ i +  = 0, A ∂u ∂ϑ ∂u

∂xi = X iA ∂ϑ A

(15.242)

and from (15.240):

Now on H




∂t∗ ∂u ∗ ∂t∗ ∂q 0 = + B B ∂ϑ ∂u ∂ϑ ∂ϑ B   j j ∂t∗ ∂u ∗ ∂ x j ∂u ∗ ∂q ∂x ∂t∗ ∂x j + = + + ∂ϑ B ∂t ∂u ∂ϑ B ∂ϑ B ∂u ∂ϑ B ∂ϑ B

(15.243)

∂− H we have µ = 0, therefore, from (15.242):
∂xi = 0 : on H ∂− H ∂u

(15.244)

Chapter 15. The Nature of the Singular Hypersurface

971

Moreover by Proposition 15.1: ∂t∗ = 0 : on ∂− H ∂u

(15.245)

Thus substituting in (15.243) from (15.242) and taking account of (15.244) and (15.245), we obtain: ∂t∗ ∂t∗ ∂q j ∂q 0 j = , = Lj + XB B ∂ϑ ∂ϑ B ∂ϑ B ∂ϑ B or, simply: ∂t∗ ∂q ν = L ν B + X νB (15.246) B ∂ϑ ∂ϑ Substituting this in (15.241) we obtain simply: m AB = h(X A , X B ) = h/ AB

(15.247)

Hence detm = deth / > 0 and the mapping (15.240) is of maximal rank. It follows that this mapping is a diffeomorphism onto its image and its image, the component of ∂− H , is a smooth embedded 2-dimensional submanifold in Minkowski spacetime. The component of H can similarly be shown to be a smooth embedded 3-dimensional submanifold in Minkowski spacetime except along its past boundary, the corresponding component of ∂− H . To analyze the behavior of H in a neighborhood of ∂− H , we shall derive an analytic expression for an arbitrary invariant curve of H in a neighborhood of ∂− H . Now, the parametric equations of an invariant curve of H , in rectangular coordinates in Minkowski spacetime, the parameter being u, are (see (15.239)): x µ = x µ (t∗ (u, ϑ), u, ϑ)

: u ∗ (ϑ) > u > u(ϑ), ϑ = const.

(15.248)

or, since x 0 = t, x 0 = t∗ (u, ϑ) x i = x i (t∗ (u, ϑ), u, ϑ)

: u ∗ (ϑ) > u > u(ϑ), ϑ = const.

(15.249)

We have:     dx0 ∂t∗ (u, ϑ) (u) = du ∂u  i   i   i dx ∂t∗ ∂x ∂x (u) = (t∗ (u, ϑ), u, ϑ) (u, ϑ) + (t∗ (u, ϑ), u, ϑ) du ∂t ∂u ∂u Substituting from (15.242) and taking account of (15.244) we obtain:     dx0 ∂t∗ (u) = (u, ϑ) du ∂u  i   dx ∂t∗ i (u) = L (t∗ (u, ϑ), u, ϑ) (u, ϑ) du ∂u

(15.250)

972

or simply:

Chapter 15. The Nature of the Singular Hypersurface



   ∂t∗ dxµ (u) = L µ (t∗ (u, ϑ), u, ϑ) (u, ϑ) du ∂u

(15.251)

We see again that in Minkowski spacetime the vectorfield L is tangential to the invariant curves. We also see that at u = u ∗ (ϑ), that is, on ∂− H , the right-hand sides vanish by (15.245), so the past end point of an invariant curve is a singular point. To analyze the behavior at the past end point we consider the second and third derivatives at this point. Differentiating (15.250) with respect to u we obtain:     ∂ 2 t∗ d2x 0 (u) = (u, ϑ) du 2 ∂u 2    i   d2x i ∂L ∂t∗ 2 (u) = (t (u, ϑ), u, ϑ) (u, ϑ) ∗ ∂t ∂u du 2   i  ∂t∗ ∂L + (t∗ (u, ϑ), u, ϑ) (u, ϑ) ∂u ∂u   ∂ 2 t∗ i (u, ϑ) (15.252) +L (t∗ (u, ϑ), u, ϑ) ∂u 2 Also, differentiating (15.48) with respect to u we obtain:    −2 ∂ 2 t∗ ∂µ (u, ϑ) = − (t∗ (u, ϑ), u, ϑ) 2 ∂t ∂u      ∂µ ∂ 2 µ ∂t∗ ∂ 2µ ∂ 2µ ∂µ ∂ 2 µ ∂t∗ − + + · ∂t ∂u∂t ∂u ∂u ∂t 2 ∂u ∂u∂t ∂u 2 Substituting for ∂t∗ /∂u from (15.48) then yields:   ∂ 2 t∗ (u, ϑ) (15.253) ∂u 2  −3  2 2  2 2  ∂µ ∂µ ∂µ ∂µ ∂ 2 µ ∂µ ∂ µ ∂ µ =− + −2 (t∗ (u, ϑ), u, ϑ) 2 ∂t ∂t ∂u ∂t ∂u ∂u∂t ∂u ∂t 2 In particular, at u = u ∗ (ϑ) we obtain, in accordance with (15.51):   ∂ 2 t∗ (u ∗ (ϑ), ϑ) = a(ϑ) ∂u 2 where:



∂ 2 µ/∂u 2 a(ϑ) = − ∂µ/∂t

(15.254)

 (q(ϑ)) > 0

(15.255)

Chapter 15. The Nature of the Singular Hypersurface

973

by the first equation of (15.45), and we denote by q(ϑ) = (t∗ (u ∗ (ϑ), ϑ), u ∗ (ϑ), ϑ)

(15.256)

the end point on ∂− H . By (15.245) and (15.254), at u = u ∗ (ϑ) (15.242) becomes:   d2x0 (u ∗ (ϑ)) = a(ϑ) du 2   d2xi (15.257) (u ∗ (ϑ)) = L i (q(ϑ))a(ϑ) du 2 Moreover, differentiating (15.252) with respect to u and evaluating the result at u = u ∗ (ϑ) we obtain, in view of (15.245):     d3x0 ∂ 3 t∗ (u ∗ (ϑ)) = (u ∗ (ϑ),ϑ) (15.258) du 3 ∂u 3        i d3x i ∂ 2 t∗ ∂L ∂ 3 t∗ i (u ∗ (ϑ)) = 2 (u ∗ (ϑ),ϑ) + L (q(ϑ)) (u ∗ (ϑ),ϑ) (q(ϑ)) ∂u du 3 ∂u 2 ∂u 3 We shall presently determine the coefficients:   ∂ 3 t∗ (u ∗ (ϑ), ϑ) and ∂u 3

∂ Li (q(ϑ)) = T L i (q(ϑ)) ∂u

In view of the fact that for any smooth function f (t, u, ϑ) we have: ∂f d ∂ f ∂t∗ f (t∗ (u, ϑ), u, ϑ) = + du ∂t ∂u ∂u hence by (15.245): 

d f (t∗ (u, ϑ), u, ϑ) du

 u=u ∗ (ϑ)

=

∂f (q(ϑ)), ∂u

differentiating (15.253) with respect to u and evaluating the result at u = u ∗ (ϑ) yields:   ∂ 3 t∗ (u ∗ (ϑ), ϑ) = b(ϑ) (15.259) ∂u 3 where:

 b(ϑ) =

∂µ ∂t

−2 

 ∂ 2µ ∂ 2µ ∂µ ∂ 3 µ 3 2 (q(ϑ)) − ∂t ∂u 3 ∂u ∂t∂u

(15.260)

Next, T L i is given by equation (3.178) of Chapter 3: T L i = aT Tˆ i + bTi

(15.261)

974

Chapter 15. The Nature of the Singular Hypersurface

where aT is the function given by (3.185) and bTi are
the rectangular components of the St,u -tangential vectorfield given by (3.189). On H ∂− H , in particular at q(ϑ), these formulas reduce to: aT = α −1 Lµ < 0  $i dH ψ L ψ(T σ ) (15.262) bTi = d/µ − dσ
the last in view of the fact that ζ  = 0 on H ∂− H . Substituting the above as well as (15.254) in (15.258) we obtain:   d3x0 (u ∗ (ϑ)) = b(ϑ) du 3   d3x i (u ∗ (ϑ)) = L i (q(ϑ))b(ϑ) + 2a(ϑ)(aT Tˆ i + bTi )(q(ϑ)) (15.263) du 3 In view of (15.257) and (15.263) we obtain the following analytic expression for an invariant curve of H in a neighborhood of its origin q(ϑ) ∈ ∂− H : 1 1 x 0 (u) = x 0 (q(ϑ)) + a(ϑ)(u − u ∗ (ϑ))2 + b(ϑ)(u − u ∗ (ϑ))3 2 6 +O((u − u ∗ (ϑ))4 ) 1 x i (u) = x i (q(ϑ)) + a(ϑ)L i (q(ϑ))(u − u ∗ (ϑ))2 2  1 + b(ϑ)L i (q(ϑ)) + 2a(ϑ)(aT Tˆ i + bTi )(q(ϑ)) (u − u ∗ (ϑ))3 6 +O((u − u ∗ (ϑ))4 ) (15.264) We summarize the above results in the following proposition. [See again Figure 1]. Proposition 15.3 The
singular boundary of the domain of the maximal solution in Minkowski spacetime is H ∂− H . Consider a given component of H and its past boundary, the corresponding component of ∂− H . The component of ∂− H is a smooth 2-dimensional embedded submanifold in Minkowski spacetime, which is space-like with respect to the acoustical metric. The component of H is a smooth embedded 3-dimensional submanifold in Minkowski spacetime ruled by invariant curves of vanishing arc length with respect to the acoustical metric, having past end points on the corresponding component of ∂− H . In a neighborhood of its origin at the point q(ϑ) ∈ ∂− H an invariant curve is given, in parametric form, by: 1 1 x 0 (u) = x 0 (q(ϑ)) + a(ϑ)(u − u ∗ (ϑ))2 + b(ϑ)(u − u ∗ (ϑ))3 2 6 + O((u − u ∗ (ϑ))4 )

Chapter 15. The Nature of the Singular Hypersurface

1 x i (u) = x i (q(ϑ)) + a(ϑ)L i (q(ϑ))(u − u ∗ (ϑ))2 2  1 i b(ϑ)L (q(ϑ)) + 2a(ϑ)(aT Tˆ i + bTi )(q(ϑ)) (u − u ∗ (ϑ))3 + 6 + O((u − u ∗ (ϑ))4 ) or, in non-parametric form, by: x i − x i (q(ϑ)) = L i (q(ϑ))(x 0 − x 0 (q(ϑ)) 23/2 1 aT Tˆ i + bTi )(q(ϑ))(x 0 − x 0 (q(ϑ)))3/2 − 3 (a(ϑ))1/2 +O((x 0 − x 0 (q(ϑ)))2 ) where:



 ∂ 2 µ/∂u 2 a(ϑ) = − (q(ϑ)) > 0 ∂µ/∂t   −2  2 ∂ µ ∂ 2µ ∂µ ∂ 3 µ ∂µ 3 2 (q(ϑ)) − b(ϑ) = ∂t ∂t ∂u 3 ∂u ∂t∂u aT (q(ϑ)) = (α −1 Lµ)(q(ϑ)) < 0  $i dH ψ L ψ(T σ ) bTi (q(ϑ)) = d/µ − (q(ϑ)) dσ

975

Epilogue

In this monograph we have studied the maximal solution of the equations of motion of a perfect fluid in Minkowski spacetime arising from initial data on a space-like hyperplane which coincides with the initial data of a constant state outside a bounded domain. We have also studied the boundary of the domain of the maximal solution and we have
shown that this boundary consists of a singular part ∂− H H and a regular part C. We have shown that in acoustical coordinates the solution extends smoothly to the boundary, but the function µ vanishes on the singular part of the boundary. On the other hand, µ is positive on the regular part and the solution extends smoothly to this part also in rectangular coordinates. We have shown moreover that each component of ∂− H is a smooth 2-dimensional embedded submanifold in Minkowski spacetime which is space-like with respect to the acoustical metric, while the corresponding component of H is a smooth embedded 3-dimensional submanifold in Minkowski spacetime ruled by invariant curves of vanishing arc length with respect to the acoustical metric, having past end points on the component of ∂− H . The corresponding component of C is the incoming null hypersurface associated to the component of ∂− H . It is ruled by incoming null geodesics of the acoustical metric with past end points on the component of ∂− H . [See Figure 3]. Now the notion of maximal development of the initial data, on which we have based our entire investigation, is perfectly reasonable from the mathematical
point of view, and also the correct notion from the physical point of view up to C ∂− H , however it is not the correct notion from the physical point of view up to H . Let us consider a given component of ∂− H and the corresponding components of C and H . The actual physical problem associated to the given component of ∂− H is the following: Find a hypersurface K in Minkowski spacetime, lying in the past of the component of H and with the same past boundary, namely the component of ∂− H , and the same tangent hyperplane at each point along this boundary, and a solution ( p, s, u) of the equations of motion in the domain in Minkowski spacetime bounded in the past by C and K , such that on C ( p, s, u) coincides with the data induced by the maximal solution, while across K there are jumps, relative to the data induced from the maximal solution, which satisfy the jump conditions following from the integral form of the conservation laws. The hypersurface K is to be space-like relative to the acoustical metric induced by the maximal solution, which holds in the past of K , and time-like relative to the new solution, which holds in the future of K .

978

Epilogue

H C

K @ H

Figure 3

We call this the shock development problem. The shock is the hypersurface of discontinuity K . The last condition of the problem is necessary for the solution to be uniquely determined by the data. We shall show below that, at least in the case of a small initial departure from the constant state, this condition is equivalent to the condition that the jump in entropy across K is positive. Thus the requirement of determinism and that of the second law of thermodynamics coincide in the present context, a fact which recalls the interpretation of entropy increase as loss of information. The shocks corresponding to different components of ∂− H will interact once the hypersurface K corresponding to a given component of ∂− H meets the component of C associated to the next outward component of ∂− H . The development of shocks and their subsequent interaction must be the subject of another monograph. We remark here that although the notion of maximal solution is not physical up to H , it does provide the basis for treating the actual physical problem, giving a barrier at H which is indispensable for controlling its solution. In concluding the present monograph we shall give a derivation of the jump conditions, establish the equivalence of the last condition of the shock development problem with the positivity of the entropy jump, and derive a formula for the jump in vorticity across K , which shows that, while the flow is irrotational before the shock, it acquires vorticity immediately after. We begin with the derivation of the jump conditions. Note that by the last condition of the shock development problem and the fact that the sound speed is less than the speed of light in vacuum, that is, η < 1, the hypersurface K is time-like with respect to the Minkowski metric g. Let then, in general, K be a C 1 hypersurface in Minkowski spacetime, which is time-like with respect to the Minkowski metric g, with a neighborhood U such that the energy-momentum-stress tensor T µν and the particle current I µ (see Chapter 1, equations (1.5), (1.6)) are continuous in each connected component of the complement

Epilogue

979

of K in U but are not continuous across K . In the following we make use of an arbitrary local coordinate system in U . Let Nµ be a covector at x ∈ K , the null space of which is the tangent space of K at X. Then denoting by [ ] the jump across K at x, we have the jump conditions: 1

[I µ ]Nµ = 0 2 T µν Nν = 0

(E.1) (E.2)

We shall presently show that these follow from the integral form of the conservation laws ∗ dual to the vectorfield I µ , that is, (1.5), (1.6). Consider the 3-form Iαβγ ∗ Iαβγ = I µ µαβγ

where µαβγ is the volume 4-form of the Minkowski metric g. In terms of I ∗ equation (1.8) becomes: (E.3) dI∗ = 0 Also, given any vectorfield X we can define the vectorfield: P µ = gαβ X α T βµ

(E.4)

By virtue of (1.9), P µ satisfies: ∇µ P µ =

1 πµν T µν , 2

πµν = (L X g)µν

(E.5)

In terms of the 3-form P ∗ dual to P ∗ = P µ µαβγ Pαβγ

equation (E.5) reads: d P∗ =

1 (π · T )  2

(E.6)

Consider now an arbitrary point  x ∈ K and let U be a neighborhood of x in Minkowski spacetime. We denote B = K U . Let Y be a vectorfield without critical points in some larger neighborhood U0 ⊃ U and transversal to K . Let L δ (y) denote the segment of the integral curve of Y through y ∈ B corresponding to the parameter interval (−δ, δ): L δ (y) = { f s (y) : s ∈ (−δ, δ)} where fs is the local 1-parameter group of diffeomorphisms generated by Y . We then define the neighborhood Vδ of x in Minkowski spacetime by: Vδ =

 y∈B

L δ (y)

980

Epilogue

Integrating equations (E.3) and (E.6) in Vδ and applying Stokes’ theorem we obtain: I∗ = 0 (E.7) ∂ Vδ 1 P∗ = (E.8) (π · T )  ∂ Vδ Vδ 2 Now the boundary of Vδ consists of the hypersurfaces: Bδ = { f δ (y) : y ∈ B},

B−δ = { f −δ (y) : y ∈ B}

together with the lateral hypersurface: 

L δ (y)

y∈∂ B

Since this lateral component and Vδ are bounded in measure by a constant multiple of δ, taking the limit δ → 0 in (E.7), (E.8) we obtain: [I ∗ ] = 0 (E.9) B [P ∗ ] = 0 (E.10) B

These being valid for any neighborhood B of x in K implies that the corresponding 3forms induced on k from the two sides coincide at x, or, equivalently, that: [I µ ]Nµ = 0,

[P µ ]Nµ = 0

The first of these equations coincides with (E.1), while the second, for four vectorfields X constituting at x a basis for the tangent space of Minkowski spacetime at x, yields (E.2). The normal vector N µ = (g −1 )µν Nν is space-like and we can normalize it to be of unit magnitude: gµν N µ N ν = 1 We must still determine the orientation of K . Let N µ point from one side of K , which we label 1 and which we call behind, to the other side of K , which we label 0 and we call ahead. Then for any quantity q, we have: [q] = q1 − q0

(E.11)

u ⊥ = −g(u, N)

(E.12)

n 1 u ⊥1 = n 0 u ⊥0 = f

(E.13)

If we define then the jump condition (E.1) reads:

Epilogue

981

The quantity f is called the particle flux . In general, if f = 0 the orientation of N is chosen so that f > 0, that is, the fluid particles cross the hypersurface of discontinuity K from the state ahead to the state behind . Then the state ahead corresponds to the past of K and the state behind to the future of K . If f = 0 the choice of orientation of N is merely conventional and the discontinuity is called a contact discontinuity. In the case of the shock development problem the state ahead corresponds to the domain where the maximal solution holds, while the state behind corresponds to the domain where the new solution holds, and everywhere on K we have f > 0. We shall restrict attention to this case in the following. Let us denote by v the volume per particle: v=

1 n

(E.14)

u ⊥1 = f v1

(E.15)

In terms of v, (E.13) reads: u ⊥0 = f v0 ,

For notational √ simplicity the enthalpy per particle shall be denoted in the following by h instead of σ : (ρ + p) √ = σ h= (E.16) n The basic thermodynamic variables being taken to be p and s, we have (see (1.17)): dh = vd p + θ ds

(E.17)

(ρ1 + p1 )u 1 u ⊥1 − p1 N = (ρ0 + p0 )u 0 u ⊥0 − p0 N

(E.18)

The jump condition (E.2) reads:

Here N stands for the vector N µ . Substituting (E.13) into (E.18) the latter reduces to: f h 1 u 1 − p1 N = f h 0 u 0 − p0 N

(E.19)

We may also write (E.19) in the form: fβ1 + p1 N = fβ0 + p0 N

(E.20)

Here N stands for the covector Nµ and βµ is the 1-form defined by (1.44): βµ = −hu µ ,

u µ = gµν u ν

(E.21)

Taking the g-inner product of (E.19) with N we obtain: f h 1 u ⊥1 + p1 = f h 0 u ⊥0 + p0 Substituting from (E.15) this becomes: p1 − p0 = − f 2 (h 1 v1 − h 0 v0 )

(E.22)

982

Epilogue

On the other hand, taking the inner product of each side of (E.19) with itself we obtain: p12 − p02 = f 2 (h 21 − h 20 − 2h 1 v1 p1 + 2h 0 v0 p0 )

(E.23)

Equations (E.22) and (E.23) together imply that whenever f = 0, as is the case under consideration, the following relation holds: h 21 − h 20 = (h 1 v1 + h 0 v0 )( p1 − p0)

(E.24)

This is the relativistic Hugoniot relation, first derived by A. Taub [Ta]. We now consider the last condition of the shock development problem, namely the condition that the hypersurface K is space-like relative to the acoustical metric of the state ahead, and time-like relative to the acoustical metric of the state behind. At any given x ∈ K , the first part of the condition stipulates that the covector Nµ belongs to the interior of the sound cone in the cotangent space at x corresponding to the state ahead: αβ (h −1 0 ) Nα Nβ < 0

(E.25)

while the second part stipulates that the same covector belongs to the exterior of the sound cone in the cotangent space at x corresponding to the state behind: αβ (h −1 1 ) Nα Nβ > 0

(E.26)

Here (h −1 )αβ are the components of the reciprocal acoustical metric. From (1.36) and (E.12) we obtain:   1 (h −1 )αβ Nα Nβ = 1 − − 1 (u ⊥ )2 (E.27) η2 therefore the conditions (E.25) and (E.26) are: u ⊥0 > 

η0 1 − η02

,

u ⊥1 < 

η1

(E.28)

1 − η12

Substituting from (E.15), these become: η0 /v0 f >  , 1 − η02

η1 /v1 f <  1 − η12

(E.29)

We conclude that the last condition of the shock development problem reduces, when the other conditions are satisfied, to: η1 /v1 η0 /v0  <  1 − η02 1 − η12

(E.30)

To relate the condition (E.30) to the entropy condition: [s] = s1 − s0 > 0

(E.31)

Epilogue

983

we expand [h] = h 1 − h 0 using (E.17) in powers of [ p] = p1 − p0 and [s]. We have:     1 ∂v 1 ∂ 2v 2 [h] = v0 [ p] + [ p] + [ p]3 + θ0 [s] + O([ p]4 ) + O([ p][s]) 2 ∂p 0 6 ∂ p2 0 (E.32) hence: h 21

− h 20

    ∂v 2 = 2h 0 v0 [ p] + h 0 + v0 [ p]2 (E.33) ∂p 0       h0 ∂ 2v ∂v + [ p]3 + 2h 0 θ0 [s] + O([ p]4 ) + O([ p][s]) + v0 3 ∂ p2 ∂p 0 0

Also [v] = v1 − v0 , is expanded as:     ∂v 1 ∂ 2v [v] = [ p] + [ p]2 + O([ p]3 ) + O([s]) ∂p 0 2 ∂ p2

(E.34)

0

hence:

    ∂v + v02 [ p]2 (E.35) (h 1 v1 + h 0 v0 )( p1 − p0 ) = 2h 0 v0 [ p] + h 0 ∂p 0       h0 ∂ 2v 3v0 ∂v + + [ p]3 + O([ p]4 ) + O([ p][s]) 2 ∂ p2 2 ∂p 0 0

Comparing (E.33) and (E.35) with the Hugoniot relation (E.24) we conclude that:       ∂ 2v ∂v 1 + 3v0 (E.36) h0 [ p]3 + O([ p]4 ) [s] = 12θ0h 0 ∂ p2 ∂p 0 0

Consider next condition (E.30). Defining the quantity:   1 q= − 1 v2 η2

(E.37)

the condition (E.30) is seen to be equivalent to: [q] = q1 − q0 < 0

(E.38)

(ρ + p) ∂n (ρ + p) ∂v 1 ∂ρ = =− = ∂p n ∂p v ∂p η2

(E.39)

From (1.18) and (1.4) we have:

hence, in view of (E.16), (E.14), q is given by: q = −h

∂v − v2 ∂p

(E.40)

984

Epilogue

We then obtain:

∂ 2v ∂v ∂q = −h 2 − 3v ∂p ∂p ∂p

(E.41)

In view of the fact that by (E.36) [s] = O([ p]3 ), it follows that:       ∂ 2v ∂v [q] = − h 0 + 3v0 [ p] + O([ p]2 ) ∂ p2 ∂p 0

(E.42)

0

therefore the condition (E.38) is equivalent for suitably small [ p] to:       ∂ 2v ∂v [ p] > 0 h0 + 3v0 ∂p 0 ∂ p2

(E.43)

0

and this together with (E.36) is equivalent to (E.31). We have therefore established, for small [ p], the equivalence of the determinism condition (E.30) with the entropy condition (E.31). The sign of the coefficient of [ p] in (E.43) is by (E.43) the same as the sign of [ p]. This coefficient can be related to (d H /dσ )0 if the state ahead is isentropic, as is the case under consideration. From equation (1.72) of Chapter 1 we have: H= Now by (E.16):



hence:

dH dh 

vh

2



dH dh

s

1 − η2 h2

(E.44)

(d H /d p)s 1 = = (dh/d p)s v

 s

By (1.76): 1 h =− η2 v



dη2 =− dp 

dv dh

 − s

 =− s

h v2



dH dp

 s

2v (1 − η2 ) h 

dv dp

(E.45)

 (E.46) s

Substituting (E.46) and its derivative with respect to p at constant s in (E.45), we obtain:       v3 h 2 d H d 2v dv − 4 =h + 3v (E.47) η dh s d p2 dp s s

Consequently: 

(vh)3 −2 η4

  0

dH dσ



 = h0 0

∂ 2v ∂ p2



 + 3v0 0

∂v ∂p

 (E.48) 0

Epilogue

985

This is the coefficient of [ p] in (E.43). We conclude that the jump in pressure [ p] behind the shock is > 0 or < 0 according as to whether (d H /dσ )0 is < 0 or > 0, at least for suitably small [ p]. We finally derive a formula for the jump in vorticity behind the shock, assuming that the state ahead is irrotational as well as isentropic, as is the case in the shock development problem. We thus have: (E.49) ω0 = 0 In the following we denote by  the g-orthogonal projection to K : µ µ µ ν = δν − N Nν

(E.50)

Let us denote by β0 and β1 the 1-forms β induced on K from the state ahead and the state behind respectively: β0µ = κµ β0µ ,

β1µ = κµ β1µ

(E.51)

Applying the projection  to (E.20) we obtain simply: β1 = β0

(E.52)

Let us then denote by ω0 and ω1 the vorticity 2-forms ω induced on K from the state ahead and the state behind respectively: ω0µν = κµ λν ω0κλ ,

ω1µν = κµ λν ω1κλ

(E.53)

ω1 = d β1

(E.54)

We then have: ω0 = d β0 ,

where d denotes the exterior derivative on K . Thus (E.52) implies: ω1 = ω0

(E.55)

Here by (E.49) we have ω0 = 0. Consequently: ω1 = 0

(E.56)

The remaining component of the vorticity on K is ω⊥0 , induced from the state ahead, and ω⊥1 , induced from the state behind, where: ω⊥0µ = κµ N λ ω0κλ ,

ω⊥1µ = κµ N λ ω1κλ

(E.57)

Here by (E.49) we also have ω⊥0 = 0. In general we can express: ω0µν = ω0µν + ω⊥0µ Nν − Nµ ω⊥0ν , ω1µν = ω1µν + ω⊥1µ Nν − Nµ ω⊥1ν

(E.58)

Here by (E.56) we have simply: ω1µν = ω⊥1µ Nν − Nµ ω⊥1ν

(E.59)

986

Epilogue

Now, according to equation (1.51) of Chapter 1, we have: u µ ωµν = −θ ∂ν s

(E.60)

Substituting (E.59) we obtain, along the behind side of K , µ

u ⊥1 ω⊥1ν + (u 1 ω⊥1µ )Nν = −θ1 ∂ν s1

(E.61)

Contracting this with N ν , noting ω⊥1ν N ν = 0, we obtain:

Nν N ν = 1,

µ

u 1 ω⊥1µ = −θ1 N µ ∂µ s1

(E.62)

Substituting (E.62) in (E.61) yields: u ⊥1 ω⊥1ν = −θ1 µ ν ∂µ s1

(E.63)

or simply:

θ1 d s1 (E.64) u ⊥1 Finally, since s0 is constant along K , as the state ahead is isentropic, we may replace s1 on the right by the entropy jump [s]: ω⊥1 = −

ω⊥1 = −

θ1 d [s] u ⊥1

(E.65)

We now consider the vorticity vectors 0 and 1 on K corresponding to the state ahead and the state behind respectively (see (1.47)). Here we have, by (E.49), 0 = 0, while by (E.59) and (E.65), θ1 −1 µαβγ µ 1 = ( ) u 1α Nβ ∂γ [s] (E.66) u ⊥1 Let, at each x ∈ K , Px be the space-like plane:  (E.67) Px = 1x Tx K where 1x is the simultaneous space of the fluid at x corresponding to the state behind. The g-orthogonal complement of Px in the tangent space to Minkowski spacetime at x is the linear span of the vectors u 1 and N at x. Let us also denote by  / the g-orthogonal projection to Px , at each x ∈ K : µ

µ

µ µ µ  /µ ν = ν + u 1 u 1ν = δν − N Nν + u 1 u 1ν

(E.68)

and, for any function f defined on K , let us denote by d/ f its derivative tangentially to Px , at each x ∈ K : d/µ f =  / νµ ∂ν f (E.69) Then only d/[s] contributes to 1 , and (E.66) can be written in the form: µ

1 =

θ1 −1 µαβγ ( ) u 1α Nβ d/γ [s] u ⊥1

We see that the vorticity vector 1 belongs to Px at each x ∈ K .

(E.70)

Bibliography

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988 [Hu] [J1] [J2] [K-T] [La] [Ln] [Lx1] [Lx2] [Lx3] [Ma1] [Ma2] [Ma3] [Mi] [Mo] [O] [P] [Ra] [Ri]

[S] [S-Y] [Ta] [Z-R]

Bibliography Hugoniot, H. “Sur la propagation du mouvement dans les corps et sp´ecialement dans les gaz parfaits”, Journal de l’´ecole polytechnique 58, 1–125 (1889). John, F. “Formation of singularities in one-dimensional nonlinear wave propagation”, Comm. Pure & Appl. Math. 27, 377–405 (1974). John, F. “Formation of singularities in elastic waves”, in Lecture Notes in Physics 195, Springer-Verlag 1984, pp. 190–214. Keller, J.B. and Ting, L. “Periodic vibrations of systems governed by nonlinear partial differential equations”, Comm. Pure & Appl. Math. 169, 371–420 (1966). Laplace, P. S. “Sur la vitesse du son dans l’air et dans l’eau”, Ann. de Chim. et de Phys. iii, 238 (1816). Landau, L.D. (1944). See Landau, L.D. and Lifschitz, E.M. Fluid Mechanics 2nd edition, Oxford, Pergamon Press 1987, pages 332–333. Lax, P.D. “Shock waves and entropy”, pp. 603-634 in Contributions to Nonlinear Functional Analysis, edited by E. Zarantonello, Academic Press, 1971. Lax, P.D. “Development of singularities of solutions of nonlinear hyperbolic partial differential equations”, J. Math. Phys. 5, 611–613 (1964). Lax, P.D. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Regional Conf. Series in Appl. Math. 13, SIAM, 1973. Majda, A. Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Appl. Math. Sci. 53, Springer-Verlag 1984. Majda, A. The Stability of Multi-Dimensional Shock Fronts – A New Problem for Linear Hyperbolic Equations, Mem. Amer. Math. Society 275, 1983. Majda, A. The Existence of Multi-Dimensional Shock Fronts, Mem. Amer. Math. Soc. 285, 1983. Minkowski, H. “Raum und Zeit”, Address at the 80th Assembly of German Natural Scientists and Physicians, Cologne (1908). Morawetz, C. “The decay of solutions of the exterior initial boundary value problem for the wave equation” , Comm. Pure & Appl. Math. 14, 561–568 (1961). Osserman, R. “The isoperimetric inequality” , Bull. A.M.S. 84, Nov. 1978, 1182–1238. Penrose, R. Techniques of Differential Topology in Relativity, Regional Conf. Series in Appl. Math. 7, SIAM, 1972. Rankine, W.J.M. “On the thermodynamic theory of waves of finite longitudinal disturbance”, Philosophical Transactions of the Royal Society of London 160, 277–288 (1870). ¨ Riemann, B. “Uber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite”, Abhandlungen der Gesellschaft der Wissenschaften zu G¨ottingen, Mathematischphysikalische Klasse 8, 43 (1858–59). Sideris, T. “Formation of singularities in three-dimensional compressible fluids”, Commun. Math. Phys. 101, 475–85 (1985). Schoen, R. and Yau, S.T. Lectures on Differential Geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, 1994. Taub, A.H. “Relativistic Rankine-Hugoniot equations”, Phys. Rev. 74, 328–334 (1948). Zel’dovich, Y.B. and Raizer, Y.P Physics of Shock Waves and High-Temperature Hydrodynamic Phaenomena, Academic Press, New York 1966, 1967, Chapter I, Section 19 and Chapter XI, Section 20.

Index 4-velocity, 23, 914 acoustical bootstrap assumption F1, F2, 150, 154–156, 163–166, 232, 233, 238, 239, 241, 267, 319, 322, 741 M / [l] , 432, 463 M[m,l] , 521, 540, 629, 649, 653, 716, 736 X / [l] , 353, 371, 463, 521, 540, 629, 649, 653, 706 X / [l] , 706, 725 acoustical coordinates, 43, 58, 105, 111, 882, 890, 927, 961 canonical ∼, 932, 964–966 acoustical curvature, 85, 905 acoustical function, 41, 932, 949, 958, 964, 968 acoustical metric, 28, 40, 42, 882, 905, 930, 961, 966 conformal ∼, 35, 72, 99 extension of the ∼, 930, 936 induced on t , 45 induced on St,u , 44, 56, 882 reciprocal ∼, 27, 33, 58, 208 acoustical structure equations, 53, 85 angular derivatives, 203 of the Ri , 345 angular momentum, 938, 965, 968 assumption basic ∼, 100 G0, 178 G1, 192 H0, 193 J, 767–769, 874, 880 of the fundamental energy estimate, 101 bicharacteristic generator, 40, 41, 140, 881, 889

bootstrap assumption basic ∼, 149 Q E / [l] , 353, 371, 462 T , 428, 462 E / [l] QQ

E / [l] , 433, 462 E / [l] , 331, 353, 371, 462 Q

E{l} , 521, 540, 629 QQ

E{l} , 597 p

5 67 8 Q... Q

E{q} , 756 E{{k}} , 756 E{l} , 474, 521, 540, 629 final ∼, 756, 880–882 borderline contribution, 767, 782, 794, 810, 822, 829 error integrals, 831 hypersurface integral, 794, 802, 822, 823, 831, 840 integral, 771, 794, 831, 846, 854 spacetime integral, 810, 829, 831, 840 boundary of the maximal development, 52 canonical acoustical coordinates, 932, 964–966 coordinates, 937 equations, 938, 939 momenta, 937 characteristic subset of the cotangent space, 27 of the tangent space, 28 Codazzi equations, 59, 61, 224, 403 regular form, 97 commutation current, 169, 175, 177, 278 commutation fields, 139, 140, 192 conformal acoustical metric, 35, 72, 99 conjugate point, 967, 968

990 connection coefficients of the frame L , T, X 1 , X 2 , 68 of the null frame L , L, X 1 , X 2 , 70 conservation laws integral form of the ∼, 977, 980 constant states, 29, 34, 888, 905, 926 contact discontinuity, 981 curvature tensor lower order part of the ∼, 90 principal part of the ∼, 90 cut loci, 881 deformation tensor, 192 general ∼, 140 of K 0 , 114 of K 1 , 115 of Q, 141 of T , 141 of the Ri , 141, 142, 147–149 determinism condition, 982, 984 development, 40 domain of dependence boundary of the ∼, 40, 881, 889 domain of the maximal solution, 40, 881, 889, 926, 927, 968 boundary of the ∼, 882, 889, 890, 977 energies of nth order associated to K 0 , 773 associated to K 1 , 773 energy associated to K 0 , 108 associated to K 1 , 109 current, 26 general ∼, 105 energy-momentum-stress tensor, 24, 978 associated to a solution of the inhomogeneous wave equation, 103, 107 enthalpy per particle, 25, 912, 926, 981 entropy condition, 982, 984 entropy per particle, 23 equations of motion, 24, 31, 37, 904, 912, 924 of state, 23, 31 of variation, 26, 35, 37, 913 error integrals, 172, 765 Euler–Lagrange equation, 32

Index flow lines, 24 flux associated to K 0 , 108 associated to K 1 , 108 general ∼, 106 fluxes of nth order associated to K 0 , 848 associated to K 1 , 833 focal points, 881 Gauss curvature, 232, 311 of St,u , 59 Gauss equation regular form of the ∼, 97 Hamiltonian, 937–939 Hamiltonian flow, 937, 938 Hugoniot relation, 982 hypothesis H0, 178, 197, 297, 353, 371, 411, 412, 414, 540, 629, 673, 674 H1, 178, 188, 196, 411, 412, 414, 540, 629, 683, 691, 692 H2, 179, 198, 411, 693 H2 , 412–414, 540, 629, 693 induced acoustical metric, 882 on t , 906 induced connection on St,u Lie derivative of the ∼, 410, 411 inhomogeneous wave equation, 99, 139, 169, 172, 278 initial data, 886, 888, 889, 903, 926 invariant curve, 932, 936, 958, 964 in Minkowski spacetime, 974 invariant vector, 949 irrotational, 31, 888 irrotational isentropic, 33, 54 isentropic, 24, 31, 888 isoperimetric constant, 876 isoperimetric Sobolev inequality, 875 Jacobian determinant, 50, 51, 882, 890 jump conditions, 977, 979 Lagrangian density, 32, 33 lapse function, 44, 47, 53 linear wave equation, 35–37

Index local simultaneous space, 25, 913 local uniqueness theorem, 40 mass-energy density, 23 maximal development, 40 solution, 40 time of existence, 903, 926 minimal surface equation, 33, 90 motion equations of ∼, 904, 912, 924 nonlinear wave equation, 31, 33–37, 886, 888, 903 null geodesic flow, 937 null geodesics ending at a singular point incoming ∼, 940, 941, 943, 944, 949–952, 956, 957 other ∼, 940, 942, 944, 948, 950–953, 956, 958 outgoing ∼, 940–942, 949–953, 956–958, 965, 966, 968 number of particles per unit volume, 23 outgoing characteristic hypersurfaces, 40, 889, 904, 961, 968 particle current, 23, 978 particle flux, 981 partitions, 332 unordered ∼, 459 past null geodesic of a singular point, 937 pressure, 23 principal acoustical part, 269, 280, 281, 283–286, 295, 297, 321, 759, 764, 765 term, 224, 269, 270, 279, 280, 284, 296, 298, 299, 321 propagation equation for χ, 57, 675 for χ  , 676 for χ AB regular form of the ∼, 97 for trχ, 58 for µ, 64, 67, 275, 717 for µtrχ regularized form of the ∼, 210 for µ
/µ

991 regularized form of the ∼, 285  , 290, 299 for (i1 ...il ) xm,l  (t, u), 313 for (i1 ...il ) xm,l

for (i1 ...il ) xl , 215, 224, 235 propagation law for κ, 64, 906 property C1, 101, 741 C2, 101, 741, 746, 752 C3, 102, 741, 753 D1, 102, 754 D2, 102, 754 D3, 102, 754 D4, 102, 754 D5, 102, 754 reciprocal acoustical metric, 27, 33, 58, 208 second fundamental form of t , 906 of St,u relative to (t ), 58 of St,u relative to Cu , 56 of the t , 53 shock development problem, 978, 981, 982 shock formation, 893, 903 singular hypersurface boundary of the ∼, 934, 935, 937 extrinsic point of view of the ∼, 933, 936 in Minkowski spacetime, 969, 974, 977 intrinsic point of view of the ∼, 931, 936 sound cone, 33, 42 sound speed, 25, 926 source functions, 139, 169, 171, 172, 758 state ahead, 981, 982 behind, 981, 982 constant ∼s, 29, 34, 888, 905, 926 equations of ∼, 23, 31 symmetric hyperbolic systems, 904 temperature, 912 time function, 39 trichotomy, 940, 956 uniqueness theorem, 968

992 variation, 26, 35, 99, 905 equations of ∼, 26, 35, 37, 913 of 1st order, 757, 874, 912 generated by time translations, 919 of higher order, 99, 139, 171, 757, 874 volume per particle, 981 vorticity 2-form, 29 jump in of the ∼, 985, 986 vorticity vector, 30 jump in of the ∼, 986 wave equation linear ∼, 35–37 nonlinear ∼, 31, 33–37, 886, 888, 903 wave function, 31, 171, 767

Index