HANDBOOK OF MATHEMATICAL FLUIDDYNAMICS VOLUMEII
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HANDBOOK OF MATHEMATICAL FLUIDDYNAMICS Volume I1 Edited by
S. FRIEDLANDER University of Illinois-Chicago, Chicago, Illinois, USA
D. SERRE Ecule Normale Supe'rieure de Lyon, Lyon, France
ELSEVIER Amsterdam Boston London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo
ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 2 1 1, 1000 AE Amsterdam, The Netherlands O 2003 Elsevier Science B.V. All rights reserved.
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Library of Congress Cataloging-in-Publication Data A catalog record from the Library of Congress has been applied for.
British Library Cataloguing in Publication Data Handbook of mathematical fluid dynamics Vol. 2 I . Fluid dynamics - Mathematics I. Friedlander, Susan, 1946 - 11. Serre, D. (Denis) 111. Mathematical fluid dynamics 532'.05'015 1 ISBN: 0 444 5 1287 x @ The paper used in this publication meets the requirements of A N S I N S O 239.48-1992 (Permanence of Paper).
Printed in The Netherlands.
Contents of the Handbook Volume I 1. 2. 3. 4. 5.
The Boltzmann equation and fluid dynamics, C. Cercignuni A review of mathematical topics in collisional kinetic theory, C. Villuni Viscous and/or heat conducting compressible fluids, E. Feireisl Dynamic flows with liquidlvapor phase transitions, H. Fun und M. Slemrod The Cauchy problem for the Euler equations for compressible fluids, G.-Q. Chen and D. Wllrzg 6. Stability of strong discontinuities in fluids and MHD, A. Blokhin und ): Trcrkhinirl 7. On the motion of a rigid body i n a viscous liquid: a mathematical analysis with applications, G.P Guldi
1
71 307 373 42 1
545 653
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Preface The Handbook of Mathematical Fluid Dynamics is a three volume series of refereed review articles that cover many kinds of fluid models, including ones that are rarefied, compressible, incompressible, viscous or inviscid, perfect or real, coupled with solid mechanics or electrically conducting. We have included many issues: for example, the Cauchy problem, boundary value problems, waves, instabilities, and turbulence. We have concentrated on mathematical questions arising from fluid models (as opposed to the physical and numerical aspects of fluid dynamics which are well-developed elsewhere). However, we have encouraged the authors to describe the physical meaning of their results. Volume I is more or less specialized to compressible issues. The table of contents for Volume I can be found on page v of Volume 2. Volume 2 contains a wide range of material with the majority of the articles addressing issues related to incompressible fluids. I t begins with a discussion of probabilistic and statistical fluid models. Such models have been used since the 19th century in attempts to describe turbulent flows. Our first article presents statistical hydrodynanlics in a modern framework. This is followed by papers in which inviscid flows are examined as volume, preserving maps, and intriguing issues of nonuniqueness of weak solutions to the Euler equations are presented. A group of papers addresses the Cauchy problem for the Navier Stokes equations and the existence of attractors for these equations. The topic of stability and instability of fluid motion is examined in a number of contexts including viscous flows, waves on the interface of free surfaces, the ubiquity of instability for inviscid fluids, and the dynamo instability in an electrically conducting fluid. The breadth of material arising in mathematical fluid dynamics is illustrated by the final article on relativistic shock waves. Once again, we are deeply indebted to the authors for their immense work in writing for this Handbook. Their articles illuminate the fascinating variety of problems that have as their source the same physical system but produce so many different mathematical challenges. We thank the referees who worked hard to ensure the excellent quality of the papers. We express our appreciation of the Editors and staff of Elsevier who were very helpful and professional and who gave us the opportunity of making available to the scientific community a collection of articles that we hope will be useful and inspiring. Chicago, Lyon September 2002 Susan Friedlander and Denis Serre
[email protected] [email protected]
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List of Contributors Babin, A.V., University cdC~rliforniuat Irvine, Irvine, CA (Ch. 6 ) Ben-Artzi, M., Hebrew University, Jerusulem, lsruel (Ch. 5 ) Brenier, Y., lnstit~ltUniversitrrire de Frunce, Nice, France (Ch. 2 ) Constantin, P., The University rf Chic~lgo,Chicugo, IL (Ch. 4 ) Dias, F., ENS Cuchun, Cuchutz, France (Ch. 10) Friedlander, S., The Univrrsity oflllinois rit Chicugo, Chicugo, IL (Ch. 8) Gilbert, A.D., University of Exetet; Exetel; UK (Ch. 9 ) Groah, J., University of Cul(f;)rniu,D~rvis,CA (Ch. I I) Iooss, G., 1t1.stit~rt Univer.situire cir Frutzcr. 1n.stifutnon-linkuirr dr Nicr, Vulhonnr, Frunce (Ch. 10) Lipton-Lifschitz, A.. Crkriit Suissr Fir.st Boston, New York, NY (Ch. 8) Renardy. M., Dep~rrtrnetrtc~f'Mrrfheinrrtic~.\; Blcrck.sburg, VA (Ch. 7 ) Renardy. Y., Delxrrtmrtzt of'M~rthemtrtic~,s, Bloi.k.shurg, VA (Ch. 7 ) Robert, R., Uni1vr.sitc; Grenohlr I, Srritrt Mtrrtin d'HPrc>.s,Frcrr1c.c) (Ch. 1 ) Shnirelman, A., University r~f'Hull.Hull. UK, E l Alfil,Unir.rrsity, TeI A19il: /.st-(re1(Ch. 3) Smoller, J., University c~f'Michiguti.Ant1 Arbor; MI (Ch. 1 I ) Temple, B., University c!f'Ccrlifi)rtzitr,Dtrvis, CA (Ch. 1 1 )
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Contents Contents of the Haizdbook Preface List of Contributors
1. Statistical Hydrodynamics R. Robert 2. Topics on Hydrodynamics and Volume Preserving Maps K Brenier 3. Weak Solutions of Incompressible Euler Equations A. Shnirelman 4. Near Identity Transformations for the Navier-Stokes Equations I? Constaiztin 5 . Planar Navier-Stokes Equations: Vorticity Approach M. Ben-Artzi 6. Attractors of Navier-Stokes Equations A. Ci Babin 7. Stability and Instability in Viscous Fluids M. Renardy and K Renardy 8. Localized Instabilities in Fluids S. Friedlander and A. Lipton-Lifschitz 9 . Dynamo Theory A.D. Gilbert 10. Water-Waves as a Spatial Dynamical System E Dias and G. looss 11. Solving the Einstein Equations by Lipschitz Continuous Metrics: Shock Waves in General Relativity J. Groah, B. Temple and J. Smoller Author Index Subject Index
v vii ix
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CHAPTER 1
Statistical Hydrodynamics (Onsager Revisited)
.
Raoul Robert
.
lnsritur Fourier CNRS Uni~~er.sife Grenoble I . UFR de muthPmatiques BP 74. 38402 Srrinf Murrin d'H2re.s cedex. France E-mail:
[email protected]
Contents I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Equilibrium problems: Self-organization of the turbulent flow . . . . . . . . . . . . . . . . . . . . . . . 2.1. Equilibrium state\ for 2 D incompressible ideal flows . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Extension of the theory to other systems. the Vlasov-Poisson equation . . . . . . . . . . . . . . . 2.3. Relaxation towards the equilihriuln and paralnetrization ot'the small scale\ . . . . . . . . . . . . . 3 . Out-of-equilibrium problems: Weak solution\ \hocks. and energy dissipation . . . . . . . . . . . . . . 3.1. Statistical bolutlons of ID invihcid Burgers equation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Energy dissipation tor 3 D How5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . 1.2~1conimenls and ncknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rekrcnceb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
HANDBOOK O F MATHEMATICAL FLUID DYNAMICS . VOLUME I1 Edited by S.J. Friedlander and D . Serre O 2003 Elsevier Science B.V. All rights reserved
3 4 6 25 31 37 37 44 50 50 51
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Statistical hjdrodynamics (Onsager revisited)
1. Introduction Though it would be soundly enlightening to discuss thoroughly how statistical methods were introduced into the field of turbulence, we intend to limit ourself to only a few preliminary comments. In its crudest form the statistical approach consists in splitting the turbulent velocity field into a smoothly varying part plus a random fluctuation. This naturally yields the notion of turbulent viscosity (Reynolds, Boussinesq). From the beginning of the last century, systematic efforts were made by prominent scientists to elucidate the enigma of turbulence by statistical methods. In most cases these efforts amount to perform more or less formal calculations on random solutions of Navier-Stokes equation, solutions which are supposed to exist, provide a relevant description of turbulent flows and moreover satisfy some symmetry properties. The main success of this approach is the famous 415 law of Kolmogorov, Von Karmhn and Howarth which gives a precise relationship between the energy dissipation and the third moment of the velocity increments (see Frisch [25]). More recently further progress was made on the mathematical aspect of this statistical approach when two important issues were addressed: ( I ) Give a precise definition of statistical solutions. (2) Prove existence results for such solutions. For these aspects we refer to [241. Our purpose here is to expound some developments of the statistical approach which are directly inspired by the original ideas of Onsager on turbulence 1461. Onsager's starting point is a clear distinction between the two and three-dimensional cases: in two dimensions the energy of the turbulent flow is conserved, and the main phenomenon to explain is the self-organization of the flow into coherent structures (large scale eddies). On the other hand, in three dimensions, the dissipation of the energy persists in the limit of vanishing viscosity and we observe a power law energy spectrum. The distinction made by Onsager is brilliantly shown by the experiments of Van Heijst 1681. In the first part we will try to carry through Onsager's program on 2D turbulence. that is, try to extend in a rigorous way to hydrodynamics the statistical mechanics approach of Boltzmann. We will also indicate some practical consequences concerning numerical simulations. In the second part we consider the 3D case. Here also we follow Onsager's ideas: we do not start considering solutions of Navier-Stokes equation and then make the viscosity go to zero (what we may call Leray's point of view) but we suppose that the turbulent flow might be correctly described by weak solutions of Euler equation which are not regular enough to conserve the energy. Despite the fact that the Cauchy problem for such solutions is still unsolved, examples of such weak solutions with energy dissipation have been recently constructed. We begin by considering the simpler case of the one-dimensional Burgers equation which retains some of the interesting features of our problem (energy dissipation for weak solutions with shocks). Due to the discontinuities in the solution the energy will decrease. It is natural to try a statistical description of the dynamics of these shocks, that is, to describe the action of Burgers flow on a stochastic process. A first answer to this issue appears astonishingly simple: the class of Levy's process with negative jumps is preserved by Burgers equation.
Of course, it is tempting to imagine a similar approach for Euler equation. Obviously, things are much more difficult. As a first step we propose a formula giving the expression of local energy dissipation due to the lack of regularity of weak solutions. This leads to a natural entropy condition for such solutions.
2. Equilibrium problems: Self-organization of the turbulent flow The most striking feature of 2D hydrodynamical turbulence is the emergence of a largescale organization of the flow, leading to structures usually called coherent structures (see references in [17,54]). Jupiter's Great Red Spot, a huge vortex persisting for more than three centuries in the turbulent shear between two zonal jets, is probably related to this general property [40,64]. Such hydrodynamical vortices, whose dynamics is governed by Euler equation or some quasi-geostrophic variant, occur in a wide variety of geophysical phenomena and their robustness demands a general understanding. Similarly, the galaxies themselves follow a kind of organization revealed in the Hubble classitication [ I I I. The dynamics of Galaxies is dominated by stars under collective gravitational interaction rather than gas or hydrodynamical processes. In particular, for most stellar systems the collisions (i.e., close encounters) between stars are quite negligible and the galaxy dynamics is well modeled by the Vlasov-Poisson equation. The common remarkable feature of these structures is that they occur and persist in a strongly turbulent environment. In the case of 2D turbulence, Onsager [46] was the tirst to suggest that an explanation might he found in terms of statistical mechanics of Euler equation. In the case ofgalaxies. the natural approach, which consists i n defining a statistical equilibrium for a cloud of stars, fails (see references in 1171); this is mainly due to the fact that the relaxation time associated with the "collisions" of stars widely exceeds the age of the universe. This lead astrophysicists to suggest that it is a much more efticient mixing process generated by the collisionless Vlasov-Poisson equation which drive the system towards some sort of equilibrium. In [ 171 we stress, at a physical level, the analogy between the violent relaxation of stellar systems and the mixing of vorticity yielding coherent structures in 2D turbulence. This analogy resides in the similar morphology of the Euler and Vlasov equations. and we try to explain their self-organization with the same theoretical tools. Let us now be more precise. In his pioneering paper, Onsager's argument on the possibility of negative temperature equilibrium states was based on the approximation of the continuous Euler system by a great (but finite) number of point vortices. This leads to a tinite-dimensional Hamiltonian system, to which the methods of statistical mechanics can be applied (see, for example, [13,451). Though qualitatively very enlightening, this approach reveals severe difficulties; for example, there are many different ways to approximate a continuous vorticity by a cloud of point vortices and different approximations can lead to very different statistical equilibrium states, so the thermodynamical equilibrium that we can associate to a continuous vorticity depends dramatically on arbitrary choices (this difficulty was underlined by Onsager). Thus, if we want to produce reliable quantitative predictions on the actual behavior of the flow we have to proceed differently.
Sttrtisticvll hydrodynamics (Onsuger revisited)
5
A natural way to define equilibrium states is to construct invariant Gibbs measures on the phase space. But we do not know how to construct such measures on the natural phase space L m ( R ) for Euler equation. Some work has been devoted to the study of Gibbs measures with formal densities given by the enstrophy and the energy [8], and also to Gibbs measures associated with the law of vorticity conservation along the trajectories of the fluid particles [I 21. Unfortunately, all these measures are supported by "large" functional spaces so that not only the mean energy and enstrophy of these states are infinite but the phase space L m ( R ) is of null measure. So, it is only at a formal level that this makes sense. Moreover this approach fails to give any prediction on the long time dynamics corresponding to a given initial vorticity function. The most common approach to overcome these difficulties is to use a convenient finitedimensional approximation of the system, possessing an invariant Liouville measure. Then one can consider the canonical measures associated with the constants of the motion and try to perform a thermodynamic limit in the space of generalized functions when the number of degrees of freedom goes to infinity. For example, for Euler equations one can consider the N Fourier-mode approximation or the point-vortex approximation. Two difficulties arise in this approach. The first is to choose a relevant scaling to perform the limit, the second is even more fundamental: generally. the approximate system will have less constants of the motion than the continuous one, so that the long-time dynamics of that system may be very different from that of the continuous one. For more comments and references on these attempts see, for example, [23,49,54]. To overcome the difficulties evoked above. our approach is based on the following points. We work on an extended phase space (the space of Young measures) on which the constants of the motion put natural constraints. We construct a sequence of finite-dimensional approximations of the Euler flow (with good convergence properties such as strong L' uniform convergence on any finite time interval), satisfying the two following properties: (i) A Liouville theorem holds for the finite-dimensional approximations. (ii) For the family of measures given by (i), we can prove the Sanov-type large deviation estimates for empirical Young measures which are necessary to take the thermodynamic limit [41,49,521. Of course, it is easy to satisfy the point (i) by considering the spectral approximation; but then, it is a very difficult issue to prove that the associated family of measures satisties (ii). Our approach here is to get a Liouville theorem for a general class of approximations, including approximations on spaces of functions which are spatially localized like tinite element approximants. For such approximations we are not able to prove directly the large deviation estimates (ii) but we can use them as an intermediate to construct the tinal approximation on the space of piecewise constant functions for which the large deviation estimates hold (41 1; so that the use of the fi nite-element approximants appears here as an essential intermediate step in order to both insure the convergence of the approximations and keep the large deviation estimates. One might worry about the fact that our approximate dynamical system retains only the enstrophy among the infinite family of the Casimir functionals which are conserved by the continuous system (in contrast with the finite mode Hamiltonian approximation
of [72,73]). Of course, it would be more satisfactory to construct approximations having in addition a large number of constants of the motion. But we think that this is not truly necessary to our microcanonical approach. Indeed, if we are interested in the long-time behavior of 2D-Euler flow, and if we believe that a statistical mechanics approach can bring some light on this issue, then we expect that we will finally have to solve some constrained variational problem: find the maximum value of some entropy functional under a set of constraints. But while we have no doubts about the set of constraints which is directly derived from the constants of the motion of the system (energy, integrals of functions of the vorticity field . . . ), it is hard to guess what the relevant entropy functional is. But in our microcanonical approach the entropy is not related to the fact that many constants of the motion are (or are not) exactly conserved by the approximate flow but it is only associated to large deviation estimates for the invariant measures.
2.1.1. 2 0 Elder equcrtion. The motion of a two-dimensional incompressible inviscid fluid in a bounded domain il is governed by Euler equation, which we write in the classical velocity-vorticity formulation:
+ div(wu) = 0. curl u = to, div u = 0 . (0,
U. n = 0
on i l i l
where u ( t , x ) is the velocity field of the Huid. ( I ) = curlu the scalar vorticity. n the outward unit normal vector to ilQ. Because of incompressibility we introduce the stream function
9 ( t .x ) :
The constants of the !notion of this dynamical system are: - the energy
-
the integrals
for any continuous function 8 . These constants of the motion which are associated to the degeneracy of the (infinite-dimensional) Hamiltonian system are usually called Casimir functionals.
Stati.stica1 hyrirodyna~?~ics (Onsuger revisited)
7
- if f2 is the ball B(0, R), we must consider also the angular momentum with respect
to 0:
The Cauchy problem. Yudovich's theorem [70] gives a satisfactory existence-uniqueness result for the Cauchy problem for (E): For any given initial datum wo(x) in the space Lm(f2), there is a unique weak solution of (E); this solution w(t, x) is in Lm(Q) for all t , and furthermore belongs to the space C([O, co[;LP(f2)) for all p, I < p c 00. We will define the flow r, of the Euler equation on the phase space LW(f2)by w(t, .) = Gwo. Furthermore, this weak solution satisfies the following useful stability property: If wf, is a bounded sequence in the space Lm(f2), which converges in the strong L* topology towards wo, then T,wi converges ~ * - s t r o n gtowards l~ two, uniformly on any bounded time interval. The turbulent mixing process. Let us now briefly describe the mechanism of turbulent mixing which is responsible for the self-organization of the flow in Euler equation. As we have seen, Euler equation can be described as the advection of a scalar function (the vorticity) by an incompressible velocity field with which it interacts via a Poisson equation. The vorticity is not passively advected by the flow but is coupled with its motion, this coupling will be responsible for the fluctuations of the stream function which will mix the vorticity at small scale and induce a self-organization and the appearance of structures at larger scales. This process is studied at a physical level in 1171. Our concern here is to introduce an entropy functional which will give a precise content to the vague notion of turbulent disorder of the flow. Of course, to detine such a functional, following Boltzmann's approach, one would have to choose a relevant measure on the phase space. It is well known that, at a formal level, Euler equation is an infinite-dimensional Hamiltonian system; but, in contrast with the finite-dimensional case, this does not imply the existence of an invariant Liouville measure on the natural phase space L m . Fortunately. it occurs that to def ne the entropy functional we do not actually need to have a Liouville measure on the infinite-dimensional phase space, we only need the existence of finite-dimensional approximations which admit invariant Liouville measures. large deviation theory will then yield the relevant entropy. This is the very root of our thermodynamical approach. Although we can find finite-dimensional approximations of Euler equation which preserves the Hamiltonian structure 172,731, this structure is broken by any kind of approximation of practical use. But for the needs of thermodynamics the Hamiltonian structure is not truly necessary, it is the Liouville theorem and the constants of the motion which are the key ingredients. In the case of Euler equation, it is well known that a Liouville theorem holds for the usual spectral approximation. We shall show that this is a particular case of a general property: there is a natural way to approximate Euler equation on any finite-dimensional space in such a way that the volume measure is conserved. The spectral approximation is only a particular case of that. Then the problem of defining an equilibrium statistical
mechanics for ( E ) amounts to the study of families of measures. For an arbitrary choice of the approximating spaces the study of the asymptotic behavior of these measures seems untractable, but fortunately we can choose spaces for which the thermodynamic limit of these measures can be carried on 14 1,521.
2.1.2. Finite-dimensional approximations and Liouville theorem. A classical way to construct finite-dimensional approximations of Euler equation is as follows. Let FN be an N-dimensional subspace of Lm(f2) and denote by PN the orthogonal projector from L2(f2) onto FN.Then we define the approximate solution wN(r) as the solution of the ordinary differential equation in F N :
= a N$N . = O o n an. where u N = c u r l $ N , and If FN is properly chosen and wo is regular enough, then w N ( t ) converges towards w(t) for the strong L~ topology, uniformly on any bounded time interval 1391. The constants of the motion of the dynamical system ( E N )are: - the energy dx. - the enstrophy (wN)' dx. Let us notice here that ( E N )is a differential system with quadratic nonlinearity so that the solution always exists o n a small time interval; but due to the conservation of the enstrophy the solution cannot blow up and it exists globally in time. Now, it is well known that if we take for FN a subspace generated by N eigenvectors of the operator - A (with the Dirichlet boundary condition), the volume measure on FN is conserved by ( E N ) .This is in fact a particular case of what follows. We consider the modification of ( E N ) which consists in replacing, in the detinition of $ N , the Dirichlet problem by the variational formulation:
5
$
E
1'
FN and
v
$
~
V
~
~
X f o r a=l l p i~n F ~~ .
~
~
For the sake of simplicity, from now on we shall also denote by ( E N )this modified dynamical system. Of course, we shall suppose at least that FN is included in the Sobolev space H(!(Q), so that for any given wN, the above variational problem possesses a unique solution q N(by the Lax-Milgram theorem). One can easily check that the energy and the enstrophy are still conserved but now we have in addition THEOREM 2.1 . I . The volume rneasurr on FNis corzserverl hg tlze dyr~u~nicul.system (EN). PROOF. FN is endowed with the L2 scalar product. Let us write ( E N ) in the form wy = G N(wN),where G N( w N )= - P N ( u N. v w N ) is a nonlinear transformation of F N . Then to
~
~
Stuti.st~calhy~irodynurnic.~ (Onsugrr revisited)
9
prove the theorem it suffices to show that the trace of the derivative G h (wN) vanishes. Let us compute the first variation of G N corresponding to a small variation 8wN:
By definition, we have tr(G/N(wN))= Ei(G/N(wN)[e;],e;), for any orthonormal basis e; of FN.Let us denote u; the vector field associated to e;, we have:
(G/N
(wN)[ei].e;) = -
S,
(u;. v w N ) e ; dx -
(UN
ve;)e; dx,
but since div u N = 0, the last term vanishes, and after integration by parts we get: ( ~ / N ( w ~ ) [ ee;) , ] ,=
I*
wN curl$;
. Ve; dx.
Let us consider now the positive definite and symmetric linear operator A defined on FN by: V$I . V q d x = (A$, (P), and take for e, an orthonormal basis of eigenvectors of A. We obviously have e, = A, $, (k; is the eigenvalue corresponding to e;), so that curl t,h;.Ve; = 0 and tr(G/N(wN))= 0. Now, it remains to prove the convergence (when N + co)of the approximate solution wN( I ) towards the solution w(t) of the Euler equation. We shall take for approximating space FNthe space Fl,( Q ) of the finite-element approximation of the Sobolev space H " ' ( R ~with ) , compact support in Q ( m is an integer > 5 and h a small positive parameter, see Appendix A). Then we have the following convergence result whose proof is classical. PROPOSITION 2.1 .2. Let w ( t ) he uny weuk .solutiorl of' ( E ) , with wO(x) in the .spcic,e Lw(Q), and let T > 0 he,fixed. Then fi)r ull E > 0, there is A ( & ) > 0, suc.11 that,fi)r trll h, 0 < h 6 h ( ~ )there , is u ~olurionwl'(t) c?f(El,) such thtit:
As we will see later, the measures PI, on Fll(Q) associated with this approximation are not easy to handle, but it appears that a slight change in the approximating dynamical system improves greatly the situation with a view to our thermodynamical purpose. Let us denote by f,"the flow on Fh(Q) defined by the system (El,). Let pi, : LI', + Fi, be the classical prolongation operator of the tinite-element method (see Appendix A), and nil = P,,'. Let us define Ll,(f2) = XI, Fl,(Q), and denote (9:'= XI, o I-," o pi,, the flow induced on Lh ( a ) . Obviously, (9: preserves the volume measure on Lh (Q). From Proposition 2.1.2 we deduce the following.
COROLLARY 2.1.3. Let w ( t ) be any weak solution of ( E ) , with wo(x) in the space L W ( Q ) , and let T > 0 be Jixed. Then for all E > 0, there is h ( ~>) 0 such that for all h , 0 c h h ( ~ )there , is w0h in L h ( Q ) such that:
<
I[w(t)- @;woh
I L~~~ < E .
for all t in [O, T I .
PROOF. By the ~ ~ - s t a b i lproperty it~ of Euler equation, we only need to prove the result for wo in C,?'(Q). Using Proposition 2.1.2, we have, for h h ( E ) : Ilw(t) - wh ( t )11 < E ,on [0,T I . Let us denote wl,(t) = x h w h ( t ) , we have: Ilw(t) - wh(t)ll < Ilw(t) - rhw(t)ll I[rllw(t)- o h ( t )[I, where rl, is the classical restriction operator (see Appendix A). But since
<
+
it becomes:
Now we have (see Appendix A)
and similarly
thus ( I w ( t )- wl,(t)l/< C ( T ) h+ C E , and the result follows. Let us summarize our results. We have constructed a flow (9: on Lh(S2) which approximates the Euler flow and preserves the measure dol, = Bi dw;, where o l l ( x )=
xi
W;
(XI
h
-
j ) (ti nite sum).
2.1.3. Baldi's large drviution theorem and ther~nodynutniclimits. In order to detine relevant statistical equilibrium states, we have to take the thermodynamic limit of the invariant Liouville measures with the conditioning given by all the constants of the motion. To perform this task we need some tools from large deviation theory. Baldi's theorem gives general conditions under which a family of probability measures on a locally convex topological vector space has the large deviation property. As we will see, it provides a powerful tool to carry out thermodynamic limits for infinitedimensional systems.
II
Stuti.stica1 hydrodynamics (Onsager revisited)
The large deviation property. Let E be a locally convex Hausdorff topological vector space. We consider a family p h , h > 0, of Borel probability measures on E. We will say (see, for example, Varadhan [69]) that the family p h has the large deviation property with constants h(h) and rate function L iff: (i) h(h) is > 0 and limb,+, h(h) = + m . (ii) L : E + [0, + m ] is a lower semi-continuous functional on E (not identical to + m ) . Moreover, L is inf-compact, that is: the set {v I L(v) 6 ) is compact for all real numbers b. (iii) For every Borel subset A of E, we have:
<
-A(A)
1 < lim inf -logpI,(A), h - t m h(h)
and
where A(A) = inf,,,A L(v). The functional L is also usually called the information functional, and - L the entropy functional. Let E' be the topological dual of E , endowed with the weak-star topology a ( E 1 , E ) . For a Borel probability measure ~1 on E , we define its Laplace transform: ~ L ( ( o=)
J; exp(((o. v)) d p ( v ) ,
for cp E E'
As it is well known, fi is a convex, lower semicontinuous and proper functional on E'. The same is true for the functional Log jL(cp). B A L D I ' STHEOREM 2. 1.4. Let p h he u jumily ($'Bore1 probul7ility rnecrsiires or7 E, .sciti.sfiing thefi)llowing ussumptions: ( 1 ) There is utunction A(h) us in (i) such thuf
I lim -Log ;I, (h(h)cp) = F(cp). ll-tcw, h(h) whew F is u convex, lower sernicontit1irou.v cirrd proper fiir~c~tiot~c~l on E' rvhich is jinite or1 u neighborhood ($the origin. (2) Compucity ci.s.sumption: For evety R > 0, there is ti compact set KK C E .such thut
1 lim sup -L o g p l l ( K ~ ) / ~ + m A(h)
<
-
R
Let us denote by L the Young-Fenchel trurz.sjorm of' F, tlzut is: L(v) = sup (((o, v ) - ~ ( ( o ) ) , ,for v E E. ql€ E'
L is a convex, lower semicontinuous, and proper functional on E. Baldi's theorem states that under the assumptions (1) and (2) the upper bound in (iii) holds. If we suppose thut L has so~neadditional strict-convexity property, we can also derive the lower hound. We will suppose thut L .suti.s$es the jbllowing condition. ( 3 ) For e v e n real number r , the set A,. = { v I L(v) r ) is the cln.sure ($the subset of the points v c$ Ar where the .subdlfferentiul 8 L(v) is nonempty and contains un element cp such that:
<
~(v'> ) L(v) + (cp, v1 - v), ,for ill
V'
# V.
Then Baldi's theorem asserts that under the hypotheses (I), (2), (3) the two bounds in (iii) hold. So, we see that the family plr has the large deviation property with constants A ( h ) and rate function L. Indeed, one easily checks that the functional L is inf-compact on E: for every real number h, the set A/, is closed and the lower bound applied to the open set Ki,'+I yields A/, c K/,+I (with the notation of (2)). Coi~lnl~nts.( I ) In practice it may be difficult to check that the hypothesis (3) is satisfied. In fact. Baldi's proof works us well with the following weaker hypothesis (3'). (3') For every 11 such that L ( v ) < +co, for every open set 0 containing and every F > 0, there is 111 E 0 such that L ( v l ) 6 L ( I J )+ c and L is strictly convex at I J I , that is: 3cp E i ) L(I]!) such that IJ
(2) L is strictly convex at v if, for example. ifL(11)is nonempty and
for all 0 < t < I . v' E dom L, v' # v. (3) In the case where only the hypotheses ( 1). ( 2 ) are satistied. as we have seen, Baldi's theorem gives an upper bound. But the functional L may fail to be inf-compact in that case. Nevertheless, we can see that the set A. = { v E E 1 L ( v ) = 0 )
is nonempty.
Notice first that we obviously have F(O)= O and since F is also the Young-Fenchel transform of L, we get: inf L(v) = 0.
I ~E E
Furthermore, we have p , ( K ) (when h + co).
+ p j ( K ; ' ) = 1 and from (2) we know that plI(K;') + O
Sturistical hyr1rodynr1n1ic.s(Onsuger revisited)
13
Now, if Ao were empty we should have A ( K I ) > 0. Then, applying Baldi's theorem we should have (/I/, ( K I ) + 0); this would yield a contradiction. Moreover, one can easily deduce that for any open set U containing A0 there is a number a > 0 such that: /I~(u(')
< exp(-h(h)a),
for h large enough.
We shall say that the family , L L ~concentrates about the set Ao. Thermodynamic limits and the concentration property. When dealing with thermodynamic limits one usually encounters the following situation, which we resume here in an abstract form. Let 6h be a family of random variables with values in a Hausdorff locally convex topological vector space E. ah generally comes from some finite-dimensional approximation of an infinite-dimensional system. If we can prove that, for h large, with a high probability, 6/, remains in a neighborhood of some points v* of E, then u* is the equilibrium state of our system, and the thermodynamic limit is performed. Large deviation theory shows that such a situation is very common. We will assume in the sequel that the family 61, (or the associated probability distributions /I,, on E ) has the large deviation property with constants k(h) and rate function L. Since Prob(GIl E E ) = I for all h, we have L ( v ) = 0 and the set A. is a nonempty compact subset of E. And, as in the above comment 3, for every open set U containing Ao, there is some a > 0 such that: ~ r o b ( 6 /E, u")
< exp(-h(h)a),
for large h
That is, the family 61, concentrates about the set A ( ) which is the equilibrium set of the system. In our "microcanonical" approach, we will study now the situation where 6/, satisfies some given constraints (it would be more correct to say that we introduce some conditioning on the random variables all). These constraints will be given, for example, by the constants of the motion of an infinite-dimensional dynamical system. We introduce the constraints in the general form E 8 , where & is some subset of E . Of course, since ?ill comes from a finite-dimensional approximation, the ideal constraints SI1E E will not be exactly satistied, but only up to some approximation given by an open neighborhood of 0 in E, W. Let us denote Ew = E W. We shall then consider E &w. Let us now give a definition.
+
DEFINITION. Let E, E* be subsets of E. We will say that 6/, concentrates about E* conditionally to & iff: (i) VW', lim infh,, Log Prob(Gh E I W>J-m, ) (ii) VW*, 3 a > 0, 3W, VW',
'Ew'E'*) &wf)
Prob(8h
E
6 exp(-k(h)or).
for h large enough.
Here W*, W, W' denote open neighborhoods of 0 in E. REMARKS. ( I ) Heuristically, this definition means that if we know that 6h takes its values in a neighborhood of E , then it will be in a neighborhood of E* with a high probability. (2) As previously noticed, we have to widen the sets E , E* into open neighborhoods. In fact, Prob(Gl, E E) is not defined for an arbitrary subset &; and even if E is a Borel subset, it can be zero. (3) The condition (i) ensures that, when h -+ co,Prob(Sh E E w ! ) cannot be too small. Now we derive the following concentration result which will be useful to carry out thermodynamic limits. C O N C E N T R A T ~TOHNE O R E M 2.1.5. We suppose that 6h has the [urge deviation property with constatzts h ( h ) and ratefilnction L. Let & he a nonempry closed subset of E and E* the subset o f & where L achieves its minimum value on &. Then 6h concentrates about &* conditionully to I . PROOF. See 1411. R E M A R KThe . set E*, in which 61, approximately remains with a high probability, is the equilibrium set of the system. If £* does not reduce to a point (the equilibrium state), we are in a phase transition situation. We shall see in the following how we can use this concentration result to derive a maximum-entropy principle for Young measures.
2.1.4. A nrci.ritn~cm-c,ntro/)y princ.iplr ,fi)r ?i)urlg t?~rtr,surc..s.It currently happens when dealing with a limit process for a sequence of bounded measurable functions that the sequence does not converge and shows an oscillating limit behavior, whereas some estimates and conservation laws hold. In such a case, the concept of Young measure has been found relevant to describe the behavior of the sequence (examples can be found in hyperbolic systems of conservation laws, homogenization, hydrodynamics . . .). Young measures can be viewed as giving a macroscopic description of the system, whereas the bounded measurable functions are all the microscopic states. We use the results of Section 2 to derive a maximum entropy principle for Young measures, that is: the macrostate (Young measure) which realizes the maximum of an entropy functional has a natural concentration property (a large majority of the microstates satisfying a given set of constraints are in a neighborhood of that macrostate). It turns out that this entropy functional is the classical Kullback entropy (see Sanov's theorem). Younx measures. Throughout this section, X , Y will denote two locally compact separable and metrizable topological spaces. Let us suppose that a positive Borel measure d.r is given on X. Let us recall that Young measures [71] are a natural way to generalize the notion of measurable mapping from X to Y: at any point x E X, we no longer have a well-determined value, but only some probability distribution on Y.
Stuti.sticta1hydrodynamics (Onsager revisited)
15
In other words, a Young measure v is a measurable mapping x --+ v, from X to the set MI (Y) of the Borel probability measures on Y endowed with the narrow topology. Clearly, v defines a positive Borel measure on X x Y (that we will also denote by v) by:
for every real function f (x, y), continuous and compactly supported on X x Y ( f E C,(X x Y)). Moreover, for f (x) E C,.(X), we have
that is, the projection of v on X is dx. It is well known [41] that this property gives an equivalent definition of Young measures. That is, for any positive Borel measure v on X x Y whose projection on X is dx, there is a measurable mapping x + v, such that the above formula holds. The mapping x -+ v, is unique up to the dx-almost, everywhere equality. To any measurable mapping f : X -, Y, we associate the Young measure Sf :x -+ 6,f(,,, Dirac mass at f(.r). We shall make two additional assumptions: (*) The measure dx is diffuse and of finite total mass. (**) There is a distance function d ( x , x f ) giving the topology of X, such that: for all F: > 0, there is a finite partition of X into measurable subsets X = {x' I i = 1 . .. . , n ( X ) ) with I X ' / = [ x . ~ I for all i. j (we shall say that X is an equipartition of X), and satisfying d(X) 6 E , where d ( X ) = supi sup,,.,,,x, d ( x , .rf) is the diameter of X. Notice that (*) and (**) imply that [ AI approaches zero when the diameter of a measurable set A approaches zero. Hypotheses (*), (**) are satisfied, for example, if X is an open convex and bounded subset of R" with dx = Lebesgue's measure, and also if we consider any image of X by a dx-preserving homeomorphism. We shall denote by M the convex set of Young measures on X x Y , and we recall some useful properties. M is closed in the space M/,(X x Y ) of all bounded Randon measures on X x Y (with the narrow topology), the narrow topology is equal on M to the vague topology (weak topology associated with the continuous conlpactly supported functions) and it is metrizable. Furthermore, if Y is compact then M is compact. In the sequel M will be endowed with the narrow topology. ( 6 , f : X + Y measurable] is a dense subset of M. This property is classical in the case Y is compact (see reference in [411). The general case follows by approximation. Approximate first (for the vague topology) a given Young measure v by vx (as in the proof of Theorem 2.1.6 below) which is constant, equal to v', on each set X' of an equipartition X, and then approximate each v' by a probability measure with compact support.
A lurge deviation property. Suppose now that we are given a basic Borel probability measure no on Y. Then with any equipartition X of X we can associate a Borel probability measure y~ on M in the following way. We take yl, . . . , y,,, n ( X ) Y-valued independent random variables with the same distribution no.We consider the random function
where 1 x, is the characteristic function of the set Xi.We denote by Jx the Young measure associated with , f i , and by px the probability distribution on M of the random variable ST. Now, we can state the main result of this section. THEOREM 2.1.6. When d (X) + 0, the family pfi hus the large deviution property with constants tl (X)/ I X I cind rute fltnctioiz I , (v), where 71 = dx @ no und I , (v) is the clu.~.~icuI K~lllhaekii~fi)rmc1tion,fi4nerioncrl (see Vurt~dhun1691). defined on M by: I, ( v ) =
S X x, Log 2 dv,
if v is ub.solut~lycontit~uou.~ with respect to T , otherwi.~e.
PROOF. The proof is an application of Baldi's theorem, see 141 1 for details.
In this Young measure framework. Theorem 2.1.5 yields the following. COROLL.ARY 2. 1 .7. Let & he tr r~onemptyc~lo.sc~t/ S N / ~ . S Pof' ~ M . &* the .srrh.srt of'& ,t9/zrrr. the,func.tiot~tilI , trc,hiei~c~.s its rnitlirnurn vcilur or1 &. Ttlrrr Ss c.orlc.c~rltrrrtr~.v rihout &* c.otltlitiorlctlly to &. REMARKS. Note that since the functional I , is inf-compact and & is closed. &* is nonempty. Theorem 2.1.6 appears as a generalization of the well-known Sanov's theorem. Indeed, apply the contraction principle to the mapping v + [I), d.x.
2.1.5. Thertnodyrzutnic.litnit of'the invcrritmt mru.nrw.sit? tho .spuc.e of Yr)ut~g~)~ocr.r~~rr.r. Lorig tittle dytzutilic.~clrztl Y ~ I U I I R~r~rci.surc~. AS we have seen, Euler system describes the advection of a scalar function (the vorticity) by an incompressible velocity field. thus the vorticity w remains bounded in L w ( R ) . The functionals
are constants of the motion (for any continuous function H). That is to say, the distribution measure of w, IT,, defined by (n,,0) = C ~ ( Wis) ,conserved by the flow. Let us consider an initial datum wo. It is well known that, in general, as time evolves, T,wo becomes a very intricate oscillating function. Let us denote r = I l w o l I ~ ~ cSince ~ , . the
<
measure n, is conserved, Gwo will remain, for all time, in the ball L y = {w: llwllm r ) . Extracting a subsequence (if necessary), we may suppose that, as time tends to infinity, r,wo converges weakly (for the weak-star topology a ( L m , L ' ) ) towards some function w*:
We can easily see that C H ( r t w ~does ) not converge towards Ce(w*) if 0 is nonlinear, whereas some other invariants can converge, as it is the case for the energy. So, much information (given by the constants of the motion) is lost in this limit process. Thus, the weak space L m ( R ) is not the good one to describe the long-time limits of our system. Fortunately, the relevant space to d o this is well known, it is the space M , of Young measures on R x [ - r , r ] , that we have just defined. We can identify the long-time limits of the system as Young measures. Indeed, M r is a suitable compactification of L y since the narrow convergence (when r approaches infinity) of towards some Young measure v preserves the information given by the constants of the motion, that is, for all functions B ( z ) :
but the left-hand side is constant and equal to
(IT,,,,,: 0 ) . so
that:
The same arguments apply to the other invariants. For example. since r,too converges weakly towards C(x) = J'zdv,(z), we have, for the energy, E ( T , w o ) + E ( Z ) ,which is the energy of the Young measure v, and thus:
We shall denote by (**) the set of constraints (associated to the constants of the motion) other than (*), that C has to satisfy: = {energy constraint, angular momenturn constraint (eventually)).
(**)
Thus we see that the constants of the motion bring the constraints (*), (**) on the possible long-time limits. Since we do not know anything (in the general case) on the long time behavior of the solutions of Euler equation, we will consider Young measures merely as a convenient framework in which we can perform the thermodynamic limit of a family of invariant measures. In the following, we will call the Young measures satisfying the constraints (*), (**) the mixed macrostates, in contrast to the small scale oscillating vorticity functions called microstates.
The random Young measure Sf,. Let Q be a bounded open subset of Rd,the space F h (Q) is composed of the functions of the form f / ~ ( ; - j ) which are compactly supported in R (see Appendix A). The space Lh (Q) = q,(Fh (Q)) is composed of functions of the (X - j ) which vanish in a neighborhood of the boundary a52 (whose width form approaches zero with h). Let us write a function of Ll,(Q): li, = f i x ( ; - j). We denote d fh =
xj
x j fix
x,ipJl,
d f i , and = f e x p ( - b / f i d x ) d f h , the probability measure on L h ( R ) , where the scaling factor l / h d is introduced in order to give a finite value to the mean f:dx)dPh(fh), in the limit h + O.We will write p h = BjeJl, dnl(f/), where dn,(y) = s e - y 2 d Y We will consider now fh as a random function with probability distribution p h . Thus Sf,, is a random Young measure on f2 x R. It follows from Theorem 2.1.6 that the family (depending on h ) of the random Young measures Sf,,has the large deviation property with constants l / h d and rate function I, (v), where we denote n = dx @3 n, A straightforward consequence of this large deviation property is that the random Young measures A / , which in addition satisfy the constraints (*), (**) are exponentially concentrated about the set E* of the solutions of the variational problem
jLl , ( Q ) (/
where I is the closed subset of the Young measures on 52 x R satisfying the constraints (*),
(**I.
Note that this variational problem has at least one solution since I is nonempty and closed and I, ( v ) is a lower semicontinuous and inf-compact functional on M. Now, let us denote no = nI Q I o , , ,and n ' =
[email protected] all v satisfying (*),one can easily get the relationship: I, ( v ) = I,,(v) 15211,, (no). Thus if I,,(no) ioo,minimking I, or I,, on & gives the same equilibrium set &*. In fact. the use of the functional is more natural since it is associated to the invariant distribution rr,,,,. To justify the use of I,, in the degenerate case I,, ( n o ) = ca,one can. for instance, modify the definition of the I measures pll, and consider p,, = dnll(,j;/),where d n i ( y ) = y e x p ( - Q l l ( y ) ) d y , and the polynomial function QI,(y) is such that TI, converges towards T O in the narrow topology when 11 -, 0. Of course, we have
+
/,I
It is not difficult to see that the proof of Theorem 2.1.6 [41 ] works for these measures, and it follows that 6f1, has the large deviation property with constants I /h" and rate function I,l(v). Notice that -I,,(v) is the entropy, that is, the functional which measures the disorder created in the fluid by the turbulent mixing.
REMARKS. For Euler equation we have d = 2. We shall consider also the case d = 6 for VlasovPoisson system. a In order to get probability measures, we replace d f h by p h d Jh with p h = I e x p ( - h j" f; dx) despite the fact that this functional is (eventually) not conserved by the flow. Indeed, we consider as an authorized trick to multiply the measures by any functional which is conserved by the flow of the infinite-dimensional dynamical system.
2.1.6. Computation of the equilibrium states, the equation of Gibbs states. Once we have identified the relevant entropy functional, the determination of the equilibrium states come down to the solution of a variational problem: i.e., find the minimum value of I,l(v) under the constraints given by the constants of the motion of the system. After that it remains to discuss at a physical level the relevance of these states. From now on (for obvious typographical reasons) we shall denote w instead of wo, n instead of rr', and K, (v) = -1,) (v) the Kullback entropy. We have now to solve the variational problem: Find the macrostates v* satisfying K,(v*) = mar(K,(v)
1 Sn
v,dx = n ,,,.
--
; (v)
= Z(w)
where w is any initial vorticity in Lw(52). We have seen in a previous remark that such I.]* always exists. We begin with the simpler case where w takes only a finite number of distinct values u l , . . . , a , , (w takes the value u; on the set 52;). Then, we have n,, = 152116,,,+ . . . + /R,,16,,,. It is clear that any mixed macrostate is of the form 11, = (x)6,,, + . . . + er,(x)6,1,,, with the constraints: I
.
.e
l=
IQ
e ; ( x ) d x = I52;I.
i = 1 , ...,t i .
The most mixed state is such that e;(x) = IQil/]52l, for every x ; that is, n , = l nI j ~ l , , . The probability distribution v , is obtained by multiplying r,by a function equal to ei(x)152;21/IQ;I for u = a , . This function is equal to dv,/dn,. Therefore. the Kullback entropy writes as:
Since the second term is a constant, this entropy is indeed equivalent to the classical Boltzmann mixing entropy: ~ ( e= )-
Jn C
e; (XI Loge; (x) dx.
As e = ( e l , . . . , e,,) must satisfy the supplementary constraint Cy=lel (x) = I, only n - 1 independent constraints FI, . . . , F,,-I remain. Let eT, . . . , ef be a solution of our variational problem. Then, by the rule of Lagrange multipliers, there are constants cu = ( a l . . . . , a,-1) and 6 such that the first variations of the functionals satisfy:
for all variations Se; such that El Sr, = 0. Straightforward computations give:
where $* is the stream function associated with the vorticity
tr,c.T(x),
SF, = /n6r; dx. Then, we easily get:
where the partition function Z is given by
and we take a, = 0, by convention. Thus $* satisfies the equation of Gibbs states: (G.S.E.)
-A$* = E;u,e:(x) =
I d
fl d*
LogZ($*).
1
So, we see that if ey is a solution of the variational problem, there are constants a, B such that eT is given by ( I ) and $* by (G.S.E.).
Conversely, for any given set of parameters a , we can consider a solution $ U . f l of the nonlinear equation (G.S.E.) (as the right-hand side of the equation is of the form , f ( $ * ) , with f' continuous and bounded, we know, using Schauder's fixed point theorem, that a solution, in general not unique, always exists) and the associated Gibbs state eff.fl given by (1). Of course, eff,P is a critical point of the functional J ( e ) = S ( e ) - / 3 Z ( e )a ; F ; ( e ) ,on the linear manifold r , ( x ) = 1. Furthermore, we prove the following result:
c:':~'
x::,
I f /3 > - h l / ( C , o,?), where Al > 0 is thejr.st rigenvalue of'rhr o p P R O P O S I T I O2.1.8. N ~ ~ r l ~-tAo r (c~ssocicltedto the Dirichlet l~olincl~zrs vrrlur condition), then ew.fl is the unique ~nuximlinzc?j'the,filnctionrrlJ ( r )on the set dejinrd by e; ( x )3 0 , i = 1 , . . . , n , e, ( x ) = 1. PROOF. We shall prove that the functional J ( r ) is strictly concave on the set defined by 0 < p;(.r) < I , i = I , . . . , ! I . For that. let LIS compute the second variation S * J for any variation 6 ~ ;Straightforward . computations give:
where we denote 6(0 = 0. it becomes:
xi tr,Sc, and S1// is the stream function associated to Sto. As S2 F; =
Let us consider tirst the case P 3 0. Then we have, by Green's formula:
from where:
thus J is strictly concave. We consider now the case p < 0. Using the well-known inequality
we get:
.4s I / r ,
,I and (x, rr,6oI )2 < (x,
112)(x, fie,)',
it becomes:
/rj
(I,?. 8' J < 0. We see that for /1 > -Al As is a critical point of .I. i t is the i ~ n i q u epoint o n which J reaches its ~ i ~ a x i ~ i l u ~ n value. cjU,t'
KI:MAKK.';. A consequence o f this result is that, fol- p 3 - A I imum of S ( ~ Jsilh.ject ) Lo the coristr.;~irits X P ,= I .
/x(/:, - the Gibbs state r ~ " . is~ a Inax-
i=
.z ( P ) = ~ ( ~ " . f iI;, ) (, P ) =
I;; ( ~ " . t ' ) ,
l,....ri.
As i ~ s i ~ ;\vc ~ l . rcplacc Ihc \tucly of ~ h constraincrl c ~ n a x i ~ l i i ~ a problem ~ion l'or S ( P )hy the *iti~cIyof' the cqll;~tiol~ ol' Ciibhs slates G.S.E. The gcncriil sti~tlyol' G.S.E. for :ill viilucs ol' tlic p:u:ui~ctcrs w , /1 is I l ~ rI'rom being obvious. hcc.:u~sc mimy hil.i~rcation plicrlornc~i~t c i u ~o c c u ~(scc ' 165 1 ). As is the I,agl.:ir~gc riiirl~iplicro t t h c c11cl.pyc o ~ ~ s t r a i li t~iat . thc invcrw o l ' ; ~tc~npcl.:~turc. iuitl the :ibovc (hcol.cn~171.ovc\the cxistcnclt ol'cquilih~.ii~m status with ;I ncgativc t c ~ ~ ~ p c r i ~(this t i ~ pr ct i c ~ ~ o ~ i ~wet~~isforcsccri or~ hy 01ih;igc1-1461), Whcri so111csi~l?plclllcntiu-y co~~sttuith ol'tlic motion occur. wu must t:ihc lhcln inlo account. This Icatls. ol'course. ro somc modifcation\ in the ctluation ol.Gihbs \latcs. For uxar~iplc,whcrl R is the h:lll N ( 0 . K). we must ~.on\itlc.rillso the i~ngulurmomentllm. The gcnelxl casc ol'iiny nicnsul-ublc and ho~rndcdirlitial vorticity ( ( 1 ccui ht. stildied in :I similar way 1401. I t yields the generalization ol' rcl;~tion ( I ) for tlic optimum state 1 1 ( w e omit the stiir that lobcls rhc optimi~rnstutc): dl,,
(,
0.) =
11 ( 1
1 - 11\
($1 i 1
1
dro(
Z(~/J(.Y))
where ~(q,) = , / ' c , - ~ ( ~ l - - Ptlno(y) ~ l l ~ and xo = mI r c , l . The 1,agl.iirngc m i ~ l ~ i p l i cwr sl . . . , cr,, arc 11owrcplaccd hy Ihc continuous t'unclion cr( Y). Notice that $0.) is thu probabili~ydensity li,r finding the vorticity ?: at position .\. Then thc genel.uli~cdG.S.E. ciui be written:
.
a ( ! ) and j3 being determined by the constraints: global conservation of vorticity:
Sn
d r = n,,:
11.~
conservation of energy:
We can prove the following: PROPOSITION 2.1.9. For j 3 r n a ~ { I~y' Equatiorl (G.S.E ) .
E
suppno) < A,, there is a unique ~olutionto
R E M A R KNotice . that since they satisfy a relationship w = J ($), the solutions of G.S.E. define stationary solutions of Euler equation. R E M A R KIsolated . vortex structures. Fairly isolated vortex structures are often observed in two-dimensional turbulent flows. As already foreseen by Onsager [46], such vortices must correspond to a negative temperature. Indeed, a vortex core corresponds to an extremum of the stream function. A$ is positive for a minimum and negative for a maxilnum. Let us assume that $ is maximum, and therefore the vorticity -A$ is positive. Assume also that $ vanishes on the walls. then $ is positive. Now, if /3 is positive, w = f ( $ ) must be a decreasing function of $ so that w would increase from the vortex core to the wall. The vorticity would be essentially located at the walls, while for j3 negative the vorticity is concentrated in the vortex core. as expected. When $ is minimum in the vortex core, we only have to reverse the sign to get the same conclusion.
2.1.7. Colzereizt structures and statistical equilibrirlri~sttrtrs. Up to now we only considered the issue of the thermodynamic limit of a family of invariant measures of approximate systems via large deviation estimates. Of course this necessary step is not sufficient to give a conclusive justification of the equilibrium statistical mechanics. Such a task would involve intricate dynamical considerations (involving rtn ergodicity assumption and a precise estimate of the mixing time for the approximations). It seems that such an analysis is out of reach at the present time. Nevertheless, we will give now some elements of discussion which may help us to delimit the field of validity of the theory. We will consider the well-known phenomenon of the formation of coherent structures in 2D turbulence. In meteorology we can observe experiments or numerical simulations that such structures form. Let us scrutinize what chain of logic would lead us to identify these structures with the statistical equilibrium states previously described. Note first that we observe the phenomenon (the formation of the structure) over some finite time interval [O, T I . Obviously, the turbulent real fluid has some very small dimen-
sionless viscosity, so that we may suppose that in our time interval the flow is well approxinlated. in a strong L' sense, by a solution of 2D Euler (we consider, for example, the case of periodic boundary conditions to avoid the problem of boundary layers fortnation). Then we can approximate (still in a strong L? sense), uniformly over (0,T I , the flow by a solution of our finite-dimensional system, taking the number of degrees offreedom N large enough. Now we have to assume that this finite-dimensional system is ergodic and comes close to equilibrium i n a mixing time T ( N ) which is less than T. Of course, to have a good approximation of the flow over 10. T 1, we have to take N very large (this is well known in hydrodynamical si~nulations)and the crucial question is: how does T ( N ) increase with N? We clearly do not have any rigoroi~sargument to insure that T ( N ) does not increase dramatically with N so that the above justification might fail. From a careful examination of the results of many tests i n various situations, the following facts (see 1.531and references therein) emerge: ( I ) If the turbulent flow reaches an eqitilibriu~nstate after a mixing process occupying the whole clomuin. then the description of the fin;il state ;IS ;I global maximum entropy state is accurate. ( 2 ) In many cases the flow reaches a kind of eclitilibrium which is not a statistical e c l ~ ~ i librium in the whole domain occupied by the flow. This indicates clc;irly that difticulties may arise with the ergodic hypothesis. ( 3 ) 111 S L I C C;ISCS. ~ iti\idc the suhclomain occupied by the cohcl-cnt \Iructurc. the rela;is\oci;ilcd with our entropy t'i~~iction;~l is I'lrirly tionship (vorticity-strc;im f'i~~ictio~i) well satislictl. This iriclic:itcs also clc:i~.ly t l i ~ r t 0111- entropy l'unction;il retains sonic rclcv:~ncceven when crgotlicity tliils. In rci11H i t iris. 1 he]-c is. I I ' I s oI / / J / / I I I S . of'course. alwiiys some viscosity. In the stutly of' two-dimcnsion~ilturhulcncc it is usually ;~ssurnetlthat the viscosity is small enough so that n signilicunt dissipation ol'cncrgy cannot occur: hut it c:ulscs a decreiisc of the cnstrophy. This is due to thc fict that the s~tppleI mcntiiry tcrln - EAco has a tiltcring cl'l'cct o n the small-scale o\cillnlions of' the vorricity ( , I ( / . .\-) and will soon get it equal to its local mean value to'. Tlicn we may wonder whether the ccli~ilibriumstate (I)* has solnc stiihility property. What d o wc get [is cqi~ilibriumstate i f ' repent the process :uid take ( I ) " as initii~lvorticity? Let II/* be the unique solution of the general G.S.E. equation given hy Proposition 3. I .9. (o* thc corresponding vorticity. Then. it can be shown that Arnold's classic;~lcstimiites 131 apply and give the following stithility result. Any hounded initial vorticity coo gives a solution ( I ) ( / ) of' Eulcr equation. which satisties the inequality:
( ( ~ ( )r - tr)*)' d . <; ~ c
(oJ,) -
(o*)' d.v.
for ~ I I I .
where C > 0 is s o n ~ econstant. Furthet-more, we can prove that if the Young measure 11 is a mixture of to* si~chthat G ( 1 ) ) = E ( w * ) then, . for all x , u , = S,,,*(,,. So, if we repeat the process. starting with (o*. we shall again get w* as equilibriuln state.
-
2.2. E,ttet~.siorzofthe theory to other .sT.stetn.\,the Vlu.sov-Pois.son eyuutiorl 2.2.1. A c.10.s.s c~fcIy17rr1)1ic~r11 sy.sterr1.s. A large class of evolution equations, coming from the modeling of various physical phenomena displaying complex turbulent behavior, can be described as the convection of a scalar density by an incompressible velocity tield. More precisely. they are of the form:
where (/(I. X ) is sonie scalar density function defined on R x R ( R is a bounded connected smooth domain of R"),u ( t . x) is an incompressible velocity tield taking its values in R",which can be recovered from L/ by solving a P.D.E. system. Thus L denotes a (not necessarily linear) integro-di fferential operator. Let us give some well-known examples of such systems. ( 1 ) The simplest exalnple of ( I ) is the lillc~rlitrtrrl.y)oi? rt/~rtrtiorl.where u(x) is a given incolnpressible velocity field o n R . ( 2 ) The t/lrtr.si-gco.str~)~)I~i~. ~rrotlclof geophysical fluid dynamics 1401:
where (1 is the so-called potential vorticity. ( . is a nonncgi~tivcconstan[. and ,/' a given l.11nctionin 1.'-(R). Obviously, Eulcr eqilation is obtained by tahing c . and ,/. eclu;ll to 0. ( 3 ) Colli>ionlcss kinetic equatio~lssuch i t the Vltr.vo~~-Poi.\.\o~r c,c/rrtrtio~~ of stellal- dyn~umicscan also he written in the tor111( I ) . Indeed. let ,f'(t.x. v) bc the scalal- density I'unction giving the distribution of the stars in I: gillaxy (in the phase space ( x = position, v = velocity)). Ii'collisions (close cncounters) arc neglected, the density is purely advected by the flow in phase s p ~ ~ cso e . that it satisties the following Vlasov-Poisson equation:
where E ( t . X ) is the gravitational field, E = -V,@. @ ( I .x ) = -G ,[ IX-'L I dx' is the gravitational potential. and p ( t . x ) = / ' ( I .x. v ) dv is the spatial density of stars. Equation ( V ) conserves the following functionals: - The total energy (kinetic potential):
1
+
-
The integrals
SS
H ( f ' ) dxdv. for any continuous function H such that H(0) = 0;
-
The linear momentuln
-
The a n g ~ ~ lmomentum ar
Let us denote U = may be written:
(k): U is a vector field o n R". which satisties divh(U)= 0, and ( V )
so t l i ~ i tEcli~ation( V ) is ol'the form ( I ) . 'The lir.st 41Cp i n 0111. progr-;i~nis to tlclinc ;I flow ;tssoci;itcd to ( 1 ) on the pti;isc \pace I a Z ( R ) .Unli)rtun:ircly. to o ~ knowlctlge. ~ r there is n o gcneritl cxi\tencc-i1niclt1~11cs\ result I'or rlic C:~i~cliy prohlc~n1)r sy\tcrns like ( I ) . Example\ I and 7 arc well known. but conthings arc Inorc intricate. cerning Vlasov-Poisson ccl~~ation. One can lind existence results lor regular i n i t i ~ i lcl~itain 161. while the cxislcncc ol'wcak solutions is invcstigi~tcclin 13.20.28.371. A nice existence theorem 01' wcah aolu~ionscorrcspontling lo all i r l i t i ~ i ltlcnsity ,ti, in I , ' ( I R " )n 1 , ~ ( 1 i k ; " ) . satist'ying ,I;/'&v2,/iIdxdv< m. is givcn by Horst iund HunLc 1281. As L I S L I ~ Ii~~licli~cncss ~. is 11iorc tricky. ;irnd we need so11ie ;1~1~lition;il ;~sst~~nnption on the initial tlcnsity. Let us mention two recent rcsi~lts. - with a Lipschit/ regularity condition o n ,I;) (Lions and Pcrthame 1371). - wirh ,/;I i n I.% ;~~nd compactly supporrcd (Robert 150)). Finally. lor o i ~ rconcern. if ,/i, is givcn in LZ wirh compact support. the siti~;itionis co~nplctcly:inalogous to tli:it givcn by Yi~dovich'stheorem I'or- Eulcr cclu;ition: We have a iiniq~lcweak solution ,/'(I.x. v ) in C'(I0. w l : I,/'(IR"))l'or all 11. I 11,< w. which defines a llow T, o n L ~ ( W " ) . Using the moment cstirni~tcsof 1371. we easily prove that this weak solution rom:iins comp:ictly supported for all time. Here we have also ;I st;ibility property: If' ,/(I is a bot~ndcdseclilence in I , ~ ( R " )all . the functions h;iving their support included in o siumc compact set which convcrgcs in the strong .L2 topology towards ,/;I.then T,,fir i~nilormlyconverges strongly towards T,,f,, on any bounded time interval.
<
2 . 2 . F I - I ~ I oI ~ /i
o f ' / ~ ~o ~ - P o . s si o ~t o We will consider weak solutions 01' ( V ) , on a ti xed tirne interval 10. TI. corresponding to ,ti, i n .LX(R").
with compact s ~ ~ p p o rMaking t. appropriate changes of scale in space and time (this of course yields a change in the gravitational constant G --. g), w e can always suppose that thc support of the solution ,f' remains in Q(, = 1 1 / 2 , 1 /2[". -
TIIP .spircScBfi,(Q(,). The space fi(Q(,) (here we suppose h = 1 / N ) consists of the re, f ; / ~ (v j ) which are Zo-periodic strictions to Q(, of the functions of the form (see Appendix A). FI, ( Q ( , )is endowed with the L ' ( Q ~ )scalar product. P(.r - , j ) = I . the function 5'0 = 1 belongs to F / , ( Q l ) ,and we shall take an Since orthonormal basis to. . . . ,4,,. . . . of this space. Then the functions ( x , v) = ~I,(xj(,,(v) lbrrn an orthonormal basis of Fh ( Q 6 ) .
x,
xi
.
D~.firlitio,~of' (V/,). Lct LI\ denote Pi, the orthogonal pro.jection from L'(Q(,) onto F / , ( Q ( , ) .The approximate dynarnical \y\tern i \ given by: (
)
,f;"
+ I-',, ( v . O , ,f"'
+ E" . V, f'l' ) = 0.
w ~ c I ' ~ ~ E ~ ' ~ - - v ,~@~ i" t,h @ I ' ( r . x ) = - ,.q/ ,V I x d v , for x' E Q7.
,,/~(I,X')
dx', and /)I1(t.x')
IX-X'I
=,I;,,
, f l ' ( t . x', v )
f ( . Here. also. ( C ' I , ) is an ordinary differential systcnl with quadratic nonlinearity. One stlaightforwirrdly checks that thc f'unctionala ,/'I1 dx dv and ,lyf,(,/"')'dxdv iirc consc~'ved.It seems that thcy irrc thc only consta~itxof 1111' notion which I-emainconserved by the approximate system. A consccluencc of the invari;unce o f thest. tunctionuls is that thc solution o f (Vl,) exicts globally in time. Let u s now p~.ovca Liouvillc t h e o ~ . e ~ n . I
Tti I : O K ~ M2.2. 1 . T l ~ o~ ~ ) l i utr~c~trsrrrc. ~rr or1 Fl,( Q h ) i.s ~~orr,rt,t-~,r~il /7\3 tl~c,,fiort, r!/'(Vl,). PROOF. Let 11swrite (Vl,) in the form: ,/;" = P~,(u" . O,f'") = G(,/"'). where U" = Using the basis C,,,, detined irbovc, we calculate -
Let u\ calculate thc lirst variation S G of
from where
(;,
corresponding to a ~ ~ i r i a t i o6nf h :
(G,).
ZX
but I!"
I
. V-(' 2
/)l/
d x tlv =
let 11s11ow conqider the first term: we have A U " = ( - ~ , , , ( ,0) ,
where 8 , , is the gravitational
and 0 potential associated with the density pl,,/, p , , , , ( x ) = Q ( ( x , v ) d v , for x r (I3, outside. It follows that p,,, = O. except f'or y = 0. Thus, the only terms that we have to consider are I)(/
arid the pl.oof is complete.
Pla)Ol.
Let a r ciiIct~IiltC$
!
I I , / ' ( I )
-
/
(
I
)
whcrc ( . ) dcnotcs the I.'( (Ic, I hci~lal-procluc~. Using div 11 = 0 imd cliv uI' = 0. wc g c ~
Then wc cstiiiia1e these two tcr~lls. First term:
we get
, = (1. I
(
.C/Ir)+ ) (/'.
(1 . C
I).
.S'rtrIi.~tic~trl i~yt/u~c/~r~cr~~~i(~,\ (Oii.~tr,qri. rc.vitilrd)
but
I1 v./'' I
c'll1.'' I H i c ,-
,,x,Q(,j
for .v > 4,
and from
I fI
('
H
If" 1
for 0
,.,Q(,,.
< ,v < I?) ,
we deduce
1 u" . v,jh1
for 3 < .\. < I t , .
('
Furthermore.
11.f
~'/1./'111 Z(Q(,,
< ll,/'
-
<
11,fIl
/)11~~/~./'~~ ('/11" [ . ~ ( ~ ,H,::~, ,)
so that the tirst terrn is boundecl by
I1 .fI1 ti;;;.,.
(./,111+$
Second term:
I ( '
- 1 ). v
I. lil- / ) I
I l l l l . , ,
-
I , ,,
t
h -
/ / , .( 1',,)
wc have
IIui
l l ~ l r - ~ l l ~ , ~ ( s , )-i)ll,,!(c~,, ~ ~ ~ l l i ~~ ~ ~ l l . ~ ~ l l - t ~ l l / . ! , ~ I1
-~l1,,?(~,,,=
so that the second term i s bounded by
./"'
1.11;
~ ~ l l ~ . / ~ l l l . ~ -~ Q (2 ( ~,, i l, j . ~ ~
Now, since ,f' is smooth. tor a11 1 i n 10.7'1 we have:
l l V , / ' I l ~ . ~ ( ~ll./~Il//;;,, ~ , ~ . 5; C 7 ( T ) . Finally, it comes
d -ll.i'(t)
dl
Let
LIS
take ,
111
<
- , / " r ( f ~ ~ ~ ~ , C'(T)(II~"-' 2
+
ll,f(t)
- ,t"'(t)~l;~).
large enough wid s = 111 - 1. i t yields: -
and the proof i s complete.
< 1i1
-
,/;jlll:?e~p(t~(~ +II(CX~(/CIT)) l)
-
I).
Let 11snow consider the case of a bounded weak solution with compact support in Q6. for all r in 10.7'1. Using a mollitier H,.. we approximate the initial datum j;) by ,fi,, = N,*,fiI. Usirig the estimates of 137). we know that the corresponding solution ,f; ( t ) rcniains also compactly supported in Q(,. The stability result also ilnplies that Il,f(t) - f;.( t ) 11 V I , , -' 0 (when F -+ 0), uniformly f o r t E [ O , 71.Then we easily deduce the following: COKOLI.ARY 2.2.3. Lrf ,f hi) ti h o i r ~ ~ c l~~l rt li i ksol/rliorl c!f'(V ), with sujll)orf c . o l ~ t i r i ~ill t~d (*o~iz/)et(.t .x~r/).xctof Q6 ,#Or (111 1 E 10.T 1. T ~ ~,for I I(ill t: > 0, tIl~ri> 1,s J ~ ( E> ) 0 ,s~tt~/z ~h(it ,/i)rill1 11. 0 < 11 11( E ), tI~ort>is t i ~ s o l i t t i of I~1 ~(t ) of ( Vll ) .v~tchtlteit: ti
<
The classical prolongation operator /711 (see Appendix A ) is now an isomor.phism from Li, ( Q 6 )(space of the Z"-pe~.iodicfunction3 ot'the form I;:x - j ) ) onto Fi,(Qh).
El
(v
As for Euler- rquation. we denote I:)-( = XI, o r," o p,,. TI, = 1)1,', the flow induced on L , , ( Q h )And with the same i~rg~lrnents as thosc used in thc proof of Corollary 2.1.3, we get;
Wc scc t l i i ~ there ill50 we liilvc c o ~ i s t ~ ~ t ;I~ HOW ~ t c d(-1; 011 l,ll(C)f,) which app~.oxinli~lcs the VIi~sov-Poisso~iflow i11ic1~ ~ C S C I . Vthc C ~IIIC;ISLI~C (I,/,, = @, (I,/,/.
2.2.3. Sttrti.\/ic.cil c~iliriliht.ii~tr~ .s/tr/(,.s,/Or tlrc I/lri.so~~-Poi.s.so~~ c~lrrit/iotl. Following thc siumc ;~ppro;~ch as !'or Eulel- cclui~lionwe II~IVC now to solve the viil.iational prohlcm:
whcrc ,/;I is given. n = d x @ rr,,,. :uid E is the. suh\ct 01' 11ic Young rncasu~,cson C)(, x IR which s;~tislythc constrain[s: .. ( * J ./,IQ,,~ ~ ~ . ~ i = l xTiIl, ,v: , . ( * * i ) $ ,/jf2 V'C(X. V ) C I X ~ I V ,[,~OCIX = E ( , / ; ) )where , F ( x )= j;), I'(x. v ) d v and is the gravitational potc~ltiolassociated to (extended by 0 oi~tsiclcQ ; ) : (*+ii) vi!(x. V ) dx tlv = Po. lineill. momentum: (**iii) x A vii(x. v ) d x d v = Mo. :lngul;u momentum. Though i t always has solutio~is.this vi~~*i:~tio~i;~l problcm is no1 physic~tllyr.clevant: bccause whcn time tends to intinity, the density does not \t:~y contined in Of, but tends to spread in the wholc space R".Thus. it secms ni~ttrralto solve the variational probletn in the whole space [ 171. For this, i t is necessary to reformulate the problem in such a way that i t can make sensc in an unbounded domain.
+
,Un
,/,rf2
Let us first SUppOSe that we work on a bounded domain where the density ,fib takes only a finite number of values ~ the value cr, o n the subset R ' . Then TI,,= 'j-Yzo1f2~ that I ) ~ = , ~ 'j-y=o /I'(x. v ) & ~., t'or almost all x , v E 52, the and I pi = 1. Now one easily sees that the constraint (*) reduces to
52 of' W" and consider the case
r= o 0. . . . , ( 1 1 . . . . , ( I , , . ,/i)taking and the constraint ((* iimplies functions /I' satisfying p' 3 O
and straightti)rwal.d calculations give
We see that the second term tends to intinity when 1 0 1tends to intinity, but we get an ecli~ivalentformulation for the constrained variational probleln it' we retain only the tirst term in thc functional. This ter111obvioi~slyrrlakes sense in an infinite domain. Then we consider the following vurii~tionalproblem ( V P'): minimi/e
11ndt.l.thc constr.:~ints: (*i) x : ' = ( , / ) ' = I , (*ii) /)'(x. v ) ~ x ~ vQ'1. = i = I . . . . . 11. ( r * i ) , (+*ii). (**iii), with V ( X . V ) = II;/)'(x.v ) .
,/A
x:'=,
This problem is stildied in 15 11. It can bc shown that there is n o solution i l l the cast f2 = EX". due t o the fact that the density ciun spread out indctinitcly in the whole space while increasing the entropy and conserving thc invariants. I t was propowd i n 1 171. on thc hasib c ~ fphysical arguments, to introduce an artificial spatial conlining and ~ h u sto solvo ( V P ' ) in a domuin -Q = H x F', where R is sonic bill1 i n the physical space. This issue is still unclciu. and riither controversial. Nevertheless, i n that case we can prove 1.5 1 ] that thcrc is always at le~istone solution fix ( V P ' ) . so that the "gravothcrmal catastrophe" which occurs i n thc cl:~ssic:~lstatistical mechanics of :I c l o ~ ~ofdst:lr\ I I I docs not occur in the case ofthe violent relaxation proccss.
In this section we contitue the study on a physical level and investigate the relaxation process of the equilibrium. On the one hand. this can bring somc light o n the dynamical mechanisms responsible for a possible lack of clgodicity 1531; o n the other hand. it provides a natural parametrization of the small scales, an important practical issue Ir)r nu~nierical
sinlulutions. We shall consider here only the case of Euler equation, similar considerations for Vlasov-Poisson system can he found in [ 171. I t is well known that direct numerical simulations ol'2D ti~rbulentflows need a mesh size o f the order of the viscous dissipation scale. This cau\cs :I dr:i\tic limitation in the Reynolds ni~mhersthat can he I-cached. In practice. this difficulty is overcome by introducing an n i l l the ccluations. in order to limit at a reasonable levcl the artificial turbulent d i f f ~ ~ s i otcrnl nu~nber.of degrees of freedo111of the systcm. This empirical recipe rests on the belief that l e the statis~icaleffect of' the small scales on the large we cut1 pnrurnclrire in a s i ~ l ~ p way nlly This assumption is a general prerequisite scale motion (in which we iu-e ~ ~ c t ~ ~itltcrestcd). l'or the feusihility of large edcly simulation (ace, tor-cxaniple. Sadoi~rny1561. Bn~devnntet SaJou~.ny17 1). The purpose of this section is t c show ~ how the st;itisticul cquilihrium tlicot-y can be i ~ s c d to p a ~ ~ m c t r i the z e elt'ect of thc snlall scalcs. This cl~lestiotiwas fir\t invcstig;~tcclby Robert ancl So~iinicria155 1. where a set of' relaxation eql~ationsleading the sy\tcm toward\ thc c c l i ~ i l i h r i ~\\as ~ m proposecl. A further t~nalysisof the reli~xationmechanism was given in liobet-I itntl Ro\icr 153 1; it yieltls ;I more cnvolvecl expression 1 ) r the turbulent diffusion lel.lns. O L I~.cI;~x;~tion ~ c~lil;itio~ls ;ire of ;I dil'f~~sio~i-convection typc. The main difference w i ~ hNavicr-Stokes c q i ~ ; ~ t i ois~ iIII;II s they conscr-vc thc cner.gy ;~nrlall the collst;lnt\ of thc nlotion of' Eulcr cyuiitio~~s. To c~bt;~in O L I S~.c.las;ilionccl~lation\\vc shall ~iccclto in~l.oduccmcthocls o l ' ; ~ marc physici~l Ilavoi~~.. insl,ircd I'scini cl;~ssicallinear I l ~ c r ~ l l o c l y ~ ~ a ~For ~ l i the c l ; . .;Ae 01' simplicity we will consiilcr the c i ~ s cwhere ~ l i cinitial vorticity l'unclion t;lkcs only 11 distinct v:iluc\. For a sclltly 01' Ihc ycncl.al ci~sc.;I dctailctl cliscwssion 01. thc ~ ~ ~ c t l i o;~nrl ~ l . r.c,.;~~lts. ; :~riilapplic;ltion lo large cdcly simul;~tions.we rcl'cl- to 1511. 1,cl 11s considcr the c;lsc LC'IIC~C ~ I ) O ( X t;~!ic\ ) 2.3.1. K o l t r . v t r l i o t r ~ ) ~ ~ ) c ~ c ~11rc~ . v . s11. I c , ~ , c ~ l . c,cr,vcJ. \. only rr tlistinct v:ilues ( 1 1 . . . . . t i , , . We \hall ;i\sulnc that during i t \ cvolutiol~tou:~~.cls a litial ccluilih~.il~m state, the x y s t c ~ call l ~ ul~.cudybe clcscribc~l~ n i ~ c r o ~ c o p i c iin~ ltcr-~ns l y ol' ;I set ol' local ~~ruhuhilitics /)I ( I . x ) . . . . . I ) , , ( / . X I . In other wonls. the system hiis ;ilrc;tdy u ~ i r l c r g o n ~ tine-scale vorticity oscillations. The lociilly ;~vcragcdvol-ticity is 6 ( t . x ) = C ,t i , I), ( t . x ) and u is the corresponding vclocily licld. Thc vorticily patc.1lc.s arc transported hy u, and we s~ipposeIllill. in i~dclitio~l, they under-go a difl'usion process. so that thc ccinscr-vation cqui~tionI'or cach vorticity pr-ohithility can he wriltcn ~1s:
I . Wc imposc thc boundary conclition J , . n = 0. s o that the total area occupied by each patch i x con.;ervod. We can asurllc (willio~ltloss of generality) J, = 0. i s . . the locally uvcrugcd vcloci~yof the Huid is u. We denote .I,,, = C, t r ; . l i . Wc shall assume that the kinetic energy associated with the diffusion cu~-~-cnts,
.I, is ~ h cliffusior~ c current ol' Ihc patch
J , : Z,,(J1 , I (to)=
,/,
,
. . . ..I,,) = 1 - . IQ q~fidx.
I
2
1'1
dx. is sn~ullcompared to the macroscopic kinetic energy:
Let us tiow compute the rate of change of the energy E and the entropy S in the convection-diffusion process. StraightSorw;ud computation\ give:
To get a closed set of equations, we need to relate the currents J; to the probability field 11;. In this aim. we shall use two different argun1ents. First, w e shall exploit our knowledge of thc entropy t'unctionnl which must encreasc. And secondly, we shall also exploit an analysis of the dynamics: tine-scale oscillations of vorticity create local fluctuations in thc velocity held which in tur,n induce a diffusion current for the Incan vorticity. I t appears that combining these tbvo arguments fortunately yields a tractable expression for J;. To carry out the f rst part o f our program let us qtate a principle of n general scope. i I o l i o i i i . . During the relaxation process towards the equilibriuru. the system telids to lniiximi~eits rate of entropy prod~rctionwhile i t satisfies all the constraints imposed by thc dyn;imioc. At first hight such u vague k)rmulution seems trivial and ~ ~ s e l e ssince s . there are generally ~~ntrnctuble difficulties to express all the constraints imposed by the dynamics. In fact. ;is wc shall see. thc M.E.P.P. can hc VCI.\;. cfficic~itil'wc have ;I r)~.eciscrccipc to LI\C it. Thc idc;~ i h lo move ti)rwaril by trial uncl error. We begin by guessing w n i e crude set of constraints that the dynamic\ pi11 o n the cu~-renls.I;.The principle then yielcls ;I co~-~.c.spondinjr s e ~of r.elax;~tio~i ccluations. which wc c;in compare with the ;~ctualb e h ~ ~ v i o r ~ o t 'hyste~ii; thc s o th;~t we can Ici~rnsomcthi~igtro~il~ l i csystclll. and deduce more realistic conztri~i~llx. ctc. In this i~pproach.thc M.E.P.P. is ;I very cl'licie~ittool to deri\;c some of thc 1iii1in f . c a ~ u r cof'thc very intricate detailed dyliar~iicalhchi~viorot'thc system. t
K I ~ M A K KOUI'M.E.P.P, . nlust not be conf~~secl with the rnillirn~~nl entropy production principle o f Prigoginc. The Inttcr. which is u I-cformulntion of conservation laws. applies to a stationary state in the linear regime. Wc reter to 1291 for a detailcd discwssion o n the subject. A l t h o ~ ~ githwas not explicitly liwm~~l:itedby this : i ~ ~ t h oOLII. r , principle is clearly in the \pirit of Jaynes' ideas 1291.
[,el us now apply OLW M.E.P.P. Thc first eviderit dynamical con\traint on the ,l;'s is giccn by thc corlservation of the
-
mi~croscopicellespy: thiil is. E ( J I . .. . . J , , ) = 0. Also i t is clew that. at :uiy point x, the density of energy associ;ited to tlic tliffusion I
I?
11.unspol-t $ cannot be urbitcirily 1;rrge. More precisely, we shall suppose that for :uny gi;en statc of thc system. there is a function I W ( xSLICII ) th;~t the following constraint hold.$:
The M.E.P.P. then gives the following variational problem (V.P. 1):
under the constraints: (Cl ) J; (x) = 0. for all x.
xi
.
I t can be shown 15.31 that this problem always has a solution J I . .. . J,, satisfying (C3) with equality; moreover. there exist a parameter P and a measurable function A(x) such that: ( A ) J, = - A ( X ) [ V ~ I ,
-
p(i)- (I,)~,v~//].
This gives a general form for the currents J I , . . . . J , , . The variable ( 3 0) ditfuuion coefficient A ( x ) is not known: but once it is given, the parameter p . which i u the Lagrange milltiplier 01' Z . ia cletermined by the conservation of the energy:
which givca
-
where tr)' = C ,( I ~ ~ J , . Ncxt we need to determine the diffusion coefticient A ( x ) . For thia. let us consider thc particular case = 0. In this case. the energy constraint is not active and .J, is merely an ordinary difl'usion current. Thus. we can compute A ( x ) by ilsing the analogy with the convection-dil't'usion of a passivc scalar. Let us introduce the notations:
is the tine-scale Huctuation of the vorticity and u the corresponding fluctuation ol'velocity. Let us suppose that some scalar density p ( t . x ) is convected by the microscopic flow u. Then the mean value p ( t . x ) of p ( t . y ) on some ball H(x. 1 . ) is collvectttd by the mecun tield u and undergoes a diffusion process created by the H u c t ~ ~ a t ~u. o nP ( t . x ) will satisfy an equation of convection-diffusion type: (;)
where the diffusion current J can be c a l c ~ ~ l a t eby d classical method5 132,33,531: J = D V p . with the diffusion matrix:
where F is thc spatial scale at which oscillaticlns of vorticity occur and where c . is a constant which is not exiictly known and depends on some mcan decorrelation time of the system. T ~ L I assuming S. thitt f o r b = O the pi ill-e advcctcd like passive scalars leads us to identify:
To summarize. we have founcl that the evolution o f the convection-cliffusioti equ ;I t 'lons:
11,
is given by the following set of
whcl-c A ( x ) is given hy ( 1 1 ) . allcl fi i \ c l c r c ~ . ~ ~ ~ iatn eci~cli d . time. by tlic conwl-vation o f thc cncrgy ( H ) . K ~ : M A K K .It1 1.551 we havc showli, ill tht. p:u.ticul:tr case ol'a vortex patch. that these cq11:1tions can he obtained by using o ~ l ythe rccipcs 01. linear thcrniodyna~~iics about the equilihrii~ll~. Unfo~.tun:~tely.this method clocs not cxtcnd to the case oft1 Icvels. RI;MAKK. To get the 1.01-mulo ( I ) ) wc havc m;tde the ~ ~ s s u l n p t i ot nh a ~the oscillations of vorticity o c c i ~ ill r ;I well-defincd spatial scale F 153 1. This is ol'course ;I great simplification sincc. on thc one halid, thcsc oacillutions Illily ~II'I'ccI ;I Iill'ye l'llllge Of ~ C L I I C S ;llld, 011 Lhe otlic~.hand, such n scnlc c w o ~ ~ v;u.y l d with spituc and tinle. I t is well known that. as the How cvolvc.;. the owillu~ionsrend to hccolne smaller and smaller.
In practice, for computational purposes. we shill1 consirlel r . ~ ' L 0 g t . i ~;in empirical cocllicient with a tixed valut'. Kate that the variable coct'licient A ( x ) given by (1)) has tlic dimension of a viscosity ancl v:inisIit.s where (I,' - r ~ , ' = 0. i.c., whc1.c thc1.c is n o mixing of the vorticity nl small sciiles. 2.3.2. Cort~~cr:qcvic.c~ tou~rr~1.v tlrc c~c/riiliht.iiitrr. In \vlii~t Collowh we shall consider eclilations ( H E I I ) with A given by:
The link between ( R E n ) and the equilibrium theory is given by the following convergence result.
P R O P O S I T I O 2.3.1. N Let us suppose that the solution p ; ( t , x ) , i = 1 , . . . , n , of' ( R E n ) r.oi1verge.s ( i n a strong enough sense), M ~ P I It tend.^ to injnity, towards u ~tertioncrry.stute 1); ( x ) . Let 11,s ( I S S L I I ~ C eriso thclt there is sorne open connected .subclorncrin A of' L? .suti.yfving A c i2 und: (*) pi* ( x ) > 0 erne1 A (p*)( x ) > 0, ,fi)r ~111x in (**) u* . n = 0 rrnd J ; . n = 0 orz i3 A. ( n is the unit vector nortnrrl to the houndcrrv i3 A, u* is the velocit~1,field rr,s.soc~iated to 6,= r r i pf , rrncl JT is the c~rrrenttr.s.socirrterl to p: (A).) TIIPII I);( x ) is CI Gihf7.s sterte on A ; thrrt is, there (ire>prlrcrrnrter.~a:. . . . , a; ( a ; = 0 ) (111c1 p * .s11(.h t l l ~ l t :
A.
PKoof:. When t + ca, we have: l ) , ( t .x ) + p,?(x). cG(t,x) -+ G),(x). $ ( t , x ) + I / / * ( x ) ~uidp ( t ) + j-l* (given by ( H ) where we replace p , ( t . x ) by p f ( x ) ) .We have also . I , ( t ) = - A ( p ( t ) ) [ V / ) ; ( t -)f i ( t ) ( ( G ( t )- t r , ) / ) , ( t ) V $ I 4 .I,*. and since / ) , ( t .x ) satisfy ( K E n ) . the functions /),*(x)will siitisfy the set o f stationary ecllr~~tions:
Now let us calculate
Integrating by parts, we see that thc first tcrrn is zero (due to V . u* = 0 ) while the second gives:
-
p*
/" x(6,
-
V
* J: dx.
r
but the last term vanishes since J: = 0 and J;, V $ * . J: dx = 0 (this last equality comes from $*V . ( ~ T u+ * J T ) d x = 0 , by integration by parts).
sA
Sttrtittic~trlhjtlu)tlyr~tr~~rir.~ (O~rwr~rr rrvi.sitrd)
We finally get
from where V
Ln p"i
~ * ( L J-, cli)V$* = 0
on A ,
for i = 1 , . . . , n . I*
+
Subtracting equation 11 from equation i , we deduce that Ln B*(cri - a , , ) $ * has a I',, constant value -0,on A . Now, using the relationship p:(x) = 1, we deduce that 1,: satisfy the Gibbs state relationships on A . 0 The relaxation equations ( R E w ) have been used for large eddy simulations 1531; they where found very efficient.
3. Out-of-equilibrium problems: Weak solutions, shocks, and energy dissipation
3.1.1. 1)qfitlitiotl r?f'.sttrti.stic.trlso1irtiot1.s~ f ' t r t lc~\~)lirtiotl cclr~cltiorr. Let 11s begin with the simple case of a finite-dimensional dynamical systern:
where ( 1 : R" + R" is a srnooth mapping. and we assume that for any given initial state ( I ) has a global smooth solution detined for all time:
The family 0,is a group of srnooth diffeomorphistns of R". Then to any given initial Borel probitbility tneasi~t-e/ t o o n R" we can associate the frumily
Thus, in this case we clearly know what we mean by statistical solution: a family of Borel probability measures satisfying ( 1 1 ) . Let us now introduce the characteristic function
38
R. Robert
Making appropriate integrability assumptions, we calculate the derivative 3,fi, (v) -- i
f
a ( u ) . r e ' " " d/tl(u),
for all v in IR".
(III)
We can easily check here that, under some integrability assumptions, (III) is equivalent
to (II). Let us now consider the more general case where (I) is an ew)lution equation, u taking its values in some functional space. Let us assume that the Cauchy problem for (I) can be solved by means of a continuous semigroup St acting on the space E (with dual E'). The same calculation as above for/it = St (P0) formally yields:
f<.,,,,.
,,, ,,
dp, (u),
for all v in E'.
(III')
We will consider the fl)rmula (III') to give a definition of statistical solutions. More precisely, since we will always consider for E a dense subspace of the space of distributions 79'(R"), we will have CI~ C E'. DI':I:INITI()N. We shall say that t*~ is a statistical solution of(1) if fl~r all ~5 in (7r fit(z~) is a C I function o f l, the right-hand side o f ( l i t ' ) is properly detined and the equality hc)lds. REMARK. At this level of generality, the question of the uniqueness ot" the solutions of (i11') is open. Indeed, even if we suppose the existence of a semigroup $I, we certainly need strong assumptions to prove that a solution of (i11') satisties t*t = & (I*{~).
3.1.2. k)'om ho,u~etzeotts to illtrilzsi~' statisti~'ai s o h t t i o l t s q / B l t r g e r s eqttatiolz.
The one-
dimensional Burgers equation (B)
iJ t II -~- iJ.v
('
-~ ll
-- ()
!
is of the form (I) with ti(ll) -- --iJ.v(-5l!2). Equation (B) has been extensively studied [58]. We know that starting with a smooth initial data tt(). a unique smooth solution exists for a short time but discontinuities may occur at a linite time. We can also prove that discontinuous weak solutions exist for all time but then uniqueness is lost. To ensure the uniqueness of such weak solutions one has to introduce a supplementary condition (Lax entropy condition which amounts to consider only discontinuities with negative jumps). Finally, Kruzkov's theorem gives a contraction semigroup in the space L I lk~r the weak solution. After integration by parts (III') gives for the general statistical solutions
~}t/~,(V)
-- i
-~u-
ei
") d p , ( u ) ,
for all v in C ~
where p, is a family of Bore1 probability measures on V1(JR).For any real number h we denote TI, : D 1+ V 1the translation operator rl1(f ' ) ( x ) = f'(x - h ) . We shall say that p, is an homogeneous statistical solution if, in addition, we have for all I : ) =p
I
for all h .
A rather straightforward but important remark is that if p1 is a homogeneous statistical solution of ( B ) . it is also a statistical solution of I
+,
('r ) -
=
for all c.
This remark brings us to the following usefill proposition:
and we now prove that (11.
p l 1 ) ( ~ll')c~l(l'."' i. d,il (11)
-
I!-%
0
For this, let us consider the I'urlction , / ' ( / 1 ) = j'(s,,rr. ~ ~ , , ) r i ( dlll(rr). ' ~ " ~ " )Since / A , is homogaieous, ,f is independent o f 11, taking the derivative at 11 = 0 yields:
Thus, we only need to prove that
which ic n straightforward consecluence of (ii):
)
( but JIp,,
I
I = ;;;.IJplJl
/
2,
L
(
(
I
d
o
arid the rehult is proved.
)
1/2
L (.ll~lill,.~. 0
We come now to the clclirlition on an intrinsic statistical solution. This notion provides u very nntuntl way to hiulclle the int't~al.eddivergence problem. Infrared divergence occurs for a random field with homogeneous inore~neritswhen the probability measure cannot he dcftiecl on 'P' but o n the quotient space 'DJ/constonts: corresponding to the fact that only the probability law o f the increment< i t ( . \ . ' ) - r i ( . r ) makes sensc. The cIii~ructc~.istic functionul ol'such tields is defined o n thc subspace .;,'C ot'thc t ~ ~ n c t i o ol' n s C';;" suc ti that ,I' I)(.! ) d . = ~ 0. Thus. in view of tt~rbulcnceproble~nsi t is iniportant to dctine statistical solutions of the hydrodyn~uniccqilations which are clclincd on the cltroticnt space 'D'/co~ista~its. Proposition 3. 1 . I Ici~dst o a ~ ; I ~ Lcleti~iition. II.~I~
fibr :illy choice of p (of' coilrsc, wc assurnc th;tt tlic right-hallti i d c ii~tcgri~l i\ ~ ~ r o ~ ~ c i ~ l y clclinetl).
+
i l , ~ il, ( I ( , U) cliv u = 0.
+ V / ) = 0.
Intl.ocluci~~g I.cl.:iy's pro,jcctor P (L'-orthogorial i71.0,jcc.toronto tlic clivergcilcc l'r-cc vector. liclds). ( E )can he wrirtcn (at Ic;lst fornlally) its ill u
+ P(il, ( r r , u)) = 0.
where we use the notation ( u @ u : Vv) = ( u ; ~ iiI;vn). ~ , The definition of intrinsic statistical solutions then straightforwardly follows: iLfir(v) = i 1 lim 1 + ~ ) for all v in
/
( ( u - (u, p l l ) )@ (u
-
(u, p l l ) ) :V P ( V ) ) P ~ ( ~ ~ . ~ ' ) ~ ~ ~ ~ ( U ) ,
(c(G)~.
We will now exhibit a class of intrinsic statistical solutions of Burgers equation. To do that, we need some ~naterialfro111 the theory of Levy processes.
3.1. A o i o L . For a clear and comprehensive study of Levy processes we refer to 110,271. We only want to give here an alternate presentation from the general point of view of probability ~neasureson functional spaces. We define directly a homogeneous LCvy process with finite variance on the line as a Bol-el probability measure ji on the cluotient space D'(W)/constants, with characteristic functional (defined on C,;) of the form
,/iZ
wherc I !I(.,-) = I ! ( . \ ) cl.,, ;iricl the f'i~rictionIl/ : W 4 C satisfies ( i ) $1 it1 C" ci!id $ I ( ( ) ) = 0. ( i i ) I/I(-II]) = $ I ( I I ~ ) . ( i i i ) (/I is conditioniilly 01' positive typc. or eclui\/~ilcntly- $1" is 01' positivc typc
+
The function is the I i v y cxponcnt of the process. Elcmcnta~.ycalculations using Bochncr's theorem yield the estimate
Let us consider the functional C(II]= ) exp
(/
$ I ( , ~ ~) )( Id ~ ) . for
111
i n C(>
C' is obviously continuous o n the space C';;". I t is classiciil to check (using the properties ( i ) . ( i i ) ( i i i ) ) that it is of positive type and thus by Minliis' theorem it defines a Radon probability measure 11 on the space 'P' (271. The linear operutor I is continuous from GIG, into C ( 7 . and its transpose ' I :'Dl 4 D'/consta~ntsis the primitive operator. Thus exp(./ $/(I I ~ ( . Y )d.\-))is the characteristic functional of ;I Radon probability Ineasure ,LL or1 the space 'Pf/constants. image of 1 1 by ' I . i t is the LCvy process with exponent $1. Using general arguments. we can get further regularity properties of the Levy process. For any open bounded interval J = I t r . 171, the irnage of v by the restriction operator to J is actually a cylindrical probability on the space L'(.I) since C ( u ! )is continuous for the L'
norm. The image of 11 by the operator I: -+ j;,' v(.s)d,s is thus a cylindrical probability on the Sobolev space H ' ( J ) . We can now use the Hilbert-Schmidt inclusion of H ' ( J ) into H V ( J (for ) s .= 1/2) to see that, by restriction to J , 1.1 gives a Radon probability Ineawre o n the space H.Y(J)/co~istants. R E ~ I A uK . Of course. as it is well known, the trajectories o f Livy processe., havc further regularity properties (at each point right and left limits exist) 110,27), but the presentation above will sut'lice to o u r needs.
3I . t
i ~s / 1it i s / i 1 1s ~ 1 i o 1 1 .I s our main result.
(HI)
S
t
o
.
Wc are now i n situation to state
i)/~// = = i $ i I ) , , ( ~ .
Pl
l:. We only givc
;I
skctcli ot'thc proof'.
F-;l.,sl ,\10/7. WC cllcch Ih;11 , / ' I l l 1 - (11.f ~ ) ~ l ~ ~ , , , , < l / l<~ W. ( t / l f'01.ilII1 illld Noricc that wc write [I i115lcaclo f [I,,.
ilII COllll?ilCI
~S[,(~O/l~/ ,S1l1/7. Wc c : ~ l c ~ ~ l ;1:~ = t e , / ' ( ( I / - (11.[I))'. ~ ~ ' ) ~ J ' ( "d~, "i !l ( / ~ )~'(II, t 1 i l l .;,'( thc illlcg~-;~I I I I ; I ~ C SI~CIISC.;111dby Lcbcsguc's theorem wc liiivc.
I ~ I C10 [he first s ~ c p ,
Notc that thc last integral convcrgcs in the spncc I . ' ( . / ) by thc first stcp.
Tllir~l.stcJ/). We calculate
IIIIC~V;II . I .
and. after straightforward calculationc and taking the limit
E
-+0, we get
where
arid
1
for .r < 0.
Easy ci~lculntiolisyield
I'or K large enough. Illus (rcplacirlg P by
\bc g c ~
I~,J,, (1.
O n thc other hancl. we gct 1)1. A,,
:tricl
[he result is proved.
U
Now it ~.emainsto solve equation ! H I ) within the class of L k \ y exponents. As i t was pl.eviously noted. the discontinuous physically acceptable solutions of ( R ) have negativc jumps. Thus, i f we want to construct physically acceptable statistical solutions, we have IO considcr o111y Levy plocesseb with negative jumps. These processes
are characterized by means 01' thc Levy-Khintchinc representition formula for the exponent 1 10,271:
where ,/;-u,cl,
is a positive measure o n 1-co. 01 (the Lkvy measure of thc proccss) satisfling .s2 (lrl(.s) i+m. And we can now state the following
11
n
PRoor:. See 1 15 1.
Rti\~lh~lis. - The assunlption i I/J,',(O) 0 corre\poncls, 1.01- the ;~scoci;rrcilproccsc, to the propcrty I I I C ~ I I IV ~ I I ~ I C( ! I ( . \ ' ) - I ( ( . \ - ) ) 2 0. 1'0r.1' > . \ . - 111 the cusc whcrc the initial p~.occssi \ the 13rowni;ul proccss on the line (i.r..l / / l r ( ~= ~>) - t l . 1 1 %'- ). the explicit l'o1.1~ of the solutio~iis given i n I 14. I5 1.
<
-
The c:~sc 01' i \ n initially Brownian / I ( , ( . \ - ) . v;lni\hing for . I 2'; 0. w~isfirst invcstig;~tcd by Sinai 16.31 ~ ~ s i r the i g explicit Hopi-C'olc constr~lctionof' the cntropic solurioll 01' Burgers ccluation. The relation will1 Lkvy procc\scs was first ~ncnriolleclin C:IH;I~O ;uld I)ircholl 111.151 ill terms ot' intrinsic statistical solutions. Clsing the Hopf-Cole solution iigain. Bcrtoin 191 proved t h i h s;lrlic relittion tor 110. a gcncrnl LCvy proccss with finite vnriancc imd 110 positivc ji~mps.v;unishing lor .v 0.
<
Here wc consider both the th1.c~-dirnensio11:ilincomprossihlc Navict--Srokcs iund Eulcr equations. For sil~~plicity we l i ~ l i i toi~rsclvcsto flows on rhe I O ~ L I 'T S = ( i K / ~ ) j i.e.. . ~~itli periodic hountlary conditions. Let us take Navicr-Stokcc equation tirht. For nn initial velo~ityficltl u~ with finite etiergy. :IS is wcll known (L,eruy (34.351).there exists at least one weah solution (i.c., in rlie sense 01' clist~.ihutions)to the Ciu~chyproblem. A prio1.i such a solution hclongs to L8%(0. 7'; L ? )n [,'(O. 7': HI)and there is not enough smoothness to i n s ~ ~the r c cl;~ssicolenel-gy equality; all we know is one can definc sollle wcnk solution satihfying in t~ddition:
As a first step we show that for any weak solution u of Navier-Stokes equation, the local equation o f energy
is satisfied. with D(u) defi lied in terms of the local smoothness of u. Thus, the nonconservation of cncryy originates from two sources: viscous dissipation, and a possible lack of smoothness in the scllution. For Euler. equation. we considcr weak solutions in ~ ' ( 0T; , L ' ) . Although there is n o gcner:~l result at prcscnt for the global-in-time existence of such solutions, solne examples are known (consider any two-dimensional wcak solution given by Yudovich's theorem 170)). According to an approach in the \tudy o f turbulence that goes back to O n ~ a g e r1461, i t [night be true that such weak solutions of 3D Euler describe the turbulent flow correctly (in the limit o f infinite Reynolds number, of course). Smooth solutions conserve energy as is shown by a qiniple integration by parts, but thix calculation does not cxtend to wcak solutions. Smne weak iolutions have been uonsrructed withour energy conservation (Schefk r IhOl. S t i n i ~ . e l ~ ~Ih?,l). i a ~ l Onsagcr had co~jccturcdthat weak solution 1 / 3 s h o ~ ~conserve ld energy. The gro;lt interest ol' this question was duly enlphasi/ccl by Eyink 1221, who iilso gave a proof of energy conservation i~nder21 st[-ongel.assumption. Thcn Const~uitin,E end 'fiti I 191 gave a si~npleand clegi~r~t ploc~l'ofcnu~py conscr~iitionuntler thc wcahcr ilncl 1110rcI ~ ; I ~ L I ~;issi~niption ;I~ that u hclonga to the Bcsov space [l',"" with u > I / 3 . I c o ~ s i d c t i o above ~ s on ~lissip;~tion i n N;i\,icr-Stok~' cqui~tionapply to wc;ik solution> o f Eulcr as well: one has :I local cclu:~tionof energy:
ilnd thc explicit form ol' I)(u) makes i t poasihle to provc cncrgy conwrvntion ~ ~ n d e;I l slightly weaker assu~nption. We then come to the problcm 01' distinguishing between weak solutions 01' Eulcl- and Novier-Stokes. which may he consicle~.edphysically acueptablc. We first see t h n ~the weak solutions of Navier-Stokes constructed by Lei-iiy 133.351 clo satisfy D ( u ) 3 0. We also show that any wcitk s o l ~ ~ t i oOI'ELIICI. n which i$ ;I strong I i ~ i l i tof s[11ooth s o l ~ ~ t i oofN;i\!ier~ih Stokcs s;~tictiesthi.; s;uiic condition. Finally we ;we led to a definition ol'dissipativc weak solutions: those satisfying l)(u) 2 0 . 3.2.1. Tlrc~loc.trl cclirotiorr c~f'e~rc'r:y~,fi)r. ~ . c , t r X .~ollirio~z.s c!f'A1tr~'ic~/-StokoL~ rrlrtl Errlrr. aclrr[ri o Our ~ ~ n a i npoint is cxp~.essedin the following two rcsults:
P ~ o r o rsl o~N 3.2. 1 . 1,vt u E ~ ' ( 07':. H ' ) n LZ(O. 7': L'). Sfokr.s c~clrrtrtiorzorr tlrr tlrrrc-cli~rzcrrsiorztrl toi.rr.r I :
I
il,u + ~ ) , ( L / , U-) I I A U +V p =(), div u = 0.
lr
r~.rrrX..sollrriorz r ! f ' i V ( r \ ' i ~ ~ ~ -
Let cp he ~ l n yitlJinite1~1 d ~ t r e ~ ~ t i ~ i h l e , f 'with ~ r ~cotnpact ~ c t i o ~ .support ~ on IR3, even, nonnegative nirh integrcil I , rrnd ( o f (6)= $ c p ( : ).
+
Plrt D, ( U ) ( . X ) = $ J'VcpP(<). ~ u ( s ud<, ) ~where 6u = u(.u 6) - u(.r). Tller~,c1.s E ci/~/~rotrch~.s 0. thr,firnc.tion.s D, ( u )(which Lire in L' ( 10, TI x 7 ) L). O I I V ~ ~ in< ~ ; S ( ' , ~ I I L Jsense (!f'di.~trih~itiot1~ 0 1 1 10. TI x 7 ,to~t~(ir~1.s u di~trihutionD ( u ) , not tlel>e~~cling on cp, crnd tlz~.,fi)llo\ving loc~ilrq~r~ition of' C ~ I I Ci.sI SLI ~ ~ti.sfie~I
PROOF. Using Sobolev inclusion of H ' in Lo, one easily sees u is in L 7 ( 0 ,T ; L') and O ;, ~ ~ 1 ' the ) ; same for p since, taking the divergence of ( I ) , therefore l r ; l i r is in L ~ / ~ ( T one gets
and if 1) is the only solution with mean zero, the linear operator 1 r ; ~ t--t ~ 1) is strongly continuous on L'l for I < c, < cm.and so 11 E L'I2 (0. T : L'I2). Now let us mollify Equation ( I ): denoting u' = cp6*u,p' = cp'*p, (i1,u)' = cp'*(lr,u). . . one has
this equation. multiplicrl sci~li~rly by u plus Ecl~~ation ( I ) mi~ltipliedby ut . gives ill ( u . u')
+div((u. U')U + /I'U + PU') I~A(U u ' .) + 2 v V u . VU' = 0.
+ E:', where
E+( t .. v ) = ;),
(i/;il/) ' I / ,
- i~~ii,;ij;ii;.
+
Since u E 1,'(0. 7 ' : 1,'). u . u' converges to u' ~und( u . u' ) u + p'u pu' converges lo 2 p ) u in the sense of distributions on 10. TI x 7 . Moreover, Vu' tends to V u strongly in I-'( 10. 7'1 x 7).thus E , ( t . . Y ) converges in the sense of distributions towards
(u'
+
Another calculation gives
But i), (11 / I ( ,)* 11; = il; ( 1 1 ,
(11 , l (
/ ) ' ).
due to the incompressibility of u .
Moreover, ij, ( u ;( ~ i , ~ l)'r , - ( l r i i r j u ,)'I tends to 0 i n the sense o f distributions on 0 10. 7 ' [ x 3 Tand thus j ' V p t ( < ) . SU(SU)"< has the same limit as 2 E , . The same reasoning applies entirely for a wcnk solution of Euler ( v = 0) and gives PROPOSITION 3 . 2 . 2 . LPI u E ~ ' ( 0 T , ; 12) AL' ' 1 ~ ' r ~~01utiotr lk ( ! f ' E u l ~~yurrtioi?. r Tiz~tl tlw , / j ~ i ~ ( ~ t iilt o i ~(u) . s r o l ~ l ~ ~it1r gt /~~, o.s~ti,soof ~ / i . s / r i l ~ i ~ i ; toot z( .1 ~(,I i . s ~ r i b ~ ~ ~DI(out j,i ~zot dcl)c~ulil?y 0 1 1 cp, trtl~lt h c , f i ) / l o ~ i ,I(1c~rr1 i t ~ ~ ~ ~ ~ / ~ l (o~j 't~i 1o1t ~l r hol(1.s: g.v
R E M A R K . I n the two previous propositions. D ( u ) nicn\ures a possible dissipat~on(or production) o f energy caused hy a lack ol'sl~ioothnessi n the velocity field u , this term being by no means relaled to the presence or absence of'vi\cosity. Now let us state a simple s~noothnesscoridition which implies [Nu) = 0.
P I ~ O O I Onc , . ha\ 1 ,I' V v ' ( 6 ).(su((Yu)''161
<:
/Cqf ( < ) ~ / i id<. u / i~ n t r g r ; ~ ~ i n g o v 10. c r 7'1 x 7
y ioltls
and putting
= c 1 1 . onc sees t I i i 5 tcnds to 0 \c.itIi r .
U
KICMARK. I t u i s i\ weak solution 01 Eulcrcq~~;\tion and s:rtisfies the ~moolhncsscondition in the proposition above, the11thc kinetic energy o f u is consrrvcd (.just intrg~.atci n .v thc locc~lc q ~ ~ i ~ toi ol ' cnn e ~ y y )This . provides a proot'of Onsagcr's co~i.jccti~rc I I9.22.40] ~ ~ n d c l an assumption slightly weaker than u r ~ ' ( 07':. with ~y > I / 3 .
BY.%)
There is still some d o ~ ~whether ht weak .;elutions o f 3.2.2. Kcl(,\yr~rcy~ 10 rrtrl 1r~t~1~~rlcirc.c~~~ Nnvier-Stoke\ equation, the ~~niqueness o f which i\ unknown. or hypothetical weak solutions of Euler equation, arc ~.rlcvantto the description o f turbulent flows at high Reynolds number. It seen15 reasonable to 1.cquil.e home extra conditions: one of them might be that
the lack of smoothness could not lead to local energy creation. In other words, one should have D ( u ) 3 0 on 10,T [ x 7. It is quite remi~rkablethat this condition is satisfied by every weak solution of NavierStokes obtained as a limit of (a subsequence of) solutions u, of the regularized equation introduced by Leray 134.351:
For uo given in L' and E > 0. this equation has a unique C 2 solution u,. The sequence (u, ) is bounded in L 2 ( 0 . T :H ' ) n L 2 ( 0 . T : L') and a subsequence conT ;H ' ) and strongly in verges to u, a weak solution of Nuvier-Stokes, weakly in ~'(0. L3(O. 7': L ~ ) Bilt . for the regul;uizecl equation, one has the local energy balance:
hcnce 11(Vu,) ? co~lve~.gc\ in the sense of'distrihutions towards
-
( / ( I . . \ - ) . the f ' ~ ~ ~ l ~ t i o ~ l i ~ l For cvcry infinitely di1f'crcnti;ihlc allti nonncg;~tivct'~~nction u ,/,]'(VU)'(, ( I ..\-) d.v dl is convex aricl lower-\c~nico~iti~iuou\ o n the wc;tk s p i ~ c c 1.?(0. 7': H ' ). itncl tIi11\
which implies lim,,lr 11(Vu,)' I ~ ( v u )= ' I ) ( u ) 3 0. This I'ilct is well kriown: ace. lor example. 1361. -
R I ~ M A H KTwo . ni1t~lr;ilclucstions iirisc ;I[ this point: ( I ) Docs there exist ;I wei~ksolutioll 01' N;~vic~.StoI\c\i l l the S ~ ; I C C lZ2(O.7.: H ) n 1,\(0. 7.:I>') with D ( u ) # O ? ( 2 ) Does the condition D ( u ) 2 0 imply uniqueness lor weak solutions of NilvierStokes'? Let 11scull such weak solutions with I ) ( u ) 3 0 "dissipative". In the case o f inviscid Burgers equation in one-space climension. I ) ( u ) 3 0 coincides with the usual entropy condition of negative jumpx. which docs imply uniqueness.
'
The following proposition shows that the condition I)(u) solutions o f Euler equation.
0 appears naturally tor weak
PROOF.The weak solution of Navier-Stokes uL'satisfies:
Since u ' tends to u in ~ ~ (T:0L '.) strong, one ha5
in the sense of tlistributions, ancl thus D ( u )
0.
U
R E M A R K . Let u t ~ ' ( 07':. L ' ) be i l weal\ solution of Eulcr. clixsipativt. in the sense I3(u) 0. Thcn it is n dissiputive solution of Euler in the sense of' Lions 1361. Indeed. every weak scllution with ,I' 411: d.r < 0 is a dissipative solution in Lion's scnsc. Notice that this last conclition c l o e not pl.t.vent ;I priori from ;l 1oc:il cl-e:~tionol' c11el.gy 111 so11lc rcgions of'tht. flow.
2
'
3 . 2 /
I / I I S Y J I I ~ I I I / ~ I I / I/ I , We have ;ilrc;rd?; sccn th;it I ) ( u ) docs 11ot d c l ~ u o~nl cy. A\si~minghome xpacc c o ~ i ~ i r i u iof t j u. we ill-c ahlc 10 c\l31-cssi t marc cxplici~lyusing ;I ri~cliallysy~nluctl.icI'unctio~iq ~161 ( ). L,cl 11sput .S(U)(.\.I.) = (u(.v 1.E) - U ( . ~ ) ) ~ ( U ( . \ - 1.6) - t1i.v)). (lZ((),where clC tlcnotcs the iu.c:~nlcasu~.con thc sphc~.c. An casy comput:ition gives
+
Now let I
,k,
I,v(~)
'-L
115 as\umc
~ J ' ( I - ) I . ' dl.
th:it.
;i.;
c.
-+
0.
+
t c ~ l d \to
;I
limit s ( u ) ( . v ) .Thc11 1)) ( u ) =
3
= -=.Y(U).
The I'OUI--fifthlaw ( v o n K ~ I I - Wi111tl ~ I Howarth. I~ Kolniogorov) say\ thal tor a st;~tion;iry. h o n l o g c n w t ~ s:111cl isotropic random turbulent vcloc-ity ficld u onc slio~1I~1 h;ivc
wht.~.eI ) is the nie:lli rate of (incl-tiul)cncl-gy difiipi~tionper unit mass and (.) dcnotcs the statistical Ilienn. Without isotropy. one proves (Monin, cf. Frisch 12.51)
integrating in ( over the ball 16 / 6
t
one gets
+ E(ltl=l
3 1 D = ----- lim 16n i - ~ l
(u(X
c()
-
U(X))'(U(I
+
F[) -
u(x)) . 6 d x ( ( )
Our expression of
thus simply gives a local nonrandom form of the above expression of the inertial dissipation.
4. Last comments and acknowledgements We have tried here to give an overview of sonic topics which we feel in.;pircd by Onsagel-'.; views. We focussed as tu-as possible o n issires which yielded rather precise mathematical dcveloplncnts. Iloing so we volunt:~rily clisci~rdedother important i \ s ~ ~ icnst~~rbulcncc theory such :14 i~iterriiittcncyor the rece~iti~iil)ort;~~~t ~Ii\covcryof' tlic i~iverseenergy ca\c;~clc i n 2D turbulence. I t is :I gl.c;~tp1casi11.cSor lilc to take the opporr~l11ityto w;u.mly thank those who I1;1ve 1i:1tl a direct corit~.il~t~tiori ill the cl;tbor-ation ol'thc prc\ent work: F. Ho~~cllct. P.H. Ch;~v;uiis. J. Duchon (who greatly liclpctl 11ieto write Section 3. I ). T. I)i11110111. J. MicheI. A. Mikclic. C. Rosier. J . Somllieria. Great t11;1nl\s also to C. H;uclos. H . Caat:~ing. I? Con\tantin. A.J. Cliorin. G. Eyirih. M. F:u.ge. U . Frisch. J.L. 12cbovit/. A. maid;^. K. M o r c ~ who . kintlly contributccl to the discussion arid diffusio~~ 01' t h c e ideas.
Appendix A ~ .briclly rcc:111 some stalli r r ~ - / ~ rr vr irr r r i o r r . For the co11iSortol'tlle I - c ; I ~ c\ye dard notations arid ~ ~ . o p c r t i 1c3s1.
A l ~ l ~ ~ n . v i r r ~ t r /c!/'tlrc, i o , r S o h o l ~ 1.sptrc.cJ ~ H " ' ( R " ) . We dc~iotcQc, = 1 - 112. 1/21". x tho characteristic function of (Ii1. and /3 = x, . . ., x ( 1 1 1 I terms). For ;r given p;~r;~~iietcr Ir > 0. we dcfinc o prolongation operator 111, : to ilny Si~nction, / j , = ,j;: x ( - , j ) ( , j be-
+
longs to
z").we ;~ssociatcthe f'unction
xi
Let 11sconrider now a compactly rupported measurable bounded function h(x) d s f y i n g
f L(x) dx = 1 and I. 0,
k
b ( x ) i ( y ) ( x- y) dxdy =
for k = 0. forO
+...+
wherc k = ( k I . . . . , kill, Ikl = k I k,/,and ( x - y ) k = ( x I - , Y I ) ~ I . . - (-.xXl l~) LI , ~ . Then we dcline a restriction operator r.1,: for f' E L;,~(R"), we denote ,/,( = $ ,[ ).(; -
z,,
j ) , f ( x )dx. and deti~lc,.I, , t = ,f,/x ( j). We have the well-known estimates (where ( . denote5 different constants which d o not depend on /I ): (1 If fj, E L'(R"), we have Ilv/,,/'II12? c.ll.f'll,... ( 2 ) I f ,fi, E ~'(11~'~). we have pi,,fj, E H"' and I l l ~ l ~l , f i ~ l <~ ~ll./ir~ Il l . ? . ~ rnoreover i.ll,fj, 111.2 jl/jl, ,fil[II,? Il/jl (Il,:, where c . > 0 does not depend on h . I t li)llows that / I / , is LIII isolnorphisr~~ from the space o f the functions ,til which are sc1irar.c ir1tcgr;tble onto a subhpace kj, o f H "I. ( 3 ) If f' E H " ' + ~ ( w " ) for . 0 k s 111 I and k n , , we havc: -
<
<
6
<
<
< < < +
C 11 = I / N . We denote H;:, the Soholev sp;lcc H"' TIIP/)i,r.iotli(.c,ir.\rl. Let LIS S L I I I ~ O S thi~t on the c/-cli~~lcnsic,~i;~l tol.us (R/z)", :uid Lj,(Q,I)the s p ~ ~ ol'thc cc I-cstsictit)nato Of/of ~ h c j i N' Iunctions o f ~ h f'o1.m c , / ; ; p ( ; - j ) which iu,c ~ " - ~ ~ c r i o c(i.c.. i i c ,/,: = ,I,, . for all , j . I is endowed with thc L' hcnlar. pl.oJuct. in it").19, ( Ohvic,usly. il' ,/' ;ulcl ,/j, are E1-pc~.ioclic.so a,-c ,.I,,/'and Ancl thc following cslinlarch holcl: ( 1 ' ) ll' ,I' E wc hitye l l t ~ ~ ~ , / ' l5:l ~(~11.1 , ~ ,lll,?(L,,,l. ~~,,~ (2') I f ,/' is z"-periodic. we have (I /I/,,ti,11 l,ly;,, 6 ll /if l / l . ~ , s , im ,. J
x,
[,iCr.
(3') I f ,I' r H $ ' .
for 0
6
< 1, < < rrr + I and I, < rri. we have .\
Kcferences 1I 1
V.A. A t i ~ o ~ i <M(I.\! w . / ~ I Y I / ? ~ ( / ? / ( pItil\(, J i / i . \ / t . i / ) u / i ( it! ~ ~ ~\ / 7 / 1 ~ 1 i t Vcbt~iikl,c~iii~gr~~iI (;(IS. 01ii\,. 7 ( IOO?). 1.35.
/;)I
111 .\/11t \ \ \ / < , I I I \
121 V.1, AI.IN~ILI. 0 1 1 1111 ( I /lrio~.i ( , , \ / I I I I ~ I / ~I ,I I t/iv / / i e i t ~ o. ~/ ' / I ~ , / ~ ~ ~ / ~ I I , I Iiit~l)i/i/>. I I ; ~ ~ ~ I A / 1 1 1 c r h~l~ttl>. Sot. 7'r:111\1. 79 ( 11160). ?(17-2(10. 131 A , Arhc~lcv.G / o / ~ ~1,\!\/1,11(~, I/ (1111 I I , < V I ~.\O/II/I'(J~J (I/'\?/(I\OI,'\ \\.\I(,III 111f,(/i~(~/ro~i\. LISSK C ' O I I I ~ .kl:ttl~.Math. Phy\. 15 11C)75). 131-143.
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(1
/ 131 E. Caglioti. P.L. Lionh. C. Murchioro and M . Pulvirenti. //~III(,II.\~oII~I/ 525.
EII/P~ (,~/I~IIOI.Y:
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1 101 I'
I.ectir~.c\ in Appl. Math.. Vol. 18. Alncr. Math. Soc..
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I111
1/11, i ~ l I l ~ l ,1y0\1 1 \ 1 ~ 1l~ll~i~lll /01~.\11/11/~01r\ 0/'/.'11/1~1,\
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\~/I/V /~II/~,I~~~II// N(II,ICI.
122 1 (i. liyinh. I
12-11
('.
~-~/II~IIIOII.)
1111117irr.l~11/~~11( 1'. (';~rnhriiIgc L l l ~ ~ \ e r -
1251 I?(>] 1271 12x1
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I ~ I ~ , ~ I I ~ ~ I ~ I / ~ \ ~ / / , ~ , ~ ~ \ ~ I Noiili~learity IIIIIII~~\. X
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1351 J. Ler;~y. E.\.\cii .sir,- Ic, 248.
II~O~II,~,III~,II!
t l ' r r ~ r /ic/ctitle ~.i.\c/irc,rr.rc~~rr/~/i.\.\ir~rt /'C\/)~I(.~,.Acta Math.
63 ( 1934). 103-
Stcrtisticcrl hyrl,.oc~namics(On.rager revisited)
53
[36] P.L. Lions. Muthenlatical Ejpics in Fluid Mechunic.~.&>I. I . 1ncompres.sibleModels, Clarendon Press, Oxford ( 1996). 1371 P.L. Lions and B. Perthame, Propagc~tiortof mommt.r and reguluriry for the 3-dimm.riona1 Vlusov-Poisson system, Invent. Math. 105 (1 99 1 ), 4 1 5 4 3 0 . (381 D. Lynden-Bell. Sruristical rrrech~rnic.~ of'violrnr wluxution in stellar systems, Monthly Notices Roy. Astronom. Soc. 136 (1967), 101-121. 1391 C . Marchioro and M . Pulvirenti. Mathemuticcrl Theory of 1ncompre.s.sibleNonvi.scou.s Fluids, SpringerVerlag. New York (1994). (401 J. Michel and R. Robert, Stuti.sticc11mech~micnltheory ($the greur red spot ofJupiter, J . Statist. Phys. 77 (314) ( 1994), 645-666. (41 1 1. Michel and R. Robert, krrge devintion.~fi)r Yormg meusures artd .statistien1 mechunic.~of infinite d i m m sionrrl dyrrcrmicnl .sy.strrtrs rvith con.ser~vrtioril w . Comm. Math. Phys. 159 ( 1994), 195-2 15. (421 A . Mikelic and R. Robert. On the ec/rrcrtion clt.scrihing [I reluxution towurd LI stuti.sricul equilibrium strrtr in the t~co-rlir?rc~~r.sio~~~~I perfi'ctjf~rid( ! ) ' ~ I U I I I ~S~I A . SM, J. Math. Anal. 29 ( 5 ) (1998). 1238-1255. (431 J. Miller, Srcrri.~tic~crl ~rtrc.hrtnic.sof Euler ec/rrcrtion.r in two rlinrm.sion.s, Phys. Rev. Lett. 65 ( 1 7 ) ( 1990). 2 137-2 140. 1441 J . Miller, P.B. Weichman and M.C. Cross. Sttrti.sric~1mechtrnic..s. Errler eyuution.~,und J u p i t r r : ~red .sl)ot, Phys. Rev. A 45 (1992). 2328-2359. 1451 D . Montgomery and G. Joyce. Stoti.\tic.oI rnc~chcrnic..~ c!f'rrr~~rtivr trnt/)rrcrture srrrt~.s.Phys. Fluids 17 ( 1974). 1139-1 145. 1461 L . Onsager. Srtrri.sric~(11 hgr1roclyt1c111ric.s. Nuovo Cimento Suppl. 6 ( 1949). 279. 147 1 T. Padrnanabhan. St(rti.stic~11 rnec.ltcrr~ic.sof'firrrvitutirrg .s~v.srrr~r.s. Phyc. Reports 188 ( 5 ) ( 1990), 285-362. (48I K . Pf:~ffelmoser.Glo1)trl c.kr.s.sic.rr1.soI~~trorr.\ of the Vlrr.so\a-Poi.s.sor, sy.stc,~trin t h r w t1irrrrtr.siotr.s,fi)r grrrrrtrl itriritrl dcrttr, J . Din'rrrntinl Equations 95 ( 1992). 28 1-303. [49I R. Rohert. A r ~ t r ~ . ~ t r t r ~orrr,q)x r~tr ~)r.trrc.il~lc~.fi)r. rn~o-tli~trnr.\io~rol EilIc,r or/rrrrrio~r\.J . Statist. Phys. 65 (314) ( Ic)91),531-553. 1501 R. Robert. Urrie.irc; dc, / ( I .s~lrtriorrfirihk, i r .src/t/~or/ i.or~r/)(rc,rtie, l'c;c/rrtr~ir~rr (k, Vl~i.\o~~-l'oi.~.~orr. C . R. Actid. Sci. Park SCr. 1 324 ( 1')')7). 873-877. 15 1 1 R. Rohert. Otr rlre ,qrrr~~ir~triorrol c~ollcrl~.\c~ o/'.\tc,ll(rr .sr.,rc,rtr.\. Cla\sical Q u i i ~ ~ l uGravity ~ii 15 ( 1998). 38273840. 1521 R. Rohert. 0 1 1 rlr~,.rrtrri.\ric.trlrrrc~(.lrtrrric..\(!/'21)i r r c ~ o r r r ~ ~ r c ~Ertlc,r. ~ . ~ i Ic~yrrtiri~~tr, )I~~ Comm. Math. Phys. 212 ( 1 ) (2000). 245-256. (531 R. Rohert and C . Rosier. Orr !/I(,irrotlellirr!: (!f.\rrrtrll .\c.trlc,.\ f i ~ 21) r ~rrrbrrlc~rtr flr)~\..\,J . Sti~ti\t.Phy\. 86 (314) (1997). 1.541 R. Robert and J . Somnleria. Stcrti.stic~tr1c,yrtilihriiorr .\tcttr.\ for- r ~ ~ ~ o - t l i r r r c ~ r flo~r..\, r . \ i ~ ~ J~.~Fluid ~l Mcch. 229 (1991). 291-310. 15.51 R. Rohert and J . Sonilner~u.Ke,ltr.\-crtiort tobrctrtl.\ o .\rcrti.\ric.ol c~yrolihrirr~rr .\rtrrc, itr t r ~ ~ o - t l i r ~ r c ~ r ~ . s/~c,rtc,c.r io~r(~I ,flrtitl c!\.rrctritic..~. Phys. Rev. Lett. 69 ( 1992). 22762279. 1561 R. Sadourny, firrl)itle~rrtel~/fir.\iorrirr I(rr~c,.\c.trlc, florv.5. Large-Scale Trilnrport Procexses in 0ce;lns and At[nosphere. Willebrand and Anderson. eds. Reidel. Dordrecht ( 1986). 1571 J . Schaeffer. (;lohi11 c,ri.\tc~rrc.t~ , / i ~ rrlrc, . Vlcrsov-Poi.\sou .\v.~trrrrr~,ithrrc,trr.!\..\rrrrrirc,tr.ic.(lcrrtr, J . Differential Equation 69 (1987). I I 1-148. 1 58 1 D. Serre. .Sy.\ti.rrrc,.\ tk, /(]I.\ (I(, c~orr.\c~n~crtio~~. 1. Diderot. ed. ( 1996). (591 Z.S. She, E. Aurell and U . Frisch. 771r irr~'isc.itlBurger? eclirerriori n,irIr irririctl cltrrtr of Brob~'rrierrrr w r . C o n ~ m . Math. Phys. 148 ( 1002). 623-641. 1601 V. Scheffer. Arc irc\,i.cc.id/ l o ~ t~t.itlr , c,otrrl~trc.t sul~l)ortit1 ,sl)crc.c,-tirtir. J . G r o m . Anal. 3 ( 4 ) ( 1993). 3 4 3 4 0 1 . 161 1 A.I. Shnirelmnn, Ltrric.c tlrc,or:r. t r ~ r t l f l o ~of r . .itl(,trl ~ i ~ i c ~ o r t i p ~ u . ~ . ~ i l Russian ~ I c ~ ~ ~ Jt .i tMath. l. Phys. 1 ( 1) ( 1993). 105-1 14. 1621 A.I. Shnirelrnan. Wc~cth.colitri~rr.,of'itrc.orrrl)r(~.s.~ihI~~ Errlc,r. ec/rrtitiorr.\ n.irlr clc~c~rc~tr.~irrg errc,rqy. SCrn~nnireEDP Ecole Polytechnique Expos6 16 (1996-1997). 16.7 1 Ya.G. Sinai. The, sttrtistic..~of.c/roc~k.sirr the .soltrt~orr.s o/ itn~i.\c.ielR r r t ~ r nc,clrrtrtiorr. , Comm. Math. Phys. 148 ( 3 ) (1992). 601-621. (641 J. Sommeria. C . Nore, T. Dumont and R. Robert. Tllc;ot-ic, .\tuti.\ticiue (1. Itr ttrc.lrr rouge ~ / Ju1)irer. r C. R. Acad. Sci. Paris S6r. 11 312 (199 1 ), 999-1005.
[65[ J. Somrneria, C. Staquet and R. Robert. Find rcluilihrium stutr ( f a two-dimmsionul .shrur luyrr, J . Fluid Mech. 233 (1991), 661-689. 1661 A. Thess. J. Somrneria and B. Jiittner, Irrrrtieil or~qurri?(rtiot~ ( f u two-dimen.sionu1 turhulrnt r20rtrx strrvr. Phys. Fluids 6 (7) ( 1994). 24 17-2429. 1671 B. Turkington and R. Jordan, Iclrcil mcigneto~uielirr two dimm.sion.s. J . Statist. Phys. 87 ( 3 4 ) (1997), 661695. 1681 G.J.F. Van Heijst and J.B. Flor, Dipole fon?1citiorr rind c~olli,sio)rin tr .str(itifieclj/uid, Nature 340 (1989), 212-215. 1691 S.R.S. Varadhan, Large c1evicrtiurr.s and cr/)/dic~ation.s,Ecole d'~t15de Probabilites de Saint-Flour 15-17. 1985- 1987. [70] V.I. Yudovich, Non-.steitiorrur:v~o~t~ (,fnrr incomprr.s.sihlr liquid, Zh. Vych. Mat. 3 ( 1963). 1032-1066. 17 l 1 L.C. Young, Crrrrrc11i:rd .srrrfilc.c,.s irr the c.crlc~ulrrsof'vuritition.~.Ann. Math. 43 ( 1942), 84-103. 1721 C.K. Zachos, Hamiltorricrrr ,fk)tv.s,SU(co). SO(co), USp(co) eirld .string.s, Differential Geometric Methods in Theoretical Physics. L.L. Chau and W. Nahm, eds, Plenum Press, New York ( 1990). 1731 V. Zeitlin. Firritr n~ocloernerlo
CHAPTER 2
Topics on Hydrodynamics and Volume Preserving Maps Yann Brenier
.
.
lrr.srirrrt Ut~iver.$ittrirc. (10 Frnt1c.r. CNRS LIMR 6621 Nice. etl rl6trrr.h~i1t~rtt tlc 1'Uttil'er.sitr;. Pc~ri.s6 Frc~ttc.e E-rt~tril:hrer~ier@nlc~th.ur~ic.r.,fr
Contetlls I . General prehentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . I . The contig~lrution\pace elan incomprc\\ihle fluid . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. The Eulcr equation\ o f inconipre\\~hlcfluid\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Ceo~nctricintcrprctatio~iol' the Eulcr crlu;~tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . I .4. T h e h o r t e \ t path prohle~ii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57 57 57 SX hl
1 .5. A model l o r [lie SPP: The Closest P o ~ n Prohle~n t . . . . . . . . . . . . . . . . . . . . . . . . . . . I .6 . N o ~ ~ e x i s t e ~ol'soIutio~i\ ~ce lol-the SPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7. The hydrostat~cl i ~ n i ot f t h c Eulcr cquatlon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I .8. The rcloxcd shortest path problc~ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I .9. Solution\ of the relaxed shortest path prohlcm . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . I 0. Consistency o l t h c relaxed SPP with tlic i u l c r equations . . . . . . . . . . . . . . . . . . . . . . . 1 . 1 1 . E x a ~ i ~ p loc f\ gcncrali~ed\ o I u t ~ o ~ i s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Measure prc\crving maps and density theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Gcncral detin~tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Smooth measure preserving mops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Dcn\ity of s~noothmeasure preserving map5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Proof o f the density theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Related density r c s u l t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . The closest point proble~nand the pcllnr factor~/stion01 map\ . . . . . . . . . . . . . . . . . . . . . . . 3.1. The Mtrngc-Kantorovich theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . Main \tcp\ ol'thc study ol'thc relaxed SPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Exihtc~lcco l ' a d ~ i i i s ~ h solutio~is le . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62 63 64
Abstract Various topics related to the mathematical theory of ideal incolnpressible fluids are discussed in relation to the concept o f measure preserving maps . H A N D B O O K O F M A T H E M A T I C A L F L U I D D Y N A M I C S . V O L U M E I1 Edited by S.J. Friedlander and D . Serre 0 2003 Elsevier Science B.V. A l l rights reberved
66 67 6X 68 hX 69 70 71 72 77 77 79 XI 83 X5
Keywords: Volume preserving maps, Incompressible fluids, Euler equations, Geodesics, Monge-Ampere equations, Monge-Kantorovich problem, Polar decomposition
7i,l~ic,.s or1 hydroc1yttur11ic.sund volunir prc..s~r~linfi mcips
1. General presentation 1.1. The conjigumtion spnce ( g u n incoinpres.siblejuid In Classical Continuum Mechanics [I], the motion of an incompressible fluid moving in a compact domain D of the Euclidean space Rd can be seen as a trajectory t + g(r) on the configuration space G ( D ) (denoted by SDiff(D) in [I]) of all diffeomorphisms of D with unit Jacobian determinant. This configuration space can be embedded in a larger one, namely the set S = S ( D ) of all maps h from D into itself, not necessarily one-to-one, such that, for all Borel subsets B of D, h-' ( B ) is a Borel subset of D having the same Lebesgue measure as B. Equivalently, we can say that a Borel map h belongs to S(D) if
holds true for all $ E C ( D ) , where d s denotes the Lebesgue measure, normalized so that the measure of D is I, and C ( D ) denotes the Banach space of all real continuous functions on D. For the composition rule, G ( D ) is a group (the identity map I being the unity of the group), meanwhile S ( D ) is a semigroup. Both G ( D ) and S ( D ) are naturally embedded in the Hilbert space L ? ( D . R") o f all square integrable mappings from D into R". One of our goals is to compare S ( D ) and G ( D ) . I t will be shown, in the simple case D = 10. 1 ld, that S( D ) is closed in L' with G ( D ) as a dense subset, for all d 3 2. In the following discussions, it will be always assumed that D is chosen so that S ( D ) is the L' closure of G ( D ) .
1.2. Tlic Eltler ryiitrtion.~c~f'iric~o1n/~rc.s.sii7le,fluirl.s An ideal incompressible fluid moving inside D is usually described by a velocity field u ( t , X ) and a pressure field p ( r , x ) , subject to the classical Euler equations 1221
with the boundary condition that u is parallel to i)D. Here we use standard PDE and Continuum Mechanics notation,
and . denotes the inner product in the Euclidean space. The flow ( t ,x ) + g ( t , x) describing the motion of fluid particles is defined by arg(t, x ) = u ( t , g ( t , x)),
~ ( 0 , s=) .r,
and can be solved by the Cauchy-Lipschitz theorem provided that v is smooth enough and parallel to D. By elementary calculations, the Euler equations can be replaced by the following equivalent set of equations
which insures that r + g ( t ) is valued in the configuration space G ( D ) , provided that v is smooth enough. These equations have been established by Euler (Opera Omnia, Series Secunda, Livre 12, pp. 274-3 15,3 16-36 I), and still receive a lot of attention from Mathematicians, mostly but not only in the field of nonlinear PDEs, as shown, for instance, by the recent publication of various books by Arnold-Khesin, Chemin, P.-L. Lions, MarchioroPulvirenti, etc., or Majda's conference at the ICM of Kyoto 1301. The first attempts to address the Euler equations as a nonlinear PDE problem (existence, uniqueness, stability of solutions) go back to the 20s [28].
A formal, nonrigorous, Riemannian structure is naturally induced on the configuration space G ( D ) by the L' norm of the ambient Hilbert space L ' ( D , R") of all square integrable mappings from I1 into R".The L' norm is defined by
where I . is the Euclidean norm. deti ne. for any pair go. gl in G ( L ) ) , their geodesic distance We may,fi~untrlI~~ S l ) ( g o . g ~= ) inf
6'
llijfg(t, .)llL,2dt,
where the inti mum is performed over all smooth trajectories t -+ g ( r ) E G ( D ) satisfying ~ ' ( 0= ) go.
g(l ) =g' .
A geodesic curve can be deti ned as a curve t is S > 0 such that if to < 11 < to 6, then
+
-+
g ( t ) E G ( D ) such that for all
If, in addition, the t parametrization of g is chosen so that Ilil,g(t. .)\IL? then (2) means that g minimizes the "Action"
to E
R,there
ist-independent,
W i c s un lzydro~lytzanzic~ and volume prrsrrving maps
among all smooth trajectories t
E
59
[to,t l ] + g on G(D) satisfying
It turns out that the Euler equations are governed by the corresponding Least Action Principle on the configuration space G(D), which means that Euler flows are just geodesic curves on this configuration space for the formal L~ Riemannian structure. This has been known for a long time, in particular, after a famous paper by Arnold [ I ] . (Among related results, let us quote [20,40,39,38]. . .) Let us give a confirmation of the Least Action Principle through the following rigorous result.
THEOREM 1.1. Assume D to be the closure ofa bounded convex open .subset oj'the Euclidean space. Let (g, p ) be a s~rficientlysmooth .~olutionto the Euler equations ( I ) and let [to,tl ] be a time interval short enough so that
holds true,fir all t E It().tl 1, x E D, y # 0 in R".Thm,j;)r all smooth curves t + i ( r ) on rho (.o~~fig~lrritioti space G ( D ) suc,h thrit
where
with eyutrlity ifand onlv ij'g and jj coincide on It().tl
1.
The proof is elementary and relies on the one-dimensional PoincarC inequality (that can be proved, using Fourier series, as an exercise): L E M M A1.2. Let to < r l . f i r all uhsolutelv coriti/iuous c.urves
such that to) = z(t1) = 0 and
Z'
is square irztegruhle,
PROOF.Let us compare g and ii; subject to (3), fix x E D,and denote ~ ( r = ) ~ ( tx ), , ( ( t ) = g ( t , .x). S ~ n c ep is smooth, there is a constant K = K ( p ) 2 0 such that
Since U is convex, the value of K ( p ) can be taken as
By using the one-dimensional Poincare inequality, we get
since ((!,)
= : ( t i ) for 1 = 0 . I . Thus
provided that
11
- to is small enough so that
Since ,y is a solution to the Euler equations. we have : " ( I )= i ~ , ' , ~ ( /a. ) = V / ) ( t :, ( t ) ) It
follows, after integrating by part. that
which leads t o
After integrating over .r E D, we get
I
(
1
) -I-i
t
,
)
'1
dr
dr.
Since both g and ,$ are volun~e-preserving.
which shows that
which completes the proof (the equality case being left as an exercise).
0
R E M A R K .The Least Action Principle is satistied only o n sufficiently short time intervals. On larger tiwe intervals. 8 is no longer a minimizer bul rather a critical point of the Action. When 1) is not convex, the proof is valid with n larger constant K ( / I ! depending on thc geolnetry 01' 13. Condition (4) is sharp i n thc following case: D is the unique disk in R?. to = 0. t l = rr. v(.r) = ( x ? . x I ). p ( . r ) = !(2 . r17 .rS) . = .rc/" (where complex - and ~ ( t.r) notation .r = .rI i x z ih used). This fairly trivial solution to the Euler equations fails in minimizing the Action as soon as 1 1 > rr (that is. after half a rotation of the disk).
+
+
From a geometrical point of view (different from the natural PDE point of view which consists i n addressing the Euler equations as an evolution equation with prescribed initial velocity ticld), i t is natural to solve the Shortest Path Problcm (SPP). or Least Action Problem (LAP). This problem amounts to minimize Action (2) among all smooth trajectories on G ( D ) connecting two given elements KO, g l . Because of the group property of G ( D ) , we can assume go to be the identity map I and denote gl by h . We can also rescale the time interval [to, t11and set TO = 0, tl = 1 . D E F I N I T I O1.3. N Given h E G ( L ) ) ,the Shortest Path Problem (SPP) consists in looking t'or acurve t E 10. I I + g ( t ) E G ( L ) ) ,such that g ( 0 ) = I . S( I ) = h , minimizing the Action
As mentioned before, the corresponding system of PDEs formally comes about as the Euler equations, written in the so-called Lagrangian form
with two point boundary conditions in time, which is different from solving the Cauchy problem where only initial conditions are prescribed, namely, g ( t = 0, x) and a,g(t = 0, x ) for all x E D .
1.5. A modelfor the SPP: The Closest Point Problem 1.5.1. Approximute geodesics. Before addressing directly the SPP, we can introduce a concept of approximate geodesics. The simplest way to define approximate geodesics on G = G ( D ) is to introduce a penalty parameter E > 0 and to consider the formal dynamical system in the Hilbert space H = L'(D. R")
where the unknown M is a time-dependent map in H , H . and t l , , ( M . G ) = inf (IM - g I I H = inf IIM ,g €
<;
g€S
-XI/,,
denotes the gradient operator in
=dH(M.S)
(6)
is the distance in H between M and G , or, equivalently, between M and S , the L' closure o f G. where 11 . 11 H is the Hilbert norm of H . This approach is similar to - but not identical with - Ebin's slightly compressible flow theory 1 191, and is a natural extension of the theory of constrained tinite-dimensional mechanical systems 1361. As the penalty parameter E approaches zero, we expect that for appropriate initial data, typically for M ( I = 0) = Mo E G and M 1 ( t = 0) = uo o M o , where vo is a smooth divergence free vector field on D tangent to the boundary, the time dependent map M converges to a geodesic curve on G . Whenever a map M has a unique closest point n s ( M ) on S (which is not necessarily true since S is neither convex nor compact in H ) , the gradient of the squared distance from M to S is well-defined at M and equal to M - n s ( M ) . Thus, we may write. at least formally. the approximate geodesic equation ( 5 ) in the new form:
A rigorous analysis of this equation, introduced in [ 1 1 1, clearly requires an analysis of the Closest Point Problem (CPP): DEFINITION 1.4. The Closest Point Problem (CPP) amounts to finding, given a map M E
H = L 2 ( D , EX"), a closest point (with respect to the L' norm) on the semigroup S ( D ) of all measure preserving maps of D .
7i)pic.son hydroc1yncmrlc.s und volume preservln~nlups
63
The CPP can also be considered as a model problem for the SPP and will be subsequently investigated in detail.
1.5.2. The tirne-discrete S P P A different approach to the SPP is to use time discretization. Let N > 2 be a given integer. We call a semidiscrete shortest path a sequence g ~. .,. , g ~in G ( D ) , or more generally in the semigroup S = S ( D ) , which minimizes
subject to the constraint
where I denotes the identity map and h is the final configuration to be reached. A necessary optimality condition for such a sequence g l . . . . , g ~is that for all 1 < i < N, g; minimizes
or. equivalently (using that S is included in a sphere. which can be checked as an exercise),
i(g,-l+
, q ; + l ) . So. we see In other words, g; is the closest point on S of the mid-point that, once again, the CPP naturally comes up. As a matter of fact, the CPP was originally introduced i n [ X I as the building block of a related numerical method to solve the Euler equations.
1.6. Notzexi.stetzc,e c!f'.solutionsji)r the SPP A local existence and uniqueness theorern for the SPP can be found in Ebin and Marsden paper 1201: if h and I are sufficiently close i n a sufficiently high order Sobolev norm, then there is a unique shortest path. In large, uniqueness can fail for the SPP. For example, in the case when D is the unit disk, go(x) = .r = - g l ( s ) , the SPP has two solutions, g ( t , X ) = x e f i n ' and g ( t , x ) = xe-'"I, where conlplex notations are used. In 1985. A. Shnirelman [40] found, in the case D = 10, 113, a class of data, that we will call "Shnirelman's class", for which the global SPP cannot have a (classical) solution. These data h are of the form
where H is an area-preserving mapping of the unit square, i.e., an element of G([O, 112), such that
(which means that the Action can be reduced if the third dimension motion is used). Indeed, let us consider a smooth curve g connecting I and h on G([O, I]'), generated by some smooth time-dependent divergence-free vector field u ( t , x ) , parallel to the boundary of D. Then, Shnirelman shows that there is such a curve j satisfying
The new trajectory 'ij can be roughly obtained in two steps. First, u is rescaled by squeezing its vertical component (with symmetry with respect to x3 = 112)
for 0 < .rj < 112. and
for 1 / 2 < .rJ< 1. Next. the new tield i i , which is divergence-free and parallel to the houndary, but only Lipschitz-continuous, is mollitied and generates i.Of course, the vertical rescaling can be repeated trtl ir!firiiturn in order to reduce the Action. This will generate infinitesimally small scales in the vertical direction. So we can already guess that a good concept of g e n e r a l i d solutions to the SPP, for such data. must be related to the limit of the Euler equations under vertical resealing. namely, the so-called hydrostatic limit of the Euler equations discussed in 129. Chapter 4.61.
Let us consider the Euler equations in a thin domain such as D = D, = (R'/z') For notational convenience, we denote the space variable by
and the velocity by
x 10,c: 1.
Ey~ii.o r ~hyti,r~clvtti~r~tic..v f ~ n t vol~rmr l presc~rvi~ig mupr
65
The vertical rescaling
leads, as F
+
to the so-called hydrostatic limit of the Euler equation 1291, to
Although these equations look simpler than the original 3D Euler equations, they actually lead to considerable analytical difficulties, as mentioned in 1291. To the best of our knowledge, only results in one horizontal space variable have been obtained for the initial value problem, under restrictive conditions on the initial conditions, related to the famous Rayleigh stability condition. Typically, it is required that the horizontal component u I of the velocity field at time r 0 satisfies the local Rayleigh condition
-
for all (.vI. :). Under additional conditions, existence and uniqueness of local smooth solutions were proved by the author in 1101. Conve~gencefrom the original Euler equations was proved by Grenier in 1241 under similar conditions. (See also 1121 for a more direct proof.) In addition, Grenier showed that sorne solutions of the Euler equations may nor converge to the hydrostatic solutions if they do not suti'rfy the local Rayleigh condition at t = 0. A key step of the analysih provided in 1101 is an appropriate reformulation of the hydrostatic equations. Let us consider a smooth solution ( 1 1 , 1 1 ) . 11) of the hydrostatic equations and define a Lagrangian foliation as a family of sheets := Z ( t , .r, a ) . labeled by tr E 10, 1 1 . where Z is a smooth function such that:
If Z ( O , x , c c ) is given and colnpatible with ( 1 I), (12). it is always possible to get a Lagrangian foliation at least on a short interval of time. Then, the hydrostatic equations become (after elementary co~iiputation~)
where the new unknowns (c.. v) are defined by: ~ ( tX ,, U ) = i),,Z(t, x. a ) > 0, t l ( fX. . a ) = u ( t , x.
Z(t.1,a)).
and p is unchanged. This change of variable turns out to be essentially the one we need to solve the SPP in a generalized sense.
1.8. The rt,irr.rer/ shor-rt~stpath prohlt~nr To solve, the SPP in a generalized sense, i n particular for data /I in Shnirelman's class, a natural idea is to introduce appropriate "Young's measures" 145,431. There are different ways to d o so 15.39,411. One approach to be used, as in 161, is closely related to the hydrostatic rescaling of the Euler equations. Given a s~noothtrajectory t E 10. 11 + g ( t ) on G ( D ) ,we detine two nieasures (respectively nonnegative and vector-valued)
detincd on Q' = 10. 1 I x I ) x D . These measures satisfy
Moreover, tti is absolutely continuous with respect to (., with n vcctor-valued density L'(Q'. dc)", so that 1n = ur, and the Action is given by
or, equivalently. A ( g ) = K ( c , . K (c..
rrl)
= sup (I.@)
I
111).
Q'
1) E
where
( F ( t ..K. (1) dc(t. . v ,
(1)
+ @ ( I ..x.
( I )
.d
~ t ~.K, ( /( ,I )).
(7.1)
and the supremum is taken among all continuous functions F and 0 on Q', with valuca respectively i n R and R", such that
pointw~se.Then a natural definition of the Relri.rerl SPP, called RSPP, is to look for pairs of measure4 (c, m ) that minimize K ( c , I N ) and are admissible in the \enbe of (20). (21). and (22), but do not necessarily satisfy (19).
Topics on hydrodynamics and volume preserving maps
67
1.9. Solutions of the relaxed shortest path problem In [6], it is shown that, for D -- [0, 1] a and each data h ~ S(D) (which of course includes Shnirelman's class), the RSPP always has solutions (c, m) and that there exists a unique locally bounded measure Vx p(t, x) in the interior of Q - [0, 1] • D, depending only on h, such that
a, (cv) + vx
(cv | v) + cVx p - o
(26)
holds in the sense of distributions on the interior of Q'. In this equation, c_ is an appropriate extension of c, allowing the (nonobvious) pairing with Vx p. More precisely, for any fixed e 6 R '1 and any nonnegative smooth radial compactly supported mollifier y on R J, c(t, x, a) -- lim -
f
+l/2
dO
f
c(t, x - 203e - Sy, a ) y ( y ) dy,
(27)
5___>0 d _ l / 2
for the weak-* topology of the dual space of L l (]XTp], C(D)) (the vector space of all IVp] integrable functions of (t, x) ~ Q with values in the space C (D) of all continuous functions on D). So, we have exactly recovered the hydrostatic limit of the Euler equations, in their second formulation, namely (14), (15), (16), as the optimality condition of the RSPP. This result is obtained in several steps. First, the existence part is easily obtained through standard convex analysis and duality theory. Next, a priori bounds are obtained for the Lagrange multiplier of constraint (20), namely V~ p, which turns out to be uniquely defined by a duality argument. Finally, the conservation of momentum (26) is obtained as an optimality condition. Still in [6], the original SPP and the RSPP are related in the case D = [0, I] 3. It is shown for any data h c S([0, 113) of the form
h(xl,x2, x3)- (H(xi,x2),x3), and, in particular, for any data in Shnirelman's class, that, for any s > 0, there is a smooth trajectory t ~ [0, 1 ] --+ g~: (t) on G (D) such that g,,(0)-/,
A(g~:) + I[g,;(I, . ) - h I]/2(D) ~< l ( h ) + s,
(28)
where 1 (h) is the optimal value of the RSPR In addition, the measures (c~:, m~:) associated with g~:, through (19), converge, as s --+ 0 to the generalized solutions of the RSPR Moreover, the g~: are almost solution of the Euler equations in the sense that their velocity field v~; weakly satisfies V 9v~: - O, as s tends to zero.
O, v~: + (v~: 9V)v~, ---->- V p,
1.10. Consistency ofthe relaxed SPP with the Euler equutions A local consistency result of the relaxed SPP with the classical Euler equations is also provided in [6]:
THEOREM 1.5. Let (g, p ) he (1 smooth solution to the Euler equations T > 0 such that AT^ < n 2 ,where A is the supremum on Q (fthe largest eigenvulue of the Hessian matrix rf p, and set h = g(T). Then, the pair (c. m ) associated with g through (1 9) is the unique solution of the relaxed SPP:
Explicit examples of nontrivial generalized solutions to the weak SPP were first described in [51. Let 11sjust quote a typical example, when D is the unique disk and h ( x ) = -x. Then, the classical SPP has two distinct solutions g+(t, x ) = eJr'x and g - ( t , x ) = ep'"'x, with the same pressure tield p = n2/.r('/2, where complex notations are used. A generalized solution (c.tn) is given by
L,,
f ( t . r , a ) d c ( t . ,r. a ) =
h,
1 1' 10. I 1 x 1 ) 0
f ( t . G(t.a.H),o)dHdtda.
(29)
t.(t.x.(l)d,,l(t..r,tl)
=
1 /I
; I , G ( I . ~ . H ) + (G I . ( t , u .H ) . ( l ) d o d t d ~ .
10. I ]x 1 ) 0
(30)
for all continuous function f ' , where
This generalized solution describes a very peculiar wave-like motion of the fluid particles. Each particle initially located at ti E D splits up along a circle of radius ( I - l t r ~ ~ ) l l ' s i n ( n twith ) center (I c o s ( n t ) moves across the unit disk and shrinks down to the point -(i when t = 1. Of course, its acceleration is still given by the pressure tield, ,I = nT)(x('/2,as expected from the theory of the RSPP.
2. Measure preserving maps and density theorems In this second section, measure preserving maps are studied in a relatively general framework. Density theorems are also discussed.
7i)pic.s o t ~hydro(ly~lrimi(.~ rind volume preserving maps
69
2.1. General dejnition DEFINITION2.1. Let X and Y be two topological spaces. Let a and B be two Borel probability measures respectively defined on X and Y. We say that a map 4 : X -+ Y transports (X, a ) onto ( Y , B ) or that /3 is the image of a by 4 if, for all Borel subsets B of Y, $-'(B) is a Borel set in X and a ( 4 - ' ( B ) ) = B(B). When X = Y and a = B, we say that 4 is a measure preserving map (MPM). REMARKA S N D E X A M P L E S . ( I ) An equivalent definition is: For all Borel functions J' which are B-integrable on Y, x -+ f (d(x)) is Borel and a-integrable on X and
(2) Of course, the definition can be extended to abstract measure spaces. (3) in the case X = Y = [O, I I, a = 0 = I . 1, where / . I denotes the Lebesgue measure, some examples of measure preserving maps are given by
@(.u)= .r
+ 21 mod I
d(x)= 1
-
-
.r
(which is discontinuous),
(which is orientation reversing).
d(.u)= rnin(2.x. 2 - 2.r) (which is not one-to-one).
(35)
(4) A remarkable theorem (see 1351, for example) asserts that if X is a separable complete metric space and no point in X has positive a measure ( a ( . r ) = 0. Vx E X), then there is a map 4 : X + Y = 10. 1 I that transports a to the Lebesgue measure on 10, I ] . (The idea of the construction is quite simple. Let ( ( I , , ) , 11 = 1 , 2. . . . be a dense sequence in X. Let rescale the distance d on X so that the diameter of X is one. To each point .r in X , we attribute the sequence d ( x ) = ( d ( x ,cl,,)) E 10. I]'. which provides a kind of system of coordinates in X. Then, we use binary coding to write cl(.r) as a point in ( ( 0 , I)')N, which is in one-to-one correspondence with ( ( 0 ,1)') and leads us back to 10. 1 1 through binary decoding. This establishes a correspondence 4 between X and 10. 11. Further refinements are needed to make it one-to-one correspondence (in the almost everywhere sense). Then, it is easy to modify @, by composition (using the property that no point in X has positive measure), to enforce ( 3 1 ). Of course, such a construction deserves to be done very carefully.) So, in a certain sense, from the measure theoretic point of view. there is no essential difference between (X, a ) and (10, 1 I, I . I), as long as X is metric. separable, complete without any point of positive a measure. E X E R C I S EDiscuss . the case of two discrete spaces X, Y, with respective elements N and M. In the particular case, when X = Y and a = @ is the counting measure, describe the set of all measure preserving maps.
2.2. Smooth memure preserving mllps Subsequently, we consider the case when X = Y = D is the closure of a bounded open set with Lipschitz boundary in R" and cw = is the d-dimensional Lebesgue measure, denoted by I . I and normalized so that I Dl = 1 . Typically, D is the unit hypercube [0, I jd. The set of all measure preserving rnaps (MPM) is denoted by S = S ( D ) , where S stands for semigroup. Indeed, by definition, S equipped with the usual composition rule is a semigroup, but not a group (due to the presence of obviously noninvertible elements, such as example (35) in the simplest case D = [0, I I). This set S can be seen as a subset of the Lebesgue space LP(D, IRd) of all p-integrable maps from D onto R" (the ambient space) and, more specifically, as a closed subset of a sphere, whatever is the value of p E [ I , +m].(Let us just recall that every Lebesgue measurable function is almost everywhere equal to a Borel function, so that there is no problem to define S as a subset of LI'.)
EXERCISE.Show that S is closed and contained in a sphere Let us now consider more restrictive def nitions of measure preserving maps, requiring some smoothness. Let us first consider the vector space V of all time-dependent Cm vector fields on D,
compactly supported in the interior of 10. 1 1 x D and divergence-free:
Let us denote by g , ( v ) ( x ) the solution at time t of the ODE d.r/dt = v ( t . . r ) with .r as initial condition at t = 0. Because n is smooth and compactly supported for all t , #q,(v)is a Cm orientation preserving diffeomorphism of D , leaving a neighbourhood of the boundary i ) D pointwise unchanged. Since u is divergence-free. the Jacobian determinant of (I!) is identically equal to one, because the general identity
is valid for all smooth vector fields v. In particular, g, ( 1 1 ) is (Lebesgue) measure preserving. because of the change of variable formula ~).f(~~x))det(i),g,(u)(.r))dr= Thus,
defines a subset of the group G of all diffeomorphisms with unit Jacobian determinant. Actually, Go is a subgroup of G . Clearly, Go is also a subset of the semigroup S of all measure preserving maps, which already contains G . E X E R C I S EShow . that indeed G O is a group. Describe G Oin the case D = [O, I], d = 1. EXERCISE.Prove (36) using that det(1
+ A) = 1 + tr(A) + o(A*).
E X E R C I S ELet . #J be a Lipschitz map D + D belonging to S. Show that #J must satisfy
for almost every y E D. (To make a detailled and precise proof, the use of 1231 I S recommanded.) Can such map be smooth (at least c ' )without being one-to-one? '
Clearly, S is a much larger set of maps than Go. (The case 1) = 10. 1 ] is a striking example, since then GO is reduced to the identity map.) However, as shown in this section, from the point of view of Ll' topologies, for / I < +a,Go is dense in S as soon as rl 3 2. To make the proof as simple as possible we assume D to be the unit hypercube. THEOREM 2.2. Lct I) = 10, 11" utltl d 3 2, tllrrr S is /Ire c,lo.slrrr of' G O in tlzr .S/>U(.P L/'(I), R"), f i ~ rtrll p E [ I , f a l . R E M A R K S(.I ) Clearly d 3 2 is needed! S is usually strictly larger than the clo(2) For topologies ti ner than the LI' ( p < +a), sure of GO.This is obvious for the C ' topology which preserves the unit Jacobian determinant pointwise. Soholev topologies, which occur naturally in the theory of incompressible elasticity, such as W I.'', preserve the unit Jacobian determinant in the almost everywhere sense, at least for p large enough. In the very special case d = 2, the C" topology is almost sufficient to preserve the unit Jacobian determinant. (This is. in fact, related to syniplectic topology.) There has been a lot of researches related to these questions (let us quote a few names among others, at least in the field of Calculus of Variations, such as J. Ball, F. Dacorogna, S. Miiller, T. Sverak, L. Tartar, and some related work by Coifman and Lions and Meyer, F. Helein, C. Viterbo, etc., as well as the book by Arnold and Khesin 121). The L' topology is too weak to preserve the unit Jacobian determinant. As a matter of fact, orientation reversing maps such as ( X I . xz) + ( X I , I - X?) on the unit square can be approximated by elements of G O in Ll' norm for p < +a,as we shall see.
2.4. P r o c f r f the density theoretn There are several possible proofs of this "folklore" density result. The following one (due to the author but unpublished) does not differ much from the one provided in Neretin's paper [33].
2.4.1. Meusure preserving maps and permutations. Usually, density results are proved using regularization techniques such as convolution. Here S is not a vector space and convolution cannot be used straightforwardly. Of course, since G o is formally a Lie group with a Lie algebra made up of smooth divergence-free vector fields compactly supported in the interior of D, a natural idea would be to look for a vector space of generalized divergencefree fields, to which convolution could be applied, that would generate S by integration. But there is no obvious space of that type (although the theory discussed in the second part of the text solves this problem in some sense). So, we are going to follow a completely different track relying on the approximation of S by a discrete group, the group of permutations. Indeed, at the discrete level, as D is a finite set of m elements with the counting measure, S can be identitied to the group of the permutations of the tirst m integers. So, to approximate S, it is natural to introduce, for each integer n 3 0, the subset P,, of all maps in S constructed in the following way: the unit cube D = 10. 11" is split into N = 2"" si~bcubesof size 2-", denoted by LI,,,;. for i = 1 , . . . , N , with center of mass x,,.;. To each pertnutation cr of the first N integers. we associate the transform 4 = G, I) + I), detined by
for all .r E Dl,,,. Such a map will he called (with a slight abuse) a permutation. These maps forni a set of N! elements denoted by Pi,, and P will denote the collection of all P,, for 1 1 3 0, which clearly is a "~*uhgroup" of the semigroup S. Apparently. Go and P are poorly related to each other. However, we claim P R O P O S I T I O2.3. N I f 1 3 = 10, 11" rvitlr cl 3 2, tl1c~rji)rrrll L/' nor-rns 1 6 1, < +m, P is c.oi1trrirrc.d in the cYosurc. c!f'Go.
2.4.2. Proc!f'r!f'Pro/~o.sitiorl2.3. Since every element of P can be written as a finite product of permutations exchanging adjacent subcubes (i.e., having a joint face), it is enough to show that such permutations can be approximated by a sequence in GO. because of the following lemma: 2.4. Let S I . S2 hc tkvo .su/~.set.s of'S c.ontuined irr the clos~rrc.r!f'G()rt.ith rc.spec,T to LEMMA the Ll' norin ( 1 1) < +m). Then this closure. (11x0 c~or1trriri.s{ s l o s ~ sl: E S I , s2 E S ? ] .
<
PROOF. Let sl
E
S l , sz
E
S2. For all g l , g2 in G O ,
/Isl 0 s2
-
gl o s 2
I/gl o s2
-
gl o g2
I ( L ~ l
l l L ~ p
= Ilsl
-
gl
I/
(since s? is MP (measure preserving)),
L,l
< Lip(gl )[Is2
-
g 2 / I L , ~ (since sl is Lipschitz continuous)
Thus, by the triangle inequality, we can make
arbitrarily small by choosing first 81and then gz, which completes the proof since gl o ~2 belongs to Go. To prove Proposition 2.3, it is now enough to approximates permutation of two adjacent subcubes by a sequence in Go. After obvious rescalings and translations, we are reduced to construct on the c ~ ~ Q b e= 1- I , + I I x 1- 112, 1 /211/-' a smooth and compactly supported divergence-free vector field
buch that g~( u ) is arbitrarily close (in Lf' norm) to the nlap
By using again thc lemmn, we can deconlpose this map and rather consider the (partial) symmetry map
and two iunalogous maps on ~ h ccubcs L)- -- 1- 1.01 x 1- I/?. 1 /2111-' ;knd L)+ = 10. 1 ( x 1- 112. 1 /211'-'I. Let us consider only the tirst map. We introduce a so-called "slream function"
+
and set
Of course, this field is not smooth, but we can already integrate it (hucause of its special structure. although the Cnuchy-Lipschit1 theorern does not apply) and get a nonsmouth Ilow (I. .r) -+ s , ( u ) ( x ) which exactly fits with our given symmetry map at a tirlle t = I. (Exercise: compute all trajectories ds/dr = v(.r) in Q.) To get ;I smooth approximation gl ( 1 1 , ) r (2,). it is cnough to mollily 11 and rather consider 11, E V defined by
where . , ) I and 0,. are suitable compactly supported smooth approxi~nations of, respectively, $ on I- I , I [ x 1- 112, -t 1 /2[ and I on 10, 1 1. (See more details for the mollification process in 1331.) Notice that r l 3 2 is clearly needed to achieve the construction.
+
2.4.3. Bistochastic measures. To prove the density theorem it is now enough to show that P is a dense subset of S. As a matter of fact, we are going to prove a richer result based on the concept of "bistochastic measures" which are probabilistic generalizations of MPM (in the same way as Young's measures are generalization of functions in the framework of Calculus of Variations and nonlinear PDEs). For the definition, we go back, just for a short while, to a general setting. DEFINITION 2.5. Let X and Y be two topological spaces with Borel probability measures a and B , respectively. We say that a Borel probability measure p on X x Y is bistochastic if its marginals are respectively a and B , namely,
for all Borel subsets A and B of X and Y, respectively. This concept goes probably back to Kantorovich and was used to provide generalized solutions to the Monge optimal mass transfer problem, which will be discussed later in the course. There is a natural embedding of the set S of all MPP into the set DS of all bistochastic measures. Indeed, to each such a map 4 from (X.a ) to (Y,P ) , we associate a unique LL in DS by setting
forall Borel subsets A and H of X and Y,respectively. or. equivalently with distributional notations,
where 6 denotes the Dirac measure. E X E R C I S EShow . that / L is bistochastic if and only if for all functions f , a-integrable on X , and for all functions g , P-integrable on Y . (.r, y ) -+ ( , f ( . r )~. ( y )is) /L-integrable and
EXERCISE. Investigate the bistochastic measures as X and Y are finite sets with discrete measures. Address, in particular, the case when X = Y with the counting measure.
2.4.4. Drtzsity of' P in S ccnd DS. Let us now return to the case X = Y = D = 10. l ]" and a = p = 1 . 1. To show the density of P in S. it is enough to show that P is densely embedded in DS, with respect to the vague topology of measures, thanks to the following lemma, which can be proved as an exercise.
?i)pic~on hydrodynattric~and volume preserving mups
75
L E M M A2.6. Let ( & ) be a sequence in S and ( p G nbe ) the corresponding sequence in DS. Then 4, converges to E S for all Ll' norm, p < -too, ifand only if ( p ~ , ,vaguely ) converges to pb. Thus, we are left to show that for a fixed given p E DS, there is a sequence of "permutations" (p,,) such that p p , converges vaguely to p . Let n > 0 fixed integer and N = n d . We split D = [0, I l d into N subcubes of equal volume denoted by Dn3ifor i = 1, . . . , N. We set
for i, j = I , . . . , N so that v is a so-called N x N bistochastic matrix, i.e., a matrix with only nonnegative entries whose every column and every row add up to one. From a classical result of G. Birkhoff, such a matrix always can be written as a convex combination of at most K = K ( N ) (where K ( N ) < C N ' ) permutation matrices. Thus, there are coefficients H I . . . . , O K 2 0 and permutations ( T I . . . . , O K such that
Let us introduce L = 2"', where I will be chosen later, and set
where 1.1 denotes the integer part of a real number and EL
CH; = I, l
! i =
sup loI, X
E
10. 1 1 is chosen so that
H;\ < LI
-
By setting
we get a new bistochastic matrix which satisfies
Up to a relabeling of the list of permutations, with possible repetitions, we may assume all coefficients 8; to be equal to I I L and get a new expression
Now, we can split again each D,., into L subcubes, denoted by D l l + l ~ j ~ for l,, i = I , . . . , N , m = 1. . . . , L, with size 2-("+') and volume 2-(11+1)d. Then, we define
for each .r E Dll+l,l~lll. By construction, (i, m ) + ( o , ( I ) , m ) is one-to-one. Thus, p belongs to Pll+l. Let us now estimate, for any fixed j' E C ( D ) , 11 - 12 =
lj2
f (x. y ) p ( d x . d y , -
d
f ( x , P ( x ) )dx
We denote by q the modulus of continuity o f f ' . I I is equal, up to an error of q(2-"+'1/2), to
l 2 is equal, up to an error of sup I , f i K I L , to
, i s equal to I s . up to 17(2-1'-1+"/2)
which is exactly I?. by definition of p. Finally. we have shown that
I I I - I?(
< sup l,f12(211-1)'1 + 3rl(2-11-1+"17),
since L = 2/", K = N~ = 221rd. This completes the proof, after letting first 1 and then +m.
t1
to
Topics OII l~:droclj~~~amic~s rrnd volumr prr.srrving maps
77
2.4.5. Proofof the Birkhoa theorem. The proof relies on the classical "marriage lemma" from combinatorics, which asserts which a necessary and sufficient condition for N girls to marry N boys without dissatisfaction is that, for all subsets of r < N girls, there are at least I. suitable boys. Let ( v i j ) be a bistochastic matrix. There is a permutation a such that infi vi,,(,, is a positive number cr > 0. (In other words the "support" of a is contained in the support of v . ) Then, we have the following alternative. Either cr = I and v is automatically a permutation matrix, or cr < I and
defines a new bistochastic matrix with a strictly smaller support and v is a convex combination of v' and a permutation matrix. Recursively, after a finite number of steps, v is written as a convex combination of permutation matrices, which completes the proof.
2.5. R e l ~ ~ t edei~sitx d results
Using the marriage lemma, P. Lax has shown that if 4 is a continuous MPM on D = 10, 1 ]'I, then there is p E PI, such that
where rl is the modulus of continuity o f 4 and C',I depends only on di~nensionof (/. A sort of Lusin theorem holds true for MPM. More precisely, if @ is an almost everywhere one-to-one MPM, then, for all e > 0, there is a MPM homeomorphism (i.e., a oneto-one its continuous MPM with its continuous inverse) such that the measure of the set where 4 differs from 4, is less than e. For this kind of questions, see the books of Oxtoby (Springer Lecture Notes. Vol. 3 I8 (1973)) and Sudakov (Proc. Steklov Institute 141 (1979)).
3. The closest point problem and the polar factorization of maps Since S = S ( D ) is not convex (because S is included in a sphere), the CPP is not trivial. (Note that the CPP on G ( D ) is even worse since this subset of S ( L ) ) is not closed.) However, since S is a closed bounded subset of a Hilbert space, i t follows from Edelstein's theorem 131 that almost every element M E L ' ( D . R"), in the topological sense of Baire category theorem, has a unique closest point on S ( D ) . However, such a result is quite abstract and one of our tirst tasks will be to address this problem more concretely. This will lead to a Polar Factorization theorem for maps in the Hilbert space L'(D, R"), involving the semigroup S( D ) (rather than G ( L ) ) ) and the "dual" convex cone K ( D ) = { M E L'(D, R"); ( ( M , I
-
h ) ) 3 0, V h E s(D)}.
where ((., .)) denotes the L2 inner product. This convex cone will be characterized as the set of all square integrable mappings from D into IEd that coincide almost everywhere on D with the subgradient of some lower semicontinuous convex function defined on IRA. More precisely, we will show, following [ 7 ] , THEOREM 3.1 . Assume that M E L ~ ( DIKd) , .sutisJies the$)llowing nondegeneracy condition: if N is a negligible subset of D, theti M - ' ( N ) is also negligible. Then there is u unique decomposition
where h belongs to S ( D ) and
(dejned up to an additive constunt) is the restriction to D IRd. Moreover; h is the unique closest point to M on D, V@ is the uniq~iereurrungement of M in the cluss K ( D ) . Q,
c.fu lower .semicontinuou.s convex,finction on
By rearrangement of M , we mean any map v from D onto lRd such that
holds for all q5 E c,.(R"). Theorem 3.1 shows that a map M E L' has a unique rearrangement as a gradient of some convex potential, which generalizes the classical theory on nondecreasing rearrangements of real valued functions 1251. Theorem 3. I can also be seen as a nonlinear Hodge decomposition theorem. Indeed, whenfi)rmtrl/y linearized about the identity map, the polar decomposition yields the classical unique decomposition of vector fields
where z is a given vector field, U I a divergence free vector field, parallel to the boundary of D, and p is a real valued function. E X E R C I S EObtain . the Hodge decomposition by a formal linearization of the polar factorization. E X E R C I S EFind . the polar factorization of a linear map when D is a ball. (Assume the uniqueness of the factorization and use the classical polar factorization theorem for real square matrices.) What happens if D is not a ball? Regularity results were obtained by Caffarelli 114,15] T H E O R E M3.2. Assume M to have c ' . regularity ~ on D up to the boundup, for some 0 < CY < I , D and M ( D ) to be strictly convex with a smooth boundaty unrf the Jucobian determinant o f M to be positive and bounded uway from :em (which insures the nondegeneracy condition). Then @ is strictly conve.r on D and both VQ, and h helong to C ' , f f ,
Topicv or1 hydrodyncrmics und volume pre.rerving mrrp.7
79
up to the boundary. Moreover; h belongs to G ( D ) and 0 can be recovered by .solving a Monge-AmpPre equation. This regularity result shows that, under strong assumptions on D and M , M has a unique L~ closest point on G ( D ) .The proof is based on the fact that the Legendre-Fenchel transform of @, namely, 9 ( y ) = sup (x . y - 0 ( x ) ) .I€ R'/
is a weak solution of the Monge-Ampere equation det ~~9= p . where p(.r)dx is the image measure of dx by M. Caffarelli shows that 9 is a solution in the sense of Alexandrov and is strictly convex. Then, he obtains both local 114) and global [ I 51 regularities.
3.1. The Moizge- K~rntoro~ich theory The proof of Theorem 3.1 relies on a relaxation technique called Monge-Kantorovich theory. Its origin goes back to Monge's Inass transfer problem addressed in the 'mCmoire sur la thCorie des dCblais et des remblais' 1321. A modern approach. based on duality arguments, is due to Kantorovich 1341, and was used in Probability theory by Vershik, Sudakov, and more recently Rachev 134.42.441. Let us quote a typical result (which does not differ essentially from Theorem 3.1 ). T H E O R E3M . 3 . Assurne po trrrd pl to h0 two i~oi~i7egtrti~~e L~~h~.sgzie int(>gr(rhl~ (.orirl~(~(.tly .vcr~~portrdjunctiot~.s on R",suck thtrt
PROOF (Sketch). Let us consider a ball B in R" containing the supports of both po and pl, and introduce the set M of all Bore1 regular probability measures 11 on B x B having po(x)dx and pl ( x ) dx as marginals, which means
for all continuous functions f on R".By using Riesz representation theorem on Borel measures and elementary convex analysis (as the Rockafellar theorem stated in [ I 3]), we obtain the duality equality .w . yu(dx, d y ) = inf
S,
[@(x)po(x)
+ ~ ( x ) (x)] p ~d x ,
where the infimum is taken over all pairs (@. ly) of continuous functions on B satisfying
Then, it can be established that the infimum is attained by a pair ( 0 ,(I/) such that @ is the restriction of a Lipschitz continuous convex function defined on R",and for po(x) dx almost every point of R", ly coincides with the Legendre-Fenchel transform of @,
L F ( @ ) ( y )= sup (.u . y
-
@(.r)).
~ E W I
Moreover. if v = I!,,~, E M maximizes ,/H
.r . y~j(d.r.d y ) , then
holds for IJ,,~,, almost every (.t-.y) E R" x R". Using well-known properties of the , ~ ~ ofthe form Legendre-Fenchel transform. one deduces that I ~ is~necessarily
which implies
for all continuous functions ,f' on IJ,,,,, is (.I- ) d.u .
R" and achieves the proof
since the second marginal of
R E M A R KI . We can def ne the Kantorovich-Wasserstein distance (see (371, for exaniple) between po and pl by setting
A ( p o . P I ) = inf Then we get
7i~pic.s 0 1 1 I ~ y c / r t ~ c i nrind i . volunir prr.vrrving rnc1p.r
Indeed,
(since po and pl are the marginals of v , , ~ ~ )
for every
LI E
M (since v , , ~ maximizes ,
1s . .r v ( d x , dy)),
(since po and pl are also the marginals of v ) . REMARK 2. The proof of Theorem 3.1 uses similar arguments and corresponds to the special case where pO(.r) = I and p , (.r ) d.r is the image measure of dx by the mapping u . However, the proof is more complicated, partly due to the assunlption that u belongs to L', which rules out the assumption that pl is compactly supported.
4. Main steps of the study of the relaxed SPP The detailed proof can be found in 161. It is ti rst deduced from the Rockaffelar theorem in classical convex analysis [ 13 1: PROPOSITION 4. 1. The ir!firnurn I * ( h ) c!f'tho RSPP is rr1rr~ry.srrc.l~iel~erl, trrlc1,fOr c),8rry e > 0 there exist ,somcJ c.olltirluou.s functiorls 4, ( t . .r ,tr ) or7 Q' (i17d pc ( t S ) 011 (2, ~ ' i t l l ( t , .r d.t- = 0, . S L ~ C / Ith(rt t o r awry o/7tirt1(11.soIr~tior~ ~ ~ o r ~ t i ~011 ~ uQ' o uC. I ~I I ~ i j r q5>, V,
Ill
(c, l t l ) .
Then, an approximate regularity result is obtained:
.
Qt
P R O P O S I T I O N 4 . 2 . Let 0 < t iT / 2 L I ~ I ~ = IT/^, T - r / 2 ] x D x A . Let x E D + I L - ( . Y ) E IW" be tr stnooth r1il'ergence:frt.e ~ ~ e c t o r j e l c parullel l, to 2D ,crnd s E R + e'li'(x) E D he the itzt~,grtrlcurve of u1p~r\siilgtllrougl~x at s = 0. Then,
Since ~ ( t. x ., r r ) is a possibly highly concentrated measure (like a delta measure in x , as in the case of' classical solutions). i t is i~riclearhow to deduce from Proposition 4.2 a bound S L I C ils ~
hy letting tirst F 4 0 (to get 1 1 instend of V, 4, ). then 6. 11 -+ 0. Such a bound would he nieaningful, if (. could bc bounded away f r o m zero. which is exactly the contrary of' ~ h c classical cusc tund cannot hc expected. anyway. becausc ol'the initial and final cotiditir~ns. Howcver, a bound on (VpI is expectable. The formal (and, of course, not rigorous) wgurrlent is as thllows. Starting horn (40), we gct
Dift'erentiating in .r. we fr~rrnallyget (26) or
Then, integrating in
+
(i)l~'
(1
E
( 1 1 .
A. V,)~~)c.(t..x.d =t r- )V , p .
and, by S c h w a r ~inequality. (/IV,111)2<
/
/ & r r , 2 d c /d ~ + [ ~ ~ , ~ l ' d ~ . / l ~ ~ , ' d ~ .
E)pic..s o p t h~drodynamicsund volume preservirig map.$
83
All these calculations are incorrect. However, the formal idea can be made rigorous by working only on the 4, and using finite differences instead of derivatives, and leads to
THEOREM 4.3. The family (Vp,) converxes in the sense c?fdistriburions toward a unique lirnif Vp, depending oniv on h, which is a locally bounded measure in the interior of Q and is uniquely defined by
f i r ALL optinzul so1~ltioiz.s(c, nz = cv). Finally, (26) is established.
In this subsection we obtain admissible solutions to the RSPP through an explicit construction closely related to the one introduced in [ S ] for generalized flows on the torus T~ (and used later in (411). We perform the construction only i n the cases D = T" and D = 10, 1 I", the latter being an extension of the former. Ad~ni.t.sit,lc.solutions on rhe toru.~. Let h E S ( D ) where D = Ti'. We introduce. for (.Y. y, :) E D 3 ,a curve t E 10, TI -, w(t. .I-. y. :) E D, made, for 0 r T/2, of a shortest path (with constant speed) going from x to y o n the torus T " , and, for T / 2 r T , of a shortest path (with constant speed) going from y to :. Such a curve is uniquely detined for Lebesgue almost every ( x , y, z ) E D 3 .Then, we set, for every continuous function f ' on Q' = lo. T I x n x D ,
< <
< <
(Intuitively, this amounts to define a generalized flow for which each particle tr is tirst uniformly scattered on the torus at time T/2 and then focused to the target h ( a ) at time T.) This makes (c, rn) an admissible solution. Let us check, fbr instance, that the continuity equation is satisfied, by considering a continuous function f' that does not depend on tr and showing that (c, f ' ) = j' f . We split (c. , f )= 1, 12 according to t T / 2 or not. For r T/2, we have, by d e f i n.q~ t ~ ow(t, n , a , y , h(tr)) =to([. y, y. h ( u ) ) . Since we work on the torus T", co(t, y, y, h ( u ) ) = y w(t, 0.0, h(tr) - y). Thus
+
+
<
(since a E D + x = h(rr) E D is Lebesgue meaqure preserving)
(by using twice the translation invariance of the Lebesgue measure on the torus) and, doing f ' . We also get for K ( c . t n ) the following the same f o r I , , we conclude that (c., j')= estimate that does not depend on the choice of h E S ( D ) :
(where ( I n ( . .
.)
denotes the geodesic distance on thc rorua).
Let us now lift the unit cube to the torus by shrinkAtltr~i,s,sihlc ,solurioir,~otr rllc ~ c t ~c.ichr. i/ ing it by a factor 2 and rcflccting it ?'/ times through each face of i ~ boundary. s To do that, we introduce the Lipschit/ continuous mapping
from 10. I 1' onto 10, 1 I", and ita 2" reciprocal rnapx, each of them being denoted by (-);I. with k E ( 0 .I)". and mapping back 10, 1 I", one-to-one, to the cube 2 - ' ( k t 10. 1 I"). Given h f S(I0. I I"), we associate &. E s(T") by setting
+
where .r E $ ( k 10. 1 I"), k E (0. I ) . We consider the admissible pair (7, ti]) associated with and constructed exactly aa in the previous subsection. Then we set
Zq)ic:s or1 11yclrodyruunic.stlncl volume pre.srrviiig rnt1p.s
85
Explicit calculations show that ( c ,m ) is an admissible solution for h and that (as in [41])
Note that it is important to use a continuous transform (9 (so that the particle trajectories are properly reflected at the boundary of the unit cube and not broken) in order to keep the continuity equation, but there is no need to use a one-to-one transform, which makes the construction very easy.
References 1 l I V.I. Arnold. Ann. Inst. Fourier 16 (1966). 319-361. 121 V.I. Arnold and B. Khesin. fi)l~o/o~qicul Methoclr in Hyclrorlynurnic~.~, Springer-Verlag, Berlin (1998). 13 1 J.-P. Aubin. Mtrtl~c~mrrtic~ul Methorlt ~ ~ ' C Utrrrd I I Ecor~or~iic I~ Theory, North-Holland, Amsterdam ( 1979). ~rrotlel.v,fi~r . s o r ~ i - , q r ~ o . s t r o ~rIvrrrr~riic..s. ~ I ~ i c ~ Phys. D 109 ( 1997). 333[ill S. Baigent and J . Norhury. Ttco cli.sc~rc~tt, 342. 1.51 Y. Brenier. J. Anier. Math. Soc. 2 (1990). 225-255 161 Y. Brenier. Mirrir~rrrlarotlesics or1 ,qroiil~.\( ! / ' ~ ~ o l r i ~ ~ r e - p r c .rrrerps. . r c ~ rColntn. ~ ~ i ~ ~ ~Pure Appl. Math. 52 ( 1999). 1 1 1352. 17 1 Y. Brenier. t'olrrr /trc.tori:trtiorr trtrtl r~ror~oto~rc~ ~ - t ~ r r ~ ~ r o o~fr~,~qcc~~ ~ r ~r o~r - ~fiirrc~fiorrs. r r l ~ i e ~ cComm. l Pure Appl. Math. 44 ( IO9I ). 3 7 5 1 1 7 . [ X I Y. Breriicr. Coriiput. Mcth. Appl. Mech. 75 (I')X9). 325-332. 101 Y. Brenicr. Tlrc, tlrrcrl Ic.tr.\r oc.tiorr prohle~r~r,/i~r ttrr itlcvrl. rrrc.orri/~rc~\ \ihlc, f/riiel, Arch. Rational Mcch. ( 1993). 1 101 Y. Brcnicr. Horrro,qt~~rc~orc.\ lir.tlro.>rotic.f k ~ t t , . sn.irlr c.orn.cz.t ~ ' c , l r ~ cpr(!/ilc,.r. .i~ Nonline;~rity I2 ( 3 ) ( 1999). 3955 12: Arch. Ratiorial Mecli. ( 1993). [ I I I Y. Brcrlicr. l)t~ri~~trriorr (!/ rlrc, Errlc~rc,clticrriorr.v /ir)rrr er t.trri(.trrrf~-i, o f Corflo~rrh irrrc,rtrc~tio~r. Coni~ii.Math. Phy\. 212 ( I ) ( 2 0 0 0 ) .03-103. 1 121 Y. Brcnicr. Kcrrrtrrk., orr tho tlc,ri~,trliorro f rlrc, Ir~tlu).\trrtie~Erilar c~t/r~rrtiorr.s. Preprint. 2002. http://wwwm:ith.un~cc.f~r/b~-cnicrl. 1 131 H. Bre/ib. Arrctlr.\e, /i~r~c~tiorrrre~lle~. 1141 L. Calfl~relli,J. Ariier. Math. Soc. 2 (1992). 11.51 L. Ca(tarelli. Horirrtlcirr. r(,,qiiltrrit\ (!f'r~~erp.s \t,irlc c.orr,~cJr~~ort~rrritrl,\, Coinm. Pure Appl. Math. 45 (1092). 1141-1 151. 1 161 J.-Y. Chcniin. E'lititle.\ ~~tr~fitir.\ irrcv~r~rl~rc~.\.r~I~Ie~.\. A>tt!ri\que 230 ( 1995). 1 17 1 M. Cullen. J. Norhury and J. Purser. Gc~~~c~r(tlisc~c/ Ltr,qrtrrr,qitrrt .solririorc.~ fol-trtr~~ovl)/rc~ric. trrr(/oc.c,corr(.,/le~~~,.s, SlAM J . Appl. Math. 51 (199 I ) , 2(k31. ( I X I R. DiPcrna and A. Majda. Co111111. Math. Phy\. 108 (1987). 667-689. I 101 D. Ehin. 7Y10rrroriorr (~f,\li,qIilh.t.olrrl~rc..\.rih/ef/iiicl.\ ~.ic,b~.c,tl tr.! tr rnoriorl b~,itIi .\trorr~c.orr,\lr-tririirr,q/i)~.e~.Ann. of Math. ( 2 ) 105 ( 1977). 141-200. 1201 D. Ehiri and J. Marden. Ann. of Math. 92 (1970). 102-163. 121 1 Y. Elioshherg and T. Ratiu. Invent. Math. 103 (1991 ). 327-310. 1221 1,. Euler. Ol~rrcrOrrrr~rcr.Serie.; Secunda 12, 274-361. 1231 L.C. Evans and R. Gariepy. Mctrsitrc, Tlit,orv ctrrtl Firre, Prol)ertic,.s of ~~rtrrc~tiorr.~. Studies in Advanced Matheniatics. CRC Prebb. Boca Raton, FL (1992). 1241 E. Grcnicr. 0 1 1 the, tleri~~c~tiorr (!flionrogc,ric,oic.v/rytlrs).vrtrric.eqiitrriorr.s. Math. ModCI. Numer. Anal. 33 (5) ( 1999). 065-970. 1251 G.H. Hardy, J.E. Littlewood and G. Polya, Irrc~qrictlitic~.~, Cambridge University Press. Cambridge ( 1952). 1261 B. Hoskins, 711r r~rccthc~~rrtrri~~ctI fheorv of,frorrto~errc,.ris,Annual Rev. Fluid Mech. 14, Palo Alto (1982). 131-151.
[271 P. Lax, Al)pro.rimtltiort c?f'frlerrsurepreserving trrrrr.sforrnution.s,Cornm. Pure Appl. Math. 24 ( 197 1 ), 133135. [281 L. Lichtenstein, Math. 2. 23-32 (1925-1932). 1291 P.-L. Lions. Mather?rcrticul Tc)~)ic..s in Fluitl Mrc~lrc~rric..~. K)I. I. 1nc~omprr.s.rihlr Modrl.~,Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, New York (1996). 1301 A. Majda, Proc.eedirr~.sof' the Ir~trn~utiorrrrlCorrgrc,.s.s of' Mtr~h~rnrrtic~itrrrs. Kyoto 1990. Springer-Verlag, Berlin (1991). 13 1 1 C. Marchioro and M. Pulvirenti, M~rtl~enrcrtic.crl T h r o p of !fncornprc..s.sihIeNo~rvi.sc.ou.sF1uirl.s. SpringerVerlag, New York ( 1994). 1321 G. Monge. M6m. Math. Phyh. Acad. Roy. Sci. Paris (17XI), 666704. 1331 Y. Neretin. Ctrtc,j,wrie.s c!f'hi.sroc./rcrsric.r?~etr.srrw.strrrrl rr1~rc.rrrtttrriot~s r!f'.sonie irtfirtitr-rlir~~rtrsior~~~l gro~r/).s. Sb. 183 (2) ( 1992). 52-76. 1341 S.T. Rachev. Theory Prohab. Appl. 49 (1985). 647-676. 1351 H.L. Royden. Rc~crlAr~nly.ti.s,Macmillan, New York (1988). 1361 H. Rubin and P. Ungar. Motiorr ~rrrclc,rn .\trorll: corr.rtrcrirrir~~,fi~rt~t~. Comm. Pure Appl. Math. 10 (1957). 65-87, 1371 L. Riischendorf and S.T. Rachev. J. Multivariate Anal. 32 ( 1990). 48-54. 1381 D. Serre. Srrr Ir /~rirrc.il)r~ ~~trricrtior~trr~l r/c,.s r;c/rrtrtiort.\ I/(. I(( t~rc;c.ru~iqur d ~ . fltriclc..~ s por.f(rir.\. RAlRO Model. Math. Anal. 27 ( 1993). 739-758. 1391 V. Shelukhin, E.ri.stc,frc.c, tlrcorc~mirr tlrr r~trritrtiorrtrlprohlr~~r fiir (~orrrl)rc..ssihlr irt~'irc.irl,fltritlr. Manuscripta Math. 61 (19x8). 495-500. I101 A. Sh~lirrlnlan.0 1 1 tlrc, ,qr,otrrcJrr.\.o/ tlrc, ,q,r)rrlr( ~ / ' c l ~ / i ~ o r ~ r o r lcrrttl ~ I ~ rhc, i . \ ~rIyirorrri(..\ ~~.s (?/'trrt i(lr,ol ~ I I ( ~ o ~ I I ~ I P . s . \ ihlc, /lrric/. Math. Sh. USSR 56 ( 1987). 7')-105. 141 1 A.I. Sh~lirel~na~l. (;orrc~r~(rli:c~(l /Itiirl P o I I . ~tlrc,ir. , o/)/~,r)\-ir)t(rrior~ (11rr1 (rl)l)li(.(rriort.\. Gcom Funct. Anal. 4 ( I1)OJ). 586-620. 1421 V.N. Stldahov. Proc. Slcklov In\lil. 141 (1970). 1431 I.. Tartar. 7111,c~or~r~~c~rr.strrc~~l (.o~rrl~tr~.trrc~.. ~rrc~tlrotl cr/~l~lic~tl ro \\..\tc,rrr.\ of c.orr.\c,r.~,trtiorrItnl..!. Syslem 01' Nonlincsr PDE. NKfO AS1 Scrlcs. Kcldcl. I)ol-dccht ( IOX3). 1441 A.M. Vcr\hih. Ku\sian Ma~h.Surveys 25 ( 5 )(1'170). 117124. 1451 I..C. Young. I.c(.tro.c,\ orr rlrc, (rl(.rtlrtr ( I / Vtrr.itrriorc.\.Chcl\ca. New Yorh ( 1080). 1401 Y.D. Zhcng and A. Mqjda. E\i.\rc,rlc.o o/',qlohctl ~~.otrX.solrrtiorr.\ ro orrc~-(or~rl~orrc~~rt Vltr.\oi.-l'oi.s.\orr tr~rrl k ~ h h c ~ r - l ' l ~ ~ ~ ~ c ~ h - l ' o\y.\/e,111.\ i . ~ . s o ~irrr orrr8.\p(1c.c2(li~~~cv~.\iorr \~.it/rIII(,(I.\I~I~(,\ r1.j ir~iri(rIrlcrrrr. Co1ii111.Pure Appl Math. 47 ( IO'J4). 1365-1401
CHAPTER 3
Weak Solutions of Incompressible Euler Equations A. Shnirelman
Corltrnts I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2. Example of nonunicl~tene\so f a weak solutiol~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 . . . . . . . . . . . . . . . . . 90 2.1. Hcurtstics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The idea o f con\tructio~i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 02 2.3. Asymptotic \elution for ~nodulatetlKoltnogorov flow . . . . . . . . . . . . . . . . . . . . . . . . . 04 2.4. The lir\t-order tcrtn olthe a\yniptotic \olut~on . . . . . . . . . . . . . . . . . . . . . . . . . . . . ')O 2.5. Thecon\tructio~i.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ')X 3. Ex:itnplc of weak \elution\ with dccrcasitig c~lcrgy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I00 3. 1 . AIIirlc;~01' an cx;inlplc t r l ; i wcah \ c r l u t i c ~ t lwith dccrca\i~lgenergy . . . . . . . . . . . . . . . . . . I00 3.2. Gcncrali,cd Il[rw\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.3. Conslrctctitrti (11'an cx;~n~plc 4. E~lcrgyhalancc in wcah \crlutiot~\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 4. I . E~icrgychanges itlid flow irrcgctl;iritic\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 4.2. Dcrsilcd cncrgy halancc ill rllc 111-cvitruscx:~tiiplc . . . . . . . . . . . . . . . . . . . . . . . . . . . I I4 Achnowlcdgcn~cnt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.5 Rclcrcnccs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 15 .---
Abstract A weak solution of the Euler equations ia an L,'-vector tield rr(.r. r ). satisfying certain integral relationa, which e x p r e s incomprcaaibility and the monienlum balance. Our co~i.jecture i h that some weak solutions are limita o f solulions of viscous and comprcsaihle fluid equations. aa both viscosity and comprcaaihility tend to zero: thus. we believe [hat weak solutions describe turbulent Hows with very high Reynolds numbers. In this article we show that there exist very different claases of weak solutions. having little The lirst is a weak solution o n ;I 7-d torus having in common. We describe two ex:~~mples. compact support in time. Thus this solution displays nonitniquene.;a in the strongest poasible sense. We show that in fact this is not a true solution: the i l ~ ~ is i d driven by "ghost" forces. having /.era space scale. Thus. these forces are orthogonal to any sniooth function. and ore
HANDBOOK OF MATHEMATICAL FLUID DYNAMICS. VOLUME I I Edited by S.J. Friedlander and D. Serre O 2003 Elsevier Science B.V. All rights re\erved
zero as distributions. But nonlinearity of the Euler equations transforms these forces into real motions of the fluid. Another example is a weak solution on a 3-d torus whose kinetic energy monotonically decreases in time. This may be regarded as a rough model for turbulent flows. The kinetic energy in this model is absorbed in the acts of nonelastic collisions and coalescences of fluid particles. The main tools in this construction are generalized flows and multiflows (multiphase flows). We show that our results agree well with the results of J. Duchon and R. Robert on the energy balance in weak solutions.
Keywords: Euler equations, Weak solutions, Nonuniqueness, Inverse cascade, Kolmogorov flow. Ghost forces, Generalized flows, Multiphase flows MSC: 76C99.7hF99.76TOS. 35D99,35435,60525
1. Introduction
One of the main problems of fluid dynamics is the description of the motion of a Huid whose viscosity and compressibility are very small. Weak solutions of incompressible Euler equations are intended to describe the Huid motion as both viscosity and compressibility are going to zero. They are defined as follows. Consider the incompressible Euler equations
au ar
- + (u. V)u
+ V p = 0:
Here rr(.r. r ) is the velocity held of the fluid; p ( x , I ) is the pressure. For the sake of simplicity we consider the flows on the 11-dimensional torus: x E M = T"= R " / Z t l ;t E 10, T I . If cp(.r, r ) E C ( y (M x (0. T I ) is a scalar test-function, and u ( x , I ) E CI;O(M x (0.T I ) is a vector test-function. such that V . 11 = 0. then, after multiplying both sides of (1.1) and (1.2) by t t ( x , t ) and cp(.r, I ) . respectively, and integrating by parts, we obtain the following identities:
t ) is sufficiently regular (say. ( ' I t C ) and s;~tisfies( 1.3), ( 1.4) for any tesr-functions cp, then i t satisfies also (1.1 ). ( 1.2). But the left-hand sides of both rclntions ( 1.3). ( I .J) make sensc for arbitrary vector tields tc(.r. 1 ) E L ' ( M x 10. T I . R " ) . This ,ustitics the fol-
If
II(.\-,
11.
lowing: DFFINITIONI . I . A vector tield u(.r. t ) r L' is called a weak solution of the Euler erluations ( I . I ), (1.2). if u ( x , I ) satistics relations (1.3), (1.4) for arbitrary test-functions t ! ( . u , t ) . v(.r, I).
Numerous experiments and measurements show that in the most cases the motion of a Huid with small viscc~sitylooks very unlike c1;lssical solutions of the Euler equations. It is natural to conjecture that they are described asymptotically by some sort of weak solutions because Equations (I .3), ( 1.4) express merely the local mass and moliientum balance. In this work we show that there exist very different classes of weak solutions, having little in comnion, and some of them are physically meaningless (or, at least. they are irrelevant to turbulent Hows). Moreover, at present we have no weak solution at hand which really describes a turbulent flow. We can draw n pirrallel with the situation in the gas dynamics where we also need weak solutions to describe compressible flows. even if they are initially smooth. Here we have an important class of weak solutions. namely shock waves. We can be sure that without shock waves the gas dynaniics would lose al~iiostall
its contents and interest. For incompressible fluids we have no weak solutions playing the role of a "shock wave"; construction of such solutions (if it is possible) remains a goal for the future work, and this article is by no means exhaustive. There is no established notion of a weak solution for the incompressible Euler equations and, consequently, there is no general theory. At present, the only path leading to their better understanding is a thorough investigation of examples of weak solutions. This article contains two examples of weak solutions. The first example is a simplification and clarification of a famous example constructed by Scheffer in 1993 [ 131. Scheffer's example is an unbounded and discontinuous vector field u ( x , t ) E L2(R2 x R ) , satisfying (1.3), (1.4) for all test-functions LI, cp, and such that u(x, t) = 0 if /x12 lt12 > 1 . We con, that struct here a simplified example of a weak solution u(x, t ) on a 2-d torus T ~ such u(x. t ) 0 for It[ > C. Thus, the zero solution is not unique in the class of all weak solutions; the same is true for all smooth solutions as well (but we do not know, whether any \ i j e ~ ksolution is nonunique!). This result and the underlying phenomena are discussed in Section 2. Section 3 contains the second example, having nothing in common with the previous one. It is a weak solution u ( s , t ) on a 3-d torus T" whose kinetic energy E ( t ) = ; I L I ( . ~ .t)12d.r decreases monotonically in time. Such behavior is characteristic for a highly turbulent flow i n the absence of external forces (i.e., i n case of decaying turbulence). But other properties of this solution are different from what we can anticipate for turbulent flows; so its physical rneaning is doubtful. Section 4 is devoted to the energy bulunce in weak solutions. I t is well known that for regul:tr solutions of the Euler equations kinetic energy is constant. For weak solutions the energy is constant. if they are not very s i n g ~ ~ l(the a r borderline is ;It the regularity about the Hiilder class ~ ' 1 ' This . was conjectured by Onsager 1 121 and proved by Constantin. E and Titi 141 and Eyink (61.).But for less regular weak solutions the energy can be n o rnore col-rstant. So. the energy change is connected with irr-egularities of the velocity field. Duchon and Robert 15 1 have found an explicit formula expressing the local rate of the energy dissipation (or production) due to irregularities of the velocity field. We check that in our example of a weak solution with decreasing energy the local rate of the energy dissipation is positive. Thus. not only the total energy decreases. but i t decreases locally, too.
+
-
./ArA
2. Example of nonuniqueness of a weak solution
The first evidence that something is wrong with weak solutions is the following result of V. Scheffer, published in 1993. He constructed a weak solution u ( x , r ) of Euler equations in L2(R' x R ) , which vanishes identically outside the ball 1.1-1' ltj' < 1. This means that this solution is identically zero for r < - I ; then "something happens", and the fluid is set in motion; but at t = I the motion stops, and for all t > I the fluid is at rest again! And all the motion is confined to the disk 1x1 < I , and all this without external forces! Such behavior is unacceptable physically, and it is clear that there should be additional conditions which, together with the formal definition (1.3), (1.4), define a physically meaningful solution.
+
Wruk solu!io~~.s ofirlcor11prr.s.sih~rEulrr rquu!ion.s
91
First we discuss the phenomenon discovered by Scheffer. His original construction is long and complicated. Here we present a simpler example, exploiting the same idea, which is in fact a physical one and has a definite name. Instead of the flows on the plane IR2, we consider the flows on the 2-dimensional torus T2, and construct a weak solution u(x, t ) E L ~ ( T Kt2), ~ , having compact support in time. Thus, we discuss only the first part of the Scheffer's example, namely, the weak solution with compact support in time, which is the most striking feature of his example. This means, in particular, that the weak solution with given (zero) initial velocity is not unique, and that the kinetic energy is not constant in time, and is not even a monotonic function. In fact, our solution, as well as the one constructed by V. Sheffer, is a discontinuous, unbounded L~-function. The physical phenomenon behind our construction is the inverse energy cascade in the 2dimensional turbulence [10,8]. If the fluid, moving in a 2-dimensional domain, is pushed at t = 0 by an external force f (x) with small space scale (i.e., the force f'(x), whose Fouriertransform .?((), is concentrated at large 1(1), then the energy is transported, because of non-linearity of the Euler equations, to other frequencies and space scales. However, for 2-dimensional fluids no considerable amount of the energy can be transported to smaller scales. The obstacle is conservation of vorticity, which is a scalar function transported by the flow (Kelvin-Helmholtz vorticity theorem). If the velocity field has a characteristic scale 1 and the kinetic energy per unit volume is E , then the mean square of vorticity (called enstrophy) is about EII'. But the enstrophy is constant, so the scale I cannot decrease. On the other hand, energy can go down the spectrum, to larger space scales, because the flow can (and. we believe, usually does) distort and mix the vorticity field, so that the vorticity becomes a highly oscillating function. Then the velocity. being obtained from the vorticity by an integral operator, becomes stnoother, and the energy moves to smaller frequencies and larger scales. This energy transformation is generally irreversible. (Of course, this picture is not justifi ed mathematically and remains a physical hypothesis. There are few rigorous results in this domain; see, for example. 1171.) As an extreme case of the above picture, we can imagine that the space scale of the force f' is infinitesimally small. At the same time, the mean square of f ' remains tinite, so that the kinetic energy of the flow excited by this force be finite. But this means that f ' cannot be a distribution, for there exist no distributions with these properties. We have to introduce a new class of generalized functions to include objects like 1'. Assume that this can be done. Then simple dimension considerations show that the space scale of solution ~ r ( s t. ) grows linearly with t. Therefore, it takes a fi nite time for the energy to reach the low-frequency range. If such solution really exists. then we have a ready example of nonuniqueness of a weak solution. In fact, assume that the initial impulse ,f(.r) occurs at t = 0. Let u(.r, t ) be the above inverse-cascade solution, defined for t 0. Continue this solution by zero for t < 0. The solution obtained satisfies (formally) the nonhomogeneous Euler equations with the righthand side (external force) F ( x . t ) = ,f(x)S(t). But the function j ' ( x ) has a tinite square mean, and its space scale is zero. This means that the force F ( x , t ) is orthogonal to every smooth vector field. Thus, u ( x . t ) satisties identity (1.3) for every test-function v , i.e., it is a weak solution of homogeneous Euler equations, defined for all t and vanishing for t < 0. In fact, this is not solution at all; there are external forces, driving the fluid, but they are orthogonal to all smooth functions. Smooth test functions are not appropriate "sensors" for
these forces, which well can be called "ghost" forces (I am thankful to S. Klainerman for this term). We have described an idea of a possible example of nonuniqueness of a weak solution. Its realization is out of reach by the methods of today. The difficulty is that by no means can we solve the Euler equations in any nontrivial case. What we really construct in this work is a substitute of the inverse cascade, imitating some of its features. We define a very complicated hierarchical, finely tuned system of forces imitating the inverse cascade. The forces are organized in such a manner that at every step we have to solve a simpler problem using an asymptotic approach. In fact, we construct a sequence of solutions of nonhomogeneous Euler equations whose right-hand sides are oscillating in space with rapidly increasing frequencies and at the same time are modulated. The fluid works as a nonlinear demodulating device, converting an oscillating force into a nonoscillating flow, which is a usual L2-field. Then we prove that the limit is a weak solution of the homogeneous Euler equations. This example, as well as the example of Scheffer, shows that the formal definition of weak solution, given above, is not satisfactory. Though every possible candidate to be called a weak solution should satisfy relations ( 1.3), ( 1.4), for they express the local balance of mass and momenturn (in fact. when deriving the Euler equations in the fluid mechanics, we start frorn the relations ( I .3), ( 1.4) or equivalent, and then, assuming sufficient regularity of the field u(.u. t ) , pass to the differential form of the Euler equations), we need some additional conditions to ensure uniqueness of solution. Such conditions, proposed by Duchon and Robert 151, are discussed in the next section. I t should be emphasized that both the Scheffer's and the author's weak solutions do not belong to LdZ.(O. T: L'(M)). So. they cannot be limits of Nnvier-Stokes solutions with bounded initial energy as the viscosity tends to zero.
2.2. The idecc c?f'cotlstruc~tiot1 An incompressible vector field u ( x ,t ) E L7 is cillled a weak solution of the nonhomogeneous Euler equations with an external force f'(.t-.t ) E D', if for every test-field u(.T, t ) E C , y , V . U = 0,
Our construction is based on the following simple lemma.
L E M M A2. 1 . Let u , ( x , t ) be 11 weak .solution of norlhor~loger1eousEirlrr eyucltions with exrernu1fi)rce.s , f ; ( x .t ) , i = 1 . 2 , . . .. Suppose thut u , -+ u .strongly in L2, while + 0 ~.otrkIyin 2)'.us i + co.Then u ( s , t ) is u weak solution o f t h e Euler eyuutions ( I .3), ( 1.4). PROOF. Is clear from the def nitions.
Wrrrh .volutiot~.sc~/'irrc.on~pre.s.sihle Euler rqutrrions
93
In our construction, we shall use the forces fi (x, t ) having the special form
where f i j (.r) t Cm. The weak solution ~ ( ; ( . rI ,) of the nonhomogeneous equations (2.1 ) with such external force is a smooth solution of the homogeneous Euler equations ( I . I) on every time interval t i , j < t < ti.,j+l satisfying condition u ; ( x , t ) = O for I < t i , and the following jump condition:
Our construction starts with an arbitrary smooth solution u o ( x . t ) of the Euler equations (-,m < t < a). Let us define the first term of our sequence as u1(.r,t) =
(
(
I
)
0.
i f 111 < 1 , otherwise.
This is il weak solution of the Euler equations with the force
and
Now we are going to describe the inductive rule tor passage from rc; to u,+l. Suppose that at the moment 10 the solution ic(x. r ) has a jump. I t is smooth and satisfies the Eulcr equations for t < to and t > to, but u ( x . to 0 ) - i r ( s , to - 0 ) = j ' ( . r ) (we omit the indices i . j ) . Thus r r ( n , t ) is a weak solution with the tbrce , / ' ( . r ). S(t - t o ) . We shall replace this force by a sum of tinitc number of 8-like impulses
+
I
x ( . r . t ) = ' y f i , , , ( x ). S ( I
so that the weak solution U(X , to U ( X . to
-
u(x,
-
t,,,),
t ) of (2. I )
with the force ,q(.r. t ) satisties the conditions
0 ) = u ( x , t(, - O ) ,
+ T + 0 ) = U ( X .to + O ) ,
where T = 11 - 10. Thus we shift the solution u(.r. t ) t'or t > to by T and insert a new piece of solution in the interval ( t o ?to T ) . The most important property of the impulses g,,,(x) is that each of them is either . s n ~ ~ I l (in the sup-norm) o r is an oscillating vector field with a high frequency. In both cases x; ( . r )
+
94
A.
Shnirelmun
are "weakly close to zero". The non-linearity of the Euler equations should transform these high-frequency impulses into a smooth field u(x, t). Let us apply this operation to the function u;(x, r) at every moment t , of ~ discontinuity. If Ti is the time delay at the ith step of construction (which we take, for simplicity, independent of j ) , then
if ti., < t < ti.,+, . At every step of our construction we take the new much shorter time delays T,+l and much higher frequencies of the oscillating forces j ; + 1j3. The function u(x. t ) = lim,,, ui (x, t ) is a smooth solution of the Euler equations (1.1) on the complement to some perfect set M on the t-axis and zero outside some finite time interval. The external force is concentrated on M , but it has zero space scale, and therefore is undistinguishable from zero as a distribution. (The rigorous sense of the last sentence is that the sequence {ui(.r, t ) ) satisfies conditions of Lemma 2.1 .) In the following section we shall describe the construction of the forces ,fij(x) and the velocity fields u; (.r , r).
The main building block of our construction is a special type of flow, called modulated Kolmogorov flow. U ( K )d . = ~ 0 ciin be detined Every velocity field v ( x ) on T ' such that V . v = 0 and using the stream function $ ( x ) :
where v'$ is V$ rotated by q . The Kolrnogorov,flo is a f l o w , defined by the stream function (2.1 1 )
$ ( x ) = k-'h(x)sink(cl. x ) .
where cr E Z2, k E Z, and h ( x ) is a given smooth function independent of k. Consider the initial velocity field
) a where v(x) is the Kolmogorov flow dependent on a large integer parameter k, u ~ ( x is smooth flow independent of k, and a is a constant, < a < I. We are interested in the asymptotic behaviour of the flow with such initial condition for large k; in fact, we are going to construct an asymptotic solution of the Euler equations with the initial velocity defined by (2.1 I), (2.12), if k + co.The Kolmogorov flow is known to be strongly unstable; so our asymptotics should be valid on small time intervals (depending on k). The asymptotic solution we are constructing has the form
4
Let us write the Euler equatioris ( I . I ) in the form
where
and P is an orthogonal projector in L ~ ( T 'R?) , onto the subspace of divergence free vector fields. Insetting the asymptotic expansion (2.13) into (2.14), we find that
If we define
then
where
,
THKOREM 2 . 1 . F2)r every a .suc~/7thrrt I < u < I , rltc .verir.v (2. 17) i.s rr.sytrll?totic,it7 tho clot~itritlIt - to( < k"; t l l i ~tneotls thrrt i f vv = KK71 711, ( I S ~ h o ) h(.x) : ~ , c~tld~ ( 1 (ire ) gi\vtl futlc~tiorl.swith hout~dcdspecstrum. T l ~ eji)r t ~ el'cry .s > 0. M > 0 thrrc~erist.\ N > 0 .su(.h thllt
+
Thus, we found an asymptotic solution for the initial conditions (2.12) on the interval k-2a. (Note that there are two possible ways to understand what is an asymptotic solution for large k . We may call a solution asymptotic if it is close to the true solution. Another way is to call the solution asymptotic if upon substituting it in the equation we obtain a right-hand side which is asymptotically small in a suitable norm. We use here the second definition.) It - to1
<
2.4. TheJirst-order term rf the clsymptotic solution
Let us find an explicit expression for the low-frequency part of the term ul (x) in the asymptotic solution (2.17). This term is the most important for our construction. Suppose that
where
Then
Using the form (2.22). (2.23) of v(.r). we obtain that
where the vector field e ( $ ) = ( o I($), c 2 ( $ ) ) , and
If $ ( x ) has the form (2.23) then
Here the functions G,,(x), H;(,T) are obtained from the amplitude function b(x) by some quadratic differential operators of orders not more than 2. The most important for us is the term P G o ( x ) . After a simple but tedious computation we find that
where
X = u' is a constant vector field with components (u2, - a l ) : we identify it with the differential operator u2 - I' 1 P is the orthogonal projector in L * ( T ~IR2) , onto the subspace of divergence free vector fields. So.
&
&;
+ H2(x,k)cosk(tr,x) + H 3 ( s ,k ) sin2k(tr,x) + H 4 ( x ,k)cos2k(rr,x ) ]
+ K ' ~ .[H1(,v.k)sink(cr.s)
+ kffG1(x.k).
(2.30)
where Hl1(x,k) (11 = 1 , . . . , 4 ) and G I ( x , k) are smooth functions, bounded in Cw for all k > I . Thus the principal nonoscillating part of vl for k + co depends only on the amplitude function h ( x ) and the wave vector tr (or X = a L ) . Let us consider the inverse problem: for a given field v(x), such that
tind an amplitude h ( x ) and the wave vector u of the Kolmogorov flow, such that
where B(x) = h2(x) and X = a-L.In general, this is ilnpossible, but the followink 1s true. '
THEOREM 2.2. There c)xi.vt two intc>gervec.tor.r X I , X2, .s~rc.hthat ji)r rr2rry.smooth vector .field v(.r ), .scrti.sf.iiing V . u = 0, u dx = 0, there e.~i.sr smooth positi~v.jinctions BI ( x ) , Bz(x), such that
Moreover; there exist two pseudodlfferential operutors 0 1 , @2 oforder (- 1) with symbols depending on1.v on <, such that
where B; are const~~nts.
2.5. The construction
We are going to construct a sequence u ; ( x , t ) of incompressible vector fields with a common compact-in-time support satisfying conditions of Lemma 2.1. Let u o ( x , t ) be an arbitrary smooth solution of the Euler equations. Set 11 1 (.Y
.f ) =
otherwise.
This is a weak solution of inhomogeneous Euler equations with external force f I ( x , t ) =uo(.rq - l ) J ( t
+ I ) - u ~ ( x .l ) & ( t
-
I).
(2.35)
Suppose that after the ith step of construction we have a function u ; ( x . t ) which is smooth but at the moments I;..; of discontinuity, j = 1 . . . . , J ; . On every interval t , , , < r < t,,,+l the function 11; is a smooth solution of the Euler equations, and it vanishes for t < t i ,1 and for t > t,,,,, . Thus, u , ( . r , t ) is a weak solution of nonhomogeneous Euler equations with the right-hand side
where
We will now describe the construction of the function lr;+l. Let T be a positive number specified below. We define the following time moments: t ,. ! .=I
+(J
-
1)T,
I,!:, = t ,,. , + j T .
First of all, we define u;+l on some intervals:
j = I . . . . . J,.
(2.39)
Wrclk .solution.sof
incompressible Euler equutio~i.~
99
In every interval r:,,; < t < tl!;, the function u;+l ( x , t ) has a finite number of moments of discontinuity ti.,$p, p = I , ..., Pi.,, t ; . ; , ~= t!'., r;.,,pj,j = t,!,,+l. Beyond these moments 1.1 u,+l ( x , t ) is a smooth solution of the Euler equations, and on the whole t-axis ui+l ( x , t ) is a weak solution of nonhomogeneous Euler equations with external force
The main point is that the force J;+I(x, r ) may be made arbitrarily close to zero in a weak sense; this means that part of impulses j ; . j . l ) ( x ) are small (in the sup-norm), while others are oscillating in x with a high frequency. Successive application of impulses j;.j.l,(x) at the time moments ti,,. I , . . . , t;..;. p,,, should transform the velocity field u i + l ( x . t!'. - 0 ) into the field u;+l (.r. t;,+l +0), or, the same, the field u , ( x , ti,, - 0 ) into 1.1 the field u i ( x , ti,, 0 ) during the time T. To explain the structure of this sequence of impulses, consider velocity field u ( x , t ) which is a smooth solution of the Euler equations for all r # to; set u * ( x ) = u ( x , to f O), and u ( x ) = u+(.r) - u - ( x ) . Then ~ c ( xr ,) is a weak solution of nonhomogeneous Euler equations with the r.h.s. f ( x , t ) = u ( x ) 6 ( t - t o ) . We are going to construct a weak solution U ( , x , t ) with the r.h.s. F(.r, t ) = P F l , ( x ) 6 ( t- t I , ) ,to < tl < . . . < t p =to+ T , so that ( r 0 = ( . r ) , U(.Y.to+ T + 0 ) =~c+(.r),andthe force F ( x , t ) is weakly close to zero. This sequence looks as follows. By Theorem 2.2. we can represent the field u(.r) as a sutn of two tertns, v = o"' u'", so that
+
x,)=O
+
where X I . Xz are two integer vectors, and B I . B? are two smooth functions, which may be assumed to be positive, for the decomposition (2.36) contains only their gradients. Let us define the functions h ( ' ) ( x )= and $"'(.r) = k ' h("(.r)sin k(tr,. x ) , where tr, = -x:. I = 1.2, and k is a big integer parameter. Now let us define vector fields v ( ' ) ( x )= V ~ $ ( ~ ' ( A - ) . We begin with a preliminary construction. The first impulse is applied at r = to; this is the force Fl (.r. t ) = k U ~ ' l ' ( . r ) f i (-r to). Suppose that the velocity U ( . r ,to - 0 ) = ~ c - ( . r ) . andapplytheforce F I ( . x . r )= k U ~ " ) ( x ) f i ( r - r ~ ) . ~Uh(e. rn. to+O) = I , - ( . r ) + k U V " ) ( . r ) . This is the initial condition for which we have obtained the asytnptotic expansion (2.13). valid for It - t01 k-'". Let us retain a finite (but big enough) number of terms: we ) obtain a solution U ( . r ,t ) , such that U ( x ,to k - ? ~- 0 ) = ~ ( x k) W v ( ' ) ( s ) v ( l ) ( s + oscillating terms of order 0 and less with respect to k smooth terms of order kPU and less. At r = r ~ + k - ' ~let usapply the second impulse F 2 ( x .t ) = ( u - ( A - ) + v ' l ) ( x ) -U(.r. to+ k2a - 0 ) kWv ' " ( x ) ) s( ~to - k P z W )Then . U(x,to kPzU 0 ) = u ( x ) u ( ' ) ( x ) k f fv ( ~ ) ( x and ) , we again apply our asymptotic solution. If we retain a finite number of + terms, we have U ( x . to 2kK2" - 0 ) = u - ( s ) u ( l ' ( x ) u("(.r) k W ~ ( 2 ) ( . r )oscillating terms of order O and less smooth terms of order k W and less = u + ( x ) u f ( x ) ,
Jm
-
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
where u ' ( x ) = oscillating terms small smooth terms. Now let us apply the last impulse F ~ ( Xt ), = -u1(x)8(t -to - 2kP2");we have then U ( x ,to 2 k - 2 f f )= u + ( x ) .Thus three impulses F I , F2, F3, together with the small force on the interval to < t < to 2kP2", transform the field u - ( x ) into the field u + ( x ) during the time T = 2k-2'Y. The function U ( x ,t ) is not exactly what we need, because on the intervals to < t < to k-'" and to k - ' ~ < t < to 2k-'" it does not satisfy the Euler equations; upon substit~~tion we obtain a small r.h.s. r ~ ( xI ). , satisfying
+
+
+
+
+
for arbitrarily big s and M ( s = 3 is enough). This force may be replaced by another force @ ( x ,t ) , consisting of a finite number of weak 8-like impulses, @ ( x ,t ) = xf,:=l cp,,,(x)S(t - till) with the same effect, i.e., if G ( x ,t ) is a solution with the r.h.s. F = FI F2 F3 @ such that U(x, to - 0 ) = u - ( x ) , then U ( x ,to + 2 k p 2 " ) = u + ( x ) . Thus we have defined a sequence of impulses transforming u - ( x ) into u + ( x ) during the time period to < t < to + 2kP'"; three of these impulses are strong (having the order k W ) and others are weak (of order k P Mfor arbitrarily big M ) . If we construct this sequence for all discontinuities of the flow u , , we obtain the next function of our sequence, u,+l; the series of impulses which we use to transform every field u , ( x , ti,., - 0) into u ( x , ti,., 0 ) form together the force f ; + l ( x , t ) . Let us construct in this way the sequence u 1, u2, . . . , taking at every step 21 different parameter k = k l . X.Z. . . . . I t is easy to prove that if the sequence k l ,X.2. . . . is growing fast enough, then the sequences ~t 1 . L I Z , . . . and j'l. , f . . . . satisfy conditions of Lemma 2.1, and u; ( x ,t ) -, u(.v, t ) in L'. where u ( x . t ) is a weak solution of the Euler equations with compact in time support.
+ + +
+
3. Example of weak solutions with decreasing energy In this section we construct an example of a weak solution which, as we hope, has a close relation to the turbulent flows, although it is still far from being physically relevant. T H E O R E M3. 1 . Let M = T~he the 3-dimeri.sior7al tor.~c.s.Thrrc. c).ri,vt,v ~'rerk,solution u ( x . f ) E L~ ( M x I T I, T? I. IR3), such that its kinetic e n ~ r g yW ( t ) rnonotonic~~lly c1ec.rc.ctsc.s on some time intc~rvul[ T I T2 , I. Note that in the previous example of a weak solution we had a different picture: on any time interval either the energy is constant, or it oscillates wildly, being even an unbounded function of time.
3.1. An ideu (fern ex~tmldeof' ti wectk .solution ~ i t dect-e(tsing h erlrrgy The simplest mechanical system with decreasing energy but without explicit friction consists of two particles moving freely along the line, which stick upon collision and form a new particle. If m 1, mz are the masses and vl , vz are the velocities of the particles before
Weuk .sol~ctio~~.s ofincomprrssihlr Eulrr rquuriot~.\
101
+
collision, then the new particle has a mass m = rn I m2 and the velocity v, so that its momentum mu = ml ul m2v2. It is easy to see that its kinetic energy mv2/2 is strictly less than m lv:/2 m2v;/2. The idea of our construction is to organize a flow so that the fluid particles collide and stick; this sticking is the sink of the energy. An essential part of the fluid particles should take part in these collisions; each particle may meet other ones many or even infinitely many times. It is clear that such flow should be very nonregular; in fact, in a smooth velocity field different particles do not collide at all. If the flow field is not sufficiently irregular (say, belongs to the Halder class C f f a , > then the above-mentioned result of Constantin et al. [4] shows that the rate of collisions is not high enough to absorb positive amount of energy.
+
+
i),
The most appropriate description of nonregular flows is based on the notion of a generalized How (GF), introduced by Brenier [ I ]; a much more advanced theory is developed in 121; see also [ 161 for a further disci~ssionof generalized flows. The idea is to decouple the points of the How domain M and the Hilid particles. Recall that in "classical" Huid dynamics the fluid configurations are identified with smooth volume-preserving diffeomorphisms: all particles are labelled by the points of M, which are their positions at t = 0, and every other configuration of Huid particles is obtained from the initial one by a smooth permutation, i.e.. a smooth time-dependent volume preserving diffeomorphism : M + M. But for irregular flows this correspondence between Huid particles and points of M breaks down, and we have to introduce a separate space Q of fluid particles; this is a measurable space with a probability measure p ( d o ) . This space, in general, has nothing to do with the flow domain M. and it may have its own natural coordinates.
<,
D E F I N I T I O3.1. N Let M be a flow domain. and I T I . T7 ] a time interval, TI < T2. A generalized flow (CF) G is a collection of the following ob.jects: ( 1 ) Probability space (Q. F.p ) , where 62 is a set. F is a a-algebra of subsets of Q, and p is a probability measure on 3; (2) Measurable mapping .r : Q x [ T I ,TI] + M. (o,r ) H .r(to. I ) , such that .r(to. t ) is a continuous trajectory for almost all w E Q. This map (w, t ) H .r(w, t ) should satisfy the following conditions: ( i ) For any Lebesgue measurable set A c M and any t ,
(ii)
2
J ( G ) = / Q p ( d w ) ~ ~ ~ i ( w ,dr r)< m .
The property (i) may be called incotnpressibility and (ii) expresses the finiteness of the mean energy, i.e., the action. Thus a generalized flow is a sort of random walk in M . The simplest example of a G F is a smooth flow & (x) of volume preserving diffeomorphisms. In this case, 52 = M, w = xo r M , and x ( w , r) = c , ( x o ) . But we shall see below that there exist GFs which are genuine random processes. We start by introducing sorne classes of GFs. DEFIN~TION3.2. G F G is called a generalized flow with definite velocity (GFDV), if there exists a vector field ~ i ( . u ,I ) such that (i) n E I.'(M x [ T I ,T2]): (ii) V . u = 0 i n the weak sense; this means that Jf ( u , V q )d x d t = 0 for every testfunction (c, E C ( y ( M x ITI. Tzl): (iii) For almost all ( w , I ) ,
The field u ( . ~ -I, ) is called a tield associated with the GFDV G. I t is clear that this field is ~~nique. This detinition generalizes the known connection between the Eulerian and the Lagrangian descriptions of a fluid motion. If the velocity field ~ c ( ~ wI ,) is not sufficiently reguliu then. instead of a flow of diffcomorphisms as in the classic~llLagr:~ngi:~npicture, wc have 3 generali~edHow, i.e., a random walk. The second important notion is the generalized flow with local interaction (GFLI). DEFINITION3 . 3 . A generali~edRow G is called a generali~edf o w with local intcraction (GFLI), if for every test-field I ) ( . \ . , I ) E C',? such that V . 11 = 0.
/,
I
B(dw)
(.ii(co. I ) . t l ( , \ . ( ( ~ ,
r ) . t ) ) dl = 0,
D~;.FINI'~ION 3.4. G F x((o, t ) i.4 called a pressureless GFLI, if (1.9) is valid for cvery testfield v ( . r , t ) E C;;V,without restriction V . 11 = 0.
This detinition makes sense, since .r(to, t ) E HI for almost all cu. t:(.r(tu.1 ) . t ) ( I ) E H I , and hence the integral in (3.4) is correctly defined. Let us define a tield f (x. t ) by the following identities:
t
H '.
for every test-field u ( x , t ) r C?. Roughly speaking, the field ,[(.r, t ) is obtained by summing up accelerations of all particles passing the point (.\., I ) . For the GFLI. t h e field f (.r, t ) is potential; for pressureless GFLI. f ( x , t ) = 0. In the last case all interactions between fluid particles which change their velocities occur only between the particles occupying the same position.
The basic connection between weak solutions and generalized flows is established in the following
3.2. I f G F G is hoth GFLl und u GFDY and u ( x , t ) is un ussociuted velocity THEOREM .field, thetz u ( x t ) is a weak solution o f t h e Elder eyuutions.
.
PROOF.We shall use the probabilistic notation: for every integrable function F ( w ) on the probability space Q we deti ne
Let x ( w , t ) be a GF, satisfying conditions of Theorem 3.2, and u ( x , t ) E C ( 7 ,V . u = 0. Then, by the definition,
E
lT
(i(a t ) , u(.r(w, t ) ,t ) ) d t = 0.
Note that for almost all (0, .r((o,t ) E H I ; this implies that v ( x ( w ,t ) ,t ) E HI and i ( w , t ) E H - ' ; so, the integral makes sense. Moreover, we may integrate by parts, use successively the fact that .r(w. t ) is an incompressible GFDV, and obtain:
This means that u ( x , t ) is a weak solution.
-
Weak solution u ( x , t ) is called a pressureless weak solution. if (1.3) is valid for arbitrary u(.r, t ) E C ( y ,without restriction V . u = 0. If a pressureless weak solution ir(.r, t ) is regular enough, then it satisfies (1.1 ), ( 1 . 2 ) with p 0; all such solutions are trivial steady shear flows on the torus, which are easy to describe. But in L' there exist a lot of nontrivial pressureless weak solutions.
3.3. If' G F G is hoth (i prc.,s.surc.I~,s.s GFLl urld ti GFLIV tirid ~r (.r, t ) i.s the THEOREM c~.s,sociureclvelocity,field, the11u ( x , t ) is ti l~rc.,s.sur-c.1es.s rvecik . s o l ~ ~ tri?of~t /~iEil1c.r ~ rquutions. PROOF. Is the same as in the previous theorem, if we omit condition V . u = 0 and take into account all test-fields u E C?.
The next relevant class of GFs is the class of Sticking GFs (SGF). Their definition formalizes the intuitive picture of sticking particles. Suppose that for every t E [ T I T , 2 ]a measurable partition C, of the particle space R is defined; this means that for every t there is a measurable space which we denote by C,, and a measurable map n, : R -+ C,. We identify an element a E C, with the set n,l(a) c f2;these sets are the elements of partition C, . Let v, ( d o ) be a direct image of measure Y, in C,. For v,-almost all a E C, a conditional measure x,,, is defined, i.e., a measure in O, such that it is concentrated on a , and for every p-integrable function cl/(w),
Suppose that the family C, satisfies the following: C O N D I T I O3.1. N (i) C, i.s N co(rr.seningji~ntily; this mrutzs thut l f t l < t2, then ,jilt- v,, -almost every a E C,, there e.rist.s sotne a' E C,, s~rc.11thirt a c a'; (ii) Tl~e,firtnilyC, is cot~tinuol~.sfrom the right:
(This tr7ccrt1.stktrt,fi)r o ~ ~ c rt? .tho pcrrtitiotr C, i.s thr c.oor.sr.vt~)ossihlcorie, nhic.11is ,fitlrr thtrtl C,,,fi)rctll t' > t . )
A Sticking Generalized Flow (SGF) is defined as follows: D E F I N I T I O3.5. N GF G is called a SGF, associated with the family (C,)of partitions of O, satisfying Condition I. I, if
( i ) For every t .x ((02,t ) ; ( i i ) For every t
E
( T I Tz1, , and v, -almost every a
E
( T I .T ? ] ,every a
E
E
C,. if
( 0 1 (. 0 2 E
a , then .r(tol.t ) =
C,. and every (00 E a,
Condition (i) means that the particles, belonging to a E C,, have got stuck together and formed one compound particle by the moment t (we may identify this "large" particle with a ) . The coarsening condition (Condition 3. I (i)) implies that these particles keep moving together for all t' > t . This implies, in its turn. that the (right) velocity R(w, t ) is the same for all o E a E C,. Hence, for almost all t , and v,-almost all a E El,the velocity is defined and is constant on a . It should be stressed that two different particles passing the same point (.x. t ) do not necessarily stick, in contrast with [3,7].
Wetrk .soluriott.soJ'inco~t~/~rr.s.sihIr Eulrr rquurior~.~
105
Part (ii) of Definition 3.5 says that the momentum of the compound particle (T is equal to the sum of momenta of "small" particles w which constitute a . We must distinguish between the trivial and nontriviul SGFs. D E F I N I T I O3.6. N SGF G is called trivial, if for every t , for v,-almost all a E C,, i ( w ,0) = const for x,.,-almost all w E a . Otherwise the SGF x ( w , t) is called nontrivial. Trivial SGFs are easy to describe. Every compound particle a consists of small particles w, moving from the beginning with the same velocities; this means that their trajectories coincide, and they stick only formally; their sticking does not affect their motion. Since there is no interaction between the particles, the trivial SGF is a free motion of noninteracting particles with constant total density. The last class of GFs is introduced basically for technical reasons. D E F I N I T I O3.7. N G F G is called an L ~ G Fif,
Now we can formulate the theorem connecting all defined classes of GFs. THEOREM 3.4. Suppose tlltrt G is ( i ) tr nontri\~itrlSGF; ( i i ) trn L ~ - G F : ( i i i ) tr GFL)\! c~~ith tr.s.soc~itrtet1 \~oloc.ity,field LI(x, t ). Tllon ( i ) u ( x , t ) is (I / ~ r c ~ . s ~ ~ cvvt'trk rc~l~ sol~ction . s ~ ($the Errl(>l-eqlctrtiot1.r; ( i i ) the kinetic energy W(t) = f l~c(x.t)lZdx is tr stric.t!\. d c c ~ r u t r . s i t ~ ~ ~ l l r$t r ~ ( ~(i.e.. tiot~ W t ( t )< O , and W ( T ) < ~ ( 6 ) ) . The proof of this theorem is longer than of the previous one. We prove first that conditions (i)-(iii) imply that x ( w . t ) is a pressureless GFLI, and then use Theorem I. I. The absence of pressure looks strange, but in fact such pressureless interactions are familiar in the fluid engineering; they are used in different devices like mixing chambers. and they are fought with in many cases where they cause "pressure losses". And at last we reach our goal. THEOREM 3.5. lf.tho,fiowdorncrin M is tr 3-dirn torus T ~thon , thew et-iststr G F G, satisf:ilitlg ull conditions c!fTheoretn 3.4. Thus. we obtain an example of a weak solution of the Euler equations with decreasing energy. Theorem 3.5 is proved by an explicit multistep construction; we describe it here, omitting some details.
3.3. Construction of an example 1. Multijo~vs. Our construction is based on the notion of a multiphusejow, or briefly a multijow, with mass exchange between the phases. Multiflows are especially simple Generalized Flows (GFs). Suppose that in the flow domain there is more than one fluid, and all these fluids are moving simultaneously. The fluids are called plzases, and the flow is called a multiphase flow, or simply a multiflow (MF). Multiflow is calledjnite if the set of phases is finite; multiflow is called countuble if the set of phases is countable. Every phase of a multiflow has its own density and velocity fields, and there may be a mass exchange between different phases. Let us denote different phases by A;, B,,Cx, etc.; let PA, ( x , t ) be the density, and L ~ A(x, , t ) the velocity field of the phase A; (we always assume that all these functions are smooth). Suppose that the particles of the phase Ai are transformed into the ) ,(i, j = 1 , . . . , m ) . The data M = particles of the phase A, with the rate L I A ~ ~ A , (t X . . (A;, PA,. u d i , NA,,A,, I , j = I , . . . , m ) define a multiflow with mass exchange between phases. It is convenient to introduce the following notations:
The equations of mass balance are the following:
Particles that change their phase, also transfer their momentum. If no other forces exist, then the momentum balance equations are
a
-(PA,
i)t
UA, ) + V . PA, UA, '8 "A,
The MF M = ( P A , , uA,. ( I A , , A , ) : ~ .defines ~ = ~ a G F G = G M .This is a Markov process with continuous time. Its states are the pairs (.x, i ) , where .r E M, and i , 1 6 i 6 m , is a number of a phase. A fluid particle is moving with the velocity u ~(.r(t), , t ) when it belongs to the phase A;; but at every moment it may become a particle of any other phase Aj. The probability that this event occurs during the time interval (t. t + dt) is equal to p d , , ~( x, ( t ) , O d t . So the particle history is described by the function i ( t ) , showing to which phase the particle belongs at every moment t . This is, with probability 1 ,
Weak .solutions c~it~c.ompre.s.siblr Eulrr rquution.~
107
a piecewise-constant function with a finite number of jumps. So, the particle space 52 may be defined as follows. Consider piecewise-constant function i ( t ) , 0 t T , where i ( t ) may assume values I , . . . , m . The function i ( t ) assumes the constant value ik on the segment (tk-1, tk), k = 1 , . . ., N , to = T I , t~ = T2, and N is an arbitrary natural number. The particle trajectory x ( t ) is defined by the initial position xo = x ( T I ) and the function i ( t ) in the following way: x ( w , t ) is continuous in t , x ( o , T I )= xo, and & - L ~ A( ,x . t ) for t i - I 6 t < ti ( i = 1 . . . . , N ) . The data (xo;t l , .. . , r ~ - I ;i l , . . . , i ~ )
< <
define the particle trajectory uniquely and form coordinates in the particle space 52. Thus the particle space 52 consists of a countable set of components 52; ,..,,,,, ( N = 1 , 2 , . . .; ik = I , . . . . m ) . In every component the natural coordinates are (xo, tl , . . . , t ~ I )- , such that 0 < tl < . . . < t ~ - 1< T . So, Oi,,,,,.; , is a direct product of M and an ( N - I )-dimensional simplex. The probability distribution in 52 is defined by the following formula: ~ r o b ( x o< ~ ( 0i )xo
+ d r ; il , . . . , i N ;
TI
< ti < rl
+ drl , . . . ,
All MFs considered in this work satisfy the condition /)A,, A , = O for J < i . In this case, the function i ( t ) has not more than ( N - 1 ) jumps. where N is the number of phases. Hence, the particle space 52 has a finite number of components. If M is a finite MF. then corresponding G F is denoted by G M . The following theorem establishes connection between multiflows and sticking generalized flows: THEOREM 3.6. If'M is ci multijlow, clnd Equrrtiot7.s (3. IS), (3.4) hold, then c,orresponrling generrllized,fk)w G
i.s t i Sticking Genercilizrd Flow.
PROOF.Consists of accurate definition of the family of partitions C, and checking equality (3. I I ).
2 . The initictl ,pow. We begin our construction with a simple MF M 1 . This multiflow consists of a finite number of phases A , . . . . , A,,,, n I > I ; the density p ~ and , the velocity u ~ of, each phase A, do not depend on ( x , t ) , different phases have different velocities,
and there is no interaction between the phases, i.e., all absorption coefficients adI.A, = 0. M I is an incompressible MF, but it has no definite velocity, because different phases have different velocities at every (x, t ) . The M F M I is considered on the time interval [0, T I . Now we start improving this simple MF, so that eventually we obtain a GF, satisfying all conditions of Theorem 3.4. 3. Phase sepurcrtion. Let us pick two phases of the M F M I , say A; and A,, and denote them for convenience by B = A;, C = A,. Choose some small L > 0, and partition the 4-dim domain M x [0, T ] into equal cubes of size L. In every such cube Q we modify the densities of the phases B and C so that they become partially separated. First of all, in every cube Q we define a smooth function h ( x , r ) , defined in L/4neighborhood of the center (XQ,tQ) of the cube Q. This function satisfies the following conditions:
(ii)
i)
-h i)t
+ (UC. V)h = 0;
(3.19)
(iii) This function is constructed in every cube Q separately and independently; its restriction on the cube Q is denoted by hy(.r. t ) . In general. it is impossible to construct such a function in 2 dimensions; in 3 dimensions i t is possible. because i n 3 dimensions there is much more freedom. Now let us replace pi;. p ~the , densities of the phases B. C. by pi; ag,p(: u(:. which i n the simplest case are defined as follows:
+
+
Here @ ( t ) E C r , 0 < @ ( f ) < I . @ ( I ) = I outside some neighborhood of points To. TI ; q ( t ) E C r , O ~ ( t ) I , q ( t ) = 0 for It1 > I , q(r) = 1 for ( 1 1 < A; $(.r. y ) , O < s. y < 1 , is a smooth positive function, homogeneous of order 1 , such that $ min(.r. y ) $(.r. y ) < min(x, y ) (for example, we may take $(x, y ) = &). As a result of this step, in the central part of each cube Q the phases B and C are concentrated in thin alternating layers (whose thickness is of order L ~ )The . configuration of these layers is chosen so that velocities u ~ u~, of the phases B. C are nearly tangent to the layers in every cube Q. Note that this near-tangency may be achieved in the 3-dim case, but is, in general, impossible in the 2-dim (there is much more freedom in 3-dim space). This is why we construct our example in a 3-dim domain.
<
4
<
<
4. Controlling phases. The change of densities of the phases I3 and C requires change of velocities u g , u ~which , restores the mass balance, i.e., equality (3.15). But this modification of velocities breaks the momentum balance (3.16). This means that the modified motion of the phases implies some noncompensated forces. Now let us use the following principle: if some phase Ai of a multiflow absorbs another phase A j , moving with different velocity, with the rate U A , , ~then , , this is equivalent to a volume force u d , , ~. (, L I A ,- U A , ), applied to the phase A i , because absorbed particles of the phase A, transfer their momentum to the phase A i . We use this principle to restore the momentum balance. We add to the multiflow 6 new phases, in addition to the existing ones; call these new phases 'DT, D l , D T , D y , 2)-T, D 3 . The phase Df moves with constant velocity &vk, where vectors 1.1, 1.2, 1.3 are mutually orthogonal. The phases Df are absorbed by the previously defined phases A,, but do not absorb each other. The absorption coefficients uDi A may be prescribed arbi'
A '
trarily; we define them so that the momentum equations (3.16) hold. The phases D: are called co~itrollirzg~>h~~ses, and this name stresses their role in our constructions. The phases D f . Al. . . . , AN form a new MF. called M2. Let us give new names to its phases; call them A-s =v:, A-4 = D l . . . . , A-I =D.T,Ao = D ~ , A I. .,. , A N , so that the phases A1 , . . . , AN retain their previous names.
5 . Cor~~l~c~~~.vtrtirz,q IJ/I(ISC.~.The total density of all phases of the MF M ?is not, generally, constant; but our final goal is to construct an incompressible GF. In order to restore the incompressibility, we have to add new phases to the MF M2. We introduce a countable number of new phases 81. &?. . . . , called (.orr~l>(~r~.s(rtiti~ 1>/1(1.so.\. Every phase EL consists of particles moving with a constant speed u~t;, without interaction with other phases. Thus, the density (.r, t ) is shifted with constant velocity rrt-, (.r. t ) . The superposition of the densities of all phases exactly compensates the nonhomogeneities of the total density of the MF M 2on some smaller time interval [In.T - h?]. To construct compensating phases, let us denote by p(.r. t ) the total density of all phases of the multiflow MI. Using the Fourier series, we can represent p as a sum of a countable number of plane waves, (ak,,,,cos(kx - wt)
p(x, t) =
+ bk ,,,,sin(kx
-
wt))
k.0)
where k E Z", w E $Z. For every (k. w ) we introduce a new phase EL,,,,, whose velocity u ~,,,, ;= x k , and density p ~,<,,, = 2(af ,,,, 6; ,,,,) ' I 2- (ak.,,, cos(kx - cot) Pk .,,,sin(k.r lh 1' w t ) ) . The phases Ex,,,, have positive density and compensate the density fluctuations of the multiflow M?.
+
+
6. The irzdlcc.tive step. Further constructions proceed by induction. At the kth step we have as an input a finite MF M k , whose phases are A -,,,, , . . . A,,, , and a countable MF & with phases &+, . . . . The sum of their total densities is constant on a time interval [bk,T - hxl.The A j of the MF M kmay absorb the phase d i , if i < j ; the phases &f are compensating, they do not interact with one another and with the phases of Mk.
&I,
.
The next steps are the following: Let us add the first Nk phases of the M F N k to the k MF M k ; Let us give them the new names An,+, = E:, . . . , A,,+N, = EN, = A,,,,, . These new phases (with the same densities, velocities, and absorption rates) form an M F M L , while the rest of the phases of the MFMk form an countable MF M;. Now we pick two phases of the MF M ; , say A,, = Bk and A,, = Ck, and perform their partial separation as described above. Then we modify the velocities and densities of all phases of the MF M i , in order to restore the mass balance of all phases. The MF obtained does not satisfy the momentum balance, and, to restore it, we introduce six new controlling phases D Z l , . . . , DL3, as described above. Let us add these new phases to the + . 1 ., . . , A-,,,, - 1 = 'DL3. We obtain a new MF M i , giving them new names A -,,,,, = D L MF, called Mk+l. The total density of the MFs Mk+,and N/ is no more constant. To restore the incompressibility, let us introduce a countable number of new compensating phases which do not interact with one another and other phases (these new phases have small and rapidly decreasing densities). If we add these new phases to the MF N;, we obtain a new countable MF N k + [ .The sum of the total densities of MFs Mk+[and Mk+l is constant on some smaller time interval [hk+[. T - hk+lI. The inductive step is thus finished. The construction we have described depends on a number of parameters; the most important are Bk and Ck, the separated phases, and L k , the size of the small cubes in which the separation of two phases is done.
7 . The rc.siilt of' c~orr.striictio~r.The MFs Mq have the following evident property : every phase of MAis at the same time a phase of Ma+l(and even has there the same name). This means that the particle space ink of the generali/.ed Row G L ,corresponding to the MF M x ,is naturally included into i n k + [ ,the particle space of the G F Gx+1. corresponding to t h e M F M l t I . ~ e i tn = U E l inx. Our main result is the following theorem. THEOREM 3.7. It is possible to define the /)her.re.sUk, Ck, the) .s/)eico.sctiIt~~ LA,trnd other I)artrrneters, rc~quirc.tlfi)rthe induc~tivc.steps c!f't~orr.str~rc~tio~~ ( k = 1 . 2, . . .). .so fhtrl thr,fi)llowing ho1d.s: ( i ) Thc rorctl e1c~rr.sitie.strnd kinetic energies c!f'thc~MF.s Nx trncl to zero trrbitrerri!\.,fir.st, U S k + 03; ( i i ) The) phtrsc1.s A; c!f'thc)MFs MAurc. crs~nrptotic~uI!\. c.o~rz/)lc~trly .s~parcrtecl.or7 sornc .smtrller rirne intervul [ T I ,Tzl c [O. T 1, t1.s k + w ; this rnecrr7.s thtrt the clotnuin G = M x [ T I .TZI mtrv be divided, modulo u set (!/'zeromeelsure, into tnea.surcrhle sets G ; (-a < i < co),so that the dc~rzsityof' the pkcrsr A, tends to 1 , tvhile the totnl den.vity c~f'crllother pha.se.s terrc1.s to 0, in G , , c1.s k + co.All .sclts G , elre/ of' c1cr.s.s Fg; (iii) The phtl.~evc~locitic).s u ~ , ' ( xt ), tend to co17ti11uoii.s 1itnit.s ~i*, (x, t ) , u.s k + w 1417ifi)rmlvin G = M x I T I , T 2 ] ; (iv) The sequence c!f'GFs G k , corresponding to the MFs M k ,converges to u G F G . surisf~ingull requirements r$ Theorem 1.4. In l?urtic.uIcrc G is a genercrlited~ow with definite velocity U ( Xt,) , .so that u ( x . I ) = lid, ( x , t ) in every G;.
Wecik so1lrtioit.s of ittc~otnpre.r.sihle Euler equations
Ill
This solution is dissipative, i.e., its kinetic energy decreases. In the next section we show that it is dissipative in a stronger sense: kinetic energy decreases even locally. This means that the equation of the local balance of energy contains a negative term, which means that there is a sink of the energy, distributed in space and time. This property reminds the main property of developed turbulence in the Kolmogorov theory, namely, the existence of the energy dissipation even in absence of explicit viscosity. But this similarity does not go too far; in turbulence theory it is postulated that the energy is transferred from the large-scale motions to the smaller-scale ones, so that we have an energy cascade. In our example the energy is dissipated in the acts of direct inelastic collisions of fluid particles; so there is no cascade. Another difference is the absence of pressure. This means that fluid particles change their velocities because they absorb other particles moving with different velocities and not because of the action on them of the pressure gradient. The third feature of these flows, which may be connected with the second one, is their excessive irregularity. The velocity u ( x , r ) is an unbounded, everywhere discontinuous field, which may be constructed so that it belongs to Ll' for any p , but not to Lm. Accurate measurements ; Kolmogorov theory they are even show that turbulent velocity fields are c o t t t i n ~ ~ o uin. ~the Holder continuous, with the HKlder exponent close to 113. The construction of example of a Halder continuous weak solution is the goal of the further work; we should understand a lot from the process of the construction.
4. Energy balance in weak solutions
Weak solutions are intended to describe turbulent flows at very high Reynolds numbers. The niost prominent feature of these flows is energy dissipation: the kinetic energy of the flow in the absence of external forces decreases; if we want to keep the energy on some level, we have to continually stir the Huid by an external force. The rate of the energy dissipation does not depend on the Reynolds number if the latter is very high. and is definitely positive. This is one of the best established results of experimental physics. For example. if the Huid is pumped through a long pipe, the energy losses via viscous dissipation should be compensated by a pressure drop. Accurate measurements show that for small viscosity (or. equivalently. for high Reynolds numbers) the pressure drop is defined by the DarcyWeisbach formula I I I I
where p is the fluid density, v is the mean velocity, L and d are the length and the (internal) diameter of the pipe, p is the fluid molecular viscosity. RP = v d p l r t ~ ~isr the Reynolds number, and k(Re) is a di~nensionlesscoefficient which is asymptotically independent of Re as Re -+c a (its limit for round pipes is approximately 0.02). I t is natural to assume that i.e., in the case of inviscid fluid, the energy should be dissipated in the limit case RE = a, as well. But this is impossible for regular solutions of the Euler equations and we should consider some sort of weak solutions.
The idea that the energy dissipation in turbulent flows is defined by irregularity of the ffow field is due to Onsager 1121. He conjectured, using some heuristics, that the natural margin of the energy dissipation is close to the Holder class C a , where a is about 113. If a > 113 then, according to his hypothesis, the kinetic energy of the flow is conserved. This hypothesis was partially proved by Eyink [6];the full proof in a Besov space instead of Hijlder (which is some refinement of the initial hypothesis) was done by Constantin, E and Titi [4]. The picture was further clarified and simplified by Duchon and Robert [5] by their introduction of a quantity which is a local rate of energy dissipation (or generation) by irregularities of the velocity field (see below). The formal defi nition of a weak solution given above is too wide to incorporate naturally the energy decay. It is sufficient to say that this definition is time-reversible: if u ( x , r ) is a weak solution then -u(-r, -r) is also a weak solution; if the energy of the first solution decreases then the energy of the second one increases. This means that we need some additional conditions which say which weak solution is "right" and which one is "wrong". It should be said at once that a mere requirement that the full energy E ( r ) = J' klu(x, t)12dx does not increase is not enough, for it may happen that increase of the energy in some part of the flow domain is compensated by faster decrease in some other part. I t is necessary to define a local quantity describing the energy dissipation. This was done by Duchon and Robert (see 151 and the article of Robert in this Handbook). They have defined a distribution I)(.\., t ) which can be regarded as ;I rate of energy dissipation (or generation) in a weak solution. I t is defined as follows. Suppose a weak solution u ( . r . t ) belongs to the space L7(M x 10. T I ) (we need this property for the next constructions. while it is not yet clear whether such solutions really exist for a given initial conditions). If i ~ ( . rt,) is ;I smooth solution then the energy balance equation looks ;IS ti)llows:
Integrating over the whole domain M , we obtain E ( t ) = lM i ( ~ r ( . rt)j2d.r . = const. Using the fact that u ( x . t ) is incompressible, we can rewrite (4.2) in the form
where p satisfies equation Ap = - ( L ~ , L , , ). We assume for simplicity that M is a 3dimensional torus (otherwise we have to deal with boundary conditions which are not so simple in this case and only bring irrelevant difticulties). Then 17 = A-'(u;u j ) , , , , ; thus, 1) is obtained from u @ u by a pseudodifferential operator of order zero [9]. If u E L3 then u @ u E L ~ / *and , hence p E L~~~ (the Calderon-Zigmund theorem). Thus, for any weak solution u ( x , r ) E L3, the left-hand side of (4.3) makes sense trs tr
Weak .solutiotts of'ittcomprr..ibe Euler equcrtions
11.7
distribution. But there is no guaranty that this is exactly zero; therefore we dr$ne the distribution D(.r, t ) as
Then the energy balance for any weak solution looks (tautologically) as
Duchon and Robert found an explicit expression for the distribution D ( x , t ) . Namely, let us denote 6u(x. y , t ) = u(.r, t ) - u(y, t). Let ( ~ ( x E) C(7 be any spherically-symmetric function in IW" such that lcp(.r) ds = 1, and for any E > 0, ( P ~(x) = e-'(P(x/E). Let us define
Then the following theorem is true 151: THEOREM 4. 1 . Tllc~ri~ c'.vi.st.s (I litt~itlim, -0 D, ( x . t ) = D ( x , t ) it7 tlw .SPIZ.SP (!f'di.strih~~tiot1.s. T/~i.clitl~itc1oe.s /lot rlq~eilrloil .s/~rc.ific, choice c!f'th~jilt~~.tiot~ (P, c1t7d it i . ~ ~.r~lct!\' 1hi.s tli.srrihuriotl C ) tllclt et1ter.s the lyfi-k(o~I side of' (4.5) c ~ t l e tl ~ ~ ~ k itc >iderltity. .s The proof is based on the following trick. For any function ,q(.r. t). we detine ,qC(.I-. t ) = I g ( y . t)cp,(.r - y ) d y . Suppose u(x. t ) E L' is a weak solution: then the field i r k ( . r . t ) satisfies nonho~nogeneousEuler equation
a simple calculation shows that the right-hand side
Thus the energy equation for the flow u' is
where Ek = (uF,V . [irF @ u'
- (11
@ u)']).
(4.9)
As e -+0, the tirst two terms in (4.8) tend to the corresponding terms of (4.5) as distributions. This proves that E,. + D in the sense of distributions. Then direct calculation shows that E, - D, + 0 as distributions. This is the end of the proof.
Now let us return to the Onsager's conjecture (see above). If u(x, r ) is a weak solution , Clx - yj", where C is independent of r and a, > 113, then it is and lu(x, t ) - ~ ( yt)j easy to see that D(x, t ) = 0. Duchon and Robert 151 present a simple and more general condition on J u ( x ,t ) - ~ ( yt ). J which ensures conservation of kinetic energy of the flow.
<
4.2. Detuiled energy balance in the previous exumple Consider the weak solution with decreasing total energy constructed in the previous section. It is easy to see that we can construct such a solution belonging to the space Lp for I , in particular we can construct solution u(x, t ) E L ~ ( Mx [ T I ,T2]).We may ask any whether the above distribution D(x, I ) is positive for this solution. The answer is affirmative:
T H E O R E4.2. M f i r 0 ~teclkL'-.solution ~c(.r,r ) con.structed it? Srcrion 3, the clistr-ibution D(.u, I ) is a nonnegative merrsure it1 M x [ TI, T2]. PROOF. Let us denote G = M x [ T I ,T?]. If u ( x , t ) is the weak solution with decreasing energy constructed in Section 3, then, in fact, 11 is a velocity tield associated with a generalized flow G; this is, in its turn, a multiflow with countable set of phases A ] . A ? . . . . . Dornain G can be divided into countable set of measurable subsets G; (modulo LI set of' measure yero) so that every phase A; occupies the set G, in the sense that the density PA, is I in G; and 0 outside G;. But the velocity tield [(A,(x. I ) ofthe phase A; is a continuous function, as a limit of uniformly converging sequence of continuous functions. Moreover. the above construction shows that tor every phase A, its velocity 114, (.v.I ) is a sum of a countable number of wave packets whose frequency is of order L:. while the amplitude is of order L:. k = 1.2, . . . . Here ( L A) is a sequence which may be made decreasing arbitrarily fast. This implies that the Hiilder norms Ij . 11(.211 , o f these wave packets decrease as '/3-& (.\-. 0 E C - . and therefore for every phase dl,its velocity . This implies that
By the Lebesgue theore~n.almost every point of .r E G , is a point of its density, i.e.. mes(G; n R(.r. r ) ) / me>(H(.v. r - ) ) + I as r- -+ 0. Further. for every point j. E G ; n B(.r. r - ) , 611(x. T ) = u d , 0,)- lid, (.v). Therefore
here we use the fact that the phase A j is absorbed by the phase Ai with the volume absorption rate UA,,A,( x , t ) . Absorption of the phase Aj by other phases is negligible because (x, t ) is the density point of the set G , . Therefore
as E + 0, and we obtain the final result: for almost all x E G;,
Thus, the local rate of the energy absorption D ( x , t ) is nonnegative.
n
Note that for our example. the pressure is absent. and the energy equation (4.5) looks as follows:
Acknowledgement Part of this article was written during my stay at the Max Planck Institute for Mathematics in Bonn. I am very thankful to this lnstitite for its stimulating atmosphere and excellent working conditions. I am thankful to Susan Friedlander for her hard work on improving my English, and to the referee for important remarks. I am also grateful to the Hermann Minkovsky Foundation for support of this work.
References 1 I I Y. Brcnicr. 71rc. Ic~tr.\r trc.tiorr /)rirri.il~lctrrrtl rlrc, ,u,lcrrc,tlr.orrc.r,/)ro ~ , ~ r ~ r r c ~ r 7 r l i : c ~ c I I firrc~or~r~~r-c~.s.\ihla lo~~~,fi~r pc,r/rc.r ,/lei
161 G. Eyink, Dlrrgy dis.silxrtion rritlro~rtviscosity irl the irlnrl hydrodynurnics, I. Fourier ut1u1y.si.surld lo(.uI rtrrrgy trur~gfrr,Phys. D 78 ( 3 4 ) (1994), 222-240. 171 W. E. Yu.G. Rykov and Ya.G. Sinai, Generrrlized voriutionul princip1r.s. global wruk .solurinn.s und hrhrrvior rvirh rcrr1r1~1111 b~iticrlrltrrrr fi)r ,s~,stern,scf corl.sevrrtion 1rnv.s urisitlg in urlhesion l)rrrticlr &numic..s. Comm. Math. Phys. 177 (2) (1996). 349-380. 181 U. Frisch, Errhulet~ce,The Legucy 4 A . N . Kolrnogorov. Cambridge University Press, Cambridge (1995). 191 L. Hiirrnander. The Arrcrl~.si.sof Lirlerrr Prrrtirtl Difrrmtitrl 0l)~rrrfors.111. P.srudotl~flkrrt~tilrl 0prrrrror.s. Grundlehren Math. Wiss.. Vol. 274, Springer-Verlag, Berlin ( 1985). [ 101 R.H. Kraichnan, It~ertirrlrrurgr.t it! tvt,o-tlin~et~.siotrul turhulet~c~r, Phys. Fluids 10 (7) ( 1967), 14 17- 1423. 1 1 1 1 L.G. Loitsyariskii. Mrc.hor1it.s of'Lic/uirl.s crrltl Gnsrs. Pergamon Press. New York (1966). [I21 L. Onsager, Sttrti.sticrr1hy(l,r~c~t~trnri~..v. Nuovo Cimento (9) Supplemento No. 2 (Converno Internazionale di Meccanica Statistics) 6 ( 1 949). 279-287. [ 131 V. Scheffer. Arr if~~~i.sc~itl,florv rt.it/r r.onr/~crc.t.srrl~/~ort it, .s/)rrc,r-lirrrr, J. Geom. Anal. 3 (4) ( 1993). 3 4 3 4 0 1 . 1 141 A. Shnirelman. 011the rrotr~rrritltr~r~rs.v qf'rr.c,~k.solrrtion ~!f'rheEulrr c,qtrtrtint~,s.Comm. Pure Appl. Math. 50 (12) (1997). 1261-1286. 1151 A. Shnirelman. Wotrk .roluriorr.s with t/t~~~rc.tr.sirt,q etrrrgs (!fit~c.or,~prc,.s.sihlr Errlrr rqrrtrfior~.~, Comm. Math. Phyh. 210 (3) (2000). 544-603. [ 1 6 1 A. Shn irelman. Getrc~rtrli:otl flrrirl &11t..r. tlrr~it-cr/)~~roxitt,rrliotr rnul r r ~ ~ ~ ~ l i c ~ rCeom. r ~ i n t ~Funct. . ~ . Anal. 4 ( 5 ) ( 1994). 586-620. 1 17 1 A. Shnirelman. 7'11c. ltrtric.c, tlreor;~tlrrtl tlrc,,/lort..\of' rrrt idctrl ~II(.OIII~I)P.\sihli, fllrill. Ru\sian J . Math. Phy \. 1 ( I ) (1993). 105-1 14.
CHAPTER 4
Near Identity Transformations for the Navier-Stokes Equations Peter Constantin Dc;r,arfmmt o/'Muthematic.s, The Univur.sitv o f Chicago E-mcril: c'on.~t@cr. uchicago.edu
Contents I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Energy dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Uniforrn bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Non-uriili~rnibounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Euler equation\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . h. Diffusive Lagranpian t r o n \ f o r a t i o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledg~ncnt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Refcrcnces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
HANDBOOK O F MATHEMATICAL FLUID DYNAMICS, VOLUME I1 Edited by S.J. Friedlander and D. Serre 02003 Elsevier Science B.V. All rights reserved
I I9 I 14, 124 126 128 13 1 139 139 134,
Necw identity tru~~.\for~t~crtio~~.\ fi)r the Nrrvirr-Stokes equution.~
1. Introduction Ordinary incompressible Newtonian fluids are described by the Navier-Stokes equations. These equations have been used by engineers and physicists with a great deal of success and the range of their validity and applicability is well established. Together with other fundamental systems like the Schrodinger and Maxwell equations, these equations are among the most important equations of mathematical physics. Nevertheless, their mathematical theory is incomplete and requires cut-offs. The present state of knowledge is such that different approximations seem to be useful for different purposes. The mathematical questions for incompressible fluid equations have been discussed in many of existence and reg~~larity books and review articles (for instance, [ 1,2 1,35,47,10,16,36,43,46] and many more). In this work I describe some results reflecting research concerning diffusive-Lagrangian aspects of the Navier-Stokes equations [I 2,141. There are two distinct classes of approximations of the Navier-Stokes equations that we consider. In one class, the energy dissipation is treated exactly but the vorticity equation is not exact. This class contains the Galerkin approximations [21.47] and mollified equations 13,351 (see ( I I ) below). The other class treats the vorticity equation exactly but the energy dissipation is approximated. This is the class of vortex methods 161 (see ( 1 2) below) and their generalizations. This class is related by a change of variables to a class of filtered approximations ( 13) of the formulation (33, 391: the models 13 1.41 are a subclass of these. The Navier-Stokes equations and their var~ ~ Lapproximations I S can be described in terms of near identity transformations. These are diffusive particle path transformations of physical space that start from the identity. The active velocity is obtained from the diffusive path transf'ormation and a virtual velocity using the Weber for~nula.The active vorticity is computed from the diffusive path transformation and a virtual vorticity using the Cauchy for~nula.The path triunsti)rlnation and the virtual fields are computed in Eulerian coordinates ("laboratory frame"). In the absence of kinematic viscosity, both the virtual velocity and the virtual vorticity are passively transported ("frozen in") by the How. In the presence of viscosity. these tields obey diffusion equations with coefficients that are proportional to the kinematic viscosity and are derived from the diffusive transformations. The diffusive path transformations are used fix short time intervals, as long as the transformations do not stray too much from the identity. The duration of these intervals is determined by the requirement of invertibility of the gradient map. If and when the viscosity-induced change in the Jacobian reaches a pre-assigned level, one stops. and then restarts the calculation from the identity transformation. using as initial virtual field the previously computed active field.
2. Energy dissipation An incompressible fluid of constant density and temperature can be described in terms of the fluid velocity u(x. t ) and pressure p ( x , t ) . functions of Eulerian (laboratory) coordinates x, representing position and t , representing time. The Navier-Stokes equations are an expression of Newton's second law, and in the absence of external sources of energy they are
coupled with the constraint of incompressibility
In the initial value problem, a velocity
is given at t = 0. The coefficient v > 0 is the kinematic viscosity. In an idealized situation, the velocity is defined for all x IR3 and vanishes at infinity. The vorticity equation is obtained by taking the curl of the Navier-Stokes equations:
where
is the vorticity. If w is divergence-free, one may invert this relation: Defining a streamvector $ that satisfies - A $ = w , using the Newtonian potential and then taking its curl, one obtains the familiar Biot-Savart law
If the initial velocity vanishes then u(.r, t ) = 0 , p(.r. t ) = 0 solve the equations. More1 ) .exists for all time, is smooth. over, if uo is close to ice = 0 then the solution u ( . r . I ) , l ) ( . ~ and converges to 0.The open question in this situation concerns the behavior of the solution for large initial data. The Navier-Stokes equations are a parabolic regularization of the Euler equations (obtained by setting v = 0 ) . Although the viscous term is important. for the study of large data one needs to consider properties of the Euler equations. The Navier-Stokes equations conserve momentum (integral o f velocity in the present setting). The total kinetic energy
is dissipated by viscosity
The dissipation of kinetic energy is the strongest source of quantitative information about the Navier-Stokes equations that is presently known for all solutions. This dissipation is used to construct Leray weak solutions with finite kinetic energy that exist for all time, u E Lm(dt; ~ ~ ( d x )Vu ) , E ~ ' ( d t8 dx) (341. This class of solutions is very wide. The
solutions have partial regularity [3] but are not known to be smooth. The uniqueness of the Leray weak solutions is not known. Nevertheless, immediately after inception, at positive times arbitrarily close to the initial time, the Leray solutions have square integrable gradients and become smooth for an interval of time. The solution is then uniquely determined and remains smooth for an interval of time whose duration is bounded below by a nonzero constant. The issue is whether the smooth behavior continues for all time. The simplest self-similar blow-up ansatz of Leray has been ruled out [37,48]. The most important task is to obtain good a priori bounds for smooth solutions of the Navier-Stokes equations. If one has good bounds then the smoothness and uniqueness of the solution can be shown to persist [43,44,47,21,35].In situations in which such bounds are not available, the study of solutions of the Navier-Stokes equations needs to be pursued by considering long-lived approximate solutions. The advantage of dealing with approximations, besides practicality, is concept~~al simplicity: one may formulate sufficient conditions for global regularity quantitatively, in terms of the approximate solutions. If one devises approximations and obtains uniform bounds for them then, by removing the approximation, one obtains rigorous bounds for weak solutions of the Navier-Stokes equations that are valid for all time. For instance, one can prove: THEOREM I . Let ~ ( he 1 ~r,fitnc.tiorr irl L ? ( I w ~that ) , .scrti.sfirs the diu~rgetrce~free c~onclition V . uo = 0 in the' S ~ I I S C(?f'distrih~ttion~. Let T > 0 he trrhitrCiq; There ~ x i ~ tCI. sLc~roycve~rk .sol~rfion(14 (.I-. I ) ,( p ( . r ,t ) ) of' the) Nervier-Stokrs c~yurltiorr.~ thtrt is rlc~fir~~el ,fi)r- t E 10. T 1, .sc1 t;.sfic.\
/ I /
r,,
IW'
?k,
I /~s(.r..s)~'d.I-dr
<
u(.r.t())l2d.r
fi)r trll t 3 to tcntl to E I c [O. TI, wht~reI is ( I S B i!f:fitll ~ merrsrtrc2I I I = T . The initit11tinlo belongs to it. 0 E I , irr other cvort1.v to = 0 is trllo,r~rt/.In trtlclirioi~.r h .solution ~ .s~rti.sfifi~.s
If'the irrititrl tlortic.iry wo = V x u~ is in L I ,
'
L tort1 rilorc.o\lc)r
/ W Od.r ~
<
oo then it rrinrritrs horrtltlrtl in
'
,fir t 2 to, to E I . If the initiul dutu is in H und the initiul Reynolds number is small then the solution is injnitely differentiuble f i r positive time und converges to 0. More precisely, ly
tlleri the solutiotl exist.sfir ull t > 0 , be1ong.s t o C ~ ( R " ) ,und converges to 0 .
This theorem combines the bound in [8j that was proved using a version o f the retarded mollification approximation procedure o f [3] with the result o f [ 3 0 ]that was proved in the space-periodic case using Galerkin approximations. T h e last statement about smooth solutions is proved b y studying the evolution o f the product o f energy and enstrophy. T h e specific constant (about 0.495) comes from the fact that, for divergence-free, zero-mean functions I I L I ( ~ ~ x < (see ( 2 2 ) below). I f the initial Reynolds number is small then it stays small and its rate o f dissipation is a well-known quantity that controls global existence (see ( 2 3 ) below). Both the mollification approximation and the Galerkin truncation approximation procedure respect the energy dissipation inequality (8) exactly but they introduce errors in the vorticity equation. T h e (not retarded) mollification equation is described below. One defines a tnollitied 11 by
fiJmJm
./iK,
Here 6 > 0 and the positive kernel J is norrnali~ed J(.r)d.r = I, smooth, and decays sufficiently fast at infinity. T w o canonical exarnples 01' such J are the Poisson kernel J ( . r ) = n - ' ( I 1x1')-' and the Gaussian J ( . r ) = (2n)-'/'p-i-'l'/'. T h e Fourier transforms o f J , J ( 6 ) = e-IEl and. respectively. J ( 6 ) = exp(-161'/2). are nonnegative. vanish at the origin, decay rapidly, and are bounded above by 1 . Because o f the fact that at the Fourier transform level one has
-+
the operator of convolution with J 8 , [ u I s= J J ( - i V ) l r is a classical smoothing approximation o f the identity. T h e mollified equation is
together with V . u = 0. Here Iul = Iuls is computed by applying the mollifier at each instance o f time. This nonlinear partial differential equation has global solutions for arbitrary divergence-free initial data uo E L ~ T.h e solutions are smooth on ( 0 . TI x iR3 and, moreover, the energy inequality ( 8 ) is valid for any to E 10. T I , t 3 to T h e vorticity of the mollified equation does not obey ( 4 ) exactly. By contrast, classical vortex methods
123
Near identity trunsf)rn~clrionsf i r the Nuvirr-Stokus rquurions
[6] respect the structure of the vorticity equation (4) but do not obey exactly the energy dissipation inequality (8). In this paper we call vortex methods the equations
with u calculated from w using the Biot-Savart law ( 6 ) ,and [ u ] = [uIs computed from u using the mollifier (9). Both equation and solutions depend on 6 but we will keep notation light by dropping the reference to this dependence: w = ws, [u] = [us]& These vortex methods may also be described by using an auxiliary variable w . One considers the equation
( M * means the transposed matrix.) A direct calculation verifies that the curl of w , V x w obeys Equation (l2), as does w. This calculation uses only the fact that [ u ] is divergencefree. The system formed by Equation ( I 3), coupled with [ u ] = J8(-iV)P(u)),
(14)
is equivalent to ( 12), (6), (9). Here P,
is the Leray-Hodge projector on divergence-free vectors. The initial u! is required to satisfy Pwo = uo. At fi xed positive 6, the solution is smooth and global. If u = 0 then these systems have a Kelvin circulation theorem: the integral $ w . dx isconserved along closed paths y that are transported by the How of [ u ] .(In contrast, the mollified equations d o not have a Kelvin circulation theorem.) The energy dissipation principle for the vortex method is
+.
One can This is obtained by taking the scalar product of (12) with I + ] where u = V x obtain the energy dissipation principle also by taking the scalar product of (13) with l i t ] . One uses the fact that Js(-iV) is a scalar operator (multiple of the identity as a matrix, i.e., acts separately on each component of a vector) that commutes with differentiation. Then the cancellation of the nonlinearity follows from the divergence free condition. The energy dissipation principle gives strong control on the mollified (or weak control on the unmollified) quantities:
124
and
Because 7-I is a positive function that grows exponentially at infinity, the inequality implies real analytic control on [ u ] :
One needs to bear in mind, however, that this bound is weaker than the one provided by the energy dissipation ( 8 ) for the mollified equation ( I I ), where ( J , y ( - i ~ ) ) ' l uis ]bounded in .'.f
3. Uniform hounds The energy dissipation principle ( 8 ) holds exactly for the mollitied equation ( I I ) and has a counterpart for the vortex method (12) in (16). (19). These are uniform inequalities, in the sense that the coefticients are 8-independent and the right-hand sides are bounded uniformly for all 8 > 0.Most uniform bounds are inherited by the solution of the NavierStokes equations by passage to limit. Some uniform bounds for Equation ( I I ) can bc summarized as follows:
THEOKEM 2. uo he u .syuure-integruhle, d i ~ t ~ t . g e t z ~ . e ~fun(.tion. free Let K > 0. TIZLJII there c/xit.ru urziyuc .solutiot~( u , p ) of' ( 1 1 ) cIt~fitzc~r1,fi~r (111 t > 0. The .rolution is rvrrl colrrlyric. ,fhrpo.ritir~ times. Thc limit lim,,o u ( x , t ) = u o ( x ) holds it1 LI wrnk serlsc. i l l L'. Tllc, Pnrrg? ineyuulirv (8) hold.s,fi,r unv 0 6 to 6 t . The un!#Ornz hound
holds with
Near idenfifyl r u ~ t . ~ / i ~ ~ - ~ f1i)r ~ u fihr i o ~Nuvirr-Stokes t.~ ryuuiiot~s
and C , cr ~miversalconstant independent of 6. If the initial vorticity Iw01 dx < oo then it rrrnclins bounded in L ' and, mnreoveq
IRj
fi)r all r
125
= V x uo is in L 1 ,
2 to. 1tz udclition, the vot-ticity r1irc.c-tion
The proof of this result sl~irtswith the vorticity equation for the mollified equation:
Here p , , ~is the signature of the permutation ( 1 , 2 . 3 ) o (i. j . k ) , and repeated indices are summed. Multiplying scolarly by [ one obtains
where Det(cr, h , ~ is) the determinant of the matrix formed by the three vectors u , h, c . Integrating in space and using the energy dissipation, one can deduce the bounds for co in l d l and the bound on the direction [. For the bound on 14 in Lw one uses the enstrophy differential inequality
(
d
)
'14
(L3IV(o(.x.t)l-d.t
This is obtained from the vortic~tyequation (20) above by niultiplication by by parts, and use of bound (see ( 2 7 )below)
w , integration
I? Constantin
126
+
Then one employs the idea of [30]: one divides by (c2 h 3 IW (x, t ) l 2 d ~ and) integrates ~ in time using the energy principle ( c > O is a constant). One obtains a bound for
in terms of the initial data. The L" bound follows from interpolation (22), the bound above, and the energy principle. We omit further details.
4. Non-uniform bounds If u is a smooth solution of the Navier-Stokes equations and if
then one can bound any derivative of u on (0, TI in terms of the initial data, viscosity, T , and D (see [ 2 11, and references therein). Similarly, if one has a bound
then one can bound any derivative of u on (0, TI in terms of the initial data, viscosity. T, and B . The quantities D and B have same dimensional count as viscosity (units of length squared per time). If one has a regularization that respects the energy dissipation and one has uniform bounds for the corresponding quantities then one can prove global existence of smooth solutions. If any of the two conditions is met then the solution is real analytic for positive time. Consider the mollified equation ( 1 1 ) at 6 > 0. Assuming (23), for instance, T one obtains bounds for l R j Iw(x, f)12d.r and for IVw(x. t)12 dx dr in terms of initial data, D, and T , directly from (2 1 ). The interpolation (22) then produces a bound for B . Vice-versa, if one has the assumption (24) then one does not use interpolation when one derives the enstrophy inequality from (20); rather, one integrates by parts to reveal u and one uses directly the assumption about IlullLx to deduce a uniform bound for the maximum enstrophy in the time interval [ 0 , T I :
h, lR,
sup
S IOJ(X,
($7' R'
2
111 dx
< & < a.
This allows to bound D. In either case, the number E depends on the numbers D (respectively B) of assumptions (23) (respectively (24)). By increasing &, if necessary, we may assume, without loss of generality, the condition
Near ide~zfityfrcmsfi,rmc~tionsfor the Nuvier-Sfokrs equution.~
127
This condition reflects the fact that we are not pursuing decay estimates. If no assumption is made then £ depends on 6 > 0. Once the enstrophy is bounded in time, higher derivatives are bounded using the Gevrey-class method of [ 2 8 ] . T H E O R E3M. Let 6 > 0. Consider solutions rf ( 1 1 ) with initial data uo E L2, wg E L2. Assume thut one cfthe inequalities ( 2 3 )or (24) holds on the interval ($time [ 0 ,TI. Then there exists a constant co E ( 0 , 1 ) depending only on the number p = p ( £ , v , T ) of (26), so that
sup ro,
k3
1'
e2"'l&((. t ) d(
< 2£
<
holds j i , r ~ ~ l l O < to T . Here h = m m i n { t o / ~c.0). ; I f ' & is unifi)rm in 6 LLS 6 -+ 0, then the solutiot~($the Nuvier-Stokes equations with initial data uo is real-analytic and obeys the bo~rtzdubove. Using the fact that 14 = V x
+, with divergence-free stream vector +, one sees easily that
holds pointwise. I t is elementary then to check that
holds for any positive A. Using the Fourier transfor~nof ( I I ) and the inequality above, one may follow closely the idea of [ 2 8 ]One . considers the quantity
with v =
and 0
< s < t 6 T and derives a differential inequality of the form
< < < +
with absolute constant c. At t = s one has by construction ~ ( s ) E. The differential inequality guarantees that y does not exceed 2& on a time interval .s t .s 2coT with c.0 a small non-dimensional constant (proportional to p-' = v 3 ~ - ' £ - ' ) .The time step 2 q ) T is uniform because of the assumption that E is finite. One starts from s = 0. The differential inequality implies a nontrivial Gevrey-class bound for the second half of the first time interval [O, 2c0T 1. One now sets s = coT and takes another step of duration 2q)T. At each
step, the second half of the time interval yields a nontrivial bound, and because the initial point s is advanced by a half-step, one covers all of [COT,T I . The pre-factor in the exponential bound is uniformly bounded below by 2covT. If to < c o T , we can advance 2t0 at a time and similarly obtain a uniform bound on [to,TI, with exponent pre-factor bounded below by 2tov. This result implies that the velocity is real-analytic. One may extend u to a complex domain z = x + i y by setting
and, in view of (27), the integral is absolutely convergent for lyl < A, r 3 to. Note also that, at fixed 8 > 0, one has a finite 8-dependent bound on E, and therefore the solutions of the mollified equations are real-analytic. Likewise, if one allows for 6-dependent bounds, then the vortex method can also be shown to have real-analytic solutions. In this case, however, because the energy principle is not strong enough, one starts from the assumption (25): 4 . Let the itiiticll vorticity w~ kelotlg to L*. Coizsider;jbr S > 0, the solution THEOREM ( I cwnstnnt
of' ( 12) cttzd u.ssurne therr (25) holds or1 ( I time iritervcil 10, TI. Then there exists c.0 E (0. I ) clcy~enclitigotzly on rlze tzlonher p = p (E, v, T ) of' (26). .so thut
SUP <7'
I(,
Ll
c * * l " d ( ~ t.
dF
< 2E
hold.s,fi~re111 0 < to < T . Herc h = ~ r n i n { r o / TC ;O ) . If'&is rrrl;fi>rtl7i,7 S t1.s 8 + 0, tlic~ti the solutiori of' the Nuvier-Stokes eyutirions ( 4 ) ,c~itliirlititil cltit~~ w~ i.5 rutrl-trrlo!\.tic. rrtltl ohevs the bo~triclcthove. The proof follows the same ideas as the proof of the corresponding result for the mollitied equation ( I I ).
5. Euler equations The three-dimensional Euler equations
are locally well-posed 126,321. They conserve kinetic energy (if the solutions are smooth enough 138,27,18]).Such smooth solutions can be interpreted I I ] as geodesic paths on an infinite-dimensional group of transformations. Despite energy conservation, gradients of solutions may grow 1421. The vorticity w = V x u obeys
Ncwr idetttity trunsformutiottsfir
129
rhr Nuvirr-Stokes rquotion.~
Because of the quadratic nature of this equation and the fact that the strain matrix
is related to the vorticity by a linear classical singular Calderon-Zygmund integral, it was suggested [36] that blow-up of the vorticity might occur in finite time. This problem is open, despite much research [5,7,11,22,29,45]. The blow-up cannot occur unless the time integral of the maximum modulus of vorticity diverges [2]. The vorticity magnitude obeys
(a, + L l . v)lwj=alwl,
( 31 )
where the logarithmic material stretching rate a can be represented [9] as
Here
.C = y/Iyl, D ( T . <(.w - y, t ) . < ( ~ I, ) ) = (f . < ( . r ,t ) ) ~ e t ( j , ( ( x- y , I ) . ( ( x , t ) ) ,
(33)
and
<
In the two-dimensional case, = ( 0 . 0 . I ) and so w = O. In three dimensions. if the vorticity direction is well-behaved locally in regions of high vorticity, then there is a geometric depletion of nonlinearity. More precisely, if <(.x-. t ) x ((.w - y. I ) vanishes in a quantitatively controlled fashion as .Y -, 0 (for instance, I<(x - y. t ) x <(y. t ) ( klyl). then a can be bounded in terms of less singular integrals (for instance, in terms of velocity instead of vorticity). This observation suggests a correlation between vorticity growth and the geometry of vortex tubes. Such a correlation has been observed in numerical studies and was exploited to prove conditional results regarding blow-up for the Euler equations (9,201, the Navier-Stokes equations ( 191, and the quasi-geostrophic model 1251. The quasi-geostrophic model (9,17,23,24,40] is an example of an active scalar. Active scalars are advection-diffusion evolution equations for scalar quantities advected by an incompressible velocity they create: the velocity is obtained from the scalar using a fixed, timeindependent formula:
<
The Euler equations themselves are an active vector system 1 12,131:
130
with
the Weber formula 14.11
Here P is the Leray-Hodge projector on divergence-free functions, and
is a solution of
The initial data for A is the identity
and the initial datum for u is ire. Thus A ( . I , I ) = lr is the inverse of the Lagrangian path ti H X ( o . I ) . The familiar Cai~chyl'or~nulain this language is
Here
is the solution of
with initial datum ((s.0 ) = cy,(x). One may use a near-identity approach to the incompressible Euler equations: One solves ( 3 6 ) for a short time, as long as VA - I is not too large. Then one stops and resets u = WI A J in place of u o , sets A = x , adjusts the clock, and starts again. This approach allows one to interpret the condition 121 in absence in blow-up in terms of VA: if
then the solution of the Euler equations is smooth [ 121.
Near identiw transj2)rrnurinns f o r the Nuvier-Stokes equutions
6. Diffusive Lagrangian transformations The central object in the Lagrangian description of fluids is the Lagrangian path transformation a H X ( u , t ) ; x = X ( u , t ) represents the position at time t of the fluid particle that started at r = 0 from a. At time t = 0 the transformation is the identity, X ( a , 0 ) = u . An Eulerian-Lagrangian formulation of the Navier-Stokes equations [I41 parallels the active vector formulation of the Euler equations [12]. In order to unify the exposition, we associate the operator
with a given divergence-free velocity u ( x , t ) , Using f , ( u , V), we associate with any divergence-free, time dependent velocity field u a transformation x H A ( x , t ) that obeys
with initial data
Boundary conditions are imposed by considering the displacement vector
that joins the Eulerian position x to the diffusive label A . This is required to vanish at infinity, and one can think of A as being computed by solving
with initial data
If u is the solution of the Euler equation and v = 0 in Equations (46) and (49) then the map A is the inverse of the particle trajectory map t i H .r = X(tr. I ) . In the presence of viscosity this map obeys a diffusive equation. departing thus from its conventional interpretation as inverse of particle trajectories. Nevertheless, continuing the analogy with the inviscid situation, one uses the map .r H A ( x , t ) to pull back the Lagrangian differentiation with respect to particle position and write it in Eulerian coordinates. This Eulerian-Lagrangian derivative is given by
where
that is,
In the case u = 0, the invertibility of VA follows from incompressibility; in the diffusive case, the determinant of VA does not remain identically equal to one as time passes. This imposes a constraint on the time of integration. We consider a small non-dimensional parameter g > 0 and work with the constraint
With this constraint satisfied we can guarantee the invertibility of VA. In order to describe the dynamics and their relationship to the Eulerian dynamics, one needs to consider second derivatives of A. These influence the dynamics because commutators between EulerianLagrangian and Eulerian derivatives do not vanish, in general:
The coefficients Cr;, are given by
Note that
These commutator coefficients are related to the Christoffel coefticients of the us~ialflat connection in IR3 computed using the change of variables tr = A ( x . t ) . With this change of variables, a straight line in x , x(.r) = m s h becomes the label path t r ( . v ) = A(.r(.v),t ) and the geodesic equation d2.r - = 0 becomes % + f!!' @! = 0 with ds d,, I , / d . ~ d.1
+
'
*
The simple geometry of IR3 is hidden behind a complicated transformation. but the transformation is the main object of study. The coefficients C;:, (but not u ) enter the commutation relation between the Eulerian-Lagrangian label derivative and f , , ( u . V):
This commutation relation is the viscous counterpart of the inviscid commutation of time and label derivatives. In the inviscid case, the map A is the main active ingredient in the dynamics. The Weber formula (38) 1431 computes the velocity at time t directly from the gradient of A using a passively advected velocity v (30). In the viscous case, the Weber formula
Neur iclentiQ trut~.!f)~rrtrution.~fOr the Nuvier-Stokes equations
can still be used but v ( x , t ) is no longer passive. Instead of (40) v obeys
that is,
with initial data
Equations (46) and (59), together with the Weber formula (58), are equivalent to the Navier-Stokes equations [ 141: THEOREM 5. Let A , v , und u solve the system (46), (59), und (58). Then u obeys the inc.ornpre.s.sibleNuvier-Stokes ecluufiorls
We describe now kinematic consequences of (46). One starts with a smooth arbitrary incompressible velocity field u , computes A using (46). then computes the inverse matrix Q = ( v A ) (at ~ ' least for a short time), and then Eulerian-Lagrangian derivatives V" and coefficients C. One then can evolve a vector v from an initial datum solving (59). and can compute its Eulerian-Lagrangian curl = V" x v. The resulting equations 114,151 are summarized below.
<
cincl ti.s.soc,iritrwit11 it T H E O R E 6. M Let LI he tin rirhitrclry divergerrc.c~~freejirtrc.tiorl A .\olving (46) crncl rr vec~tor,fieltlv solving (59). Then 111 clc
= (8;A'") ult,
The E~rleriuticurl o f ' w , V x
ti
r)irrl'
(62)
U J , obcy.s the
The Eulerian-Lugrcrngiriri curl of' v,
equatio~i
< = v A x v , obeys
The Eulerian curl of w,V x w ,and Eulerian-Lagrangian curl of v, 1 = vAx v, are related by the formula
The determinant of VA obeys
These considerations apply to arbitrary u without having to impose the equation of state (58). When v = 0, the cotangent equation (63) appears in Hamiltonian formalisms [33,39] for the Euler equation in various gauges. The numerical merits of these have been analyzed critically [4 11. When u is related to w by a filtered Weber formula, a gauge of the cotangent equation appears as a model of formally "averaged" Euler equation [ 3 I] and, in a viscous case, as a model of Reynolds' equation 141. Note that, when v = 0, Equation (65) is just the pure advection equation (a, + u . V ) < = 0. If v > 0, < obeys a linear dissipative equation with coefficients C;:, . Using Schwartz inequalities only, one obtains
where
are squares of Euclidean norms. The evolution of the coefficients C;';, defined in (56) is given by
At time I = 0 the coefficients vanish, C;',(x. 0 ) = 0. Note that the linear equation (64) is identical to the nonlinear vorticity equation (4). I t is the relation (58) that decides whether or not we are solving the Navier-Stokes equation: if u is a solution of the Navier-Stokes equations then (58) means u = Pu1 and consequently V x 11 = V x ui implies that the Eulerian curl of ui is the fluid's vorticity. The formula (66) relating the vorticity to is then a viscous counterpart of the Cauchy formula.
<
T H E O R E 7M. I f 11 solves the Nuvier-Stokes eylrtrtiori. A .sol\~e.s(46). tir~cl1, so1vr.s (59) then the Euleritrn curl o f u , w = V x u , is rc.ltrtecl to tho Elrlericir7-Ltigrtinfin curl of LJ, < = V" x v , b y the Cuuch.yji,r~nulu
Because of the linear algebra identity
one has
I11
two dimensiotls, (70) and (7 I ) become
reflecting the fact that, for v = 0 , (LJ = < in that case. A consequence of (70) or (7 1 ) is the identity
that generalizes the corresponding inviscid identity. Let us consider the expression
dctincd for any pitir ( ( 1 . IM ). where y E R', M E L ; L ( R ~are, ) respectively, a vector and an invertible matrix. This expression, underlying the Cauchy formula. is linear in (1 and quadratic in M ,
The quadratic expression in the right-hand side is detined for any matrix M . I t is easy to check that
and
hold, so C describes an action O~G'L(IW') i n R'. A third property follows from the explicit quadratic expression (75)
Here N is any matrix, and the meaning of C ( q . N ) is given by (75). If we consider. instead of vectors 11 and rrlatrices M , vector valued functions c/(.r.t ) and matrix valued M ( . r . t ) and use the same formula
then the properties (76). ( 7 7 ) ,and (78), as well as ( 7 5 ) ,obviously still hold. Denote by
the total instantaneous energy dissipation rate, and by
the total kinetic energy. We consider functions u(.r, r ) that satisfy
and
For solutions of the Navicr-Stokes equations and for solutions of the molliticd equation ( I I ) , the constants K O , KROdepend only on the initial kinetic energy, viscosity. and time. For- vortex ~ncthods.howcvcr, these constants depend on the cut-off scale 6. Thc displacement Y satisfies certain bounds that follow from the hounds above ancl (49).
THEOKEM 8 . A.s.srtr~le//?tit t/w t3pc.tor ~~trlitrtl,fitr~c~tio~r Y ohc~!.s (49) ,fill. t E I t l . 7"1 (111tl t ( , ,1 1 ) = 0, ,/Or ,sortro 1 1 2 0.A . S , Y U U117~11 T ~ 1 / 7 0 i ~ ~ l o c1((.r. ~ t ,t ~) is ( I ~ l i ~ ~ ~ ~ t ~ ~ o r ~ ~ ~ ~ ~ ~ f i ~ tl~rir.rtrti.sfies r/?e 1)ourrtl.r (80) trrrti (8 I ) or? I ~ tittle P irlrcn-ol 10. T I . Tlrrlz t sccti.yfio.v tlw ir~c~yric~lity
togc.11ier with
Neur idettriry tr~~n.sformutions /i)r the Nuvier-Stokes equution.~
Let us consider the analytic norms
We will use p = I and
13
= 2. One can prove
T H E O R E9M. Let the vector valued function L solve ( 4 9 )j2)r t 3 tl 3 0 with C(., tl ) = 0. Assume rhtlt CI velocity u(.v, t ) , dejned,fi)r t 3 to, to t l , is a divergmce7freefuncrinn that
<
sat is fie.^
Theri there esists
trri
crbsolutr corlstunt ((1 pure number) c such t h t ~ t
Note that (27) reads 1/2
~ ~ L ~ ~ 1~1 { A . A ,
Note also that. if r < h then
and
hold. Combining Theorems 3 or 4 with the preceding result we obtain therefore T HEOK t~ 10. Cor1sidc.r .solutiori.s r!f'tlic~m o l l ~ f i ~rclirrrtion.~ tl ( I I ) o r c!f'thv L70rterrrietliotl ( 1 2 ) tr.ssoc.irrrecl n'itli tr,filtrr (9). A.s.sur?lcthe irrifirrl tlerter trrc. clivc~r,qt.r~c~r~frw rrtid l ~ o l o t to r~ H I , LIO E L?, V x uo = w~ E L ~ Consider . 0 t T , trnd let E rlc.notc. t r bountl f o r tho en.vtrophy on the time inrc~rvtrl10, T I :
< <
sup 0<1<7'
S
Iw(.~,t)(~dx<&.
IR'
Consider crri crrhitrclry trttnsience time 0 < to < T aritl the lerlgth .sctrle
h = $min{t(l;
clV ' E - ~ }
with C I N certain ubsolute constant. Then the velocity u obeys the bound ( 8 7 ) f o runy r < A, with U, given by
and cz = 2 4 2 7 i . Consequently, ffo any tl 3 to, the solution e of Equation ( 4 9 ) with initicrl dcrta e(., t l ) = 0 obeys the bound (88) und, fi)r urbitrury rl < r < A,
These bounds depend on the cut-off scale S of the filter only through the bound on enstrophy I. In particular, if the enstrophy is bounded uniformly for small 6 on a time interval then the above results apply for the Navier-Stokes equations on that time interval. If we measure length in units of and time in units of T then the enstrophy bound for U(X.I) =~ 1 1 ( ~r/ T /) becomes ~ . sup
/IV U ( . ,.s)/ < G L2
()<\
with G the non-dimensional number given by
Note that the number p o f ( 2 6 ) is just 0 = (3'. In these units G is the only solutiondependent parameter that we do not control. In terms of G and in these units, the detinition of h becomes
with arbitrary .so
E
(0, 1 ) . The bound ( 9 2 ) ,with for 0 < 7 < i, r = 4 3 7 , becomes
For fixed .SO and large G we may take 0 < y < 114, we deduce
h -- G-%o
that if we consider 7 = ( 1 - y)i with
- 6 c3G-. 3
U;
Choosing rl = (I
-
IIV~(..
f)
y ) r in ( 9 3 )we deduce from ( 9 3 )
I1
I,
<
Necrr identity tmnsfirmations for the Nuvirr-Stokes equution.~
for to
< tl < t < tl + T T with
Choosing
r2
= (1 - y ) r l we deduce that
holds on the same time interval.
7. Conclusions The viscous Navier-Stokes equations and their approximations can be described using diffusive, near-identity transformations. The velocity is obtained from the near-identity transformation using the Weber formula and a virtual velocity. The vorticity is obtained from the near-identity transformation using the Cauchy formula and a virtual vorticity. The virtual velocity and the virtual vorticity obey diffusive equations which reduce to passive advection formally, if the viscosity is zero. Apart from being proportional to the viscosity, the coefficients o f these diffusion equations involve second derivatives of the near-identity transformation and are related to the Christoffel coefficients. If and when the near-identity transformation departs excessively from the identity. one resets the calculation. Lower bounds on the minimum time between two successive resettings are given in terms o f the maximum enstrophy.
Acknowledgment Research was partially supported by NSF-DMS98026 I I.
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c t t ~ c l~rrcctlrc~~rrct~ic.ctl ~IIc,oI?.o/
i t r c ~ o r t r ~ ~ r r ~ . \ . \,flrtitl,//o\~~. il~/(~ C o m m . Pure Appl. Math. 39
1371 J . NeEas. M . RdiiEka and V . Sverak. On Lc,rccy'.\ .sc~lfl.sittriltrr.\olrrriorr\ (!frlrr Ntnrirr-Stok.s c,qrrtrtiotr.\. Acta Math. 176 ( 1966). 283-294. 1.381 L . Onsager. Srttri.sric.ttl Irydroc/~~rcit~ri(~.s, Nuovo Cimento 6 ( 2 ) ( 1949). 279-287.
Nrcrr identity rruns$~r~nutiot~.sf~)r the Nuvirr-Stokes equutror~.~
141
[391 V.I. Oseledets, On u new way ofnjriting the Nuvier-Stokes rquution.~.Thr Humiltoniun ,formuli.sm,Comrn. Moscow Math. Soc. (1988). Russian Math. Surveys 44 (1989). 210-21 I. [40] S. Resnick. Dynrrnricul prohlerri b~nonlinecrr crdvec.tive purtiul dlffrrentiul equations. Ph.D. Thesis, University of Chicago, Chicago ( 1995). [411 G. Russo and P. Smereka, Irnp~rlseformulutio~~ of the Euler equution.~:Grnrrul propertir.~und numeric.ul n1ethocl.s. J . Fluid Mech. 391 (1999). 189-209. 1421 D. Serre, Lcr c.roi.s.sn~tcede la vorticite rltms I t s ecou1rmmt.spurfirits inc.omprr.s.sihle,s,C.R.A.S. 328 ( 1999), 549-552. [43] J. Serrin. Mothemrrticcrl princ,iplrs of c.Ias.sic~crl~uic1 nlrc.hunic.v. Handbuch der Physik, Vol. 8, S. Flupge and C. Truesdell. eds ( 1959). 125-263. 1441 J. Serrin. T11einitial t ~ ~ lproblem ~te for rhr Notier- stoke.^ rq~tution.~, Non-Linear Problerns. R.E. Langer, ed., Univ. Wisconsin Press. Madison (1963). 69-98. (451 J.T. Stuart. Nonlinecrr E ~ l c rpc~rtirrldiflrrerlticrl equutions: Singu1rrritie.s tn their .solurion, Proc. Symp. in Honor 0fC.C. Lin. D.J. Benney. C. Yuan and F.H. Shu, eds, World Scientitic, Singapore (1987), 81-95. 1461 R. Temam, Some rlevr1o1ptn~nt.s or1 Nuvier-Stokes rqurrtiotr.~in the .s~c.onrlholf'ofthr 20th c.enturv. Developpement des Mathernatiques au coura de la seconde moitie du XXeme siecle, J.P. Pier, ed., Birkhiiuser, Basel. (471 R. Ternani. Nuvier-Stokes Equtrtiorts. 3rd Edition. North-Holland. Arnaterdam (1984). 1481 T.-P. Tsni. On Lrrtry'.~.self-,sir~tilar sn1ution.s of t/t(,Navii,r-Stok(,.s rqrrarion.~.s~rti.sfj.in,yk)c.erl ~nrrg,vp.stimuti,.c, Arch. Rational Mech. Anal. 143 ( 1998). 29-5 I .
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CHAPTER 5
Planar Navier-Stokes Equations: Vorticity Approach Matania Ben-Artzi lristit~rreof'Mir!hr~iitrtic..s, Hrbrr~i,Urlivrr\ity, J ~ r ~ r . s i r l91904. r ~ r ~ I.srirc~l E-rriiril: I I ~ / ~ ~ I ~ ~ : ~ ~ ~ I ~ I ~ ~ I / I . / I L ~ I . ~ ~ ~ ~ . I ~
C0tlfetlf.~
I . Introduction . . . . . . . . . . . . . . 2. The caw of'smooth initial data . . . . 3. Some elilllate\ lor s~nooth\elutions 1. F.xlen\ion 01' the \olulio~~ operator . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14') . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 5. Me:i\ures ;I\ 1ntt1:11tlala . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.58 h. Asynlplotic hchavior k)r I:trgc linlc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I0 l 7. Co~~cluding rcnl;irk\ : I I I ~ope11prohlcnl\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Kctcrcncch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Oh
HANDBOOK OF MATHEMATICAL FLUID DYNAMICS. VOLUME I I Edited by S.J. Friedlander and D. Serre O 2003 Elsevier Science B.V. All right5 reherved
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1. Introduction In this survey we review the existence, uniqueness, and regularity theory of solutions to the Navier-Stokes equations when they are formulated in "vorticity form". We also discuss the large-time asymptotic behavior of solutions for sufficiently small initial data. In fact, the three-dimensional case has hardly been studied (we refer to the article by P. Constantin in this book), and we shall therefore concentrate on the two-dimensional case. We recall the basic equations [26,1 I]. Throughout the paper, we use bold-face notation for vectors and vector-functions (in IR2 or IR3). Their components are labeled as w = (w', . . . w") (11 = 2, 3) and jw12 = C y = l (wi12. The scalar product is denoted by (I' . b'. If a! E Z : is a multi-index, we let V u = a .b= a';: and loll = EL1a'. Denoting the velocity by u(x, t ) , the pressure by p(x, I ) , and the (constant) coefficient of viscosity by v ( v > O), the Navier-Stokes equations in a domain R R" are
.
The equations are supplemented by an initial condition
and, if R # R". by boundary conditions (such as u = 0, the "no-slip" condition) on the boundary ifR, for all r 3 0. If 52 = R". growth (or, rather, decay) condition must be imposed on u at inf nity. In the case that uo E ~ ' ( 5 2(or ) uo E H '(a)). the well-posedness of the problem with suitably defined weak solutions (strong for H 1 ( R ) )is well known since the pioneering work of Leray 1271 (see also 1291 for the case of the full plane). The strong well-posedness is only local in time if n = 3. We refer to [ 12.25.361 for full accounts of this theory. In what concerns well-posedness of the system ( I . I ) beyond the L'-framework, we refer to 1241 and references therein. as well as earlier works by Kato and Ponce using commutator estimates in various Sobolev spaces [ 19,2 1-23,341. Our interest here is to study well-posedness of the flow. in "rough" spaces, by using the vorticity forniulation. We recall this formulation in the general three-dimensional case. Taking the curl of the first equation in (1.1), and denoting by w = V x u the vorticity, we get
The connection between u and w = V x u is given by the "vector potential" A,
Under mild growth assumptions, one can take
where G is the fundamental solution of A . Note that the fact that V . o = 0 for all t 3 0 (a "structural assumption" that must be verified for any solution of (1.3)) implies that V . A = 0; hence, indeed, from (1.4).
Remark that when u is given by (1.4), then automatically V . u = 0, so that (1.3) and (1.4) is equivalent to ( I . I), at least in the case of sufficiently regular solutions. The system is supplemented by the initial condition
From the point of view of hydrodynamical phenomena, an interesting case is that of the evolution of vorticity (and its associated velocity field) when it is initially given by isolated vortices, vortex filaments or sheets. Since, in the "zero viscosity limit" (i.e., v = 0, leading to the Euler equations) the circulation is preserved (Kelvin's theorem), the use of vorticity in numerical methods has become very popular. In particular, in "vortex methods" [ 131, even smooth initial data are replaced by a distribution of singular "vertical objects". Mathematically speaking. we need to study the system ( 1.3) and (1.4), (1.7). when wo(x) is a measure. This will be the main focus of this article. Indeed, since very little is known in the three-di~nensionalcase. we shall deal here with the two-dimensional situation. We refer to the article by P. Constantin i n this volume. concerning approximate solutions to the vorticity equation i n the three-dimensioniil case. Also for simplicity. we have avoided adding a source term (external force) in the case of Equations ( I. I ) or ( 1.3). In fact. for issues considered here such as existence, uniqueness, and regularity, the results can be extended to the non-ho~nogeneouscasein a rather standard way. When 52 # R" the system ( 1.3) and ( 1.4) rnust be supplemented with boundary conditions on o ( x . t ) . x E if52, t 3 0 . The most comlnon physically plausible boundary conditions are stated in terms of u (such as the "no-slip" condition). Casting these conditions in terms of o is quite involved and. in fact, has hardly been treated in theoretical studies. On the other hand, in nurnerical works, the methods used for the implementation of vorticity boundary conditions (or, in the hydrodynamical language, "generation of vorticity") are quite diverse. Some of them could perhaps prove instrumental in the rigorous treatment of the problem. However. i n this survey we shall not touch upon this topic, and refer the reader to the book [ 131 and to 121 for more details. In order to avoid the boundary problem, we shall concentrate in this survey on the case o f the full plane. 52 = R'. The velocity u(x. t ) is obtained from ( 1.6). In fact, in the two-dimensional case we can easily obtain a convolution integral connecting u t o w as follows. The velocity field is now two-dimensional u(x. t ) = (rr (a'.a'. t ) , u2(.r'. .w2.t ) ) and the vorticity is given by w(x, t ) = w(x, t ) k ,
'
Pkancrr Nuvier- stoke.^ equations: Vorriciry approach
147
Furthermore, the term (o. V)u vanishes identically, so that Equation (1.3) reduces to a (nonlinear) convection-diffusion equation for the scalar vorticity w ,
Carrying out the operations in (1.4) and (1.5) we obtain
where the "Biot-Savart" kernel K is given by
Note that V . K = 0, implying (by ( I . lo)) the incompressibility condition V . u = 0. In what follows we shall study the well-posedness of (1.9) in various functional spaces X. This means (at the least) that, given the initial vorticity w ~the , solution evolves along a continuous trajectory in X . The paper is organized as follows. In Section 2 we recall the derivation of solutions for smooth initial data. As is appropriate for parabolic equations, the "maximum principle" plays a fundamental role. In Section 3 we derive space-time estimates for smooth solutions. They are the main tools used in the extension of the solution operator to initial vorticities in L ' (R'), as is done in Section 4. In Section 5 we discuss the further extension to measure-valued initial data. We shall see that uniqueness is still an open problem (for measures with large atomic part). In Section 6 we discuss the asymptotic behavior of the vorticity for large time. We conclude in Section 7 with remarks concerning various related open problems. Notctriotl. The norm in L"(R"), 1 6 p < co,is denoted by
with the usual (ess-sup) modification for p = +a. The space W.'.ll(IW1')(.r positive integer) is the LI' Sobolev space, normed by
c
If X is a Banach space, normed by 11 . Ijx, and I R+ is a finite or infinite interval, we define the following spaces of X-valued functions f' : I + X . C ( I , X ) Continuous functions (not necessarily bounded), topologized by uniform convergence over compact subintervals of I . LP(I, X) Strongly measurable functions, normed by 11 f'(t)l]$ dt)Ii", 1 P < GO, with the usual modification for p = m.
(h
<
L ; ~ , ( IX, ) Strongly measurable functions such that $f E LI'(1, X ) for all $ E C ( Y ( 1 ) . If X I , Xzare Banach spaces, then X = X I fl X2 i~ normed by
11. ]Ix
= 11
./Ix,
+ 11 . IIx,.
2. The case of smooth initial data Our first theorern is a theorem of McGtath (3I]. It imposes rather strong regularity assumptions on r q ) ( x ) Set, . for 0 < X < 1,
c~(Ps') = 1 j'
t C(PS')n L ~ ( I W ' )f, is uniformly (I-) Hijlder continuous,
/ . f i x ) - f i y ) ) 6 b i , f ~x ylA, X, y E R 2 } ,
ckA(IWZ) = {,fE c~(Iw'), Vaf.E cA(IR2), I C YI < k ) . THEOREM 2.1 (McGrath). A.FSLIN?C tkut,fi)rsome 0 < h < 1 , wr,(x) E L 1(JR2)n c2.'.(IW2). Tllen there exists 0 .solution to ( 1 .9) ~lrld( I . 10) .such thut (a) The solutior~is ~ ~ I L I S Sull ~ Cderivutivt'~ LI~; uppeuritig in ( 1.8) urr corltitzuous it1 Kt2 x (0, 00). (b) w ( x . I ) . U(X, r ) L ~ W ~ ~ o ~ ~ r i und t r ~ irnifOr~trlv ou.~ b o u i ~ d ~it1dIR2 x 10, m). ( c ) ~ ( x.) .E L W ( [ O m). . L'(R')). ( d ) f i r . every T > 0 , lu(x, t ) ) + 0
sup 0<,<1,
1x1
R
+w
.t?
PROOF(Outlinr). Fix T > 0 and let Q T = IRZ x 10, T I ,
x I = C ~ Q Tn) L " " ( Q T )n ~ ~ ( [T Io. L', (IW')). where Il(o(x,I)IIx.l = llwll/.%cv,1
+ s u p o ~ , ~IIQ(..7 .
I.
Let Ho
c X I . be the ball
F o r t e B ~ , o n e d e t i n e s t h e m a p A ~ ( = by v m e a n s o f ( l . l O ) , i . e . , v = K * ( , O < r < T. I n particular, it is easily seen that v E C(Q.1.)n L m ( Q T )and by standard facts concerning linear parabolic equations the equation
has a unique classical solution in Q r , and in particular H E X T . We let A : Bo + X r be the map H = A( (where v = A'$' in (2.2)). Using the maximum principle and its dual statement
Planar Nutrier-Stokes t.quution.s: Vorticify upprouch
149
in L 1 (note that V . A ] ( = O), it follows that A Bo Bo. Now the assumptions on elements of X T imply that { v = A l ( , 6 E Bo} is uniformly bounded and equicontinuous in Q T . The regularity hypothesis on wo (and its decay at infinity) imply, therefore, that A Bo, the set of all solutions of (2.2) with v E A I Bo, is uniformly bounded and equicontinuous (in fact, A Bo w ' - ~ ( Q ~ )Furthermore, ). the elements of A Bo vanish uniformly as 1x1 + oo,0 r < T. Thus A Bo is compactly imbedded in Bo and, since A is continuous, the Schauder fixed point theorem yields o E Bo such that w = Am. This w is a solution to (1.8) with 17 u = A1 o. The uniqueness is shown by a similar argument.
'
<
R E M A R K2.2. The maximum principle can be applied to Equation (2.2) and its dual (since V . v = 0). We can therefore conclude (for the solution of (1.9)) that Ilw(., t)l]1 < Ilw~ll1 and Ilo(., t)lloo < [lwolloo, t 3 0, and by interpolation,
Observe that the interpolation argument used above is based on the linear theory. Indeed, once the solution to ( 1.9) is obtained, the velocity field u(x, t ) is "frozen" and Equation (1.9) is treated as a linear convection-diffusion equation. The Ll' estimate (2.3) is then obtained for all solutions of Equation (1.9). including the original vorticity w . A similar reasoning is applied to justify the duality argument, and will be used also in the sequel (see the proofs of Equation (3.4) and Theorem 6. I ). If we limit further wo E C;;X'(IR2),the solution w E cM(lR2 x R+) can be obtained as a limit of a sequence of solutions to linear convection-diffusion equations. We refer to I I ] for details. In fact. certain basic estimates are easily derived in this case and then extended to more general spaces. We designate by S :I;(c
(R') + cW(R2 x
IW+ )
(2.4)
the solution operator to ( 1.9) and ( 1.10). w = Soo. The corresponding velocity field is given by ( 1.10). and we denote it by
(when there is no risk of confusion we shall write w ( t ) instead of o(.. t)).
3. Some estimates for smooth solutions I t is convenient to establish some of the basic estimates for the solution operators S . U, assuming that wo E C ~ ( I W ~ ) . Multiplying (1.9) by w and integrating over R' we obtain
since lRl w(u . Vjwdx = 0 by V . u = 0. Recall that, by the Nash inequality [14,8], if a smooth decaying function in R '. then, for some q > 0,
4 is
11411; < v-I 11411l l l ~ 4 l l 2 . Using this inequality in (3. I) and noting (2.3) with p = I we get,
hence
By duality (using again V . u = 0).
s o that 0
t
To cstimate IJu(.,t)JI,.
6
1
-
I
Ilwol1.
note that IK(y)l 6 ( 2 x ) F 1 l y l ' .so that,
Note that in view o f the Hardy-Littlewood-Sobolev inequality 128, Chapter 41 o r the Fact that V K is a Calderon-Zygmund kernel, we also have
with C = C,,. We shall now refine these estimates by looking more closely at Equation (1.9). Using the heat kernel
the solution w ( x . t ) can be written as
c
Our first aim is to derive uniform estimates for solutions having initial data wo E K c?(!R2), where K is precornpact irl the L (R') topology. We use the following notational convention. The constant C > 0 stands for a generic positive constant and 6 ( t ) stands for a rnonotone nondecreasing, unifor'nily bounded, generic function defined for I 3 0, such that lim,,06(t) = 0. Both C and 6 ( t ) may depend on various parameters ( p , v . . . .) but not on the solution functions. However, they [nay depend on certain subsets of initial data. We sometimes indicate specific dependencies by adding parameters, e.g., C ( p )or S ( r ; K ) . Since ~ ' - l i l ' ~ , (t ). *, is a boun
'
<
*
<
( S ( t ;K ) depends on 11, 1 1 ) Next we note that
' we Inserting (3.6). ( 3 . 8 ) .:ind ( 3 . 9 ) in ( 3 . 7 ) ancl using the Young and Hiilder inequ*,I I '111es. get
where I/(/
+ I / r = I //I + I .
1/17
+ I//-
= 313. 1 < 1 ) < 7. Setting
and noting that since cy) is slnooth, M l ) ( t )is continuous, M I , ( 0 )= 0. we infcl- from (3.10).
hence M , , ( t )
< A(1; K ) ( I < 1) < 2) and, interpolating with (3.4).we have
(The case p = w is obtained by duality as in (3.4).)
Finally, we note that the estimate (3.5) can be strengthened to yield
) Indeed, this follows by replacing in (K c?(IR2), precompact in the L ' ( R ~ topology). (3.5) the term ( % ) ' I 2 by (+)'I2 2nLi(t.K ) and using (3.1 1). R E M A R K3.1. Note the similarity of the estimates (3.8) for the heat (linear) equation and (3.11) for the vorticity (nonlinear) equation. In what concerns the L ' - La decay estimate, we have (3.4), where q > O is the "best constant" in the Nash inequality. As pointed out in [8], r,~ 3.67n, whereas the corresponding estimate for the heat kernel is llG,(., t)ll, = ( 4 ~ r v t ) - I .The estimate (3.4) was derived in [1,20] and was improved by Carlen and Loss [9], replacing by 4rr. Thus, quite surprisingly, in spite of the nonlinearity, the Lm estimate for a(., t ) (in terms of Ilw~ll1 ) is identical to that of the linear heat solution. Observe, however, that rndiul solutions of (1.8) are also solutions of the heat equation, since the nonlinear term vanishes identically. It follows also 19, Theorem 21 that q in (3.5) can be replaced by 4 n .
4. Extension of the solution operator We shall now study the extension of the solution operator S, U (see (2.4) and (2.5)) to initial data i n L ' (iR2). Our goal is to show that the system ( I .9)and ( I. 10) is well-posed in L'(R?). As in the case of the heat equation. the solution "regulari~es"for positive time. Thus, estimates over time intervals Ic. oo),c > 0. are easy to obtain, using data at t = c . I t is convenient to introduce an "intermediate" space
(where c ~ ( R ' ) consists of continuous functions tending to zero at infinity. normed by 11 . I),). The space Y has actually been used in the study of vorticity by Marchioro and Pulvirenti 1301 in their treatment of "diffusive vorticies" (approximation by finite-dimensional diffusion processes). In addition to the interest in Y as a "persistence" space for vorticity, some basic estimates in this space serve in the study of "zero viscosity" limit. being independent of v > O [ I I. I t is easy to see that the convolution operator K* : Y + co(IR2) is bounded. We have the following lemma. L E M M A4. I . (a) (Existence). The opertrtors S, U con he c.ston~letlcontinuoii.sly trs
Illdeed, the r t l ~ ~ V p sS uncl VU curl be extended oontinuou.sl~,:us
vs: Y vu : Y
-
c(IW+,Y ) n L/:,(PS+. Y ) , c (R+. co(R')) n L [ (R+, ~ c()(IR~))
< <
for U I I J I p 2. Further-tnorr, the functionr o = Swo, u = Uwo = K * Swo xive a M ~ C Usollitiori ~ to ( l .Y) ~ n (1.10). d ( b ) (Uniqueness). Let H ( x , t ) , v(x, t ) = K c H he u wruk sofutiot~in IW? x R+,c?f
~cherc.. ,fi)r. some I < p < 2,
-
i);vu.~: Y C(!R+,Y ) . if vffu : Y -+ C(R+. co(IT?'))
P ~ o o r(Ourlitrc~: . .\PO I I I for tli)tcril.\). Diffcrentiatillg ( 3 . 7 )we obtain
In view of (2.3) and the boundedness of K*. we have
s o that. w i n g ( 3 . 9 )in (4.4) and denoting N ( t ) = ((V(o(.,t)ll,,
A similar inequality is obtained for IIVw(., t ) l l ~ We . deduce
If O ( x , t ) is another solution to ( I .9), H ( x , 0) = & ( x ) E C?(iR2), a similar derivation yields
The conclusion of the proof of (a) is now standard. For w() E Y we take a sequence {w:/');"=,& c ~ ( w ' ) converging to in Y , and using (4.8) and (4.9) we obtain o ( t )=
,,
lim ~ o ( ( / ' ( tu)(,t ) = lim,,,, Uw,,C j ) ( I ) . To prove the uniqueness assertion, we note that v = K s H E co(IRZ),so that (3.7) holds, with (0,oo,u replaced by 0 , 00, v. We can then derive an estimate analogous t o (4.8). Finally, the regularity claim follows tmm standard arguments concerning parabolic equations 1351. We may now proceed t o (he main rcs~iltof'this section. THF.OK~:M 4.2. ( a ) (Existence). Tllc o/)c,t-c~tot:s S. U
c ~ r t rhc
c>.~trt~(/rrl c~or~tir~~rorr.v/~~ cr,v
u ~ i v (o1 w c ~ k.volutiot~to ( I .9). Furl/7ort11i)rt~, 1 1 7 ~c~.vti111~~1~~,s ~ ( 3 . 4 )(rrld (3.5) ore \-trlitl. (b) (Uniqueness). Let H(x, t ) , v(x, t ) = K * H /XJ LI C I ~ P ~ I X~01irtiot1 . 1 0 ( 4 . 3 )it1 R' x R+. Assume. thor U I I ~OJ*
n7ell H ( . , t ) = Swo(t),fi)rull 0 6 t w. ( c ) (Regularity). For evrrv wo E L'(TW'), tIru firncrions w ( x . t ) = Swo(t), u(x, t ) = Uoy,(t)crre in crn(IR2 x W + ) utld Ey~r(rtiot1( 1 . 9 )i.s .sari.\;ji:fdin t l c.lrrs.~icnl ~ ~ ser7.v~.
Furthermore, fi)r e1'ei-y integer k artcl double-index ct, the mups
ore contirzunus.
PROOF. Let K g C ~ ( I W n ? )L'(IW2) be precompact (in the L' topology). We first show that the family of maps
is equjcontinuous. Indeed, it follows from (3.7). using (3.9) (with r = I ) and (2.3), (3.12), that
( ( s w (-~w) ( (,<, 116~* WI
-~JOIII
+S(t; K ) ,
(4.13)
which converges t c ~0 (as r + 0) uniforrrlly i n (q)r K . Next, it follows from (3.11) and Lemma 4. I(c) that, for any B > 0. ct = (a!',a 2 ) , sup 6 < I <W
sup ( ( j v a ~ w o ( 0 I I} < G o . c~~I€K
which implies, by (3.5)and ( 1.9). sup
sup
)
( ( J ~ w o ( t ) (-t( ,I ( i ) , ~ 0 4 , y , ( f )< ( ) m. ,
c < I
The estimates (4.13)-(4. IS) imply the equicontinuity o f (4.12) in L ' (R'). Now let { ~ L I : ~ ' ( x )C) ~c?(R2) , converge to wo E L'(IR2) in L ' . Taking K = { w ~ ' ( x ) ,}the ~ ,foregoing argument yields the equicontinuity (in L ) of the tri~jectories ( 1 ) ( 1 ~ ) ( I ) = ~ a , : ) ) l ' ( t In ) . what follows we prove thc uniform convergence of these trajecto-
'
ries. Writing u ( " ) ( t )= ~ o , : : ' ( t ) we have,
Let p t ( 1.2). Clearly,
~\fl(..t)\< ~ /~, l - ~ + ~ ! / ~ \ l < o ; ; ) - w ; ; ' ) \ \ ~ . I n view of (3.12) we obtain in 1 2 , for 0 < .F
< t,
1
I [ u ( ~ ~ 's)(w(l1](.% (., $1- ( ~ ( ~ ' l ~.s)) ( . , 1)
< I ~ U ~ ' ~7)' /(I r n~ ,1 (w?.,
.$I - d m ) ( . , 7))
/I/)
,< 6 ( t : k') . s-'l2II (wol)(..S ) - o("')(., 7)) Il, and using (3.6), (3.1 I ) , we have in 13,
1 ( u ( ~ ' ) ( , s-)
/I/,
U(~~~)(.S))(O(~~~~(.~)
< CIIU(~~+.V) - U ( ~ ~ ~/Iq) ( . S ) < &(I.K )
spl!'
I[ O(~"
/I2
- (o0'"(.s) Ill,.
Inserting (4.17)-(4.19) in (4.16) we have
1 < ~ r - ~ + ' ! / ) l (,>; l
~ ~ ( ~ ~-~ ( ,~P '1 '( ( ff ) )/,
+6
; k'(
-.
~
)
..
l
~
Denoting N ( r ) = \upo<,<, s '-li/'(lw(")(r) \
Nit)
< ~llw,!;'
which implies, for 0 < t
I/w"''(r)
W ( ~ I1([)
i
-
-
~
-
< r* = t * ( K ) ,
~ ' " ~ ~ (
- "011
( ~( ~~' ' ~) ~ ( . ~ ) / l ~ ~ d . ~ .(3.20)
(o'"')(r)III,, (4.20)can be rewritten as
1
(r~ri
-
1,
- w,(,""Il, + 8 ( r ; K ) N ( t ) ,
Turning back to (4.16) we now obtain,
1
- o+',"l
1
I
(t) I
.t-'+li1'.
I < p < 2.
(4.2 1 )
Take p = 413 and use (3.6),(3. l I), (4.2 1 ). to estimate
Inserting these inequalities i n (1.22) yields, for 0 < t < I * ,
We can now conclude the proof of the theorem. In view of (4.14) and (4.15), (4.23) and Lemma 4.1, tho sequence ( c l ' " ) ( r ) = ~(ol,"'(t)converges in c(E+, I,' (!R2)) to a funcL'(IR2)).Ler~lrlla4.1 now implies that the sequence converges also in tion w ( t ) E c@+, C(R+. Y ) (in fact, with all derivatives), hence the regularity claim. To establish the uniqueness claim, we note first that, in view of the equicontinuity (4.12), the estirni~te(3.1 I ) extends to
1,'(IR1) is precompact. Assume first that Ho E L'"(R'). Then H(x. I ) satisties when K (3.7) (with to, (00. u replaced respectively by H . 41, v ) , and repeating the argument leading up to (3. I I ) we get
-
Setting N ( . , t ) = SHo(r), we get, as in (4.20), for I < p < 2.
< <
which implies that H ( . , t ) = H ( . , t ) , 0 t t * , for some t* > 0. One can then proceed stepwise in time to obtain H(., t ) = 0 ( . . I ) , t > 0. This proves uniqueness if No E L' (R') n LW(iR2).Dropping the assumption Ho E L ~ ( R ' ) ,wc still have by hypothesis. for any s > 0, that H ( - , s ) E fdW(LR2). Invoking the foregoing argument (with H(., s ) as initial data). we obtain
158
M. Ben-Arrzi
Also, since 6(., t ) E c@+, L'(IR2)), the set K = { 8 ( . ,s), 0 < s 6 1) compact. Hence, combining (4.24) and (4.27),
c L 1(IR2) is pre-
Letting s + 0 in (4.28) we have, with t > 0,
and, in particular, we obtain (4.25) and (4.26). We can now repeat the first part of the proof to obtain 6(., t ) = 8(., t ) = S60(t). t > 0. REMARK 4.3. The existence of a solution to the vorticity equations (1.9)-(1. lo), when wo E L ' ( I W ~was ) , first proved by Giga, Miyakawa, and Osada [ 161, using a delicate esti-
mate for Green's function of a perturbed heat equation. The constants appearing in their treatment are unspecified and depend nonlinearly on Ilw~lll,in contrast to the linear dependence in (3.4) and (3.5). The proof given here follows [ I ] and the uniqueness part relies also on 161. Observe that only the classical estimates for the heat kernel have been used. A similar approach has been used by Kato 1201, using also the classical heat kernel but different fi~nctionalspaces. Kato derives (3.4), but not (3.5), (3.1 I ), and (3.12). which are essential in the uniqueness proof here. We refer to the implications o f this uniqueness proof to nonlinear parabolic equations in Remark 5.3 below.
5. Measures as initial data Let M be the Banach space of finite (signed) measures on EX2. normed by total variation 11 . [IM, so that naturally L'(R') M. Following Kato 1201, we shall now extend the solution operators S, U to M. The estimates obtained in Theorem 4.2 and the weak density of L ' in M lead to a straightforward result concerning existence and regularity. However. uniqueness remains partially open. For simplicity we henceforth assume v = 1 and write C = G I for the heat kernel. A measure q E M can be decomposed as
where r/,. is continuous and qcl is the atomic part of q . The decomposition is "orthogonal",
In what follows we write h = ( h i , bz, . . .) and
We can now state the extension theorern for initial data in M.
+
T H E O R E5M . 1 . Let wo = ( w ~ ) , . (wo),, r M . Then the system (1.9) and (1.10) hus solution a)(., t ) E C(R+; n w ' . ),~u(.. t ) = K s w ( . , f ) . O U C ~thut ( a ) w ( . ,t ) + w o U S t -+ 0, it1 the weak" ropolog-y t?j'M. (b) For every 1 p m, IIlo(., t ) Ill, is u dec.r<~u.~ingJunction o j ' f E R+, und
(1
wlsl
< <
sup rl-I"J~l(o(..r)ll,' C W . 0<1<00
(c) Lor (too),, = h = ( h l .h.. . . .). For ctrc.l~413 < p < 2 thrrc. fire c.ot1.sr(irlt.sS I , , .\uc.h thtit it I l hll,, < S,, rhc11
F/,
r 0,
This c.ondition (tirld ( a ) ) clc~tc~rftiir~es utiiyue!\' tile .sollrriotr tr). 111 /~arric.i4/~11; ~f ( c l y ) ) , , = 0 thetl the c,ottdition ( 5 . 5 ) (fijr c1t1y.sitiS~lc~ 4/3 < 11 < 2 ) r1~t~rtnit~r.s ro iwiyirely. R E M A R K5.2. Since for r > 0. o ) ( . ,t ) L ' ( R ' ) . Theoreni 4.2 can be applied to t 3 t . Thu5 w ( x . 0 ~ " j ( l W ' x R+)and c\tirnate$ like (5.4) follow for r 3 r ( w e (3.1 1 ) ) and need t o be establ~shedonly in (0, r ) . PROOF. Using a standard mollitier, we constroct ;I sequence { ~ ) : / ' ] ; C I _ 5 ~ L'(R'). (1) c,; Iln)()llm.and ( t ~ ( ; 4 uo (in the weak* topology 01' M). Let (1)"' be the soIIt,),, 11 I
<
lution given by Theorem 4.2, wii)(., 0) = w : , ' ) . Using the estimates (2.3). (3.4) we see that ~ ~that, ~ for . any fixed r > 0. there exists a subsequence. which we relabel as { w ' ~ 'such w c j ) ( . r. ) converges to a function ( o ( . ,5 ) in Y (in fact, with all derivatives. see Lemma 4. I ). In particular, we have also co("u'J' 4 tou, u = K * (0. I t is easy to verify that ((0,u ) constitutes a solution to ( I .Y) and ( I . 10) and satisties 12.3) (with p = I ) and (3.4).with Ilooll I replaced by I l w ~ l lThe ~ . estimate (5.4) follows by interpolation. To prove (a), it is clearly sufficient to show (compare (4.12)) that the family
is equicontinuous in the weak* topology of ,U. Taking $ E C ~ ( I W * ) ,
((,. ,) is the (jM. C ( ~ ( R ' )pairing). ) Usirig the estimate
(see the derivation preceding (4.23)) we obtain the equicontinuity of (5.7)from the uniform integrability of 3, ( w ( j ) ( . . t ) , (1) in [0, T I . Finally. it remains to prove (5.5) and the uniqueness part (c). We use the integral equation (3.7). Note first that the heat kernel G satisfies, for any 7 E M,
and. with the atomic part h = q,,.
With (.,I > 0 t o be determined, take 8,) = t:,,/(2(.,,).The hypothesis I(h(l,,iA,, then irnplies, by (5.10).that there exists T > 0 such that for the sequence
(4,''-
8
sup I ' - ' ' ~ ~ ~ ~ G ( ~ , ~<) -,* ~ ~ ~ ) " ~ ~ , ~ 2
O
where we have used the fact that w:/' are obtained from r q , by mollification (which commutes with G*). Arguing as in (3. lo), with 6 ( 1 ; K ) replaced by a function h ( t ) p , we obtain a solution to the integral equation (3.7)for 0 t T and 4/3 / I -= 2, if F,, is suf-
< <
<
<
ficiently small. Since (0:;' is smooth, the solution is necessarily w ( ~ ) (I .) , as constructed above. As in the derivation of (3.1 I ) , we have. for some p' < s,
hence also
where w ( . , r ) is the solution constructed in the first part of the proof. Now if 8(., I ) is another holution of Equation (3.7),satisfying (5.12). we may proceed as in the uniqueness part i n the proof of Theorem 4.2 (the argument following (4.26)) to obtain @(.,I ) = co(., t )
for t
< T,
Plunur Nrrvi~r-Sfokrseqlrutions: Vorticify upprouch
161
provided that E , satisfies
We can then prove the identity 0 = w for all time by proceeding stepwise. Finally, (5.5) follows readily from (5.12). R E M A R 5.3. K The smallness condition (5.6) has been extensively used in proving uniqueness for solutions of nonlinear parabolic equations (see "note added in proof" in [6]), and is commonly referred to as the "Kato-Fujita" condition. In Theorem 4.2 (i.e., for initial data in L ' (JR2))we have avoided it by assuming that the solution 0 is in c(&,L ' (JR2))n C(JR+. L"(IW~)),thus obtaining (4.28).The requirement 8 ( . , t ) E Lm fort > 0 can be considerably relaxed, still avoiding a "Kato-Fujita" condition. We refer to [4,7] where similar uniqueness arguments have been used in the study of nonlinear parabolic equations. When the atomic part of the initial measure is not small, a suitable uniqueness condition is still unknown. R E M A R5.4. K We refer the reader to 151 where the results of this section are extended to well-posedness for initial data in functional spaces beyond M (IR2).In fact, these functional spaces are detined by suitable restrictions of the action of the heat kernel on the initial data.
6. Asymptotic behavior for large time We assume now that wo (1.10) satisfies
E
L'(R'). Then, for any t > 0, the solution w(x, t ) to ( I .9) and
Thus, in general, there is no decay (for large time) in L ' norm. On the other hand, by (3.4). the vorticity decays in all L'' norms, 11 E (0. co].As mentioned earlier (see Remark 3.1), the constant q in (3.4)(and the subsequent inequalities) can be replaced by 4rr, thus equalizing the L ' - L,' estimates for vorticity with those of the heat equation. The proof of this improvement (see (9, Theorem 51) is obtained by using a logarithmic Sobolev inequality instead of the Nash inequality used in (3.2). As in the case of uniqueness arguments (see Remark 5.3), the methods used in the study of the vorticity equation (1.9) can be successfully applied in the study of various types of nonlinear parabolic equations (and vice versa). This is certainly true in what concerns large-time decay estimates. The study of such estimates for Navier-Stokes equations is well established ( I 10,15,33] and references there). We refer to [3,9] for L' decay estimates of solutions to "viscous" Hamilton-Jacobi and conservation equations. In particular, in the
two-dimensional case, the fact that Equation (1.9) is scalar renders the vorticity a convenient object of study. A solution to the heat equation in R'l decays in L' norm if the integral of the initial value vanishes. It is remarkable that a similar fact holds for vorticity in the two-dimensional case.
THEOREM 6.1 . Cotzsidrr the .\\t.\terti (1.9) and ( 1.10) und o.s.sumr [hut wo c ~ ) ( dx x) = 0. Then (a) lirn,-, Ilw(., t)ll I = 0. (b) S L L ~ ~ M JinWud~iitio~l, , f h f ~ f(14)is L ' O W I I ) L I C I~ ~ ~u p p ~ r f eThen d.
E
L ' (IR2) and
JR2
Iim t l ' / " I I ~ o ( .t)II , = 0. I+CX)
1)
1) E [ I . ~ 0 1
PROOF.We refer to [9,Theorem 41 for a proof of ( a ) . The proof of (b) follows [ 15, Theorem 2.41. Replacing ( 1.10) by
we get uo E 1,'. From the L' theory of the Nilvier-Stokes equations it follows that Vu(x, t ) E L'(R' x R+),hence so is ro(x,r ) . In view of (3.1), the function Ilr,,(.. 1)II2 is decreasing in t . so
which proves (6.1 ) with /J = 2. To prove the cast. 1) = I . use the integral equation (3.7). Since the decay is known for the heat equation, we need only estitnatc the second term in the RHS of(3.7),in the L ' ( R ' ) norm. Denoting F ( S ) = .sli'/~w(.. ,v)I12 we have
In view of (3.9) (with r = I ), we conclude
and the RHS tends to 0 as t + cm by the Lebesgue dominated convergence theorem. By inrcrpolation we get (6.1 ) for 1 6 11 < 2 . The conclusion for 2 < p 6 a,follows by duality, as i n the proof of I I Equation (3.47)1. 17
.
R E M A R 6.2. K The conclusion in part ( b j of the theorem can be considerably improved. In fact, under the same assumptions (in fact, only exponential decay of q,is required), we have sup l'/211w(.,t)11, < 00 o < t
and combining this with (3.4) we obtain sup t3'2)(w(.qt ) 0 < 1 <ns
llm '00.
These estimates are identical to those obtained for the heat equation. We refer to 19, Theorem 4 ) for details and sharp constants. The asymptotic behavior of solutions to the vorticity equation (1.9) can be studied in detail in tenns of "scaling variables" [ 10,17, IS]. They are defined by
4 = ( 1 + t ) - l / 2 x,
t = ln(l
+t).
Defining new functions v. H by
and setting for sinlplicity
11 =
1. Equation ( I .9) is tramformed into
(spatial derivatives are now with respccc t o 4 ). Clei~rly.the rclation v ( . , s ) = K * H ( . . r ) is still valid. The results of Section 4 yield readily the well-posedness ol' Equation (6.2) in I.'(R'). as well as decay csti~natesin s. Howcvcr. the interest in this Iransfor~ncdcclu:ition lics in its well-posedncss in a scale o f weighted-1,' spaces defined as follows:
We refer to 115, Section 31 for a proof. Observe that L'..' L ' if s r I . By Proposition 6.3 it is a "persistence" spuce for the vorticity, in analogy with thC space Y in Lemma 4. I.
The spectrum a (C) of C in L2-', for any s > 0 , is given by (see [ 15, Appendix A])
In particular, for a fixed s > I , the finite set of real nonpositive numbers
consists of isolated eigenvalues of C if k < s - I (in the space L2.'). Gallay and Wayne [ 151 construct finite-dimensional invariant manifolds for the semiflow of Equation (6.2) (which can easily be translated to the solutions of (1.9)), for sufficiently small initial data. It is based on this spectral structure and on methods used in the study of dynamical systems. The construction can be described as follows. Fix k E N and s 3 k 2. Let 'Ht S L2.' be the finite-dimensional subspace spanned by the eigenvectors associated with A ( k ) , and let Jk = L ~@ .X k~ be its orthogonal complement. For r > 0 we denote by f3,. the ball of radius r in L ~ . '(centered at 0 ) .
+
T H E O R E6.4. M Fix r > 0 .suffzc.ic~ilr!\' sn~eillcind s, k t1.s rrhove. Let p E ( ?t- . k+l - ). ( a ) There e-rists a glohtil!\. Li/).sc.llit: C ' intrp ,q : 'Hn --+ JL.sirc.h thut g(O) = 0, L>x(O)= 0. rrnd S ~ C / tI h ~the t rntrtli/~)/el
is loc~ullvinvuricitzt in the~0llowitlgsense. There exi.sts 0 < r l ir .such rhcit the ser?riflo~t~ trs.socitrter1 rrith (6.2),co~nmetzc.irlg ur uny point $0 E ?;, fl B,, , stciys in ?;, n B,. j i ) r rill r 3 0. (b) This invcirirint m t i r ~ i f i ~ "cittrrict.~" ld rill trcijec,tot-ies htrl~irzg.stntill inititrl rlcitci. More explicir,~,,for every 00 c ?;, n B, there e.ri.sts t r m t r n ~ f i ~SO,,, l l suc.h thcrt (111trcrjectories hexinning cit poitzr.s of' SO,,fl Br, (with H o re.stricted trlso to B,, ) uppt-oric.17 the trajectory H ( . . 7 ) stctrting of 4). We hrive, i f $ ( . , 7 ) is t r .solirtiorr t o ( 6 . 2 ) ,with $(., 0 ) = 40 E SH,, n B,.,,
( c ) The munifi~fdSH,,is u continuous mup of Jh. I t intersec.ts 7,n B, on!\. at O0 unrl the fumily
is a foliation
of B,, .
We refer to [15, Section 31 for a proof of the theorem.
REMARK 6.5. Observe that the decay rate in Equation (6.4) corresponds to a decay rate of t-1' for solutions of the vorticity equation (1.9). Thus, for sufficiently small initial data in weighted-L~spaces, the asymptotic behavior of the vorticity is determined, to any order, by "finite-dimensional dynamics". REMARK 6.6. In analogy with Theorem 6. I , if
then I[@(.,r)ll, --, 0 as s --, 0 (see [15, Theorem 3.2]), and Theorem 6.4 can be applied to determine its asymptotic behavior. Note that in this case the velocity field is squareintegrable (assuming s > I ). The Gaussian
is a stationary solution of (6.2) and an eigenfunction of L: (with zero eigenvalue). In terms of the original vorticity, it corresponds to the solution of ( I .9) obtained by the heat kernel with singularity at t = - I . I t is called the "Oseen Vortex". Taking k = 0 and .s = 2 i n Theorem 6.4, it is easily seen that %Ois the one-dimensional subspace spanned by G and coincides with the invariant manifold 7 (i.e., g 0).Thus, combining Theorem 6.4 and the conservation of ,fw2H ( 4 . r ) d t . we get
-
COKOLL.ARY 6.7 (Stability of Oseen vortex). Fix 0 < / L < 112. There. (..I-istsr > 0 .such rhtrt (f'H(4.T ) i x ( I solution to (6.2) with IIHolll,~.z iI- ~rrz~l ,/wz H o ( ~ d( ) = ( I the11
We refer to [ 15, Section 41 for a detailed analysis of this convergence.
7. Concluding remarks and open problems I t is common to say that the case of the Navier-Stokes equations in two dimensions is "resolved". Admittedly, the situation here is much better than that of the 3-D case. Furthermore, the L' theory of existence and uniqueness is complete. However, as we have seen, there are important problems, related to "rough" initial data, that remain yet unresolved. Rather than "purely mathematical", they touch upon very relevant issues of fluid dynamics and numerical simulations of singular flows. Even in the (weighted)-L' context, the asymptotic results discussed in Section 6 show that the two-dimensional case still carries much interest. Another aspect of this interest is the (relatively) recent interplay between the methods used here and those used in the study of various classes of nonlinear parabolic equations.
In what follows we list a number of yet unresolved problems. ( 1) Uniquenessfor measure-valued initial dutu with large uromic part.
As was mentioned in Section 5, the uniqueness of the solution to (1.9) and (1.10) when wg is a measure with large atomic part is unknown. It seems that tools developed in this context could prove useful for other classes of nonlinear parabolic equations. (2) Un$orm estimutrs with respect to v and Eulrr eyuutions. It is known that for smooth initial data one can obtain estimates which are uniform in v E (0, I], where v is the coefficient of viscosity (see [ 18,3 I]). The solutions converge, as v + 0, to the unique solution ("zero viscosity limit") to Euler equations with the same initial data. When the initial data is not sufficiently smooth (say, in L ' n L f ) , p > 2 ) we can still obtain the convergence of a .subsequence to a solution of Euler equations. However, the uniqueness of such a solution is not known. Thus, one might try to establish at least the uniqueness of the "zero viscosity limit". (3) Tile cuse c?f'houncledclotncrins. In this case, there is no existence theory for solutions of the Navier-Stokes equations in vorticity form (with "no-slip" boundary conditions), if the initial vorticity is only known to be in L 1 .We refer to 1321 for the case of measures as initial data, but with homogeneous boundary condition on the vorticity. As already mentioned in the Introduction. this is a case of prime importance in applications. Indeed, if this problem is ill-posed. then the numerical procedure of approximating singular vorticities by smooth ones needs to be justified.
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121 (31 141 15 1 161 171 [XI 191 110)
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1 121 P. Constantin and C. Foias, Ntrvior-Stokes
Eclucrtiorr.~,The University of Chicago Press ( 1988). [ 131 G.-H. Cottet and P.D. Kournoutsakos. Vortc..r Methods. Throrlv und Prut.tic,r, Cambridge University Press,
1 141 1 151 [ 161 [ 171 ( 181
[ 191
1201 ( 21 1 1221 1231 1241 1251 1261 1271 (281 1201 1301 13 1 1 132 1 1331 1341 13.51 1361
Cambridge (2000). E.B. Fabes and D.W. Stroock. A ~ r n vproofof'Mosrr's porrrholic. Hurtruc.k ineyutrlity using the old ideas of Nosh, Arch. Rat. Mech. Anal. 96 (1986). 327-338. T. Gallay and C.E. Wayne. 1111vrritr1r/ r ~ ~ a n j f i ~and l d . ~the k)ng-time osyrnptotic..~of the Nrn~ier-Stoke.s crtrd ~ ~ o r ? i cryricrtior~.~ iv 011 LQ2. Arch. Rat. Mech. Anal. (2002), in press. Y. Giga. T. Miyakawa arid H. Osada, Two-tlirnr~n.siorrulNtrvi~r-Srokr.s.fk)n~ with tnrtr.surt,s 11s initirrl vorticity, Arch. Rat. Mech. Anal. 104 (1988). 223-250. Y. Giga and T. Kambe. L t r y e titrte bt,ha~iorof tho ~~ortic.itv c?f'two-dimr~n.sio~ruI vi.sco~r.sfk)wtrntl its u l ~ p l i c.trtiorr to ~'orter,fi~nntrtiorr. Cornrn. Matli. Phys. 117 ( 1988). 549-568. K.K. Golovkin. Wrtrishing vi.sc.o.sit\. irr C~r~ilrrc.11~ l),nhlem ,for livtlronrechtr~iit~,\ rqutrtio~r.~. Proceedings of the Steklov Institute of Mathematics. Vol. 92. O.A. Ladyrenhkaja, ed. ( 1966) (Amer. Math. Soc. Transl. ( 1968)). T. Kato, Renrrrrk.\ orr tlrc Errlt~rtrrrrl Nrr~'ier-Stokr.s Equrrtinrr.~in R2. Proc. Sympos. Pure. Math. 45 (2) (1986). 1-7. T. Kato. Tlrr Ntr~~irjr-Stokr.cryirtrtiorr f i r l r r r i r r ~ . o ~ ~ r / ~ r . r ~ . s . \ iirrI ~ R2 I r ~ ,~virh f l ~ ~rri (rrrpo.srrre / r r s thr irritirrl ~ w r t i r i ! r : Ditferential Integral Equations 7 (1994). 949-966. T. Knto and G. Ponce. Woll-po.s~~tlr~c~.\.\ of tlrr E~rlertrrrtl Ninjirr-Stokr.s ~rlrrtrtiorr.sin the Leho.s~rrr.\prrc.r.\ L)(IR2). Revihta Mat. Iheroarnericann 2 (1986). 73-88, T. Kato and G. Police. 0 1 1 ~rorr.ctertio~rtrr:\. ,flokt,.\ c~f~~rvc~orrs r r r r t l irlt~trl,flrritlsin L ( ' ( R ~ )Duke , Math. J. 55 ( 1987). 4 x 7 4 9 0 . T. Kato arid G. Police. Cot~rr~rrrt(rtorc,.\/irrrrrtr\ rrrrd rhc Grlcr lrrrtl Ntn,ir,r-Stokr.\ er/rrtrtio~r.\.Colnrn. Pure Appl. Matli. 41 (1088). 801-007. H. Koch and I). Txtarci. Il/c.ll-l~o.\c~(/rrc.\.\ for 1111,Nen,ii,+.Srokr,\ c~clrrrrtiorr.~. Adv. in Math. 157 (2001 ). 22-35. O.A. L2;idy/hetiskayu. 771c.M(rtltc~trr(rti(.rrl171cv11:~ o/'Vi.\r.r~rr.rI~rc-r~~rrl)r.r~.\.\iI~I(~ F'Iott.. Gordon and Hrci~ch,New York ( 1960). k:ngli\h triiti\I:ition. I,.D. I.andau and E. M. l.ilbh~t/. I.'lrtic/ Mcc.lrctrric.s, Pcrgamon P r o \ . Ncw York (1050). J . Ixray. F~rrctlo(I(, tli~,c,r:\a\ iyrctrrrorra irrtt;,qrtrlo.~IUIII litrt;tri~.c,.\1.1 tk,.! clrrc,lyrr~,.\ ~~,r~hl(,trrc\ qrrc /XI\<' I ' l r \ t l r r ~ c l v r r t r ~ r ~ iJ~. ~Math. ~ r ( ~ . Pure\ Appl. Sir. 0 12 ( 1033 ). E.H. I.ich ;11ic1 M. I.o\I\.Atr(rlv.\i\. Amcr. Math. Soc.. Providence. Kl ( IOOh). J.I.. 1,iolis nrid G. Prodi. Orr /lrc~orc;rrrc~ tl'o\i.ster~c.c,t.1 tl'rrrric.irc; d~rtr.\It,\ c~yrrtrtiorr.\(I(' Ntr~,icr.Str~X(,.\t J r r tlirrrc~rr.\iorr2.C. K . Ac~id.Sci. P;~rih248 ( 1059). 35 1')-3521. C. M;~rcliioroii~idM. Ptilv~rcriti.H ~ ~ l r o e l v r r ~ ~int r111 r i(~I (~/ ~. ~ I I I ( ~ I I \ I ~ (I I \I I ~ t,ort(,.\ ~II(,oI:Y. C O I I ~ IMath. ~ I . PIly\. 84 (10x2).483-503. F.J. McCrath. Norr.\/tt/iorr~rr:\.111~ore //OI\, r!/' ),i.\c,orc.\trtrtl itli,trl /lrritl.\. Arch. Rat. Mcch. Anal. 27 ( 1068). 328-348. T. Miyah:iwn nnd M. Yuniada. /'l~rrr(o-N~r~~ior.-S/r~kc~.\ /Ilt~t..\irr o horrrrtk,(l tlr~rtrrrrrr~t.irlrrrrc~r\rrn,,\( I \ irriti(r1 cltrttc. Hiroshiriia Math. J . 22 ( 1002). 4 0 1 4 2 0 . T. Miyakawa and M. Schonbek. Orr ol~tirrrctlt l ~ , c ~rtrtc,.\ t r ~ f i ~ tt.c,trX r .\r~lrrtrorr.\t(1 rlrt, Nrr~,ic,~.StoXr.r c,qrrtrtiorr.\ irr R". Preprint (2000). G. Police. Orr r ~ t . o - t l r t ~ r c ~ r r . \ i o ~i ~~( rrI ~ ~ ~ ~ r r r ~ ~ ~ - ~C~o.~r i~i ~i Partlill hn l. t ~ , /1)lfkrcntial I r ~ i t l ~ . Equation\ 11 ( 1086). 48351 1 . J . Scrr~n.0 1 1 / / I ( , irrtc~t.iorr~~~,qrtltrrr/~ ~t.c,(tX.solrr/iotr.\ rlro Nlr~.icr-S/oXc,.\c,yrrotiotr.\, Arch. R;it. Mcch. Anal. Y (1962). 187-195. R. Telnam. Ntr~.ior-Stoko.sE~lrr(rtiorc.s.North-Holland. An1sterd;tm ( 1079).
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CHAPTER 6
Attractors of Navier-Stokes Equations A.V. Babin
.
Dc.~'~t'ri~~etlr q"Motheinorric.t. Utrir.erviry of Cor/ifhniirr Irr Irvinr . Irvinr CA 92697.3875. USA E-iiitril: trhtrhii~r@i~ior~h.uc~i.~ol~r
Cor1terzt.s Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . Solution ernigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Dynii~~liciil syste~nsin function space\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Hau\dorfl' ar~dfractal d i ~ ~ ~ e n \ i oofn attractor s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. BaGc detin~tiona. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Ba\ic thcore~l~s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. More a\pects ol'finite din~ensionnlity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . Attractor ol'the 7L) NS systefn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . I . Ili~llc~lsio~l of i~ttri~ctors for thc 211 NS \ Y \ ~ C I I I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 .? Periodic houndary c o ~ ~ d i t i o.~ .i s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. I.owcr c\tiinatcs for the di~ncnsionol'iitrr;ictor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Equation., in an unhoundcd dolllain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . The 3D Ncivicr-Stokes equatio~ls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. The dinlension of regular invariant scts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Equations ill it rotating frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Thin d o ~ n i ~ i ~.l r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Generaliled attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modilications ol' NS and related hydrodynamic cquat~on\ . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgc~nents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R e k r c ~ ~ c c s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
HANDBOOK O F MATHEMATICAL FLUID DYNAMICS . VOLUME I1 Edited hy S.J. Friedlander and D . Serre 02003 Elsevier Science B.V. All rights reserved
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I XX 188 I01 194 I06 I06 I00 200 203 206 206 207 210 212 212 214 214
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Attractors ofNuvier-Stoke
rquurions
Introduction Dynamics of fluids is governed by the equations of hydrodynamics. The long-time behavior of fluids can be adequately described in terms of attractors of the equations. Here we consider mostly attractors associated with the Navier-Stokes system of hydrodynamics (for brevity we will refer to it as the NS system). We consider two-dimensional (2D) and three-dimensional (3D) Navier-Stokes systems with either no-slip or periodic boundary conditions. The theory of the NS system has to deal with non-scalar equations and with the divergence-free condition that leads to non-local operators. Therefore, mathematical methods developed for the NS system have to be rather general and usually can easily be extended to more complicated physical systems. The Navier-Stokes system takes the form atu = F ( u ) , after the pressure is excluded; it describes the evolution of divergence-free vector fields u ( t ) = u(x, 1 ) in an appropriate function space. In many problems of hydrodynamics, the influence of the initial data has vanished after a long time has elapsed. Therefore permanent regimes are of importance. The simplest permanent regimes are described by time-independent vector fields that are solutions of the equation F ( u ) = 0, such a solution is an equilibrium of the dynamical system. The local theory of equilibria and their perturbations is very rich and includes their stability, bifurcations, theory of local invariant manifolds through them (see Chossat and looss [521, Guckenheirner and Holmes 11031, Marsden and McCracken [ 1551, Sattinger [ 173 1, Stuart [ 18 1 1, and Section 2 of this article). Nevertheless, time-independent regimes are very special and i t is widely believed that time-dependent permanent regimes are of importance. To describe spatially and temporally chaotic turbulent flows one has to consider non-trivial time-dependent regimes (see Chorin et al. 15 1 1. Ruelle and Takens 1 17 1 I). Time-dependent regimes may include time-periodic. time-quasiperiodic, and chaotic regimes; their common feature is that they are defined for all times, both positive and negative. A mathematically rigorous description of such regimes and related questions of asymptotic behavior and stability are given by the theory of attractors. There are aspects of the dynamical theory approach that are common to finitedimensional and infinite-dimensional systems. They are related to bifurcations. to the chaotic temporal behavior of solutions, and to the fractal structure of attractors (see Lorenz 11481, Barnsley 1351, Falkoner [701, Marsden and McCracken 1155], Guckenheimer and Holmes [ 1031). We consider here questions that are specific for infinite-dimensional dynamical systems, in particular for PDE. The theory of global attractors of partial differential equations (PDE) in general and NS equations in particular is developed in the works of Babin and Vishik [321, Hale 11041, Ladyzhenskaya [ 1.341. Temam [ 185 1. Vishik [ 1891. The theory applies the ideas of fi nite-dimensional dynamical systems to intinite-din~ensional dynamical systems described by PDE. Many aspects of the theory are specific to an infinite-dimensional case. In particular, the boundedness of energy does not mean that a blow-up of solutions is impossible and does not imply the global solvability of the 3D Navier-Stokes system. There are completely new phenomena, for example, the dimension of attractors tends to infinity when viscosity tends to zero; such behavior and its asymptotics make sense only in an intinite-dimensional situation. Another phenomenon that has no analogue in the finite-dimensional case is the
presence of a spatial variable in addition to the time variable. Relations between spatial and time variables manifest themselves in trivialization of dynamics on the attractor of the NS system in unbounded channels near spatial infinity. Another example is higher regularity with respect to spatial variables of functions on the attractor compared to a generic function from the function space. Many aspects of the theory of attractors are important for applications, in particular to geophysics and meteorology (see Lions et al. [I 43-1 451). Remarkably, the global attractor is a finite-dimensional set. Dimension may be large, global attractors include high-dimensional invariant sets. Examples from the theory of finite-dimensional dynamical systems (the Lorenz attractor) show that dynamics on such sets may exhibit a chaotic temporal behavior. One of approaches to turbulence is based on this observation (see Ruelle and Takens [ 17 I]). On the other hand, the attractor is embedded into a function space and the points of the attractor are functions that represent spatial patterns. Since the attractor has a large size and dimension, its different points correspond to different spatial patterns. Therefore the dynamics on the attractor generates the dynamics of spatial patterns that may lead to a spatial chaotic behavior. Attractors are very interesting objects from a computational point of view. Since the attractor is a tinite-dimensional object in the inti nite-dimensional phase space of the system, the dynamics on it is subjected to restrictions that are additional to the original NavierStokes system that describes dynamics in the whole phase space. Understanding these restrictions may lead to better computer simulations of physical flows. Dimension of the attractor may serve as an estimate of number of degrees of freedom required to describe the long-time behavior of the Navier-Stokes system in detail. A global attractor also contains all the information on the instability of the dynamical system (see Babin and Vishik 130,321). Quantitative expression of the instability is given by Lyapunov exponents. which are closely related to the dimension of the global attractor. The purpose o f this paper is to give a sketch of the core of the classical theory of attractors with minimum technicalities and to point out major directions in which the theory develops. When exact formulations of results are too technical, we refer to the literature. The Navier-Stokes equations for viscous incompressible fluids have the form po(i),u
+u . V u )
-
u A u = ,f
+Vp
(1)
with the divergence-free condition
Here, u = ( u I , u?. u j ) is the velocity field, u = u(.r. t ) = u ( x l . x2, x j . r ) , v > 0 is the kinematic viscosity, j' represent volume forces; the density po of the fluid is constant and we set po = 1. We use the notations
When u , f d o not depend on x3 and u3 = J:?= 0 we obtain the 2D Navier-Stokes system. On the boundary a D of the domain D, no-slip boundary conditions are imposed
An important case, in which boundary layer effects are absent, is the case of periodic boundary conditions
Usually, we take UI = u? = u j = I . In the case of periodic boundary conditions, we impose the following zero average condition on u and f ' :
Condition (5) for u ( t ) holds for all t if it is satistied at t = 0. To introduce classes of solutions for the Navier-Stokes system, Sobolev function spaces are commonly used. We recall that the norm in the Sobolev space H, = H,(L)) of a function ~ c ( . r defined ) in a domain D c R" is given by the formula
where .s is an integer, i)" = i j y ' . . . il;" are partial derivatives of order la1 = a1 When s = 0, we obtain the L7 norm
+ . . . + a,,.
We denote by H , . .s 3 I , the space o f vector tields with a tinite norm (6) which satisfy (2) and either (3) or (4). The space H I is obtained as the completion in the HI-norm of infinitely smooth divergence-free vector fields u = ( u 1.. . . , u,,) that satisfy the boundary conditions. We denote by Ho the closure in the space H = ( Ho)" of H I .Clearly Ho c H. The space H- I is dual to H I . We denote by the orthogonal Helmholtz-Leray projection
Gradient tields are in the null-space of the Leray projection. Applying the projection we can rewrite ( I ) in the form
174
A. V Bubin
where
The space H 2 coincides with the domain of the Stokes operator A and the norm ) J u ) ) $is equivalent to J J A U11;; ~ l u11; is equivalent to JIAIl2u 1: = (Au, u)o = I ~ V1 ;U . We consider here the no-slip boundary problems in either bounded domains or unbounded domains with exits to infinity of bounded width. For periodic boundary conditions we impose zero average requirement (5), therefore Poincark inequality J I V U J J3~ A1 JJu1: holds with h l > 0 (see ( 19)) and JJVu1: is a norm. The space H- c Ho is dual to H I . The bilinear Euler operator B(u, v) has the following important skew symmetry property ( ~ ( uu), , u )= ~ 0 for all u. u E H I .
(11)
For basic properties of the spaces H, and the bilinear operator B , see, for example, Babin and Vishik (321, Constantin and Foias [ 5 7 ] ,Ladyzhenskaya 11281, Lions 11411, Temam [1831.
1. Solution semigroup Solution .semigrou/~fi)rthe 21) NS system. The two-dimensional Navier-Stokes equations for viscous incompressible fluids have the form
together with the divergence-free condition (2). Here, u = ( u l . u z ) is the velocity field, u = U ( X , t ) = u ( x I , XI, t ) , v > 0 is the kinematic viscosity. and f ( X I , x2) represents a volume force. The Euler nonlinearity and divergence for 2D Navier-Stokes equations are
On the boundary i j D of the domain D no-slip boundary conditions are imposed
We assume that the boundary is smooth enough (see Ladyzhenskaya 11351 for a nonsmooth boundary). A special case of interest is periodic boundary conditions
In this case we impose the zero average condition ( 5 ) on u, f ; therefore the functions from the space Ho satisfy (2), (14), and (5). (In the periodic 2D case the integral in x3 in ( 5 ) is skipped.) Applying the Leray projection I7 we rewrite the NS system (1 2) in the eql~ivalent form (9).
We consider the initial value problem
The existence and uniqueness of solutions of the 2D N S system with periodic or no-slip boundary conditions are well known (see Constantin and Foias 1571, Ladyzhenskaya [ 1281, Lions [ I 4 11, Telnam [ I 831). T H E O R E M1.1. Let .f E H - I . For unp 1r0 E Hopthere exists u unique solution u ( t ) ofthe 2 0 NS system ( 12) with iriiti~rldutu ( 15). This solution belongs to HOj i ~ rull t 3 0. The solution mapping
, 2 0. One also can solve the system with initial determines a family of operators ( S T } T , ~ solutions ~ ~ l H~ data at t = T I and find a solution v ( t ) , t 3 T I . If ~ l =~u l , ==~the ~ l , = and ~; uI~= ++ ~U I ~ = ( ~ ~ give - ~ the ' , ) same result ~ l = ~U ~ = ~ = (~) .~Therefore, -T, the operators ST form a semigroup that acts in the space Ho. Many important properties o f the NS equations can he formulated in terms of the so. basic properties are described in the following theorem (see lution semiproup { S l ] These Babin and Vishik 1.321 for a detailed proof).
=
THEOREM 1 .2. Let ,f' E Ho. Then, tlic~.sonigmrrp { S I] thtrr c~orrr,s/>orlcl,v to the 21) NS .sy.sterri ( 1 2). ( 1 3 ) it?t r /~orirrd~~l tlorr1uirl or to the pt.rioclit. /)roh1~111 ( 1 2). ( 14) 1rtr.s t l i t ~ , f i ~ l l o ~ ~ i r i , ~ l~ro/~ortie.s. Thc .sol~rtiori.su ( t ) = S l ~ r otrw hoioitk,cl iri Ho ~ol;fifi,rrn!\. bt.her~t 3 0 trritl 110 i.v houriclc~clirl Ho. Thc opercttors Sf ttrc c.orltitlirous,fiu)rt~ Ho to Ho fi)r t 3 0. Morro\,c.~ if 0 t T trritl 11ir0 110 R, tI10ri Sl L ~ O tIt~~~orit1.s ori L ~ O ririiji)r-rrrl~c~or~tiri~ro~r.s/y ,for .fi.rot/ R , T . Tlir ,fi)llo~c~irig .srnoorhirlg property 11olcl.s:I / ~ Pfknc.tion,s Sl ir rrrc. hourirlccl iri H I trrzd Hz urlifi)rrr~lyit1 r trrid 110 ~7hetlf 3 to > 0 crr~tlIluo 110 R. The oprrtrtors Sl. t > 0,trrp cori~l>crc't.
< <
<
<
Since the statements of the theorem contain basic information on the semigroup and their proofs include important estimates. we sketch the principal points of the proof and give basic estimates on the solutions. We deduce the boundedness statements in this theorem from the fundamental energy estimates which we give below. Multiplying (9) by rr and using the skew symmetry ( I I ) , we obtain the energy equation
and a differential inequality for smooth solutions
~
176
We have
where h l > 0 is the first eigenvalue of the Stokec operator A . Therefore, we get from ( 18)
Inequality (20) implies for a tirne-independent ,f that for T I 3
is bounded when llu('7i))(l;is bounded. Therefore, S,uo are bounded in Hence, llu(Tl Ho uniformly in r 2 0. Equation (17) yields one Inore firndurnental estimate which includes the time-averaged norm 1101111;of the gradient Vu. Integrating ( 1 7 ) in time, we conclude that regular solutions of the 2 D NS systern satisfy the following energy equality
and from (22) we obtain
To establish the smoothing property we multiply ( 9 ) by r.411. After integration by pi~rts. one gcts
According to ( 2 3 ) the time integral of I I V L ~isI bounded ~; by a constant which depends only on ~l~r(,ll;.lntcgrating in t , applying Sobolev inequalities and using (23), one obtains an estirnare of the form
and 11 ( t 1 = S,LIOare bounded i n H I for I c Iro. 7'1 when to > 0 and Iluoll~< R. Since thc embedding HI c Ho is compact. S, are compact for t -, 0. Similarly one obtains the boundedness of r Ilillu (1; and I 11 A L 1 ;~ which iri~pliesthe boundedness in H z . To derive the continuity of operators S t , we consider the equation for the difference 11) = u I - uz of two solutions of NS system ( 12) written in the form (9):
After multiplication by w and application of ( I I ) one obtains
Using Sobolev embedding theorems, we estimate (B(u2, w ) , w ) in ~ (25) and get in two space dimensions:
Therefore
Since (23) gives an estimate of the exponential term, this inequality implies the Lipschitz dependence on the initial data and the Lipschitz continuity of operators S t . Thus, Theorem 2.1 is proven. Loc~rl31) .settti~roul~ In three space dimensions. the Navier-Stokes system has global weak solutions (see Constantin and Foias 1571. Lions [ 141 I, Temam I 1831). There exist weak solutions (may be non-unique) that satisfy the energy estimates we derived above fix the 2D N S system. The formal derivation is the same.
One can prove the local existence and uniqueness of strong solutions if the initial data and the forcing are smooth enough. THEOREM 1 .4. L e t s > 1/2trtzd,/ E H,-1. Forel~rr\'~roE H.F~ I I P I Y r.ri.sts T I trtltltr utliyue .sfrotrg .solirfiotl ~r ( I ) c!f't/le 313 NS systc)nl dltif is rlt 112. These norms cannot be estimated by (28). Avail-
.
able estimates of higher norms are local in time. For instance, when d = 3, s = I, one can get an estimate of the norm
<
which implies boundedness of llu(r)ll; o n a small time interval 0 t allows a blow-up as t + T I . Consequently, one can get an estimate
T I(llu(O)ll I ), but
i
That irnplies the existence and the continuity of operators Siu in H I locally in time. I t is not known if S t / / are differentiable (or continuous, or single-valued) in H, when s i1 /2. Also, it is not known if there exists il solution which belongs to H,, with .v 2 1/2 for all t , 0 < t < cw, for arbitrary u ( 0 ) E H , .
2. Dynan~icalsystems in function spaces r i o t s 1 1 r o r i Here we describe a general framework of intinitcdimensional dynamical systems that is, in particular, ~ipplicnbleto the 2 0 Navier-Stokes system. A first order i n time partial differential equation can be written i n the form I
where I.' is ;I (nonlinear) differential operator. It' the problem is globally well-posed. thc initial condition
determines u unique solution ~ c ( r )= i r ( . r . 1 ) . t 3 0 . For every fixed t , we get rr :is a function of . r ; this function belongs to an appropriate function space E. So we obtain a vector u ( t ) E E which depends on time t and initial data ~ r o .I n concrete situations, the operator F ( u ) has to be specified as well as the classes of' solutions ir and initial data rro. Thc specitication o f u solution should include appropriate boundary conditions and smoothness conditions. Since the solution is unique. the solution mapping is well-detined:
As in (16). we denote this mapping by Sf,u ( t ) = Slrrcj:the mapping is defineti on the set of initial data. When the equation does not contain thc explicit dependence on time I . the mappings Sf satisfy the semigroup identity
We call the family of the operators (S,)= { S f , t evolution equation (29).
2 0 ) a setnigroup associated with the
Absorptiotz and attraction. Let E be a complete metric space with distance dE(u, v). In many examples E is a Banach space and
Let a semigroup of operators {St, t 3 0 ) act in E
Throughout we assume that operators S,LLare continuous with respect to u in the metric E. We introduce a (nonsymmetric) distance l j E ( B I ,B2) from a set BI to a set B2, ~ E ( B B2) I , = sup inf d E ( x l ,x2). 1 , c B , .r2EB2
A set B is i i ~ ~ ~ ~ ~ifr iS,u B t z tC B for all t 3 0. A set B is .strictly invuriunt if S, B = B for all t 3 0. A set Bo is called an rrhsorhing .set if for every bounded in E set B there exists T such that St B c Bo for all t 3 T. When (5,)has a bounded absorbing set Bo and the operators St are uniformly bounded on bounded sets and continuous, the set
is also absorbing and is invariant. Therefore the existence of a bounded absorbing set is equivalent to the existence of a closed bounded invariant absorbing set. A set Bo is called an crttrc~c.ting.set (in a space E ) if for every bounded in E set B lim
1-M
aE(s,( B ) , Bo) = 0.
(34)
D E F I N I T I O2.1. N A set A is called the ~ l o h ucittr[ic.tor l of ( S , ) in E if ( i ) A is compact; ( i i ) A is strictly invariant: S,A = A ; (iii) d is an attracting set for ( S t ) in E , that is ~ , C ( S ~ ( A) B ) .+ 0 as r + co for every bounded in E set B.
The following properties characterize global attractors. A global attractor is a minimal set among all compact sets that attract all bounded sets. A global attractor is a maximal set among all bounded strictly invariant sets (see Babin and Vishik 1321, Hale [ 1041, Ladyzhenskaya [1341, Temam [185]); it has a maximal domain of attraction. Sometimes, global attractors are called maximal attractors or minimal attractors. Here we often call a global attractor simply an uttrc~ctor.Note that if a global attractor exists, i t is unique.
180
A. C! Babin
R E M A R K The . above constructions and theorems in the next subsection are applicable to the case where E is a metric space. In addition to a normed topology of a Banach space, a metrizable weak topology of a linear separable Hilbert space or a metrizable topology of convergence on bounded sets can be used. Another important example is the case where a semigroup is defined on a bounded invariant subset of a linear Banach space. See Babin and Vishik [ 3 2 ] , Hale [104], Temam [I 851 for details. R E M A R KAn . absorbing set for the 2 D NS system was constructed by Foias and Prodi [ 7 5 ] . The attractor of the 2 D Navier-Stokes system was constructed and many of its important properties were established by Ladyzhenskaya [ 129,1301. Upper semicontirzuiry. If Equation ( 2 9 ) depends on a parameter 8 , atu = F ( u , B), we have a semigroup that depends on the parameter and the global attractor of this semigroup also depends on 8 , A = A ( $ ) . Under very mild assumptions A ( 0 ) depends on 8 upper semicontinuously. This means that S E ( A ( H ) ,A ( $ ( ) ) )-+ 0 as 0 -+ 80 (since the distance S E from one set to another is not symmetric, this does not mean that SE(A(Ho),A ( 8 ) ) + O ! ) . For example, it is proven by Babin and Vishik 1281 that the attractor of 2 D NS system in a bounded domain depends upper semicontinuously on the domain, see Babin and Vishik [28,32] for details. ~ i ( t )-CO , < t < CO, is called a t r u j ~ t . t o r yof {St } if Strict i t i r ~ ~ ~ r i r r t i cA . ~curve . ii(t1 t2) for all -cm < t? < CO, 0 < f l < CO.
+
Sf, u (12) =
The following important property of a global attractor is equivalent to its strict invariance. For every point t r E A there exists a bounded in E trajectory r r ( t ) of { S fdetined ] for all -CO < r < +CO such that u ( 0 ) = tr. Strict invariance is a strong property. We show it on an important example that we will use later. If u ( t ) is a bounded trajectory of the 2 D NS system, inequality (21) gives
+
<
~ ( ~ ~ ~p -1i l *1l ( T:l - h l )11~(7i1)11:
for any To, T I 3 f i . We assume To +
This inequality and ( 2 3 ) imply
-GO:
(1
-
t,-l'i~(r~-7i))) v2hy
11.fII&
since ~ l u ( f i ) l l l ?is, bounded we get for all T I
Arrrucro,:c
(~Nrrvirr-Srokr.~ ryuurions
18 1
Letting T I -+ -co we obtain lim
v
1
?
sup llVul[odt T I + - ~ T - T ~TI
llf 11; < --.
11 v
The right-hand sides of (36) and (38) do not depend on T and u. Therefore, if an attractor
A of the NS system exists (we prove the existence of the attractor of 2D NS system in the next section), then every trajectory u(t) = Slug on the attractor satisfies (36) and the estimate sup lim sup I I O E A T-+c*~
I
1
IIVL 11; ~dt
11 f 1 ;
<X1v2
'
Here, f' is the body force, v is the viscosity, and 1, is the first eigenvalue of the Stokes operator; depends only on the spatial domain.
DEFINITION A .point :is an ccliri/if>riiu~~ poiilt of (St J if Sf,:= :for all t Since ,: does not depend on I. for semigroups detined by (29). ,: satisfies the equation F ( : ) = 0. D~:E'INII'ION. An itrr.stohlc rrrtrr~ifi)ltlM+(:) through an equilibrium point :of .TI is the set of all points v E E such that S I v is detined for all t 6 0 and S,I]+ :in E as r 4 -a. I f { S t )has a global attractor A and :is an equilibrium point of { S , ) .then M + ( : ) c A.
DEFINITIOA N .sftrhle n~nrr~fOltl M-(:) through an equilibrium point :of .TI is the set of all points 11 E E such that Stu is detined for all t 3 0 and Srv + :in E as t + +a. The behavior of a dynamical system near an equilibriuni is described by the theorem on stable and unstable manifolds of semigroups in Banach spaces; this theorem is fairly sirnilar to the ti nite-dimensional theorem. We formulate the theorem skipping technical details. i n particular the differentiability conditions (see Babin and Vishik 1321. Henry 11061 for details; see also Bates et al. 1.361, Chen et al. 1441 for details, generalizations, and more references). Let S, be a nonlinear differentiable (of class C", cr 3 1 ) semigroup in a Banach space E. Let :be an equilibrium of .TI, that is Sf: = :for all t 3 0. The differentials S,I(:) form a seniigroup of linear bounded operators in E. The properties of this seniigroup play important role; the behavior of Sf near :is in many respects similar to that of S:(:). The most important assumption is the existence of a circular gap in the spectrum of S,I(:). Namely, we assume that the spectrum of Si(z) does not contain a ring = p' in the < ( p s)' is not in the spectrum if E is complex plane. Therefore, a ring ( p - 8)' < small. We conclude that the spectrum is divided by the ring into two parts: external a+
+
and internal a_.Therefore,the Banach space E splits into two complementary invariant subspaces E + ( p ) and E - ( p ) , S , ! ( z ) ( E - ( p ) )c E - ( p ) , S : ( z ) ( E + ( p ) = ) E + ( p ) for all t 3 0. W e assume that E + ( p ) is finite-dimensional. Under these conditions, the nonlinear semigroups have locul invariant manfolds M+(z, p ) and M - ( z , p ) through z in a neighborhood o f z (the local manifolds may be non-unique). A set M is called local invariant (in a neighborhood o f z ) when Slu E M i f u E M and STu stays in the neighborhood o f z for O < r < t . When p = 1, M+ ( z . p ) is called a local unstable manifold o f S t , M ( z ,p ) is called a local stable manifold o f St. When I < I = I is in the spectrum and p < I, M+ ( z ,p ) is called a center-un.stuble mcinifi)ld o f S, . T H E O R E 2.1 M . There exists u mc~rz~fi)lcl M+(z, p ) thcrt is in the neighhorhoodof'z u gruph of'tr,firnc.tion(!f'~~lu.s.s CUfrotn E + ( p ) to E ( p ) , M+ ( z ,p) is tangent to E+ ( p ) ut z. There t>xi.st.sN ma111fi)ldM ( z . p) that is in the neighborhood oJ'z u gruph c$u function c?f'clus.s CUfi-on1 E ( p )to E+ ( p ) . M ( z ,p ) is tungent to E- ( p )ut z. The intersection M+ ( z ,p ) n M ( z . p ) = z . In the t7righhorl7oodc!f'z, M & ( z ,p ) rirc. loc.crlly invr~rirrntwith rt>.spPt.tto S l . The,fi)llo~:irlg c~ttrrrc.tione.stitntite d i s t ~ ( S , uM+(:. . p ) ) < c f ( p- E ) ' , 0
(40)
than u ( t )E M - ( : . p ) . When /I 2 1 , SI is evtt~ntlctlin.siclo M+(:, p ) to nr,ytrti\~~ t trntl
d i s t c ( S l ~ c:) ,
< c"(p +c ) ' .
t
< 0.
(42)
I f there are many circular gaps I < ] = p: with points o f the spectrum between then), one can construct many different local invariant manifolds o f Sl near :. Intersections o f these manifolds are also smooth local invariant manifolds. Therefore. the nonlinear dynamics near z is in many respects similar to the linear dynamics o f S:(:).
The Nervier-Stokr.~systetn netrr c~yuilihriutn. The 3D and 2D Navier-Stokes systems in a bounded domain with a time-independent force always have time-independent solutions (equilibria). Namely. i f D is a bounded domain. for any ,f E Ho. there exists a timeindependent solution :(.r) E H-, (an equilibrium) o f the 3D or the 2D NS system ( I )
see, e.g., Lions [I411, Temam 11831. W e consider here the 3D case since the 2D case is simpler. The simplest example is f = O and then z = O is the only equilibrium. The long-time dynamics described by the NS system is the simplest in this case, every solution tends to
zero as t + co.According to Leray theorem, every weak solution of the 3D NS system u ( t ) becomes regular for large t and u ( t ) -+ 0 as f + cm. Therefore when f = 0 the global attractor of 2D and 3D Navier-Stokes system consist of one point z = 0. We consider a Ho and consider regular solutions that have a limit, more general case f u(t)+:
a s t + cminH,,
s >
112.
The linear semigroup S:(:)vo is generated by the variation equation (57); this equation takes the form
The operator S:(,-)is compact when r > 0. Its spectrum is discrete and has only one accumulation point, namely. zero. Therefore almost any circle j ( I = r lies in the resolvent set of S,'=, ( z ) and one can choose a sequence of circles [ ( I = r , in the resolvent set of S; (,-) with r l + O as j + co. Let a sequence of positive numbers I.,,j = I . . . . , determine the circles ] ( I = r It in the resolvent set of S,!(:). r l + 0 as j + cm. According to Theorem 2.1, we have invariant manifolds M + ( r l ) and M - ( r , ) through :that are tangent to the corresponding invariant subspaces o f S,!(:): M + ( r j ) are finite-dimensioni~l.As we discussed before (see Theorem 2.1 ). the behavior of St near an equilibrium is similar to that ofthe linearization S,'(z). In particular. the following theorem (see Babin and Vishik 1321) describes the trucking property.
Here, we can approximate an infinite-dimensioniil solution u ( r ) of the NS system by a solution i l ( r ) that lies in a finite dilnensional manifold. the dynamics of i ( t ) is described by a system of ODE.
R E M A R K . I n the case of potential forces when ,/' = 0 and := 0. results of Foias and Saut 182,831give more detailed information on the dynamics. A normal form of the NS system is given and the structure of the normalization map is described. The normalization nlap reduces the NS system near zero in the non-I-esoniuntcase to the linear problem and i n a resonant case to a more complicated normal form.
Exponentiul attructors. Another important notion of the theory of dynamical systems is an exponential attractor, also called an inertial set (see Eden et al. [66,68] for a construction). D E F I N I T I O NA. set E c E is called exponential attractor of the semigtopup (S,] in the Banach space E if (i) E is compact and has a finite fractal dimension; (ii) E is (not strictly) invariant S , l c I for all t 3 0; (iii) there exist positive constants (. and c' such that for all r 3 0 and for every bounded set B c E
Note that points (i) and ( i i i ) of this definition are more restrictive than the corresponding points of Definition 2.1 and point ( i i ) is less restrictive. For the definition of a fractal dimension see Section 3. If an exponential attractor exists, it always contains the global attractor, A c E . An exponential attractor is non-unique. Existence of an exponential attractor for 2D NS system is proven by Eden et al. 1661; see also Eden et al. 167,681. Trrijrc.tot:\. rrttrcic~tot:c.. When the forcing j' depends on t . ,f' = , / ' ( I ) .Equation (29) takes the form i),u = & ( L O , f ' ( t ) .I t becomes non-autonomous and operators S, do not form a semigroup anymore. The function u ( t .s), t 3 0.is a solution of a shifted equation i),u = & ( u ) , f ( t s ) . Following Sell 1174.175], Daferrnos 161 1, Chepyzhov and Vishik 1491 we define an operator y, : ( ~ ( t )f (. t ) ) H ( ~ ( t .s). ,/'(I .s)). We can consider functions ([I(!). , f ' ( t ) ) . t 2 0, as elements o f a topological space (-)+ of time-dependent functions. The topology in this space is detined by the convergence in an appropriate norm on bounded intervals I t l . t?], for example. ( / ' I 2 ~ l u ( s ) l l : d s ) ' l ~for 11 all positive tl. tz. This convergence can be described by a metric, so results of Section 1 are applicable. The closure of shifts ,/'(I s ) is called the hull of f ' . Under appropriate conditions, one can prove the existence o f the global attractor of the semigroup {ST.).The attractor of this semigroup is called rrtijoc,tory trttr(rc.tor (see Chepyzhov and Vishik 1481). This attractor consists of the solutions u ( t ) detined for -co < t < co and corresponding to an element f ( t ) of the hull that is defined for -w < t < co.The construction is directly applicable to the 2D NS equations with time-dependent forcing term , / ' ( I )and can be easily generalized to more general non-autonomous equations. This construction is also useful for treating equations without uniqueness. in particular the 3D Navier-Stokes equations. For details of the theory of trajectory attractors see Sell [ 1761 and Chepyzhov and Vishik 148-501 where existence of trajectory attractors for non-autonomoi~s2D NS system and non-autonomous 3D NS system is proven. For a related method of "short trajectories" see Mdlek and NeEas [150], Milek et al. [1521.
+
+
+
+
+
+
+
A toy model.
To illustrate the above concepts, we consider a linear equation
-
where Ao is a self-adjoint operator in a Hilbert space H with an orthonormal eigenbasis Rj
One may take Ao = - A + c I , where A is the Laplace operator with appropriate boundary conditions and c is a constant. The numeration of eigenvalues is in increasing order, hi+l 3 A,. Let No be the number of non-positive eigenvalues
Equation (46) can be written as an infinite system of uncoupled equations
for the coefficients u , ( t ) of the expansion of ~ c ( tin) the basis g , . The general solution of ( 4 6 )is given by
I t can be split into two parts
Obviously, this estimate is uniform when the initial data r r ( 0 ) belongs to a bounded set B in H. The linear subspace EN,,with basis S I . . . . . SN,,
is strictly invariant with respect to the linear semigroup Sf =
and is finite-dimensional. It satisfies the attraction property
for every bounded set B. Therefore, we may consider Eo+ an unbounded global attractor of the linear semigroup e - A " t . The only condition in the definition of a global attractor which is not satisfied is the compactness (so an unbounded global attractor is not a global attractor in the itsual sense). The set Eo+ is strictly invariant and attracting; it is not compact, but it is locally compact, that is every bounded part of it is compact and it is finite-dimensional. Nontrivial examples of unbounded attractors of nonlinear PDE are given by Chepyzhov and Goritskii [47]. When AN,, > 0, the subspace Eo+ is an unbounded exponential attractor with the rate of attraction
If one takes a larger subspace with the basis g l , . . . , g ~ with , N I > No, one gets a higher rate of attraction
More interestingly, one may increase EN,,just a little, taking
with arhitwry small r., and still (49) remains true (with a different (.' ). so it is an unbounded exponential attractor. too. The set t: is invariant. S,& C & but it is not strictly invariant. St& # &. This toy example shows that by expanding the global attractor a little and increasing its dimension, one may drastically increase the rate of convergence to i t and obtain an exponential attractor. A nontrivial generalization of this idea leads to the proof of existence of an exponential attractor of a dynamical system, see Eden et al. 166-681. The set Eo+ can serve as an illustration to the notion of an unstable manitold. Clearly. zero is an equilibrium of the linear semigroup. If AN,, < 0, the subspace EN,, = Eo+ is the unstable manifold through zero of the linear semigroup S, generated by (46). When h N , ,= 0 the subspace EN,, is a center-unstable manifold of St. The center-unstable manifold of this semigroup is non-zero if h i < 0 , it has the basis X I . .. . . g ~ , with , A I . . . . .AN,, < 0, NO+ I > 0 . A stable manifold of the linear semigroup through zero is a linear subspnce with the basis RN,,+ I . . . . . This toy model illustrates the basic properties of attractors: the set Eo+ is an attracting set, it is locally colnpact; moreover, it is finite-dimensional. It is invariant, that is on Eo+ every trajectory u ( t ) can be extended to -a< t < +co. When A ( ) = - A ( . I ,where A is a negative operator, then for c = 0 the attractor consists of only one point: zero. The nontrivial attractor Eo+ is a result of the perturbation ( . I , with sufficiently large c > 0 that creates instability. Of course, the Navier-Stokes system is much more complicated than this toy model. Still in many cases one may consider the global attractor of the NavierStokes system as a result of perturbation of the linear Stokes system by the nonlinear term
+
and the external forcing. The attractor is no longer a linear subspace. Finite-dimensional nonlinear models (for example, the Lorenz system) show that the structure of such set may be very complicated and this set does not look like a smooth manifold neither locally nor globally. Nevertheless, many observations are still true. The global attractor is a compact, finite-dimensional (in the sense that will be discussed below) set. This set is invariant, that is the dynamics on the attractor is invertible. Remarkably, it is a compact bounded set, that is, in a sense it is smaller than a linear subspace (unbounded). And this set uniformly attracts all the solutions of the dynamical problem, in particular of the NS system. This toy model illustrates another general property of attractors: the upper semicontinuous dependence on parameters. Let the parameter be B = IN,,+^ > 0. When 0 + 0 the attractor changes continuously (in fact, it does not change at all when 0 > O), but at the limit value B = 0 the attractor EN(,= E N ~ , +changes I its dimension: it becomes larger. At the same time d i ~ t ~ ( EENo+1) ~ ( , , = 0. E-ristence c?f'uttructors. Here we give basic existence theorems from the theory of at-
tractors. More details are given in Babin and Vishik [32], Ladyzhenskaya 11341, Temam [ 1851. A very detailed treatment of general aspects of the theory of existence of attractors of operator semigroups is given by Hale 11041; see also Ball 1.341. We consider an operator semigroup { S t} in a complete metric space E . The operators S, are assumed everywhere to be continuous bounded (nonlinear) operators in E . A semigroup { S t } is called asymptotically compact if, for any 140 E E and a sequence t , + w. the sequence St,lco has a convergent subsequence. THEOREM 2.3. Let ( 1 serrligroul~( S t} 11tri.eti horirltlrtl trh.vorl7ir1ghtrll / l K . U,> ( ) ( S/,l K } he houi~tletl,trrrtl (SI } ht) tr.syt?r/~totic.trl!\' t.ot~~l~trc.t. TIlrr~{ S t} 11tr.v tr ~ l o h t rirt/rurc.tor l A. T l ~ r ccttrtrc.tor corlttri~~s trr1 e y ~ i i l i h r i ~ r1 1r0 ~ , ~Slrio = 110 ,fiw trll t . If' St i.s t.orltirluou.s it1 t , tile trttrtrc.tor A is ( I c.orrr~ec.retl sct. The attractor A is detined as an omega-limit set by the formula
A=
nE(T),
where
E(T) = closureE
7'>0
All the properties of A can be derived from the detinition. see Babin and Vishik 1321. Ball 1341, Hale 1 1041, Ladyzhenskaya [ 1.341, Temam I I85 1. C ~ R O I . L A R2.Y1 . Let { S ,} htrve tr l ~ o u r ~ d t ~ d t r h ~ o rhtrll. l ? i ~Let ~ g t11ool,ertrtor:s S t , t i ) rt 3 0, he c~ot~rir~uous trtld utlifi~rr~ily hountled urld c~otry~trt.tfi)r et7et;\>t 0. TIIOII { S t) 1ltr.s cr glohcrl trttructor:
The existence of the attractor for the 2D NS system is based on fundamental properties of its semigroup ( S T )which are stated in Theorem 1.2. Using Theorem 1.2, we obtain the following theorem.
THEOREM2.4. Let f E Ho. Let ( S T }b e the semigroup in the space Ho that is generated by the 2 0 Navier-Stokes system in a bounded domain. Then, { S t )has a global attractor A. The attractor contains Lrn equilibrium uo, is u connected set, and is bounded in H 1. PROOF. Inequality (21) readily implies the existence of an absorbing ball
Note that Boo is bounded in Ho, but is unbounded in H I . We set
When ST B c Boo, we have ST+[B C Bol. Therefore, the set Bol is absorbing. The smoothing property from Theorem 1.2 implies that the functions S,u are bounded in H I , if t > 0, and I j ~ ~ < l l ~Ro; ) this is true in particular for t = I. Therefore, the set Bol is absorbing and is bounded in H I . The compacteness of the embedding H I c Ho implies that Bol is compact. The smoothing property and the continuity of St in H I also implies that operators S, are compact. Hence the global attractor exists by Corollary 2.1.
A stronger property of attraction ( ( H o .H2)-attraction)holds (see Babin and Vishik 127, 321): THEOREM 2.5. Lct ,f E Ho. The trttr.trc.tor.A i.5 c.ornptrc.t in Hz trntl S H , (3,( H I . A) + 0 fi)r trtljl hoioltlotl it1 Ho set R . R E M A R KUsually, . the smoothness of functions on the attractor is determined by the smoothness of the forcing term and by the boundary. Interestingly. the attraction can be in a stronger norm than boundedness of solutions (see Babin and Vishik 1.321) since difference of two functions from H I may belong to H z . The regularity of functions on attractors and attraction in stronger norms are studied in Babin and Vishik 127.321, Ghidaglia and Temam 1971. Temam [ 1851. In particular, when ,f' and i ) D are infinitely smooth, the attractor A consists of infinitely smooth functions (see Temam 1 1 851. Foias and Temani 1891).
3. Hausdorff and fractal dimensions of attractors
A fundamental characteristic of an attractor of a dynamical system is its dimension. The physical meaning of the dimension of an attractor is roughly speaking the number of degrees of freedom required to describe the large-time dynamics of the dynamical system. The attractor may be a very complicated set, so a definition of dimension has to be applicable to general sets. First, we give the definition of Huusdofldirnei7sior~of a set in a Banach space.
If K is a compact set, we consider finite coverings C K of K by balls B,, (xi) of radius r, centered at x , , BrI (xi) = { u : lu - x; I E < r;]. We denote by ICK I the maximum of r for the covering C K . Let P ~ . ~ ( K ) =inf x r , ~ .
(53)
ICKI<~
Let
The Hausdorff measure pff( K ) equals co for small dorff dimension is defined as
ci
and equals 0 for large a. The Haus-
Note that since E is intinite-dimensional, dirnH(K) may be equal to infinity. Now, we detine the,fr~r(.ttrlclirne/z~ionof a compact set K. We consider ti nite coverings of K by balls B,.(.r,) of radius r centered at .x, with a fixed radius r. We denote by n ( r , K ) the minimum number of balls in such a covering. The box-counting dimension (fractal dimension) of K is dim[:( K ) = lim sup
,.-o
log 17 (r. K ) log(l/r)
'
We always have
Note that if K is a smooth compact tl-dimensional manifold (or a piecewise smooth manifold, or a manifold with a boundary) that lies in E. then d = d i m H ( K )= d i m F ( K ) . An important property of the fractal dimension is the existence of Mafie's projection (see MafiC [ 1541). Namely, if a set K has dimension d i m F ( K ) ,there exists a projection onto a linear subspace with dimension d < 2 dimF(K ) 1 which is one-to-one on K . This follows from the following Mafie's theorem (see Eden et al. [6XI. Foias and Olson 181 I).
+
THEOREM 3. 1 . Let H he tr .veptrrtrhle H i l h ~ r .sl)erc.cJ t t r ~ l t lX he tr c.or?lptrc.t.suh.vlltrc,eof' H with Htrr~stIoYff dii?rotr~iondimH X k err?(/dimH(X x X ) kt. Let Po be trrl orthogorrtrl prc!jec.tion of' rcrnk c~yuttlto kt I. Then ,fi)r nJer:v 6 E (0. 1 ) tlwrc. exists trn orthonorrntrl projection P = P ( 6 ) c!f'the samcJ rtrrlk in H .such tlrtrt 11 f i - P ( 6 ) 11 < 6 trnd P restric.ter1 to X is one-to-one.
+
<
<
Note that the fractal dimension has the following natural property: if dimF X 6 k" then dimF(X x X) 6 2k" (the Hausdorff dimension does not have this property). Since dimH(X x X)
< dimF(X x X),
the conditions of Mafie's theorem are fulfilled if dimF(K) is finite.
I90
A . I! Buhin
REMARK. Conditions for the inverse Mafie projection to be Holder continuous from P X to X are given in Eden et al. [68],Foias and Olson [U 11, Hunt and Kaloshin [ I I I]. A fundamental problem is to estimatc the dimension of the attractor or, more generally, of an invariant set of a semigroup {ST).Methods for estimating its dimension essentially use the properties of the linear operators S; obtained by the differentiation of STUO with respect to i r ~ These . operators are the solutions of the variation equation. When
where F(i0 is a nonlinear operator, which is differentiable in an appropriate sense, with differential F 1 ( u ) we . write the variation equation as
In most cases. and in particular for the NS system, a family of linear time-dependent hounded operators
is generated as a holution of' the linear variation equation (57). Since u ( 1 ) = Sl~co,the solution t 7 ( t )= S:I~Oof (57) depends on rco, the linear operator S: = S,'(lro). In particular, fi)r the NS system. F(lr) = I ~ A I+I R(rr. 11) wherc H(rr, r r ) is ;I bilinear operator and
The operators S,rro are Frkchet-differentiable with respect to rro. i d the differential of the variation equation (57) obtained by formal differentiation o f (12) (see for details Babin and Vishik 132)). Let H be a Hilbert space. I f E,/ c H is a (1-dilnensional linear subspace, then a linear operator L maps o (1-dimensional ellipsoid B,/ C Eli into a (1-dimensional ellipsoid I,(Bll) C L(EII).I n a Hilbert space, a volurne \ / O I , ~ ( Rof , ~ )a ti-dimensional ellipsoid is well-defined. For a hounded operator L in a Hilbert space H. the quantity w,,( L ), which rneasures the changes of tl-dimensional volumes under action of L , is defined as ~ ' ( 1= ) S ~ ( U O ) IisI O ;I solution
c,~[l(L)= sup R,,
vol,i(L ( B,I ) ) VOI,/(
'
the supremum being over all tl-dimensional ellipsoids. If 8,1 is a ball. L O , / ( L= ) a1 . . . a , / , where a , is the length of jth axis of the ellipsoid L(B,,). Clearly, cyf coincides with jth eigenvalue of the operator L*L when the spectrum of L*L is discrete. For details. see Babin and Vishik [321,Temam [ 185).We only note here that when L is compact, u ) ( / ( L )-+ 0 as d -, co.
DEFINITION. An operator S is uniformly quasidifferentiable on a set X C H and a linear operator S ' ( I I )is a quasidifferential of S on X at a point u IG X if IISLI- S I J- S ' ( ~ ) ( I-! u ) l l H
sup
11 i!
II.IIEX
- 14 11 H
-+ 0
a4 E
-t
0.
In applications. S 1 ( u ) usually coincides with the operator S' obtained by the formal differentiation of S. We call S 1 ( u )a quasidifferential because v is not an arbitrary element of H. Note that i n some cases estiniate (59) holds for u , v E X C H (and not for arbitrary u , I! 6 H ) since X in applications consists of rnore regular functions than a general function from H (for example, we apply it when H = H o ( D ) and X is bounded in H I ( D ) ) . We say that S has quasidifferentials S t ( l i ) uniformly bounded on X when
where IIS1(u)IIL(H)is the operator nortn. A global version of r o [ / ( S 1 ( u )is)
w,/= sup L O [ / ( S ' ( ~ ~ ) ) . / l t X
Here we give theorems on thc Hnusdorff and fractal dimensions of strictly invari~untsets.
To cstiniate the di~nension,one has to consider coverings of A by sinitll balls. The possibility to prove the finite dimensionality of the global attractor tor :I system in an inti nite-
dimensional spacc, in particular. for a system corresponding to the 2D NS systcm. is based on three fundamental facts. Thc tirst fact is the invariance property SIA = A. which implies that if one has a covering of A by a family of balls 8,. ir then S, (H,. + i f , ) is also a covering of A. The second fact is the ditt'erentiability of the operators Stir with respect to 1 1 . Thanks to the differentiability, one can locally approximate the action of S! on a ball B,- i t , by S,'(ic ,)I?,.. The differential S:(iro)tll)= I : ( { ) sotisties the variation equation (57) which. for the N S system. takes the forni
+ ,
+
J
4-
(
I I
)
)
+(
i t )I
)
+A
= 0,
I,(())
=
(60)
with r c ( l ) = S,tc(,. Thc third fact is that thc linear operators S;(lr,,) have thc smoothing property and therefore are compact. The eigenvnlues fif'. /3:'. . . . of the operator S,'*S, are positive and. since S: is compact, PI, + 0 as t? -t m. Theretore li)r a 11-tiilnensiond nieasure / ~ , / ( S : ( i r r , ) H , . ) C p i P i . . . P f l , ~ , / ( l l), and. i f (1 is lul-ge, ~ . i [ ~ ( S / ( iBI.) co)< C'P1,~,,(RI.) with p < I. Therefore, for large r . / ~ , / ( S : ( i c ( ~ )
<
<
The following theorem that allows to get quantitative estimates is proven by Douady and Oesterle [64]. THEOREM 3.2. Let SX = X , and let S be ungi)rmly quu.sidr~erentiubleon X , with uni,firmly bounded on X q~lcrsidiffrrentiu1.sS 1 ( u ) .Let d he such thutfi)r some k < 1
Then, the Hu~~sdorff dimension rf X is.finite and is not greuter than d .
The following theorem on the fractal dimension is proven by Constantin et al. [58], see also Constantin and Foias [561.
Then, the.fr~lc.tu1dimerlsion of'X is finite ancl not grrrrtrr thcrn d .
Chepyzhov and llyin (461estimate the fractal dimension under conditions similar to that of Theorem 3.2: THEOREM 3.3'. Under the /iyl~othr.sc~.s c!f'Throrern 3.2 .srr/?/~o.s(~ thtrt the cjr4cr.sitl~frrr11ti(1I.s S ' ( u ) c.ontin~co~r.s!\' tl(~l)c~ntl on 11 E X in the oportrtor riortii. Tl7c.n tlir frrrc.ttrl rlitiieri.siori o f ' X i.s 17otgrrutrr thcol (1. The statements of Theorems 3.2 and 3.3 can be illustrated as follows. Inequality (61) means that the mapping S strictly decreases the d-dimensional Hausdorff measure with a coefficient k < 1. Since SX = X , this is possible only when the Hausdorffnieasure of X is zero. This sketch can be made rigorous (see Babin and Vishik 1321. Temam 11851). Here, we follow mainly Temam IIX51, skipping some technicalities to show the most important ideas. If SI(uO)is the solution operator for the variation equation (57). then using a Liouville type formula for solutions of linear equations of the form (57). we get the estimate m,,(~:(uO)) < supexp E,I
[I'
1
tr(~'(u(~))n,l(r))d~ +
where nE,(,, is the orthoprojection in H onto E , / ( r ) = Si(iro)E,,, the supremum being taken over all d-dimensional subspaces and tr( F ' ( ~ ( r ) ) f 7 ~ , , (being , ) ) the trace of the finite-dimensional operator F ' ( u ( r ) ) n ~ , , ( ~ ) . From (6 1 ) and (63), we obtain that dimH X < d if, for some t > 0 and k < 1 , for all trajectories u ( t ) on X
Note that by (58)
where A is the linear Stokes operator. According to a result of Metivier, its eigenvalues and the trace can be estimated similarly to the eigenvalues of the Laplacian in a N-dimensional domain. namely A , 3 ~ ,j 2 j N ~ andh ~
A typical estimate of the trace of the operator S:(uo)(t), where u(t) = Slug takes values in the attractnr A = X , has the form
Here. hA is a constant that depends on the attractor and the parameters of the problem but ) on the attractor. (In the next subsection we show how not on a particular solution ~ ( tand (68) i h derived using (39).)Therefi~re,from ( 6 3 ) we get the estimate l i t m sup sup Inco,l(~l'(lrcl)) 6 - I , C ' ; , ~ ~ +'IN ~I'
+44.
l-'"srf,lCX
When rl is large enough, nanlely when
(we can take the minimum integer d that satislies this condition) we have (63). lim sup sup l n c o , , ( ~ ; ( ~ i , ~ )< ( t0, )) ~+WI~,,CX
and condition (6 l ) holds. One can apply Theorems 3.2 and 3.3 to S ( t ) . t -, x. I n addition. the right-hand side of (68) is concave with respect t o ti. This allows to lipply the observation of Chepyzhov and llyin (451 that (69) is sufficient f o r both Hausdorff and fractal dimension to be not greater than (1. Therefore, we obtain the following theorem.
3.4. Let S I X = X . Sl he c ~ u a s i d ~ ~ r u r ~ t or! i ~ i Xh I(/I?(/, c ~ ,fi)rr v ~ t qt , //lo yurr.sid(fli~rc.t~titrl.s S ; ( u o ) he unifi)rtnl.v bountlrd on X. Lvt un irltvgvr d .scr/i.!fj. (69). T!len tlrr H ~ ~ i c . v t k ) ~ f t / i r ~ i ~oft ~X. s is i o rlot t ~ grcjccter than ri. Tiiu frirc.tul rlir~rrtl.vio~i r ? f X i.r ,lot grrrrtrr thon tl. THEOREM
R E M A R K .The above results can be extended to non-integer values of d, see Babin and Vishik 1321, Chepyzhov and llyin [451,Temam 1 1851.
R E M A R K .Estimates of dimension of attractors can be made in terms of the Lyapunov exponents of the operators S:(uo)(t). We define, for j 3 2, a global jth Lyapunov exponent by
(see Temam [ 1851 for more detail). One can derive from (63) p1
+ . . . + p,,+l <
SUP litn sup - supexp II(O)EX / + c u r EO
[I'
1
t r ( ~ ' ( ( ) ) , , ).
(70)
Condition (6 1 ) is replaced by the following condition:
THEOREM 3.5.
Let Sf X = X . St h r qi~rr.~iclifirrtititibl~~ot~ X , Irt
(7 1 ) h o l d a r ~ r l , j h revery t ,
the t~~rtr.sitlifi~re~~tirrI.s S,! (110) ( t ) be ~~tlifi)rr~l!\' boiitidod on X . T I I P I I ,tho H t ~ ~ . s d o r f f ' d i t r ~ ~ ~ ~ ~ ~ s i o r i o f ' X is riot grc'llter thtrtl
The ti 1st statements related to tinite-dimensional parametrimtion of attractors were given by Foias and Prodi 175j and Ladyzhensliaya I 129.1301. Foias and Prodi 1751 proved that the asymptotic behavior as t -+ co of the solutions of 2D NS system in many cases is determined by the asymptotic behavior of their tinitedimensional projections. Ladyzhenskaya 1 129,130]constructed the attractor of the 2D NS system and proved that a trajectory on the attractor is completely determined by its orthogonal projection onto the space spanned by the tirst 11 eigenfunctions of the Stokes operator. There are several ways to parametrize attractors. The first one is to project the attractor onto a ti nite-dimensional linear subspace. and if the projection is injective, the subspace gives a parametrization of the attractor. By Mafie's Theorem such a projection always exists. A related question, which arises in connection with spectral nunlerical methods is the following: Do lower Fourier modes give the parametrization and how Inany modes does Ptrrtr~llc~tri:trtior~ c!f'trttrtrctor.r.
one have to take? A similar question, which arises in connection with finite-difference numerical methods, is the following: Can one parametrize functions on the attractor by their values at given points (determining nodes) and how many nodes does one need? More general question is: When do values of a finite number of functionals determine the longtime dynamics? Such problems, important for computational applications, are addressed in papers by Chueshov [53,54], Cockburn et al. [55], Constantin et al. 1591, Foias et al. [79], Foias and Temam [87], Jones and Titi [121,122], Ladyzhenskaya 11331, Shao and Titi [ 1791). A global attractor is a very complicated obAp11ro.uirnntion(?f'~tfroctor.s L I I I dyl7~1i)lics. ~ ject lying in an infinite-dimensional functional space. Exact analytic description of attractors is very difficult, even for low-dimensional systems of ordinary differential equations. Therefore, it is important to develop methods of tinding approximate attractors and approximate dynamics. The simplest approach is to approximate infinite-dimensional system by a finitedimensional one and study the attractor of the finite-dimensional system. Babin and Vishik 1281 proved that the attractor of Nth order Galerkin approximations for the 2D N S system lies in a small neighborhood of the attractor of the original system when N is large. This follows from the fact that attractors upper semicontinuously depend on a parameter (see Babin and Vishik 128,29.32.]).Therefore. the attractor of approximations lies near the exact attractor. But this property does not exclude that the exact attractorcan be much larger than attractors of approximations. Therefore. one has to consider the original equation and find ways to approximate its attractor and dynamics o n it and near it. 1 1 i / I I I I I / S . The ktct that the global attractor A is finite-dimensional is quite remarkable. But one has to take into account that this set may have a non-regular structure. To describe the dyn:unics on the attractor and near it one may try to include it into a larger. but still ti nite-dimensional exponentially attracting nlanifold M which is invariant with respect to S,. S, M c M. and contains the attractor. If this is done. the dynamics on M is given by a system of ODE and such a system could be used for numerical simulations. The best possible solution is to describe M as a graph of a smooth (or. at least. lips chit^) function defined on a tinite-dimensional subspace of the functional space. Such an object is called an inertial manifold. An inertial manifold is a global variant of a local invariant manifold in Theorems 2.1 and 2.2. Inertial manifolds exist when there is a wide enough gap in the spectrum of the linear part of the equation. Existence of inertial manifolds is proven for a number of systems, in particular for the Kuramoto-Sivashinsky equation (see Foias et al. (80.84.851. Tema~n11851 for detail). Unti)rtunately. it is not known if the 2D NS systern possesses an inertial manifold. An approach to approximate long-time dynamics constructively is based o n the construction of approximate inertial manifolds, see Foias et al. 178,84,85]. Foias and Te~nam [871, Foias and Titi 191 1. Temam IIX.51, Chueshov 153.541 and the references therein. For the approach based on construction of approximating algebraic and analytic sets. see Foias and Temam 1901.
4. Attractors of the 2D NS system 4.1. Dimension of nttractors,for the 2 0 NS system If the phase space of a mechanical system is a manifold, the dimension of the manifold is the number of degrees of freedom of the system. The attractor represents the phase space for permanent regimes. Though it is not a manifold, one can determine its Hausdorff or fractal dimension (or at least obtain an estimate of this dimension). Of particular interest is to estimate the dimension in terms of physical parameters of the problem. Methods of deriving such estimates that are applicable to the 2D NS system were developed by Ilyashenko [I 13,l 141 (for the Galerkin approximations), Babin and Vishik [23,24,26], Ladyzhenskaya [I 3 1,1331, Foias and Temam [86], Constantin et al. 1581, Constantin and Foias [56] on the basis of a theorem of Douady-Oesterle. Constantin et al. [60] developed the method of upper estimates of the Hausdorff and fractal dimensions of PDE in its modern form. This method allows to obtain estimates which are physically consistent and are shown to be precise for important examples (see Section 4.3 for details). In particular, estimates of the dimension for 2D NS can be given in terms of the Grashof number
where A I is the first eigenvalue of the Stokes operator A (or the bottom of the spectrum when the spectrum of A includes a continuous component as in the case of an unbounded domain). Note that 1~l1' can be taken as a typical length and ]lfjlbl'has the dimension of the velocity. so the Grashof number is dimensionless. As shown in the preceding section, to get an upper bound for the dimension. i t is sufticient to estimate the trace in (68). Here we give a sketch of an estimate of the trace in (68) with X = A (see Babin and Vishik 1321, Constantin et al. 1601, Temam IIXS] for details). We take an orthonormal in H basis cp;(r) = cp,(.r, s). j = I , . . . d that spans E,/(s). We have
.
After a straightforward computation, we get
According to the Lieb-Thirring inequality for an orthonormal basis pi, with Ilp,
1; = I,
with a dimensionless constant C L TThus
and
According to (67),with N = 2
Theref'ore
k[l 7'
lim sup sup
I
t r ( F f ( i l ( r ) ) n , , [ , ) )dr
7 . - u rt0cA
c;,~. < --V C2. , ~r ~l 2 + lirn sup sup 2v T-wrt,,c.A Finally. we deduce (68)from (39),namely
f
[l I
lirn sup sup tr(~~(u(~))nb,,(~) I-+wI~~~cA
2
[I?
+ h-4,
(7.5)
where hd=2-
c2 F L7' . '
I ~ ~ A ,
F
= "A1 lim sup sup
~ + W I , ~ I ) ) EIA
~;~~llfll;
~ ~ ~ i d ( r )
1j3A
(the quantity E is similar to the energy dissipation flux in turbulence theory. see Te~nam [ 1851). The right-hand side of (75) is negative when ~ C ; ~ E / ( L J ' A) < csi,hld'/2. that is
Therefore, (69) holds when an integer d satisfies d > c'G. Here, c', c are dimensionless constants which depend only on the shape of the domain. Therefore, we can use Theorem 3.4. S o we obtain the following result of Constantin et al. 1601, see also Constantin and Foias [56], Temam [ 1841, Chepyzhov and Ilyin 14.51. THEOREM 4.1. Let d sutisfi either (76) o r d > c'G. Then d-dimensional volumes on A decuy exponentiully us t + oo.Hu~i.sdofldimen.sion dimHA of the uttructor A .suti.sfie.s the estimcrte
R E M A R KIntroducing . the macroscopical length Lo = A;'/' and the dissipation length L,, = ~ ~ / ~ / 1one , ~can / ~rewrite , condition (76) in terms of the ratio of different spatial scales. following Teniarn I 185 I:
Note that. since ~l,fll:,~'has the dimension of a velocity, a generalized Reynolds number can be introduced
This number is related t o the Grasshof number: Ro = GI1' R E M A R KThe . definitions of the Hausdorff and fractal dimensions of attractors include the metric of the space, so dimensions computed using the HI-norm instead of the HOnorm could be different. But this does not happen. Clearly, balls jlulll r. are smaller than [lv[lo r. Therefore the number of HI-balls needed to cover A is larger than the number of Ho-balls needed to cover A and therefore H I -computed dimension may be only larger. When f' E H , the solution operator S,=I is a smoothing operator, in particular for positive t it is Lipschitz from H into H I with a Lipschitz constant Lo. Since S IA= A. the Lor). Therefore, covering by balls {Ilv - v, 110 6 r ) induces covering by { I I r l - S l v; 11 I the dimensions computed using the metrics in H and H I are the same.
<
<
<
In this case, there is an additional fundamental inequality which includes enstrophy. When periodic boundary condition<;are imposed. the following well-known identity taker place ( B ( L ~v ), , A U ) , , = 0
for all c E H1.
(For the proof see, for example, Babin and Vishik 1321, Eden et al. 1681, Temam 1 1 8.51.) Using this identity and multiplying the 2D NS equation ( 1 2) by A u , we deduce similarly to (22) and (23) that solutions of the Navier-Stokes system satisfy the inequality
and the enstrophy equality
For solutions o n thc :~ttl.acto~., since /' is tinle-indepenclent. we obtain similarly to ( 3 0 ) . (36) sup sup v l l ( 0 ;< 7
,,,,€A
sup
'
lit11
,c,,tA
ll,/~ll;,
sup 7.--%7
I'
9
air^^^ d/
< -.ll,t'll; 17
A better cstimate lor. the solutions o n the urtractor cornpal-cd w i ~ h(36).(39)allows t o get bcttcr cstirnate of its clilncnxic~ri.N;~~iicIy, the following theorem is proven by Conslantill ct nl. [601. Sce also Doering ;und Gibbon 1621. where a sinlplitied pl.oot' is given. We give the formulation in the most important case when the Grasshof number G 2 I. ;I
As in Theorem 4.1, the main part of the proof is to gCt a good estimate of thc trace in (68). Now, one can use (80). For technical reasons. the metric of H I is now used. After a
technical calculation, we obtain (68) in the form
where
x = vhl lim
:ITI~AU(S)
sup sup -
T - t o o u ~ ~ A
1 2dr < l
l
l = v -7A3I G2 .
-
v
The right-hand side of (81) is negative when
with an absolute constant t . 3 . Therefore, if
then condition (69) holds and Theorem 4.2 follows from Theorems 3.3 and 3.4. A similar result can be formulated in terms of X . THEOREM 4.3. If' (82) llol~/.s, tllol d-dirnetl.siorltr1 ~~olrrt?~c~.s or1 A erl~or~~tltitrl!\. dec,cy trs t + a.Tho Htru.stIot,f ditlretisiotl of' tkr trttnrctor A .strti.yfi~.s dimHA < d . T110 ,f'rtrc.ttrl tlit~ler~.siotl ofthe trttrtrc.tor A strtisfies dimFA < d. R E M A R KThe . ratio (X/(v3h:))'13 in (82) can be interpreted (Temam I 1851) as a square , a macroscopical length Lo = I;'" and a of the ratio of length scales ( L ~ / L ~ ) 'with microscopical length L, = (v31X ) '1".
4.3. Lower estitntrtr.~ filr the dit~lcrlsiotlrfcrtrrrrc.tor The method of the previous subsection gives upper estimates on the dimension of the attractor, which allow the dimension to grow according to the given forlnulas as the parameters vary. Though the estimates are physically reasonable, one may ask the question: Does the dimension really grow, and do the methods yield precise estimates? A positive answer is based on the following observation of Babin and Vishik 1231. Their idea (see also Babin and Vishik [ 3 2 ] )is to estimate the dimension of the attractor from below by using the inclusion M+(zo) c A, where M+(:O) is the unstable manifold of the semigroup S, through an equilibrium point (time-independent solution) zo The unstable
manifold M+(eo) contains a local invariant manifold M+(zo, r ) , r I (see Section 2). The local invariant manifold M+(zo, r ) is a smooth manifold through zo. The tangent linear subspace for this manifold at the point -70 is the invariant subspace of the linear operator S:=, (zO)corresponding to eigenvalues 5 with 151 r > I (unstable invariant subspace). Note that S:(zo)vo is generated by the variation equation (60) which takes the form (43). To prove that the dimension of the attractor is large it is sufficient to find a steady-state solution zo with a high dimension of the ~ ~ n s t a binvariant le subspace. The unstable invariant subspace coincides with the invariant subspace of the operator Lv = B(v, zo) + B(zo, v) + vAv corresponding to eigenvalues k with Rek > 0, s = eA. We consider the 2D NS system with periodic boundary conditions U(*I
+ 271/@0.~
2 =) I I ( X I ,~ 2 ) .
u ( x I ,x2
+ 271) = L,(xI, x2),
(84)
with cro < I, that is, in an elongated periodic box. We take a small CYO > 0. Let the forcing term have the form
with a 2n-periodic ,gl(.\-2).The steady-state 2D NS system
has a solution
where a periodic function U(.r?) is tound from the equation
We take the parameter y of the form y = 11)'. A fixed number I is chosen so that
where H(.r?) is a periodic function that satisfies i ) ; ~ ( . k ? ) = g l ( . r ~ )I.t is proven in Babin and Vishik (23,321 that the unstable manifold Mi(:()) c d of the semigroup St through this point has dimension not less than (./ao with a positive constant (.. The proof uses the analysis of the Orr-Sommerfeld equation made by Yudovich [ 191 I. This implies the following theorem of Babin and Vishik 1231.
THEOREM 4.4. The dimension (fractal and Hausdotf) of the global attractor of the 2 0 NS system in the periodic box [O, 2 n l a a l x [0,27r] with the special force f is estimuted from below a.s,fillorvs:
Now, we deduce a corollary for a square (or almost square) periodic box. When m is an integer, solutions u which satisfy (84) can be periodically extended; they form an invariant subspace in the space of solutions that are 2rrm-periodic with respect to x2 and 27r/ao with respect to x I . We consider the 2D NS system in 2 n / a o x 2rr[l/ao]-box, where [ I l a ~ = ] m is the integer part of I /a() (for simplicity one can take I /ao integer, so that [ 1 /a()] = I /ao). The solutions which form M + ( z o ) and satisfy (84) lie in the attractor of the system in the square 2rr/a0 x 27r[I /a0]. Therefore (85) is true for this system too. Since f' = lv2g where g = g(x2) is 2rr-periodic, we have the norm in L2([0,27r/a0j x [O, 27r[ 1/a0j])
and the first Stokes eigenvalue is k I = is equal to
ui . Therefore, for this problem the Grasshof number
and we obtain from (85) the estimate
R E M A R KThe . original lower estimate (85) was given by Babin and Vishik 1231 for an elongated periodic box 10, L/cuoI x 10. LI in terms of a;'; in terms of the Grasshof number this estimate implies that d i m d 3 c.'G'jS. The above elementary argument based on repeating the space period [ I/aol times yields (86) from the original estimate (85). Estimates in terms of G of the form d i m d 3 cG7'.' were obtained by Liu 1 146.1471 by a direct treatment of 277 x 2 n periodic box. R E M A R KThe . upper estimate of Theorem 4.2 differs from the lower estimate in Liu \ 146. 1471 by a logarithmic factor ( I I O ~ ( G ) ) ' /One ~ . would like to obtain lower and upper estimates of exactly the same order by improving either the lower or the upper estimate.
+
Ziane [ 1921 considers the case of an elongated periodic box 10, L / a o ] x [O, L ] for which the lower estimate (85) was given. He obtains the following upper estimate of the dimension o f the attractor which is sharp.
203
Atrrcictors cf Navier-Stokes rquution.~
THEOREM 4.5. The,follorving estirnate holds for the fractul und Hausdotf dimensions of the attractor of the 2 0 NS system cr'ith periodic boundary conditions (84) in an elongated box:
-
h here G = ( u ( ~ 1 2 ~ 2 ~ , f . [ [ L ~ I ~ ~ ~ l aand , , ~c'x is ~ ~an ) ,absolute ~ ~ ~ / u 2constant; f has zero averuge with respect to x2.
This estimate is exactly of the same order with respect to a()as the estimate (85) of Babin and Vishik 1231. R E M A R KDoering . and Wang 1631 consider 2D NS system in an elongated box with periodic horizontal and no-slip boundary conditions on top and bottom boundaries. They have c r ( ) the same obtained upper estimate of dimension of the attractor dim(A) c . ~ e ~ / ~ / with kind of dependence on the aspect ratio a()as in (87).
<
Flow o f a fluid in a pipe or in a channel can be described by the NS system i n an unbounded domain. When the domain L) is unbounded. the operators S, are not compact Lunymore i n the usual Sobolev metric. One can prove the existence of attractors of such semigroups in a weak topology. but generally speaking the attractors can be infinite-dirnension;~l(see Babin and Vishik 1331). This can also be seen from the lower estimates of the dimension of the attractor of 2D NS system in the previous section that show that the dimension of the attractor tends to infinity when the size of the domain tends to infinity. Nevertheless. when no-slip boundary conditions are imposed, and the domain is not too wide at infinity (like a channel or a pipe) and the forcing term is spatially localized, one can prove the existence of a finite-dimensional attractor. We consider here for simplicity a rectilinear channel D = ( ( X I . .m): 0 < .rl < h. -m < .r? < oo}along axis. is. This case was studied by Abergel I I (in Lero flux case) and by Babin 15-71. Similar results on the existence of the attractor in a curvilinear channel are proven by Babin 15.61. We consider a strongly perturbed Poiseuille flow, that is. we assume that the velocity field has Poiseuille component
This component gives the leading contribution as .rz + CXI since it does not tend to zero; the flux through the cross-section generated by the Poiseuille flow is non-zero when V()# 0. Clearly, V(x) is a steady-state solution of the 2D NS system
We consider velocity fields V(X.
I )=
V(S)
+ 11(,v, t ) ,
where u(.r. r ) has a ti nite energy
for every t 2 0. The flux through the cross-section for such flows is the same as for the Poiseuille flow V. The equation for 14 takes the form
where
To Jescribc the spatial localiz;~tion of forces and solutions, it is convenient to use weighted Sobolev spaces. When the weight function has the form
the norm
ill
the Sobolev space H l . y . y
2 0. is given by
+
We take here y > 0. Note that 1, E H,,, if and only if ( I ~ . r ~ ~ " ~ ! ' (rc HI,(). In particular, condition 11 E H o , ~means that the arnount of energy , / ; v L , , I , /1.(l2d.r in rernote parts of the channel decays as LpY. The body force f' E Ho,, is spatially localized in the channel. I t can be arbitrary large and excites turbulence in the channel. The turbulent regimes are described by the attractor o f the NS system. We set
A"
-
int
1
- IIEHI
( B ( u .V ) ; u ) o JJVuJI,,
1
The quantity A:) is proportional to h', where h is the width of the channel, and to Vo. The following inequality holds for all u E H I:
where
F
is arbitrary small and C is independent of s
THEOREM4.6 (Babin [7]). Let v" = v - :A > O und f' E HI,^ with )/ > 0. Then the 2 0 NS system (88) with no-slip boundaty conditions ( 3 ) determines u semigroup St of bounded continuous operutors in H . This semigroup hus a global uttractor A. The uttructor is bounded in the spaces H2. ( D ) and H I . ( D ) und has finite Huusdot$ and fructul dimensions ~ t ~ h i c.suri.qfi h tlze estimate
The condition v" = v - 1:)> 0 imposes a restriction on the magnitude of the flux through the channel. Without this kind of restriction the basic Poiseuille flow with a very large flux is unstable everywhere in the infinite channel and one cannot expect existence of a tinitedimensional or compact attractor. R E M A R KWhen . the flux through a cross-section is zero ( V = 0), existence and tinite dimensionality of the attractor for the case of rectilinear channel is proven by Abergel l l 1. For a curvilinear channel, existence and finite dimensionality is proven by Babin 151. When the forcing is not assumed to belong to a weighted space, the existence of the attractor and estimate of type (90) for f' E H-1.0 is proven by Rosa 11701. Ju 11231 extended the compactness and convergence results of Rosa [ 1701 to the H I norm for more regular ,f' E Ho,(,. These results are proven in the zero-flux case. Moise et al. 1 161 I considered a flow in a rectilinear channel past an obstacle. They consider flows that are perturbations of a constant flow U, at the spatial infinity. Compared with the Poiseuille flow such a flow has a simpler behavior at infinity; namely, because of the absence of shearing for the constant flow. it is stable for any value of U,. Existence of a global attractor is proven in Moise 11(>1 1 for perturbations that are not assumed to lie i n weighted spaces and for arbitrary largc U,. Linear stability analysis shows that small disturbiunces of the Poiseuille flow d o not propagate along the channel if the tlux is not too large. The next theorem proved by Babin 171 shows that arbitrary large disturbances do not propagate along the channel, too: at the spatial in tinity any time dependence dies out. So the turbulence is spatially localized.
. fi?.,
68).
THEOREM 4.7. Lct f' E Ho, n3itl! > 0,tttrtl 1 1 - h0 > 0. Tllc~rl,thrrt>erist.~tr tirncitrtit~~)e~n~it~nt soi~ttiotr':(.r) E A//>,.so/rttion1/(.1-. t ), -a< I < +a, tlrut /ic>.s on the trttruc.tor A , trt11nit.scttr tt.s\,tnptotic c,.x/~trn.viorrtr.s I.\- I 4 a:
206
and
R E M A R K If . the original channel is rectilinear only at infinity and has a local narrowing, or an obstacle, then a change of coordinates that makes the channel rectilinear everywhere results in a localized force as considered above (it also results in a spatially localized linear term which can be treated similarly). One can consider in a similar way the 3D NS system in a cylindrical domain (in a pipe). In this case, existence of attractors and non-trivial solutions defined for all t , -co < t < +co, is not proven. But one can prove that if such solutions exist and if the flux through cross-section of the pipe is not too large, then time-independent asymptotic expansion of form (92) holds, see Babin 171. So, in the 3D case turbulence does not propagate to infinity if the flux is not too large. R E M A R KNavier-Stokes . equations in domains with a finite area are studied by Ilyin [ 1 191, he proves the existence of the attractor and its finite dimensionality.
5. The 3D Navier-Stokes equations
I t is not known if a solution of' the 3D Navier-Stokes system with regular initial data stays regular for all times. One can construct weak solutions defined for all times, but it is not known if such solutions are unique. This situation makes it impossible to apply the standard theory of global attractors which is applicable in the 2D case. An approach to a general 3D NS system is to consider regular invariant sets not requiring attraction to then]. Though dynamics is not defined for arbitrary initial data, one may assume (cases when this assumption is proven to be true are discussed below) the existence of bounded strictly invariant sets. One may try to estimate the dimension of such sets. A regular invariant set Z of the 3D NS consists of values of regular solutions r r ( t ) that are deti ned for all t , -co < t < co,and are bounded in Ho
Z = { u : u = ~ c ( t ) .-GO < r < co}.
(92)
It follows from (36) that such solutions are uniformly bounded in HI)
For the 2D NS system with f E Ho, such solutions are also bounded in H I and in Hz. But for the 3D NS system uniform boundedness in a norm better than Ho is not known, even for regular solutions, and only the energy estimate is available to estimate time averaged
Ilu(t) l:, see (38). This is not sufficient to get estimates of the dimension in the 3D case by known methods. Therefore we consider regular invariant sets of 3D NS system of the form
-
Z = { v : u = ~ ( t )-co , < t < co}n {additional boundedness conditions).
The simplest boundedness condition is boundedness in H 1 . One can estimate the Hausdorff and fractal dimensions of such sets. Such sets always exist, for example, when the force ,f E Ho does not depend on time, there always exists a regular time independent solution ;7 E H2, see Lions [ 1411, Temam [183]. Such a solution gives a one-point regular invariant set. Estimates of the dimension of regular invariant sets bounded in H I can be given in the following form (see Constantin et al. 1581, Temam [ 1851). THEOREM 5 . l . Let y C H I be rr h o w ~ r l r in ~l H I c/~rrrirtit!E he cl
~ I I V L I T ~ L I.set I I ~ of' the ,fi)nn
( 9 2 ) . Let t l ~ e
-
~<*lrclr ( ( ( 1 ) is rr rqgirltrr soliitio~rtlrcrt lics ill 1tri~tlr r (0)= ~ i , , .Her(,. 11)) is thc. 1301rtr)rec?f'thc~ tlolrrtrirr D. Lct I r r hc (111 illtegcr. tlrtrt .strti.~fjo.s
-
Tlrorr, tllc sct Z lrtrs ,fitlitc~Htrrrsclotf/' crrrtl ,fi.trc.ttrl rlirirrn.sion.s tIr(r/ strtisf~itile o.sti~rrtrtr.s
dimH(?) 6 1 1 1 , dimr:(y) 6 1 1 1 . The proof of this theorem is similar to that of Theorem 4.2. The question of the existence of a non-trivial (that is, more than one-point). locally attracting regular invariant set of the 3D NS system in the general case is open. The existence of nontriviitl regular invariant sets of the 3D NS system. in fact of global attractors of the 3 D NS system, is proven in two cases: by Raugel and Sell ti)r domains that are thin in one direction and by Babin, Mahalov, and Nicolaenko for 3 D NS system with a large Coriolis force. We discuss these important cases below in more detail.
The 3D Navier-Stokes equations in a rotating frame are:
where U = (U ' , u', u 3 ) is the velocity field, R e 3 x U is the Coriolis force term. The literature on rotating fluids is large, see Greenspan [ 1011, Pedlosky [ I 651. The axis of rotation is along the x3-axis. We impose periodic boundary conditions (4) and zero average condition (5) over the periodic box D = [0, 2 n u l ] x [0,2 n u 2 ] x [0, h a 3 ] . We take u~ = 1. The case of a small Rossby number corresponds to large R . Applying Leray projection I7 onto solenoidal vector fields, we rewrite (95) in the form
(;, ;;) 0 I
being the rotation matrix. where S = nJn.J = The linearized version of (95) was studied by Sobolev [I 801, who continued the analysis of PoincarC [I661 (cf. Arnold and Khesin [2]). The extension of this analysis to the nonlinear equations (95) was done by Babin et al. in [ 10,l 1,131. First results on regularity of Euler and Navier-Stokes systems in rapidly rotating frame were obtained by Babin et al. [ 1 1.131. First results on the regularity in the context of three-dimensional geophysical flows were obtained by Babin et al. 112,151. Mathematical papers on this and related equations include works by Grenier [ 1021, Embid and Majda 1691, Gallagher 192,931. First, we give the results of Babin et al. [ 161 on the global regularity of solutions of the 3D NS system with a large Coriolis force in a simplified form that is sufficient in view of the application to attractors. The proofs of these results use techniques based on a detailed study of resonant three-wave nonlinear interactions of dispersive PoincarC inertial waves. Note that the skew-symmetric Coriolis force term does not affect energy estimates at all. The regularization cfkct of the Coriolis force is pill-ely nonlinear. since it does not affect energy of the Fourier modcs of the linear Stokcs problem at all. 0
5 . 2 . Let u > 112. THEOREM llU(O)ll, Mu. lc~t
<
11
> 0.Lt~tM,. M,r hc trr1,itrtrt:~Itrt~c~,fi.rrtl tllrl~lh~t:~. Let
trrld let $2 3 Ro(M,r. Mu. v ) . T/IEII,thore>ceri.vt.s rr ru,yrrlerr soilrtiotl U ( t ) . 0 (93) s~rchthtrt
where CA del>et~tls o11ly otl M,, Mu/.., v, trt~d011 tr.sl,t.c.t urtios (12,
(13
< m, r?f
ofthe pc'riod box.
<
THEOREM 5.3. Let (97) holtl rvith a = I. Let Ro hr trrhitrtrt:\ Itr,~c).Lrt llU(0)llo Ko t r t d T = R i / v . Tl~erl.,for cvet:\.,fi.wc~l 52 3 R', tt,h~rc.52' is rr rrur~lhcrI-vl1ic.hdty,c~rltl.sor,!\'otl M,r, v. (17, (13 ((rtld (1oe.s 1101 clcy~t~tlcl or1 U ( 0 ) ) .trtlr1,fi)rtrrly n.c)trk solrtfioil U ( t ) ofthe 3 0 rottrtirlg Nuvier-Stokes eclucrtiotz.~(95) that is tlejir~erlor1 10. T ] colt1 .scrtisjie,v the cl~r.ssicul energy inequrllity (28) on [O. 71,thefollowing propositiot~is true: U ( t ) ccln he exter~cledto O < t < +m a n d is regu1arfi)r T t < +m; it br.lot7g.s to H I N I Z IIU ~ ( 1 ) 11 1 C I(M,F. V ) fi)r evc)ry r 3 T . A
A
A
A
<
<
<
Now, we define the semigroup and the attractor. Let Bo = ( U : IjU 11 1 C I( M u F , v ) ) . According to Theorem 5.3, Bo is an absorbing set for all weak solutions that satisfy the energy inequality. We take 0 6 large enough and according to Theorem 5.2 for 0 3 52; solutions with initial data in Bo are regular and bounded in H I for all t . Since such solutions are unique, we have operators St U for U E Bo. The set X = UT20ST BO is invariant, S , X c X , and the semigroup S, is generated by the 3D NS system on the invariant set X . Note that S, on X is a semigroup of continuous operators which are compact for t > 0. Formula (50) defines, according to Theorern 2.1, the global attractor of the semigroup; it is bounded in H I according to (98) with a = 1 . So, we obtain the following theorem.
5.4. If' f' E Ho c111cl0 2 R'(1l.f 110, v , el?, L Q ) , there exisr.~a glohrrl crtfruc.tor THEOREM ofthe 3 0 Ncn-ier-Stokes equution.~,kvhich i.s hounded in H I and H2. Every weak .solution thcrt strri.yfies the c~lcr.s.sic~a1 energy inequerlity is attracted to the global Lrttruc.tor LIS r + +a. Q~~crnfity (93) btvllere I Dl = (217 1 ~ 1 2 ~ ix,finite. 3, The crttruc.tor hers Hcrusdofl dirncv~.sion dimH(A) m ntzd ,frnc.rnl rliinension d i m ~ ( A ) m , tvhere m i.s cr minimrrl integer rhcrr .sati.sfie.s
<
<
PROOF. The boundedness o f f . follows fro111the boundedness of the attractor i n H I and related boundedness of
To get estimates of the dimension we use Theorem 5.1. The conditions assumed in Theorem 5.1 (boundedness of the invariant set in H I ) and boundedness of c are satistied for the 3D N S system in rapidly rotating frame according to Theorem 5.3. The estimates used in the proof of Theorem 5.1 (see the proof of Theorem 4.1 or Section V11.2.3 of the book Te~narn1185) for more details) are directly applicable since the Coriolis term does not affect energy estimates at all and the trace of the skew-symmetric operator R S U is equnl to Lero. Therefore, all the computations can be done without any change and we get estimates of the dimension. RI:MAKK. In the inertial coordinate system. after the change of variitbles
(95) turns into the Navier-Stokes system ( I ) without Coriolis term but with a moditied initial data
+ v A U ' + ,f" + V p ' . 52 curl(U )'((I) = curl U (0) + -e3. 2 i), U' = - U ' . V U '
V . U' = 0.
When U(0) = 0, ,f" = 0. the solution U' represents the solid body rotation. We perturb the initial data for this basic solution with an arbitrary large general periodic field U ( 0 ) and include an arbitrary large periodic forcing 1'. For a very small perturbation one would obtain the global regularity by classical methods. Conventional wisdom says that when the basic solution is increased, Q ;2 m, one should expect that the magnitude of admissible perturbations would shrink. But our analysis (see Babin et al. 1171 for details) shows just the opposite: the nlagnitude of admissible perturbations tends to infinity.
The 3D Navier-Stokes system possesses invariant sets on which the dynamics is welldetined in several cases. The following types of invariant sets on which the dynamics is well-detined are well known: ( I ) Eq~~ilibrin (time-independent solutions): ( 2 ) Solutions independent of one of variables ( 2 D NS system); ( 3 ) Radial solutions (see Lndyztienskayn 1 1 2x1): ( 4 ) Helical solutions (see Mahalov ct al. [ 1491). The tlynamics near an equilibriu~nwas discussad i n Section 2 (Theorem 2 . 2 ) , i t can he studied for a general 3D prohlem. In the remaining ex:umplcs. the system posscsscs a syrnmetry that allows to I-cduct.thc dinie~i\ionality.The dy nwmic\ ol' thc sy riinictric. soli~tionhcan he dcscribcd by equations with two iridepcndcnt spatial variuhlcs. Global rcguliu-ity of solutions 01' such cqu~ttions is proven. Thcrcforc. global dynanlics o n the invariant scts fr,rmcd by the symmetric solutions is well-detined. Nevel-thclcss. the existence of global dynamics near these intinitedimensioni~linvariant sets when the symmetry is broken by a perturbation is fhr from being trivial. For example, when tht: force ,/' and initial data are independent of.^, and periodic bounda[-yconditions are imposed, we obtain the 7,D N S systc~ll.Its solutions can be considered as solutions of 3D NS system: such solutions t'orm an invariant subsut I'OI the 3D dynamics. One may try t o perturb initial data slightly and sludy 3 D dynamics of t~lmost?D vector fields. I n the genetxl case, thc existence of global solutions for gencral (even small) 3D perturbations of Iiirgc 2D data is not known. The difficulty to prove glohal existence eve11 for small perturhations originates from instability o f the dynamics. Global existence of perturbed helical Hows is proven fur special classes vf perti~rbaticms (see Ponce et al. 11671 ['or detail). Existence of'global solutions of the periodic 3D problem for large 3D pc.rturbations was proven by Rnugel and Sell [ 1681 in the case where the spatal domain is thin enough; we hl-iefly discuss the results below. Raugel arid Sell 11681 consider the 3 D N S syhteln with per-iodic boundary conditions in the box
with fixed U I . (12, and a srnall (13. They proved existence of global dynamics for 3D N S system when initial data are bounded. This allows to construct a large bounded invariant set and an attractor of the restriction of 3D NS system to this set. Clearly, when the forcing terrn is sl-independent, the subspace ~ , f " of I?-independent functions L ~ Ois invariant with respect to the action of Sl and, on this subspace, the dynamics is defined for all t < cw. If an 13-dependent perturbation is small, then, for a finite time 1 T, solutions stay close to Hi1' and 7' a cx, ns the perturbation tends to zero. One has to take into account the fact that the dynamics in H(;", though well-defined, is unstable. Therefore, generally speaking, even if S,llo is close to H(:" at t = 0, it may go far from as t --+ m and eventually may blow up in tinite time. But this blow-up does not happen if rr3 is srnall enough. A rough explanation of this phenomenon lies in properties of the Stokes operator A . The space Ho splits into a surn of two orthogonal subspaces H$' and HIiL.The subspace Hi1' consists of vector fields that are s3-independent and N(: is its orthogonal complement. The projection onto H;" is denoted by n'". The spectrum of A restricted to the subspace H(: has the smallest eigenvalue hf 2 (./rif, ( . > 0. Therefore, the linear semigroup expl-IA I contracts H,: at least as expl-tc./rr-:]. When tr? is small, the contraction balances the impi~ctof a boundcd x3-dcpendcnt fi~rcingf ' and of a bounded initial data 140, so Sllln slays near Hi1' li)rzver. To make a proof' out of this observation. onc has to overconie s e r i o ~ ~ t cs~ h ~ l i c~lil'fici~lties. i~l in p;u.ticular to lind thc relation bctwecn (1.7 and thc magnitude ol'the H,:- uncl ~ i ~ ' - c o n ~ ~ o nof' c nI t~ sOand ,f' (see Iftilnie and Rnugel I 1121, Moise et al. [ 1621, Raugel and Sell [ 16x1, Telllam and Zinne 11871 for details). We give si~tficientcoriditiona froni a recent papcr by Iftimic and Raugcl I I I?) for I ( ( / ) to bc I-egi~lar (to bclong t o ~'"(10.+w): Hl,?(I))) for-all I 3 0) when (13 is smiill enough:
<
"'
with 0 < ,' < 112. Hcrc ~ ( o i j> . are thc third co~nponentsofvectors 110. ,/'.These conditions tillow the perturbations to bc largc; lor instance thc fi~nction
with a large ti xed C satisfies the conditions with fi = I /1. Notc that the imposed conditions are anisotropic. For example. the energy 11(1 fi"")~(l)ll~~~,,,, of the .t3-dependent part ( I n V ' ) u o is small for slnall t t s , namely it is of order ~ ' c i l whereas the .r3-independent part f 7 " ' ~ , ~may be taken in the form 10, C o ~ ' i 2 s i n ( . r l01 ) . and has energy of order c'. and thus is large. -
-
R E M A R K The . 2D NS system in a thin domain with combined periodic no-slip boundary conditions is considered by Iftimie and Raugel [I 121, Moise et al. [ 1621, Temam and Ziane [I 871. R E M A R KNo-slip . boundary conditions in a thin domain for the 3D NS system with initial data bounded in an appropriate norm stabilize the dynamics. The dynamics on bounded sets is defined for all times and the attractor consists of one point, see Avrin 131 and Montgomery-Smith [ 1631 for details.
M ~ l l t i ~ ~ ~~l o~ ri ~e idg r o ~ ~One p . ~of . the ways to overcome the difficulty of non-uniqueness of weak solutions is to use theory of semigroups of multivalued operators. The theory of attractors of multivalued semigroups was started by Babin and Vishik 1271. Further works in this direction are by Babin (91,Ball 1341, Melnik 11561, Melnik and Valero [ 1571. In these works, the existence of different types of generalized global attractors for 3D equations is proved. These attractors do not have as good properties as the attractors ofthe 2D NS system. I t is not known if they are ti nite-dimensional in general. The global attractor of the 3D NS system with a large Coriolis force in Theorern 5.4 attracts solutions that may be multivalued for t < T. Therefore it can be considered as an example of a finite-dimensional attractor of a multivalued semigroup.
-
The construction of trajectory attractors (see Section 2 ) can be applied to treat equations without uniqueness, in partici~larthe 3D Navier-Stokes equations. For details of the theory of trajectory attractors, see Sell 1 1 761, Chepyzhov and Vishik 148. 491, Kapustyan and Melnik 11251. Feireisl (711. Trc!jcc,toi-ytrttrtrc~on\.
Metr.r~trc. trttrtrc.toi-,s. Kuksin and Shirikyan in 1 1271 prove the existence of a unique invariant measure for the stochastic nonautonomous 2D NS system. The support of the measure is called a measure attractor. Schmalfuss [ 1781 studies relations between measure attractors and random attractors. Flandoli and Schmalfuss 172.73 1 prove existence of a generalized stochastic attractor for a stochastic 3D NS system.
Capinski and Cutland 14.31 considered attractors of the 3D NS system in the framework of non-standard analysis.
Nor?-sttrr~durdtrtltr1y.si.s.
Modifications of NS and related hydrodynamic equations The methods originally developed for the NS system are applied successfully to many different problems. Here, we refer to some of related papers; the list of papers presented here is inevitably not complete. We did not intend to give a complete bibliography but rather
to show the vast scope and some of directions of continuing mathematical research on attractors of problems related to the Navier-Stokes system. First we briefly mention papers which treat the 3D equations of hydrodynamics which include additional physical effects.
3 0 11rob1stn.s. Lions et al. [142-1441 study the primitive equations of geophysics. The equations in many respects are similar to the 3D NS system but are more complex since they take into account more physical effects, in particular rotation and stratitication, and contain more unknown functions. The authors have built a general theory of such equations and have analyzed their properties; in particular, they have estimated the dimension of regular invariant sets (see also Lions et al. [ 1451). The existence of global strong solutions of the primitive equations is not known in general case and the existence of the global attractor in the classical sense is not known, too. Babin et al. [ 12,161 consider the primitive equations (Boussinesq system) under periodic boundary conditions. They prove that when stratitication or rotation is strong enough, the equations have global regular solutions; the dynamics in corresponding function spaces is well-defined and the global attractor exists, consists of regular solutions and, therefore, has a ti nite fractal and Hausdorffdirnension. (These results are similar to Theorems 5.2-5.4 of this paper.) In a number of papers, moditications of the classical 3D NS system are considered. Moditied equations often have better regularity properties than the original 3D NS equations and in many cases existence of a linite-dimensional global attractor can be proven. Ladyzhenskaya in 1 136.137) considers moditied Navier-Stokes equations which admit global regular solutions. She proves the existence of the global attractor, studies its properties and, in p:~rticulnr.gives estimates of its dimension. Milek and NeCas 1 1501 consider a syste~iiof Navier-Stokes type which has i~niqueweak solutions i n appropriate spaces and prove the existence of a finite-dimcnsion;~lglobal ;ittractor. Milek et al. I 1521 study Boussinesq approximation in three dimeriaions with a moditied stress tensor. They prove existence of a finite-dimensioniil global attractor. Milek and Prakik ( IS I I consider non-Newtonian fluids and prove existence of the global attractor with a ti nite fractal dimension. Bellout et al. 1371 consider non-linear bipolar viscous fluids. Upper bounds are obtained for the Hausdorff and fractal dimensions of the global attractor. Foias et al. (741 consider three-dir~iensionaIviscous Camiissa-Holm equations (also called Navier-Stokes alpha-equations). They prove existence ol'n finite-dimensional global attractor and give an estimate of its dimension i n terms of thc physical parameters of the equations. 21) ~ ~ w b l e ~ i lThe s . theory of attractors for the 2D NS system can be extended to flows on 2D compact manifolds. Such questions were considered by Cau et al. 141 1 and by llyin I I 15-1 I XI. They proved the existence of the attractors and obtained upper estimates of their dimension. The estimates of llyin [ 1 181 have the same fomi as the sharp estimates folthe 2D NS system with periodic boundary conditions (see Theorem 4.2). Cao et al. 141 1 estimate the number of determining functionals (such as determining modes and determining nodes) tor the 2D NS system on a rotating sphere.
214
A. V Buhin
Foias et al. [77] consider the BCnard problem. In the 2D case they prove existence of a global attractor and give an estimate of its dimension. In the 3D case they estimate dimension of regular invariant sets. Ghidaglia and Temam [98] consider the equations of slightly compressible fluids proposed by Chorin and Teniam. They prove the existence of a global attractor and estimate its fractal dimension. Hoff and Ziane [109,1 101 prove the existence of a compact global attractor for the Navier-Stokes equations of compressible flow in one space dimension. This equation does not fit into standard framework due to a lack of compactness. Since properties of the semigroup do not allow to prove finiteness of the fractal dimension, they describe the properties of the attractor in terms of determining nodes. llyin [ 1201 considers a non-autonomous 2D NS system with a rapidly oscillating almost periodic forcing. He proves that its attractor tends to that of the averaged equation when the frequency tends to infinity. Ladyzhenskayaand Seregin [ 1381 study the 2 D equations of the dynamics of generalized Newtonian liquids. They prove existence of attractors and estimate their dimension. Miranville and Wang [ 1581 consider the 2D NS system with a tangential boundary condition (see also Brown et al. 1401).They give an estimate of dimension of the global attractor. They also estimate dimension of regular invariant sets for 3D channel and Couette-Taylor flows. Miranville and Ziane I 1601estimate the dimension ot'the attractor ot'the BCnard problem with free surfaces in an elongated rectangle. Njamkepo 11641 studies a thermohydr:iulic problem for the slightly compressible 2D Navicr-Stokes cquations. Bounds on the fractal dimension of the global attractor are given in terms of the physical data of the problem. Ziane 119.31 considers 2D NS system with free boundary condition in elongated rectangular domains and gives an upper bound on the dimension of the attractor.
Acknowledgements The effort of the author was supported by AFOSR grants F49620-99- 1-0203 and F4962001-1-0567. The author would like to thank Professor E. Titi and Dr. A.A. Ilyin for very useful discussions and valuable remarks.
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I 1901 I I9 I ]
1 1921 1 1931
plicutiot~.s,Collkge de France Seminar, Vol. VII (Paris, 1983-1984). 10, Res. Notes in Math., Vol. 122, Pitman. Boston. MA (1985). 272-292. R. Temam. I~zfitlite-Ditnen.sionulDynutniccll Sy.stems in Mrchanic..~and Phy.sics, Appl. Math. Sci., Vol. 68, Springer-Verlag, Berlin (1988). R. Temam. Al)~)ro.rittlcrtionof'nttrcrc.tor.s, /urge eclclv .simu/r1tion.sund multi.sr~c~le methods. Turhulmr,~ rmcl .stochnstic. proc.e.sses: Kolttro'qoroi<'sideels 50 yeerrs on. Proc. Roy. Soc. London Ser. A 434 ( 1890) ( 199 1 ), 23-39. R. Temam and M. Ziane, Novier-Stokes rylccltions BI rhrre-din~m.sior~u/ thin dotnuin.~with vuriou.~houndur,. c,ortclition.s. Adv. Differential Equations 1 (4) (1996). 499-546. E.S. Titi. Une vclriPtP crl?proximcrntecle I'crttrcrc,teuruniver.se1 rlrs Pquutiorl.~ de Nuvirr- stoke.^. 11on Iiniuire, cle clittrt,n.tion finir. C. R. Acad. Sci. Paris Ser. I Math. 307 (8) (l988), 383-385. M.1. Vishik. Asvmptotic. Behcnjiour of So1utio11.sof Evolutionary Equrrfion.~,Lezioni Lincee, Cambridge University Press, Cambridge ( 1992). Y. Yan, 1~i1~tett.siorr.s c ~ rrttrcrc.to,:c f fijr cli.sc~,r~ti:c~tio~~,fi)r Nuvier-Stokr.~r~qucrfion.~, J . Dynamics Differential Equations 4 (2) (1992). 27.5-340. V.I. Yudovich. 111,stclhili~ ~fpcrrtrllrl/foit,.sofcr vi.sc.ou.s inc~~~n~pre.s.sihle ,fluid with rr.spec.1 to prrturhrrrion.~ pc,riodic. in .sl)crc.r.Zh. Vychisl. Mat. i Mat. Fiz. Suppl. 6 (4) ( 1966). 242-249 (in Russian). M. Ziane. Olitit~rcrlhounc1.s or1 rho rli~?~rr~.sion of 1/10 trttrrrc.tor of the Nuvier-Stokc,.s pqrrotiotl.~.Phy\. D 105 (1-3) (1997). 1-19, M. Zinne. On the t r t ~ o - d i ~ i ~ c ~ r rNtrvier-Stokr,.c .tio~~(~/ rc/rrtrtio~t.s icir11thYfrr,r holrnclcrr~i~ontlitior~. Appl. Math. Optim.38(1)(1998). 1-19,
CHAPTER 7
Stability and Instability in Viscous Fluids Michael Renardy and Yuriko Renardy I)o/)~rtt?l(.II~ of'M~1the117cttic.c. B/rrr~!i.rhur~. VA 24061.0123. USA E-lll~l;/: ~l~ll
Cor1tet1t.s I . Il~troduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 2 . M:ithrnlatic;il atl;ilysi\ o f stability a ~ l dhii~irc;itio~i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 2.1. 1.i11cnr\tahility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 2.2. N o r ~ l ~ ~\tahility ~ c a r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 2.3. Inv:iria~it t ~ ~ ; ~ ~ i r l ' o. l d. \ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 2 . 4 Siinplc hiiurcariori\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 2.5. Bil'urcarion\ with sy1111nctry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 2.6. A ~ ~ ~ p l i t uc qduca ~ i o ~for l s \pari;~lly inlinilc \ysrc'lil\ . . . . . . . . . . . . . . . . . . . . . . . . . 23X 2.7. A h \ ~ l u r cand co~ivcctivcinsrahiliry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 3 . C o ~ ~ v c c t i llow\ o ~ i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 3.1. The BC11;ird prohlc~ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 3.2. M i i r n ~ i g octr~~vcction ~~i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 3.3. Ilouhly d ~ l l u s i v cCOIIV~C~IOII 3.4. Two-layer tlowx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 3.5. Maglietic liclds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 4 . Flow hcrwccn rotating cylilldcrs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 4 . I. The T.iylor p r o b l m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 4 7 . The Dean and Giirtlcl- prohlcn~s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 4.3. Countcrrorntinp cylinder\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 4.4. Viscoclatic llows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 4.5. Two-l;iycr flow\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750 5 . Parallel \hear llow\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 5.1. Tlic cigc~ivaluc~ ~ r o h l cSoll l ~parallel shear flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. No11-lllcrdalgrowth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Nearly parallel flow .; and secondary instahilirics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Viscoclatic flow\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Two-layer flow\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Capillary hrcakup o i jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Linear brahility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Onc-di~iicnsional~iiodels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H A N D B O O K OF M A T H E M A T I C A L F L U I D D Y N A M I C S. V O L U M E I1 Editcd by S.J. Fricdlandcr and D . Scrrc O 2003 Elsevier Science B.V. A l l rights reserved
6.3. Finite time breakup . . . . . . . . . 6.4. Similarity solutions . . . . . . . . . 6.5. Suppression of breakup by elaticity References . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
277 278 281 282
"He is going round and round", said Roo, much impressed. "And why not?" said Eeyore coldly. "I can swim too", said Roo proudly. "Not round and round", said Eeyore. "It is much more difticult. I did not want to come \witnming at all today", he went on. revolving slowly, "But if, when in, I decided to practise a slight circular tnovernent from right to left - or perhaps I should say", he added. as he got into another eddy. "From left to right. just as it happens to occur to me. it is nobody's business but my own". (A.A. Milne, The House at Pooh Comer)
1. Introduction The systematic study of hydrodynamic instability began roughly a century ago, with the experimental investigations of Reynolds [ 1391 on parallel shear flows and BCnard [ 101 on convection. and the theoretical studies of Lord Rayleigh. Since then, instabilities in fluid motion and the new flows arising as a result of such instabilities have continued to pose a challenge to mathematicians, which has influenced and inspired progress in several fields of mathematics, e.g., dynamical systems, partial differential equations, functional analysis and asymptotics. A comprehensive review of all the work which has been done on hydrodynamic stability - or even all the important work which has been done - would be impossible in the space of this article. and we necessarily had to be selective. We have focussed our exposition on problems in simple geometries and under simple boundary conditions; these are the type of problems which are most accessible to mathematical i~nalysisand have been most thoroughly studied. Our review begins with a general introduction to mathematical issues arising in the study of stability and bifurcation. We discuss the connection between linear stability and spectrum, nonlinear stability. the derivation of reduced equations near the onset of instability. and the analysis of bifurcations. For the remainder of the article, we discuss hydrodynamic applications grouped around four general topics: thermal convection, flow between rotating cylinders. parallel shear Hows. and capillary instability ofjets. For each topic. we have tried to strike a balance between providing a basic introduction and reviewing some of the "classical" questions and results and discussing more recent research and problems of current interest. A number of books on hydrodynamic stability have appeared over the past forty years, covering many of the topics in this article in more depth and many other topics which we have not discussed: we refer in particular to the books of Betchov and Criminale 1 1 31, Chandrasekhar 1251, Chossat and looss 1281. D r a ~ i nand Reid 1.781. Godrkche and Manneville 1481, Joseph 1701, Joseph and Renardy 1721. Lin 1851. and Straughan 1153l.
2. Mathematical analysis of stability and bifurcation
We consider the nonlinear equation
.i = F(x.t),
226
M. R e t ~ u nc~rld l ~ Y Rrnurdv
where x and F take values in some Banach space, and we assume some solution xo(t) is known. We then consider the equation for perturbations. With x = xo(t) u, we have
+
li = F(xo(t)
+ u. t )
-
F(xo(t), t ) .
(2)
The question of stability is concerned with whether u remains small if u(0) is small.
D E F I N I T I O NWe . call the solution .ro(t) stable if for every E > 0 there is a 6 > 0 such that [lu(t)ll < E for all t > 0 if Il~r(O)ll< 6 It is called asymptotically stable if, in addition, jlu(t)jj + 0 as t -+ cc as long as ((u(O)(jis sufficiently small. It is reasonable to expect that "small" perturbations are governed by a linearized equation (we shall say more about the rigorous justification of this assumption in the next section). We assume that F is, in some sense, differentiable with respect to x and we denote the derivative by D F. The linearization of (2) is
For the rest of this section, we shall focus on the case where F and xr) are independent o f t . With A = DF(.\-(,).we then have the equation
For tinite-dimensional systems, i t is well known that the zero solution of ( 4 ) is stable if all eigenvalues of the matrix A are in the open left half plane and unstable if there is an eigenvalue in the right half plane. In infinite-dimensional systems, the situation is more involved. The general abstract formulation concerns operators A i n a Banach space which are inti nitesimal generators of a CO-semigroup(see 1961 for an introduction to operator semigroups). In this case. we can deti ne the spectral bound
where a ( A ) denotes the spectrum of A, and the growth abscissa or "type" of the selnigroup
The appropriate generalization of the ti nite-dimensional stability result is then that o(A) = r ( A ) . This assumption is implicit in all studies of hydrodynamic stability which infer stability from calculations of spectra. The assertion that w(A) = r ( A ) is known for a broad class of evolution problems (see [S] for a review). For instance, it is known to be true if e"' is continuous in the operator norm for t greater than some to. This includes analytic semigroups. Since i t is well known that the Stokes operator generates an analytic semigroup in any LI' space (see, e.g., Section 1.6 of [ 174]),this result covers in particular all linear stability problems for the incompressible Navier-Stokes equations.
On the other hand, there are many hydrodynamic stability problems which involve other physical effects for which diffusive mechanisms may be negligible. For instance, mass diffusion is seldom considered in the mass conservation law for compressible fluids. The same can be said about viscoelastic stresses, and, depending on the relevant time scales, it may also be true for solute concentrations, temperature or any of a number of other quantities. As a result, the coupled equations for problems of this type involve a hyperbolic part which precludes the application of abstract results to the effect that w ( A ) = r ( A ) . Indeed, for hyperbolic partial differential equations it is generally not true that w ( A ) = r ( A ) . A counterexample is given in [I 171. The example considered there is the equation
with 2 ~ - p e r i o d i cboundary conditions in both directions. This problem is associated with a Co-semigroup in the space H ' x L' for the pair ( u , u , ) , with the operator A detined as
I t is shown in [ 1 171 that r ( A ) = 0, but w ( A ) = 112. Even in finite-dimensional situations, the assessment of stability based only on spectra can be misleading. As an example. consider the system
with initial condition . r ( O )= I . ~ ( 0=) 0 . The solution is
Even though ~ ( t decays ) exponentially as t + co as stability theory says i t should. ~ ( t tirst increases and reaches a maximum o f IOO/i) = 36.79 at t = 1 (see Figure I ). This value at the maximum is substantially larger than the magnitude of the initial condition. Transient growth similar to this example plays an important role i n transition to turbulence i n parullel shear flows. The notion of pseudospectrum (see the review in 1 1 6 1 1 ) is helpful in understanding examples like this. The e-pseudospectrum of A is detined as the set of all k for which
We can detine an F spectral abscissa r-, ( A ) as the supremum of the real parts of the e-pseudospectrum. If A is a normal operator in a Hilbert space. the E-pseudospectrum consists of all points which are within distance E of the spectrum, and r-,.(A)= r ( A ) + c . If, however, A is not normal, the e-pseudospectrum can be much larger. For instance. for the example above, it is easy to show that
)
m.
and hencc thc F-pseudospectrt~~ii includes a disk centered at - 1 with radius There is ;I link between transient growth and "bad beh;~vior"ol'the pseudospectrum. More precisely.
For thc cxample ahove. we have r, ( A ) 3
-
I
+ m,so that
which reaches its maximum at 25 for F. = 0.03. Another aspect determined by the F-pseudospectrum is how large a per~i~rhation to rl niust be for a point to be in the spectrum of the perturbed operator. Indeed. the F pseudospectrum is precisely the set o f all 1 for which there is an operator A A such that 11 A A I l 6 E and A is in the spectrum of A A A . In the example above, 0 is in the I / 100pseudospectrum of A , and. indeed, it is an eigenvalue of
+
In inti nite-dimensional Hilbert spaces, it is known [45,61,102,651 that w ( A ) = inf r, ( A ) . f
>o
(16)
In contrast to the finite-dimensional situation, the right-hand side need not equal r ( A ) . Indeed, the counterexar~iplesin the literature rely on nonnormality of the operator A . The well-known example of Zabczyk 11751 consists of an infinite matrix which contains an infinite string of Jordan blocks. The Nth block by itself produces algebraic growth, like
but in the limit N + cc exponential growth is obtained. The PDE counterexample 1 1 171 cited above is stri~cturallysimilar to the Zubczyk example. It is not known whether cases where w ( A ) > r ( A ) actually occur in hydrodynamic stability, but neither are there general results to preclude this in situations where hypert s viscoelastic Rows (to the effect that bolic PDEs are involved. Some partial r e s ~ ~ l for ( ! ) ( A ) = r ( A ) in restricted situ:ltions) are obtained in 1 1 17.121 1.
111 general, wc expcct that solutions which arc lincarl y stable arc also stable solutions ol' the nonlinear equations. as long as the initial perturbations are sufficiently small. For systems of ODEs, the idea to prove this is rluitc simplc. Considcr :I systcnl of [he form
where .v takes values in R" and N is a smooth fi~nctionfrom R" into R",which is oi' quudr:~ticorder near thc origin: 11 N (.I-) 11 < C 1I.v 1)'. We can construct the solution oi. ( 18) as ;I tixed point of an itcrotion:
I f all cigenvulues o f A are in the left half planc. it is not hard to hhow that the l i m i ~of the iteration inherits the exponential decny from the linear problcm. provided that Il.roll is small enough. The snme idea also works for the Navie1.-Stokes ccli~alion\;. For inslarlce, it i~ shown i n 1 1741 that exponential linear stability of a stationary flow also implies nonlinear stability to s~nalldisturbances in the space L'' for 11 > 3. Hyperbolic equations are Inore complicated. For instance. if wc considcr the equation
with periodic boundary conditions. we obviously have exponential stability of the linearized problem
We have exponential stability of the nonlinear problem, too, but we cannot infer this by simply treating the nonlinear term as a perturbation. Indeed, if the equation were
then we would lose stability and even well-posedness. The typical proof of stability for (20) proceeds by multiplying the equation by u and integrating over one period, which leads to
This, of course, implies exponential decay of u in L*. Unfortunately, the range of situations where such energy estimates are readily available is rather limited. As a consequence, there is an extensive mathematical literature on stability of the rest state of, for instance, viscoelastic fluids, but relatively little on stability of other flows 155,121 1. We note that, in the preceding example. the conclusion about exponential decay does not depend on smallness of the initial data, as long as a sufficiently regular solution exists (which is actually not the case in this example). Indeed, energy estimates are often useful for establishing stability with respect to large perturbations. Consider the Navier-Stokes equations in a bounded domain a. for a flow driven by a prescribed body force f:
with Dirichlet boundary conditions v = 0 on i)Q.Let (vo. po) be a stationary solution, and let v = vo u, 11 = /I() y. The equation for the perturbation (u. y ) is
+
+
If we now multiply by u and integrate over Q, we find
From this, we can infer exponential decay to zero as long as IVvoj remains less than rl/p times the first eigenvalue of the problem
In terms of dimensional analysis, if V denotes a typical velocity scale and L a length scale for the region, then IVvol is of order V/L and the first eigenvalue is of order I / L Z , hence the condition for the energy argument to apply is that V/L is less than some constant times r 7 / p ~ 2 in ,other words, p V L / q must be less than some constant. This combination is known as the Reynolds number. We refer to the book of Straughan [ 1531 for a review of the use of energy methods in hydrodynamic stability. One approach to proving instability when there are eigenvalues with positive real part is to use an invariant manifold theorem. We shall discuss such theorems in the next section.
This approach is applied to the Navier-Stokes equations in 11741. It requires a spectral separation condition: The spectrum of the linearized problem is required to be the union of two parts a1 and 02, where al lies in the half plane R e 1 > a > 0 and 02 lies in the half plane R e 1 < b < a . For problems in bounded regions which have discrete spectra, this always holds, but for unbounded regions the spectrum may be continuous. However, as pointed out in 11741, instability for the Navier-Stokes equations can be proved even without a spectral separation condition by adapting the proof in 135, Section VII.21.
An invariant manifold for a differential system is a manifold with the property that solutions which start on the manifold and follow the evolution prescribed by the differential equation remain on the manifold. Invariant manifolds are particularly useful if their dimension is much smaller than that of the original system and the dynamics of interest occurs on the invariant manifold. For the study of instability and bifurcation, the center manifold is of particular interest. We shall begin with a discussion of the simpler case of stable and unstable manifolds. Consider a system of ordinary differential equations
<
where .r E R". and N is smooth and of quadratic order near the origin: j l N (.I-)/I C11.1-j 1 2 . Assume that A has eigenvalues of positive and negative real part. but n o purely imaginary eigenvalues. Then we can decompose R" into two subspaces Y and Z. both invariant under A , such that A ( y has only eigenvalues with negative real part, and A li: has only eigenvalues with positive real part. Let .r = ! :be the corresponding decomposition of .r, and let P and Q denote the projections of R" onto Y and Z. We can then write (28) in the form
+
Suppose we have a solution of (29) which decays to Lero as t rewrite (29) in the form
For "small" solutions, this suggests the iteration
-+ m.
Then we can
It can be shown that all snlall solutions which converge to zero as t + ffi can be constructed as fixed points of this iteration. As a result, these solutions are parametrized by ~ ( 0 The ) . corresponding initial values ~ ( 0lie) on an invariant manifold z = z(v) which is called the stable manifold. Likewise. small solutions which tend to zero as t i -cc can be constructed as tixed points of the iteration pA('-.')
PN
(!I"
(.v)
+
:ll(s))
ds,
These solutions lie on a manifold y = y(z), known as the unstable manifold. Versions of the stable and unstable manifold theorems for the Navier-Stokes equations and applications to stability studies are discussed in 1174). For situations near the onset of instability, we need t o consider situations where the spectrum of A has a part on the imaginary axis, in addition to a stable and unstable parts. Let .r = u 11 trl be the decomposition into the stable, unstable and neutral parts, and let P . Q. H be the correspunding prc!jections, Then we have the system
+ +
Supposc now that we want to lind solutions o f ( 3 3 ) which arc ho~rnclcdboth for t -. w and for I + - w . We can rewrite the first two equations in the form
In analogy to the discushion above.
il
i~ natural t o try solving ( 3 3 )by the iteration
The problem with this iteration is that, in the case of neirtral spectrum, we have no control of the growth of solutions when nonlinearities are included, and hence we cannot obtain
adequate bounds for w"+' in (35). The cure for this is the introduction of a cutoff function, ) a smooth function which truncates terms nonlinear in w unless w is small. Let ~ ( w be which is equal to 1 for Ilui/l < 1 and equal to zero for Ilwjl > 2. Then we replace (33) by
For this modified system, it is possible to construct an invariant manifold of the form u = u(u1). u = u(w). Of course, in a sufficiently small neighborhoodof the origin, the truncated
system is equal to the original one. THEOREM Cot~.sicler . LI .SJ-S~PI)I (?f'ODE.sc?f'th~,fi)rtn (33) with the u.s.sun1ptiot7.sr1.s strrted a h o \ ~ Then, . in sortlc) nrighhorhoocl of'thc origin, there wist.s rrn invrrrirrnt nirrnifolrl c!f'thp
+
fi)nn I ( = 45(111), ti = $(711), n~hc)rc.q5 (rnd 1// rrrr>.s~nooth,fi~n~~tion.s rrtlrl Il$(ui) 11 l]$(ui) 11 = 0(11 111 /I2),fi)r .sni,rII 11 UI 11. Tho in\~rrirrnt111rrni/i,lr1,( . r r I I ~ dthe r.rn/er tnrrnif?)ld, hrrs t11eji)llo,t.ing ~ ~ r o l ~ c ~ r t i r ~ s : ( I ) If'(/,sol~~tion i.s ,s1?1rrI/,fi)r (111 t , it tn~(.stlie on thr>(.rvltr>rtn(rt7if~)Irl.
(2) I/' t110 ~ ~ ~ r , s t ~1)crrt r l ) l r ~(i.r>..tlle r~~/rirrtiotl ,fi)r
11)
i.s
O / I . S P I I ~ ,t11r>t1 ,strrhi/ity
of' (1 .st11r11/
. ~ o / r ~ t i oi.snrlr~tr~t~~trinr~(l 11)..str~/)i/it\~ ~ , i t / r i n/ / I ( ,( ~ , t l t (n~rr( r ~ ~ ~ f i ) / ( l .
The center ~nanifoldis not unique. because it depends on the choice of the cutoff function x employed in the proof of the result. However. small solutions lie on every center manifold. Moreover, if' the nonlinear terms in the differential equation are analytic. then the Taylor expansion o f the center manifold c:un be computed uniquely. i.e., Taylor expansion of different center manifolds agrees to a11 orders. I n studying proble~nsof onset of instability. one ~ ~ s ~ ~considers a l l y differential systems depending on it parameter .i.= A(/i).\- + N(.t-. j i ) . Here A ( / L )is such that, for some critical value / i = / L O , there is an eigenvalue of zero real part. We can put such a systeni in the context described above by simply adding 11 to the list of unknown variables. with the i= 0. trivial differential equation i A discussion of versions of the center inanitold theorem which are applicable to infinitedimensional systems such as the Navier-Stokes ecluations can be found in 116.11.
2.4. Simple h1fiirc.crtiot7.s
After the reduction to a center manifold. the next step consists in characterizing the behavior of solutions on the center manifold. This is the ob.jective of bifurcation theory. We consider n differential system depending on a parameter:
i = A ( ~ ) . u+ N ( x . p ) ,
(37)
where N is smooth and of quadratic order near .x = 0: 11 N ( x , p ) j / 6 C I I . ~ I I ?We assume that A ( p ) has eigenvalues of negative real part for LL < 0 and at p = 0 one or several
eigenvalues cross the imaginary axis. Consequently, the solution x = 0 is stable for p < 0 and loses stability if p > 0. The consequence of the instability can be any of the following three phenomena: ( I ) Evolution to a stable state which is a small perturbation of the zero solution. (2) Evolution to a stable state far away from the zero solution. (3) Evolution to a singularity. We shall see examples of all these possibilities in our discussion of hydrodynamic instabilities below. Bifurcation theory is concerned with the systematic use of perturbation methods in case the tirst possibility applies. We refer to [88,29,54,50,51,67,79] for introductions to this subject. In this section, we shall briefly discuss the two simplest cases: (1) A simple real eigenvalue crosses through 0. (2) A pair of simple complex conjugate eigenvalues crosses the imaginary axis. We shall tirst discuss the case where a simple eigenvalue crosses through 0. By the center manifold theorem, we can reduce the problem to the case where x is one-dimensional. The differential equation then has the form
We assume that cu > 0, i.e., as /L goes through zero, the eigenvalue crosses zero with nonvanishing speed. If we look for nonzero stationary solutions, we divide by .\- and tind
If # 0. the implicit function theorem guarantees the existence of a branch of solution!, of the form
This case is known as transcritical bifurcation. since solutions exist for both positive and negative values of p . On the other hand, many problems in applications have syrnrnetries where the right-hand side of (38) is an odd function of .r. In that case, the equation for nonzero stationary solutions is
We can solve this for p as a function of x:
Near the origin, solutions exist only for one sign of p , nariiely for p > 0 if y < 0 and for p < 0 if y > 0. This type of bifurcation is known as a pitchfork bifurcation. Linear stability analysis of the bifurcated solutions shows that, both for the transcritical and the pitchfork case, supercritical branches ( p > 0 ) are stable, and subcritical branches are unstable.
If a complex conjugate pair of simple eigenvalues crosses the imaginary axis, the center manifold reduction yields a two-dimensional problem. We can write the reduced equation in terms of a complex amplitude 2 . If we truncate at cubic terms, we find a system of the form
i = iwz + a p z + 8 ,z2 + p2z? + p 3 j 2
+ y l z 3 + y2z2i + y3zz-- + y4i3+ 0(lz14+ 1z2p1 + jZp2j). 2
(43)
We assume that the eigenvalues of the linearized problem cross the imaginary axis with positive speed, i.e., Recu > 0. This equation can be simplified further using the theorem on normal forms (see [28, p. 331). This result implies that there is a nonlinear transformation z = u, + a , w 2 + u2wG + u32Ll2+ . . . , such that the transformed equation has the form
If we now truncate the equation,
we can tind explicit periodic solutions of the form
U I=
R exp(i I J ~ ) , where
Perturbation methods can be used to show that the solution of this system is the leading order term for a branch of time-periodic solutions of the full equations. These solutions are supercritical and stable if Re 7 < 0 ; they are subcritical and unstable if Re y > 0 .
Symmetries both complicate and simplify bifurcation problems. On the one hand, they force degeneracy of eigenvalues. leading to higher-dimensional problems than the simple bifurcations discussed in the previous section. On the other hand. symmetries also reduce the complexity of the resulting bifurcation equations (e.g., by forcing certain terms to vanish), and this simplifies their analysis. We consider an evolution system
which has a group U of symmetries. That is, there is a group U of linear operators such that, for every f E U , we have
We note that if x o is an eigenvector for A ( b i ) ,
the11 Txo is also an eigenvector for every r E U . This forces multiplicity of eigenvalues, unless xo happens to be invariant under U . In this section. we shall discuss two examples of bifurcation with symmetry: bifurcation of steady solutions on the hexagonal lattice. and Hopf bifurcation with O(2) symmetry. Bifurcation on the hexagonal lattice leads to a problem in six dimensions; we shall write , and z?. The original the bifurcation equations i n terms of three complex amplitudes, z ~ :,z, physical problem concerns finding doubly periodic functions with respect to a hexagonal lattice, and we can think o f ; 1 multiplying a Fourier component proportional to exp(ikx), ,-2 nlultiplying a Fourier component proportional to exp(-ikx/2 i k v & / 2 ) , and z3 nlultiplying a Fourier component proportional to exp(-ik.r/2 - i k y f i / 2 ) . The symmetries required of the equations are: ( I ) Translatio~~ in x: (:I. :z, :?) + (pi@:l, P-'@'~:,~, ( 2 ) Translation in y: (:I. , - 2 . : j ) -+ (,:I, ~'d';.~, r-i'b:3). (3) Reflection across the origin: (,- 1 . :2, :3) + (?I. 52. :Z). (4) ReHection across the .t- axis: (,-I, :. z 3 ) -t (,-I, ~ 3 :?). , ( 5 ) Rotation by 120 degrees: (;.I. ,-2. : J ) + ( ~ 1:3,. :I ). The followir~gequ;~tionsclescribe the dynamics o n the ccntcr manifold when tcrms up to cubic order are included:
+
:I
=u(ll):l
52
I
i.3
=u(~):7
ff(j1):2
+ p(/l):2:3 + y1(/1)1:1 12:1 + y 2 ( j ~ ) ( l ; ? l ~ + l:)~~);~. + P ( / l j ? ? ? l + YI )I:?I':Z + ) ( I z ~ I '+ I;[ l'):?. + P ( / l j ? 1 5 + ~ I ( / o I : ~ I ~+ : ~y ~ ( j i ) ( ~ : ~+l I~: ~ I ? ) , - ~ . (}l
(50)
The coelficicnts u ( p ) . p ( j l ) , YI ( p j 3 and ~ ' ( 1 1 ) arc real. We ;)rc interested in the onset of instability. so we assume u ( 0 ) = 0 . u ' ( 0 ) > 0. We remark that there seems to bc an intrinsic inconsistency in including both qundr:~tic and cubic terlns, since for sniall solutions the quadratic terms should dor~iinateover the cubic terms. There are two reasons for including the cubic terlns in the ~ini~lysis: ( I ) There is a class of solutions given. for instance, by ,-2 = ,-3 = 0 for which the quadratic tcrlns vanish. ( 2 ) I t can be shown that no stable solutions of small iimplilude exist unless the coefficient of the quadratic terms is assumed srnall of the same order as the magnitude of the solution. Hence it is reasonable ro analyze this case for which ( S O ) is the appropriate system to study. The symmetries can now be used to classify stationary solutions of ( S O ) . We ~iiultiply - - the ith equation by i;. Setting r , = I;~ 1.' L/ = ;.I :2:3, we f nd
In particular, this implies that y is real as long as B ( p ) # 0. Subtracting the first two equations. we ti nd that either r.1 = r l or
We can repeat this for any other pair of equations. As long as we rule out special cases (specifically yl ( p )= y l ( p ) and yl (/L) = - y z ( p ) / 2 ) , we can conclude that at least two of the r;. are equal. This leads to three classes of solutions: ( 1 ) Rolls: Only one of the r; is nonzero. (2) Hexagons: rl = r2 = r.3 # 0. (3) Rectangles or "patchwork quilt": r.2 = r7 # rl . For each of these classes, i t is easy to do the reat of the algebra to solve (5 1 ). Each class of solutions can be classified according to their symmetry. Rolls arc invariant under translation in one direction and reflection across the axes. Hexagons ;ire invariant under reflection across the axes arid rotation by niultiples of 60 degrees. Rectangles have lesser symmetry: they are invariarit only by reflection across the axes. Rolls and hexagons art. cxamplcs of the equivariant branching lenima. We can obtain the roll solutions (~noduloii symmetry transformation) if we set :I = .\ real, ~2 = ,3 = 0, and we can obtain the hexagons if we set ,-I = .-2 = :j = .v reill. In either case. ~ h cqualions c ~ x d u c eto the siniplc eigelivulue bifur. reducecl problcnl ic n o t \ullicicnt cation discussed in the prcvioi~ssection. ( 0 1 ' c o u r ~ ethe to clrtrl'riiine stability ot'solutions.) Rectangles. o n thc o ~ h c hand, r havc less synimctry ancl rccluirc two independent amplitudes for their description. The Hopf hit'i~rc:ttionwith O ( 2 ) syni~nctryis itssociutctl with a complex conjugiitc pair ol' double eigcnvulucs crossing the imaginary axis. Wc can describe the evolution o n thc ccntcr ni:uiil'old by two coniplcx :rrnplit~ldcs.:I and , - 2 . In the physical prohlcni. we can think ol' :1 as beirig associated with :I tilode proportional lo cxp(ik.r.)anrl :? with a motlc proportion;~lto e x p ( i k . ~ The ) . synililctrics arc: ( I ) Tri~nsl~ition: (:I, :?) + (oi"'; 1 , c-'~':?). ( 2 ) Reflection: (:I. :?) -+ ( - 2 . :I ) . Transformation to normal torm leads lo the additional symmetry (:I . ~ 2 4) (iJ"/',I. rid':?) . Up to cubic ordcr. the ecluations o n the centor manifold. written iri no~.malform. are
We LII-eintercstcd in the onset of instirbility where t r ( 0 ) = iro. Re ~ ' ( 0 >)0. Wc can immediately identify two rcduccd situ:rtions l ' o ~which Ihc cqlr:itions reduce lo the simplc Hopf bif'urci~tiondiscussed previously: ( I ) :?=O(or:l = O ) . ( 2 ) , I = ,2. The tirst case corresponds to ~ravelirigwaves, i.e.. the reculting periodic solutions satisfy
238
M. Rrr~arL!\.und Y Ret~ardy
for some s. That is, they are invariant under simultaneous translation and time shift. The second class includes solutions that are invariant under reflection; they are called standing waves, since they result from interaction of right- and left-traveling waves of equal amplitudes. The stability result for simple bifurcations that supercritical branches are stable does not carry over to bifurcations with symmetry. For instance, in the case of Hopf bifurcation with O(2) symmetry, it can be shown that if both bifurcated solutions (traveling and standing waves) are supercritical, then one of them is stable; if at least one of them is subcritical, both are unstable.
2.6. Amplitucie ecjuatiorzsfi)r sputiallj injinite systems In spatially infinite systems, spectra are generally not discrete and the mode causing instability is part of a continuum. At any point above the stability threshold, there is therefore a whole continuum of unstable modes, which allows for a much more complicated dynamics than the interaction of finitely many modes considered in classical bifurcation theory. We refer to the article of Fauve in [48]for an introduction to this subject. In this section, we shall discuss the derivation of a m p l i t ~ ~ dequations e and some of the instabilities they exhibit in the simplest contexts. We consider n partial differential equation involving the independent variables of time t and a spatial variable .r; the dornain is the entire line -co < .r < co,and we assume invariance under translations in x. We write the equation in the form
where L contains linear terms and N quadratic and higher order terms in r r . We have indicated in our notation that both L and N involve differentiations with respect to .r. Due to the translation invariance of our problem. we can use Fourier transform for the linearized equations, leading to an eigenvalue problem of the form
for a Fourier mode proportional to exp(ika).The simplest situation to consider, analogous to the simple bifurcations discussed above, is that of an eigenvalue h ( p . k) which reaches either zero or a purely imaginary value at a critical value k = ko. We shall consider the following two situations: ( I ) The equation is invariant under reflection in .r and there is a simple eigenvalue h(1-1.k) of L ( p . ik) such that h(0. +ko) = 0. The rest of the spectrum of L(0. & ) is stable. Near p = 0, k = ko, we have h ( p , k) = a p - 6 ( k - ko12 +0(1p12 IpIIk - kol Ik - ko/'). Here a and S are real and positive. (2) The equation is not invariant under reflection in x . There is a simple eigenvalue h ( p , k) of L ( p , ik) such that h(0, +Lo) = k i w and the resf of the spectrum of L ( 0 , & ) is stable. Near p = 0, k = k ~we , have h ( p , k) = i o + u p ic(k - ko) -
+
+
+
+
+
+
O ( I / L I ' Ipllk - koI Ik - ko17). Here w and c are real, while CY and 6 itre complex with pmitive teal parts. be the eigenfunction of L(0. i k o ) : L ( 0 , i k o ) u o = O or L ( 0 , i k o ) u o = rwuo, reLet spectively, Formally. we can reduce the equation in a manner which is analogous to center manifold reduction by making the ansatz S ( k - ko)'
At leading order i n
E,
this leads to the Ginzburg-Landau equation:
The only differcrlce between the two cases is that in the tirst case the coefficients CY, P , and S arc real. while in the second case they are complex. The further discussion will focus on the case ol'supercriticul bifurcation where P has a positive real part (note that we have already assulned CY and B to have positive real parts). Although the derivation 01' the Ginzburg-Landau cquation has much formal analogy hel ; ~ ~ l a l o g oto ~ls with ccntcr ~n:uiilOldreduction, there is no theorem known which w o t ~ l ~ the center ~nanifoldtheorcn~.Thert: ha.; been significant recent progress in es~nblishinga rigorous connection hctween solution5 of the Ginzburg-1,andau equation and solutions to 1. All thcsc rssults eatahlish the the full dil'lrential equation I 1X.33.40,7~~.147.119.157.1~~5 validity ol' the Ginzhurg-Lnndiu~cquation :IS a n approxirn;~tionon a finite. although large. time interval. Qucstiorls rcgiuding thc behavior I'or 1 + remain to be answcrtcl. The Gin~hurg-Landauequation ( 5 9 ) has the solutions mX
where
Here a,.,a, denote the real and imaginary parts of a . I t is natural to consider the stability of thehe solutionx. Without loss of gencri~lity,we assume that A. is 1-et11and sot
and obtain the following lineariled equation for the perturbation C':
Let us first consider the case where p and 6 are real. In this case, we rewrite the problem as a system for real and imaginary parts and take the Fourier transform. The resulting linear
stability problem is governed by the eigenvalues of the matrix
The trace of this matrix is negative. and the determinant changes sign when
Positive eigenvalues occi~rtherefore (for k small enough) if B A ; < 2 S q 2 . In view of (61), we can rewrite this as 3 S y 2 > a . The instability in this case is known as the Eckhaus instability [39]. For 6 and S complex (case 2 above), we focus on the special case y = 0, when linear stability is governed by the eigenvalues of the matrix
The trace of the matrix is negative, and the determinant is
An instability occurs if 6 6 + p8 < 0, which is known as the Benjamin-Feir instability I I I 1. The Eckhaus and Ben.janiin-Feir instabilities are the sin~plestinstabilities which result from the interaction of a continuum of modes. Such instabilities itre known as sideband instabilities, since they concern the instability of a spatially periodic solution with respect to slow modulations. Other types of sideband instabilities can arisc from interactions with long wavelength nodes in systems which have a neutral mode at wavenuniber 0 (typically due to conservation laws) and in systems which are infinite in two spatial di~nensions.
We consider a linear system of the form
for x on the entire real line. We can use Fourier transform in .r:
and we obtain the equation
Onset of instability occurs when L ( p , i k ) has a purely imaginary eigenvalue for some k
We now consider the question whether instability actually leads to growth of solutions at a given location in space, or it propagates out to infinity at the same time as it grows. Let us consider (68) with an initial condition of compact support. By superposition, we can reduce this problern to the initial condition
The solution of the initial value problem is obtained by Laplace transform in time, leading to u(.\. t ) =
(h
-
-
L i p . i k ) ) p ' u o p ' " ' + n f d k dh.
Here y is chosen to bc any s~ifficirntlylarge real number. If the problem is stable. then ( h I , ( / L . ; A ) ) - ' exists in the entire closed right half plane. and (under appropriate assumptiotns on the behavior t'or large k and h ) we can show exponential Jccay of by shifting thc integration contoirr for h in (72) into the left half plane. If. on the other hand. there is an instability. then ( h- L ( p . ; A ) ) - ' has a singul~uityin the right half'pl:~nc for some rei~lk . Hence we cannot shift Ihc contoill- t'or h to the left hall' plane it' we keep the k intcgrntion o n the real axis. We 11nily. however. still be able to shil't the contour for A i f we ~ i ~ l l ~ ~ l t i ~shift ~ ~ ethe o lCOIIIOLII. ~ ~ l y for A . To ;~ssessthis possibility. spectrun~".i.c., we fix h iund considcr thc vi~lucsof A for which we considel. the "spi~tii~l (3, - I , ( / i .i k ) ) - I is si~igiil:~~.. I f the real part o1.h is I;~rgce~iougli.the spilti;ll spectr~inldocs consist ot'two separate parts. C + ( h ) not intersect the real axis. and hence i t will gener~~lly in thc upper hull'pliunc 2nd X - ( h ) in the lowcr half pliunc. Suppose now that C + ( h )iuncl C - - ( Aremain ) SepiKiitc t i ~ every r A in the closed right half plane. I f this is thc case, wc may. under appropriate conditions it( infinity. shift the contour for into thc left h:tll'pl:unc. while at thc s:umc time shifting the contour l i ~ X.r in such n way that i t continlies to sepiuate C+(A.) and C P ( h ) For . any fixccl .v. we thcreforc expect thc solution in ( 7 7 )to decay us r 4 CG. I n this case. we say the instability is convective. If, on the other hand. thcre is :I h ill the right hnli'plunc for which C f ( h )and C p ( h ) rncrge. then shifting the contour is n o longer possiblc. Indecd, i f the riiergirig arises t'rclm generic liuigrncy of two curves. then the siiddle point method can be used to cvaluate the integral in ( 7 3 )and show that there is indeed growth of the solution :IS t + m. We rel'cr to the rcvicw article ol'Huerre and Rossi i n 1481 for xpecilics. To illustrate these ideas, consider the equation -
Wc huvc L(lc.i k ) = - c i k - k 7 t 11. Hence onsot of instability occu~-sat Consider now the equation
EL = 0,
k = 0.
for fixed A. This equation has two roots for k and the two roots coincide if
This happens in the right half plane only if 11 > c2/4. Hence the onset of absolute instability is at LL = c 7 / 3 .i t . , at a threshold strictly greater than the onset of convective instability. Let us consider (73) on a finite but large domain, x E (0. L), with Dirichlet boundary conditions. We may ask when onset of instability occurs in this case. If L is large, we expect the infinite problem t o be an approxirl~ationfor the problem on the finite region, so if the problem on the infinite domain exhibits growth of solutions, this should be reflected in the solution on the ti nite region as well. The question, however, is whether this growth is transient or persists as t 4 m. We may expect that a convective instability eventually leaves through the end of the interval, while absolute instability persists. Indeed, this is the ciise. An elen~entarycalculation shows that onset of instability on the tinite domain occurs at
which converges to the absolute stability threshold as L Consider now a probletn with varying cocf'licicnts
-+
cui.
Wc may consider the crlir:rtion with thc cocflicients hcld f xed.
In general. there is no conneclion between the stability of ( 7 7 ) and (78). but wc may hope for such a connection if the coefficients vary slowly enough. The rcvicwh of Huerre and Monkewit~1661 and Huerre and Rossi (481discuss usymptolic results tor such cases in the situation where (78) is unstable f'or .vcr in a finite rangc. Heuristically. we would expect convective instabilities to grow in a transient fashion. but eventually leave the unstable region and subsequently decay. On the other hand, absolute inslability for (78) would be expected to lead 10 unstable modes tor (77) which are localized within the 1-ugiot1of' abzolute instability. Asymp~otic:uld nulne~.icalresults contirming such nn expectation are reviewed in 1661 and 1481.
3. Convection flows
Instabilities due to the onset of thermal convection were first observed by Thomson 1 1 591. and investigated more systematically by BCnard 1101. Ironically. the irlstability observed
by Benard was not the one which now bears his name; his convection cells were due to the Marangoni effect (temperature-dependent surface tension), which will be discussed in the next section. The Benard problem is concerned with a horizontal layer of fluid between parallel plates, subject to a vertical temperature gradient. The first theoretical analysis of the problem is due to Rayleigh [ 1061. The governing equations are the Navier-Stokes equations and the heat equation. In the Boussinesq approximation, temperature differences are assumed small enough so that all fluid properties can be considered independent of temperature, except for the density in the gravity term, which is assumed linear in temperature. Moreover, viscous heating is neglected. With these approximations, the governing equations are
div u = 0,
Here u = ( l r , v. 111) denotes the velocity, p the pressure, and H the temperature. The fluid parameters are the density p , viscosity 11. thermal diffusivity K , and thermal expansion coefficient cu. The boundary conditions are H = HO at ,- = 0,H = H I < 4) at := d , and either n o slip conditions I ( = u = M I = 0 or "free" boundary conditions u~ = 1 1 ; = 11; = 0. A simple solution of the equations is given by the pure conduction state:
We linearize at this state and nondimensionalize by scaling length with (1. time with t 1 2 / ~ . velocity with ~ l t ltemperature , with Ho - H I . and pressure with ~ ' p l r 1The ~ . resulting linear stability equations are i )u -
ilt
=-Vp
+ KPHe3 + P A u .
div u = 0.
with boundary conditions H = 0 and either u = u = 111 = 0 or ul = u ; = v, = 0 at := 0 and := I. The dimensionless parameters are the Kayleigh number
and the Prandtl number
Equations (8 I ) can be combined into the single equation
with boundary conditions w = u ~ ; = , w,,;; = 0 in the free case and
in the case of no-slip conditions. The case of free boundary conditions is particularly easy to analyze. We assume infinite plates and separate variables: u~= W(z) exp(iax ipy cr).For free boundary conditions. we have the explicit eigenfunctions W,,(z) = sin(nnz), and we obtain the eigenvalue relation
+
+
The onset of instability occurs whcn
+
p2 = n 2 / 2 = 4.935; in The smallest critical value of K occurs whcn rl = I and cr' this case (87) yields R = 2 7 n J / 4 = 657.5 1. For no-slip boundaries. the eigenvalues and eigenfunctions need to be determined numerically; the results are qualitatively similar. The critical Rayleigh number in this ci~seis R = 1708 and the corresponding wave nulnber is c r 7 + f i ' =9.716 1381. The Benard problem between inti nite parallel plates is invariant under translations and reflections in the .r and J directions, us well as rotations about the :axis. This high degree of symmetry allows for a multitude of patterns which may develop as a result of the instability. The classiticittion of all patterns and the mechanisms by which some patterns are selected over others poses a formidable problem. Mathematical analysis has focussed primarily o n the following cases: ( I ) Solutions periodic with respect to a lattice in the .r. J plane. where the period of the lattice is given by the critical wavelength. (2) Solutions periodic with respect to a lattice, but the period of the lattice is longer than the critical wavelength. (3) Solutions which arise from modulation of periodic solutions. On a lattice, we are considering solutions with an .r, J dependence of the form U)
=
w,,,,, exp(i/,lkl . x
+ i n k r . x).
where k l and k2 are given vectors, and x = ( x , y). We choose k l such that its length corresponds to the critical wave number. On a general lattice, fk l will be the only vectors in the lattice which correspond to a critical mode, and the bifurcation problem reduces to a twodimensional center manifold. The bifurcating solutions are two-dimensional flows known as rolls. Because of the translation invariance, there is a one-parameter family of these solutions. If lkzl = Ikl 1, then the center manifold becomes four-dimensional, and in addition to rolls, there are rectangular patterns where the modes proportional to exp(ikl . x) and exp(ik2 . x) have equal amplitudes. Lattices with k l = k2 are called rhombic, the square lattice where in addition k l and k2 are orthogonal is a special case. If the angle between k l and k2 is 120 degrees, we have jkl I = lk21 = Ikl k21, and the center manifold becomes six-dimensional. Bifurcating patterns in this case include rolls, rectangles ("patchwork quilt"), hexagons, and triangles. In the case of the hexagonal lattice, there is a fundamental difference between the problem in the Boussinesq approximation and the problem without the Boussinesq approximation. Without the Boussinesq approximation, the bifurcation is as discussed in Section 2.5. The Boussinesq approximation, however, introduces an additional symmetry. Narnely, the system (79) is invariant under the transformation ,: + 1 - :, ui + - 1 1 1 , H + 20, - 0 , where 4, denotes the temperature of the steady conduction state. This additional symmetry forces P ( p ) in ( 5 1 ) to vanish and thus alters the bifurcation picture. With the Boussinesq approximation. analysis shows that rolls are preferred over hexagons 1 145.491. I f the Boussinesq approximation is dropped, however, then a rangc ol' Rnyleigh numbers appears where stable hexagons exist, and a typical bifurcation diagram looks like Figure 2 (from 1371). see. e.g.. [Y3,20.94.7-41.The strongest non-Boussinesq cfkc1 i n practice is the tclnperaturedependenceof viscosity. This effect is opposite i n liquids. where the viscosity decreases with increasing temperature, and in gases. where it increases. As a result, stable hexagons in gases have the fluid rising at the center of a hexagonal cell and falling around the outer part of the hexagons. The opposite is the case in liquids. Periodic solutions with a longel. period than the critical witvelength were tirst studied in 175);they are investigated more systematically in 136,371. Convection rolls can be subject to a number of secondary instabilities. Sideband instabilities are instabilities to long wave modulations: for small amplitude rolls they can be
+
Fig. 2. Typical h~turcationdiagram for n o n - B o u \ ~ i n ~ \BCnord q prohlem
described by amplitude equations as discussed in Section 2.6. The Eckhaus instability was already discussed in Section 2.6; this instability breaks the spatial periodicity of the rolls but preserves the translation invariance of each roll along its axis. Another type of sideband instability, the zigzag instability, breaks this translation symmetry and deforms the rolls into a "meandering" shape; the relevant amplitude equation is the Newell-Whitehead equation [91]. Both instabilities are discussed in some detail in the article by Fauve in 1481. A number of other secondary instabilities of BCnard rolls have been investigated; we refer to 12 1 ] for an overview and references to the literature.
The Marangoni instability is introduced by the temperature dependence of surface tension. In general, surface tension is a decreasing function of temperature. Consider a liquid layer heated from below and bounded by a free surface above. If Huid rises at a certain point in the layer, this will create a hot spot on the free surface. Consequently, surface tension decreases, arid this induces a How which diverges at the free surface and causes the fluid to rise further. Hence there is a mechanisms for instability; as in the case of buoyancydriven convection, the instability is opposed by the dissipative effects of viscosity and heat conduction. The Marangoni instability was observed in BCnard's experiments 1101, for more recent experiments see. e.g.. 1781. The first theoretical investigation ot'the problem is due to Pearson I97 1. The governing equations are basically the same as t'or the BCnard problem, except that the gravity term is absent. We state them in dimensionless form:
div u = 0. i)H -
i)t
(89)
+ (U . V)H = AH.
The boundary conditions at the bottom wall (: = 0 ) are H = 1 and u = 0. At the free surfi~ce := 1 . we have a prescribed heat Hux i ) H / i ) : = - 1 . we ignore surface deformation 711 = 0 . and we have a shear stress which balances the effect of the surface tension gradient
Here M is the Marangoni number
where -a denotes the derivative of the surface tension coefficient with respect to temperature.
The base flow of pure conduction u = O , G = 1 - z , p = 0 loses its stability at M = 79.6, with a critical wave number of approximately 1.99 1971. The bifurcation problem at the onset of instability is analogous to the Btnard problem; because of the different boundary conditions at top and bottom, there is no up-down symmetry. Hexagons turn out to be the preferred pattern [ 144,30,191. Both the Benard and Marangoni problem can be modified in a number of ways; for instance, effect of side walls, horizontal in addition to vertical temperature gradients, several fluid layers, and addition of other physical effects (solute gradients, surfactants, electromagnetic fields, viscoelasticity etc.). We shall not give a comprehensive review, but the next three sections will discuss some such problems.
We consider a layer of fluid between parallel plates. The fluid contains a solute and the density depends on solute concentration as well as temperature:
We assurne that cu and p are positive. i.e.. the solute is heavier than the fluid it is dissolved in. At the plates. we have prescribed temperatures and solute concentrations: S = So, H = Ho at the bottom and S = SI,H = H I at the top. In the Boussinesq approximation, in dimensionless form. the equations o f motion are
div u = 0. ilH -
ilt
+ ( U . V)H = AH
Here S and fl are nondimensionalized relative to SO - SI and 4) - H I . The boundary conditions are H = S = I at ,- = 0, and H = S = 0 at := 1. For the velocity. as in the Btnard problem, we consider either no slip or free boundary conditions. The new dimensionless parameters are the solute Rnyleigh number.
and the Lewis number.
where K S is the dit'r'usivity of the solute.
-
The purc conduction state is given by u = 0, 8 = S = I - z. If R > R , then as in the B6nard problem, the density of the fluid increases towards the top, leading to an instability. There is. however. a more subtle mechanism of instability which can arise if the solute stratification is unstable and the temperature stratification is stable (i.e., H and R are negative). In that case. the instability arises because the diffusivity of temperature is much higher than that of the solute. Hence. if there is a perturbation to the flow, tempernture variations will equilibrate much more rapidly than concentration variations, and hence the urnstablc stratitication of concentration can cause instability even though the overall density stratitication is stable. If, on the other hand, temperature stratification is unstable and solute stratification is stable. then the stabilizing effect of the difference in diffusivities can cornpcte with the BPnard instability to set up an oscillatory onset of convection. Doubly diffusive convection with stabilizing temperature and destabilizing solute gradients was first observed by Jevons in 1857 1681. The first theoretical analysis is due to Stern 11521. Stern also noted the possibility of oscillatory onset i n the reverse case where the solute is stabilizing and temperature is destabilizing. A review of further experimental and theoretical investig;~tionsas well as applic:~tioncto oceanography can be found i n 11361. The explanation of pattern f'or.11lation in salt tinger convection is still an open problern. Proctor and Holyer's analysis I I01 I suggests that rolls are the prcfcrred pi1ttel.n. This is not in ug~.eementwith experiments I 1461. N o rcally convincing explanation for the discrcpancy has been given: a tli.;cu\sion of non-Boi~ssincsyeffects and nonuniform gr;~dientsof tc~n~pcrature iund salinity i n the base state is given in 1 1371. but the intlucncc ot'thcse cff'ects appeilrs to be small. Thc hifu~.cationproblem for oscillatory onsct 01. convsclion ha\ hccn studied by Silbcr t~ndKnobloch 1150) f'or the square Ii~tticc.and Koherth. Swift. iund Wagncr [IJOI 1I)r the hcxogon;~Ilattice. The simplc\t solutions to cl;~ssit'yarc bifurcated periodic solutions which havc cnoi~ghsy~nnict~~ics so that the bifurci~tionproblem ciin bc ~.crluccdto a simpls Hopf hil'ul-cation. Mnthcmiitically. this rcquil-es identifying the isotrupy s i ~ b g r o ~ ~ofp s(; x .s' (whcrc (; is the symmetry group ol'thc lattice considered) which havc ;I two-climcnsional lixcd point space. We discussed the iun:~logoi~s problem tor the simplel- situi~tionof O(2) symmctry in Section 2.5. On the squiue lattice. the patterns which result from thi\ analysi\ :ire two type.; of traveling waves (rolls and squ;~res),two types 01' st~~ntling w:~vcs (rolls :uid sqi~iu.e>). and a pattern ol'alterni~tirlgrolls which ptriodically xwitch their direction at 90 degree angles. On the hcxngonal lattice. there are four types of standing waves (rolls. hexagons. tr-iitnglcs. and patchwork quilt). three types of traveling wavcs (1.011sant1 two kinds oftri~velingpatchwork quilt), one solution ot'oscillating 11-iangles.and three solutions of periodically alternating direction (Robel'ts, Swift. ~undWagner call which have p:~ttc~'ns thcsc twisted patchwork quilt and wavy rolls 1 and 2 ) . Applications of bifurcation theory t o the specific problern of doubly diffusive convection are given in the papers of Ni~gata and Tho~nas1901 a ~ l dRenardy [ 1331. Thc preferred pattern.; depend on thc p;irameters.
-
The BCnard problem for two fluid layers can also lead to oscillatory onset of convection. Indeed, there are two quite distinct mechanisms for this.
Sftrhili~ycord irr.\/crhili!\, irr ri.\c,ou.r,/l~cidc
249
( I ) Flows with a strongly stable interface: In this case interface deformation is negligible, but the two fluid layers are coupled through the interface via the continuity of horizontal velocity, shear stress. temperature, and heat flux. Oscillatory onset can result from resonant mode crossings in situations where the onset of convection in The eipenmodes have separate convection cells both layers occurs sim~~lt;uieously. in each layer. ( 2 ) Flows with a weakly stable interface: In this case, oscillations result from the competition of the Benard instability with a stable interface. Convection cells extend through both layers. For a strongly stable interface. oscillatory onsot occurs when the "effective" Ruyleigh numbers in the two layers are near-ly equal. Thc oscillatory onset of instability is studied in 1471 and 1311, ant1 experimentally in 1104.22,41. The problem of pattern selection on the hexagonal lattice is studied in 11351. Traveling rolls. wavy rollsf I), and oscillating triangles arise as preferr.ed patterns in this analysis. Since oscillatory onset occurs only in a narrow window ill parameter space. it is natural to study the instability its a Tnhens-Bogdanov bifurcation [ 1381 (this is the zero f'recluency limit, where two complex con.jugnte eigenv;tlucs merge into a zero eigenvalue) or consider the even more degenerate situi~tionwhere two real cipcnvalucs cross over 174.1221. Computations tor the TakensRogditnov bil'~~rc:~tion 1 1 381 reveil1 a rich variety of periodic. qunuiperiodic, and chaotic xolutionu. Kcn;tt-cly imd Joscph I 178 1 poi~itcd~ L I the I possibility o f oscill~ttoryinstability in a twolayer system with a weahly s ~ ; ~ h i~ltcrlr~cc. lc Thih oscillation arihcs from the competition between the Rhiarcl inxtithility ;tncl a sti~hlcintcrl.itcc. The primary tilcto~~s influencing thc stithilily o f the i~iterf'c~cc arc clcnsity clilP'c~.e~lcc. xurl:~c.cIcl~\ion.irrlcf thermal c.ontluclivity dil'lCrc~~cc 1 1201. Tlic hif'i~~.cittio~i pl-ohlcrn fo~oscillatoryonset is htuclictl in I 1 131: no stahle piitterns iu-c pr-cdictcd. Expct.i~l~cnts in this rcpimc ilppcar-to bc litcl\irig.
Convcctiori ill co~~ducting Iluidx auhject to niagnotic lieltls is ~.elcvi~nt to pli~~~ctilry itlid stell ~ u intcriol-s. I11 this \cction. we xh;lll dihcuss the ximplcxt such problem: tlic BCn:~rtlp~.ohlem with n constant vcrtic;~Iin~poscdmttgnetic field. For a morc comprehensive tliscussion. we r c k r to the hook o f C'h:tnd~tschhar1751 o~ the review hy Proctor i~ndWeiss I1001. I n the Ro~~ssincscl approximation ancl i n di~iicnsionlcssl'orm. the cclu;ttionr, of' motion
div u = 0. i)H
i)t
+ (u.V)H = AH.
The new dimensionless parameters are
where nz denotes the magnetic permeability and Ho the strength of the imposed magnetic field, and a magnetic Prandtl number p, which represents the ratio of magnetic to thermal diffusivity. We linearize about the base state u = 0, B = 1 - z and H = (0,0, I), and obtain the perturbation equations,
div u = 0,
ijH
-
au i):
i t
PV x (V x H).
The simplest boundary conditions to consider are the "free" boundary conditions for the Benard problem together with the condition of zero tangential magnetic tield at the boundary. With some algebra, the equations can be combined into the single equntion
The boundary conditions are ul = D2n1 = ~ ' 1 1 1= ~ " u = l 0. For a mode proportional t o exp(iax i v y a t ) sin(n:), we obtain the eigenvalue relation
+
+
+
Here a 2 = a 2 p 2 . For stationary onset of convection. a = 0, we find
The effect of a magnetic field is therefore to inhibit convection and increase the critical Rayleigh number; the critical wave number is also increased. Equation (100) also allows for oscillatory onset of instability, but only if < 1. Under typical laboratory conditions (e.g., in liquid metals), P is greater than one by several orders
-
of magnitude. In astrophysical applications, where heat transfer is primarily by radiation < I can be relevant. The pattern rather than conduction, on the other hand, the case formation problem at the onset of instability has been studied by Clune and Knobloch [3I]; for steady onset, rolls are the preferred planform, but for oscillatory onset, a variety of stable patterns are possible.
4. Flow between rotating cylinders
For the purpose of measuring the viscosity of liquids, Couette 1341 in 1890 designed a device consisting of two rotating cylinders, of which the outer one was rotating at a given angular velocity. The viscosity could then be inferred from the values of the rotation rate and the measured torque. A few years later, Mallock 1871 tried a similar device, but with the difference that the inner cylinder was rotating instead of the outer one. However, no stable shear flow was ever established in the range of Mallock's experiments, and the device was useless for viscosity measurements. In 1923, Taylor 11581 showed that, when the inner cylinder rotates, the flow becomes unstable at a critical rotation rate. The instability is driven by the centrifugal force. If the inner cylinder rotates faster than the outer cylinder. then particles on the inside experience a stronger centrifugal effect than fluid particles on the outside and n disturbance which exchanges fluid particles between the inside and outside is able to grow. Over the past forty years. the problem has been investigated thoroughly, both experimentally and thcoretically. Many types o f flows and bifurcations can bc observed. making the problem a rich source of applications for the methods of bifurcation theory. Most analyses of the problem assume the cylinders to be infinite and look for solutions periodic in the axial direction. This is of course an idealization. The effect of the end plates in real devices has been studied by Benjamin [ 121. In cylindrical coordinates, the NuvierStokes equations read
where p is the density,
11
is the kinematic viscosity, and
The incompressibility condition is
The flow domain is given by R I < r
R?, and the boundary conditions are
i
a t r = R;. Couette flow is a steady purely azimuthal flow, which is given by
and a pressure p ( r ) which satisfies 1 PVO.)' 1) ( r ) = -. r
When both cylinders rotate in the same direction, the tirst instability is to axisymmctric disturbances. Let (it:.. I!;,. tr!) denote the disturbance velocities. The linearized problem for i~xisyninietricdisturbances can be reduced 1381 to the pair of coupled equations
( 1 OX)
with boundary conditions
at r = R I and r = R?. We look for normal modes
and nondimensionalize. The radial coordinate is nondimensionalized with respect to the gap sized = R2 - R I .We set Ro = ( R I R z ) / 2 and .r = ( r - Ro)/rl,so that the boundaries are at x = f112. Moreover, we set a = kd and a = .rtl'/v. We introduce the dimensionless ratios
+
and we represent the velocity profile in the baqe flow as V ( r ) = r Q ( r ) = r Q I g ( x ) , where g(.v) is given by
The resulting stability problem then has the dimensionless form (381
where
is known ;is thc Taylor number. A t'tirthcr simplification arises in tlic narrow gap limit I ) -- I . i n which g(.r) simplifies to ( I Ih)
and I ) , = I ) . If we rcplnce g(.r) by its nvcragc ( I -t j 1 ) / 2 , then the coctticicnts in thc cfifii.rcnti:~l equations are constant. iuid indeed the linear stability problem hccomts ccl~~iviilcnt to the Bknnrd probleln. In this approximation. the onset of instability occurs i ~ t 7' = 3415.5/(1 1 1 ) and wove number u = 3.1 17. Although this approxi1i1;ition is C X act only i n the limit 11 --+ I . i t works well its long as 11 3 0 ( i t . . the cylinrlurs art. rotating ill the sitme direction): if only thc illncr cylinder rotates. i.c.. 11 = 0. the :~pproximation yiclds the critical Taylor number 7;.= 3415.5. while the cxact value is T,. = 33XO.O. For counter-rotating cylinders. on the other hand. the averaging approxitnation does not work. We note thiit if 11 < 0, then g ( . r ) changes sign. :I feature which is missed by averaging it. In addition. if < 0, thc range ol'pariuiictcrs in which the first instability is ;~xisymlnetric is q ~ ~ ili~nited te (see (281). Thc bifurcation resulting from the instability is super.critic.al and leads to a steady axisylnmetric Row known as Triylor vortices. The vortices have toroidal shape and fill thc space bctwccn thc cylinders; on one side of the vortex, fluid is carried from the inside to the outside and vice versa on the other side. Figure 3 (from 1281) shows a photogrctph of
+
t"ig. 3. 'I':1yl01- vortex
IIO\V.
Fiy. 4. Wavy vo~.tcxIlow
Taylor vortices. As the Taylor number increases further. the Taylor vortices also become unstable. In the case where only the inner cylinder rotates, a Hopf bifurcation leads to wavy vortices (Figure 4, from [281).These waves are traveling in the azimuthal direction.
the flow is time periodic in the laboratory frame, but steady in a rotating frame. This fact can be exploited to analyze the next bifurcation [ 103,l 1 11. It leads to modulated wavy vortices (Figure 5, from [28]), which are periodic in the rotating frame and quasiperiodic in the laboratory frame. For still higher Taylor numbers, chaotic solutions appear. The Taylor problem with only the inner cylinder rotating therefore exemplifies the Ruelle-Takens scenario [ 1421 for transition to turbulence.
4.2. The Dean
L I I I ~Giirtler problems
A number of other shear flows with curved streamlines have centrifugal instabilities analogous to the Taylor problem. In Dean flow the cylinders are at rest, and the flow is driven by an azimuthal pressure gradient. Of course, one cannot strictly do this in an experiment, since pressures cannot be multivalued, but if the gap is narrow, then the flow can be realized over a stretch of circumference which is long relative to the gap. From (102), we can determine the velocity profile
where denotes the imposed pressure gradient, and A and H have to be determined to satisfy V ( K I) = V(K2) = 0 . In the narrow gap approximation. the flow can be npproxi~ n i ~ t eby d n qilildratic profile:
where V,,, is the mean velocity.
The line~irstability equation for axisymmctric disturbances can then be put in the ti)rm
where
I f we replace I - 4 x 2 by its average 213, then this problem is equivalent to the adjoint of the Taylor problem for / L = - 1 , with A playing the role of the Taylor number. This crude approximation predicts a critical A of 56000 with (Y = 4. while more accurate computations yield A,. = 46458, a,. = 3.95. See 1381 for more details.
The Giirtler problen~is that of a boundary layer on a concave wall. It is assumed that the boundary layer thickness 6 is small relative to the radius of curvature Ro of the wall and that the bahe flow is parnllel to the wall. The linear stability problem, in dimensionless form, can he shown to havc the ti)rtn
where r r is the pel-turbation velocity parallel to the wall, v the perturbation velocity perpendicular to the wall, I/ is thc dirrlensionless base tiow (length is rescaled with the boundary layer thickness and velocity with the velocity U , outside the boundary layer). The parameter A is the equivalent of the Taylor ~iuniberand is given by
When the cylinders in the Taylor problem rotate in opposite d i r ~ c ~ i o nthe s . onset ol' instar i c 11 less than -0.78 (in thc n;~rrowgap limit). As /1 i \ bility b t c o ~ n e sn o ~ i a x i s y ~ n ~ n c tt'or clccrcnsetl further, the ; ~ / . i ~ ~ l i ~wave t h u l ~iulllbcrincrcahes in riipid s ~ ~ c c e s s i oFor n . finite gaps. :I qi~:~litutivcly similar behavior. is found (see 138.18I). T h e onset of' nonaxi\y~nmct~-ic illstability Ici~dsto a Hopf'hil'i~rc;~tio~l with a110(1)synlniclry (corrc<pondingto translationx und reRections in the axial direction) and :in additional SO(?)
~ > i ~ t t c~~' -ului l t i n gI'roni the i~lstiibility01' T,~ylorvol-tcx flow in the c ; ~ s cwhcrc o ~ l l ythe inner 01' two nonaxisymmctsic m o d 0 proclucch new partcrlh cylintlcr- rotates. Tlic intcr~~ction known as intcl.pcnct~'i~ting \piri~lsi ~ r i t lsul~crposcdI-ihhons. A p i c ~ i ~ rofe intcspcnctl;~ting sl)i~.;~ls is show11 i l l Fig~11.c7 (I'rot11 131).
Viscocl:~stic fluids allow ~ O illst:~hilitics I. in shear Ilr~wwith cur\;cd strc;urnlinch which i ~ r c driven hy elastic clfi'cts. Over the past ten years. thcrt. has been an cxknsivc study of these instabilities. for thc Ti~ylorprobleni us well as other Hows. such as HOW between pi~rilllcl i ~ the ~ i viscoel;lstic plates o r between n cone and a pliitc. We shall give a b r i c f ' d i ~ c i ~ s sof Tayloi- P I - o b l c n ~wc ; I-efcr to the rcvicws of' L~uxon1831 ;und S h u q k h I 1481 lol-additionill i~~lorniation. The possihility of instabilities /,ei.o Kcynoltls number in the Taylor plnhlcni w;ls first investigated by Larson et al. I X 1 1 . They considered the Oldroyd B fluid. For this fluid, the stress tensor is given by
T = rl., (Vu where
+ ( vu
)")
f TI,,
If inertial forces are neglected, the dimensionless parameters are the aspect ratio E = d l R I , the Deborah number
and the retardation parameter q = 11.,/q,,. With u denoting the perturbation to the axial velocity, the linear stability problem for axisymmetric disturbances in the narrow gap limit reduces to (8I ]
with boundary conditions 11 = 11' = 0. Here
and
where a is the dimensionless growth rate. It is immediately clear that the eigenvalue problem requires A and hence (: to be purely imaginary. and hence rr cannot be real. The onset of instability is therefore always oscillatory. Moreover. since Ilr appears only as 1 k 2 ,only the magnitude of the difference in rotation speed o f the cylinders matters for the instability: i t makes noditt'erence whetherthe inner or the oiltercylinder is rotating. For 11 = 0 . we have
The onset of instability occurs when a = * i and i: = ii.At this point. I)r must be such that A is the tirst eigenviilue of ( 127). Larson, Shaqfeh. and Muller tind a critical value l e 5.92. with a corresponding cu of 6.7. For nonLero 1 1 . the critical value of the of ~ ' / ? - f of Deborah number increases. I t turns out that actually the onset of instability in the viscoelastic Taylor probleln is not axisymmetric. Moreover. in the narrow gap limit, the critical Deborah nuniber is alniost independent of the azimuthal wave number. Hence the problem is somewhat analogous to the Newtonian Taylor problem with counterrotating cylinders, which suggests that the patterns tlcveloping after onset of instability might be described by Hopf bifurcations or perhaps mode interactions resulting from degenerate Hopf bifurcations. Analysis shows, however, that branches are subcritical, and no stable solutions were found near the bifurcation point 1155.1231. Co~nparisonswith experiments 1891 agree fairly well in the critical Deborah and wave numbers, but not in the frequency of the oscillations. Moreover. the experiments suggested a supercritical bifurcation. Some experiments ( X I appeared to totally contradict theoretical studies; the onset of instability is axisymmetric and stationary, and the critical Deborah number is rnuch lower than the theoretical value. This puzzle has recently been clarified (21: the instability observed in these experiments is actually due to a totally different mechanism which involves temperature variations created by viscous heating.
If two liquids are placed inside the space between rotating cylinders, the shape of the interface hecornes of paramourrl interest. Parameters determining the stability c.)f the interface include the ratios of densiticc and \~iscositicsbetween tho two liquids and surface tension. Experiments 1721 show many possible arrangements: horizontal str;ititication due to gravity. circular Coucttc flow, sheets coating one of tho cylinders, tingering, rollers, emulsions etc. We shall review results of one case which is particularly accessible to iuinlysis. naniely the case ~vlienboth cylinders rotate at the siume rotation speed and gr~ivityis negligible. The ;rnalysi\. 01-igir1;rllydue to Joseph and P~.eziosi171 I. is reviewed in Chapter I I of(721. It' both cylinders rotate at the same rotation speed. then one possihlc solution is concentric arrnngemcnt where fluid I occupies the region K I < I . < I) and fluid 2 occupies the region I ) < r. i/?,. I t is not clifficult to show that this arrangement is st;rhle if it i \ ;I loci~lriiinirnuni o f a potential erlergy which co~lsistso f a surl'irce tensiorl 1cr1i1and ;I ccntl-il'ugal Icrm:
Herc K ( H . :) is tllc pcrti~rbcdpositiorl o l ' ~ h eintcrl;rcc. /I, is tlic tlcnsity ol' lluicl i , a ~ rr~ isd Ihc coct'ficic~itol'h~~rl'ircc ~ c i ~ \ i oLV ~ci .L I W tlic iiol:~~io~i
Tlic important di~ncnsionlcsscliiilnt ity in the prohlc~liis
Surf'crcc tension tlcstahili/cs ;I concentric intcrl'i~ccbccal~hc;I cyliniicr is no1 a mininli~l surt'i~ce.Thc cent~~il'ugi~l force is stithili/ir~gil'p? > [)I. i.e.. .I > 0. The cylindrical intcrflrcc can he shown to he :I local rnir~i~nurli ofthc potcntiiil cncrgy i f .I 2 I ;uid n gloh;rl minimum it' .I 3 1 and il global minimum it' .I 2 J( 1 HI/f.))-'. Jo.;cph irnd Prc~iosidcrivc thc Eulcr-Lirgri~ngc ccluation for axi\yrnnie~r-icsolutions 01' tlic c o r i s t ~ ~ i ~ niiiii~iii/i~tio~i i~~ed p1.ob1c111 i111tls t ~ ~ i lits y s o l t ~ t i o ~With ~ s . 1. = H/l), tlic Ei~lcrLagrangc cquiition rciids
+
Here r' denotes the axial derivative (the axial coordinate is rescaled with D) and h is a Lagrange multiplier. The constraint is
The transformation
transforms ( 135) to
which can be solved in closed form for v ( r ) . If r.1 and 1.2 denote the minimum and maximum of r , then
One now needs to verify which of these solutions sati\fy the volume constraint. I t turns out that. other than the constant solutiort I . = I . there are no solutions with > H I . i.e.. all solutions nus st havc an interface which intersects the inner cylinder. We shall t i ~ c u so n the lirnit whcre the inncr cylinder has zero racliits: K I = 1 - 1 = 0.In that case.
the volulne constraint reads
and the period in the axial direction is
By exploiting the fact that
we can put the volume constraint in the form
Moreover, the integral in (142) can be evaluated in terms of elliptic functions. We omit the somewhat lengthy expression. We can then solve for r2 by setting the two expressions, (142) and (144) for h, equal to each other. The results are more conveniently expressed in terms of j = J r i3 and A = h/r2. The integral in (142) is then of the form 2r24(j), so that A = 2 4 ( j ) . Moreover, we find
I
Fig. 0. Axial wavclcnglh ,4
;I\
a function of i .
Tile Ii111iti11g v i ~ l c ~ 1'c01.\ , j arc: ,;= 4. :it wIii~.llpoi111A r ~ i i c t ~ ciliti11ity s i 1 r 1 ~1.2 l rc;~cI~cs I. and j = -8. at which point 1' cc:~sc\ Io he ii Iiiollotollc I'utlction 01. I . . Thc conscllucncc ol' lhis i s t h a ~thc i111crhccis no longer of' thc tor111I . = I . ( : ) . The c.orrc.spo~lditi~ lTv;~luco l ,f i s -5,42285. I:igi~rc\ 8 ;IIICI 9 show 1.2 :IIICI A ;is l - ~ ~ t i c l i 01' o ~,~j ,sJoseph :111cl Prc/io\i :~lhocoli\idc~.soliitio~is01' "~iodoid"type lor which the inter~f';~cc i s not ;i griiph I . ( : ) : they lind that such solutions exist for -5.42285 2 .I :: -8. 18834. 11' .I i X . I X 8 3 4 , ~i~rf.:~cc tension i s n o longcl stl.ollg enough to holtl tlrops :I[ the inncr cylinder. and fluid gel\ tI1row11 o i ~ tro the outside. Figi11-c I 0 (I'rom 171 1 ) shows ;i c~)n~parison o f calcul:~lctl sh:~pcs with cxpel'ililcn~s.
5. Parallel shear flows
<
We c o n ~ i d e the r linear stability of:\ pi~filllelIlow u = ( ( l i ( : )0.. 0)i n thr strip - I :& I Two cases o f p:~rticuliu- interest are Couettc flow [ I ( : ) = :and Poiscuillc flow U ( : ) = I - ?. Other situations of interest include comhin:~lionso f Coilettc and Poiseuille How. as well as vurioi~s"houndi~ryloycr profiles" which ;icrll;~lly do not satisfy rhe Navicr-Stokes equations. To this base How. we add a small disturbance of ~ h form c
The linearized Navier-Stokes equations, in dimensionless form, are
and the no-slip houndary condition requires that li = G = 1;) = 0 at , : = f1 . Here D stands f'or d/d,-. Squire [ I S 1 I showed in 1933 that this threc-dimensional stability problem can be reduced -+t o the two-ciirnensional c;ise. Namely, set Ly = ,/& = cui pi,. p/G = b/a and W R = a R . Then o w tiiids
+
-
Since iV > LY. we have K iK . 50 the tlirce-tlimenxiotial stability problciii is always equivaletit to :I two-ditnciisiotial pi-ohlcm at ;I Io\vcr Rcynolds number. We notc that Squire trans1'oriii;itioii rlcgcncl'atcs f o r ii class o f modes which huvc L Y ~ = 0. For such moilcs. o n e ciisily Itncls that /;I = 0 iititl
+
i t i \ ciisy to show that nontriviiil solutions ciiii cxisl only if Im(c,) is neptivc. Hcncc this class of triotles leads o i i l y to stable cigeiivalucs. Ncvt.1-thuless. i I i h precisely this cliixs o f modes which plays ;I criicinl role in the n o n m o d d growth discussed i n the next section. We continue to study ( 14X).Thc problem c;in he simplitird further by introducing the strcam !'unction 4: i, = / I d > . 17) = -ii?d>, One then obtains the Oi-r-Sommcrfcld cclu;ition iind
0 1 1 (Y and K . Thc hoiiiidi\ry coiiditioiis :itc 4 = 114 = 0 . The Ow-Sommci-fcld cquiition hiis hccn the xub,jcct o f extensive Linalytical nnd nutiierical invcstigations. We refer to DriiAn and Reid (381 i'or :I review of results. In 1973. Romiinov 1141 1 proved that foi- plane c'oucttc flow. ll (:,) = :. the imaginary part o f ( . is always negative t.cgiirdlcss ot Reynolds nunihcr. i.c.. no instabilities occitr. R o m u i o v ' s prool. exploits the closed torni solution of (1.50)i n terms ot Airy fiinctions which c;in hc found f'or U ( : ) = :, For Poixcuillc How. U ( : , )= I - ,-?. 011 thc othcr hand. an instability occiirs for K = 5772.23. The tirst prediction of this instability is due to Heiscnberg 1571. based on asymptotic approximations at high Reynolds nutnbcr. The bifurcation resulting from the i ns ta hi I it y i x a sit hcr i t icul Hc J pf hi fuI-cati o n . The I I-ii vc I i 11g w avc so Iu t ion s rex u It i 11g from
Here we have dropped the tilde
this Hopf bifurcation are known as Tollmien-Schlichting waves. They do not form a stable branch; indeed, as we shall discuss later, the ultinlate result of the instability is not a twodimensional flow. The stability of parallel How in a circular pipe (Hagcn-Poiseuille flow) leads to an eigenvalue problem similar to the Orr-Sommerfeld equation. Like in the plane Couette flow case. no instabilities have been found, but a proof of stability is still lacking. Since instabilities occur only at high Reynolds number, it is natural to study the high Reynolds number litnit of (150). The inviscid limit is obtained by simply setting ( i a R ) ' equal to zero, leading to the reduced equation
This reduced equation describes the high Reynolds number limit except in boundary layers near the wall. and "critical Inycrs" whcrc L1 - (, is close to zero. An approximate solution for high Reynolds number can thus be based on a matched asymptotic expansion which matche., inner solutions in the boundary and critical layers to an outer solution based howcvcr, which makc this problem f i r from easy: on ( IS I ). There are two fei~ti~rcs, ( I ) Thc location ol'thr critical points may approach the wall as R + oc,and in thc cases of most intcrcst. it actually does. Consequently, the boundary and critical Iaycrs cannot be considrreci independently of e;lch other. ( 7 ) CJ~iIccs(1'' = 0 :II thc criticitl point. the rcducctl ccluation ic \ingulnr at the critic;iI point. s asymptotics i:, given by pliine Coucttc A fairly simple case of high R e y ~ ~ o l driumbcr Ilow. 11 ~ L I I . I Iout S that c . I ~ I I C I S 10 either 1 or 1 as K -t w. with tliu ~lil'l'r~rcncc prolx)rtion;tI to h'-li3. Lct us s c (. ~ = - I ( U K ) ' / ' X . .Thc csscnti;~lpart ol' the pn)blcm is to consider thc combincd boundary and critical layer at = - 1. We rcac:~lcIhc indepcndcnt v:~~.i:~hlc ancl set y = (uH)'/'(: I ) . At le;~dillgorder thih Iei~dsto t h c~q i ~ ; ~ t i o ~ i
+
+
-
,
We necd t o lind solu[ionx such that d ( 0 ) = I ) $ ( ( )=) 0 and / I $ ( ! ) + 0 for !+ ca.T h e solution o f ( 152) which satistic5 the boundary conditions at Lcro is given by
The condition that I)$(>,)
-
0 for y -+ cx, can be reduced t o
That is. k mu5t be - exp( - i r t / 6 ) times a root of the equation
where the integral follows a path parallel to the real axis. T h e first solution, as given by Drazin and Reid 1381 is s = -4.107 f 1.14421. The leading contribution to the outer inviscid solution is given by
where c.1 i \ d e t e n n ~ n e db y the m a t c h ~ n gc o n d ~ t i o n
with @ given hy ( 153). The limit K 4 ca for ti xed u is riot the only one of interest. For instance. the inslability ot'pl:~nc Poiseuille How c m n o t be predicted in this fashion, because a c t ~ ~ a l all l y modes are stable if R -+ ca fol- fixed ( Y , I115tei1done needs to look at limits where K -+ CG and at thc same time w -+ 0. with sonie relationship hetween K iuid u . It till-11so i ~ t l l i ~ tfor 1a1.g~K instability occur\ I'or C I K-I" iu < L ' : K " " with certi~inc o n w n t s kind ( ' 2 . These scalings can he ~~ndcrstoocl ;IS I'ollows: The critical eigenvalue is of order wZ and the width o f t h c viscous hoitndary layer near the wall is of order ((c.(wK)-I:' = O( ICY'KI ' I Z ) . The cxpi~~ision of' the eigcnvaluc now in\~olvesu 2 ;IL( well ;IS the h o i ~ ~ i d ; ~li~ycr ry thicI\~~ess. We call cxpcct cunccllutiona f ' o ~ncutr~ll.;lability to occur il' both terms iirc o f thc silmc o~.clcr.Thi\ leads 10 w K-'!' 0 1 1 thc "~~lq)er" hr;~nch.tlic I c i ~ d i ~ order ip p c r I ~ ~ r h ; ~ t i10 on ltic c~gcnv;~luc ia purely inviaciti. iu~cl(cu3/-1- ' I 2 is ol' Ilia \i\nic ortle~.;IS w 4 . Wc rc1i.r to I)l.a/iti ant1 Kcid 1381 I'or :I clclailctl cxpoaition ol the analysts.
,.
-
Tlic ~.csult\on linciu' stability o f p;u;~llcl shear flows discussad it1 the prcccding scction d o not ;~dccli~:~tcly explain oh\c~,v:~tionain cxpcrin~cnts.Extremely ciircl'itl cxpcrinicnts (C)?] ;[re ~iccdcd.for i~ist;tncc.to oht;~ini~grcciiiciitwith the linci~r\lability :lnalysis of plane Poiacuillc flow. In more conimon ci~~curnst;uict.s,the Ilow becomes ~ ~ n s t ; ~ hatl c:I Rcynolda ~ ~ u n i hsotncwlicrc c~. hctwccn I000 ancl 1500. 1;tr It.\.; thitn the ~hcorcticalstahilily limil of 5771.23. Similarly. pliuic Couetle flow i111d Hagt-n-Poiscuillt- flow in a circular pipe bccoti~citnstcthle even though linear \tithilily throt-y alwayb pl.c~iicts~ h c mto hc \lable. The do~nitiant.;tr.uctut'c\ evolving from thew inati~biliticsd o 11o1look like the Ir';tvciing wuvcs corrc\poncling to linear cigenl'itnc~ion.;.Instcad. unstable ("turbulent") (low i~ ch;~ractcri/ccl by the appearance of' s t r c i ~ t i i w i ~streak>. c Those are nlterna~ingregions ol' ~ ~ I S I C i111Cl I. slowel. flow in the region near thc w:~ll. In~crmittently.these stl-eaks arc dcstroycd by a rapid in5t;tbility known ila ;I turhulrnt bu~-\t.After a burst. the streaks slowly reform. While nonlinear et'fccts play n I-ole in the dynamic\. they d o not by Ihe~iisclvcsexplain the instability. Indeed, it is easy to show that the nonlinear tcrriis in the equations protiuce n o growth in (he energy of a diaturbances. Rather. it is ;I linear effect which amplifes disturbances to a level where they become large enough for nonlinearity t o become important.
This linear effect is an instance of non-modal growth as discussed in the example of Equation (9). We reexamine the linearized Navier-Stokes system, making no a priori assumption on the time dependence of a disturbance. Instead of (147), we obtain the system
Motivated by Squire transformation, we set & = (YIJ)/&. This leads to the system
d m ,i = ( a u + f i u ) / & .
i) = (fiu
-
We note that the first. third and fourth equation of this system do not involve 6 and hence can be considered independently of the second equation. In the second equation. however, there is the term (fi/;)RU1ni, which links the evolution of i, to nl.This kind of one-way coupling is precisely analogous to the coupling in the example (9).
where the equation for y depends on .v. but the equation for .v does not depend on J . The coupling has no effect on eigenvalues. However. it leads to linear growth in the inviscid limit 144.801 and to transient linear growth in the viscous case. We note that the coupling term in ( 159) is strongest when fi = 6 and w = 0. This is precisely the class of modes which wits disniissed as "always stable" in Squire theorem. Physically. these disturbances correspond to streamwise vortices. The coupling term represents the effect of a streamwise vortex on streamwise velocity. Where Huid is advected away from the wall by the vortex, the How is slowed down, where fluid is advected towards the wall. the flow speeds up. This leads to the formation of streaks. Over the past decade, this linear growth of disturbances has been investigated extensively in the literature. and energy growth and pseudospectra have been investigated in detail. The basic result is that a disturbance can grow linearly by a factor of order R. Among the many papers on this topic, we cite those of Butler and Farrell 1231, Boberg and Brosa 1 171, Gustavsson 1561, Reddy and Henningson 1 107j. Henningson and Reddy 1581
and Trefethen et al. [ 162,163 1. While the linear mechanism of streak formation resulting from strearnwise vortices is well understood, ideas on the eventual breakdown of streaks and particularly on the regeneration of streamwise vortices are much more speculative at this point. Model problems have been proposed by Baggett, Driscoll and Trefethen 161, Gebhardt and Grossmann 1461 and Waleffe [ 1661.
Another approach to transition to turbulence is to study the behavior of shear flows perturbed by an unstable or nearly unstable eigenmode or that of tinite amplitude states resulting from bifilrcations. In plane Poisei~illeflow. the instability at R = 5772.23 leads to a subcritical Hopf bifurcation. Bifurcated solutions exist down to R = 2900, where the b ~ i n c hturns around (see F i g ~ ~ rI eI . from Chapter 2 of 1481). If the flow is constrained to be two-dimensional, then there are indeed stable traveling wave solutions of tinite amplit that s~lchstates are three-dimensionally unstable. The instabiltude. I t turns o i ~ however. ity is inviscicl and therefore occurs much more rapidly than the growth of unstable twodimensional w;tvcs. There is a broad spectrum of un.;table modes. with maximal growth
Fig. I I . Finite a~iiplirudctr:~vcllng wave\ lor pl:t~icPoi\cuillc flow. the axe\ reprc\cnl Reynold\ number. ompli01 C;c~i~britlgc LI~iiver\ilyPI-c\\. rude and wave nurrlhcr. Kcprilltcd wit11 the perr~li\siol~
Fig. 12. Suhhar~~ioliic A vortice\. Reprinted with the pcrmi\\iot, oI'C;~mhrillgeUniver\ity Pre\\.
rate at ;I spunwise wavelength which is comparable to the streamwise wavelength ot'the two-dirnensiont~lbase How. The secondary instabilities lead to flows with streakline p a terns in the shapc of a A : they are therefore called A vortices. Two kinds of modes exist: Fundamental riiodcs have thc same strealnwisc periodicity as the two-dirnensionitl wave. while subharlnonic modes have twice the period. Figure 17 (from Chapter 2 of 1481)shows a pattern of subhiumonic A vortices observcd i n a boundary layer How. In experiments such as this the triuisition is induced by a forcing which excites an initially two-dimensional wave. For surveys of work on transition we refer to the reviews of Herbert 159.601. Bayly and Orszag 191, Reed and Soric [ 109. I 101, and Huerre and Rossi 148 1. The results on nonmodal growth discussed in the preceding section suggest the study of secondary instabilities in Hows which are perturbed by streamwise vortices itnd streaks rather than by Tollmien-Schlichting waves. Secondary instabilities in such flows have been studied by Zikanov 11761 and Reddy et al. I 1081.
Viscoelastic effects can substantially lower the critical Reynolds number of parallel shear flows. The Maxwell and Oldroyd B fluids are the most thoroughly studied. For plane Poiseuille flow. the critical Reynolds number can be lowered to approximately 1700 by elasticity [99,156]. A long standing question is whether there can be purely elastic instabilities in parallel shear flows such as those in Taylor-Couette flow discussed in the previous chapter. Instabilities have been found in the presence of wall slip [98,1 15,141. and for fluid models where the shear stress does not depend monotonically on the shear rate,
but other'wise results have been mostly negative. Akbay et al. I I 1 found an instability for the lower convected Maxwell nlodel which is linked to resonant mode crossings in "supersonic" flows [I201 ( i t . , flows for which the fluid speed exceeds the propagation speed of shear waves), but the instability does not seem to persist for more realistic models [ 1 141. For plane Couette flow of the upper convected Maxwcll and Oldroyd B fluids, and for plane Poiseuille flow at low Reynolds numbers, only eigenvalues with negative real parts have been found [ I 12.1561. The potential role of non-modal growth i n viscoelastic flows requires further study. For the upper convected Maxwell model, it is not even known whether spectral stability implies linear stability at all (for the Oldroyd B fluid, a proof is given i n 1 1 171). An interesting mathematical aspect of viscoelastic Hows is that their spectral properties are very different from the Newtonian case. Even for bounded flow domains, a part of the \pectlum is continuclus. As on exalnple, we consider plane Couette How of the upper convected Maxwell fluid. The flow domain is the region 0 < y < 1, and tht. equations, i n dimensionless forni. are
(161)
div u = 0 .
The basc How is given by
For the uppet- convcctcd Maxwell niodel. :I Squirc t r ~ ~ n s f ~ ) r ~ n cxists a t i o ~ i1 1601, although this is not the caw for viscoelastic tluids in gcncrul. The study 01. eigenvalucs ciui hcncc be limited to two-dimensional perturbations. Wc nssulne propor.tionality to cxp(icu.v nr ) . The analogue o f the Orr-SommerfeId equation cad\ I 1 17 1
+
Here I ) denotes d/dy, and
I n contrast t o the Newtonian case. a meaningful stability probleln arises even in the case K = 0, because of the time derivativcs of the stress tensor appearing in the constitutive law in (161). If we set K = 0, then (163) can bc solved in closed form 1521. Two linearly independent solution5 are given by (cr iu!.)c x p ( f av). and the other two are given by
+
exp(-ia W ? fa J m v ) . The eigenvalues are then determined by inserting the solution into the boundary conditions @(O)= D@(O)= @ ( I ) = D @ ( I ) = 0 . This yields a quadratic equation for a , the roots of which have the form a = -icrc, where
Gorodtsov and Leonov 1521 verify that A < 0 and B < - 1 /4 if u > 0 . As a result, all eigenval~~es are stable. In the lirnit a --+ oo.the two eigenvalues become
and the eigenfunctions become locali~edat the walls. In addition to discrete eigenvalues, there is a continuous spectrum given by the equation S = 0 , i.e., a = - I / W - iuy,where y E 10. I]. If S = 0 , the coefficient of the highest order derivative in ( 163) vanishes, and the differential equation becomes singular. The presence of continuous spectra, even for bounded How domains. is a general feature of viscoelastic Hows, which is linked to the memory of the Huid. The decoupling of the continuous and discrete spectra, which occurs in plane Couette How of the upper convected Maxwell Huid is highly unusual. In general, we cannot expect the discrete eigenvalues (even if there are finitely many) to be roots of a polynomial and we should also expect situations where discrete eigenvalues bifurcate from the continuous spectrum. Indeed. this happens for more general flows 1 1 701. In plane Poiseuille How of the upper convected Maxwell Huid, there are six discrete eigenvalues if a is small, but only four if a is large. For the Oldroyd B Huid, there is an additional line of continuous spectrum and an additional family of discrete modes. The characterization of essential spectra in more general flows is a problem of significant mathematical interest. It is of practical interest. too, because the approximation of the continuous spectra poses major difficulties for numerical stability calculations. Even for parallel shear How, we need to distinguish between the essential spectrum of the one-
dimensional problem which arises for a fixed wave number and the essential spectrum of the two-dimensional problem. For instance, the value
is in the essential spectrum for the two-dimensional problem, because it is a limit of eigenvalues for a + oo. In [I251 it is proved that for two-dimensional flows without stagnation points, the essential spectrum contains two parts: one is located on the line R e a = - I / W, and the other is characterized by the short wave limit of Gorodtsov-Leonov wall modes.
Two-layer flows lead to new instabilities and bifurcations associated with deformation of the interface. In the simplest case, we may consider plane Couette flow of two superposed fluids. We state the linear stability problem i n dimensionless terms. Let the flow domain be given by 0 < ,- < I . and let the interface be at ,- = I I . Fluid I with viscosity 11I and density /)I o c c ~ ~ p i the e s region 0 < :< 1 1 .and fluid 2 with viscosity /L? and density pr, occupies the region I I < :< I. The ratios of fluid properties are 111 = ~1 I / / L ~and r = p l / p ? . The velocity profile of the base f o w is given by
The linear stability problem (for modes proportional to exp(ia.r is governed by the system
I -(I)' Kt
7
PI.
1
- a - - fl-)tl
-
-/a/?
Pi
-
+ i P y + a t ) ) in each fluid
U ' ( : ) l l ) - U ( : ) ; ~ iZtf 7 1 4 ,
Here R, = U * l * p , / p , , where U * and I* are the maximum velocity and channel width in dimensional terms. At the walls we have the boundary conditions 11 = u = 111 = 0,and at the interface := I I we have the conditions of continuity of velocity,
co~ititiuityof shear stress,
balance of normal stress
and the kinematic free surface condition
Here /J is the perturbation to thc interface position, [ . 1 denotes the jump of a quantity across the interface (value in fluid I rrlirii~svalue in fluid 2), and .Y= .$*/(PIU * ) . where S* is the surface tension coefficient. and F 2 -- ( U * ) ' / ~ I * . whcrc I: is the acceleration due to gravity. If'both fluids arc equal and S = I / F ' = 0.the system above has the obvious solution 1 1 = v = u: = 1) = 0. h = 1 and (T = - i u L / ( I I ). This neutrillly stablc mode is referred to its an intertl1ci;il ~iiodc,it corresponds to sitnply perturbing the interface without ch:uiging the tlow. It' fluid propel-ties dil'li31.. however, then a perturbation of the intcrflwc will cI1:inge the flow. .l'hcre iu-c two tcr~usin the intcrlhcc coliclitions i~hovc.which intlucc such :I coupling. Thc tc1.111 i~ppc:iringin the balancc o l ' n o ~ ~ ~stre\\ i i i ~ l i~i\,ol\,cs g~*;~vity ;11i(1s i 1 r t 1 1 ~ ~ ' tension. the other tc1.111 ~ I P I ) C ~ I I . ~i nI Ithc ~ continuity 01' volocity involves the ,iilmp in (1'. which is ;I I-cault of viscosity atratifici~tioti.Thc cl'l'cctx 01' gra\;ity :inti surl'lcc tension o n stability of thc intcrt~ccarc ri~thc~.ohvious, hut viscosity stri~tilic;itionis ;I more interesting cf'fkct. Scluirc tnlllsfOrllliltion ;~ppliesto nlultilaycr flows. 3rd so it is suI'fi~i~'t~t to ~ 0 1 1 sider two-tlimcnsio~i:iIdistu~~hanccs. Thc first stl~dy01' thc cf'f'cct of viscosity stl-~itilic~ition is due to Yih 11731. who developed a rcgular perlur-bation cxp:inslon tor the cigcnvaluc in the l i l i i i t of s1ii;11ILY.The gcncn~lresult is quite complicatcil. but some trends can be and nearly ccllral densities and vi\;cosities, I f wc seen fr.orn the limiting c;iac of srn:tll set 111 = I - F I T / . 1. = I - c.; and assume I I sm:ill. then we have. i ~ tleuding order i n F and l l I1301:
The effect of gravity is stahili~ingif the heavier Huid is at the bottoni ( F < 0). as expectcd. The effect of viscosity stratification is stabilizing if l i t > 0, i.e., the thin laycr at the hottom is occupied by thc less viscous fluid. Moreover. it' I I is small. thc cffect of viscosity slratitication (O(1:)) is stronger th~uithat ol' gravity ( ~ ( l : ) ) .Many papers have appeared which extend Yih's analysis to more Ihan two layers or other Rows. e.g., Hugen-Poiseuille How in a paper. We refer to 1771 for a rcvicw and references to the literature. Hooper and Boyd 1621 considered the short wave limit LY + m. In this limit, the contribution to the eigcnvulue which results from viscohity stratification is of order LY-'. and
Fig.
13.
Balnhoo w;tvc\
ill
~ ~ p w ; ~Ilow r i l o l o i l nil w;lrcr.
in the case of e q ~ ~ densities al i t is always destabilizing. Other asymptoric limits which have been investigated include the long wave limit in n semiintinite geometry 1631. similar liquids 11301. arid high Reynolds ~ i ~ ~ n i b 164.16). ers During the 1080s. the lincar stithility 01' two-layer flows wits also invcstig;tted numel-ically: we r c k r to Blcnncrhassctt 1 15 1. Joseph. Rcr1;trcly. ancl Rcniu.cly (691. Kenrtrdy 1 1271. and Yiantsio\ and Higgins 1 1721. Hil'i~~.cations resulting fro111instabilities in two-l;tyer shear flows arc mostly supcrc~-itical itncl leiid to tr~tvelingiriterfitcial waves 1 15.133 1. These interfltciitl wave itrc subject to sitlcb:~ricl iristobilities 1 1 161. For expcri~ncntalstudies 01.11-rtvcling intcrfrtcial wi~veswe refer to 127.1431. Direct numerical simi~latioris183 I show that the applicability ofwcahly nonlinear ~uialysisc;ui be quite limited. in particular at low Reynolds numbers where the growth of' C the mean she;tr Ilow is strong. interfltcinl waves is weak and the distortion ot'wavcs ~ L I to In those situations. fingers are observed to I'orm. Core-:tnnul:tr How in pipes is ol' signilicant practical import~uncc.c.g.. in the pipeline transport 01' visco~iscrude oil. We refcr to the recent article by Joseph ct al. 173 1 li)r rt review of research in this field. Experiments carried out by Joseph's group 171 show patterns of "bamboo waves" and "corkscrew waves" resulting from axisy~nmctricand nonaxisym~ n c t r i cinstubilities. B~uiiboowaves have pointed crests ancl are almost symmetric. they itre roughly pcriodic. but individual waves have the appearance of solitary waves (see Figure 13). They appear in upward How of oil and water with the oil at the core. The shape of bamboo waves has been successfully reproduced in recent simulrttions 1841. I n downward How, corkscrew waves appear, in which the oile core has a spiral shape. The bifurcation leading to these waves is a Hopf bifurcation with O ( 2 ) sy~nmetry.which poses a problem of pattern selection between (azimuthally) traveling waves (the corkscrew waves) and
standing waves (for which the core has the shape of a snake). The pattern selection problem is studied in [136]. In two-layer flows of viscoelastic flows, stratification of normal stresses leads to a term in the shear stress condition at the interface which can lead to instabilities. Early investigations of the problem were missing precisely this term and erroneously concluded that the jump in the normal stress was not important. The first correct analysis is due to Renardy [ I 3 1 ], who considered the short wave limit. Chen [26] analyzed the long wave limit. Numerical and experimental investigations have been carried out by Khomami and his coworkers [ 154,168,1691. Bifitrcations resulting from the instability are studied in [ 134, 771.
6. Capillary breakup of jets
We consider a cylindrical column of fluid at rest with radius R and free surface conditions at the boundary. Since a cylinder is not a minimal surface, surface tension causes an instability which leads ultimately to breakup of the column. In this section. we consider the linear stability problem. The inviscid case for this problem was studied by Rayleigh 11051, thc viscous case was iunalyzed by Wcber 11671 in the long wave limit: the analysis was extended to the full Navier-Stokes equations by Chandrasekhar 125 1. Since only axisymmetric modes cause instability. we confine our attention to these. Let u and I I J denote the radial and axial velocity, p the prcssure. and S the perturbation to the interface position. For a disturbance proportional to exp(icu: r r t ). the lincari~edNavierStokes equations read
+
The boundary conditions at
1.
= K are the vanishing of shear stress,
the balance of normal stress by surface tension.
and the kinematic free surface condition
To nondimen\ionali~ethe problem, w e scnle x , 7 and S with R , GJ with I / K , cr with w ~ t h1 7 / ( p K ) . and 17 with r l ' / ( , o ~ ' ) The . nondimensionulized equations are
1 7 / ( 1 1 ) ~ 2 )11.
and the bouncl;uy conditions at r = I are
Here the Ohnesorgc number i s defined hy 'I Oll = -
7'/, H
To solve ( 179). we s1iu.1 1'1.ornLapliicc's equation for thc prcssurc.
U p to a constant l'i~ctor,the solulion which is I-cgulitr at I . = 0 is 11 this in [he lirst equutiorl uI'( 179) itlid xolvc (or It. Thc I-oxult is 110.) = -
+ c711(rd'(yl+=).
~ I I ( W I ' )
Km
=
lo(ur.).\Vc then use
(
1x3)
Fil~nlly.we can detcrlllirlc 111 I'rolli the 1hil.d equation 01. ( 170). By inserli~lgthe result into the boundary conditions. we obtain a tr:~nscendrntalequ:~tion for rr.
Fig. 1.5. Growth rate at medium Ohnesorge number ( I ).
Fig. 10. Growth rate at high Ohne\orge ilun>her ( 10')
The Figures 14-16 show the growth rate n as a function of u for various values of the Ohnesorge number. Regardless of the Ohnesorge number, linear instability prevails for 0 c a < I. In the inviscid limit (Oh + 0), the growth rate has a rnaximum near u = 0.7. This maximum shifts to lower wave numbers as the Ohnesorge number is increased. In the Stokes How limit (011-t oo),the maximum disappears and the fastest growing waves are long waves.
One-dimensional models are based on the assumption of a slender jet where the scale on which the jet radius varies is sufficiently longer than the jet radius and the variation of axial velocity across the jet is negligible. When a jet breaks up into spherical droplets, these assumptions are valid in the necks between the drops where breakup takes place. We find it convenient to formulate the one-dimensional equations in a Lagrangian formulation. We consider a reference configuration in which the jet has uniform thickness 6. Let X denote the position of a fluid particle in this reference configuration. and let .r ( X . t ) be the actual position. The stretch is defined by
Let u ( X , t ) denote the axial velocity. The equality of mixed partial derivatives leads to (1 8 5 )
.St = 14 y, . The cross section of the jet is
A =n62 1 s . The balance of axial momentum yields
Here T,, denotes the axial stress component, p is the pressure, and 2 n T 6 / f i is the product of the surface tension coefficient T with the circumference, i.e., the axial force of surface tension. For a Newtonian fluid, we have
With
11
denoting the radial velocity. incompressibility implies that
and within the slender body approximation
11
is proportional to r . so
The stress condition on the lateral free surface of the jet now implies that
By inserting the resulting expression for
/I
in ( 186). we obtain
We now consider the one-dimensional equations in the case of Stokes flow, i.e., p = 0. In this case ( 19 I) reads
M. Ret~urllyund L: Rmurdy
278
where h ( t ) is an undetermined integration constant. Moreover, if we consider periodic disturbances, then ( 1 85) implies that
where L is the period. After rescaling variables, we may assume that L = T / S = 377 = 1 , so the governing equation becomes
where h ( t ) is determined by the constraint
The following result was established in [ 1241.
THEOREM. Con.sitl~rtho trhovc~prohlerti tvith fhc irliti~rlc.otlditiorl s ( X .0) = .s, ( X ) , 1.~8h~ro s ; i.s ~wrioclic1~9ith /)oriotl I . .s~rti.sfie.s fhr c~ori.strtiir~t trrirl L I S . F L I I ) I L ~ir.s S u~iique rncr.ritnutn s,,,;,,
crt X = 0. Supposc~rhtrt ill
tr
riei~hhorlloodof' X = 0 . Ivr htrt~c
The result shows that, unless the maximum in the initial condition of s is very flat, then breakup will always occur in finite time. A formal description of the asymptotic approach to breakup is given by the similarity solutions which will be discussed in the following section. I t would be desirable to have a mathematic~iltheorem which links the asymptotics near breakup to the behavior of the initial condition near the niaximum of .s.
We now look for self-similar solutions of ( 19 I ) and ( 185). Such solutions were found by Papageorgiou 1951 and Eggers [41,42[.These solutions compare well with numerical simulations and experiments as the breakup point is approached. We begin with the case of Stokes flow, analyzed by Papageorgiou 1951. The analysis was subsequently simplified by Eggers [43].As in the preceding section, we can scale out parameters to tind the equation
We shall look for solutiorls of the form
Here t = 0 is the breakup time. IF we assume that large velocitier are confined to the selfsimilar region, then ( 19 1 ) implies that
arid hence
Inserting ( 198) into ( 197). we tincl
Kcgular bchnvior of'thc solution at 6 = 0 irnposcs thc conditions
This lead5 to
We next make the substitution
We can then find the solution of the differential equation ( 2 0 2 ) in the implicit form
Since changing the constant only aniounts to a rescaling of 6 , we can assume C = 1 without loss of generality. The constraint (201) now takes the form
The integrals can be evaluated (see [53, p. 299, # 3.259,3]), resulting in the equation
The snlallest positive root of this equation is at 8 = 2.17487. I t follows froni (19 I ) that the velocity for the similarity solution has the form
where
If we now examine the inertial term in (191), it scales like ( - 1 ) -'+", while the righthand side o f the equation scales like ( - t ) ' P . With 8 < 5/2, this implies that the inertial term would dominate as t + 0. I t follows that the assumption of Stokes flow becomes invalid very close to breakup regardless of how viscous the fluid is (Lister and Stone 1861 show that this is not so if the jet is surrounded by another viscous liquid). The balance of inertia, viscosity, and surface tension leads to another similarity solution for which = 5 / 2 . We set
The resulting equations (after scaling out irrelevant constants) are
Unlike the Stokes case, we do not have an analytic procedure to solve these equations. Numerical results of Eggers 1421 show the existence of a solution with acceptable behavior at both zero and infinity. Unlike the Stokes solution above, Eggers solution is highly asymmetric. Indeed, the observed breakup of jets is very asymmetric. with necking occuring between a large drop and a much smaller satellite drop.
Stclbilijy and instability in viscou.~ j7uid.s
28 1
6.5. S~lppressionof breakup b y elasticity The addition of polymers can suppress or at least significantly delay the breakup of a liquid jet. Linear stability has nothing to do with this, indeed it can be shown that fluid elasticity actually promotes the instability. Rather, the stabilizing effect is felt in the later stages of the evolution, where polymeric fluids tend to evolve into a "beads on string" pattern where approximately spherical drops remain connected by thin filaments. A review of literature on experiments and computations on viscoelastic tilaments up to the early 1990s can be found in [ I 7 11. We shall consider the one-dimensional equations for the Oldroyd B fluid, in the inertialess case. We need to modify our equations above to account for the elastic stresses. We have
where the elastic stresses C and Y satisfy the equations
Here I / A is the relaxation time. and /L/A is the polymer contribution to the viscosity. I t is convenient to make the substitutions C = /,,s2 - L L , Y = y / , - / L . This leads to the equations
where the integration constant called f ( t ) corresponds to A([) in earlier sections. We shall consider a periodic situation, with the constraint that the average length does not change, I.C..
In [ I 181, it is shown that breakup in finite time does not occur, and a continuous solution exists for all times. The only assumption on the initial data is that p. y , and .s are continuous and positive. Thus the Oldroyd B model is an example where elasticity can completely suppress capillary breakup. In reality, breakup is usually delayed by elasticity, but occurs eventually. Mathematically, such behavior is found, for instance, with the Giesekus model. In this model, numerical computations [ 1 191 show breakup in finite time, and the asymptotic approach to breakup can be described by a similarity solution [ 1261.
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M. Renardy and Y. Renardy
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M. R m u r b und Y Rmurd.)
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1 1321 1 1331
I 1341 1 I35 1 ( 1361
11371 11381
1 1301 11401
1 14 1 1 11421 [ 1431
1 1441 1 1451 1 1461 [ 1471
1 14x1
Ncwf. Fluid Mech. 28 (1988). 99-1 IS. Y . Renardy. Wc,crkIv rtorrlirrc~or-/~c~lrtn~ior o/'l)c,rroilic.(li.\frrrl~trrr(.c,.\ irr fb~~o-leryr,r Corrc,rfc,/IO\I.. Phys. Fluid\ A I ( ICJX9).1660-1 670. Y. Kc~lardyand Y. Kena~dy.14rtfc~r.rrtc,lc,(.fiorr/or. tlrc. r~.\~~illtrtor:v orr\c,t in flrc~r-~~ro\oloftrl ~~orr~~c~c.tior~. Pliy\. Fluid\ A 6 ( 11)93). 1376-1381). Y. Rcliardy. Wc~rXlyrrorrlirrc~trr./)c,hcrt.ior.o/'/)c,r.~otlr(. cli.\trrr.h~rrrc~~~.~ irr r\~~r)-ltrvc,r. ~)ltrrrec.lrtrrrrrv1 /IOII. o/ rrl)/)c'r-l.orn.c,(.fc,clMcr.\tt,c~llli(/rti(l\. J . NOII-Newt.Fluid Mcc.11. 56 ( 1005). 101-176. Y. Ke~ii~rdy. IJcr//c,r.rr /i~~~r~rcrfrorr ,/iw t~.\c~il/(rtor~~ l ~ ~ ~ l k - ~(~i~rrr/)c,/irior~ ~ ~ o c l c ~ 111 ( 1 /~t~o-Icrvc~r 1~c;rrcrr~cI ~)r.c)l~/c~r~~, %. Angew. Math. Phyh. 47 ( 1006). 5h7-500 and 48 ( 1097). 171 Y. Kc~lardy..Srrtrke,.\ (or(/(.o,i(.\t.,r~tt..\ rrr co,r,-rrrrrrttltrr/1o11, o/ f~~~o,/lrrit~s. J. Fluid Mcch. 340 ( 1907) 207-3 17. Y.Y. Kc~i;t~-dy and M. Rcnardy. I r ~ / I r c c ~ r~!/'rrorr-Norr.\\rrrc,\y ~c~~~ c,/fc,c.t.\ 0 1 1 pottc'r.rr\ irr \olr /irr,yr,r-c orr~.c,c./iorr, %. Arlgcw. Mi~tIi.Phy\. 49 ( 1'198). 774-250. Y.Y. Rclial-dy. M. Rcnnrdy and K. Fujimurn. 7irkc~rr,s-Rog~lrr1rot~ hi/r~r.c.rrriorron the, Irc~\tr,yo~rcrlkrrtrc.c, /or ek)rthlc~-lrr\c~r c.orn,c,(.fiorr.Phyh. D 129 ( IVY')). 17 1 -201. 0. Rcy~~old\. AII c~~l~c~r.irrrc~frrcrI irrt~c2.sti,qcrriorr ofrlrc, c,ir-c~r~rrr.\ttrrrc~c..\n.lric.11tk~tar~~rirr~~ \t.hc,rIrar. /lr(,rrroriorr of ncrtc,r- \Irctll hc) tlir.c,(.ror sirrr~ort.~, trrrtl r!ltlrc, Itnt, ( I / rc,\i.\rtrrr(.c, trr ~~trr-crllc,l c.lrtrrrrrrl.\. Philob. Tram. Roy. Soc. Londc~n174 ( 1x83). 035-987. M. Roberts. J.W. Swift iund D.H. Wagner. Tlrc, Hol~f'h~/i~rr~~rriorr or1 rr Irc~.ur,yorrctlItrttic~c~. Multipararnctcr BIf'urcafion Theory. M. Goluhif\ky and J.M. Guchcnheir~~cr, cd\. Aliicn Mttfh. SOL,.Series: Confc~lip.Math.. Vol. 56. Alncr. Math. Soc.. Providence. R1 (1986). 7x3-318. V.A. Ro~iianov.Sftthili!\. ofl)ltrrrc~-l)trrtrII(,ICorrc,tto & ~ n , .Functional Anal. Appl. 7 ( 1073). 137- 146. D. Rucllc and F. T:I~cII\, Or! !Ire, rrccfctrc~o~fcrrl~ctlc~~rc~c~, Coliini. Math. Phy\. 20 ( I971 ), 167-1112. M. Sangnlli. C.T. Gallagher. H.C. Chang and M.J. McCready, Firritr trrrrl)litrrtfc~n,trt,r.s trr rlrc, ~rrrrr-firc.c, hct\c.c,err flrtitl.\ u,itIr c1~ffi~w111 t'i.vc.o.\i!\.: T/7e,orv t111tI c~.vl)~~.irlir,~~t.\. Phyb. Rev. Lett. 75 ( 1905). 77-80, J.W. Scalilol~and L.A. Scgel. Firrift, crrrr/~litrrclr~ c.c~llrrltrrc,orn~c~c~firm irrrlrrc.ccl I)y .\irrfirc.c, tc,rr\iorr. J . Fluid Mech. 30 (1967). 140-161. A . Schliiter. D. Lor[/ and E Bu\\e. Orr rlr~.srtrhilir,.of.\fc~trtly,hrritc~ crrrrl)lrrrrclc, c.orn~c~c.fiorr. J . Fluid Mech. 23(1965). 129-144. R.W. Schmitt. Dorthlc tliflir\iori irr oc~c~ttr~o~qrtrl~I~y. Ann. Rev. Fluid Mech. 26 ( 1994). 255-285. G. Schneidcr. The ~~trlitli!r o?f,y~rrc~~-crlCetl Girr:hurg-Ltrtrrltrrr c,c/rrtrtiorr.s,Math. Meth. Appl. Sci. 19 ( 1996). 7 17-736. E.S.G. Shaqfeh. Prtrc,(v eltrsric. irr.trcthilitic~.sirr visc.or~rc,rric.,fioir..\, Ann. Rev. Fluid Mech. 28 ( 1996). 1201x5.
[ I491 A. Shepeleva, 011r11(,vtilitli~of'fhc ( / ( , ~ P I I C ) Z I ~ CGinzhurg-Ltitlderu ryuution, Math. Meth. Appl. Sci. 20
( 1997). 1239-1 256. 11501 M. Silber and E. Knobloch. Ho/?f"hi/itrc.rifiottor1 tr syucrrt Itiffic.c,. Nonlinearity 4 (1991), 1063-1070. 1 15 1 1 H.B. Squire. 011r/to .sfcihility of' rltwe-tlitttrrt.sion(11r1i.srurhrrtt~~r.s 01' vi.s(.ou.sflow hofb~.(,(,~t ])(iruIIt~I~ ~ 1 1 l . s . Proc. Roy. Soc. London A 142 ( 1933). 621-628. 11521 M.E. Stern, Tlte ".\elf fi~utrr(iiit"titit1 fl~c~rittol~crlit~~ (.ott~'~(.riotl. Tellus 12 (1960). 172-175. 11531 B. Straughan. Tile Etrc,,~?.Mc~fhorl.Sf(ihili!\: tir~rlNottlii~e~itCort~~~c.fion. Springer-Verlag, New York ( 1992). 1 1541 Y.Y. Su and B. Khomami, Plrrc~lyol(r.crit. ittrc~r/2rc.irilitt.sttihilitic~.~ ill .strl)c,rl)o.srtl,porv of'/~olvt~~rric. fluir1.s. Rheol. Acta 31 ( 1992). 4 13420. 1 1 551 R. Sure\hkumar. A.N. Beris and M. Avgousti, N o ~ t - r r \ - i . r y t i t ~ t ~ .\~rhcrific~til i~fri~~ hi/irrc~trrioil.sin ~~i.s~.oc~lti.sfic. E i y I o r - C o t f f flow. Proc. Roy. Soc. London A 447 ( 1994). 135-1 53. 1 1561 R. Sure\hkulnnr 2nd A.N. Berih. Littc~tirsftrhilify (rtrrrly.tis (?/'~,i.\e.orltt.\ric. Poisrrrillc,flo~rrrsi~t,qerri At-ttolrliho.tc,tl o~/io,qo~rtrli,-r~fio~~ til~oriflttn.J. Non-Newt. Fluid Mech. 56 ( 1995). 15 1-1 X2. 1 1571 P. Takii.. P. Rollerman. A. Doellnan and E.S. Titi. An(t!\.rii.ify o/'(~.\.\oirit~l!\. hott~rtlrtlsoltrfi~)t~.s ro .\~~nrilittc~iir ~)t1rt~1?011t~ \y.\r(,ttr.\ ontl 1,(11i(/;fy o/ f11c~ ; i t ~ , - / ~ i o ~ q - ~(,(littirio!t. i ~ i ~ l ~ i i SIAM ~ J . Math. Anal. 27 ( 1996). 424448. ( 158I (3.1.Taylor. Sftrhili!\ ( I / ti ~~isc~~~rr.c liqr(rd ~~onttri~t(~tl h(,r~l.c(,n111.0 ,vftrritt,q ~ ~ ~ I I ~ IProc. ( / ~ ~Roy. I : sSoc. . A 223 ( 192.3).28')-343. 1 1501 J. Tholllsori. 011( I c~lrott,qirt,qfc~.s.\c~l~ifi~rl .\frit<.ti(,rin i.c,rrriitt lir/iri(l\. Proc. Philos. Soc. Gla\gow 13 ( 18x2). 404468. I I O O l G. Tliipii ~ I I R. I ~ Bcr~iski~ .Yti~l)i/if~ ~. o/'ir t ~ ~ ~ / i t . ~ ~ ~ ~f i~. \~~ ~t to -~ r~,/Il(iil y/ (~i . )\ rI~I i, I(~I /~I \li,qltr (,l(t.sri~~ify, Phys. Fluid\ 13 ( 1970). 505-568. I I01 1 I..N. Tl'clctlicl~.I'\c~ri(lo\l~c~(~t,ur 01 littc~rro/~~~,rrfot-\. SlAM Kcv. 39 ( 1907). 3x3400. 1 1621 1. N.'l'rcl'cthc~l.A.E.'l'relcthe~l.S.C.Kcddy ;~licl'I'.A.I)r~\colI./ i \ ~ , I t ~ o ~ ! \ . t.~f(rl~i/i!\. t ~ ~ ~ ~ )~i.irl~oiir i(. ~~i,q~~t~~~tiIt~~~ Sciellcc 261 ( 1093). 578-584. ( I63 1 A.t<.Trckthcn. I ..N. Trekthen i111cI P.J. Sch~ii~cl. .S/)(>(.rr(i (rtt(1 ~ ) . \ ~ , I I ( ~ ) . \ //~o(t ,l I) .i Il ~ I'oi.\c~irillc~ ( ,I /Io~i,.( ' ~ I I I I X I ~Mcth. . AppI. Mcch. Icngrg. 175 ( 100c)).413470. 1 lh4l A. xi~ltlc~~h:~i~wllcclc and < i . loos\. ('oirfot.~ricoti/ol~l rltc,orv itr iti/nrir~~ clittic~tr.\tot~.\.I)YII;IIIIIC\ Kc~~ortcil I ( 1007). 175- 103. 1 1651 A. vt111 Hiirtcli. 0 1 1 flt(,~~trli(/ifv /!/'rIic(;iit:/~iit~q -/~in(l(ii( ~Y/II~I~ J . ;No111i11c;ir OII. Sci, I ( I00 I 1. 307-427. 1 I001 E'. Wi~lcll'c./ / ~ ~ / r o ~ I \ .\t~iI~ilify ~i~~i~ (in(/ t i f~i~~ t ~ l ~ ~/ t~l( ,~~~o ~ tfrii~i.\i~~nf.\ t (t li ~ ~ ~ ;f o ( 1 .\(,lf-\ir\r(i;~~ii~,q /)r01.1,\\. S ~ L I ~ , Appl. M;IIII.95 (1005). 310-343. [I071 C. Wchcl-. on^ %ct./itllc,irtc,.\ I.lti.\.\i,qXcir.s.\rr(iI~Ic~\. %. Angcw. M;itl~.Mcch. 11 ( 103 I ) . 135154. 1 I O X 1 G.M. Wilso~ii1t1d B. K l i ~ ~ ~ ~ AII i ; ~i~~ i. i~i/.~ ~ ~ t ~itt~~i~\fi,qiifio~t ; ~ t i i ~ ~ i r ( i01/ ~ I I I ~ I : / ( Iin\f(iI~iliri~~\ (?(I/ ;I!t~ittlfil~ivi~r /ION, of~~i.~(.oc~lti.\fic~ /Iiri(l.\; 1'tit.f I. /tic-ottiltctril~lc. ~x)lvttrc,)\v.\rc,/n\. J . Noli-Newt. Fluid Mcch. 45 ( 1007).355-384, I I601 G.M. Wil\oli iilld H . Kholliilll~i.,411o~l)(~ritncttt(iI ~ I I I . ( , . \ ~ ~ ,01~ (ittrc~./(i<.i(ll I I I O I I iti.\f(ihilirio\ iti ~nt(lfikivot. /IO)I. ( I / ~~i,\(~o~~l(i.\ri(~ /Iiti~/.\: /+it.! I/. Elit.\fi~.(i11(1t t o t t l i ~ t ~1,[/(,(,1\ ~ ~ i r f i t ~ ~ t ~ ~ ~ ~ ~l~o11~1ttc1t ~ l ~ ( \v\ti,tti\: i r i / ~ l l'(it.f ( ~ Ill, (i)itrl~otihlc,~)olvtrtc~r.\y.\fc,nr.\. J . Rheol. 37 ( 1')")3). 3 15-354. 1 1701 H.J. Wll\on. M. Kcllitl-dy and Y. Kenal-dy.Sft~o(~rttt.(~ o/'/lt(,.\/)(,(.rrrt~ti itr :eprr) Ki,~~tokl\ t~irrtil~~~i.\lt~,tir-/6)11 o/ tltc, (l('M ~rnclOkltr,r.tl-H lit/rritl\. J . Noli-Newt. Fluid Mcch. XO ( 1000). 25 I-2hX. [ 17 I 1 A.1,. Y;irin. 1~'t.wl,it/~~ii/ .l(,r,\ (i11t1Fi/tir.\: H v ( l r o ( l ~ ~ ~ ( i (ot(1 t t ~ i (k'Ite~o10,yy. ~\ l,o1ig11ii11i. H;irlow ( 1003). 11721 S.G. Yiti~lt\io\ii~idH.Ci Higgilib. l~rtre~it~ \ f ( i l ~ ~ 0l i/r/>l(iti(, ~ I ' o i ~ ~ ~ i t/101i, i l I ~o/ ~ ft1.0 \rt/)(,t~)o\c(/ //I(;(/,\.l'liy\. Fluid\ 31 ( I'IXX). 3225-3238. 11731 C.S. Yih. ltr\rtrhilifv rlirc, to i.i.\(,osirv \rrtift/i~~t~rioii. J . E'luitl Mcch. 27 ( 1067). 337-352. 1 I74 1 V.I. Yudov~cl~. 7 1 1/.itt~~~rri:cifiot~ ~ Moflioil I I I H y t l r o ~ l ~ ~ ~.St(il~il;r~ ~ i t ~ t i '/71(,oi:v. ~ ~ ~ ~ l A11icr.M ; I I ~Soc. . 'l.ri~~~\l,. Vrll. 74. A I I I ~ Math. ~ . Soc.. Providence. RI (IOX')). ( 1751 J . %abc/yk. A r r ~ f oti ( ~ ('o .\c~tirigrr~ttl~.\. Bull. Aciid. Pololi. Sci. (Math. Altr. Pllys.) 23 (1075). 895-898. 1 1761 0. Zikitl~ov.On flrc, i~i.\r(tl~ilify of'l~il~c, Poi.\c~ttillc~ /Io~i..Phy\. Fluldx X ( 1006). 2073-7033.
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CHAPTER 8
Localized Instabilities in Fluids Susan Friedlander Dol~crr?rt~er~t of'Mirtho~ncrtic.s.Stirti.ctic.c,crtid Coml)rrt<.r Si.ictlc.c,. Thr Uttivrr.sity rf1lli11oi.s ut Chic.cr,qo. 8-51 S. M o w t r r l Street. C l l i i v r ~ oI L 60607.7045 USA E-t11cri1:. Y I I . S ~ I I I @ I ? I < I I / I..i.c/~i .II~~
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I . It~troductio~l. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 7 . F o r t ~ l ~ ~ l i ~.l i.o .~ l. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 7.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 2.2. The Eulcrian equalton\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 2.3. .She l.iy t..tngiiitl cq~tiitiotl\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 2.4. Stability and instahil~ty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 2.5. Kelvin-Hclnlholt~ itlstahility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 7.6. Riiylcigh-Taylor in\t;thility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 3 . Equilibria and other exact solution\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FIX 3 . I . Bachgroutld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2% 3.2. One-dtt~~cnionaI cquilihria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20X 3.3. Two- and three-di~nenhioniilcquilihriii and exact holut~on\. . . . . . . . . . . . . . . . . . . . . . . 790 3 . 4 Bclrranli tlows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
5.2. 5.3. 5.4. 5.5.
Kclvi~ll n o d c . . . . . . . . . . Kapid di\tortiotl theory . . . . . tlchhofl '5 approach . . . . . . . The gcotnctrical optics approach
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'CSFB d o e not necchsnrily agree with or endorbe the content of this article HANDBOOK O F MATHEMATICAL FLUID DYNAMICS. VOLUME I I Edited by S.J. Friedlander and D . Serre O 2003 Elsevier Science B.V. All rights reserved
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5.6. The unstable essential spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7. Growing perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8. General properties of the amplitude equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9. Viscous instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Flows with stagnation points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Integrable flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Secondary instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Nonlinear instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. A nonlinear instability theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Astrophysical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Differentially rotating stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Riemann ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Conclusion and open proble~ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Abstract We study the effects of localized instabilities on the behavior of inviscid fluids . We place geometric optics techniques independently developed by Friedlander-Vishik and Lif'schitzHatnciri in the frurneworh ol'the general stubility/instability problem o f fluid dynamics . We show that. broadly put all laminar flows are unstable with rehpect to locxli/.ed perturbations .
.
320 323 324 326 328 328 331 332 334 334 335 340 340 340 345 346 348 348
Localized in.str1hi1itie.sin fluids crre equrrl hut sonle rmima1.s aw more ~ q ~ r thun u l other.^. All ~t~itna1.s George O r ~ r e l l ."Animol Frrrm"
1. Introduction The foundations of hydrodynamic stability theory were laid down by Helmholtz, Kelvin, Lyapunov, PoincarC, Rayleigh. Reynolds, and Stokes in the nineteenth century. For more than a hundred years this subject attracted the attention of a great number of researchers. A vast body of literature on this subject exists and encompasses the work of mathematicians, physicists, engineers, astrophysicists, geophysicists, meteorologists, etc. Much background material on the classical approach to fluid stability can be found in the substantive general texts by Lin [130j, Chandrasekhar [23], Joseph [87],Drazin and Reid 1391, Swinney and Gollub [ 1591, as well as several more specialized monographs, reviews, and collections of papers, such as Yudovich 1821, Holm et al. 1851, Arnold and Khesin 171, Godrkche and Manneville [74],Dikii 1341, etc. In addition to hydrodynamic stability, the twentieth century saw the birth of the sister discipline of magnetohydrodynamic stability which was developed in order to address important practical questions occurring, i n particular, in thermonuclear fusion, astrophysics and dynamo theory, see, for example, Frieman and Rottenberg 1701 and Lifschitz [ 1171 The bea~~tiful synergy and cross-fertilization between these topics was instrumental in accelerating advances in both ol'them. The key question of hydrodynamic stability theory can be formulated as fhllows: What happens to a given fluid flow under the influence of small disturbances. If the How is robust under the influence of all small disturbances, it is called stable and can be expected to occur i n nature. Il'there are perturbations which start to grow, we call the flow unstable and expect i t to break up o r otherwise change its character. These possibilities were experimentally demonstrated by Reynolds 114.5 1. The topic of stability is important both mathematically and for practical considerations. Generally, it is much easier to establish the instability of a given flow rather than to prove its stability. In fact, in this article we will argue that. in a certain sense. all nontrivial inviscid Hows are unstable. There are two well-established techniques for the analysis of hydrodynamic stability1 instability, namely. spectral methods (normal modes) (see, for example, Chandrasekhar 1731. Drazin and Reid 1391) and energy methods (see. for example. Arnold 141. Holm et al. 1851. Arnold and Khesin 171, Vladimirov 1 1731). Recently. a modification of the spectral method called the pseudospectral method was proposed by Trefethen 11651, Trefethen et al. 1 1661. Waleffe 1 1761 and others. These conventional techniques proved to be very useful for studying the stabilitylinstability of some steady flows with relatively simple structure but their applicability to more complicated flows has proved to be limited. In this paper we emphasize a third technique, namely, the geometrical optics method which is capable of probing the instability of general classes of three-dimensional inviscid flows. (By its very nature this method cannot prove stability.) In contrast to spectral and energy methods, the geometrical optics method is specifically designed for studying highly localized short-wave perturbations of an arbitrary background flows. These perturbations are localized wave envelopes moving along the trajectories of fluid elements. The evolution of a
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particular envelope is governed by a characteristic system of ordinary equations along the relevant trajectory. In the language of geometrical optics the characteristic equations consist of the eikonal equation for the wave vector and the transport equation for the velocity amplitude. Needless to say, the system of ordinary differential equations is more tractable than the full system of partial differential equations governing the dynamics of general perturbations. Broadly speaking, the flow is unstable if the magnitude of the amplitude of the perturbed velocity grows in time without bound along at least one trajectory. As we will describe later, this observation produces an effective tool for detecting fluid instabilities. Four review articles by the present authors covering some of the material discussed below are Lifschitz [I 221, Friedlander [58], Friedlander and Yudovich [69], and Friedlander and Shnirelman (611.
2. Formulation
There are two complementary ways for describing fluid motion. We can either analyze the distribution o f the velocity, the pressure. and the density at every point in space; or we can investigate the history o f every particle, see Lamb 1941. Accordingly, there are two c o n plementary descriptions, namely, the Eulerian description and the Lagrangian description. and two sets of governing equations: the Euler equations and the Lagrange equations. Euler and Lagrange published their investigations in mid-eighteenth century, see Euler 147.48). Lagrange 1931. I t is interesting to note that Euler actually developed both space-based and particle-based approaches to describe Huid motions.
lnviscid incompressible fluid motion whose density is normalized to unity is governed by the Euler equations describing the distribution o f the velocity V and the pressure P in space:
where V(x,r ) is the velocity, and P(x,r ) is the pressure of a general flow in EX3, and
is the advective derivative along the velocity field V. If the flow is considered in a finite (or semi-finite) domain 'D, equations ( I ) . ( 2 ) are augmented by the impenetrabrlity condition on the boundary iIV,
where n is the unit vector normal to 3V.Other physically plausible and mathematically appropriate boundary conditions are periodicity conditions or the free space problem with suitable decay at infinity. The initial velocity field Vo naturally satisties the same conditions. The vorticity of the flow is given by SZ = V x V. The vorticity equation is
The basic flow is denoted by U(x. t ), fi(x. I ) , 9(x. t ) = V x U(x. t ) , its perturbation is denoted by v(x, r ) , p ( x . r ) . o(x, t ) . The evolution o f a perturbation is governed by
If. I'or exalrlplc. the problcm is imposed.
ih
considc~.cdin a finitc domain. ~ h oimpenetrability condition
Thc linearized cquutionx I'or the perturbation are
-
plus thc appropriate boundary condition. Taking the curl of Equntlon ( 1 I ) gives the equation for the evolution of the perturbation vorticity w V x v:
or, equivalently, i )o -il r
( U ,W ) + { v . S Z }
= 0.
(15)
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where {. , .] denotes the Poisson bracket of two vector fields, i.e.,
We can introduce the linearized operator:
where l7 is the projection operator on the space of divergence-free vector fields satisfying the appropriate boundary conditions. In this notation the governing linear equation assumes the form
When the basic flow U is stationary, we can introduce the corresponding spectral problem
A considerable portion of this paper will be concerned with the spectrum of the operator C. 2.3. The h g r u n g i u n equations
The Eulerian equations introduced in the previous subsection are the most popular but not the only set of equations governing the evolution of fluid motions. Lagrange discovered a complementary set of equations which emphasizes the local aspects of fluid motion. Let a = ( a l ,u2, u3) be the labels of a fluid particle, and x = ( X I , x2, x 3 ) its Cartesian coordinates. Lagrange considered x as a function of a and t , and wrote the equations for x ( a , t ) in the form
where xo = ( x l o x20, , x3") are the initial coordinates of a fluid particle at time t = 0, and we use the notation a ( x ) / a ( a ) for the Jacobian of the mapping a + x. The first set of equations follows from the second law of Newton for fluid particles, while the second equation is a consequence of the incompressibility. When the initial positions of the fluid particle xlo, x20, x3o are used as its Lagrangian labels, Equation ( 2 2 ) simplifies to
Localized instabilities injuids
The Eulerian velocity
295
V(x, t ) can be recovered by inverting the mapping a -t x,
By eliminating the pressure from Equations (2 1) and integrating them over time, Cauchy presented them in the form
where the vector C = ( C I ,C2, C3) is an integral of motion, C, = Ci ( a l ,az, ag) which is closely related to the vorticity of the flow. Recently, Abrashkin et al. [3] discovered an elegant way of representing the LagrangeCauchy equations. They introduced two matrices
and represented the equations of motion in the matrix form
det R = det Ro.
(28)
where the subscripts and superscripts T denote the differentiation and matrix transposition, respectively. These equations have to be augmented with the integrability conditions which express the fact that R is a Jacobian matrix,
The vorticity of the flow has the form Q=-
I RC. det ILJ
A recent review of the matrix approach to Lagrangian fluid dynamics is given by Yakubovich and Zenkovich [ 178 1. It is clear that some aspects of the fluid motion are easier to understand in the Eulerian framework while others are easier to describe in the Lagrangian framework. In particular, as we will see below, some interesting classes of exact solutions of the equations of motion have a very natural description in terms of R and C.
In this paper we adopt the general definition of stability developed by Lyapunov in the late nineteenth century and used by Yudovich [ 1821, Holm et al. [85], Holm [84], among many others. We specialize it to problems at hand as need occurs. To start with, we define the spectral stability of the basic flow (which only makes sense for steady flows U, U = U(x)) as followc. We say that a steady basic flow is spectrally stable provided that the discrete spectrum of the operator defined in an appropriate function space X has no strictly positive real part. We also define the linearized stability of the basic flow. We choose a function space X of vector fields with a norm 11 . Ilx where problem (1 8). (1 9) is well-posed. The basic flow U is linearly stable provided that for every E > 0 there is a 6 > 0 such that the inequality IlvoIIx < 6 implies the inequality I l ~ ( t ) (<( ~s. To put it differently, if there exists an initial condition v[l E X such that j)v(r)))%, . is unbounded on the whole r-axis. we call the basic flow linearly unstable. otherwise we call it stable. Under such a definition, the basic How U is not restricted to be steady. belt the issue o f stnbilitylinstability ot'time-dependent flows is more delicate since the question arises as to what is a change in the basic flow and what is growth in the perturbation. The matter is clear if the basic tlow is steady, periodic or quasi-periodic. Finally, we clef ne the nonline~u-stability ofthe basic How. As hetorc. wc choose a t'unction apace X with n norm 11 . 11 such that the problem (7). (9) is well-posed. We call the basic flow U nonlinearly stable if for every c > 0 there is a d > 0 such that the inequality Ivollx 8 implies the inecluality Ilv(t)llx < P . otherwise we c;~llit unst;~hle.We give ~i more precise detinititrn i r Section ~ 7.2 below. dependent o n thc choice We emphasize that thc stability of a given tlow ia cr~~cially of the function space X and. even Inore importantly. the nor111 on this spiice. A detailed discussion can be found i n the book by Yudovich 1 1821. Several representative examples are considered below. The thrust of thc present paper is the subject of irlsttrhi/ir>~which ic the converse of the above definitions of .srrrhilif~. We now give a brief qualitative description of two classical instabilities. niunely, KelvinHelmholtz and Rayleigh-Taylor instabilities which involve I-athcr simplc basic flow patterns, i.e.. particles moving in straight lines or circles. These instabilities are caused by two complementary physical mechanisms which are the ultimate causes of the vast mqjority of hydrodynamic instabilities.
Kelvin-Helmholtz instabilities were qualitatively described by Helniholtz 18 I I and quantitatively analyzed by Kelvin 1891. Instabilities of this kind frequently occur in nature. two such manifestations are the so-called "wind-over-water" and "clear air turbulence" instabilities.' Consider two parallel streams of inviscid incompressible fluids superposed 'On c~ccasion,thew in~tahilitiehc i ~ nhare 11-agic consequence, for traveller\ by boat or airplane
one above the other and assume that the upper and lower streams have positive and negative velocities, respectively. Since at the interface there is a discontinuity in the velocity, the vorticity is a nonzero delta-function-like distribution which can be modeled as a vortex sheet. Consider a small sinusoidal perturbation of this sheet. For the two-dimensional flow in question the vorticity is conserved under the motion of the fluid particles. Thus the vorticity has to induce a velocity in the positive (or negative) direction in parts of the sheet displaced upwards (or downwards). At the undisturbed points of the sine wave, the vorticity induces a rotational velocity which amplifies the wave and causes the instability to grow. As a result, the vorticity sheet develops vortex rolls and eventually breaks in a turbulent fashion. Kelvin-Helmholtz instability is global in nature and flowspecific. Since the basic mechanism is strongly affected by specific features of the underlying flow such as its shear, as well physical forces such as gravity, viscosity, etc., it is very difficult, if not impossible, to describe Kelvin-Helmholtz instabilities of generic flows.
Rayleigh-Taylor instabilities were first theoretically described by Rayleigh 11441 whose complemented by theoretical and experilnental analysis by Taywork was suhseq~~ently experimental apparatus a Huid moves between two lor [ 1621. I n a typical Taylor-Co~~ctte concentric cylinders rotating with different angular velocities. In steady motion the centrifugal force at any radius is baliinced by the pressure gradient. Consider a small radial displacement of a fluid ring. Conservation of angular niomentum causes a change in the angular velocity. which now may or may not be sufficient to oft'set the pressure force. If the balance is maintained, the ring moves back to its original position thus initiating an oscillatory pattern. If the balance is violated. the ring moves away from its original position. the initial disturbance grows and the instability develops. Thus a necessary and sufticient condition for instability with respect to i~xisymmetricdisturbances can be established in terms o f the radial derivative of the angular velocity R (the so-called Rayleigh discriminant),
Experiments in which the differential of the angular velocity of the corotating cylinders is gradually increased show that above a certain threshold a pattern of small counterrotating Taylor vortices is set up. This pattern subsequently bifurcates into azimuthal travelling waves, twisting regimes, quasi-periodic regimes. and so on. until the flow becomes turbulent. Bifurcations from primary to secondary instabilities and beyond are studied by Chossat and looss 1261. The Rayleigh-Taylor instability is local in nature and very robust, it is weakly affected by the specific features of the underlying flow. As we will see below, Rayleigh-Taylor instability can be observed for generic three-dimensional flows in the form of localized geometrical-optics type instabilities.
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S. Friedlander and A. Lipton-Lifschitz
3. Equilibria and other exact solutions
3.1. Background The problem of finding equilibria is notoriously difficult, accordingly, very few nontrivial equilibria in more than one dimension are known. However, it is easy to find an equilibrium depending on one spatial variable, for example, plane-parallel shear flows and differentially rotating cylinders. Equilibria depending on two spatial variables are more difficult to find, still, there are several classes of examples. Fully three-dimensional equilibria are exceedingly difficult to describe. The same observation is true for time-dependent exact solutions of the equations of motion. The governing equations for a steady-state solution U(x) of the Euler equations have the form
or, alternatively,
+
where the scalar field H = ~ ' / 2 P is the so-called Bernoulli function. Equation (33) shows that both streamlines and vorticity lines lie on the level surfaces of H. Very little is known about general solutions of Equation (32). However, an interesting observation was made by Arnold 151 who proved that for Equation (33) all compact noncritical level surfaces of H (i.e., surfaces which do not contain zeroes of V H ) are diffeomorphic to a 2D torus. 3.2. One-dimensional equilihriu The simplest solution of the above equations is plane parallel flow of the form
where U I is an arbitrary function of its argument. A simple generalization of this flow known as plane parallel shear flow is
The axisymmetric analogue of the plane parallel shear flow is known as rotational shear flow. In cylindrical coordinates r , 8 , z this flow is given by
As before, this flow has a two-component generalization
Localized instabilities injuids
3.3. Two- and three-dimensional equilibria and exact solutions We now turn to strictly 2D flows and assume that the velocity field has the form
It is clear that the vorticity of the flow which is perpendicular to the 2D plane has a scalar magnitude Q(x1, x2). By virtue of incompressibility, we can introduce the so-called stream function 9 (x 1, x 2 ) , such that
The name obviously comes from the fact that streamlines of the flow are the level curves of the function 9. Simple manipulations show that 9 satisfies the PDE
A special class of solutions is such that the Bernoulli function depends only on H(IL), and q satisfies the PDE of the form
(I/,
H=
and appropriate boundary conditions. In general, this PDE is nonlinear (and has to be solved numerically), however, in some special cases, its analytical solution can be found. The simplest case occurs when H' is a linear function of @, H ' = -A@, so that Equation (4 1 ) becomes the eigenvalue problem
plus appropriate boundary conditions. For instance, if @ is bounded at infinity, the corresponding solution describing a cellular How has the form
A less symmetric variant
is illustrated in Figure 1. This flow has a "cats-eye" type pattern with a periodic cellular structure that contains hyperbolic points and elliptic points as well as oscillatory regimes. A somewhat more difficult but still tractable case occurs when H' is piecewise constant, so that the equilibrium equation can be written as
Fig. I . A typical :i\ymmetric "car\-eye" How pattern. Here (1 = 2.
III
= 1.5.
where x g c , , is the characteristic function of a moving bounded domain D ( t ) , which is equal to one when (.rl.x?)E Di)(r),and zero otherwise. If D ( t ) is an ellipse rotating with constant angular velocity, then the corresponding solution is given by Kirchhoff and Lamb 1941. An interesting generalization of this solution is given by Kida 1901. see also Neu 11361, Bayly, Holm and Lifschitz [ 131, and others. Kida's solution describes an elliptic vorticity patch superimposed on a field consisting of pure strain and pure rotation. The background tield has the form
The patch remains elliptic for all time but its aspect ratio and orientation change in time. The area of the patch and its vorticity remain constant. and can be normalized to n and unity, respectively. The geometric characteristics of the patch, namely, its aspect ratio (ratio of the major to the minor axis) ~ ( tand ) the angle H ( r ) between the major axis and the .rl axis, are governed by a system of ODES:
Inside the patch the combined velocity field is linear,
Kirchhoff's solution can be obtained frorn Kida's solution by putting s and w to zero, = )lo. H = qclt/(rlo 1 )' Ho. The full dynamics of Kida's vortices are fairly complex; the actual details can be found in the references above. A natural generalization of p~rrely2D flows are cylindrically symmetric flows of the form
+
+
As before, we can introduce the stream function 9 (r. 7 ) such that fJ =
1 i)W r i3:
<
I ij4' W = --. r ilr
Thc corresponding equilibri~~nl condition is
The best known solution of this cquation is called Hill's spherical vorrex. Hill 1831. Lamb 1941. For si~chvortex
Here rr is the spherical radius of the vortex corc iund Ci/o is the uniform velocity al infinity. The corresponding H is piecewise linear and can be 1i)und if need occurs. The stream function W ( r . ,-) is shown in Figure 2. I t is natural t o gencrali~ethe above flows by introducing the velocity component V(r. :) i n the H direction. I t is easy to show that
where f is an arbitrary function of W . The corresponding equilibrii~mequation has the form
S. Friedlander and A. Lipton-Lifschitz
Fig. 2. The mttridional cross-section of o typical H i l l ' s vortex.
In the fluid dynamics context this equation was derived by Bragg and Hawthorne 1181. Independently, it was introduced to describe plasma equilibria by Grad and Rubin 1761, and Shafranov [ 1541. Qualitative properties of this equation have received much attention in the MHD literature, see, for instance, a detailed account in Lifschitz [ 1 171. Its quantitative analysis is very complex and depends on a particular judicious choice of profile functions H , J', see, for example, Lifschitz et al. 11291 who took H and J' as powers of @ in order to exploit the intrinsic symmetry properties of Equation (55) and describe a rich family of fluid equilibria that are called vortex rings with swirl. A typical vortex ring belonging to this family is illustrated in Figure 3. Among other things, this figure shows that, in agreement with Arnold's observation, all compact noncritical level surfaces of H are toroidal. Generically, there are critical surfaces outside of the vortex core, which separate surfaces diffeomorphic to torii, and surfaces diffeomorphic to cylinders. For general discussion of vortex rings (without and with swirl) see Moffatt I 1351, Turkington [ 1671, Sharif and Leonard [ 1551 and Saffman [ 15 I ] . Some authors prefer to write the equilibrium equation (55) in the form
where y = r2/2.
Lncalized insrahilirie.~in fluids
Fig. 3. A typical vortex ring with \wid. Sptral \trcarnlinc\ winding around \Ire;trn \ctrt;tcc\ arc clearly visible. Produced in The Lahorntory tor Advanced Computing ;it The University of Illino~sat Ch~cagoby R. Gro\\Phy\ic\. Vl)lut~~e 120. by A. I.il'\chit~. mtn. J . Leigh. and A. Lit'\chit7. Reprinted from Journal ofCr)~npc~t;itional W.H. Suters and J.T. Bealc. Thc Onset of Instability in Exact Vortex Ring\ with Swirl. pp. 8-29 (1906). with permission l'rot~iEl\evier Science.
I t is possible to construct flow equilibria in the whole space with physically acceptable properties. In the MHD context this was recently done by Bogoyavlenskij [ 171. His results translate directly to the Euler equilibria. We assume that H' = p@. p > 0, t' = 44'. and write the equilibrium equation as
Separation of variables, cL = +(y)[ucos(o:) for +:
+ bsin(wr)], yields
the following ODE
which can be recognized as a generalized Laguerre equation. For "resonance" y Z = 4 n f i w' Equation (58) has a solution of the form
+
Fig. 4. The mcridionitl cro\\-\cction of a typic;~lBogoy;tvlenshi; How.
where ldj,"(.)is a generalized Laguerre polynomial. see Abramowit~and Stegun I I 1. Equation (22.6.17). The corresponding solution of the equilibrium equation which is rapidly decaying in the r direction and is periodic in the :direction. has the form
A typical "resonance" equilibrium is show i n Figure 4. By virtue of linearity, we can superimpose these solutions (while keeping 17. y tixed), and construct equilibria, quasi-periodic in the :direction. Most of the corresponding stream surfaces are diffeomorphic to cylinders. An important class of exact solutions of the Euler equations constitute the so-called linear Hows with the velocity and pressure tields which are linear arid quadratic in the spacial coordinates, respectively,
where M ( t ) is a symmetric matrix. For this velocity and pressure to satisfy the Euler equation ( I ) , the 3 x 3 matrices I C ( r ) , M ( t )have to satisfy the following conditions
Among linear flows, two-dimensional elliptical flows of the form
and hyperbolic flows of the form
rrl 3 (12 > 0, play a special role since they encapsulate important local features of more general flows having points of stagnation. We emphasize that for flows (64) the pressure is negative at infinity but this difficulty can be disregarded for most cases of interest. The ellipticity of the flow (63) is characterized by a non-dimensional parameter S = (~1; -(I:)/ (rrf a ; ) . In the limiting cases S = 0. S = 1 . this flow reduces to the circular and linear flows. respectively. An interesting class of two-dimensional rotational Ptolemaeuc flows was discovered by Ahrashkin and Yakuhovich 121. In Lagrangian variables these flows have the form
+
where F(.).G ( . ) are analytical functions of their arguments, nnd i1. 1 1 are real constants. For these Hows trajectories of Huid elements (which are either epicycloids or hypocycloids) are the same as trajectories of planets in the Ptolernaeic system. In principle. thcsc flows can be considered in bounded domains :mtl matched with appropriate irrot~itionalflows. In particular, Kirchhoff vortices can be viewed as LI special type of Ptolemaeus flows. The Eulerian description of these flows is implicit and very complex.
where I ; ' . G-I are the inverse functions of F. G . and the overbar denotes the complex conjugate. Subsequently. Abrashkin et al. 131 f'ound a three-dimensional generali~ationof PtoleInaeus flows by effectively separating variables in the matrix equations of motion (27)(29). A typical solution has the form
where / I ( ~ Iis~ an ) arbitrary function such that I/h1(tr3)# 0. For h(tr3) = 0. flows (67) become Ptolemaeus flows. There is an intricate relation between linear flows (61) and flows discovered by Abrashkin et al. 131.
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S. Friedl~nderand A. Lipfnn-Lifichitz
A class of solutions which are complementary to integrable flows, such as the vortex rings described above, constitute the celebrated Beltrami flows for which the velocity and vorticity fields are proportional, i.e.,
see, for example, Yoshida [I 791 and references therein. For such flows the Bernoulli function H is constant. Beltrami fields U can be interpreted as eigenvectors of the curl operator V x supplied with appropriate boundary conditions. In general, Beltrami flows have regions where streamlines are chaotic. An important example of Beltrami flow is the so-called Arnold-Beltrami-Childress (ABC) flow of the form
Henon 1821 investigated numerically a particular case A = B = C = I and found that the Lagrangian trajectories appear to till some open domains of the three-dimensional torus. Following this indication that ABC Hows exhibit the phenomenon of Lagrangian chaos, such flows became the subject of considerable study. Dombre et al. 1381 investigated ABC flows both analytically and numerically. They showed that for certain parameter values resonances occur which disrupt the so-called KAM surfaces and the remaining space is occupied by the chaotic particle paths. Stagnation points may occur and when they d o there is numerical evidence that they are connected by a web of heteroclinic streamlines.
4. Linearized Euler equations 4.1. The .vl~ecrrtrlprohletn We now return to the linearized Euler equations given by ( 18). ( 19). For a given equilibrium velocity U(x), the classical approach to linear stability is based on an investigation of the spectrum of the operator C given by (17) in a function space X of vector tields where ( 1 8). ( I 9) are well posed. However, the operator C is a degenerate, non-self-adjoint, non-elliptic, non-local operator and for an arbitrary steady flow U(x) the structure of spectrum a is remarkably little understood. DEFINITION I. We adopt the following definition of spectral points for any Banach space B and operator T E a ( B ) . A point z E a ( T ) is called a point of discrete spectrum if it satisfies the following conditions: z is an isolated point in a ( T ) ; z has finite multiplicity; the range of (z - T ) is closed, which implies there is a subspace Q c B where ( z - T ) is invertible.
Loca1i:ed instabilities injuid.7
307
On the contrary, if z E u ( T ) does not satisfy the above conditions, it is called a point of essential spectrum. For the Euler operator C we have, as we will discuss in Section 5.6, partial information concerning the essential spectrum for general classes of flows U. However, the existence of discrete eigenvalues is at present too difficult for any general results. Much of the discussion in such standard texts as Chandrasekhar [23] or Drazin and Reid [39] concerns properties of eigenvalues of C , but only in cases of specific, relatively simple flows U. Because of its "non-standard" nature, there are no general theorems that may be applied to prove the existence of unstable discrete eigenvalues for C,i.e., discrete points z E a ( C ) with Rez > 0. However, in certain rather special examples it is possible to show that unstable eigenvalues exist.
4.2. Some 2 0 and 3 0 examples of unstable eigenvalues The spectral problem for the linearized Euler operator is considerably simpler in 2 dimensions rather than in 3 dimensions. In particular, i n 2 dimensions we can define a scalar stream function to replace the divergence-free vector field. We write
U = e3 x V W ( X I x?), ,
v = e3 x V 4 ( x I .x?, t ) .
(70)
Hence,
Here en is the unit vector perpendicular to the 2-dimensional plane with Cartesian coordinates ( X I . .r?). The 2-dimensional equilibrium equations will be satistied when W satisties an elliptic equation of the form (41). In general, the second Poisson bracket on the RHS of (15) is very difficult to analyze. However, in 2 dimensions the problem greatly simplities because e3 . V = 0. The vorticity equation ( 1 5) gives the spectral problem
where i)w/ijt is replaced by hw. We consider the eigenfunction 4 and the eigenvalue h for Equation (72); after substituting (70). (7 1 ) into (72) we obtain
We take the boundary conditions to be 2ir-periodicity in ( X I . .q). A simple and very classical example that has received much attention in the literature of the past 100 years is plane parallel shear flow (see, for example. Chandrasekhar 1231. Drazin and Reid (391).In this case U = (U(x2). 0) and (73) becomes the so-called Rayleigh equation:
where we have written
The celebrated Rayleigh stability criterion [I431 says that a sufficient condition for stability is the absence of an inflection point in the profile U(x2).This criterion follows from a simple application of the so-called "Energy method" for stability in which an energy integral is constructed by integrating (74) multiplied by the complex conjugate of @ ( x 2 )to give:
A related sufficient condition for linear stability was discovered by Fjortoft 1541. In fact a stronger stability result, namely nonlinear stability, follows from a method developed by Arnold 141. sornetirnes called the "Energy-Casimir method", which is based on the existence of two different integrals of the motion described by the nonlinear equations of motion. Arnold's methods prove that for plane parallel shear flow i n 2D the Rayleigh criterion guarantees not only spectral stability but also nonlinear stability in a space J l where the norm
is fi nite (see Arnold and Khesin 171 for more details). In the case of plane parallel flow this sufticient condition becomes the definiteness of tho quadratic form
Here W(.I-1. .r?) is an arbitrary 2 n l k periodic in .rl and 277 periodic i n .I-? function having generalized second derivatives that are square integrable over the rectangle 1 0 , 2 n / k ] x 10, 2n 1. The classical treatment of the instability problem for the Rayleigh equation was based on formal asymptotics of a perturbation of o special "neutral niode" (see Tollmein 11631. Lin 11301). However. the formal treatment does not give complete information about the asymptotic behavior and does not exclude the possibility that the neutral mode is isolated (see Drazin and Reid 1391). Howard 1861 proved that for continuous profiles with inflection points there were no more unstable eigenvalues than the number of inflection points. Rosenbluth and Simon [ 1481 used an alternative perturbation approach to establish a sufficient condition for instability, however, no explicit protiles were exhibited for which this condition is satisfied. The first rigorous proof of the existence of unstable eigenvalues was given by Fadeev 15 1 I for monotonic profiles with inflection points. In the case of monotonic profiles, Fadeev observed that the spectral operator could be written in terms of an operator A belonging to a class known as the Friedrichs' model. He remarked that the problem for non-monotonic prof les is more difficult since the spectrum of A becomes multiple.
Meshalkin and Sinai [132], followed by Yudovich [180], Frenkel 1561, and Zhang and Frenkel [I831 investigated the instability of a viscous shear flow U(x2) = sinmxz using techniques of continued fractions. More recently Friedlander et al. [62,16,60] showed that these techniques could be used for the inviscid equation (74) with U ( x 2 ) = sinrnxz (so-called Kolmogorov flows). Eigenfunctions are constructed in terms of Fourier series that converge to Cm-smooth functions for eigenvalues A that satisfy the characteristic equation. We write
The recurrence relation equivalent to (74) yields the following tridiagonal intinite algebraic system
where
p For each integer , j = 0. I . . . . . l111/21. we construct il scqucncc tl,+~,,,,. the recurrence relation (80). Let
E
Z. thitt satisfies
tl, = I.
where
The intinite continued fractions in (84), which are functions of functions for :E D = {: I Re: > 0, - I < Im: < I). Furthermore p~
--
lim pl+/,,,l=: -
/I+%
m.
:,
converge to analytic
(85)
hence for 7. E L) we have Ip,l < 1 . Thus the sequence of coefticients d,+,,,,, given by (84) decay to zero exponentially as / I + cm.Matching the algebraic system (80) across n = j
with the coefficients given by ( 8 3 ) leads to the characteristic equation relating z and the wave numbers k , j, namely,
Thus for each eigenvalue z satisfying ( 8 6 )we have constructed a Cm-smooth eigenmode @ given by ( 7 9 ) . In [I61 properties of the continued fractions are used to prove the existence of roots z E D for wave numbers such that k 2 j 2 < m 2 . This approach constructs all the unstable modes of the Rayleigh equation with a sinusoidal profile. However, it cannot be used in order to construct stable eigenmodes for which Rez = 0, since such values of z lie outside D. The existence of unstable eigenvalues for shear flows with a general rapidly oscillating profile U(t?z.u2),nz >> I , was demonstrated in 1161 using homogenization techniques to compute the spectral asymptotics. Gordin 17.51 has solved numerically an interesting problem of finding a "maximally unstable" profile U(.r,), provided its enstrophy /' I U ' ( X ds' ~)~ is ~fixed. We note that for the Rayleigh equation with smooth profiles there is no unstable c.011tit~uo~rs spectrum. Results concerning the stable continuo~~s spectrum include those of Dikii 1331. Case 121 1. Rosencrans and Sattinger I149I. There are just a few results concerning ~ ~ n s t a beigenvalues le for flows with more spatial structure than shear flows. Friedlander et al. 1671 examined 2D flows that are a natural extension of Kolmogorov flows to allow for oscillatory structure in more than one dimension. A formal asymptotic construction of a branch of unstable eigcnvulues is given ti)r a flow with strearn function (44) with 111 >> I and 0 < lo1 < I. The cigcnvalue problem ( 7 3 ) is a PDE with oscillatory coefficients where the basic operator is a product of a skew symmetric operator and a symmetrizable operator. In general. the infinite set of Fourier coefficients are non-separable and it is not possible to obtain the characteristic equation as in the case of shear How. However. when there is one "fast" variable ( i t . , ltl >> 1 ). homogenization techniques lead to an "averaged" ODE whose spectrum contains unstable eigenvalues with eigenfunctions in L'. The "averaged equation" associated with the homogeni~ationprocedure is
+
where @()(.xl) is 2n-periodic. For the modes with j = 0 the "averaged equation" (87) becomes
Provided that
1
l ~ if
I . this equatioti can be written in the standard Sturm-Liouville form as
This spcctral probler~lhas a complete family of orthonormal, square-integrable functions and corresponding eigenvalues ( A o N } of multiplicity no greater than two. The corresponding Rayleigh quotient is
Thus the cigcnvalues for the j = 0 modes arc purely real. In a rccent paper Li 1 1 121 considers a similar problem for the 2D extension of'the Koloniogorov problcm. He uses a Galerkin approximittion t o truncate the infinite-dimensional system that arises f't,om Ecl~~ation (73) and cxnniincs the cigcnvalues that for this approxitni~tc~noclcl.Howcvcr.. there is no rigorous j~rstilic;~tionfor the connection between the cigcnvulucs of thc truncatctl system and unstable eigetivi~lucsfor the Euler cquationx. For solne very speci:~l flows. such as linci~rflows in ellipsoidal domains. one c;111analy/e thc spcct111111 ol'thc lincori/.cd opcrator C in great dcti~il.Flows ol'tliis kill11 ; ~ ~ t r i i11~lot t ~oI' d attcntion, ace. I'or cxi~mplc.(ireenspun 1771% G l c d ~ c i111d r Potio~ii;~rcv 173 1. Vl;~ditiiirovi111~1 llin 11711. Vlaclinlirov and Vostretsov 1 1 741. con side^.. fol- cxarnple. i~ simple basic flow of the Sorrii (63). which ciun bc considered us a stcnrly solu~ionol' the Euler equalions in thc cllipsoid:~ldotn:~in 'Ddefined by the Sollowing conclition
,I-) = { x : . ! : / ( I ;
+
.1:/(1:
+
.Y,:/(!,:
< I
).
(92)
I t turns out that the spectral problcm for the corresponding operator C is exactly rcduuiblc to u linitc-dimensional matrix spectral pl.oblcm. To accomplish this 1.cduction we choosc a bask of' incompressihlc vcctor fields tangential to the hou11diu.y i)'D of the ellipsoidnl domain I 7 in the h r m o f vcctor p o l y n o ~ n i i ~ l ~
w here
is u ~nonotnialin .r 1 . .rz. . v ~ .and e is n unit coordinate vector. we. e.g. Lebovitz ( 101. 1021. I t can be shown that if 4 is a vector polynomial of' degree 1 1 , then so is C(. Thus. vector subspaces S'") spanned by vector polynomials of degree 11 of the form (93) are invariant with respect to the action o f the operator C. Accordingly. the usual Gi~lerkin 11
Fig. 5. The discrete spectrum of a linear flow in an ellipsoid.
truncation of the spectral problem is exact. The truncated spectral problem for the subspace
S'")has the form ~
~
(
1
-1
)
~
= 0,
(95)
where
M
=(
)
q!Y';' = (L(;"', (:"'),
and (. , .) denotes the scalar product defined by
Even though the corresponding matrix problem cannot be solved analytically. it can easily be solved numerically. Figure 5 shows the union of spectra of the low-n truncated spectral problems, 0 6 n 10, for a typical ellipsoidal domain. In summary, the problem of the existence of "strong" instabilities associated with discrete unstable eigenvalues of the operator C given by (17) is almost untouched in two dimensions and at present completely inaccessible for any fully three-dimensional flow except for linear ones.
5. Localized instabilities
In contrast with the paucity of examples we possess showing the existence of unstable discrete eigenvalues. there exist many examples corresponding to a somewhat different type of instability, namely instability to localized perturbations which can be viewed as high frequency wavelets. The asynlptotic methods for investigating such instabilities are analogous to geometrical optics in the theory of light rays. It is widely believed that such shor't wavelength instabilities are responsible for the transition from large scale coherent structures to 3D spatial chaos; (cf. a review by Bayly et 31. [ 141 and references therein). In order to describe such instabilities one can use solutions with complicated time dependence. The idea of using these solutions in hydrodynamic stability can be traced back to Kelvin 1881 and Orr 11371. For many years little attention was paid to this idea until geometrical optics was introduced i n the study of compressible fluids by F.G. Friedlander 1571 in 1958 and Ludwig 1131 1 in 1960. Later Eckhoff 141,421 iund Eckhoff and Storeslctten 144.451 s t ~ ~ d i ethe d stability of azimuthal shear flows and more general symmetric hyperbolic systems using an approach based on a generalized progressive wnve expansion. Eckhoffshowcd that local instability pl.oblems for hyperbolic syste~nac:tn be essentially reduced to a local :~n:~lysis involving ODES. Wc note that the i r i c ~ o r ~ i l ) r . c ~ . s . ~li11c;uized iI~/i~ Euler ccluations ( l X ) clo ~ i o tlimn :I strictly hyperbolic system and the results of' Eckhol'l'ci~~~not be dircctly applied to these equations. Craik ~ ~ Crimn d inale 131 1. CI-nih 12'91. Craili and Allen )301. Forstor ; ~ n dCrilik 15.51 revisited the idcas of' Kclvin for the i~lco~~lp~.cssible Eulel- cquntions i1nJ cxliihitcd instabilities associ;itud with specii~lex:ic( solutions of thcsc equations. Gcncr;~lloc;lliLcd instabilities (11' this type arc liriown under thu niuiic ol' broadband institbilitics. They h;ive becn usetl by Bayly 191 i n order to contirm nunlcricol rcsulta o t Pierrehumher1 I 13x1 describing the behavior of twodimension;~lHows with clliptic strci~rnlincs.A similar Icchniqi~ew~tsusetl by Lagnado ct ~ 1 1 .1921 to invcstigatc the stnbility 01' flows with hypcrbolic strciuiilincs.
We bricfly summnri/.c thc idcas developed by Kclvin nncl his fi)llowers. To this end we cons~der:I linear How of thc I'o~vn(61 ) . The corresponding linearized Eulcr equations have the form
I t turns c~utthi~tthcsc equations have
:I
fr~niilyof solutions of the torm
which arc generalized Fouricl. modes. A direct calculation shows that the equations describing the temporal evolution of the pair k(r). a([) have the form
with ka . a0 = 0. The evolution of this pair strongly depends on the choice of the matrix K which is independent of x but {nayor may not be a function of r . Below we discuss several specific examples where these equations can be solved explicitly. The general heuristic conclusion is that the underlying linear flow is unstable provided that Equation (101) has a solution which grows with time. Chandrasekhar made the following interesting observation: the superposition of the linear background flow and a single perturbation of the forin (99), i.e.,
where h:(t) is a spatially independent matrix satisfying Equation (62) and a ( t ) and k ( t ) are spatially independent vectors satisfying the ODE system (100). ( IOI), actually satisfies the full nonlinear Euler equatic~ns(7). This is due to the fact that in this very special situation the nonlinear term v . V v vanishes identically due to incompressibility. More generally, we can consider solutions o f t h e form
which represent penerali~edmodes centc~.c'dat x ( t ) . A simple c:ilculation yields the following equation for x ( t ) :
I t is easy to sce that Equations ( IOO), ( 104) preserve thc scalar product k ( l ) . x ( r ) = k,, . xu. so that solutions ( 103) can be obtained from solutions (99) by a simple phasc shift. As we will sce later. in general this relation does not hold.
In a different context ol' rapid distortion theory (RDT) of ti~rhulencethese ideas werc independenlly discussed by Prandtl 1 142 1. Tr~ylor1 16 1 1. Batchelor and Proudman [ 15 1. Townsend 11641. Cambon et al. [ 191, and c~thers.An excellent ;iccount of KDT is given by Pope 1141 1. Let us briefly summarize their tindings. We start with the Reynolds decomposition of the velocity fcld V ( x ,t ) into its tncan ( V ( x .t ) ) and fluctuation v(x. t ) . thc pressure P ( x . r ) is clccomposed in n similar wily. In is well known that the mean velocity held and pressure are govc~,nedby the Reynolds. or Inran momentum, equations:
A(P)+
if(V;)i~(V;) --+dx, ilx;
;)x; ifx,;
= 0,
where
is the advective derivative along the mean velocity field (V). Here v is the kinematic viscosity and s is the celebrated Reynolds stress tensor or the matrix of velocity covariances, 5;, = ( U ; V , ) . 111 homogeneous turbulence, the fluctuating velocity and pressure are governed by the followitlg equations:
where the terrns v . V ( V ) atid 2(i)(V,)/i).\./)(i)l!, / i ) x , ) represent interactions between the tur~bulcnceheld v and rnean velocity gradients V ( V ) . RDT assumes that these gradients are sp:~tially unifc)r~ii( i t . , that the mean velocity field is linear in the spacial coordinates. cf: Eq~tation(61)) and th;~tthe nlcan rate-ol-.str;rin tenuor
is so large that the tc13msrepresenting inrcroctions bctwccn v ~uidV ( V ) domini~tcall o~hcrs. The rcxulting KDT equations have the form
They arc clciirly equivalent to Equation (98). B~~tchelor ;uid Proudman [ 151 wcre thc firs1 to construct solutions of RDT equations in the form of Foi~ricr-Kelvin modcs. Nccdlosx to say that their solu~ionsnrc equivalent to Kelvin's. Howcvcr. the cmphabib ot' RDT is not 011 the hehaviorof:~nindividual mode and instabilities associated with huch it modu. but r-~\~hcr on the cvolurion of the spcctr-um of'a vclocity tield omp posed of lnany independent stable modes. We do not discuaa thc latter prohlem here and r c k r the reader to the rcfcrcnccs quoted above.
In a seminal paper. EckhoR'showed that local instability problelns for hyperbolic systenis can be essentially reduced to a local analysis involving ODEs. Consider a linear symmctric
hyperbolic system of the form
supplied with the initial condition
and appropriate regularity conditions, or. symbolically,
In order t o address the question of instability of its zero solution ~ ( xt ). = 0, we consider a family o f rapidly oscillating solutions of the form
and analy7e their asymptotics for phase function S i x , I ) :
f:
4
0.I t is easy to derive the cikonal equation for the
which is a fa'arnily of N PDEs of order 1 and, :IS such, can be considered us ODES along the corresponding characteristics. cf., for instance, Courant 1281. Individual members of this t i n i l y have the form
N
where (-)(x. I . S,, , . . . , S , , ) is an eigenvalue of' the symmetric matrix A"S ,,,. Let (-1 be a parricular eigenvnluc of tixed multiplicity 1 1 , and E l . . . . . E,, the corresponding orthonormal system of eigenvectors:
where I ri p , q
<
11,
or, equivalently.
C d1kJl ("I, =
where a;, = #El, . E,. Here and below we use the notation k = VS, k,, = S,,,. By using the fact that (9 is a homogeneous function of k of degree I , we can show that
Thc amplitude a()can be written as
where the scalar functions a,, are to be determined. The corresponding transport equation governing the evolution ot' all has the fh1.m
I t is 3 systenl of' PDE\ of' order I , or, equivalently. ;) sy\tcm of ODES and is rela~ivclyeasy to stutly. Eckhot'f(4 I I provetl that zero solution ol'thc originid systcm ol' PDEs i h uns~ahlc whcn. for a ccrti~inchoicc of' (-1. PDE ( 1 13) hiis a solution which grows in time. We can rcpliicc Equations ( I 18).( 173) hy an equivalent systcm ~ ~ ' O D of EF thc form
provided that thcsc ODES d o not break up in finite timc. or. i n othcr words. caustics do no1 exist. (For(uni~tely.in many physically interesting situalion\ they do not.) I t is clear that (-) plays the role of the Hiuniltonian governing the evolu~ionof Ihe pair ( x . k ) . The situation hccomcs pm.ticula~.lytransparent whcn (-) i5 a linear function o f k, (-)(x. k ) = w(x). k. In this cusc the corresponcling Haniiltonian system for ( x ,k ) is
and no caustics occur.
We note that the incompressible Euler equations (1 8) d o not form a strictly hyperbolic system and the Eckhoff's results cannot be directly applied to these equations. However, his results point in the right direction.
5.5. The geometrical optics upl>rocrch In a series of papers that we will summarize below, Friedlander and Vishik [63-65,17 I], Lifschitz and Hameiri [80,126], and Lifschitz 11 18-1221 pursued complementary approaches to geometrical optics techniques as applied to questions of instability of the incompressible Euler equations. A forerunner of these results was the work of Bayly [ 1 I ] in 1987 who exhibited a system of "geometric optics" equations with some explicit computations concerning quasi-2D inconlpressible flows. Using WKB asymptotics based on short wave length perturbations, Friedlander and Vishik 1641, Vishik and Friedlander 1171 ] showed that a geometric quantity A , which can be considered as a "fluid Lyapunov exponent", carries considerable information concerning the instability of an Euler flow. This exponent is defined as follows. We consider an initial localized wavelet disturbance of the form
The disturbance. as it evolves under the linearited Euler equations ( I X), has the form
The evolution of the localized dihturbance is graphically shown i n Figure 6.
Fig. 6. The evolution ol a locali~edHuid blob (after 1.581)
The leading order terms in the asymptotics give the "geometrical optics" equations for the wave number vector k =. V S and the amplitude vector a: dx - = U(X,t ) , dr
(133)
~ ( 0=) xo.
where
and
(g)(")
=d + iu) . v .
tlt
=
it[
it consequence ol' V . v = 0 (incompressibility). In dynamical systems terniinolopy. ( 134) is the cotangent eq~lirtionand ( 135) the bich;ir;tcterisric aniplitude eqi~ationover a trajectory o i t h e Row U. Clc:trly Equa~tio~is (133)-(1.35) ~ ~ c d u ctoc Kulvill ctluarions (104). ( 100). (101) i n rhc apccial ciisc when the background flow U ( x , I ) is Iinca~ri n X. Of caul-sc. thcsc: cqu;itions are closcly reliltcd lo thc system ol' Eckhoffccluations. O n cvc1.y tri!jcctory U of the bitsic tlow we ~issociutc;in exponent A whcrc
Thc cor~clitiona(,. k()= 0 i \
I
[)I.
A ( x o , ko. a d = lim - logla(xo. ko. a(). 1 .--!L t
=
(
138)
sup A . ~~I.~,I.uII
and we call A,,,,,, :I fluid Lyapunov cxponenr. A parallel thcory was developed in the past t'ew decades in connection with inst:tbilitics in MHD that arc localized near magnetic field lines. 'This topic is discussed i n Jetail i n Lihchitx 1 1 171. I t can be shown that. in the f~-;i~mework 01' the ideal M H D modcl. magnetic lines of force fctrm a fctniily of rays for so-called "ballonning" modes which ilrc consideretl to cnwe the most dangerous inst;tbilitics (cf. Connor ct al. 1271. Dewar and Glasscr 1321. Hameiri (791, Liischitz [ l16,l 171. Eckhot'i 14.71).More recently Vishik and Friedlandcr ((161gave a rigorous treatmcnt of geometrical optics applied to MHD. They showed that the growth rate of a localized instability is bounded f~.ombclow by the growth rate of an opcrator given by a system of loc,ctl hyperbolic PDEs along each magnetic line ol' force.
5.6. The ut~.stuhlee s s e n t i ~spectrum ~l Let the unperturbed flow U be a steady Euler equilibria. Friedlander and Vishik [63-65, 171 ] used WKB asymptotics to develop a useful tool for studying the unstable rs.vmticr1 spectrum of the Euler operator C linearized about U (see also Lifschitz [ I 15,l 181). One of the main ideas in these papers is to replace the study of the spectrum of C by the study of the spectrum of the evolution operator p t L (i.e., the Green's function for (2.8), (2.9)). This permitted the development of an explicit formula for the growth rate of a small perturbation due to the essential spectrum. Roughly speaking, "high" frequency perturbations can produce instability in the essential (or continuous) spectrum and "low" frequency perturbations are associated with discrete eigenvalues. The following theorem proved by Vishik [ 1701 gives an expression for the essential spectral radius r,,,(rl') (i.e., the maximum growth rate in the essential spectrum) in terms of the maximum fluid Lyapunov exponent A . The results are proved for free space or periodic boundary conditions and are valid in any spatial dimension although, of course, dimensions 2 and 3 are physically the most relevant. We consider perturbations with vo E L 2 , V . VO = 0 and assume that U is CX-smooth.
This theorem is proved by writing the evolution operator c t L as a product of a pseudodifferential operator and a shift operator along the trajectory of the equilibrium flow U. This allows the growth of the evolution operator to be studied to precise exponential ;\symptotics. The result of Theorem 1 provides an information concerning the stability spectrum for inviscid flows, namely the maximum growth rate of instability i n the essential spectl-urn. Moreover. it implies that any point :in the spectrum a(rl') such that I:] > e A l l l ~ ' \is t necessarily an eigenvalue of ti nite multiplicity. A positive lower bound for the value of the Lyapunov exponent A can be explicitly computed in many examples. Furthermore, Theorem 1 provides an effective sufticient condition for instability of large classes of inviscid fluid flows. Since expression (140) involves the supremum over initial conditions (xo. ko. a()). it is only necessary to show there exists one set of initial conditions for which A b'11ven by ( 138) is positive to conclude that A,,,,, > 0 and hence the unstable essential spectrum is nonernpty. Thus the existence of an exponentially growing amplitude vector a is an effective device for detecting L' instability in the essential spectrum of e t L where C is the Euler operator associated with the flow U. We note that the connection between exponential stretching and fluid instability was first observed by Arnold 161. In the following sections we will describe many classes of flows for which an analysis of the ODE system (133)( 1 36) shows there exists a positive fluid exponent A .
Loc~rr/i;c,clirr.stcrhi/iries irr fluir/.s
32 1
Friedlander and Vishik 16.31 considered the effect of norms with higher derivatives. They showed that for initial conditions such that vo is in the Sobolev space R',the analogous fluid Lyapuno\l exponent '4, is
I A , = lim - l o g ( ( l 1-w t
+ k-) 7
\
12
a1
Thus there are different "degrees" of instability for different degrees of smoothness of the initial data. As Lifschitz 1 1 181 observed, the system of ODE (133)-(136) o n each trajectory of the flow U can also be used to obtain information about the spectral bound of the operator L itself as opposed to the evolution operator r r C .Detine from ( 133)-( 136) sup
ill =
A.
s ~ l . k ~ ~ . a (CI=O ,.k~,
i.e.. A l is the maximal exponent si~bjectto the restriction that ko is perpendicular to U on a given ~ritjectory.Lalushkin and Vishili 1961 prove that
whcl-c S ( C ) = ';i~p(Kc:: :E r r ( L ~This ] . ~.csultis proved using an clcmcntal-y dyn;~miL ciil system construction to ~.clatcthe hpcctriil bounrl S(&)to ~ h 1.cstr.ictt.d c lluitl Lyr~punov exponent A ] . . Scla iund Goldhirsch I1531 clcscribcd the csscntial \pcctrilm I'OI i~nbo~rndccl clliptic~rl Hows (63)directly, while Lifschitz II23.1741 a n a l y ~ c dthcir s p c c t r u ~ ~;ISi . well ;IS the spechyperbolic Hows, viit gconlet~.icaloptics tcchniquc. Here we briclly trum t r ~ r~lnboi~ndccl summ:irize Lifschitz rcsi~lts.First wc consider c l l i p t i c ~ ~Hows. l 1 12.71. Thc Eulcr e q ~ ~ a t i o n x lincori;lcd in rho vicini~yo f such ;I flow have thc form (98). with
It is convenient to use non-dirncnsional variables
, = .v; -.
y. - I
((1
whcrc c i l =
I' = mt.
1
I
1
t'i
= . mti;
!
/ I = -
I' mzti; '
JG. and rewrite Equations 198) in the ~ n m - d i ~ l ~ e n s i ofor111 ~lal
(135)
where
and 6 is the ellipticity parameter. Here and below primes are omitted for brevity. The corresponding spectral problem has the form
The spectrum of this problem is denoted by a ( 6 ) . By separating variables in the tlon,
x3
direc-
we obtain the spectral problem in the form
where Vh, = ( i ) , , , ill,, iX
1).
We denote the corrc\ponding\pectrum by rrh,(fi). I t i \ clear that
A simple rescaling suggests that nh, ( 6 ) is independent o f X 3 provided that X-3 # 0 (the case k 3 = 0 can be studied separately). Thus. the spectrum of problem (14%)is infinitely degenerate and. consequently. essential. Without loss of generality we put X-3 = I . In order to solve spectral problem (150) we use the Fourier transform with respect to . \ - I , .Y'. A tedious algebra yields the following Fourier-transformed spectral problem written in terms of i l ( k I . k 2 ) :
where the subscripts I denote projections o n the (.{-I. .\-?) plane. The corresponding u 3 = -VI . k ~ We . introduce polar coordinates ( p . + ) in the ( k l ,k?) plane and rewrite problem ( 152) as
where N' is a periodic matrix function of $, I -fi?
p'(
I-Hcos21jl)+I-fi? &,I? "in
21/,
/,'(l-~co521jl)+l-fi?
This matrix depends on p parametrically, so that we can fix p = po and study the correspondingspectral problems separately. It can be shown that their spectra, which are denoted by crl,,.,,,(S),consist of two series of eigenvalues of the form
hi.,, ,,.,,=*l~,,,+itz,
/ t = O . f I , k 2....,
(155)
where
A,.,,, is the monodrorny nlatrix for Ecluation ( 153) with h = 0, and Ln is the principal branch varies from 0 to cm.the corresponding h*,,,,,,o cover a cross in of the logarithm. When 0,) the complex planc described by the following conditions
whcrc A(6) = InaxoG,,,,<, Rc I,,,,,.The entire spect~wnlnl (6) is obtained from this cross via sliif'tz by irr in the colliplex pliunc. I t h:~a :I 1xthc1.cxotlc "\kcleton" structure. The spectrilm of hyperbolic flows is even more exotic. In 11241 it is shown that [hi\ spectrilm occupies it strip in the complex pliulc along ~ h ilnaginary u axis which is dcfincd hy thc contlition
This s p c c t ~ . u is ~ lintlcpcnclcnt ~ of the gcomctry of the basic tlow.
In the p~.cvioussubsection we tliscussed rcsults concerning thc .s/~rt./t.i~l problem associated with the linctu*i/cclEulcl equi~tionsthat can be obtained using thc ODE (133)-(136). Another colnple~ncnturyiippl-oach t o geonletrical optics t c c h n i q i ~ eilpplied ~ to the lineari~ccl Eulcl. C ~ L I ~ I ~ ~wus O I Ltaken S by Lifschitz ~untlHa~nciriI 1261 anrl 1,ilachit~I I 19-1771. This i.; bascd on :I 1nc:Isure of'the growth of a perturbation itself as a solution of the linc~u-ized Euler equations. The growth may be exponential or algebraic in time in a norm thi~thas physicid meaning (c.g.. growth i n the energy or the enstrophy of LI perturbation). Lifschitz and Hi~mciri11261 used energy inequalities to show that the formal W K B asymptotic solutio~lslijr V(X. 1 ) const~.uctedvia ( 130)-(136) arc ~-loscto thc actual solutions of the line a r i d Eulcl equations ( 18). Hence the growth rate of the ;~ctualsolutions to these equations with initial data ( 19) that is si~tficientlysmooth and nonzero only in some ball. could be estimated in terms of the hehavior of the leading order terms of the asymptotic solutions.
2 . Tl~efu.~tesr g ~ n n ~ rate t h of'the vekocity perfurbutinn v(x, t ) is bounded from belous by the firstest g r o ~ l t hrute o f t h e antplitildes a. The $ow is unstable in the energy rzortn if THEOREM
-
Iirn
"n
sup
la(xo. ko. ao, r ) l
+
m.
laOl=l. l k L l l = ~a.. k0=0
(159)
If the exponent A defined by (138) is positive, then this instability is associated with exponential growth in time. We note that, frv)n~( 1 3 1 ). the leading order amplitude of the perturbation vorticity w is a vector b where
Thus the existence of vectors x ~ko, , a0 such that
implich "cn!,lrophy" instability. In the following subsection we discuss relations between growth in the energy nol-111and growth in the enstrophy norm due to high frequency perturbations.
As we discussed. locali~cdinstabilities of n basic Euler Ilow U are associated with growing solutions of the amplitude equations ( 134). ( 135) on it tri\jrc~oryof I!.Thcce equations have a rich geometric structure and certain gencral properties can be used to prove that large cl:~sscs of Euler Rows are unstable. see Section hclow. We sum~narizethe main general properties: ( A ) I t is irnruediate from ( 134). (1 35). (1 36) that
(this is consequence ol'the fact that thc flow is incompre.\sible). ( B ) In 2D it follows from (134). (13.5). and (162) that
or. equivalently,
Hence the product Ikl la1 is constant on a trajectory of U. Thus in 2D there exists a growing amplitude a if and only if there exists a decaying cotangent vector k. In pa~.ticular,we
conclude that the only nondegenerate, steady 2D flows for which there is an exponentially growing a are flows with a hyperbolic stagnation point. It follows from (163) that in 2 D the classical Lyapunov exponent (i.e., the exponential growth rate of a tangent vector) and the fluid Lyapunov exponent A are equal. As we stated in (160), the leading order amplitude b of a high frequency perturbation of the vorticity is given by k x a. Hence Equation (163) shows that in 2D there is never high frequency growth in the vorticity. This is an obvious consequence of the conservation of vorticity in 2D. Applying the vorticity equation analogue of Theorem 1, it follows that the unstable essential spectrum of the evolution operator of the vorticity equation in 2 D is always empty. Hence any instability in the 2D vorticity equation must come from unstable eigenvalues. As we now discuss, the situation in 3D is very different. (C) In 3D Equations ( I 33)-( 135) plus volume conservation imply
where a1 and a? are two linearly independent vectors satisfying ( 1 35) for a given cotangent vector k. Thus we conclude that the existence of a decaying cotangent vector k is sufticient to imply the existence of a growing amplitude a and hence instability. Furthermore ( 165). together with volilmc conservation and the duality of tangent vectors and cotangent vectors, proves that the existence of a positive classical Lyapunov exponent (i.e., an exponentially growing tangent vector) iliiplies the existence of a positive,/li~iclLy;tpunov exponent A . In 3D this condition is suflicient tor exponential 11i1idinsti~bililyhilt not necessiu-y. as we will discuss later. ( D ) In 3D thc vorticity amplitude b is not constant: it satisfies the ODE db dt
-= (%)b-
k x b tk.52)Ikl'
As Lifschitz [ 12 1.122 1 observes. Equation ( 166) is closely connected to the tangent equation,
A particular pair of solutions to ( 166) with k . 52 = 0 is
with the corresponding velocity amplitudes
where
It follows from (168) that the existence of a growilz~cotangent vector k with k . $2 = 0 iniplies the existence of a growing b. Since the flow is volume-preserving, a decaying k must be matched by a growing k. It follows from either (165) or ( I 6 9 ) that the existence of a decaying k implies the existence o f a growing a. Hence the existence of a decaying k implies at least weak (algebraic) instability i n the Lyapunov sense in both the velocity and the vorticity norms. For flows U with rx/)orrrtltial stretching. volume preservation plus duality between tangrowing and decaying gent and cotangent vectors implies the existence of c.x/~onenfi~rl!,. vectors k. Hence in this case the exponential growth rates of instabilities i n both the velocity and vorticity norms are positive. However, the maxinium rates are not necessarily the sarne. A sirnple exiirnplc that illustratec this is a 3D flow with a hyperbolic tixed point at which the eigenvalues of the niatrix (iIU/iJx) are ( I . - 112, 112). I n this case. the maximal exponent for the growth in la1 is 1/2 but the maximal exponent for the growth in Ibl is I . In the case of steluly 3D ilows U with exponential stretching. we have additional information from Thcoreni I (and its analogue for the vorticity equation). nanitly the maximal exponential growth ~'atcsdelern~incthc' cssenti;~lspectral radii ofthc vclocity and vorticity evolution operators. -
I t is interesting t o invcstigatc the in~pactof viscosity o n short wuvclcngtli institbilities. Thc Nnviel--Stokes equations describing the evolution of a vihcous fluid havc the form
Here 11 is thc kinematic viscohity of thc Iluid, wid F is the 10rcing terrii. We follow [,ifhchitx I I 191 and assume that viscosity is vanishingly small. so that
where F ia :I small parameter and 3 is thc normalized viscosity chosen ia such a way that the corresponding Reynolda number is of order unity. ( I t is clclir that the decomposition (171) is not unique.) Let (U(x, f ;
4.Po(x. I ; 2))= ( 6 ( x . t ) . &(X,
t ) ) +()(I..?)
be an exact solution of the Navier-Stokes equations whose stability we want to analyze. Here the leading-order terms describe a flow of an inviscid fluid. We assume that on any tixed time interval 10, T] the viscous flow converges to the inviscid flow uniformly in E . T h c linearized Navier-Stokes equations have the form
We chocrce vo in the t0rm (130). We emphasize that in contrast t o the inviscid case, the magnitude o f the small parameter t: cannot be chosen arbitrarily and is determined by the mugnitude of viscosity. On a fixed time interval 10, T I the solution of the linearized problem has the form ( 13 1 ), ( 132). It is easy to show that the eikonal equations ( 1 33). (134) are ~ ~ n a f f e c t cby d viscoxity while tho amplitude equation (135) has thc for111 cla il CJ - - - (-\2+2
(i)U/i)x)a.k
k
-
a
a ( 0 ) = ao.
lkl'
we c;111rcwritc the viscous ; ~ ~ ~ i p l i t i cqi~;~tiotl ldc ( 1 7 0 ) in the invilr~~id form (135). Thus. the viscous flow is unstahlc i n Ihc velocity norm proviclccl that -
linl xup exp[' + ~ ~ i t , , ~l = k ~l l. = l ~. I Oko=(I
I'
c k ( x o . ko. 1 ) 1'
dlf
I
It is unstable i r thc ~ vorticity norm provided that
Wc etnph;~si/.ethat i d c d inslabilitics can be s t a b i l i ~ c dby viscosity tor some valucs but remain unstable for other values ot'this parameter. A complete viscosity stabili~ationoccut.x only when the growth rate of the wave vector k is sutficicntly lurgc. Thc nhove stability conditions are given by Lifschitz 1 1 191, see also Dobrokhotov ant1 Shafarevich 136.371. they are in agreement with the necessary conditions f o r the so-called fast fluid instability given by Fricdlander and Vishik 1631, who use geometrical optics techniques to prove that
a necessary condition for instability in the Navier-Stokes equation as F approaches zero is an instability in the underlying Euler equation. Landman and Saffman 1951 analyzed the impact of \liscosity on the stability of elliptical flows. There are interesting internctions between the above analysis a r ~ ddynamo theory, see, e.g., Vishik [ 1691, Bayly [ 101, Dobrokhotov et al. 1351. and the article by Gilbert [72] which appears in this volume.
6. Examples
I t is natural to apply the geometrical optics stability theory to the analysis of Hows with stagnation (or fixed) points. Consider a background flow U(x, r ) with a stagnation point xo such that U(xo, r ) = 0. The stability equations (133)-(135) at this point reduce to Kelvin equations (100). (101 ). When the background flow is steady, these equations have constant coefficients. We all-eady know l'rom Section 5.8 that Hows with hyperbolic stagnation points (which implies exponential stretching at this point) are unstahle. Now we consider elliptic points. Under generic conditions the presence of LIII elliptic point implies the existence of a hyperby the same token. However. bolic point and hence flows with elliptic points are ~~nstnble for the purpose ofillustr.ation and to pi11our analysis into historicill prospective, i t is useful to discuss instabilities rclatcd to stagnation points in some tletnil. Furthermore, the nature ol'the instahilitics ilssociated with hypcrbolic and elliptic points can he somewhk~tclit9c.rcn~. We bcgin with hypcrholic points. Thc simplcst way for a Ho\v to have c x p o ~ ~ ~ ' n t i i ~ l stretching (i.e.. positive clitssici~lLyapunov exponent) is t'or i t to occi~r; ~ tonc point. i.c.. ;I point x , at which U ( x , )= 0 and thc matrix (;)U/i)x)at x, has i1n eigenvaluc with pohitivc real pa1.t u . AS wc Clisci~ssed111 Section 5.8. the Huid Lyi~punovexponent is then positive i~ndthe exponcntial growth rate ot'an instability is bountlcd l'rom below by u . An cxplicil exaniple o f a 2D flow with a hyperbolic point is a ccll~rlarHow with stream function (43). or the less symmetric variant (44).Many ofthc 3D equilibria discussed i n Section 3 contain hyperbolic (iund elliptic) points. Hills spherical vortex (see Equation (53)) has hyperbolic point.; at :# 0. r. = kt!.The so-called ARC Row given by Equation (69) has hyperbolic C' 2 A ? . stagnation points for all values of A 3 B 3 C such that H' We note that in 3D. us opposed t o 2D. tlows niay exhibit exponcntial stretching without having a stagnation point. For example, Friedlander et al. 1591 and Chicone 1151 proved the existence of hyperbolic closed trajectories !'or ABC flows i n cortain parameter ranges wherc there are no stagnation points. Furthermore. conceptual flows. rather than physical. such as Anosov Hows have been constructed to have the property of exponcntit~lstretching on every Lagrangian trajectory. Clearly, from our prcvious discushion. all such Rows havc a positive fluid Lyapunov exponent and are exponentially unstahle. We now turn to instabilities associated with elliptic points. For strictly 2D steady Rows. elliptic stagnation points (where the eigenvalues of (i)U/ijx) are pure imaginary) d o not give rise to exponentially growing instabilities, although any shear in U implies algebraically growing instabilities. However, if we allow 3D perturbations. the mechanism of "vortex tube stretching" permits exponential instabilities. The most classical of these is
-+
the Rayleigh centrifugal instability to 3D perturbations of a circularly symmetric rotating flow where the angular momentum decreases with radius. This prototypical instability was generalized to elliptic columnar vortex structures by Pierrehumbert and Widnall 1 1391, Leihovich and Stewartson [ I 101, Sipp and Jacquin [ 1581, and othcrs. In this context. Bayly 19, 121 used Floquet analysis to study the growth of a localized perturbation transported o n an elliptic streamline. He reduced the problem to the consideration of a system of ODE closely related to the system given by Equations ( 133)-( 135) and obtained sufticient conditions for "elliptic" instabilities. Under the condition that the magnitude of the circulation decreases outwards. instability with exponential growth is a generic property regardless of the symmetry (or asymmetry) of the How U . Bayly's work was extended by many authors, see. for example, Landlnan and Saffman 195). Waleffe 11751 and Fukumoto and Miyazaki 17 1 1 . Lifschitz and Hameiri [ 1261 exploited the geometrical optics treatment for detecting locali;red instabilities to ohtain sufticient conditions for exponential instabilities of general steady 3D flows U with elliptic points. They used the Floquet method t o analyze the ODE (133)-( 13.5) with time-periodic coefficients. Thcy obtained sufficient condition'; for the corresponding ~nonodrolnyrnatrix to have cigenvulues larger than unity. Numerical calculations indicated that elliptic stagnation points in 3 D flows arc unstt~ble.For nonsteady flows arialogous co~npututionscan he ~ n a d eol'clussical Lyapunov cxponcnts in thc 11eighhol.hoodof ;in elliptic point to detnonstr.~ktcinstability. Hcrc wc hriclly hulun~uri/ctlic corrcsponcling result\ in tho I'orm which is consis~cntwith our ;~n;llysisofthc spcclru~liol'cl1iptic;il tlows. Consider :In elliptical Row ( 6 3 ) .I n i~dditio~l to non-dimcnsion:~lv:uiahles given hy Ecli~ution( 145). wc introduce the non-~linic.~i.;io~iiil wave vcctor k' = (t11I \ ] .tr2L2.c1~X.3). alld write the geometrical optics (or Kelvin's) ccli~i~tions ( 100).( I01 ) in rhe fol-111
dai - = -Jlal tlt
+
2J1al . k l G;lk,
.k l
+ kli. kllG i l k ~ .
whew the subscripts 11 i111cl Ir c k r to vector ant1 ~iiatrixco~nponcntsparallcl to thc .\-; tixis and tht: ( . \ - I . .\.,) plane. respectively. and ~iiutriccs:T. I; arc given hy (147). 11 is clc:~rthat
wherc /[ is the cosine of' the angle betwccn k and e; at t = 0 . I t is ohvioi~sthat we c;in concentrate on the ti rst cquation ( 184). We introduce new variables c = ( c . 1 . c.?). where
and after sonie algebra represent the first equation (1 84) in the form
where
?J
I t is clear that 2 x 2 matrices A' given by ( 1 54) and given by ( 1 88) are closely related. 3. The matrix I S T-periodic so we can use the standard Floquet theory for solving the stability problem. We introduce the monodrorny nlatrix ,G, i t . , the value of the fundamental matrix solution at I = n . I t is easy to show that d e t M = 1 , \o that the eigenvalues of M satisfy the equation A
A
where A = T r w ~ is t l LIII analytic f'unction ol' 6,id. Thus, one of the roots is real and cxcccds unity in absolute v~lluewhen ( A ( > 2 which is rhc necessary and sut'fcient contlition tor exlx)ncntial instability. When lAl < 2 wc hi~vrstability. Finally, when ) A / = 2 wc have cithel- stability (whc11 M is diagonal) or algchr~~ic instability. I t is very easy t o computc A nun~cricallyand to show that f'or every h' > 0 there exists a n interval p,,,,,, < 11 < E I , , , ; , ~ such that ( A ( &11 . ) ( > 2, so that all ellipticiil flows are unstable. Poshible niecht~nis~ns for suppression of elliptic instabilities attract ~iil~cli attention. In particular. it was reulizcd that effects of rotation c ~ u ihave strong stabilizing impact. see Craik 1291. Bayly et al. 1131. Lebovit~and Lifschitz I IOhl, Lcblanc 1971, and Leblanc and Camhon (Yc).lOOl. The Euler-Coricllis equations oS motion written in a coordinate t'ra~i~e rotating, with respect t o inertial fra~nc,with angular velocity 29 have the I'ol-m A
The linear~~ccl equations for the perturbation ( F , 11) of'a ba\ic How ( U , 4)) arc
The corresponding geolnetrical optics equations consist of the eikonal cquation ( 133) and the amplitude equation of the form
For linear flow the above equation reduces to the Kelvin's form
see Lifschitit 11221. Consider an elliptical flow (63) in a coordinate frame rotating with angular velocity ro around the .\-3 axis. The general solutioli clt the corresponding eikonal equation has the form ( 1 85). The amplitude equation reduces to the form ( 188) with the coefficient matrix JQ which is a rr-periodic function o f t and parameters 8 , p , and f., where
This stability problem can be solved via the standard Floquet method. The stability diagram for LI rotating elliptical flow shows that for ,f .- -2 elliptical Hows :Ire stable. For tirne-dependent elliptical Hows. for instance for Kirchhoft-Kida's vortices, the stability problem is much more complex bec:ulse i t requires solving ODES with quasi-periodic coefficietits. Bayly ct al. 1131 developed n practical method for solving the stability problcln bi~scdon its reduction. see also Forqter :~tidCrnik 1551. In p:lrticuI;~r.Bay ly et i11. 1 131 ~howcclthat Kirchliot't-KiJit'4 vortices arc typically u~~st:rblccxcept when the interior the ncgative o f the background vorticity. Their rt\ults are in vorticity is :~pproximi~tely ;rgrccmcnt with the l~esultsobtiiir~cclhy Robinson and S;tlf'~ni~n 1 1471. Miyazaki and Fuku111oto1 1331. Le Di/cs et nl. I 1081. LC I>i/.cs i~ndEloy 1 107).Eloy and 1.c L>izcs 1301. Lcwekc ;lnd Williiumson I I 1 I I.
The best known clilss o f ititegr:~blc flow\; consists of vortcx rings with swirl. Recalling thc cqi~ilibrii~ equ;ition (33). we observe two extreme cases. Firstly, the so-calletl Reltl.iuni flows where V x U = hU and V H 0.In general. such LI "ch;~otic"How is not integrable i~ndanalysis of explicit Hows. such i w A H C flows. strongly suggcsts that all Bcltrnmi Rows arc exponentially unstuble. On the other cxtrclnc we have intcgri~blctlows where O H # 0 ancl the surklces H = Ho are integrals ol'thc motion. As we rcmarkcd. Arnold 1.51 p1.ovcc1 ihut such compact surfices arc ncstccl tori :ind these f uicl cquilibri:~col'rcspond to stcady vortex rings with swirl. The clussic:~l Lynpunov exponents for :in intcgr;lhle Row
-
are itll /era. Hence wc cannot invoke the general result of Section 5 . 8 t o conclucle that vortcx rings with swirl are unstable. A more detailed analysis of the system of ODE ( 133)( 1 35) is necessary to prove instability rehults in this situation. There are a number o f papers i n which such analysis is given. Friedlander and Vishik 1651 used thc system of ODE to
study instabilities of axisymmetric toroidal equilibria of the form (195). They obtained a sufficient condition for a Floquet exponent of the monodromy matrix associated with the system to be greater than unity. This produced the following (non-sharp) geometrically sufficient condition for exponential instability of a vortex ring with swirl, namely
where K is the curvature, r , the geodesic torsion, and i? the principal normal of a helical streamline as it wraps around a toroidal surface H = Ho with period T. A similar approach to the problem was taken by Lifschitz and Hameiri [ 1271 who used the system (133)-(135) to obtain a simple necessary condition for the stability of the core of a vortex ring with swirl. For general vortex rings the transport equations either have periodic coefficients or coefficients that are asymptotically periodic. Accordingly, two types of instability were distinguished: (A) Instabilities having Floquet behavior and exponential growth rate. (B) Instabilities growing algebraically in time. In general. computation of the Floquet exponents can only be done numerically but in the simpler situation of vortex rings without swirl the analysis leads to a sufficient condition for stability thnt is a generalized Rayleigh criterion iund is consistent with (196). These results were extended by Lifschitz et al. [ I291 where the growth rates of locali~ed disturbances predicted by the WKB-geometrical optics method were compared with nu~nericalsolutions of the full time-dependent Euler equations thnt simulated the evolution of a vortex ring. The solutions to the perturbed proble~nwere obtained using a 3D vortex method i n which the (low is represented by a collection of Lograngian vortex elemcnts moving according to their induced velocity. I t was h u n d that the WKB analysis did a reilsonably good job of predicting the growth of instabilities i n comparison with the vortex method calculations on the full Euler equations. The growth rates coming from the WKB analysis were approximately twice the growth rates from the treatment of the full equations. This can be explained by noting that the asymptotic analysis predicts the maxinium possible growth rate while the vortex method calculi~tionproduces a rate for a particular disturbance not chosen necessarily to represent the niaxi~num.I t is also important to note that the WKB analysis excludes non-local interactions, whereas the vortex method includes the full nonlinear effects but introduces a discretion error.
As we know, the sum of a linear flow and a Kelvin mode is an exact solution of the Euler equations. Thus, it makes sense to analyze the stability of such a sum with respect to localized short wavelength instabilities. Since Kelvin modes can be viewed as primary perturbations of the linear flow, the corresponding instabilities can be viewed as secondary. The analysis of secondary instabilities of linear flows was initiated by Lifschitz and Fabijonas (1251, and continued by Fabijonas et al. 1501, Fabijonas (491, Lifschitz et al. 11281.
Miyazaki and Lifschitz [I 341, and others. Here we briefly summarize the corresponding results. Let U be a primary linear flow of the form (63). We introduce non-dimensional variables (145) and write the Euler equations in the form (146). Next, we consider the same equations in a rotating coordinate system and write
where S solves the equation dS/dr = SJ, S ( 0 ) = 2,and in the rotating system,
S=
[(i:;r 8) . .st
G2 =
[+ 1
G2 = STGSis
6 ~ ' - .~~ 2 1 ~ .s2, 1 - 61.2. 0 0
I:)
,
the metric tensor
(198)
I
where c.,, .s, denote c o s r , sin r . Omitting primes for the sake of brevity, we write the governing equations in the form
I n the rotating coordinate system the equilibrium solution is trivial by construction. U = 0. fi1= const. For 6 = 0 the governing equations ( 199) arc the st;lnd;lrd Euler-Coriolis equations while for S # 0 they can be considered the generalized Euler-Coriolis equations with time-dependent nletric tensor. Kelvin modes which are exact solutions of Equations ( 199) h;~vethe i'orm
where k = (.so.0 . c . , ~ ) is ~ the wave vector of the standing wave (which is time-independent). A ( / ) is the normalized alnplitude, and Y is the scaling factor. The corresponding Kelvin equations can be written as
Following the same logic as before. we rotate the coordinate system around the unit vector e2 in such a way that A. and k turn into the unit vectors el and e3, respectively. In the rotated coordinates the governing equations (199) assume the form
where
The corresponding standing wave has the form
c ~S2c;), ~ and ( A 1, A2) solve a 2 x 2 system where A3 = 0, cw = 2(1 - s ' ) . s ~ A ~ / (-I s . v ~ of periodic ODEs similar to ( 1 87). Thus, in the rotated coordinate system the flow is inde, Since A is a solution of a Floquet problem, it can be either periodic, or pendent of x ~ x2. quasi-periodic, or growing in time. We now apply the geometrical optics technique to study the stability of the Kelvin mode (204). The geometrical optics equations have the form
A simple algebra reduces Equations (205)-(207) to a 2 x 2 system of ODEs of the form ( 187). However, the coefti cient matrix can be either periodic. or quasi-periodic, or exponentially growing. The stability problem has to be solved for various parameter values. The corresponding solution is very difticult and time-consuming. In general. secondary instabilities are always present. 7. Nonlinear instability
Problems connected with stability and instability of the full nonlinear Euler equations ( 1 )(4) are even more complex than those related to the linearized Euler equations that we discussed in Sections 5 and 6. Hence many questions remain open. However. there have been "small steps of progress'' which illuminate the challenges of the nonlinear problem and we will now describe some o f these results. Loosely speaking, a How is called nonlinearly stable if every disturbance that is initially "small" generates a solution to the nonlinear Euler equation which stays "close" to the original flow for all time. There are several natural precise definitions of nonlinear stability and its converse, nonlinear instability. These definitions reflect the crucial dependence of
a stable or unstable state on the norm in which growth with time of disturbances is to be measured. As we remarked and illustrated with a simple example, even for a lineur problem the answer as to whether or not a solution to a PDE is stable or unstable can depend on the norm. There are very few known explicit solutions for the time-dependent, nonlinear Euler equations. There is even, as we discussed in Section 3, only a limited selection of known explicit steady equilibria. Hence it is worthwhile noting the following two explicit examples of "growing" Euler flows: (A) As we remarked in Section 5.2, Chandrasekhar has observed that the sum of a "linear" flow and a single Kelvin mode of the form (102) is an exact solution to the nonlinear Euler equations. Hence a basic flow of the form U = K ( t ) x is nonlinearly unstable to a Kelvin mode perturbation provided that the ODE system admits a growing amplitude a ( / ) . In Section 6 we discussed how this can occur. In particular, if K is constant, then the existence of eigenvalues of K with positive real part implies the existence of a vector a ( t ) that grows exponentially. Of course, a drawback of the flow (102) is that in general it does not satisfy physically appropriate boundary conditions. However, this example motivated the work o n secondary localized instabilities for the linear problem discussed in Section 6.3. ( B ) Yudovich 1 180, I X 1 1 observed that there exists a class of exact solutions to the nonlinear Euler equations which imply that all non-constant 2D steady shear flows are unstable with respect to 3D perturbations in any norm which incll~desthe maximum of the vorticity moclulus. Consider the plane piu-~~llcl shear flow U = (,f'(.r?). 0. 0 ) with .r? E 10. 2n 1. From Rayleigh's cl;issical result this flow is linearly (spectrally) stable i n L' il' there are no inHcction points in the protile ,f'(.r2).I t is ciisy to check that the tollowing is iun exact solution to the nonlinear Euler equi~tions( 1 ) for any smooth functions ,/' and W :
The corresponding vorticity is
Hence no matter how small the magnitude of the initial vorticity, the magnitude of the vorticity of the flow (208) grows (linearly) with time provided only ,f"(.t-.) # 0 and W' # 0 . This set ol' exact solutions to the nonlinear Euler equation can be easily generalized to suitable .v3-inclepcnclentperturbations ot'iuny 2D steirdy flow. However. again. flows of'the form (208) do not satisfy 3-di~nensionalphysical boundary conditions.
One reason why proving nonlinear srt~hilityis a very difticult proposition is that to date there are n o results of existence and uniqueness for all time of the 3D nonlinear Euler equations with appropriate boundary conditions and initial conditions in a suitable function space. It is mathematically reasonable to consider a detinition of nonlinear stabilitylinstability in function spaces for which there is at least local in time existence and would still run into the major obstruction of the existence uniqueness. Claims of stt~hilir.~
in 3D of global in time solutions. However. the proof of the converse, i.e., instability, is not so restricted since finite time "blow-up" would be one special case ot'instability under the following definition which we formulate for a general nonlinear evolution PDE.
DEFINITION 2. We define t ~ o t ~ / i t ~ r a r . . ~ t for ~ ~ /a~general i f i t v evolution equation of the form
where C imd ; I are ( respectively the linear and nonlinear terms. Let X and Z be a fixed pair of Banach spaces with X C Z being a dense embedding. We assume that for any uo E X there exists T > 0 and a unique solution u ( t ) to (210) with
in the sense that for any 4 E D ( 0 , T )
The initial condition is assumed i n the sense of strong convergence i n %:
The trivial solution uo = 0 ol'(2 10) is called nonlinearly st;~blcin X ( i . ~ .Lyap~~no\. st;~hlc) if for all c. > 0 there cxists rC > 0 such that IIucrIl c: 8 implies that wc can choose 7' i n ( 31 I ) to be 7' = m; and J l u ( t ) l J x< c' f'or n.c. t E 10, m). The trivial solu~ionis called rrorllirrrtrr/~rrrr.v/r~blrin X i f it docs not si~tisfythe conditions stated below. In the contcxt of the Euler equations the natur:il function s p ~ Xc is the Soholev space H Ywith v. 3>11/2 + I. I t is well known (cf. Wolibner 11771. Lichtcnstein I 1131) that solutions t o the Euler equations exist locally in time in such spaces and the loci11 property will result. We first forniulute the relevant theorem concerning be sufficient for an i~~sttr/?ili!\~ nonlinear instability in a general setting. We consider the stability ol' the zero solution of an evolution equation (210). wherc C and , I f are ~.cspectivclythe lineal* and nonlinear p;lrts of thc governing equation. Once the spectrum of the linci~rpwt C is analyzed :~ndshown to have a n unsti~hleconiponent (i.e.. the zero solution is linearly unstable). the question ~ ~ r i s whether cs the zcro solution is nonlinearly unstable. I t is well known (see, for example. Lichtcnberg and Liehcrrnan 1 1 141) that the linear instability implies nonlinear instability in the finite-dimension~ilcase (i,e.. if (210) is an ODE). In the infinite-dimensional case (PDE) such a gener:~l result i~ not f PDEs i t has been shown that linear known, although for some particular types c ~ e\iolution instability implies nonlinear instability (e.g., such a result for the incompressible NavierStokes equations in a bounded domain has been proved by Yudovich [ I 821). Difficulties with deriving the nonlinear instability from the linear one usually appear whenever thc
essential spectrum of ,C is non-empty. As we discussed in Section 5.6, this is generally the case for the linearized Euler operator. Friedlander et al. 1621 proved the following abstract nonlinear instability theorem under a spectral gap condition. THEOKEM 3. Fir (I prrir of' Bntlnclr .sl)clc.r.c.X C , Z with rr d r t z s ~rr?zhcddina. Let Eqi~ucitzd C .soti.yf~thr,fi)llowiitg cottditioil (2 10) orllnit LI 1oc~11 uii.steilcctheorrnr ill X . Let tio11.q: (A) II,!L"(u)llz ColIullx /lullz, for- u E X with IluJlx < p,fi)r .seine p ;- 0. (214) (B) A .v/~rc.tr~// "gclp " cw~dition,i.r. ..
<
The main itlcn ot'thc proot'ofthis theoreln is as I'ollow\. We a w l m e the contrary. n;tl~lcly that the trivial solutioll u = 0 is nonlinc;~rlystnhlc. Lcl c. > 0 sufficiently sm:~llhe given: i t will be spccilicd Iittcr. From the definition ot' nonlinear stability it 1i)llows th;tr thcrc cxists :I global solution u(t ) . I t 10. m ) such that Ilu(1)lls < F ~wovidcdIlu(0)llx 6(c.). We pro-ject u ( t ) onto two subspiices using thc \pect~-algap condition (116). ( 7 17). Wc denote by P+ the Ricsz pro.jcction corresponding lo the p;lrtition of thc spectrunl c r e ~ ~ t c d by the gap and inti-oducc a ncw norm 111 . 111 on %. For iuiy .I E % Ict
---
The norm 111
. 111 is cquivalcnt to 11 . 11%. i.e.. there cxists C'
Since u ( t ) i h a solution to ( 2 14) it can be shown that
for any interval 0
< 1 1 < tz
>0
silch that
We choose the initial condition uo = 6wo, where wo
E
X is an arbitrary vector satisfying
Since jluoIIx > 6, our assumption of nonlinear stability implies Ilu(r)
1
<E
for a.e. t E [O, cm),
(222)
and from condition (22 I )
Now the inequalities (222), (223) plus Gronwall's inequality give
provided E chosen so that E < rnin(c-\.,'. p ) . Since M > 0, for sufficiently large t the inequality (224) contradicts our assumption that Ilu(t)llx < E . Hence the trivial solution to (223) is nonlinearly unstable in X. We now consider Theorem 3 in the context of the Euler equations ( 1 ). We write
thus the notation of the general theorem applies to instability of the steady now U. The local existence requirement and condition ( A ) of Theorem 3 are easy to satisfy by making the natural choice for the spaces X and 1,namely X=H',
I1
.r>-+I
2
and
%=I,?
(226)
with the restriction to vector tields that are divel-gence-t'ree and satisfy appropriate boundary conditions. However, the spectral gap condition is much more dif'fici~ltto verify for i~ given steady solution because, as we have discussed i n Section 5.6. the essential spectrum of elc is generally non-empty but its exact structure is not known. An item of information we have about the structure of the spectri~mis the essential spectral radius theorem discussed in Section 5.6. In some examples the "Ruid Lyapunov exponent" A can be explicitly calculated. Also. Theorem I implies, in particular, that ) [:I > e A ' is a point of the discrete spectrum (i.e., an isolated point any :6 ~ ( c ) ' ~with with finite ~nultiplicitywhere the range of (: - elC) is closed). Any accumulation point of ad,5c(c'' ') necessarily belongs to ( ~ , , , ( e ' ~ )Thus . if
then there exists a partition
satisfying the gap condition (216). There are several examples of 2-dimensional flows where A and discrete unstable eigenvalues can be calculated to show that (2 16) holds. These are the examples of discrete unstable eigenvalues discussed in Section 4.3. As we remarked, in 2D the fluid Lyapunov exponent and the classical Lyapunov exponent are equal. Hence A = 0 for any plane-parallel shear How. It therefore follows from Theorem 3, and from the results of [ 161, that there exist unstable discrete eigenvalues for any shear flow with a rapidly oscillating protile, that all such shear Hows are nonlinearly unstable in H ' with .Y > 2. Other recent results concerning nonlinear instability of 2-dimensional shear flows include the work of Grenier (781 who proves nonlinear instability in Lr' for piecewise linear ~ uniform profiles. Koch 191 I proves in 2 dimensions that nonlinear stability in c ' .requires boundedness of the derivatives of the flow map, which implies that all steady shear flows are nonlinearly i~nstablei n c ' . ~ . A more general 2-dimensional How than parallel shear How that can be shown to be nonlinearly unstable is the "cats-eye" flow studied in Friedlander et al. 1671. see Figure 1. In this case, the existence of hyperbolic stagnation points implies A > 0. The exact value ol' A can be calculated a the positive eigcnvi~lueof the gradient matrix of U at the hyperbolic point. The results of' Friedlander et al. 1671 show that there exist discrete unstable eigcnvalues with real part greater thiin A , hence again we can invokc Theorem 3 to prove that such "cats-eye" Hows :ire nonlinearly unstable. We recall the remark in Section 5.8 that i n 3D there is n o i~nstableessential spectrum for the line:~ri/.cd1~or.tic.iryecluation. Hence iilny I<' instability in the 71) vorticity equation must arise from discrete i~nstablccigcnvulucs. Thus Theorem 3 ciui be immedintely applied in this context to conclude that linear instability implies nonlinear instability i n H.' t'or the 2D \-o~./ic.if\. equation. A recent paper of Bardos ct al. 181 proves ii stronger resi~lt.n:unely that the existence of an unstable eigcnvalue for the vorticity equation with real part greater than the Lyapunov exponent for U implies nonlinear instability in L' tor the 2D \,ortic.;/\. equation. Following the approach of the paper of Bardos et al. 181. Friedlander and Vishik 1681 h~iveproved the analogous result ti)r nonlinear instability i n L' for the ZD \.cloc.i!\. equation. Wc note that the l,' energy 1ior111is the most nati~rtilphysical norm i n which to consider the issue of tluid stability or instability. To date we do not have enough infor111:ttion about the structure of the spectrum of the line a r i ~ e dEuler operator for general 3D Hows. to apply Theorem 3 or other known techniques to prove nonlinear instability in the most "natural" function spaces t'or the ecluation. There is also a considerable body of literature concerning nonlinear stability/instability i n spaces which are not the "correct" spaces for the Euler equation in the sense that the possibility of,fi/lite time blow-up in such spaces has not been ruled out. This includes the celebrated stability results developed by Arnold 141 and applied by Inany of his followers. As we remarked in Section 4.3, Arnold's methods apply to stability in a function space we denote by J I . More precisely, he showed that the steady solutions of the Euler equations
are vector fields on a certain infinite-dimensional manifold M , which are the critical points of an energy functional E restricted to M. If the critical point is a strict local maximum or minimum of E, then the steady flow is nonlinearly .stuble in a space J I whose elements are divergence-free vectors v having a finite norm (77). The theory for proving nonlinear stability is mathematically elegant but has limited applicability to fluid flows for several reasons, in addition to the fact that the space J I is not a "correct" space for the Euler equations. Firstly, there may be no critical points of E at all on the manifold M. Secondly, the natural way to prove that a critical point is a strict local maximum or minimum is to prove that the second variation of E is negative or positive but definite at the critical point. However, it was shown that for the 3D Euler equations this quantity is never definite (see Sadun and Vishik [ 1 501). Another approach to nonlinear stability/instability with respect to growth in the space J I was given by Shnirelman 11.571 who introduced so-called "minimal" flows where the vorticity of the flow is effectively mixed through transportation via the Euler equations so that further mixing is impossible. Shnirelman conjectures that all generic 2D Euler flows have a similar asymptotic behavior as t -+ cc and every flow tends to some minimal flow. Such minimal flows are not stable but rather "compactly unstable". It is an open and challenging question as to the relationship between Lyapunov stable flows. Arnold stable flows, minimal flows and spectrally stable flows (i.e., no unstable eigenvalues in Spec L). It is conjectured that those flows which are neither Arnold stable nor spectrally unstable, are nonlinearly unstable in the space J I , but the nature of their instability is different from that of linearly unstable flows.
8. Astrophysical applications
Hydrodynamic stability questions naturally arise in the astrophysical context and had attracted a lot of attention of mathematicians and ~~strophysicist alike. In order to build an adequate astrophysical stability theory, one needs to incorporate effects of rotation. compressibility, and gravity into the model. There is a vast and growing literature on the subject which can be traced back to the works of Riemann 11461, PoincarC 11401, Lyapunov, and others. Limitations of space make it impossible to give even a brief overview of all the relevant astrophysical stability problems and their solution methods. Accordingly. below we restrict ourselves to two representative examples. For a general discussion we refer the reader to the well-known books and review articles by Cox 1221, Schutz 1 IS?], Tassoul [ 1601, and Unno et al. [ 1681.
Axisymmetric differentially rotating stars play an important role in astrophysics. In this section we briefly summarize their stability analysis presented by Lebovitz and Lifschitz 1103,1041, and subsequently extended by Faierman et al. 1521, and Faierman and Mijller 1.531.
General equations governing the evolution of a compressible star occupying a finite domain V,can be written in the form
are the fluid velocity, density, pressure, and gravitational potential, reHere V, p , P, spectively; y is the adiabatic exponent, and r is the universal gravitational constant. These equations are supplemented with the usual initial conditions for V, p, P, and appropriate boundary conditions on the free boundary ilV,. The exact nature of the boundary conditions is not known. In many cases of interest we can use the simple impenetrability boundary condition of the form
Steady axisymmetric differentially rotating stars are described by solutions of Equations (229)-(232)of the form:
where ( r .0 . :) are the standard cylindrical coordinates. The corresponding equilibrium conditions are:
I R
-,.n- + - h,,.- w,.= 0. 7
1 P,,;- W;= 0 . R
-
Closed-form solutions o f Equations ( 2 3 5 )are available only in exceptional cases: in general, they have to be solved nutnerically. Below we assume that a certain equilibrium solution, representing a differentially rotating star, is given and concentrate o n studying its stability. To a n a l y ~ ethe stability of a differentially rotating star we have to investigate the evolution of the small perturbations v, jj. 11, $J o f the velocity. density, pressure, and gravitational potential. However, rather than doing so directly, in is more convenient to express 6,p in terms of the so-called Eckart variables ti1 and 17:
p=
+
R(m C
11)
11 = R C n . '
Jm is the local speed of sound (see Eckart 1401).The linearized equations
where C = forv,m. n, @are
+
where D/L)t = i)/i)t Qa/ao is the convective derivative along the equilibriutn velocity field n(l-, :)en, while the functions ~i are given by
and K
iq
the integral operator of the form
Tht: corrcaponding initial and b o l ~ r ~ d icondition\ i~y are
Symbolically wc can wrile the linearized problem in the form
where e ia n tive-component vector Function. c = ( v . 1 1 , . t r ) = ( I : " . o". 1%:. 1 1 1 , 1 1 ) . and C is a sylnlnetric hyperbolic integro-dift'esenti~~l operator dctined by the differential expression o f the form
and boundary conditions (244). Here
[;;; 1 ci2]
AH= -
n
o
0
0
O
0
n
O
O
C
O
O
R
The ;lpp:~rent asymmetry ot- the matrix A" is due to the fact that we use contravari:int components (11". 1 1 : ) ol'the velociry fielcl v. Since the coct'ficicnts of' the evolutio~icrpc~.;~tor; ~ n ~~ lh bouncl;iry c conditions arc timeancl angle-indepcndcnt. wc ci111corisiclcr pi~rticuli~r. solutions of the form' 11".
whcrc y = ( I . . :) arc coo~xJi~ia[es it1 rhc poloidul cross-sccrion %,, ol.thc domain 'P occupied by the sku.. i~nclX = 0. ir I . f2. . . . . For brcvi~y.hclow we onlir prirncs. Thc col,rcspoilding specrl.ul problem for e ( y ) has thc form
where Cn is w i intcgro-differential operator defined by thc dii'l'crential expression ol' ~ h c t'ol-in
and boundary conditions (244). 'LC~O~II/,
itnd l,~l'\chiu[
1031 LISC
I[II
~itllcrIII~III A
i15 ;I
\pcclr~il~xirki1~1cLc1.
Although a complete spectral analysis of the operator Ck is far beyond our current technical capabilities, we can find some points belonging to its essential spectrum by constructing the corresponding Weyl sequences of quasi-eigenfunctions (cf. Lifschitz [I 171 for a detailed discussion of the Weyl method). Let yo = ( r o ,zo) be a point in the poloidal domain V p , and I = (cos v , sin v ) be a unit poloidal vector. We define the 2 x 2 symmetric matrix of the form
By constructing a properly oriented Weyl sequence localized in the vicinity of yo, it can be shown that the following dispersion relations
determine non-isolated spectral points h associated with yo, I. In general, these spectral points can be complex. We introduce the following notation
and distinguish the following cases: ( A ) The purely real case. the matrix M is non-negative, 0 < /~,,,i,, (B) The purely imaginary case, the matrix M is non-positive. jr,,,;,, yl, 0; (C) The mixed case. the matrix M is neither non-negative nor non-positive. p m i n < 0 < l ~ l l l iy,,, ~ . I t is clear that the spectrum of the problem (253) can be purely imaginary only if the matrix M ( y o )is non-negative for all points yo in D,,. By using the energy estimate derived in Lebovitz and Lifschitz 11031. one can prove that the non-negativity of the matrix function M is necessary for stability. By using the classical Sylvester criterion we can represent the corresponding necessary stability condition in the form of two inequalities,
<
The latter condition is the generalization of the classical Rayleigh stability condition (31) for differentially rotating fluid masses with explicit ,:dependence. If at least one of the conditions (259), (260) is violated, the star is unstable with respect to localized perturbations growing exponentially fast. The above stability conditions can also be derived (and, in fact, strengthened) via the geometrical optics stability method. We refer the reader to Lebovitz and Lifschitz [I031 for further discussion.
Instabilities described in this section are partly due to the interplay of the effects of con~p~.essibility and gravity. For further discussion of the compressibility effects in the purely hydrodynamic context, see Eckhoff and Storesletten 1441, Leblanc 1981, Le Duc arld Leblanc [ 109). among others.
The Riernann ellipsoids are the only known Family of incompressible rotating stars that depart from axial symmetry. Due to this fact they played, and continue to play, a consistently important role in the theory of rotating stars. Their stability had been the subject of many investigations by Riemann [ 1461. Poincare 11401, Cartan 1201, Chandrasekhar 1241, Lebovitz [101,1021. and others. In this section we briefly sum~narizerecent tindings of Lebovitz and Lifschitz 1 105,1061, who extended the classical normal mode stability analysis of the Riemann ellipsoids to ellipsoidal harmonics of order tive, and complemcnted i t with thc geometrical optics stability analysis of these ellipsoids. General equations governing the evolution of an incompressible star occupying n finite rlonlain 'Dl can be written in a coordinate franie rotating, with respect to an inertial frame, with angular vclocity I?, in the form
Thcse equations are supplcnicnted with thc U ~ L I Linitial ~ conditions for V, P , ond kincmatic and physicill boundary conditions on thc Il.t'rt boundary ; ) D r The . kinematic condition requires that particles which are initially on the boundiiry rzniain on the boondory; (tic physical condition requires that thc pressure vanishes there:
V ( x . I ) . n(x, t ) = 0 and
P(x. t ) = 0
on i)'Pl.
(264)
The 3-type Kie~nannellipoidb lijrni a family 01. stcady s o l u t i o ~ ~ofs Equation (261 )(264) for which the domain ;T, is an ellipsoid with the semiilxes ( 1 1 . (I?. 113. the velocity U is a linear function of x.
the pressurc 4)is a quadratic function of x,
and the angular velocity 19 is directed along the xn axis, tP = 19en.~The non-dimensional ratios (u2/al, LIR/UI) define the constants 8, m , noup to an interchange andlor a simultaneous sign change of I9 and m . The interchange 19, m + m , I9 replaces an equilibrium by its adjoint equilibrium which is characterized by the same shape but a different velocity field. Configurations with ( m / 8 ( < I , j m / 6 ( > 1, (m/O/ = 1, are called direct (Riemann), adjoint (Dedekind), and self-adjoint, respectively. Without loss of generality we may assume that I9 0. Not all choices of senliaxes ratios correspond to an S-type ellipsoid, the 1, admissible configurations occupy a horn-shaped domain in the unit square 0 < u2/ul 0 < c ~ ~ / L I ~ I . A detailed description of S-type ellipsoids, including an extensive bibliography, is given by Chandrasekhar [241. The stability properties of the Riemann ellipsoids are governed by the linearized equations of the form
>
<
<
Here v and are the usual Eulerian perturbations of the velocity and pressure. while <,, is the normal component of the Lagrangian displacement 5 . For convenience, the unit of time is chosen as ( n G p ) - ' / 2 .By necessity, the linearized equations involve the Lagrangian displacement. In principle, i t is possible to eliminate v in favor of 5 via the relation
see Lebovitz 1021. The boundary condition of vanishing pressure has the for111
The geometrical optics stability method augmented with an appropriate energy estimate is powerful enough to handle problem (267)-(27 1 ). In fact. the Lagrangian term does not cause serious practical difficulties. The actual calculations follow the general pattern of Section 6.1: details are given in Lebovitz and Lifschitz 11061. In addition. the normal mode analysis which we used in Section 4.2 for studying the stability of linear flows in ellipsoidal domains can be generalized for studying the stability of Riemann ellipsoids, although the actual technical details are very involved, see Lebovitz and Lifschitz [ 105j.
9. Conclusion and open problems
In this article we have discussed how WKB asymptotics related to geometrical optics techniques can be used to detect instabilities in ideal fluid motion. We have described how 3 ~ e h o v iand t ~ Lil'hchit~1105.1061 use the notation
A. 4, rather than
(0.
cY. n . no
Ln)ccili:ed it~strrbilitir.~ it1 fluids
347
this approach, which is based on localized high spatial frequency disturbances, tits into the classical framework of spectral and energy methods applied to fluid stability theory. These methods complement each other and make it possible to claim that in some appropriate sense "all Euler flows are unstable". It is clear that there are several different kinds of instability. In fact, the differences are so big that one could argue they deserve to be regarded as different phenomena although they all satisfy the basic concept of instability, namely the ultimate growth in some norm of a disturbance that is initially small. We have emphasized that the norm in which such growth is measured is a very important ingredient in evaluating the instability. When there exist discrete unstable eigenvalues in the spectrum of the Euler equation linearized about a given steady flow. there is "fast" linear instability and a disturbance grows exponentially with time. The non-standard nature of the linearized Euler operator for general flows (namely, it is degenerate, non-elliptic, and non-self-adjoint) rules out the application of standard general theorems to deduce the existence of eigenvalues. Rather, the problem must be tackled on a case-by-case basis for particular flows. To date there are only very few examples in 3D and none for fully 3D flows where the existence of discrete unstable eigenvalues has been exhibited. Clearly it would be important to obtain such results for more examples, particularly those that model physically realistic fluid behavior. On the other hand, geometrical optics techniques reduce the instability problem to the consideration of the growth rate of an "amplitude" determincd by a system of ODE. As such the problem becomes considerably more tractable than the original problem posed in terms of PDE. This growth rate detects "slower" instabilities associated with the linearized Ei~lerequation. The growth rate rnny be exponenti~rlor [nay be "very slow" algchraic growth. A positive exponential growth rate implies that the unstable continuous spectrum of the lincari~cdEuler equation is non-empty. Such o positive exponential growth rate occurs in rather generic flows (e.g., any flow with ;I hyperbolic point). In this casc the spectrum o f the evolution operator tills an annulus. For each point of the spectrum we ciun construct :I solution of the linearized Eulcr equation which grows in time but not necessarily monotonically; rather, it may have "bursts" at some moments i n time and yet be srnall no st of the time. Hence such an instability is a different and weaker phenomenon than an eigenvalue instability. I t is natural to ask if instability in the linearized Euler equation implies instability in the full nonlinear Euler equation. We have described results that give a positive answer to ~r this question when there exist discrete eigcnv;~lucs.i.e.. the casc of'"t'i~st"l i n e ~ instability. However. in 3D nonlinear instability is an open and challenging question when linear instability is of the weaker type associated with the continuous spectrum and detectable by geometrical optics ~nethods. Localised instabilities can also be used to investigate the growth of a perturbation of an unsteady state: although, except in the context of time periodic basic states. the issue arises as to how to differentiate between an evolving basic state and the growing perturbation. We have discussed certain specific related problems including the connections between the "amplitude" ODE and Kelvin modes. rapid distortion theory and secondary instabilities. These are further indications of the power of geometrical optics techniques to probe the intricate behavior of ideal fluids.
348
S. Friedlander and A. Lipton-Lifschirz
The stability/instability of fluid flows is a classical subject with many impressive achievements and well developed methods. However, there remain significant unsolved problems. The very different types of instability discussed in this article suggest that the whole concept of instability should be revisited and physical intuition incorporated to classify a "graduated scale" of instabilities and to examine their implications for fluid behavior.
Acknowledgements We are deeply grateful to our collaborators on our work in connection with fluid instability, namely Misha Vishik in the case of the first author and Eliezer Hameiri in the case of the second author. We thank Stephane Leblanc for his helpful comments concerning this survey article. Susan Friedlander's research is supported in part by the NSF through grants DMS-9970977 and DMS-0202767.
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[ 1 121 Y. Li. On 2 0 Eulrr eqrctrtioir.~:The Energy-Cusimir irr~tuhiliti~.~ untl rhr .sprcrru,fir linrtrrizrd 2 0 Euler
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[I731 V.A. Vladirnirov and K.I. Ilin. 0 1 1 Aurolrl'.~~~trritrtiorrrrl prirrc.i/)lr,.s in j u i d mzc,htmir..s,The Arnoldfest: Proceedings of a Conference in Honour of V.I. Arnold for his Sixtieth Birthday, E. Bierstone, B. Khesin, A. Khovnnskii and J.E. Marsden, eds. Amer. Math. Soc.. Providence. RI (1999). [ 1741 V.A. Vladimirov and D.G. Vostretsov. Irr.\ttrhility o/'.srrrrrly flows icitlr (~orr.srtrrrt~,orrir,ity irr ~ ~ e . t . s(!/'t~lli/)tic. ~ls c,ro.s.s-.tc,ctio,r. Prikl. Mat. Mekh. 50 ( 1986). 369-377. [ 1751 F. Waleffr. 011tlre t l ~ , r c ~ - t l i ~ ~ r c ~ rirr.sttrhili?\. r . ~ i o ~ ~ t ~(!f'.\trrrir~rd l ~,oritic.c,s,Phys. Fluids A 2 ( 1990). 76-80. [ 1761 F. Waleffe, HH\rotlyrrrrrrtic. .\ttrhiliry trrrtl trrrhrrlorrt~r:Beyorrd trrr~r.sirrrt.sto r r .srl/'-sustrrrrirrl: /)roc.r2.s.\.Stud. Appl. Math. 95 (1995). 3 19-343. [ 177 1 W. Wolibner. UII t l r e o r r ~ r ~.cur ~r~ I'c~.ri.ctc~~r(.c~ tlrr I ~ I ~ I I I ~ P I I/)l(rrr I ~ ~ I d'lrrl I~ ,fltrit/e /)rrr/rrif, I r o r r r o ~ ~ ~i~rr~orrr/)rr~ssr~r~, ihlr. /)r~rrtltrr~t r r ~ rtr,rrt/).s ir!firrirrtr~~~r lorr'q. Math. Z . 37 ( 1933). 698-726. (17XI E.I. Yakubovlch and D.A. Zenkovich. A rrtrrtri.~rr/)/)rooi'/r to Ltrgrtrrr,yitrrr ,flrrid c!\'rrerr~ri~.v, J . Fluid Mech. 443 (2001 ). 167-1 96. [ 1791 Z. Yo~ihida.Hc~Itrtr~rri lirrrc.tiorrs: A /~tr,rrrli,qrrrof'.strc.rrrrrli~~r c.11r1o.s.Curr. Topics Phyh. Fluids 1 ( 1994). 155178. [ 1801 V.I. Y~~dovich. E.I(IIII/)/~, ~~f.\r,r.ortt/(rrv ~t~1ti011~11~v f i o ~o~-/)r,riotlii.,fio\~~ , rr/~/)r,orirr,yb~,lriIo11 1t1111irrtrr f l o t ~(!/'(I , ~,is(.orrci r r ~ ~ o r ~ t / ~ r cflrrid ~ c . ~/(fic3.c i l ~ l tit.\ ~ .\trr/)ility. Appl. Math. Mech. 29 ( 1965). 4 5 3 4 6 7 . fijr l\rc2 Errlrr c,f/rrr~rioi?.\.Dinamika Sploshnoi Srcdy 16 ( 1074). 7 I( I81 V.I. Yudovich. 011/f~.\.s of'\rirr~otlrr~r\. 78. [ I82 [ V. I . Yudovich. 7'llc~Li~~r~rrri:rrtio~r M r , t l r ~ ~irtt l H\nlrr~c~\~rr~rrrric~r~I Strrhili!\. Wc,or:v. A~nct-.Math. Soc.. Providence. R1 ( 1989). [ I83 [ X. Zhanp 21nd A.L. Frenkel. Ltrr#c,-\r.rrlc, irr\t(rI)ilrt\~r~/',yr~rrr~r.oli:l..d ovr~ill(rtirr,qK ~ ~ l r ~ r o , q o r - o ~ , SlAM f l o ~ ~ ~J..\ . Appl. Math. 58 ( 19')X). 540-SM.
CHAPTER 9
Dynamo Theory Andrew D . Gilbert
.
Sc.lrool of Mtrtlrc~rr~otie~rrl S(.ic~rrc.c,.\.Urri1.c.v.$i!\ c!fE\c~/rr: EX4 4C)E UK E-rricril: (r.tl.~ilhc.,~Cn'c.~.oc ..rrL
COllt~11t~ I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 2 . Goverl~illgcqu;itielli\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 2.1. Maxwell'\ cqu;~tiolh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 2.2. Kcl;~tivi\tic ~IIV;I~~;IIICC ;111dII;II~\~~)I-III;I~~OII propertic\ . . . . . . . . . . . . . . . . . . . . . . . . 302 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 2.3. TIlc i ~ ~ i l u c r i ccqu;~rioli lr~ 2.4. I3ou11tlal-yc o 1 1 d i 1 1 \ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 . .I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 2.5. 1 c i 1 1 i 1 1Ii I I I I I I ~. ~ ?.(I. Ollicr g c o ~ l ~ e t r i c \. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 3 . I)cc.iy . ;III~~-L~~II;IIII(I IIICO~CIII\ . s~~ III>II~~ l l I hu\111d\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 3.1. M;~gnctic licld ~lc.co~lllx,\~ 115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 3.2. 1)cc;ly I I I O ~ ~ \ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 . . . . . . . . . . . . . . . . . . . . . . 275 3.3. AII~I-C~~II;III~(I~IIC(II-CIII\ in ~>criocI~c (';irtc\i;~~i~COIIICI~Y 3 . 4 A l ~ t i - c l y ~ l~IIC(II.~III\ i ~ ~ ~ ~ o ill \IICI.IC;~I g e o ~ ~ ~ e.t r.y . . . . . . . . . . . . . . . . . . . . . . . . . . 378 . . . . . 3.5. IIlll>cr h o u ~ l J \ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3x1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3x3 4 . I.it111i11.1r ~I~II;IIIIII\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3x3 4 . I . P u n c ~ ~ i l a r c ~tlyna~llo ~hrl 4.2. Sl~lootliP o ~ l o ~ i ~ a l I- cdyli;lll~o ~ ~ h ( . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 4.3 G . 0 . Kohcr[\ ~ l y ~ ~ ; i l lM i cu~l r: ~ l l l c\vale ;111:1ly\i\ . . . . . . . . . . . . . . . . . . . . . . . . . 302 4.4. G.O. Kohcl-t\ r1yn;1111olor l;irgc K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .307 5 . 7'11~alpha cllcct ;i11r1tlylla~lirlIII~CICIIIII~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 5.1. Tlic a l p l ~ ;cllcct ~ 5.2. A l p h a - c ~ ~ ~ ~~I~II;IIIIO\ cga ;III~ cly11;11110W;IVC\ . . . . . . . . . . . . . . . . . . . . . . . . . . 407 5.3. The Solar dy11;1111o. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . JOX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 5.4. ('o~~vcctivc ~~II~IIIIO ;111d \ 1Iic ~~(I~~~II;IIII(I 5.5. Galactic and ;~ccrctiol~ d ~ \ cdylialnc~s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 l h (3 . d y ~ i a ~ ~ i.o .s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 0 . 1 . D c l i ~ ~ i t i oa~~i l\t upper l hou~lrl> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .417 (1.2. Stretch . tw15r and told . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .418 lo ill flow\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 6.3. F;I>I d y n a ~ ~;1crio11 6.4. Up[xl- ho~llltls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 6.5. Flux coyjcctu~-c\. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
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H A N D B O O K OF M A T H E M A T I C A L F L U I D D Y N A M I C S . V O L U M E I1 Edited hy S.J. Friedlander and D . Scrrc O 2003 Elsevicr Scicllcc B.V. A l l right5 rc\crvcd
356
A. D. Gilherr
6.6. Nonlinear fast dynamos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 7. Open issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
Abstract Dynamo theory concerns the generation of magnetic tield from the flow of an electrically conducting fluid, relevant to the magnetic tields of the Earth, Sun, planets, stars and galaxies. The review focusses on fundamental dynamo mechanisms. The induction equation is derived as an approximation from Maxwell's equations and boundary conditions are discussed. Antidynamo theorems and upper bounds are considered, followed by discussion of some basic models, the Ponomarenko dynamo and (3.0.Roberts dynamo. The application of dynamo theory to the Solar dynamo and the geodynamo is then reviewed, with the introduction of the alpha effect. and the concepts of alpha-squared and alpha4mega dynamos. Finally fast dynamos are considered. Keywords: Magnetohydrodynamics, Magnetic tields, Induction equation, Dynamo theory, Electromagnetism. Fluid flow
L)?namo throrv
1. Introduction Sir Joseph Larmor [ 1301, at a meeting of the British Association for the Advancement of Science, posed the question 'How could a rotating body such as the Sun become a magnet?' He gave three suggestions of how this might be achieved. In the first he refers to the motion of an electrically conducting fluid, plasma in the case of the Sun, and what happens if this flows through magnetic field lines: "Such internal motion induces an electric field acting on the moving matter: and if any conducting path around the Solar axis happens to be open, an electric current will flow round it, which lnay in turn increase the inducing magnetic field. In this way it is possible for the internal cyclic motion to act after the manner of the cycle of a self-exciting dynamo, and maintain a permanent magnetic field from insignificant beginnings, at the expense of some of the energy of the internal circulation." This quotation summarises the basic idea of dynamo theory, that in the flow of an electrically conducting fluid magnetic fields may be ampli tied i n the manner of an instability, hence the term .splf-ercitit~g.The explanation requires some seed magnetic field to begin with. but indicates the possibility of growth of field and currents, sapping energy from the fluid motion. The growth will occur until finally the tield is strong enough to affect the fluid Now through the Lorentz force. The growth must then saturate, and we are left with an equilibrated ~nagnetohydrodynarnic(MHD) system i n which the field and flow evolve dynamically on an equal footing. Such a system has Inany degrees of freedom, and wc are now fl~miliarwith the idea that such systerns can show complex hehaviour. In hydrodynamics, onc has only to think of 1111.bulence. or the weather. or just the strange attractor in Loren/.'s trio of ordinary dit'ferentiiil equations. derived as a truncated model of Huid convection. This complexity is evident in the behaviour of Solar and terrestrial magnetic tields. which show coniplex structure i n space and time, including complete reversals of polarity. This point was well appreciated by Larmor, who remarks that with a fluid mechaniciil origin changes in the rnagnetic held can be achieved by a flow causing the rearrangement of the electrical currents through the fluid. On the other hand with an explanation based on the microphysical material properties of the Earth or Sun i t would be difficult to account for the observed time-dependence. We now know that as well as the Earth and Sun, some planets. galaxies and Inany stars possess magnetic tields. and Lannor's explanation above is generally believed to be correct. However putting it on a fi nn mathematical footing took many decades. for a number ot'reasons. First, this explanation is not local, as it requires consideration o f how electric currents flow throughout the region containing the conducting fluid. Mathematically. this means the solution of a partial differential equation, the induction equation. in this region with suitable boundary conditions. The Sun and Earth have a high degree of spherical symmetry in terms of their material properties and so the electrical conductivity is essentially a filnction of radius. Thus any region of conducting fluid will take the form of a sphere or a spherical shell, in which we may approximate the conductivity as a constant. The geonletry is thus simple, but then to obtain amplification of magnetic fields. the fluid motion has to be complicated, making solution of the induction equation challenging. This should be contrasted with the situation in the laboratory, where it is not difficult to construct a self-exciting dy-
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A.D. Gilbert
namo, without the use of a fluid, by an appropriate arrangement and motion of coils of wire; an attractive thought experiment is the disc dynamo (see [ 144,1731). However, this results in the electrical conductivity being a complicated function of space, which is unrealistic for an astrophysical dynamo. We shall always have in mind homogeneous fluid dynamos, consisting of a region with simple geometry, containing fluid of constant conductivity. Secondly, it is very difficult to conduct experiments on such homogeneous fluid dynamos in the laboratory. The Sun and Earth are characterised by large length-scales. To allow a fluid dynamo to function at laboratory or industrial scales, it is necessary to stir a liquid metal. over volumes of the order of I m3 and with velocities of the order of 10 m/s. With considerable expense and engineering expertise it is now becoming possible to achieve growth of magnetic fields using liquid sodium as the conducting fluid 1421. Dynamo theory has certainly suffered from a lack of experimental input compared with other areas of fluid dynamics, and these experiments will be very welcome in stimulating future theory and testing ideas, particularly concerning the nonlinear equilibration of dynamos and the role of hydrodynamic turbulence. However, the immediate mathematical problem with developing Larmor's explanation was one of symmetry. or rather the need for symmetry breaking in a fluid dynamo. As we discussed above, the geometry of astrophysical objects is very simple, and this means that the fli~idflows required to amplify tnagnetic fields have to be complicated. Dynamo action, that is the amplification of magnetic fields. cannot be obtained if too much sylnlnetry is imposed on the magnetic field or on the fluid Row. The first result i n this direction was obtained by Cowling 1571, who showed that an axisymmetric magnetic tield cannot be susti~inedby fluid motion. This early anti-dynamo theorem suggested that Liu-mor's idea quoted abovc was probably doo~ned,especially since the Earth's tield is approximately axisymn~etric,with a dipole form. Perhaps it woi~ldbc only a matter of time before a generul result was proved prohibiting dynamo action i n all Hilid flows in regions o f simple geometry. I t was not until the nineteen fifties that positive results in dynamo theory began to emerge. Bullard and Gellam 1361 expressed the dynamo problem in a sphere using :I spherical harmonic expansion for the flow and magnetic tield. This leads to coupled ordinary differential equations in radius for the field, which after truncation may be solved numerically. Bullard and Gellam 1361 obtained dynamo action i n a simple flow, which acted as an important stilnulus to dynamo theory even thoi~ghthe results were later discovered to be incorrect: this How is not a dynamo, and the numerical resolution used was insufficient (Gibson and Roberts 1851).The first real proofs o f dynamo action were obtained by Herzenberg [ 108 I and Backus 19 1, who employed inter~nittencyin space and time respectively to allow mathematical analysis. In Herzenberg's model rotors, that is cylindrical regions of rotating fluid, are imagined embedded in a stationary fluid. If the rotors are small compared to their separation, then at each rotor the magnetic tield due to the other is approxi~nately uniform. The problem then becomes tractable, and magnetic held growth may be proved: it was also demonstrated in the laboratory. although using solid metal rotors in a metal block rather than a fluid (Lowes and Wilkinson [136.137]). In 191 flow tields that are intermittent i n time rather than space were introduced. During periods when the whole fluid is at rest the magnetic field decays with higher harmonics. having finer scales, decaying most rapidly. When the fluid motion occurs, this is sufficiently rapid so that diffusion may be
neglected. With careful tuning, it is possible to focus attention on just a few, large-scale tield modes, and with this simplification again dynamo action could be proven. The results of Backus and Herzenburg are important in establishing rigorously the existence of dynamos in fluid flows. The flows chosen are however rather artificial, and to explain astrophysical magnetic fields it is necessary to develop models closer to realistic convective fluid flows. Such models were developed initially by Parker 11521 who considered plumes and down drafts in the convection zone of the Sun. Hot fluid would rise and twist under the action of the Coriolis effect, leading to lifting and twisting of magnetic field lines. If one averages over many such eddies, one obtains a new transport effect with a coefficient denoted r by Parker ( 1521. This effect, combined with differential rotation, leads to magnetic tield growth and travelling waves of dynamo activity, in agreement with Solar observations. These ideas were developed independently by Steenbeck et al. 1 1971 who used the coefficient a ti)r the same transport effect, a terminology which has stuck to give us the rrlphrr qflfi~ct.At the sarne time Braginsky 124,251 developed related averaging methods for nearly axisynimetric flows and magnetic tields. These averaging ideas are precisely the right tool to use to simplify the dynamo problem. The true flows and magnetic fields are three-dimensional. and so Cowling's anti-dynamo theorem is not applicable. This three-dimensionality would make the induction equation very difticult to solve, but instead it is averaged away at an eiuly stage. to leave much simplified equations which can be solvetl numerici~llywith modcrate effort. or i n a t'cw cases ;unal ytically. These i~verugingideas were extremely succcssf~rland led to a surge ot' activity in dythc filndament:tl ideas clucidatcd by namo theory, which continues to this day. Altho~~gh of the Solar rlyni~mo. Parker 1 1.521 and Stcenbeck et ill. 1 1971 explain thc basic bchavio~~r it has proved much niorc difficult to refine these modcls. precisely b c c a ~ ~ the s c underlying motions and magnetic tields are really three-dimensional and complicated. The Solar convection lone is highly turbulent and basic ideas. as to how the dyniumo opcratcs ~uidwhere the magnetic field is loc:tted. hove been revised in only the last two decades Sollowing the results of hclioseismology. which probes the fluid motions below the Solar surfncc. The Earth's dyniumo presents specii~lproblems, as the fluid flows ;\re dominated by the Coriolis Moving effect. leading to thin boundary layers which are difficult to resolve ni~merici~lly. further afi eld. observutions f'ro~nsatellites have given inti)rm;~tionabout the magnetic tields of planets and the moons 01. Jupiter. Magnetic fields 01. nearby stars have been measured. and some have star spots covering signifi cant portions of' their discs. On a larger scule again. Faradi~yrotation of polari~ationplanes in the presence of magnetic fields allows measurement of galactic fields, :und this has led to the development of dynamo theory bascd on fluid tlow in thin discs, relevant also to accretion discs. Modelling the dynamos in thcse astrophysical bodies presents serious difticulties for the analysis of all but the most idealised problems: even numerical exploration is very limited and will remain so with foreseeable computer power. On the theoretical side. there has been niuch interest in fast dynamos, where the field adopts tine-scale structure and grows rapidly in time. The other ~najortheoretical problem concerns the nonlinear equilibration of dynamos. where there remains much uncertainty. What happens when the magnetic tield becomes strong enough to suppress or modify convective fluid motions? Finally with dynamo action now established in laboratory experilnents. new avenues are opened for theory and computations.
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It is impossible in a review such as this to cover all the areas mentioned above in any detail, and the reader will no doubt find many omissions. The focus of this review is on fundamental generation mechanisms, rather than applications and numerical modelling. We begin with basic material, deriving the induction equation from Maxwell's equations, and then discussing anti-dynamo theorems and upper bounds. We go on to consider the Ponomarenko and (3.0.Roberts dynamos: these show how generation of field can occur in simple flows and this leads to the introduction of the alpha effect. We survey application to dynamos in the Earth, Sun and galaxies, and discuss convective dynamos. We then return to basic dynamo mechanisms, giving some examples of fast dynamos and highlight the role of Lagrangian chaos in fluid flows. Finally we consider the nonlinear equilibration of dynamos and give a list of some open questions in dynamo theory. There are many excellent books and reviews that the reader may wish to consult. Books on dynamo theory and M H D include 158,126,144,153,170,218], and the collection [ 1661. For reviews of dynamo theory in general see [ 172-1741, while for Solar and stellar dynamos see [27,195,2141. The special features of the geodynamo are discussed in (72,l 10,1891. For more information on fast dynamos see 1 16.52.1921 and the monograph 1531.
2. Governing equations We begin by setting out Maxwell's equations and discuss their invariance under Lorent/. transforms (scc, for example. Landau and Lifshitz 1 1 381). For most astrophysical and geophysical applications, the time scale over which rhc electromagnetic held evolves is much longer than the time taken for light to cross the system. For example. the Sun's tield evolves over months and years, and the Earth's over decades and longer time scales. We may therefore approxilnate Maxwell's equations to obtain the induction equation. i n which electromagnetic waves are ti ltered out and the magnetic tield plays a central role. We then discuss boundary conditions and set out the dynamo problem. Note that in this review we use i), , i),., etc.. to denote derivatives. whereas subscripts, whether numbers or the letters .r, y . :, denote components of vectors or other labels. We occasionally make reference to [ I 1; for succinctness these references are given in footnotes as AS followed by the section or equation number.
2.1. Mtrxwell '.v eyutrtion.~
Maxwell's equations are:
Here E is the electric field, B the magnetic field, J the electric current, and p the electric charge. We will assume here and below that the magnetic permeability p and the dielectric constant E are constant, taking their free space values. The speed of light (. is defined by -2 (. - = E P . We also have the equation for conservation of charge
which may be deduced from (2. I), (2.3). We will eventually require a further equation, namely some constitutive relation between the electrical current J and the electromagnetic tield, for example, Ohm's law, but we defer consideration of this until we have further simpli tied the equations. From Maxwell's equations it is possible to isolate the energy and momentum of the electromagnetic field. Suppose we have a continuum distribution of charge with density p and velocity v. The corresponding current density is J = pv and the Lorentz force per unit volume is
The rate of working of the Lorentz force on the charge distribution is
Note that we could have several species of'charges. tor exi~mple,ions and electrons. with densities and velocities. sity / I , . v,,. In this case we would incorpodifferent (overli~pping) rate a s u ~ nover the different species labelled by cu. The energy density of the elcctro~nngneticfield is given by
and obeys the energy equation
+ v . P = -W
i ) , ~
(P = /
~ - I E
x
B).
(2.9)
This equation represents conservation of energy; the Poynting vector P gives the flux of energy in the field. while W ia the rate of transfer of energy to mechanical degrees of freedom. The momentum of the electro~nagnetictield may be identified with c . P ' ~ and there is a corresponding transport equation
where
is the electromagnetic, or Maxwell, stress tensor.
2.2. Relcrtivistic invariance and transformatinn properties In dynamo theory we are interested in the induction of magnetic fields in a moving, fluid medium. The properties of the medium will be important, in particular its electrical conductivity. As mentioned above, at some point we will need to relate the electrical current J at a point to the local electric field. To do this we need to move into a frame that is comoving with the local fluid velocity u at a point and apply Ohm's law (and possibly derive other physical effects) in this frame. We thus need to know how electric and magnetic fields transform in moving frames, and this is far from clear at the outset. Maxwell's full theory is relativistically invariant. rather than Galilean invariant, and so to derive the transformation laws for the fields we will in this section set up electromagnetic theory using 4-vectors. Let us consider contravariant 4-vectors with space-time coordinates written xi' = ( c t . x). where Greek indices run from 0 to 3. We also define the metric
and the covariant and contravariant derivatives
Suppose ;I 4-vector .\-Ii = ( c t ,x ) is given in a frame S. Then in a f'rame S' having a velocity u relative to S with aligned nxes coincident at t = 0. the vector has components .rfl' = (~1'. related by the Lorentz transformation XI)
with
Here subscripts I[ and Idenote components parallel and perpendicular to u. To set up the electromagnetic field in a relativistic form we tirst detine potentials A given by
C#J
and
(using (2.2). (2.4)).We can then introduce 4-vectors for the potential and electrical current
The electromagnetic field 4-tensor is defined by
and contains the electric and magnetic fields as components, explicitly
We may also compute Fl"' using the form (2.12) of the metric and write these tensors both in a convenient compact notation as F,, I , = (E. CB),
F1"' = (-E, cB).
(2.20)
Using this same notation the tensor dual to Fl"' is given by -
F I / U -
1Elr150 r F,, = 2
(-CB.
-E),
F,:,, = (CB. -E)
(2.2 I )
(with F / " ' " ~ the completely antisymmetric tensor satisfying F " ' ~ ' = I ) and here electric and magnetic tields become interchanged. The electromagnetic tield has two relativistic 5calar invariants F ( F I " , = 2 ( ( . ' ~ ' - E' 11
)
I.;,,,F*'"' = -2c.E . B.
(2.22)
Finally, Maxwell's equation5 become i l l , F1'" = .I "It'.
ill, F*Ii1'= 0.
while conservation o f charge (2.5) hllows from the antisymmetry of F1"
It'the Lorentz. gauge ill, All = 0 is adopted, then we ;1rc lel't with the wove-ccluntion ij,,
i ) I f A"
= .I
"/F.
(2.25)
Equations (2.9). (2.10) f'or energy and momentum may be si~mmarisedin another 4tensor equation. The electromagnetic stress 4-tensor is
or explicitly, using obvious notation with i . ,I = 1 . 2. 3.
A. D.Gilbert
364
and so incorporates the electromagnetic energy E (2.8), Poynting vector P (2.9) and Maxwell stress tensor Tij (2.1 1 ). Equations (2.9), (2.10) amount simply to
a,, rlLU = -,f ".
(2.28)
where f " = (Wlc, f) is the 4-force acting on the fluid. From applying the Lorentz transformation (2.14) to the electromagnetic field tensor Fl"', we finally obtain how the electric and magnetic fields transform:
+
EiI = Eli.
E; = y ( u ) ( E _ ~ u x B),
(2.29)
BiI = B I I ,
B; = y ( u ) ( B ~ -u x E/c~),
(2.30)
P' = Y(U)(P- J . u / ( . ~ ) ,
J;, = y(u)(JII- u p ) ,
(2.3 1 )
J; = J ~ .
Note that the transformation properties of E and B are complicated because they form the components of the 4-tensor Fl"' rather than parts of individual 4-vectors.
We will not he sing the full weight of Maxwell's equations, but a simpler system in which electromagnetic waves are filtered out. The key approximation is that velocities in the system under consideration should be much smaller than the speed of light (.. We suppose that the system has typical spatial scale C. time scale 7, and velocities u with
where -- is here used to denote order of magnitude. In this case Equation (2.4) gives the estimate
This approximation means that the displacement current / ~ s i l , E= c.-'ijf E may be dropped smaller than V x B. This leaves us with the from (2.3) as it is negligible, a factor v2/c2
prr-Mtrxwell rqc1tltio1l.s
Using (2. I), (2.3) gives the estimates
L ) y t t c r ~ t ~theory o
which means that (2.5) may be approximated by
and the Lorentz force reduces to
The key estimate (2.33) means that the electric tield drops out of the electromagnetic energy and the stress tensor, leaving us with
The energy equation remains unchanged,
but the Poynting vector drops out of (2.10) and this becomes sadly nothing more than a vector identity. In this approximation the electromagnetic tield possesses energy. but carries n o momentum. Now we return to Ohm's law. We have i n mind a moving fluid with a velocity field U ( X , I ) . At :I given point and time. we go into :I comoving fr;imc, velocity u. where we mensure modified fields given by
1 . In this frame we within the current approximations. using (2.29)-(2.31 ) with y ( 1 1 ) wish to relate the current to the ambient electric and magnetic fields. and the simplest relation to impose is Ohm's law J' = rf F,'. where rr is the electrical conductivity of the medium. Back in the original frame we have J = n ( E + u x B). This is Ohm's law in a moving medium, and a key part of the dynamo process: motion across a tield can generate a perpendicular current. This relation is used very widely in dynamo theory, but is only an approximation. whose validity depends on the material making up the moving fluid. We will not do so, but at this point one could include other effects. For example, the process of arnbipolar diffusion, important in galactic dynamos. gives a term of the form (J x B ) x B on the right-hand side of (2.45) (see, for example, 130.2231). The Hall effect is considered by Galanti et al. [sol, which gives rise to an extra term proportional to B x J.
A. L). Gilbert
366
From Ohm's law (2.45) and the pre-Maxwell equations (2.36), (2.37), we may deduce the induction equation
&B=V x
(U
x B) - V x (qV x B).
(2.46)
This is the fundamental equation studied in astrophysical and laboratory MHD, and
is the rnagtletic d1fiit.sivits. It is important to note that high conductivity corresponds to low diffusivity and vice ver.sa. We must still impose V . B = 0, although note that if satistied at some time t it must be satistied subsequently, by taking the divergence of (2.46). Equations (2.34). (2.45) give the electric charge with V . ( u x B) = -PIE, but this is now decoupled and of little importance to us. Let the total magnetic energy in a bounded volume V be written as
then using (1.42). (2.45)we obtain iI,Ev + j i 1 L L , n ( E x B ) d S =
-/II~
J; I J ~ ' ~ Vlru - J x BdV. -
(2.49)
where i ) V is the surface bounding V and n is a tield ofoutward. normal unit vectors. On the right-hand side it may be seen that magnetic energy is lost through Ohmic dissipation (tirst term) and exchanged with kinetic energy through working by the Lorentz force (second term). The surface integral on the left-hand side represents the Poynting flux of energy out of v . The induction equation is closely analogous to the vorticity equation. If we assume incompressible How
as we shall do from now on, and constant diffusivity r/ or conductivity under consideration, then we may rewrite (2.46) as
rr
in the region
This is similar to the vorticity equation with B replacing vorticity w = V x u, and now decoupled from the Huid flow. In the equation are terms representing transport, stretching and diffusion o f the magnetic tield. In the limit 11 = 0 of a perfect conductor (a = oo) the magnetic flux through a material surface is conserved, as similarly are circulations in inviscid fluid flow (see, for example, Batchelor [ 141). Mathematically the terms u . VB - B . Vu represent a Lie derivative, and indicate that the magnetic tield is Lie-dragged in the fluid
flow for 11 = 0: magnetic vectors evolve like the displacements between infinitesimally close fluid elements. Note one difference between vorticity and magnetic fields, that dissipation at the microscopic level comes from currents J = p-' V x B in the case of magnetic ijiui) fields, but through the square of the symmetric rate-of-strain tensor r,, = ;(ij;u, for fluid flow. Finally note that if B is written in terms of a magnetic potential B = V x A, then (2.46) may be uncurled to yield
+
where F is a scalar field reflecting the gauge freedom in defining A. Note that uncurling an equation such as this in general only gives a locally defined scalar field F. However, in a space for which all closed curves may be continuously contracted to a point, such as a sphere or spherical shell, F rnay be defined globally (that is, as a single-valued function). In a space that is periodic in one or more directions not all closed curves are contractible, and this requires care in defining such potentials; in dynamo theory a global definition of such a scalar is usually possible provided there is no mean magnetic tield in the direction of periodicity.
I t is often necessary to consider situations in which the medium undergoes a change i n physical properties across i t surttce S . We consider only the simplest situation in which S is a tixed closed surlhce. We take there to be no flow across it. n . u = 0.whcr-e n is a lield of o u t w a r d - p o i normal unit vectors o n S. We s ~ ~ p p o the s e discontin~~ity occurs only in the conductivity cr. or equivi~lentlythe diffusivity 1 1 . which is much the most impo~.tiuit situation in dynamo theory. Exccpt forju~npsat such surhccs. we take rr or rl to be constant in each region o f space from now on. If we integri~tethe pre-Maxwell equations (2.34)-(2.37) over hmall volumes and contours cutting S. we obtain the jump conditions
Here [ . I represents the value just outside S minus that just inside, and ps and J,y are possible surfitce charge and current distributions. Let subscripts 'i' and 'e' denote values inside and outside S. and to fix ideas suppose that the conductivity inside S is tinite and non-zero, 0 < a, a < m. The nature o f the boundary conditions then depends on the conductivity cr, outside. We will also let V, be the volume inside S, and V , the volume outside, leaving V to represent a general volume. specified in the context used.
-
If the external conductivity a, is finite, then it is impossible to maintain a surface current
J s (Such a surface current could only arise from the limit of an external perfect conductor, in which the current J cc just outside S as a, -+ 0.) Given Js = 0, we are left with five scalar boundary conditions (2.54)-(2.56). Of these only four are independent: in the original pre-Maxwell system, if (2.35) is satisfied initially it is subsequently from (2.37) and this dependence is carried over to the boundary conditions. In the absence of such surface currents, boundary conditions (2.55), (2.56) guarantee the continuity of the normal component of the Poynting vector field P across S (see (2.42)), and so the conservation of energy across S . It is useful to write down an equation for the total magnetic energy. The magnetic energy in any volume V is still governed by (2.49), which was derived allowing for variable conductivity. Provided the magnetic field falls off sufficiently quickly at large distances that the surface integral tends to zero as V is enlarged to include all of space, then we obtain
2.4.1. I t ~ . s ~ ~ I ~ i thI 'ot i~g t ~ ~ k ~ t : \ ' ~ ~ o t ~Ad iparticularly t i o t ~ . ~ . important limit is that of an insulator outside S, for which n, = 0 or 17, = m. modelling, for example, the poorly-conducting mantle outside the Earth's liquid metal core. I n this case J = 0 in V , from (2.45) and so B is a potential tield, determined from a scalar potential B = V X with V'X = 0 from (2.35). Any How u in V, is irrelevant. Conditions (2.54). (2.55) (with J s = 0 ) require all components o f ti eld to be continuous across S.
If S is a sphere of radius t i and the exterior tield is required to decay at intinity, as we shall always assume for insulating boundary conditions, then ): may be written as a spherical harmonic expansion in spherical polar coordinates (r-. 0 . 4 ) .
Here Y/,,, ( H , 4) are spherical harmonics
Y I ~(~0, . 4 ) = ( - 1 )Ir1
4n(l
+m ) !
normalised so that
(fi g
--
SinSn
) ~ . ~ / g sin H do d d . &I=o e=o
P,,,, (COS H )ol"'@
The term I = rn = 0 in (2.59) is omitted to avoid a net flux of B out of S , which is forbidden by (2.35). Note that at large distances
If on the other hand S is the plane external potential field is given by
,: = 0.
with exterior z > 0, then the spatially decaying
r where we have taken the layer to be periodic in x and y with periodicity length 2 ~ L. In considering conservation of energy we have to be careful, as in the limit of an insulator the current J and the conductivity a, = ( l l e P ) - ' tend to zero together i n V,, the region outside S , making the tirst integral on the right-hand side ot'(2.57), the Ohmic dissipation, potentially ill-def ned. However, if we consider the evolution ot' the magnetic energy i n the external region V, by differentiating (2.38) for V = V, with respect to time, and using J = 0 . (2.36) and (3.37). it nay be checked that the rate ot' change ot' energy i n V, is equal to the Poynting flux of cncrgy into S.with no contribution from Ohmic dissip;~tion.Since this flux is continuous across S as discussed above. combining i t with (2.49) for V = V, yields an equation for the total magnetic energy t: in the system
Here the sources and sinks of energy o n the right-hand side ;ire only non-zero inside S. in V,. whereas the energy I v includes that from tield inside ancl outside S.
2.4.2. Por-:fi~.r/y c~orrtluc~titrg boirrrtltrr:~c.orrt1itiorl.s. At the other extreme. the region V , outside S could be a perfect conductor with a, = GO. or equivalently 11, = 0 . There will then generally be a surface current Js on S determined by (2.55). which leaves (2.53). (2.56) as the only useful boundary conditions. Obviously ;I magnetic field can be trapped in the perfect contluctor outside S,iund will be static if there is n o How. from (2.5 1 ) with i l c = 0 . If however there is n o magnetic field outside S initially. then this will remain the case and from (2.45) there must also be no electric tield. E = O outside. Then using n . u = 0 on S and (2.45). the boundary conditions (2.54). (2.56) to be applied to the tield just inside S become
in general, or
if S is the plane
z = 0, and
if S is the sphere r = u . For the perfect conductor we again have to treat energy with care, as we have a limiting surface current in a region in which the conductivity tends to infinity, giving an u priori ill-defi ned Ohmic dissipation in (2.57). Here we should use Equation ( 2 . 4 9 ) for a volume V = Vi lying just inside S, that is, to exclude any surface currents. Since the electric field E must vanish outside S, n x E must vanish just inside S by ( 2 . 5 6 ) and so the Poynting flux out of Vi vanishes to leave the energy equation
where V, is the interior of S. Any surface currents are not implicated in the energy dissipation.
We shall first give a very specific definition of a kinematic dynamo and then relax the geo~iietricalframework. while considering the essential features that need to bc retained. Let S bc the sphere 1. = tr containing conducting fluid, with volumc-preserving Row tield U ( XI.) , cond~ctivityCT i~nddiffusivity 11. Suppose all 01. space outside S is insulating. iund that the tield decays as r 4 co.From the above discussion we have to solve the induction equation ( 2 . 5 1 )with some initial condition and the boundary condition ( 2 . 5 8 )that the tield be continuous with the external potential field (2.59). We may then evolve the magnetic tield in time and measure its energy
The Row U ( X 1. ) is a clyr~trrnoif. for some initial condition and diffusivity > 0 . the energy ) does not tend to zero as t + co.We can usually define a dynamo growth rate ~ ( 1 1 by
and say that the flow u is a dynamo if y(l1) 3 0 for some 17 > 0 . Here Bo is the initial condition. taken to have finite energy, that is, i n L'. Note that ~ ( ' 1 )= 0 is the marginal case: in a full nonlinear system with fluid motion coupled through the Lorentz force this would represent a bifurcation from a non-magnetic state with a flow u and y ( q ) < 0, to some other state, mediated by magnetic tield growth. Our focus at the moment, however, is linear, kinematic dynamo theory. When fully nonlinear dynamos are considered, the flow u is no longer provided and some mechanism is
needed to drive a fluid flow, for example, convective motion or a prescribed body force. This can introduce complications; for example, for a given driving there may be several possible stable fluid flows u, some of which may be dynamos and others not (see 1351, discussed in Section 6.6).
The above definition is very natural for astrophysical bodies having spherical symmetry, such as the Earth or Sun. However spherical geometry and insulating boundary conditions are not always easy to handle. Frequently one wishes to consider cylindrical, planar or periodic geometry, and other boundary conditions. In this case it is useful to have some guiding principles as to when a flow is a dynamo. We suggest the following: 1. The total magnetic energy El, ( r ) should always be tinite. 11. In the absence of a flow, u = 0, the energy E v ( t ) + 0 as t + oo for any 11 > 0 and any allowable initial condition Bo. 111. When the flow is present, E v ( t ) does not tend to zero as t + co for some initial condition Bo and some diffusivity 11 > 0. Condition I is designed to eliminate any sources of tield at infinity. Condition I 1 is there to avoid possible sources of magnetic excitation from field that is so~nehowtrapped because of the geometry o f the system and the boundary conditions. This sometimes may be expressed as a constraint on which initial conditions are allowable, as in the case of' periodic boundary conditions discussed below. The problem of the evolution of trapped magnetic fields is a sub,jec( in its own right lying largely beyond this review. We only mention that processes such as flux expulsion 1 155.21 31. topologic:~l pumping 1641. and generally the study of magnetoconvection 1165.1671 are important in understanding the behaviour of stellar and Solar magnetic fields. In some contexts the issue arises of how a dynamo operates when there is :in additional iumbient mean held present 1 1801. an example being possible dynamo action in the moons o f Jupiter. which are immersed i n Jupiter's magnetic field. The structure of magnetic fields in the presence of'a mean lield is also used to define the alpha e f i c t ; see Section 5.1 below. The geometry discussed above. of a sphere of conducting fluid bounded by an insulator. satisfies I given that the field is required to decay at large distances in the insulator (see (7.50)).111 Section 3.2 we shall see that in the absence of motion all mi~gneticfields decay. so that I 1 is also satisfied. Let us consider briefly how these principles apply to some other geometries. If the f o w u is n o n - ~ e r oonly inside a sphere S of radius r = r i but the conductivity is unifr)rm in all of space. then it is required by I that the tield decay at large distances. If the sphere is surrounded by a stationary pcrtkct conductor then to skttisfy I1 there must be n o field trapped in the perfect conductor. and so boundary conditions (2.67) must be applied o n S. A commonly used idealisation is to assume space is periodic in 1 , 2 or 3 directions. In this case the integral giving the total energy is taken over one fundamental domain V , and Poynting fluxes i n any direction of periodicity cancel out in the energy balance equation. For example, in a cylindrical geometry with coordinates (I-. 0 . :) (as in the Ponomarenko
372
A.D. Gilbert
dynamo, Sections 4.1 and 4.2) it is natural to take flow and field to be periodic in z.Similarly in a plane layer configuration with 0 6 z h the geometry may be periodic in x and with period 2 n L . If the layer is bounded above and below by an insulator the field is required to decay outside at large distances by I. For a perfect conductor outside the layer, the field outside has to be zero by 11; in addition the total fluxes through the sides of the periodicity volume,
<
J J ,
are constant under the induction equation. The only initial conditions that are therefore allowable for the dynamo problem are those for which these fluxes are zero, to avoid a trapped magnetic field which is forbidden by 11. If the flow is periodic in all 3 directions, then there are 3 constant fluxes, and these must all be set to zero for the same reason 171. Finally we mention that we have a s s ~ ~ m ewe d are working with a flow in Euclidean space. However flows and also maps in other spaces are sometimes considered: for example, even a uniform steady flow can be a dynamo, if the space chosen is complicated enough [61. All such models are of interest, but have to be assessed carefully as to what they tell us about fundamental dynamo processes in physical space.
3. Decay, anti-dynamo theorems, and upper bounds In this section we explore the basic dynamo geometry o f a sphere S containing conducting fluid inside. and an insulator outside. We prove some anti-dynamo theorems, which are of two types. The first restricts the kinds o f magnetic tield that can be amplified by fluid motion. and the second the classes of fluid motion that allow magnetic tield amplification. We also give some upper bounds on growth rates.
First, we introduce the toroitlal-l>oloi
of B, Here T(r. H . 4) is the toroidul sculur, defining the toroi~ltrlc.o~rrpo~~e~lt
and P(r. 0 . 4) is the poloidul.sculur, defi ning the poloitltrl corrlpor~enrof B,
Dytrurtro theory
373
where L2 is the angular momentum operator, with - L 2 = ( s i n 8 ) - ' & s i n 0 tlo = r-2;3
I r'2 Ia '
-
r-2
+ sin^)-^$,
L2 .
The toroidal component BT has no radial component, but the poloidal component Br has all three components. Note that a function of r alone can be added to T or P without affecting B in (3.1). Taking the curl interchanges poloidal and toroidal fields, with
Also there is the important property that
Given the scalars T and P, Equation (3.1) certainly detines a divergenceless tield. The converse is true since
and 1,' may be inverted. Let us consider thc 7' field; the P tield is similar. By cxpitnding in spherical hurn~onicsYi,,, and using I,'YI,,, = / ( I I ) Yi,,,.we obtain
+
(recall (2.61 )). The I = 111 = 0 term is not present in r . B because there can be no net flux of B ovcr a sphere, ah V . B = 0 . Correspondingly the above detinition of T has i t zero average ovcr spheres,
Given this, note that L'T = 0 implies that T = 0 , and similarly for P. The insulating boundary conditions (2.58) of continuity of B across the boundary amount to. from (3.2). ( 3 . 3 ) .
r = tr
The perfectly conducting boundary conditions (2.65) or (2.67) are satistied provided P = if,.(rT)= V ? P = 0 on the boundary. This assumes there is no tield trapped in the perfect conductor. a condition we will assume from now on. However, it may be checked from the induction equation that if B,. = u,- = 0 on the boundary for all time, then so must (v'B),.
(as (V x (u x B ) ) , is zero there). But r ( V 2 ~ )=, L ~ v ~from P (3.7) and so V * P = 0 automatically. We are left with
It is also worth briefly noting the Cartesian equivalent of the toroidal-poloidal decomposition. Suppose fluid is confined to a plane layer 0 < z < h . Then we may write
with
The conditions for insulating and perfectly conducting boundaries are, respectively,
and
Principle I I in Section 2.6 above requires that in the absence of fluid flow, u = 0 . the magnetic energy should decay away. We will check this for a sphere r- = t i containing stationary conducting fluid. surrounded by an insulator. I n this case the tield satisties the diffusion equation i),B = r l ~ with'boundary ~ conditions (2.58) matching onto a potential external held B = V x (2.59). In terms of the toroidal-poloidal decomposition (3.1). inside the sphere the scalars T and P themselves obey the diffusion equation by (3.7). Outside. the potential tield (2.59) has J = 0 and so corresponds to T = 0 and V' P = 0 by (3.5) and (3.6). If we focus on one spherical harmonic, with
then the boundary conditions to be imposed at r- = rr amount to
from (3.1 I ) and (2.59).
375
Dyttnmo rhrors
Straightforward calculations (see, for example, [ 144,1721) give the decaying modes as
with decay rates
Here ,jl are spherical Bessel functions' and k l is any zero of jl. This gives a discrete set of decaying modes. We have 1 3 1 only in the toroidal-poloidal decomposition above and so the slowest decaying mode is poloidal, corresponding to I = I and ko as the first zero of ,jo, which is n.The corresponding magnetic field is given form = 0 by
'
YI'
rln -
-- Yclccay = -- - . (I
,/I(:) =
sin :- :cos : - -7
inside r = ( I , ancl is ;I decaying dipole: the degeneracy 111 = - I . 0. I of this I = I modc corresponds to the degeneracy in direction 01' the dipole moment. Note that the toroidal field is zero outside r = tr uimply as ;I conseqilence o f the field being potentiiil there. This mciuns that the order 01' mi~gniturlcof the toroidal field cannot be ascertained directly fro111outside a pliunct. such as the Earth. or n s t x . Only the poloidal field can be observed from I . > o and extrapolated down to I - = t r . 7 The result for the slowest decaying modc y,lc,;,y = - / I T - / ( Ii~lso gives a L I S C ~ Lineqi1;~1I~ ity. Given any magnetic field defined inside the sphere r = tr and matched to a spi~tially decaying potential field B = V X outside. with IBI = 0 :icross the sphere and of course V . B = 0 everywhere. we have 7
This may be proved by expiunding iun arbitrary magnetic field in the set of'decay motles identified above: we onlit the deti~ils.This inequality is LIS~I'LII i n determining upper bounds on the strength o f the flow licld needed to overcome difl'usivc decay: see Section 3.5 below.
The early development of dynamo theory was dominated by ernti-c!\~ltrr~lotI~ror~~~?l,v. which show that for dynamo action and growing magnetic fields. the field and flow have to be sufficiently co~nplicnted,breaking certain symmetries. The most famous of these theorems
is that of Cowling [57],which shows that an axisymmetric magnetic field cannot be maintained by dynamo action. We first discuss two theorems in Cartesian geometry (see, for example, [144]), which much simplifies the argument. We take q > 0 and always consider incompressible flow, with V . u = 0. Anti-dynumo Theorem I. The first result we prove is the Cartesian version of Cowling's theorem, that a magnetic field B(x, y , t ) , independent of z , cannot be maintained by dynamo action. In this case the flow must take the form u(x, y , t ) as any z-dependence would be transmitted to the magnetic field. With this fluid motion B, field can be generated from B., and B,, components, by vertical flows u; that depend on ( x , y ) . However, there is no source fo; the B., and B,, components, and this will mean that a dynamo is impossible. We will consider Cartesian geometry with periodicity 27r L in x , y and z , letting V denote one periodicity box. To simplify matters, we shall assume a steady flow field U(X.y ) and that B takes the form of a normal mode
+
B(s. y, t ) = b(x, y)e11 c.c.,
y = Reh
Here 'c.c.' means the complex conjugate of the previous term. This is put in to give a real field as h and the eigenfunction b(x, y ) will generally be complex, since the right-hand side of the induction equation (2.46) does not represent a self-adjoint operator. We will restrict allowable initial conditions to those with no mean magnetic tield in the fundamental volume V (by principle 11 above in Section 2.6). If there is no flow, u = 0, and the tield is expressed in a Fourier series, it is clear that the slowest decay rate is y,l,,;,y = - , l / ~ s ' . Correspondingly we have the inequality (cf. (3.22))
for any periodic, zero-mean ti eld B with V . B = 0. or similarly for any zero-mean. periodic scalar ti eld 4
We will show that for any flow u(x, y ) and normal mode (3.23). the growth rate is no greater than the slowest decay rate, y 6 ydccay.and hence all magnetic field modes ydccay Since decay. Suppose the contrary is true, and we have a normal mode with y V . B = i), B , + i),,. B ., = 0, we may write
It is important to note that the potential A(x, y. t ) is periodic in .r and y because there is no mean tield in V . One may check that the three components of the induction equation are satisfied provided that
8, B; + u . V B , = ((8,A)a, - ( a , ~ ) a , ) u ;
+T~v*B;.
The potential A thus obeys a scalar transport equation with no sources. This potential must therefore decay: to show this multiply (3.27) by 2 A and rewrite it as &A2
+ V . (uA') =2 q V . ( A V A ) - 2q)vA12.
(3.29)
We integrate this, use periodic boundary conditions to discard the surface integral and then apply (3.25) to give
~ have A = 0. With Thus any growing normal mode (3.23) with y > ydecay= - q / ~must A zero, there is no source term in Equation (3.28) and so B, obeys a scalar transport equation. Repeating the above argument shows that B; = 0. Thus we are left with only the )'decay trivial field B = 0. We conclude that all normal modes have y
<
A l l t i - r ~ l l u l n oTlzeorrt~z2. The second result places no restriction on the form of the magu = ( 1 1 , (.r, s. :, I ) , LI,.(x,s , :, !), 0) cannot amplify netic field and is that a pI~i~znr,florr~ a magnetic field 12171. For this flow, the B , and B,. components can be forced by the :-dependent flow tilting B, field; however there is no source term for the vertical H , component. We shall only :tddres.s the simplified form when u is steady. and the rnagnetic field takes the form of a normal !node. now
--
Suppose there is a normal mode with y > yc~cc;,y - I ) / L { In this planar flow. there is now no source for the verticitl tield H ; . which obeys (3.28) with 11; = 0, and h o must be identically zero, by arguing as we did above. B must then toke the form B = V x ( i A ) . where the periodic potential A = A(.r. y. : , I ) obeys (3.27) (given H ; = 0) and again can only be zero. We fi nally obtain only the zero tield and so conclude that all normal modes ydecay have y We see that when too much symmetry is imposed on the Row or the field, the flow has no influence in reducing the decay of nornial modes. However these results hide the transient iunplitication that can and does titke place. Mathematically. in the presence of fluid Row. the time-evolution operiitor in the induction equation is non-norrnal, and normal nodes are not generally orthogonal with respect to the energy or norm. This allows strong transient amplification of fields before inevitable diffusive decay. For example, while A is transported above as a passive scalar in (3.27).and so cannot grow. the magnetic tield B is related to the gradient of A : in an initial value problem B can show transient growth while contours of constant A are pressed close together by the fluid flow. before diffusion takes effect. In the limit 11 + 0 this transient growth becomes more and more long-lived. We have taken a steady flow and normal mode for the magnetic tield above: this is a major simplification and assumes that the normal modes form a complete set. This follows for steady flows from results of hydrodynamic stability theory (see, for example.
<
378
A.D. Gilbert
[53, Section 9.21). For time-periodic flows or more complex flows, general results have been obtained using functional analysis to show that fields tend to zero and to give bounds on decay rates. Further references may be found in Iverq and James [ 1 141 and Fearn et al. [73]. Before leaving this section, we note that perhaps the simplest situation in which magnetic field amplification is possible, that is not excluded by the above, is the case of twodimensional but non-planar fluid flow u ( x , y , t ) with u , # 0 and three-dimensional magnetic field B(.x. y , z , I ) . Examples will be discussed in several sections below, for example, Section 4.3. 3.4. At~ti-dytzunzotheorenzs in .splzericul geotnrtry The above two results for Cartesian geometry can be translated into spherical geometry; indeed this is where they originated. Spherical geometry introduces a number of complications, which are best dealt with using the toroidal-poloidal decomposition described in Section 3.1 above; good sources are Moffatt 11441 and Roberts 11721. We will work within a sphere of r a d i ~ ~r s= ( 1 , use insulating boundary conditions and show that for a steady flow all magnetic field modes decay no more slowly than the decay rate in the absence of fluid flow, y < ydcc.ly= -'1372/"? To do this we assume we have a normal mode with y > ydccilyand show that we are left with only the trivial tield B = 0. A t - l o T o t 3 . We begin with Cowling's theorem 157 1 i n the t'onn proved by Braginsky 1241. From the Cartesian forn~ulation:is replaced by 4: an axisymmetric magnetic held B(r. H . t ) cannot be maintained by a fluid flow (which is necessi~rilyaxisymmetric). Wc use the toroidal-poloidal decomposition for B in ( 3 .I ) and similarly for u . For fields independent of 4. u7. x H I . = 0.and so the toroidal component of u x B is up x B p , giving the poloidal component of the induction equntion as
Now we may write Bp = V x A-/.where A7. = A$ is a toroidal tield and uncurl the above equation to yield an equation for the 4 component,
There is no contribution from a scalar gradient V F here as the problem is independent of'
4 and so ( V F ) d l= 0 (cf. (2.52)).Explicitly we obtain tor A and 80.
where s = r sine and ~ 7 =. $ s Q , so that Q ( r . H ) is the angular velocity of the fluid Row. These equations are analogous to the Cartesian case: A obeys a scalar transport equation with no sources, and B is generated from A by variations in angular velocity, that is differential rotation. often called the omegcl qjfiect for brevity.
Consider first (3.34): if we define a flux function ported by
x
= s A , then this is materially trans-
where the Stokes operator D' is given by
The problem is to show that the right-hand side of (3.36) implies diffusive decay of X , as it surely rnust. If this equation is nlultiplied through by 2 x , it can be written in the form
-
This holds inside the sphere S of radius r- = u with magnetic dit'fusivity q q,, and insulating boundary conditions. Outside the field is potential and so (v2- s - ~ ) A= 0 , which similarly arnounts to
Across r- = r r , u . n = 0 and I X I = [ i ) , . I~= 0, to guarantee continuity of B. If we integrate (3.38) over V,, and 11 times (3.39) over V. then the divergence terms give surf'ace integrals over S which cancel to leave
(There is no contribution from the surfl~ccintegr~tlat infinity since x = 0 0 . - I ) from (2.62)). This equation does not involve the flow u and applies cqually in the case of no 7 7 flow, when we know that the slowest decay rate is ydcc;ly = -tin-/(I-. I t therefore follows' that for an eigenmode to satisfy y > ydccLIythe tield A = x /.r must be zero and so must Bp. Now there is no source term .sBI> . VQ in the Rg Equation (3.35) above. Let us set W = . \ - I H d . i n which case Equation (3.35) may be written
Multiplying by 2 W . this can be written as
On r- = ( 1 , H$ = 0, since there is no toroidal tield outside and I B $ ) = O (see (3.1 I )). and so integrating leaves
ore
torlnally here. and similarly helow. one could derive ;in inequality analogou, to ( 3 . 3 5 ) from the decay rates ( 3 . 2 0 ) obtained for the purely diffusive prc~hlem(will1 u = 0) ;ind then apply it a h in ( 3 . 3 0 ) ;~hovr.
Again this equation does not involve the flow u and so plainly a mode having y > ?-'decay would contradict our results for decay rates. We conclude that all magnetic eigenmodes must have growth rates y Ydecay.
<
Anti-dyncrmo Theorem 4. We now consider the spherical analogue of the second antidynamo theorem discussed in Cartesian geometry in the previous section. This showed that a flow on plane surfaces u = ( u , , u,,, 0) cannot maintain any magnetic field. We now replace z by v, so that the theorem becomes that any purely toroidal flow u = u r ( r , 0, $, t ) , that is flow on spheres with zero radial component, cannot maintain a magnetic field B(r, 0 , @ ,t ) (see [9,36,69,144]). Again we show that for a steady flow and a normal mode, y ydecay.From the induction equation with r . u = 0 it may be checked that
<
-
and so Q r . B = L 2 P (see (3.1)-(3.3)) satisfies an advection-diffusion equation. From this we may deduce that
valid inside Vi and, outside V 2 Q = V'L' P = 0 (as v2P = 0 there) and so 0 = 2V. (QVQ) - 2 1 ~ ~ 1 ' .
(3.46)
Across the boundary ( (II = 1 i),. (II = 0 by (3.1 1 ). Integrating (3.45) over V, and 11 times (3.46) over V,., and adding to cancel the boundary terms gives
Thus a normal mode with y > ydCcaymust have zero radial tield and so zero poloidal scalar P , leaving only toroidal field B = By.. This obeys
With By. = V x ( r T ) we may uncurl this equation to yield i),(rT) = uy. x By.
+ r l v Z ( r+~ V) F = -r(uy. . V T ) + ~ T , V ' T + VG.
(3.49)
where VG = V F + 2qVT is some unknown gradient. The 0- and $-components of the equation indicate that G is only a function of v. We may therefore replace VG by rg(r. t ) to leave
a,T
+ u r . VT = ~/v'T + g ( r , r ) .
(3.50)
Now T averages to zero over spheres, by Equation (3. lo), and so multiplying by 2 T and integrating yields
Dynamotheory
381
(3.51)
at /v T2 dV -- -2rl /v i IVTI2dV, i
using the condition T = 0 on r = a. This shows that the toroidal field decays with at least the decay rate }"decay, and no normal modes exist that decay more slowly. This concludes our discussion of anti-dynamo theorems. At first sight it looks as though these theorems should generalise not only from Cartesian to spherical geometry but to any orthogonal coordinate system. This is not the case. The decomposition into toroidal and poloidal fields in spherical and Cartesian geometry has the key property that the diffusive operator V 2 maps poloidal field to poloidal and toroidal to toroidal (see (3.7)). This does not carry over to other coordinate systems. For example, whereas in spherical geometry with coordinates (r, 0, 4~) there is no source term for the radial field Br from Bo or B,/, through the Laplacian, in cylindrical geometry with coordinates (r, 0, z) there is, from Bo, and this can allow a dynamo to function as we shall see below in Sections 4.1 and 4.2. A question related to anti-dynamo theorems is whether a flow can maintain a magnetic field with a zero poloidal field P = 0 [116] or a zero toroidal field T = 0 everywhere in space [117]. In the former case the dynamo is said to be invisible as the whole magnetic field exterior to the sphere r = a would be zero. It appears that with 'reasonable' fluid flows such dynamos do not exist.
3.5. Upper hounds in the absence of fluid flow, u = (), we have seen that field decays, with the slowest decaying mode having Yde,,'ay -- - t l 7r2 / a2 ill spherical geometry. Since we expect the growth rates of normal modes to behave continuously as we increase the flow u from zero, it is clear that we will need to reach some tinite strength of flow before dynamo action can occur, with ?, ~> (). There are a number of upper bound results which make this precise. In each case we consider the energy equation in the form
z~a,&--~fv Bl2dV-fvU.(V
xB)xBdV
IV x
i
i
( v = v~ u v,~),
(3.52)
from (2.64), and give an upper bound on the term which inw:)lves the flow field u. We shall work in spherical geometry with insulating boundary conditions, but there are analogues in Cartesian geometry.
Upper bound !. For the first upper bound [49] we let Umax be the maximum value of lu[ in the domain V and use the Cauchy-Schwartz inequality and then inequality (3.22) to establish that
-L,
u . (V • B) x B d V <~ Umax
IV x BI 2dV i
~< a U m a x 7/"
/V
IV x B[ 2 dV. i
[B[ 2dV i
(3.53)
Putting this expression back in ( 3 . 5 2 ) shows that
~ / equivalently K, when the magnetic and so growth can only occur when 7 ( L ~ U , , , ~ or Reynolds nutnher
This dimensionless parameter measures the strength of the flow field, compared with the effect of diffusion, and is a key parameter in MHD. It is defined as a velocity scale multiplied by a length scale. divided by the magnetic diffusivity; its precise definition depends on the problem at hand. The above bound is based only on the magnitude of velocity. However, magnetic fields are intensified not through motion so much as through stretching, by the term B . V u in ( 2 . 5 1 ), and this is brought out in the next bound.
r t o t2 The next bound of Bnckus 191 (see also 1 1641) makes the role of stretching explicit, but requires the additional 110slip condition that u = 0 on the boundary S of V,. First write ( V x B ) x B = 9 . V B - ~ 4 1 and ~ integrate 1 ~ by parts to show that
u . ( V x B ) x BdV =
+
Now R, H , ~ ) , I , . ,= H I Hie,, where e l , = A ( i ) , t r , i f , i r , ) is the sym~netricrate-of-strain tensor. If el,,;,,is the largest eigenvalue o f this tensor in the volume V. at any time. then u . ( V x B ) x BdV
<
~ I , ~ ; I X
Putting this back in ( 3 . 5 2 )and applying ( 3 . 2 2 )then shows that the energy satisties
A necessary condition for growth is that
giving a new def nition o f the magnetic Reynolds number R . Note also that for a normal mode, Ev grows at a rate 2 y and so the above inequality gives a bound on the growth rate,
with the above definition of R .
These are necessary conditions, but far from sufficient: it is clear they do not capture the requirement that the fluid flow be sufficiently complicated, especially in view of the toroidal flow anti-dynamo Theorem 4 above. Another upper bound has been derived by Roberts [ 170, p. 75 1, while Busse [39] and Roberts [ 1721 give a bound on the magnitude of the poloidal flow needed to overcome the toroidal flow anti-dynamo theorem.
4. Laminar dynamos We now move on to discuss some working kinematic dynamos, in which flows are given and shown to amplify magnetic fields. There are two matters we should note: tirst, there are very few elementary examples in this field, perhaps unsurprisingly in view of the antidynamo theorems and the 40-50 year gap between Larmor's suggestion and the tirst working examples. Secondly, we will now become more cavalier about the geometry. The above discussion is fora sphere of conducting fluid surrounded by an insulator, for example, modelling the Earth's liquid iron core surrounded by the largely insulating mantle. Few ~nodels can be embedded naturally in this geoinetry. and we will now generally consider spatially periodic situations, but with an eye on the principles stated in Section 2.6 above as to what constitutes a dynamo. This will allow us to understand fundamental dynamo mechanisms without thc distraction of gco~netricalcomplications.
Perhaps the si~nplcstdynamo is that of1 1.581 in which the given flow is written in cylindrical polar coordinate5 ( I . . H . :) as @1.52+iU
(r
(r > ti).
where the i~ngulnrvelocity 52 and axial velocity U are conhtants. This Row is depicted in Figure l(n). Plainly there is solid body rotation inside kund outside the cylinder r = 11. and solid body motion alone cannot give a dynamo. by upper bound 2 above. The dynamo is therefore intimately connected with the discontinuity of the flow at r = t i . A magnetic as Reynolds number can also be deti ncd, tor cxi~~mple
however we will not formally non-di~nensionalisethis proble~n.I t is useful also to define the parameter x = U / t i R . which determines the pitch of the helical motion. At tirst sight this is a bizarre flow, with a vortex sheet at r = t i , a highly unstable contiguration. Nonetheless it forms the basis of ongoing dynamo experiments being conducted in Riga, Latvia, and in which dynamo action has recently been detected (see, for example. 177-791). These involve a vessel depicted schematically in Figure I(b) leading, at a basic level o f approximation, to flows that are piecewise constant in radius.
A.D. Gilbert
u=o
(no flow)
Fig. 1 . ( a ) The Ponomarenko dynamo flow field. and (h) a highly \chernatic picturc of'the Riga dynamo experiment. Liquid \odium i h dr~venround n ve\srl by a propeller. ;I\ \hewn.
To obtain growth rates we consider a normal mode B = b(r)ex'+"""+"~.There is no stretching in the system except at r = t i . The induction equation in r < t i or r > tr siniplities to the form
+
where A,,, = ij,? r - ' i ) , . - m'r-' is the horizontal Laplacian operator. Vertical diffusion and motion are absorbed into the definition of the constant p. which takes two values: inside r = ( 1 , p = pi and outside 11 = p,, where ~
~
= ~ h1
+; itnQ + i k U + rlk2.
7
=h
+k
( R e p , . Re 11, 3 0 ) .
(4.6)
We must impose V . B = 0, which amounts to
The solution is also subject to [BI = [ n x El = O across r = ~ i although , of these tive jump conditions only four are independent (see Section 2.4). To solve these equations detine hi = h, i i h d . which obey
385
Dynurno theory
with p = pi or p,. These are satisfied by modified Bessel functions,'
building in the condition that [B] = 0 and correct behaviour at the origin and infinity. The three constants, A * , A,, are fixed by imposing V . B = 0 in the two regions and the continuity of electric field [ n x E] = [n x ( q V x B - u x B)] = 0 across r = a , which gives two conditions,
or, bearing in mind that [ . I represents the value as r + [ I + minus that as r + N - ,
Of this total of four conditions only three are independent. I t turns out that the fastest way to proceed 11581 is to note that (4.7) and [BI = 0 imply that lij,-h,.] = 0: this, with the tirst electric tield condition i n (4.12). gives two independent equations
Using these with (4.9),(4.10) and (4.6).straightforwardly gives the dispersion relation as two simultaneous equations fi)r pi and p,
where
The dispersion relation has to be studied numerically for general parameter values. Despite its complexity it is probably the simplest exact solution we have in dynamo theory! Fortunately some of the asymptotic approximations used in this and other models give rather more manageable formulae. The above formulae do simplify significantly when n752 XU = 0 and so the tield lines are aligned with the shear across r = e l . In this case pi = pu = 11. say. and the Bessel functions reduce to
+
The dispersion relation becomes the single equation 211 = i ( i 2 Q ( ~ l l l(LII))K,,,-I -~ (~111)- lIn+l(UP)K,,+I( u p ) ) ,
(4.17)
or, with more manipulation, i Ill] = (lln 52 (1,,, (~11)) Kt;, (up)
+ I,;, (up)K,,,(up)).
(4.18)
This again cannot be solved in general; however approximate formulae can be obtained in the limit o f large magnetic Reynolds number (4.2). Informally think of the limit 1) + 0 for fixed flow field and geometry and in this limit it turns out that u p + oo.Substituting asymptotic expansions for Bessel functions ~ , (, z,) and K,,,(z) for large z = u p in (4.17) or (4.18) then gives
and so the complex and real growth rates (neglecting the small term growing modes.
-1)k'
in (4.6))are, for
These tire correct for fi xed tr, a. U, 111 and X . in the limit of small ,I. and so. in n o n dimensional terlns. tor thc Ponomarenko dynamo at largc magnetic Reynoltls numbcr K (4.2). The real growth rate y in (4.20) increases as a t'i~nctionof wave number 111 and to lind out which is the fastest growing mode we need a tormula that is i~niformlyvalid for 111 large in (4.18). Using expansions for Bessel functions" I,,,(~rrz), K,,,(II~:)with z = trplirl and 111 >> I gives eventually
(recalling that
):
= UltiQ and m i 2
+ kU = 0). The fastest growing modes are given by
Thus at large magnetic Reynolds number the maximum growth o f magnetic tield occurs on or t r / U . independent of the magnetic diffusivity the order of the turn-over time-scale R - I
This makes this dynamo a,first clvnunlo, which we shall discuss later; however the fields ~ limit. amplified are of rather small scale, of order u / ~ ' in/ this Some further comments are in order. First of all note that the growth rate is zero if rn is zero, or if k is zero (since r t l R = - k U ) . If rn is zero B = B(r, z , t ) is axisymmetric and cannot be maintained by anti-dynamo Theorem 3 . If k = 0 the magnetic field is independent of :and falls foul of anti-dynamo Theorem I. In addition note that we have set mi2 + kU = 0, and that flows with i2 = 0 or U = 0 cannot be dynamos by anti-dynamo Theorems 2 and 4. respectively. Secondly note that all the generation is located at the discontinuity in the How field. The dynamo is a boundary phenomenon, and the above growth rates at large R can be obtained by a boundary layer analysis 186,1721. The mechanism involves diffusion of B0 field to give radial field, and the 5tretching out of radial field at the discontinuity to regenerate Bf/ field. This will be seen more clearly in the next section when we smooth out the discontinuity. Similar generation of field at discontinuities occurs in the rotor dynamo of Herzenberg I I08 I and the experimental dynamos of Lowes and Wilkinson 1 136,1371, the latter using solid metal rotors embedded i n a conducting block. Obviously further work can be done in tuning the Ponomarenko model for the experimental configuration depicted i n Figure I (b), for example. optimising the geometry so 21s to minimise the critical magnetic Reynolds number. allowing an outer vacuulii region. and requiring the instability to be absolute rather than convective 178 I. 11.
In thc above discontinuo~~s Ponomi~rcnkodynamo. thc generation is hiclden in a thin 1t1yt.r of infinite she~ir.More relevant to astrophysical and geophysical fluid dyn;umics would generally be the smooth flow
which we consider next. Thc fundamental issue here is: how do swirling. helical Hows generate ~iiagneticfields? Such flows can occur in convection and other instabilities. ;uncl are ;I very natural building block of dynamos i n astrophysical Hows. For this ex:uiiple there are no obvious exact solutions and we have to assume a limit of large magnetic Reynolds number K to obtain approximate analytical results (86.1781. or study examples o f Ilows nu~nericallyI I83,I841. We shall use a dimensionless version o f the induction equation, non-di~nensioni~liscd using ;I sc;~leof the flow and ;I velocity sc;~lc. written in the form
where E - ' = K is a magnetic Reynolds number. To recover dimensional results we simply replace c by 11.
We may put B = b(r)eh'f'"'H+'kto obtain from the induction equation
Again we have coupling terms between 6, and bHbecause of diffusion in cylindrical geometry, and also generation of bHand 0, from b, by differential rotation Q 1 ( r )and axial .;hear U 1 ( r )respectively. We shall seek a magnetic mode localised near any given radius r = u , and to obtain growth rates we expand all spatial quantities in powers of & ' I 3 << 1, setting =&-'I'M.
k =&-'/'K.
r = LI
+&'13.y,
a,. = & - 1 / 3 i j .
(4.28)
We have chosen this wave number scaling to obtain the most useful results. We scale the growth rate as
and for the tield.
These expansions are then substituted into the induction equation (4.25)-(4.27). and Q ( r ) and U ( r ) are also Tay lor-expanded about r. = t r . The ti rst two orders in the expansion simply tix
Of these three conditions the second is crucial, and is that magnetic tield lines must be aligned with the shear in the flow at this radius. This may be thought of as a resonance condition. Physically, for a given M and K , as we move away from this critical radius. the pitch of the stream lines changes and this enhances diffusion of this particular normal mode, as we discuss further below. At the next order we obtain the equations for h,. and hH(dropping h;, which can always be reconstructed from the condition V . B = 0). and these can be written compactly as
which are coupled parabolic cylinder equations, with
These coupled differential equations can be rewritten as
~ a linear combination of The operators P+ and P- commute so that the solution for b , . is solutions to the two equations P*hro = 0 or, in canonical form,7
Solutions that decay for s + fcc exist only if c.+ = -j - I for j = 0, 1 , 2 , . . . and this gives eigenvalues of the original dynamo problem.x Untangling all the previous changes of variable and expansions gives finally leading order growth rutes.
[86.1781. These growth rates were determined for t r r . k = O(E-'/'), but i n l k t are valid for all 111 and k . This scaling (4.28) was chosen as i t gives the richest fol-mula tor y itnd includes the case o f lnaxirnurn growth rate y = O ( c l / ' ) . I t is equally valid for rn = 0(I ) , except that the last term in Equation (4.38) is now subdominant, and y = o(F'/'). For 111 > O(F-'1'). growth rates are negative. The eigcnf~~nctions take the form of a (complex) Gaussian ~nultipliedby a Hermite polynomial h, . h,,.b; a He, (CT) c - ~ ' / ' .
(4.39)
In the equation for the growth rate (4.38). the tirst term. with the lower sign, represents amplification of field through the interaction of ditf'erenti:tl rotation and diffusion i n cylindrical geometry (cf. (4.25), (4.26)). The second term is always negative and represents enhanced diffusion: although the shear in the flow and the field are aligned at I. = tr because of the condition (4.32), as we move away from r = rr the pitch of the stream lines changes. and the flow begins to advect field across lines of constant tield. increasing the effect of diffusion. The tinal term is si~nplywhat is left from ~nolecul;trdiffusion of tield in this geometry. with the flow playing no role. As in the discontinuous Ponomarenkodynamo. tield decays if ti1 = 0 or k = 0 in keeping with anti-dynamo Theorems 3 and I. The first case is obvious, for the second note that if
390
A.D. Gilbert
-
k = 0 then by the resonance condition (4.32), we have m R f ( a )= 0 and again there is no positive term in (4.38). Note also that if there is no axial flow, U ( r ) 0, there is no dynamo as we then have 111 = 0 from (4.32), and in this case the enhanced diffusion caused by the motion leads to the mechanism of flux expulsion [155,2 131 by which field is removed from regions of closed stream lines in the plane on a time scale of order Interestingly the first two terms in (4.38) possess the same scaling with rn and e. For dynamo action at small e it is required that the sum of these two terms be positive for j = 0, and this can be rearranged (using also (4.32)) as the geometrical condition that
at the given radius r = ( I . This condition states (roughly) that the rate of change of pitch of the stream lines should not be too great: if it is then the enhanced diffusion will dominate over the regeneration of field. and the flow will not be a dynamo for small e. An example of I: family of Hows that satisfies this condition is spiral Couette flows, for which v v 2 u = 0,
and the left-hand side of (3.40) is 3 (assil~ning/I and r / are non-zero). Dynamo action can o c c ~ ~ r iany t t r i t d i ~;tt~ ~\~~fficiently , large magnetic Reynoldc number K = c : ' . On thc other hand i n spiral Couette-Poisei~ille flows. for which 1,v2u - V P = 0 with ;I constant axial pressure gradient P = 1):. we have
and not all radii support dynamo action at large R . These Hows hitve been studied numerically IIX3,1841. The smooth Ponomarenko dynamo is the simplest ex~umpleof a class of smooth Hows that defeat the anti-dynamo theorems. In this dynamo the How lies o n cylindrical surtiices. whereas the anti-dynamo Theore~ns2 and 4 rule out Hows on planes and spheres. The key point is that in i t cylindrical geolnetry BO field. parallel to the stream surtitces. can diffuse field. - This. together with shear. which regenerates Bf, from B,.. to give perpendicular. /I, gives a closed dynamo loop antl can amplify tield. I n spherical and planar geometry there is no analogous process by which diffusion generates poloidit1 field from toroidal. The tield structure is given sche~naticallyin Figure 2, which shows spiralling tubes of field (top row) for In = I and IU = 2. The tubes have a roughly fish-like cross section (bottom row). and the trailing fins of the fish are a result of the pitch changing with radius, reducing the radial scale and enhancing diffusion, as mentioned above. The Ponornarenko mechanism may also be thought of as the tirst of a number of generic dynamo mechanisms, here amplifying field for flows in which the stream lines lie on closed surfaces. These will arise for general flows of the form u(.r, in Cartesian coordinates. for example, in convection near to onset [ 140.1591, or of the for111 u(r. 0 ) in spherical polar coordinates ( r ,0.4) (for example, 1671). Asymptotic growth rates have been found in these cases 1921 and agreement obtained with simulations of Plunian et al. 1157j of
1)y11~11?io theory
'
for111\\p1~111i11g 1:1g. 7. M;igl~c[iclicld i l l tI1c I ~ O I I O I I I ; ~ ~ Cdy11;l111o II~O ;I{ I;lrgc ~ l i ; ~ p ~ cKcy11ol11\ tic I ~ L I I ~ I ~KC = I. 8 ruhc\ o i licld Ioc;lli\cd I I ~ ; the ~ I .~-;idiu\r = 11 ior wl~ichI I I R ' ( ~+) XO'(rr) = 0 . 'I'hc hottr~rnrow \Ilowr ;I cro\\ \IrLlcttlrc I)! !lie licl~lfor I I I = I 111odc\(left) \cctiorr o l ' l l ~ ct[~hc\.w111Ictlrc top row \llow\ tllc ~lircc-cli~~ic~i\io~~:~l ;111d111 = 1 111o
flows modelling nuclei~rreactor cores. Other applications include modelling of galactic jets 11821. A very general fl-arnework for such approximations has been given by Soward 11881. and some results on dynamo siituration. with dynamic:il feedback on the now tield through the lo rent^ force. [nay be found in 112.621. A related family of dynamos was introduced by L o r t ~[ 1351, and involves a helical coordinate system in which the dynamo problem can be reduced to a finite set o f ordinary ditl'erential equations; see. for example, 12 1,22,47 1. Ponornarenko dynamo Hows are helical: the 11c~lic.ity doti~ity
392
A.D. Gilhrrf
in [ 1431 is generally non-zero. This gives some indication of the complexity of the flow and will play an important role when we shortly consider large-scale fields. However it is hard to make a more concrete connection at this point, as some helical flows will satisfy (4.40) and others will not. If a flow does not satisfy this condition, with r ( l log l Q 1 / U ' l ) ' / > 4 for all radii, then it appears that for all small c2perturbations to the flow for which the stream lines remain on surfaces, the flow will remain a non-dynamo, at least at large R. This opens the question of more general perturbations, which could introduce regions of chaotic particle paths into the flow; flows with Lagrangian chaos will be considered when we discuss fast dynamos in Section 6 below. 4.3. G.O. Roberts bnumo: Multiple s a l e unu1ysi.v
The smooth Ponomarenko dynamo shows that dynamo action can occur in a laminar flow, in which spiral stream lines form a single eddy. The fields are localised about stream surfaces. As a next step in complexity we consider laminar, spatially periodic flows; these might, for example, model convective cells. This introduces two additional elements that we have not yet considered. First, a spatially periodic flow will generally have stagnation points. Secondly, there is the possibility to generate large-scale fields that span many cells, and to consider transport effects governing such fields, important in modelling dynamo action i n convection and turbulence. The How field we consider is a member of the family of ABC,fk)rv.s 14.20.501
These have the Bcltrtrtili propcJr!\. that V x u = ku for a constant k ; here k = 1. If and A = L? = I , the How is independent of :, u = (cosy. sinx. sin!
+ c o s x ) = (if,.+,
C =0
-if, $. $1.
+ =sin! + cosx. and has the cellular structure shown in Figure 3. This and similar flows are considered numerically by (3.0. Roberts [ 1681 and analytically by Childress 148.501 and G.O. Roberts 11691. We will follow their analysis i n a low Reynolds number limit for a large-scale tield, using standard multiple-scale asymptotic analysis. The above flows are already non-dirnensionalised and so we will use the induction equation in the form
where R is the magnetic Reynolds number based on the scale of the cells and magnitude of the flow, and is now taken to be small, R << I. We consider a magnetic field B that depends both on the scale x = (x. y , :) of the cellular flow, but also on a larger scale x = o ( R - ~ ) . Diffusion of field across a single cell occurs on a short time-scale t = O ( R ) , and diffusion on scale x = O(R-') over times t = O ( R - ~ ) .
Fig. 3. The G.O. Roberts flow lield
ill 11,
+
(4.45). Thc and - \ign\ \how the direction oi'lhe motion in = 0 or] the nctworh o I ' r p ; ~ r ; ~ l r i c c .
:.
with
We therefore set t = Kr. x = K'x. t = K 3 7 ' and replace
in the induction equation. with R,,= B, ( x . r. X. 7'). O n the small sciilc ( x , r ) . we will require that fields B , are periodic in space and time. and this means the imposition of solvability conditions that will eventually govern the behaviour of licld on large scales ( X . 7'). The velocity field nioy be taken to be any How that is strictly periodic in (x. r ) and has zero mean over these frlst variables, (u) = 0. From the induction equation at the first few orders we have
and from V . B = 0,
304
A.D. Gilhrrr
At leading order in (4.48) the only solution that is periodic in the fast variables (x, r ) is a constant field, and so Bo = Bo(X. T) depends only on large space and time scales (and (4.53) is satisfied automatically). At the next order Equation (4.49) for B I is a diffusion equation, forced by the periodic function on the right-hand side, and may be solved using Fourier series. Note first that V, x (u x Bo) = BO. Vxu as BOis uniform on the small scale. If we write
of which the above ABC flows are special cases, then (4.49) may be solved in terms of Fourier tnodes of B I as A
B l . k , n= l (k' - i u ) - l ( i k . BO)uk
(4.57)
The flow thus drives a small-scale field B I , by churning up the large-scale field Bo. Equations (4.501, (4.5 1 ) for B2 and B3 could be solved i n a similar manner. These equations also have solvability conditions, that the right-hand sides must be 7.ero when averaged over the fist scales (x, r ) , so that in Fourier space the factor k' - iw is non-zero for all terms present and ij, - v,' may be inverted. These conditions for (4.50),(4.51 ) are satisfied a~~tomnticnlly. sing the tr~ctsthat (u) = 0 . (Bo)= Bo, and (V,(.)) = 0 for any quantity ( . ) . However the solvability of Eqi~ation(4.52) for B4 gives the evolution of the large-scale held as
the latter condition follows from averaging (4.55). f i ) ~ ' (c,rr7f: ~(, The key term here is the rircJclnc ~ I c c ~ / m r t i o / i ~ ~ ~or~ ,riir(1ri
which may be written, using (4.56). (4.57) and averaging over space and time, as
To simplify this expression note first that by incompressibility k and uk,,,,are perpendicular and so u:,(,, x u ~ .is~parallel , to k. We may therefore write
Here g ( k . w ) is the contribution to the helicity per unit volulne of the Huid Row from the mode (k. o), H = (U . V x U) =
H^(k. w).
One can easily check that fi(k, w ) is real and that f i ( k , w ) = HI(-k, - w ) . With a little further simplification (combining the contributions from modes (k, w ) and (-k, - w ) ) we may write the mean emf as
This is an important result: the mean emf is written as a tensor a,,,,, multiplying the largescale tield. The generation of a mean emf by small-scale fluid flow is known as the alpha ~$i.c.t and was tirst introduced by Parker [ 1521 and Steenbeck et al. [ 1971. We will discuss it further in Section 5.1 below. For the (3.0. Roberts flow (4.45) the alpha tensor may be computed as
With this alpha effect now written down explicitly, the mean field satisties
Consider a large-scale field mode Ho = /3ciKZ+"" with this has growth ratc
/3 . i = 0 . and
Vx x Ro = * K H o :
The field nodes and the corresponding vector potentials ;re, up to tr~unslationin %. given by
We see that from (4.66) that the (3.0. Roberts flow. which has positive tluid helicity density h = u . o,amplifies large-scale magnetic fields having negative 11rtrgrlctic.h ~ l i c . i t yclc,~r.rit~ llM = A . B . If the flow is rewritten in dimensional units with
and the induction equation used with 11 replacing K-'. then the result is that large-scale modes of wave nurnber y hiivc growth rates
The fastest growing mode has
396
A. D. Gilbert
This example highlights the role of fluid helicity in dynamo theory. The alpha effect, a transport effect that at large scales dominates over diffusion, is given by a weighted sum over the helicity of the fluid flow. We will discuss alpha effects further below in Section 5.1. Helicity has a topological interpretation in terms of the linkage of vorticity lines for h = u . o in a fluid, or magnetic field lines for h M = A . B [ 143,146,2 151. If the alpha tensor vanishes completely, for example in mirror-symmetric flows for which H ( k ,o)= 0, it is possible to rescale and pick up an eddy diffusivity term as a transport effect involving second derivatives of Bo. This term can be negative for suitable choices of flows and magnetic Reynolds numbers, corresponding again to the destabilisation of field at large scale [ 123,1291; see also [ I 181. A transport effect analogous to the alpha effect can also occur in forced fluid flows [761: in this case the averaging is taken over Reynolds stresses and large-scale helical fluid flows can be destabilised. The above theory involves the limit of low magnetic Reynolds number, and it is important to note that this is based on the scale of the periodic flow field, not on the scale of the magnetic field itself. Indeed to contain the fastest growing mode given above, the whole system needs to have a scale of order L = c/,;;f,, and so a Reynolds number based on this scale would be R,, = ucrL/rl= 2 ~ - ' This . diverges in the limit we are taking and shows, tirst that there is no contradiction with the upper bounds derived in Section 3.5 above, and secondly that these dynamos are not particularly efficient for small R. The motion u has only a weak effect o n the field Bo. as small-scale diffusion is so dominant, and thus only large-scale fields can be sustained by this mechanism. The (3.0.Roberts flow ti)rrns the basis of dynamo experiments taking place in Karlsruhe, Germany 142.198.203I. To obtain a suititble two-scale flow liquid sodium is constrained to flow in a series of pipes and helical channels, which mimic the Row depicted i n Figure 3 for finite ranges of :and x' !'. The height of the ~tpparatusis approximately I metre. diameter 2 metres and i t contains 52 helical cells. Dynamo action has been observed experimentally and has heen studied in detail numerically: the field adopts a configuration that within the limitations of the finite geometry may be identitied with a large-scale helical mode. Finally, it is possible to extend the above theory for low magnetic Reynolds number to include dynamical effects, if the underlying flow is viscous and driven by a prescribed body force (93.125.126l. In this case the alpha tensor (4.63) becomes a nonlinerir function of the mean field. If the forcing, in the absence of magnetic tield, is chosen to drive the (3.0.Roberts How, Equation (4.65) becomes (after a suitable rescaling from Bo to S. say) A
+
Here c u is ~ the value of the alpha coefficient in the kinematic regime when B is weak. When this system is simulated beginning with a seed magnetic field, initially the kinematically most unstable field mode grows. However, as nonlinear interactions become important the magnetic energy is driven to the largest scales of the system, through a series of meta-stable states, in what is known as an inverse cuscudc 193). Such cascades have been observed in MHD turbulence 11621 and in other nonlinear dynamos (see, for example, 12131).
4.4. G.O. Roberts dynamo.fir lurge R We studied the (3.0.Roberts dynamo when the Reynolds number R , based on the scale of the cells in the flow, is small. In this limit only large-scale magnetic field modes are destabilised and growth rates tend to zero. It is natural to consider also the opposite limit of asymptotically large R , as we did for the smooth Ponomarenko dynamo in Section 4.2. The dynamo has been studied numerically by G.O. Roberts 11681 and in this limit by Childress 15 11, Anufriyev and Fishman [ 3 ] ,Perkins and Zweibel [ 1561 and Soward [ 1871. For large R a Ponomarenko dynamo will also generally occur in the closed stream lines of each cell of the network depicted in Figure 3 whenever a criterion analogous to (4.40) is satisfied 192, 1881. We will now howeverconsiderthe possibility of large-scale fields: at high H magnetic modes tend to localise on stream surfaces and so large-scale modes must be localised on the network of hyperbolic stagnation points and separatrices shown in Figure 3. I t is ti rst convenient to rotate axes and rescale space, and so write the flow in the form
u = ( s i n s c o s y , -cos.r siny, h sinxsin y) = (i)!.$, m i ) , I//, K $ ) ,
(4.72)
with ~1
= sin.r sin F.
K =J2.
(4.73)
We will solve the induction equation i n the dimensionless form
with c. = K-' << I ~iow.We note the 11iinor point that the rcscaling of spacc implies a redefinition of the ~nagncticReynolds number, iund that now V x u = &u. The How is shown in Figurc 4(a).
Fig. 4. The G.O. Robcrth Ilow licld written in the lor111(4.72). ( a ) shows xtrcam line\. ( h ) shows the btcady xtatc with a uniforln ilnposcd i-directed ~ilogneticticld. 1 7 , = I . h , = 0 with F << I . M;ignctic field lines arc \hewn, which arc alxo lirlcx otconbtnnt \cal;~rit.
A. D.Gilbert
398
To obtain growth rates we will follow Childress [51] and again use an alpha effect to make progress. We divide the magnetic field into a mean and a fluctuating part, simultaneously taking out a normal mode in z and t ,
Here ( . ) denotes an average over I and y and b is a constant which we refer to as the mean field. Averaging the induction equation and solenoidal condition (4.74)gives
(h
+ & k 2 ) b= i k i x (U x b').
i k i . b = 0,
(4.76)
using (3, (.)) = (a,.(.))= O and (u)= 0. The fluctuating, zero-mean parts of these equations are
(h
+ ~ k ' ) b '= ( i j , , ij,., i k ) x (U x b + u x b') -
i k i x (u x b') + & ( i f ;
+ a?)bf.
.
(ij., ij,., i k ) . b' = 0. These equations are exact and we see a similar structure to the previous ~nultiplescales framework. The mean tield b is driven by a mean emf I: = (u x b ' ) . The fluctuating tield b' is governed by a linear equation with a source term from transport and stretching of the mean tield. We ti x s << I and consider the limit of small, positive X , that is of tields with large-scale dependence on :. (For X = 0 the magnetic tield is independent of :and so cannot be a dynamo by anti-dynamo Theorem I .) Assuming that the combination h + sk' is small as X + 0 with E ti xed (veritied later in ( 4 . 1 0 2 ) )the . leading order approximation to (4.77)is
0 = (2,. ij,.. 0 ) x
(U x
b
+ u x b') + ,(a; + i);?)bf.
(8,.i),., 0 ) . b' = 0 . This is just the steady induction equation for b' in the presence of a L. riven. constant mean field b. The fluctuating field b' generated will depend linearly on b and so the mean emf can be written
where or is the alpha tensor. Since b; = 0 from (4.76).only the 2 x 2 block of (.r. y ) components of the or tensor is relevant, and this block is isotropic, which follows from the symmetries of the flow u. For example, the flow is invariant under rotation through x / 2 about the axis x = y = n/2. All the symmetries of the flow are evident from Figure 4 but a more formal discussion is given in 1881 and Section 5.5 of (531.
The framework is now very similar to that of anti-dynamo Theorem 1 in Section 3.3 above, except that there is an imposed constant mean field b maintaining magnetic energy in the system. We decompose the full magnetic tield in the form
where the scalar potential a incorporates the mean field b, and may be written in the form
Here [r1(s,y ) is required to be periodic in space, and the definition of 2 is such as to make zero at the centre ( ~ 1 2r ./ 2 ) of thepr-itrrnt;vct2llO x. y z in the development below. The steady equations (4.79) for (1 and h, are (using 11; = K $ )
<
(1
<
u . Vrr = F V ~ L I . u.Vh,=-Ku.vrr+~v~h, (cf. (3.27). (3.28)). with :I prescribed mean gradient in the flow u. and so The equation for o governs a sc:~l:~r transport 1 175.1 8 I . I87 I. For cxamplc. let 11stakc h , = I . is ol'interest i n the theory ofsci~l:~r h, = 0. and then we can interpret the results in two ways. First. the problem is one of the steady state of n unilbrm .\--directed field i n this How licld. in which case tlic licld lines (pro.jccted down on thc (.I-. !)-plane) of'constant rr ;Ire shown schematically in Figure -l(b). For small c. the lield is expellctl from the rccirculoting eddies and sits in boundiu-y I;~ycrs along the separotrices. In this case we are interested in computing the alpha el'fcct. with
(using (4.72)).Replacing h,. = -i),tr in thc last term tuid integrating by parts gives
Secondly, we can interpret the field lines (without arrows) in Figure 4 ( b ) as contours of constant passive sci~lartr in a steady state obtained by imposing a uniform gradient i n y. In this case the quantity of interest is the net flux of scalar in the -!-direction. down the gradient. which gives the effective diffusivity K of the scalar i n this flow tield,
The second term o f (4.86) appears in (4.87) for K . and in tact the tirst term is also closely related. as we shall see.
To make further progress we assume that the magnetic field is confined to boundary layers along the network of separatrices as shown in Figure 4. Inside the eddies the field is suppressed by the process of flux expulsion. Let u s focus on one boundary layer, 0 x ~r and y = 0. Here we replace coordinates (s,y ) by (s,$) with s = x. It then follows that a, = a, and a,. = u,a*. We assume that the scale of the field is & ' I 2in the $ coordinate, but of order unity in s. In this region u, = O(1) and u,. = o(E'/*) and, at leading order, Equations (4.83), (4.84) become
< <
~ u., 2 sinx = sins is the fluid velocity on this separatrix. A furwhere y = (112 + L ~ : ) I /2 ther transforn~atio~ to von Misrs cwordinate.~,
leaves us with
ijm17:
= ij:bc
-
(4.92)
K ijmt i ,
heat diffusion equation for t i and a similar but forced equation for h,. This is correct for the separatrix 0 < .r n. y = 0 . for which 0 < a < 2. but can be extended all the way around the boundary of the primary cell 0 < .r. y < ir. with 0 < a < 8. as depicted in Figure S(a). For exarnple. y is replaced by u y on 2 < a < 4. After one ;I
<
o (time) 0=4
St a=O
0=2
insulate
4
insulate
8
Fig. 5. Magnetic boundary layer5 in the G.O. Roberts flow field. (a) Boundary condition\ in the cellul;~rgeometry. (b) The boundary layer problem us a heated, cooled and insulated semi-intinire rod.
traversal of the primary cell we must identify u(O,[) = u ( 8 , t ) and b,(O, 6) = b,(8,[). At the corners that connect the separatrices. the boundary layers thicken to be of width &'I4 in x and y and diffusion is subdominant to advection, with the result that the fields u and h, are continuous at a = 0, 2 , 4 and 6 (see [5I] for further discussion). In the eddies u is constant as there is no magnetic field, and in view of (4.82) u = 0 in the centre of the primary cell. Looking at the symmetries evident in Figures 4 and 5, and bearing in mind that 3 = y - 1112, the boundary conditions on cr and I?: are
and that (1, 6 , tend to zero for large 6. that is, towards the centre of the primary cell. With these boundary conditions, we note that given a solution r r ( a . 6)of (4.91), then the corresponding solution to (4.93) for / I , is just h, = $ K6iIErr. The problem for tr takes the form of a diffusion equation (4.91) with n , the coordinate around the primary cell. taking the role of time and spatial coordinate 6 3 0. The boundary conditions correspond to cooling the end of a semi-infinite bar to temperature o = TI/^ for time 0 < a 6 2. insulating it ti)r 2 n 3. heating the end to tr = n / 2 for 3 < n < 6. insulating it f i ~ r6 < a 8. and then repeating this process; see Figure 5(b). What is required is the periodic solution in time. corresponding to o periodic solution in n itround the primary cell. This can be solved by the Weiner-Hopf technique and is discussed in 1187).A key quantity that emerges t'rom the analysis is
<
< <
with
The integrands in (4.97) are constants as a varies over the ranges given (using (3.91). (4.94) and (4.96)). and correspond to the total heat in the bar during periods of time when the end is insulated. Now we need to obtain K (4.87) and a (4.86). The average (trrr,.) is dominated by the 8, which give equal contriboundary layers corresponding to 2 a 4 and 6 a butions. On 2 6 a 4, y = u ,. and also y d.1- d~ = d.s d$. Thus r r l c ,.d.v dy = tr ds d$ =
<
< <
< <
E'/*u ds d t . Now the integral of u over 6 is given above in (4.97) and is a constant on this separatrix, which is of length rr in s. Thus we obtain
We also require (b,u,) for u in (4.86). Now
using integration by parts, and so (b,u,.) = - K (uu,.). Finally we obtain the results for thealpha effect and effective scalar diffusivity,
For a passive scalar, the flux through molecular diffusion is E , and so the transport through boundary layers represents an enhancement of diffusion. As far as magnetic fields are concerned, the alpha effect given implies a growth rate for large-scale tield modes of wave number k. with
fro111(4.76) and (4.80). the lower sign correspontling lo modes of negittivc magnetic hclicity. The last term is of course subdominant. Reassuringly. K # 0 is recluired for dynamo i~ction.in keeping with anti-dynitmo Theorem 2 above. amplifying the tield may be seen from Figure 4. Given rnean The physical ~nechanis~n tield b i n the .r-direction, the flow draws out tongues of tield along the septtratrices i n the !-direction. The upward and downward flows are such as to lift +!-directed tield in the tongues up in the :-direction. while pushing down -!-directed tield. If the I;, tield depends on a large scale us 6 , = cosk:, the effect of the generation and motion of y-tield is to reinforce the tield component 6,. = sink:. This corresponds to a magnetic mode of negative helicity. This mechanism. identified by G.O. Roberts 11681. only lifts tield a distance of order F ' / ' in :. The reinforcement of large-scale fields is rather poor thr small F . and indeed the growth rate (4.102) increases as the scale in is reduced, by increasing X. The form of the growth rate (4.102) suggests that the maximum growth rate is A,,,;,, = 0(1 ). achieved at Xmax = O(F-'1'). However this is not a tirrn estimate. and further analysis has been done by Soward ( 1 87 1 to show that the rnaxi~nulngrowth rate is
All,,,,
=0
(
log log c : ' o g l ) .
,,.
c- 1/ 2
= o( J W )
This analysis involves a rather more delicate treatment of the boundary layers in which the vertical :-dependent structure of the tield now intrudes.
Dwicltno theory
403
The maximum growth rates for the G.O. Roberts dynamo decrease only very slowly as
+ 0 and so this dynamo is very close to being a fast dynamo, for which k,,, would remain of order unity as E + 0. The dynamo can be made into a fast dynamo by modifying
E.
the flow so as to introduce singularities at the hyperbolic stagnation points [ 1871; these have the effect of tearing fluid elements apart in a finite time at the corners of the cells. Further studies of the alpha effect and scalar transport in space-periodic steady flows u(.T,y ) may be found in 155,1941. In a numerical study of a space-periodic flow 11.571, modelling the core of a nuclear reactor, there is a competition between Ponomarenko modes, localised on closed stream surfaces, and (3.0. Roberts modes, localised on the network of separatrices. A similar competition is found in simulations of dynamo action in convection and driven fluid flows 11591; see Section 5.4 below.
5. The alpha effect and dynamo modelling In this section we leave systematic asymptotic analysis of dynamos, and discuss dynamo nlodelling and the magnetic fields of the Sun, Earth and galaxies. We return to the more mathematical topic of fast dynamos in Section 6 below.
We first met the nlpho ef'fect while studying the (3.0.Roberts dynamo. Using dimensional quantities, we may write down a general expression involving a weighted sill11 o f the helicity i n the 111odcsof the flow field.
which is valid when the magnetic Reynolds number K = r c o / k ~ l based on the scale of' the small-scale f o w is small. In a continuum limit of w w e vectors ( k . (0) and when the s~nall-scaleflow is isotropic. the tensor is isotropic. involving an integral over the hc.lic.itj, .sl~c~c.tr?H ~ l ,(~A . (0). A
11441. We also met the alpha effect for large K in the (3.0. Roberts I 1681 flow, but this was closely connected with the geometry of the separatrices. and unsurprisingly no formulae are known tor general flows. An alpha effect also appears in the theory of Braginsky 124.25) in which a magnetic tield is dominated by a strong axisymmetric component. and this is maintained by a flow with strong toroidal and weak poloidal components. In this theory the magnetic Reynolds number is large, and the non-axisymmetric parts of the flow and tield, while not dominant.
A . D. Gilbert
404
mean that the anti-dynamo Theorems 3 and 4 do not apply. We will not develop this theory here; see, for example, [26,144,188] and references therein. The idea of the alpha effect has been extremely important in the development of dynamo theory. While analysis of dynamo models such as those of Backus [9], Herzenberg [I 081, Lortz [I 351, Ponomarenko [I 581 and G.O. Roberts [ 1681 show that in principle dynamo action can occur, the fluid flows proposed are far from those in astrophysical bodies. However, the alpha effect involves averaging over small-scale flows, which could include turbulent convection. This new transport effect can both short-circuit detailed study of fluid flows and also bypass the anti-dynamo theorems. The introduction of this effect by Braginsky 124,251, Parker [I521 and Steenbeck et al. 11971 represented a breakthrough in the modelling of real astrophysical dynamos. An alpha effect may be argued on general grounds as follows (see, for example, [ 126, 1441). Consider a situation in which there is a magnetic field B on large scales L, and turbulence u on small scales I. We write the full magnetic field as B = B B' and take (B') = (u) = 0, where ( . ) represents an average over scales intermediate between L and I. Then the induction equation may be separated into a mean and a fluctuating part,
+
if, B' = V x (u x
B) + V x
(u x B' - (u x B')) + q ~ ' ~ ' .
(5.4)
The equation for B' is linear, with a source term involving B. Plainly B' and so the emf are linear functionals of B. If the field B is of large scale then an expansion of the form -
-
E; = a;,;H ;
+ J.liix if,:q, + . . .
is suggested. Here the alpha tensor a,, and the rclc/s t/jfi~r.vioritrrlsor J.lijl are in fact pseutlotelz.sors.
Without being too formal, we recall that vectors (and tensors) may be classitied according to how they transform under mirror inversion M : r + -r. This transformation has Jacobian 3= i ) ( M ( r ) ) / i ) ( r )= -I. Vectors are called poltrr if under mirror inversion they transform by multiplication by 3 = -I, so that they change sign: such vectors transform 'in the natural way' under M . Vectors are called tr.rirr/ if they transform by nlultiplication by -3 = I, so that they do not change sign. if it acquires an Similar deti nitions apply to tensors; a tensor is really a p.s~~ctlo-t~ri.sor extra factor of - I when it transforms in the natural manner under mirror inversion. In particular ~ ; , j k is a pseudo-tensor, and thus so is the binary operation x and the operator V x . Since electrical current is a polar vector, corresponding to the velocity of individual charged particles and V x B = pJ, it follows that the magnetic field is an axial vector. Plainly then, the mean emf = (u x B') is a polar vector, and so the tensor ol relating E (polar) to B (axial) in (5.5) above is a pseudo-tensor. The (second rank) alpha tensor a is a property of the underlying fluid flow u, and for this pseudo-tensor to be non-zero it is necessary that u should itself fail to be invariant under mirror inversion. The flow should thus lack purits invuriutlce, and the simplest way this can occur is if the flow possess helicity H = (u . V x u), another pseudo-scalar. The
flow can still be isotropic and homogeneous: indeed one can imagine a box full of isotropic homogeneous turbulence, helical in nature, generated by small propellers locally imparting helicity to the turbulent eddies [144]. Note that the third rank tensor3! , need not be zero for a parity invariant flow, as it could take the form Bjjx = B&jjk where fl is a scalar rather than a pseudo-scalar. Thus the connection between the alpha effect and helicity may be firmly cemented. The most natural way a flow can fail to be parity invariant is if it possesses helicity, and helicity can arise very naturally in rotating convecting bodies through the Coriolis term in the equation of motion. This tends to impart rotation to diverging or converging masses of fluid, for example, in convective plrrmes, and so to create correlations between vertical flow u and vertical vorticity w = V x u. Helicity however is not absolutely crucial to the alpha effect. One can write down flows for which all modes lack helicity, g ( k . w ) = 0, but for which an alpha effect can arise (at higher order than (5. I ) ) as the flow lacks parity invariance; see 176,941. The above general formulae for the alpha effect are limited to low R . Another approach is to study random, isotropic. hon~ogeneousflows u(x. t ) varying on a time scale T and length scale I . I n the limit of a short correlation time s u l l << 1 ,
with
I 197).Again there is a clear connection between the alpha eI.fect and the helicity of the How. The derivation of these formulae requires ensemble-averaging over random flows. which raises a number of issues of how enselnble averages may differ from typical realisations (for exiumple, 153.1 12.2 191). However. these forn~ulaeiippear to work well and the tensors have been computed numerically i n random flows by Kraichnan I17JI. The quantity cx in (5.7) has the dimensions of velocity and P of diffusivity, and so modelling of. tor example. the Solar dyn~umoit is natural to estimate cx -- VH and fi CV where V = Cl'T and I are typical velocity and time scales in the Solar convection zone. and H is now some dimensionless measure of how helical the flow is. Molecular diffusion '1 would then be negligible compared with P as in the Sun the magnetic Reynolds number R = C V I I Iis very large. Obviously cx may also vary in space. depending on the strength of the flow and the level of helicity. and we will discuss this further in Section 5.3 below. Let us however suppose for the moment that we are working in an infinite fluid flow and that cr and /3 are constants. Then, as we found above for the (3.0.Roberts flow, magnetic nodes have growth rates
-
where k is the wave number and the upper sign is for positive magnetic helicity. the lower for negative. This represents an a2-c!\ncr,no as in a magnetic mode with, say. k = k z . it is
406
A.D. Gilbert
solely the alpha effect that is responsible for generating x-field from y-field and vice versa. This will be contrasted with aw-dynamos below. Note that if a -- V H and B CV with H = 0 ( 1 ) , R >> 1 then it is clear that the most unstable magnetic modes will have length scale I l k C and growth rate A 7-I.These scales are independent of molecular diffusion, being based only on the length and time scales of the flow. However, none of the analytical models we have considered in Section 4 for large R have this property! The discontinuous Ponomarenko dynamo (Section 4.1 ) has growth rates of order I - ' ,and so does the (3.0. Roberts cellular flow, to within logarithmic terms. However the magnetic fields amplified are on very small scales, of order R - ' / ~ L . The smooth Ponomarenko dynamo has maximum growth rates of order R-'137-' on spatial scales of order R - ' ~ ' C . Of the dynamos we have met, none amplify fields at spatial scales C on time scales 7 .The reason is that diffusion is playing a crucial role in the amplification processes so far identified, in generating field components transverse to stream surfaces (most clearly seen in the smooth Ponomarenko dynamo of Section 4.2). For large R this mechanism becomes weak on large scales: large-scale modes grow slowly and the maximum growth rates occur for small-scale modes. We will take up this topic again when we discuss fast dynamos below in Section 6; for the present we simply note that to obtain fast amplification of large-scale fields a smooth flow u must have regions of chaotic stream lines, rather than laminar stream surfaces. Our focus has been on kinematic dynamo theory, in which the alpha effect is solely a functional of the flow field u. However, as a magnetic field grows, it will begin to affect the flow through the Lorentz force J x B, and in a simple two-scale picture, it can do so either by generating large-scale flows, or by changing the small-scale flow u. In the latter case it will modify the alpha effect, which will now be a function of B, as will other transport effects, including eddy diffusion and magnetic buoyancy. A typical alpha effect might be, for example,
-
-
-
where c is a constant. This formula reflects the notion that the alpha effect will be suppressed when the magnetic energy density of the mean field f p - l 1B12 is comparable with the kinetic energy density i p u 2 . Similar forms can be obtained analytically in certain limits, as Equation (4.7 1 ) mentioned above. We refer the reader to the books of Krause and Radler [I261 and Moffatt [ 1441 for references and discussion. The advantage of this modelling is that it allows dynamos to be constructed and simulated, building in many interesting and important physical effects, but without reference to the detailed and difficult fluid mechanics of highly turbulent flows interacting with magnetic fields. The corresponding disadvantage is that there is little to guide the choices of these transport effects, which in general are inhomogeneous, anisotropic and, of course, nonlinear in B. In the early nineties it was pointed out that the quenching of the alpha effect should not really depend so much on the large-scale field B and the corresponding Lorentz force J x B but instead on the Lorentz force for the fluctuating fields J' x B', which can be many times larger if the magnetic Reynolds number is large [45,107,206,208].The reason is that to reconnect magnetic field the scales need to be reduced to o ( R - ' /and ~ )if this is done
by drawing out loops of field, then this may lead to a much enhanced Lorentz force, with B' R'/?B.The alpha effect suggested by Vainshtein and Cattaneo [206] then takes the form
-
the extra factor of R meaning that the alpha effect is strongly suppressed for rather weak large-scale fields. This form has been supported by numerical simulations for large R by Brandenburg 1281 and Cattaneo and Hughes 1441. Since R is extremely large in many astrophysical bodies, for example, the Sun, this leads to many questions about how the dynamo can operate there, from the traditional viewpoint of modelling the Solar dynamo as an crw-dynamo, which we introduce below. These issues, of the strength and time scale of the Solar field, remain to be resolved. Also unclear is the dependence of the alpha etTect on the hydrodynamic Reynolds number, which is also extremely large in astrophysical objects, and the role of boundary conditions [23,29,3 1 , 2 121.
The alpha effect introduces a term (u x B') = crb into the induction equation, in the simplest. isotropic case in which cr is a scalar constant. This can amplify magnetic fields on its own. as above. or interact with differential rotation to give clylltr~rlorc.tr1.c.s 1144.152]. Let us drop the overline. and let B denote the mean field in this section only; then the induction equation with an alpha effect becomes
+
Consider a flow o f the form u = U i and field B = H i B / J .with BI, = V x ( A i ) (cf. (3.26)), and U , R and A all independent of y . The induction equation then becomes
(cf. (3.27), (4.84)).We now have in mind 11 as a turbulent diff'usivity, neglecting the molecular value at Iiu-ge R . The relative magnitude of the two source terms on the right-hand side of (5.13) is given by
If this ratio is small, then the BI) . V U term may be neglected and we return to an cr2dynamo, with U playing no role. If the ratio is large, then the a B r terms may be dropped in Equation (5.13) but crB retained in (5.12) to give an crw-dynamo. Here U generates a
408
A.D. Gilhrrr
strong toroidal field component B from Bp, while the alpha effect, necessary to avoid antidynamo Theorem 2, regenerates B p . Assuming a constant V U as well as a , the dispersion relation for magnetic modes,
in an aw-dynamo is
The presence of a fully complex frequency indicates the presence of waves and their direction of propagation is crucial in comparing with observations [152,2 161. Suppose we have a growing mode, with the upper sign taken so that Reh > 0, and let us take the fastest growing mode so that k is parallel or anti-parallel to 9 x V U . Then, bearing in mind (5.15), we see that if Im h > 0 the mode propagates in the -k-direction. The sign of Im h is the same as the sign of a k . j x V U ,and so we see that a growing dynamo mode propagates in the direction -ai x V U . In spherical geometry with 4 replacing s and ( r , 8 ) replacing ( x , z ) . this becomes -a4 x V O . where O ( r .H ) is the angular velocity. Here the flow U and dominant magnetic field B are toroidal. Note however that if there is also a strong poloidal flow up. which we have excluded. then the direction can be changed, simply by advection o f the magnetic field.
5.3. Tilo Solar c!\,rrtrrrro
The above theory of ao-dynamos is immediately relevant to the Sun. The Sun shows an I I-year dynamo cycle. although the period is strictly 22 years. with polarity changing at each half-cycle. This is evident in the distribution of Sun spots. Although these appear irregularly in space and time, when plotted as a function of latitude against time in a hrrtrrr~fi ditrgi.trr?l(for example, [ 144.2141) waves of Sun spot activity are evident. originating at latitudes of roughly 25'-3O0, and then propagating equator-wards in both hemispheres. The above modelling then suggests an ao-dynamo wave with these fields breaking thl-ough the surface because of magnetic buoyancy [ 1521. The toroidal field (given by B in the above analysis) is not directly visible from outside the Sun, but there is evidence that there is a strong coherent field in the Sun because of the Hale polarity laws. Regions of strong magnetic activity on the Sun's surface appear in pairs of opposite polarity, angled to lines of constant latitude. The magnetic polarities of the leading regions (in the sense of the rotation of the Sun) have. almost without exception. the same sign in the Northern hemisphere. and the opposite sign in the Southern. All the signs are reversed each dynamo cycle. This suggests two opposing and very coherent belts of field deeper in the Sun which propagate equator-wards. to be replaced by new. oppositely oriented fields each dynamo cycle. The traditional view of the Solar dynamo was that it is of a w type. with a > 0 in the Northern hemisphere and a < O in the Southern, and aS2lar < 0 throughout the convection zone, to give the correct migration of Sun spots towards equator (for example, ( 126,144,
1531). Numerical simulations showed waves, and the parameters can be adjusted to give an I I -year dynamo cycle matching the observations. There are many open questions, however. The alpha effect is usually thought to arise because of the influence of convection and the Coriolis effect. For example, at the base of a rising plume of hot fluid the fluid flow will be convergent and so will tend to rotate under the action of the Coriolis effect, giving Parker's famous picture of rising twisted loops of field, and helicity u . o.However, as a plume rises to the top of the convection zone it will expand, the flow will become divergent, and the rotation and helicity will be reversed. Thus it is not clear what the sign ofcr should be in each hemisphere, although by symmetry it should certainly be odd about the equator. Similarly on the surface of the Sun the angular velocity increases from the poles to the equator. In a rotating fluid sphere such as the Sun there is a tendency for motion to be independent of a coordinate along the axis of rotation, in so far as the Taylor-Proudman theorem applies, in which case this would suggest that i ) f 2 / 4 r > 0 inside the Sun. These concerns with the functioning of the Solar dynamo came to the fore with the numerical simulations of convection in spherical geometry by Gilman 1961, Gilman and Miller 1971 and Glatzmaier 198-1001. These studies did not include an alpha effect nor eddy diffusion. thus avoiding the modelling problems mentioned above. The simulations gave flows with i)R/i)t-> 0. in agreement with the surface rotation of the Sun but in contramigration diction to the theoretical framework. Together with this went a robust pol~-b~~rircl of magnetic fields. As 21 possible resolution ot' this paradox. these authors suggested that the Solar magnetic field might instead be generated at the base of the convection zone. at about 70% of the Solar radius, where convection overshoots into the stably stratified, radiative interior. At about the same rime, direct information about the internal structure of the Sun was becoming avail~tblethrough the science of helioseismology. involving the inversion of frequencies of the five-minute oscillations of the Soliu- surface. The rotation of the Sun leads to the splitting of frequencies between Eastward and Westward travelling modes of oscilIntion and so to information about the profile of differential rotation inside the Sun. The results (see. for example. 134.68.1061) are shown schen~aticallyin Figure 6. In the convection zone R = R(H).largely independent of radius r.. and i n the radiative Lone f 2 is approximately constant. At the interface of the two Lones is a thin layer. of no more than 5% of Solar radius, in which the angular velocity has large radial gradients. This is known as the ttrc.lioc.lirlc I I95.1961. The tachocline now seems a likely location for the Solar dynamo. iilthough the issue is far from settled. in particular how fields and fluid rnotions i n the tachocline and convection zone interact. In the tachocline there is intense shear and an alpha effect can arise frorn overshooting convection or other instabilities. These processes can provide an crw-dynamo giving equator-wards propagating dynamo waves. Indeed at lower latitudes and the equator i ) R / i ) r > 0. and so a negative alpha effect cu < 0 in the Northern hemisphere would then give the correct direction o f propagation. Another reason to think that rnuch of the magnetic field is localised in a thin layer is its coherence as evidenced by the Hale polarity laws. The new and surprising results from helioseismology have, by overturning the previous view of the Solar dynamo, acted as a stimulus t o theory and numerical studies. We can only mention a few current directions of research and a few references. One important issue is: how is the tachocline maintained. and what is its structure'? Spiegel and Zahn 1196) give
I pole (sub-rotation)
Fig. 0. Schctiia~tcSolar rotaltoti profile. Cotitotlr\ o i co!i\talil R ( r .0 )21-c\hewn: f? i\ la~-ge\l;II the equator and \ni;~lle\t;it the pole\.
a theory in which turbulence enhances hori~ontaltransport of angular momenturn at the top of the stably stratified radiative zone and this is responsible for maintaining the thin tachocline. On the other hand, Garaud 1841 argues that the observed protile is stable and so the enhanced transport will not take place. and Gough and McIntyre 1105 1 suggest that the presence of the tachocline indicates that there is a magnetic field trapped in the radiative interior of the Sun. Numerical simulations of convection in rotating spheres modelling the Sun and aimed at studying the structure of the tachocline are now being undertaken by Miesch et al. [ 1421. Assuming the existence of the tachocline, alpha effect dynamos have been constructed both as analytical models, and involving large-scale numerical simulations [59.60,154,163, 1761. A nonlinear tachocline dynamo represents a complicated fluid system. with stratification, shear, convection and magnetic tields all interacting. One important issue is, assuming the dynamo is of cuw type, how is the alpha effect generated? One. traditional, possibility is simply convection. However, other ways of generating complex fluid flows could include hydrodynamic instabilities of the stratitied shear flow, and even instabilities of the belts of toroidal magnetic field themselves 174.1 13,2021. In the latter case the resulting alpha effect would be a nonlinear function of the field. vanishing for fields below some instability threshold, and so would not explain the growth of magnetic field from a seed field, but instead might explain its present-day structure and behaviour. Numerical simulations building in and assessing the effects of these dynamo mechanisms are under way (for example, [61]). Another approach is to consider idealised fluid flows modelling the tachocline, but without building in any turbulent transport coefficients such as the alpha
effect; see, for example, [27,32,159,204],and the discussion of convective dynamos in the next section. Finally, the study of rro-dynamos in thin layers with space-dependent and nonlinear transport effects involves much interesting mathematics; see, for example, 113, 141,2051.
5.4. Convective dynamos und the geodynumo The most natural way to drive an astrophysical or planetary dynamo flow is through fluid convection. The scientific literature on convection is vast, and there are many possibilities that can be investigated. Perhaps the simplest relevant situation is a plane layer of fluid, heated from below, and rotating about a vertical axis with angular velocity f2.This geometry was considered in the limit of rapid rotation by Childress and Soward [54], Soward 11851 and Fautrelle and Childress 1711. These authors use multiple scale techniques, exploiting the natural scale separation between the depth, say h , of the fluid layer, and the width of the cells ~ ' / ~where h , the Ekman number E = v/2f2h2 << 1. A dynamo of rr2type is then obtained. A related study of Busse 1381 includes an imposed shear, to give instead an cuw-dynamo. The role of rotation is important here to make the convective flow sufficiently complicated for dynamo action to take place, in particular to obtain flows with helicity. I n a rotating layer even two-dimensional convecting rolls u(x. z , t ) can act as a dynamo just beyond onset, provided the nonlinearity o f t h e flow is taken into account so that the stream lines do not lie on plane surfaces, and so anti-dynamo Theorem 2 of Section 3.3 is avoided 1 1401. Other studies o f dynamo action in rotating plane layers include 127.1 15.201.2041. From this basic situation many further ingredients can be included. The geometry can be f'ully spherical or a spherical annulus (for example, ~109,220,2211). or a cylindrical annulus. For analytical studies of convection the intermediate configuration of the Busse ~ i n t ~ i i lhas i i ~ many advantages: here the fluid is confined to a cylindrical annulus with sloping top and bottom walls 137.41 1, and dynamo action has been studied by Kiln et al. [ 1 191. Many of these studies have been aimed at modelling the geodynanio, and other planetary dyna~iios.The most recent simulations, by Glatzmaier and Roberts [ 101-1041. are beginning to resemble observations of the geomagnetic tield, including reversals of the tield. However, the parameter space to be explored is very large. and the parameter values actually relevant to the Earth are uncertain, while at the same time unobtainable with brute force computations using current and foreseeable technology. For example the Ekman number is perhaps E = 0(IO-") 1721. Of course similar remarks apply about the accessibility of the parameters relevant to Solar and galactic dynamos, and even laboratory dynamos (which are characterised by a high Huid Reynolds number and so are highly turbulent). In geodynamo theory a number of important issues arise. The tirst is the role of the Taylor-Proudman theorem. which tends to constrain a non-magnetic flow to be twodimensional, independent of the coordinate along the axis of rotation. and the extent to which the magnetic field releases this constraint. A second issue is the source of energy for the geodynamo, whether arising from thermal convection, compositional convection, or from the precessional motion of the Earth. There is also the question of what role the solid iron inner core plays in the dynamo process, and its rotation rate. Finally we mention the problem of how the convection in the liquid core is coupled to the temperature
412
A.D. Gilbrn
Fig. 7. Geometry of the planar Ruid \ystern. (a) The plane layer: gravity a c t in the -:direction. The rotation vector S2 l i e in the (T. :)-plane, making an angle IY with the vertical. A velocity Uo in the r-direction i ~mposed at the ba\e of the layer. (h) Rotation o f ' a x r from (.r. y ) to (.i. ,?). with \. aligned with the roll axes.
distribution at the base of the mantle. We refer the reader to reviews by Braginsky 1261, Busse (401,Fearn 1721, Hollerbach [ l 101, Malkus 1 1391, Roberts [ 17 1 1, Roberts [ 1721 and Soward [ 1891 for further information on these and other topics. To illustrate convective dynamos. and how the kinematic dynamo mechanisms that form the heart of this review can be realised in physical fluid flows, we will here briefly discuss a kinematic dynamo model studied by Ponty et a]. I 1591. The fluid system is set up broadly to model the Solar tachocline at a co-latitude 1Y. and involves convection over a layer o f concentrated shear. A plane layer geometry is used with axes .r (East). ?. (North) and :vertical. depicted i n Figure 7(a). The plane layer is of depth h , with periodic boundary conditions in .\- and y. and rotates about the direction
corresponding to co-latitude 3 . The layer is heated from below with a temperature difference A T across the layer. If the system is non-dimensionalised using the layer depth and the thermal time-scale the governing equations nlay be written in the form
The problem is thus far specified by the five dimensionless parameters
413
& I I ~ I I I I ~ theory
which are the colatitude, Rayleigh number, square root of the Taylor number, Prandtl number and Roberts number. K is the thermal diffusivity, v the viscosity, 17 the magnetic diffusivity, and a the thermal coefficient of expansion. The Boussinesq approximation has been employed, and 6' is the deviation of the temperature field from a purely conductive state. In this dimensionless framework the layer is bounded by 0 6 :6 1 . At top and bottom boundaries H = 0 and the magnetic field is taken to obey insulating boundary conditions. S o far the system is subject only to convective instabilities. However to model the tachocline a layer of shear is introduced by imposing a bottom boundary condition that u = U o i in ; other dimensional units. In the dimensionless system this velocity becomes u = U o h / ~ xin words, we impose
with the Reynolds number Re as a further. sixth parameter in our problem. This parameter can take either sign. depending on the sign of Uo taken. Now with these boundary conditions there is a htrsir, .stutr u = u(:), H = 0 and B = 0, corresponding to a conductive, shear flow. For large rotation r >> 1 (and 9 # 1 ~ 1 2 )the . shear becomes localised in an Eknian layer a distance of order r - ' I 2 from the base of the plane layer.
There is no corresponding layer at := 1. There are then two bi~sicmechanisms by which this basic state can bifurcate to a more complex flow I1601. The tirst is through fluid convection. when the Rayleigh number Ktr is increased to a sufficiently large value. The second mechanism is an Ekman instability. which is a purely hydrodynamic instability that can occur when a Reynolds number KP* = Kr(2/s cos 8 ) ' 1 2 , based on the thickness of the Ekman layer, is sufticiently large 170,133. 1341. At onset of linear instability, since the basic state is independent of n and y, the most unstable mode will have a dependence exp(ikl.r ik2y A t ) with hori~ontalwave vector ( k I .k7), consisting of rolls whose axes are in the direction (k.. -kl ). If we detine new axes (.\-. .?. :) with .? along the roll axes then the fluid flow at onset is two-dimensional. u = u(.T. :. t ) , independent of .?: this is shown schematically in Figure 7(b). In this study the fluid flow is restricted to take this two-dimensional form also into nonlinear regimes. with the angle E between the x-axis and i-axis and the wavelength 2n/k,. in the .?-direction fixed from the linear stability analysis. The magnetic tield then can be taken to be of the form
+
+
allowing high-resolution, two-dirnensional computations at low ~iiagneticdiffusivities. Finally note that the magnetic Reynolds number R has to be defined, not as an input parameter to the problem, but as a diagnostic, since the velocities in the fluid layer depend
A.D. Gilbert
Fig. 8. Kinematic dynanio ill n flow generated by a saturated E k ~ i i a i~ihtitb~lily ~i with parameter\ give11 in (5.36). (5.27). In (a) the How V = u ; is shown. with the dark contour !liarking r r r = 0 . ( b ) the \tream function $ in a cornoving frame is \hewn. with the arrows marking velocity vector\, and ( c ) the ~nagnitudeIbl of the ~ilagnetic field. Reprinted from Journal of Fluid Mechanics. vol. 435 11591. With permi\\ion of Cambridge University Pre\s.
non-trivially on the driving from the temperature gradient Rrr > 0 or the shear Re > 0. If the measured dimensionless velocity is U = (u2)'1'. then we may define R = q U . The limit of large magnetic Reynolds number may be investigated by increasing q for a given flow field.
Fig. '1. Klncni;~ticdynarrlo in :I Ilow gcliel-ntcd hy it co~lvccti\cillhlitbilily \ * ~ t hpal-anlctcr given in (5.38).( 5 . 1 0 ) . In ( a ) the How V = o , is \howrl. with the dark contoul- ~iiorhingo ; = 0. t h ) rhc rrrc;im function i in a comovvelocity \,rc.tor\. :ind ( c ) Ihr ~ii;~gnitudc Ibl of the magnetic field. ing fr;ime I \ \hewn. with the arrow\ ~llarhi~lg Rcprlntcd froni Journal o f Fluid Mechanic\. vol. 435 1 1591. Wirh pcrml\\ion ofC;lmhridgc Univrr\iiy Pre\\.
Some results are shown in Figures 8 and 9. In the first of these,
A.D. Gilbert
The flow field is shown in Figure 8(a), (b) and is generated by a pure hydrodynamic instability, since Ru = 0. (a) shows the velocity u , along the roll axes, and (b) the stream function and velocity vectors in a frame moving with the instability. In this frame the flow is steady, and it can be seen that there is a hyperbolic stagnation point and separatrices. Note the strong Ekman layer at the base of the domain. For these parameter values the magnetic Reynolds number is measured to be R 2 3660, and the magnetic field for 1 = 1.2 is shown in Figure 8(c). with Jb(.f.z. t)l plotted against i and z. This hides somewhat the three-dimensional nature of the field as it averages over the 7 direction, but it is clear that the growing mode is connected with the separatrices, as in the (3.0. Roberts dynamo for large R (Section 4.4). Other steady flows or values of I can also give Ponomarenko type modes. localised within the recirculating eddies. In Figure 9, the parameter values are
t. = 2.33".
k,. = 4.3.
(5.29)
This is priniarily a convective instability, modified by shear, with Ro approximately 2 1 times the critical value for this rotation rate. The flow is now unsteady in any frame, and a snapshot of the flow fcld is shown in Figure B(a), (b). Thc magnetic Reynolds number is measured as K 2 5480 and the magnetic field is shown in Figure Y(c) for I = 8.5. This now takes the form of sheets of field accumulating near to the convective roll boundaries. This is typical of a fast dynamo operating in an unsteady flow (see Section 6.3 below) and there is evidence that the growth rate for I = 8.5 re~nainsof order unity as y, or equivalently R. is increased 11591. Another study of fast dynamo action in convective flows. this time in a Busse annulus geometry, is 1 1 191.
I t is known from studies of Faraday rotation. the rotation of polarization of electromagnetic radiation as it passes through a magnetic field, that galaxies possess magnetic fields. The issue of whether these fields are primordial, created when the galaxy was formed, or have been generated since remains a vexed one [I 271. Nonetheless, dynamo theories have been developed for galaxies of the crw type. Here differential rotation in galaxies provides the w-effect, while small-scale helical turbulence. arising from a combination of rotation and motion generated by supernovae explosions, provides the cr effect. At the level of crw-modelling. it is natural to start with a prescribed differential rotation and alpha effect, with the extra ingredient that now the geometry is that of a thin disc [ 199,2001. Such models have been analysed in detail by Soward [ 186,190,191 ] and are now being used to model the observed magnetic fields in galaxies; see reviews in [ 19,1791. Another situation i n which a disc geometry is relevant is the generation of magnetic fields
in accretion discs (see, for example, [33,43]); such fields can contribute to instabilities, turbulence and so eddy viscosity [ 101. These processes can modity the structure of the disc and the inflow of accreting material.
6. Fast dynamos We now return to kinematic dynamo theory in prescribed flows u and discuss fast dynamos, which have the property that the growth rate remains on the turn-overtime-scale of the flow at large magnetic Reynolds number. An extensive review of this topic i q given in [53].
Let us consider the dimensionless magnetic induction equation in the form
where H = F - ' is the magnetic Reynolds number and u is a How that is already known to be a clynamo. We let y ( ~be) the maximuni growth rate for a given c.. ni;txiniiscd over all sensible initial conditions. Thcri the flow u is a , f i l . v t c ! \ . r t ( ~ ~ ~ t o if y ( r ) is bounded above zero as F + 0 12071.If the growth rate tends to zero in the limit. the dynamo is . s l o n , . More tomially. we define
Here Bo is the initial magnetic ficld. B ( x . t ) is the tield at time I . and I I B J / '= ( B . B) gives the usual L' or energy norm. with
The initial condition should lie i n I-'. that is have tinite magnetic energy. The limiting f t r . v i ~ ! \ . I I ~ I I I I O,g~.o\~.tll rurtcJ is then given by yo = lim inf y ( c ) . 8
-0
<
and the dynamo is fast if yo > 0 and slow if yo 0 . I n terms of the dynamos we have already discussed. the discontinuous Ponomarenko dynamo (Section 4.1) is fast. The smooth G.O. Roberts dynamo is very close to being fast (see (4.103)). It is fast in the modified form studied by Soward IIX71. but this now tield now possesses singularities and is not smooth. Note that although these two flow fields are fast dynamos, they only amplify magnetic fields on small scales of order ~ ' 1 ' .and so have
4 18
A.D. Gilbert
been called intermediate dynamos by some authors 11471; we shall however not make this distinction. The key issues then are whether there are smooth flows that are fast dynamos, and whether fields on the scale of the flow can be amplified in this way. As we discussed in Section 5.1 below (5.8), assuming that an alpha effect and turbulent diffusivity are independent of molecular values but are based on the length and time scale of the flow implies that the flow is a fast dynamo. It is therefore incumbent on the theorist to find examples of flows u with this property. By means of additional general motivation, we note that in the Solar dynamo R 2: loX,and the field evolves on a time scale of months and years, very much shorter than the time scale of molecular diffusion. Although the Solar dynamo is nonlinearly equilibrated, whereas we are studying kinematic regimes, it is nevertheless necessary to understand fundamental mechanisms giving fast growth and fast evolution of magnetic fields.
The most famous picture of how a fast dynamo can operate is the .stretch-twi.sr,fi)ld (STF) dynamo. This goes back to Vainshtein and Zeldovich 12071, Zeldovich et al. 12 181 and even Alfven 121, and is depicted in Figure 10. We consider an initial flux tube (a) containing magnetic flux 00and energy Eo. This is stretched out (b), thus quadrupling the magnetic energy. I t is then twisted ( c ) and folded (d), to give a doubled tube. This has energy 4Eo. and the flux through a cross section o f the doubled-up tube is now 2 0 0 . Plainly, if the process
Fig. 10. The stretch-twist-fold procehs: an initial Hux tuhc ( a ) is stretched (b). twisted (c) and folded (d). to obtain a doubled Hux tube. A quadrupled Hux tube after 2 cycles (e): for clarity. only the centre line of the tube is \hewn.
Fig. I I . Folding o f magnetic tirld in the plane lead5 t o opposing tield which i \ vulnerable to diffucion, however weak.
is repeated, then after tl iterations the energy will be 2'" Eo and the total Hux through a cross section of the bundle of 2" tubes will be 2"@0. If we can ignore weak diffusion, the growth rate would be y = log?. where the time for one complete cycle is taken to be unity. There are many comments to make. First. in drawing this picture we have not written down the flow field explicitly, although this can be done 1145,209.210]. We have just used the fact that with F = 0 field lines are carried materially in the How. and stretched so as to conserve Huxes. Secondly, what is the effect of diffusion and why might we expect it to be unimportant in the limit F --, ():'The key aspect of the STF picture i \ that through threedimensional motion. magnetic field is brought into alignment. The tield i n the extended tube (Figure IO(d), ( e ) ) has a consistent direction, and any weak diffusion, causing the tield distribution to be smeared out. should rnake little difference to the total Hux across the bundle of tubes. This should remain 0 2 2"00. corresponding to a growth rate yo 2 log2. The important role of Hux here was stressed by Finn and Ott 17.51 and Bayly and Childress 117,181. This should be contrasted with the situation for Hows and magnetic fields l y i n ~~n a plane with u(x. y , I ) = ( u , v , 0 ) . B(.v. F.I ) = ( B , . H,.. 0 ) . for which the folding is always so as to bring opposing field together and there is no amplification of a coherent iund robust magnetic Hux; see Figure I I. In fact we know already that such a contiguration cannot be a dynamo, by anti-dynamo Theorems I and 2. We can write B = V x ( A i ) (cf. (3.26)) where the potential A(x. J, r ) obeys (3.27) (with 11 replaced by F). Suppose 0 < F: << I . then from some initial condition A is advected by (3.27). with diffusion negligible at the outset. As A is transported materially gradients of A increase, and so does the corresponding magnetic field. I t is important to note that although the tield increases. magnetic Ruxes through surfaces are bounded: this is depicted in Figure I I , where Huxes are given by the difference in values of A , bounded by the initial condition. This fi ne-scaling process continues until A varies on a scale of order E'''. at which point A disappears from (3.301, together with the magnetic tield. Physically. at this point weak diffusion is acting to eliminate the opposing bands of field. Note that if however diffusion is strictly zero, e = 0, then the increase in B and the magnetic energy continues indetinitely for any non-trivial flow: we can certainly have the growth rate for zero diffusion. y (0) > 0. while the limiting growth rate for weak diffusion. yo 0. This highlights the singular limit 7
<
'
420
A.D. Gilhrrf
of E + 0, and also the care which must be taken in applying the two non-commuting limits of large time t and small diffusivity E . In studying fast dynamos, we first must take lim,,, before applying lirn,,(). The third comment is that the STF picture is very simplified and idealised. Obviously the flow has a sense of helicity in view of the arrows giving the stretching and rotating parts of the motion in Figure IO(a), (b). This leads to twisting of field lines within the tube and local magnetic helicity, although the total magnetic helicity is conserved for E = 0; see [90, 1461 for further discussion. Thus the field structure is more complicated than that depicted. The other problem is that after one STF cycle the doubled-up tube will not exactly fit into the original tube. since we have in mind a smooth volume-preserving flow with V . u = 0. Instead with the tube of flux will be entrained flux-free fluid. As the STF process is iterated the cross section of the bundle of 2" tubes will increase until the hole in the centre of the bundle disappears. At this point it becomes important just how the STF flow is defined more globally. This problem arises in Vainshtein et al. 12091, where a flow modelling the STF moves is simulated numerically. Thus the STF dynamo is at this point only a picture of how fast dynamo action can occur. I t is nonetheless a very attractive picture, with all the correct ingredients. I n particular it stresses the importance of alignment of field lines so as to build up coherent magnetic flux, which is robust to the effects of weak diffusion. I t also highlights the importance of chaotic stretching in the How: the length of the tube doubles on each iteration of the STF moves, and this corresponds to a litlc-.stte~tc~llitrg e\porl~rlthli,,, = log 2.
Betore discussing further theory and co~~jectures o n fast dynamos. we will consider some examples that have been studied numerically. We will assume that Lagrangian chaos. that is the exponential stretching of magnetic field i n finite volumes. is important ~uidthis means that the flow must depend on at least three of the coordinates (.r. y. :. t ) . The class of Hows that are best studied, and easiest to simulate numerically. are those taking the form u(.r. J,. I ) . We can think of such a How as comprising two components: the horizontal part U H = (it,. 1 % . 0) and the :-independent vertical component L r ; . Examples of such Hows have been studied by Childress and Gilbert 1531. Galloway and Proctor 1831. Hollerbach et al. [ I I I I. Klapper 1 1301. Otani 11501 and Ponty et al. 1 161 1. and numerical evidence for fast dynamo action obtained. We shall focus on an example of Otani [ 1501. which we refer to as the MW +,flot~..
short for 'modulated waves. positive helicity'. This flow is obviously closely related to the (3.0. Roberts dynamo flow given in Equation (4.45) (with a translation in y). However, whereas the two Beltrami waves composing u in (4.35) are steady. now they are smoothly switched off and on, that is modulated. Otani's example is based on earlier dynamo models of Bayly and Childress [17,18], in which Beltrami waves and diffusion were applied alternately to the magnetic field.
Dyttrrnzo theory
Fig. 12. Po~ncarc'seclioll ol' ~ h cflow MW+. with 0
< . t . j. C 2n. Thc
hot.i/o~~ralline \hewn i\
('.
usetl fir
averaging licld.
Foci~ssingonly on the horizontal component of these flows, the effect of introducing weak timc-dependence to thc steady (3.0.Roberts flow (4.45) would be to break up the network of separatrices shown in Figure 3. These would be replaced by a network ot'chaotic layers. When the time-dependence is as strong as in (6.5).the network is rather wide. and the only obvious remnant of the original cellul;tr flow is modest islands of I-egi~larity. This is shown on ;I Poincark section in Figure 12. where the positions o f advected markers arc plotted every period of the How. Beci~usethe flow is independent of :we car1 titkc 0111 a normal ~ n o d ci n the magnetic tield with :-wave number k , and since the flow field is time-periodic a growing magnetic mode will take the Floquet form
where c is periodic in t with period In.and h is the kinematic dynamo growth rate. We s underlying flow. and take the magnetic held to have the same periodicity i n space i ~ the Figure 13 shows miiximu~nrnagnetic growth rates y = Re h plotted against logloE - ' with k = 0.8 153,l 501. The growth rate appears to saturate at around j/o 2 0.39 as the dil'l'usivity c + 0. There is thus cleiu. evidence of fast dynamo action. with growth rates apparently bounded above x r o in the limit of large magnetic Reynolds number K = F - I + co.Note also that this is for ti xed wave number k . and so the tield amplified has a large-scale component. Although the maximu~ngrowth rate saturates cleanly as E -+ 0. the eigenfunctions become ever more complicated. Figure 14 shows the eigenfunctions at a given instant. for E = 5 x 10p3 (left) and c = 5 x I O - ~(right); the numerical resolution of 512' is becoming evident on the second picture. Plotted is ( ~ h1' , lh,.l')"' (see (6.6)) as a grey-scale graph with white for zero and black for the maximum value.
+
122
A.D. Gilbert
Fig. 13. Growth rate y ( r . X ) plotted against l o g l o ~ . Ifor MW+ with X = O.X. Also \hewn are f (circle), hl,,, (square) and A [ , (triangle).
As the diffusion is decreased the eigenfunctions gain finer and tiner structure. The field takes the form of piles of sheets, which accumulate in the form ot'tcndri1.r about what was originally a hyperbolic stagnation point of the (3.0.Roberts flow, in the centre of Figure 12. From further analysis of the data (see 153. Section 2.3 1) it can be seen that the field direction in the central pile of sheets is largely coherent and directed along the sheets, although there are some sheets of opposite sign there also. The flow is thus bringing largely like-signed horizontnl held together. The vertical component 11, of the flow is playing a n important role here in largely avoiding the accumulation of tine opposing tields that would occur in its absence. The folding and vertical motion d o not give perfect alignment of field, as in the idealised STF picture, but still do rather well. In fact the dynamo mechanism is rather like that for the steady (3.0.Roberts flow diss l i ~ ~ ( ~ r 1 17.IXJ. Recall cussed in Section 4.4. and is known as the . s t r r t ~ ~ h ~ f i ~ l t l - .nirchrrni.nn from Figure 4 that i n this How tongues of tield are pulled out and then sheared by the component I ( , to reinforce a large-scale :-dependent tield. in fact a helical wave. These tongues are narrow, of order K-'/' = E " ' . Now in the chaotic MW+ flow folds of tield are also pulled out. but now by the chaotic stretching and folding. These can then be sheared in :. The difference. though. is that the chaotic stretching and folding operates on space and time scales of order unity (independent of E ) and so the tield that is reinforced has scales of order unity i n ;, and growth rates of order unity: a fast dynamo with a coherent large scale field. The value o f k can be tixed to optimise this effect, k 2 0.8. and give the maximum growth rate. In addition there are fluctuations. which contribute to the complexity of the field seen in Figure 14. However these are really just a distraction: the actual mechanism is reinforcing a large-scale held and these fluctuations are part of what is essentially a passive cascade of magnetic energy to diffusive scales of order E ' / ' 1 \ 4 5 ] . What determines the asymptotic growth rate y o ? The markers on the right-hand side of Figure 13 give some points of comparison. The lowest, triangle marker gives the Liapunov exponent A L of the flow, measured in the main chaotic region. and this, surprisingly, appears to lie below the asymptotic dynamo growth rate. The uppermost, square marker is
Dynamo theory
Fig. 14. Eigclifi~,~ction\ tiir k = 0 . 8 ant1 ( a ) r. = 5 x lo-.' (left) and (h) F = 5 x 10 (111, 1: + / h , 2 ) 1 / 4i \ plot~sd~ o t . 0< . A .Y 2n.
423
(right). Thc qlli~nttty
the rate of stretching o f material lines in the chaotic region, I i l i I l c ,and this exceeds yo, in keeping with u rigorous upper bound that we discuss below. The n~iddle.circle miirkcr in Figure 13 gives the growth riitc f of magnetic llux in the absence o f diffusion. F = 0. Recall that tot- the S T F model above i t was helpful to think in terms of the iunplilicntion of coherent fluxes. as these w o ~ ~ he l d robust to weak diffusio~i.For the M W + flow with c. = 0 iund ;I stlitable initial condition. the flux @ ( . ( I ) was measured through a surfr~cemarked by the line C' in Figure 13 i t ~ l extetiditig ~l in the :direction (see 1531. for further details). This Hux shows exponential growth and its growth rate f. obtained with :err) diffusion. appears to agree with the asymptotic growth rate yo in the liltrit of weak diffusion. in agreement with conjectures of Finn and Ott 1751. Bayly and Childrcss 117.IXI. Bayly 1151 and Childress and Gilbert 153). I t is important to stress that this agreement, although physically reasonable, is mathemntically non-trivial. With c. > 0 any suitable measurement of the field will give the satme growth rate y ( ~ ) . because the licld takes the limn of a growing eigcnfunction ( 6 . 6 )with iun underlying timcperiodic spatial form. This eigenfunction is smooth o n scales of 01-der c.'" 11451. given by the balance o f exponential stretching iund dil'fusion. However. for E = 0. the held never settles down. but is cascaded to finer and finer scales in the flow: in this case dit't'cl-ent quantities will generi~llyhave different growth rates. For example. for p l a n ~ ~flow r energy can grow exponentially. but Huxes cannot grow as they are bounded by initial conditions. as discussed in Section 6.2. We will return to discuss conjectures about magnetic flux and dynamo mechanisms below; here we finish the section by remarking o n other studies and simulations. The above flows of the form u ( x . y , t ) have received most attention as they are easy to simulate; furthermore the geometry of the How field is not too difficult to underst;und. being based on UH. The first flows to be studied for fast dynamo action however were steady threedimensional ABC flows u ( x . g , z ) , which we defined in Equation (4.44). Growth rates for A = B = C = I are summarised in Figure 15. Arnold and Korkina IS] studied this case and
1
2
3
Fig. 15. Dynamo growth rate y plotted against l o g l o R fix the A = B = C = I flow. The dnt;~points for R < 550 (times signs) are from Galloway and Firsch 1x21 and Arnold and Korkina IS], and the R = 1000 point (star) i \ from Lau and Finn [ 1.121. The flux growth rate f is \hewn by a circle and I I ~ ~ ,by , , a \quare.
exploited symmetry to reduce the scale of the numerical eigenvalue problem for dynamo growth rates. They found a window of dynamo action for 8.9 5 R 5 17.5 (with R = F ' ) ; further simulations by Galloway and Frisch 1821 found a second window of dynamo action for R 2 27 which has been traced up to R = 1000 by Lau and Finn 11321. Growth rates appear to asymptote around yo 2 0.77: which is less thim measurements of the linestretching exponent /I,,,,, (square on Figure 15). and glrtrtrr- than measurements of the flux growth rate r (circle) [53,871. This apparent lack of agreement appears to exist because the limit K --+ GO is numerically very delicate for this flow. The bands of chaos happen to be very fine in scale 1631, and it is far from clear that even R 2 1000 is yet in the asymptotic regime of large R . Clearer evidence for fast dynamo action in a steady three-dimensional flow, a Kolmogorov flow, has been obtained by Galloway and Proctor 1831, with reasonable agreement between flux growth rates and yo 15.71. Other studies of possible fast dynamos include a steady Beltra~niHow in a sphere by Zheligovsky 12221 and a How u(.x-.?., , . t ) . which is of ABC type but with additional time-dependence, by Brummell et al. 1.351 (see Section 6.6).
6.4. Upper c'lounds
Taking the limit of large magnetic Reynolds number brings the presence or otherwise of chaos in the fluid flow u into sharp focus. In the perfectly conducting limit R = GO, magnetic tield is carried as a passive vector tield, and magnetic tield lines are stretched in the same way as are material lines. This suggests that there may be upper bounds on the fast dynamo exponent yo involving the chaotic properties of the fluid flow. The first of such bounds was proved by Oseledets [ 1481 and Vishik 12 1 1 ] (see also [ 13 11) and is that yo cannot exceed the maximurn Lici/)unov exporlont A,,,;,, of the flow. To define this quantity
Dyncr17ro theors
425
consider a vector vo located at some initial point xo in the flow. It is carried, stretched and rotated by the flow and at time t is v(r), having length Iv(t)l. We then define A,,,,
= sup limsupt-' log(v(t)(. Vll.Xl1
ti00
Now as this is maximised over all initial vectors vo and initial points xo, a positive value does not indicate the presence of chaos in the flow. Indeed in the (3.0.Roberts cellular flow depicted in Figure 3, A,,,,, is positive because a vector vo located at a hyperbolic stagnation point xo, and pointing in the stretching direction there, increases exponentially in time. This bound does therefore not necessitate a chaotic flow u for fast dynamo action. Chaos is however required by the tighter upper bound, conjectured by Finn and Ott [ 7 5 ]and proved by Klapper and Young 11221, that for smooth flows yo is no greater than the ro,t>ologic~ilerltr-opyh . This is an appealing result, because for two-dimensional flows u(x, v . t ) , such as MW+ above, the topological entropy is equal to the line-stretching exponent, 11 = hl,,,,. To define hl,,, more formally. let Lo be an initial material line of finite length in the fluid How. We let it evolve, carried passively, for time t to give a line L ( t ) of length jL(t)[.Then we may detine I II~,,,, = SLIPl i m s u p t log(f,(t)I I.,)
I-%
This bound. yo 6 /I = II~,,,,.for smooth two-dimensional flows indicates that strong magnetic ticld linc stretching is an essential piut of any fast dyniumo process. In three dimensions /I is whichcvcr is greater: the rate of growth of lines or of surfi~ceareas. in the fluid flow. We may also detine the most c o ~ n ~ n o n used l y measure ofchaos. the Liapunov exponent A , , . Here we take an initial vector vo detiried at it !\'pic.trl point x,) and measure A,, = sup ~ i r n s u ~ t l-o' g ( v ( t ) ( . V,]
1
-
1J
Because this is not maxilnised over all initial points xo it is obviously less than (or possibly equal to) A,,,;,,. The Liapunov exponent A[, is also less than or equal to the line-stretching exponent /rI,,,,: the reason is that different averaging processes are occurring 1751. For Al, the notion of n typical point involves an average, over points in the chaotic volume under consideration. For hli,,,. we take the average length of a line L ( t ) which we can consider as composed of many vectors. Because we compute its length we are weighting the longer, more stretched vectors. This extra weighting gives a greater growth rate, hli,,, 3 A[, (with equality only if the stretching is uniform). Alternatively note that in computing log IL(t)l we surn the lengths of many vectors composing the line L ( t ) and then take the logarithm. whereas in computing Al, we take the logarithm of lengths of vectors tirst, and then average. The fact that the operations of averaging and taking logarithms do not commute explains why /I~,,,, and AL generally differ.
426
A.D. Gilb~rt
The result that h l i n ,is an upper bound on fast dynamo growth rates is confirmed for the MW flow in Figure 13 above and for other models (see [53]). On the other hand a positive Liapunov exponent AL tells us nothing about yo.
+
6.5. Fl~lxconjectures We know that the topological entropy h is an upper bound on yo, or for two-dimensional flows u(x, y, t ) simply the line-stretching exponent hline.Measuring h or hlinedoes not involve diffusion, just the chaotic properties of the flow field. The question then is: can the fast dynamo exponent yo itself somehow be related to the properties of the chaotic flow, without explicitly considering the limit E + O? Bearing in mind the STF picture, the natural quantity on which to focus is magnetic flux. The intuition here is that fluxes should be robust to diffusion, which will tend only to smear out magnetic field locally. There are two approaches: in the first, we take the asymptotic growth rate of magnetic flux
rr
through an open surface S in the How. If this growth rate fsis maximised over initial conditions Bo and surfi~cesS.then we obtain a Hux growth exponent f.The con.ject~~re of' Finn and Ott 175 1 is that f = yo. The second, related approach of Bayly and Childress 1 1 7.181 and Bayly 1 161. is to measure smooth linear functionuls of the field. Let B(x.0 be an evolving tield. from some initial condition Bo(x). Let us take a fixed smooth test function f(x) and define
( V being the volume in which the flow is defined). We then may define a flux growth rate by
f = sup lim s u p t I lop /of(t)l. f.HO r-w
One may then conjecture that provided f > 0, f = yo for smooth flows. There are a number o f other con.jectures, depending on whether the flow is smooth or not (see Section 4.1 of (531). Little has been proved mathematically, although Klapper 1\211 has used the idea of shadowing to prove results for hyperbolic systems. Finally, given that the fast dynamo growth rate is less than the topological entropy, it is of interest to try to quantify the difference /I - yo 3 0 in terms of the flow properties. Clearly, this difference is related t o the folding of field in the fluid Row, and how far short the flow falls of the idealised STF picture. which we recall had perfect alignment with h = yo = log2. There have been two approaches: one is to define a ccrncellrtiotz exporzetzt, which captures the tendency of the magnetic field to have alternating directions on fine scales for E = O and large time r [65,66,15 I ] .
Djntrmo rheorv
427
The second approach is to use methods based on periodic orbit sums. In a chaotic system there are many periodic orbits, and studying orbits of increasing period reveals increasingly detailed information about the properties of the flow. It is possible to recast the dynamo problem in terms of sums over periodic orbits [8,11,193];there is little rigorous theory as yet, but good numerical agreement in some examples. This approach has also led to some interesting conjectures by Oseledets [I491 about the relationship of yo with the ergodic properties of the fluid flow. Much of our limited theory on fast dynamos comes from the study of evolution of magnetic field under mappings, rather than fluid flows. Unfortunately we do not have space to do more than list a number of studies here. The simplest maps considered are baker's maps, and these can be used to model the STF dynamo picture 1751. To model the fast dynamo seen in Otani's flow above, Bay ly and Childress [ 17,181 introduced the idea of the stretchfold-shear (SFS) dynamo nlechanisnl and illustrated it with a very simple model based on a baker's map in the ( x , g)-plane and shear in the z-direction. This model is tantalisingly difficult to analyse, but progress has been made using the setting of Rugh 11771; see 191 I. Other dynamos that have been studied include maps on a two-torus 11491 and dynamos based on pseudo-Anosov maps with vertical shear (891. Further studies are reviewed in 1531.
While much is understood about last dynamos in kinematic regimes, little is known about how they eclililibrate in n nonlinear framework. Indeed i t is uncle~trwhether there is any meaningful distinction between slow and fast dynamos in a nonlinear regime: thc dctinitions rcfcr to kinematic growth rates, and in a saturator1 regime the growth rate i h zero regardless of how the field grew in the first place! Nonetheless. the study of fast dynamos has highlighted the importance of Lagrongian chaos in Ruid flows and consideration of dynamo mechiinisms at large K , and these form a basis for understanding nonline;u regimes. Here we highlight the issues and discuss the results of two recent numerical studies. The first issue is of setting up a suitable model. The flow is no longer prescribed but nus st be somehow driven. Possibilities include convective driving (see Section 5.4) or imposing a given body force, which is convenient especially when periodic geometry is used. Once the model is specified, a flow u will be driven: if it is sufficiently complicated and K is large enough. a magnetic tield will increase until the L o r e n t ~I'orce becomes important and the growth ceases. We may list a number of questions that can be asked: ( I ) What is the saturation level of the magnetic energy'? I f the magnetic field has structure over a range of scales, how much energy is in the large-scale tield. and how much at the smallest scales'.'This latter issue is important in comparing with observations. for example, the magnitude of the observed large-scale galactic and Solar tields [95,206,2081. (2) What is the mechanism t'or the equilibration of the magnetic field? For- exarnple. does equilibration occur because the magnetic tield suppresses the Huid How entirely, or because the stretching in the Huid flow is suppressed 146.1381. or because the folding of field in a fast dynamo is moditied to be less constructive? Or is it some messy combination of these processes?
(3) What is the time scale of evolution of a large-scale magnetic field in a saturation regime? This is a crucial question, and perhaps the possibility of fast evolution of a large-scale magnetic field is the only sensible counterpart to a fast dynamo in a nonlinear equilibrated regime. Can an equilibrated magnetic field substantially change its structure through reconnection on a tist advective time-scale? The question is important as the Solar field evolves on a rapid 1 I-year time-scale, and in this sense is a nonlinear fast dynamo. (4) Tied up with these earlier issues is: To what extent can the evolution of the saturated magnetic field be described by a nonlinear a effect (for example, (5.10)) and eddy diffusion, or other transport effects [44,107]? (5) Another set of problems involves spatially extended systems, again relevant to modelling astrophysical dynamos. For a fast dynamo, magnetic field is amplified with most rapid growth for fields on the same scale as the fluid flow (sometimes this is called the small-sctrlr dyr~nrno).If this fluid flow is of small scale in a spatially extended system (for example, small-scale convection or turbulence), the question then is how do fields evolve over these larger scales. One mechanism that has been observed is an inverse cascade [28,93,162],whereby after initial saturation on the scale of the flow, fields evolve to larger and larger scales until finally the dominant field is at the largest scale available in the system. Again there are important issues of time scales and saturation levels in such an inverse cascade. ( 6 ) The final question is: How do all these mechanisms and results depend o n the magnetic Reynolds number K and the Huid Reynolds number Kc.. bearing in mind that both are very large in astrophysical applications, with 1 << K << Kc for the Sun and I << Ke << R for the galaxy? Also: How much d o they depend o n the model and the forcing adopted? The problems of handling nonlinearity, as well as spatially extended systems and large parameters, R and Rr, make this research area mostly the domain of numerical simulations. We shall focus on two studies to illustrate recent results and some of the problems o f dealing with nonlinear dynamos. Our ti rst study is that of Brummell et al. 135). who solve the equations
The flow is driven by a prescribed body force F which is chosen so that in the absence of magnetic field, the How is
+ E sin R r ) + cos(y + c sin R t ) . sin(x + E sin R t ) + cos(: + E sin R t ) . sin(? + e sin R r ) + cos(x + E sin Q t ) ) .
uo = (sin(-
(6.16)
In fact the force is simply F = (a, - K r - ' v 2 ) u o (as the flow is Beltrami, uo x oo = 0). Note that this force is Reynolds number dependent; for steady flow E R = 0, F = o ( R ~ - I )
for all Re, while for unsteady flow el2 # 0. F = o ( R ~ - ' ) in the viscous limit Rr -+ 0 and F = O(1) in the inviscid limit Re -+ oo. The flow is parameterised by E and R , and reduces to a steady ABC flow (4.44) for ER = 0. Other studies of nonlinear ABC and related flows include 18 1,206,46,138]. This steady flow has only thin bands of chaos and small Liapunov exponents; however, as E and R are increased, the chaos fills out more and more of the ( 2 ~ ) domain ' and fills the whole region for E = R = I. These flows appear to show fast dynamo action, given the numerical limitations on the magnetic Reynolds number, R 6 170, and the growing magnetic field has a tube structure with a magnetic length-scale
as R is increased. The authors then explore the dynarnical regime for R and Rr varying up to 100. It is found that members of the above family of flows exhibit hydrodynamic instability for noderate values of Rc; however this instability is slowly growing compared with the kinematic evolution of the magnetic field. and so need not be considered. Runs show rapid growth and equilibration of magnetic field for F = R = I. When growth saturates, in physic~tlspace the magnetic field takes the forni of tubes and sheets with a greater tilling factor than in the kinematic regime. The length scale of the field increases, but still scales as IH cx K-'I2 (independently of Kc). This indicates that the tine scales of the magnetic tield are still controlled by a baliunce between ndvection ant1 diffusion. Onc important quantity to measure is the magnitude of the equilibrated magnetic energy. This shows two rcgiincs:
'
C? cx K1"Kcl~-
(K
< SO),
t' cc K r
'
( K 3 SO).
( 6 .I 8 )
In both cases the scaling is with Kc-'. suggesting thnt the primary balance is between the Lorentz force and the viscous term in the Navier-Stokes equation (6.13). The O ( K r p ' ) scaling law is very reasonable in the viscous limit Kc 4 0 in view of the scirling F = O(Re-I ) o f the driving force. However. a triunsition to itti Kc-independent scaling. in which the Lorentz force balances inertial terms, would be expected at some point as Kc, is increased to large values: recall that for large Kr the forcing is F = O( I ) for the case of an unsteady driven How. A similar scaling to (6.18) is found i n the theoretical study of the Ponomarenko dynarno 1 1 21, with & a R ~ / ~ Rfor, 'I << K << Re. resulting from the same balance. Numerical simulations of this Ponomi~renkodynamo show a siinilar behaviour but with shallower power laws 1621. As for the dependence on R in (6.18). in the first regime the energy increases with K suggesting a bnli~ncebetween the flows driven by the lo rent^ force on s~rtallscales of size Ill and the driven How u ~ with . J x B Re-'V'U and V 111' ~ ' 1 ' .In short. the field is acting primarily to suppress the fluid flow and this is the equilibration mechanism. In the second regime, the saturation independent of I? is interpreted by the authors as the magnetic tield acting to suppress the Lagrangian chaos in the flow. In nonlinear regimes. the magnetic field lines must be considered as elastic and resist stretching through the Lorentz force feedback on the flow. This in turn suppresses the separation of neighbouring fluid
-
-
--
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A. D. Gilbert
elements and so the Liapunov exponents of the flow. (Note that this refers to Lagrangian chaos, the separation of material particles carried in the flow, and is independent of the notion of Eulerian chaos, whereby the components of the flow at a given point in space may be chaotic functions of time.) Similar suppression of chaos has been studied by Cattaneo and Vainshtein [45] and Cattaneo et al. [46]. Maksymczuk and Gilbert [ 1381 found this mechanism in a truncated, viscous model but also found evidence of an inverse cascade, whereby after kinematic growth, large-scale magnetic field modes tended to become dominant and saturate through flow suppression rather than suppression of chaos, in the framework of Gilbert and Sulem [93]. It remains to be seen whether these results carry over to more realistic models. Another situation studied by Brummel et al. [35]is the case E = I, l2 = 2.5 in (6.16) and Re = R = 100, in which kinematic growth is seen and there is an initial phase of magnetic saturation. However, the field then decays exponentially. As a result, the simulation is left with a flow field ul forced by F, which is not a kinematic dynamo for these parameter values. The system has evolved from one purely hydrodynamic branch to another, mediated by a transitory magnetic field. The fact that nonlinear dynamos are nonlinear magnetohydrodynamic systems with many degrees of freedom means that there can be many solution branches and bifurcations, and this is what makes their study in any generality difficult and numerically time-consuming, especially in the demanding and important limits of large R and large Re. The second study we discuss is that of Brandenburg 12x1, who considers evolution of magnetic field i n a compressible flow. The aim is to study the effects of scale separation between the kinematic dynamo modes and the scale of the system. as far as this can currently he achieved numerically. The fields are governed by
Here p is the density, A is the vector potential for the magnetic field, c, is the sound speed. and pv,,, the dynamical viscosity. The domain is a (2nI3 periodic box. and the forcing employed is
Without going into the details, we mention that k(t) is a time-dependent wave-vector which varies in direction but whose magnitude is i n the band 4.5 < IkJ < 5.5, and the forcing is designed to be maximally helical, with ik x fk = lklfk at any time. In this way energy and helicity are injected into the flow field at wave numbers k = Ikl 2 5 . In one set of runs the magnetic and fluid Reynolds numbers are increased from R = Rr = 16 to 120, these values being based on the scale and magnitude of the forcing at k 2 5; the values are R = Re = 8 0 to 600, based on the scale of the system. The kinematic growth
rate increases as R = Re is increased. During the kinematic regime most of the magnetic energy is in scale k = 5 and smaller. However, as the Lorentz force becomes important, the magnetic energy transfers to larger scales in a (non-local) inverse cascade, and the energy in the smaller scales declines. This is even more pronounced in a further run when the forcing wave number is increased to k 30; there is a clear wave packet of magnetic energy travelling to larger scales. In both cases the field in the final state has a strong helical component on the scale k = I, and smaller scale excitation around the forcing wave numbers. The final state is similar to an alpha-effect dynamo in which a large-scale field is maintained by a mean emf. Measurements of the alpha effect suggest strong R-dependent quenching as in (5. lo), together with quenching of the turbulent magnetic diffusivity. The magnitude of the large-scale field in the final state appears to increase slowly with R. Although the kinematic growth of magnetic field occurs on rapid time-scales, during the saturation phase the large-scale tield evolves much more slowly. For increasing values of R, the time scale of evolution of the large-scale field increases. This has been interpreted as a consequence of a magnetic helicity constraint. Magnetic helicity is an invariant for a perfectly conducting fluid, for which R = co,and can only be destroyed by diffusive processes. In the limit of large R it is found that helicity is increasingly well conserved in MHD systems. Thus to generate a dominant helical large-scale field, which appears to be the ultimiite steady state of these systems. much helicity must be dissipated and this takes an increasing length of time. (The alternative that helicity of the opposite sign is stored in other scales ofthe field does not seem to be realised.) We refer the reader to 128.29.3 1.90. 2 12 1 for further discussion. These and other simulations give interesting pointers to phenomena in the nonlinear regime. However. the reader should always be aware that Reynolds numbers may be fir from asymptotic even in the largest current simulations. There is still much uncel-tainty about dynamo bchaviour f'or large K and large Kc. us studied in the computer laboratory. We only have observations of stellar and galactic dyniunos. One important point is that in a simulation such as 1281. the large-scale tields. once generated, appear to be dominant and evolve very slowly over scales closer to the ditl'usive time-scale R - ' than the turnover time-scale. There is little evidence yet of strong magnetic tields showing significant dynamical behaviour o n an advective time-scale in such simi~lations,even though this appears to be the case in the Sun. with the held reversing each I I years. Note that tor a comparison between different numerical studies the precise definition of the magnetic Reynolds number R = CV/r1 (and K e ) may be important: for example, the length scale of a 2ir-periodic box could be taken as C = I (the inverse of the smallest wave number) (for example, 13.51) or C = 2 n (for extunple. 1281). and this can niake a large and subjectively important difference i n quoted values of R or Rr. Also there may be different ways of def ning V. To further complicate matters. in a nonlinear simulation with a body force, it may be convenient to detine a magnetic Reynolds number based on This 11 is .based on the Row that would be the magnitude 3 of the forcing, R,: = 3 ~ ~ / , ~ driven in a balance between viscous terms and the forcing. and is a control parameter for the experimentalist. Alternatively, one could detine RF = based on a balance between forcing and inertial terms; in hydrodynamics the strength of a given forcing is similarly sometimes measured by a Grashof number, G = 3 C 3 / v 2 ;see 1561.
432
A.D. Gilbert
For the same forcing there may also be many different flows (for example, with or without a magnetic field present), giving different values for an effective magnetic Reynolds number R = CVIrl, based on the magnitude V of the actual flow u that is realised. Such an effective magnetic Reynolds number is now a diagnostic to be measured in the simulation. Similarly in a convective simulation, the control parameters might include a Rayleigh number, and the effective magnetic Reynolds number is then a diagnostic that depends on the form of the convection realised in a given run.
7. Open issues In this final section we set out a few open issues, based mostly on fast and Solar dynamos. ( I ) Leitnit~urdyr1crrno.s: Are there general formulae for alpha effects and growth rates at
large R for laminar two-dimensional flows with hyperbolic stagnation points joined by a network of separatrices? What about networks of separatrices in a bounded plane layer, as depicted in Figitre 8:' This would generalise the results ti)r the (3.0. Roberts ( 1681 How (Section 4.3) to stream line configurations that arise naturally in flows driven by convection or other instabilities. (2) Frrst c1yrrtrt~1o.s: Can fast dynamo action be proved for a given smooth How or class of flows? Recently Dr. Oleg Kozlovsky (personal communication) has proved rigorous results for fast dynamo action i n a steady three-dimensional How with a pretzel configuration. This stnooth How is carefully designed to allow tield vectors to be always brought into alignment, using rows of' stagnation points to cut and glue together field. and create what is basically a baker's map as a return map. A picture of how such :I dynamo might operate is given in Figure 7.1 1 o f 1531: as with STF. i t is easy to draw the picture but much more difticult to go from this to a precise mathematical result. Can fast dynamo action be shown to be a generic property in cer( 3 ) Ftrst c1~tltrr11o.v: tain classes of Hows u ? For example. if a two-dimensional time-dependent How u(.x.J. I ) (Section 6.3) fails to be a fast dynamo, with yo 0. is there a nearby How which is a fast dynamoq?Thishas been conjectured (see 153.751) and is true for many simplified models but, mathematically, little is known and obviously the above word 'nearby' needs to be specified. (4) Ftrst c1yltrrrlo.v: What can usefully be said in general about the nonlinear saturation of fast dynamos, or is this process highly model-dependent? What are the answers to the questions in Section 6.6. in particular how do saturation processes depend on the hydrodynamic Reynolds number? ( 5 ) Folditla: in the STF picture (Figure 10) a tube of tield is doubled up at each iteration to create a bundle of 2" subtubes arranged with a consistent orientation. Plainly this cannot be done indefinitely, as n + cm,for an initial tube of,fiirirc 1rolur7r' and volume-preserving periodic flow u. As the STF process is iterated, field-free fluid is entrained and the bundle of field becomes thicker until at some point the picture breaks down. However, can the STF process be achieved repeatedly with a .sirlgle magnetic field line? In other words, is the following possible? There is a volume preserving periodic flow u (or if preferred a volume-preserving map M ) detined inside the interior V
<
I)wtrrrrro theory
433
of a sphere S, a solid torus T c V and an initial closed curve Co c T which wraps once round the hole in T. Let the image of Co after n periods of the flow u be called C,,. Can i t be the case that for all n , C,, lies wholly inside T and wraps 2" times around T with consistent orientation? This is a question about the kinds of folding that are allowed in three dimensions, and we note that if the condition that the flow is volume-preserving is dropped, then this can be achieved by a map known as the solmoid. (6) Solrrr ~ i y n o m o :Why is the Solar magnetic field so coherent, obeying the Hale polarity laws to such a high degree, when usually at large magnetic Reynolds numbers dynamos tend to have strong fluctuating fields? Is this because the tachocline is a thin layer dynamo, with an effectively rather reduced value of the magnetic Reynolds number? (7) Soltrrclytltr~tio:How does the Solar field evolve so rapidly, on the 1 1 -year time-scale'? Although fast dynamo theory shows that kinematic dynamos can evolve on the turnover time-scale of the fluid motion, there is some evidence that nonlinearity through the Lorentz force tends to suppress reconnection. The very dramatic suppression of the alpha effect that has been proposed (see (5.10)) would tend to slow down the evolution of a large-scale equilibrated magnetic tield. Put differently, the Hale polarity laws indicate strong coherent belts of toroidal field. of opposite signs in the two hemispheres, presu~n;lhlylocated deep down in the tachocline. How are these destroyed in a matter ot'a few years. every I I years when the fields i n the dyn;umo cycle reverse? One possibility is that the dramatic i~pprcssionof the alpha eRi.ct that has been proposed and observed at large magnetic Reynolds number becomes moditied when thc fluitl Reynolds number is also large. In this case the magnetic field may drivc snlall-scale fluid flows that can tap magnetic energy, lead to accelerated reconnection processes and so ;in enhanced alpha cfl'ect. This regime is. however, diflicult to study analytically or nu~nerically. (8) P('trrtrtlig~ns:How do we go beyond the alpha effect'? This is beginning perhaps to outlive its usefulness, and it is now clear that to model the Solar dynamo and geodynamo it is necessary to take into account the actual Huid motions and the form of the convection. This is now becoming possible, in restricted regimes, in numerical simulations but we are lacking the theoretical tools to really understand the results. For example. consider the Solar dynamo: is it best to think of this as an cuct)dynamo. para~neterisingturbulent convection and large-scale circi~lations?Or are we now reaching the point where we should be considering a given. organised largescale How, for example a few giant convective cells overshooting into the tachocline. and trying to understand the effect of these flows o n the tield? Here the philosophy is more aligned with the study of laminar and fast dynamos, but based on realistic Hows and i~ltirnotelybringing in the nonlinear feedback through the Lorentz force. On the positive side, such a fluid dynamical viewpoint may get around issues such as the suppression of the alpha effect (see (5.10)): however, the mathematical analysis or numerical study of such models remains challenging.
Acknowledgements I arn grateful to Axel Brandenburg, Steve Childress, Chris Jones, Isaac Klapper, Paul Matthews, Helmut Moritz, Yannick Ponty, Andrew Soward, Andreas Tilgner, Steve Tobias and Keke Zhang for useful references and discussions. The pictures in Figures 8, 9 and 14 were obtained using the parallel computer 'ceres' at the University of Exeter, supported by the HEFCE JREI. Also, Figures 8 and 9 were kindly provided by Yannick Ponty whose work with the author and Andrew Soward was funded by the Leverhulme Trust. References 1 II
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[ 141 I N . Meunier. M.R.E. Proctor. D.D. Sokoloff. A.M. Soward and S.M. Tobias, A.vyrrrl)rorir. ~)rol)c,rfie.sc!fo r ~ o r ~ l i ~ rco~t ~ o r - ~ l y r rN.(~IY,: ~ r ~ ~ rPOI-iotl. o tir~r~~litirclc~ tlrrd I o t i ~ ~ r ~ l~i rI ~q ~) cr l~ ~ ~ rGeophy\, l ~ ~ ~ ~ cAstrophys. ~c~. Fluid Dynam. 86 ( 19977,249-285. 11121 M.S. Miewh. J.R. Elliott. J. Toolnrr. T L . Clune. G.A. C l a t ~ m a i e rand P A . Gillnan. T11rc.r-tlinrc~rrsiortc~l cl)l~eric~trl .sir~rrrltiriorr.\of Soltrr c.orn~c,c.fiorr. I. I ~ i f l c ~ r c ~ ~,ntcrfiorr ~ f i r r l trrltl ~)crftrrrro~oltifiorroc.hic,~ynani. 73 ( 1093). 133-145. [ 1501 N.F. 01:tlii. ,t k i ~ ~ ~ ~ r r ~IYII~IIIIO ( r f i c . III ~ I I ~ ~ - ~ I I I ~ I ~I ~~ II II .~ \ ~I ~~ I- I~ I /~~~~I ~ ~ ~J. ~ Fluid I I ~ / Mcch. ~ ~ I I I253 / / ~( IIL)93). I ~ . ~ . 327-3-10.
1 1 591 Y. Pol~ty.A.1). (iilbcrt ;11ii1 A . M . Sow;iriI. K i r ~ ~ ~ r r r(I\.r~o~rrro l r f i ~ ~ o1c.ft1111 irr /11r:y1, III(I,~II~~I~(. Rc,\.~~o/(l\ rrr~1)11~~~1~ ,PI\\.\clr-i~.~rr hv .\lrolo-crrrcl ~.orr~~c~c-rir~rr. J. I-luld Mcch. 435 (2001 1. 261-287. IIOOl Y. Pol~ty.A.1). (iilhert i111i1 A . M . SOW;II-~. 7710 o ~ r . \
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M.R.E. Proctor. (.orn>cc.riorr orrtl r ~ r c r g r r c ~ f o c ~ o r c i ~irr ~ ~ ,rrl~idlv ~ ~ f i o ~ r ,r)flrrir~,q.\~)lrc,~-c,.\.L.ecture\ on Solar ;111d Planetary Dynalnos. M.K.E. Proctor and A.D. Gllhcrt. cd\. C;ili~hrtdge University Pre\s. C;rmbridgc (1994). 97-1 15.
440
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Nicolis, eds. Reidel (19x7). 73-133. 1 1731 P.H. Roberts. Frrr~tlrrt~rrr~ttrl,\ of'i!\.tr~rttrothror:v. Lecture\ on Solar and Planetary Dynamo\. M.R.E. Proctor and A.D. Gilbert, eds. Cambridge Univer\ity Press. Cambridge ( 1994). 1-58, 11741 P.H. Roberta and A.M. Soward. I?\rrirtrro tltror\.. Ann. Rev. Fluid Mech. 24 (1992). 459-512. 117.51 M . N . Rosenhluth. H.L. Berk. I . Doxns and W. Honon. Efll,l.ri~,r tliflirriorr i t 1 Itr~tritrot.(.o~r~wc.til~t, /1011..\. Phys. Fluids 30 ( 19x7). 2636-2647. 1 1761 G. Riidiprr and A. Brandenburg. A Solrrr t!\.tt(rtrro i ~ thc r o~,or.shoorIrryrr: Cvr.1~/)criorl rrtril brrrtt~r;fl\ilirr~rrrttr.A m o n . A.;trophys. 296 ( 199.5). 557-566. 1 1771 H.H. Kugh. 711c. c.orrc~ltrtiotr.s/~o~.trrrttr l i r r Ir\./~~~rholic. rttro!\,/ic. 111tr1)s.Nonlinearity 5 ( 1992). 1237-1263, 117x1 A . A . Ruz~naikin.D.D. Sokolofl'and A.M. Shuhurov. A Iryrl,r)r~rrr,gtreti~. rc.w~~(ivt~rrtrro. J. Fluid Mrch. 197 ( 19x8). 3CJ-50. 117CJ1 A . A . Ru/nlaikin. D.D. 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Soward. 7'1ri11i l i s t . hirlorricrfic. uit)-rr1irt1rl.s I.LOIIS Ic~trgflrrt,iilts ~rlodc$,\. Geophy\. A\lrophy\. Fluid I > y r ~ a ~64 i l . ( 1002). 103-199. 1 1') I I A.M. Soward. 791111tli.rc. kirrc~t~rcrtic. a c o - t ~ r i ~ t / 11. ~ ~ Short l.\ 1c~r1,yrhsc.trlr, rrri~i/c,s. Geophys. Astrophys. Fluld D y n a ~ n64 . ( l9Y2). 201 -225. 11021 A . M . Soward. kc.\/ rl~t~ctttro.~. 12ect~~rt.s on S o l i ~ and r Planetary I>yn;~~no\. M.R.E. Proctor and A.D. Gilbert, c d . Ca~nhl.idgcUniversity Press. Cnn~hridge( 1993). IX 1-7 17. [ 193 1 A.M. Sowc:rd. 0 1 1 the trrl(, of .stngrrc~tiori~ ? ~ i rc t r r t l ~~c,riotli(. 1ror.ric.1~~ 11trr11.si r i tr r ~ ~ . o - c l i r r ~ c ~ t ~ .~\ ~i ir)t tl .~\ cl ,~r /l /lo~$,,/lr.\t (I~tr(rt~ri) ttri1(1~~1. Phys. D 76 ( 19'J4). IX 1-201. 1 1941 A . M . Sowt~rdand S. Childrehs, O r r ~ cttrtr,qt~c~ric~ , K(~ytroltl\trrt11111c~r tl~rtrtt~ro oc.tie~rtirr .\l)trtierl(~~>er.iotlic. flon ~ , i t hrtrc~otrrr~otiotr,Philos. Trans. Roy. Soc. London A 331 ( 1990). 639-733. [ 19.51 E.A. Spiegel, Tlic, t.lrtrotic. Soltrr c.yi.le. Lectures on Solar ;~ndPlanetary Dynamos, M.R.E. Proctor and A.D. Gilbert. eds. Cambridge University Press. Cambridge ( 1994). 245-26.5. [ I961 E.A. Spiegel and J.-P. Zahn, Tlrc, Sokrr rcrt~lzoc.liiic~. Astron. A\trophyh. 265 ( 1992). 1 0 6 1 14.
1 197 1
I I981 11991 12001 120 1 1 12021 12031 120.11 [2051 120hl (2071 12081 12001 12 101 12 I I
1
11I ?I 11131 I2IJI 121.51
11101 12 I7 1 1718I I2101 [2201 [ 22 I
1
12721 12231
M. Steenheck. F. Krause and K.-H. Riidler. A c~erlc.rrlertiorrof'thr rltnrrr rlec.tromotii~c~,fi~rc.r in erti elec~tric~oll~~ c.o~rtl~re.tittg flrtitl irr f~rrl~~rlrrrt trrotio~r,~orelrrtlrr irrf/urrtc.r r~f'Corioli.s~i),u~e.s. Z. Naturforqch. A 21 ( l966), 369-376. R. Stieglitz and U. Miiller. Er~~e~rit~rc~trfed tlc~r~tot~.st,rrtior~ of cr Irortro,qr~tc~or~~ f~~~o-sc.erlc, elytrtrr~ro.Phys. Fluids 13 (2001 ), 561-564. M. Stix. Tlrc, ~erlerc.tit.clyrrtrrrro. Astron. A\trophys. 42 ( 1975). 85-89. M. Stix. Errertrrrtr: Tlrc, ,qtrlerc.tic. el\rrertrro. A\tron. Artrophys. 68 ( 1978). 459. M.G. St. Pierre. TI~PI ~ I I J I field ,~ hrtrtrc.11of !Ire, CI~iIclrc~.~.s-So~~~crr-cI clv~~cr~~r~), Solar and Planetary Dynamo\. M.R.E. Proctor. P.C. Matthew\ and A.M. Rucklidge. eds. C;ltnbridge Univrr\ity Pre\s. Canlbridge (1993). 295-302. 1.-C. Thelen. /\ rrrc~err1c~le~c~fr~rr~~oti~.c. /21rc.c,irrclrtc~rtlh\. rtretnrphy>.1:luid I)yrl;~~ii. 7 3 ( 1003). 2 17-2.54. k.G. Zweihcl. Artrl~ilroleocli[fit.\r~~tr c1t.ifi.jotrcl (/\trtrnro\ ill t~r~-l~rrl~~rrr ,qcr.\c,.\. A\trophys. J . 329 ( 1088). 384391.
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CHAPTER 10
Water-Waves as a Spatial Dynamical System Frkdkric Dias
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I . Introductiot~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 440 ;i revel-\thlc Oy11;1tllic;11\y\te111 . . . . . . . . . . . . . . . 7 . ~I'w~-~I~III~II\IoII;IIII.. ~ ~ c l I ~w:~ter t i g W;IVC\ 2I I i I I I i yw I w i h ~~ I c c c ~ ' I It I I I I I - e . . . . . . . . . 446 7 .1.11~ c;~\e o l ; ~Iluitl 1.1yc.r hclow ;ill clastic. ice pl:ttc . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 7.3. TIic cit\e ol'two l i l y c ~ \w i l l ~ ( ~\II~(;ICC c~t tctisio~i . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 . with se~rliiee1c11\io11. . . . . . . . . . . . . . . . 7.4. I'IIcC;I\C o l ' o ~ i clitycr 01 i ~ l l i l ~ i depth tc . . . 2.5. .l'hc ciisc (11 cwo I.~ycr\. one hcitig t ~ ~ l i t i t t c tlccp l y . u i l l i o c l ~\clrfl cc or i ~ i ~ c r l t c itcll\lon ;~l 3 . TIIC lillc;~ri/cd proh1c.111. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Spcctretllt ol' tllc lil~c;~t-i/ctl opefittor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Phy\lc;~l \itu;ltioll\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Ccntcr l i l ; ~ n ~ l o lrcJt1ctio11 d (the linitc depth c;isc) . . . . . . . . . . . . . . . . . . . . 4 The litiitc tlcplh c;t\c via I-cvcr\lhlc t ~ o r t n i l~ol r ~ n s . . . . . . . . . . . . . . . . . . . . . . . . 4.1. The c;l\c c~l'otlclayer witliout \url';icc tctlsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. The cilsc (11 otic litycr with \ ~ l r I i ~ c~CII\IOII e 4.3. Tlic care (I l t w o I i ~ y c r willioul \ \urLicc tcn\ii~ti . . . . . . . . . . . . . . . . . . . . . . . . . 475 5 . The c;~\c o l ' i t ~ f i t ~ idepth tc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 5.1. Periodic wa\c\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .477 5.2. Nort11;tl (i)l.lil\ 111it~litiilctItt~ieti\~on\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4x0 5.3. Result\ (or the (icl )' rc\onancc with continuous \pcctriltn . . . . . . . . . . . . . . . . . . . . . . . JX I 5.4. A new rcvcr\ihlc h ~ f u r c . a [ i c I~?~~~i (11 r: cigc~ivaluc\tlivirig it1 rhc e\\cri!iiil \pcclrurll ~ I ~ r o i t glllc l i (11-igi~i487 6 . Stratitied lluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 7 . ~ r l ~ t - c c - d i ~ ~ ~ c ~11.ilvcllirlg i s i o t ~ ; ~water I wavcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4x5 7.1. Fortnul;~ticrn ;is a dy~iatnicalsy\le111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 7.2. S p c c t r u ~01' i ~ the liticari/cd opcrutor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4XX
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H A N D B O O K OF M A T H E M A T I C A L F L U I D DYNAMICS. VOL. U M E 11 Edited by S.J. Fricd1;indcr and D . Scrrc O 2003 Elsevier Science B.V. A l l right5 rcerved
7.3. Three-dimensional travelling waves periodic in the direction of propagation . . . . . . . . . . . . . 7.4. Dimension breaking bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. Two-diniensional standing wave problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Abstract The mathematical study of travelling waves, in the context of two-dimensional potential flows in one or several layers of perfect fluid(s) and in the presence of free surface and interfaces, can be formulated as an ill-posed evolution problem, where the horizontal space variable plays the role of "time". In the tinite depth case, the study of near equilibria waves reduces to a low-dimensional rc.ver.sil~lror-tiitittry tlifrret~riul ryutrtion. In most cases, it appears that the problem is aper-turl7ation of'ctn itltrgr-crhlr .yy.strrn, where all types of solutions are known. We describe the method of study and review typical results. In addition, we study the intinite depth limit. which is indeed a case of physical interest. In such a case, the above reduction technique fails because the linearized operator possesses an rs.sentiu1 sprctrrirn tilling the whole real axis, and new adapted tools are necessary. We also d i s c ~ ~ the s s latest results on the existence of travel1 ing waves in strati tied fluids and on three-dimensional travelling waves, in the same spirit of reversible dynamical systems. Finally. we review the recent results on the classical two-dimensional >tanding wave problem.
490 490 49 1 494
1. Introduction The present article focuses on classical problems in the theory of water waves. The topic of water waves is an old one and one can say that the theory of water waves was initiated by Stokes 11081 in 1847. What do we mean by the classical problem of water waves? We mean the problem consisting in solving the incompressible Euler equations in a domain bounded above by a free surface (the interface between air and water) and below by a solid boundary (the bottom). The bottom can be at any depth (even infinite). The driving force is due to gravity. The effects of surface tension might be equally important and can be included in the analysis. There may be several superposed layers of immiscible fluids, with free interfaces between them, and with or without interfacial tension there. What makes the water-wave problem so difficult is not its governing equation which is linear (Laplace's equation), but its two nonlinear boundary conditions on each free surface and interface. For a lot of coastal engineering applications. solutions given by the linearized water-wave problem are accurate enough (see ( 1051), but thr a number of practical applications the fully nonlinear problem rnust be solved. Moreover, the water-wave problem has attracted mathematicians for almost a century because of its extremely rich structure. I t is innportant to emphasize here that the present review is restricted to a mathematical point of view. The most recent reviews on water waves. which are less ~nathematical, are those of Hammack and Henderson 1491 on resonant interactions among surt'acc water waves. Banner and Peregrine 1 1 2.1 o n wave breaking i n deep water. Dias and Kharif 1301 o n the bifurcation. stability and evolution of w;iter w;\ves, Perlin and Schult/. I9XJ o n capillary effects on surface waves. Dunci~rn(391 on spilling bre;ikcrs. Recent rcvicws on numerical aspects :\re those of Tsai itntl Yuc I 1 171 ;ind Sc~trdovelliiuid Zalcski 1 1031 o n the direct numerical simulation of free-surfice and interfacial Hows. For ;\ complete bibliography o n the numcricnl coniputatiori of ~h~.ec-di~iiensio~i;~l water waves, one c;un rcti.r to the paper 1341. Our review is more in the spirit ol' the section entitled "Existence theorenis" in Wehausen and Laitone's contribution to the Encyclopedia ot' Physics I 1201. Since the woterwave problem is ;\ difficult ionl linear problem to solve. i~pproxi~natc theories have been developed. Most of these approximate theories are based o n perti~rbi~tion expi~nsionsand today perturbation expansions :ire still commonly used. When such approximate theories are used, it is tacitly assumed that there is an exact solution which is being approximated. Therefore existence and uniqueness prootk are an essential part of'exact water-wave theory. But such proofs h;~vcgenerally been difficult to establish. and have usually been obtained for only rather restricted. although physically important. situutions. N o attempt will be made t o give an exposition of all mathematical methods which have been used i n establishing the various existing theorems. Instead. the purpose of the present review is to show how c!\~rttrt~ric~trl .sJ.sfrttls rrrc~tllot1.s can be used to obtain result5 on the .vl~trtitrlhc~lltrl~ior c!f'trti11rllirlg ~ . t i ~ ~rlcrir c ~ . srltc. htr.sic ~rtltli.vtirr-1,ctl.frc.e sitt:firc.c .sttitc2.The water-wave problem can be viewed as a bifurcation problem which overlaps several important subjects: ( i ) elliptic partial differential equations in unbounded domains like strips, ( i i ) the theory of reversible systems in infinite dimensions, ( i i i ) the normal form technique, and (iv) the methods of analysis available for systems close to integrable ones.
The idea to use "dynamical" arguments for solving nonlinear elliptic problems in a strip was developed in the 1980s, pioneered by Kirchgassner 1731 (see [74] for a review on the water-wave problem). We said above that our point of view is mathematical. But even within this point of view, our review is rather restricted. Indeed only dynamical systems aspects of the "small solutions" of the steady water-wave problem are considered, except for the section on standing waves, where the flow is unsteady. When dealing with progressive waves, we d o not review the following aspects: Existence proofs based on methods of local analysis of the type of implicit functions theorem, including conformal mappings for reducing the problem to one of existence of a harmonic function satisfying nonlinear boundary conditions (see, for example, [8 1,94]), integral equation formulation (see, for example, [43]), Lyapunov-Schmidt method (see, for example, 1291); Existence proofs based on variational formulations (see, for example, 16,481); Mathematical results on approximate models or amplitude equations for water waves (see, for example, [ 181 for a review on model equations); Results on large-amplitude waves (see, for example, 17]), and results which rely on numerical arguments; Global results (see, for example, [77,1 16,SI); Stability results (see. for example. 188.2 I ] ) ; Existence (or non-existence) of solutions to the Cauchy problem for water waves (see the work of Wu [ 122.123 1). This is a still widely open problem. Existing results are valid o n a ti nitc (small) time interval and impose restrictions cither on the size or on the smoothness of the initial data; Fully rigorous derivations of nonlinear amplitude equations (see. for example, 1 103 1). Finally we deliberately present the problem in the,frrrriic~u.ork~ f ' r n ~ c ~ r - s ~~cc~tor,fic~lrl.s. ihlc thus not taking advantage of the Ha~niltonianstructure of the .sprrtitrl bc,irtrr-rc,tr12rproblen~. This Hamiltonian structure has been fully justified and is more and more used in the mathematical theory of water waves (see, for example. [47,481 and their bibliography). Our point of view allows a simpler presentation and the mathematical arguments we provide can be used in more general settings than just the Hamiltonian ones.
2. Two-dimensional travelling water waves as a reversible dynamical system 2.1. Tllc cu.se c!f'orlr Icryer ~ i t l oi r without .s~rr:firc.c)tori.\iori trc.tirlg ort tliefior .srtr:firc.r Consider tirst the case of one layer (thickness h ) of an inviscid fluid of density p under the influence of gravity g , with or without surface tension T acting on the free surface (see Figure I). The flow is assumed to be potential. We are interested in steady waves of permanent form, i.e., travelling waves with constant velocity c. In a reference frame moving with the wave, these solutions are steady in time, and we intend to consider the unbounded horizontal coordinate as "time". Natural scales are c. for the velocity, h for the length, and pc2 for the pressure. Once the equations are written in dimensionless form, two important parameters, one based on gravity and the other on surface tension, occur in
<
Fig. I . Sketch o f a wave travelling along the free wrt'ace o f a Ruid layer of' thickness h .
the equations:
h , (inverse of (Froude number)') '?
h=
and
('-
T 1) = - (Weber number). c,'
The choice of scales is not i~nique.For example, one could have chosen the capillary length 7'/(/)(,') or the gravity length as length scale. What is import~lntis that the equations depend only on two dimensionless parameters. In dimensionless ti~rm,the complex potential is denoted by I / ) ( [+ ill) and the complex velocity by I / ] ' ( [ + 111) =g-ic. The free surf~tccis denoted by 11 = %([). The Ei~lerequations are expressed here by the fitct that 111 is analytic in t = [ + ill and by the boi~ndary condition
-
II -
=0
at
11 =
- I (flat botto~ii).
the kinematic condition on the free surface 1rZ (0- = 0 -I
-
at
11 =
-
%(()
(free surface).
and the Bernoulli first integral on the free surface. expressing the condition that the pressure iump is proportional to the curvature.
I
,
+$)
+A?
-
hZ" -
( 1 + 2/2)3/'
=C
at
17
-
= Z([)
(free surface).
where C is a constant, and h = 0 i n the case of no surface tension. In order to formulate the problern as a dynarnical system. we tirst tlxnsform the unknown domain into a strip. There are different ways tor perfor~ningsuch a change of coordinates. We choose the one used by Levi-Civita 18 11.
R E M A R K .We could have chosen another change of coordinates. for instance the twodimensional version of the change of coordinates used in Section 7.1 for the three-
dimensional case or the intermediate choice made in [74]. The main advantage of the Levi-Civita variables is that they lead to a weakly nonlinear problem, instead of a true quasi-linear partial differential equation (PDE) problem. Moreover, a big part of the system stays linear, which is helpful.
+
The new unknown is cu i B as an analytic function of w = x the potential and the stream function respectively, and
+ iy, where x and y are
The free surface is given by y = 0 and the rigid bottom by y = - 1. The function cu represents the angle of the streamline with the horizontal axis, while B represents the logarithm of the velocity modulus. Observe that fixing p = 0 along the upper surface means that we impose the volume flux to be I in the moving frame, while we leave free the Bernoulli constant C. Our formulation uses an unknown vector function denoted by U , which satisfies a differential equation of the form
where
11
represents the set of parameters
2.1.1. Tllc c~r.ccr\'itllorit .sicr-:firc,ctetlsiolr ( h = 0 ) . In this case, U in ( I ) is detined by
where J . l ~ ( . r=) P(.r. 0). while
11 = h
and
where c u ~= cu 1 ,=o and a [ \ . = I = 0. The ti rst component of the system ( I ) is obtained after taking the derivative of the Bernoulli first integral with respect to .r. I t leads to an apparent loss of information. which in fact explains the arbitrary constant which appears in the analysis, corresponding to the Bernoulli constant. We observe that the free surface may be deduced from U by the formula (which detines Z(.r ) )
and
449
Water-waves as a .watial c!vnamical system
Equation (1) has to be u n d e r s t o o d in the Hilbert space 2-1I= IR × { L 2 ( - l , 0)} 2, and U ( x ) lies in D = ]R × {H I ( - 1,0)} 2 A {fl0 = / 3 ] y = 0 , Ody=-I = 0}, where H I ( - 1,0) denotes the Sobolev space of square integrable functions with a square integrable first derivative on the interval ( - 1,0). So, for a fixed x, the right-hand side of (2) is a function of y, and U (x) is required to satisfy the b o u n d a r y conditions indicated in ID. A solution of the waterwave p r o b l e m is any U 6 C°(D) f3 C 1 (]HI) which is solution of (1), where (e.g.) C ° m e a n s "continuous and b o u n d e d for x c IR". It is clear that U = 0 is a particular solution of (1), which c o r r e s p o n d s to the flat free surface state. Notice that we have a o n e - p a r a m e t e r family o f constant solutions U = (rio, O, rio) t with arbitrary rio. They give a flat free b o u n d a r y Z = e -t~° - 1 (:~ 0 in general), and c o r r e s p o n d to varying the Bernoulli constant C. An important property of (1) is its reversibility. Indeed, let us define the s y m m e t r y : S U = ([%, - ~ , 13)'. It is easy to see that the linear operator S a n t i c o m m u t e s with F()v, .). This reflects the invariance under reflexion s y m m e t r y x --+ - x of the problem. 2.1.2. The case with sur/ace tension (b :fl 0). by
[u(.,-)]t,,) = (o,,,c, i. o,(,, :,),
In this case, the variable U in ( I ) is defined
,'))'
and /, -I sinh/4o + ) v h - ' e
F(It, U ) =
at~ ~
}
IG'f!!l (e-t~coscv - I ) d y - t i e
- I < v <(),
t~o, (4)
-;57' w h e r e / , = ()v, b, cl ), hcl = C - I / 2 and fill = fl(x, 0). Equation ( 1) has to be understood in the Hilbert space H = IR × { L 2 ( - I , 0 ) } 2, and U ( x ) lies in D = R × { H L ( - I , O ) } 2 A {oql = O:ly=O, Otly=-i = 0}. The free surface is given by fornmla (3). Here again U = 0 is a particular solution of ( I ) for Cl = O, which corresponds to the ltat free surface state and ( 1 ) is reversible t, nder the synTnTetfy defined by: SU = (-m), -~, ~)'. Notice that for ct > ~ ( 1 + 2)v - 3L2/3), there are two families of " c o n j u g a t e " constant solutions U,.~ = (0. O, fill) I, where (I-
,,"")p,,'-"-
I / 2 ( l + , ' " " ) ] = ;,,',.
These constant solutions are such that the free surface is flat and Z = e -t~° - I (g: 0 in general). Fixing cl = 0 would mean that the velocity scale c we chose a priori corresponds to/31) = 0, i.e., U = (), while the conjugate flow has a different velocity given by
+ 8h)'I2- I I. Since we only study solutions of Equation (1) near U = 0, the occurrence of this conjugate flow would then only matter when h is close to 1. This means that for h close to I, h is not well-defined, since we arbitrarily choose one of the two conjugate flows for the choice of the velocity scale. This is why we prefer to keep cl as an additional parameter. efio = 2-I [(I
2.2. The cuse of ufluid layer helow, an elastic ice plute This case takes place, in particular, when a relatively thin layer of ice covers a large lake. The equations are the same as in the case with surface tension, except for the dynamic condition on the free surface. Assuming that the terms involving the ice thickness can be neglected. using Bernoulli's equation and following the approach of Forbes 1421 and Il'ichev and Kirchgsssner 1551, who model the ice sheet as a Kirchhoff-Love plate, one finds the following dynamic condition
f(e' + V') + ~2 + D
=C
-
at rj = Z ( 6 ) (interface),
(5)
where C is a constant and 11 is a dilnensionless parameter, given by
where E is the Young's modulus of the ice, / I , the ice thickness. and v the Poisson's ratio for ice. The scales tor length. pressure and velocity are the same as before. Again one ciui introduce the change of coordinates used by Levi-Civita
The interface is given by y = 0 and the rigid bottom by y = - 1. The variable U in ( I ) is defined by
and v,-Po,
I
wc-p,, cos a(,.
where p = (A, D, c l ) , D c l = C - 112, and Do = B(x, 0). The boundary conditions for U are a ( x . - I ) = 0 and a ( x , 0) = ag(x). The interface is given by formula (3). Here again U = 0 is a particular solution of (I ) for ( , I = 0, which corresponds to the flat ice sheet and ( I ) is rever.sihlr under the symmetry defined by:
The system ( I ) , (6) has to be understood in
W = IR3 x I L * ( - 1 , 0)l2,and U ( x ) lies in
2.3. The (,rise of'tkt301cryrl:s n~ithout,su~f;ic~~ trti.sion Let us now consider the two-layer case (densities p l , p,). assuming that there is no surface tension, neither at the free surface nor at the interface. At rest, the thickness of the upper layer is h2 while it is 11I for the bottom one (see Figure 2). The dimensionless parameters we find convenient are
The domain can he transformed into two superposed hori/ont;ll strips and we may i ~ s e the siumc type ot' variables as above, except that now the length scale is /I?. We use 11, os length scale in order to be able to take the limit 11I 4 cx, below. One difficulty is that the .v coordinate is not the s;umc in each strip! We choose as the basic .I- coordinate the onc' ,i" vcn by the bottom layer. This choice introduces a fr~ctori n the Cauchy-Ricmann eqi~;ttionsot' the upper layer sincc d.v2/d.vl = op20pPlcl,where F,O is the trace o f p i at y = 0. In such a forlnulation. the unknown is defined by
Fig. 2. Two \upcrpo\ed Hu~dlayer\ of tinire dcpth.
and the system has the form ( I ) with p = ( p , e , h) and
where we denote by ale the trace of a1 at = 0, and by a 2 1 , 8 2 1 the traces of a2 and at = I. Let W l l . ' ( l ) be the Sobolev space of integrable functions on the interval I . having integrable derivatives up to pth order on this interval. The basic Banach space W = R' x ( w I.' (-e, 0))' x ( w (0, 1 )}' requires more regularity than in the previous in the tirst component and the domain case, in order to be able to define the term $I,=() of the vector field F is now:
'. '
while the rrr~c,l:\il)ilit>..\\trlt~lctt:\. reads:
.
Notice that we do not use here Hilbert spaces: this is precisely due to the "bad" tr.1ce term ilu? ;r;I,.=o in thc fi rst component of the vector field (7). Another choice of space. such us H
'
instead of W 1 . I .would not lead to a good estimate of the resolvent operator (see below). The new system ( 1 ). ( 7 ) must be completed by the following two Bernoulli tirst integrals (interface and free surface):
which give the first two components of ( I ), (7) after differentiation. In principle we might choose t o treat this problem on a codimension-7 manifold. instead of expressing these tirst two components. It appears that it is easier to work as we do if we keep in mind that there are two known first integrals. This freedom on the choice of the two Bernoulli constants is due to the fact that the system ( 1 ), (7) has a two-parameter fiimily of constant solutions U = (PI(),PzI,0, Plo,0, PzI)' corresponding to uniform flows in each layer. with different velocities.
2.4. The cuse of one luyer ofinjinite depth, with surjuce tension Let us now consider the case of one infinitely thick layer of an inviscid fluid. The flow is still assumed to be potential (see Figure 3). The notation is the same as for the finite depth ~ . dimensionless parameter case, except that we choose a new length scale / = T / ~ cThe occurring in the equations now is
-
The free surface is denoted by rj = Z ( t ) ,and we have the boundary conditions
and the same (kinematic and dynamic) conditions on the free surface as for the one-layer case treated above. Using again the coordinates of Levi-Civita [ X I 1, the free surface is given by y = 0 , and the flow region by y < 0.In our formulation, the variable U is detined by
and i t is straightforward to obtain the system under the form ( I ). with
where EL = ( / L 1 . (.I ) and (.I is the Bernoulli constant. The system ( I ). ( 9 )has to be underlies) in D = R' x ( w ' . ' ( R - ) } '{~U O = stood i n the space = R2 x { I , ' ( R - ) ) ' .and u ( . ~ U J,.=()I. As above. a solution of the water-wave problem is any U E CO(UD)n C 1(Hi) which is solution of ( I ). ( 9 ) .
w
Fig. 3. One layer of infinite depth
It is clear that U = 0 is a particular solution of (I), (9) for cl = 0, which corresponds to the flat free surface state, and the system (I), (9) is again reversible, under the symmetry
SU = ( Z . -a(), -a, ,3)' Note that in the unbounded strip y < 0 the system (I), (9) is linear: this is very helpful for the study of solitary waves (see below and [65]) because it makes the choice of function space easier when the behavior at infinity in the x-coordinate has to be mixed with the behavior at infinity in the y-coordinate. Another remark for the one-layer case is that the problem can be reformulated into an integro-differential equation (after solving the linear problem in the region y < 0). However, one needs to specify from the beginning what type of solution is studied, and the choice of solutions is then restricted a priori. For instance, for periodic solutions one might use Fourier series spaces (see, for instance, 171 I), and for solitary waves one might use a Hilbert transform in the half space (see a remark in 1651). This possible reduction is not used here, since we want to apply our method to more general problems and solutions, especially to problems with several layers.
R E M A R K . We can consider the above case as the limit of the case of one layer as the depth tends to inti nity. For that it is necessary to write the system ( 1 ), (4) with the length scale I instead of h . The parameters now are / L I and b. This modifies the vector field (4); in particular, the strip ( - I . 0) is replaced by ( - Ilh. 0), and h + 0 :IS h + m.
We go back to the case with two superposed immiscible tluid layers. without s~~rfttce or interfacial tension, but we assume now that the bottom layer is infinitely deep. i.e., cJ = m (see Figure 4). The formulation is then identical to the one above with 11 = ( p . h ) and the interval ( k c . 0) replaced by R-.The domain of the vector tield is now detined by
and the new vector held is denoted by (7,). However. even though there are t~1.ocrthitrurr:~ Berno~llic.ori.vtcrrzts,there is only a ot1r-/~trrtr~r~~trr,fi1117iIy U = (0. fiz I . 0. 0. 0.Pzl 1' of constant solutions. This is due to the fact that we implicitly impose the boundary conditions ul andfll -+ Oas y + - a .
R E M A R K . The case of more than two superposed fluid layers. with or without surface and interfacial tensions, may be easily formulated in the same way (see. for instance. 1581 for two layers and various surface and interfacial conditions). Finally, it has to be understood that problem ( 1 ) is not a usual evolution problem: the irziticrl v c ~ l ~ ~ ~ l ~ r oish ill-l~osc~l! lc~til This is in fact an elliptic problem in the strip R x (- 1 . 0) for problems ( I). ( 2 ) and ( 1 ), (4)and ( I ) , ( 6 ) ,R x (-e. 1 ) for problem ( I), (7), R x R for problem ( 1 ), (9),R x (-a.1 ) for problem ( I ) . (7,). However, we treat this problem by local techniques of dynamical systems theory, following the idea introduced by Kirchgissner
Fig. 4. Two layers. one being intinitely deep
in 1731. In the,finite deptll ccl.se, the key feature is the possibility to apply a center ~?~c~tlifi)i)(ri into a r r ~ ~ r r s i horrlit~clry le d~for~'11fi~11 equution for the study of solutions staying close to 0. In the infinite depth case. we are still able to isolate a finite-dimensional-like filrnily of solutions, whose behavior as J.wJ -. co is different from the corresponding behavior in the finite depth case. Moreover. this introduces new types of bifurcations, not governed by a reduced finite-dimensional reversible ordinary differential equation (ODE). The aim of the next section is to show the properties of the linearized operator near 0, which clarify the study of local solutions in all cases and, in particular, allow the use of S L I C a~ center manifold reduction i n the ti nite depth case. rerluction
3. The linearized problem
Since we iu-c interested in solutions near 0. it is n a t i ~ r to ~ ~study l the li~lc;~ri/ed problem near0. Thih linc:iri~edsystcln rci~ds dU = I,,, U d.r in W.I n all problems rt'itlr ,/i~ritc-dc~ptlr Itr\c~t:s.i t can be shown that the spectrum of' the unbounded (closed) linear operator I.,, which is .\!.nrnretric.rtitlr riJ.~/)cc.t to hot11tr.\.iJ.s ot'thc complex plane beci~useof reversibility is composed of isolated eigcnvalucs of finite mult i n gat infinity. More precisely. denoting by i k thcsc eigcnvalucs tiplicities i ~ c c u ~ n ~ ~ l aonly (not necessarily purely irnagin~lry),one has the classical "dispersion rcl;ttion" s:itisficd by the eigenvalues. under the ti)r~iio f a complex equation A ( p . k ) = 0, analytic in ( p . A ) . The linearization of'the vector fields (2). (4). (6).(7) leads to the fi)llowing t/i.sl)c~~:siotr rolrrrion.~, respectively: k
-
A tanh
k
-
(A.
k
-
= 0.
+ bk2) tanh k = 0 . (A + ~ k ' )tanh k = 0.
p(h' - k') tanh(ke) tanhk
fork # 0. for k # 0. -
(k
-
Atanhk)[k
-
A.tanh(kc)]= 0 .
Fig. 5. Position of the four "critical" eigenvalue\ ik. close lo the imaginary ;)xis. for the vector tield lineari~ed from (4). depending on the parameter values ( h .A). Dashed curve\ correspond to the existence of'double nonLero real eigenvalues.
Note that we take t.1 = 0 for linearizing ( 4 )and ( 6 )since ( . I plays the role o f an additional parameter to be considered later. Any real solution k o f A ( p . k ) = 0 gives a pure imaginary eigenvalue ik o f the linear operator L,, . For a reason that we shall explain later. we are o.sl>et.irrl!\.irltrrc..stotl itr c~igrtl\~trlrrr.v rchic.h lic r~etrro r otl the ir/rtrgirltrr:\. tr.ri.\.. Indeed, there is only a small number o f eigenvalues on (or close t o ) the imaginary axis, the rest o f them being located in a sector ( i k E @: Ik,. < plk; I r ) o f the complex plane. For the system ( 1 ), ( 2 )A ( p . k ) = 0 is given by ( 1 I ) and we have the following situation: for h < I . the only (simple) eigenvalue on the imaginary axis is 0; for h > I, we have 0 and a pair of simple eigenvalues lying on the imaginary axis: and for h = 1 . O is a triple eigenvalue. All other eigenvalues are not close to the imaginary axis. except for h near I where t w o symmetric real eigenvalues tend towards 0 as h + I . For the system ( I ) , ( 4 ) A ( p , k ) = 0 is given by ( 1 2 ) and the eigenvalues close to the imaginary axis are described in Figure 5 (see 1741).T h e left side o f the curve f and the line A ( h = 1 ) in the parameter plane ( h . A) correspond to the occurrence o f double eigenvalues on the imaginary axis or at the origin, while the point (17. h ) = ( I / 3 , I ) corresponds to the occurrence o f a quadruple eigenvalue 0. T h e dashed curves correspond to the occurrence o f pairs o f double real eigenvalues, the most important being the curve closest to A (see Section 4.2.4). For the system ( I ) , ( 6 ) A ( p , k ) = 0 is given by ( 1 3 ) and the positions o f the "critical" eigenvalues are described in Figure 6 . T h e curve in the parameter plane ( D . h ) ( D is defined in Section 2.2) corresponds to a pair o f double eigenvalues on the imaginary axis (this pair tends towards 0, when D + m),while the line A ( A = 1 ) corresponds to a double eigenvalue at the origin, in addition to a pair o f simple eigenvalues on the irnaginary axis, tending towards 0 as D + m.
+
Fig. 6. Position ol'the tiur critical eigenvalues i k , close to the imaginary axis, for the systetn lineari~edfrom (6). depending on the paratnelrr v:~lues( I ) .A )
Fig. 7. l'osittot~oftlie crittc:~lci~c~~vitluc\ for thc hy\[c111litlcitri/,c~ll'rotll ( 7 ) .i l c p c ~ ~ co11 l ~the ~ i ~~ : I ~ 0 at lc~t\l~lt)uhlc.11 i \ ~ I I U ~ , . I , / > /011 for fixed v:tlt~c01. ,I. The ci~c~~villuc P the c~l!-ve\ A:.
I I ) ~ ~ (I c~ . IA- )\
For the system ( I ). ( 7 ) A(i4. k ) = 0 is given by (14) and 0 is always a doubleeipenvalue. except on the set given by
where 0 is n quadruple eigenvalue. The positions of the critical eigenvalues are shown in Figure 7. We observe that 0 is always an eigenvalue for the linearized vector fields ( 2 ) and (7). This is due to the freedom i n the choice of the Bernoulli constant, while for the fields (4) and (6) this happens only if h = I (we fixed the constant ( . I = 0). The roots k of the dispersion equations A ( p . k ) = 0 give the poles n = ik of the resolvent operator (01- L,, ) - I . In the case of an infinitely deep layer, for vector fields L,, U linearized from (9) and (7,) where ( . I = 0 in (9),the spectrum of L,, is as follows (see the proofs in 1651 and [SX]):
(i) there is a discrete set of isolated eigenvalues ik of finite multiplicities, which are given by the roots of the dispersion relation A(@,sgn(Re k)k) = 0; (ii) the entire real axis constitutes the "essentiul spectrum". Moreover, 0 is an eigenvalue embedded in this continuous part of the spectrum for (7,). More precisely, for (9) and (7,), respectively, we have for Re k > 0
A(/*, k ) = (h - k){Cp(h
+ k) - h.] tanhk + k],
which should be completed for Rek < 0 by A(/*, -k) in order to obtain the symmetric spectrum. The roots k of the dispersion equations A ( p , sgn(Rek)k) = 0, which are not purely imaginary or 0, give the poles a = ik of the resolvent operator (all - Ll,)-'. The nature of the point 0 of the spectrum is more delicate. When 0 is not an eigenvalue, the structure of the resolvent operator near O is the same as near the rest of the real line; in particular, the linear operator (all - L,,) for 0 real has a non-closed range, and the closure of the range has a codimension one for (9). When 0 is a simple eigenvalue, as for (7,), we are able to build a projection operator (on the one-dimensional kernel) commuting with Ll, and such that 0 is no longer an eigenvalue of L,, in the complementary invariant subspace. We are thus coming back to the previous case (9) (see details in 158)). Some natural questions arise. What really happens in the spectrum for the vector tields ( 4 ) and (7). when the bottom layer thickness grows towards infinity'? What are the physically realistic cases? I t is not difticult to see that as the depth is growing, there are more and more real eigenvalues trc.c~~r~ilultrri~lg rc~~qctlarly on t l ~ c~ ~ ~ h reti1 o l c tr.ris. the eigenvectors being bounded but not tending towards 0. In the limit, us we choose a basic space such that (cw. p ) -+ 0 as v + -a, all real eigenvalues disappear (except 0 for (7,)) but the spectrum stays as an e.s.se~ltitr/ .s/)ec.tricttl.Concerningthe physical relevance of studying the infinite depth case. we need to consider what are the characteristic scales of the problems.
In this section we consider typical physical situations. in connection with the theoretical results we mention in this paper. All waves are travelling in the physical direction 6 with :I constant velocity c.. We mention p~riotlic.w ~ r l ~ ~which . s , are indeed periodic in t,solittrry ~ ~ t 1 1 ~which es are waves loctrlizc~lin space, i.e.. tending to a flat pattern at infinity, and grllcr(rli:ed solittrry bvcules which are a sort of superposition of a periodic wave at infinity and of a localized wave at a finite distance. In the following examples we take a typical value of surface tension for an air-water interface: T / p = 74 cm3/s2. Ctrses in the,frume ofsvstern ( 1 ), (2).
(i) Tsunami: h = 4000 m (depth), L = 100 km (wave length). Hence k = 0.25, and the corresponding value of h. given by linear theory (I 1) is h = k/ tanhk 2 1. Note that
( I I ) corresponds to h = 0, which is a good approximation since here h = 0(1 o - ' ~ ) .
Solving for c gives c = 195 m/s and we are close to a solitary wave generation. (ii) Solitary waves in a wave tank: in this case, h = I0 cm, L = 250 cm, hence k = 0.25, h 2: 1 (as above), and this corresponds to c = 98 cm/s, while h = 0(1oP4). ( i i i ) Wind waves, generated by a storm in the ocean: h = I000 m, L = 150 m, hence k = 42, and the dispersion relation gives h -- k, corresponding to c = 15.2 m/s, while h = O ( l 0 " ' ) . We are then far from the solitary wave generation. Cuses irz flzefrcrmr of sy.stetns ( I ) . (4) o r ( I). (9). Here we note that fixing h fixes the ratio h / h since h/h = ~ / ~ g hThen, ' . once k is known, the dispersion equation ( 12) gives h and 17, and the corresponding velocity c of the waves. The dispersion relation ( 1 2) can be rewritten in physical variables as
For realistic water depths, the plot c ( L ) exhibits a minimum, say at L = L*. Waves with L iL* are LISU;III y referred to as c.cll~illtrryrc9trv(>s.while waves with L > L* are referred to as ~rrr\,ifyrc~rl~c,.v. For example, for h = 3 cm (and consequently b/h = 0.0084), one finds that L* 1.7 cm. Consequently, using the superscript * to denote values at I, = I * , we have k * = I I . and h* -- 5.5, 17' -- 0.046, c* = 23.2 cm/s. This point represents i n fact the occurrence o f double imaginary eigenvn1ue.s and belongs to the curve f in Figure 5. On the capilliuy side, a typical value is 1, = I cm. Hence k = 14, and h 4.7. 17 -- 0.04. ( . = 25 cm/s. The point i n the parameter plane (17. A ) is below the curve T . On the gravity side, a typical value is I, = 5 cm. Hence k = 3.8 and h 3.4. h 0 . 0 3 . (, = 29.5 cm/s. The point i n the parameter plane lies again below the curve f. Note that although u depth of'3 cm may appear relatively shallow, i t is in fact close lo the deep water case introduced in Section 2.4 when one is interested in the ( i y )' resonance (occurrence of double imaginary eigenvalues)! In deep water. the (icl)' resonance occurs for a speed (. = 23.2 cm/s and it wavelength I, = 1.73 cm. that is essentially the same values as for 11= 3 cm. This corresponds to the critical value / I I = 114 of the parameter / I I detined in (8). This is due to the fact that the capillary length scale I = T / ~ ( . 'is equal to 0.14 cm at the (icl)' resonance, which is much smaller than 3 cm. Experi~nentshave been performed by Longuet-Higgins and Zhang 1 1 26.861 in deep water near the (ill)' resonance. They show good agreement with the theory developed in Section 4.
--
-
-
-
Co.sos irr tlrc,fi.trtrre c?fs~.stcrrr( 1 ). (6). Several experiments on waves in an ice plate are reported in the book by Squire et al. 11061. ( i ) For the experiments o f Takizawa [ 1151. which took place in Lake Saroma in Hokkaido, Japan, the water depth is Ir = 6.8 rn. and other parameters are E = 5.1 x 10' ~ / m ' , v = 113, h i = 0.17 in. Then h / D is tixed. The speed (. was in the range 10-91 m/s. This range includes the speed (. = 6.09 m/s, which corresponds to the occurrence of the ( i q ) 2 resonance (point on the curve r in Figure 6 ) ( k = 2.27, h = 1.8, D = 0.02), as well as the speed (. = 8.2 m/s, which corresponds
to the occurrence of generalized solitary waves (point on the line A in Figure 6) (h = I, D 0.01). The wavelength of the ( i q ) 2 resonance is 18.8 m. (ii) For the experiments of Sq~lireet al. 11071, which took place in McMurdo Sound in Antarctica, the average water depth is 350 m. The other parameters are E = 4.2 x 10%/ni2, v = 0.3,/i1 = 1.6 m. The speed c was in the range [0-281 m/s. This range includes the speed c = 18.5 m/s, which corresponds to the occurrence of the (iq)2 resonance (point on the curve f in Figure 6 ) ( k = 0.038, h 10, D The wavelength of the ( i q ) 2 resonance is 165 m. The observations of Takizawa [I 151 and Squire et al. [I071 are in good agreement with the theoretical results developed in Section 4.
-
-
-
C~isesirl tlie fr(l111eof ( I ) , (7). A lot of experiments have been performed in the configuration of two superposed fluids. In the experiments of Michallet and BarthClemy [90], the fluids are water and petrol. The density ratio is p = 0.78. The total depth is 10 cm and the thickness ratio e = h I / h2 varies between 0.25 and 10. Depending on the initial conditions, both types of solitary waves (see Figure 7 ) (the 'fast' one bifurcating along h = A(;, and the 'slow' one bifurcating along h = h): can be observed experimentally. Taking P = 0.25 gives h: = 2 1.9 (that is c. = 18.9 cm/s) and h(7 = 0.83 (that is c. = 97.2 cm/s). Taking c = I0 gives h i = 4.9 (that is t . = 13.4 cm/s) and h, = 0.093 (that is r . = 98. I cm/s). In the case h = .A: there is an additional i111aginal.y eigenval~leik. For (.= 0.25. X. = 2 1.9, that is a wavelength of 2.3 cm. For c = 10, k = 4.9, that is a wavelength of 1.2 cm. Note that although Michallet and BarthClemy devoted their experiments to the 'slow' waves. they did not observe the inHuence of the extra imaginary eigenvalue (see Dias and Il'ichev 1331 for a discussion). Recent experiments in o three-layer configuration cle:u-ly show the presence o f generali~edsolitary waves (see I89I). Ccisc~.sill t/rc,fi.tirrlc~of' ( 1 ), (7,). The obvious example is the open ocean with a layer of warm water above the cold one. Take, for example, p = 0.998 for the density ratio and 112 = 100 n~ for the thickness of the upper (warm) layer. Then a critical value of h is I / ( 1 - p ) = 500. which corresponds to c. = 1.4 m/s, and the wave length of the ripples (corresponding to the pair of eigenvalues i i k . with k = h ) is 1.3 m. As far as we know. generalized solitary waves have not been observed in the open ocean. However. i n the course of their investigation of internal-wave disturbances. generated by stratitied Row over a sill, Farmer and Smith 141 1 observed waves resulting from the interaction between a long "internal" mode and a short "external" mode with the same phase speed.
After having discussed the physical relevance of the waves studied in this paper. we now focus on the reduction procedure. For all these problems one can obtain an estimate on the resolvent operator, of the form
for large Ikl, k E R, where L(W) is the space of bounded linear operators in MI (see, for example, [64,58]).It is fortunate that in all these problems the resolvent (ikII - L,,)-' can be obtained quasi-explicitly. especially in the problem (I), (7), because the "bad" trace term in the first component of L,, niakes it difficult to obtain an estimate such as (17) with a choice of basic space orher than the space W we chose. This estimate appears to be essential in our method of reduction to a (small-dimensional) center manifold. For the study of the nonlinear problem ( I ) the idea is now to use,,fi)r r h r j n i r r depth cuse, a cc.rrter r ~ ~ u r i ~ freduction i~ltl which leads to un ordirlury difibrrnrial ryu~rtionof small dimension. Let us assume that, for values of the (mu1ti)parameter p near po, the eigenvalues of L,, lie either in a small vertical strip centered on the imaginary axis, of width tending towards 0 for 11 -+ LLO. or at a distance of order 1 from the imaginary axis. Then the estimate (17) allows us to find a center manifold (see 173,92,1 19,761). Indeed, the nonlinear part N ( p , .) of the vector tield F ( p . .) maps analytically D into D for the vector tields ( 2 ) and (4). Hence, in such cases, the simple version of the result proved in 1 1 191 applies. For the vector field (7), the operator N ( p , .) ~napunalyticallyID into JHI,which is not a Hilbert space (however. still u Bannch space (which is not reflexive)) and is less "regular" than ID; so in this case we need the improved result of 1761 where it is needed to replilce, for instance. the space CO(ILl)by C 0 , ' ' 2 ( ~ )i.e.. , continuity in space D is replaced by Hiilder continuity (exponent 119) in this space. Rol~ghlyspe;lhing. all "small" houncled contini~oi~s solutions taking values in D of the systeni ( I ), for values of the (~nulti)p;~r;~~mcter 11 ncar / I ( ) . lie on an invariant manifold ,M,, which is smooth (howcvcr. we loosc the CZ rcgul:trity) and exists in a ncighborhootl ol' 0 intlcpcridcnt of /1. The dimension of M , , is equal to the sum of climcnsions of i~ivari:~n( subspaces belonging to plrrc im:rginary eigenvalucs for (he critic:~l vitluc /lo of the (multi)pnrametcr.In other words. the ~iiodcscorresponding to cigcnvi~lucsf;u from the imaginary axis are functions ("sli~vcs")ol'the modes belonging to cigcnvalucs near or o n the i~iiagiri;~ry axis. In addition, the rcvcrsibility property lerttls to a ~~itrrij/i)ltl \\.lric.lris irr\.trr.iorir irrrtlt~r.tlrc ru~~~c~r:sil~ilir\ .s\rrrrrrcJtr:\. S. The trace of the system ( I ) o n M , , is ;~lso ,u,\,c,r.\il~lc, irrrtlo- tlrc ,r.stric.tiorr SOol' the sy~iimctryS. At this point we shoi~lde11iph;lsi~ethat the physical relevance of this reduction process is linked with the distance o f the rest ofcigenvalues to the imaginary axis. So. this vi~lidity decays to 0 when the thickness of the bottom layer incrcases, and in such a case we have to think of illlother technique.
4. The finite depth case via reversible normal forms The aim ot'this section is to present a systematic method of study. valid in cases with linite depth layers: for instance. we niny use this method for solving cares with more than two superposed layers. with or without si~rfaceand intert'acial tension. We consider the tinite depth cases (2). (4), (6), ( 7 ) in studying the reduced ODE which gives all solutions staying near 0. These solutions lie on a low-dimensional center manifold. and this ODE is still reversible and its linear part contains the "critical" eigenvulues. Then. we use the rromnlrl ,fi)rrntec~hrriyic' (see, for instance, the book 1621. especially for reversible normal forms) to simplify the form of the leading orders of the Taylor expansion of the reduced vector tield.
We shall use a terminology of resonances due to Arnold [ I I] for describing the form of the reduced linearized operator which corresponds to the eigenvalues lying on the imaginary axis, at the critical value of the parameter.
4.1. The case of one luyer withollt surfLrlce tension Let us consider the system (I), (2). When h < I, and h not close to I, the center manifold is one-dimensional, and the only "small" solutions near 0 belong to the one-parameter family of constant solutions U = (Po. 0, Po)' = where 60 is the eigenvector belonging to the eigenvalue 0 of the linear operator L,, . When k > I , h not close to 1, the center manifold is 3-dimensional. A one-parameter family of periodic waves bifurcates from every constant solution. Along this branch, the amplitude increases, starting with amplitude 0 (the constant solution), and the wave length depends regularly on the square of the amplitude. This is a result analogous to the one given by the Lyapunov-Devaney theorem (see 1871 for Hamiltonian systems) for tinitedimensional reversible systems, despite the occurrence of the 0 eigenvalue. The analysis below gives all "small" solutions for h close to I , with some details. In particular, for h 2 1, we still obtain the above family of periodic solutions. The first rigorous results on periodic solutions, the so-called Stokes waves ( 1 847). are due to Levi-Civita 181 1 and Nekrasov 1941, and for the solitary wave (for h < I ), they are due to Lavrentiev 1801, Friedrichs and Hyers 14.31. Beale 1141. Below we leave free the Bernoulli constant, which allows a better understanding of thc family of solutions. The linearized operator obtained for h = I is denoted by L,,,,. Recall that 0 is a triple eigenvalue and there is no other eigenvalue on the imaginary axis. Let us denote by 40. 61.$2 the vectors in ID such that
we have
In the terminology of Arnold [ 1 I 1, this is a 03+resonance. The 3-dimensional center manifold is denoted as follows:
where P , , ~ , , =iso at least of second order in (A. B. C ) and PI,vanishes at 0. Here v ( p ) is a scalar regular function of the parameter defined below, with the help of the normal form, such that v = 0 for h = 1. It then results from normal form theory (see, for instance, 162, p. 25 and p. 3 I]) that we can choose the coordinates A, B , C by finding a suitable form for P,, up to a certain order,
such that for any fixed 11, the system reads
dC
- = B @ , , ( A ,B' - 2AC)
dx
+ B R c ( A , B',
C,v),
(20)
where @,, is a polynonlial in its arguments, of degree p in ( A , B , C ) , and R I ~Rc , are even in B , due to reversibility: the vector field anticommutes with (A. B ,C ) H ( A , - B . C ) . Moreover
holds. We may compute the principal part of the polynomial @,, (see. for instance. a similar computation in the appendix of 161 1):
In dl,we consider all coefficients ( ( 1 . h. c. . . .) as functions of 11 instead o f A . tor u better comfort. A nice property of ( 18). ( 19). (20) is that. if the higher order terms K R . K c . which arc sy.s/c,rlri.s irrtc:q,rrhlc. Indeed, we not in normal form are suppressed, then the "trrrrrc~trtrrl" have the two first integrals
where
For ( H . K ) f xed. all trajectories i n the ( A , H . C') space are given by
Fig. 8. Different gr~iphsof A
H
f ~ , u ( A for )
IJ
> 0 (left).
and
11
< 0 (right)
where
The corresponding curves LI2 = f H , K( A ) are deduced from Figure 8. depending on the sign of 1 1 and on the values of the frst integrals ( H . K ). In all cases we have a family of equilibria implicitly given by
which correspond to the curves in the ( H . K ) plane. These equilibria may be elliptic or hyperbolic depending on the branch T,, or GI where ( H . K ) is sitting. On the branch on the right, for v > 0 and H = K = 0, there is one solution homoclinic to 0.On the branches in the ( H . K ) planes for v > 0 or v < 0, the equilibria are lit~ritpoirlts denoted by TI, r ~ f ' l ~ o r n o c ~ l i tc~orrc..sl~or~tlir~~q lic~.~ fo .solittrry rtvr\3c'.s.Other s~nallbounded solutions are peri~ the curve intersects odic (cnoidal waves), corresponding to the positive part of f h . when transversally the axis R = 0. We can check that the solitcrt:~rc3trvc>.\ctre c?f'rle\~rfiorl: for I) > 0. H = K = 0.we have indeed for the principal part
hence
Fig. 9. Shape of the holitary wave in the case of one layer without surface tension.
Now, we need to prove that what is true on the normu1fi)rm of the reduced vector j e l d i s still t r ~ u ~ , f the i ) r fill1 vpc.torjeld. In particular the curve of symmetric equilibria persists (by implicit function theorem) and the two nonzero eigenvalues of the linearized operator are either purely imaginary (elliptic case), or real symmetric (hyperbolic case). There is one equilibrium for which 0 is a triple eigenvalue. There is a homoclinic connection to every hyperbolic equilibrium, as may be shown by taking the one-dimensional unstable manifold of each of these hyperbolic equilibria, and showing that this curve intersects transversally the plane of symmetric points B = 0,as for the normal form. Hence the trajectory obtained by completing in a symmetric way is an homoclinic curve. A complete proof of such a persistence in the three-dimensional phase space may be found, for instance, in 161 1. We can sum up these results by the following theorem:
THEOREM I . A.s.srrrnethrrt rr 3-t/it?reti.siotrrr/rr\~er.si/?/~ rl~c.tor,fir/r/ hrtviir,~t r , f i u ~ fl?oitrt / rrt 1irr.s ( 1 03+ro.~orrt~trc~e~. T/I(JII /I10 pIrr~.sr) 1)ot-frrlit/rc)rli.0, of tho \ ~ r ~ ( ~ t o r , f i t ~(l,fi.re~/ / ~ / , / i ) r\Y,IL,P (ir(~trr 0 ) (?ft110I ) ~ / i r t ~ ~ r tl)r~r(ritte~t(,t; iot~ is ~qotr(~ri(~~rI/y the, . s ( / t t r ( ~rr.s,fi)t.the) t r o r t ~ ~ ( ~ / , f(i )Is), rti~ ( 19). (20) tnit~c~rtc,t/ trt c/r~rrr/rurtic~ orulo:
0
C O K O I . I . A K Y2. 771~' trl)o\'c,tlrc~or~~ttr ol)/)lic~.s,/i)r tk~.rc,ri/~itr,q tlrc ~ I Y I I Y ~ IHIYII I\ ~, O~.\olr~tiotr.s I ~ I ,s/)(r(~, ~ J D ~ / ' t I r cn ~ r t b~1 r7 1 \ ~Prol~l(~ttr 2. I
. I uitrite, (lc,,~)il~, tro .srrt;jit(~~ t(~ir.siotr) trc~rrt11o
c~t/itiliht~irrttt. , / i ) t . h 11tJ(0.I. R I : . M A K K .Note that U = 0 corresponds here to a uniform flow of velocity (.. and that other constant solutions near 0 correspond in fact to i~nifol-mflows moving at a velocity slightly different than c.. Since c . was chosen arbitrarily as the velocity scale. solitary waves corresponding to solutions homoclinic to nonLero constant solutions are in fact honioclinic solutions to 0 . with the right choice of the velocity scale. Therefore. there is a unique form of solitary waves (see Figure 9), duc to the arbitrariness o f the delinilion of the panumeter h (with an arbitrary (.!).
Let us consider the system ( 1 ), (4). The eigenvalues of the linear operator L,, (for ( . I = 0) are given by rr = i k , with X satisfying (12). and Figure 5 gives the position of the foulclosest eigenvalues to the imaginary axis. For the study of solutions o f ( I ) near 0 , there are three main "interesting" cases to be considered: ( i ) (1). h ) is near A + = {h.= 1 . h > 1/31: in this case. L,,,, has only a double 0 eigenvalue on the imaginary axis (02+resonance), ( i i ) ( b ,h ) is near A - = {h = 1 . 0 c h < 1/31: in this case. L,,,, has a double 0 eigenvalue, and a pair of simple imaginary eigenvalues +ic/ on the imaginary axis ( 0 2 + ( i r l )resonance),
(iii) (b, h) is near f (left part); in this case L,,,, has only a pair of double imaginary eigenvalues hi y on the imaginary axis ( ( iy)2 resonance). The system ( I ) , (6) leads to cases (ii) and (iii) as well, and can be treated similarly. When (19, h) is close to ( 113. 1 ) a specific study is needed, because at this point the eigenvalue 0 is quadruple (see 1561 for this case). There are other interesting cases, for instance when one has two pairs of resonating eigenvalues on the imaginary axis (the ( i q ) ( 2 i q ) resonance is the most special because of the occurrence of heteroclinics between periodic solutions, see [13]). However, we shall not detail their study here, since it is always in the same spirit, and we restrict our presentation to the most typical cases. 4.2.1. Case (i): 02+ wsnr1nnc.e. This case was first solved by Amick and Kirchgiissner (41. I t is also studied in particular in the papers [74,101,64]. The method used in 1641 is the ~ 0). Here the center manone we present here (however without fixing the parameter t . = ifold is two-dimensional. Let us define by ( A , B) the (real) coordinates (or "amplitudes") associated with the choice of eigenvectors
E o = ( 0 . 0 . I)'.
El
=(-
1.-(y+
l),O)'
Then. we need to know how the revcrsibility symmetry So acts on ( A . 8 ) . There are two theoretical possibilities: ( A , H) + ( A . -H) or ( - A . B). Here, as in all water-wave problems, the first case holds. This is the 0" resonance. Then the normal form (see. for instance. 1621).truncated at leading orders. reads
where one can compute explicitly the coefticients (see. for instance. 1641) as functions of the parameters:
We notice the blow-up of the coefticients when b tends towards 11.3. due to the change of dimension of the central system at this point ( i t becomes 4-dimensional). Here, the two conjugate equilibria (both corresponding to a flat free surface) are denoted by A- < A + . They exist provided that ( . I > -(A - 1 )'/(617). The equilibrium A - is hyperbolic while the equilibrium A+ is elliptic. The vector field (24) is integrable. and its phase portrait is given in Figure 10. For any fixed C I , there is a one-parameter family of periodic solutions. and a solution homoclinic to the hyperbolic equilibrium. All these solutions disappear after the
Fig. 10. Phase portrait of' the 2D vector held (24) [casc(i)l. for 1 . 1 z ( A - 1 ) ? / ( 6 h )
saddle-node bifurcation when c.1 < - ( h - 1 ) ' / ( 6 h ) .For c.1 = 0 and h > I . the homoclinic Irolution ot' the truncated IryIrtcm is given hy
Because our system is two-dimensional iuid revcrsihlc. i t is easy to show that these phase portraits fully persist for the complete ay.stem. We sum LIPthese results in Ihe following theorem:
We observe that A (.r ) > A - , hence the homoclinic solution corresponds to a ",solittrr~~ 1 1 ) for the problem ( 1 ). (4). whose principal part follows directly from (25). wcrvr " r!f'~le/)rc..s.sior~ (see Figure
R E M A R K .Fixing C I = 0 leads to an artificial distinction between the cases h > 1 and h < I, since this is just a matter of choosing the suitable constant flow for the velocity scaling (the one which is an hyperbolic equilibrium). 4.2.2. Ccrsr (ii): 02+(ill) rrson'lnce. This case was treated in the spirit of this review in the work [64]. Here the center manifold is four-dimensional. Let us denote by fi q the by2) tanhq, and define pair of simple eigenvalues depending on b, such that q = ( 1 and C the complex one, corresponding to the oscillating by (A, B ) the (real) amplit~~des mode. Then the reversibility symmetry So acts on ( A , B, C , C) as follows: ( A , B, C , C) + ( A , - B. C. C ) . This is a 02+( i q ) resonance. The normal form, truncated at quadratic order, reads
+
where the (real) coefficients I!, LI. 8 are the same as for case (i), and r., vl , d m;hy also be explicitly computed i n terms of the parameters (1.h, r.1) (see 1641 where they are computed for c.1 = 0 ) .We notice that rr > 0 and we have
I(
( . = ( 1 / 3 - / ~ ) ~I where ( h - I .
c.1)
+-sinh 2y Y
is close to 0. This system is indeed integrable. with the two first integrals
Fixing ( . I = 0 to simplify the discussion, we see again on Figure 8 (after an obvious scaling) I ~ A ' 2 . HA K depending the various graphs of the functions ,fH,K ( A ) = (2/3)tr on ( K . H ), for 11 > 0 (left). and for v < 0 (right) (11 has the sign opposite of h - I . since b < 113). In this case, for the normal form vector field. the curves TI, and T,, in the ( K . H ) plane correspond to families of periodic solutions, where the C component is not 0. except for H = 0,where this gives the conjugate constant flow (as above). Now. we have other types of periodic solutions and quasi-periodic solutions corresponding to the interior of the triangular region in ( K . H ) plane. The curves F, correspond to the existence of homoclinic solutions, one hornoclinic to O for h < I ( v > 0 . H = K = 0 ) .and all others homoclinic to some periodic solution. Figure 12 gives in the (A. B) plane the phase portrait of all small bounded solutions (left side) for O < c.H < v2/4tr, r.1 = O, and for H = O (right side). Notice that the hornoclinic solution to A + corresponds here to a ~rtirnrli:erlsolitrrry nvr~,rfor -~.H/LJ the problem ( I ) , (4), tending at infinity towards a periodic wave. Note that A +
+
+
+
--
Fig. 12. P h a e portraits in the ( A . H ) plane o f the vector field (26) li)r 0 < c.H < 1$'/4tr,right side: H = 0 .
= 0.
IJ
> 0
(A < I ) ; lef side:
when [ H I is very small, meaning that oscillations at infinity are then very small in this for the problem ( I ) , (4). case. For H = 0 this corresponds to a .soliflrq*L V ~ I I ) P (!f'~I~v~rfior? For h > I we have analogous phase portraits where, for instance, for H = K = 0, we have a solution homoclinic to A + # 0.This limit equilibrium corresponds to the flow conjugate to 0,and might be chosen a priori as the origin (instead of the previous origin) if'we change the scale c for velocities (see the discussion for case (i)).Then A would become A' with the new scaling. and h > I would become A' < I. The natural mathematical problem consists now in proving persistence resi~ltswhen consiclerinp the full system, not only reduced to its normal form. In summary. the l)c~r:si.strrlc~~ (!f /)t~r.iorlic ,solrrfiorr.sof the normal form c;ul in general be performed. through ;in adaptation (!/'clrrtr.si-l)c,r.ioclic..solrrtior~.s of the Lyupunov-Schmidt technique IO4.821. The ~)c~r:si.\/crrt.c~ is 111uchmorc dclic:ctc. i111dcan only be pcrtormcd in a subset of the 2-dimensional spi~ce ol' first integrals. where these solutions exist for the normal form. For a lixcd value of the hifi~rc:~tiorlparameter 11. cli~asi-periodicsolutions of' the perturbeci reversible vector field exist f'or ( H . K ) lying in n region which is locally the product of a line by a Cantor set (see 1641).The persistence ot' pairs of reversible solutions (invariant under the rcversibility symmetry) hornoclinic to periodic aolutions, provided that they are not too srnall. is proved. lor instance. in 1109,641 (see Figure 13). For the normal form, there is a family of orbits homoclinic to a tiunily of periodic solutions whose amplitude can be chosen ;~rbitrarily srnall. Such a case ( i i ) has been invehtigatcd by many authors (see. for instance. 1 1 5.1 13. 821). There are homoclinic solutions to oscillations at infinity whose size is srn:~ller than itny power of the bifirrcation parameter. corrc.sponding to the fact that we ciuinol avoid such oscillations when we consider the full untrunc;~tedsystem. The extremely deliciitc aspect srritrll trritl still cl.ri.sti~lgo.sc~il1otiorl.s was proved by Sun and Shcn I 1 13 1 on of c~.vporit~rititrl!\. the water-wave problem ( 1 ), (4), and is being thoroughly studied by Lombiu-di for a wide class of problems (including the wuter-wave problem) in 1821. Moreover, despite the fact that a solution ho~noclinicto 0 exists for the normal form (26). this is not true in general f i ~ r the full system (see 1831). even though one can compute a n asymptotic expansion up to any order of such a homoclinic (non-existing)"solution"! (see 1841 for a n extensive study of the phenomenon). The difficulty comes from the fact that there is only one unstable direction and one symmetric stable direction for the origin ( 1 1 > 0). Indeed. the two-dimensional unstable manifold of a periodic orbit near 0 becomes one-dimensional when the amplitude of the oscillation vanishes. In fact, this two-dimensional unstable manifold (identical to the two-dimensional stable manifold for the normal form) intersects transversally the
Fig. 13. Shape of the generali~edwlitary waves in case (ii).
two-dimensional subspace invariant under the symmetry reversibility ( B = 0, C real) ( 2 intersection points) for the 4-dimensional normal form vector field. For a large enough size of the periodic orbit for the perturbed vector tield, its unstable manifold is shown to intersect transversally the plane ( B = 0, C real) in two points, as for the normal form. This shows the persistence, for the full vector field, of two reversible solutions homoclinic to this closed orbit. Now, it res~lltsfrom [84]that, as soon as the radius of the periodic orbit is smaller than a critical value, there is a loss of transversality for the perturbed vector tield, '/~]), and that the generic minimal size of the limiting oscillation is ~ ( ~ ( l ) e x ~ [ - l ~ / vwith I < II (1 = II would be the optimal result here, but not yet obtained, see 1841). While the result o n non-existence of solitary waves is generic here, there is a precise proof that there are no true solitary waves (of elevation here) near h = 113 (see [ 1 101). and the result for h < 1 / 3 (not near 113) is not known, although a not completely rigorous analysis suggests that there are no such solutions (see [ 124.271). We summarize these results in the following rough theorem: THEOREM 5 . A.\.\cirtic tlltrt o 4-tlitrrc~11.sio/1trl rc~\~rt:sihlc ~ ~ c . t o r , / i r11ri.s l t l o ,$.I-otl 11oir1tt i t 0 trtltl 11cr.s tr 0 2 + ( i c l ) rtl.sorltrtrc.c,fi)r its lirlc~crr-l7trr.t. T/~c.rl,irl ( I t ~ c ~ i ~ q l ~ h o r 01' l ~ o0. o rtlotrt. l
t170 c r i t i ( ~ i 1v ( i / ~ ioft110 < ~ /7~/~irc(itiotl / ) ( i r r i i ~ ~ o t11<> t ~ ~,ts;t t ~ t i l/l w t ~ i o d i.so111tio11.~ <~ ( ~ / : / ~ ~ ~ ~ I ~ ~ ~ ~ I (~/o,so to 61 g c ~ ~ ~ o r gi\vtl i ~ ~ ~ I)! i / /the \ r ~ o t ~ t t ~ ~ i I ,(26). / i ) r r tciit(1 ~ I;<>it1 (I o t ~ < ~ - l ~ ( i r ~ i t t t c ~ t ~ ~ r , / ~ i t t i!\: T/lcrc. i.s it1 titklitioti o trt~o-~~tircrt~ret~~r,firtt~iIy r!f'~)rriotlicr i t ~ t c/litisi-l~c~r-ioc/ic. l .sollitiot~.s. Morc.o\lc)~:c)trc.h /1j.l~c,t-i7olic. lwriodic. .sollitiorl r!f:fiP~/i~ct~c.\. c.1o.s~ to (1 11ti.sf r r v rp~*c~t-.sihle hot t ~ o c ~ l i tr ~ oi tt~~ t i c ~ ( ~/t~i or ~ )~i .~\ ,i (f11tit / o ( t/ / ~ (!/'tlli.s/)c,t~io(lic.so/iiriot~is Iut;y<>rt11(it1(111 ~ ~ . r ~ ~ o t ~. s~t t ~ ~ r ti / qu(it~titj. ~ / t i t i / I ~
More details [nay be found i n 1841.
COROLLARY 6. The crho~)etllcorct?~ c i l ~ l ~ l i e . s ,cl~.sc.rihir~,y f~r t r t r ~ ~ c ~ri7tr1zr.s l l i r ~ ~r!f'tllc~\i7rrtrrwtive Prohlrttl 2.1.2 ,fi)r A. t~crirI citltl h < 113 (.srlitrll .slit:jiic.p rrr~.sior~). I t tilso tipplies to the proh/ett~( ! f ' t r ( i ~ ~ t , l l irc,trixe.\ t ~ g 2.2 ~itzdert r r l elti.stic.pltrte (6),fOrA tlrcir I (see tllc litlr A in Figure 6 crtld .ccc 1971).
r
of the 4.2.3. Ctrse ( i i i ) : ( i y )? rc.sotltrr1c.e. This case occurs for (h. A ) near the curve parameter plane i n Figure 5 , and was first treated in 16.1). Here the center manifold is fourdimensional. Let us denote by i i y the pair of double eigenvalues at criticality, and define by ( A . B ) the complex amplitudes corresponding respectively to the eigenmode and to the generalized eigenmode. This case is often denoted by "I: I reversible resonance". We can The normal always assume that the reversibility symmetry So acts as ( A . B) + (A, form (see, for instance, 1621) reads at any order (making c.1 = 0. which does not restrict the
-s).
study, since h is not close to I):
where P and Q are real polynomial of degree one in their arguments, for the cubic normal form. Let us define more precisely the coefficients of Q:
which means that for 11 > 0 the eigerivalues are at a distance fi from the imaginary axis, while for v < 0 they sit on the imaginary axis. Values v > 0 correspond to points in the plane (11. h ) above the curve f. and v is of the order of the distance to this curve (the precise expression of I! in terms of the pararneters is given in 1341). The vector field (28) is integrable, with the two following first integrals:
I t is then possiblc to descrihe all small bounded solutions of ( 7 8 ) .111decd.we obtain
where
We show in Figure 14 various grxphs for the functions ,fh.,// depending on ( K . H ). for 0 . 47 < 0 (right). and for 1 1 < 0 . (12 > 0 (left). which are the most interesting cases. The change ((12. H. 1 1 ) r-t (-42. - H . -10 Ie:tves , / K , / / ( i O itnchiunged. I t results that the relevant graph (we need I A ~ '> 0 ) of ,/h.,,/ for I ] < 0 , and (12 > 0 corresponds to the side H > 0 of the left part of Figure 14. while for I ) i0 , (12 < O we need to consider the side H < 0 of the left part of Figure 14. Notice that for LIZ> 0. I ) > 0 there is n o small bounded solution other than 0 . Looking at these graphs, where in particular- any tangency to the 11 axis on the positive side corresponds to a periodic solution of frequency close to (1, it is clear that we obtain for solutions and. for clz c: 0. a fixed 11 two-parameter families of periodic and cl~~asi-periodic v > 0, ;I circle of solutions hotnoclinic to 0 with exponentially damped oscillations at infinity, while for q2 > 0 , v < 0, we have a one-parameter farnily of circles of solutions homoclinic to periodic solutions (as in case ( i i ) ) where the amplitude is minirnum at .t- = 0 . 11
472
E Ditrs cmd G. looss
Fig. 14. Different graphs of rc
H
f ~ , ~ ( r depending r )
on the parameters H and K
Fig. 1.5. Depre\sion wave for caw ( i i i )
The computation of the coefficients o f the normal form (28) corresponding to the system ( 4 ) is performed by Dias and looss in 1341; i t is shown that (12 < 0 holds all along f. For the ice problem (6). the present case ( i i i ) occurs along the curve f ol' Figure 6. iund the coefficient (12 can have either sign (see 1971). depending on the water depth. The mathematical problern of persistcncc of thc ~tbovesolutions of the normal firm system for the full vector field is done in an analogous way as for case (ii). This means in particular that we have a one-parameter family of pairs of reversible homoclinics to periodic orbits. For the homoclinic to 0, it is in fact simpler than case ( i i ) . This is due to the fact that the unstable manifold of 0 (identical to the stable manifold, for the normal form) is two-dimensional. and intersects transversally i n two points the plane of symmetric points ( A real, B pure imaginary). It gives the persistence of two reversible homoclinic orbits. corresponding to two different "bright" solitary waves, with exponentially damped oscillations at infinity: one has a crest in the middle (elevation wave), and the other has a trough in the middle (depression wave) (see Figure IS for the depression wave). Note that this type of solutions has been experimentally observed (86,126,107.1 IS] at least when some forcing is present. The forcing can be an obstacle at the bottom. wind on the surface, a moving load on the surface in the case of ice experiments. The complete proofs on persistence for periodic and homoclinics can be found in 1681. and for quasi-periodic solutions, it is shown in 1671 (the method applies directly here with very slight modifications), that persistence holds true i n a subset. locally looking like the product of a curve by a Cantor set, of the region of the ( K , H ) plane where these quasiperiodic solutions exist for the normal form. We sum up these results in a rough theorem: THEOREM 7. Assutne t / ~trt 4-dir?zen.sionulrrver.sihle vec~torjeltl/rti.s ti fixed point ot the origin, trnd has u ( i )? resoncrnce f i ) r its linecrr ptirt. Then, in tr nrighhorhood of' 0, urzd
neur the criticul val~ieof the bf~ircutionpurumeter (possibly only on one side cf criticrrlity), there is N one-~)crr~lmeter,fumily of' periodic solutions c!f',frequency necir q and u t~!o-prrr~r~neter,filnzi/y of other periodic urzd quasi-periodic solurions. Moreover; we huve genericcrll~one of the two CCISCS, clepmclitlg on the sign (?fa certain nonlinrur co&ficient ( q l in (29)): Case I : ,fi)r hifurention purcrmeter vcrlues rvhich lerrd to,fi)urnon-purely imugineiry eigenvcrlue.s,fi)rtlw lirze~1ri;7ed operntor; crnd,fbr q2 < 0, there urr two reversible 0rhit.r homoclinic to 0; Case 2: ,fhr /~ifirrc.crtiot? p f ~ r c n n r t V~Nr/ I ~ P . S~l*hich /rod fo~fi)urpure~/.y itnciginurv eigenvulues , f i r rhr 1ine~rri:c~dol~ercrtor;rrnrl jilr q? > 0, there L I ~ P tbtpoo n ~ - ~ ~ ~ r e i ntiilnilies ~ e t r ~ rof' re~~~ersihlr or1)it.s Ilornoc~linicto the "Il?perl)olic" periodic .solution.s (?f~frequ~ncie.s ner~r '1. C O R O L L A R8. Y The trhovc) theor~tntrpl~lie,s,fi)r cl~.sc,rihin,y tr~rvellin~ wcrves of'thp wrrteru'trve /wobletn 2. 1 .2,fi)r-( h . h ) tlprrr thc~c.irrvr. f of'Fig~iro5 (r.ri.s~1 ) (S(V 1341).This t h ~ o r r > ~ n rrlso cil1p1ie.sto the)pro/?le~t~ of't~zit~c~lling rt1rr13r.s~/tlc/r>r (in ~Icr.stit.l ) l ( ~2.2 f ~,fbr ( D. A ) t ~ ~ o r tho c.Lrr\v f r!f'Fig~rrc. 6 (~.(I.scs I ~ i n d2 ) (.SOP [97]). R E M A R K S .For these results on homoclinics. i t should be ~nentionedthat the decay at intinily is r,.vponc~nficil. Thcrc are dcgencrnte cases (codimension two situations) where this is no longer true. For ex:umplc. when the coefticient 47 is close 1 0 0 (see 1571).there exists in general (for I J = 0 ) an homoclinic to 0. with LI ~)o!\~rronritrl tlcc~r! trt infini!\.. This casc may occur in examples having rnore pitr:~rricters.such ;is with scveral supcl-poscd laycrs. I t should be noticed that this phcno~ncnonis in fact different t'rom the similar property of polynomial decay that we shall ~ncctfor cilscs with ;un inlinitely deep layer. Both phenomena are due to different causes. For proble~nswith several bounded superposed layers. with surfuce and (or) intert.;~cial tension. there iirc always vi~lucsofthe parameters where cases ( i ) , ( i i ) . ( i i i ) occur. They can be treated in the same way. More co~nplexbifurcations may occur. tor example in casc ( i i i ) when ql changes sign. Such a casc is a codi~nension-2sing~~larity. and is partly studied i n (351 and completed i n 1 12 1 1.
4.2.4. H(flrrc~rfion~f'l~lc~tllortr t~f'.soli/eri;~ rr.er\,cJ.s. So far. we hove discussed the solitary waves that one can obtain via the normal form technique. In fact. this technique provides only a small portion ofthe existing solitary waves. i n particular in cases ( i i ) and ( i i i ) above. What happens is that it is possible to combine several solitary wave solutions together and still obtain a solution of the problem. a so-called multibump soliti~rywove (see Figure I6 for the profile of such a wave - this type of profile has been obtained numerically in 1.771 o n the f11ll water-wave equations). This can be done as soon ;is one is sufticiently far from the bifurcation curve. In practice. this distance can be exponentially small. The formation of rnultibump solutions has been studied in 126.23.25.24) (the first three works deal with a model differential equation and all of them use a Hamiltonian formulation), We d o not intend to describe this process in detail here (it would make the paper much longer!). Rather, we follow the formal approach used in 11251. and we concentrate on case (iii). Similar results occur for case (ii).
Fig. 16. Multibump solitary wave resulting from the superposition of two depression waves near the ( i y ) 2 resonance.
In the last subsection, we showed that at each order the (iq)* resonance normal form admits two reversible homoclinic solutions provided certain coefficients have the correct sign. The corresponding solutions of the water-wave problem are modulated wavepackets whose envelopes are symmetric and decay exponentially to zero at infinity. In the middle, one wave has a central crest (elevation wave), while the other wave has a central trough (depression wave). As is well known, the normal form (28) yields the nonlinear Schrodinger (NLS) equation to leading order, and of course the NLS equation admits two symmetric envelope-soliton solutions. But one can also construct small-amplitude asymmetric solitary waves, by translating the crests of a symmetric solitary wave relative to its wave envelope. The problem is that such asymmetric waves do not persist when considering the full system. Exponentially small ternis come into play! Shifting the carrier oscillations relative to the envelope leads to the appearance of growing oscillations of exponentially small amplitude o n one side of the wave packet. However. due to nonlinearity, this growing tail evolves into a new wavepacket and it can he shown that, for certain values of'the phase of the carrier oscillations, the whole disturbance terminates. resulting in a solitary wave with two wavepacketa. Otherwise. a third wavepiicket is generated and the process continues. The main result is that there exists a countable intinity of symmetric and asymmetric multibump solutions. But, unlike the solitary waves obtained in the previous subsection. each of these ~nultibumpsolitary waves bifurcates at a certain tinite amplitude. When the parameters h and h. are close to the critical point (17. A) = (113. I ) . which corresponds to the occurrence of a quadruple eigenvalue 0, it can be shown, via center manifold reduction and a normal form argument (see 156)). that the problem essentially reduces to the fourth-order differential equation
where y is directly related to the elevation of the free surface and .r to the horizontal coordinate. When the parameter P is equal to 2, one is along the curve f in Figure 5. When the parameter P is equal to -2, one is along the dashed line in Figure 5. Equation (32) has been studied excessively in 123,261. Using the fact that (32) is a Harniltonian system, these authors proved the existence of an intinity of homoclinic orbits and the presence of spatial chaos. They showed rigorously that (32) admits a unique (up to translations) -2 (i.e., in-between the dashed curve and the half line A of homoclinic orbit for P F , 2) it has at least two small-amplitude homoclinic orbits. Figure 5 ) . while for P in (-2 What happens at P = -2 is that the unique orbit can bifurcate into a countable infinity of multimodal homoclinic orbits. As P is increased towards 2 (i.e., one goes from the dashed curve towards the curve f in Figure S), the domain of existence of each orbit reaches a
<
+
limit (turning) point before the value P = 2, except for one orbit which can be followed all the way towards P = 2. This orbit is nothing else than the depression wave found earlier (near the half line A)! Some of these multimodal homoclinic orbits have been computed numerically for the full water-wave problem in [37]. As said above it was found in 1231 that, close to each turning point of a branch of homoclinic, for P -- 2 (i.e., next to the curve f in our context), there is a bifurcation into a branch of asymmetric homoclinic orbits.
4.3. The c ~ l s eof trco 1tryer.v rt'itholtt .cur-filce tension In this case [see system ( 1 ) . (7), and the dispersion relation (14)1, 0 is always an at least double eigenvalue of L,, with the two independent eigenvectors
.
which satisfy S(O = & ISt(') . = 6;. When there is no other eigenvalue on the imaginary axis, this just gives near 0 the twoparameter family of stationary solutions of ( I ) , (7): U = a(()+ fit;, corresponding to the freedom on the horixontul velocity in each layer. When there is a pair of' simple eigenvalues on the imaginary axis, in addition to the double 0 eigenvalue. we have near 0. a family of periodic waves, hil'urcating from any of' the above equilibria. This situation is similar to system ( I ). ( 2 ) . tor h > I (not close to I ). and to system ( I ) . (4). ti)r h < I . ( 1 1 . h ) not close to A . When h = h t ( c , p ) , 0 is ;I qi~i~dri~pleeigcnvaluc with two i~cltfitioni~l (generoli~ed) cigenvectors t I and t2such that
/ t l = to. I>/,,, t, = < I .
S
.
St, = (7.
If there is no other eigenvalue on the imaginary axis. which is the case lor h = h i (see Figure 7). this is a 003+resonance. This case leads to ;I four-dimensioti:~lcenter manifold U = A t ( ) + B
+ I)((; + W , , ( A . B . C . I ) ) .
where the dynamics is ruled by an ODE ofthe form (see ) t 2 ) )
where 4,! is a polynomial in its arguments, of degree p in (A, B, C , D ) , and R B , RC, RD are even in B, due to reversibility. Moreover
holds. The normal form here is analogous to the system ( 1 8), (19), (20), except that there is the additional first integral D. We recover easily the two-parameter family of reversible equilibria: B = 0. C = Co(A, D), where A and D are arbitrary. The study of the normal form is the same as for ( I 8), (19). (20) (see Figure 8), after a small change in the parameter v (at leading orders). The persistence of the phase portraits in the four-dimensional space, for the full vector field, may be proved with the same argument as in Section 4.1, because the linear subspace of symmetric points is now three-dimensional ( B = 0). For example, periodic and hornoclinic orbits cross tl-ansversally the space B = 0 for the normal form. Then it can be proved that they persist for the full vector field. Now, we have the other possible situation A = :A (see Figure 7) which is analogous to case ( i i ) of Section 4.2. with a pair of simple eigenvalues on the imaginary axis, in addition to the quadruple eigenvalue at 0. This is a 0 0 3 + ( i q ) resonance. The values of the coefficients of system (33)-(36) are computed in 133 1. The discussion of various solutions on the normal forms is nearly identical to the case ( i i ) above, except for the two additional first integrals. occurring as new constants in the discuusion. The proofs for persistence are large dimension of the space of symmetric points. the same us before, due to a s~~fficiently So, here again, we have solutions homoclinic to exponentially small periodic waves. Let us notice that the ithove case was tirst treated in 1 1 141. The shape of the generalized solitary waves corresponding to these homoclinics is shown in Figurc 17. See 19 I ] for a ni~merical computation of such waves. RE.MAKK.If we add surPLtceand/or interfacial tension, we have additional para~iieters.
Bernoulli constants may appear as in system (4). and the sum of the number of these Bernoulli constants plus the dimension of the kernel of the linearized operator, is equal to the number of layers. This allows more complicated bifurcations. with more eigenvalues appearing near the imaginary axis (in addition to the multiple 0 eigenvalue. or (and) to the corresponding freedom on Bernoulli constants). For example. in cases when a pair of double eigenvalues appears with a pair of simple eigenvalues on the imaginary axis. we cannot avoid, here
Fig. 17. Shape of the generali~edholirory w;ivc\ tor a two-layer \y.;tem.
Water-waves as a spatial dynamical system
477
again, exponentially small periodic oscillations at infinity for a class of generic situations (see [84]). Moreover, in cases analogous to case (iii) (00(iq) 2 resonance), for q2 < 0, we have a large family of solutions homoclinic to each member of the family o f reversible equilibria. In such a case there are solitary waves with exponentially damped oscillations at infinity, tending to constant flows, with the freedom on the velocity in each layer.
5. The case of infinite depth The aim of this section is to present typical results/'or two-dimensional travelling waves in fluid layers when one layer (the bottom one) is infinitely deep. We consider in more detail the systems ( 1), (9) and ( 1), ( 7 ~ ) . We observed in Section 3 that the spectrum of the linear operator L/, contains the full real line, and that, on it, the only possible eigenvalue is 0. There are other eigenvalues ik in the complex plane, solutions of the dispersion relation A[lz, sgn(Re k )k l = O. In the case of system ( I ), (9), we note an interesting situation where we have a pair of double eigenvalues ik = 4 - i / 2 on the imaginary axis fi)r/21 = I / 4 [see ( 15)]. This is again a 1 : I resonance, but the rest of the spectrum is the full real line, which indeed crosses the inaagillary axis, so we cannot use a center manilbld reduction into an ODE (no gap between the imaginary axis and the rest of the spectrum). Notice that the point 0 in the spectrum is "resonant" with the purely imaginary eigenvahles, which may lead to problems even for
the existence of periodic solutions. In the case of system ( I ), (7-v), we note that 0 is always ;,In eigenvahle with eigenvector ~1'1 [see (16) and Section 4.31, and we have a pair of simple eigenvalues ik = q-iX on the inlaginary axis, and for ( I -- [)))v ~ I there arc i1{7 other eigenvalues on the imaginary axis. For (1 - p ) k > I we have another pair i i k l of simple eigenvahles appearing on the ilnaginary axis, elnerging froln the continuous spectrum, and this new pair does not 1heel the other pair. For this system, interesting cases are the strong resonances when k l/)v = I / 2 or 1/3, and a particularly interesting one is when (I - p))v is near l, when one pair of eigenvalues disappears (is melting) in the continuous spectrum.
5.1. Periodic waves For periodic solutions, we fl~llow the L y a p u n o v - S c h m i d t inethod, except that the presence of 0 in the spectrunl gives some resonant terms. It appears that we can formulate all these problems in a way that there are no such resonant terms. As a result, there are as many iwriodic solutions ( with period near 2zr / ko) as in the trum'ated normal f o r m (see [58 ]). In this section, we consider the case of system ( 1), ( 7 ~ ) when there is a pair of simple imaginary eigenvalues + i k o of the operator Lj,, such that other pairs 4-ikl which might be on this axis satisfy k] :¢ nk0 for n = 1,2 . . . . . This condition is satisfied in our case for k0 = )v, and is in general (fl)r other similar problems) satisfied because there is only a finite number of eigenvalues on the imaginary axis, whose positions depend on the parameter set #. The method developed below assumes that the basic pair i i k o is not close to 0, because
478
E Dius and G. 1oos.s
this would imply that other pairs of eigenvalues would be close to some multiples of this pair. It results that in general the only point in the spectrum of L, in resonance with our pair is 0. The "exotic" character of this point of the spectrum leads to a specific difticulty we are dealing with below, by adapting the classical Lyapunov-Schmidt method for periodic orbits. Let s = (ko y ) x , where y is close to 0, and ko y is the wave number of the periodic ) space of (21r-periodic) functions of solution we are looking for. We denote by H ~ ( Ethe s, such that their derivatives up to order 1, are in L ~ ( R / ~ ~ taking z ) , values in the Banach space E. For such a space we use the norm defined by
+
+
which gives a Banach space structure. Let us detine, in the space ear operator T,, = k(,$ inversion of
-
L,,, with domain ID3 = H.?(w)
W: = H ~ ( W ) the ,
lin-
n H ~ ( D )The . basic tool is the
where V is given in W:. and where we look for U E ID-.Now expanding in Fourier series V and U we have ti)r n # 0. I . - 1
which. with the resolvent estimate (17). insures that if we define
~,,d"" ED :.
U =
then
1 ~ ' l l , , <~ (.I
I1 V IHI: holds.
(38)
~ l € Z \ { O . I.- I \
It then remains to study the equations (irlkoII - L,,)U,, = V,, for 17 = 0.I . - 1. For n = 0, the compatibility condition on VO and the additional condition for being in the range of L,, are auto~naticallysatisfied by the 0th Fourier component of the nonlinear term. for rc.~~er.sihlr .sol~ttiorl.s(in particular the nonlinear term has 0 components on -co < y < 0). Then the 0th Fourier component UOof the solution U in (37) is uniquely determined in D. up to an arbitrary multiple of the eigenvector 6;);;. For 11 = I . - 1 this is a classical Fredholm alternative (one compatibility condition for VI and for V- I ) . Let us now consider the system ( 1 ), (7,) rewritten as follows
where we look for solutions in D:,and
We observe that G(.. ., .) : R ' x that
D1+ WDis analytic in the neighborhood of 0, and is such
Now, let us define the symmetry F i n C(W:) n C ( D c )by
It is then easy to check that
holds. For solving (39), (40) we use a classical Lyapunov-Schmidt method. We are then able to prove the following (see proofs in 1581). where we denote by t,, the eigenvector of L,, belonging to iko:
THEOREM 9 . For. ( I ( ( ) , I A l ) !\'itrg it1 tr r~eiglrhorliootlof' 0, thorp i.s rr ,firririI~c?f'pcrioelic ,so/i(tio~~.s (!f ( I )< (7%), />~/~rrt.crtir~'g ,fro1110, rt~11il~I1 /)o,s.sc~.s,s t ~ r ~ ~ , f O / I ol~ot~\~e~r~qir~,g r ~ ~ i ~ ~ , q /)ori,tJr. .scric,.sirr i r ~ A. . 2:
For other similar problems, the proof's are ;tnnlogo~ls.the number of arbitrary constants depending o n the dinlension of the kernel of L,, . For cases of strong resonances, the additional difliculty is the same as in finite-dimensionnl reversible systenis. See. for instance. 17 1 ) for the study of periodic solutions for the 1 : 2 reson;uice. and 1581 for the I : I resonance. Roughly speaking. there are as many periodic solutions as would be given by the analysis of the corresponding generic reversible bifurcation in finite dirnension. A new difficulty occurs when one considers solutions with very large periods, for example in the case when the basic frequency is given by u pair of eigenvalues of L,, close to 0. The difficulty is that O also belongs to the spectrum (since it is on the real line). and that any other pair of imaginary eigenvalues is also quasi-resonant with the basic pair. This problem needs further investigation. I(()
5.2. Norma1,firtns in injnite dimensions Since we cannot reduce our infinite depth problems to finite-dimensional ODES, we still would like to believe that eigenvalues near the imaginary axis are ruling the bounded solutions. This is a motivation for developing a theory of normal forms in separating the finite-dimensional critical space from the rest (the "hyperbolic" part of the spectrum, including 0). This leads to "partial normal forms", where there are coupling terms, especially "bad" in the infinite-dimensional part of the system (see [35,65]). Indeed, there are some additional difficulties: (i) when 0 is an eigenvalue embedded in the essential spectrum, we need the explicit ,fi)rtnof'tlze resolveizt operlrtor n e w the red uxi.~, to extract the corresponding eigenmode from the rest of space, by a suitable projection (see [58,66]), ( i i ) in space W the linear operator does not have an "easy" (even formal) adjoint. This adjoint and some of its eigenvectors are usually necessary for expressing projections on the critical finite-dimensional space. In our problems, we use again the explicit form of the resolvent operator near the (double) eigenvalues, to make the projections explicit (see 1721). Let us give below some details on these infinite-dimensional normal forms. Consider our system under the form
where F ( p . 0) = 0 and L,, U denotes its linear part. Assume that for / A = 0 . one has a and is spectral decomposition of the ti)r~iiW = Eo $ El, where EO is finite-dirnensio~ii~l, spunned by all eigenvectors and generalized eigcnvectors belonging to eigenvalues of Lo lying on the imaginary axis. Thcn Eo C ID.The space El, is a complementary subspacc invariant under Lo which, in this subspace. has no eigenvalue on the imaginary axis, and is such that its spectrum contains the full real line (with no eigenvalue in 0 , and close to 0). and is bounded by a double sector in @ centered o n the real axis. In all the waterwave problems. with an infi nitely deep bottom layer. we also have the estimate ( 1 7) for the resolvent of Lo on the imaginary axis, far from 0 (this indeed implies the sectorial bound of the spectrum of L,, for p near 0). The above spectral decomposition of W implies that the 0 eigenvalue which occurs, for instance, in problem ( I ) , (7,) is extracted from the continuous spectrum, by using a suitable projection for being incorporated into Eo (this is done explicitly in [581 and 1661). The result (see 165.351) is that one can find a polynomial change of variables i n W such that U = V + W + @ ( p , V , W).
V E El). W E El,.
where @ ( p ... .) is a polynomial with values in ID.of degree 1 in W and degree 171 in V, with regular coefficients in p . such that (41) reads
where L0 = L O [E,,, L" = LolE,,, and N ( p . V ) corresponds to the usual finite-dimensional polynomial (degree m ) normal form, i.e., satisfies the following additional symmetry (see [621)
Now the remaining coupling terms satisfy the following estimates
which means in particular that we have no linear term in W in (42), and no linear term in V in (43). Moreover, we observe that in general, there are quadratic terms in V in (43), which determine the size of W in this method. These quadratic terms are due to the rrsor r e r t r c . c ~of the pure imaginary eigenvalues of Lo with the point O i n the continuous spectrum. Notice now that we simplify the system (42),(43) in using Bernoulli tirst integrals which allow to eliminate the coordinates belonging to the 0 eigenvalue. Indeed, this elimination is in general not singular with respect to p , however it becomes singular in the case of rc.\~rr.siblt~ problem ( I ), (7,) when ( I - p ) h is close to 1. Section 5.4 dealing with a 11e2n3 hj/irrc.citiorr describes below what happens in such a case. I t then appears that the above helief' that eigenvalues o n the imaginary axis govern bounded solutions, becomes wrong at least in this case. In Rict our system (42).( 4 3 )still contains all small bounded solutions of (41). but n scaling based o n solutions corresponding to a truncated normal form in V implies :I ".sler\~irl,q"for W which wily not be sntisfied for other types o f solutions. such as the ones which occur in Section 5.4.
With the method we use now. we need to give tr l ~ r i o rthe i type of solution we are looking fir. This is i~malor difference with the case where ii (center manifold) reduction to an ODE is possible. For periodic waves. with periods close to the basic one. one may use a method si~nilor to the one given for the nonresonant case. as i t is done in (581. We nlny also use the formulation (42). (43). which is more transparent for such solutions, easily found o n the reduced normal form in the variable V . The result is roughly that there are as many periodic solutions as in the tinite depth case. we tirst invert the infinite-dirnension~il For solutions homoclinic to O (.solitctt:y ctler\~r.s). part (43) in W. ~rsingFourier transforn~.Indeed. the linearized Fourier transti)rm uses the resolvent operator of L,,. The fact that the resolvent operator is not analytic near 0 ( 0 is not o pole, since we eliminated it, but there is still a jump of the resolvent in crossing the real axis 1651) leads to the fact that this "hyperbolic part" of the solution cannot decay y The finite-dimensional part (42) exponentially. but instead c1ec.eiy.s l ~ o ~ r ~ o t n i ctrr lt l irlfirlity. where W is replaced by its expression in terms of V , is an integro-differential equation,
because of the non-local term coming from W . The principal part of this equation comes from the usual normal form. In the case of I : I resonance (for instance, as in problem ( I ) , (9) for ~1 = 114, with double eigenvalues ik = fi/2), the normal form is given by (28) and in case of problem (l), (9), (11< 0 holds (see [(is]). In other problems, with several layers. the 1 : I resonance may occur [except in problem (I), (7,)] with 42 > 0 or < 0 or even cancels (see [ 3 5 ] ) ,leading to solutions similar to the ones given by the finite depth case, except that the convergence at infinity towards 0 or to a periodic wave is no longer exponential. For problem ( I ) , (9). and ~ 1 12 114, a fixed point argument in a space of polynomially decaying functions leads to the existence of two reversible homoclinics like in the finite-dimensional case, except for the decay rate which is indeed proved to be in I/.r2 in [ I I I I. See also [ I I where this decay is checked numerically. The principal part of the solution at finite distance still comes from the finite-dimensional truncated normal form, but it decays faster at infinity than the other part of the solution, which makes this tail part predominant at infinity. This is the main difference with the finite depth case, where the principal part corning from the normal form is valid for all values of .v [see [651 for the proofs related with problem ( I ) , (9)).
I t appears that the technique descrihed uhove may miss important solutions. This occurs precisely for the problem ( I ), (7,) when the parameter F = I - ( 1 - p ) h is close to 0 . In such a case the singularity of the resolvent operator (icll - I*,,) - I is a little worse when c. + 0 . Indeed the pro.jection operator on the eigenvector helonging to the 0 eigenvalue bcconies singular. having c: in its denominator! One may then suspect that this changes relative orders of magnitude for various components of the variable U . The dispersion relation ( 16) in this case shows a first factor giving two isolated eigenvalues ik = fi h , and )] the 0 eigenvalue, and a pair for X. real near 0. a second factor I ~ I I F . plX.1 O ( 1 k 1 ~giving of imaginary eigenvalues near 0 only for F < 0 . For c. > 0 . the factor of I k ) corresponds to the linear part of the dispersion relation for the Benjaniin-Ono model (see 1 16.3 1 .Y5 1). Indeed, a suitable scaling here allows to find trri tr.s~vrll~roric e\pirrlsiori iri 1>ort7or-s c?f' c. of a formal solitary wave. with a 1/.v2 decay at infinity [66], with the same principal part as the solitary wave solution of the Benjamin-Ono equation. I n addition. as shown above. we also ohtain a two parameter family of bifurcating periodic waves (their amplitude is one of the parameters) with periods close to 2 n l h . We can use the form of the family of periodic solutions (given in Theorem 1 ) to construct @ in the change of variables leading to the normal form (42). (43). I t then appears that the manifold W = O contains all the family of periodic solutions. and that the W part of the system possesses an approximate homoclinic to 0, close to the Benjamin-Ono solitary wave. Finally, the formal solitary wave is not a solution of ( I ) . (7,). because of the additional frequency h. I t is shown in 1661 that there are two reversible solutions homoclinic to each of the above periodic wave. provided that their amplitude is not too small (exponentially small in E as it is proved in [as]). The principal part at finite distance of all these "generalized solitary waves" is given by the approximate homoclinic described above (see Figure 17). One difficulty here
+
+
Wtrter-trnr\.es trs
(1
s ~ ~ t r t iduvl t ~ c r ~ ~ ~sysrrn~ icuI
483
is that it is not possible to "morally" reduce the problem to a finite-dimensional one, as in 182,84,109,1 131, because of the essential spectrum filling the real axis (especially near 0). Another difficulty, not appearing in previous work, is that the decay at infinity, towards the periodic solution (with a shift opposite at both infinities), is only polynomial, instead of exponential. This implies the use of refinement techniques for being able to use a fixed point technique in a good function space. See 1961 for a numerical computation of periodic as well as generalized solitary waves. Notice that if we consider the problem of two superposed layers, the bottom one being, as here, infinitely deep, but the top layer being bounded by a rigid horizontal wall, we may formulate the problem as a dynamical system, in a simpler way than here, and the spectrum of the linearized operator is simplified, in the sense that we do not have the pair of eigenvalues fi h o n the imaginary axis. This avoids the problems above provoked by the additional frequency. and one ti nds a solution homoclinic to 0, close to the Benjamin-Ono solitary wave (see 13,l 121 for proofs. not relying on a dynamical system formulation).
6. Stratified fluids A large class of 3,D tr;~vellingW;IVCS in perfect fluids appears ns internal waves i n a stratitied medi~~m A. simplitied moclcl of a stratified layer consists in considering two (or more) superposed immiscible fluids of different dcnsitics. Thc interlncc between two fluids is then an unknown iuid LIII iuialysis in the t i ~ r m01' o rcvcrsiblc dyna~iiicalsystem may be pcrfor~iicdas we indicated in Section 2.3. whcrc we repl;ice the ~ ~ p p free e r s ~ ~ ~ - fby a c ea rigid boundi~ry(the samc boundiuy coridition as at thc bottom. see. for instance. 19.101 and see 1931 for :I t'ormulation as :I dynamical system). 111 I;~ct.a more realistic version of thc problem is when the density p increases continuo~~sly flow top to bottom. ;IS a known function of the streium I'unction W ( d p l d r = 0 and incompressibility. i.e.. div U = 0 ) .Basic papers in this respect :we 13,. I I X. 171. There we various ways of f'ormulating this problem as a dyniuiiical system. For instance. u popular onc consists in considering the equation ~l,y (DJL) 1381 satisfied by W .named the I ) l r l ) r u ~ i l - . l t r c ~ o t i ~ ~ - L oequation
in the strip (0< :< I ) . which may be written easily in a dynamical system form. and in which a bifurcation similar to case ( i ) of Section 4.3,. 1 occurs (see 1751). and a bifurcation to fronts (see 1701) connecting two uniform states. resulting from the cancellation of the quadratic coefficient o in (24). There are results o n periodic intcrnal waves and large amplitude solutions in 17x1. and a study of 3D internal flows. with :I derivation of the K-P model (see I I X I ) . and a dimension breaking bifurcation in 1791. in the spirit mentioned in Section 7.4 below. An interesting problem arises us one tends towards the limiting case when the density stratification becomes discontinuous. i.e.. when /) has a large gradient in a small region near a certain height h , and tends to a piecewise constant p,,, having a
discontinuity in h. We see on the DJL equation that the factor p l / p becomes singular, and the limit is to be understood in the distribution sense. A unified formulation, including the discontinuous case (a free interface separates two fluids with different constant densities) as well as the continuous stratification case, is proposed by James [69]. He studies the limit p ( P ) + po, where po(z) = I for z E (0, h), and po(z) = q < 1 for z E ( h , I), and p ( P ) is a decreasing function such that p(0) = I , p(1) = q , with lpl($)l becoming very large in ( h - E, h E). His formulation uses the coordinates (x, $) and the unknown function Y = Z(x, $) - $, where by definition P [ x . Z ( x , $)I $. The system becomes
+
-
with boundary conditions Y I$=(),I = 0. In (44), U = pv (v is the vertical component of the velocity), and = v/u is the slope of a streamline. This system is reversible under the symmetry (Y. U ) H (Y, -U). In the case p = po, we have p;) = - ( I - y)6/, where SIl is the Dirac distribution i n $ = h . Here the basic spaces are
where
and where we notice that D is not dense in W. Jalnes shows that a solution in c"(R. ID)n C I (R. W) of this system leads to a classical solution in the continuous stratification case. and leads to the standard discontinuous case (no interfacial tension) in the p~ case. This formulation is weak in the sense that the dependency in p of operators is weakly continuous, despite the tact that J;: Ipl($)l d$ = 1 - y is a finite constant. The author is able to prove an estimate like ( 17) on the resolvent of the linearized operator. uniformly in p . This allows the existence of a family of center manifolds in a ball of D centered at 0 , this ball being independent of p close to po. Here the result of Kirrmann for finding center manifolds is needed 1761, because one works in Banach spaces. The dependency in p on these center manifolds is only continuous in W (not in D). However, if one considers a typical bifurcation problem like the one studied in case ( i ) of Section 4.2.1, one obtains a two-dimensional amplitude equation whose coefficients are functions of the stratification p. James shows that the vector field as well as these coefficients, at any order, are continuous functions of p , as p + po (topology of L 2 ) . These technical results are proved in 1691.
7. Three-dimensional travelling water waves We tirst consider 3D water waves that bifurcate from the state of rest, and for the sake of brevity, we only expose here in detail the case of waves which are periodic in the direction Y orthogonal to the direction x of propagation, which is the reversible version of the paper of Groves and Mielke [47] based on a Hamiltonian formulation (see also Bridges 1191). In the cases where the waves are periodic along the direction of propagation 145,531, the technique is similar. We assume that surface tension is present, which seems to be fundamental in the present formulation for allowing a finite-dimensional reduction process. We thank Mariana Hlirrigu~for giving us the 3D formulation indicated below. Most papers which present a dynamical system approach use a Hamiltonian formulation of the problem 145117). Notice that no rigorous mathematical results exist so far on the Euler equations in the case of travelling water waves which are localized in all horizontal directions. Notice also that for the study of travelling waves which are periodic in both horizontal directions, there is no need for a dynamical system formulation (see [ 100,291). In Section 7.4 we give some results on the "dimension breaking" bifurcations, which consist in the bifurcation of 3D waves from 2D waves of finite amplitude.
One of the difticitlties is that. even for one-layer systems (one rigid bottom. one free surI'i~cc).there are no simple variables providing a f1;ittcning of the f'ree surface. as i t is for the 7D case. Formulating the problem in the moving flamc. with velocity - ( . along the .v direction. we arc interested in steady solutions close to the unil'orm flow of speed ( . ( 1 i n ditnensionless viiriablcs) i n the .I-direction. Let us ~ l s ethe velocity potential .\. 4. where 4 is the perturbation potential. We define as bet'ore for system ( 4 )thc dimensionless parameters h and h. We denote by the vertical direction. Then the rigid bottom is located at = 0, while we denote by := 1 + I/(.'. j.)the equation of thc free surf~icc(the depth 11 is the length scale), where 11 is close to 0. Then the problem satisfies the following system
+
,
-b[
(1
11 I
+ I].? +
1/;)Il2
on: = 1
+ q(.x-.y).
where C is the Bernoulli constant on the free surface. and we notice that the factor of - h is twice the mean curvature of the free surface. Let
E Ditrs crt~dG. Ioo.\s
486
Hence our system may be rewritten a s follows (where A denotes the Laplace operator in the (?I, 2 ) plane)
4Y . = 11. Ll.r = - A d ,
for 0 < z < I
+ i l ( x , y),
where ( . I = - C / b , and with the boundary conditions
We now flatten the upper boundary in making the change of coordinates
I t follows that the primes,
6'
2' E
1
(0, I). In the new coordinate system ( . r . y. : I ) w e find. after dropping
- -(ul;=l
- h
+ hq) +
with boundary conditions
We need to define suitable function spaces for the dynamical system formulation, and then we shall need to modify the boundary condition, to put i t into a linear form. Define
+ P ) = f'(y), for almost all y E R ] . H,"(z) = {J' E H~:,~[IW x (0. I 11: , f ' ( . + ~ P. Z ) = jxv, z ) , H:
H.
p =
( f E HI:,,(R): ,f'(,*
for almost all (y. :)
E
R x (0.I ) } .
where H ' is the classical Sobolev space. and in which P > 0 is the period of the waves in Y . C = (0. P ) x ( 0 . I ) and we assume 0 < s < 112. Now set
The above systeni is ol'the form
with
-
+ W, is o .\lrrootlr function with domain (cotlimcnsion-two manifold i n a and F : neighborhood o f 0)
Remark that H;,,, and H! ( Z ) are Banach algebras I'or .s > 112. I > I . and that H':,'-.
.
H,',, . . c H;,/,, and H , ' ~ ' ( c ) . H-'(C) c H,'(C) if.s > 0. One of the difficuliies here is ?he incorporation of a nonlineiu boundary condition in the function space! Let us now transfor~nthe boundary conditions into linear ones. To do this. -, H)+' ( C ) by we define in a neighborhood of 0 a smooth function H : W,+)
Then the boundary conditions become
We make the change of variables near 0 , using G : W,+I + W , y + ~given by
where cp E H,'+'(c)
is the unique solution of the linear boundary value problem
We now define
Hence the boundary conditions become
It can be shown that G defines a change of variables near 0 . by checking that its (bounded) inverse at the origin is [dG(O)l-I = 1. I t can be shown (as in 1471) that the operator d G ( U ) : W,,+l + HI,+' extends to an isomorphism d G ( U ) : IHI, -+ W, for every U near 0 in W,,+I. Now set W = G ' ( U ) in (45). and tind the new system tor U = G ( W ) which reads
-
where U = G ( 4 . u . 11. ( ) = (+. u . 11. ( ) as detined above satisties the boundary conditions $, = 0 on := 0 . I. Finally, our system may be written as
as before, where C( = (A,h, ( . I , P ) and F is s~noothacting from D, = {($. l r , I / , [ ) E IHI,+l; $:l:=o,~ = 0) into IHI,. Notice that D , is dense in W, since s < 112 (this density is useful to avoid "technical complications" in the reduction procedure).
Let us proceed as in 1471. The linearized operator at the origin (solution for c.1 = 0 ) reads L,, = d F ( 0 ) .
and acts in
IHI, with domain ID,.The linear operator L,,
is then defined by
with boundary conditions
The eigenvalues cr of L,, satisfy the dispersion relation
(A - hs,f)rl,sin r,, - c r 2 cos s,, = 0. where
For 11 = O and cr = ik we naturally recover the 2D dispersion relation (12). We may observe that a = 0 is a double eigenvalue with the eigenmode ( 1 . 0 . 0 . 0 ) , and the generale invariance of the system under the shift 4 + I$+ ized eigenmode (0. I . - I /A. 0) d ~ to~ the const arid to the freedom on the Berno~~lli constant. which leads to a two-panunetcr family of trivial solutions. The eigenrnode Inuy be eliminated by tixing an addition;ll linear condition o n 4, and we may tix the Bernoulli constant for cli~ninatingcompletely the eigenvalue 0. Moreover. thc system has an O(7) sy111111etryi ~ i ~ ; ~ r i i ~due ~ l ctoe .translational invariance in y. addcd to pcriodicity. and to the reflection sy riimetry y + - \ . . I t resi~ltsthat for n # 0.
Fig. 18. The dot5 and cjrclc represent respeclively 7-D itlid 3D eigcnv;~lues.Crosses ;ire double (2D)eigenv;~lues, double circles arc quadruple (3D) cigctlvalurs. Col~iparcwith Figure 5 (2D case).
all eigenvalues are at least double (with a factor e*2'nn)'lPin the eigenmode). Moreover, as shown in [47], the spectrum of L,, consists entirely of isolated eigenvalues of finite multiplicities (the above ones), and we also have the estimate (I 7) on the resolvent (the proof in HIo is simpler than in (471). The location of "critical" eigenvalues is summarized in Figure 18 (see [50,47]). A center manifold reduction is possible as indicated in Section 3.3, using the method of [921 or [76]. The study of the solutions given by the normal form and of their persistence under remaining high order terms is made in [47]. In addition to waves periodic in x and y , waves periodic in y and quasi-periodic in x , and generalized solitary waves, periodic and even in y , tending at infinity to periodic waves in the propagation direction x, can be found. We refer to this paper for getting the detailed form of these solutions, whose study is analogous to the two-dimensional case, except for the high dimension of the center manifold (restricted to even solutions in g in 1471) which leads to discussions on high-dimensional reversible ODES. A typical study of the normal form (in a ten-dimensional reduced space) without restrictions on the symmetry of solutions is made by HSrSgu$-Courcelle and Il'ichev in [SO]. It corresponds to the bifurcation occurring near the curve f,-of Figure 18 where a simple pair of eigenvalues and a pair of quadruple eigenvalues ( 2 x 2 Jordan blocks) sit on the imaginary axis.
7.3. Tlrrc.e-tli~i~er~sio~~~il trcii~cllirlg\c3clvc.sperioclic. irl the tlirection c~f'l~ro/)tr~trtio~z For waves periodic in the direction of propagation .r, with an arbitrary profile in s, the formulation may follow the same method as above. This was done by Groves in 1451, who used a Hamiltonian formulation. I t can also be done in the framework of reversible vector tields like in 1531. Here again we have an O(2) symmetry, due to .r --t -.r invariance of the problem, and 0 is still at least a double eigenvalue, for the same reasons us above. The picture for critical eigenvalues near the imaginary axis depends on the same set of parameters (A,17. P , (.I), but it looks quite different fro111 the one given in Figure 18. see 1451 for details (case (.I = 0). I n 145,531, the authors study the tirst nontrivial 3D bifurcation, where the reduced space is 6-dimensional due to a pair of real double (3D) eigenvalues meeting at 0 (already double). The full normal form is studied in 1531. I n 1451 the author obtains large families of periodic and quasi-periodic solutions. Restricting the study to n l r n solutions in .r, 145,531 reduce the dimension to 2. This leads to an analysis similar to the one made in Section 4.1 (the shift invariance 4 + 4 const may be eliminated easily). with a parity symmetry in addition which cancels quadratic terms in the normal form. They can then prove the existence of an homoclinic solution corresponding to a wave periodic in .I-, and localized (and even) in y . We refer to 145,531 for details of the proof. In addition, there is in 1531 a discussion on the next bifurcation (eight-dimensional reduction), which reduces, for even solutions, to the same case (ii) as in Section 4.2.2.
+
7.4. Ditnerzsion breaking bjfurcatiori This topic tirst appeared in Hiiriigug-Courcelle and Kirchgissner 15 I ] . The idea is to look for bifurcations of 3D travelling waves from a nontrivial 2D travelling wave. The difficulty
is to determine the spectrum of the linearization about the 2D wave, since the linear operator has x-dependent coefficients, so that the spectrum cannot be calculated explicitly except in some very special cases involving model equations (see I32,52,54,20]). The only existing work at this time on the "real" water-wave problem, is by Groves, HL5gu~-Courcelleand Sun [46]. The authors start with the 2D solitary waves occurring when h > 113 (see Section 4.2. I), and consider bifurcating waves, periodic in the transverse direction y. They study the linearized operator (this part is technical and more complicated than in model equations), and show the existence of a pair of pure imaginary eigenvalues close to 0, 0 itself belonging to the essential spectrum. It is then possible to show the bifurcation of a family of solutions periodic in v , in a way analogous to the one mentioned in Section 5.1. The family of solutions they obtain cannot be obtained via the analysis of Section 7.2, in starting directly from the 0 solution. We should notice that the above method does not give any fully localized solutions.
8. Two-dimensional standing wave problem In contrast with the other sections, we consider in this section the standing wave problem, i t . , non-travelling waves, which are supposed to be periodic in the horizontal direction and in time, with a vertical mirror sylnmetry. This old problem, which goes back to Stokes 11081. has received considerable interest recently. The aim of this section is to indicate the no st recent results. The notations here are selfconsistent. Let us denote by -& < !< rl(.v, t ) the region occupied by the liquid, where 11 is the height of the free surfrice ( 0 when the Huid is at rest) and /I the avcragc depth; we assumc that the flow is potential. and we denote the potential by $(.\..!. t ) . We look for time periodic (period T ) . and .\.-periodic flows (wave length A). Choosing respectively T/277. A/277. AlT, h ' / 2 7 7 ~ as scales of time, length, velocity and potential. we obtain the dirnensionless system of equations (below &= IrA/2rr)
where the velocity components ( 1 1 . t i ) satisfy 11 = i ) $ / i f . v . 11 = i)$/i)y. and where / L = x ~ ' / 2 n h , being the acceleration due to gravity. Restricting the study to solutions with 11 even in t and in x , the linearization of (47). (48). (49) near 0 (a flat free surface) gives solutions of the form
$(x. y , t ) = -
'
p sinh(ph)
sin y r cos 11.r cosh p ( y
+ h).
provided that the dispersion relation
is satisfied. Assunie that we choose and ho such that (50) hac a unique positive solution ( p , q ) = (PO,'10) E Pi2. Then, we observe that even though Ipptanh(ph) - q21 # 0 for ( p . '1) # (PO.qo), this quantity may be arbitrarily small (leading to a small divisor problem, mentioned below). The structure of the nonlinear problem may be set into the form
where X lies in a suitable function space ( 2 ~ - p e r i o d i cin t and in x ) , and where F is analytic in its arguments, taking values in another suitable function space. The tirst step is to transform the original problem into such a formulation (especially fixing the domain). This was done in [991 by using Lagrangian coordinates, while in the infinite depth case it is done via a conformal mapping in [ 104.81. The known solution X = 0 corresponds to the fluid at rest. and we have a linearized operator
with a one-dimensional kernel. I f the problem was an ordinary bifurcation problem, there would be a nontrivial continuous branch ( X ( F ) , / L ( E ) , h ( ~ )of) solutions ot' (51 ), where p ( 0 ) = /LO.h(0) = ho. In fact. the pseudo-inverse L o ' of Lo (inverse on its range) is not bounded. because of the small divisor problem. We then need to restrict the parameter set to values where suitable diophantine conditions are realized, for having a bound for L ~ J(controlling ' the loss of regularity ). Moreover, since we need to use the Nash-Moscr implicit function the or en^, we also need to be able to invert (with a uniform bound for the inverse) the lineari~edoperator t i t t i 11011-:ov X . which is much more difticult, and in particular. again requires diophantine conditions to be realized. All this is done in the work of Plotnikov and Toland 1991. Their result is valid for ( / A . h ) near ( / l o . h o ) satisfying conditions which are described below. First. one defines for u E ( 1 , 2 ) .
-
-
and one deti nes for any fixed p the set C ( p )= {h > 0: q' - /LI) tanh(ph) = 0 has at least one solution (12. (1) E N']
Then it is shown that N(u) has full measure. and that for /LOE N ( ~ Jthe ) . set C(p0) is ) N(u) countable with only one limit point, namely 0. Now we detine the subset h { i ; , ( v c such that the equation q 2 - pp tanh(ph) = 0 has at most one positive integer solution for any p o E f i ( u ) and any ho. It is shown that the complement of A()(u) in (0, oo)has zero measure. The result is the following (see [99)for a more precise statement):
Wurrr-bvrrvt.ctrs tr .spotrtrl ti,vt~umiculsystem
493
T H E O R E M10 (Plotnikov-Toland). Choose v E (1,24/23), p" E Nb(v), ho E C ( p " ) , nnd denote by ( p . (1) the urziy~tesol~rtionof q 2 - p o p tanh(ph0) = 0. Then there exists (in injinite set E c Rf with 0 us u limit point, such that, fi)r uny E E E n (0, EO), the .systm~(5 I ) has u solution X, p , h, with u regulur X, und estimute.~]I X 11 = o ( E ~ ) , I P - pol Ih - hol < CE.
+
Notice that the existence of standing waves is shown on a set of points in the twodimensional parameter plane, which might be considered as distributed on a curve passing through the point (PO.ho), and that there are infinitely many such points accumulating near ( P O .ho). Moreover. there are infinitely many points like (PO,ho) in the parameter plane. For the inti nite depth problem, the dispersion relation is
which leads to solutions of the linearized problem under the form
. t ) = cosy t cos
?)(s
/?X
This gives solutions as soon as
11
is a positive mtion;tl number r/.s. Then
give infinitc~l~ I I I ~ I I I T.solirtion.s.In what follows we shall consider the case where / L is near I . since all other cases reduce to this case after a suitable rescaling: dividing the scale of time by t . , and the length sc~tleby rs. ~nultiplies11 by sir. The fact that at any rational value of 11. there is an i ~ ~ f i r ~ i t c ~ - t l i r r r c ~ ~ ~ ~ s i o ~for i e i lthe k c ~linr?~c~I e a r i ~ e doperator Lo creates great difticulties, known as "infirlitc~l,~ ~ncrn,~ ru.so~~trr~c~r,v ". In the paper 181, Atnick and Toland justified the algorithmic approach conjectured by Schwartz and Whitney [ 1041. Looking for solutions synirnetric under rellexion .I-+ -.v and even in tirne t , they proved that if one chooses the dominant mode as II(.Y. t ) = cost cos.r. the resonances do not arise at any stage of the computation of the expansion in powers of c.. where E = 2 m . The system ( 5 1 ) (without 11) is expressed in the form of' an infinite system of coupled ordinary differential equations in the time-periodic spatial Fourier series components of the standing wave. Another way to look at this problem is to use a Hamiltonian formulation adapted for spatially periodic solutions. introduced by Zakharov in 1 1 271. Dyachenko and Zakharov 1401 have indicated that not only the quadratic terms of the vector field can be rernoved by a canonical change of variable, but also the resonant cubic terms of the normal form. are such that the truncated system is integrable. This was veritied in detail by Craig and Wolfork 1301, who showed that the integrability is lost at the next order. Moreover, Craig 1281 studied the two-mode standing wave solutions of the vector field truncated at order 4. In principle, one can tind the formal expansion of all possible standing waves, as they are described below, also by starting with their results.
494
F: Dius and G. looss
The infinitely many resonances problem is solved in showing that there are infinitely many formal solutions which may be obtained in expanding in powers of E = ,2with no difficulty at any order, and where the leading order is given by
The number of basic modes given by I c N may be injnite, and (fI)(, = fl (free choice for any ( I ) , so the result of [8] is recovered for I = { I ) . This result justifies previous numerical computations (with hundreds of terms) made with I = ( 1 , 2 ) and I = ( 1 , 2 , 3 ) by Bryant and Stiassnie [22]. The method of Iooss in [59] uses the formulation of [104,8], while in [601, the problem is directly formulated in the framework of analytic functions. A suitable near identity explicit churige cf variable allows to modify the system in cancelling the quadratic part, which simplifies greatly the analysis for a Lyapunov-Schmidt method (not using normal forms). The itlJitzite-dimen.~ionul bifurcation equation hu.7 all unc.ouplecl c.riricu1 modes L l r le~ldingordrr.~(as might be also found with the results of 130]), and allows to detect all solutions whose power series begins as mentioned in (52). The problem of convergence of such series i n a suitable space is still open today. The problem due to infinitely many resonances, in addition to the loss of regularity already present in the finite depth case. leads to serious complications in applying the Nash-Moser theorem. We might however expect that these solutions exist for a restricted set of E E (0,E O ) , with positive measure.
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CHAPTER 1 1
Solving the Einstein Equations by Lipschitz Continuous Metrics: Shock Waves in General Relativity Jeff Groah I ) c ~ l x r r t u ~ r~f'iCl~rtlrc~t~rtrti(~,~, ~ut Urii~.c.,:\.rt~ ( ~ f ' C o l ~ / i ~ ~ -I)(n~r.s. r i i ~ r . CA '9.5616. USA E-tirrril: ~ t ~ ~ , ~ r r ~ ~ ~ l ~ @ ~ , ~ ~ t i ~ l i t ~ ~ . ~ ~ . v i ~ . ~ . ~ ~ ~ / r ~
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C0111~'111.~ I . I ~ ~ t r o i l u c t i o. ~.i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 I . I . Spaccti~i~c allil the gr;ivltatio~lal metric tc11\or . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 1.2. Ir~troil~~cticrri to tllc P:~ri\tciocqtration\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 1 .3. Shock waves ill gcilc~.;~lrelativity i111dthe t i i i i \ t c ~ ncqu;llion\ In S c h w i ~ r / \ c I ~c~oI ~ o Ir d i ~ l ; ~ t.~.\ . . . 5 13 2. Wcah \olutiollr crithc l i n \ r c i l l equation\ when the ~iicti-ici\ o ~ l l yI.~p\cliit/coiilinuo~l\;Icrt)\\ ;In ~ n t c r l i ~ c5c 10 . 2.1. The general prohlc.111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 2.2. The \pllcric;illy \y11l11lctr1cc;~\c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 3. Miitcl1111g;111 f;KW to ;I.I'OV illctrlc i~cl-o\\ii \h~rcl, wa\c . . . . . . . . . . . . . . . . . . . . . . 541 3.1. The pctlcral FKW-TOV illatchiilg prohlclli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 1 3.2. The co~iscrvatiotlcon\trailit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 4 A c l i ~ \ \o l ' \ o l u ~ i o i ~illotlclillg > hl;l\t waves in G R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 'Supported ill p;irt hy NSI: Applied Mathcniatlc\ Grant Nuinher IIMS-XO-05705 ;inti hy the I~l\titlitcof Thcorclicol Fyilanlics. llC-l>:1vi5. 2 ~ u p p o r t c di n piu-1 hy NSF Appllcd Ma[hcm:ttlc\ Grant Numhcr DMS-Xh- 134.50. by the In\litutc ofThct)rt.!ici~l Dynalnics. UC-Dnvih. and IHES. H A N D B O O K O F M A T H E M A T I C A L F L U I D DYNAMICS. V0I.UME I1 Edited by S.J.Friedlander nnd D . Scrrc O 2003 Elsevier Scierlce B.V. A l l right5 rewrved
4.1. An exact solution of TOV type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. An exact solution of FRW type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. A class of exact shock wave solutions of the Einstein equations . . . . . . . . . . . . . . . . . . . . 4.4. The Lax shock conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. A shock-wave fornmulution of the Einstein equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. The Einstein equations fi)r a perfect fluid with spherical symmetry . . . . . . . . . . . . . . . . . . 5.3. The spherically symmetric Einstein equations formulated a\ a system of hyperbolic conservation laws with sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Summary of the weak formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Wave speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
564 565 566 569 576 576 576 577 584 589 591 594
1. Introduction These notes address mathematical issues that arise when one attempts to incorporate shock waves into Einstein's theory of General Relativity. At the start, one is led to consider solutions of the Einstein equations when the space-time metric is only Lipschitz continuous, and this is the rnain topic of this article. In Section 1 (which is taken from [36]), we introduce General Relativity and the Einstein equations, and then we begin the discussion in Section 1.3 (taken from [91),by writing down the Einstein equations for a perfect fluid assurning spherical symmetry, and assuming standard Schwarzschild coordinates. We point out that the equations imply that the metric components of solutions can be at best Lipschitz continuous functions of the coordinates when shock waves are present (that is, the metric components, viewed as functions of the space-time coordinates, lie at best within the class cO.'of functions that are continuous with Holder exponent 1 , 161). We then write down a system of conservation laws with source terms that is weakly equivalent to these equations, and this helps explain the less than expected regularity of the metric. A rigorous derivation of the equivalence of these equations is the topic of Section 5. Now the Lipschitz continuity of the metric components is interesting because the curvature tensor, a quantity determined by second derivatives of the metric tensor, must remain free of delta function sources i n order to be ;I bon;~fida weak solution of the equations. This motivates the discussion in Section 2 (taken frorn 129I), which presents the general theory of matching space-time metrics Lipschit/ continuously across smooth shock s ~ ~ r f i ~ cIne sSection . 3 (which is takcn from 13 1,301). we dcvelop :I theory for matching o Fricdmann-Robertso~i-Wi~Ikermetric ol' cosmology. to :I Tolm:~r~-Oppcnhci11ncr-Volkoff' metric for a static fluid sphere. cross i~ shock wave intcrf~cc.iund in Section 4 (tttkcn I'rom 1301). we use this theory to clerivc :I class of cxact shock wavc solutions of the Einstein ccluntions that model blast w~tvesin Gcncral Kcli~tivity.In these exact solutions, the Nix Htrrl,y singularity ot'the FKW metric is repliiced by :I shock wave explosion, and the outgoing shock wavc lies at the lending edge of what is interpreted as the expiunsion of the galaxies in the cosmological intcrprctiition of' the FRW metric. The construction of these exact solutions takes advantage of being able to . Section 5 (taken f'rorn ( 9 ) ) .we work with metrics in the lower srnoothncss cl:tss of c ' ~ ' . ' In show that the spherically symmetric Einstein equations written in standard Schwtu-/.schilcl coordin:~te.s(that is. the equations which bcg;in the discussion in Section I ) . arc weakly equivalent to u system of conservation laws with source terms. This reformulation of the equations shows that we can expect solutions with shock waves to exist. and helps explain the LipschitL regularity of the metric components when shocks are present. The system of equations derived i n Section 5 is also the starting point ti)r the existence theory given in [ 101. The main theorem in 1101 is stated in Section 1.3. This result confirms what is indicated by the equations derived i n Section 5. and demonstrates rigorously that the initial value problem f i r the Einstein equations (assi~nlingperfect fluid and spherical symmetry). is consistent for initial density and velocity protiles that are discontinuous functions that are only locally of bounded total variation. Said differently. the result demonstrates that the Einstein equations of General Relativity are meaningful in the presence of arbitrary numbers of interacting shock waves, of arbitrary strength. The class cO.'is one derivative less smooth than the Einstein equations suggest the metric components ought to be, and i n fact, the singularity theorems in I I 1 ] presume that
metrics are in the smoothness class C ' . ' , one degree smoother, cf. [ I I, p. 2841. One of the remarkable features of the results of Section 2 is that for smooth shock surfaces, there always exist coordinate transformations that smooth the components of the gravitational metric to c'.'.and these coordinates can be taken to be the Gaussian normal coordinates of the surface. However, the Gaussian normal coordinates break down at points of interaction of shock waves, and thus it remains an open problem whether general Lipschitz continuous solutions of the Einstein equations can always be smoothed by coordinate transformation. This leads to the following interesting dichotomy: If such a coordinate transformation does nor always exist, then solutions of the Einstein equations are one degree less smooth than previously assumed; and if such a transformation does exist, then it defines a mapping that takes weak solutions of the Einstein equations to strong solutions. In the latter case, it follows that the theory of distributions and the Rankine-Hugoniot jump conditions for shock waves need not be imposed as extra conditions on the relativistic compressible Euler equations in General Relativity, but rather must follow as logical consequences of the strong formulation of the Einstein equations by themselves.
In Einstein's theory of General Relativity. all properties of the gravitational tield are determined by the g~a~~ittrtiotlol 11lc~rric tctr.sor g . a Lorentzian metric that describes a continuous field of s y ~ n ~ n e t rbilinear ic forms of signature ( - I . I . 1 . 1). detined at each point of a fourdimensional manifold M called "spucc-time". Frcefltll paths through the gravitational ticld are the geodesics of the metric; the non-rotating vcctors carried by an observer in freefall are those vectors that are parallel transported by the ( ~ ~ n i symmetric) q~~e connection determined by ,g: spatial lengths of objects correspond to the lengths of the spacelike curves that deti ne their shape - length measured by the metric ,y: and tilne changes for an observer are determined by the length of the observer's timelike curve through space-time. as measured by the metric g . The length of a curve in space-time is computed by integrating the element of ~irclength ds along the curve, where, i n a given coordinate system on space-time, d.s is detined by
Here we adopt the Einstein summation convention whereby repeated up-down indices are assumed to be summed from 0 to 3. A coordinate syste~no n space-time is a regular map that takes a neighborhood U , of space-time to RJ, .r : U , -+ EXJ. Since space-time is a manifold, it can be covered by coordinate charts. We let s = (.r0. .r , .r2. a3) denote both the coordinate map and the coordinates of a point .I-( P ) E RJ. The functions x;,(.r). i, j = 0, 1.2. 3. are the x-components of the metric g . At each point .r. the matrix g ; , determines the lengths of tangent vectors in terms of their components relative to the x coordinate basis (i)/i)xi}.That is, in x-coordinates. the tangent vector to a curve x ( < ) (as parameterized in x-coordinates), is given by X ( < ) = .ii (dot denotes d / d t ) , so that along the curve x ( < ) , the increment dx' in the .r'-coordinate, in the direction of the curve,
'
&
is given by dx' = .i' dc. Thus, according to ( I . I), the increment in arclength along a curve ~ ( 6is) given in terms of the increment in the parameter 6 by
so that, the length of an arbitrary vector X = X'
& is given by
where again we assume summation over repeated up-down indices. We conclude that the length of a curve is just the integral of the x-length of its tangent vector along the curve. according to the Under change of coordinates .r + T , a vector X' ;fi; transforms to Xu tensor transformation laws
&
(Our slightly ambiguous notation is that indices i. j , k. . . . label components i n .rcoordinates, and cu. p . y . . . . label components in y-coordinates. So. for example, X' is the .vl-component of the tangent vector X. Xu is the \."-component of X. etc. This works cl~~itc well. but tensor.; must be re-l:tbcled when indices are evitluatecl.) I t follows that the metric t e n o r transtor~iisaccording to the tensor transformation law i1.v' i1.v
1
sup = 'Sl / -i ) \ U i)>.fi
That is. at each point. g transfor~nsby the matrix tr~unstorm~ttion law
e matrix A = i).\-//i)yPtransforms the vector components of for a bilinear form, b c c ~ u ~ sthe the \.-basis (i)/i)yU)over to their components relative to the .\--basis (il/il.\-'I. The Einstein su~n~nation convention kecps track oi' the coordinate tritnstor~iintionlaws as in ( 1.2) and ( 1.3) so long as we keep the indices on coordinate fi~nctions"up" (as in .r' ). coordinate basis indices "down" (as in it/i).\-' ). indiccs o n vector components " L I ~ " (as i n X' so that X = XI f$ ). indices on basis I -forms "up" (;IS i n d.rl), and indices o n components ol' I forms down (as i n (0; so that to = to; d s ' ). In general, a tensor o f type ( X . I ) is said to have k-contravariant indices (up) and I-covariant indices (down) if the components in a given coordinate system transform according to thc tensor transformation law
Here the (matrix) Jacobian satisfies i).\-/i)\ = (i)y/if.r)-', and by letting
we can raise or lower an index by contracting the index with the metric; that is, for example,
raises the index i . In the modern theory of differential geometry,
.
TI: ;';:,:
.
are viewed as the
@J . . . @ &- @J dxil @ . . . @J d x ~ ' which ) form a components on the tensor products { * fl.1 O.r1L basis for the set of operators that act linearly on k copies of T * M and I copies of T M , cf. [4]. Freefall paths through a gravitational field are geodesics of the space-time metric g . For example, the planets follow geodesics of the gravitational metric generated by the Sun (approximated by the Schwarzschild metric outside the surface of the Sun, and by a TolmanOppenheimer-Volkoff(T0V) metric inside the surface of the Sun), and according to the standard theory of cosmology. the galaxies follow geodesics of a Friedmann-RobertsonWalker (FRW) metric. In spherical coordinates x = ( t , r, H,d), the Schwarzschild line element is given by
the TOV line element is given by
and the FRW line element is given by dr2 = -dl2
dr' + ~(~)z(------I - kr2
+r.dR~).
The line element determines the metric components g , , through the identity ( 1.1). Here 5 denotes Newton's gravitational constant, Mo denotes the mass of the Sun (or a star), M ( r ) denotes the total mass inside radius r (a function that tends smoothly to Mu at the star surface), B ( r ) is a function that tends smoothly to 1 - 2GMo/r at the star surface, H = d ( t ) / X ( t ) is the Hubble "constant", and ~ S Z '= do2 + s i n 2 ( ~ ) d @denotes ' the standard line element on the unit 2-sphere. (Here 2 s M = ~ s M / c ' . and we take (. = 1 . 141.) Each o f the metric5 ( 1.4)-( 1.6) is a special case of a general spherically symmetric space-time metric of the form
where A , B, C, D are arbitrary, smooth, positive functions. A spherically symmetric metric is said to be in standard Schwarzschild coordinates (or the standard coordinate gauge). if it takes the simpler form
Solving 111eEirlsrrin ryuufior~s h! Lipschirz c.ontir~uou.s)nrtric,.s: Shock wuvr.s in ~ m r r u rrlativiry l
507
It is well known that, in general, there always exists a coordinate transformation that takes an arbitrary metric of form (1.7) over to the simpler form (1.8), 142).In Section 1.3 below we carry out this reduction with an eye toward anticipating the regularity of the metric components A and B, [42,9]. The geodesics of a metric are paths x ( s ) of extremal length, determined by the geodesic equation d2xi - . dxl dxk - r ! -1"s ds '
ds'
where the so-called Christoffel symbols or connection coefficients f ; k are defined by
qk
(Here, "k" denotes the classical derivative in direction x k . ) The Christoffel symbols are the central objects of differential geometry that do not transform like a tensor. Indeed, they fail to be tensorial by exactly the amount required to convert coordinate differentiation of vector components into a tensorial operation. That is, for a vector tield Y . let Y' denote the .r'-component of Y . The covariunt derivative V is dctined by
where. letting semicolon denote covariant differentiation. Z defines a vectctr field with .rcomponents
For arbitrary vector fields X and Y . one delines the covariant derivative V x Y by
We say that a vector field Y is parallel along a curve whose tangent vector is X i f
all along the curve. I t follows that the covariant derivative V x Y measures the rate at which the vector tield Y diverges from the parallel translation of Y in the direction of X . In a similar fashion, one can define the covariant derivative V T of any ( k . I ) tensor T as the (k. I I ) tensor with components
+
For example, for a (1, 1 ) tensor T .
More generally, to compute V T for a ( k . I ) tensor T, include a negative term for every contravariant index (contract the index with f as above), and a positive term (as above) for every covariant index in T . We say that T is parallel along a curve with tangent vector X if V x T = 0 all along the curve. It follows that V x T measures the rate at which T diverges from the parallel translation of T in direction X . For a (2,O) tensor T we define the covariant divergence of T to be the vector field detined by div T = T'" .;i;,
if
The covariant derivative commutes with contraction and the raising and lowering of indices, [42], and by (1.13). V reduces to the classical derivative at any point where the Christoffel symbols T,!~vanish. = 0 at a point in a coordinate system where g;,.~ = 0, all It follows from ( I . 10) that i . j. k = 0. . . . , 3 . The existence of s ~ ~ coordinate ch frames at a point follows directly from the fact that the metric components g,,, Lire smoothly varying, and transform like a symmetric bilinear form under coordinate transformation. If i n addition. g,, = diag(- 1 . I , I . I ). then such a coordinate system is said to be locally inertial. or locally Lorentzian ;it the point. The notion of geodesics and parallel translation have a very natural physical interpretation in General Relativity in terms of the locally inertial coordinate frames. Indeed. General Relativity makes contact with (the Rat space-time theory of) Special Relativity by identifying the locally Lorent/,ian fr:uiies at o point as the "locally non-rotating" inertial coordinate systems in which space-time behaves as if it were locally Rat. Thus physically. the non-I-otating vector tields carried by an observer in freefall should be the vector tields that are loc,cilly c.o/~stcrrrtirr the Ioc~cillyirrortitrl c.oorrliritrtr , f i - r i r ~ r t ~rl
So11,irtgthe Eitr.steitl equfltiot~.~ Lipschic c.on1inuou.s mrlrics: Shock u3rrve.sin ,qr~tc,rirlreltrlivilv
509
coordinate systems related by the transformations of Special Relativity; that is, in Special Relativity, the 10 parameter Poincare group replaces the 10 parameter Galilean Group as the set of transformations that introduce no accelerations. The Poincare Group is obtained from the Galilean group by essentially replacing Euclidean translation in time by Lorentz transformations, and this accounts for time dilation. The space-time metric can then be viewed as a book-keeping device for keeping track of the location of the local inertial reference frames as they vary from point to point in a given coordinate system - the metric locates the local inertial frames at a given point as those coordinate systems that diagonalize the metric at that point, g, - - diag(- I . I . I , I ), such that the derivatives of the metric -. components also vanish at the polnt. Thus, the earth moves "unaccelerated" in each local inertial frame. but these frames change from point to point, thus producing apparent accelerations in a global coordinate system in which the metric is not everywhere diagonal. The fact that the earth moves in a periodic orbit around the Sun is proof that there is no coordinate system that globally diagonalizes the metric, and this is an expression of the fact that gravitational tields proditce nonzero space-time curvature. Indeed, in an inertial coordinate frame, when a gravitational field is present, one cannot in general eliminate the second derivatives of the metric components at a point by any coordinate transti)rmation, and the nonzero second derivatives of the metric that cannot be eliminated, represent the gravitational field. These second derivatives are measured by the Riemann Curvature Tensor associated with the Riemannian metric ,g. Riemann introduced the curvature tensor i n his inaugur;~lIect~ire01' 1854. In this lect~lrc he solved the longstanding open problem of describing curvature in surti~cesof dimension higher than two. Although the curvature tensor was first dcvcloped for positive dctinite "spatial" nlctrics, Einstein accounted for time dilation by letting Lorcntz tr;~nsSormations play the role of rotations in Riclnnnn's theory. and except Sor this. Riemann's theory carTensor K ' l k l ( x ) is n quantity that ries over essentially unchanged. The Rie11n;unn Curv;it~~rc involvcs second dcriv;itivcs of g , , (x). but which tr;unsforms like a tensor under coordin;itc trunsSormation; that is, thc components transform like a sort of four component version ol' a vector field. even though vector fields ;we constructetl csscntiolly from lirst order deriv;itives. Thc connection between General Relativity iind geometry can be si~mmarizedin the statement that the Riemiunn Curvature Tensor associiited with the metric ,g gives an invariant description of griivit;itional accelerations. The components of the Riemann Curvature Tensor are given in terms of the Christoff'el symbols by the formillo, 141 I.
'
R',L/ = r;/k-r,'L, +
One can interpret thi4 ;I\
;I
{r;r:L -r;r:11.
"cLI~I" plus a "cot~itnnut;itor"
Once one makes the leap to the idea that the inertial coordinate frames change from point to point in space-time, one is immediately stuck with the idea that. since our non-rotating inertial frames here on earth are also non-rotating with respect to the fixed stars. the stars must have had something to do with the determination of our non-accelerating reference
510
J. Grouh ct 01.
frames here on earth (Mach's Principle). Indeed, not every Lorentzian metric can describe a gravitational field, which means that gravitational metrics must satisfy a constraint that describes how inertial frames at different points of space-time interact and evolve. In Einstein's theory of gravity, this constraint is given by the Einstein gravitational field equations. These field equations were first introduced by Albert Einstein in 1915 after almost ten years of struggle. The Einstein equations can be written in the compact form
Here G denotes the Einstein curvature tensor, T the stress energy tensor (the source of the gravitational field), and K is the gravitational constant. In a given coordinate system x , the field equations ( I . 15) take the component form
where
denotes the x-components of the Einstein curvature tensor, and T,; the .r-components of the stress energy tensor. We let 0 < i , j < 3 refer to components in a given coordinate system, and again we assume the Einstein summation convention whereby repeated updown indices are assumed to be summed from 0 to 3. The components of the stress energy tensor give the energy density and i-momentum densities and their fluxes at each point of space-time. When the sources are modeled by a perfect fluid. T is given (in contravariant form) by
where w denotes the unit 4-velocity vector of the fluid (the tangent vector to the world line of the fluid particle), p denotes the energy density (as measured in the inertial frame moving with the fluid), and / I denotes the fluid pressure. The velocity w has four components ni' = dxi/d.s when the fluid particle traverses a (timelike) path .r(.s) in .r-coordinates, and s is taken to be the arclength parameter ( I . I ) determined by the gravitational metric g . It follows that w is a unit timelike vector relative to g , and thus only three of the four coniponents of w are independent. The constant K in (1.15) is determined by the principle that the theory should incorporate Newton's theory of gravity in the limit of low velocities and weak gravitational fields (Correspondence Principle). This leads to the value
Again, c denotes the speed of light and
denotes Newton's gravitational constant.
Newton's constant first appears in the inverse square force law
In (1.19), M is the mass of a planet, Mo is the mass of the sun, and r is the position vector of the planet relative to the center of mass of the system. The Newtonian law ( 1.19) starts looking like it is not really a "fundamental law" once one verities that the inertial mass M on the LHS of ( 1.19) is equal to the gravitational mass M on the RHS of ( 1.19) (Equivalence Principle). In this case, M cancels out, and then (1.19) (remarkably) becomes rnore like a law about accelerations than a law about "forces". That is, once M cancels out, the force law (1.19) is independent of any properties of the object (planet) whose motion it purports to describe. Thus, in Newton's theory, the "gravitational force", which is different on different objects of different masses, miraculously adjusts itself perfectly so that every object (subject to the same initial conditions), traverses exactly the same path. Thus Einstein was led to suspect that the Newtonian gravitational force was some sort of artificial device, and that the fundamental objects ot'the gravitational tield were the "freefall paths". not the forces. From this point of view, the tield equations (1.15) are more natural than (1.19) because they are. at the start, equations for the gravitational metric, and the gravitational metric fundamentally describes the paths of "freefillling" ob,jects by means of the geodesic ecluation of motion ( 1.9) - which just expresses "local non-accclcriition i n locally inertial coorclinate frames". In Newton's theory ot'gravity, the non-rotating framcs here on eorth itre aligned with the stars because there is a global inertial coordinate system that connects us. In contrast. according to the modern theory ot'cosmology. which is bitsed on Einstein's theory of' gravity, the non-rotitting incrti:tl frames here o n earth arc irligncd with the htars because the FRW metric ( 1.15) maintains this alignmcnt, and ( I. 15) solves the Einstein equations for an appropriate choice of K(/).(This is still a bit unsatisfying!) In the limit that a ti nite set of point masses tends to a continuous mass distribution with density 0 . Newton's force law is replaced by the Poisson equation for the gravitational potential 4,
Indeed, in the ciise of a compactly supported density p ( . t - ) .one can use the funda~iiental solution of Laplacian to write the solution of ( 1.70) as
so the Newtonian acceleration at a point x is given by
Thus we recover ( 1.19) from (1.22) by approximating p in ( 1.22) by a tinite number of point masses.
The Einstein equations play the same role in General Relativity that the Poisson equation (1.20) plays in the Newtonian theory of gravity - except there is a very significant difference: the Poisson equation determines the (scalar) gravitational potential q5 given the mass density p , but in Newton's theory this must be augmented by some system of conservation laws in order to describe the time evolution of the mass density p as well. For example, if we assume that the density evolves according to a perfect fluid with pressure 1, and 3-velocity v, then the coupling of Newton's law of gravity with the Euler equations for a perfect Huid leads to the Euler-Poisson system
The f rst four equations are the compressible Euler equations with the gravitational forcing term on the RHS. The fi rst equation, the continuity equation, expresses conservation of mass. the next three express conservation of i-momentum, i = 1 , 2 , 3 (for a perfect fluid this really says that the time rate of change of momentum is equal to the sum of the force of the pressure gradient plus the force of the gravitational field; e' denotes the ith unit vector in R'). and the last equation expresses the continuum version of Newton's inverse square force law. Note that for the Huid part o f ( 1.33). information propag;ites at the cound (and shock) speeds. but the gravitational potential q5 is updated "instantaneously". depending only on the density p ( x . t ) . according to the formula ( 1.21 ). I n contrast. tor Einstein's theory of gravity. the tirne evolution of the gravitational metric is determined simul~aneously with thc time evolution of the sources through system (1.15). and all of the components of the stress tensor directly influence the components o f the gravitational tield s,, . This principle is the basis for the discovery o f the Einstein equations. Indeed. since the 0-column of the stress-energy tensor ( I. 18)gives the energy and Inomenturn densities. and the i-colu~nn gives the corresponding i-Huxes (in the relativistic sense). it follows that conservation of energy-momentum in curved space-time reduces to the statement
where (capital) Div denotes the covariant divergence for the metric x, so that it agrees with the ordinary divergence in each local inertial coordinate frame. In this way equations (1.24) reduce to the relativistic compressible Euler equations in Hat Minkowski spacetime. Since the covariant derivative depends on the metric components. the conservation equation (1.24) is essentially coupled to the equation for the gravitational field g . But the stress tensor T is symmetric, fi,, = T , ; , and so the tensor on the LHS of ( 1.16) must also be symmetric, and therefore the Einstein equations ( 1.16) supply ten independent equations in the ten independent unknown metric components ,y;, . together with the four independent functions among p and the unit vector tield w . (Here p is assumed to be determined by an equation of state.) But (1.16) assumes no coordinate system. and thus in principle we are free to give four further relations that tie the components of G and T to the coordinate system. This leaves ten equations in ten unknowns. and thus there are no
further constraints allowable to couple system (1.15) to the conservation laws (1.24). The only way out is to let (1.24) follow as an identity from ( 1 . 1 S), and this determines the LHS of (1. IS), namely. the Einstein tensor G i j is the simplest tensor constructable from R I x l such that ( 1 2 4 ) follows identically from the Bianchi identities of Riemannian geometry (R~,~,,= , , ,0. , where [ k l . 1111 denotes cyclic sum, cf. [ 4 2 1 ) .Thus, ~ the simplest and most natural field equations of form ( I . 15) are uniquely determined by the equation count, 1421. The next simplest tensor for the LHS of ( I . IS) that meets ( 1.24) is
for constant A . In these notes we always assume A = 0. One can show that in the limit of low velocities and weak gravitational fields, the equations ( 1.24) reduce to the first four equations of ( 1.23). and the ( 0 . 0 ) component of the Einstein equations (1.16) reduces to , This establishes the the Poisson equation (1.20), thus fixing the choice K = 8 ~ r G / c . ~1421. correspondence of Einstein's theory of gravity with the Newtonian theory.
I n Einstein's theory of gravity. based o n ( 1. IS), the conservation of'cnergy and momentum I .24) are not i~nposecl.but follow ;IS differenti:tl identities from the lield equations ( 1.15). 111 ;I specified system of coordinates, (1.15) determines a hyperbolic system of equ;ttions that si~ni~lt:uncoi~sly describes the timc evolution ;uicl interaction of local incrtiitl coordinate frunlcs. as well ah the tirr~cevolution of the tluid according to ( I .24). Since GR is coordin;tte indepcntlcnt. we can always view the time evolution ( 1.15) in local incrti;tl coordinates at any point i n space-time. in which case (1.73) reduces to the clnssicol relativislic Eulcr cqi~:~tions at the point. This tells us that, heuristically. shock witves must form in the time evolution of ( 1. 15) hecause one could in principle drive a solution into ;I shock while in I: neighborhood where the cquntions remained ;I small perturbation of h he cla,~sic;~l E~rlcr equtttions. (This is much easier to say than to demons1r;tte rigorously. ant1 as h r as we know. such a demonstration remains a chitllenging task.) We conclude from this that shock waves are its I'unda~nentalto the timc evolution of solutions of the Einstein equations for a perfect fluid. as they are tor the time evolution of'the classical compressible Euler e q u I.1011s ( 1.23). Notice that ( 1. 15) At n shock witvc. the fluid vuriitblcs p . w and 17 are discontini~oi~~. implies that the Einstein curvature tensor C; will be discontinuous at any point where 7' is discontinuous. Since G involves second derivatives of the metric tensor
CI
j ~ h i ,is the silnplebt known route t o the licld equations ( 1 . 15).01.courhe. since ( 1 .IS) represents a new starting tollow\ that thcre mu\[ he a "cc~~~ccptual Icap" at some stage of ally " ~ ~ I - I V ~ I ~ I O of I I " ( 1 . 15).
point. it
Using MAPLE to put the metric ansatz (I .8) into the Einstein equations ( 1.15) produces the following system of four coupled partial differential equations (cf. (3.20)-(3.23)of [9]),
where the quantity @ in the last equation is,
@=----
BArB, 2AB
B 2(:)
B
A'
AB'
A A'
'
AA'B'
-r+x??(a) + ? a ~
Here "prime" denotes i)/i)r, and again K = 817~/(.'is the coupling constant, 5' is Newton's gravit~~tional constant. ( . is the speed of' light, TI-'. i , j = 0. . . . , 3. ;ire the components of the stress energy tensor, and A = A(r. t ) . R B(r. t ) denote thc components of the gravitational metric tensor ( I .X) i n stundartl Schwarzschild coordinates x = (.I.", .rl. .t-'. . v 3 ) = ( t . r. 0 . @ ) . The muss function M is defined through the idcntity
-
In terms of the variable M . Equations ( 1.25) and ( 1.26) are equivalent to
1
M' = - K 2
~ ' A T"".
and
respectively. Using the perfect fluid assumption ( l.l8), the components T r l satisfy
S o h i ~ r/lc, z ~ Eitl.src~irrrclucrrro~~s by Lipschit: c.onti~tuou.smetrics: Shock waves it1 ,yrtlerul rc.lutiviry
5 15
where T: denote the components of T in flat Minkowski space-time. The components of TIMare given by
where a 2 = p / p , cf. [28,91. Here v, taken in place of w, denotes the fluid velocity as measured by an observer fixed with respect to the radial coordinate r. I t follows from ( 1.30) together with ( 1.35)-( 1.37) that, if r 3 ro > 0, then
and it follows directly form ( 1.35)-( 1.37) that
so long as a < c. Equation ( 1.38) shows that M ( r . 1 ) can bc interpreted as the total mass illside radius r at time I . Now we are interested in solutions o f ( 1.25)-( 1.28) in the case when shock waves are present. A shock wave in thc compressible Euler equations leads to discontinuities in the fluid density, pressure and velocity. and thus in light of ( I. 18). it follows that a shock wave would produce a discontinuity in the stress tensor T at a shock. But when T is discontinuous, Equations ( 1.25)-( 1.27) above imply immediately that derivatives of the metric co~nponentsA and H are discontinuous at shocks. Moreover. if' A irnd H have discontinuous derivatives when shock waves are present. it follows that ( 1.28). being second order. cannot hold classically. and thus Equation (1.28) must be taken in the weak sense. that is, in the sense of the theory of distributions. From these considerations. we see that the metric components A and H can be at best only Lipschit/ continuous. that is. c".'.at shock waves. That is. A and R are one degree less s~iiooththan the general theory suggests they should be. I I I I. The general problem of making sense of gravitational metrics that are only Lipschitr. continuous at shock surfaces was taken up in 1301. The analysis there identities conditions that must be placed on the metric i n order to ensure that conservation holds at the shock, and that there do not exist delta-function sources at the shock. 1 121. When these conditions are met, the methods in 1301 imply the existence of a c',' coordinate transformation (to Gaussian normal coordinates), that improves the level of smoothness of the metric components from c".'up to c'.'at the shock. All of this is the subject of Section 2.1. However,
516
J . Groah et ul
these results apply only to smooth interfaces that define a single shock surface for which G = K T holds identically on either side. For general shock wave solutions of (1.25)-(1.28) (that can contain multiplicities of interacting shock waves), it is an open question whether there exists a coordinate transformation that can increase the level of smoothness of the metric components by one order, because the Gaussian normal coordinate system for the shock surface breaks down at points where shock waves interact. We conclude this section by showing that the mapping (r, t) -+ ( F , 7) that takes an arbitrary metric of form (1.7) over to one of form (1.8), implies a loss of one order of differentiability in the metric components when shock waves are present. This argues that our with the existence of such a smoothing coordinate transformation, results are cor~,si.sterzt but still leaves open the problem of the e,~i.stmceof such a transformation. We review the construction of the mapping (r, t ) + (7, i ) , [42,9] with an eye toward keeping track of the smoothness class of the metric at each stage. To start, one must assume that the metric component C ( t . r ) in (1.7) satisties the condition that for each tixed t , C increases from zero to infinity as r increases from zero to infinity, and that
(These are not ilnreasonable assumptions considering that C measures the areas of the spheres of symmetry.) Define
Then the determinant of the Jacobian o f the mapping (r. t ) + (7. t ) satisties
in light of ( 1.41 ). Thus the transfortnation to (7. t ) coordinates is (locally) a nonsingular transformation. and in (7. t ) coordinates the metric ( 1.7) takes the form
(Here we have replaced 7 by r and A , B and E stand in for the transformed components.) I t is easy to verify that. to eliminate the mixed term, il suffices to define the time coordinate i so that, cf. 1421,
In order for (1.44) to be exact, so that i really does detine a coordinate function. the integrating factor @ must be chosen to satisfy the (linear) PDE
Soh~irt!:tiit Eirlstrbr rquutiorr.~
Lil,,sc~l~it: cori/br~rorrsmc,trir,.s: Shock wuvr.s irz ,yrrfmr/ rc.lrrtivir!:
5 17
But we can solve (1.45) for d ( r , t ) from initial data @(r,to), by the method of characteristics. From this it follows that (at least locally), we can transform metrics of form (1.7) over to metrics of form (1.8) by coordinate transformation. To globalize this procedure, we need only assume that Cr(r, r ) # 0, and that C takes values from zero to infinity at each fixed t . Now note that in general d ( r . t), the solution to (1.45), will have the same level of differentiability as A(r, r ) and E(r. r); and so it follows that the components of dt and d r in (1.44) will have this same level of differentiability. This implies that the 7 transformation defi ned by (1.44) preserves the level of smoothness of the metric component functions. On the other hand, the 7 transformation in (1.42) reduces the level of differentiability of the metric components by one order. Indeed, the level of smoothness of the transformed metric component functions are in general no smoother than the Jacobian that transforms them, and by (1.42), the Jacobian of the transformation contains the terms C,. and C, which will in general be only cO.' when C E c l . ' .Thus, if we presume (motivated by 129]),that for general spherically symmetric shock wave solutions of G = K T , that there exists a coordinate system in which the metric takes the form (1.7), and the components of in these coordinates are C ' . ' functions of these coordinates, then it follows that we cannot expect the transformed metrics of form ( 1.8) to be better than c".',that is, Lipschitz continuous. In Section 5 we show that when A and R are Lipschitz continuous functions of ( t , r ) . and T is bounded in L W , system ( 1.25)-( 1.28) is weakly equivalent to the system obtained by replacing ( 1.26) and ( 1.28) by the system Div T = 0 in the form.
(We use .V in place o f r when the equations are expressed as LI system of conservation laws.) This is a nice formulation of Div 7' = 0 because the conserved vilriables 11 = (T:', T;' ) are the Minkowski energy and momentum densities (cf. ( 1.35). ( 1.36)).and thus do not depend on the metric components A ( A . 8 ) . Note that all terms involving A , . B,- and R, in the equation Div 7' = 0 have been eliminated by substitution sing Ecluations ( 1.25). ( 1.26) and ( 1.27). However. Div T = 0 also contains ternis that involve A , = 0. and there is no A , equation among ( 1.25)-( 1.28) - so some change of variables is required to eliminate such terms from Div 7' = 0 in order to close the equations (cf. (5.21) (5.22) below). I t turns out that it suffices to choose and T i ' as independent variables; that is. when we substitute for T:) and T$ in favor of the original conserved quantities To" and T " ' , all terrns involving A, in Div T = 0 (remarkably) c~trrlc~c~l o r ~ tthus , allowing the formulation ( I .46), ( I .47). When T " and T" are expressed in terms of u = (u". u l ) ( T ~ ). " ) in ( 1.46). ( 1.47). (1.25) and ( 1.27). the equations close, and what results is a syste~iiof conservation laws
-
72'
-,
with source terms that takes the compact form
The first equation in (1.49) is (1.46), (1.47). and the second equation is (I .25), (1.27), so that
L ~ = 00 ( ,T~ ~i l ) = ( 0u. u i ) , A = (A, B),
and g = (go. g i ) is determined from the RHS of (1.46), (1.47), while h = (hO,h ' ) is determined from the RHS of ( 1.27). ( 1 2 5 ) upon solving for (Af, B f ) , respectively (cf. (5.49) below). Note that (1.48), (1.49) do indeed allow for CO-'metrics with discontinuous density and velocity based on the conservation law structure of these equations, and such solution correspond to gravitational metrics that are in the smoothness class CO.I.In general, they tlo /lot admit solution metrics smoother than Lipschitz continuous. Solutions to Equations (1.48). ( 1.49) have recently been constructed in 1 101 by a fractional step Glimm scheme that is loc.r~l!\~ irtcrtirrl. The main result of that work can be stated as follows (we refer to [ 101 for details). Assume that
where a, the sound speed, is assumed to be constant, a < (.. (Examples of this. including the case 'rc = 113, and the case o f an isothermal sphere. are important physically. but here we view ( 1 .SO) as a natural model problem for genernl relativity because ( I .SO) keeps wave speeds subluminous, and prevents the formation of vacuurn states, 1101. The assumption of spherical symmetry together with ( 1 . S O ) detines the simplest possible setting for shock wave propagation in the Einstein equations.) The assumption ( 1 .SO) implies that the scalar curvature R is proportional to the density,
For the existence theorem, assume the initial boundary conditions p(r. 0 ) = p o ( r ) .
~ ( r0), = uO(r). for r > ro.
M(r0, 0 = M,,,,
v(rO,t ) = 0.
fort 3 0.
( 1.52)
where ro and M,.,, are positive constants, and assume the no-black-hole and finite-totalmass conditions, 2M(r, r ) < 1, r
lirn M ( r , t ) = M , < c o ,
r-'XJ
Solvitlx the Einstein rq~rct1iotl.sby Lil,sc/iit: c.orltirlrrous trrrtric.~:Shock wrivrs it1 , q e ) ~ ~ r (rirI l u f i v i t ~
5 19
hold at t = 0. For convenience, assume further that lim r 2T,,,,00 ( r , t ) = 0 ,
r+m
( 1.54)
holds at t = 0. The main result of [ I O] can be stated as follows:
THEOREM I . Assunze thtlt the initit11 boundury dutu satisfy ( (1.52)-( 1 .54), and Llssutne thclt there esist positive c.onstr~nt.sL , V , and ij .such that the initi~llvelocits and density p~!file.suo ( r ) ~ t z dpo ( r ) . s ~ t i . ~ f i
<
r < co. \rI~~,rc T Vll,,l,l.f'(.) d ~ ~ n o t rt~l .ls~totti1 11riritltion~ ! f ' t h o , f ~ i ~ ? (, f. tovr'r io~~ thr. intc>nvrl[ t i ,h 1. T l ~ c ~tr nh o l l n d ~ d\t't~ik(.shoc.k ~ ' r r ~.sol~,tion v) ( 1.25)-( 1.28),.srrti.sffkirig ( 1.52) tirlrl ( 1 .53), eri.vt.s up to .sonlp po.siti\v titno T > 0. Morco~vr;the r ~ l ~ t r i c , f u r ~ c t A io~~.s rrtltl B tire Li/).sc~llit,c ~ o r r t i i l ~ r o ~ r . s , f i o l c ~ tof' i o ~( lr. .st ). rinrl ( 1.55) ( . o n t i n l i ~to~holt1,fnr t < T \t,ith rirIjlr.\.tctl \'rrl~rc~.sji)r V rrncl II thrrt trrc. rlctcri,~ir~c~rl.f'ro~i~ the rrnr/ls.si.s.
,fi)r- t11l 1.0
Note that we cannot expect bounded weak solutions fi)r LIII time 7' + oo because black holes can tonn i n finite time. ant1 the metric component H = ( I - 2 G ~ / r ) - '+ oo at a black hole r = 2GM i n standard Schwiu-zschild coordinates. By ( 1.5 1 ). the case p 4 co as / 4 7' would correspol~tlto the fi)r~ni~tion of a rltiXc~tl.vin,g,rrloritj~. Note that by ( 1 ,251 and ( 1.27). the metric co~nponentsA and H will be no smoother than Lipschit/. continuous when shocks are present, and since (1.28) is second order i n the metric. it I'ollows that ( 1.28) is only satisfied i n the weak sense of the theory of ctistributions. Note finally that in 10. 7'). corlsixtcllt with the conclusion that (1.53) says that the total nlit.\s is c.o~r.\tc~nt there d o not exist delta function sources o f mass at shock waves. or at points o f shock wave interaction, in these solutions. Theorem I contirlns what is indicated by Equations ( 1.48) iuid ( 1.49): that the Einstein equations are consistent at the level of‘^'^).' metrics. :uid i r e meaningful in the presence of arbitrary numbers of interacting shock waves. of arbitrary strength. A c~trefulderivation of ( 1.48). ( 1.49) is given in Section 5. but Theorem I will not be discussed in these notes. The interested reader should consult I 101 fl)r a detailed proof of Theorem 1.
2. Weak solutions of the Einstein equations when the metric is only 1,ipschitz continuous across an interface In this section we consider a general four-dimensional space-time manifold with nietric tensor g having signature = diag(- I . I . I . I). We look to characterize solutions of the Einstein field equations ( I . 16) that are only Lipschitz continuous across a smooth 3dimensional surface C . To start. recall that
is the Einstein curvature tensor, where Rij and R denote the Ricci curvature tensor and Ricci scalar curvature, respectively, formed from the Riemann curvature tensor of the metric g. The Riemann curvature tensor, with components R i L , , is given by
and Rij and R are obtained by the contractions R1.. 1.- - R ?I O. J.
and
R=RZ.
The Einstein tensor G satisfies the condition div G = 0, where div denotes the covuriunr divergence defined in terms of the covariant derivative V of the metric connection for g. We reiterate that since divG = 0, it follows that for solutions of (1.16) we must have div T = 0. The distinction here is that div G = 0 is a geometric identity, independent of the Einstein equations, and holds as a consequence of the Bianchi identities, while div T = 0 relies on both the identity div G = 0 and the Einstein equations (1.16). In later sections we will assume the stress tensor for a perfect fluid, which is given in covariant components as
In the ciise of a barotropic equation of state, 1) is assumed to be given by a function of p alone, 11 = p ( p ) . In this case, div T = 0 gives four additional equations which hold on solutions of ( 1.16). In the case when shock waves are present. the Rankine-Hugoniot jump conditions
express the weak formulation of conservation of energy and momentum across shock surfaces. see 1281. On solutions of the Einstein equations, (2.4) follows from the jump conditions
(From here on, I . I always denotes the jump in a quantity on either side of an interface.) The jump condition (2.4) involves the fluid variables. while the jump condition (2.5) is independent of the fluid variable and involves the metric tensor g alone. In the following sections we will generalize the Oppenheimer-Snyder model by ~natchingtwo (metric) solutions of the Einstein equations (1.16) in a Lipschitz continuous manner across a spherical shock surface. I t is not so easy to verify the Rankine-Hugoniotjump relations (2.4) directly in these examples because (2.4) involves the fluid variables in (2.3). so a direct verification of (2.4) requires using div T = 0, which is rzot an identity, and so cannot be managed without invoking the full Einstein equations ( I . 16). However, in the next section we bypass this problem with a general theorem which implies that (2.4) follows as a geometric identity from the corresponding identities div G = 0 together with geometrical constraints on the
second fundamental form on the shock surface, once one knows that the metric is Lipschitz continuous across the shock surface. The second fundamental form K : T z + TE on a co-dimension one surface C with normal vector field n, imbedded in an ambient Riemannian space with metric tensor g , j , is a tensor field defined on the surface in terms of the metric g , and describes how the surface is imbedded in the ambient space-time. Here, T x denotes the tangent space of C. The second fundamental form K is defined by the condition
K ( X ) = -Vxn, for X E T z . When the metric is only Lipschitz continuous across a co-dimension one surface, the second fundamental form K is determined separately from the metric values on either side. In the next section we give necessary and sufticient conditions (the Israel conditions) for conservation to hold at a Lipschitz continuous shock wave interface, the condition being given in terms of geometric conditions on the jump in the second fundamental form across the surface. The conditions are that [tr(K2) - (tr K
)'I
=0.
[div K - d(tr K )] = 0 .
(2.8)
where tr clenotes trace, div denotes covariant divergence. and d denotes exterior differentiation i n the surfi~ce.We conclude that the physical conservation laws (2.4) turn out to be a consequence of geometrical constraints built tr prior; into the Einstein tensor. together with geometrical constraints that describe how the shock surt'itce is imbcdclecl in the ambient space-time manifold. We note that a sufficient condition for conservation is that I K J = 0 everywhere across the surface. In fact, this implies that i n Gaussiwn normal coordinates , these coordinates. where 11 denotes the metric will then be in C ' because K,, = s , , . ~ in differentiation in the direction normitl to the surflice. (See 112.3-2.41.421.) As we point oilt in the next section, the transformation to Gitilssian normal coordinates is in general only it C, I . 1 coordinate transformation. but once this transforrnation is made. the C'% coordinate transformations alone are sul'ficient to describe the locally Lorentzian properties of the space-time. (Recall that by C".' we mean C" with Lipschit/. continuous derivatives.) In the case of rnetrics that are only Lipschitr. continuous, the natural class o f coordinate transformations is the class of C ' . ' transformations. Indeed. if the mapping .t- + J is C ' . ' . then il.\./it>' and i l ~ / i l . \ . are lips chit^ continuous. and thus Lipschitz continuous tensors are riiapped to Lipschitz continuous tensors under the mapping .v + >..iuid this is the least smooth class of transformations that preserves this mapping. Note that by allowing c1.1transtormations. we allow derivatives of i ) . r / i ) ~and i)~/i).\. to jump, and this allows LIS to
;idjust the jump in the derivatives of tensors itcross a shock surface. For- example, if
g = S ' . ~ g K .then
so the jumps in the derivatives of i t y u / i t . r i change the jumps in the derivatives of x i , across C , and Israel's result states that within the class of C ' . ' transformations, we can match the
derivatives in g across C if and only if [ K ]= 0, the map to Gaussian normal coordinates Now in the Einstein equations G;; = K T , ~G;; , is the image of a second order being differential operator on the metric entries g i j , and thus in general we expect metrics that are Lipschitz continuous across C to have delta-function sources in G, and hence in the fluid variables T, on C . It is natural to ask, first, when do such delta-function sources appear at a shock wave C given that the metric is only Lipschitz continuous across C , and secondly, what is the physical significance of such delta-function sources when they d o appear? For the first question, we present a proof in the next section that if g = g L U ,gR is Lipschitz continuous across C in a coordinate system x, then delta-function sources appear in G on C in x-coordinates if an only if [ K ]# 0 (cf. [ 2 2 ] ) For . the physical interpretation of the delta function sources in G. and hence in T , when I K 1 # 0 at C, we comment that , the weak formulation the equivalence of the jump conditions [ ~ ; ] n=, 0 = [ ~ j l n ;and of divG = 0 at a point P in space-time is based on the existence of locally Lorentzian coordinate frames at P: i.e., coordinates in which ,g;;,x(P)= 0. In such coordinate frames, space-time is loc~rllyjat,and the physical principles of special relativity can thus be identifi ed locally. In particular, the covariant divergence agrees with the classical divergence in = 0 of special locally Lorentzian frames, and the global physical conservation laws relativity can be reduced in local form to div T = 0 in curved space-time. (It is well known that, except in special cases, global conservation laws in General Relativity d o not exist.) In the next section we show that. within the class of C ' . ' coordinate transformations, there do not exist locally Lorentzian coordinate frames in a neighborhood of a point P E C where G I , has a delta function source. Thus. space-time is not locally flat at points on ii Lipschit1 continuous shock wave where G has delta function sources. In Section 5 we show that for spherically symmetric shock waves. I GI 1 1 1 , = 0 implies I K ] = 0. and thus conservation implies that there are no delta function sources i n the shock waves we construct as generalizations of the Oppenheirner-Snyder case. and thus these solutions itre locally Lorent~inn at each point on the shock. I t is an interesting open question as to whether general Lipschitz continuous shocks can evolve from smooth solutions i n the time evolution of G = K T . ~
'
3
'
.
LlQ
In this section we give the proof that the jump conditions (2.5) hold at a Lipschitz continuous shock surface if and only if (2.7) and (2.8) hold. We formulate the theorem in 11-dimensions for a nonsingular metric ,y of tixed signature ~1 = d i a g ( ~,,. . . . c , , ) where each F , = 5 1 . Before stating the theorem. we introduce some notation. Thus let y = ( y ' . . . . . y") be a smooth coordinate system detined on an n-dimensional manifold M , y : M + R t ' , and let C be a smooth hypersurface in M. Assume that C is given locally by +(?) = 0, where $ is a smooth function satisfying
Let L and R (for "left" and "right") denote the two sides of M defined by the surface C , and let g L and R K denote smooth metrics defined on the left and right side of C. respectively. (It suffices to assume that g L and g R are at least c', with derivatives unifor~nly
Sohing the Einstein equations by Lipschitz continuous metrics: Shock wave.y in general relutivity
523
bounded at C , and we assume this from here on out.) For completeness, we give a proof of the following theorem due to Israel, see [12,22]. THEOREM2. Let g = g L U g R denote a tzonsingular metric of arbitrary signature whose conlpotzents g,, in y-coordinates are smooth on the left and right sides of C , separately, and Lipschitz continuous across the surjiace. Assume that C is given locally by cp = 0, where cp is smooth, assume that (2.9) holds, and assume that the normal vector n is nonnull relative to the metric g, so that (without loss of generality) we may take n to be a unit vector g,, 12't~ = 1. Then J
at a point P
E
C i f and only if both
[(tr K ) -~ tr(K2)] = 0 cltzd [div K - d(tr K ) ] = 0 hold. (Here, the invariant oprrntions div. trace andd on K ure restricted to the surjiuce C . )
Note that by a smooth transformation of the coordinates in a neighborhood of a point P E C we may assume that the surface C is given by cp = y" = 0, so that n = a/i)v". In this case, the invariant conditions (2.1 I ) and (2.12) reduce in J-coordinates to
[(K: ( y (PI)) - (K: (?( P)))'] = 0 and [ K ~ , , ( v ( P )-) K ; : ~ ( Y ( P ) =o. )I where the summation in (2.13) and (2.14) is assumed to run from I to n - 1 . The proof of Theorem 2 will follow as a consequence of several lemmas. The idea is to construct Gaussian normal coordinates for the surface C. these being coordinates in which . We the components of the second fundamental form take the simple form K;,, = then use this identity to write the Einstein curvature tensor G and the jump conditions (2.5) in terms of the K;, and obtain (2.13) and (2.14). 112.221. We will use the following identities for the components of the curvature tensor G; in an arbitrary coordinate system.
-Igij,,,
L E M M A1 . Thecomponent.sc?f'G uregiven by
where the brackets [.] around a set of indices indicate that surnrnatiorz is to be taken only over the increasing sequences of indices occurring itzside the brackets.
PROOF. To prove (2.15), we have
But
R! = Rri = I
R'!
TI
r l
rji
because R$ is antisymmetric in
(up) and (yfi). Moreover,
To prove (2.16) we have
We now construct a Gaussian normal coordinate system (111'. . . . . 111") associated with the surface C in n neighborhood of 4) E C . 141 1. To this end. assurne that ,g has !.components x i l . and by making a smooth coordinate tl-ansformation we may assume with= 0. For each P E C let y,,(s) out loss of generality that C is defined (near PO)by !" denote the geodesic satisfying y,, (0) = P .
y 1 ~ ( 0=) n.
where n is the normal vector to C at P , .s is arclength, and for convenience we assume that n points into the right side of C. We detine the uir'-coordinate in a neighborhood of E C as the "distance from C" as follows: if y,,(.s) = Q. then set uil'(Q) = s. In this way, u!I1 < O on the left side of C. and ui" > O on the right side of C. Now define the ur'-coordinates for i = 1 . . . . . ,I - I . by w l (P ) = (!' P ) for P E C. and detine u!' i n a neighborhood of C by taking 111' to be constant along each yp(.v): i.e., '
Q =1
'
)
if and only if
Q = yp(.\.),
for some P and s , i = 1, . . . , 11. The coordinates u! = ( 1 1 1 ' . . . . . u!") are called Grr~r.s,sitrtl rlorr~lcilc~oordir~cites in a neighborhood of E C. Note that the Gaussian normal coordinates UI are in general only C ' . ' related to the original !-coordinates because the geodesics normal to the surface C are in general only C ' curves since the f' can in general .I have jump discontinuities at C when is only Lipschitz continuous across C. (Indeed. . . , x u - ' ) and ( 7 . 0 ) E R" is the coto see this, consider the curves y v ( s ) where ,V = (!I,. . ordinate value of the point P o" C such that y p ( s ) = Q has y-coordinates y , ( . ~ ) Thus,
y ( P ) = G ( P ) for P E C. But being constructed from families of geodesics on each side of C, y ? ( ~=--) q(', S) is a smooth function of ?, and s on each side of C separately. It remains to check continuity of derivatives at yf7= 0. But, at .s = 0,
because
,!= ( j ,0) at
s = 0.
Moreover,
where n f denote the y-coordinates of the normal to C at P. Since the metric is continuous at C, this latter derivative is continuous across C as well.) Gaussian normal coordinates satisfy the following well-known lemma, cf. 1291.
Note that Lemma I implies that thc surfaces ri!" = const are orthogonal to the coordinate directions i ) / i ) u l i .for i = I . . . . . 11 - I . For a sn~oothmetric g , the components of the second fundamental form arc given by the following lemma:
PKOOF. We have, for every vector field X I .
so that
But
526
J. Groah rt a/
where we used the fact that in Gaussian normal coordinates, gjn,k = 0, i = 1 , . . . , n . Thus
as claimed. In the Gaussian normal coordinates w associated with a given co-dimension one surface C and a Lipschitz continuous metric g = g L U g R (where we assume as usual that g L and g R are smooth), the metric g is determined on C , but the first derivatives of the metric suffer a jump discontinuity at C. Thus the second fundamental form K , which depends on the first derivatives of the ambient metric g , also suffers a jump discontinuity at C. In this case it follows from Lemrna 2 that K and K R , the second fundamental forms on C for the metrics X L and g R . respectively, are given by (2.25), for g = g L , g R , respectively. Thus the following corollary of Israel is immediate. COROLLARY1 . The metric cornpot1ent.s c ? f ~= g L U g K in Guussiun normtrl c~oordinutr.\. trre C ' ,filt~ctiot~.))l~ ofthe c~oort/ititrte\~rrriah/e.s iftrnd only if [ K ] = ( K - K 1 - ) = 0 trt ruth poirlt or1 the .rutfirc.e C .
The next lemma expresses the components of the connection coefficients fix the ambient metric s in Gaussian normal coordinates in terms of quantities intrinsic to the shock surface. We state this for a smooth metric. and see that i t applies to each side g = g L and g = S K separately when the metric is only Lipschitz continuous. LEMMA 4. The c.ornl)onetlt.sit1 Gtrit.s.vitr~i r~orrnerlc.oortlir~trtrs r!ftlic' c.orir~c~c.tiori c.oc:fic.icr~t,s ,fi)r( I lrletric. g tit ( 1 poitlt P E C ctrtJgitlerr I?
-
Here, tlenote.~ the (t7 - I )-ditnc~r7sior1trl c.orlrlrt.tior1c,oclffic.ierltsc.or~il,rttrel,frot~~ the intr-insic metric g o n C with w-co1npor1et1t.s k i i . i , j = 1 . . . . . r 7 - I. PROOF.To obtain (2.26), use ( I . 10) to write
Since gX" = 0 when a = n and k # n , it follows that
Solvbrfi thr Einstein rqucltiorts I? Lil~.sclrit:.continuou.\ nlrtrics: Sh0c.k waves in firnrr~ilrrlativiry
527
which is (2.26). Similarly, statement (2.27) follows from
statement (2.28) follows from
r."= -1g k n ( - g rn
2
1rr.n
+ Rni.11 f g11n.i 1;
and statement (2.29) follows froni
upon noting that in Gaussian normal coordinates w we have grfU= 0 unless a = n . and g ,,,, p = O f o r a , B = I , . . . , 1 1 . The next lemma uses Lemmas I and 4 to expresses the components i n Gaussian normal coordinates of the Riernann curvature tensor for the ambient metric g in terms of quantities intrinsic to the shock surface (Gauss-Codazzi equations). Again we state this for a smooth metric. and see that it applies to each side g = ,gl. and g = ,yK separately when the metric is only Lipschitc. continuous.
PROOF. For (2.35), write
528
J. Crotrh rt ul.
Thus, since only a can be 11, we have
which by (2.26) gives (2.35). Statement (2.36) follows because gl" = 0 for i # n . For (2.37), write
which gives (2.37) on applying (2.27). In this case as before, (2.38) follows from (2.37) because girl = girl when i # n. The next lemma uses (2.36) and (2.38) to expresses the components in Gaussian normal coordinates of the Einstein curvature tensor for the ambient metric g in terms of quantities intrinsic to the shock surface. Again we state this for a smooth metric, and see that it applies to each side s = S L and g = g R separately when the metric is only Lipschitz continuous (cf. [221). LEMMA 6. T/IPc ~ o t t ~ / ) o t ~ 111 ~ ~G t ~t tr,~s~ . ~ . s ti t?ro t~ r t t ~ t~~~/o o r f / i t ~ t r tof o.s t / ~ cE i t ~ . s t ~ ~t i.t ~ i i r \ ~ ~ t ~ i r ~ g trf tr / > o i t ~Pt E C rrrc' ,givrt~
terl.sor,fOr( 1 rnetrit.
PROOF. To prove (2.39). use (2.22) to write
so that by (2.36)
where the sum must be taken over indices a < T. But by definition,
and
Using these i n (2.4 1 ) yields (2.39) To prove (2.40), use (2.23) to write
where we have applied the antisymrnetry o f the curvature tensor. Thus b y (2.38),
from which (2.40) follows at once. We can now give the
PWOI- Ot: THEOKI:.M 2. Assume that g = U ,yK. where thc metric g is smooth on cither side o f a co-dimension one shock surfitce G , and is Lipschitz contin~rouaacross the surfi~ce.Let 111 denote the Gaussian norrnal coordinates associated with the surfiice C nnd the ~netricx. Then we can apply (2.39) and (2.40) o f L e m m ~6t to from the left and ,4K frorn the right o f C. respectively. to ohtain
=
1 -
2
{(tr K K ) ?
-
tr((KK)')]
I
- - { ( t r Ki,)' 2
-
tr((Ki.)')}.
and
Here we use the fact that
a and ~ ; I I (
-
K i i R 1 j j l : ] itre equal o n Z for n L and
SK because they depend only on intrinsic properties o f the metric
n = i ) / i f n l 1 I . and so the jump conditions (2.5) i n Gaussian normal coordinates reduce to the condition
Now since G transforms like a tensor under arbitrary cl-coordinate transformations, the conditions (2.46) are equivalent to the statement [G;]n, = 0 in the original y-coordinates. Thus, in light of (2.46), we conclude that (2.1 I ) and (2.12) of Theorem 2 follow directly from (2.44) and (2.45). In view of Corollary 1 of Lemma 3. we can also conclude the following corollary of Theorem 2 (due to Israel), which gives a global criterion for conservation across C , 1 12, 221. C O R O L L A R2. Y If [ K 1 = 0 at errch point of' C, then the Jutnp conditions [ G F ] n ,= 0 rnust hold rit the point P . Moreo~~er; since in this u s e the metric. is C 1 in C ~ ~ u . s ~ inormrrl utz coorclinrrte.~,the cotidition [ K ] = 0 is UISO u nece,s.sury and su&c.ienr condition ,fi)r the origitzrtl Lipschit: coiitit1uou.s inetric corr1ponent.sg,p in the y -coordinate.s to he eyuivrilent to ( I C ' rnetric unrler u C ' . ' -trrrn.sf~)t-t~zurio/z of'the coordinrrte vtirirrhle.~. PROOF. The sufficiency is clear. and the necessity of this condition follows because, if the metric is equivalent t o a C ' metric under some regular C 1 . ' coordinate transformation, then the mapping from these coordinates to the Gaussian normal coordinates is a C* mapping. and thus the metric in Gaussian normal coordinates will be C 1 , which implies that the second fundamental form is continuous across the surface. (Note that I K I = 0 at a point is not sufficient for conservation I G ;In; = 0 at the point.) We now show that K , , and G I , , viewed as second order operators on the metric components g,,. have delta-function singularities at a point P E C if and only if I K J # 0 at P. Thus, let g = U x K be Lipschitz continuous across a shock s u r f ~ ~ cCe in .r-coordinates. The strategy is as follows: we first do the case when .\- is a Gaussian nortnal coordinate system detined in a neighborhood of P E Z. We then show that delta function sources appear at P E C in .r-coordinates if and only if they appear in any coordinate system related to x by a c'.'coordinate transformation. Since any coordinate system in which is Lipschitz continuous is related to the Gaussian normal coordinates by a c ' . ' coordinate transformation, it follows that delta functions appear if' and only if [ K 1 # 0 . We then show that when delta-function singularities appear in G;,j at P E C in a given coordinate system x . the metric is not locally Lorentzian at P in sense that there does not exist a C ' . ' coordinate transformation that takes .r-coordinates to coordinates in which the metric is locally Lorentzian at P , more specifically, such that g;;.r ( P ) = 0 . Finally, we show, surprisingly. that delta-function singularities never appear in the scalar curvature R at any point on a shock wave discontinuity on either side of which g is smooth, but across which g is Lipschitz continuous, and this is due to a cancellation of delta-functions in the sum R,". L E M M A7. Let .w he the Gart.ssitrn normcrl coortlintrtr.~c.ot~ttriningtr point P E C. ,t,herr C is trny .stlzoorh .vurlfic.e, .so thtrt i)/i)ri is the ~rornitrlc/irrc.tion on C . Then the) src.ond ortlrr ti-derivrrtives r!f'g,,jthtrt trplwtrr in the]Or~nul~~,fi)r the Ricri tensor R;,;. occur only in the terms R,, , i # n , ,i # n . trnd in R ,,,, , trnd the,vc)nt-c.gi~let~ hs
I R;, = - -g;;.,r,r lower order n-derivcrtise.~, i # n . 2
+
J
# n.
Solvitl!: tllr E i t r s t r i t t erluutions hy Lipscltitz cot~tinuousmrrric.~:Shock wuvec in generul re1ativit.y
53 1
und
1 RII,I= -RmJ(RmJ(,,III 2
+ lower order n-derivatives,
lvhere the surn in the 1n.stformula is tuken over a , B
# n.
PROOF. From (2.29). assuming Gaussian normal coordinates, we have
Consider Rij = R:,, , which is given by the formulas
Now, since g is Lipschitz continuous across C , and R,, involves second derivatives of g . i t follows that delta-functions in R ; , can arise at P E C only i n the second order 11-derivatives appearing in the formula for H i , . To see this, note that in Gaussian normal coordinates. g,,, = S,,,. and g , , are arbitrary for I , j = I . . . . . 11 - 1. Thus the tirst derivatives in k # 11 are Lipschitz continuous across C because = ,r:' on C . and thus ,y,,,x,, involves rrt t t ~ ) , : ~ / jump discontinuities for k # 11. Now from (2.5 1 ), the second order 11-derivatives can come olrlj. from or T;'T,, . In the tormer. this can only happen when a = t l . so consider
$7,"
,
But g"" = 0 unless 0 = 11. which implies
Thus we conclude that when i = 11 or j = 11. there are no nonzero second order 11derivatives in $:,,,. and when i , j # 11. $I, gives , rise to only one second order 11derivative. namely. iI , y , I .,,, i.e..
r7 11."
I 2
= -g..
+ lower order ti-derivatives.
J,l.rlil
Consider now f ~ ~ ,,, which ,, can have second order 11-derivativesonly for j = 11:
The f rst two terms, g,,,r,l and g,;.,,,. inside the braces in (2.54). can have second order 11derivatives only when a = n or r = n , in which case a = 17 = r (because g,,, = 0 ) .which
532
J. Grotrh rt trl.
implies that both of these terms are zero because r;; = 0. But the third term g,,,,, in the braces in (2.54) has second order n-derivatives only when i = n , and thus we have
,
I c:n3,r = -RffPgUp,,l,l + lower order n-derivatives, 2
: is a lower order n-derivative if i # n or j # n. Thus we conclude that the second and $* order n-derivatives in the Ricci tensor occur only in the terms Rij, i # n, j # n, and in R,,,,, and these are given by (2.47) and (2.48). We now consider the scalar curvature R and the curvature tensors R,,, and Gij as second order distribution derivatives of the metric components gi, in Gaussian normal coordinates when g is only Lipschitz continuous on C. I n general we expect that second order distribution derivative of g will introduce delta-function singularities on C. The following corollary gives necessary and sufficient conditions for the appearance of such delta-function singularities on C. C O R O L L A R3Y. Let g = g L U g R be L I I I ~lnetric t h ~ is t Lipschitz co~ztinuou.~ ucro.ss N shock s1lrjiic.e C. ~ i r lslnooth ~l 011 either side of' C . Then in G~lussicrnnormrrl c.oorc/inrrte.s the scrrlnr c ~ ~ r l ~ u t R, u r evierrvd LIS LI .secolld order di.strihutior? derivutivc of' the t n ~ > t r i ~ . r~o~ly>ot~rlit.s g , , . IILISa t )c3or.stLI , ~ L L di.s(.ol~til~~~i!\' I ~ I L I P~ C I C ~P E Z:the Ricci u11d Eill.st~iil (.L~rt~Litllrc ~ P I I S O I : Rii ~ ~ 1 1 dG;, I I L I delt~i~fi411ctio11 ~J~ .sil~~~il~iri/io.s L I ~P E C if'(111d0111~jf I K I # O tit P. PROOF. Assuming Gaussian normal coordinates, we have from (2.47) and (2.48) that
R = ~ ~ ~ g c r r . ,-r ,gl'sii.,r,r r
+ lower order 11-derivatives.
and thus the formula for R in terms of contains no second order IT-derivativesin Gailssian normal coordinates for any Lipschitz continuous shock wave, and hence R is at most dis, i. j # 11. and continuous on C. Moreover. in Gaussian normal coordinates K,,, = g,,,.,, hence if [ K I # 0 at P E C , then g,,.,, must suffer a jump discontinuity at P for some (i, j ) . i, j # 11. Thus by (2.47), Rii is given by the delta function ~,,,,,,,plus a discontinuous function. Conversely, if I K ] = 0 at P E C. then ,~,,,,,,,is at most discontinuous at P , and thus R,,, is at most discontinuous at P. Since G I , = Rii - ?I R g ; , , and R is at most discontinuous, we conclude that in Gaussian normal coordinates. R,,, and G;,,contain delta-function singularities if and only if [ K I # 0. Now let
R = R:,, ,...
denote the components of the full Riemann curvature tensor in .r-
coordinates, and let = $ Y A denote the components in a coordinate system y related to .r by a c'.'coordinate transformation. Note that in any coordinate system, the components of the curvature tensor are given by (2.50), and hence are determined by the same second , 2 = LIE].We order differential operator L on the metric components, thus R = L [ g ] and note that the highest order derivative terms in L are of the form a function of the unknowns g;,, times litzeur second order differential operators. Thus it is possible to define solutions
g that have only weak (distributional) derivatives of second order. The following lemma demonstrates that curvature tensors defined from L in the weak sense continue to transform by the tensor transformation laws under arbitrary c'.'transformations of the coordinates.
L E M M A8. Let R be a weak solutiot~o f ' R = L [ g ]in x-coordinates. Then
= R!& is a
crveak .soI~(tionof R = L [ j ],$)r crny coordinate system y related to x by ~r C ' . ' coordinate trar~.sfi)rtnation,where ct,e use the shorth~indnotation
und ttl~lltiplicationby rr$lnction is tcrken it1 the wr~rk.sense.
PROOF. For smooth R and smooth test functions cp, let
where L * l g , cp] denotes the expression obtained from L I , y ) by integrating the second order derivatives in 'y once by parts. Since the second order derivatives in L are given by4
i.e., are of the form ~ " " ' ~ y , , , it~ ~follows . that L * I g . cp] contains at worst products o f the metric entries g ' , . the test function cp, and their first derivatives. Thus the integral in the l,*lg. (PI is finite for any Lipschit1 continuous metric ,y and any weak formulation Lipschitz continuous test function cp (of compact support). Now assume that R = R',Ll is a weak solution of R = L l s ] . i.e.. R is a linear functional on the space of Lipschitz continuous test functions (a distribution). that solves
for every Lipschitz continuous test function cp. Note that if i l . \ . / i ) ~is lips chit^ continuous. then the derivatives are bounded, and thus if we let ,jj = *y% be shorthand notation for
then L * [ R % , cp) is bounded for any Lipschitz continuous test function cp.
ere "l.o.t." denote5 "lower order terms". i.e.. terms th;11 contain lower order 11-derivatives
S o to prove the lemma, let g be an arbitrary (non-degenerate) Lipschitz continuous metric, let cp be an arbitrary Lipschitz continuous test function, and assume that the coordinate systems x and y are related by a c'.'coordinate transformation (so that in particular, both a x l a y and a y l a x are regular, Lipschitz continuous maps). Let j F denote a smooth regularization of the metric gap, and let x E ( y ) denote a regularization of the coordinate map ~ ( y SO ) that .rF(y) is smooth and has a smooth inverse. We can clearly choose these regularizations so that 'j P + jab in cO-', x F ( y ) + x ( y ) in c ' . ' , -+ %().) in c"', ff
and & ( x r ) + % ( x ) in
in
F(v)
c'.'. Then
cO. I . Define ?zt
= L(it'),
and
(2.59)
Now, i t follows directly from definitions that
But (2.61) simply says that R'$ is the curvature tensor for the metric *y' p , and since everything in (2.61) is smooth, we know from the fact that the curvature transfi)rms as ;I tensor that R' must be the curvature tensor for the metric R': i.e.. since everything in if and only if (2.61 ) is smooth. we know that (2.61 ) holds for every cp E cO.' ij
Since g' + ,y in c ~ ' .(2.62) ' , implies that. as E holds for every cp E cO.'. the sense of distributions to the distribution T . where T satisfies ( T .p) =
k,
-+
\
0. R' tends in
L*l&!. ~ 1 .
Therefore (2.63) demonstrates that T = R as a distribution. Thus, in the limit E + 0, we conclude from (2.63) that R' + R,from (2.61 ) that @ -,R, and hence from (2.60) that . the sense ot'distributions. This completes the proof of the lemma. R = R zi f v ~n T HEO K E M
3 . As.surne that
g = gl- U
i.r .smootl~or1 either .side of
rr 3-rlinle1lsior1~11 s1zot.k
.surjiic.e C , und is Liljschitz continuous cicro.ss C. Ther~the sctilur curv~itureR . whet1 viewed us rr second order operutor (in the ,vecik ,reri.se) on the rlletric c.ornl7orlerzt.sg,,
.
produces at most u jump discontinuity (i.e., no delta-function .singuluritie.s) ut P E C ; and the t~urvcrturetensors R ) k l , R i j , and G,,; produce no STfunction .singuluritie.s at P E C if and only $the jump in the secor~tlJi~tzdtrrnmtul,fi~rm K .sati.sje.s 1 K ] = 0 ut P . PROOF. By Corollary 1, the theorem is true in Gaussian normal coordinates x , and thus by Corollary 3 and Lemma 8 it holds in any coordinate system y which is c'.' related to x . Since for any metric g = g L U g H which is smooth on either side of C and Lipschitz continuous across C the transformation to Gaussian normal coordinates is an invertible c'.' coordinate transformation. the theorem follows at once. As a direct corollary of Theorem 3 we see that there exists a locally Lorentzian coordinate frame in a neighborhood of a point P on a Lipschitz continuous shock surface C if and only if [ K I = 0 at P; namely, we have U g K i.s smooth on either .side ofcr 3-ditneii.~iont1l C O R O L L A R4. Y A.s.srr~nethur g = .shoc.k .slcr;firc.c C. rrnd is Lip.sc.hit,- c~ontirluori.~ crcro,ss C in rr c.oortiinnte S ~ S ~ P I sI I tlqfineel in n neighborhood of P E C . Then t h ~ r oe.rists tr rogulrrr C ' . ' c.oordintrte trun.yfi)r~ncrtior~ y + .r .such t11crts i.s /oc.crl!\'L o r t ~ i i f : i ~ ~ ~g, f(11 i ) rP (g;, = rljl ~ n t gl ; ; , ~= 0 trt P ) , ifcrrid on!\. if1 K I = 0 crt P .
PKOOI..Assume I K I = 0 at P . and choose locally Lorentxian coordinates at P for the smooth metric obtained by restricting ,? to the surf'ace C in LI neighorhood of P in the surface C. Extend these coordinates to Gaussian normill coordinates .r based on these sill-face coordinatea, the .Y coordinatea being defined in an 11-dimensional neighborhood of P. Then i n .\--coordinates the metric components g ; , satisfy x;, = rl;, and K;, = g;,,,, = 0 at P. and so .r is locally Lorentzian at P. Conversely. assume that [ K I # 0 , but that s,, = rl;, and there exists a coordinate transformation y + .v S L I C ~(hilt. i n .I--coordini~tes. g , , , ~= 0 at P . Then in .r-coordinates, g is c.' at P . and hence there are no delta-ful;ction singularities in the components x;, of g in .v-coordinates. Thus by Theorem 3. I K ] = 0. and hence the locally Lorentzian coordinates .\ cannot exist when I K I # 0 . The next result partially validates the statement that real shock waves cannot form in solutions of the source free Einstein equations R;, = 0 . or equivalently G;,, = 0. by demonstrating that "shock waves" in solution metrics are only coordinate anomalies i n the sense that they can be transfortned away by coordinate transh)rmation. Note that the theorem allows for the possibility that discontinuities can form in solutions (which we expect because the equations are nonlinear quasilinear in nature). but asserts that if the solution ~uetricis Lipschitz continuous across a smooth surface. but smooth on either side, then there is a coordinate transformation such that in the new coordinates the metric is smooth across the surfxe. COROLL.ARY 5. A.s.sutne rhtrt tlic c~otr1ponertt.srtf'g = gl. U S K in tr c~oortliritrtr.sy.ste17iy trrc. Lipsc.hit: c.ontin~roustrc.ross ( I snootk 3-tlir~ler~.sioritrl .shoc.k sut-f'1firc.e C . crr-c. C' ,fiirlctiorl.sof' y on either .sicle c!f'C. trtlcl tissutrzr that ttll k deri~vrtivrsurt) c.orltinuous to the hoirndtrry C froin either .side of' C . As.s~rrrlecilso thtrt g i.s ( I )L'(>CIX. .sol~rtio~i ($Rub = 0 or C u p = 0 )~'/7fircvi
viewed [IS second order operators on the metric components gap. Then in Gaussian normal coorditzutes x (which are c','related to the original coordinates), the metric components g;, are uctucrlly C' Jlrnctions of'x c~cros.sC . PROOF. Assume first that ,g = g L U ,gR is a weak solution of Rap = 0. But R;., = 0 in the weak sense across C implies that there are no 6-function sources in R;,, on C , and thus by the previous theorem, [ K ] = 0 across C . Thus Israel's result implies that g;,.k are all continuous across C , and since G;, 0, the jump conditions are automatically satisfied across C. It follows from (2.47)and (2.48) that in Gaussian normal coordinates,
--
I R;] = -pi, 1
+ lower order n-derivatives,
i # n, j # n,
and
(2.64)
+
R,lII= -gl' gjr,rlr, lower order n-derivatives. 2 But since the ,gIl.k are continuous across C , it follows that the lower order terms in (2.64) and (2.65) must be continuous functions across C , our assumptions implying that the and g K . But since R,, = 0 for both g L derivatives of g in the surface C are the same for and g K . we can solve for g;,,,,,, in (2.64)and (2.65)in terms of lower order derivatives that are continuous across C. and conclude that s;,,,~,~must rrl.so be continuous across C for all i. j = I , . . . tr. (Recall that g,,,= const in Gaussian normal coordinates.) This shows us that kth order derivatives of g,,/ which :we up to second order in .Y" are, in fact, continuous functions of .r across C in Gitusaian normal coordinates. Now differentiate (2.64)and (2.65) with respect to .I-". Then the differentiated lower order terms in (2.64)and (2.65) are continuous across C , and hence again we can solve for ,4;/.,,,,,,in terms of functions that are continuous across C. Thus we conclude that Xth order derivatives of s;,, which are up to second order in .rU are. in k t , continuous functions of .Y across C in Gaussian normal coordinates x . Continuing, we see that all the kth order derivatives of g ; , are continicous across C in Gaussian normal coordinates. Since, by Corollary 3, the scalar curvature never contains delta-function aingularities on C , the result for R;, implies the same result for C ,; .
.
The same argument establishes the following Inore general version of this corollary. C O R O L L A R6. Y Assurne thtrt g = U g K is s~nootlr0 1 1 c.ither sitle ofrr 3-tlir~~r1~.sior~t11 shock .srrrfirce C , crnd is Lil).sc.hit: c~ontit1irou.stlcross C in .so~nc~ t.oorclint~tc,.v\..str~try . Asslrrtre thtrt g is (( wetrk solution c!f'G,p = K Tap thtrt c.ot~ttrinsno tlr/ttr~firnc.tion.sir1gu/trriries on C . Than in Guu.s.sicrtz norrntrl c~oort1intrte.sthe rnrtric c.ortlportc~17t.s g;, ore C' ~ i t t ~ c ~ t i o n . ~ of.r ~ftrnclonly if'[C ]= O crc.ros.s C .
Sumnlclry. The results o f this section are summarized in the following theorem.
4. Let C denote rr smooth, 3-dir~~erl.sionrrl .shock surfirc,r in spcrcr-time wit11 THEOREM spacelike normal vector n. Assutnr that the c.ompot7e11tsg;; ofthe grnvit~rtiotzcrlmetric g crre smooth on either side o f ' C (continuous up to the bo~rndcrryon either side sepumtely),
Solving the Eintrritl rqiitrtior~sI)! Lipschic ~.otltinuou.s~nrrric.s:Shoci rr'uvrs in ,qrtrrrtrl rc,ltiii~~ity 537
atzd Lipschit2 continuous across C it1 some $xed coordinate system. Then the ,following .stclternents ure equivrdent: (i) [ K ] = 0 at each poitrr of' C . (ii) The curvature tensor.^ R;,., atrd G ; , viewed as second order operator.s on the metric Lwn/?onent.s g;, produce no rleltnTfunctionsource.s on C . (iii) f i ~ reclch point P E C there e.xi.sts a C I coordinate tmn.~ji)rmationclejined in LI neighborhood of P , such that, it1 the new coordinates (which can be taken to he the Gau.ssinn normal c.oordinutesfi)r the .surjiace),the metric components rrre C I ji1nction.s of' these coordinntes. (iv) f i r cuch P E Z, there e,ri.rt.s cr c.oorclinrlte,frcrmrthcrt i.v locul!\1 L.ormfziun it P , LIIICI CCIIIhe roachecl ~vithinthe c1u.s.s of CI. coordinate trun.sfi)rmtrtion,s. More-eo~vr;if cinq c!f these eq~iit~ci1et1c.ie.s holrl, then the Runkine-Hugoniot jump conditiotr.c.. [G]ytzfl= 0 (ii1hic.hr~x/~re.~,s the rcvak.fi)rtn c?f'cori.srrvrrtior~ c?fenerg,yL I ~ I Cmonientutn ~ ucross .E rt>henG = K T ) ,hold cit each point or7 C .
.
.
'
Here [ K 1 denotes the jurnp in the second fundamental form (extrinsic curvature) K across C (this being determined by the metric separately on each side of C because g,, is only Lipschitzcontinuousacross C),and by c'.'we mean that the first derivatives are Lipschitz continuous. Theorem 4 shoilld be credited mostly to Israel, 1121, who obtained results (i)( i i i ) in Gaussian normal coordinates. Our contribution was to identify the covariance class of' c'.'trunsform:ttions, and to thereby obtain precise coordinate independent statements for ( i i ) and ( i i i ) . its well as the equivalence with (iv). A4 a consequence of this, we obtain the result that the Ricci scalar curvature K rrelvcJrhas delta-function sources at a Lipschit/. continuous matching of the rnetrics. as well as the results in Corollaries 5 and 0 which validate the statement that shock wave singularities in the source-free Einstein equittions R , , = 0 or G , , = 0 can only appear as coordinate anomalies. and can be transfi)nned away by coordinate transformation. Note that when there are delta-function sources in (; on a surface C. the surface should be interpreted as :I s11rf:tce layer (because G = K T ) . and not a true fluid dynarnical shockwave, 112.22). In Theorem 5 below. we show that for spherically syrn~netricsolutions, IGI,rnor~r = 0 alone implies the absence of surface layers (and hence the other cquivalcncies in Theorem 4). so long as the areas ot'the spheres of symmetry rnatch .s~noothlyat C . We use this result in o i ~ rconstruction of the shock waves that extend the Oppenheimer-Snyder model to the case of nonzero pressure. The following counter-example shows that in general the above equivalences can fail even when IGf j r r , = 0 holds at each point on C.' For the counter-example it suffices to show that there exist Lipschitz continuous shock waves which satisfy the Israel jump relations (2. I I ) and (2.12) across a shock wave interface, but which cannot be transformed to a metric that is C ' in a neighborhood of each point on the shock. By Corollary 1. it suffices to construct a shock wave interface across which the Israel conditions are satisfied, but such that the second fundamental form K is not continuous across the interface. To this end, let g,, denote the coordinates of a metric in 'see 1/21 where wch
arl
ex;l!l~plci given in which G
-
0 o n both sides of Z
538
. I . Grouh rt (11.
Gaussian normal coordinates, such that the spacelike normal to the shock surface is given by n = a/i3xfl, and g,, is of the form
Assume now that the hi, are given by q,, q,
+ clij.rn +,
I
if .r" > 0 if x u c 0 '
where L I ; ~and h i , are constants to be determined. Thus by Lemma 3, the second fundamental forms K I. and K on the left and right of the shock surface are given by K; = r1;j and K; = h i. j = 1 . . . . n - 1 . Since K h and K: are constant,
.
,,,.
for K = K I - , K '. Thus the Israel jump conditions (2.1 I ) and (2.12) reduce to
Hence to satisfy the Israel jump conditions it suftices to tind a (tra)'
-
tr(a2) = 0 = (trb)'
-
= ( I ; , and b r h,, satisfying
tr(b2).
But in the simplest case where a and b are 2 x 2 matrices. tra = (I 1 1
+ (172
and
t r ( a 2 ) = a f l + 2 t ~ 2 ~ t ~ ~ 2 +atn~d ~ s o2 ,
(tra)'
-
tr(a2) = 2 det(a).
Thus we can satisfy the Israel jump conditions by choosing a and b to be any 2 x 2 matrices with zero determinant. If i n addition t ~ , # , h i ; , then / K I = K - K'- # 0, and so by Theorem 4, conservation I = O holds across the interface .r" = 0, but, in view of Corollary I , the metric cannot be transformed to a metric that is globally C ' across the shock.
2.2. The .sl~hericcrlly.vvtntnetric. c.ctsc) In this section we restrict to spherically symmetric metrics. The theorem to follow states that in the special case of spherical symmetry, the jump conditions I G I J ] t z i n ,= 0 that express the weak form of conservation across a shock surface, actually are impl~edby a
Sol~irrgthe Eitlstein equcitions by Lipschit r ~ ~ n t b l u ot?~rrric.r: u.~ Shock waves it1 ~ ~ r i c , r urc.lutivity l
539
single condition, so long as the shock is non-null, and the areas of the spheres of symmetry match smoothly at the shock and change monotonically as the shock evolves. Note that in general, assuming that the angular variables are identified across the shock, we expect conservation to entail two conditions, one for the time and one for the radial components. Thus the fact that the smooth matching of the spheres of symmetry reduces conservation to one conditions can be interpreted as an instance of the general principle that smoothness in the metric implies conservation of the sources.
5. A.s.surne thut ,g und g ure two .sl>hericully.symmetric metrics thut march LipTHEOREM schit: co~~tin~rou.sIy ucross (I three-dimensional shock intetjiuce C to ,form the matched tnetric g U j. Thut is, u.s.rumr tlitrt g and C: (ire Lorent7iun metrics given by
t ~ n t rhcrt l thore n-i.st.s cr smooth c.oordirztrtc. trcrn.sfOrmt1tio17 P : (1, r ) -+ (7,J ) , tI
(2.70)
in tin 0 / ) ~ 1 1n ~ i g I 1 1 ~ o r h o (o!df ' t l l ~~ I ~ o c~11rfi:filc~ k C . .so thrlt, in j)rrrtic.l~ltrl;the (rro(r.s (!f't/tc~ 2-sl~hcrcsof sytntncJtry in thc htrrrcd crrlcl u n h t r r r ~ trnc2trir.s l ogrc.c2 0 1 1 the shock .srrr;firc.r,. A.s.sirrnr trlso thtrt tho slloc~ksrtrf~rfilc.er = r ( t ) in rorhtrrlucl c~oolr/irrtitr.si.s rrrtrl>pc~Ito //I(' .s~rrftrc.c7 = ? ( I ) I? (1.r ( 1 ) )= P ( t . r ( t ) ) .A.s.s~rrnc~, jincil!\; thtrt thc 1101711(11 n to C is 11011null, t l l l t l tl1tit
~vherc. n(c.) denotcs thcl t l e r i ~ ~ t r tc?fthe,fiinc.tion i~~c~ c . in thc tliruc,tiorl of'the \lrc.tor n.' The11 t h c ~ $ ) l I o ~ v i(IIY n ~ t ~ q ~ i i ~ ~ to r r lt ~hn~~ t t u t e t t ~ etllrrt t ~ t the c.otnponolt.s (!j'tllr rnrtric. g U jj it1 crny Guic.s.sicrt~ irortntil c.oort/irlcitc .sy.~totntrrc C I . I fiinc.tion.s r!f'th~.src~oort1inrrtr.srrc.r7).ss t h ~ surf21c.eC :
Here. [ , / ' I = ,f - ,f c1~~nvte.s the jutnp in the yutrrztit~,f ' t i c ~ l n . s . s C. tird K i.s the. .s~c.ond fut~clcrtnc~t~tul,fi~r~~~ 0 1 1 the slzock int(~t+ce t / r < f i n ~hy d (2.6).7 ('1.e.. we nssunle thiit the oreas o l the 2-spheres of .;ylnmetry change monotonically in the direction norn,al lo the surface. E.g.. if (. = r 2 . then $(.= (1. so the as\uinption n((.)# 0 is valitl cxccpt when n = ;)/;It, in whlch case the rays ofthe shock surface would be spacelike. Thu\ the shock speed would he fi~sterthan the \peed o f light rays if our assumption n(c,)# 0 tailed in the case (, = r 2 . his does not contradict the spherical shell exalrlple of Israel in 1131 hec;iusc (3.70)fails i l l that cxa~~lple.
540
J. Grotrh rr ol.
PROOF. Let ( w ' , w 2 ,w" = ( z ' , 8, cp) be a smooth coordinate system on C , and let z = (zO,. . . z3) denote the extension of these coordinates to a Gaussian normal coordinate system in a neighborhood of C (where we let a/azO = a/azl' when we restrict to spacetime, cf. [29]). Then by Lemma 2, n = a/azO, and T = a / a z t is tangent to the shock surface. Now in light of Corollary 2 of Theorem 2 it suffices to verify that (2.73) implies (2.74). By Theorem 2, in w-coordinates we have
.
But in Gaussian normal coordinates the metric ,q U i is diagonal on the surface C. To see this, note that the restriction of the metric (g U g) to the surface C is diagonal because the off diagonal cp and 8 components are zero in both (2.68) and (2.69), and the metric components (,q U g ) ~ , ,for , j # 0, are zero throughout any Gaussian normal coordinate frame in a whole neighborhood of C. Thus, by Lemma 3,
Therefore. since g U j is diagonal on C. K is also diagonal, and so the only nonzero components of ti are
K
-
I 2'
-
and
(2.78)
But, since c and (. (defined in (2.68) and (2.69)) transform like functions under arbitrary ( t , r)-transformations. (2.70) implies that c and (. define the same invariant function in a neighborhood of C . Thus, by (2.70) and the fact that (. = ~ 2 = 2 (. = jjZ2 on C. we see that ~ 2 2 . 0= n ( c ) = j 2 2 , 0 # 0 and ~ 3 3 . 0= n(c.)sin2 H = jj33.0 # 0 on the surface C , and hence Iti221=
0 and
It i 3 3 I = 0 across C. Now we have O = [G;,, Inin, = [Goo]= [tr(ti2) - (tr ti)'] = -21 K I r
and since ( t i 2 2 (2.82) imply
](ti22
+ K33).
(2.82)
+ K3j) # O (by the assumption n ( c ) # 0), we conclude that (2.73) and
Since K,, is diagonal, (2.80), (2.81), and (2.83) imply (2.74), so (2.73) implies (2.74), and we are done.
3. Matching an FRW to a TOV metric across a shock wave In this section we apply the theory of Section 2 to the general problem of matching a Friedmann-Robertson-Walker metric (FRW) to a Tolmann-Oppenheimer-Volkoff(T0V) metric Lipschitz continuously across a radial shock wave. We will tirst show that given any such metrics. one can always in principle construct a coordinate mapping between the FRW and the TOV coordinates such that, under this coordinate identitication, the FRW and TOV metrics match Lipschitz continuously across an interface that is implicitly determined. In order for the matching to describe a true shock wave, the further constraint of conservation must be imposed, and this restricts the possible FRW and TOV metrics that can be matched across a shock wave. An application of Theorem 5 demonstrates that ~ and this allows for the possibilthe conservation constraint reduces to a s i n g l ~c.orlditiori, ity of nontrivial examples. (The constraint can be viewed as a restriction on the possible equations of state on either side of the shock.) The conservation constraint is shown to ic in the densities and pressures on either side of the shock. An reduce to a q ~ ~ a r tequation application of MAPLE shows that the quartic,firc.ror.r.and we use this to show that two possible types of pre5sul-e jumps are allowed. and one of thcm can be ruled out by physic;~l consideration.;. We use thi\ formulation of the concervation constraint in the next section to constriIct i~ si~nplecli~ssof exact FRW-TOV shock wave solutions ilnder the ussumption that the FRW and TOV equations of' state are each of thc form 17 = a / ) . where n is constant. (That is. i n these examples. we ;tssumc that the so~rndspeed fi is (a different) constant on either side o f the shock w ~ I v ~ . )
3.1. Tlic gctlrrcrl FK W-TOV rrrtrtc~llirlg~ I I Y ) ~ I ~ J I ) ~ The FRW metric describes a spherically symmetric space-time that is homogeneous and maximally symmetric at each lixed time. (421. In coordinntes. the FRW metric is given by.
where t = .\-". r = . r l . H r .r2. cp 3 .r3. R = R ( t ) is the 'cosmological scale frtctor'. and d ~ =' do') sin') H dcp' denotes the standard metric on the unit 7-sphere. The constant k can be normalized to be either + I 1 . or O by appropriately rescaling the radial variable, and each o f the three cases is qualitatively different. This induces a rescaling of R(r). and so alternatively, R ( t ) can be rescaled to any positive value at a tixed time (say R = I at present time). in which case only the sign of k is unchanged. The sign of k gives the sign of the curvature in the constant curvature surfaces at each tixed r. and so froni (3.1) it is clear that the 3-space at r = const is unbounded when li 0. and when k > 0, r = I /fimarks the outer boundary of the coordinate system in (3.1). I n standard theory of cos~iiology.
+
.-
<
the case k = 0 corresponds to critical expansion, k > 0 to a closed universe, and k < 0 to an open universe. Current estimates of the Hubble constant H = R / R argue for an open universe. To obtain the equation for R ( t ) implied by the Einstein equations, assume that the stressenergy tensor is of the form
for a perfect fluid, and that the fluid is co-moving with the metric, [ 4 2 ] .The fluid is said to be co-moving relative to a background metric g;j if u' = 0 , i = 1 , 2 , 3 , so that g is diagonal and g i j u'uJ = - I imply that
Substituting (3.1) into the Einstein equations (1.16), and making the assumption that the fluid is perfect and co-moving with the metric, yields the following constraints on the unknown functions R ( t ) ,p ( t ) , and p ( t ) [42,30]:
together with
Equation (3.6) is equivalent to
Substituting ( 3 . 4 )into (3.5)we get
Since p and 17 are assumed to be functions of t alone i n (3.1), Equations (3.7) and (3.8) give two equations for the two unknowns R and p under the assumption that the equation of state is of the form p = p ( p ) . It follows from (3.7)-(3.8). cf. 1301, that ( R ( t ) . p ( t ) ) is a solution if and only if ( R ( - t ) , p ( - t ) ) is a solution. and that
Thus to every expanding solution there exists a corresponding contracting solution, and conversely.
Solvit~xthe Eitls~eirlecluutions /JY Lil~.sc.l~it: c.otttBluous r~trtric.stShock brwvc,s irl jior~[,rrrlrc.lutii~ity
543
The TOV metric describes a time-independent, spherically symmetric solution that models the interior of a star. In coordinates the components of the metric are given by
We write this metric in bar-coordinates so that it can be distinguished from the unbarred coordinates when the metrics are matched. When M(F) r Mo = const and B = A - 1 , the metric reduces to the (empty space) Schwarzschild metric, and the singularity at F = 2GMo is referred to as the Schwarzschild radius for the mass Mo, and represents the edge of a black hole. (See (33.341 for proof that black holes cannot form in smooth TOV metrics that solve the Einstein equations with nonzero sources.) Assuming the stress tensor is that of a perfect fluid which is co-moving with the metric, and substituting (3.10) into the field equations ( I.16), yields (cf. (421)
-
where M M(F), 6 = p(F), and fi = p ( J ) satisfy the fi)llowing system of ordinary diff'erential equations in the unknown functions (6(F),/I(?)*M ( J ) ) :
Equation (3.13) is c:tlled the Oppenheimer-Volkoff equation. and is referred to by Weinberg as tlzcfiuztltrttrentol ccl~rtrtiotlt!f'No~i.totlit~ti t~sfn)l~liy.\.i(.,s. 142, p. 30 1 1. In this section we assume the case of a barotropic equation of state fi = p(6). i n which case Equations (3.12). (3.13) yield a system o f two ODES in thc two unknowns (6. M). We always assume that
and that the sound speed is less than the speed of light dl, o
1.
The total mass M inside radius F is then defined by
(.
= I.
The metric component B
= B(F) is determined from
and M through the equation
In the special case when the density 6 is assumed to be constant, one can solve the Oppenheimer-Volkoff equations for the pressure, and the resulting solution, first discussed by Schwarzschild, is referred to as the Interior Schwarzschild metric. We remark that for any given FRW and TOV metrics, there are maximal domains of definitions for the variables. We assume that the FRW metric is defined on the maximal interval t- < r < t+ and 0 < r- < r < r + , and the TOV metric is defined on the maximal interval 0 < ?- < F < F+. For example, if k > 0, then we must have r i1 /A,t must be restricted so that p ( t ) and R ( t ) are positive, and by (3.8) we must require y P ( t ) ~ ( t -) ~ k 3 0. We now construct a coordinate mapping (t, r ) + (f, F). such that, under this coordinate identification, the FRW metric (3.1 ) matches the TOV metric (3.10) Lipschitz continuously across an interface r = r ( t ) that arises inlplicitly from the matching procedure. That is, we define a coordinate mapping that takes the unbarred frame of the FRW metric over to a barred TOV coordinate system that leaves fixed the H and cp coordinates. In order to apply Theorem 5 of Section 2, we require that the areas of the 2-spheres of symmetry of the FRW (3.1 ) metric agree with the areas of the 2-spheres of symmetry of the TOV metric (3.10). Thus to start. assume that
so that
That is. we define the tirst component of the coordinate mapping (t. r ) -. (7. I-) by
Note that at this stage the trr~nsformationr = R I is defined globally, which is important in order to apply Theorem 5 of Section 2. which requires that (3.16) hold not just at the shock surface, but in an open neighborhood of the shock surface. We next use (3.16) to rewrite the FRW metric in ( t , ?)-coordinates. We have from (3.16) that
SoIvirl,g t/re Eir~,stoit~ eqlcertiorl.t I)! Lipsclrit: c.ontinlrou.sntrtric,.~: Shock wcn1e.s it1 aotercrl rrl(rti~ity
545
and thus
Thus, the FRW metric (3.1) is given in ( t , ?)-coordinates by
which, using
becomes
We can now cotnplete the definition ofthe coorclinatc identification ( 1 . r ) 4 (i.7) by detining i = T ( r . r . ) so as to eliminate the cross term dt dF in (3.22).We do this tirst for n general metric of the f o r ~ n
I t is not hard to verify that if
VJ = $ ( I . ? ) is chosen to satisfy thc equation
then
is an exact differential. Since (3.25) defines the coordinate i as a function of ( 1 , 7). and we already have ? = R(t)r.. it follows that (3.25) defines i = i ( t . r ) . thus completing the definition of the sought after coordinate transformation ( t . r ) + (7. ?). Assuming (3.25). the (7. i )line element for (3.23) becomes
Now in terms of the metric
which appears in (3.22), C , D, and E are given by
D = R',
and
E = -RR;.
Thus, using (3.27). the FRW metric in (f. ;)-coordinates becomes
But from (3.28)-(3.30) we obtain
Now. equating the dl.' coefficients in the TOV solution (3.10)and the FRW solution (3.3 1 ) and using (3.32), we obtain the equation for the shock surf'~ice:~
which, using (3.8).si~nplitiesto
Hence (3.34) defines the shock surface, and the shock surface in ( t , r)-coordinates can be obtained from (3.34) by making the substitution 7 = K(t)r. (Of course, additional assumptions are required to insure that the shock surface as defined implicitly by (3.34) is reasonable, for example, stays within the domain of definition of the FRW metric. namely. I - k r 2 > 0, when k > 0, etc.) I t remains only to determine II/ from (3.24) so that the d i ' terms in the TOV and FRW metric agree on this surface. To obtain $. which determines the coordinate i in terms of the ( I , r ) coordinates of the FRW metric in a neighborhood of the shock surface, we solve Equation (3.24) subject to initial data on the shock surface ' ~ o t ethat. interestingly. the d?' coefticienta are independent of $
which is forced upon us by the condition that the d i 2 terms match on the shock surface. So, equating the d i 2 terms in (3.10) and (3.3 I), our assumption is that
holds on the shock surface (3.34). Rewrite (3.24) in the form of a first-order linear partial differential equation for $,
Here, C and E are functions o f t and F given by (3.28) and (3.30), and thus we can solve the initial value problem (3.36) in (t, ?)-coordinates with initial data (3.35) given on the shock surface (3.34), provided that the shock surface is non-characteristic for (3.36). Now the characteristics for (3.36) are given by
so that the function $ is obtained by solving the ODE
starting with initial values on the shock surface (3.34). where d / d p denotes differentiation in the ( E . C)-direction in ( t , ?)-coordinates. Solving (3.35) for !/I gives the initial values of $ to be met on the shock surface: namely.
Thus, if dF/dr denotes the speed of the shock surface in ( t . ?)-coordinates, then the condition that the shock surface be non-characteristic at a point is. by (3.37). that
If (3.40) holds at a point on the shock surface (3.34). then we can solve (3.36) uniquely for in a neighborhood of the point. thereby matching the FRW and TOV solutions in a Lipschitz cc~ntinuousmanner in a neighborhood of such a point on the surface in the (i,F ) coordinate system. Since we need only to detine local coordinate systems in order to define a space-time manifold, the shock surface (3.34) detines a complete Lipschitz matching of the metrics FRW and TOV at each point of the surface where the non-characteristic condition (3.40) holds. It is interesting to observe that one need not explicitly solve the PDE (3.36) for $ in order to determine the shock surface equation (3.34), and the solution of (3.34) can be calculated even when we do not have a closed form expression for i
+
as a function of t and r . That is, we find it somewhat remarkable that, other than it's existence, we do not require any detailed information about the transformation i = i ( t ,r ) in the subsequent developments. We shall discuss the condition (3.40) further below in Propositions 2 and 3, but first we discuss the equation for the shock surface (3.34). This is necessary in order to obtain an expression for the shock speed, and to motivate the conditions in Propositions 2 and 3 below. Note first that we have not made any choice regarding whether the FRW metric is on the "inside" or the "outside" of the TOV solution. For the case of a star, the FRW metric is on the inside (at srnall values of f within the shock surface), and the TOV is on the outside of the shock surface. For defi niteness, we will only consider this case, although the discussion we give below applies equally well to the case when the FRW metric is on the outside. The shock position is defined implicitly by (3.34). Note that (3.34) allows an interpretation of a global principle of conservation of mass in the special coordinate 7. Indeed, M(ro) is the total mass that would appear inside the radius 6 were the Tolman-OppenheimerVolkoffsolution continued to values of F < h. Thus, M ( r ) represents the total mass that is generating the TOV solution outside the radius i = 70. This describes the left-hand side of (3.34). The right-hand side of (3.34) can be interpreted as the total mass inside the sphere of radius FO at a fixed time r in the Freidmann-Robertson-Walker solution. That is, if we interpret 4 n p ~ ( t ) ~ as r ; the total Inass behind the shock at fixed t in the FRW metric, then (3.34) says that this is equal to the muss M ( r o ) observed by the TOV metric outside the shock. when the shock is at position FO= R(t)ro. Thus (3.34) says that the "total mass" is conserved as the shock propagates outward. Therefore. the total mass i n the TOV solution that an observer- sees out at infinity is fixed. and this equirls the total mass in the inside FRW metric plus the total mass in the outside TOV ~netric.As an application of this global conservation of Inass principle, we note that since in a "physically relevant" model tor a star. the density p ( i ) for the TOV metric should be a decreasing function o f f . the global conservation principle cannot hold when /, - p = Ipl = 0 across the shock surfitce. Indeed. if d p l d r < 0 f o r ? < to. and p(ro) = /,(ro), then
cannot lie on the shock surface: the global conservation and so by (3.34). the point (to. 6 ) of mass principle implies that if d p / d i < 0 . then Ipl # 0 across the shock. With this motivation, we can now calculate the shock speed under the condition Ipl # 0 . Indeed, by the irnplicit function theorem, the shock surface (3.34) is given by 7 = f ( r ) provided that
But, using (3.12), (3.42) becomes
Solving rllr Einstein equurio11.s11y Li/).scltir:cor1tinuou.s mrrric..c:5lroc.k nrrnlcJ.rin grnrrul rrlutiviry
549
-
at a point on the shock surface. Thus, as we have shown above, if we assume that d p l d r < 0, this condition is always valid on the shock surface. We can now calculate the speed of the shock s f (where "dot" denotes dldt). Using (3.34), which we write in the form
and differentiating with respect to t , we tind
Since [ p l < 0 (we are assuming that d p l d ? < 0), the shock speed is negative if b > O and positive if b < 0. Observe that, from (3.37), the condition on the shock speed (3.45) that guarantees that the surface be non-characteristic at a point is (cf. (3.39))
where we have used (3.8). (3.16), (3.28), (3.30). Note that in the classical theory of shock waves, the stable shock waves always advance toward the side of the shock where the fluid pressure is lower, and the corresponding shock waves that move into the higher pressure side are unstable. and are referred to as rrrlq/irt.tiori shot.ks, 1271. This means that if d b l d ? > 0.then the shock is stable if s 0 (/i < 0 ) .and unstable if s < 0 ( p > 0 ) .We remark that all of the above development is independent of the equations of state 17 = p ( p ) and /., = /.,(p). The famous example of Oppenheimer and Snyder 12.51 is obtained in the limit when the pressure 1) 0, and the TOV solution is replaced by the Schwarzschild nietric, (3.10) assuming a constant mass function M(F) M = const. B = A - I . In this case the FRW solution satisties P ( t ) ~ ( r ) 3= ~ ( 0 )and - so for a particular solution satisfying R ( 0 ) = 1 . d ( 0 )= 0, (3.8) implies that k = XnG/3. Thus (3.34) reproduces the well-known result that the radius of the star at time t = 0 in the Oppenheimer-Snyder model is given by the relation (see 142, p. 3461)
-
-
Note that in the Oppenheimer-Snyder limit. the interface must be interpreted as a contact discontinuity rather than a shock wave because tr = const and thus no energy or momentum is transported across the interface. The following proposition gives identities that hold at the shock surface as a consequence o f (3.34) and the coordinate identitication ( 1 . r.) + (i.F). These will be useful in later developments. PROPOSITION 1 . 011the .shock sutjucr (3.34), thr,fi)llo~~irig idrriririe.~hold
*
PROOF.The transformation that maps the (r, r)-coordinates of the FRW metric to the (T, ?)-coordinates of the TOV metric is given by
d r = d r d r + Rdr, d i = $Cdt
-
$ ~ d = r ($c- $ E d r ) d t - $ E R d r ,
where we have used (3.25) together with the fact that ? = *2(f, r ) = R(t)r. From these it follows that
where in this section we use the notation x = ( t , r ) , x = (7, F ) , and 2 = ( r , ?), and we supress the (0, cp) coordinates. (Here, the upper i and lower 1 on the right-hand side of (3.52) denote the (i. ./)-entry of the matrix.) From these relations i t follows easily that
Now in the tr-coordinate plane, the FRW and TOV metrics have components w R: i n x- and x-coordinates given, respectively. by
and
where A = 1 - 2GM/r, B satisfies (3.IS), and the upper i j denote the (i. J ) entry of the the metrics I.RW and 1.1s agree, by which matrix. Now on the shock surface M = we mean that
FPf3,
Rather than calculate this out directly, we use the fact that the FRW and TOV metrics must have components that agree on the shock surface in the i-coordinates. Thus we calculate -d r -d2r2+(1-kr2)
,
and
(3.57)
(Again, the superscript c-wB on the RHS of (3.57) and (3.58) denotes the ( a ,B) entry of the matrix.) Equating the (0. I)-entries in (3.57) and (3.58) we obtain (3.49). Equating the ( I , I )-entries in (3.57) and (3.58) we obtain (3.5 I), and this together with (3.49) gives (3.50). Equating the (0,O)-entries in (3.57) and (3.58) gives the first equality in (3.47), and applying (3.5 I ) gives the second. Finally, (3.48) follows from (3.49) together with (3.30), E = - R dr. This concludes the proof of Proposition 1 . Alternatively, we can derive (3.47)-(3.5 I ) directly from (3.1 1 ), (3.8), and (3.34), together with the expressions (3.28). (3.30). and (3.35) for C. E , and B, respectively. To obtain (3.51). solve (3.34) for p , solve (3.1 I ) for M , and substitute these into (3.8). To obtain (3.48). ~nultiply(3.8) by r'. solve fi)r 9 p R 2 / ? . and substitute this into (3.28). Using (3.48) together with (3.30) gives (3.49). The identity (3.49) together with (3.51) yields (3.50).St~iternent(3.35)together with ( 7 4 8 )gives I = I - kr'). Using (3.48)
21
together with (3.30) and (3.5 1 ). in the expression 1 (3.47).
+ A EJ'/C'
gives the Inst equality in
We end this section by giving conditions under which the shock surface is n o n characteristic: i.e.. that (3.39) holds. We assume here that the shock surface lies within the domain of definition ol'the FRW metric if k > 0 . The first proposition gives conditions on the equation of state p ( l , ) that guarantees the shock surtitce (3.34) is non-characteristic provided it does not interscct the Schwarzschild radius. A = I - 3,CjMlr = 0 . ot'the TOV solution.
e ~ l c ~ r y ~ , /on ~ etho r o skoc.k s~rrfirfirc.e(3.34). tlzetl thc .sAoc.k.s~rt:firc~r i.v 17orl~l7c/rc. c,htrrtrc,trri.vtic. PROOF.We already have (cf. (3.37). (3.30))that C
C
E
R(-R)?
A=-=-
,
and
From the Oppenheimer-Volkoff equation (3.13) for dj/dF, we see that the sign of dp/dF is positive inside the Schwarzschild radius and negative outside. Thus sign([p]) = sign(dp/dl-) = sign(dj/dF) = - sign(A). But on the shock surface, we also have by (3.48)
and so sign(A) = sign(C). Finally, we also have from (3.9) that
~b < 0. Thus,
We shall also need the following proposition: PROPOSITION 3. If.R = 0 c~ndA # 0 at ( I point on the .shock surjiicr (3.34) (i.e., the point rtrdius) thm, if the .shock .spe~di.s,finitrrrt the point, the .shock is not oil the S~~l1rt'cirz.schi(cl/ slrtj2rc.e i.s c11.so~ ~ o i ~ - c I ~ e r r ~clt ~ ~rhr ~ tpoint. ~ri.sti~~ PKOOF.By (3.36). the characteristic surfaces satisfy
where we have used (3.48) and (3.49).Therefore, if R = 0. the characteristic is tangent to t = const, and thus any ti nite speed .v = dF/dt is a non-characteristic speed. Surnmctry The results of this section can be summarized as follows. Let (3.1 ) and (3.10) denote arbitrary FRW and TOV metrics that solve the Einstein equations for a perfect fluid. (We make no restriction on the equation of state at this point.) Then we have identified the following conditions under which there exists a smooth regular coordinate transformation P : ( I , r ) + (f,F), and a corresponding shock surface I. = r ( t ) in FRW ( t . r)-coordinates (which maps to the . that, when writcurve F = r(i) in TOV barred coordinates by (i,F ( T ) ) = P(t,~ ( 1 ) ) )such ten in the same coordinates, the lnetrics (3. I ) and (3.10) agree and are Lipschitz continuous across the shock surface which is given implicitly by the equation M = 4npF3. For example, the metrics agree on the shock surface when both are written in either the barred or unbarred coordinates. We summarize most of the results of this section in the following theorem: THEOREM 6. Assume that the .\hock .surjuc.eF = l-(t) i s dqfit~edirnplic,it!\. by
Solvitl!:
lilt.
Eirl.slc,irr eqrrl~liorzs1)y Lil).scl~irzcotrrit~uou.~ nirtric..s: Shock wrrvrs in jimrrrrl rrlfrtit,ity
553
in a neighborhood o f u point (to, Fo) thut sutisjies (3.62).Assume that
so thut tlze spheres of symmetry ugree in the burred und unbarred frumrs, and the shock srr$Crce in ( t ,r)-c.oordinatesis given by r ( t ) = i ( t ) /R ( t ) .A.~sumefinullythut both
and the tzon-ch~ir~lcteristic conclitiorz
hold at t = to (cj:(3.28),(3.30)crrlcl(3.49)).Tlzrn the coordinnte f = W I ( t, r ) c.un be d ~ f i n e d .snloorhIy crnd it1 .such CI L V C I ~thcit W = ( P I ,W2)is one-to-one und rqulur in u nrighhorhood oftho poirlt (to. rg) (c:f: (3.36)).~lricltllr t?~rtric.s ( 3 .1 ) C U I ~(3. 10) n7trtc.h it1 CI Lip.rchitz c.ot~titluolr.s,fil.sI~iot~ L1cro.v.s the .sl~oc.kxr~rfiic.~ r = r ( t ) in CI tl('ighborhood of (10. ro). By the implicit function theorem. a sufficient condition for (3.34) to define a surface locally through ( 1 0 ,?()) is that
By differentiating (3.34)directly, we obtain the alternative sufficient condition,
3.2. The c,on.ser\vltior~c.orlstruint Assume for this section that we are given smooth FRW ( 3 .I ) and a TOV (3.10)solutions of the Einstein equations (1.16) such that Theorem 6 and (3.62)-(3.65) hold for all t E - ( 1 - , I + ) . r E ( r - , I.+), and r = r ( t ) / K ( t E ) ( r - . r+).That is, assume th at the shock surface F = r ( t ) is detined by (3.34) and that the metrics agree on this surface throughout this range of variables, when the unbarred coordinates (3.1) and barred coordinates (3.10)are identified by the transformation (7, i )+ ( t ,r ) constructed in the last section. Thus r = F(t. r ) is given by
and the transformation i = T(t,r ) is assumed to exist throughout this interval in light of the non-characteristic assumption (3.65).Other than it's existence, we do not require any
detailed information about the f transformation in the subsequent development. The following theorem gives conditions under which the matched FRW and TOV metrics define a true shock wave solution of the Einstein equations: that is, a weak solution such that all of the equivalencies (i)-(iv) of Theorem 4, Section 2, are true, and conservation of energy and momentum hold at the interface. THEOREM7 . Let g U jj denote a tnetric obtained by matching an FRW metric g and a TOV metric jj Lip.schiti continuously Lrcros.s the interjkce dejined implicitly by (3.34), such thut Theorem 6 lzolds. Assume that at euch point ofthe inreduce the condition
~vherei-, denote the shock speeds d r l d t , d r l d t , re.spectively, diflkrmtiution being tuken ~ : i t hre.spec,t to the unhnrrecl FRW time c.oordinate t , holds; rind 1 . ] denote.7 the FRW-TOV jutnp in CI rj~1~lntity ricross the itrtetfiice, cis ccilculuted in the .same coordincrrr system. Then the re.sultitzg tnetric g U clefittes N true shock wave .solutiotz (?/'theEinstein equations in tlzr .sm.ve thut N I I ofthe ~ q ~ ~ i ~ ~ ~ I (?fTheorrm e t r c . i ~ . s 4 holrl, N ~ ~~L~ IP . imp!\' F P thut the RrinkineHlr~otriotjump (.on~Iitiotr.s
hold rrt the> .shock. P R ~ O FBecause . r = R ( t ) r holds in a neighborhood of the shock surface, conditions (2.70) and (2.71) of Theorem 5 is met. Thus. according to Theorem 5. all of the equivalencies of Theorem 4 follow from the single (invariant) condition
which is equivalent to
in light of the fact that both the FRW and TOV metric are assumed to satisfy the field equations G;, = K T ~on , either side of the shock. We emphasize that the indices i , j must refer to components in the sutne coordinate system, where coordinates on either side of the shock are identified through the coordinate trinsformation ( I . r ) + (i,r ) . To start, use the Einstein equation G = KT, the condition [cijln;t7, = 0 for conservation across the shock (cf. Theorem S ) , and the assumption that the source fluid is co-moving with respect to the metrics on either side of the shock (cf. (3.2)), to rewrite the condition for conservation as
Solving fhr Einsteb~equutiorr.~by Lipschitz continuous metrics: Shock waves in gmerul rrlufivity
555
Here n ; and ii; denote the i-components of the normal vector n to the shock surface (3.34) in unbarred (FRW) and barred (TOV) coordinates, respectively, and ln12 = g ' j n ; n j . (Note that ui = 66 in (FRW) coordinates, u' = ~ - ' / ~ 6in 1(TOV) coordinates, thus giving rise to the factor B.) Since n ; = 0 = i i ; , i = 2 , 3 , we need only pay attention to the 0- and I -components of n. To verify (3.72), note that, for example, in the (FRW) unbarred frame, (3.2) gives
Moreover, we need not choose the vector n to be of unit length, so long as n ; and ii, are the components of the same vector. Since the LHS of (3.72) is an invariant scalar, so is the RHS. In order to evaluate n ; and i i ; , let (3.34) (formally) define the surface r = r ( t ) , which we can write as the level curve of the scalar q ( t , r ) = r - r ( t ) = 0. Then we can choose n ; dx' = d q , so that
which yields no = - k ,
and
nl = I . To obtain t i l , we write the function cp in (?, ?)-coordinates: q (i,r ) =
r
R(t (i.r))
Then
so that
But using the fact that
r = Rr, together with (3.25),
-
r (t (i.f ) ) .
we have
which implies
Putting this into (3.77) yields
Using the identity (3.47) of Proposition 1 we obtain
where - B and A - ' = ( I - ~ G M / ? ) - ' are the coefficients of dt' and d ~ 'in the TOV metric (3.10). Finally. using the FRW metric (4.57) to compute lnl', we obtain
Now substituting (3.73),(3.79). and (3.80) into (3.72) yields [ ~ ' j ] n ; t , = (/'
+ p);'
-
(ji + 3)
( I - kt-')
A R'
2?
t-
+
7
I
kt-' - 0 R-
-
(3.81)
which is Equation (3.69). Equation (3.69) gives the additional constraint imposed by conservation across the shock in terms of the quantities r = r ( t ) (the shock position), and the values that p . p. p . /, and R take on the shock surface. The following proposition explains why the pressure in the Oppenheimer-Snyder model must be taken zero:
L E M M A9. I f ji = I, = 0 idet~ticully(so that the TOV .rolutior7 t-c.duc.o.sto the Sc~h~wr:schi/d solution), und p 3 0 and p 3 0 e\vr?,where, the11 (3.69) inil~liesp = 0 rir~dr ( r ) = const nll cllong the shock. PROOF. When
6= I, = 0, (3.69) reduces to
Solvittg the Eihsteb~eq~ccrtio,r.sby Lipschit: cot~tinuousn2rtnc.s: Shoc.k wuvr.s in general relutivit~y
557
Since (1 - k r 2 ) / ~ > * 0 in the FRW metric, the lemma follows at once. We now derive an equivalent formulation of the conservation constraint (3.69).
L E M M A10. The conservation constmint (3.69) has the equivulmt,fi)rmulution
Before giving a proof of (3.82). we first note that, assuming (3.63) and (3.64) hold, the condition 0 < H < I is equivalent to the condition
to the condition that the shock surface lies within the coordinate restricthat is. eq~~ivalent tion of the FRW metric. To see this, use the shock surface equation M = 4npr3 to simplify (3.84) as follows:
.,
(
8 ~ 5, 1 2G -k = r 2 - - kI ? - ]. 3
K- = - - - p ~ -
and so
This can be written as
Thus the condition
is equivalent to (3.84), in view of our assumptions (3.63) and (3.64). Moreover, since we are assuming (3.62)-(3.65) hold throughout, it is clear that (3.88) is equivalent to (3.84) when k < 0,as well. When making general statements about FRW-TOV shock waves, we always assume that (3.88) holds.
PROOFOF L E M M A10. Differentiating (3.34) with respect to t and applying (3.12) yields
Solving for b in (3.7) yields
Combining (3.89) and (3.90) thus gives
Differentiating F = Rr with respect to t . using (3.91), and solving for i- we get
Substituting (3.9 I )and (3.92) into (3.8 l ) , we obtain the following equation, which is equivalent to the conservation condition [T'' lrlini = 0:
Equation (3.93) expresses conservation at the shock surface (3.34). But by (3.9 1 ),
holds on the shock surface, and using this we can transform (3.93) into the tinal form (3.82). For convenience, we summarize the results of this section in the following theorem:
THEOREM 8. AS.YUITIO tl~(itFR W (111dTOV t i l ~ t r i crrw ~ ~ I I ' P I I. Y I ( ( . / I tllrrt ril(itc.17 Lil>s(.llit: c.orltitluou.s!\.trc.ro.s.sthe shock sut:firc,e (3.34) trtld thtrt (3.62)-(3.65) holtl. Tl1rr7 (i)-(iv) of' T17eorrm 4 hold or1 tho s1iot.k .s~rrji~c.c~ if' und otllx I / ' c>itllrr-(3.69) or- (3.82) 1zolrl.s otl tl70 .shock surjuc.e. We now use the conservation constraint to solve for p as a function of p . p , and Solving (3.82) for 11 we obtain
/7.
Solvirrfi rhr Eitrsrrirr eqlrarior~sby Lil~schihc.ontir~uou.v rnr1ric.s: Shock wuves in firrtrrul relurivit~v
559
where
Thus we conclude that every TOV solution determines two possible FRW pressures at the shock through the conservation constraint. Since the FRW pressure is constant on the r = const surfaces, these implicitly determine the FRW equations of state p = p ( p ) from the TOV density and pressure. Now the terms in the numerator of (3.95) combine as follows:
where we use the notation that the brace (.I-is taken zero unless we take the minus sign in (3.95) (and correspondingly minus sign in (3.97)).Using (3.97) in (3.95)gives
which upon multiplying the numerator and denominator by H / ( 1
-
0 ) yields
We can further simplify 1)- as follows. First, we can verify the identity
Substituting this into the numerator of (3.100) yields
Thus, if we define the variable
where
then the pressures I)+ and 1)- take the similar forms
In Section 5 we will prove that
which leads to d p l d p < 0 . and so can be ruled out as physically unlikely possibilities. An easy calculation gives the equivalent formulation of (3.103) in terms of the TOV pressure /,,
where
The following two theorerns follow directly from (3.103). Let us now interpret an FRW-TOV shock wave as the leading edge of an explosion in which the FRW solution is on the inside, expanding outward into the static TOV solution. In this case, we can take i / p < I as an entropy condition for such a shock wave; that is, the density should be greater behind the shock. The following theorem states that p / p < 1 implies / I > /I as well, when we take the pressure to be /I = /)+ in (3.103), and there is a constraint on the allowable values of 8.
cind assume
Sol\ing the ECr.stein equc~!ion.s/I? Lipschitz continuous merrics: Shock wuvrs in grnerul relativity
Then p+ > 0 f u n d only if p - p > 0, und this holds shock, where
if und only if 81 < 0
56 1
< 1 at the
Since the FRW pressure is determined by the TOV solutions according to (3.103), we now ask what possible pressure jumps can be assigned at an FRW-TOV shock wave at a given position. The final theorem of this section shows that all possible pressure jumps can be assigned as we vary the value of 8. The pressure jumps that can be assigned at a point can be viewed as possible initial conditions for the subsequent dynamics of an FRW-TOV shock wave solution. n d z < 1. Then j2)r every choice of positive THEOREM10. Assutne (3.62)-(3.65) ~ ~ [hut ~~ulue.s,fi)r p , ,G, und p , the pressure p+ tnonotonicully tokes on every vulue from I j j , +m), utzd the pressure cl~feretlce( p + - fi) motzotonical1.y tukes on everv vulue,from 10, +m), u,s H rutlpJ.st t ~ o t ~ o t o t ~ i c ~ ~ ~ l l y1, If r,oHtI t).z
PROOF.When p > 6, it follows immediately form (3.102) and (3.103) that p+ > 0 if and To see this, note that the numerator in (3.103) is always negative because only if H > 81.
when :< I . Thus by (3.103). I)+ > 0 if and only if y H > I. Furthermore, if ,5. /,. and p are ti xed, then p varies monotonically from /, to cc as H varies from l to H I because / I , , < 0. and when H = 1 .
+
We can perform a similar analysis on the difference (pi
-
P). because, as is easily shown,
This completes the proofs of the Theorems 9 and 10. Another direct consequence of (3.103), (3.104) is that if A > O and H < 1 then, when p > p , the only shock waves with positive pressure must satisfy 1.' = p+ and (-)=yo>
I.
In this case, (3.102) implies
(3.1O X )
4. A class of solutions modeling blast waves in GR In this section we use the theory developed in Section 3 to construct a class of exact, spherically symmetric, shock wave solution of the Einstein equations for a perfect fluid. The solutions are obtained by matching a Friedman-Roberson-Walker metric (3.1) to a static Tolman-Oppenheimer-Volkoff metric (3.10) across a shock wave interface. This is in the spirit of the Oppenheimer-Snyder solution, except, in contrast to the Oppenheimer-Snyder model, the pressure 11 is nonzero. These shock wave solutions can be interpreted as simple models for the general relativistic version of an explosion into a .static, singulal; i.sotherma1 sphere. It is interesting to keep in mind that shock waves introduce time-irreversibility, loss of information, decay, dissipation, and increase of entropy into the dynamics of a perfect fluid in general relativity. The FRW metric is a uniformly expanding (or contracting) solution of the Einstein gravitational field equations, which is generally accepted as a cosmological model for the universe. The TOV solution is a time-independent solution which models the interior of a star. Both metrics are spherically symmetric, and both are determined by a system of ODES that close when an equation of state 11 = p ( p ) for the fluid is specified. In the solutions that we construct below, one can imagine the FRW metric as an exploding intzer core (of a star or the universe as a whole), and the boundary of this inner core is a shock surface that is driven by the expansion behind the shock into the olrtor, static. TOV solution which we imagine as the outer layers o f a star. or the outer regions of'the universe. I n these solutions, the shock wave emerges from = 0 at the initial (Big Btrnx) singularity in the FRW metric and so. broadly speaking. one can interpret these examples as providing a scen~u-ioby which the Big Bang begins with a shock wave explosion. The outer static TOV solutions that appear beyond the shock wave i n the examples below. are the general relativistic version of a .sttrtic. i.sotllrr71rtrl sl~lrorubecause the metric entries are time-independent, and the constant sound speed can be interpreted as modeling a gas at constant temperature. I t is .sirigirltrr because it has an inverse-square density profile. and thus the density and pressure tend to co at the center of the sphere. The Newtonian version of a static singular isothermal sphere is well known. and is relevant to theories of how stars form from gaseous clouds. 121. The idea in the Newtonian case goes as follows: a star begins as a diffuse cloud of gas which slowly contracts under its own gravitational force by radiating energy out through the gas cloud as gravitational potential energy is cone s the gas cloud reaches the point verted into kinetic energy. This contraction c o n t i n ~ ~ (Inti1 where the mean free path for trans~nissionof light is small enough so that light is scattered, instead of transmitted. through the cloud. The scattering of' light within the gas cloud has the effect of equalizing the temperature within the cloud. At this point the gas begins to drift toward the most compact configuration of the density that balances the pressure when the equation of state is isothermal; namely. it drifts toward the configuration of a static, singular. isothermal sphere. Since this solution in the Newtonian case is also inverse-square in the density and pressure, the density tends to infinity at the center of the sphere, and this ignites thermonuclear reactions. The result is a shock wave explosion emanating from the center of the sphere, and this signifies the birth of the star. One can interpret the exact solutions constructed below as general relativistic vet-sions of such shock wave explosions.
Solving rhe Ei~~stein erl~rnrio~~s 1,v L~pschicc o ~ ~ t i n u ornrrric..~: u.~ S11oc.k wuvrs ill grnrrul rrluriviry
563
Fig. I . A plot of 5 v\. rr
In the construction we assilrne that the FRW and TOV solutions both have isothermal equations o f state, but ilt different temperatures. That is, we assume p = n p in the FRW solution. and = 66 in the TOV solution. where both the inner FRW sound speed fi and the outer TOV sound speed fi are assumed to be constant. Here p denotes the fluid pressure and p the mass-energy density, arid again we let the unbarred iund barred variables refer to the standard coordinate systems lor the FRW and TOV metrics (3.1 ). (3.10). respectively. We assume throughout that the speed of light c . = I. Thc construction is based on exact solutions of' FRW and TOV type that exist li)r these special equations ol' state. In Section 3. (3.34), we showed that in general the shock position f = ? ( I ) is given implicitly by the eyuatiori M ( 7 ) = q p ( t ) r 7 . where M ( f ) denotes the total TOV mass inside radius 7. and p ( t ) is the FRW density at the shock. For the exact solutions with constant sountl speed constructed here, the shock surface condition implies that p = 3 p across the shock. Moreover. i n order that conservation of energy and momentum hold across the shock, we show that the sound speeds must be related by an algebraic equation of the form 6 = H ( n ) . where H 1 ( a ) > 0 . H ( 0 ) = 0 , and H ( a ) < a. cf. Figure I. Since. at the shock. the inner FRW sound speed and density exceed the outer TOV sound speed and density. we conclude that the outgoing shock wave is the stable one. and the larger sound speed in the FRW metric is interpreted as modeling an isothermal equation of state at a higher temperature (consistent with the heating of the fluid by the shock wave). In the limit cr + 0. the model recovers the Newtonian lirnit of low velocities and weak gravitational tields. 0.458 < rr. = 3 1 3 % We verify that there exist two distinguished values of a. nl 0.745, such that, if 0 < cr < 1 . then the Lax characteristic condition (that characteristics impinge on the shock, [141) is satisfied if and only if 0 < cr < a l : and the shock speed is less than the speed of light if and only if 0 < a < a ? . A calculation gives 6, = H ( a l )% 0.161. and 6 2 = H(a2) 0.236. We conclude that for a between a1 and 02. i\ new type of shock wave appears in which the shock is supersonic relative to the fluid on both sides of the shock. Thus, in this theory, a fluid with a sound speed no larger than f i
a
J. Groah er 01.
5 64
can drive shock waves with speeds all the way up to the speed of light. The time-reversal and stability properties of these shocks when a, < a < a2 remain to be investigated. Since Lax type shock waves are time-irreversible solutions of the equations due to the increase of entropy (in a generalized sense, cf. [27]) and consequent loss of information (effected by the impinging of characteristics on the shock), we infer from the mathematical theory of shock waves that when 0 < a < 0 1 , many solutions must decay time asymptotically to the same shock wave. Thus, in contrast to the pure FRW solution, in these models one should not expect a unique time reversal of the solution all the way back to the initial Big Bang singularity when the sound speed lies in the range 0 < a < al. Note that the TOV solution when p = 56 is, by itself, of limited physical value because p = co at F = 0. One can interpret this as saying that this exact solution is unstable because it requires an infinite pressure at ? = 0 to "hold it up". In contrast, the shock wave solution here removes the singularity at r = 0 (for times after some initial time), and so the construction demonstrates that a shock wave in the core can supply the pressure required to stahilizr a TOV solution by holding it up.
We now construct exact solutions of TOV type which represent the general relativistic version of static, singular isothermal spheres. First assume the equation of state
for the TOV metric, and assume that the density is o f form
for some constant y . In this case, an exact solution of TOV type was first found by ~olniun" 1401; namely, by (3.14).
M ( V ) = 477 y?.
(4.3)
Putting (4.1)-(4.3) into (3.13) and simplifying, yields the identity
From (3.1 I ), we obtain
'1" the case
(T
=
113. this solution
was re-discovered by Misnerand Zapolsky. cf. 1.12.p. 3201
S o l \ i n ~flze Einstein C ~ L I ( I ~ I ' O I I . S hv Lil~xchitzC O ) I ~ I ' I ? U ~ U . S mr!ric.s: Shock bvuv~sirl gr,~rrulrrluriviry
565
To solve for B, start with (3.15) and write
which simplifies to
This equation has the explicit solution
By resealing the time coordinate, we can take Bo = I at to
= 1 , in which case (4.8) reduces
We conclude that when (4.4) holds. (4.1 )-(4.5) and (4.8) provide an exact solution o f t h e Einstein held equations ( 1.16) of TOV type. Note that since is the sound speed of the tluitl. (4.1)-(4.3) provide exact solutions for any sound speed 0 < 6 < I . Note also that, 7 when 6 = 113. the extreme relativistic limit for free particles 1421, (4.4) yields y = .hn!; (cf. 142, Equation ( 1 1.4.13)l). These exact solutions by themselves are not so interesting physically because the density and pressure are intinite at 7 = 0 at every value of time. Oiir shock wave construction, given below, removes the singularity at F = 0 in these solutions, after some initial time.
4.2. A I I eutrc.t .solutio~lof' F R W type
We now construct exact solutions of FRW type. We restrict to the case k = 0 in (3.1 ), so that the metric takes the simple (confonnally flat) form
Now assume an arbitrary equation of state o f the form p = p ( p ) . We will obtain a closedform solution of the Einstein equations (1.16) in this case. By (3.7)-(3.8). it suffices to solve the system of two ODES and
(4.1 1 )
Rewrite (4. I I) as
and substitute into (4.12) to obtain
(The upperllower plus-minus signs will always correspond to the two cases represented by the upperllower plus-minus signs in (4.13).) The point to be noted here is that when P = p ( p ) is assigned, (4.14) is independent of R , and thus we can integrate it explicitly; namely, since
we obtain
Formula (4.16) gives t as a function o f p . and we can use this, together with (4. I I ) . to obtain a cloxcd-form expression for R as a function of 1).Thus since
if we combine this with (4.1 I ), we get
which has the explicit solution
H
= Ro exp
I'
S
(1,)
-
1
3 6 +17(0)
d6.
We now use the theory developed in 1301 to match the above TOV and FRW type metrics at a spherical interface across which the metrics join Lipschitz continuously, and such that the conservation constraint (3.34) holds at the interface. The resulting solution is interpreted as a fluid dynamical shock wave in which the increase of entropy in the fluid drives a time-irreversible gravitational wave.
Assume now that the equation of state for the TOV metric is taken to be -
/I=
(Tp
for some constant 6 ,and that the tixed TOV solution is given by (4.2)-(4.5) and (4.8). Then, given an arbitrary FRW metric, our results in 1301 imply that we can construct a coordinate mapping (f,?) -+ ( t , r ) such that the FRW metric matches the TOV metric Lipschitz continuously across the shock surface (3.34). This applies, in principle, to uny equation of state p = p ( p ) chosen for the FRW metric. Using (4.3) and solving for p on the shock surface ? ( I ) = r ( t ) R ( t ) gives:
To meet the additional conservation condition, we restrict FRW metrics to k = 0. and we use (3.82) to determine the pressure. Substituting (4 = A = 1 - 8 ~ r G y= const into (3.82), we see that the resulting equation is homogeneous of degree three in the p , and p , variables. Since /, = 6 6 , and
on the shock surface, it is clear from holnogeneity that (3.82) can be met if and only it' 1) = 00 for so~iiecol~stanta. Substitilting this into (3.82) gives the following relation between a and (T (cf. Figure I ):
Alternatively, we can solve for a in (4.2 1 ) and write this relation as
This guaralltees that conservation holds across the shock surface, and thus Theorem 5 holds. and the results of Theorem 4 apply. Note that H (0)= 0.and to leading order.
as a + 0. I t is easy to verify that within the physical region 0 6 n. 6 < I . H ' ( o ) > 0 and 6 ia. as would be expected physically because p = 3 p > jj at the shock surface. One can verify that when a = 113, we have
568
J. Grouh rt ul.
and when a = 1 , we have
We now obtain formulas for p(t). R(t), and the shock positions r ( t ) and F(t) = r(t)R(t). Substituting 11 = a p into (4.15) and (4.18) yields
and
Using (4.20) we obtain
Putting this into (4.24) gives
Integrating Equation (4.27) gives the tormula for the shock position:
Thus (4.20) gives p in terms o f t :
Finally, we can use (4.25) to obtain R ( t ) and the shock position r ( r ) = F(t)R(t)-I :
S o l v i n ~the Einsteirr ec/ucitions by Lil).vc./fit,-coilrinuous metrics: Shock wuvrs in grrwrul rrlutivity
Differentiating (4.28) and (4.3 1) gives the speeds of the shock ( r , r)-coordinate systems, respectively:
569
and i- in the ( t , F ) - and
where again, 6 = H ( a ) is given in (4.21). Note that the solution (4.28)-(4.3 I )contains two arbitrary constants, ?(), Ro or ro, Ro, as it should from the initial value problem (4.13), (4.14). Note also that for an outgoing shock wave, we choose the plus sign in (4.13) and (4.28), and in this case there is a singularity in backward time
As t + t,, it is clear that F + 0. p , p , p , /, all tend to intinity, and R , r tend to zero. If we take this as a cosmological nod el, then t = t, represents the initial Big B~rtrgsingularity in which u shock wnvc emerges from r- = 0 . We surnrn~~rize these results in the following theorem: THEOREM I I . A.\.s~orrc~ trt7 ryutrtiotl c?f.\ttrtclr?f'thcji)rrlr/, = 6 p j i ) r the>TOV ttlrtrie., trt1t1 11 = CII) ,fi)r tlre' FRW ttlc~tric.tr.ssLtt?rcJ (4.2 1 ) I~olel.s,ertltl terkr k = 0 . Tlretl the. TOV .solr/tiotl gi~,c)trhjl (4.2). (4.3), (4.5). (4.8) r~,illtrrtrtc.11the. FKW .solrrtiot~~i1,evlI? (4.29), (4.30) crc.rr).s.\. t l ~ c.slroc.k ~ .s~rt:fircx~ (4.28), s~rc.17tl~trtc.on.scr1~trtior7 r!f'cwcr~j' rrrrcl ti~otitrt~trrtir hold trc,ro.s.stlrr s~trfirc~e. Tllc c~oorclitlcrteitletrt~jit~crtior~ ( t .r ) -+ (i.r ) i s ,qi,,rtl I?\.i = Rr, toxc)thrr wit11 tr stllooth jut~c.tianT = i ( r .r ) w~lrosec)xi.vtc~trc.c (it7 tr rleighhorl7ootI of' tllc. .s/~oc.k. S L I + I L . P ) is ~lerr~ot~.str~cteel it1 1.301.
By Theorem 5 , all of the equivalencies in Theorem 4 hold across the shock surface. In the next section we show that the shock speeds are less than the speed of light, and we determine when the Lax characteristic conditions hold.
4.4. The O i x .slloc~kc~otrt1iriotr.s To complete the analysis of our shock wave solution discussed in the last section, it remains to analyze the shock speed and characteristic speeds on both sides of the shock. In classical gas dynamics, characteristics (in the same family of a shock) impinge on the shock from both sides, leading to an increase of entropy and consequent loss of information. This is also the source of the well-known time-irreversibility, as well as the stability, of gas dynamical shock waves. This interpretation carries over to a general system of hyperbolic conservation laws. Indeed, this characteristic condition has been proposed by Lax, 114, 271, as a stability criterion for shock waves in settings other than gas dynamics. This "Lax
characteristic condition" can be easily applied in general systems where either a physical entropy is difficult to work with, or it has not been identified [ 2 7 ] .Since in gas dynamics the density and pressure are always larger behind (stable) shock waves, and in our example p = 3fi (cf. (4.20)), we restrict our attention to the case of an outgoing shock wave in which the FRW metric is on the inside and the TOV metric is on the outside of the shock. This is equivalent to taking the plus sign in (4.13) (and the corresponding upper sign in Equations (4.14)-(4.16)). The goal of this section is to show that, in this case, there exist values 0 < (TI < a2 < 1 (alx 0.458, a2 = 4'313 0.745) such that, for 0 < o < I , the Lax characteristic condition holds at the shock if and only if 0 < a < a , ;and the shock speed is less than the speed of light if and only if 0 < a < 0 2 . We conclude that our gravitational shock wave represents a new type of fluid dynamical shock wave when a2 < a < 1. For the outgoing shock waves with a in this interval, the shock speed exceeds all of the characteristic speeds on either side of the shock, because both the fast and slow characteristics cross the shock wave from the TOV side to the FRW side of the shock. Our tirst result is the following lemma: LEMMA1 1 . For- 0 < o < I . tlre shock .s/?ertl,relutive to thr FR W,fiuidpcrrtic,lr.s, i.r given l7.Y
The function s ( a ) is plotted in Figure 2. By numerical calculation we obtain that I - .\ ( a ) is monotone for 0 < a < I , and becomes negative above a = a?,where. using computer algebra, we ti nd
0
02
04
7
06
Fig. 2. A plot of the hhock speed .\
08
vs. n .
a
1
S o h i n ~the Einstein equations hy Lipschit? continuous metrics: Shock wuve.s in ~ e n e r u relativity l
57 1
Therefore, by general covariance, the following theorem is a consequence o f Lemma 1 1
THEOREM 12. For 0 ia 5
i1,
the shock speed is less than the speed of light i f and only if
<02.
To prove Lemma 1 I, we recall that the "speed" o f a shock is a coordinate-dependant quantity that can be interpreted in a special relativistic sense at a point P in coordinate systems for which g i j ( P )= diag(-I. I, 1 . 1 ) . ( W e call such coordinate frames "locally ) 0 as Minkowskian" to distinguish from "locally Lorentzian" frames in which g i j S k ( P= well. Since we are dealing only with velocities and not accelerations, we do not need to invoke the additional condition gij,k ( P ) = 0 for a local Lorentzian coordinate frame in order to recover a special relativistic interpretation for velocities.) In such coordinate frames, a "speed" at P transforms according to the special relativistic velocity transformation law when a Lorentz transformation is performed. W e now determine the shock speed at a point P on the shock in a locally Minkowskian frame that is co-moving with the FRW metric. To this end, let ( t ,r)-coordinates correspond to the FRW metric with k = 0 in (4.10). Let ( t ,?)-coordinates correspond to a locally Minkowskian system obtained from ( t ,r ) by a transformation o f the form r = cp(F),so that
-,
Choose cp so that cp'/c2 = I and R-(cp )-r-/cp- = I at the point P : i.e.. at P = P ( t . r ) . set cp(r)= ? and cpl(r)= I / K ( t ) .Thus, in the ( t .?)-coordinates, 1
/ ? - 7
at the point P. and so the ( t ,;)-coordinates represent the class o f locally Minkowskian coordinate frames that are tixed relative to the fluid particles o f the FRW metric at the point P. (That is, any two members o f this class o f coordinate frames will differ by higher order terms that do not affect the calculation o f velocities at P.) Therefore,the speed dF/dt of a particle in ( t . ;)-coordinates gives the value o f the speed o f the particle relative to the FRW fluid in the special relativistic sense. Since
we conclude that i f the speed o f a particle in ( r ,r)-coordinates is dr/dr. then its geometric speed relative to observers tixed with the FRW fluid (and hence also tixed relative to the radial coordinate r o f the FRW metric because the fluid is co-moving) is e q ~ ~ to alR$. Now consider the shock wave (4.3 1 ), which moves with speed ( c f .(4.33))
572
J. Grooh rr trl.
Then by (4.37), the speed of the shock s relative to the FRW fluid particles must be given by (4.35). A graph of s(a) is given in Figure 2, from which we conclude that the shock speed moves with a speed less that one relative to the FRW fluid if and only if D < 0 2 holds; and for 0 < 0 < 1 . s ( a ) = I if and only if a = a 2 holds, where numerical symbolic algebra gives 0 2 = &/3 0.745. This completes the proof of Lemma 1 I . We next determine when the Lax characteristic condition holds at the shock. To this end, we first determine the speed of the characteristics relative to the fixed FRW fluid particles. By (4.37), the characteristic speeds on the FRW side of the shock must equal the sound speeds && in the (t, ;)-coordinate frame, because the FRW fluid is co-moving with respect to the ( t , F)-coordinates. (The characteristic speed is obtained from the fluid speed and sound speed by the special relativistic summation formula for velocities [29j.) We conclude that the FRW characteristic speeds, h ~h:Rw ~ (the ~speeds, of the characteristics relative to the FRW fluid) are given, respectively, by the formula
Thus, since the ( I . I-)-coordinatesare also co-moving with the fluid. the sound waves in the ( t . r)-coordinates of the FRW metric must move at coordinate speed
+
We refer to the -, characteristics as being in the 1.2-characteristic families, respectively. Now in the one space-one time-dimensional theory of conservation laws, the Lax characteristic condition states that the characteristic curves in the family of the shock impinge upon the shock from both sides, while all other characteristic curves cross the shock, cf. 1271. Since in our example, the shock is outward-moving with respect to I- and 7. it follows that on the FRW side only the 2-characteristic can impinge on the shock. and thus we must identify the shock wave as a 2-shock. Thus the Lax characteristic condition must be interpreted meaning that the following inequalities hold: s c
h:Kw.
and
(4.40)
refers ,, to the speed of the faster characteristic on the TOV side of the shock Here i& as measured in the ( t , ?)-coordinate system, which is related to the (i.?)-coordinate system through the (t, r ) + (7, F ) coordinate identitication. By (4.35) and (4.39). (4.40) is equivalent to the condition
Solring tile Einstein equc~tionsby Lipschit; cotttinuous mrtrics: Shock wuvr.~in g~rtrrulrrlutivity
573
Fig. 3. A plot of the difference between the inner characteristic hpeed ~ F H Wand the ahock speed .s, ah functions of 0 .
A numerical plot of the function A(cr), given in Figure 3, shows that A ( a ) changes from positive to negative at the unique point a = a , , where
We are now ready to prove the following theorem:
Since (4.40) follow5 fi-om (4.42) and (4.43). the proof' ot' Theorem 1.7 will bc complete once we prove the following lemma which immediately implies (4.41 ).
The next theorem is another immediate consequence of Lemma 4.4:
Note that when a1 < a < 0 2 , (4.45). (4.46) describe a new kind of shock wave in which the 1- and 2-characteristics both cross the shock because the shock speed exceeds the characteristic speeds on both sides of the shock. This occurs even though the sound speed and shock speed all remain less than the speed of light. In words, Theorem 14 states that
574
J. Grouh et a1
a
can drive the shock speed all the way in General Relativity, a sound speed f i up to the speed of light. It remains only to give the proof of Lemma 4.4. Let u denote the velocity vector for the fluid on the TOV side of the shock, and let cr = 0, I refer to components in the ( I , F)coordinate frame, and i = 0, I to components in the ( t , r)-coordinate frame. Then a velocity vector tangent to the particle paths of the fluid on the TOV side of the shock is given by (LO, u- 1 ) = (1.0) in barred coordinates, because the fluid is co-moving relative to the barred coordinate system on the TOV side of the shock; for brevity we write ia = (1, O ) a . (Since our aim is to compute the characteristic speed, which is a ratio of two vector components, Then a tangent vector of any length will suffice.) Let x i = (r, r)' and F a = ( I ,
Thus the speed of the TOV fluid as measured in the FRW coordinates (r, r ) is given by
But, ilt =(?,F) ij r
1
= 7, $(t. r )
Since
and this holds in a neighborhood of the shock surface, we have
But by (4.30),
S o l t d n ~the Eitlstein equcrtiot~.tI)! Li/).schit: cont1nuou.s mrtrics: Shock wcir'r.s in g~rlc,rnlrc.lrrtivit?,
575
Thus by (4.37),
and this gives the TOV fluid speed in the locally Minkowskian frame which is fixed with the FRW fluid particles. But & is the sound speed for the TOV metric; thus & is the sound speed as measured in the frame obtained from the ( r , F ) coordinates by a Lorentz transformation for ii. Therefore, to obtain the TOV characteristic speed h:ov in the frame ( t ,J). we use the relativistic addition of velocities formula:
and this implies that
We now calculate
hLv.By (4.54). we have
where again we use (4.2 1 ) to eliminate 17 i n favor o f a . A numerical plot of h;ov(cr) vs. a is given in Figure 4. This verities that < 0 for 0 < cr < 1 . and thus completes the proof of Lemma 4.4 in light o f the inequality h ~ , , < h;,.
i&,v(cr)
Fig. 4. A plot olthe outer characterislic speed >isa I'unclion of n .
. I . Groah et a/.
576
4.5. Concluding remarks Note that these examples provide a theory of inherently strong shock waves because the condition p = 3; implies that [ p ] + 0 iff p + 0, the latter being a singular limit, cf. [27]. Note also that when k > 0, the FRW-TOV shock-wave solutions described in Section 3 reduce to the well-known model of Oppenheimer and Snyder (0-S) when j = 0. It is interesting to note, however, that the 0-S rnodel reduces roflut Minkowski spuce when M'P tuke k -+ 0 in the 0-S solution (see Weinberg, [42, p. 3441, Equations ( l 1.9.23) and (I 1.9.21)). Moreover, when we take 8 -+ 0 in our solution (4.28)-(4.31), we also get flat Minkowski space. However, the first limit is singular (because R = 0 implies R const when k = 0, cf. [42, p. 3441, Eqilation ( I 1.9.22));the second limit is the only way to impose 6 = 0. Indeed, we can obtain a rzebt9,time-reversible 0-S type contact discontinuity for the case k = 0 by noting first from (3.82) that ,5 = 0 = p implies p = 0, and thus we can integrate (4.16) and (4.19) in the case p = 0 to obtain the formulas
-
The shock surface is then given by
-
where M const when we assume empty space. ,ij = 0 = /,. We conclude that (4.58)(4.60) detine a non-trivial, time-reversible general relativistic model that corresponds to the exact shock wave solution given in (4.28)-(4.31). and thus detine a new 0-S type model o f gravitational collapse, cf. 142, p. 345 1, Equation ( I I .Y.25). We note also that once values for D and 5 = H ( a ) are specified, the formulas (4.28)(4.3 I ) determine a ~lnicl~tc shock wave solution despite the appearance of two free parameters. say Ro and &. To see this, note that after tixing the shock position Fo. the freedom in RO is only a coordinate freedom due to the fact that K(t) + cr-' R ( t ) under the coordinate rescaling r. + cur in the FRW metric (4.10) when k = 0.
5. A shock-wave formulation of the Einstein equations 5.1. Introduction In this section we show that Einstein equations ( I .25)-( 1.28) are weakly equivalent to the system of conservation laws with time-dependent sources ( 1.48), (1.49), so long as the metric is in the smoothness class c".',and T is in Lm. Inspection of Equations (1.25)-(1.28) shows that it is in general not possible to have metrics smoother than Lipschitz continat shocks), when the metric is written in the standard uous (that is, smoother than c~'.'
Solvit~fithr Eitlstrin equtrtiotls I)? Li/).sc.llit: c.orrtinuous mrtrics: Shock wLivrs it1 firrlrrrrl rr1utivit.y
577
gauge. Indeed, at a shock wave where T is discontinuous, A , , B , , and Bt all have jump discontinuities. As stated in Section I, a space-time metric g is said to be spherically symmetric if it takes the general form [42,4 I, l 1,221,
-
-
where the components A , B . C, and D of the metric are assumed to be functions of the radial and time coordinates r and t alone, df12 do2 sin2(6)d@2denotes the line element on the 2-sphere, and x (xO,. . . , x 3 ) = ( t , r , 0 , @), denotes the underlying coordinate system on space-time. In this case we assume that the 4-velocity w is radial, by which we mean that the .r-components of w are given by
+
respectively. for some functions 111" and u l ' . Now in Section I we showed that, in general, there always exists a coordinate transformation (I., t ) + ( i .i) that takes an arbitrary metric of form (5.1) over to one of the t o r ~ n142 1,
A metric of form (5.3) is said to be in the standard Schwarzschild coordinates (or standard coordinate gauge). and it is our purpose here to establish the weak formulation of the Einstein equations (1.25)-( 1.28) for ~netricso f the torn1 (5.3) in the case when A and H are ti nite and satisfy A B # 0 . In Sections 2 and 3 we introduce and verify the equivalence of several weak formulations of the Einstein equations that allow for shock waves. and that are valid for metrics ot'form (5.3), in the smoothness class c'~'.'.In Section 4, we show thi~tthese equations are weakly equivalent to the system ( 1.48). ( 1.49) of conservation laws with time-dependent sources. This is the starting point for the existence theory set out in I101.
In this section we study the system of equations obtained from the Einstein equations under the assumption that the space-time metric g is spherically symmetric. So assume that the gravitational metric g is of the form (5.3), and to start, assume that T'' is any arbitrary stress tensor. To obtain the equations fvr the metric components A and B implied by the Einstein equations (1. IS), plug the ansatz (5.3) into the Einstein equations (1.15). The resulting system of Equations ( 1.25)-(I .28) is obtained using MAPLE. Equations (I .25)(1.28) represent the (0, O), (0. I ), ( I , I ) , and ( 2 . 2 ) componentsof G'; = K T ' ; . respectively (as indexed by T on the RHS of each equation). The (3,3) equation is a multiple of the
578
J. Grotrh pt (11.
(2,2) equation, and all remaining components are identically zero. (Note that MAPLE defines the curvature tensor to be minus one times the curvature tensor defined in (1.14).) We are interested in solutions of (1.25)-(1.28) in the case when shock waves are present. Since A and B have discontinuous derivatives when shock waves are present, it follows that (1.28), being of second order, cannot hold classically, and thus Equation (1.28) must be taken in the weak sense, that is, in the sense of the theory of distributions. To get the weak formulation of (1.28), multiply through by A B~ to clear away the coefficients of the highest (second) order derivatives, then multiply through by a test function and integrate the highest order derivatives once by parts. It follows that if the test function is in the class c(;.'(that is, one continuous derivative that is Lipschitz continuous, the subscript zero denoting cornpact support), and if the metric components A and B are in the class c O . ' , and ~ ' is jin class L m , then all terms in the integrand of the resulting integratedexpression are at most discontinuous, and so all derivatives make sense in the classical pointwise a.e. sense. In order to account for initial and boundary conditions in the weak formulation, i t is standard to take the test function q5 to be non-zero at t = 0 or at the specified boundary. In this case, when we integrate by parts to obtain the weak formulation, the boundary integrals are non-vanishing. and their inclusion in the weak formulation represents the condition that the boundary values are taken on in the weak sense. Thus, for example, if the boundary is r = r-0 3 0 . we say q5 E c(:"( r 3 r ~t .2 0) to indicate that q5 can be non-zero initially and at the boundary r = 1.0, thereby implicitly indicating that boundary integrals will appear in the weak fortnulation based on such test functions. We presently consider various equivalent weak fi)rmulutions of Equations ( 1.25)-( 1.28). and we wish to include the equivalence of the weak formulation of boundary conditions in the discussion. Thus, in order to keep things as simple as possible. we now rc>.vtr-ic.1to the case of weak solutions of (1.25)-( 1.28) detined on the domain r- 3 r~ 3 0 . t 3 0. and we always assurne that test functions q5 lie in the space q5 E c(:"( t 3 0. r- 3 ro) so that initial and boundary values are accounted for in the weak formulation. (This is the simplest case rigorously demonstrating the equivalence of several weak for~nulationsof initial boundary value probletns. More general domains can be handled in a simil~umanner.) Note that because ( 1.25)-( 1.27) involve only first derivatives of A and B , and A . B E c O . ' , it follows that ( 1.25)-( 1.27) can be taken i n the strong sense. that is. derivatives can be taken in the pointwise a.e. sense. The continuity of A and B imply also that the initial and boundary values are taken on strongly in any c".' weak solution of ( 1.25)-( 1.27). On the other hand, Equation (1.28) involves second derivatives. and so this last equation is the only one that requires a weak formulation. The weak formulation of (I .2X) is thus obtained on domain r 3 0. r 3 ro 3 0, by multiplying through by a test function q5 E c,;.' (r 2 ro, t 3 0) and integrating by parts. This yields the fi>llowingweak formulation of ( I .2X):
S o l v i n ~ihr Eitr.siri~~ rquution.~by Lil~schii:c o t ~ i i r ~ u otnrrrics: u.~ Shock wuvrs in gmrrul rrlutivifv
+ -@rA4B2 + T2Kr4 T
579
drdr
Our first proposition states that the weak formulation (5.4) of Equation (1.28) may be replaced by the weak formulation of the conservation laws div T = 0, so long as A and B are in and T'J E La.
cO*
PROPOSITION 4. Assume thut A, B E c'.' ( r 3 ro, t 3 O), TIJ E Lm ( r rO, t 2 0) ~ l n dthut A . B , L U Z ~T solve ( 1.25)-( 1.27) strongly. Then A , B , and T solve T !: = 0 (the 1 -component of'Div T = 0) ~ve~lklv i f ~ l n dotzly $ A , B, and T .suti.sfy (5.4). PROOF.The proof strategy is to modify (5.4) and the weak form of conservation using (1.25)-(1.27) as identities, and then observe that the two are identical at an intermediate stage. To begin, substitute for B, and A' in several places in (5.4) to obtain the equivalent condition
+-r AIR 2 8 + %T2']} B
drdt
Now, the weak form of conservation of energy-momentum is given by
Here, we have used the fact that T2* = sin*^^^^, TiJ = 0 if i # j = 2 or 3, and f& = sin2 Q f!?. -- Next, we calculate the connection coefficients f : k using (1.10) to obtain,
Substituting the above formulas for into (5.6) and using ( 1.25)-( 1.27) as identities to eliminate some of the Ti.; in favor of cxpressions involving A. B. and r . we see that (5.6) is equivalent to:
~
=
{
T
o
~
[ f ( :~ T~ K ~
+
381)
+ - +-B -
BI rAB2
-
2 B' A' -
r-Bf + ( B
B
-
I)) + 2 ~ ~ T * * ] ) d r d t
After some simplification, it is clear that (5.5) is equal to (5.8). This completes the proof of Proposition 4.
Solvit~gthe Einstein equations bv Lipschitz continuous metrics: Shock waves in general relurivity
58 1
We next show that the Einstein equations (1.25)-(1.27), together with Div T = 0, are overdetermined. Indeed, we show that for weak solutions with Lipschitz continuous metric, either (1.25) or (1.26) may be dropped in the sense that the dropped equation will reduce to an identity on any solution of the remaining equations, so long as the dropped equation is satisfied by either the initial or boundary data, as appropriate. The following proposition addresses the first case, namely, for weak solutions in which the metric is Lipschitz continuous, the first Einstein equation (1.25) reduces to an identity on solutions of (1.26), (1.27), so long as (1.25) is satisfied by the initial data. T H E O R E M15. Assume that A , B E c'.' and T E Lm solve ( 1.26), (1.27) strongly, and solve Div T = 0 weakly. Then if A , B , and T sutisfy (1.25) at t = 0, then A , B , and T also solve (I .25)for all t > 0. PROOF. We first give the proof for the case when A , B, and T are assumed to be classical smooth solutions of (1.26), (1.27). and Div T = 0. This is followed by several lemmas necessary for the extension of this to the weak formulation, which is given in the final proposition. S o to start, assume that A , B, and T are all smooth functions, and thus solve Div T = O strongly. For the proof in this case, define
Because (1.26) and (1.27) hold, H O '
- H"
0 . Since by assumption T I : = 0 and since
G'" = 0 for any metric tensor as a consequence of the Bianchi identities. it follows that
In particular, setting j = 0,
By hypothesis, HiO= 0 when i # 0. In addition, the connection coefficients unless i o r k equal O or I. Therefore, (5.1 I) reduces to the linear ODE
5:'are zero
at each fixed r . By hypothesis, HO" is initially zero, and since we assume that HO" is a smooth solution of (5.12), it follows that HO" must be zero for all t > 0. Next, assume only that A , B E cO.' and T E LC" so that (1.26). ( 1.27) hold strongly (that is. i n a pointwise a.e. sense), but that Div T = 0 is only known to hold weakly. In this case, the argument above has a problem because when g E c O . ' , the Einstein tensor G, viewed as a second-order operator on the metric components A and B. can only be defined weakly when A and B are only Lipschitz continuous. It follows that the Bianchi identities, and hence the identity Div G = O (which involves first order derivatives of the components
J. Groah et al.
582
of the curvature tensor), need no longer be valid even in a weak sense. Indeed, G can have delta-function sources at an interface at which the metric is only Lipschitz continuous, cf. [29]. However, the above argument involves only the 0th component of Div G = 0, and the 0th component of Div G = 0 involves only derivatives of the components G" and, as observed in (1.25), (1.26), these components involve only thejrst derivatives of A and B. Specifically, the weak formulation of GPi = 0 is given by
and since, by (1.25), (1.26), G"' involves only first order derivatives of A and B, it follows that the integrand in (5.13) is a classical function defined pointwise a.e. when A , B E c'.'. But (5.13) is identically zero for all smooth A and B because Div G = 0 is an identity. Thus, when A , B E c~',', we can take a sequence of smooth functions A,, B, that converge to A and B in the limit E + 0 (cf. Theorem 16 below), such that the derivatives converge a.e. to the derivatives of A and B. It follows that we can take the limit E + 0 (5.13) and conclude that (5.13) continues to hold under this limit. Putting this together with the fact that Div T = 0 is assumed to hold weakly, we conclude that
in the weak sense, which means that HO" is in Lw and satisfies the condition
Therefore, to complete the proof of Theorem 15, we need only to show that if A , B, and T solve (1.26), (1.27) classically, and Div T = 0 weakly, then a weak LDOsolution HO" (i.e.. that satisfies (5.14))of (5.12) must be zero almost everywhere if it is zero initially. Thus i t suffices to prove the following proposition. PROPOSITION 5 . Assurne that H, ,f initial value l~rohlem
with initial dutu Ho
-
E
Lzc(R x
R). Tlzerz every LCc ~,veaksolutiorz to the
O is unique, and identiccilly equal to zero (1.e..,fi)r all t > 0.
PROOF.We use the following standard theorem 161.
Solving the Einstein equations by Lipschifz continuous metrics: Shock waves in general relativity
583
THEOREM 16. Let U be any open subset of Rn. Then u E W~:?(U) i f a n d only i f u is locally Lipschitz continuous in U , in which case the weak derivative of u agrees with the classical pointwise a.e. derivative as a function in L g c ( U ) . COROLLARY7 . Let u and f be real valued functions, u , f : R + R, such that u , f Lm[O, T I , and u is a weak solution of the initial value problem
on the intewal [O. T I . Then u ( t ) = O f i r all t
E
E
10, T I .
PROOF O F C O R O L L A R YStatement . (5.16) says that the distributional derivative u , agrees with the Lm function f u on the interval [ 0 ,T I , and thus we know that u E W,;;:(O, t ) . Therefore,by Theorem 16, u is locally Lipschitz continuous on ( 0 , T ) , and the weak derivative u , agrees with the pointwise a.e. derivative o f u on ( 0 , T ) . Thus it follows from (5.16) that on any subinterval [ a ,hj o f 10, TI on which u # 0 , we must have
Moreover, since I ( is Lipschitz continuous. both u and In(u) are absolutely continuous on h l . so we can integrate (5.17)to see that
[ti,
for all t E la. hl. But u is continuous, so (5.18) applies in the limit where ci decreases to the first value o f t = to at which u ( t o )= 0 . Thus (5.18) implies that u ( t ) = 0 throughout [ a .h l . and hence we must have u ( t )= O for all t E [O. T I . and the corollary is proved. The proof o f Proposition 5 now follows because i t is easy to show that i f H is an LW weak solution of (5.15).then H ( x . .) is a weak solution o f the scalar ODE H, ,f'H = 0 for almost every x . (Just factor the test functions into products of the form 4 , (t)$?(.r).) Using Proposition 5 , we see that i f Equation (1.25)holds on the initial data for a solution o f (1.26), (1.27), and Div T = 0 , then Equation (1.25) will hold for all I . By a similar argument, it follows that i f (1.26) holds for the boundary data o f a solution to (1.25), (1.27),and Div T = 0 , then (1.26)will hold for all r and t . W e record this in the following theorem:
+
THEOREM 17. A.s.sume that A , B E cO.land T E Lw solve (1.25), (1.27) strongly, and solve Div T = 0 weakly, in r 3 ro, t 3 0. Then $ A . B. and T sati.sfy ( I .26) at r = ro. then A, B , and T also solve ( l.25),fur all r > ro.
584
J. Groah rt
a1
5.3. The spherically symmetric Einstein equationsformulated as a system of hyperbolic conservation laws with sources Conservation of energy and momentum is expressed by the equations
which, in the case of spherical symmetry, can be written as the system of two equations:
Substituting the expressions (5.7) for the connection coefficients ( I . 10) into (5.19) and (5.20), gives the equivalent system
+ -I ( A- ' + 2,'+ 2 A
,)
T 1 , + - A' ~ " " - 2 r ~ - -7 3 2B B
Now if one could use equations to eliminate the derivative terms A , . A', B,. and B' in (5.21) and (5.22) in favor of expressions involving the undifferentiated unknowns A , B, and T , then system (5.21), (5.22) would take the form of a system of conservation laws with source terms. Indeed. To" and To' serve as the conserved quantities, T and T I ' are the Ruxes. and what is left, written as a function of the undifferentiated variables ( A . B, To". To' ). would play the role of a source term. (For example, in a fractional step scheme designed to simulate the initial value problem, the variables A and B could be "updated" to time t,, A t by the supplemental equations (1.25) and ( 1.27) or (1 2 6 ) and (1.27) after the conservation law step is implemented using the known values of A and B at time 1 , . The authors will carry this out in detail in a subsequent paper.) The system then closes once one writes T I ' as a function of ( A . B. To", To'). There is a problem here, however. Equations (1.25)-(I .27) can be used to eliminate the terms A,.. B,, and B,., but (5.21) and (5.22) also contain terms involving A , . a quantity that is not given in the initial data and is not directly evolved by Equations ( 1.25)-( 1.27). The way to resolve this is to incorporate the A, term into the conserved quantities. For general equations involving A , . this is not possible. A natural change of T variables that eliminates the A, terms from (5.2 I), (5.22) is to write the equations in terms of the values that T takes in flat Minkowski
"'
+
Solving thr Ein.ttrin c,qucitions hy Lil~sclrit:cotrtirruou.~rnrtrics: Slrock w0vr.s in grnrrul relcrtivity
585
space. That is, define TM in terms of T , by
where the subscript denotes Minkowski, cf. (1.32)-(1.34). It then follows that TM is given by
where u denotes the fluid speed as measured by an inertial observer fixed with respect to the radial coordinate r, cf. ( 1 35)-(1.37). (We discuss (5.24) in more detail in the last section.) Substituting (5.23) into (5.2 1 ), (5.22). the A, terms cancel out, and we obtain the system
The following proposition states that system (5.25). (5.26) is equivalent (in the weak sense) to the original system Div T = 0. P R O P O S I T I O6N . If A (rnd B trre ~ i v o r lLil~schit,:continuou.~ furic.tiot1.s ~Iejinc~I on the domcrin r 3 ro, t 3 0 , rhrri TM i.r tr weuk .\.oIution of (5.25) id (5.26) jf'trrzd o n b jf T is N rt3euk~ o l u t i o of'Div t~ T = O in this do~n(rirl. PROOF. For simplicity, and without loss of generality, take the weak formulation with test functions compactly supported in r > ro, t > 0. so that the boundary integrals do not appear in the weak formulations. (Managing the boundary integrals is straightforward.)
The variables T i solve (5.25)weakly i f
Set I/, = Acp, whereby Acp, =
=
{
-
T
+, ++. Using this change o f test function, (5.27)becomes -
-
T 01
+' + [;
-
('A"' + ;) -
-
T
00
which is the weak formulation o f (5.2 1 ). W e deduce that TM solves (5.25) for every Lipschitz continuous test function cp i f and only i f T solves (5.28)(the weak form of T'.): = 0) for all Lipschitz continuous test functions That weak solutions of (5.26)are weak solutions o f T ' ; = O follows by a similar argument.
+.
It is now possible to use Equations ( I .25)-(1.27) as identities to substitute for derivatives o f metric components A and B , thereby eliminating the corresponding derivatives o f A and B from the source terms o f Equations (5.25).(5.26). Doing this. we obtain the system o f Equations (1.46), (1.47), which was our goal. However, depending on the choice o f equation to drop, either ( 1.25) or ( 1.26), it is not clear that i f we use the dropped equation to substitute for derivatives in (5.25), (5.26). that the resulting system o f equations will imply that Div T = 0 continues to hold, the assumption we based the substitution on in the first place. The following theorem states that (1.46). (1.47) is equivalent to Div T = 0 in the weak sense:
S o l i i ~ ~tile g Eirlstrir~rq~rtrtiotr.cI?! Lil)sc.lrit: c~o~rtinuous metric..\: S17oc.X w~ri~r.s ill grtrrrul rrlativitv
5x7
T H E O R E M18. Assume tllrlt A , B are Lipschit? continuou.~ functinn.~,and that T E LIW, on the domain r 3 ro, t 3 0. Asattne also rhnt (1.25) holds ut t = 0, and thut (I .26) hold.^ ut r = ro. Tl~ewA , B , T are rvenk solutiot~sc$ (1.25), (1.26), (1.27), und Div T = 0 $and only ~fA . B,TM are rcqeak solutiotls ($either system ( 1.25), ( 1.27), ( 1.46), ( 1.47), o r system (1.26), (1.27), (1.46), (1.47). PROOF. Without loss of generality, we consider the case when we drop equation (1.26), and use (1.25), (1 27). and Div T = 0 to evolve the metric, and we ask whether we can take the modified system (1.46) and (I .47) in place of Div T = 0. In this case, we must justify the use of ( 1 2 6 ) in eliminating the B, ternis in going from Div T = 0 to system (1.46) and (1.47). That is, i t remains only to show that Equations (1.25) and (1.27), together with system (1.46) and (1.47). imply that (1.26) holds, assuming (1.26) holds at r = ro. (If so, then by substitution. it then follows that Div T = 0 also holds.) Note that we can almost reconstruct (5.2 1 ), the first component of Div T = 0, by reverse substituting ( 1.25), ( 1.27) into ( 1.46). To see this, first note that we can add ( 1.25) and ( 1.27) to obtain
Equation (5.29) is an identity that we may add to (1.46) to obtain
Adding and subtracting
to and from the RHS of (5.30). and using
(cf. (1.26) and (5.9)).we have
Note that all but the last term on the RHS of (5.33) are equal to the first component of Div T , and so
Therefore, if A , B, and TM are solutions to (1.25), (1.27), (5.33), and (5.22), it follows that
because G':. = 0 is an identity. But HoO= 0 holds because we assume (1.25), and hence (5.34) implies that
where f
= qil+ 2 f d l -KT
r = r ~ it. follows from
,.fi?TIl
E Lm. Since we assume that H"'= 0 on the boundary Corollary 7 that H"' = 0.
I t remains to identify conditions under which
TL' is a function of (T;),
T i 1 )assuming
tIi;~t T has the forrn of a stress tensor for a perfect fluid, (5.24).A calculation shows that,
in this case, the following simplifications occur:
Using (5.35)and (5.36) we see that only the first terms on the RHS of ( 1.46).( 1.47)depend on v. and the only term that is not linear in p and p is the third term on the RHS of (1.47). We state and prove the following theorem: THEOREM 19. A.\.\urnrthat 0 < 1, < pc2, O < d p l d p < c7. Thrrl i s tr,fitrlc.tior~ r?f.T:) trnd TL1 so long 0.5 ( p , u ) lie in the tlornain D = { ( p .1 1 ) : O < p. 1 1 1 1 < c.]. PRUOF. We may write (5.35) and (5.36) in the form
Since d,fll d p = c2 - p' 3 c7 - a 2 > 0, it follows that the function f'l is one-to-one with respect to p . Also, df'ldp = l)'pc2 pc2 3 /7(.' > 0. so the function ,f- is also one-to-one in p . Consequently, the function h = f 2 . f'll is one-to-one. and thus
+
Solvirrg the Einstritt rqrrcrtions t)y Lipsc.hit: c.o~~tir~uou.s metrics: Shock wtrvrs it1 grtlrrul rc.1utivit.y
Now introduce the linear and invertible change of variables x = T:) - T:, z= whereby (5.39) becomes
TL',
589
01, y = TM
Equation (5.40) is quadratic in z , and so we may solve it directly, obtaining
From (5.41), we conclude that for any (x, y) there are two values of Z , though only one of them will correspond to values of p and v in the domain D. That is, since
and := T / > 0, it follows that there is at most one solution of (5.41) in the domain D, namely
We conclude that if ( p . u ) lies in the domain 0 , then for each value of T:' exists precisely one value of T;' .
A calculation shows that in the case of (T:), T i ' ) is given by
17
and T i ' there
= u Z p .0 = const. the formula for
n
Ti'
in terms
where
Our results concerning the weak formulation of the Einstein equations (1.25)-( 1.28) assuming spherical symmetry given in Theorem 18 can be summarized as follows. Assume that A , B are Lipschitz continuous functions, and that T E Lm. on the domain r 3 ro, t 3 0. Then (1.25)-(1.28) are equivalent to two different systems which take the form of a system of conservation laws with source terms. I n the first case, we have shown
that weak solutions of the system (1.25), (1.27), together with Equations (5.25), (5.26) (for Div T = O), will solve (1.25)-(1.28) weakly, so long as (1.26) holds at r = rg. This reduces the Einstein equations with spherical symmetry to a system of equations of the general form 11,
+f
(lr.
A . B), = j ( 1 4 , A , B, A', B,, B', x ) ,
(5.46)
(TP.
T:;) agree with the conserved quantities that appear in the conservation where u = law div TM = 0 in flat Minkowski space. (Here "prime" denotes i ) / a x since we are using x in place of r.) It is then valid to use Equations (1.25)-(1.27) to eliminate all derivatives of A and B from the RHS of system (5.46). by which we obtain the system ( 1.25), (1.27), (1.46), (1.47). a system that closes to make a nonlinear system of conservation laws with source terms, taking the general form
which reproduces ( 1.48). ( 1.49) of Section I . Weak solutions of (5.49) will satisfy ( 1.26) so long as (1.26)is satistied on the boundary I- = 1-0. In the second case. we have shown that weak solutions of the system (1.26). (1.27). together with Equations (5.25). (5.26) (for Div T = 0).will solve ( 1.35)-( 1.28) weakly. so long as (1.25) holds at t = 0.This reduces the Einstein equations with spherical symmetry to an alternative system of equations of the general form
A , = hO(u,A . B. .r). I B, =/I,(((,
A. R,x).
I t is then valid to use Equations ( I .25)-(I .27) to eliminate all derivatives of A and B from the RHS of system (5.50). by which we obtain the system ( 1.26). ( 1.27). ( 1.46). ( 1.47). a system that closes to make a nonlinear system of conservation laws with source terms, taking the general form
Weak solutions of (5.53) will satisfy (1.25) so long as (1.25) is satisfied at t = 0.
Solving the Einstein equations hv Lipschiti continuous metrics: Shock waves in generul relativity
59 1
5.5. Wave speeds In this section we conclude by calculating the wave speeds associated with system (1.46), ( 1.47).
The easiest way to calculate the wave speeds for the fluid is from the Rankine-Hugoniot jump conditions in the limit as the shock strength tends to zero. To start, note that the components of the 4-velocity w for a spherically symmetric fluid (1.18) are w0 = dtlds, w L= d r l d s , w2 = w 3 = 0. Since - 1 = g ( w , w ) , the components wo and w ' are not inde~ We define fluid speed u as the speed pendent and, in particular, - I = - ( w ' ) ~ A + ( w ' )B. measured by an observer fixed in ( t , r ) coordinates. That is, the speed is the change in distance per change in time as measured in an orthonormal frame with timelike vector parallel to a, and spacelike vector parallel to a,. It follows that the speed is given by v = x / u , where
Taking the inner product of w with ill and then with il,, we tind that u = wOx = w' and hence
m,
and
whereby,
Using (5.57) and (5.58) in ( 1 . I s ) , it follows that the components of the energy-momentum tensor take the following simplified form, which is valid globally in the ( t , r)-coordinate system:
T 0 1 =-
I
+
('U
(I? PC2) ( . 2 - ,)2 '
Note that these components are equal to the components of the stress tensor in flat Minkowski space times factors involving A and B that account for the fact that the spacetime is not flat. Using ( 1.32)-(1.34) we can write the Rankine-Hugoniot jump conditions in the form
From (5.59), (5.60), we deduce that wave speeds for the system (1.46), (1.47) are times the wave speeds in the Minkowski metric case, and this holds globally throughout the ( I , r)-coordinate system. (See [28].) Eliminating s from (5.59) and (5.60), yields
, and the right fluid state to Now take the left fluid state on a shock curve to be ( p ~UL), be (p, v). For a spherically symmetric perfect fluid, (5.61) defines the right velocity v as a function of the right density p . Then to obtain the fluid wave speeds, just substitute this . this procedure, function into (5.59), solve for s, and take the limit as p -+ p ~ Following (5.6 I ) si~nplifiesto
Note that Equation (5.62) can be written as a quadratic in v, iind hence there are two solutions. The '+' solutions will yield the 2-shocks, and the '-' the I -shock. Dividing both sides of (5.62) by ( p - pr,)' and taking the limit as p + p,., we see that
Solving (5.60) for s we obtain,
and taking the limit as p + p ~we , obtain
(Here the plus-minus on RHS is determined by the two possible signs of v' = civldp as allowed by (5.63).)After substituting for dvldp using (5.63). and simplifying, we obtain
Sol\.ir~gthe Einstein ec/iccrtions by Lipschit: c.o~ltinuousmrtric.~:Shock wtrves in gettrrul relurivitv
593
+ p'] B [up1f (c* + u 2 )J7 + c 2 v ] [v' f 2 v J 7
T h i s gives, PROPOSITION 7 . The r t w r speed.^ lsftlle genrrul rrlutivistic Euler equutions (5.49) ure
T h e following proposition demonstrates that the system ( 1.46), ( 1.47) is strictly hyperbolic whenever the particles are moving at less than the speed o f light:
PROOF. To determine where the wave speeds are equal. set h equal to h+ and solve for 1, to obtain 11' = c 2 . Next, substitute v = 0 into h- and h+ to verify that h < A+ when v' < C ? A / B .Proposition 8 follows directly. A s a final comment, we note that Proposition 8 is true because it is true in a locally inertial coordinate system centered at any point P in space-time. Indeed, in such a coordinate system, the connection coefticients vanish at P , and the metric components match those o f the Minkowski metric to tirst order in a neighborhood o f P . A s a result. the general relativistic Euler equations reduce to the classical relativistic Euler equations at P . Since i t is known in Special Relativity that the Euler equations are strictly hyperbolic for timelike particles, 1281, it follows that the same must be true in General Relativity. Other pointwise properties, such as genuine nonlinearity and the Lax entropy inequalities, 127.15 1, can be verified for the spherically symmetric general relativistic equations in a similar manner. Because A and B enter as undifferentiated source terms, it follows from ( 1.46). ( 1.47) that for spherically symmetric How, the only wave speeds in the problem will be the characteristic speeds for the fluid. Loosely speaking, the gravitational field is "dragged along" passively by the fluid when spherical symmetry is imposed. From this we conclude that there is no lightlike propagation (that is, no gravity waves) in spherical symmetry, even when there is a matter present. For the empty space equations, this is the conclusion o f Birkoff's theorem, 1421.
References (I
I
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Soh.ing the Einstein e,qucitiorrs by Lil).sc,lrit: c,o~rtinuou.smrtrics: Shoe,&wrrvr.~in grrrrrul relurivitv
595
[3 1 ] J. Smoller and B. Temple, Ge~rc~rrrl rc~lertii~i.stic~ shock wave.s that extend the Op/)~nh~ir?rc,r-Snyde,rmodrl, Arch. Rat. Mech. Anal. 138 (I997), 239-277. 132 1 J . Srnoller and B. Temple. S/toc.k-wm~c,.sollrtio~rcirr c~lo.sc,elfivm crnd rlre Ol)l>c~rrh~irrrer-S~~y(Ie~r limit in Grric,rcrl Reluti\i?,, with J . Srnoller. SlAM J. Appl. Math. 58 ( I ) (1988). 15-33. (331 J. Srnoller and B. Temple. 0 1 1 the O~)/~errl~eimrr-Volkov c~yuotiorr.~ in Genc,rtrl Rrltrtii~i!r.,with J . Srnoller. Arch. Rat. Mech. Anal. 142 (1998). 177-191. (341 J. Srnoller and B. Temple, So1~rtio11.s ofthe Ol~~~er~hrir~rrr-V,~Ik~~ff'equutio~r.s irrsirlr Y/X 'ths of'tlic, Sc.hwtrr,-.sc~lrildrtrdires, with J . Smoller. Comm. Math. Phys. 184 (1997). 597-617. (351 J . Smoller and B. Temple, Co.\rnolo,q~it.ith (1 .thocA bvtrvr. Comm. Math. P h y ~210 (2000). 275-308. (361 J. Smoller and B. Temple. Slroc.k-~~rt~r .solrrtiorr.s of' the Eirr.stc,irt equeition.~:A gp)rprul theory u+th c.xcrrnl)/c~s.Proceedings o f European Union Rehearch Network's 3rd Annual Summerschool. Lamhrecht (Pt'alz), Germany. May 16-22 ( 1999). to appear. (371 A. T ~ u hAl)/)ro.rii~rotr . Solrrtio~i.\ofthc, Ei~rstr~irr ecl~rrrtio~i.s,fi)r i.snrt,r)l)ic.~notiotr.~ ofplrrrrc~-.sy~ri~~ic~tri(~ tlistrihutiorr.~ofpe~;fi(.r fl~rie/.\.Phyh. Rev. 107 (3) (1957). 884-900. 138I K. Thompson. The .cpe~.icrlrc.l(rtii,i.\ticslrock trrhe. J . Fluid Mech. 171 ( 1 9X6), 365-375. 1 391 R. Tolrnan. Rrlertii,ity, T/rc~r~r~o(lyr~cir~~ic~.~ crrirl Co.srrrology. Oxford Univer4ty Pre\s. Oxford ( 1934). (301 R. Tolrnan. Strrtic. .solrrriorr.c (!f'Eirr.stc~irr'.s fic,lcl ~e/rrrr~io~rs.fi)r .s/)/IP~~,s of'fluirl. Phys. Rev. 55 ( 1939). 364-374. 141 I R.M. Wald. Gc,rrrrol Hrlcrtii~ity.University of Chicago P r e ( 1984). 142 1 S, Weinherg. (;r(/i,itl~tiorr(11re1 Co\r~roIo,qy:Prirrc.~l)Ic,.\rrrrcl A ~ ~ ~ ~ I i ~ ~(!/'I/Io ~ i t i Ge,rrc,rrrI o r r . ~ Tlr(,o~;r~ ~ f ' H ~ ~ / r i t i ~ ~ i t y . Wiley. New York ( 1972).
This Page Intentionally Left Blank
Author Index Roman nunibers refer t o pages on which the author (or hislher work) is mentioned. Italic numbers refer to reference pages. Numbers between brackets are the reference numbers. No distinction is made between first and co-author(s).
Aberpel. F. 203. 205. 214 1 1 1 ah rut now it^. M . 304. .14S 1 11: 360. 434 1 1 I Ahrashkin. A.A. 295. 305. .148 121: 348 131 Acrivw. A. 482.495 13 1 1 Adelmeyer. M . 46 I, 462. 466.470. 475. 48 I. 496 162) Akhay. 11. 269. 282 1 I I A k y l a . T.R. 470. 473.482. 494 1 1 1: 499 1124): 490 [I251 Al-Muhn~yedh.U.A. 258. 282 121 Alcr~lany.A. 390. 403. 4.10 1 157 1 AlI'v&i. H. 418.4.33 121 Allen. H.R. 313. .340 130) Anlick. C. 446. 466. 483. 402404, 494 121: 494 131: 494 141: 404 1.51: 404 161: 494 (71: 494 1x1: 494 (01: 494 ( 101 Andcrcck, C.D. 240.257.282 131: 282 141 Ando. H. 340. .15.< 1 168 I Anilc. A.M. 594 1 l l Anttrnov. V.A. 3 1. 51 1 l 1 Anufriyev. A.P. 397.4.14 131 Arino. 0. 220. 282 1.5 1 Arnold. V.I. 24. 5 1 121: 57. 59. 71. 8.5 1 1 ) : 85 12): I 19. 128. 1.19 I I I: 7-08. 214 121: 201. 2'18. 308. 320. 331. 339. 348 141: .348 151: .348 161: .148 171: 372. 392. 423. 424. 4.14 141: 4-11 151: 4.34 161: 434 (71; 462.494 1 l l I Arsencv. A. 26. .5/ 13 1 Auhin. J.-P. 50. 52 141: 77. 8.5 131 Aurell. E. 5.1 1.501: 427. 4.14 181 Avgousti. M . 258. 287 1155) Avrin. J.D. 212.214 131: 214 141
215 1 1 1 1 ; 215 1121: 215 I I R ] ; 215 114); 215 1151: 21.5 I 161; 215 1171: 215 IIXI: 215 I I 9 I : 215 1201: 215 121 1; 215 1221; 215 12.31; 215 1241; 215 12.51: 215 1261; 216 1271: 216 1281; 216 1291: 216 (30); 216 131 1; 216 1321: 216 1331 R:lckus. G. 35X. 380. 382. 404. 4.14 IL)l B;lggetr. J.S. 267. 268. 282 101; 28.7 I 1081 B;ii. K. 273. 282 171: 284 1731 Baigenr. S. 8.5 141 Balhus. S.A. 417. 4.34 1101 B;tIcli. P. 52 151 Ball. J.M. 187. 212. 716 1341 Balmlor~h.N.J. 427. 4.14 I I I I H;inncr, M.I.. 445. 495 1121 B;irdos. C. 26. .52 161: 33'). .{.IS IXI Rarnslcy. M . 171, 216 (351 Barrandon. M . 406. 49.5 1 13 1 Batlh6lcniy. E. 460. 497 1901 B;lrtIicler. P. 273. 28.3 1271 B;~sdcvant.C. 32. 52 171 B;l\\om. A.P. 391. 41 I. 420.434 1121: 4.14 113) Batchelor. G.K. 313. 315. .?JX ( 151: 366. 4.34 I 141 Bale\. P.W. 181. 216 1361 Baumert. B.M. 258. 282 (81 Boyly. B.J. 268. 2S7 101: 300. 3 13. 3 18. 328-33 1. .348 191: . M Y ~ I O ~ : . 3 4181 1]:.348 ll2]:.<48 1131: .I48 1141: 360, 419,520. 422. 423. 426. 527, 3.34 115): 434 I I h I : 4.34 1171: 4.34 1181 Benle. J.T. 129. 130. 1.19 121: 302. 332. 252 1 1291: 362. 569. 49.7 1141: 49.7 (151 Beck. R. 4 16.4.14 1 101 Belcnhny;~. 1,. 309. 310. 339. 348 1161 Bellour. H. 213. 216 1371 Beltrami. E. 302. 4.34 120) Ben-Artli. M . 146, 149, 152. 153. 158. I6 I. 162. 166 1 l 1; 166 121: 166 131: 166 [ 4 ] Ben;rrd. H. 225. 242. 246. 282 1 lOl
598 Benfatto, G. 5, 52 181 Benjamin, T.B. 240, 251,282 1111; 282 1121; 482, 495 [161 Benoit, J.-P. 314, 3491191 Benton, E.R. 391,434 [211; 434 [22] Beris, A.N. 258, 268, 269, 286 [1231; 287 11551; 287 [ 1561 Berk, H.L. 399, 440 [175] Bernstein, B. 269, 287 [160] Bertoin, J. 41,42, 44, 52 [9]; 52 [10] Betchov, R. 225,282 [13] Biagioni, H.A. 161,166 [5] Binney, J.J. 4, 52 [11] Black, W.B. 268,282 [141 Biackman, E.G. 407,435 [23] Blennerhassett, P.J. 273,282 115]: 282 [16] B loom, F. 213, 216 [37 ] Boberg, L. 266, 282 [I 7] Bogoyavlenskij, O. 303,348 [17] Boldrighini, C. 5, 52 [12] Bollerman, P. 239, 282 [ 181; 287 [ 1571 Bona, J. 446, 483,495 [17]: 495 [18] Bondarevsky, V.G. 216 1381 Bose, D.K. 483,495 [! 7] Boyd, W.G.C. 272, 273,284 162]: 284 164] Bragard, J. 247,282 [191 Bragg, S.I,. 3()2, 348118] Braginsky, S.l. 359, 378,403,404, 412,435 [24]: 435 1251:435 [26] Brandenburg, A. 360, 365,391,396, 4(}7, 4 I(), 411,416, 417,428-431,434 [ 19]: 4,r [27]: 435 1281:4,t5 129]; 435 1301; 435 1311; 435 1321:435 1331:436 1621:44011761 Brcfort, B. 216 1391 Brenier, Y. 62, 63, 65-68, 78, 81, 83, 85 151: 85 16l: 85 17l: 85 181:85 191:85 I lOl; 85 l l ! I: 85 [121: I01, I04, ! ! 5 [1]://5 [2]: 1/,5 13] Brezis, H. 80, 81,85 [13]: 158, 161, /66 16]; 166 [7] Bridges, T. 446, 485,491,495 [19]; 495 [2()]; 495 121] Brosa, U. 266, 282 [I 7] Brown, R.M. 214, 216 1401 Brown, T.M. 4()9, 4.t.5 [341 Brummell, N.H. 371,41 !, 424, 428,430, 431, 435 1351:441 12041 Bryant, P. 494, 495 1221 Buffoni, B. 473-475,495 1231; 495 [241 Bullard, E.C. 358, 380, 435 [361 Buryak, A.V. 473, 49,5 1251 Busse, F.H. 245,246, 249, 282 1201:282 121 I: 282 1221; 285 11041; 286 11451; 358, 381,383, 396, 41 !, 412,435 1371:435 [381:435 1391;
A uthor Index 435 I401; 435 141 l; 435 1421; 438 I109]; 43811171; 441 12201; 441 [2211 Butler, K.M. 266, 282 [23]
Buzano, E. 245,282 [24] Caffarelli, L. 78, 79, 85 [14]; 85 1151; 119, 121, 122, 139 [31 Caglioti, E. 4, 521131 Cambon, C. 314, 330, 349 i191; 351 1991; 351 I I001 Campbell, C.G. 417,435 [431 Cao, C. 213,216 1411 Capinski, M. 212, 216 1421; 216 1431 Carlen, E.A. 150, 152, 161-163, 166 181; 166 191 Carpio, A. 161, 163, 166 I !01 Carraro, L. 44, 52 1141; 52 1151 Cartan, H. 345,349 1201 Case, K.M. 310, 349 1211 Cattaneo, E 371,406, 407, 424, 427--431, 435 1351; 43,5 1441; 435 1451; 435 1461: 441 1206] Cazenave, T. 161, 166 171 Champneys, A. 470, 473-475,495 1231:495 1251: 495 1261:495 1271:49811211 Chandrasckhar, S. 52 1161: 225,249, 274, 282 125 I: 29 I, 307, 345,346, 349 [231:349 124 I Chang, H.C. 273,286 11431 Charru, F. 273, 283 127] Chavanis, P.H. 4, 7, 3()-32, 52 1171 Chemin, J.-Y. 85 1161 Chcn, K. 273,274, 282 171:282 1261:284 1731 Chcn, P. 391,435 1471 Chcn, S. 119, 134, I,t9 141 Chcn, X.-Y. 181,216 1441 Chcpyzhov, V.V, 184, 186, 192, 193, 198, 212, 216 1451:216 1461:216 1471:216 1481: 216 [49]: 216 1501 Chicone, C. 328, ,r 1251 Childrcss, S. 129, !,r 151: 360, 378, 381,392, 397,398,401,403,405, 41 I, 417, 419-424, 426, 427,431,432,434 1171:4,r l l8l: 4,r 1481:435 1491:4,r 1501:4,r [51l: 4,t5 1521:435 1531:4,r 1541" 436 1551: 4.t6 1561:4.r 171 I: 4,r 1951:44011941 Cho, J. 407,431,441 [2121 Chorin, A. 52 1181: 119, 123, 139 161: 145, 166 1111:171,2171511 Chossat, P. 171,217 1521; 225,235,253-256, 28,r [281; 297, 349 [261 Chow, S.N. 234, 283 1291 Christen, M. 383, 4,r 179] Christensen-Dalsgaard, J. 409, 435 1341 Christodouiou, D. 562, 594 121 Chueshov, I.D. 195,217 153]; 217 1541
Cifersons, A. 383.436 [791 Clarke. M . A . 274. 284 [77] Clever. R . M . 246.282 [2 1 1 Cloot. A. 247.283 1301 Clune, T. 25 I . 283 13 1 1 Clune. T.L. 410.41 I . 439 11-12];441 12041 Cockburn. 6. 195. 217 [55j Colinet. P. 249. 28.3 1321 Collet. P. 239. 283 1331 Colova\. P.W. 249. 282 [4I Connor. J.W. 3 19. -149 1271 Constantin. P. 45. 47. 52 [ 191: 90. 101. 112. 115 1-11: 119. 121. 122. 126. 128-131. 133. 139 171: 139 1x1; 140 (91: 140 1 101: 140 [I 11; I401 121; 140113): 14OIIJI; 140[151: 140[16): 140 1171; 140[IXI: I40 1191; 140 1201; 140 121 1: 140 1221: 140 1231; I40 1241; 145. 167 [ 121: 175. 175. 177. 192. 1%. 196. 198. 199. 207. 217 1561; 217 (57):217 1581; 217 1591: 217 1001 Cordoba. D. 12'). 140 1 171: I40 1251 Cottet. G.-H. 146. 167 1131 Couetlc. M. 25 I . 28.1 1341 Courant. R. 3 16. .340 128 I; 504 131 Cowling. T.G 358. 360. 376. 378.4.16 1571: 4.16 15x1 Cox. J.P. 340. .140 1271 Crnig. W. -146. 485. 403. 404. 49.5 1281: 40.5 1201: 40.5 (301 Craih. A.D.O. 313. 330. 331. .140 1201: .NO 1301: .{40 13 I 1: -1.50 1551 Criniinnlc. W.O. 225. 282 1 1 31: 3 I . .{JO13 l I Crow. M.C. 5.1 1441 Cullen. M. 8.5 1 17 1 Cutlal~d.N.J. 212. 216 1421: 216 1431 CvitanoviC. P. 427. 4.14 l l l l Dolermos. C.M. 184. 217 161 1 Ilalccki. Ju.1.. 231. 28.1 135) Davis. K.E. 482. 49.5 13 1 1 Dcgcn. M.M. 240. 2cS2 141 Degontl. P. 26. 52 Ihl Dcl,ucu, E.E. 410. 4.16 (501: 4.16 (001 Dclncnt'ev. S. 383. 436 1701 Denn. M.M. 268.28.5 1991 Derks. G. 49 1.49.5 1201 Dewar, R.1.. 3 19. .14Y 1321 Dia~nond.P.H. 406. 428. 4.17 1 1071 Dias. F;. 445. 460, 4 7 0 4 7 3 . 475. 476. 480. 482. 483. 49 1.494 1 11: 49.5 1321: 405 1331: 49.5 1341: 495 1351: 49.5 1361: 49.5 1371: 406 1441: 497 19 I): 498 1961: 4YX 197) Dikii. L . A . 201. 310.349 1331: 349 1341 Dikpati, M. 4 10. 4.16 ( 61 1 Dionne. B. 245. 28.3 1361: 28.1 1371
DiPerna, R.J. 26,52 [20]; 85 [ 181 Dobler, W. 391,407,429,431 , 435 1291: 435 1311; 436 [62] Dobrokhotov. S. 327, 328, -149 1351; 349 1361: 349 [371 Doelman, A. 239,287 11.571 Doering. C.R. 199,203. 217 1621; 217 1631 Dombre, T. 306. 349 1381; 424, 4.?6 163) Douady. A. 192,217 1641 Doxaa. 1. 399. 440 1 1751 Drazin, P.G. 225, 244, 252, 253, 255, 256, 263, 265, 283 1381: 291. 307, 308, -149 139) Dri\coll. T.A. 267. 282 161; 287 11621: 201. 35.1 1 1 66 1 Drobyshevqki. E.M. 371.436 164) Du. Y. 426.4.16 [651: 436 1661 Dubreil-Jacotin. M.L. 483. 495 (381 Dubrovin. B.A. 506. 594 141 Duchon. J . 44. 52 1141: 52 1151: 52 121 1: 00. 92. 112-1 14. 115 151 Dudley. M . L . 390, 4-36 1671 Dumont. T. 4. 5.1 1641 Dl~nciln.J. 445. 49.5 1391 Dung. I-. 217 lh51 l ~ y : ~ c h c ~ A.1. ~ h o403. . 49.5 1401 Il/iemhow\ki. W.A. 401). 4.15 1341: 4.16 (681
P;. W. 45. 47. .52 IIOI: l)O. 101. 104. 117. 115 141: 116 171: 128. 14OIlX) lihin. I). 50. h2. 63. S.7 1101: 8 5 1201: 128. 140 I2hl Echart. C'. 3-12. .QO 1401 Eckl~:~u\. W. 239. 240. 2X.l 1301; 2cS.l 1-40] Echholf. K.S. 313. 317. 310. 355. .I49 141 I: .349 1421: .149 1431: .149 1441: .I49 1451 Eckrn;lnn. J.-P. 230. 2S.1 1331 Eden. A. 184. 186. 1x0. 100. 100. 217 166l: 217 lh7): 217 IhXJ Egg"\. J. 278. 280. 28.1 141 I: 2H.l 1421: 2~S.11431 Einstein, A. 504 151 Elia\hh~.rg. Y. XS (21 1 Ellingsen. T. 245. 206. 2S.l 1441: 28.5 1041 Elliott. J.K.410, J.(Y 11421 Elli\. G.F.K. 503. 504. 515. 577. 504 1 l I ) Eloy. C. 33 1. .?4Y (461: .151 1 1071 Elsas\r~-.W.M. 380.4.16 1691 Emhid. P.F. 308. 217 1601 Etllcr. L . 57.85 1221: 292. ZJY 1471: .I49 1481 Evilns. L.C. 71. 8.5 1231: 503. 587. 594 161 Eyinh. G.L. 5.45.47.52 1271: 52 1231: 90. 112. 116 161: 128. 140 1271 Faheh. E.B. 150. 167 I I41 Fabijonir. B. 332. .MY I49I: 349 150l: .152 11251: 3.52 ( 1 78 I
600
Author Index
Fadeev, L.D. 308, 349 [5 I] Faierman, M. 340, 350 [52]; 350 [53] Falconer, K.J. 171,217 [701 Faller, A.J. 413,436 [70] Farmer, D.M. 460, 496 [41 ] Farrell, B.F. 266, 282 [23] Fautrelle, Y. 4 ! 1,436 [71] Fearn, D.R. 360, 378, 41 i, 412, 436 [72]; 436 [73] Fefferman, C. 129, 140 1191; 140 1201 Feir, J.E. 240, 282 [111 Feireisl, E. 212, 217 [71] Ferriz-Mas, A. 410, 436 [741 Field, G.B. 407,435 1231 Finn, J.M. 419, 423-427, 432, 436 [751; 438 [1321 Fishelov, D. 146, 166 I2 i Fishman, V.M. 397,434 131 Fjortoft, R. 308, 350 [541 Flandoli, F. 212, 217 [72]; 217 [73] Flor, J.B. 3, 54 [68] Foias, C. 3, 52 [24];119, 121, 122, 126, 127, 134, 139 [4]: 140 [21 ]: 140 [28]: 140 [30]: 145, 167 [12]: 174, 175, 177, 180, 183, 184, 186, 188-190, 192, 194-196, 198, 199, 207, 213, 214, 217 [56l: 217 [571:217 [58]: 217 [59]: 217 [6()1; 217 [66]; 217 [67]; 217 [68]; 217 [74l; 217 [75]; 217 [761; 2181771; 2181781:218 1791:218 18()1; 218 181 I: 218 1821: 218 1831:2/8 1841:218 1851; 218 1861: 218 1871:2/8 1881:218 1891:2/8 1901:2/8 1911 Fomenko, A.T. 5()6, 594 141 Forbes, L.K. 450, 496 1421 Forster, G.K. 313,331,350 1551 Fraenkel, I,. 446, 494 [71 Frciberg, Ya.G. 383, 387,436 [781 Frenkel, A.L. 3()9, 350 1561; 354 11831 Friedlander, F.G. 313,350 157] Friedlandcr, S. 292, 3()9, 310, 318-321,327, 328, 33 I, 337, 339, 348 1161:350 1581:,t50 1591: 350 1601:350 161 l; 350 1621; ,t50 1631: 350 1641:350 1651:350 1661:350 1671; 3501681:3501691:353 11711 Friedrichs, K.-O. 446, 462, 496 1431:594 13] Frieman, E. 29 !, ,t50 1701 Friesecke, G. 483,497 [78] Frigio, S. 5, 52 I121 Frisch, U. 3, 49, 52 I251; 53 1591: 91, 116 181: 306, 349 1381: 396, 405,424, 428, 4,16 [631; 436 1761; 4,17 1821; 437 1941; 4,1911621 Frischmann, G. 269, 282 111 Fujimura, K. 249, 286 [I 38] Fukumoto, Y. 329, 331,350 1711; 352 !133] Gailitis, A.K. 383,387,436 [77]; 436 [781; 4,t6 1791
Galanti, B. 365,429, 436 [80]; 437 [81] Gallagher, C.T. 273,286 [143] Gallagher, I. 208,218 I921; 218 1931 Gallay, T. 161-165, 167 [151 Galloway, D.J. 420, 424, 437 [82]; 437 [83]; 43811111
Garaud, P. 410, 437 1841 Garcfa-Archilla, B. 218 [94] Gariepy, R. 71, 85 [23 ] Gazzola, F. 218 [951; 218 1961 Gearhart, L. 228, 283 [451 Gebhardt, T. 267,283 146] Gellman, H. 358, 380, 435 [361 Gerbeth, G. 383,436 [791 Gershuni, G.Z. 249, 283 [47] Ghidaglia, J.-M. 188, 214, 216 I39]; 218 1971; 218 [981; 218 1991; 21811001 Gibbon, J.D. 129, 140 I291; 199, 217 1621 Gibson, R.D. 358,437 [851 Giga, Y. 158, 163, 167 !161; 167 117] Gilbert, A.D. 328,350 1591; ,t50 172]: 360, 378, 387, 389-391,396-398, 403,405,411-417, 420-424, 426-432, 434 181:4,t4 1121:4,15 1531: 436 1561:437 1861:437 1871:437 1881: 437 [891; 4.t7 [901:4.t7 191 ]; 4.t7 1921: 437 [931:4.t7 [941:4.t7 [951: 4.t,';1138]: 4391159]: 4,19 11601: 4401166] Gilman, P.A. 409, 410, 436 1591:436 1601: 436 1611:437 1961:437 1971:439 11421 Gjcvik, B. 245,285 1941 Glasscr, A.H. 319, ,t49 1321 Glatzmaier, G.A. 409-411, 4,t7 1981:4,17 1991: 4.t7 I lOOl: 4.17 Ii()! I: 437 11021:4.t7 11031: 4.t7 11041; 4.t9 11421 Gledzer, E.B. 31 I, 350 1731 Glimm, J. 594 171 Godrbche, C. 225,238, 241,242, 246, 267, 268, 283 1481: 291, ,t50 1741 Goldhirsch, I. 321,353 [ 1531 Gollub, L. 291, ,t5,t [ 159] Golovkin, K.K. 166, 167 1181 Golubitsky, M. 234, 245,282 1241; 283 I361: 283 1491; 28,t 1501; 283 1511 Goode, R 409, 4,t5 [341; 436 1681 Gordin, A. 310, ,t50 [751 Goritskii, A.Yu. 186, 216 147] Gorodtsov, V.A. 269, 270, 28,t 152] Gough, D.O. 409, 410, 435 134]; 437 [1051: 4.17 11061 Grad, H. 302, 350 [76] Gradshteyn, I.S. 280, 283 [ 53 ] Graham, M.D. 268, 282 114] Gramchev, T. 16 I, 166 [5 ]
Greene. J.M. 306, -149 [38j; 424,436 (631 Greenspan, H.P. 208.218 [ I 0 1 1: 3 1 1. -150 1771 Grenier. E. 65. 85 [24]; 104. 115 (31; 208. 218 11021: 339.350 [7X] Grilli. S. 445. 496 [44] Grimshaw. R. 482. 404 [ I ] Groah. J. 503. 507. 5 14-5 16. 5 18. 5 19. 577. 594 [XI; 594 191; SY4 [ 101 Grossmann. S. 267.28.3 1461 Groves. M.D.446.473. 485. 488. 490. 49 1, 49.5 1241; 496 (451: 496 1461: 496 1471: 4% 14x1 Grulinov. A.V. 406.428, 4.17 11071 Guckenheimer. J. 17 1.218 [ 1031: 234.283 1541 Cuillopi.. C. 122. 126. I 4 0 (301; 230. 28.3 (551 Gundruni. T. 383. 4.36 1791 Guo. Y. 339. .34H [ X I Gu\tavsson. L.H. 266. 28.1 1561 Cuyenne. P.445.496 (4dJ
Hale. J.K. 171. 170-181. 187.210 1441: 219 (1041: 219 11051: 234. 28.1 1291 Hameiri. E. 318. 31'). 323. 329. 332. 350 1791; .I30 I XOI: .152 1 1261: .352 1 1271 Ha1111l)ach.J.L. 445. 496 1491 Hiinel. H . 383. 4.M 170) HZr5gu)-Courcellc. M. 4x5. 490. 40 I. 40.5 1321: 406 1461: 496 1.50): 496 151 1: 406 1521: 496 153): 496 1541 tla~.dy.G.H. 78. 8.5 1251 Haskcll. T.G. 460.472. 498 11071 Hastic. R.J. 310. .WO 1271 Hawking. S.W. 503. 504. 5 15. 577. 5'1.1 l l l l Hawley. 1.F. 417. 4.U 1101 Hawthorne. W.R. 302. 348 1181 Hcisenherg. W. 263. 28.1 1571 Hel~iihc~lt'.H. 296. 3.51 1x1 1 Henderson. D.M.445,496 (4Yl Hc~iningson.D.S. 266. 268. 28.f 1581: 2NS 11071: 2x3 ( Iox I Hcnoli. M. 52 1261: 306. .{4Y 13x1: 351 1x21: 424. 436 163 1 Henry. D. 181. 219 11061 Herbert, T. 268.284 (501: 284 1601: 3 13. .{48 1 141 Hcrhst. I. 228. 284 161 1 Her/enherg. A. 358, 387. 404. 437 1 10x1 Hida. T. 41.42.44.52 1271 Higgins. B.G. 273.287 11721 Hill. A.T. 219 11071 Hill. M.J.M. 301. 351 1x31 Hirsching, W. 41 1. 4.18 IIO9I Hoff. D. 214. 219 11091; 219 [ I IOI Hollerhach. R. 360. 4 12. 420. 438 1 l 101: 4.18 1 1 l l 1
Holm. D.D. 1 19. 134. I 3 9 141; 140 1311; 213, 217 1741; 291, 296, 300, 330-332, 348 1131; 349 [50]; 351 (84): 351 185 1 Holmes. P. 171. 218 11031: 234, 2 W 1541 Holyer, J.Y. 248,285 ( I O I ] Hooper, A.P. 272, 273, 284 1621; 284 1631: 284 1641 Hopf, E. 219 (1081 Hiirmander, L. 1 12, 116 [91 Horst, E. 26, 52 1281 Horton, W. 399,440 1 175 1 Hosking, R.J. 459,498 11061 Hoskins. B. 85 1261 Howard, L.N. 308, 309,.350 1601; 351 1861 Hoyng, P.405. 438 [ 1 12 1 Huang. F. 228, 284 1651 Huerre, P.242. 284 (66) Hugheh, D.W. 407, 4 10, 4 1 1, 4 16, 427430, 4.1.5 (441: 4.1.5 1461: 4.18 I I 131: 4.18 I l 101 Hunt. B.R. 190, 210 1 l l I ] H u n ~ e .R. 26.52 12x1 Hyer\. D.H. 440,402,496 1431 lcliih;~w;~.Y. 205. 28.5 192 1 lerley. G.R. 12"). I.)15): !, 427. 4.14
ll l1
Iftiniie.D.211.212.2IV~112~ lid;^. S. 365. 28.5 (021 Il'ichc\, A. 450. 460. 470. 400. 49.5 1331: 400 150): 496 155] l l i t ~ .K.1. 101. 31 l..1.5.+ 11721: 3.74 11731 Ilyashenho. Yu.S. It)(>. 219 1 1 131: 2IY I I I41 Ilyin. A.A. 102. 103. I O X . 206. 213. 114. 216 1451: 2/13 1461: 21Y 1 1 151: 2 l Y [ 1 Ihl: 219 1 1 171: 219 1 11x1: 219 1 1 101: 21Y [I201 100ss. ti. 171. -717 1521: 225. 333-235. 353-256. 28.1 [?XI: 2X4 1671: 2x7 1 1641: 207. 3JY (261: 454. 457. 458. 4h1463. 405. 466. 468.475. 377. 3 7 9 4 x 3 . 404. 49.5 (341: 49.5 1351: 496 1561: 4Y6 (571: 496 1.581: 4Y6 (501: 4Yh 1601: 406 161 1: 496 1671: 496 (631: 406 1641: 496 1651: 496 Ihhl: 497 lh71: 497 1681: 497 IX.51: 4YX 1 1 191 I\r;iel. W. 515. 521. 523. 530. 537. 539, .7Y4 1 121 Iver\. D.J. 378. 4.1,s 1 1\41
Jncquin. L. 314. 320. .14Y 1101: 35.3 115XJ James. C. 483. 484.497 1691: 497 1701 James, R.W. 37X. 3YO. 4.16 1671: 4.l8 I I 141 Jayneb. E.T. 33. .52 1201 Jevons. W.S. 248. 284 16x1 Johnso~i.M. 5Y4 1131 Jones. C.A. 371.41 I.4.38 1 I IS]; 440 IIXOI Jones. D.A. 195,2171551: 2 l Y [121]: 2\41 11221
Jones. M.C.W. 454.479.497 [7 1 1 Jordan. R. 52 1301: 54 (671 Joseph, D.D. 225, 234. 249. 259, 262. 272, 273. 282 [71; 284 1671; 284 1691; 284 1701: 284 171 1: 284 [72]; 284 [73]; 286 [12XI; 291, -351 1871 Joyce. G. 4. 5.3 [45 1 Ju. N.205.219 11231 Julien. K.A. 249.284 1741 Juttner. B. 54 1661 Kaiser. R. 3XI.438[1161;4381l 171 Kalantarov. V.K. 219 [ 17-41 Kaloshin. V.Y. 190. 219 [ I I I I Knmbe. T. 163. 167 [ 171 Kapustyan. A.V. 212. 219 11251 Kato. T. 128-130. I.jY 121: 140 1321; 145. 152. 158. 167 I 191; 167 1201: 167 121 1; I67 1221; 167 1231: 480.497 1721 Kaylor. R.E. 413. 436 170) Ka/an~srv.A.P. 396. 438 I I 181 Kelvin (Lord) 206. 313. 3.51 1881: .351 1x91 Kerr. A.D. 450. 498 1 106 1 Ker\wcll. R.R. 431. 4.M 1561 Kli:~sil'.C . 445. 49.5 1301 Khe\il~.H.A. 71. 85 121: 208, 214 121: 201. 308. .{.IS 1 7 1 K~IOIII;IIIII. H. 258. 274. 282 121: 284 1771; 2x7 11541: 287 I IhXI: 287 I 1001 Kido. S. 300. .{51 1001 Kiln. I;.-J. 4 1 I. 4 16. 4 1'). 420. 427. 429. 430. 4.33 1461. 4.18 1 I I C) 1; 44 I [ 200 1 K11ig. I.K. 52 131 1 Kil-chg:i\>ncr. K. 245. 284 1751: 446. 448. 450. 455. 456. 46 1 . 466. 4hX470. 483. 4x5. 400. 491.494 [ 4 [ : 4 9 6 [ 5 l [ : 4 9 6 1521: 496 1531: 406 1551; Jell, 1031: 496 1641: 407 1731: 497 1741: 407 1751 Kirrlnann. P. 239. 284 1761: 454. 457. 46 I . 480487. 484. 400. 406 16.51: 407 1761 Kli~ppcr.I. 420. 425. 426. 438 1 1701: 4.38 1 I2 I 1: 4.18 11221 K1cecr1-1n.N.305. 4.M 1801 Knohloch. E. 245. 248. 25 I. 28.3 131 1: 28.1 1491: 287 11501: 41 1. 441 12051 Koch. H. 145. 161. I66 131: 167 1241: 33'). .<51 191 1 Kohn. R. I 19. 12 1. 122. 139 131 Korkina. E. 423. 424. 4.W 151 Koxhmiedcr. E.L. 246. 284 04781 Kostin. I.N. 219 11261 Koulnoutsakos. P.D. 146. I67 [ 131 Kraichnan. R.H. 35. 52 1321: 91. I16 1 101: 396. 405.4.W 1 123 1: 4.18 [ 1241
Krasovskii, Yu.P. 446,497 [77] Krause, F. 359, 360, 395,396, 4 0 4 4 0 6 , 408, 4.&5'[125]; 4381126];44/ 11971 Krein, M.G. 231. 28.3 1351 Kubo, R. 35. 52 1331 Kuk~in.S. 212, 219 11271 Kulsrud, R.M. 416.438 1127) Kuzanyan, K.M. 41 1,434 1131 Kurrnin. G.A. 119. 134. 140 1331 Kuznetwv. Yu.A. 234.284 179) Kyba. P.J. 460. 497 1891 Ladyzhenskaya. O.A. 145. 167 1251; 171, 174. 175, 179. 1x0. 1x7. 194-196. 210. 213. 214. 210 112XI; 220 11291; 220 11301; 220 1131 I; 220~132];220~1331;220~134~;220~135~: 22011361:220~137];220[138] Lagnado. R.R. 3 13, .351 102 1 L:igr;lnge. J. 292. .<.51 1931 Laitone. E.V. 445. 498 1 1 201 L.an~h. H. 292. 300. 301. 3.51 1041 Lnndahl. M.T. 266. 2~S41801 l.:ind;ltl. 1..1>. 145. I67 1261: 360. 4.18 1 1 28 1 I.ar~tl~n;ln.M.J. 338. 320. .35/ 1051 1.anghornc. P.J. 450. 400. 472. 4YS [ IOhl; 4%S 1 107 1 1.al1hcl.r. K. 483. 497 17.51: 497 17x1: 407 170) 1.aliollc. A. 300. 4.18 1 1201 l,:~ri~ior. J. 357. 4.W 1 1301 I.;lr\on. R.G. 257. 258.2S.t [ X I 1: 2x4 1821: 2S.i 1 80 I l.~iI~~\liki~i. Y11.D. 321. .3.51 IOOI: 424. 4.38 [ I31 1 I,:ILI.YT : . 424. 4.18 11.321 I.;~vrcliliev.M.A. 402. 497 1801 I.ax. P.D. 86 1271; 170. I40 1211: 563. 500. 593. 594 II4I: .5Y4 [ 151 1-c Di/c\. S.331. .<40 1461; 3.51 [l071; .{.?I ~ l O 8 ~ 1.c Iluc. A 345. .151 11001 Leal, L.G. 313..1.51 1921 I.cbl;lnc. S. 330. 345. 351 1071: .I51 1081: .<TI 1001: .<.?I [ IOOI. 3.71 1 1001 L.ehon, G. 247. 28.3 1301 I,chovll/.. N.R. 3 1 1 . 330, 340, 343-346, . U / 1 101 I; 3.51 [ 1021: 3.51 [1031: 3.51 [ 1041: .<.51 [105]: 3.5 1 1 106 1 I,cgros, J.C. 249. 28.1 1321 1.cihovic.h. S. 210. 220 1 I4OI: 320. .IS/ I I 101: 413. 4.18 11.131 1,clc. S.K. 413. 438 1 1331 Leonard. A. 302. 35.3 11551 Leonov. A.I. 269, 270. 283 1521 LCoraf. J . 396, 428. 4.39 1162) Lesay. J. 44.35.48.52 1341: 52 1351; 120, 140 1331: 135. 167 1271; 220 1 1391: 220 1 1401
Author Index
Levi-Civita, T. 446, 447,453,462, 497 [81 ] Leweke, T. 331,351 [1111 Li, J. 273,284 [83]; 284 [84] Li, Y. 31 I, 352 [I 12] Libbrecht, K.G. 409, 436 [168] Lichtenberg, A.J. 336, 352 [I 14] Lichtenstein, L. 58, 86 [28]; 336, 352 [113] Lieb, E.H. 150, 167 [28] Lieberman, M.A. 336, 352 [I 14] Lielausis, O.A. 383,436 [79] Lifschitz, A. 291,292, 300, 302, 318-321,323, 325-327, 329-333,340, 343-346, 348 [13 ]; 349 50]; 350 [52]; 350 [80]; 351 [ 103]; 351 104]; 351 [105]; 351 [106]; 352 [115]; 352 116]; 352 [117]: 352 [I 18]; 352 [119]; 352 120]; 352 [121]; 352 [122]; 352 [123]; 352 124]; 352 [125]; 352 [126]; 352 [127]; 352 128]; 352 [129]: 352 [134] Lifshitz, E.M. 145, 167 [26]: 360, 4,./8 !128] Lilly, D.K. 413, 4,t8 [134] Lin, C.C. 225,284 [85]; 291,308,352 [130] Lin, X.-B. 2 / 9 [105] Lions, J.-L. 145, 167 129]; 172, 174, 175, 177, 182, 2()7,213, 22011411: 22011421: 22011431: 22011441:22011451
15ons, P.-L. 4, 26, 3(), 48, 49, 52 1131:52 1201: 53 [36]: 5,'/137]; 64, 65, 86 [29]: 119, 121, 140135]
Lister, J.R. 28(), 284 1861 l,ittlewood, J.E. 78, 85 [25] IAu, S.S. 257,282 [3] l,iu, T.-P. 594 [16] IAu, V.X. 202, 2201146]: 2201147] l,oitsyanskii, L.G. I i i , //6 111] Lombardi, E. 469, 47(), 477,480, 482,483, 496 [661:497 [82]" 497 183] 497 1841" 497 [85] Longuet-Higgins, M.S. 459, 472,497 [86] Lord, G.J. 470, 495 127] Lorenz, E.N. 171,220 [1481 Lortz, D. 245,286 [145]; 391,404, 4381135] Los, J. 472,497 [67] Loss, M. 15(), 152, 161-163, /66 [81:/66 [9]: 167 128] Lowes, F.J. 358, 387, 4,./8 11361:438 11371 Lu, K. 181,216 1361 Ludwig, D. 313,352 113 i ! Luskin, M. 594 1171 Lyapounov, A.M. 462,497 1871 Lynden-Beil, D. 53 1381 MacKay, R.S. 446, 497 1881 Mahalov, A. 208, 210, 213,214 141; 215 I I0]; 215 I!11; 215 1121; 215 1131; 215 l l4l:
603
2151151; 215 ll61; 2151171; 2151181; 215 !191; 215 1201; 22011491 Majda, A.J. 58, 85 II81; 86 1301:86 1461: 119, 129, 130, 139 [21; 140 1201; 140 1221; 140 1231; 140 136]; 208,217 1691 Makino, T. 594 [ 181 Maksymczuk, J. 427,429, 430, 438 [I 38] MSlek, J. 184, 213,220 [ 1501; 220 [ ! 511; 22011521 Malkus, W.V.R. 412, 438 11391
Mallet-Paret, J. 191,220 11531 Mallock, A. 251,284 1871 Marl& R. 189, 22011541 Manley, O.P. 3, 52 [241; 172, 195,213,214, 217 1591:217 1761; 218 1771; 218 1781; 218 1791; 22011451 Manneviile, P. 225,238, 241,242, 246, 267, 268, 28,t 1481:291,350 1741 Marchesin, D. 594 1191 Marchioro, C. 4, 8,52 1131:53 1391; 86 1311: 152, 1671301 Marsden, J.E. 59, 63, 85 [20]; 119, 128, 134, 140 [261:140 [31 ]: 145, /66 [11]: 171, 217 [51 ]: 221 [1551; 234, 284 [88]: 291,296, 3.s/1851 Marry, P. 390, 4()3,439 1157] Matthcws, P.C. 390, 41 I, 439 [140] McCracken, M. 171,221 1155]: 234, 284 [88] McCready, M.J. 273,286 [I 43] McGrath, F.J. 148, 166, 167 [3 ! ] Mclntyre, M.E. 41(), 437 [105] McKee, C. 594 [ 13] McVittie, G.C. 594 12()] Mehr, A. 306, ,r [38]: 424, 4,t6 [631 Mehta, A.P. 46(), 497 [891 Meinei, R. 396, 4381125] Melnik, V.S. 212,219 [ 125]: 221 1156]; 221 [157] Menasce, D. 473,475,495 [37] Mennicken, R. 340, 350 1521 Meshalkin, L. 3()9, 352 [I 32] Meunier, N. 411,439 [141] Michallet, H. 460, 476, 497 [90]: 497 [91] Michel, J. 4, 5, 8, 14-16, 18, 25, 53 [40]: 53 [41] Mielke, A. 239, 284 [76]: 446, 461,483,485,488, 490, 495 [21 ]: 496 [471:497 1921; 497 [93] Miesch, M.S. 410, 4391142] Mikelic, A. 53 142] Miller, J. 53 [43]: 53 [44]: 409, 4,t7 [97] Milovich, J.L. 391,435 [47] Miranville, A. 214, 221 [158]; 221 [159]: 221 [1601 Misner, C. 521-523, 528, 530, 537, 577,594 [21 ]; 594 [22]
Miyakawa. T. 158. 161. 166. 167 1161: 167 1321: 167 1331 Miyazaki, T. 329. 331-333. 350 171 1; 352 [128]: 352 [133]; 352 11341 Mizohata. K.594 [ 181 Moeller. M. 340. 350 152): 350 1531 Moffatt. H.K.302. 33 1. 351 [ 10XI; 352 [ 1351: 358. 360. 372. 375. 376. 378, 380. 392. 396. 40310X.419.420,422.423.439 11431: 4-19 11441; 439 [145]; 439 11461 Moise. 1. 205. 21 1. 212. 221 [161]; 221 11621 Molchanov. S.A. 405.418.4.19 11471: 4-11 1219) Monge. G. 79. 86 1321 Monkewitz, P.A. 242, 284 1661 Montgomery. D.4, 53 1451 Montgocnery-S111ith.S. 2 12. 221 1 163 1 Moss. D. 416. 4.14 1191 Mullet.. S.J. 257. 2.58. 282 181: 284 [ X I 1: 28.5 [ X Y I Miiller. LI. 390. 441 1 19x1 Nagata, W. 248. 285 (901 N e k . J. 121. I40 1371: 1x4. 213. 216 1371: 2-70 1 1 50 1 Nekr;~\ov.A.I. 446. 462. -197 IOJI Ncrct~n.Y. 72. 73. Xh 1331 Ncu. J.C. 300. .3.52 I 1361 Ncwcll. A.C. 246. 28.5 I Y I I Nich(~ll\.D. 440. 485. 49.5 1201 Niccllac~iho.H. 184. 186. 180. 190. 195. 199. 208. 210. 213. 214 141: 21.5 1101: 21.5 1 1 11: 21.5 1121: 21.5 1131: 21.5 1141: 215 1151: 215 1161: 215 1171: 21.5 11x1: 21.7 1101: 215 1201; 21.7 121 1: 217 1651: 217 1061: 217 1671: 217 1681: 218 I X O I Nirellherg. I.. I IY. 121. 122. 1.19 131 Nishida. T. SO4 1231: 594 124) Nishiclho. M. 7-65. 28.5 1921 Nja~nkepo.S. 214. 221 [ 1641 Norhury. J. 8.5 141: 8.5 1 171 Nordlund. A. 41 1. 417. 4.15 (321: 4.15 1331 Nore. C. 4. 5.f 1641 Noulle/. A. 306. 4.18 II20I Novikov. S.P. 506. 594 141 Novo. J . 218 1941 OehtcrlC, J . 102. 217 1641 Ohkitani. K. 129. I40 1291 Olive. V.M. 328. .14Y 1351 Ol\on. E. I 19. 134. 1.19 141: 189. 190. 218 [ X I I Onc~.H.482.498 1951 Onsager, L. 3. 4, 22. 23. 45, 47. 53 1461: YO. I 12. I16 (121; 128. 14013XI Oppenheimer. J.R. 549.594 125);594 126) Orr, W.McF. 3 13,352 11371
Orszag, S.A. 268. 282 193; 313,348 1141 Osada, H.158, 167 1161 Osaki. Y. 340. -153 ( 1681 Oseledeth, V.I. 119. 134, 141 1391; 424, 427, 439 [ 1481; 4.19 1 1491 Otani, N.F. 420, 42 1 , 427, 437 (951: 439 1 1 501 Ott. E. 419,423,425427,432,436 16.51; 436 [hh];436 1751; 439 [I511 Padmanabhan. T. 5.7 (471 Paes-Leme, P.J. 594 1191 Palm. E. 245. 266.283 1441; 285 1931: 285 1941 Papageorgiou, D.T. 278,285 1951 P5riu. E. 470,472,473,483.498 1961; 498 (971 Parker, E.N. 359, 360, 395. 404, 406408, 410, 427.439 ( 1521; 439 (153];4.19 11541; 441 (20x1 Parker, R.L. 37 1. 390.439 1 1551 Pata. V. 218 1961 Pazy. A. 226.285 I961 Pear\on, J.R.A. 246, 247, 268, 28.7 197);28.5 1981 Pedlo.;ky. J. 208. 221 I65 I Peg". R.L. 40 1.406 1541 Peregrine. D.H.445.49.5 1121 Perkin\. F.W. 307. 439 I15hl Perlin. M. 445. 408 [')XI Pcroul:me. M.C. 472.497 [OX1 Perry. P.A. 114. 216 1401 Perthame. B. 2h. 30. S.l 1371 Pelrie. C.J.S. 268. 285 / O X / Phfiln~osrr.K. 53 (481 Phan-Thicn. N. 313.351 1021 Pico. P. 5. 52 1 X I Pierre. M.G.St. 41 1. 441 1201 I Pierrehunlhert, R.T. 3 13. 320. 252 1 13x1; .1.52 [ 1391 Pilyugin. S.Yu. 21.7 1221 Plataci\. E. 383. 436 17Y1 Plotnikov, P.I. 492. 498 IYY] Pluni:ln. E 300. 403. 4.19 11571 Poincilri.. H.208. 221 [Ihh]:340. 345. 3.52 11401 Polya. G. 78. 8.5 1751 Ponce. G. 145. 167 [Zlj: I67 1221: 167 1211: 167 1.341: 210. -721 11671 Ponomarenko, Yu.B. 383. 385.4U4.439 I 1581 Ponomarev. V.M. 3 1 1 , .150 (731 Ponty. Y. 390. 397. 403. 41 I-Llb. 420. 4.17 1021: 439 1 159 1: 4.19 I 1601: 4.19 I l 6 l 1 Pope. S.B. 314. 3.72 1141 1 Porteous. K.C.268. 28.5 1Y91 Pouquet. A. 396, 405. 420. 428. 429. 4.17 181 1: 427 1941: 4.19 [ 1611: 4.10 1 Prahl. S.A. 246.284 178I Prandtl. L. 314. 35.1 11421 Pr;lut~sch,T. 4 10. 4.19 I 16.31
la
605
Author Index
Pra~ik, D. 213,220 [ 151 ] Preziosi, L. 259, 262, 284 [71] Proctor, M.R.E. 248,249, 285 [ 100]; 285 [101 ]; 360, 371,382, 410, 411,419, 420, 422-424, 437 [831; 438 [111 ]; 438 [I 13]; 439 [141 ]; 439 [ 145]; 439 [164]; 439 [165]; 440 [166]; 44011671; 441 [205] Prodi, G. 145, 167 [29]; 180, 194, 217 [75] Proudman, I. 314, 315,348 [15] Prtig, J. 228,285 11021 Pulkkinen, P. 411,435 1321 Pulvirenti, M. 4, 5, 8, 52 181; 52 I131; 53 1391; 86 1311; 152, 1671301 Purser, J. 85 [17] Rachev, S.T. 79, 80, 86 [34]; 86 [37] Racke, R. 210, 221 [ ! 67 ] Rfidler, K.-H. 359, 360, 395,396, 404-406, 408, 43811261; 441 [1971 Rammaha, M.A. 213, 216 [41 ] Rand, D. 255,285 [I 03] Rasenat, S. 249, 285 1104] Ratiu, T. 85 [21 ]: 119, 134, 140 [31 ]; 291,296,
35/1851
Raugel, G. 210-212, 2/9 [ 105]; 2/911121: 221 l i681 Rayleigh (Lord) 243,274, 285 [ 1051:285 11061; 297,308,353 1143]; 353 ll44] Reddy, S.C. 266-268, 28,r [581; 285 [ 1()7]; 285 [1081:287 1162]" 291, ,r162 [166] Reed, H.L. 268,285 [109]; 285 [1 I()] Rccder, J. 485,498 [ 10()] Rehberg, 1. 249, 2851104] Reid, W.H. 225,244, 252, 253,255,256, 263, 265,283 138]; 291,307, 308,349 [39] Renardy, M. 227,229, 230, 248, 249, 255,258, 268-271,273,278,281,284 [691:284 [83]: 285 III! l; 285 l! 12l: 285 I! 13l: 285 II 14l; 285 [I 151:285 [! 16]; 285 [I 17]; 285 I118]: 285 [i 19]; 286 [ 12()1; 286 [1211; 286 1122]; 286 [123]: 28611241: 28611251; 28611261: 286 [ 137]: 286 [ 138]; 287 [1701
Renardy, Y. 225,248, 249, 258, 259, 269, 270, 272-274, 284 169]; 284 [72]: 284 [73]; 284 [77]: 284 [83]; 284 [84]: 285 [I 12]: 285 [I 131; 285 [I 14]: 285 [I 16]: 286 [ 1231: 286 1127]; 2861128]; 286 [ 129]; 286[130]: 286 [131]; 286 [1321; 286 [133]: 286 1134]; 286 [1351; 286 [1361; 286 [1371; 286 [ 138]; 287 11701 Resnick, S. 129, 141 [40] Reynolds, O. 225,286 [139]; 291,353 [145] Ricca, R.L. 396, 420, 439 [146]
Riemann, B. 340, 345,353 [146] Robert, R. 4, 5, 7, 8, 14-16, 18, 22, 24-26, 30-32, 34, 35, 37, 52 [17]; 52 [21 ]; 53 [40]; 53 [41 ]; 53 [42]; 53 [49]; 53 [50]; 53 [511; 53 [52]; 53 [53]; 53 [541; 53 [55]; 53 [64]; 54 [65]; 90, 92, 112-114, 115 [5] Roberts, G.O. 392, 397,402-404, 432, 440 [168]; 440 [ 169] Roberts, M. 248, 286 [140] Roberts, RH. 358,360, 372, 375,378, 383,387, 41 I, 412, 436 [73]; 437 [85]; 437 [101]; 4371102]; 437 [103]; 437 [104]; 4381115]; 4401170]; 4401171]; 4401172]; 440 [173]; 4401174]
Robinson, A.C. 331,353 [147] Robinson, W.H. 460, 472, 498 1107] Rogachevskii, I. 365,436 [80] Rogers, C.A. 221 [1691 Romanov, V.A. 263,286 [141] Rosa, R. 3, 52 [24]; 205,217 [76]; 221 [161]; 221 [170] Rosenbluth, M.N. 308,353 [148]; 399, 440 [175] Rosencrans, S.I. 31 (), 353 [ 149] Rosier, C. 24, 31, 32, 34, 35, 37, 53 [531 Rosner, R. 4()6, 419, 420, 427,441 [2081: 441 1209]: 441 [2101 Rossi, M. 331, .r [ 108] Rottenberg, M. 29 !, 350 [701 Royden, H.L. 69, 86 [35] Rubin, H. 62, 86 [36]: 3()2,350 [76] Rtidiger, G. 410, 4401176] Ruelle, D. 171, 172, 221 1171]; 255,286 [142] Rugh, H.-H. 427,440 I1771 Rtischendorf, I,. 80, 86 [371 Russo, G. 134, 141 [41] Rfi2i(:ka, M. 121, 140 1371; 184, 213, 2201152] Ruzmaikin, A.A. 328, ,r [35]: 360, 372, 387, 389,405,416, 418,434 [6]; 434 [7]: 439 [147]: 440 [178]; 440 [179]: 441 [2181; 441 [219] Rykov, Yu.G. 104, 116 [7] Ryzhik, l.M. 280, 283 [53] Sachs, R.L. 466, 498 [I 01] Sacker, R.J. 221 [ 172] Sadourny, R. 32, 52 [7]; 53 [56] Sadun, L. 340, 353 [I 50] Saffman, RG. 302, 328, 329, 331,351 [95]; 353 [ 1471:353 [ 1511; 446, 497 [881 Sagdeev, R.Z. 419, 420, 441 [209]; 441 [210] Sangalli, M. 273,286 [143] Saric, W.S. 268,285 [109]; 285 [110] Sarson, G.R. 371,440 [180] Sattinger, D. 171,221 [173]; 310,353 [149]
Saut, J.-C. 183.218 1821; 218 1831; 230.28-1 [551; 446.483.495 [ 181 Scanlon. J.W. 247.286 [ 13-11 Scardovelli. R. 445. 498 [ 1021 Schaeffer. D.C. 234. 28.3 [501; 283 [SI 1 Schaeiier. J . 53 157 1 Schefirr. V. 45. 53 1601; 90, 116 1131 Schlichting. H. 35.1 11561 Schliiter. A . 245. 286 11451 Schmalfuss. B. 2 12.217 1721: 217 1731; 221 11781 Schmid. P.J. 267. 268. 285 [ 1081: -787 (1631 Schmitt. B.J. 381. 1-18 [ 1171 Schrnitt. D. 410. 436 1741 Schmitt. R.W. 248.286 [I461 Schneider. C. 239.282 11x1: 2X4 [76l: 286 11471; 446. 498 1 103 1 Schonhek. M. 161. 167 1331 Schult/. W.W. 445. -198 1981 S c l ~ u ~ ~M. l r r 410.4-16 , (741 Schut/. B.F. 340. .<.3i' 1 152 1 Schw;~rt/. L . 4 9 2 l 9 4 . 498 I 1041 Segcl. \..A. 247.286 11441 Scki~.T. 409. 437 IlOhl Scla. N. 22 1. .< 1 1531 .i.< Sell. G . R . 1x4. 195. 210-212. 218 1801: .?IS 1841: .?IS(8.51; 221 I16XJ:221 11721: 221 11741: 221 11751: 221 11761: 221 (1771 Seregin. G.A. 2 14. 220 1 1381 Scrre. I). 38. 5.1 15x1: 50. 86 13x1: 128. 141 1421 Scrrin. J. 119. 121. 130. 132. I 4 1 14.71: I 4 1 1441: 154. 167 1351 Sliaf:~rcvicIi. A. 327. 328. .<40 1351: -14') 1301: .<40 137 1 Shal'ranov. V.U. 302. .i5.1 1 1541 Shao. I.. 3 14. 349 ( 101 Shao. Z.D. 195. 221 1 1791 Shaqtkh. E.S.G. 257. 258. 2X4 1x1 1: 28.5 (891: 286 II4XI Shnril'. K. 302. .i5.1 11551 Sharp. D. -594 121 1 She. Z.S. .5.i 15Y1: 306. 405. 4.16 1761 Sheluhhin. V. 50. 66. 86 (391 Shen. M.C. 460.476.483. 498 [ I 131; 498 ( 1 141 Shen. Z.214. 216 1401 Shepeleva. A. 230. 287 1 1491 Shibahashi. H. 340. 35.1 [ 1681 Shinhrot. M . 485. 498 11001 Shirikyan. A. 212.219 [ 1271 Shnirelmnn. A.I. 4.5, 5.1 161 I: 5.1 162): 59. 63. Oh. 83. 85, 86 1401: 86 141 1: 91. 101. 116 1141; 116 1151: 116 [161; 116 (171: 292.340. -150 161 1: 35.3 11571 Shrairnan. 6.1. 399. 440 [ I81 I
Shukurov, A.M. 387,389, 391,429,429,434 1191; 4.76 1621; 440 11781: 440 [179]; 440 11821 Sideris. T.C. 210. 221 1 1 671 Silher. M. 245, 248,283 1371; 287 11501 Simon, A . 308, 35.3 11481 Sinai. Ya.G. 44, 5 3 163); 104, 116 171; 309. 352 11321 Sipp. D. 329, 353 11581 Skeldon, A.C. 245, 28.3 1371 Smale. S. 171, 217 151) Smerrka, P. 134, 141 141 1 Smith. F.T. 273, 282 ( 161 Smith, J.D. 460, 496 141] Smoller, 3 . 503, 5 15, 5 17, 520, 525, 540, 542, 543. 549. 564. 566. 567. 569. 570. 572. 576. 582. 592.593.594 1241; 594 1271; 594 12x1; 594 1291; SY4 1301; SY.5 1311: 59.5 (321; 59.5 (331: 59.5 1341; S9.5 (351; 59.5 (361 Snyder. J.R. 549. -594 1251 Soholev. S.L. 208. 221 IIXOI Sokoloff, D.D. 300. 372. 387. 389. 391. 405. 41 1. 416.418. 4.14 {h(;4.<J (71: 4.<4 [11)1;4.19 (1411: 4.19 1147): 440 117x1: 440 11701: 440 11x21; 441 121x1: 4-11 12191 Solovyev. A.A. 387. 390. 440 IIX3): 440 I184) So~nmrria.J. 4. 5. 7. 22. 30-32. 35. 52 1171: S.3 1.541: S.1 1551; 5., [h4]: .54 l051: .54 I061 Sommcrmwin. G. 240. 282 122) Souplet. Ph. IOl. I 6 6 ( 4 ) Sohi~rd.A.M. 300. .149 1381; 360. 378. 300. 301. 307. 301). 401404. 4 I I 4 17. 424. 427. 4.W 1131: 4.36 1541: 4.16 1.551: J.<6 1631: 4.q6 1731: 4.
Stix. M. 416.441 [199]: 441 [200] Stokes. G.G. 445.49 1,498 [ 1081 Stone, H.A. 280,284 [861 Storesletten. L. 313. 345. 349 1441: 349 145) Straughan, B. 225. 230.287 (1531 Strauss. W. 309. 337. 339. 348 181; 350 [62] Stroock. D.W. 150. 167 [ 141 Stuart. J.T. 129, 141 1451: 171, 221 [ l X l I Su. K.C. 274.284 (771 Su, Y.Y. 274.287 11541 Suhramanian. K. 407.43 I . 435 13 1 1 Sudakov. V.N. 79. 86 (421 Sulern. P.-L. 396. 405. 420. 428430. 436 (761: 437 [811;4.<7[93[;4.{9(1611 Siili. E. 219 [ 1071 Sun. S.M. 469, 470. 476. 480. 482. 483. 485. 491, 496 1461; 496 1661; 498 [ 1091; 498 1 1101: 4YX[III~;498~112~;4Y8~113~;4YX~II4~
Sure\hhutnnr. R. 258. 268. 269. 282 121: 286 11231: 287 [1551: 287 11561 Suters. H. 302, 332. 3.72 11291 Sutherland. B.R. 460. 497 1891 Sverak. V. 12 1. 140 1371 Swift. J.W. 245. 248. 28.1 l401: -786 11401 Swit11lc.y.H.I.. 257. 282 131: 291. .1.5.1 1 150] Tiihah. E. 120. I40 1231 TiihGC. P. 23'1. -7S7 11571 T;~kc~l\. F. 17 1. 172. 221 I 17 1 1: 755. 286 [ 142) T;I~I/;Iw;I. T. 450. 460. 472. JON 1 1 15) Tiin. B. 181. 216 1441 Ttirtar. 1.. 66. 86 1431 Tiissoul. J.-L,, 340. .1.5.< 1 1601 Tutaru. D. 145. I67 1241 Tnub. A. .5Y.5 (371 Taylor. (3.1. 251. 287 [ISXI: 207. 314. 35.1 I lhl 1: 35.1 1 1621 Tiiylor. J.B. 319. 349 1271 T6I. T. 42h. 436 1661 Tci~iatil.R. 3. .72 1241; 1 10. 121. 122. 126. 127. 140 1281: I40 1301: I41 (461: 141 1471: 145. I67 1301: 171. 172. 174. 175. 177, 170. 1x0. 182. 184. 1x6- 190. 192-200. 207. 209. 21 1-214. 216 1391: 217 ISXI: 217 1.591: 217 1001: 217 16x1: 217 1761: 21s 1771: 218 17x1: 218 1791: 218 (XOl:218 (841: 218 ( 8 6 ) :218 1x71: 218 1881: 218 IXOI: 218 1001: 218 (971;218 [YXI: 218 1991: 2201142~:220[143~:220[144~:220(145~: 221 11621:221 (1821;221 [1831:222 11x41: 222 11851: 222 11861: 222 11x7) Temple. B. 503. 507, 514-520. 525, 540. 542. 543. 566. 567. 569. 572. 577. 582, 502. 593.
594 191; 594 [lo];594 1171; 594 1281; 594 1291; 594 1301; 595 1311; 595 1321; 595 133); 595 1341; 595 1351; 595 1361 Thlter, G. 184.2 13,220 [ 152) Thelen, J.-C. 410, 441 12021 Thehs. A. 54 1661 Thomas, J.W. 248. 285 [90] Thompson. K. 595 1381 Thomson, J. 242,287 11591 Thorne. K. 521-523,528,530,537,577.594 122) Tilgner. A. 396,441 [2031 Titi. E.S. 45.47,52 1 19);90. 101. 1 12. 115 141; 119, 128. 134. 139 141; 140 [IX]; 195.210,213, 216 141 1; 217 1551; 217 174);2IKlX51; 218 1911; 218 1941: 219 I121 1; 219 11221; 220 11491: 221 11671; 221 11791; 222 IIXX]; 239.287 11571 Tlapa. G. 269, 287 I 1601 Tobias. S.M. 37 1. 4 1 1 . 424.428. 430. 43 1 . 435 1351; 4.19 1 141 1; 441 12041: 441 (2051 Toland. J.F. 446.473475.492494.494 151: 494 10);494 17):494 (XI:49.5 123);49.7 1261: 496 14x1: 498 1991; 498 1 1 161 Toll~nien.W. 308. .15.1 ( 1631 Toltnan. R. 504. 59.5 1301: CY5 1.101 Toomre. J . 410. 41 1. 4.19 11421; 441 12041 Torkclsson. U. 417. 4.<5 1331 Townscnd. A.A. 314,353 Il64l Trachtenhcrg. S. 146. 166 171 Trekthen. A.E. 707. -787 11021: 287 11631: 301. .1.5.1 1 166) Trcl'cthcn. I..N. 227. 267. -78-7 161: -787 I lhl 1: 2x7 Il62): 287 I 1631: 201. .1.5.1 1l651: .1.5.1 1 1661 Trcm;iine. S.D. 4. 52 [ I I ) TrCvc. Y. 105. 218 1701 Twi. T.-P 121. I41 1481 T\i~i.W. J45.4YN 1 1 171 Tuominrn, I. 41 I . 4.1.7 1321 Turhington, B. 54 167 1: 302. .1.5.1 1 1671 Turner. R.E.I.. 483. 494 101: 494 1101: 49.5 1171: 49X)I 1x1 CJLai. S. 594 11x1 Ungar, P. 63.86 1361 Unno. W. 340. .1.7.1 1 168I Vnimhtcin. S.I. 406. 407. 417420.477.429. 430. 4.15 (451: 441 ]706]: 441 1207):441 JZOX]: 441 12091; 341 12101 Valero. J . 212. 221 1l.57) van Harten. A. 239.282 I 181: 287 1 1651 van Heijsi. C.J.F. 3. .54 16x1 Vanden-Brtxck. J.M. 470. 473. 475. 49.5 127): 49.7 1371
608 Vanderbauwhede, A. 233,287 [164]; 461, 4981119] Varadhan, S.R.S. l l, 16, 54 [69] Vattay, G. 427, 434 [I I]
Velarde, M.G. 247,282 [19] Vergassola, M. 396, 438 [129] Vershik, A.M. 79, 86 [44] Vishik, M.I. 171, 172, 174, 175, 179-181, 183, 184, 187, 188, 190, 192, 193, 195, 196, 199-203, 212, 215 [23]: 215 [24]; 215 [25]; 215 [26]; 216 [27]; 216 [28]; 216 [29]; 216 130]: 216 [31]: 216 [32]: 216 [33]; 216 [48]; 216 [491; 216 [50]; 222 [189] Vishik, M.M. 309, 310, 318-321,327, 328, 331, 337, 339, 340, 350 [591; 350 [621:350 1631; 350 [64]; 350 [65]; 350 [66]; 350 [67]; 350 [68]; 351 [96]; 353 [150]; 353 [169]; 353 11701; 353 [171]; 424, 441 [2111 Vishniac, E.T. 407, 43 I, 441 [212] Vladimirov, V.A. 291, 31 I, 353 [172]; 354 1173]: 354 [1741 VolkofL G.M. 594 126] Vostretsov, D.G. 311,354 1174] Wagner, D.H. 248, 286 11401 Wald, R.M. 5()9, 521,524, 577,595 1411 Waleffe, E 267, 28711661: 291,329, ,t541175]: 354 11761 Wang, S. 172, 213,220 11421: 22011431: 220 [ !441:220 [ 145 ] Wang, X. 2()3, 2()5,214, 217 1631:221 11581: 221 11591:221 11611 Wasncr, J. 269, 282 [11 Wayne, C.E. 161-165, 167 1151: 446, 498[!()31 Weber, C. 274, 28711671 Wehauscn, J.V. 445,498 11201 Weichman, P.B. 53 1441 Wcinbcrg, S. 507,508, 513,516, 521, 541-543, 549, 564, 565,576, 577, 593,595 1421 Weinstein, A. 291,296, 351 [851 Weiss, N.O. 249, 285 [ I()()]: 36(), 37 !, 390, 408, 4401167]: 441 [213]: 441 [214] Weissler, F.B. 161, 166 [4] Wheeler, J. 521-523,528, 530, 537,577,594 [221 Whitehead, J.A. 246, 285 191] Whitney, A. 492-494, 49811041 Widnall, S.E. 329, .t52 1139] Wilkinson, !. 358, 387, 4.t8 11361; 43811371 Will, G. 383,436 [79] Williamson, C.H.K. 331,351 I1 I I1 Wilson, G.M. 274, 287 11681:287 11691 Wilson, H.J. 270, 287 II 701 Wirth, A. 396, 438 11291 Woltork, P. 493,494, 495 130]
Author Index
Wolibner, W. 336, 354 [I 77] Woltjer, L. 396, 441 [215] Woods, P.D. 473,498 [121] Wu, J. 129, 140 II71; 140 I241 Wu, S. 446, 498 ! 122 !; 499 [ 1231 Wynn, S. 119, 134, 139 141 Yakubovich, E.I. 295,305,348 [2]; 348 [3]; 354 11781 Yamada, M. 166, 167 132] Yan, Y. 22211901 Yang, T.S. 470, 473,499 [1241; 499 [125] Yarin, A.L. 281,287 II 711 Yiantsios, S.G. 273,287 [1721 Yih, C.S. 272,287 II 731 Yoshida, Z. 306, 354 11791 Yoshimura, H. 408,441 [216] You, Y. 221 !177] Young, L.-S. 425, 4,t8 [I 22] Young, L.C. 14, 54 [71 ]; 66, 86 [45] Young, W.R. 129, 1.t9 151 Yudovich, V.I. 7, 45, 54 1701:201,222 1191 I: 226, 229, 231,232, 287 11741: 291,292, 296, 309, 310, 335,336, 339, 3481161:350 1671: 350 1691:354 11801:354 11811:354 11821 Yuc, D.K.P. 445, 49811171 Yuferev, V.S. 371,4.t6 1641 Zabczyk, J. 229, 2871175] Zachos, C.K. 6, 7, 54 172] Zahn, J.-P. 409, 440[I 96] Zakharov, V.E. 493,495 [4()]: 4991127] Zaleski, S. 445, 4981102] Zeitlin, V. 6, 7, 54 [73] Zeidovich, Ya.B. 360, 372, 377,405, 417, 418, 434 [6]: 4,t4 [7]: 441 [2()7]: 441 [217]: 441 [218]: 441 [219] Zeng, C. 181,216 [36] Zenkovich, D.A. 295, 305, ,t48 [3]: ,t54 [178] Zhang, K. 371,41 I, 4401180]: 441 [220]: 441 [221] Zhang, X. 309, ,t54 [ 183]: 459, 472,497 [86]: 4991126] Zheligovsky, V.A. 424, 441 [222] Zheng, Y.D. 86 [46] Zhou, Y. 215 [20] Zhukovitskii, E.M. 249, 28,t [47] Ziane, M. 202, 211,212, 214, 2/91109]: 2/9[! 10]; 221 [160]; 221 [162]: 222 [ 1871; 222 [ 192]; 222 [193] Zikanov, O. 268,287 [176] Zweibel, E.G. 365, 397, 435 [30]; 439 [156]; 441 [223]
Subject Index condition divergence-free. 172 -entropy. 48 - Israel jump, 538 - "KattrFujita", 16 1 - Lax xhock. 569 - no-xlip boundary. 173. 174. 203. 2 12 - Rayleigh \tabili~y.344 - zero average. 173. 208 connection. 504 - coetficients. 507 constant - griivil~~t~on;~I. 510 - Newtoll'\ gr;lvit;~tional.506 constraint -
absorbing - ball. 188 - xet. 179 anti-dynamo theorern. 358. 375-38 I . 387. 389, 390. 398,402. 404.41 1 - for planar now. 377 - for toroidal How. 3HO!-383 a\yniptotic behavior. 147. 161. 163. 105 aIlri!clor. 180. 190. 108-200. 203 - cxlx)ncnl~al.1x4. I Xh - gloh~~l. 179. 187. 1x8. 202. 205. 209. 2 12-2 I 4 n~,~xin~al. 179 - rn1n1111:tl.I70 - unhountlcd glohal. I X O -
d~~ncnxitrn brcahing. 490 Hopi', 236. 137. 248. 254. 256 p~tchhrh.234 - rcvcrtihlc. 4X I - Taken-Bogdanov. 240 - tranxcritical, 234 - with \ymlnetry. 235 boundary condition - free. 2 14 Bouxxinc\q - npproxin~ation.243 - xyxter11. 2 I3 butterfly diagram. 408 -
-
Camasha-Holm. 7 13 channel - curvilinear. 203. 205 - rectilinear. 203. 205 chaos - Lagranginn. 420. 425. 427. 429 Chri\toltl xyn~bolx.507 coherent xtructures. 3, 23 compressibility. 340
ct)nlr;tvari;~nt.505 cc)ord~natc\ - G;lu\\ian nornli~l,57 1 - loc;~llyinertial. 508 - \tandard Schwar/schild. 506 Coocttc-Ti~ylor, 2 14 countcrrota~ingcylil~dcr\.256 covarlant. 505 - dilfercntiation, 507
delta-function xingularitic\, 530 dc~noduli~tion. 92 drterri1ining nodeb. 105. 2 13 dillcrentially rotating \tar\. 340 domain - thin, 210 - unbounded. 203 doubly diffusive convection, 247 tlynomics - tinile-d11nen\iona1. I05 dynnmo - accretion disc, 41 7 - alpha xquared. 305, 31 1 - alpha-omega, 407-4 1 1. 4 16, 333 - convective. 359. 3 1 1-416
-experimental, 358, 383, 387, 396 fast, 359. 387, 403, 406. 4 1 6 4 3 2 - galactic, 359. 416 - in MW flow. 420-423.426 - mechanism - - (3.0.Robens, 402 - - Ponomarenko, 390 - - stretch-fold-shear, 422, 427 - nonlinear. 39 1 , 396. 427A32 - slow, 4 17 - small-scale. 428 - Solar, 357. 359.408-4 1 1. 4 18. 43 1,433 - stretch-twist-fold. 4 1 8 4 2 0 - theory, 328 - wave. 407.3 1 1
equations
- Einstein, 509 - Kelvin, 3 19,328, 329,333
-
+
Eckhoff's approach. 3 15 effect alpha. 3 5 9 . 3 9 5 4 1 1.43 1 4 3 3 - - suppre\sion. 396,406, 407. 428 Marangoni. 243 - omega, 378. 390. 4 0 7 4 1 I elastic ice plate. 350 'ncrgy - balance. 1 l I . 1 12. 1 I4 - in weak solution. 90 density. 5 10 - dissipntlon, 3. 37. 44. l l 1 estiniare. 175. 206, 230 - magnetic. 36 1 , 365, 366, 368-370. 38 1. 427 potential. 259 enstrophy, 01. 3 2 cquarioi~ amplitude. 238. 324 - - hicharacteristic. 3 19 - Burgers. 3. 37. 42 - cotangent, 3 19 cikonal. 292. 3 16 Euler. 57. 58. 61. 62. 64. 65. 67. 89. 120 - Eulcr-1,agr;ingc. 259 - Eulerian. 292 - Ginzburg-Landau. 239 heat. 152, 158. 162. 163 - hyperbolic partial differential, 227 - induction. 357. 364-367 - Mongc-Alnpere. 79 - Newcll-Whitehead, 246 Oppenheimer-Volkoff, 543 - Orr-Solnlnerfeld, 263, 269 - Poisson, 5 1 1 - pre-Maxwell, 364. 367 - Rayleigh, 307 variation, 183. 190, I92 - Vlasov-Poisson, 4, 25, 26, 30 -
-
-
-
-
-
-
-
-
-
equilibrium, 181-183, 188, 210, 298
- problems, 4 - state, 6, 13, 19, 24 equivariant branching lemma, 237 essential spectral radius, 320, 326, 338 exponent, 194 - cancellation, 426 - flux growth. 426 - Liapunov, 425 - line-stretching, 420,425, 426 exponential stretching, 326, 328 exponentially small oscillations, 469 tixed point -hyperbolic, 326 Floquet -analysis, 329 exponent, 332 - theory, 330 How - Anosov. 328 -
(ABC). 306. 328. 33 1 . 392.423.429 - Beltranii. 306. 33 I. 302. 420. 424. 428 - Bogoyavlcnskij, 304 - "c;ils-eye". 300 - compre\sible, 2 14.430 core-annular, 273 - Couette. 252. 262 -elliptical. 305. 321. 329. 33 1 - generalized. 101 L'-generali~ed, 105 --with definite velocity. 102 - - with local interaction. 102 - hyperbolic. 32 1 , 323 - integrable. 33 1 Koliiiogorov, 95. 97, 309, 424 - modulated, 04 linear. 304. 31 1. 313. 332, 346 minimal, 340 rnultiphase. 106 - partillel shear. 262 - plane parallel. 298 - Poiseuille. 203. 205. 262 - Ptolernaeus. 305 - rotational shear. 298 sticking generalized, 104 two-layer, 248. 259. 27 1.45 1 - viscoelastic, 268 -
Arnold-Beltrami-Childress
-
- -
-
-
-
-
-
-
-
fluid -conducting. 249 - incompressible. 57 - Oldroyd B, 28 I - stratitied. 483 hrce - centrifugal. 25 1 - Coriolis, 207, 209. 212. 330. 333 - Lorentz. 361, 370. 391.406. 407.427. 429. 43 1 -oscillating. 92, 93 forces - "ghost". 92 formula - Cauchy, 1 19. 130. 134, 135 - Darcy-Weihbach. I I I - Wether. I 19. 130. 132- 134 fractal. 202, 203, 2 I3 - dimension. 189. 191-194. 108-200. 205, 207. 209. 2 13 freefill1 paths. 504 gap -circular. 1x1. 1x1 - narrow. 253. 255. 258 \pcctral. 337. 338 gcode\ic. 50. (32. 504 gcodynamo. 350. 4 1 1. 4 12 gcol~lclricaloptic\. 20 1. 3 13. 3 1 X. 3 1'). 327. 378 344. 346 Cil~mnlhchcnic. 5 1 X gravity. 340 group - G;~lilc;in.500 Poincaw. 500 growth. 266 - abscissa. 226 - 11on-modal.265
- Lieb-Thirring. I97 - logarithmic Sobolev, 16 1 - Nash, 150, 152, 161 instability. 335 240 - algebraic, 323, 326, 330 - Benjamin-Feir, 240 - broadband. 3 13 - convective, 240 - Eckhaus, 240. 246 - Ekman. 4 13 - Kelvin-Helmholtr. 296 - localired, 3 13 - Maranponi, 246 - nonlinear. 334-336 - Ray leigh-Tay lor, 297 - \econdary, 245, 267, 332, 334 - hideband, 240, 245. 273 - viscou\. 326 - ~ i g ~ a246 g. invariant. 179 - \trictly. 179 invcr\c - c;l\ciidr. 306. 42%.43 I - absolute.
-
-
Hausdorff. 1')l. 103. 202. 205. 207 IXX. 1x0. 192. 108-200. 203. 200. 21 3 - Iiicnhure. I Xc) hclicity - H u I ~ .301. 305. 4 0 3 4 0 7 , 409, 420 - magnetic. 305.402.43 1 hcliosei\mology. 4 0 0 4 1 1 hexagon>. 237. 245. 248 high frcqucncy wavelcth. 3 13 horlioclinic. 464. 467. 47 1 - to periodic solutions. 47 1 homogeni~ation.3 10 - dilnen\ion.
inequality -
Hardy-Littlcwood-Sobolev.
1 50
1,ngranpi;tn cqu;~lionh.204 - h)rrnul;~tion.276 I;irgc deviation - c\timatcs. 5 - property. 16 I;~tr~ce. 744 - hcxagonal. 236. 245. 248 - \quare. 245. 248 law - So~lr-hfh. 49 - Hale polarity. 408. JOY. 433 - Ohm'\, 3hl. 365 layer - boundary. 264 - - ~n;ignetic.4oO403 -critical, 264 - Ekman, 413 - intinite depth. 453.477 -
limit - Oppenheimer-Snyder.
549 -zero viscosity. 146, 152. 166 Lyapunov -dimension, 194 - exponent. 194. 3 18-32 1. 325.338.339
Navier-Stokes equations, 119, 172, 207, 209. 326 normal form, 235,445. 461.462, 466, 468.470 - i n intinire dimensions, 480 number - Deborah, 258 Grasshof. 196. 198 Lewis. 247 - Marangoni, 246 - Ohnesorge, 275 - Prandtl. 243 Rayleigh, 243. 247 - Reynolds. I I I, 198 - Rossby. 208 - Taylor. 253 -
magnetic - diffusivity.
366 flux. 366.4 18.419.423.426 - Reynolds number, 382.41 3.428.43 1 magnetic ti eld. 249 - boundary conditions. 367-370 - - perfect conductor. 369. 370. 373 - - inhulator. 368. 369, 373 decay modes. 374. 375 - translent amplification. 377 manifold - approximate inertial. 1 % - center. 23 I. 233. 245.460, 484.490 center-unstable. 182. I86 - finite-dirnen\ion~~l invariant. 161 - inertial. 195 tnv;lri;~~~t. 161. 1h5. 230. 23 1 - lociil ~ ~ l v a r i a ~1x2. l t . 183. 201 \t;~hle. l X I. I86. 23 1 unblablc. IX I. 200. 23 1 ~ll;~tchcd ;~hyniptoliccxp;~~~hioll. 204 M;txwcll'h cqualio~ls.360-304 - relativi\tic invariance. 302. 304 nIea\ure -as initial data. 158 - :1111-:1ctor\.2 I 2 - linltc (signed). 158 -
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operator multivalued, 2 12 - resolvent, 460 oscillating triangles. 248 -
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spherically \ymmetrlc. 538
- Tolil1nl1-Oppenhalner-Volkoff( T O V ) . 506
mode ballooning. 311) detertnini~lg.2 13 i~~tertacial, 272 - Kelvin. 31 3. 332. 334. 335 model - Be~!ja~~~in-Ono. 4x2 - Giesekus, 28 I - Maxwell's. 269 modulation, 92 monodromy matrix. 329. 330, 332 ~ ~ ~ u l t i t l o106-1 w . 10 -
patchwork quilt. 237. 245. 248 twi\ted. 248 periodic - boundary condit~on\. 171. 199. 201. 208. 2 10 - box. 20 1. 202. 208 - \elution. 468. 460. 47 I - w:rvc\. 158. 477 ph;l\c -
- compcn\a~inp, I 0 0 controlling, I 0 0 pipe. 203, 206 pol~rrlactori/;ition, 77. 7X pressure. 5 10 prcssurclcss weak \olulion. 103 p r ~ m i l i v cequ;~lion\ o f gcoplly\lch. 2 13 principle - Mach. 5 I 0 - maximuni. 147- 140 - maximum-entropy. I 4 problem - Bcnard. 7 14. 317 - Dean. 255 - Giirtler. 355 - Tuy lor. 25 1 - - viscoeli~s~ic. 257 process -Levy. 3.41 - Markov. 106 projection - HelmI1olt~-Leray, 173 -
Leray. 173, 208 Matie. 189 - Riesz, 337 -
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- naked, 5 19 solution
- asymptotic, 94 stationary, I 6 5 statistical. 3, 37 -weak. 37, 89 - - with decreasing energy, 100 solution mapping, 175, 178 spatial dynamical system, 443 rpectrum, 226,455, 488 - continuous, 270, 3 10, 477, 48 1 - discrete, 306. 338 -essential, 271, 307, 320. 444,458, 480, 4x2, 483.49 1 - pseudo. 228 - spatial, 24 1 spiral, 256 - interpenetrating. 257 stability - hydrodynamic, 29 1 - linear. 296. 306 - Lyapunov. 296 - magnelohydrodynamic. 29 1 - nonlinn~r.229. 296. 336 - speclral. 200 stagnation point. ?2X - ellip~ic.328. 320 - hyperbolic, 325. 328. 330 \t;itic. \i~?gul;lr.i s o t h e r ~ ~ sphcrc. ~al 561 \tati\lic;ll cquilihr~um\late\. 23 slicking parliclcs. IOU \trcanl I'unclion. 200. 307 strcamwi\c s~reah\,265 \trict invar~ance.I80 \uhgroup - isotropy. 248 Sun. 257 \urficc tenrion, 446 syrnlnelry - O(2). 237. 756, 480 - SO(?). 256 \yhtem - Eulcr-Poisson. 5 I 2 -
quasi-periodic, 255 - solutions, 468. 469, 47 1 quasidifferentials. 19 1 random flow. 405 - walk. 102 Rayleigh stability criterion. 297. 308. 335. 344 regularity. 208 relations - di\persion. 455 - Rankine-Hugoniot jump. 520 relaxation tc~wardsthe equilibrium, 31 resonance\. 461 - infinitely many. 493 rever.;ihility. 449. 452 -symmetry. 361 rcver\ible. 44'). 45 1. 454 - dynumical sy\tem. 344.446 - nornlal t ' o r ~ n 40 . I rihhon\. 256 Ricci - curvature. 520 - \calar curvature. 520 Rienl:o11 ellipsoid\. 345. 336 rolls. 237. 245. 248 -wavy. 248 relating. 207. 33 1 - cylinder\. 25 1 - f'rrlrnc. 207 rough initial data. 165 Ruelle-Taken\ scenario. 255 -
salt finger. 248 \caling voriuhle\. 163 second fundan1cnt;tl form. 52 1 self-organi~aticr11.3. 4. 7 helnlgroup. 175. 177. 178 -analytic. 226 - Co. 226 - identity. 178 - operator. 226 set - attracting. 179 - o~ncgcl-limit. 187 - perfect. 94 - regular invariant. 206. 207. 2 13. 2 14 - strictly invariant. 206 similarity solutions. 278 singularity
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tachocline. 400. 410. 41 2 tensor. 50.5 - Eimtein curvature, 510 - g r i t ~ ~ ~ t ; ~metric. t ~ o ~ ~.504 ;~l - pseudo. 404 - Riemann curvature. 509. 520 -\Ires\ energy. 5 I 0 theorem - Baldi'\ I;trge devi;~tion. 10
-Cowling'\, 358, 376, 378-380 - Mace, 1x9. 194 theory - Monge-Kantorovich, 79 - rapid distortion. 3 14 topological entropy. 425, 426 toroidal-poloidd decomposition. 372-374. 378 trajectory. I80 - attractor. 184. 2 12 transfornmation - Lorentz. 36'2. 364 - near-ident~ty.I I 9 - Squire, 263. 266. 269. 272 transient. 266 g r o w t h . 227 transport - equation. 292, 3 17 - parallel. 504 - \calnr. 390. 402 turbulence. 3 1 1 turbulent - bur\[. 205 - flow\. X')
- Kirchhoff's, 305 - Kirchhoff-Kida's. 33 I - modulated wavy. 255 - Oseen. 165 - rings. 302, 33 1 - streamwise. 266 - Taylor. 253 -tube stretching. 328
- twi\ted. 256 - wavy. 254.256 vorlicity. 91 -equation. 146. 293. 307. 325. 339. 366 - formulation. 145 wall \lip. 268 wave - bamboo. 273 cnoidal. 464 - corkscrew. 273 dynamo, 407, 4 1 1 - generali~edsolit;~ry. 458 i n :In ice plate, 459 - intcrhcial. 273 tnultihurnp \olit;lry. 473 - \hock. Xc) aolilalry. 458. 407. 1 x 3 - ataniling. 49 I .Toll~ii~cn-Schl~cIlrt~~ig. 264 - tra\,ell~ng.444 - thrce-dinicn\iot1;11. 4x5 watcl-, 443 W K B a\ylliptotlcs. 31X. 323. 332 -
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vortex - tlilfu\ive.
152 c l l ~ p l ~colutiitial-. c 32') - Hill'\ \pherical. 301. 328 1,' - 1.1' e\titii:tte\. 161 - A . x1x Klda'\. 30 1 -
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