NONLINEAR EVOLUTION EQUATIONS
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Series on Advances in Mathematics for Applied Sciences - Vol. 10
NONLINEAR EVOLUTION EQUATIONS Kinetic Approach Nina €3. Maslova Institute of Oceanology St. Petersburg Branch
vpy r l d Scientific
~ngapore New Jersey* London Hong Kong
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NONLINEAR EVOLUTION EQUATIONS. KINETIC APPROACH Copyright O 1993 by World Scienufic Publisl~ingCo. Pte. Ltd. All rights reserved. This book, or parts thereof; may nor be reproduced in any form orby miy means, electrorzic or ~~tccha~iical, iticludingphorocopyi~ig, recordi~igorany information storage m ~ drefrieval system now known or to be invented, without written permission from rhe Publisher.
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Acknowledgments
I am indebted to my colleagues from seminars on mathematical physics in St.Petersburg and in Moscow and on dynamic systems in Moscow, from mathematical Departments of Universitat Kaiserslautern and Universite' Paris VII and to students in St.Petersburg and Kaiserslautern Universities. In particular, I would like to express M.S.BIRMAN,JA.G.SINAY, my hearty thanks to Professors O.A.LADYZHENSKAYA, H.NEUNZERT,C.BARDOSand R.ILLNER.Last, but not least my gratitude goes to my daughter, LENAfor the tedious work of preparation the manuscript.
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Contents 1 Introduction 2
1
.
Kinetic approximations of nonlinear evolution equations Formal constructions 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Kinetic approximations of hyperbolic equations . Formal constructions 2.1 Scalar hyperbolic equations . . . . . . . . . . . . . . . . . . . 2.2 Hyperbolic systems and the Euler equations . . . . . . . . . . 3 Parabolic equations. Formal constructions . . . . . . . . . . . . . . .
.
3 Boltzmann equation Collision operators
1 2 3
Boltzmann equation . . . . . . . . . . . Nonlinear collision operators . . . . . . . . Linearized collision operators . . . . . . .
4 Transport operators 1 Stationary transport operators Nonstationary problems . . . 2 3 Velocity averages . . . . . . . 5
.............. ............... ...............
5
5 12 12 14 16 21 21 24 32 37
. . . . . . . . . . . . . . . . . . . . . . 37 . . . . . . . . . . . . . . . . . . . . . . 44 . . . . . . . . . . . . . . . . . . . . . . 48
Steady-state solutions of the linear Boltzmann equation Solutions of the Boltzmann equation in the whole space . . . . . . . . 1 1.1 A priori estimates . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Asymptotics for small 151 . . . . . . . . . . . . . . . . . . . . . 1.3 Asymptotics for large 1x1 . . . . . . . . . . . . . . . . . . . . . The incompressible limit and the Hilbert series . . . . . . . . . . . . . 2 Stationary solutions of the linear Boltzmann equation in a bounded 3 domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The basic function spaces . . . . . . . . . . . . . . . . . . . 3.3 Main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steady-state solutions in an unbounded domain . . . . . . . . . . . . 4
vii
...
vlu
5
Contents
4.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Asymptotics . . . . . . . . . . . . . . . . . . . . . . 4.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-dimensional problems . . . . . . . . . . . . . . . . . . . . . . 5.1 Setting of the problem . Main result . . . . ......... 5.2 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . 5.3 Sources and asymptotics . . . . . . . . . . . . . . . . . . .
73 76 78 79 84 84 87 92
6 Nonlinear steady-state problems 95 1 Bounded domains . Small perturbations of Maxwellian distributions . 95 1.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 1.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2 Steady solutions in unbounded domains . Asymptotics in the trace regions105 2.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2.2 Solutions in R3 . . . . . . . . . . . . . . . . . . . . . . . 108 2.3 Boundary value problems . . . . . . . . . . . . . . . . . . 115 3 Steady solutions near vacuum . . . . . . . . . . . . . . . . . . . . . . 120 3.1 Three-dimensional problems . . . . . . . . . . . . . . . . . . . 121 3.2 The Couette problem . . . . . . . . . . . . . . . . . . . . 126
7 Initial-boundary value problems for t h e B o l t z m a n n e q u a t i o n 1 2 3 4 5
6 7
8
9
The Boltzmann semigroup . . . . . . . . . . . . ........ Near equilibrium global solutions of the Cauchy problem . . . . . . Periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solutions in La . . . . . . . . . . . . . . . . . . . . . . . . . Initial-boundary value problems . Global solutions . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Linear problem . . . . . . . . . . . . . . . . . . . . . 5.3 Nonlinear problem . . . . . . . . . . . . . . . . . . . . . Local solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global solutions with large isitial data . . . . . . . . . . . . . . . . 7.1 Relaxation in a spatially homogeneous gas . . . . . . . . . . . 7.2 Povzner's theorem . . . . . . . . . . . . . . . . . . . . . . .................... 7.3 Di Perna-Lions theorem Incompressible limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The Cauchy problem . Uniform estimates . . . . . . . . . . . . 8.2 Periodic initial data . . . ................... 8.3 Convergence results . . . . . . . . . . . . . . . . . . . . . . . Long-time behavior problems . . . . . . . . . . . . . . . . . . . .
131 132 139 144 146 150 150 151 157 158 161 162 165 167 168 168 170 170 172
Contents 8 Statistical solutions of Euler equations
1 2 3 4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Fridman-I<ellerequations . . . . . . . . . . . . . . . . . . . . . . Kinetic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mainresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography
ix 177 177 178 180 183 189
Chapter 1 Introduction The primary focus of these notes is a description of some mathematical results concerning the properties of kinetic equations. Therefore one may ask for a compelling reason to use the broad title 'nonlinear evolution equations'. The following remarks are intended to give a motivation. The kinetic equations have a long and fascinating history dating back to papers by Maxwell and Boltzmann, where some fundamental ideas of probabilistic reasoning have been introduced in physics. An understanding and development of these ideas brought to appearance a wide class of nonlinear equations, similar in structure. The modern list of physical situations, where these equations are used is rather impressive. They arise as reasonable models in the Fluid Dynamics, the dynamics of rarefied gases, Nuclear Physics, Plasma Physics, Semiconductors etc. This provides one of the reasons to investigate this type of equations, though this reason is an external one with respect to mathematics. The following three lines of development correspond to more essential connections of the theory of kinetic equations with other domains of mathematics. The first direction is connected with problems of statistical mechanics. Basic objects in this domain are probability measures on the phase spaces of Hamiltonian systems. Traditionally, statistical mechanics is divided into equilibrium and nonequilibrium parts. In equilibrium statistical mechanics one studies properties of a special class of invariant measures, distinguished by the Gibbs postulate. This is a well developed field of mathematical study with well established mathematical tools. The most outstanding achievements in recent years are connected with applications to the study of dynamic systems with hyperbolic properties. Evolution of probability measures is a basic problem of nonequilibrium statistical mechanics. Kinetic equations appear in studying this problem as approximations of moments of the probability measures and are intended to describe a long-time behavior of the corresponding dynamic systems. The history of this topic is connected with the names of Zermelo, Poincare, Gibbs and is related to the known disputes on the relationship of reversibility and irreversibility. The problem of mathematical
2
Nonlinear evolution equations
study initiated in recent years consists of mathematical understanding and estimating the accuracy of the existing physical models, in particular, justification of the kinetic equations applied in ~hysics.The most outstanding results were obtained by M.Kac and 0.E.Lanford. The second direction is a study of initial-boundary value problems of kinetic equations applied in physics. This direction has its own history and great names. The first exact results on the spectrum of collision operators were obtained at the beginning of the century by Hilbert. The first theorems were proved by Carleman in 1932. Global solvability of the Cauchy problem for a wide class of nonlinear kinetic equations was proved recently by R.J.Di Perna and P.L.Lions [20]. At the heart of the third direction is a deep connection between kinetic equations and fluid dynamics theory. A 'derivation' of the fundamental fluid dynamic equations from the Boltzmann equation, a hundred years ago, was a strong argument in favor of validity of kinetic approach. Basic physical arguments for this derivation were given by Maxwell and Boltzmann. It was again Hilbert who suggested the first mathematical setting of the problem. However, both physical arguments and the results obtained by Hilbert were based on the hypothesis that there exist smooth stable solutions of the fluid dynamic equations. In the cases where this hypothesis is true, the Boltzmann equation has a smooth unique solution and the physical arguments are already confirmed by mathematical theorems. But now it is well known that the most interesting solutions of the fluid dynamic equations are not "good" in this sense, that is,. they- are certainly unstable, and at least some of them are known to be nonsmooth. In this sense the fundamental problem of realizing connections between the kinetic and fluid dynamic equations is still open, even though a very general existence theorem is proved. On the other hand ~hysicalarguments and numerical experiments give convincing evidence, that kinetic approximations reproduce the main features of the 'bad', but challenging solutions discussed above [53, 541. As an unexpected marginal result of investigations in this direction, two remarkable facts that became clear, that might turn kinetic equations not from just one of the special cases of nonlinear equations into a tool for solving some general problems. First, it turned out that fluid dynamic equations can be approximated by kinetic models, not motivated physically at all, but relatively simple from the mathematical and computational points of view. Thus we have a wide class of new approximations to the Euler and the Navier-Stokes equations, ensured by convergence proofs for simple cases and by numerical experiments for the hard ones. Secondly, reasonable kinetic approximations are available not only for fluid dynamics, but for a wide class of nonlinear evolution equations beside this field as well. The very idea of the kinetic approach is in a sense opposite to that of the averaging methods. That is, simplifications are expected to arise from enlargement of dimension of a problem. More precisely, for a complicated low-dimensional process its more simple and clear high-dimensional counterpart is looked for. The basic scheme of this type of analytical models is as follows:
Chapter 1. Introduction
3
To study a given partial differential equation for a function m(t,x), t > 0, x E the additional parameter v E Rv in such a way thal the averages
R c Rd we look for a function f (t, x, v) which depends on
(0.1) is a good approximation for m. Vector funclions m are treated similarly with appropriate weight functions in (0.1). Borrowing terminology and ideology from the kinetic theory, the function f is interpreted as a distribution of particles in coordinates x and velocities v. Kinetic equation for the function f describes changes of the 'distribution function' f under the influence of free motion and collisions. Another approach (not discussed here) is based on stochastic models simulating hydrodynamics flows [18]. One of the most remarkable features of the kinetic approximations is that at least some of them are uniform with respect to time. Therefore, they provide a chance to describe the relevant complicated long-time behavior of solutiolls of the Navier-Stolces equations. A more traditional and widely accepted approach to this problem makes use of statistical solutions, i.e., probability measures in function space, describing solutions with random initial data. In this field kinetic equations also naturally come into play, with another physical background, but with a similar mathematical structure. Particularly, they were used to prove existence of global statistical solutions of the three-dimensional Euler equations [49]. I should say, that practically the rather optimistic expectations described above are not yet confirmed by theorems. Moreover, the kinetic equations have not yet provided information about the limiting equations, not available by more traditional approaches. Maybe, the first result of this type is due to P.L.Lions, B.Perthame and E.Tadmor [39], who applied kinetic tools to describe some regularity properties of nonlinear hyperbolic equations. This book is by no means intended to review all the mathematical results in the kinetic theory. In particular, it doesn't describe the last results on strongly nonlinear kinetic equations and global solutions near vacuum. These topics are discussed in detail in [ l l , 121. There also exists a wide class of kinetic equations not considered here (see, for example, [32, 10, 261). The main aim of this book is to present some results that I hope to be useful to clarify the role of kinetic equations in nonlinear dynamics. General formal constructions that allow to find kinetic approximations for a wide class of hyperbolic and parabolic systems are described in Chapter 2. In Chapter 8 it is shown, how kinetic equations could be used in statistical hydromechanics. The main result of this chapter is a proof of global existence of statistical solutions for the three-dimensional Euler equations.
4
Nonlinear evolution equations
The central part of the book presents a rather detailed investigation of the boundary value problems for the Boltzmann equation. I tried make it clear that only a small number of particular properties of the Boltzmann equation is relevant for the results presented in the book. The only exception is Chapter 3, where nonlinear Boltzmann collision operators are investigated in detail and generalization of Carleman's deep results is given. The subject of Chapter 4 is hyperbolic operators describing free transport of particles [transport operators). Their properties are used to construct a family of integral kinetic equations and to study the semigroup, generated by the collision operators. The velocity averages (0.1) represent macroscopic quantities we are concerned with to construct kinetic approximations mentioned above. The regularity properties of .. these quantities are discussed in details. In Chapters 5 and 6 we discuss steady-state solutions, since complete investigation of steady states seems obviously necessary to understand evolution ~roblernsand these questions are practically not resented in the existing books on the Boltzmann equation. Existence and uniqueness theorems regarding these solutions are proved. The most remarkable phenomena in this field are connected with exterior problems. Particularly, steady solutions for a flow past an obstacle have very nontrivial structure. First, Stokes paradox is valid for rarefied gases in the form similar to the linear problems for incompressible fluids. Second, the typical solution has a slowly decreasing component, supported in a union of very thin layers in the trace of the obstacle and in the neighborhood of the Mach cone. Initial boundary-value problems are discussed in Chapter 7. In particular, global solvability of the Cauchy problem with nonsmooth initial data is proved. Some new results regarding fluid dynamics limits are presented. Chapters are divided into sections, and sections into subsections. Theorems, Lemmas, Remarks and Propositions are numbered in one list within each section. For example, Lemma 3.2.1 means Lemma 2.1 of Chapter 3, and it is referred to simply as Lemma 2.1 within the same chapter. Similar system of numbering and references is used for formulas. I didn't succeed to avoid tedious calculations, so the book contains a lot of estimates with various constants. Each formula including C with or without indexes is to be understood in the following sense: there exists a positive constant C , independent of the parameters involved into consideration. I hope that this notation will not generate confusions.
Chapter 2 Kinetic approximations of nonlinear evolution equations. Formal constructions 1
Introduction
Consider an evolution eauation
for the function m(t, x), t 2 0, x E R, c Rt. We look for a more simple equation for a scalar nonnegative function f ( t ,x, v), depending on an additional parameter v E Rt in such a way that the average
with some weight function cp would be a good approximation for function m. Assume that the function f evolves according to
The first term on the right-hand side describes the influence of the free motion of the particles along the path dx dt = 0, while their interaction is governed by the nonlinear operator J. It is supposed that this operator acts on f as a function of v only. A function f can be considered as the density of particles in coordinates x (x E R,) and velocities v (v E R,). More precisely, this function describes the density of a distribution of particles in the phase space 0, x R, at the time moment t.
Chapter 2. Kinetic approximations
6
Denote the average with respect to the variable v for any function cp by
((P):
In the kinetic approximations of the fluid dynamic equations the following functions have special meaning:
These functions are related to the physical laws of mass, momentum and energy. The operator J is constructed in these problems in such a way that the relations
are satisfied. These relations imply (at least formally) that the solution of (1.2) satisfies tlie 'conservation laws':
a
dt(f$j)= -div,(fv$i)
( j = O,.. .,d
+ 1).
(1.5)
The averages (f $j) have a clear physical interpretation, which connects the equation (1.5) with fluid dynamic equations. More precisely, set
The functions p, u , 0 can be viewed as the density, velocity and temperature of the fluid. Denote by $ the vector with components lClo,. . . ,$d+l and introduce the corresponding averaging operator
The operator m connects the distribution function f with its fluid dynamics moments. In papers of Maxwell for the first time in literature an infinite chain of equations for the moments ( p , (v) f ) with polynomials p, appeared and tlie problem of the fluid dynamic foundation was formulated as a problem of 'closing' this chain. Equations (1.5) are the first equations of the above mentioned chain. Together with the functions (f$,) the higher moments (f$,v) figure in the system (1.5). Averaging (1.2) with the weight GJv leads to equations with the higher moments (f $,v 8 v), (f $, ,,lv12v). One should expect that a 'synchronization' of higher moments occurs when t -r oo , i.e. for sufficiently large t the evolution is governed by the change in time of the fluid
7
1. Introduction
dynamics moments m f . The physical and mathematical nature of this process is not clear so far. Hilbert's idea is to study the asymptotics of the scaled kinetic equation (1.2)
with a small positive parameter e . Hilbert considered the Boltzmann kinetic equation with 0, = Ri, R, = Rt. The Boltzmann nonlinear operator enjoys the following property: J(f) =0
u f = exp
Eaj$j(v)
1
with constant coefficients aj. The function f in (1.9) is the Maxwellian. It can be written in the following form:
with the parameters p, 8, u defined by (1.6). The equation (1.8) is obtained from (1.2) by the change of variables t + &-It,
x -+ c-lx.
(1.11)
The scaling (1.11) can be viewed as a transition from the 'microscopic' scale, associated with the kinetic equation (1.1), to the 'macroscopic' scale, corresponding to fluid dynamics. Denote the solutions of the equation (1.8) by f('). Formally, the limit passage E -+ 0 in (1.8) implies that the limit function f ( O ) is defined by (1.10) where the parameters p, u, 8 depend on the time and space variables t , x. For this limit function f ( O ) the higher moments are determined by the following relations:
These relations 'solve' the closing problem, the limit passage E -+ 0 in (1.5) leads to the following system for the fluid dynamics moments
where A is the column vector t(Ao,. . . ,Ad+l) with
8
Chapter 2. Kinetic approximations
The system (1.12), (1.13) is the Euler equations for compressible fluid with a pressure given by (1.14). Assume that we can validate the limiting passages used above. It follows that we can use the Boltzmann equation as an approximation to solve the system (1.14). We are going to show that
(i) it is possible to use more simple kinetic equations instead of the Boltzmann equation to approximate the system (1.12), (ii) there exists a large class of nonlinear equations (1.1) which admits the similar kinetic approxiniations. The formal constructions for hyperbolic equation are discussed in Section 2. To describe the approximations of parabolic equations we consider equation ( 1 . 1 ) with the operator J, which enjoys the property ( 1 . 9 ) . The Maxwellian distribution (1.10) with constant parameters p, u, 0 is an exact solution of ( 1 . 1 ) . Consider solutions of ( 1 . 1 ) which are close to
Let H be the Hilbert space of complex-valued functions f ( v ) on Rt with the inner product
(1.16) Linearization of (1.1) M leads to the equation
with a linear operator L which operates on f as function v only. If .I is the Boltzmann's nonlinear operator then L is a selfadjoint negative semi-definite operator on H with the null-space, spanned by the functions (1.3), i.e.: KerL = span ( $ 0 , . We change the definition of the functions in I<erL: $0
and set
$j
(1.18)
to deal with the orthonormal basis
= 1,
$. = vj $d+l
. . ,G d + l ) .
( j = l , ..., d ) ,
= (2d)-1/2(1v(2- d )
9
1. Introduction
The fluid dynamic moments in this "quasilinear" framework are defined by the projection of the function f on the subspace Poll, while the orthogonal space (I- Po) H contains all "kinetic information". The operator
H corresponds to the fluid dynamics operator (1.7). The components mif = (f, $ J ~ )are related with the deviation of the density, velocity and temperature by the following linear counterpart of (1.7):
Nonlinear equation (1.2) leads to the following equation for the deviations from the Maxwellian (1.15) (1.19) ft = -v. vxf ~f r ( f ) ,
+ +
where the nonlinear operator
r satisfies
due to (1.4). The equation (1.19) is equivalent to the system of equations
In this framework the "closing" problem is to find the connection between the kinetic and fluid dynamic projections of (I- Po)f and Pof . The Hilbert scaling (1.11) does not change the equation (1.20) for the fluid dynamics projection Pof, while equation (1.21) is transformed into
The large parameter e-' in this equation is the possible source of the synchronization mentioned above. However, one must introduce another scaling to obtain kinetic approximation of parabolic equations. To describe the underlying reasons let us consider the linear equation (1.17). Let Fxdenote the Fourier transform with respect to x E Rd with the dual variable E Rd Then equation (1.17) implies
Chapter 2. Kinetic approximations
10
Under rather general assumptions regarding the operator L the operator L(J) generates a strongly continuous semigroup exp { t i ( ( ) ) in H. fo can be viewed as a solution of the Cauchy problem
In the problems we are concerned with the following properties of the semigroup exp {tf?(())are important. For any ( E Rd the semigroup exp t i ( ( ) has an invari-
{
1
ant subspace H(E) in H which attracts all solutions exponentially. More precisely, there exists a projection P ( t ) in H such that
for some positive constants C1, C z . This type cf behavior is typical for dissipative equations. In nonlinear problems the subspace H(() has usually very complicated structure. But now we deal with linear problem and this space is spanned by the eigenfunctions of the operator L ( 0 . For the linear Boltzmann equation (1.22) the following facts are established [4,24, 611: (i)
H = P([)H, dimP(E)H = d + 1, SUPP
with a finite constant
ej
P ( 0 = {
to,where
being the eigenfunctions of the operator
with the eigenvalues A(():
The function el(() is a divergence free vector in Rd:
while the functions e z , . . . ,ed+l are scalar with simple eigenvalues. (ii) Taylor espansions of the functions Aj, ej can be expressed as
1. Introduction with constant coefficients A:),
(1
< 1 5 n - 1 ) and a bounded function j;?';
(iii)
span {ey), . . . ,efi1) = KerL,
The function
x
in (1.26), (1.27) is the indicator of supp P ( < ) .
It is quite clear, that due to (1.24) the asymptotics of the semigroup exp t l ( [ )
{
does not depend on the detailed information regarding the projection (I - P ( [ ) )fo. On the other hand the properties (1.26), (1.27) show that the long-time behavior
{
1
of the semigroup .Fllexp t L ( < ) is governed by partial differential equations. The type of these equations depends crucially on the choice of space-time scale. In fact, let us consider the following scaling:
where the parameter I describes the ratio of spatial and time scales. The corresponding transformation for the operator C(<)and its eigenvalues is defined by
if 1 = 1. This implies Observe that the transformation (1.31) does not change )A: that the evolution is determined by the Laplace operator. This is the reason why the scaling (1.30) with I = 1 leads to the Navier-Stokes equation. This fact was observed in the context of the nonlinear kinetic equations in [58, 7,
171. Notice that the source of the complexity of the spectrum structure described above is that all conservation laws are taken into account by the Boltzmann operators. In particular, if KerL is spanned by constants only, the corresponding invariant space P ( < ) H is one-dimensional subspace of H. This example is considered in 2.3, where kinetic approximations of scalar parabolic equations are described.
12
Chapter 2. Kinetic approximations
2
Kinetic approximations of hyperbolic equat ions. Formal constructions
2.1
Scalar hyperbolic equations
Consider a scalar conservation law:
with given functions A : R 1 + R d , G : R: x R: + R1. We are concerned with a kinetic approximation for this equation. For this purpose we introduce the following equation for the function f ( t ,x, v ) , ( t ,x, v ) E R: x Rt x R t , depending on a small positive parameter E :
with a prescribed function g. We define the operator M by
N being a density of a probabilistic measure: N
> 0,
N E L ' ( R ~ ) , ) ~ N ~ ~ L I=( R1.$ )
Assume in addition that v N E L1(R,d). Then
Averaging (2.2) with respect to v one obtains the "conservation law":
Assume that the formal limiting passage in (2.2), (2.4) can be validated. Then for the limit function f(O) the equations (2.2), (2.4) lead to
Thus, we obtain (2.1) if ( g ) = G. The two simplest examples of functions M are:
(i) N l ( v ) = exp {--lvI2/2} C l ( d )
2. Hyperbolic equations
13
x being the indicator of the unit ball in Rd, while Cj(d) are the normalization constants. A different construction was introduced by B.Perthame and E.Tadmor [55]. Assume that A E C1(Rd). Consider a function f (i, x, w), t E &+,x E R:, w E R1 and denote its average with respect to w by (f): ( f ) ( t ,x) =
kl
f ( t , x, w)dw.
Let f evolve according to
a
-f (t, 2,w) = -div,A1(w)f ( t ,x, w) at
+
E-'
[M(f, w) - f (t, x,w)],
(2.5)
where M is the indicator of [0, (f)]. Then
It follows that (2.5) implies (2.4) with v = A'. Formal limit passage e + 0 in (2.4), (2.5) leads to
due to (2.6). Thus we have obtained (2.1) with g = 0 formally. But in this case B.Perthame and E.Tadmor have proved the validity of the formal procedure described above. Moreover, R.L.Lions, B.Perthame and E.Tadmor [39] have obtained new regularity and compactness results for (2.1) with initial data from L1 n Lm, using the kinetic approximations. Similar results are proved for some hyperbolic systems. The important property of the model (2.5) is that the solutions satisfy the entropy condition, i.e. the following holds:
for any convex function S and q defined by q' = S'A'. To see this observe that
This last equality shows, that solutions of (2.5) satisfy
The entropy condition (2.7) is a consequence of this very special feature of the uniform distribution M .
Chapter 2. Kinetic approximations
14
2.2
Hyperbolic systems and the Euler equations
The formal constructions described above can be used to obtain kinetic approximations of hyperbolic systems
Consider, for example, the system
a
-fi at
= -div,v fi
i=1,
..., 1,
+ &-'[Mi(f ) - f;]+ gi f="fIl
...,fr)
with the following operators Mi:
The assumptions (2.3) imply the conservation laws
Formal limiting procedure described above leads from (2.9) to (2.8). Let us return to Compressible Euler Equations (1.12)-(1.14) to discuss variants of kinetic approximations. One of its classical kinetic approximation is given by equation (1.2) with (2.10) J(f) = M(f) - f , M ( f ) being the Maxwellian distribution (1.10) satisfying
These conditions imply (1.5), which is invariant with respect to transformation (2.11). Therefore formally equations (1.8), (2.10), (2.11) lead to (1.12)- (1.13) as & + 0. It is possible to use different isotropic distributions on R:. More precisely, let J(f) be given by (2.10) with M = cuN(P, v - u). Assume that N depends on and Ivl only. Choose the parameters a > O , P > 0,u E Rd to satisfy the conditions (2.11). The above formal arguments show that the limiting moments satisfy the Compressible Euler equations. But the Maxwellian
2. Hyperbolic equations
15
distribution is a unique distribution satisfying the physical entropy conditions. More precisely, define
Lemma 2.1 [14]. The minimization problem
has a unique solution
f = p ( 2 n ~ ) ~ / {~- el ~x1~~ / 2 0 .}
It follows that the solutions of the problem (1.8), (2.10), (2.11) satisfy the entropy inequality
a at
-(flnf)+div,(vflnf)
(2.13)
On the other hand, it was proved by B.Perthame [54] that the problem (2.12) has a unique solution for any convex function h under mild additional assumptions. Assume that fo is a solution of the problem (2.12). Consider the kinetic equation (1.8) with = fo(p,O,Iv-ul)-f (2.14) J( J( f( )f ) & ) = 0, j = O , ...,d + l ' The solutions of the equations (1.8), (2.14) satisfy inequality
(
It follows that the limiting solutions satisfy the entropy inequality
with S(m) = (h(fo)),v(m) = (vhfo). Using the equations with uniform distribution fo we obtain (2.15) for any convex S. It is also possible to construct kinetic approximation of Incompressible Euler equations: du (2.16) - + V,(u 8 u) V,p = 0, div,u = 0.
+
at For this purpose consider the equation (1.8) with
M ( f ) being the Maxwellian distribution (1.10) satisfying
Chapter 2. Kinetic approximations
16
The conservation laws (1.5) have the following form
a
= e-'[l-(f)l,
-(f)+div,(vf)
at
a %(Mj) + divz(v$jf)
= 0.
The limiting equations are given by (2.16).
3
Parabolic equations. Formal constructions
Consider the equation
for m : R: x R:
+ R1
with given functions GI, Go satisfying
G~ E c'(R', R ~ )G~ , E c l ( ~ lR'). , IJet H be the Hilbert space with the inner product (1.16). Define the collision operators L and r by:
where a , p are positive constants. It is clear that L is a bounded selfadjoint operator in H, KerL -Re(f L f )H
+
= span{l) =
11Lf. ;1
Hence, we have Po = L I for the projection Po on Ker L. Let 3,be the Fourier transform with respect to x,
Then (1.19) implies
17
3. Parabolic equations
Lemma 3.1 (i) The operator k ( ( ) is a generator of a strongly continuous contraction semigmup exp t i ( [ ) for any ( E Rd. (ii) The sernigmup exp { t y f ) ) h a s an invariant space P ( 0 H "={ f lP(E)f = f 1 , with a projection P ( 5 ) given by
{
1
x([) being the indicator of Bo = {EllEl < l o ) , to is a positive constant. (iii) The function e(E) is the eigenvector of L ( t ) with a simple eigenvalue A(€): e E Cm(Bo, H ) ,
X E Cm(Bo),
+
X ( 0 = [1E12X(2) O(IE1)3]x ( 0 (iv) There exist positive constants C l , C2such that
It1
+
0.
uniformly in t . The proof of Lemma can be found in Chapter 7, where general properties of the Boltzmann semigroups are considered. Note that having established the assertion (iv) we are free to compute the Taylor expansion of the functions X and e by perturbation theory. It follows from (3.4) that
-ilJl(w v ) e - e Thus X depends on
+ ( e ,1
) =~Xe,
only, while e is a function of
€
=
iTi'
and (w v ) . Let
where Ei denotes the Taylor expansion asymptotic to all orders. We have by equating powers of
Itin
18
Chapter 2. Kinetic approximations
Without loss of generality we choose e(O) = 1. Then e(') = -iw. v.
A solution of (3.7) exists if and only if the right-hand side of (3.7) is orthogonal to the kernel of L. This orthogonality condition is satisfied if and only if ~ ( 2= )
-1.
Lemma 3.1 implies that equation (3.3) is equivalent to the system of equations
with 4 = F z r ( f , f ) . Put ~ ( t =) I ~ ( I - p ( t ) ) j ( t ) l j H .
~ ( t =) I I ( I - P ( 0 ) 6 ( t ) l l ~ .
Due to (3.5) we have y(t)
5 Cl y ( 0 ) exp {-dl +
[
dl
I
z ( T ) exp {-c(t - T ) ) ~ T.
It follows that y2(t)
+
cbt
y 2 ( ~ ) d T5 CI [c-l
z2(r)dr
+ y2(0)] .
(3.10)
On the other hand, by Lemma 3.1
for some positive constant cl uniformly with respect to J, t . The scaling (1.30) with I = 1 transforms equation (3.3) into the following equation
a.
zf
+ t-',
= L ( o ~
kE(J) =
E-'
(3.12)
[L - ~ E .Jv].
The equation (3.9) implies that the function G(t, J ) = (f ( t , J , .), ~ ' ( E J )satisfies )~
3. Parabolic equations Set cr = E ~/? ,= E in (3.2). Then (3.6), (3.8) imply
where
with some positive collstant C uniformly with respect to Dcnotc by fk thc kinetic part of f:
1(1.
Let j be a solulion of the Cauchy problem for (3.12). The estimates (3.11), (3.10) imply the following apriori estimate of f in 3-1 = L2(R:, H ) :
with some positive constant C uniformly with respect to
E.
Hence
Therefore, passing to the limit E -+ 0 in (3.13) we obtain (3.1) under rather reasonable assumptions regarding the functions Go, GI.
Chapter 3 Bolt zmann equation. Collision operators Boltzmann equation
1
Classical Boltzmann equation has the following form:
The operator J represents the influence of binary collisions between gas particles. It is assumed that every collision between two particles with velocities v , vl results in the instant change of their velocities:
v ,V l
+ v', v ; ,
(1.2)
while collisions of more than two particles rarely occur. The probability of transition (1.2) is determined by a prescribed nonnegative symmetric function P :
P ( v , v1, v', v;) = P ( v l ,v ; ,v , V l ) ,
P 2 0.
The collision operator at a point t , x, v is defined by
J ( F ,F ) =
Lg
[FIE;' - FFl] P ( v ,vl, v', v;)dvldv'dv;
(1.3)
with
F' = F ( t ,x, v'), Fi = F ( t ,x, v ; ) ,Fl = F ( t ,x, v ) . Due to conservation laws P is a distribution with the following support
(1.4)
22
Chapter 3.Boltzmann equation. collision operators
After integration the operator (1.3) transforms into
Here S2is the sphere { a E R31 IaJ= 11, endowed with its Lebesgue measure da. We shall use the notations (1.4), but now {vl,v;) is a solution of the algebraic system (1.5), parametrized by a : v1 = v - a(v - v1, a ) ,
v; = v1
+ a(v -
211,
a).
(1.7)
The rotational invariance of the collisions laws implies that Bo) B depends on Iv - vll and I(v - vl, a ) ! only. The precise form of the function B is uniquely determined by the interaction potential between the particles. We consider below the class of hard potential, introduced by H.Grad [30]. This class is determined by the following conditions:
where y E [O,1] and 6; ( i = 0 , l ) are positive constants. Two classical examples are
(i) the hard spheres model with B(v, a ) = I(v, a ) 1, (ii) the inverse power potential U = const. r-' with
where b E C([O, 11) and bl(z) = Z Y - ~ .
From the physical point of view the source of the singularity at z = 0 is the interaction at large distances. If this type of interaction is essential for the problem under consideration, then the Boltzmann equation gives a bad approximation. Replacing the function bl by a constant ('angular cut-off') guarantees the fulfillment of GradYs conditions for s 2 4. For slowly decreasing potentials (s < 4) the conditions BZ,B~ are violated. However for most interaction potentials in a neutral gas these tions are satisfied. In particular, the functions B obtained in quantum n ~ e c l ~ ~ ~ computations enjoy the conditions B2, B3. We shall consider some boundary problems for the Boltzillann equation with the space R x Ri, where R E R:. Let a R be a boundary of the domain $2 with
23
1. Boltzmann equation
the outward normal n(x) at a point x E d o . Let us denote by y + F and 7 - F the densities of distributions of falling and reflected particles on dR respectively. More precisely, set . y + F = Fx(v. n(x)), y - F = F ( l - ~ ( vn(x))), x being the indicator function of (0, a ) :
x =1 x=0
for v . n(x) > 0 x E dR for v . n ( x ) 5 0 x E aR.
Boundary conditions on the surface dR lead to a connection between the functions y*P: R ( y + F ,y-F) = 0 with a prescribed function R . If the domain R is unbounded one adds to the boundary conditions on dR the condition at infinity
with a prescribed function F,. Hence, the following list of boundary problems is associated with the Boltzmann equation (1.1): 1. The internal stationary problem
for a bounded domain R.
2. The exterior stationary problem: to find a function F , satisfying (1.8), (1.9), where R is the complement of the closure of a bounded domain. 3. The nonstationary boundary problem
with a prescribed function Fo. 4. The Cauchy problem (1.10), (1.11) with 0 = R3. Notice that the well-posedness of problem 2 is not clear a priori even formally. Variants of the conditions at infinity are discussed in Chapter 5. We shall also discuss some problems with a prescribed symmetry with respect to spatial variables x, while full three dimensional velocity dependence is allowed. A special role of these problems stems from their applications in the boundary layers problems for complicated flows. One of the classical problem consists in finding a solution of the Cauchy problem, independent of X. Solutions of this problem describe a fast process of approaching local thermodynamic equilibrium state (i.e. Maxwell distribution with variable
Chapter 3.Boltzrnann equation. collision operators
24
2
Nonlinear collision operators
We describe in this section some properties of the nonlinear collision operators, which are essential in investigations of the boundary-value problems. Through this section we consider the functions f : R: + R1, since the collision integrals operate on f as a function of v only. Define the following bilinear operators:
The last equality determines the frequency of collisions associated with the distribution function g. The following notations are used below:
Due to Bo (see Section 1) the following classical identities are valid for
with compact support:
with the following traditional notations:
f' = f(v')r f; = f(v:),f
= f ( v ) , f i = f(v1).
Due to (1.5) we infer from (2.5)
for the functions $0 = I,$, = vi(j = 1,2,3),$4 = 1vI2. These functions are called collision invariants.
2.Nonlinear collision operators T h e following symmetry property
is a direct consequence of (2.4). Setting in (2.5) cp = In f one obtains
Since (z- 1) ln x 2 0, the last inequality implies
for any continuous positive function f with compact support. Thus any solution of the Boltzmann equation satisfies the local conservation laws (2.1.5) and the entropy inequality (2.2.7). The following remarkable Carleman theorem [14] allows to describe solutions of the equation J ( f , f ) = 0: Carleman's theorem. Any measurable solution of the equation
is a linear function of collision invariants (2.7). Thus the Boltzmann collision operator enjoys the property (2.1.9). All these facts were observed by Boltzmann in his first publications. The next step was made by Carleman. To obtain exact statements regarding solutions of the Boltzmann equation he described function spaces invariant under the action of the collision operators for the kernel B(v, a ) = [(v,a)l, i.e. for the hard spheres model. Below we discuss some properties of the collision operators, generalizing Carleman's results. Since the collision integrals are linear with respect to the kernel B , it is sufficient to consider the function B satisfying
with z = I(v,a)llvl-l, 7 E [Or11, bl > 0 to describe the operators with Grad's potentials. Below we assume that (2.10) is fulfilled. An important property of the collision frequency is its unboundedness for large IvI: ~ ( f= ) O((v17) as lvl + oo. To be precise, the following statement is valid:
Chapter 3.Boltzmann equation. collision operators
26 Lemma 2.1 Assume that f
2 0 , ( 1 + Iv12)f E L 1 ( R i ) ,
for some positive constant bo. Set a = ( f ) , b = ( ( 1 (i) bo(alvlY - b) 5 v ( f ) I bl(alvlr
+ Iv12)f),H = ( f
f ) . Then
+ b)
(ii) there exists a positive constant vo depending on H and a only such that
Proof. The inequality ( i ) follows from definition (2.2) of v( f ) since IIv - vlIY - Ivl-'l I lullr I 1 IvI2.
+
To prove (ii) note that
,-
]R:fIlnfIdv 5 H + b + C for some positive constant C. To observe (2.12) set
Then
Obviously these relations imply (2.12). To proceed further we use the inequality
xy I
$1
In 21
+ ey-'
This implies that
v ( f ) > 2nlr for any 1, X
> 0. Due to (2.13) one gets
x>O,y>O.
(2.13)
2. Nonlinear collision operators
For any fixed X choose I ( X ) in such a way that
then (ii) follows immediately. Following Carleman, we consider the properties of the collision operators in the space of functions with inverse power decreasing for large Ivl. Set
L? and denote by 11 f
[Ir
= { f Ivrf E LW)19 = cp,(v) = (1
the norm of f in
Lemma 2.2 If f , g E Lp, r
>4+7,
+ Iv12)T'2
Ly. then J * ( f , g ) E L & and
with some constant C = C ( r ) independent o f f and g .
Proof. The estimate of J - is a direct consequence of Lemma 2.1. To obtain the estimate of J + we make the change of variables v l ra + p = v - a ( v - vl,cr),q = v1
+ cr(v - v l , a ) .
Then J + (f , g ) is transformed into
where EVpis the plane {ql(v - p, v - q ) = 0) with the Lebesgue measure du. The assumption (2.2) implies for q E E,,
due to Iv - pl 5 Ip - ql. Define operators
Due to (2.14) we have IJ+(f,g)l L I/z-, (IflG(lsl))
Chapter 3.Boltzmann equation. collision operators
28
It is clear that we can consider only nonnegative functions f , g. Moreover, due to symmetry property (2.8) it is sufficient to consider the case f = g. We shall use below repeatedly the following inequalities sup G(f) 5 2=
l0
R h ( R ) d R if f ( v ) 5 h(lv1)
V i ( f ) ~ L pif f ~ L y , k < 3 , r > 6 - k . To verify (2.17), set
Since lpl
> f lvI for p E A , one has
The change of variables p
+z
= (pl(v1-' gives
Since IV
and sup
Iul
<
- P I - * ~ ( P ) ~ PI I ~ I I ~ C I V I - ~
/
~v- P I - * Y ; ~ ( P ) ~ P< m
one gets (2.17). To proceed further for any fixed v we define
f; = f x ; , i = 1,2,3, xi being the indicator function of D;.For any q E E,, we have
Hence
Jql
2 2-'I21vl if p
IP
- ql
E
Dl. Therefore (2.15) implies the equality
2
IP
- vl.
1912 2 lvI2 - Ipl2.
(2.16) (2.17)
2.Nonlinear collision operators Due to symmetry property (2.8) this inequality implies
It follows from (2.16), (2.17) that
for some positive constants C1, Cz. The last inequality completes the proof. The bound stated in the lemma cannot be improved. Therefore the operators J* are unbounded in L r However, the following lemma provides a possibility to control the unboundedness. Lemma 2.3 There exists a positive constant C such that
for f,g E L r with r
> 5.
Proof. It suffices to proceed as before but it is necessary to estimate the terms in (2.19) carefully. We use (2.18), (2.16) to obtain
with
Simple calculations show that
for some positive constant C. The real source of the unboundedness is the last term on the right-hand side of (2.19). We use again (2.16), (2.17) to verify that:
To complete the proof it suffices to notice that
- Iv - pI2] f(p)dp
I (lvI2 + 1)-1+7'2
IlfllLl + C llf llr.
The estimate stated in the lemma will be used in the proof of the existence of a global solution of the Cauchy problem for a space-homogeneousgas and local solutions of some initial-boundary problems. The key estimate in these problems is given by the following lemma
Chapter 3.Boltzrnann equation. collision operators
30
L e m m a 2.4 Assume (2.10), (2.11). Then there exists a positive constant C that ~ + ( f ), [.(f) 11-'II7 I 2b1bi1(r - 2)-I I l f 11, llfllf
11
S U C ~
+ cz-'
+
for all 1 > 0 and f E L g . We have established that the collision operators J* are unbounded in Lp. The source of unboundedness is the collisions generating the particles with high velocities. On the other hand the collision operators bring some regularity in compact sets of the velocity space. The precise sense of this statement explains the following lemma. Define
L e m m a 2.5 For any convex function h we have
Vk ( h J + ( f 7 g ) )5 C ( 2 - k)-' [ h ( h f ) l / k - - , ( g )+ h ( g ) V k - - , ( h f ) ] with some positive constant C . Proof. We use (2.4) with
p ( v ) = h ( v ) ( w- V I - ~ . The change of variables f f -+
q = v - f f ( v- v * , f f )
transforms (2.4) into
with
I
=
J
-V I - ~ ~ O .
Iivul
Here I(,,, is the sphere
Due to (2.10)
14 - v l - ' B
i bilv - vllr-'
Therefore
Vk ( hJt ( f , g ) ) I bl
/
RJx R3
for
q E I<,,.
f (v)g(vl)lv- v l l ' - ' h d v d ~ ~ ,
2.Nonlinear collision operators where
Since
IW
- ql-'do
5 C(2 - k)-llv - v112-k
the convexity of h implies the Lemma. Let E2 b e a plane in R3, denote by d(w) the distance between a fixed point w E R3 and Ez. Using cp = (:)'I2 exp { -ad2(w)) in (2.20) we deduce taking the limit a + w, that
If initial and boundary conditions are near a Maxwellian distribution, one can expect an exponential decreasing of solutions for large Ivl. However, if f = exp {-slv12}, then J + (f , f ) = fv(f). Then Lemma 2.1 implies that J + ( f , f ) = 0 (Jvlrexp {-~lv1~})for large Ivl. That is the reason to consider the space
with the weight function
Lemma 2.6 If f E LTr, s > 0,r positive constant C such that
> 2, then J + ( f )
E Lr7. Moreover, there ezists a
II J+(f, f)ll,, 5 c llf ll:, . Proof. We proceed as in the proof of Lemma 2.2. For Ipl 5 Ivl/& one has
Thus On the other hand J'(f2,f~) Since G(f3)
5 cp-'(v)(l+ l ~ l ) - " + ~ + ~ .
5 Ccp-I 11f ]I, , one obtains
Chapter 3.Boltzmann equation. collision operators
32
These estimates yield the lemma. We conclude by the following simple LP estimate. Set x r
= {f
I ~ s , r Ef Lp)
3
Ilf ll,
= IIVs,rf IlLP .
Lemma 2.7 Assume that
B(v, a ) I b11~1', p E (1, rn), s > 0, r + p C(s, r p, 7,p) such that
+
+7
11J*
Y
> 0,
(2.25)
2 0. Then there exists a positive constant C =
5 C IIf IIo
IIf
IIr+p+7
(2.26)
for all f E X,+p+? Proof. Set
f = fy,o.
By Holder inequality we have
Due to (2.25) we obtain
5c~~s,T
(2.28)
for some positive constant C1. Using (2.4), (2.27), (2.28) we conclude that
Then (1.7) implies (2.26) for J+ immediately. Since
IJ-(f,f)I
5 f(v)(lvl+ 1)7C2 I I f I I O
the inequality (2.26) for J- is obvious.
3
Linearized collision operators
A Maxwell distribution (2.1.10) for any constant values of parameters p , u , 9 is an exact solution of the Boltzmann equation due to (2.1.9). A large area of physical applications and mathematical investigations is related to study of perturbations of the Maxwellian. Any Maxwellian with constant parameters can be normalized. Thus it is sufficient to deal with perturbations of
3. Linearized collision operators We change an unknown function in (1.1)
F
-t
M(1
+f).
Then the Boltzmann equation is transformed into
where the collision operators L and
are defined by
+
M L f = J ( M , M f ) J ( M f ,M ) M r ( f ,g) = J W f M , g ) J(Ms' M f ) .
+
r.
We collect in this section some basic properties of L and Assume that the Grad's conditions Bo - B3 are satisfied. Let H be the Hilbert space H introduced in (2.1.5). Recall that
( f ~ I H= 7
l l f llH
=
J
f (v)g*(v)M(v)dv,
(f,f)z2. R3
(3.4)
Lemma 3.1 1. There exists a positive function v such that the operator I i = L+v is a compact selfadjoint operator in H .
2. The function v depends on Ivl only and satisfies
0 < Yo 1 V ( V ) 1 v1(1
+ IvI)-',
v E CI,,(~)
for some positive constants vo, vl .
3. ICerL = span ( 1 ,v , l ~ ) ~ } .
4.
Let
{$j)j=o
be the orthogonal basis in I'lerL.
There exists a positive constant 1 such that
5. Set cp, = (1
+ Ivl)'.
The operator cp,+llzKcp, is bounded in H
6. L commutes with rotations in R:.
(3.5)
Chapter 3.Boltzmann equation. collision operators
34
Exact formulae for v and It' can be obtained from (3.2). Set
where
Y
is defined by (2.2). Then
Thus the assertion (3.5) in the Lemma is a direct consequence of Bo - B3. The exact expressions for the kernel of It' are calculated by Hilbert for the hard spheres model:
for some positive constants C I , C 2 . These kernels are majorants for kernels of the operators I i l , Ii'2 generated by L with the functions B from the Grad's class, (3.12) It'j(v, v l ) 5 It'J(v, 1 ) . Define the functions
where It'j(v,vl) are the kernels of the operators I$. Then due to (3.11)-(3.13)
Let X R be the indicator function of { v llvl a compact operator in H and
+
5 R } . Due to (3.11)-(3.13)xRIt'xR is
11(1 - X R ) I { ~ IIIt' ~ ~ ( 1 - X R ) It'll -+ 0 as R
+ w.
Therefore K is a compact operator in H. The inequality (3.12) implies the assertion 5.
Remark 3.2 It follows from the proof and will be used later, that the operator lIi'1 = It', + I h is compact in H .
3. Linearized collision operators
35
Using (2.5) one obtains
This relation and Carleman's theorem imply the statement 3 and the inequality
To prove (3.8) set
Compactness of K and (3.14) imply
X <
1, ( ( I - p o ) f , I ( ( I - P ~ ) f )5~ X ( l v 1 / 2 ( I - ~ o ) f ( l L
This completes the proof. The assertion 5 in the Lemma is a direct consequence of (3.11)-(3.12). The assertion 1 was proved by Grad. He also proved some other useful estimates. The generalization of his estimates was found in [62]. In particular, he proved the following Lemma Ukai. The operator M - ' / ~ ~ ~ + ~ I is ~ bounded ~ ~ M -from ' / ~L4 into P ifl
Lemma 2.6 implies the well-known Grad's estimate for the nonlinear operator I?:
+
for p, = ( 1 I V ~ ) ~ M ' / ~r (>V2. ), On the other hand Lemma 2.7 gives the following LP estimate for this operator
for p E ( 1 , co), r
> 0, p + 1 > 0, cp,
= (1
+ I V ~ ) ' M ~In/ ~particular, it follows that
Chapter 3.Boltzmann equation. collision operators
36
+
for cp, = (1 Ivl)',r 2 0. The estimates of this type were observed and proved by A.Lukshin 1401. The conservation laws (2.6) imply the relations
for f E Cl,,(R:) with a compact support. Due to Lemma 3.1 and the inequality (3.17) these conservation laws are valid for any function f satisfying the condition
We shall use below some properties of the resolvent R(A, L) = (A particular, since R(A, L) = (A v)-'II'R(A, L) (A v)-',
+
- L)-'.
In
+ +
R(0, L) = -L-l is continuous in ( I - Po) H n {f Ip, f E H } for some r 2 0.
Lemma 3.3 L-' is continuous in (I- Po) H n { f Ip, f E H } for any r 2 0. Set
Due to Lemma 3.1 cp,b,
cp,a E H
Vr E R1
while the assertion 6 of Lemma 3.1 implies that the following representation is valid:
with some scalar measurable functions a , P. The quantities 7 '( f y b ' ) ~ , ?! = ( f l a ' ) ~ are called the stressed tensor and the heat vector correspondingly, while the positive quantities
are called the viscosity and heat conductivity coefficients respectively. Notice also that due to (2.1.9)
Chapter 4 Transport operators The linear transport operators
are the basic differential operators in all kinetic models mentioned above. Integrating over characteristics of the operators D, T reduces the boundary problems for kinetic equalions to a system of integral equations. We use the system to construct approximations for the solutions and to prove existence theorems. Properties of the integral operators involved are discussed in Sections 1,2 for stationary and time-dependent problems respectively. We deal in the boundary problems discussed in 111.1 with families of the operators D, T depending on a 'random7 velocity. Regularity of the velocity averages (3.2.3) of solutions of hyperbolic linear transport equations plays a key role in proving existence theorems for nonlinear problems. Formulation given in [29] are perhaps the most successful and reflect in the best way the essence of the matter. A series of propositions concerning this property are described in Section 3.
1
Stationary transport operators
Let us associate with the boundary problem (3.1.9) the following linear problem
with prescribed nonnegative functions h,g, f-. The reduction to integral equations is determined by the choice of the operators h = h( f ) and g(f ) = J ( f , f ) fh(f). For any fixed x, v (x E Rf, v E R;) define
+
38
Chapter 4. Transport operators
Assume that R ( x ,v ) admits the following representation for almost all x , V :
with some integer N = N ( x , v), 's = s f ( x , v ) ,
For a convex bounded domains this is valid with N = 1. Denote by ffl the following restriction of f on O ( x ,v ) :
Then problem (1.1) is transformed into
ff(s;,x,v)= f-(s;,x,v)
for
ST>-00.
One adds to these relations the condition
if s; = -03. The solution of the problem (1.2)-(1.4)is determined by
where W and U are the solution operators for the problems
(W) (U)
Df+hf=O Df+hf=g
y-f=fy-f=0.
There are no problems with explicit expressions at this step. First, if s E ( S T , s?) we have
where
1.stationary transport operators
39
If ST = -00, we set
The functions (1.5)-(1.7)are solutions of the problems (W), (U) in the sense of distributions. We will describe their properties assuming that g E where
with a positive measure p in R;. The norm in the Banach space
LPgq
is defined by
To introduce the suitable space for the boundary functions consider the orthogonal projection R ( v ) of R along v:
Then the Fubini theorem gives for f E L1~'
where dC stands for the Lebesgue measure of the plane R(v). We define
n ( x ) being the unit outward normal at x E an. Denote by dS the surface measure on an. Then one obtains
if
Iv.
nl f E L1(S,dSdp). Therefore it is reasonable to describe the boundary functions in terms of the spaces
Chapter 4. Transport operators
40 with the measure a defined by d a = 10. n(.x)ldS. We shall also use subspaces of L!/
defined by
L P ~= * {
f
E LP," l r f f = f
)
Leln~na1.1 Assume that
h E LE (R:,P; ~ " ( o ) ) , h 2 0
Then the following inequalities are valid
Proof. Use the following elementary inequalities. Suppose, that
Then
Apply (1.14), (1.15) to (1.2) and use (1.9), (1.10). The result follows. Remark. It is useful to notice that the function f = W f Ug is a solution of the problem div,vf+hf = g X E R x E 8 0 , v . n(x) < 0. f =f-
+
1.stationary transport operators
41
Assume that 52 is a convex bounded domain with smooth boundary. Integrating over R gives
=
1
gdx
+
1
~ 5 2 l vn(x)ly. fdS.
Droping the first positive term on the left-hand side we obtain (1.11),(1.12) for p = 1. We shall construct solutions of nonlinear problem
satisfying the following conditions
h-l1ptg(f) E L P * ~ ]. Lemma 1.1 implies
+
W& = ( f lh-'~~'(Df hf) E Lpvq,7-f E Lgq] The functions from W,;, enjoy the following properties f f l is absolutely continuous on R(x, v) n (1x1 < R} for any R < ca and for almost all x, v
Due to (i) the limits lims,,;t f H lim,,,; , f f l exist for almost all x, v. Thus the traces are determined for the functions from W i qin a natural way. However to deal with the general boundary conditions (3.1.9) we need a description of the traces r+f in terms of Lg9 spaces. The following lemma gives such description. L e m m a 1.2 Let the assumptions of Lemma 1.1 be valid. Then we have 117+wf-
/Iq.
5
llf - 11~).
(1.16) (1.17)
42
Chapter 4. Transport operators
Proof. Use (1.14) to obtain (1.16), (1.17) for p = W . In virtue of (1.15)
Applying (1.9), (1.10), we get (1.16)' (1.17) for 1 5 p < W . Define Af=Df+hf, A'f=-Df+hf,
Lemma 1.1 yields
hlIpf E LP",
if f E W L ~ .
In virtue of Lemma 1.2 we have
~ + Ef L y , if f
E Wi,.
By the divergence theorem we obtain the Green's formula in the following form. Assume that 1 1 flEW;, ~ z E W , ; ,- + - = I i
P
P'
Then
( A f l ,f2)x - ( f l ,A'f2)x = (7+fl,- / + f 2 ) ~ - ( y - f i , 7 - f z ) ~ , (1.18) where X = L2v2, Y = s . For later purposes we state the following strengthened variant of Lemma 1.1. Define pr = ( ~ r ( x , v=) ( 1 (xI2 1 ~ 1 ~ ) r ~ E~ R1. ~ ,
+ +
Lemma 1.3 Let the assumptions of Lemma 1.1 be valid. Assume in addition that
vrf- E L y ,
p,g E LP99.
Then llhl'pprwf-
llLPsq
5 C ( r ) llv7f-
with some constant depending on r only.
43
1.stationary transport operators
To prove this Lemma it is sufficient to repeat the proof of Lemma 1.1 taking into account the inequality
Similar variations are valid for the estimates (1.16), (1.17). Namely,
The following consequences will be used in the construction of the solutions of stationary problems in unbounded domains. Let Rd \ 0 be a bounded domain, SR = {x E 0 11x1 = R ) . Denote
Lemma 1.4 Assume that vrg E L212,
(pr f - E
L > ~ , inf h > 0.
Then Ilvr7if
11,
+
as R
0
+
m.
The arguments mentioned above show that it is sufficient to consider the case h = 1, r = 0. It follows from (1.20), that
where X R is the indicator of the set {x 11x1 > R ) . Therefore
Applying the Green's formula in the domain
shows that the function
1:
F(R) = II7,+f
+ Ilrif 11;
44
Chapter 4. Transport operators
has a finite limit as R -t oo,since F ( R 1 )- F ( R 2 ) -t 0 as R1,R2 -t co. To show that F ( R ) + 0 we find a sequence R , + oo such that F ( & ) -t 0. Notice that f E L212 due to Lemma 1.2. Therefore there exists a sequence R, T oo such that
If
I2dSdp(v) -t 0 as R,,--+ co.
In virtue of (1.19), (1.20) it implies that F(R,)+O
as R , - t o o .
This proves the Lemma.
2
Nonstationary problems
It is quite clear that the approach introduced in Section 1 can be applied to nonstationary problems. We just present the analogues of Lemma 1.1 and Lemma 1.2. First consider the Cauchy problem.
flt=o=
fo,
a
A=-+v.V,.
at
(2.2)
Introduce the function f f l by fl(s, x,v ) = f (s, x
+us,
V )
s E
R1.
Then (2.1), (2.2) transform into
The solution is given by
f
= Wfo
+ ug
with
( T ,s)
Let
= exp
{- [h n ( s l ) d S l }.
2.Nonstationary problems Assume that h E LEG((R, P , L" ([O, TI x R:)) , h
L0
p,r,q E [l,ooI. Then
In addition IlhU~llL-,P,q5 llgllLm,Plq (2.11) if h does not depend on t,x. To prove the estimates (2.7)-(2.10) it suffices to repeat the arguments used in the proof of Lemma 1.1. Note that if the function H in (1.13) doesn't depend on t, then F(t) = F(0) exp {-Ht}
+
I'
G(s)exp {-H(t - s)} ds.
Thus if F(0) = 0 we have HF(t)
5
(41
Gp(s)exp (-H(t - s)} Hds
Applying this inequality to (2.3) we obtain (2.11). Let R be a domain in R t with a regular boundary aR. Consider the problem
with a given operator Set
R and A
=$
+ v . V,.
Q = {t,x,vlt E (0, T ) , x E R, v E R ~ .) In problems with a prescribed incoming flow 7- f the distribution function f is given on a part of a&, defined by = sous-, (2.12)
as-
where
Chapten 4. Transport operators
46 Let W and U be solution operators for the problems
Operators W , U are used to construct solutions of nonlinear problems with g =
d f ) ,f -
=f-(f).
For any point x , v denote the backward stay time by s - ( x , v ) and the forward one by s+(x,v ) . By definition
The problem (2.3) transforms into
It follows from (2.15), (2.16) that
where
aQ+ = dQ \ dQ-. Define a measure v on d Q f by
du=
{ Iv . n(x)ldSdtdp(v) dxd~(v)
if ( t ,x , v) E S f if ( t ,x , v ) E So,
where d S is the Lebesgue measure on 80. By Fubini theorem
Define the Hilbert spaces X, Y* with the inner products
The Green's formula for A
+
+ h has the following form
( ( A+ ' ~ ) f i , f z ) x ( ( A - h ) f I , f ~=) (~f l , f , ) , +
- ( f , , f,),,-.
2.Nonst ationary problems
47
This follows directly from equation (2.1) written in the form
af + div, f v + hf = g . at
(2.17)
Assume that f 2 0, g 2 0. Multiplying (2.17) by p f p - l , we get
where n(x) is the outward normal to a0 at x. We describe some useful properties of operators W,U in the spaces
In order to describe the boundary values of the function f we use the spaces
where the measure X is determined by
The list of properties similar to those described above for the Cauchy problem follows from (2.18). Put = LP,P,q, y , L2P.9, = Lc-J,P,q.
x
Setting in (2.18) g = 0, 7 - f = 0
where Lplq is defined by (1.8), f = Wfo. In order to obtain estimates for f = Ug set in (2.18) fo = 0, y- f = 0. By Holder inequality Ilh1IPflln5 Ilh-l'p'gll,.
Chapter 4. Transport operators
48 Using this estimate we have
Ilrtfll,
(2.21)
Similarly, it is possible to estimate the distributions f = W y - f generated by incoming flow y- f. Setting in (2.18) g = 0, f lt=o = 0 we have
l l f l l z 5 Ilr-fllY II?+f Ill, 5 Ilr-f llY pllP llhllPf 5 Ilr-f
llx
3
(2.22)
llu
Velocity averages
We discuss in this section some results on regularity properties of velocity averages ( f ) for the solutions of the transport equations
It is expected that the averaging brings some additional regularity with respect to spatial and time dependence. The theorems below can be considered as a refinement of this general expectation. We start with the stationary problem
v.V,f + h f =g,
y-f = O
with the solving operator U . For simplicity we consider here the functions h, depending on u only. Assume that the measure p in the velocity space has a density: d p = pdu
E L'(R:).
Denote by E the following averaging operator:
T h e o r e m 3.1 Let R be a compact domain in R,d, 11
E LEc (R:)
p E L' (R:)
inf h
> 0,
V
n
L' (R:) for some r > 1.
Tlaeia EU and U E are compact operators in L P = L P ( R , L P ( R : , ~ ) for )
i
49
3. Velocity averages
Proof. In view of Lemma 1.1 the operators EU and U E are continuous in LP. Therefore it is sufficient to prove the compactness in L2. Due to duality property ( E U * )= U E we may consider the operator U E only. Set f = UEg for some g E L2. The Green's formula (1.18) implies
where
L; = ~ $ 2 , L ~ ( R=) L2,2(R). Since
( E g ,f ),2 = (9' EUE9),2 it is sufficient t o prove that the operator E U E is compact. In this case EUEg(") + 0 strongly in L2, if g(n) + 0 weakly. Since Ug(")is a bounded sequence, (3.1) shows that U E ~ ( "+ ) 0 strongly in L 2 ( R ) , 7 + ~ ~ g ( -t n )0 strongly in L i .
(3.2)
To prove that E U E is a compact operator define in L 2 ( R )the operator Z by
In other words Z is a restriction of EU to the functions depending on x only. Let G be an extension of G in R::
G XER x$R.
0
In virtue of (1.6) we have
We make the change of variables
t o obtain that Z is a convolution
with
~ ( x=) Note that
P
s - exp ~ { - s h ( x / s ) ) cp(x/s)ds.
z E
Lk if cp E L~ and
Chapter 4. Transport operators
50
To verify (3.3) set ho = inf h and observe that
By the classical theorems (3.3) implies that the operator Z is compact from LP(fl) into Lq(R) if 1 1 1 l
--9 P 6' In virtue of (3.1) we may choose k to satisfy the condition (3.4) with q = p = 2. This completes the proof. The following consequence follows immediately from interpolation theorems for compact operators and (3.2). Proposition 3.2 (i) The operators EU and U E are compact in LPIP and Lp ( f l , Lr (Rd,P)) for any P, r E (1,031. (ii) The operator 7+UE is compact from LPJ into LPJ"S). Similar arguments show the compactness of the operators E W and y+EW with W defined by (1.5), (1.1). More precisely:
EW y+EW
is compact from LPvr(S-) into LPpP, is compact from LPpr(S-) into LP.'(S+).
Another approach to study velocity averages relies on a Fourier analysis of the singularities of the transport operators [29]. Consider the equation
The solution is given by
f
= ug
where the operator U is defined by (1.6) with
12
= l , s , = -00.
T h e o r e m 3.3 Assume that there exists a positive constant C such that sup e s s p { v ~ ~ d : I v . e 1 5 5e C ) a. e~Sd-1
Then the operator EU is continuous from L2 into H ' ~ ' ( R ~ ) .
(3.5)
3. Velocity averages Proof. Let
f = Fzf
51 be the Fourier transform with the dual variable t . Then
S = ~ f i j , fi
= [I + i t . v]-'
By Cauchy-Schwartz inequality
It is sufficient to prove that
~lfil"
clltl-'
for some positive constant C l . Due to (3.5)
Since
the result follows. The following proposition is a consequence of Theorem 3.3. Proposition 3.4 [29] Let the assumptions of Theorem 3.1 be valid. If I< is a weakly compact set in L 1 , then E U ( I < ) is compact in L:,,.
Obviously, the compactness results of Theorem 3.1 can be obtained as a consequence of Theorem 3.3. Several extensions to more general operators were obtained by R.J.Di Perna, P.L.Lions and Y.Meyer [22]. In particular, the functions g = (1 - .Vz)T(l- .Vz)mi with E LP, m 2 0, T > 0 are considered. We shall now show that the same type of results are valid in the time-dependent case, though with some additional assumptions. Consider for example the equation
Take any e E Sd-' and form the measure
with the density p,. The solution operator U for (3.6) is given by
Chapter 4. Transport operators
52 T h e o r e m 3.5
Assume that
(3.7) some positive I . Then the operator EU is L2 = L2 ( R 1 x R:, L d ( ~ Pd ),) into H S ( R x R t ) with s = !j&.
for
continuous
from
Proof. Consider the Fourier transform with respect to t , x with the dual variables T ,[. Then the equation (3.6) transforms into
f=Oi,
O=[l+i(~+v.()]-'.
Following the proof of Theorem 3.3 we have to bound
By change of variables w + T
+ I[lw = y we get E1fi12 I 1 5 1 - I SUpPe.
the integrand on the right-hand side of (3.8) can be bounded by I.rl-' If 1[1w < We use the assumption (3.7) to obtain
1
Set r =
If
[TI" >
&. If [ T I '
< 151
~ e ( w ) d w5 ~ ( 1 ( ) I ~ I I T I - ~ ) ' wllE1>7/2 the inequality (3.9) gives
(3.10)
we deduce from (3.10) that
E I U I5~ C ( l ) l ~ l - ~ . Thus we prove that
E
5C (
+
T
)
with
r
5
1
-
1+1 with some positive constant C 1 . This inequality completes the proof. It is quite clear, that the analogue of Proposition 3.4 follows. More precisely, the following theorem is true.
T h e o r e m 3.6 [21] Let g and G be elements of a weakly compact set A in L1([O,T ] x R3 x R 3 ) satisfying the equation Then the set { ( G g ) ,g E A ) is compact in the space L1([O,T]x R3) for all
LW([O,T]x R3 x R 3 ) .
II,
E
Chapter 5 Steady-state solutions of the linear Boltzmann equation Detailed analysis of stationary linearized Boltzmann equation is the main aim of this chapter. One may hope for additional insights from information about steady-state solutions near the Maxwellian. Existence and uniqueness theorems regarding these solutions are proved. Some asymptotical properties are considered. There are two simple key points in the approach described below. First, Theorem 4.3.1 provides a possibility to reduce a boundary-value problem in a bounded domain to an integral equation with a compact operator in an appropriate Hilbert space (see Lemma 3.3). In fact this theorem reduces study of steady-state solutions in a bounded domain to some rather technical questions. To understand behavior of steady-state solutions in unbounded domain it is necessary to understand asymptotics of solutions in the entire space with a localized source. This gives a key to a reasonable setting of the problem for an unbounded domain. Remark. The results of this chapter seem to require some comments. I began my work in this field during my contacts with translators of C.Cercignani's book [15] into Russian. The proof of existence theorem for a bounded domain was clearly incomplete. That's why I looked for another approach. One was proposed by J.P.Guiraud [33], who claimed that the tenth degree of the operator UIi' from Lemma 3.3 below is compact, without proof. Then I proved the assertion presented here as Lemma 3.3. As mentioned above this lemma is one of the key points for all results regarding steady-state solutions. The proof was published in Appendix to the Russian translation of C.Cercignani's book [43] and thus wasn't available for some Western authors (cf. [6, 641). We start with a detail description of the solutions in the whole space in Section 1. The representations of the solutions described in this section are crucial for the sequel. The validation of the Hilbert approach in the linear problems is given in Section 2. The aim of Section 3 is to prove the existence and uniqueness theorems for a bounded domain. Stationary solutions describing a gas flow past an obstacle are
Chapter V. Steady-state solutions
54
discussed in Section 4 for three-dimensional and plane problems. The last section is devoted to the Icramers and Milne problems concerning a description of the solutions in a half-plane. The results concerning solutions in unbounded domains rely on the connection between the solutions of the Boltzmann equations and the fluid dynamics equations. We prove that the asymptotic behavior of solutions at large distances from the boundary is described by linear fluid dynamics equations (i.e. the Stokes system for the average velocity and the heat equation for the temperature).
1
Solutions of the Boltzmann equation in the whole space
Consider the problem of propagation of perturbations in R3 caused by a source of intensity g. The problem is to find a function
satisfying Df=Lf+g
D=v.V,
(1.1)
with the operator L defined by (3.3.2). We are concerned mainly with two problems. First, we would like to describe the asymptotics of the solutions for large 1x1. It is quite clear, that this asymptotics determines a reasonable setting of the boundary-value problem in unbounded domains. Particularly, we shall show that there are boundary value problems, which have no solutions satisfying the condition (3.1.8) with a prescribed function f, at infinity. Second, it seems reasonable to discuss the fluid dynamics limits in this simplest situation.
1.1
A priori estimates
We start with some formal and rather technical consequences of (1.1). Note that there are two sources for a priori estimates of stationary solution. The first one is the dissipative property of the collision operator L, which leads to the estimate (1.31) below. The nontrivial kernel of the collision integral L is a very typical property of all kinetic equations. Therefore, we need an additional source to estimate the projections on Ker L. The main aim of this section is to describe a method to obtain some reasonable estimates for these projections. Note, that the conservation laws are not sufficientfor this purpose. That is the reason to use the equation for some higher moments (see (1.8), (1.9)). The Fourier transforni
1.Solutions in the whole space leads to the equation
The properties of the operator L, described in Section 3.3 are essential in the sequel. Particularly, it is important now that (i) # = L + v is a compact selfadjoint operator in the Hilbert space H with the inner product (3.3.4);
(ii) Pobeing the projector (3.3.7). The conservation laws Po D f - g
-) = O
is a consequence of (i) and (ii). We have mentioned above that the higher moments ~ ( f ) = . = (fl(I-P~)v@v)H q(f) = 9 = ( f 7 ( I v$I)H
(1.4)
figure in (1.3) together with fluid dynamics moments m = (f, $)H. To obtain an additional source for a priori estimates we use the projections of (1.2) on span{a, b), where the functions a, b are defined by (3.3.19). These projections together with conservation laws lead to the following system:
Here 3(C)denotes the strain-tensor generated by the velocity field u:
We use the notations
fi
= nio+6, 8 = r n r n 4 6 = (nil, 6 , 7 7 5 3 ) mj = ( f l $ A H .
The positive constant and n are determined in a unique way by the kernel B of the operator L. These constants are connected with the viscosity and the thermal conductivity respectively.
Chapter V. Steady-state solutions
56
Remark 1.1 The following identities have been used to deduce equations (1.5)-(1.9):
These relations follow immediately from the definitions of the quantities involved. More precisely, any function f satisfying (1
+ I V I ) E' ~H
for some r E R enjoys the identities (1.11)-(1.15) with C = C( f ) , $ = j(f) etc. If one deletes away the last two terms on the right-hand side of (1.8), (1.9), these relations give Newton and Fourier laws respectively, i.e.
+ = -p6(C)
4 = ~mit8.
(1.16)
These equations lead to the linear hydrodynamics directly. More precisely, denote by II the projector to divergence-free velocity space:
The equations (l.lG), (1.6)-(1.8) imply
In other words, the functions u and T = m4 solve the equations -pVu
+V j -KVT
= ( g , ~ ) ~divu , =0 = (g,& -
m) H
with some pressure j. On the other hand, the exact formulae (1.8), (1.9) give
57
1.Solutions in the whole space
where
fi (ij,v),
(1.20)
fiCc1) = -ifi(ij, b . w ) ~
(1.21)
fit(" =
fid2) =
((I
- PO) f lb .w(v - w))
mlH
(1.22)
(id4-
(1.23)
m(1) 4
- -i ( & a .W)H
(1.24)
my)
= - ((I- po) j , a . W(V. w ) ) ~ .
(1.25)
@I =
Due to (1.5),(1.6) the exact expressions for the pressure p and the potential (longitudinal) component of the average velocity are given by
where
Therefore we have proved the following Lemma:
Lemma 1.2 Assume that for some r E R1 ( 1 + 1 ~ 1 ) ~Ei jH, If
~f
= Lf
(1 + I V I ) ' ~HE
+ ij, then the relations (1.5)-(1.9), (1.20)-(1.30) are valid.
The second source for the a priori estimates is the inequality (3.3.8), which implies
for some positive constant 1 uniformly with respect to [. There is an obvious corollary of these equations. Note that the strain tensor a determines in a unique way the velocity field due to identities
Chapter V. Steady-state solutions
58 Hence the equations (1.8),(1.9)give
uniformly with respect to IJI. The similar estimate is valid for the pressure p due to (1.26), (1.29), (1.30). Therefore
Combining this estimate with (1.31) we conclude that
Thus, the uniqueness theorem for (1.2) is valid in H
1.2
Asymptotics for small
We are ready to present some exact statements regarding the asymptotics for small 151. Due to Lemma 1.2 the moments r f i j = m j (f ) admit the following representation:
where the terms fi:) are defined by (1.19)-(1.30). We shall prove that formly bounded. Set
rfi?)
is uni-
0 0.) (Note that p o ~ j = Theorem 1.3 Assume that v-'I2j E H , ( # 0 . Then equation (1.2) has a unique solution in H . There exists a positive constant C such that
Remark 1.4 Note that the function f(O) has the following form
1.Solutions in the whole space
59
where p + T = 0, while u,T solve the Stokes system (1.17) and the heat equation (1.18) respectively. In view of (1.36), (1.20)-(1.30) it holds
Thus, Theorem 1.3 is an exact assertion regarding asymptotics of solutions for small
ISI.
Proof. Equation (1.2) has a solution
1 in H if and only if f solves the equation
Since inf v = vo > 0, the compactness of IC implies that fiIC is a compact operator in H. It follows from the Fredholm alternative that we only have to prove the inequality (1.37) for any f E H. Define the function F by
Let us prove that there exists a positive constant C such that
Due to (1.35),(1.36)
k is a solution of DP=LP+&
with &' = j - ~ jApplying ~ . to this equation (1.5)-(1.9) we obtain that F enjoys the property (1.41), if
On the other hand, (1.42) is a consequence of (1.35), (1.36). This implies (1.41). The next step is to estimate ( I - Po) F. In view of (1.31) we have
Due to (1.42)
($6)H = ( v 1 f 2 ( ~ - ~ o ) 6 , ~ 1 f 2 ( ~ - ~ o ) & ' ) Hence We conclude by the following
Chapter V. Steady-state solutions
60
Corollary 1.5 Under the assumptions of Theorem 1.3 the representation (1.34) is valid with the following estimate
uniformly with respect to
[.
Thus due to Parseval's formula
if lljllHE L2(R3). Note that the first two terms do not belong to L2(R3) if lljllH= O(1) as + 0.
1.3
Asymptotics for large 1x1
We are now ready to describe the asymptotics of the solutions of (1.1) for large class of the sources g. Particularly we will describe the solutions generated by the functions g depending on d spatial variables only, while the full three-dimensional velocity dependence is allowed. These problems will be refer as d-dimensional. Due to applications in the boundary-layer theories these problems play an important role in the theory. Set
The matrix {Sap) determines the fundamental solutions of the Stokes equations (1.17). We start with three-dimensional problems where
Corollary 1.5 implies the following representation of the average velocity
a where Sap,,= &Sap, S$ = GS. The last term on the right-hand side of (1.45) is a function from L2(R3), while the first two terms do not belong to L2(R3),due to their slow rate of decreasing for large 1x1. Particularly, we conclude from (1.44) IlgllH E L2(R3), Supp llgll~is then u = O(lxl-l) as 1x1 -t w.
1.Solutions in the whole space Similar representation is valid for the temperature:
where m f ) E L 2 ( R 3 ) . Thus m4 = O ( I X ~ -if~the ) conditions (1.46) are fulfilled. To investigate the general sources, let us introduce the spaces
with the norms
Ilf
llnr = IIyr
Ilf
= (1
llHllLz(p)
+1 ~ 1 ) ~ .
The index r can be considered as a measure of the rate of decreasing of f E If the conditions (1.46) are fulfilled, we have f(O)
+ f ( l )E H,
with
r
1
> -, f 2
a,.
E Ho
Theorem 1.6 Assume that
Then the equation (1.1) has a unique solution in 7-1-1. Any solution of (1.1) in N-2 has the form 4
f
=xCj$j+f(0)+f(1)+F7
(1.47)
j=O
where C, are constant, U ' / ~ F E Ho, while the functions are defined by (1.35), (1.36).
Proof. Set f = .FL1], where is a solution of (1.2), described in Theorem 1.3. In order to describe the functions f ( j ) ( j = 0 , l ) we use the Ladyzhenskay inequality [37]:
if u E V ( R 3 ) . In view of (1.34),(1.19)-(1.25) and (1.48)
for some positive constants C l , Cz. Similarly we obtain
llf'l'lln-l 5 c3 l 1 ~ 1 1 ~ 0
Chapter V. Steady-state solutions
62
Due to Parseval's formula v1f2FE N o . Thus f E 'H4, due to inclusion
l-lk c l-l, for k > r. In virtue of Theorem 1.3 the condition g = 0 implies that f is a polynomial Pn with the coefficients from H. The condition Pn E l-lk implies n k < 1 - $. Thus n = 0 if k = -2. Due to (1.37) this completes the proof. Consider now two-dimensional problems. Then
+
Therefore the solutions may grow as In x as 1x1 + w.
Theorem 1.7 Let d = 2. Assume that for some 6 > 0 P0g E 'Fl1+s,
v-'I2 (I - Po) g E l-lo
Then equation (1.1) has a solution in 'Flwz.Any solution has the form (1.47).
Proof. We follow the proof of Theorem 1.6, using the following estimates
I ' 1 %-. m"'
5 C IIPosll%,+, Vr > 1
II'P~UIIL~ 5 C llvxull~z
(1.49)
+
for u E D ( R 3 ) , ~ = , (1 1x1)-",r > 1. The last estimate is a consequence of Ladyzhenskay inequality [37]. Consider now one-dimensional problems. For d = 1 we have
Therefore the solutions may grow linearly for x + m. Moreover the homogeneous equation (1.1) has nontrivial solutions which grows linearly. Set
Here
2. The incompressible limit and the hilbert series
63
Proposition 1.8 Any function f E P is a solution of (1.1) with g = 0. Moreover, any function f E 3t-, for some r, which satisfies (1.1) with g = 0, belongs to P.
In order to prove this assertion note that due to Theorem 1.3 f is a polynomial
Then The properties of the operator L imply immediately that f E P. Proceeding as before we obtain Theorem 1.9 Let d = 1. Suppose
%(g, $ 3 1 ~ 9 cpr(g, b j l ) ~ r~( g , a l ) HE L2(R1) with cp, = (1 Ixl)-', r > 312. Then equation (1.1) has a solution in 3-1-, if s > 312. Any solution of (1.1) in 1-I-, has the form
+
where d I 2 F E NO,f ( 2 ) E P, while the functions
f(j)
are defined by (1.35), (1.36).
Perhaps the most important consequence of the theorems above is the hard restriction on the possible conditions at infinity in the linear setting. It is quite clear, that the unique possible fusction f, in the condition
is f, = C Cjt+bj. However we shall prove that only a condition of boundedness of the solution determines some of the constants Cj uniquely. These problems are discussed in Section 3 of this chapter.
2
The incompressible limit and the Hilbert series
The aim of this section is to describe the connection between the Hilbert series and the incompressible limits in stationary linear problem. The Hilbert scaling leads to the equation 1
Df,(x, v) = -Lf,(x, v) E
+ G,(z, v)
&
E (0,l).
(2.1)
Since Theorem 1.3 gives the uniform estimates with respect to (, it is possible to obtain some conclusions regarding the limiting behavior of the solutions.
Chapter V. Steady-state solutions
64
Suppose G,(x, v ) = ~ g ( xv ,) . Then (2.1) transforms into
Theorem 1.3 suggests the approximation f(O) by (1.35) does not depend on E , while
+ f(') for f,.
The function
f(O)
defined
with m(') determined by (1.21),(1.24),(1.29). Note that
with a positive constant C. Suppose
j E N = L~ ( R ~L~, (R:, M ~ v ) ) then (1.37) gives fc
+ f(O)
strongly in
N.
In virtue of (1.37) the moments ( ~ ( O ) , V ) ~ (, f ( n ) , $ q ) Hsolve the Stokes system and the heat equation respectively. Thus, the first term in the Hilbert series
is exactly the incompressible limit. Theorem 1.3 makes it possible to validate the Hilbert representation of the solutions in the following sense: T h e o r e m 2.1 Suppose that
then the solution of (2.2) /has the form
with a function
k(")such that
unifoi-inly with respect to [ E R d , E E ( 0 , l ) with some positive constaitt C .
65
2. The incompressible limit and the hilbert series
Proof. Note that Theorem 3.1 proves (2.4),(2.5) with n = 1. To proceed further, an additional information regarding the terms of Hilbert series is required. Formal substitution of the Hilbert series (2.3) into (2.2) gives L f ( 0 ) = 0 L f ("+I) = D f ("1 - g6n1 n 2 0. (2.6) In virtue of Lemma 3.3.1 the solvability conditions for (2.6) are Po ( Df(") - g6n1) = 0. Any solution of (2.6) has the form f (n+l) = ~ - (1D f
(0)
(2.7)
- gbnl) + pof '"+".
Set
+
+
p(n) . u(") 4 4 m 0 ( n ) p(") = p(") + ($4.
pOf(") =
The functions p("), u("),0(") are determined by (2.7) as follows. In view of (1.11)(1.13) the equations (2.7) have the form
div,u(") = ( g ,l ) ~ 6 , , ~ V p ( " ) V,T(") = ( g ,~ ) ~ div, (9'"'
+ +m u ( " ) )
6
~
~ (2.8)
= ( g ,h ) H 6 n l
with T ( " ) = T ( f ( " ) ),q = q ( f ( " ) ) determined by (1.4). Due to (1.14),(1.15), (2.6) we have = ( u ( " ) ) bnl(g, b ) -~( D ( I - po) f ("I, b)H q("+') = - I E @ V ~ ( " )- ( D ( I - Po) f ("1, a )
+
(2.9)
-6nl(g, a ) ~ . (2.10) Due to (2.8)-(2.10) the moments u("),0(") solve the Stokes system and the heat equation with some source terms, defined by f(O),. . . ,f("-'). Let F(") be the function, determined by the equality (2.4). This function solves the equation f)j(") = s - ' ~ j 3 " ) + 9^(") E -1 with j(") = j6n-1 Due to (2.7) po$") = 0. Moreover, (2.7) implies that the function j(") satisfies
f)p(").
fi(ij("),b.w),=O
(j("),a.~)~=O
Therefore by Theorem 1.3
11
1lH
IIv1~2@(n)JlHc ullz "(4 uniformly with respect to (. The obvious estimate
completes the proof.
66
Chapter V. Steady-state solutions
3
Stationary solutions of the linear Boltzmann equation in a bounded domain
3.1
Introduction
Consider the problem
with the linearized collision operator L and a prescribed operator 72. The approach we use below is a reduction of the problem to an integral equation for the incoming flow 7-f . More precisely, consider the problem
We shall prove that for a prescribed function f- the equation (3.3) has a unique solution in a suitable Hilbert space. Thus the solution operator V : 7- f + f is well defined. The function Vy- f is a solution of the problem (3.1),(3.2), if the incoming flow y- f solves the equation
Typical boundary conditions (3.2) allow the reduction of equation (3.4) to an equation
N=AN+B for the average incoming flow
with a compact operator A and a prescribed function G. Particularly, the classical example of the operator R is the diffuse reflection. In the linear setting this model gives
with a prescribed function f-. It is clear that the method allows to include small perturbations of the operators involved. For example, we consider in Chapter 6
with a small parameter
E.
3.Stationary solutions
3.2
The basic function spaces
We shall construct solutions of the problem ( 3 . 3 ) in the function space
M being the normalized Maxwell distribution
~ ( v=)( 2 ~ ) - ~exp / ' {-lvI2/2) To describe the traces 7 f = y+ f
fi
Hi
+7-f
we use the spaces
= {f IIV.~~'/~M E L2 ' / ~(do ~ x Rd)) = {f €fi17ff= f } .
(3.5)
The inner products in the spaces H, f i are defined in a natural way:
To deal with the boundary conditions a more detail description of the boundary functions will be used. Fix a point x E dR with a unit outward normal n ( x ) . Set
Introduce the inner product in H+ by
Suppose that the operator
R has the form
with a prescribed function f - . Define a specular reflection operator S by
S f ( x ,v ) = f ( x ,v - 2 ( n v .n ) ) . It is natural that properties of solutions depend on the structure of the kernel I - S R in H + . From the physical point of view this structure is determined by the conservation laws, governing the interaction of the particles with the boundary. The assumptions below is a formalization of the physical assumption that a unique conservation law
Chapter V. Steady-state solutions
68
a t the boundary is that of mass proposed essentially by C.Cercignani [15]. Set
&=
M(vilv . n l d ~ ) - l ~ ~ ?
(So,.
Po+f = 4: (f*:),,
.
In other words Po+ is an orthogonal projector on the subspace, spanned by constants, (f, =N. Suppose that S R is a self adjoint operator in H + . (3.7)
,o*+)
Ker(1- S R ) = P;H+.
Il(I - PC?)~RII"+
(3-8)
< 1.
Note that the assumptions (3.7)-(3.9) imply that
Several extensions to more general operators are available with the same strategy of proof. Particularly, it is possible to deal with any finite dimensional kernels instead of one-dimensional one. The assumption (3.9) is also too restrictive. The method works, if we only assume that
3.3
Main theorems
Set W - = {f Iv1f2f E 3 - 1 , ~ - ~ " ~Ef 'H1Sy-f E ~ ~ ( 5 ' ) ) .
Theorem 3.1 Suppose that S f - E 3 - 1 + , ~ - ' /E ~ ~3-1. Then the problem (3.3) lhas a unique solution in W -
Theorem 3.2 Let 8 R be a Lyapunov surface. Suppose that the assumptions (3.7)(3.9), (3.11) are valid. The problem (3.1), (3.2) has a unique solution in W - if and only if (1,g)x
+ ( l , S f - ) x + = 0.
The difference between two ar.bitrary solutions is a constant.
(3.12)
3.Stationa1-ysolutions
3.4
69
Proofs
We start with the proof of Theorem 3.1. The following two a priori estimate are essential in the sequel. First, observe that due to Green's formula 1
1
27 -
t
-
5s
f -f
f
= (f,g)n
(3.13)
( f + , l).n+ - ( S Y f , l ) n + - (Lf, l)n = (g, 1)n. In virtue of (2.3.4) we deduce from (3.13)
To proceed further we need a bound for Pof . Let h be a solution of
Due to Theorems 1.6, 1.7, 1.9 we have
Applying the Green's formula to the functions h, f we get
with a positive constant C. Due to (3.15), (3.16) this gives
A direct consequence of (3.17), (3.13) is the following estimate
To proceed further we use the operators U, W defined by (4.1.5), (4.1.6) with h = v. The function f is a solution of (3.3) in W - if and only if (3.19) + Wf- + Ug. Note, that it follows from (4.1.11), (4.1.12) that Wf - + Ug E 7-t. The next step is to
f
= UIi'f
prove the following assertions. Lemma 3.3 The operator U I i is compact in 7-l.
Chapter V. Steady-state solutions
70
Proof. This is a direct consequence of Theorem 4.3.1 and the compactness of 1Ii1 (see Proposition 4.3.2). Indeed, denote the kernel of the operator I i by k . Set k n u v=
{
i
u
v
if if
Ik(u, v ) ] 5 n lk(u,v)I > n
Then I(, -t K in the strong operator topology in H. This implies the convergence U K , -t U I i in the strong operator topology in 7-1. In order to complete the proof it is sufficient to observe that the Theorem 4.3.1 implies the compactness of UIi,,. Now Theorem 3.1 follows from the estimate (3.18) and the Fredholm alternative. We shall use below also the following L e m m a 3.4 The operator y+UIi is compact from 7-1 into N+ To prove this lemma it suffices to repeat the proof of Lemma 3.3, using Proposition 4.3.2 instead of Theorem 4.3.1. R e m a r k 3.5 To prove that (3.18) is valid for f E 7-1 it is suficient t o consider the operators L with truncated kernels
B,(v, a ) = max { B ( v ,a ) ,n ) . Taking the limit n -+ oo we obtain (3.18). Proof of Theorem 3.2. Assume that g = 0. Denote by V the solution operator for the problem (3.3). By Theorem 3.1 the operator V S is continuous from 'FI+ into 7-1. Set f+ = S y - f . In view of (3.19)
By Lemma 4.1.2 the operator y + V S is continuous in I f + . The function f = S f + solves (3.1), (3.2), (3.6) if and only if f+ solves the equation
In view of (3.20) it reduces to
f+ = B f + S f with B = Bo
+ B1,
It follows from Lemma 3.4 that the following Lemma is true. L e m m a 3.6 The operator Bo is compact in li+.
3.Stationary solutions
71
Lemma 3.7 The operator P$~+WSP$ is compact in H+.
Proof. Set Pi7+WSP$ = G, cp = P$ f+. Recall the definition (4.1.5) of the operator W. Set for a fixed y
We have y+WSP$cp(x) = cp(z) exp {-v(v) (s; - s f ) ) Iv . n(x)lcp(z)exp {-v(v) (s; - s f ) ) dv. Change the variables v - t z = y + s f v , t = Ivl to transform the operator G into
where G(z, x) = 1x - Z [ - ~ - ' 1(x - Z,n(x))(x - Z,n(z))I O(x, z) with a bounded function 0. Since dR is a Lyapunov surface there exists a positive constant S > 0 such that sup G(x, z)(x- zld-' < m. The compactness of G follows immediately.
Lemma 3.8 The operator (I- B ~ ) - 'is continuous in 'H+. Proof. Consider the equation
with a parameter a E (0,l) and a given function Zt E H+. By Lemma 4.1.2
II~+w~ll,t5 1. Hence by (3.10) IIBrll,+
51 and
Il.CllH+5 (1 - a)-'
IIZ+IIn+ .
We shall prove Lemma by showing that there exists a positive constant C such that
Chapter V. Steady-state solutions
72 The function f, = W S f,+ solves the problem
The Green formula gives
Therefore by (3.23)
< IIs~r+f.+ z+llR+.
II~+f. l l;+
(3.24)
Use (3.8), (3.9) to deduce from (3.24) that
uniformly in a for some positive constant C1. In order to bound P: f,+ note that SRP: = P: and
with
Z: = ( Y P : ~ + W S ( I - P:) f:
+ P:Z+.
Set g, = W S P : f,+. This function solves the problem
+
g,+ = P,+-,+~, Z:X E aR.
It follows that g, = 0, if Z: = 0. By Lemma 3.7 this implies that equation (4.28) has a unique solution for all a E [O,1] and
[[I(' -
f:l H+
+ IIP$Z+llH+l
IIPo'f:II 5 ' 2 (3.26) uniformly in a E [O,l]. The estimate (3.22) is a direct consequence of (3.26), (3.25). In virtue of Lemma 3.8 equation (3.21) is equivalent to the following equation By Lemma 3.6 the operator ( I - B1)-' Bo ( I - B1)is compact in the proof we need only to describe the kernel of the operator
X+.To complete
Assume that h solves (3.27) with f - = 0. Set f + = ( I - B1) h+, f = V S f+, where V is the solution operator for the problem (3.3). Use (3.14) with g = 0 to deduce that
1
5 117+f[I:+
-
1
5 p 7 - f l :+ + 1 11~1'2
( I - Po) f
[In+
2
= 0.
Due to (3.10) we obtain that ( I - Po) f = 0. Thus Df = 0 and f is constant because of (3.8). Thus dim I<er A = 1. The orthogonality condition gives (3.12). This completes the proof.
73
4. Steady-state solutions
4
Steady-state solutions in an unbounded domain
4.1
Main results
In this section we discuss the boundary-value problems
Df
r-f
= Lf
XER
= Rr+f+f-
in an unbounded domain R c R:, d = 2,3. It is assumed that Rd \ R is a bounded domain with a Lyapunov boundary. The full three-dimensional velocity dependence is allowed in either problem. In order to describe a class of solvable problems with the traditional condition at infinity
we describe all solutions of (4.1), (4.2) satisfying
where
Zk= { f 1(1 + I X ( ) ~ Mf "E~L2 (0 X R:) ) , 7-1 7-10.
(4.5)
The condition (4.4) restricts the rate of increasing of solutions at infinity. However, any bounded solution belongs to 'H-2. We shall describe a subset of the solutions, satisfying (4.3) in the following sense:
f - fm
E
d
7ik with some k 2 --.2
More generally, we shall write
if there is k 2 - d / 2 such that cp E 7ik.Below we suppose that the conditions (3.7)(3.9) are fulfilled, whenever a more weak assumption is not specified explicitly. The space 7i+ defined by (3.5) is used to describe the boundary functions. We will also use the spaces
with a fixed normal n(x) on C. In order to define a solution of (4.1), (4.2), (4.4) in a unique way we introduce the following integral characteristics:
Chapter V. Steady-state solutions
74
where *j = V+!Jj - 4@6j4, while the functions V+!Jj are defined by (3.3.6). In particular
determine integral fluxes of the mass, momentum and energy on do. Note that the vector (&I, . .. ,Qd) is a force acting on the obstacle Rd\R, while the pressure p = p( f ) is defined by
Asymptotic behavior of solutions for large 1x1 is determined by functions from the following two classes:
S are the fluid dynamic potentials, defined by (1.43), Cj, B j are constants. where Soor Set W- = { f ( u " ~f E 3-1-2,v-lf2Df E 3 - 1 - 2 , ~ - f E '1.3 ) . Theorem 4.1 For any given constants Bo, . . . ,B4 the problem (4.1), (4.2) has a unique solution in W- such that
The solution has the following form
where h(O) E
p(O),
h(l) E ?(I),
Remark 4.2 W e shall prove that the function erties:
f in (4.8)
enjoys the following prop-
4. Steady-state solutions
75
Theorem 4.1 says that for a given condition on a0 it is possible to find a solution with any prescribed fluxes Q j . On the other hand in view of this theorem it is impossible to prescribe the function f, in (4.3) arbitrarily. The class of admissible functions depends on the dimension d of the problem. Suppose that d = 3. Then the function la(') + f i n (4.8) converges to zero. Therefore, the condition (4.3) is fulfilled with f, € P(O). The constants Cj in f, determine the limiting values of the pressure, temperature and temperature at infinity. The arbitrariness in the choice of constants Bj provides a possibility to solve the problem (4.1)-(4.3) with any given constants C j . This fact allows to prove the following theorem.
Theorem 4.3 Let d = 3 . For any constants Cj the problem (4.1)-(4.3) with
has a unique solution in W . The situation is completely different for the plane case d = 2. If Q j # 0 for some j 2 1, the solution (4.8) grows as In 1x1 as 1x1 + w. In order to fulfill (4.3) or even only a condition of boundedness of the solution one needs to choose Bj = 0 in the conditions of Theorem 4.1. But then the theorem implies that this choice determines the limiting values of the velocity and temperature in a unique way. Thus, we observe the phenomenon which is completely similar to the Stokes paradox in the fluid dynamics. It is impossible to specify in an arbitrary way the velocity and temperature of a fluid at infinity because these properties are uniquely determined by the requirement of boundedness of the solution. The following theorem is a two-dimensional analogy of Theorem 4.3. Assume that 0 @ R. Set
Theorem 4.4 Let d = 2. For any given constant Co the problem (4.1), (4.2) has a unique solution in W ( P ) satisfying
Clearly, the requirement f E X ( p ) excludes the solutions that grow as In 1x1, while the constant Co determines the pressure at infinity.
Chapter V. Steady-state solutions
76
4.2
Asymptotics
Theorem 4.5 Let f be a solution of (4.1), (4.2) in W . Then f has the form (4.8). Proof. Define f l , fo by
Recall that the operators W , U are defined by (4.1.5), (4.1.6) with h = v. The functions f l , f2 solve the problems
with f3 = I i U I i W f Since (1 IV()'/~I~' is bounded in H, we deduce from Lemma 4.1.1 that
+
By Lemma 4.1.3
c p k f j e ' H for j = 1 , 3 , k > O cp = ( 1 1xI2 lvI2)(1 lvI2)-I
+ +
+
(4.13) '
Let Rs be a boundary layer
Rs = { x E R : dist ( x , a R ) 5 6 ) .
(4.14)
Let ( be a function from C 1 ( R d )with properties
C=O ( = 1 The function
4 = f2(
for z € R d \ R for X E R \ R ~ , O I C I ~
solves the equation
In view of (4.12), (4.13) the function g satisfies the conditions of the theorems ( 1 . 5 ) , (1.6). Thus 4
where the functions f('), f ( ' ) are determined by (1.35), (1.36), while
77
4. Steady-state solutions
for the spheres (4.9). It follows from (4.13), that the function g in (4.16) decays rapidly. This provides a possibility to simplify the functions f (O),f('). Set
+
7;. =
vorQa(g)sao 44&4(9)5 0
,P
where the functions Smp,S are defined by (1.43). Simple calculations prove that
for sufficiently large k. In order to complete the proof we only have to show, that
Q j (f ) being defined by (4.7). Let x be the indicator function of the boundary layer (4.14). Then
where (-, .) denotes the inner product in Integrating by parts, we get
H.
It follows from (4.11), that
Therefore
Qj(g) = Qi(f2)
+ ( $ i , f3).
Due to (4.10), (4.13)
( $ j , f3)n = - ( $ j , D f i ) x o = Q j This proves (4.18).
(fi).
Chapter V. Steady-state solutions
78
4.3
Uniqueness
The uniqueness problem is discussed in this section. What is remarkable in this class of problems is that answers depend on dimension of the problem crucially. The theorems below are valid, if the operator R satisfies the following two conditions.
Ker ( I - S R ) r7 y+PoH C P+?-l+,
(4.20)
where P + is the projector on the subspace of H ' : consisting of constants. These are weaker conditions than (3.7)-(3.9). The conditions (4.20) excludes the conservations laws other than that of the mass. Theorem 4.6 Let f be a solution of (4.1), (4.2) in W . Suppose, that d = 3 , y - f = 0 ,
Pof E
3 2
'Hk for some k 2 --.
(4.21)
Then f = 0 . Proof. Define
BT =
Choose r in such a way that R
{X
E R311x1I r)
c B,.
C, = aB,
Using the estimate (3.14) in B, n R we deduce
In view of Theorem 4.5 f has the form (4.8). Since
H (h(O),C,) = 0 , H (h('),C,) = 0
(4.23)
we deduce from Theorem 4.5 that
Thus, (4.19) and (4.22) imply
( I - Po) f = 0. Set cf, = 6f with the function
defined by (4.15). Then @ solves the equation
Applying to @ the equalities ( 1 . 6 ) , ( 1 . 8 ) , (1.9) we conclude
In view of (4.21), (4.24) this completes the proof.
(4.24)
4. Steady-state solutions
Theorem 4.7 Let f be a solution of (4.1), (4.2) in W , d = 2,3. Assume that 7-f = 0, p € 3-Ik
>1
Q j ( f ) = 0 for j with some k 2
d 2
--.
(4.26)
Then f = 0.
Proof. We follow the proof of Theorem 4.6. The key point is again the representation (4.8). In view of (4.23), (4.19) we obtain (4.24) and (4.25). Let 6 be the truncation function defined by (4.15). Applying Theorem 1.3 to the equation
we conclude that f is a constant in R \ Rh for any 6 that f = 0.
4.4
> 0.
The condition (4.26) shows
X E R 7 - f = f-
(4.27)
Existence
We start with the problem
Df =Lf
for a given function f - E 7 f . The first step is to construct a solution in a bounded domain
with the boundary I?, = dR U E,, E, = {XI Consider the problem D f =L f 7-f =f7-f= O
1x1 = r}. XE a(') X E ~ R XECr
The dependence on r is suppressed for notational simplicity only. Moreover, we set f = 0 for x E R \ R(') and use the spaces 7 i k to describe solutions of (4.28). However, the main problem is, of course, to control the dependence on r. These estimates are described in the following lemma. Recall the definition (4.27) of the class W.
Lemma 4.8 The problem (4.28) has a unique solution in W. There exists a positive constant C such that for any r > 0 the following estimates are valid:
where
4 2 ) = (1
+ XI)-^
LPZ =
(1x1 ln 1x1
+ I)-'.
(4.29)
Chapter V. Steady-state solutions
80 The next step is to take limit r
-+
co. It leads to a solution of (4.27).
Lemma 4.9 The problem (4.27) has a solution in W such that
with
( ~ ddefined
by (4.29).
Proof of Lemma 4.8. By Theorem 3.1 there exists a unique solution of (4.28) such that vl/' f E 3-1, v-'/'D f E Fl for any r < co. The first uniform estimate is given by (3.14). We apply this estimate in R(r) to obtain
In order to bound Pof we use results of Section 1. Introduce again the boundary layer ns= { X E R(") : dist (x, r,) 5 6 ) . The function $ = f ( with the truncation function C defined by (4.15) solves the problem D$=L$+g XER~;$EX where g = -f DC. Thus, in view of (1.40), (1.41) we have
with the uniform estimate. IIPoF117i I
c llvl/'
(I- P0)llE .
Define
a
= (f (0)7 v ) ~ ,Sarp,r(~)= -S
8x7
7
Sapbeing the Stokes matrix (1.43). Unless otherwise specified, Greek indices below assume the values 1,2,3 and summation over repeated indices is understood. Using (1.20) we get pu:'
* SOP
= (g,
Since g = L4 - 0 4 ,
POL= 0,
we conclude that pur' = (-D$, v,),
* Sap
4. Steady-state solutions
Integration by parts gives PU;)
Since
= ( f C, v o ~ ?*)Sap,r ~
1 P O V ~=V60731v12, ~ So0,7607 = 0,
we conclude that the function u(O) converges to
strongly in L2 (O(')) as S -4 0. Similarly, the function
(f (O)
7
+1)
converges to e(O)
strongly in L2 (O(r)). Here
= ( ( I - P,) K
f, ~
1
~* s ,4, m~. - )
is a constant and
Consider the moments of Po f('). Set
Due to (1.29) this function is given by P(') = (D4,~
*
7
)
~
Since suppc is the boundary layer, we integrate by parts in Rs on the right-hand side of (4.32). Note that for any fixed r we have the estimate
This implies that ( v - , ( v . ~f )) H, ( ~ ) S n (-x Y ) ~ C ,
Similarly we deduce
Chapter V. Steady-state solutions strongly in L2(R(')),where
where the following notations are used:
A, = ( f( v . n ) ,a,),
,
B,, = ( f ( v . n ) , b , , ) , ,
c
a, = -L-'(I - Po)+rv, b,,=-L-'(I-Po)v,v,
= ( f ( v .n ) ) , .
Thus we have proved that the moments mi = ( f ,$ j ) H have the following representation where sup llmj2)llH r
< C II(I - Po)f llx
while m y ) and m y ) are determined by d o ) ,8 ( O ) and P('), d l ) ,e(') respectively. We will use the representation to complete the proof. First, we deduce that
with some constant, depending on r. The following estimate is uniform with respect to r:
By Lad~zhenskayinequalities (1.48), (1.49) [37] these estimates imply
Note, that the functions m y ) are essentially the surface fluid dynamics potentials. Thus classical estimates give:
In view of (4.31) this completes the proof. Proof of Lemma 4.9. Denote the solution of (4.28) by fr. Recall that fr = 0, if z $ $I('). By Lemma 4.8 IIpfrIIH 5 r
4. Steady-state solutions
83
+
with cp = ( 1 1 ~ +1 1 ~~ 1 ~ ) - Thus l. there exists a sequence r, converges weakly in 'H. The function fr is a solution of
+ oo
such that cpf,,
where X , is the indicator function of R('). Taking weak limits we obtain that there exists a function f such that
f
= uA-f
+wf-,
$of E 'H.
(4.33)
Due to Lemma 4.8 the function f satisfies (4.29). Thus f is a solution of (4.27). Similar results are valid for the problem (4.1), (4.2) for general boundary conditions. Lemma 4.10 Assume, that f - E 1-I-. Then the problem (4.l), (4.2) has a solution in W satisfying (4.30).
Proof. Let f be a solution of (4.27) satisfying (4.30). Denote by V the corresponding solution operator V : f - I+ f . The problem (4.1), (4.2) reduces to the equation f + = SR7+VS f + S f - in 'H+. (4.34)
+
Due to (4.33)
V S f + = W S f + +UI{VSft Denote by X , the indicator function of
a(').Set
By (4.35) equation (4.34) is equivalent to
Let us show that there exists a solution with arbitrary C j . Choose G o , . . . ,C 4 . Consider the problems
with
By Lemma 4.10 both problems have solutions in 'H-*. Define f = fi This is a solution sought. Uniqueness follows from Theorem 4.6.
+ + C Cj$j. f2
Chapter V. Steady-state solutions
84
Proof of T h e o r e m 4.1. Let d = 3. Denote by C and B constant vectors with components Co,. . . ,C4 and Bo, . . . ,B4 respectively. By Theorem 4.7 the inverse matrix A-' exists. Set C = A - ' ( B - B(O)) and apply Theorem 4.3. This gives the result. Let d = 2. By Lemma 4.10 theorem is valid for B j = 0. Let any constants B o y .. . ,B4 be given. Define g = O for x E R 4
g=
~ ~ ( 1-1 bj4)a-' , ~ f for x E R2 \ R , j=1
a being the measure of R2 \ 0. By Theorem 1.3 there exists a solution of the problem
f3 E 'H-2 In view of (4.17) this implies the existence of a solution f3 with
with any prescribed Bo. Consider the problems
Dfl = Lfl X E R = R r f fi f; Q j ( f r ) = 0 j 2 1 p ( f i ) E 'H-~ r-fi
+
with
f; = f - , fi- = R r f f 3 - r-f3. By Lemma 4.10 problems (4.36) have solutions in W. Set f = fl f2 f3. This is a solution with required properties. In view of Theorem 4.7 this proves Theorem 4.1.
+ +
5 5.1
One-dimensional problems Setting of the problem. Main result
One of the classical boundary-value problem in the boundary-layer theory is to describe a flow in a half-space x > 0 with prescribed fluxes of the momentum and energy. The kinetic theory analogue is the following problem: to find f (x, v ) , x > 0, v = (vl, v2,v3) E R3 such that
5. One-dimensional problems with a given function f - = f-(v) and given constants Bj. The fluxes Q j are defined by
where
$j
= $j -
($)'"
6j4, the functions
$j
are defined by (3.3.6),
Since (Lf,$ j ) ~= 0, the fluxes Q j are constants. The traces ?* f are defined by
The functions 7 - f , Y+f determine the distribution of incoming and incident particles respectively. The function space 'H+ for the traces reduces here to
We assume that the operator R satisfies the conditions (4.19), (4.20). The projector P+ reduces to P+ f = ?+ (?+f , I ) ~ (+2 ~ ) ' / ~ . We shall construct solution of the problem (5.1)-(5.3) in
+
where 'Hk = { f : (1 x )f ~E H } , ' H o 'H, Sf (vl, vz, va) = f (-vi, vz, ~ 3 ) . In view of Proposition 1.8 equation (5.1) has a family of exact solutions
where Cj, Bj are constants, the functions pj are determined by p1 = -1 pj = p - l ( t j - x'pj) j = 2,3 ~4 = ~ - 1 ( $ j ~ 9 4 ) 'pj = L-'$jvl j -
Chapter V. Steady-state solutions
86
Note that due to (3.3.16), (3.3.17) the functions cpj enjoy the following properties Vs E R Ivlacpj E H = vjvlP(IvI) j = 273 cp4 = $IVIQ(IVI) Ipj
for some measurable functions depending on Ivl only. Clearly, pj E X k for k < -312. The main result of this section is the following theorem: T h e o r e m 5.1 Assume, that k < -312, Sf- E X + . For any given constants Bj(j = 1,. . . ,4) the problem (5.1)-(5.3) has a unique solution in W k . This solution has the form
f
=p+f(x,v)
(5.6)
where p E P , vl"f
E N,
E C([O, m ) , H),
Let us discuss connection between the problem (5.1)-(5.3) and classical problems with astrophysical origin, i.e. the Milne and Kramers problems. In the Milne problem the incoming flow at x = 0 is specified, i.e. R = 0 and the distribution is required to be bounded for large x. Theorem 5.1 says that there is a family of solutions, which are bounded in the following sense:
These solutions are determined by the conditions
Therefore, the Milne problem has a unique solution if the flux of transversal momentum is specified. The necessary condition of boundedness is Qj( f ) = 0 for j = 2,3,4. The Kramers problem is to find a solution of (5.1), (5.2) with R = 0, which may grow linearly for large x. By Theorem 5.1 any solution from 'Hkrk < -312 with nonzero fluxes grows linearly. There exists a unique solution of the I
5. One-dimensional problems
5.2
87
Proof of Theorem 5.1
The proof is similar to that of the theorems for many-dimensional problems. Particularly, we shall use the following integral equation for the solutions of (5.1):
Here I< = L
+ v , the operator U is the solution operator for the problem
the operator W is the solution operator for the problem
Assume that g is extended to be zero at x
'
( U S ) ( X . V ) = v1
-m
)v ) =
P = (1
G
12
g ( ~v ,) ~ X {P - v w }
< 0.
+ 1x1 + Ivl)"(l + Ivl)'
a,p E R1.
>0
vo > 0,
we conclude that
This implies that
I l x v f llu I C ll9PllU
x
dy
>0
Let f be a solution of (5.7). Then for vl
Since inf v
YI
lm- Y I
for v1 Set
Then
Ixg ( y , v ) U(P { - v T }
for v1
9
< 0.
7
being the indicator function of the set { u E R3 : u1 > 0 ) .
dy
Chapter V. Steady-state solutions
88 Set F ( x ) = SVl, cp2vIv1lf 2 M d v . Due to (5.9)
sup F ( x ) I C llgcpllk 3 x
+
J,,
By Lebesgue dominated convergence theorem
if gcp E 3-1. Similarly the case vl
< 0 is considered. Thus we have proved the following lemma:
Lemma 5.2 ( i ) . The operator vcpU9 is bounded in 3-1. ( i i ) The operator I ~ ~ ( ~ / ~ v ' / is~ cbounded p ~ c p from 3-1 into L W ( [ 0 , o o ] , H ) . (iii) ( ( ( v 1 ( 1 / 2 c p v 1 / 2 + ~ g 0( ( as H x -+ oo if cpg E 7-1. Here the weight function cp is given by (5.8). The operator W is given by
( w ~ - ) ( xv ,) =
I-(o, v ) exp { - v t }
o
for v ~ >
( W f - ) ( x ,v) = 0 for v1 < 0. Setting
f = W f-,
F(x)=
we deduce, that
I
f2(vl(~(v)dv,
as x + co if Sf-cp E H+. Therefore the following lemma is valid.
Lemma 5.3 (i) The operator c p ~ ' / ~ ~ Siscbounded p from 7-1+ into H . p from 'H+ into Lm([O,oo],H ) . ( i i ) Tlte operator I ~ ~ l l / ~ c p W Siscbounded (iii) If S f -cp E a+,then I I I v ~ ~ ' I ~ ~ ~ v ' f-llH I ~ w s -+ O as x + oo.
5. One-dimensional problems
89
We use these lemmas to describe the solutions of the problem (5.1), (5.2).
Lemma 5.4 Let f be a solution of (5.1), (5.2) in Wk for some k < -312. has the form (5.6) with B j = Q j ( f ) , ( j = 1,2,3,4).
Then f
Proof. In view of Lemmas 5.2, 5.3 we may follow the proof of Theorem 4.5 to deduce that f has the form (5.6) with some constants C j ,Bj. To determine constants B j we derive from (5.1)
Due to (5.4), (5.10)
Thus ( f 7
Pjv1)H
= ( f ? L P I V I ) H( O ) - ~
1'
& j ( f
(5.12)
Therefore
( f , P j v l ) H ( 2 ) = (f,Pj"-'t)H( 0 ) - x Q j ( p ) - x Q j ( J ) . Since Q j ( f )+ 0 as x -+ m, we obtain Q j ( p )= Q j ( f ) . We use the representation described to prove the uniqueness part of Theorem 5.1. Lemma 5.5 Let f be a solution of (5.1)-(5.3) in Wk with some k < -312. If S f - = O , Q j ( f ) = O(j = 1,. . . , 4 ) , then f = 0.
Proof. Apply the estimate (3.14) in R = { x : 0 5 x 5 r ) to obtain
x
being the indicator function of R. By Lemma 5.4 the function f admits the representation
where
4
p(x7V ) = Note that
C cjJj711 l ~ , l ' ~ ~ f
+
0 as 1x1 + 00.
Chapter V. Steady-state solutions
90
Therefore, the first term on the left-hand side of (5.13) converges to zero as r The condition (4.19) implies that
+ m.
Therefore ( I - Po) f = 0. Use (5.10), (5.12), (5.5) to obtain that the function Pof does not depend on x. In view of Lemma 5.4 this implies
Use the condition (4.20) to obtain Cj = 0. Lemma 5.6 Let S f - E 'H+, R = 0, Qj(f) = 0 if j = 2,3,4. Then the problem (5.1)-(5.3) has a solution in Proof. A modification of the method used in Section 4 is needed to control the flux Q1. Set 4: = vlc for vl > 0, 4: = 0 for vl < 0. Choose the constant c to satisfy the condition (4:)' vl ~ d =v1.
Set
p f: = 4: (f,*:vl)H7 yr- f = f ( r , v ) for v1 < 0, r,+f = f (r, v) for v1 > 0. Consider the problem vl-afr = Lf,
ax
if x E (0, r), r
y- fr = fr- if x = 0 yr-fr = SP:$ fr if x = r.
<m
(5.14) (5.15) (5.16)
Extend fr to be zero at x > r. By Theorem 3.1 the problem (5.14)-(5.16) has a unique solution in 7-1 for any r < m. The behavior off for large r is our main trouble. Apply the estimate (3.14) to R = {x : 0 < x < r } to obtain
5. One-dimensional problems
91
In order to estimate Pof, it suffices to use (5.10), (5.11) with f = f,. Taking into account (5.5) we obtain
where C is a positive constant independent of r . The estimates (5.17), (5.18) show that the following inequality is valid
with some positive constant C1. This implies that there exists a sequence f,., that converges weakly in Taking weak limits in the equation
we obtain a solution f of the problem (5.1), (5.2) in 7-t-1. By Lemma 5.4 this solution has the form (5.6). But the functions which grow linearly do not belong to Therefore, Qj(f) = 0 for j = 2,3,4. Due to (5.16), (5.10)
Hence
(f,~:), = O. Thus, f is a solution sought. It is not hard now to construct a solution of (5.1)-(5.2) with general operators R, satisfying (4.19), (4.20). Denote by V the solution operator for the problem (5.1)-(5.3) with R = 0, Qj(f) = 0, j = 1,. . . ,4. Due to Lemma 5.6 VS is bounded from H+ into 'H-1. The function f = Vf- solves the equation
By Lemma 5.2 the operator y+VS is bounded in H+. Set f + = Sy- f . To satisfy (5.2) we have to find a solution of the equation
+
f + = S R ~ + U K V S ~ +sf-.
(5.19)
(Note that y+Wf- = 0.) By Proposition 4.3.2 the operator y+UIcVS is compact in H+. I f f + is a solution of (5.19) with f - = 0, then VSf+ is a solution of (5.1)-(5.3) with f - = 0, Qj( f ) = 0. In view of Lemma 5.5 f = 0. Thus the uniqueness theorem is valid for (5.19) in H+. This implies the existence of a unique solution of (5.19) in H+. Set f = VSf + to obtain a solution of (5.1)-(5.3) with Qj = 0.
Chapter V. Steady-state solutions
92
To solve the problem (5.1)-(5.3) with arbitrary B j consider the following two problems:
D f i = L f f if x > 0 1 = 1 , 2 7 - f i = R Y + f 1 + f ; if x = 0
We know that these problems have solutions in 7-L2.The function f = fi B j p j is a solution required. This completes the proof.
cQ=, 5.3
+ fi +
Sources and asymptotics
Consider equation (5.1) with a given source term g:
Since the fluxes are not constant in this problem we look for a solution with property
Following the proof of Theorem 5.1 we obtain the following result: Theorem 5.7 Iieep assumptions of Theorem 5.1. Suppose in addition that
v-lI2 ( I - Po) g E R, Pog E
( g ,(P,) E 7--@ for some ,B > 1
'
Then for any given constants B, the problem (5.20), (5.21), (5.2) has a unique solution in W k with k < -312. This solution has the form (5.6). This theorem may be used to obtain some additional information regarding the function f in the representation (5.6). Note that the functions fn = x n f solves the equation
d v1-fn
ax
= Lfn
t gn
with
gn = n ~ i f n - 1 . Thus for n
> 1 we obtain fn
= U K f n t Ugn.
5. One-dimensional problems Suppose that for some n
+ + +
+
g(1 1x1 lvl)" E 'X, ( 1 IvI)"Sf - € 'X+ Pod1 2)n+l+P E 'X for some P > 1.
+
+
Assume in addition that the operator ( 1 I v I ) ~ S R ( ~Ivl)' is bounded in 'X+ for any s E [0, n ] . Then the function f in (5.6) satisfies the condition
Remark. Various results regarding asymptotics of the one-dimensional solutions can be found in [6, 161.
Chapter 6 Nonlinear steady-state problems This chapter concerns with some nonlinear steady-state problems for kinetic equations. The subject is clearly in its infancy. However, there exist two more or less well-studied limiting regimes. The first of them is the flows near a global Maxwell distribution. The second one deals with flows near vacuum. The existence and uniqueness theorems based on perturbation methods are proved for bounded domains in Sections 1 and 3. Note, that we deal with unbounded nonlinear operators, so standard perturbation methods are not available even for bounded domains. The external stationary problem, in particular, the problem of a flow around an obstacle, is more difficult. We show in Section 2 that typical solution has a highly nontrivial asymptotics in the trace of the obstacle and in a neighborhood of the Mach cone. The results are similar to those for compressible Navier-Stokes equation. The existence and uniqueness theorem is proved assuming that perturbations caused by incoming flows are small enough. Note that the asymptotic results mentioned above do not depend on these assumptions. Remark. There are only a few results regarding steady-state solutions far from equilibrium [3, 601. Results, similar to those described in Section 1 were obtained by A.Gejnts under more restrictive assumptions regarding boundary conditions. On the other hand, A.Geints has developed variants of approachfor highly irregular surfaces. The results of Section 2 were obtained in 1981 [&I. Similar results are proved by S. Ukai and Ic.Asano [62]. Related physical problems are discussed in detail in [58]. The main results of Section 3 are published in [@I.
1 1.1
Bounded domains. Small perturbations of Maxwellian distributions Main results
The aim of this section is to provide a foundation for perturbation methods for a class of stationary boundary-value problems in a bounded domain R with a Lyapunov
Chapter 6. Nonlinear steady-state problems
96
surface 30. It is quite clear, that the contraction arguments make it possible to study the problems:
with a small parameter s, if g = g ( f ) is a bounded operator in H = {f : M ~f E/ ~ L 2 ( Rx R:)). Similarly, the function f - can be replaced by an appropriate bounded operator. However, the collision operator I? defined by (3.3.3) is an unbounded operator in 3-1. The purpose of this section is to discuss the possibilities to construct the solutions of non-linear problems with some unbounded perturbations g. We look for the solutions of linear problems with Lm dependence on spatial variables. That is the main reason to consider the problem (1.1) in the following spaces:
The corresponding spaces for the traces are
The main linear result is Theorem 1.1 Assume that R = 0 , s > 0,
v-llzg E LP,7 - f E LP- for some p E [I, m]. Then the problem (1.1) has a unique solution in
Consider the nonlinear problem
Since v-l/'r is bounded in L" the contraction arguments lead to the following theorem: Theorem 1.2 Assume, that 7 - f E LC"'-,R = 0. There exist constants so, c such that zf E E [O,E~] the problem (1.4) has a unique solution in Wm- n { f : 11 fllLm C E ) .
<
The key point in the proof of Theorem 1.1 is Lemma 1.11, i.e., smoothing properties of the operator U I i . Let us discuss the problem (1.1) with nontrivial operator R. Assume that SR is a bounded operator in L2+ and Lm+. Assume in addition that
1. Bounded domains Here S is the specular reflection operator
S f ( x ,v ) = f ( x ,v - 2 n ( x ) v .n ( x ) ) . Note that the equation Df = L f has nontrivial solutions
~ by (3.3.6). Thus, with arbitrary constants Cj and the collision invariants $ J defined the homogeneous problem (1.1) with g = 0, f- = 0 has nontrivial solutions for some operators R satisfying (1.5). The structure of the solutions depends on the kernel of the operator I - S R . We restrict ourselves by consider the following two cases: Ker (I - S R ) Ker(I - S R )
n n
{y+$j);=o = . 4 { y+$l)j,o
0
= { Y + C , C ER 1 ) .
(1-6) (1.7)
Any constant C is a solution of the problem (1.1) with g = 0, f- = 0, if the condition (1.7) is fulfilled. The simplest operator R that satisfies the conditions (1.5), (1.7) is the operator R = SP:, where Po is defined by.
x + ( z ) being the indicator of the set { z > 0). This determines the linearized diffuse reflection. We consider perturbations of this operators:
with the following class of operators R1:
1 max {llSRlllLz.+, IISRIIIL-~+ 1 < 1+,
(1.9)
With these assumptions the analogues of Theorems 1.1, 1.2 are valid. T h e o r e m 1.3 Assume, that the conditions (1.3), (1.5), (1.6)) (1.8)-(1.10) are fulfilled. Then the problem (1.1) has a unique solution in WP-. T h e o r e m 1.4 Let f- E Lm- Keep assumptions of Theorem 1.3. Then the conclusion of Theorem 1.2 is valid.
Chapter 6. Nonlinear steady-state problems
98
The situation changes if we replace (1.6) by (1.7). As we mentioned above the homogeneous problem (1.1) has a nonzero solution under these assumptions. The orthogonality condition is
with N = L2, N+ = L2+ T h e o r e m 1.5 Assume, that the conditions (1.3), (1.5), (1.7)-(1.10) are fulfilled. The problem (1.1) has a solution in WO- iff the condition (1.11) is satisfied. The difference between any two solutions is constant. Since the nonlinear operator enjoys the condition
the orthogonality condition (1.11) for the nonlinear problem reduces to
Theorem 1.6 Assume, that the conditions (1.5), (1.7)-(1.10) are satisfied. Assume in addition, that Sf- E Lm+ and the orthogonality condition (1.12) is fulfilled. Then there exist constants EO,C, such that if E E [0, EO] the problem
has a unique solution in Wm-
n{ f
11 llLm
: f
5 CE) such that
Note that the condition (1.5) is satisfied for typical linear problems. But the situation changes crucially when we consider nonlinear problems. Consider for example the diffuse reflection. The nonlinear form R is
where p = p(x, v) is a normalized deviation from the Maxwellian distribution. The conservation of mass leads to the following condition: P 2 S p = 0. It is not hard to verify, that R11%+ > 1 for any nontrivial function p. However, it is possible to include such operators into consideration, if the deviations from Maxwellian generated by the interaction with the boundary are small enough. Consider the problem
11
Assume that the operator R is bounded in Lm+. Theorem 1.3 and simple contraction arguments lead to the following theorem.
99
1. Bounded domains
Theorem 1.7 Let the assumptions of Theorem 1.3 be fulfilled with p = ca. Then there exist constants eo, C , such that if E < a0 the ~roblem(1.13), (1.14) has a unique solution in W w - n { f : 11 f llLm 5 C E } .
However, i f the homogeneous problem (1.1) has a nonzero solution the orthogonality co9ditions lead to the following requirement concerning the admissible perturbations R: P ~ S = R P:. (1.15) Theorem 1.8 Keep assumptions of Theorem 1.5. Assume in addition, that the condition (1.15) is valid. Then there exist constants EO,Csuch that if a < a0 the problem (1.13), (1.14) has a unique solution in W m - n { f : 11 fllLm C E } , such that Jan (P,+f)do, = 0.
<
1.2
Proofs
The key point in the proof of Theorem 1.1 is the following property of the velocity averages. Let U be the integral operator defined by (4.1.6) with a positive function h depending on v only. Assume, that inf h = ho > 0. Define the averaging operator E by
Lemma 1.9 The operator EUE is bounded from LP into Lq if
(i)
1 < p 5 d,
q
< dp(d - p)-l,
d>1
Proof. Let g be a function from LP extended to be zero at x E Rd \ C.! In view of (4.1.6) we have
g(x - sv, v ) exp {-hs} ds. Set G ( x )= Eg(x,v). Then EUG(x) = J
exp {- h(v)s}G(x - sv)M(v)dv.
(1.16)
Rt x ( 0 , ~ )
The change o f variables s, v + r, y with z = Ivl, y = x convolution EUG=G*cp
- sv transforms (1.16) into the
Chapter 6. Nonlinear steady-state problems
100 where
Assume that d
> 1.
Since M(v) k(r, y)
< Cz exp {-C31vl),
5 c4rd-' exp
{
-:r)
we have
exp {-h0r-l
- -r
2
A.
with some positive constants C4,C 5 . This implies that cp E L, if p < The result follows from the Young inequality if 1 < p < d. B y Holder EUG E L" if p > d. If d = 1 we use the inequality
to deduce that cp E L, for any p
EUG E L",
< co. Using again the Holder inequality, if
1 1 G E L,I with - + - = 1. P'
P
This completes the proof. Proof of T h e o r e m 1.1. We use Lemma 1.9 to deduce the following statement. L e m m a 1.10 Let f be a solution of the problem ( 1 . 1 ) in W 2 , - . Assume that f - E E Lq for some p E [2, a]. Then f E Wq- . ~ q , Proof. In view of Lemma 3.3.1 the operator Ii = L the operators IC, with bounded kernels
+ v is compact in H.
Define
kn(v1,vz) = max{(sgnk)Ik(vl,vz)l,n), where k (vl, v2) is the kernel of Ii in H:
In view of Remark 3.3.2 the operator 1 11 1 ' is compact. This implies that I(, converges to Ii in the operator norm. Fix a small positive 6 and choose n(6) such that llIi-Ii,llH<6
for
n>n(6).
(1.17)
1. Bounded domains
101
Write I( in the form
K = K'
+ I{",
I(' = I(,.
(1.18)
Let U be the operator (4.1.6) with h = v. Any solution of (1.1) solves the equation
This implies the following representation of the function f:
where the operators Z j enjoy the following properties
with some constant C depending on q only. These estimates follow directly from Lemma 4.1.3. Choose 6 in such a way that C6 < 1 to deduce that
with some positive constant Cl. Note that for any J > 0 we have III('UI<'fllLq 5 C2(6)IJEUEf ]ILq. The result follows at once from Lemma 1.9. Theorem 1.1 is a direct consequence of Lemma 5.3.3 and Lemma 1.10. The contraction arguments prove Theorem 1.2. To prove Theorem 1.3 we need the following lemma.
Lemma 1.11 Assume that the conditions (1.5), (1.G) are satisfied. I f f solves (1.1) with g = 0 , f- = 0 in W 2 - - ,then f = 0 .
Proof. Use the inequality (5.3.14) to obtain
In view of (1.5) this implies ( I - Po) f = 0. Therefore D f = 0 as well. Hence it is possible to extend f in Rd \ R in such a way that the relations
Df = O ,
( I - Po) f = O
<
are also fulfilled for the extension. Then by (5.1.33) j(t,v ) = Fzf = 0 if # 0. Thus f is a polynomial and ( I - Po) f = 0. The assumption (1.6) implies f = 0. We will prove now the existence of a solution in WP-. Denote by V the solution operator for the problem (1.1) with R = 0. Clearly V has the form
Chapter 6. Nonlinear steady-state problems
102
where V o , K are solution operators for the problem (1.1) with y- f = 0 and g = 0 respectively. The function f = Kg q7- f solves the problem (1.1) iff it solves tlie equation (1.19) 7 - f = R-y+Kr-f f Rr+%g.
+
+ +
Let us prove that this equation has a unique solution in LPl-. Our proof relies on the following statement.
Proposition 1.12 Let B be a bounded operator in a Banach space. The Fredholm alternative is valid for B if B admits the following representation
where Bo is compact, I - B1 is invertible. Write the operator
6 in the following form
where the operator W is defined by (4.1.5) with h = v. Setting f+ = S y - f , we obtain from (1.19), (1.20), (1.9) the following equation for f+:
where B = Bo
+ Bl + Bz,
Proposition 1.13 Assume (1.8)-(1.9). Then the Fredholm alternative is valid for the operator B in LP+ with p E [2,w). First we prove that Bo is compact in LP+ for 1 < p < w. Use the definitions of the operators &, W to obtain that for some s ( x , v ) > 0 such that x sv E a R the following holds
+
B O ~ + (vX) ,=
u.n>O ~ ( x=)
y(x
1 vl
We make the change of variables
+ sv)lu
n ( x ) l exp { - v s } ~ ( v ) d v .
1.1 .n l f + ( x , v l ) M ( v l ) d v .
.n>O
(1.25)
1. Bounded domains
103
to transform (1.25) into
with bounded function 0. Since a R is a Lyapunov surface the compactness follows from classical results. The next observation is that the operator B2 is compact in LP+, p E ( 1 , m ) . TO prove this statement it suffices to recall that y+UIf is compact from L2 into L2+ in view of Lemma 5.3.3. On the other hand we know that y f U K is continuous from L" into Lw+. The compactness follows from the classical interpolation theorems. In view of Theorem 1.1 this implies the compactness of B2. Let us prove that the operator B1 is invertible. For this purpose consider the equation f+ =PBlf+ +Z+ (1.27) with a given function Z+ E L2+ and a parameter P E ( 0 , l ) . Note that by (4.1.16) Ily+WSIILpt 5 1. On the other hand IIRIILp+5 1 in view of the assumption (1.5). Therefore IIBlllLP+5 1 and PBl is a contraction in Lp+. We shall prove that the assumptions (1.8), (1.9) imply a uniform estimate
with some positive constant C. Note that it follows from (1.22), (1.27) that
IIBlf+llLP+5 a II(I - P+>f+IIbPt + IISR~~~LP+ Ilf+ll~ - ~t Therefore, the assumption 1.13 leads to (1.28) with
(1.30)
The estimate (1.28) provide the possibility to take the limit 6, -+ 1 in (1.28). Thus the operator (I- B,)-' is continuous in LP+. By Proposition 1.12 this completes the proof. Assume that f + = B f + . Then the function f = UI(VSf+ + W S f t solves (1.1) with g = 0, f - = 0. By Lemma 1.11 f = 0. In virtue of Proposition 1.13 this proves Theorem 1.3 for p < oo. Assume that v-'Izg E Lw, f - E Lw-. Use the decomposition (1.18) to write (1.21) as follows:
where
Chapter 6. Nonlinear steady-state problems
104 First we show that
(1.33) for some positive constant C and p > d . Since 80 is a Lyapunov surface, it follows from (1.26) that
Ilf+llLp
(IBof+llL, I C ( P )
,
P
> d.
(1.34)
Consider the second term on the right-hand side of (1.32). Note that
To prove this inequality denote by jl the following extension of h:
Then we have
Since inf v = vo
1
03
(
xv ) =
e - u 3 ~ (x s v , v)da.
> 0 the following estimate is obvious
where H ( x ) = sup, Ih(x,v)l. Change the variable v + y = x - s v to get
Set h = I i ' V S f+ and use the definition of the operator Ii' to obtain
By Theorem 1.1 the operator V Sf+ is bounded from LP+ into LP. Therefore
In view of (1.10) and (1.34) this proves that Z+E Lm+. The estimates (1.30), (1.17) show that llB311L-+ < 1
2. Steady solutions in unbounded domains
105
for the appropriate 5. Therefore, it follows from (1.31) that
By Theorem 1.1 this proves Theorem 1.3. Proof of Theorem 1.5. Following the arguments of Lemma 1.11 we prove the following statement.
Lemma 1.14 Assume that the conditions (1.5), (1.7) are satisfied. I f f solves (1.1) with g = 0, f - = 0 in W2-, then f is constant. By Proposition 1.13 the Fredholm alternative is valid for B in LP+n{f + : ( f +,l ) ~ z , = t 0). Orthogonality conditions lead to the requirement (1.11). This proves the theorem for p < w. Denote the solution by f. The estimates (1.35), (1.33) are valid for f . Thus the theorem is valid for p = w. Proof of Theorem 1.6. Let f be a solution described in Theorem 1.2. This function solves problem (1.1) iff f is a solution of (1.21) with g = r ( f ,f ) . The = 0 ) proves contraction-mapping principle applied in the space L P + n { f + I(f + , the theorem. Similar arguments prove Theorem 1.7.
2 2.1
Steady solutions in unbounded domains. Asymptotics in the trace regions Main results
We consider in this section the three-dimensional exterior problem for the nonlinear Boltzmann equation (3.1.1). Physical interpretation of the problem is as follows. The gas with a prescribed constant velocity c passes by an obstacle. It is assumed that the gas is in equilibrium at infinity. Thus we have to solve the problem (3.1.8)-(3.1.9) with a prescribed Maxwellian distribution at infinity
M, We shift the velocities v
D,f
= ( 2 ~ ) - exp ~ / {-lv ~ - cI2/2) ,c E R3. +v
+ c to obtain the problem
= J ( F , F ) x E 0,
R (-~,-F,~:F)
=0
where the following notations are used:
D, = (c + V ) . V,, 7,-f
=
x,f
Do= D
7:f = (1
-x3f
x E 8R,
Chapter 6. Nonlinear steady-state problems
106 X ; is the indicator function of
{ ( x ,v ) E d R x
:
n(x) .( v
+
C)
<0),
n ( x ) is a unit outward normal to d R . Having in mind to study asymptotics for large 1x1 we define f by
F = M(1+ f). The problem (1.1), (1.2) reduces to
The operators L and I? are defined by (3.3.2), (3.3.3). We shall use the spaces LP defined by (1.2). The corresponding spaces for the traces related with the operator D, are defined by
+
LP* = { f : M ' / ~ ~ ( VC). n ( x ) l l l pE L2(R:, L p ( d R ) ,f = 7ff ) )
.
(2.3)
Note that condition f E LP restricts the rate of decreasing for large 1x1. Typical solutions of the linear fluid dynamics equations decay as 1x1-'. These functions belong to LP(R) if p > 3, but do not belong to LP(R) if p 5 3. The results of Chapter 5 show that solutions of the linear problem (2.1), (2.2) have the same asymptotic behavior if c = 0, r = 0 . The essential difference between the cases c = 0 and c # 0 is shown by the following property: the solutions of the linear problem (2.1), (2.2) belong to L P for any p > 2 if c # 0. This property permits us to use perturbation methods to study the nonlinear problem (2.1), (2.2). The same approach to the problem with c = 0 leads to divergent series for arbitrarily small c. One of the most important properties of the solutions discussed is the fact, that nontrivial solutions do not belong to L2. Particularly, the fluid dynamics moments mj = (f,lC,j)Hdo not belong to L 2 ( R ) . To be precise consider the integral fluxes of the mass, momentum and energy on the surface of the obstacle R3 \ R. These quantities are defined by
where du is the Lebesgue measure on 80. Set
2. Steady solutions in unbounded domains
107
Typical boundary conditions imply that the mass flux Qo is equal to zero. The quantities Q j ( j = 1,2,3) determine the force, acting on the obstacle, while Q4 determines the heat flux. We shall say, that a function f is nontrivial solution of the problem (2.1), (2.2) if Cj=, # 0. Set
~3
Theorem 2.1 Let f be a nontrivial solution of the problem (2.1), (2.2) in WP- n Wm- with p < 1215, c # 0. Then f @ L2. We shall see, that typical solution has a slowly decreasing component. The support of this component is a union of 'very thin' layers in the trace of the obstacle and in the neighborhood of the Mach cone. The following theorem describes this component more precisely.
Theorem 2.2 Under the assumptions of Theorem 2.1 the solution of (2.1), (2.2) has the following form f = f ( 0 ) + f(1) where u ' / ~f ( l ) E L2, (I- Po) f (O),f (O)
L2,f (1' E Lp for any p > 2.
Recall that the projection Pois defined by (3.3.7). It follows that the asymptotics is determined by the fluid dynamics moments. To compute (f('), +,) we only need to know the fluxes Qj. The exact formulae are presented below (see (2.40)), Note that if c = 0 the general solution of the linear problem does not belong to L2 as well. The property of the solutions we use here to state nonlinear results is that nonlinear perturbations, generated by linear problem with c # 0 decay rapidly enough. The following theorem shows that the nonlinear problem possesses the solutions with properties that are assumed in the theorems above. Consider the problem
with a given function f-.
Theorem 2.3 Let f - E L2- n Lw-. There exist constants 6, C such that if (11f-IILz11 f-IILm-) < 6 the problem (2.5) has a unique solution in WP- n Wm-, p E (2,12/5), such that
+
Ilf llLP + Ilf
llLm
5 C6.
The limit value p = 1215 excludes nonlinear perturbations which decay slowly. The similar theorem is valid for general boundary conditions
Chapter 6. Nonlinear steady-state problems
108
if the operator R satisfies the conditions of Section 1 in the spaces LP- defined by (2.3). In particular, these conditions are fulfilled for the nonlinear diffuse reflection under plausible assumptions regarding the parameters of the surface. In this case
Here the function h is determined by
where Mw(x) = P,(x) exp {-lvIZ(2ew(x))-')
.
The function supz h E H if infzEanO,(x) > 112. The perturbations generated by the obstacle are small if the functions p,, 6, have small gradients and the velocity c is small enough. Our approach is similar to that used in Chapter 5. We start with study decay properties of the corresponding linear problem in the whole space. These properties are discussed in 2.2. The proof of the theorems is given in 2.3.
2.2
Solutions in R3
We consider in this section the problem
with a given function g and a small parameter a. All estimates below are uniform with respect to E . Theorem 2.4 Assume that
for some r , p such that r
< 6/5,p 2 2, (ii)
(1
+ 1xl)Pog E L2.
Then the problem (2.6) has a unique solution in
This solution admits the following decompositioiz
2. Steady solutions in unbounded domains
109
Proof. Our aim is to extract the terms that decay slowly for large 1x1. We start with the Fourier transform
Equation (2.6) gives
( X - L1.f = g
4 E R3
where A=ic.(+~,L=~-i[.v.
To obtain the decomposition (2.7) we study projections of (2.7) on span {KerL n { a , 6 ) ) ,
(2.8)
where a = -L-' ( I - Po) y$q,6 = -L-' ( I - Po) v@v. The projections give equations (5.1.5)-(5.1.9) with j - Xf instead of j . Set m i ( f )= ( f i h ) ~ i u = ( ~ , v ) H , P = ( f , t l O + (f)1'2$4)H T=(f7(I-P~)v@v)H, q=(f,(z-P~)v$4)H. ' Write the projections on the subspace (2.8) in the following form:
where p , K are constants, 6 = d ( Q )is the strain tensor defined by (5.1.10). Note that (2.10) implies (2.14) iltl" = (6 - A ] , v . t )- it(? 4). We will exploit the connection between this system and the system of the compressible viscous fluid, which is the system (2.9)-(2.13) without the last two terms on the right-hand side of (2.12), (2.13). This last system is a closed system for the moments m j ( f ) It is convenient now to introduce the column five-dimensional vectors
Chapter 6. Nonlinear steady-state problems
110
The fluid dynamics equations are given by
where G(O) = m(4) and T is 5 x 5 matrix given by
On the other hand the exact expressions (2.12), (2.13) lead to the fluid dynamics equations with a source term, generated by the last two terms on the right-hand side of (2.12), (2.13). Thus we infer from (2.9)-(2.13)
where
The resolvent ( A - T)-' is investigated in detail in [5, 50, 511. We use these results to study connections between Pof and (I - Po) f for small [ . In particu) the indicator function of the set lar, we shall use the following result. Let ~ ( f be { E E ~ 1 IEI ICI. There exists a constant Co such that if < Co the following representation is valid uniformly with respect to f :
Here c,,C,A?) are some ~ositiveconstants while B and P,!O' are 5 x 5 matrices. Actually, Pj(O)and I [ I X ; ' ) - 1[12Ay) are the first terms in the Taylor series for the
2. Steady solutions in unbounded domains
111
eigenvalues and eigenprojectors of T. On the other hand it follows from (2.12), (2.14) (5.1.32) that uniformly in 151 > Co > 0 the following estimates are valid:
for any r valid:
> 0 with
some constant C = C ( r ) . Therefore the following proposition is
+
Theorem 2.5 Let ( 1 Ivl)'j E H , ( 1 that f solves (2.7) for ReA 2 0, supe
for cp, = ( 1
+ Ivl)'
+1 ~ 1 ) '
( I - po) ./ E H for some r E R. Assume
< m. Then it holds uniformly in [, that
and some positive constant C .
The behavior of A m ( j ) is described in the following proposition. Set A =( O a = A ( g , ~ C ) ) L Z , lC) = ( $ 0 , . . .,1C)4) 2 112~. Qj(g) = (g,4j)L2, =4 - ( ) 94.
'
4
(2.19)
Proposition 2.6 Assume that
The function a enjoys the following properties a E L q ( b ) for any q
>2 4
Proof. First note that there are positive constants C I ,C2 such that
for all
E.
Since
I C31[I-2 and x(Ol[l-' E L 2 ( R 3 )the inclusion (2.20) follows.
Chapter 6. Nonlinear steady-state problems
112
To prove (2.21) consider the functions
where the functions di are defined by (2.16). We shall prove that it holds di E L"R3)
V q > 2 V j , dj $! L 2 ( R 3 ) if j = 1 , 4 .
Due to Hausdorff-Young inequality it suffices to prove that
Set
= - p , c = (Ic1,0,0). Then
Using the polar coordinates I-,0 , cp such that
one obtains
We make the change of variables
to transform (2.25) into
with
1'
1
+
cp(y) = ~ ~ ~ - [P~ 9, z ~'
Assume that 1
< q1 5 2. Then cp is continuous in
] - "d ~, ~ ~
.
(0, Icl-') and
Since suppdl = {[ 11(1 5 1 ) we derive that dl E L " ( R ~if) and only if g' < 2. Similarly we prove ( 2 . 2 2 ) , (2.23) for j = 4 (note that A?) = 0 , < 0 in this case).
xY)
2. Steady solutions in unbounded domains
113
Let us consider the cases j = 2,3, for which and set z = r [ l f Iclc;' cos B] to obtain
A ):
=
fc,, c, > 0. Use (2.24) again
with M a = Iclc;' and some positive constants Cj. It follows that the integrand has a unique singularity at z = 0. Assume that q' > 1. Then as r + 0 we have +m
pj(r)
N
I cj+ lm
r3-2q'
- 1
z2
1-q1f2
dz for M a > 1
m
r3-2q1
IC, + z21-q"2 dz for ~a
Ipj(r)l 5 ~
=1
r for ~M a < - 1. ~
~
~
Hence dj E L2(n3) for M a < 1 dj € L ~ ' ( R " ) ,$~L2(R3) ~ for q' < 2, M a 2 1. Due t o E-Iausdorff-Young inequality and the Parseval's theorem it holds that
It follows from the results of [50], that suppdj with j = 2 , 3 is a neighborhood of the Mach cone, while suppdj with j = 1,4 is the wake region. Proposition 2.6 follows immediately if we take into account the following formulae for the eigenprojectors 7'(0) 7'(0). 1 7 4 '
t
W=iTi'
This completes the proof. We conclude the study of the Fourier transform by the following ~ ~ The equaLemma 2.7 Assume that ReA > 0, # 0, IIrnXIItI-' < O O , V - ~E/ H. tion (2.7) has a unique solution in H. The following decomposition holds uniformly
Chapter 6. Nonlinear steady-state problems
114
where f^(O'=+.Am(j),
1C)=t($01...,$4).
(2.27) (2.28)
The rnatriz A is defined b y (2.15). Proof. The equation (2.7) has a solution in H iff there exists f E H that solves the equation f ^ =( ~ + v - i t . v ) - ' [ ~ j + i ] where the operator I( and function v are defined in Lemma 3.3.1. Since K is compact, the Fredholm alternative is valid. By (3.3.1) we have
uniformly in (. Since estimate:
< C)t1-2, we deduce from (2.29) and (2.18) the following
Substituting this estimate in (2.18), we get
In particular, it follows from (2.30), (2.31) that f^ = 0 , if j = 0. This completes the proof. Proof of Theorem 2.4. Set f = f(O)+ f ( ' ) , f ( j ) = ~ ; ' j ( j ) , where f ( j ) are defined by (2.26)-(2.28). To prove that v1f2f(') E L 2 , note that for any p E Lr with r < 615 we have
To verify this inequality observe that by Holder
where
g,
Since I is finite for g' < (2.32) follows from the Hausdorff-Young inequality. In view of (2.28) we have v'I2 f E L2. It follows from (2.27), (2.15)-(2.17), that
2. Steady solutions in unbounded domains By Ladyzhenskay inequality
+
for some positive constants Cl, C2. This proves that v1f2(1 Izl)-'f complete the proof it suffices to show that
E L2. TO
The function f ('1 solves the equation
In view of (2.32), (2.27) V,m
(f(O))
E L2 r) Lm if g E L'
n LP, r < 615. Hence
We claim, that it implies that f(') E L2 n LP as well. The proof of this statement is identical to that of Lemma 1.10. This completes the proof of Theorem 2.4. It follows from the proof, that the function f (O) in (2.8) has the form 3i1+.Am(j). In view of Proposition 2.6 this proves the following statement. Proposition 2.8 Keep assumptions of Theorem 2.4. Assume in addition that
Then the function
f(O)
in the decomposition (2.8) has the form
where the vector a is defined by (2.19). It holds that
2.3
Boundary value problems
In this section we prove Theorems 2.1, 2.1, 2.3, using the results discussed above. First consider the linear problem
with given functions g, f-.
Chapter 6. Nonlinear steady-state problems
116 Proposition 2.9 Assume that (i)
v-1/2g E Lr n L p with some r,p 6 such that 1 5 r < -,p E [2,m ] , 5 (ii) (iii) Then the problem (2.33) has a unique solution with properties
W e prove Proposition 2.9 by showing that solutions of the problem
with a small positive parameter
E
converge to solutions o f the problem (2.33).
Lemma 2.10 Assume that v-'fZg E L 2 , f - E L2-, (2.35) has a unique solution in L2 = L 2 ( R ,H ) . Proof. Put OR = R
n {XI 1x1 5 R < m ) .
E
> 0. Then the problem (2.34),
Consider the problem
Using the method of Section 5.3 we reduce the problem t o
where A is a compact operator in L 2 ( R ~H,) , while G is a known function from L 2 ( R ~H,) . Use the estimate (2.38) - ( L f , f )I~ I I l v l f 2 (~ PO)^ to obtain that any solution of the ~ r o b l e m(2.36) satisfies
11;
2. Steady solutions in unbounded domains
117
with some positive constant C. Therefore, the problem (2.36) with e > 0, R < oo has a unique solution in L2(RR,H). Since the estimate (2.39) is uniform with respect to s we get a solution of (2.36) by taking weak limit R + 0 in (2.37). This proves Lemma 2.10. We describe now some estimates of solutions (2.34), (2.35) which are valid uniformly with respect to E . First we conclude from (2.38), (2.39), that
We need an additional estimate for Pof . Extend f to be zero in R3 \ fl and consider the problem
Set
'FI-1 = {f l(l
+ Ixl)-'f
E L2} .
It follows from Theorem 2.4 and Proposition 2.8 that
uniformly with respect to conclude from (2.41)
uniformly with respect to
E.
E.
Applying the Green's formula to the pair {f,h ) we
Set
It follows from the estimates (2.40), (2.42), that
is defined by (5.4.5). By standard arguments we conclude, that where the space the problem (2.34), (2.35) has a solution in 'FI-1. In order to prove the uniqueness, assume that g = 0, f - = 0, in (2.33). The properties of I< imply that
Chapter 6. Nonlinear steady-s tate ~ r o b l e m s
118
Let C be the truncation function defined by (5.4.15). Extend f to be zero a t R3 \ and set = fC. This function solves the equation
I
with 3 = fD,C. It follows from (2.44) that v1I2 ( I - Po) f E formula to obtain
L2. Apply the Green
-
Denote by
x
the indicator function of supp DC. Then we have
Integrate by parts to obtain
It follows from (2.46), (2.47), that
Hence D , f = 0 and f = 0. This completes the proof of Proposition 2.9. The following proposition is the key point for the proof of Theorems 2.1, 2.1, 2.3. Proposition 2.11 Keep assumptions of Proposition 2.9. The function f admits the following decomposition f
= f ( 0 ) + f(1)
where
Here Q = Q ( f ,d o ) is the vector in R5 with the components Q o , .. . ,Q 4 defined by (2.4), $ = ($0,. . . ,$4). The matrix A is defined by (2.19). Proof. Consider again the function = fC with the same truncation function I . This function solves (2.45) with
'
Apply to this equation Theorem 2.5. Put
2. Steady solutions in unbounded domains Due to (2.18), (2.32), (2.45), we have
On the other hand Proposition 2.6 implies that
where A is defined by (2.19). Let x be the indicator function of supp DC. Then
where ( , ) stands for the inner product in L2. Integrating the second term on the right-hand side of (2.49) we get
Therefore Proposition 2.11 is valid for p = 2. To complete the proof we show that
Note that the function f(') solves the problem
D,f(') = ~ f ( ' + ) g - D,f(O), where f(O) is defined by (2.48). It follows from (2.48) that D,~(O)
E
nL ~ .
Let U be the solution operator of the ~roblem
D , f ( ' ) + v f ( ' ) = g(') x E R %-f(l) = 0 Z E B R . Fkpeating the proof of Lemma 1.10 with this operator we obtain (2.50). Th'IS completes the proof of Proposition 2.9. We can now prove Theorems 2.1, 2.2. Let f be a solution of (2.1), (2.2) in WP- n Ww-. Then by (3.3.17)
Chapter 6. Nonlinear steady-state problems
120 and
Hence the function g = I'(f , f ) enjoys the assumptions of Proposition 2.11 if p < 1215. The function f solves (2.33). Therefore, f has the decomposition (2.38). Since (+, = 0, we have
Recall, that the matrix A is defined by (2.19). By Proposition 2.6 f ( O ) 9 L2 if C Q; > 0. On the other hand, v1l2f(') E L2. This proves Theorems 2.1, 2.2. In order to prove Theorem 2.3 consider the problem
Let h E LP n Lm with p
< 1215. Then by (3.3.17)
~ - ' / ~ r ( hl ~, E) L'
n Lm
with
r = p/2.
The contraction mapping considerations are available, i f f - E L2is obvious.
3
n Lw-.
The proof
Steady solutions near vacuum
We construct in this section some steady solutions of the nonlinear Boltzmann equation describing the flow near vacuum. The problem is to find f (x, v) such that
with a small positive parameter E . We prove existence and uniqueness theorems for bounded domains R. Observe, that the influence of nonlinear term is significant in a neighborhood of zero in the velocity space. That is the reason for some authors to introduce some unphysical truncations. Our aim is to show that the collision operator J has properties which makes it possible to avoid truncations. For simplicity we consider the case of hard spheres with the diffuse reflection only, although generalizations are available. Note that the role of spatial dimension is significant in the problems under considerations. To clarify this fact we consider two boundary-value problems. The first one is the problem (3.1), (3.2) with a bounded domain R in R3. This problem is considered in 3.1. In 3.2 the classical Couette problem is discussed, where R is the interval (0,l). The estimates of the collision operator, key to later developments, are given by (3.2.21) and Lemma 3.2.5. These estimates are used to construct a function set, in which the contraction mapping arguments are available. Due to singularity in the velocity space mentioned above this set has a very special structure.
3. Steady solutions near vacuum
3.1
121
Three-dimensional problems
Assume that 0 is a bounded domain in R3 with a Lyapunov surface an. Denote by n(x) the unit outward normal at x E do. The diffuse reflection is determined by the following operator R:
where M(x, v) = (2~)-'h2(x)exp {-l~1~h(x)/2) x (3.4) being the indicator of {v : v . n(x) < 0). Physically, the function h in (3.4) is determined by the temperature of the wall an. Assume that
x
h E Lm(aR) inf h = ho
> 0.
(3.5)
Consider here the collision term J defined by (3.1.6) with the kernel
Recall that
and note that
J
M(x, v)dv = 1.
v.n
It follows from (3.1)-(3.8), that
J
qxan
(v . n(x))f (x, v)dvdu, = 0
where du is the Lebesgue measure on 8 0 . It will be shown that the conditions (3.1), (3.2) do not determine a solution in a unique way. It is necessary to introduce an additional normalization condition to determine the full mass in R. We construct solutions normalized by the condition
The value 1 has no particular meaning; it can be replaced by any positive constant. We start with the problem
Chapter 6. Nonlinear steady-state problems
122 with a given function f -
Recall that
where the collision frequency v ( f ) and the operator J+ are defined by (3.2.2), (3.2.1). Let W and U be the operators defined by (4.1.5), (4.1.6) with h = v ( f ) s . Note that in the stationary case these operators can be rewritten as follows:
where
We shall use the reduction of the problem (3.1), (3.2) to the integral equation
f
=A(f
(3.13)
where
A(f)= W f - $&UJ+(f,f). Introduce the following weight function:
p ( v ) = exp {
S I V ~ ~ ) (1 + 1 ~ 1 ~ s ) 2~ 0 , r 2 0 .
Set
Consider the Banach space
L- = { f : ~f E L ' ( R , L-(a)) with the norm
11.11
defined by (3.17).
(3.15)
3. Steady solutions near vacuum L e m m a 3.1 There exists a positive constant C such that:
c llf ll llsll
II~+(f79)ll-~ 5 for all f , g E Lm.
This Lemma is a direct consequence of Lemma 3.2.5 with k = -1, 7 = 1. Assume that the function f - satisfies the following conditions
sup w
/
Iw - vl-'cp(v)
[If-(., v)IIL..(an, < m
inf v(f-) = vo z,u
> 0.
(3.20) (3.21)
Let a j ( j = 1,2,3) be some positive constants. Consider a set d C Lm defined by
L e m m a 3.2 There exist positive constants I a4
aj
( j = 1,. . . ,4) such that A d C d if
E
Proof. Obviously Af 2 0 i f f 2 0. Set
It follows from (3.14), (3.11), (3.12), that
In view of Lemma 3.1 this implies that there exists a4 > 0 such that llAf 11 L a1 if
E
L a4.
(3.22)
It follows directly from the definition of v(f) that
with some positive constant y. Note that for any b > 0 we have lwl>b
( I w l S l)f (x, w)dw
5 b-I If 11
( I w ~ +l)f(x, W ) ~ W5 b-l
If 11-1 .
(3.23) (3.24)
Chapter 6. Nonlinear steady-state problems
124
Set I = {wlb-' < 1w1 < b). We deduce from (3.23), (3.24) that for any 6 exists b = b(lvl,S)such that
> 0 there
Obviously, for any f E A the following holds:
where
+
h = 1 - ~lvll-'(Iv'1 l)ul 11 f l l . We deduce from (3.20), (3.26) that there exists a4 such that
In order to estimate IIAf
11- 1
observe that for f E A we have
with
1
d
C = sup E 2,c
/v/-' exp { - a ~ ~ l v l - ' rdr. )
Use Lemma 3.1 and the assumption (3.25) to obtain, that there exist positive constants C1,Cz such that
Put a3 = Ci complete:
+ C2a:ai1, a4 = ~012. In view of (3.22), (3.26). (3.28) the proof is
Lemma 3.3 There exists a positive constant C such that
llAf - Asll 5
(3.5
Ilf
- sll
for all f,g E A, E 5 a4. The proof is simple and thus omitted. It follows from Lemmas 3.2, 3.3 that the equation (3.13) has a unique solution in A. Moreover, the sequence f ( O ) = 0, f(") = A f("-') converges to this solution in Lm and
where
x = L'
(elL - ( ~ R ) ),
3. Steady solutions near vacuum
125
C , C1 are some positive constants independent of
E
E
( 0 ,a 4 ) . By Lemma 4.1.2 we
have a similar estimate for the traces:
The function f(') describes a flow in vacuum. Let us return to the problem (3.1)-(3.3).Denote by V the solution operator for the problem (3.10). The function f = V ( y - f ) solves (3.1), ( 3 . 2 ) ,if the function 7 - f solves the equation r - f = R r + V ( r - f 1. (3.29) Denote by W o ,Uo the operators (3.11), (3.12) with
II = 1. Observe, that
Write (3.29) in the following form:
We seek a solution y- f in the following form
with unknown function N . It follows from (3.30) that
where operators K , N are defined as follows
where 9 is a bounded function depending on R. The property (3.7) implies that
J ( f , f )dvdx
= 0.
(3.31)
On the other hand by Lemma 3.2.7 we have 2
llGNlI~-(an)5 C1 IINll~-(an) for some constant
Clindependent of N
(3.32)
Chapter 6. Nonlinear steady-state ~ r o b l e m s
126
Note that the operator IC is compact in L2(aR). By (3.8) we have
ln
~ ( xY)dY , = 1.
Thus any constant solves the equation N = ICN. On the other hand
In view of (3.31) the operator ( I -K)-'G is continuous in LW(dR).Thus the problem (3.1), (3.2), (3.9) reduces to the following equation
with C equal to measure of aR. In view of (3.32) the operator & ( I- IC)-lG is a contraction in a ball {N ( ( ( N ( ( L<--m2 C ) . Thus, there are no problems with (3.33). The function f = V y - f with y-f = M ( v , x ) N solves (3.13), (3.9). Recall that (1 Ivl)-'J( f , f ) E Lm if f E Lm. Thus, f belongs to
+
+ (vl)-'Df E L W ) E L1 (R,L W ( a n ) )
W = { f E LW : (1 cp7-f The following theorem follows:
(3.34)
Theorem 3.4 There exists ~0 such that the problem (3.1)) (3.2)) (3.9) has a unique solution in W i f &< EO, 0 5 s 5 !j inf h, r 2 2.
3.2
The Couette problem
The Couette problem with diffuse reflection is to find a function f ( x ,v ) ,x E (0, l ) ,v E R3,v = (v1,~ 2v3) , such that
af
"1%
=&J(f,f).
Here
The functions M ( i , v ) are defined by
M ( i , v ) = ( 2 ~ ) - 'exp { - l ~ ( ~ h i /h: 2)
(3.35)
3. Steady solutions near vacuum
127
with some positive constants hi. These functions satisfy the conditions
Thus we have
We shall show that for any given constant Co there exists a unique solution of (3.35)(3.37),satisfying the following condition
Again let us suppose at first that the incoming flows are given. Thus we consider the problem
af
v l & = ~ J ( f , f ) xE(0,l).
with given functions f - ( j , v ) ,j = 0 , l . The next step is to reduce the problem to an integral equation. For this purpose we use the operators W , U , defined by (4.1.5), (4.1.6) with h = v . Note that these operators in one-dimensional case admit the following representation
where x(v1) = - sgnvl), H(Y) = exp { - e v l l J : "(2,v ) d z ) v ( x , v ) = v ( f ) ( x ,v ) Therefore the problem reduces to the integral equation f =Af, Let E be a plane in
e.Set
A =Wf-
+eUJt(f, f ) .
.
Chapter 6. Nonlinear steady-state problems
128
where the sup is to be taken over all planes E in R3,. Our proof relies on the following properties of the collision operators
Here C is some positive constant independent of f , while s is the parameter in the weight function (3.15). The estimate (3.41) is a direct consequence of the inequality (3.2.24). To prove (3.42) note that it is sufficient t,o follow the proof of Lemma 3.2.5 with h = cp, using the exact value of the integral
It follows that
which implies (3.42). The norms 11 f 11 , I [ f 11- 1 are defined by (3.16), (3.17) with R = ( 0 , l ) . Let Lw be the Banach space defined by (3.18) with R = ( 0 , l ) . Consider the following set in Lw:
A = { f E Lw : f 1 0 , l l f
11 5 a l , ~ ( f>) a27 I l f 11-1 5
l l f IIz 5 a41 .
Assume that the conditions (3.19)-(3.21)are satisfied with L w ( a R ) = ( 0 , l ) . Suppose in addition that (3.43) i
E
Lemma 3.5 Let s > 0 , r 2 1. There exist positive constants aj (j = 1,. . . , 5 ) such that Ad c A if a < a s . Proof. Obviously Af
2 0, i f f
1 0 . Set
Define for any 6 > 0 the set I as follows IA
In view of (3.19)-(3.21), (3.43) there exists 6 = b(lv1,6) such that (3.25) is satisfied with I defined by (3.44). This implies (3.27). Thus we can define a 2 by vo/2. Obviously, we have IlUf 11-1 I a;'.
Il~+(f,f)lJ-,
3. Steady solutions near vacuuin
Due to Lemma 3.1 this implies
Thus we can put a3 = a112
+ CaTa,'.
Similarly, we obtain
llAf llz 5 a3. We have to bound IIAf
11.
Note that
where
h =
E I v I I - ~exp ~ ~{ - E I V , I - ~ T ) d
~ .
Obviously
It follows that
llufI1 5 11 ~ + l l , [c + 21. In view of (3.41), (3.42) this implies that llAf 11 proof.
:]+ 11 E'
Jtll
.
< a1 for small e. This completes the
Lemma 3.1 There exists a constant C such that
I*
1
( A f - Ag)ll I C EIn ; Ilf - 911
The proof is similar to the proof of Lemma 3.5. It follows from Lemma 3.6 that the equation (3.39) has a unique solution in A for small c . To solve the problem (3.35)-(3.37)we seek the functions f - (i,v ) in the form
where Ni are constants we need to find. Let V be the solution operator for the equation (3.39). We deduce from (3.36) the following relations
NO = Nl - c G ( N ) , Nl = No
+EG(N)
Chapter 6. Nonlinear steady-state problems
130 where N is the vector (No, Nl), while G(N) =
/ l1 I.'
s>o
(V f-,Vf-)dzdv.
0
In view of (3.38) we have
Since IGI 5 Cl (max {No,N ~ ) for ) ~ some constant Cl independent of N, this system has a unique solution for small E . Let W - be defined by (3.34) with
Thus, we have proved the following theorem:
Theorem 3.2 There exists a positive ~0 such that the problem (3.35)-(3.37), (3.38) has a unique solution in Lm, if& < EO, s E (0,mini ;hi).
Chapter 7 Initial-boundary value problems for the Boltzmann equation This chapter is devoted mainly to the initial-boundary value problems with initial data in neighborhood of non-zero Maxwellian distribution, except for Sections 6 and 7, where some local and global results for general initial data are described. The subject of Section 1 is a semigoup associated with linearized Boltzmann equation. The results obtained in this section give the main tool for investigation of nonlinear problems, in particular, the problem of connection between the fluid dynamics equations and the kinetic ones. As mentioned above, a similar approach leads to approximations of other classical equations in mathematical physics. The Cauchy problem with smooth initial data in L2 is discussed in Section 2. Similar results for periodic initial data are described in Section 3. b s u l t s of the same type were obtained independentljr in [59, 411. Theorems presented in Sections 2, 3 and the method used there differ from those of the papers mentioned. The method used here has two advantages. First, more general initial data, namely, arbitrary functions from L2 instead of the bounded ones with prescribed decaying estimates turn to be admittable. Second, the method makes it possible to control to some extent smoothness of the fluid dynamics moments. It is worth noting that the obtained estimates are uniform with respect to Knudsen number E introduced in Hilbert scaling (see Chapter 2). In Section 4 smoothing properties of the Boltzmann semigroup are used to obtain soltuions with nonsmooth initial data. The method of this section is used in Section 5 to solve the boundary problems in the bounded domains with the Lyapunov boundary, where no smoothness can be expected. The consideration is restricted to the problems with prescribed incoming flows. The approach described in chapters 5, 6 rnakes it possible to include a wide class of reflection operators. The first results in this direction are due to A.Geintz [27, 281. His results allow to consider nonsrnooth boundaries. Similar results are available for some unbounded domains. Since for the unbounded domains there is a general stability result in [63], this case is not
Chapter 7. Initial-boundary value problems
132
considered here. As already mentioned, one of the classical problems of the kinetic theory is the problem of fluid dynamics limits. A result in this field obtained in [9] is strengthened in Section 8, using the theorems of Sections 2, 3. In [8] connections between weak solutions of the Navier-Stokes and the Boltzmann equation are discussed and some new estimates are obtained, but the proof isn't complete yet. Section 9 deals with some problems of long-time behavior of the solutions. This section presents investigation in progress, rather than completed results. There are several topics, omitted here but worth mentioning. Foundation of Hilbert serics for small time intervals and smooth initial data is given in [48, 381. Analysis for vacuum perturbations is developed in [34, 11, 121.
1
The Boltzmann semigroup
We discuss in this section the properties of the semigroup associated with the Cauchy problem for the linearized Boltzmann equation (3.3.1): = Lf + g
ft+v.V,f
t>O
(1.1)
Z E R ~ , VCR!
f lt=o
= fo
(1.2)
with given functions g and fo. The main aim is to collect the basic information necessary to study global solutions of nonlinear problems. The most essential properties discussed below are valid for a wide class of selfadjoint operators L in the Hilbert space H. More precisely, assume that L is a nonnegative semidefinite selfadjoint operator in H with the following properties: L1) L has the form L = Ii' - v with a compact operator IKI and a multiplication operator v . L2) The assertions 2 and 5 of Lemma 3.3.1 are valid. L3) dim I<erL
< co.
L4) Ker L = span ($0,.
. . ,$,I,
qhj(l
I~U'/~(I
L5) -Re(f, L f ) 2~I - pO)f onal projection on Ker L. For any
+ Ivl)' E H for some r > 0. where 1 is a positive constant, Po is orthog-
E Rd introduce the nonselfadjoint operator A
A
L = C(() = L - i6.v. The following properties of the operator Boltzmann equation.
(1-3)
k have been proved in the context of the
1. The boltzmann semigroup
133
P r o p o s i t i o n 1.1 [57,4,24] Suppose L l ) . Then E(J) generates a strongly continuous contraction semigroup in H. P r o p o s i t i o n 1.2 [61]. Assume L1)-L5). For any Jo stants Cl(to), Cz(t0) such that
> 0 there exist positive con-
) . point of {A E ClReA > -vo) is either P r o p o s i t i o n 1.3 [24]. Assume ~ l j - ~ dAny in the resolvent set of L ( t ) or is on eigenvake of finite multiplicity. Thus, to prove the assertions (2.1.23)-(2.1.29) it suffices to deal with small t only. This was done by using the perturbation theory by A.A.Aresenyev, R.S.Ellis and M.A.Pinsky. The assertions (2.1.23)-(2.1.29) are consequences of these results and Propositions 1.1-1.3. A simplified version of this investigation is sufficient to prove Lemma 2.3.1. In studying nonlinear problem we shall use the following strengthened version of the contraction property: P r o p o s i t i o n 1.4 Assume L1)-L5). Set
If fo E H, then
p(0
l'
l l ~ o(s)IIZ f
5 C llf (0)llZ
(1.5)
uniformly in t E [0, oo),t E Rd, f . Proof. It follows from L5) that
Clearly, this implies (1.4). To prove (1.5) define a truncation function b(t) such that
4 E C1(R1), ~ U P =P [O,Tl, ~ 4 = 1 in [6, T - 61 for some fixed positive 6 and T . P u t
i= f b
134
Chapter 7. Initial-boundary value problems
Then the function
f
solves
fi=Ef+h,
t€R1
with h = f&. Let F be a Fourier transform with respect to t with the dual variable 7. By (1.8) we have F = l?z, (1.9) where
+
Z = F ( L ~h ) , fi = (i7+ it . v)-l. Let x be the indicator function of the set {vJ 17 t .vJ < a) with some positive constant a to be chosen later. Due to L4)
F = 3-1,
+
+
where cp, = (1 Ivl)". On the other hand (1.9) gives
Obviously we have
Setting a = l l c p - s ~ l l i119-aZII, ~
we conclude that
uniformly with respect to F and 2. It follows from (1.10) that
Therefore
Consequently, the following estimate is valid
1. The boltzmann semigroup
Using Parseval equality, we get
Since L.f = L ( I - PO)^, we can estimate all terms on the right-hand side by the aid of (1.4). This completes the proof.
Remark 1.5 Note that (l.4), (1.5) imply Proposition 1.2. To prove this we use the following inequality. Assume that y E C([O,oo)), y 2 0, y ( t ) + a l y ( r ) d r S B y ( s ) , a,B>O,
t,s>O.
(1.12)
Then ~ ( t5) CI exp {-Czt) for some positive constants CI, C2. Using (1.12) we have
(1.13)
rm
lm
Define a function V ( s ) = y(r)dr. Then V 1 ( s )5 -y(s) I -a//3V(s). Therefore V ( s ) 5 V(O)exp {-ta/P). Integrating (1.12) over s c [t/2,t ] we have
Since y is bounded, this implies (1.13). On the other hand, (1.4), (1.5) imply (1.12) for y(t) = ~ l (t)lli f if
> & > 0.
A similar result is valid for solutions of the nonhomogeneous problem ft
=
e(t)f+gl
tE(OIT)
f lt=o = fo. More precisely, for any T > 0 we define the quantities as follows
where
+ Itl)--'.
cP = cp(0 = I1l2(1
(1.14) (1.15)
Chapter 7. Initial-boundary value problems
136 P r o p o s i t i o n 1.6 Assume L1)-L5). If fo E H,
A(g)
< oo,
then solutions of ( l . l 4 ) , (1.15) satisfy
with some positive constant C independent off, fo,g,T and J. R e m a r k 1.7 I t will be seen from the proof that to bound B(Pof ) it sufices to have a bound for B(Q-,(I - Po)f) with some Q, = (1 Ivl)', s > 0. More precisely.
+
Proof. In view of Proposition 1.4 it suffices to consider the problem (1.14), (1.15) with fo = 0. It follows from (1.6), that
In order to estimate Pof we introduce again the function (1.7). It solves (1.8) with
Following the proof of Proposition 1.4 we obtain (1.11) with Z = F ( L+~h). This implies (1.20). Obviously we have
Using (1.22) and (1.20) we infer (1.19) from (1.21). This completes the proof. The estimate (1.19) suggests a description of solutions in terms of the following decomposition: H = PoH$ ( I - Po)H. Another possible approach is based on the alternative decomposition
where P(J) is the eigenprojector, determined by (2.1.25). We will prove the analogue of Proposition 1.6 for the corresponding representation of solutions.
1. The boltzmann semigroup
More precisely, set
+
with cp = 1[12(1 I[[)-'. Note that L2) implies
for cp, = ( 1 + Ivl)' and r, s E R1 Particularly, v
1
f
2
5
p
(
)7
Ilp(t)y-1/211H 5
IIP(t)IIH '
We shall prove that solutions of (1.14), (1.15) satisfy S P l l f (t)ll;
+ ~ (1 5f c [lull:, + m]
with a positive constant C independent of f , fo,g and T. First note, that (2.1.28) implies
fo, it follows Since the function P ( [ )f solves (1.14), (1.15) with the initial data P([) from (1.25) that (1.24) is valid if ( I - P ( [ ) ) g = 0, ( I - P ( [ ) )f = 0. Note also, that (1.24) follows directly from Propositioll 1.6 if I[) > lo. In order to obtain general result we shall use the estimate (2.1.24). Define the operators A = A ( [ ) , B = B([) by
where
Q =Q(0 =I
-P(t).
Proposition 1.8 Assume, that Q f o E H,
A ( g ) < m.
Then it holds uniformly in [ E Rd, T 5 m, that
Chapter 7. Initial-boundary value problems
138
(i) V ' / ~ Ais bounded from H into L2([0,TI, H). (ii) V'/~BV'/~ is bounded from H into L2([0,TI, H). (iii) Bv'/~ is bounded from LZ([O,oo), H ) into Lm([O,oo), H). Proof. Set f(t) = Afot. Then f solves the problem -af + z [ . v. f + v f at
f lt=o It follows that
1
= Kf,
t€[O,T]
= Qfo2
1
(t)llH 5 5 ~ ~ ' i f ( t ) ~ ~ & -2 llf (t)ll& + IlvlJ2f Since K is bounded in H , (i) follows directly from (2.1.24). Similarly, we prove that V ' / ~ Bis bounded in L2([0,T],H). Put f(t) = Bg(t). Then f solves (1.14), (1.15) with fo = 0, g = Qg. Therefore, f satisfies (1.21). Consider the problem
Applying the Green's formula to the pair { f , cp) we conclude, that
for some positive constant C. Use (2.1.24) in order to estimate the function cp. Due to (1.21), (1.26), we obtain
Then (i), (iii) follow from (1.27), (1.21) directly. Thus, Proposition 1.8 is valid. This gives the estimate (1.24) immediately. We conclude this section by a description of solutions of scaled linear problem (1.14), (1.15). Recall, that the scaling
transforms (1.14), (1.15) into the following form
2. Near equilibrium global solutions
Set
Proposition 1.9 Assume (1.90) satisfy SUP
L1)-L5).
I l f (t)ll&+ B.(f
Then solutions
+
of problem
I c [ ~ l f o l l ~AC(g)]
(1.29), (1.31)
uniformly with respect to e and T provided the norms on the right-hand side are finite. Proof. First put fo = 0 in (1.19). Changing the variables according to (1.28) we obtain B c ( f )I C A C ( g ) , SUP I l f (t)ll; I CAe(g)e2-
Assume that g = 0. It suffices to change the variable t according to (1.28). Then we obtain from (1.19) that (1.31) is valid. Since the problem is linear with respect to fo and g, this completes the proof. Similarly, we deduce from (1.24), (1.23) that under assumptions of Proposition 1.9 the solutions of the problem (1.29), (1.30) satisfy
where
2
Near equilibrium global solutions of the Cauchy problem
We prove in this section the existence of global solutions for equation (3.3.1) with small initial data. The problem we are concerned with is to find function f ( t , x , v ) , t > 0, x E Rd, v E R:, such that
Chapter 7. Initial-boundary value problems
140
with a given function fo. It is assumed that the linear collision operator L satisfies the conditions L1)-L5). Recall that the operator r is unbounded in the Hilbert space
In virtue of (3.3.18), (3.3.17) the following conditions are fulfilled
Here Po is the orthogonal projector on Iier L, v is the collision frequency determined by (3.3.9). Recall that the function v is unbounded for large Ivl. Define the spaces H" = { f M ~f E / L~~ ( R ;H, ~ ) ) ,
I
where H" = H"(R2) is the Sobolev space. The norm in H" is defined by
Set
We shall use below the notation 11 f
llp,s
for the norm in LP([O,TI,H s ) .
Theorem 2.1 Assume that fo E H s , s > $, d 2 2. Then there is a constant a0 such that if 11 foil, < aol the problem (2.1), (2.2) has a unique solution in W .
The following proposition describes an essential property of the solutions
where B is defined by (1.16), (1.18). Proposition 2.2 Iieep assumptions of Theorem 2.1. There exists a positive constant
C such that
I I ~ I+I I, ,I, ~I Ic II I ;~ ~ .I I Z
(2.8)
Note that due to Parseval equality (2.8) implies that
This means that the function v 1 f 2 ( I- Po)f converges to zero as t -+ oo in a natural generalized sense.
2. Near equilibrium global solutions
141
Proof. Recall the definition of the operator I? (3.3.3). Let operator
rRbe the collision
I? with the kernel
Due to (3.3.15)
I l r ~ ( f , f ) l l H5 CRIlf l ; for some positive constant CR. Therefore
+
Note, that the operator L = -v . V, L generates in H s a strongly continuous semigroup } F;' exp { ~ E ( o } , exp { t ~ = where E([) is defined by (1.3). This is a direct consequence of Proposition 1.1. It follows that the problem (2.1), (2.2) with I? = rRhas a unique solution in C([O, to],H") for small to. We construct a solution of the problem (2.1), (2.2) for general unbounded operator I' by showing that the limit passage R + m is admittable. Moreover, we shall prove that any solution of the problem (2.1), (2.2) with I? = rR,R < m satisfies (2.8). Obviously, this implies the existence of global solutions. The following proposition is a key point in the later development.
Proposition 2.3
llf llH < m,
.
There exists a positive constant C such that if f E Lm(Hs),
then
Ilv-1'2r(f7f)(12,s 5 C Ilf
llm, s Ilf
llX
(2.10)
.
In order to prove Proposition 2.3 we use the following estimate:
Lemma 2.4 Let u, w E H'(R:),
Proof. Set
where
11. 1 1
= 11' 1 1L2(Rd).
II = IIuwII
,
s
> $, d 2 2.
Then
Due to Plansherel's theorem
12
= lllEIsfi
*
,
I3
* .
= lllEIsc fill
By Minkowski inequality
Since 1161(Ll(R3)
5
IIwIIHd
'-'d2
Chapter 7. Initial-boundary value problems
142 we have 1121
5 c IIvzuIIHa-I
1131
I C IIV~WIIH~-I IIuIIH..
Similarly,
IIwIIH*
In order to estimate Il we use the classical multiplicative inequality:
The estimate (2.11) follows immediately. Proof of Proposition 2.3. Set
I(( 5 1). Note that Ilf 1: Z llv112f2112,s+ II~zflll;,,.
where x is the indicator of {[I
Using the estimate II.wII. we get lIv-112r(fl
5 c IluII, 1 1 ~ 1 1 , ,
+ f 2 , f2)II, I C [Ilfzll, + IIv1I2f1llJ
By Lemma 2.4
llv-llzr(fl, fl)lls I c llv112fl Note that it follows from L4) that
-
I(~~'~f211.
lla llV.flIls-1
.
Thus, we have
The estimate (2.10) follows directly from (2.13), (2.14). Proof of Proposition 2.2. Let A(g) be defined by (1.17). Set g = Fzr. By (2.5) A ( g ) 2 l)v-1'2g)lH ' Therefore Proposition 1.6 implies
It follows from Proposition 2.3, that "P
llf (f)II. + 1 I11: 5 C [llf
111.~ Ilf 1: + llfoll:]
.
2. Near equilibrium global solutions
143
Proposition 2.2 follows immediately. Using standard arguments we deduce from Proposition 2.2 that the problem (2.2), (2.3) has a unique solution f E Lm([O,m ) , Ha) such that I(f 11; < oo. On the other hand, the estimate
implies that f C([O,m ) , Ha).
u-'l"
Proposition 2.5 Let f l , f2 be two solutions of the problem (2.1), (2.2) in W . There exists a constant a0 such that
Proof. By Proposition 1.6 we have
with
rj = I'(fj, f j ) .
It follows from (2.14) that
Therefore Ilfj(t)llf 5
c 11f011f
WP {C
61
l ~ v l / ~ f j ( ~d ) l~l .: }
We conclude from this inequality that for any a0
> 0 there exists to such that
The function f = fl - f2 solves the problem
Set D ( t ) = 11(1
+ I ( ~ ~ ) ~ ' ~t)ll,,,,, B(I,
with ~ ( j , tdefined ) by (1.16), (1.18).
,9
= r(fi+ fz, f)
Chapter 7. Initial-boundary value problems
144 It follows from (2.14), (2.15), that
By Proposition 1.6 we obtain
Ilf
(t)llf
+
D(t) [l -
5
1'
Gail
11f(~)11'[I/.'/'
+
[ f l ( ~ ) fZ(.)lll:ds
+ llf(o)ll:]
Proposition 2.5 follows. Proposition 2.5 implies, that the problem (2.1), (2.2) has a unique solution in W, if the initial distribution satisfies the condition 11 foil, < a0 with sufficiently small ao. This completes the proof of Theorem 2.1.
3
Periodic solutions
The method of Section 2 can be naturally applied to the problem (2.1), (2.2) with periodic initial distributions. We outline below the necessary changes. Let R be a cube [2naId. Set
Define the space
I
Hs = { f M1/'f E L 2 ( a ,Hs), (Pof) = 0 ) with the norm
J
1111 =1,
I I ~ ( . , V ) I I~, ,( u ) d v .
R:'
Consider the problem ft
f lt=o
= -v.Vzf+Lf+r(f,f), = fo.
t€[O,T],
(3.2) (3.3)
To describe the time dependence of solutions we use the spaces
respectively. Let X be a Hilbert space The norms will be denoted by 11 f ll,,,, 11 f l, with the norm llf 1 ; = - ~ 0 ) f + IlPof l12,a+1/2 Set
l :,s,
3. Periodic solutions
145
Theorem 3.1 Assume that fo E H s , s > %. Then there exists a constant a0 such that if Ilfoll, < ao, the problem (3.2), (3.3) has a unique solution in W for any T 5 oo. There exists a constant C such that for all T the following inequality holds
Proof. Let
t E Zd be a lattice point.
A periodic function of x is written in the
form
f ( x ,v ) = 3i1f(t, v), where
~ ; ' f ( { v, ) =
exp {ia-'t
+
x ) f(t,v ) .
E
Set k ( t ) = L - ia-lt . y. Since I< is compact, k ( [ ) is a compact perturbation of multiplication operator A ( J )= -v - i a - l t . v. By classical theorem in semigroup theory [36]k ( t )generates a strongly continuous semigroup in H. The main tool in proving Theorem 3.1 is again the decay estimate of the semigroup exP { t k ( t ) } Using (1.13) on that
Assume that
t E Zd we conclude that there are positive constants C l , C2 such
5 = 0. Put h ( t , v ) = f(t,O, v ) .
Since f ( t , 0, v ) = ( f ( t ,.,v ) ) ,we conclude that
This implies that
Setting L = F J , we deduce from (3.5), (3.6) that Ilexp {tL}lls 5 C I exp {-Czt)
'~'s2 0
(3.7)
for some positive constants C l , C2. Consider the problem
f, = L f f I,=, = fo,
+g,
t > 0, x E [2aaId, x E [2aald.
(3.8) (3.9)
Chapter 7. Initial-boundary value problems
146 Set
+
z(fO7g)= llv-112(~- P O ) ~ I I ~ llP0gll2,a-l/2 ,~ +llf~ll~.
Apply Proposition 1.6 on [ E Zd to obtain Proposition 3.2
If fo E H a , g E L2([07T],Ha), then the problem (3.8), (3.9) has a unique solutions in W . There exists a positive constant C such that
uniformly in T , f , fo, g.
It is straightforward to deduce from Proposition 3.2 that Theorem 3.1 is true.
4
Solutions in L"
The results of Sections 2,3 ensured us that the Boltzmann equation has a unique solution globally in t if the initial perturbations are small enough. Moreover, one has to assume that the spatial derivatives of the initial data are small. The latter assumption does not hold in the problems regarding high frequency oscillations. Typical boundary conditions do not ensure smoothness of solutions either. We describe below another approach to the problem, which provides a possibility to construct solutions in R: for nondifferentiable initial data. The method of this section will be used to prove the existence and uniqueness theorem for initial-boundary value problems. Let the space LP be defined by (6.1.2). Put
The norm in X is defined by
llf llx = IIvf llL- + IIvf llL2 Note that a function f belongs to X only if the rate of decay for large 1x1 is sufficiently large. Recall the definition of the spaces Lr+'*g(4.2.19). Define a measure p in the velocity space Rt by
Set
with the measure p. The norm in Y is defined by
llf lly = Ilf
llL-.z,z
+ Ilf
ll,p.m.z
.
Assume, that the collision operators L and satisfy requirements L1)-L5), (2.4), (2.5). Suppose in addition that the operator IKI is compact in H and
for some positive constant C. The estimate (4.3) follows from (3.3.16) for the Boltzmann operator r. Thus, r is bilinear symmetric operator in Y with the following property: II.-'r(f,f)lly 5 llf 11;. Theorem 4.1 Suppose that
c
fo E X .
There exists a positive constant a0 such that if llfollx < the problem (2.1), (2.2) has a unique solution in Y n 2.
Proof. Consider the linear problem (1.1), (1.2). Assume that u-lg E X. Since the semigroup exp { t l ) is a contraction semigroup in L2, the solution f : [0, oo) -t L2 is given by
f (t) = exp {tL} fo +
UP {(t - s)L} g(s)ds.
(4.4)
Thus the problem (2.1), (2.2) reduces to the following equation in Y:
f
= Nf,
(4.5)
where N is the solution operator of (1.1), (1.2) with g = r ( f , f). We will prove Theorem 4.1 by showing that N is contraction in a ball of X with radius dependent on perturbations fo and G. The next two lemmas on the solutions of the problem (1.1), (1.2) are key points of the proof.
Lemma 4.2 Assume that
f o E L2,
g E Lrn([O,oo), L2), Pog = 0.
Then the solution of problem (1.I ) , (1.2) satisfies the following inequality: sup 11f(t)llL2 5 C
[llfollu
+ S ~ (lls(t)llLz P + lls(t)llL1)]
for some positive constant C independent of fo and g.
Chapter 7. Initial-boundary value problems
148
Lemma 4.3 Suppose that V-lg E Y, fo E X . Then the solution of
(4.4)
has the following property
To prove Lemma 4.2 we use decay estimates of the semigroup exp { t l ) given below. Proposition 4.4 Let q E [1,2],m = 0 , l . Then 1le.p {t.c) ( I - P0)"llL2
5 C ( l + t)-" [ l l f
11L2
+Ilf
llLJ
(4.6)
9
where
Proof. Use the properties of the semigroup exp (2.1.26) we have IlexP { t t ( t ) ) l l ,
<
CI
[exp {-Czt) ( 1 - ~
Here C j are positive constants independent of the set {[ E R3 1 151 5 1 ). Using Parseval equality we have
.
In view of (2.1.23)
(0 + ~)X {-CzlJ12fI P Itlrnx(J)I.
t, while x is the indicator function of
Split the domain of integration into two sets Dl = ((1 I J I denote by 11,I z the respective integrals. By (4.6)
< I), Dz
= R3 \ Dl and
where
h ( J ) = exp {--2CzlJI2t)
1t12rn~(5).
By Holder inequality
where ll.llp stands for the norm in Lp(R3). The norm Ilhll,, can be majorized by C ( l t)-m-3/2'71.By Hausdorff-Young inequality
+
These estimates prove (4.6).
Remark 4.5 If we replace R: by Rf, then the same proof gives (4.6) with
Proof of Lemma 4.2. Using Proposition 4.4 we conclude from (4.4), that
hm(l+
Lemma 4.2 follows, since T ) - ~ /< ~ m. ~T Let U be the solution operator for the problem
af
-+v.V,f+uf at
= g,
f lt=o =
t>O,
xER3,
fo.
We use smoothing properties of the operator U to prove Lemma 4.3. More precisely, define the averaging operator E by
with a density w E L1(R:)
nLm(R:). The following analogy of Lemma 6.1.1 1 is valid.
Lemma 4.6 The operator E U E is bounded from
into L"([O, m ) , L9) if q < 3p(3 - p)-', 1 5 p I 3 or q = 00, p > 3.
Proof. Set h = Ef (x) and write EUg in the form
Introducing new variables y = x - v(t - T) we get
where
k(t,x) =
exp {-v(t - 7)) t - 3 ~ ( x / t ) if t > 0 0 if t I 0 '
Chapter 7. Initial-boundary value problems By Young's inequality we have
if 1 < p < p', 1 / q = 1 / p - l / p l , 1 < p < 312, l/p' = 1 - l / p . Hence the Lemmais valid for p 5 3. By Hiilder inequality (4.7) is valid with p > 3 , q = 00, p = p'. Since Ilh(t)ll, 5 C Ilgll,, the Lemma follows. To prove Lemma 4.3 we have essentially to repeat the proof of Lemma 6.1.10. In order to complete the proof we return to equation (4.5). Applying (4.3) and Lemmas 4.2, 4.3, we obtain that
IIN(f Ill y IlN(f1) - N(f2)llY
5 5
c [ l l f 11; + z ( f o , G ) ] c llfl + fzll; - fzll; llfl
7
where
Z ( f o , G ) = llfoll,
+ IIV-~GII,, ,
C is a positive constant independent of f , f l , f2 E Y . It follows that there exists a ball B in Y of radius depending on Z ( f o , G ) such that N is a contraction in B. This proves the existence of a unique solution in B. To prove that there are no other solutions with fo and G small enough we only need to repeat the arguments similar to those used in Proposition 2.5.
Remark 4.7 A similar theorem holds for problems of dimensions d < 3 but with the additional condition
5 5.1
Initial-boundary value problems. Global solut ions Introduction
The methods of Sections 2-4 can be applied to study some boundary value problems. Since Grad has shown that the ~roblemin cube with specular reflection can be reformulated as the Cauchy problem with periodic initial data, we have an example, where the results of Section 3 may be applied in a straightforward way. We return now to general case (3.1.10)-(3.1.11). The operators W,U described in Section 4.2 make it possible to reduce the problem to an integral equation. Due to properties (4.2.20), (4.2.21), (4.2.22), the traces of solutions are well defined. It follows that we
5. Initial- boundary value problems
151
can use the same approach as in stationary problem in order to reduce the general setting (3.1.10)- (3.1.11) to the very special type of the reflection operator R:
with a given function @-. Another essential new element is that the typical boundary conditions do not ensure the smoothness of the solutions. Therefore, it is impossible to write a nice equation for spatial derivatives and to work with Sobolev spaces freely. The third problem is connected with the decay estimates. Note that the results of Sections 2-4 describe solutions of the problem with a unique trivial steady state. Clearly, some modifications are necessary in order to include solutions with nontrivial long-time behavior. In this section an approach to these problems is described for solutions which are close to a Maxwellian distribution M with constant parameters. Changing the unknown function F = M + M f in (3.1.10)- (3.1.11) we arrive at the problem
with a prescribed function f-. Here
while aQ- is the hyperbolic part of the boundary, determined by (4.2.12). Throughout this section it is assumed that R is a bounded domain with a Lyapunov boundary. The main result we are going to prove is that the small bounded incoming flows and initial data generate global in time bounded solutions (see Theorem 5.7). It is essential, that there are no restrictions on the behavior of spatial and time derivatives. Generally, solutions can display a complicated long-time behavior.
5.2
Linear problem
Consider the problem
where
t=-v.V,+L. The existence and uniqueness theorem for the problem (5.3), (5.4) is valid under rather general assumptions regarding the incoming flow f- and the source term g.
Chapter 7. Initial-boundary value problems
152
To observe this it suffices to reduce the problem (5.3), (5.4) to integral equation by the aid of the operators introduced in Chapter 4. More precisely, let a be a positive constant. Put! = f exp {-at). Use the operators U, W defined by (4.2.13), (4.2.14) v . V,, h = v a to reduce (5.3), (5.4) to the following equation: with A =
at +
+
where g = gexp {-at), f- = f - exp {-at). Since K is bounded in H, we conclude from (4.2.8) that the operator UIi' is a with contraction on any ball B = { f 1 11 f 11 < R < a ) , where (1. 1 1 is the norm in LPJ'T~ dp = Mdv, p E [I, oo]. However, this very naive and general approach doesn't provide a possibility to work with the Boltzmann nonlinear collision operator for general intermolecular forces. The first obstacle is that the operator r is unbounded in any of spaces mentioned above. The only result we can obtain by means of this straightforward approach is a local in time solvability of the problem (5.1), (5.2) with small data. To find a more fruitful approach, consider the space
P r o p o s i t i o n 5.1 The operator L associated with the boundary condition y-f = 0 generates a strongly continuous semigroup in L2. Proof. Consider the stationary problem
+
Let U, W be the stationary transports U, W defined by (2.1.8), (2.1.9) with h = X v . Assume that Re X > 0. It follows from Lemma 5.3.3 that the operator UIi' is compact in L2. Let L'* = { f 1 I v . n(x)lMlt2f E ~ ' ( 8 0 R:)) , . The norms in the spaces L2 and LZf are defined by
It follows from (3.3.8), that -
f
1
f
-
1
f
1
+ 1 2 ( - o
f
2
(5.6)
5. Initial- boundary value problems
153
where the projection Po is defined by (3.3.7). In view of (5.6) we obtain from (5.5), that ReAllflllllgll if ReX>O. By Hille-Yosida theorem this completes the proof. Proposition 5.1 suggests an alternative integral equation corresponding to the problem (5.3), (5.4) with 7- f = 0:
it
f ( t ) = exp { t ~f )( ~ ) + exp {(t - s ) ~ ) g ( s ) d s .
(5.7)
On the other hand, the problem (5.3), (5.4) with a general incoming flow 7- f reduces to the problem with 7- f = 0. For this purpose we can use the operator W, defined by (4.2.13). Another possibility exists for incoming flows, which generates steady-state solutions, described in Chapter 5. More precisely, assume that the function f, solves the problem ~ f=,0, E n; -,-fa = ~ - f - , E an. Obviously, the function f - f, solves the problem (5.3), (5.4) with 7-( f - f,) = 0. Therefore, it is possible to reduce the problem (5.3), (5.4) to equation (5.7) with the semigroup exp {tL), satisfying the condition
In order to use the methods described in Sections 2-4, we need much more, namely:
with some positive constants Cl, C2. Let f be a solution of (5.3), (5.4). It follows from (5.6) that
where (f19) = ( f , 9 ) ~ 2 . In particular, this implies that if Pog = 0, we have
Chapter 7. Initial-boundary value problems
154 Set
llf llz = Ilf llL2([0.T,,L2) 9 ( f g)2 = ( f ~ ) L ~ ( [ o , T ~ , L z ) ~
Ilf
llz*
=
Ilf
llLz([o,Tj,~l*) .
The estimate (5.10) gives the following result: Proposition 5.2 Assume that
Then the solutions of (5.3), (5.4) satisfy
This is the first decay estimate, because for T = w (5.11) gives
The next step is to describe perturbations which provide a similar property for Po f. For this purpose we use the results of Section 3, concerning the periodic solutions. Let cp be a solution of the problem
with some function h, which is 2na-periodic with respect to space variable x. Assume, that Poh(x,v)dx = 0.
J
[2ma]d
In virtue of (3.7) we have
for some positive constant C independent of h and cp. Choose the parameter a in such a way that
Then it is easy to prove that
llr-cp11~-+ ~ ~ r + c5pC~llhll~ ~~+
5. Initial- boundary value problems
155
for some positive constant C. Restrict the problem (5.12) to R and put
where
Then it follows from (5.12), (5.3), that
This implies that
Using the estimates (5.9), (5.13), (5.14) we infer llhll: 5
c [I(f,g)zl + llsll;] .
In order to estimate (Pof) consider the problem
It follows from (2.1.24) and Lemma 4.1.2 that
+
II7+~11~I I ~ ' / ~ c p5l l ~ CI(Pof)12.
Replacing h by (Pof ) in (5.15), (5.16) we obtain
Combining (5.9), (5.17), (5.18) we infer that
for some positive constants C1,Cz independent of T, f and g. The following consequence follows immediately:
Chapter 7. Initial-boundary value problems
156
Proposition 5.3 Assume that
Then the problem (5.3), (5.4) has a unique solution such that
The following estimate is valid SUP l l f (t)1I2 t
uniformly with respect to T . The estimate (5.8) is a simple consequence of this proposition. It suffices to note that (5.19) is valid for any T > 0. Set g = 0 , y - f = 0 and use (1.12). Clearly, it proves the following assertion. Denote the semigroup described in Proposition 5.1 by exp { t C ) .
Proposition 5.4 Tliere exist positive constants Cl,C2 such that the estimate (5.8) is valid. Applying this result to equation (5.7) we conclude that i f f solves (5.3), (5.4) with
then
IIf(t)lI 5
[ I I ~ ~+I II I ~ I I L ~ ( [ ~ , T ~ , L ~ ~ ] (5.20)
uniformly with respect to T, fo,g. The last property of the semigroup exp { t l ) we need to deal with nonlinear collision operator is Lhe following analogy of Lemma 4.3. Recall the definition of the (4.2.5), (4.2.6). Define a measure p in the velocity space by (4.2). spaces LP", LrvPvq Set x ,L ~ J ,y = ,5m1m92. L e m m a 5.5 Suppose that
x,
fo E v-lg E Y. Then the solution of the problem (5.3), (5.4) with y- f = 0 Iias the following property:
The proof is quite similar to the proof of Lemma 4.3. We are ready now to deal with the nonlinear problem (5.1), (5.2).
5. 111itial- 6ou11dar.y value problems
5.3
157
Nonlinear problem
Solutions of the problem (5.1), (5.2) constructed below belong lo the following sel of functions:
Note that the derivatives ft, V,f are defined for f E W in the distributional sense only. Let W be the solution operator for the problem (4.2.14) with h = v. Set f = fl f i rwhere f i = W f - Then (5.1), (5.2) i~~lplies the following problem for fi:
+
a zfz
jz
= lfi = 0
+ ~ ( J+I fz, + fz) + I< fi
fl
if ( 1 , x,v) E Q , if (t, :c, V) E aQ-.
This type of equation arises also in a natural way in studying stability problems. Consider the following more general problem:
Theorem 5.6 There exists a coi~sta~tt a0 such that if
the problent (5.22), (5.23) has a uiiique solutio~tin W JOT ally T 5 oo. It is quite clear that the estimates (5.20), (5.21) provide all tools used in the proof of Theorem 4.1. Therefore, it is enough essentially to repeat it to prove Theorem 5.6. r u t jl = h, g = K ~ I r ( f i , fl),
+
The condition (5.4.6) is salisfied if
for some suficiently small conslant bo. Therefore we have the following result: T h e o r e m 5.7 Tltere exist a co~lsta~at a0 such that if
the pr.oblenz (5.1), (5.2) has a unique solutiot~in W
Chapter 7. Initial-boundary value problems
158
6
Local solutions
A brief discussion of theorems on the existence of local solutions of the problem (3.1.10)-(3.1.11)is the aim of this section. The main difficulties in constructing local solutions are due to the fact that the operators generated by the collision integrals in "natural" function spaces are unbounded. The unboundedness is caused not only by the quadratic dependence of Jf on F, but also by the growth of the functions Jf ( F ) for large velocities. There are no problems in proving local theorems for bounded collision kernel B. In particular, the following theorem is valid. Set
where the weight function cp is defined by (3.2.23). Theorem 6.1 Let the condition (3.2.10) be satisfied with y = 0 . Then there exists T I , such that the Cauchy problem (3.1.10), (3.1.11) has a unique solution in
Proof. Recall the definition (3.2.2) of the collision frequency v ( F ) . Consider the integral equation F = WFO U J + ( F ) ,
+
where the operators VV, U are defined by (4.2.4) with h = v ( F ) . Lemmas 3.2.2, 3.2.3 imply that in an arbitrary space X(p) satisfying (6.2) the operator
enjoys the following properties
where
11.11
=
Il.l x
Furthermore, the operator V is a contraction in the ball
if T is sufficiently small. Hence, Theorem 6.1 holds. The following theorem is valid for general kernels B satisfying (3.2.10). T h e proof below makes it possible to validate approximations based on various truncations of the kernel B. Moreover, we will show, that there exist problems, where local solutions described in this theorem can be extended to an arbitrary time interval.
6. Local solutions
Theorem 6.2 Let the conditions (3.2.10), (3.4.11) be satisfied. Set
There exist a constant ro depending on the kernel B only, such that for r > ro the following holds. There exists T I > 0, such that for T < T I the Cauchy problem (3.1 .lo), (3.1.11) has a unique solution in
Proof. Replace the collision kernel B by B, = min {B,n). In view of Theorem 6.1 a unique solution exists if T I T,, where Tn is a positive constant depending on n. We will obtain uniform estimates of the solutions with respect to n. Put g = F exp { - a t ) , where a is a positive constant to be chosen later. The function g is a solution of the problem
Without loss of generality we can assume that T n 5 a-' . By virtue of the inequality (4.2.8) o < g 5 sup Fo sup J + ( ~[) v ( ~ )ae-0~1-l e (6.3) x
+ x,t
+
Lemma 3.2.1 implies that
The following statement is a key to constructing uniform estimates. Under the assumptions of Theorem 6.2 there exists constant d E (0,l ) ,such that
By Lemma 3.2.3 this condition is assured for large Ivl by a proper choice of r. Indeed, this condition is connected with the assumptions r > ro in the formulation of the theorem. For small Ivl the inequality (6.4) is assured by a proper choice of the constant a . It follows from the inequalities (6.3), (6.4) that
This inequality leads to the uniform estimate:
for T 5 e(1 - d ) ~ ~ p ~ o l l - l .
Chapter 7. Initial-boundary value problems
160
By virtue of (6.5), the solutions guaranteed by Theorem 6.1 can be extended t o a time interval [0,T I , whose length does not depend on n. Thus
AF, = J,(F,),
05t5T
(6.6)
where J, is the collision operator .I with the kernel B,. The existence of the limit as n -t oo in this equation follows from Theorem 4.3.6. Indeed, the sequence F, is weakly compact in L2. Theorem 4.3.6 implies that the sequence E F , is strongly compact in Lkc([O,TI x R3). From the estimate (6.5) we see that the sequences (ll,Fn) are strongly compact for 141 < 1 1 ~ 1 ~Hence . the sequences ($J*(F,)) are also compact for bounded 11, and
+
if Fn -t F in L2- These relations are sufficient to take the limit n -+ oo in equation (6.6), and to construct a weak solution of problem (3.1.10), (3.1.11). We now prove the uniqueness of the constructed solution. Let F l , F2 be two solutions in X . Put g = (Fl - F z )exp {-bt), where b is a positive constant to be chosen later. The function g satisfies the equation
Setting 11, = ( 1
+
we obtain
Reasoning exactly in the same way as in the proof of (6.4) we conclude that the constant b can be chosen so that for some q E ( 0 , l ) the right-hand side of the last inequality is not greater than q IIpgllx Consequently, g = 0. This finishes the proof. The proof gives one possible method for constructing the solution. A second method is based on the application of the following integral equation for the function g = F exp { - a t ) :
Let BR be a ball in X of radius R = 2 I I ~ F o ~ ~ ~ , ,Using , ( ~ ~ ,(6.4), we can prove that for small T the constant a can be chosen so that the relations
are satisfied for any gl,gz E BR with some constant q E ( 0 , l ) independent of g1,g2. From these relations it follows that the iteration sequence F(n) = V a ( ~ @ - l converges )) to a solution of equation (6.7) in X ( $ p ) and that the constructed solution is unique.
7. Global solutions
161
The following theorem shows that the Cauchy problem has solutions decaying exponentially in Ri. Let X = X(cp) be defined by (6.1) with the weight function
Theorem 6.3 Let the condition (3.2.1 0) be satisfied with 7 5 1. Then Theorem 6.1 holds for cp = cp,, with s > 0, r > 2. The main tool in proving Theorem 6.3 is provided by Lemma 3.2.6. The arguments in the proof of Theorem 6.2 go through with evident simplifications. Replacing the operators W,U in the integral equation (6.2) by the operators (4.2.13), (4.2.14) we obtain similar local theorems for the initial-boundary value problems with prescribed incoming flow. Using methods described in Chapters 5, 6, it is possible to extend these results to include some reflection operators. The main difficulties in this approach are related with the corresponding ~ r o b l e mfor the free-molecular flow, i.e. with the equation AF = 0. Note, that there exists a class of problems, where this part is trivial. A simplest example of this type is the exterior problem (3.1.10)-(3.1.11) with convex domain R3 \ 0.
7
Global solutions with large initial data
This section contains a very brief description of theorems which guarantee the existence of global solutions of the Cauchy problem for initial data satisfying only "natural" physical conditions. The first results of this type were obtained by Carleman [14] in the case of a spatially homogeneous distribution F. Carleman considered the case of hard spheres. However, his results can be generalized for a wide class of potentials. The next step was made by Povzner [56]. He considered the "smoothed" (in author's terminology "smeared") Boltzmann equation. In the spatially homogeneous case this equation coincides with the Boltzmann equation. Povzner's methods allow one to obtain exhaustive results regarding existence of solutions of the homogeneous problem. On the other hand, two theorems below (see Theorems 7.1 and 7.3) provide tools for studying long-time behavior and smoothness of the solutions. In general, Povzner's equation appears to be very useful, and it is widely used in numerical methods. Povzner's theorem and its applications are discussed in Section 7.2. We conclude this section by a description of results by R.J.Di Perna and P.L.Lions. These authors proved that the Cauchy problem for 'real' Boltzmann equation has a global solution for large initial data.
Chapter 7. Initial-boundary value problems
162
7.1
Relaxation in a spatially homogeneous gas
The problem consists in finding a function F ( t , v ) , t > 0, v E R3, that satisfies the conditions F,= J ( F ) , t e ( O , T ) , F~,,o=Fa. (7.1) This ~roblem,and its generalizations to the case of a mixture of reacting gases, is of interest in many physical-chemical applications. Other applications of problem (4.1) are connected with the asymptotic study of the full nonlinear equation. The asymptotic analysis shows that the process of achieving complete thermodynamic equilibrium consists of two stages. At the beginning, in a short interval of time, a local equilibrium is reached in the system and then in a slow process averages are equalized. The initial process is described by (7.1). A solution to problem (7.1) has to describe the transition from an "arbitrary" distribution to the Maxwell distribution M with parameters p, u,0. The solution is then used to correct the initial data in order to describe the slow process, which, according to the assumptions of physicists, must be described, in principle, by the hydrodynamic equations of a compressible fluid. We recapitulate the existing mathematical results, assuming that the function B satisfies the conditions (3.2.10), (3.2.11) with y > 0. T h e o r e m 7.1 [I] Assume that
Then for any T problem (7.1) has a solution in C([O,T],L1(R3)) satisfying the conditions (i) sup,((l < oo (7.3) (ii) sup, (Fln F) < oo.
+
Here the averages (.) are defined by (3.2.3). T h e o r e m 7.2 Let conditions (7.2)(i) and (iii) be satisfied fork > 4. Then for any T problem (7.1) has a unique solution in C([O, T I , L1(R3)) satisfying condition (7.3)(i). Theorein 7.3 There exists ro > 0 depending on the kernel B only, such that the following holds: If ( I I V ~ ~ ) ' / ~ E FL1(R3) O for r 2 1-0, FO 0, then problem (7.2) has a unique nonnegative solution in C([O,T ) ,Lm(R3)) satisfying the condition
+
uniformly with respect to T
>
7. Global solutions
163
Theorem 7.2 follows immediately from Povzner's theorem discussed below. Theorem 7.3 is used in [2] to construct global solutions of the full Boltzmann equation that are close to spatially homogeneous ones. We describe below only the key ideas in proving these theorems. The results presented in Section 3.2 imply the apriori estimates ( F ) = (Fo), (lvI2F) = ( I v ~ ~ F o ) ,( F l n F) = (Foln Fo);
(J*( F ) ) I ((1 -k I V I ) ~ F ) ~ . In fact, these relations follow directly from (3.2.4) - (3.2.6), (3.2.9). For a bounded function B it follows from (3.2.4) - (3.2.5) that
From these estimates the existence of global solutions of (7.1) in
for a bounded function B follows in an obvious way. We now consider arbitrary functions B that satisfy the conditions (3.2.10), (3.2.11). Replace B by B = min{B,n) and denote by F(")the sequence of corresponding solutions. By (7.3) and (7.4) the sequence {F("))is weakly compact in L1([O,T] x R3). The estimate (7.5) guarantees the possibility of taking the limit in any of the integral forms of the Boltzmann equation. In order to prove that the moments (7.3)(i) are bounded we can use Povzner's inequality [56]:
+
if p = (1 Iv12)k/2,F 2 0 and condition (3.2.10) is satisfied. To prove Theorem 7.3 observe, that Theorem 6.2 implies that a solution exists on some finite interval [0, TI depending on the initial data. Our aim is to show that this solution can be extended to any interval of time. To prove this we first observe that the following assertion is valid.
Lemma 7.4 Iceep assumptions of Theorem 7.3. Then for any c , to, T to I T 5 oo) there is a positive constant C such that F ( t , v) > exp {-C(1
+
,
if
(E
> 0, 0 I
t € [to,T].
This lemma was proved by Carleman for hard spheres. The arguments of Carleman's proof go through for general potentials with simple modifications.
Chapter 7. Initial-boundary value problems
164
Physically, it follows from this lemma, that particles with large velocities arise in the system under consideration almost instantly. The next consequence of this Lemma is that for t 2 to we have the system with a finite entropy. More precisely, we can assume without loss of generality, that We denote below by C any positive constant depending on initial data only. It follows from Lemma 7.4, that IH(F)I=I(FlnF)I
for t E [ t o , T ] .
Therefore we can assume that IH(Fo)I < C. Then we infer from Lemma 3.2.1 that the collision frequency is bounded from below: inf v ( F ) 2 vo
> 0 for t E [O,T].
v,t
(7.6)
This gives the main tool for a very nice apriori estimate. More precisely, we conclude from (7.6),(7.1), that Ft
+ VOFI J+(F, F ) ,
t E [O,T].
(7.7)
At this stage the estimates described in Chapter 3 give everything necessary to complete the proof. By Lemma 3.2.5 we obtain
< C with h(v) = 1 + lv12.
sup V,(hJ+(F)) t
Repeating application of Lemma 3.2.5 gives sup & ( h J + ( F ) ) t
< C,
if
k E [0,2), 7 > 0.
Due to (3.2.24) we obtain sup E2
J
~J+L(F)~< C C.
E2
Then it follows from (3.2.15), (7.7), (7.8), that sup F
< C.
t
The above results together with (3.2.19) show that the following lemma is valid. L e m m a 7.5 Let F be a solution of problem (7.1) in Lm([O,T] x R3) that satisfies the condition 0 5 cp,F 5 C(T). Then, under the conditions of Theorem 7.3 there exists a constant C not dependent on T , such that 9 ° F 5 C . Obviously this lemma enables us to extend the solution to any interval [O,T]. This completes the proof.
7. Global solutions
7.2
Povzner's theorem
A modification of collision integrals based on 'smearing' at the point x at which collision occurs was proposed by D.Morgenstern [52] and A.Povzner [56]. Let X = (x, v) be a point in the phase space I? = R: x Rt. Assume that collisions between two particles with coordinates X, XI results in the following transition:
The probability of the transition is determined by a nonnegative function P on Assume that P ( X , XI, X', X;) = P(X1,x ; , X, XI)
r4.
and define the collision integrals
where F', F;, F , Fl mean F ( t , .) evaluated at X', Xi, X, XI respectively. From the mathematical point of view the appropriate kinetic equation
with smooth functions P is a simpler object than the Boltzmann equation. Povzner proved the existence of global solutions of the Cauchy problem to this equation. It is perhaps more essential that the existence of a probabilistic interpretation simplifies the construction of numerical solutions for this model. In some cases it is possible to prove the convergence of solutions of the smoothed equation to solutions of the 'real' Boltzmann equation. The most essential feature of equation (7.9) is that it implies a reasonable equation for the measure p on r defined by
where v is the Lebesgue measure on r. Under mild assumptions regarding the function P the following analogues of properties (3.2.4), (3.2.5) are valid for all compactly supported continuous functions cp:
Chapter 7. Initial-boundary value problems
166 where
Integrating (7.9) we obtain
+
( p ,p ( t ) ) = ( S - t p , ~ ( 0 ) )
1'
(~r-tcpJ+(~)?v)d~
In view of (7.10) this gives an equation for the measure p. This equation was solved in [56] under some reasonable additional assumptions on the function P. Assume, that the conservation laws ( $ j J, v ) = 0 are satisfied for the functions
Suppose in addition, that
( J*, v ) 5 C ~ : ( P ) is fulfilled, where m k ( ~ )=
((1
+ lv12)k'2,P ) ,
Then the following theorem holds.
Theorem 7.6 (i) Suppose that the measure po satisfies the condition
If condition (7.12) holds for k 2 2, then for all t there exists a weakly t-continuous measure p satisfying equation (7.11) for all compactly supported $ and having finite moments: m k ( p ( t ) )< m. (7.13) (ii) Equation (7.11) has a unique solution in the class of weakly continuous (with respect to t ) measures satisfying condition (7.19) for k 2 4. (iii) If the measure po has a density, then the measure p has a density for all t .
7. Global solutions
7.3
167
Di Perna-Lions theorem
Perhaps the most interesting result of recent years is R.J.Di Perna and P.L.Lions theorem on existence of global solutions for the Cauchy problem
for the nonlinear Boltzmann equation (3.1.lo). Since the complete proofs and detailed discussions of these results are published we give below a short review only. The collision kernels satisfying conditions Bo), B1)of Chapter 3 are considered. A unique additional assumption is
for all R < w. Of course this assumption is fulfilled for the kernels with property B3). The authors use the notion of renormalized solutions. These are defined as solutions of the problem
+
with P(t) = ln(1 t). More precisely, the following definition is used. Set
Definition. A nonnegative function f E C([O, w ) , L1(R2 x R:)) is a renormalized solution of (7.14), (7.15) if
for all T, R E (0, oo) and (7.16) is satisfied. The transition from the Boltzmann equation to equation (7.16) appears very useful from a technical point of view. Besides, it is shown that if f is a renormalized solution of (7.14), (7.15) then for a11 P E C1([O,oo)) such that ,B'(t)(l t) is bounded on [O,m),
+
The main result is the following
Chapter 7. Initial-boundary value problems T h e o r e m 7.7 Let fo 2 0 satisfy
Then there exists a renormalired solution of (7.14), (7.15). It seems that the crucial tool in proving this theorem is the compactness result described in Theorem 4.3.6.
8
Incompressible limit
In this section the scaled Boltzmann equation aft = -v.V,f
1 + ;Lf
+r(f,f)
with a small positive parameter E is discussed. Essentially the very first results regarding global solutions were uniform with respect to this scaling. What is really remarkable is that this scaling leads to the incompressible Navier-Stokes equation. This was recognized recently by several mathematicians [7, 17, 581. The results of Sections 2 and 3 give some useful estimates regarding the dependence of the solution on the parameter a. First, these estimates show that even relatively large perturbations of the kinetic part of the solutions are admittable. Secondly, the smoothness of the velocity averages follow from these estimates immediately. However, we cannot avoid the smallness condition on the initial data.
8.1
The Cauchy problem. Uniform estimates
Consider the Cauchy problem aft
f lt=o
= - ~ . V e f + ~ L f + r ( f , f ) , t>O, = fo,
x€Rdr veRdr (8.1)
assuming that d f o € H S , s > -2,
d22.
It is assumed also that the operators L and r satisfy conditions L1)-L5), (2.4)-(2.6). Throughout this section the notations of Section 2 are used. Recall Proposition 1.9. Let F be the Fourier transform with respect to x with the dual variable [. Define the Hilbert space Y with the following norm:
8. Incompressible limit Note that
x being the indicator function of the set
We will show that solutions of the problem (8.1) satisfy the following estimate:
uniformly with respect to E,f0 and T. More precisely, the following theorem is true with W defined by (2.7). Theorem 8.1 Assume (8.2). Then there exists a positive constant a0 such that for any E E (O,1] and any fo with 11 foils 5 a. the problem has a unique solution in W for any T > 0. This solution satisfies (8.3). Proof. Obviously it suffices to obtain the estimate (8.3). In fact, we can start with the truncated kernel BR defined by (2.9). We know that the corresponding Cauchy problem has a unique solution if t E [0,T] with some T depending on R, E and fo. The estimate (8.3) provides the possibility to take weak limits with respect to R. Moreover, this estimate shows that the solution can be extended to obtain solutions in [O,oo). Let the functions fl,f 2 be defined by (2.12). Then
Using the estimate (2.14) we get:
Since PoI'f, f = 0, Proposition 1.9 implies SUP t llf(f)II:
+ llf It:
5 C [llfoll:
+ ~lY-~~~r(f.f)ll:.] .
Combining (8.5) and (8.6) we get SUP Ilf (t)ll:
+ IIf 1:
5 C [llf l,.
Ilf 1:
+ llfoll:]
uniformly with respect to E,f, fo and T. Theorem 8.1 follows immediately.
(8.6)
Chapter 7. Initial-boundary value ~ r o b l e m s
170
Periodic initial data
8.2
Consider the problem = -v.~,f+:~f+r(f,f), x E 0 , V E Rd,
t E (o,T),
XER, v E R ~ ,
f lt=o = fo,
(8.7)
with periodic boundary conditions in a cube [2aId. Using the results of Section 3, we obtain the following theorem with Hs and W defined by (3.1) and (3.4) respectively.
Theorem 8.2 Assume that fo E H s , s > $. Then there exists a constant a0 such that for any E E (O,1] and any fo with 11 foil, < a. the problem (8.7) has a unique solution in W. There exists a constant C , such that uniformly with respect to e and T the following holds: Ilf(t)ll: + Ilf 1 ; I C llfoll: 7 (8.8) where
Ilf 1 ;
x
=
l:, + IIPOIF-~XIF~l:,m
11y112(~ -~ 0 ) f
being the indicator function of
8.3
{El
< &-I).
Convergence results
The convergence problem can be solved by using the estimates (8.3), (8.8). Consider first the problem (8.7). Let A = { f', 0 < E 5 1) be a set of solutions. The estimate (8.8) implies, that
The last two estimates follow directly from Ll), L2), (2.6). Thus taking a subsequence ~ -E~,/ ~ I ' ( ~fE) E , converge weakly in L2([0,T],HS). Morewe obtain, that f',U - ~ I ~ L V over, since sup, 11 fElls is finite, the nonlinear operator is continuous in A: ~ - l / ~ r ( f fr) =, +~ - ~ / ~ r ( f O , f ~ ) , where f 0 is the weak limit of { fE). It is sufficient to apply the convergence theorem by C.Bardos, F.Golse and D.Levermore. It follows from this last theorem, that the limit function f 0 has the form
8. Incompressible limit and the coefficients p, u = (ul, . .. ,u d ) , 0 satisfy the equations
where v and rc are positive constants determined by the operator L. In fact the estimate (8.7) says directly, that
strongly in L2([0,m), Hs). Similarly,
strongly in L2([0,w),Ha). On the other hand, it follows from (8.7), (8.8) that the set {Po7-1X7fE, 0 < e 5 1) is compact in L2([0,TI, Hs). Recall that x is the indicator function of {[I 1[1 < I/[). Similar results are valid for the problem (8.1). A unique nontrivial point in this case is that the fluid dynamic moments ( $ j , f )H are not bounded in L2([0,co),R t ) . However, the functions
are bounded in L2([0,oo), Ha) uniformly in c . The estimate (8.5) implies that the nonlinear term is bounded also. Hence, there exists a sequence E, -+ 0 such that the functions (8.4) and the collision integrals converge weakly in L2([0,w),Hs). Notice that the equations for the projections
lead to similar results even more directly. In fact, the Boltzmann equation is equivalent to the system
where
It follows from the estimates above that one can pass to the limit in (8.12) (in distributional sense). The expansions (2.1.26) show that the limit equations have the form
Chapter 7. Initial-boundary value problems
172
(8.10), (8.11). Notice that two of the eigenvalues Xj determined by (2.1.26) have the form Certainly, the corresponding eigenprojections ( f , e ; ( ~ [ ) ) ~converge to zero in the distributional sense. Stronger estimates are valid as well [9], but nevertheless these modes form the main obstacle in studying solutions with large initial data. On the other hand, the estimate (8.8) shows that
This inequality implies that the smallness condition in the theorems above can be weakened as follows. There exists a constant a0 such that if
IIPofoll,
+ l l ~ - l / -~ ( ~
PO)fOll,
< ao,
then the conclusions of Theorem 8.2 are true. Similar assertion follows from the estimate (8.3).
9
Long-time behavior problems
This section is devoted to extremely hard problems regarding long-time behavior of solutions of kinetic equations. What is the relevant long-time behavior of the solutions of the Boltzmann equation? May it be expected that this behavior is really more simple than that of the solutions of fluid dynamic equations? One of the possible approaches to these questions is discussed in this section. In order to avoid technicalities, the problem is considered in an extremely simplified formulation (a more detailed description can be found in [47, 461). Consider the Cauchy problem for the Boltzmann equation with periodic boundary conditions. Moreover, assume that the collision frequency v defined in Lemma 3.3.1 is bounded. It follows from the last assumption that the collision operators L and r are bounded in H. In order to take into account the influence of perturbations caused by interactions with the boundary, - . which is one of the main sources of nontrivial behavior. we consider the problem with a given source term. The problem is to find a function f ( t ,x,v), t > 0, x E R = Rd/Zd, v E Rd, such that
where All estimates below are uniform with respect to a E [O,R], R < oo, but limit passage E + 0 isn't conside~ed.Recall the definitions of the projections P(E[) on the
9. Long-time behavior problems
173
invariant space of L, (2.1.25). A natural modification of P(eE) for the periodic case is considered below (see (9.10) for the exact definition). Set
and introduce the functions P = P f , q = Qf. Notice that the operators t , and P commute. It follows that the problem (9.1) is equivalent to the following system
with Qpo = 0, Pqo = 0. Assume g E Lm([O,oo), Ha),
Q lt=o E
HS
for some s > d/2. Since the index s will be fixed, we will use the notations II. I I, and 11.11 for the norms in Lm([O,m),H") and H a respectively. Moreover, assume there exists a solution of the problem (9.1), such that p E Lm([O,oo),Ha), s
d
> 5.
The question we are concerned with is the behavior of the kinetic part q of the distribution function. Note that equation (9.3) can be considered as an equation for the fast variables in the framework of the ideology of the averaging methods.
Theorem 9.1 There exists a positive constant C, such that if
then the problem (9.3) has a unique solution in Lm([O,m),H3).
Proof. Denote by S ( t ) the semigroup, generated by L, in H". Write the equation (9.3) in the following form:
+ + +
where G ( T )= r ( p q, p q ) g. It follows from the estimate (2.1.24), that
Chapter 7. Initial-boundary value problems
174 Using the estimate (2.6) we obtain
The contraction mapping considerations show immediately, that the theorem is true. Moreover, let q('), q(2) be two solutions of the problem (9.3) with different initial data, but with the same p and g. Set u = q(l) - q('),
Then
Z ( T ) -+ 0 as T -+ co. In fact. we have
with v ( t ) = exp { - C l t / ~ ' } . Hence
This implies that e2
Z ( T ) 5 CT
+ C & ~ ( T ) ( l l P l l+, 11~11,).
(9.5)
The conditions of Theorem 9.1 imply that
for some positive constant C . We deduce from (9.5) and (9.6) that under assumptions of Theorem 9.1 the assertion (9.4) is valid. Note that we do not need any information about the behavior of the fluid dynamic part of distribution function for large t, except for its boundedness. The analogues of the simple assertions formulated above are valid for the Boltzmann equation under general assumptions in functional spaces with more complicated structure. The consequences of these facts are as follows. First, Theorem 9.1 shows the possibility of a reduction of the Boltzmann equation (9.1) to the finite-dimensional problem. In fact, we may eliminate the function q to obtain a closed equation for p of the following form:
9. Long-time behavior problems
175
where q = Q(P,t7 qo) is a solution of the problem (9.3). Observe, that this solution depends on the initial data for equation (9.3). On the other hand, it follows from (9.4) that for large t the solution of the problem (9.4) does not depend on q(0). More precisely, it follows from (9.4) that the kinetic part of the distribution function for large t is determined in a unique way by fluid dynamic moments. This might be one of the possible refinements of the concept of 'normal solutions of the Boltzmann equation' introduced by Hilbert. Theorem 9.1 is conditional, as we have assumed existenceof a solution. The results described in Sections 2-7 certainly allow to prove that the assumptions of this theorem are actually fulfilled, but only under additional assumptions regarding smallness of perturbations. The results of Section 8 imply that in these cases the solutions of equations (9.7), (9.8) are close to the solutions of the equation for incompressible fluid. One of the main obstacles in the problems with large perturbations are the acoustic modes. Another approach to constructing approximations for equations (9.7), (9.8) is as follows. Assume that there exists a solution of the problem (9.7), (9.8), that has the following form: q(t) = F ( P ( ~ ) ) . (9.9) Recall that the function p has the following form:
where t E Zd Denote by pl,. . . ,p, the Fourier coefficients f^(t,[) in (9.10). Assume that F E C1(Rm). Then the equations (9.7), (9.8) transform into the following hyperbolic equation for the function F:
If a solution of this equation exists, it can be shown that for any solution of the problem (9.3) the following holds
It means that the Boltzmann equation has an inertial manifold [25]. On the other hand, additional assumption that the function F admits representation FN E~F(~).
C n>O
leads directly to the Chapman-Enskog series. A foundation of the Chapman-Enskog method is presented in [47] for small initial data. Notice, that the first terms in these series correspond to the equations for the compressible fluid, if g = 0 ( 1 / ~ ) .
Chapter 8 Stat ist ical solutions of Euler equations 1
Introduction
Many physical phenomena are modeled by nonlinear differential equations with unstable solutions, i.e. small variations in initial data produce very significant changes in long-time behavior of the system under consideration. These difficulties have made many researchers search new tools for investigating the long-time behavior of such systems. One of the classical approaches is connected with the concept of statistical solution. In the mathematical setting, we assume that a probability distribution of initial data is known. The problem is to obtain statistical information about the long-time behavior. Taking into account that the initial data are not explicitly given, it seems natural to consider the evolution of the probability distributions. It should be mentioned, however, that probability distribution of initial data is usually unknown as well. Therefore, it is reasonable to consider the probability measures evolution, if one can predict their long-time behavior. There exist examples, where the probability distributions converge to some limit measure P and P does not depend on the choice of initial distribution. This limit measure may be considered as an important characteristic of the system under consideration. There exist also a large class of systems for which this measure describes the average in time behavior. We will discuss some mathematical problems in this field. It is shown below, that three-dimensional Euler equations have a space-periodical statistical solution, while no similar result is known regarding 5ndividual' solutions. The corresponding measures are obtained as limit measures describing the evolution of the solutions of Navier-Stokes equations as the viscosity is going to zero (or, Reynolds number is going to infinity). The proof of this result shows in particular that some analogues of kinetic equations arise naturally in the field (see (3.4)). Another essential tool provides the theory of measure-valued solutions developed by R.J.Di Perna and A.J.Majda [19]. These solutions can be identified with first
178
Chapter 8. Statistical solutions
moments of statistical solutions. Note that another interpretation which doesn't contradict this assertion is discussed by R.Illner and J.Wick [35].
2
The Fridman-Keller equations
Consider the Cauchy problem for the Euler equations:
av + -v,va a t ax,
-
+ Vp = 0 ,
divv = 0
on the torus R = Rn/(27rZn). Let H be a space of divergence-free square-summable velocity fields. Assume the initial velocity field to be random and determined by the probability measure Po on the Borel a-algebra B ( H ) . Then the Cauchy problem for the system (2.1) may be formulated as a problem of construction of statistical solution, that is, a set {Pi)of probability measures on B ( H ) describing evolution of the measure Po. For n = 2 the problem is solved in [66]. Construction of the statistical solution for n = 3 is the main result of this chapter. Existence theorems for this solution are proved below. However, the formulation of these theorems requires some preliminary considerations. First, it is convenient to deal not with the set {Pt}, but with a probability measure P on the Borel sets of the space H = L2(0,T ;H ) . In fact, the theorems on solutions of (2.1) formulated below provide information about asymptotic properties of measures PEon H, where P" are statistical solutions of the Navier-Stokes equations with viscosity E [65]. In order to describe the properties of PEto be used in the sequel, we define the following spaces
where H-" = ( H 3 ) * ,H s = H
n W 3 . Norms in these spaces are determined by
Write Navier-Stoltes equations in the form of integral identity
2. The fridman-keller equations
179
where cp is a smooth function from H satisfying the condition It=o
(2.5)
= cplt=T =
Below we will assume the initial measure Po to satisfy the condition
The following properties of the set of measures {PC,0 < E 5 1) are necessary for further consideration (they are described in detail in [65]):
where m
+ 1 5 ko;
An essential role in the sequel belongs to the moments Mk = Mk(P) of measures on B ( H ) . Denote by
k @J
H a tensor product of spaces H and set
k times
By y = {x, t} and
yk
= {yl,. . . ,yk} we will denote points in QT and
Q$.The inner
k
product in 8 H is determined by
where (f,g)k is the inner product in L2(Q$). A moment Mk(P) of the measure P is k
an element of the space 8 H, satisfying the condition
k
k
Here and below @ v is a function from @ H assigning to y k a tensor with the components ( I ) i Y j E Qt, ij = 1,273.
180
Chapter 8. Statistical solutions
We will use the chain of Fridman-Iceller equations for the moments Mi of statistical solutions for the Navier-Stokes system (2.4). Let us write this chain in the form of integral identity:
k
where p is an arbitrary smooth function from 8 H, satisfying condition (2.5) for t = t j , j = 1,. . . , k; V j is a derivative with respect to the variable xj; the operator assigns to the function p ( ~ , , .. . ,yk+l) defined on &$+.+'the function p(y1,. . . ,yk, yj), j I k defined on Q$. The Fridman-Keller chain related to the Euler equations (2.1) has the form
rj
3
Kinetic equations
A given nleasure P on B(H),uniquely determines all its moments Mk. However, to 1 < oo it is enough to have describe moments included in the chain (2.11) for k information about an object much simpler than P More precisely, it is possible to find a measure v' on CI = Qk x Rn',such that its moments will coincide with M k ( P ) for k 5 I. The measure is described in Lemma 3.1. Note at first that calculation of the moments reduces to calculations of the averages of the functions f : H -+ R' such that: f (v) = (v, h), , where v E C ( Q 9 , (3.1) while the function h satisfies the relatioils
<
h
c(R"~),
rj
2 0.
L e m m a 3.1 Let P be a measure on B ( H ) , such that
Then there exists a unique bounded nonnegative measure vk = vk(P) on B(Ck), such that
3. Kinetic equations 'P
h c ( R n k ) , rj < pj, v k = {v,, . . . ,vk) ,vj E Rn.
E C(Q!),
Proof. Let $J~Jbe smooth functions in
*={
Rn:
1 for lvl 5 N/2, 0 for (vl > N,
bounded uniformly with respect to N. Denote by f N the function resulting from the change v(yj) --+ (*~(v)v)(yj)in f for j 5 k. The functions f N are continuous on H. In fact, set for u, v E H , N > 0, 6 > 0 BN = {Y E QT l lv(y)I l N) , As = {Y E QT I Iu(Y)- V(Y)I< 6 ) . The function f~ is determined by the values of v ( ~ on ) BN only. On the set { v k E Rnk ( (vjl 5 N, j 5 k ) the function IL is uniformly continuous. Thus, for any E > 0 there exists Sl = 6 i ( N , ~ )such , that
A'= Q!\ A!,. On the set A' the inequality
is fulfilled for some j estimate is valid:
5 k. Therefore, for some positive constant C the following
I f ~ ( u) f ~ ( v )I l C
[E
+ 6;' 1 1 -~ vllH] ,
U , VE H ,
that obviously implies continuity of f N on H. Let us now consider the integrals
1
F ~ ( v i h ) = f~(v)P(dv). By virtue of (3.1), (3.2) there exists a positive constant C, such that I f ~ ( ~5) CSUP l ~ ' P ~ s ul h lPr
IFN('Pi h)l
5 C s u p I'PI
lhl
for all tp from C(Qk) and h from C(Rnk). These estimates and the definition of f~ imply, that for any N there exists a unique nonnegative measure V ~ , Non B(Ck), such that k
Chapter 8. Statistical solutions
(see, for example, [ 6 5 ] ) . The Prohorov theorem implies that there exist a measure vk on B(&) and a sequence Nm f co,such that for rj = 0, N = N,,,
It follows from the estimate (3.3), that this relation is valid for all other hand, the estimate
rj
< p j . On the
and the Lebesgue dominated convergence theorem imply that FN(p,b ) +
/
f ( v ) d ~ ( v ) . for
N = Nm.
The proof is complete. It follows from (2.7), (2.9) and the inequality
that the statistical solution of the Navier-Stokes equations satisfies the condition
Thus, the conditions of Lemma 3.1 are fulfilled in this case for pj 5 3. The FridmanKeller equations (2.11) are equivalent to the following linear relations for the measures v; :
where
4. Main results
Setting formally E = 0 we obtain the 'kinetic' equations corresponding to the Euler equations (2.1):
We shall use below the following lemma:
Lemma 3.2 There exist nonnegative measures vi on B ( C k ) , satisfying for k < ko the Fridman-ICeller equations (3.4) and the following condition:
4
Main results
The following theorem guarantees solvability of the Fridman-Keller equations (3.5) corresponding to the Euler equations.
Theorem 4.1 If k < ko equations (3.5) have nonnegative solutions v i from C * ( C k ) , satisfying the condition
The moments M i = Mk(u;), rjMf+l = (r,Mk+l(uE))satisfy fork < ko the equations (2.12) and the following relations:
There exist a sequence ej 1 0, such that the moments of the solutions of the NavierStokes equations converge to the moments M i . More precisely:
Chapter 8. Statistical solutions
184
Proof. The relations (3.5), (2.12) for v i and M i are the result of limit passage in (3.4), (2.11). Set
Let C*(Ck)be the dual space of C(Ck). It follows from (3.6) ([23], ch.IV), that there exist v: E C*(Ck) and the sequence cj 1 0 , such that
(.?, f ) Passing in (3.4) to the limit for
3
E = ~j
(8,f ) , f E C(Ck). 1 0 and setting
we get (3.5). The relation (4.2) is a special case of (4.3). By virtue of (4.3)
Therefore, the relations (4.1) are valid. The theorem is proved. Remark. Let rba(Ck) be the linear space of regular bounded additive set functions defined on the field generated by the closed sets. There is an isometric isomorphism between C8(Ck)and rba(Ck). Thus, Theorem 4.1 does not guarantee that vi are countably additive functions. On the other hand, since Qk is compact, the Actually under the moments M i and rjM:+, are a-additive functions on B(Q";.. conditions of Theorem 4.1 there exist a bounded measure vi, on B(Ck), such that
but it doesn't describe the moments r j M i + , connected with nonlinear terms in equation (2.1). On the other hand, the structure of the nonlinear terms with respect to v makes it possible to define the function v i by two measures: v;, on B(&) and vi2 on B(Qk x Sk), where Skis a unit sphere in Rn [19]. The following two theorems describe asymptotic properties of the measures P E . They allow to obtain a strengthened version of (4.1). Put
T h e o r e m 4.2 Tltere exist a probability measure Po on B ( H ) and a sequence ~j .J, 0, such that for f : H -t R1 satisfying the conditions
4. Main results
the following relation is valid:
The measure Po has the following properties:
k 5 k,, Mk(P0) E c([o, Tlk, H-"), k I ko, s > 512.
Mk(Po) E L"([o,T]~,&I H),
6
(The function
Fk
is given by (2.8).)
Proof. It follows from the Dubinsky theorem, that an operator embedding Zz into Z is completely continuous. The estimate (2.7) and the Prohorov theorem imply, that (4.6) is valid for f E C(Z-l). The same estimate (2.7) guarantees, that
This obviously implies, that (4.6) is fulfilled for all f satisfying the conditions (4.4), (4.5). The relations (4.7), (4.9) may be proved in just the same way as their counk
terparts for the measure PC1651. The function f = (9, €3 v ) for ~ k conditions (4.4), (4.5), if
< ko satisfies the
&I
9 E LZ([O,TIC, Hr).
That's why (4.8) follows from the equality
and Theorem 4.1. The proof is complete. To show that Po is a statistical solution of the Eu1er:quations and its moments satisfy (2.12), it's enough, along with the estimate (2.7), to prove the following statement. For any 6 > 0 there is N(6), such that
for some sequence ~j J. 0. If this condition isn't fulfilled, then the measure Po (as well as the corresponding finite-dimensional distribution vil) possibly doesn't describe the moments rjM;+,. However, it is possible to construct a nonnegative function Po E rba(H), so that its
Chapter 8. Statistical solutions
186
moments are a-additive functions and satisfy the Fridman-Keller chain corresponding to the Euler equations. 0 < E 5 1) is weakly* compact on Zl. Therefore, In fact, the set of measures {PE, there exist Po E rba(Z,) and a sequence ~ j 10, such that for f E C(Z1)
It follows from the estimate (2.7) that (4.10) is valid for all f satisfying the condition
In particular, the relations (4.10), (4.11) are fulfilled if ko functions:
> 2 for 'the following
This list of functions makes it possible to pass to the limit in each of the equations in the Fridman-Keller chain (2.11) for k 1 < ko and to prove the relation
+
[Ao(p, u)]' PO(du)= 0 for
ko > 4,
(4.12)
where Ao(cp,U) is given by (2.4) for s = 0. The last relation implies, that for any positive cr
Thus we have proved the following statement:
Theorem 4.3 Let the condition (2.6) be fulfilled for ko > 4. Then there exist Po E rba (21) and a sequence s j 5 0, such that (4.10)-(4.12) are valid. In addition
187
4. Main results The moments of Po satisfies for k conditions:
+ 1 < ko the equations (2.12) and the following
Mk+l ( P o ) = Mk+l(Po), rjMk+l(PO) E L r n ( [ 0 , ~ l c*(Rk)), k, rjMk+l(pO) = rjMk+l(v;),
kgH ) ,
Mk+l(P0) E L-([0, T]L+', Mk+l(PO) E c([o, Tlkfl,
H-'),
s
> 512.
The analogous results are valid for the synchronous moments mk given by the equality mk = PtMk, where the operator pt assigns to the function $(yl,. . . ,yk) defined on Q";,he function $(t, xl, . . . ,xk) defined on [O, TI x Rk. The moments m:, corresponding to the functions P , satisfy the conditions
Note, that the Kolmogorov hypotheses on the spectrum of turbulent energy may be formulated as hypotheses about asymptotics of mk on the diagonals of Rk. The theorems considered are still valid even if the initial distribution of velocity vo on H is determined in a unique way, so that the measure Podegenerates. However, in this case the theorems also guarantee only the existence of solutions of moment equations. The measure PO, constructed in the proof of Theorem 4.2 doesn't necessarily degenerate!
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