Lecture notes on the mathematical theory of the Boltzmann equation

LECTURE NOTES ON THE

MATHEMATICAL THEORY OF THE

BOLTZMANN EQUATION

This page is intentionally left blank

Series on Advances in Mathematics for Applied Sciences - Vol. 33

LECTURE NOTES ON THE

MATHEMATICAL THEORY OF THE

BOLTZMANN EQUATION Editor

N. Bellomo Politecnico di Torino, Italia Politecnico di Torino, Italia Contributors Contributors L. Arlotti Universita di Udine, Italia L. Arlotti Universita di Udine, Italia N. Bellomo Politecnico di Torino, Italia N. Bellomo Politecnico di Torino, Italia

M. Lachowicz M. Lachowicz University of Warsaw, Poland University of Warsaw, Poland

J. Polewczak California State University, Northridge, USA

W. Walus

University of Warsaw, Poland

World Scientific Singapore • New Jersey • London • Hong Kong

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Library of Congress Cataloging-in-Publication Data Lecture notes on the mathematical theory of the Boltzmann equation / editor, N. Bellomo ; contributors, L. Arlotti. . . [et al.]. p. cm. — (Series on advances in mathematics for applied sciences, Vol. 33) Includes bibliographical references. ISBN 9810221665 1. Transport theory. I. Bellomo, N. II. Arlotti, L. III. Series. QC175.2.L37 1995 530.1'38--dc20 95-10544 CIP

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PREFACE

The lecture notes presented in this volume deal with mathematical prob­ lems related to nonlinear kinetic equations, with special attention to the Boltzmann equation. They are based on a series of lectures delivered at the Department of Mathematics of the Politecnico di Torino during the aca­ demic year 1993-1994. Four lectures, whose common theme are nonlinear problems, make up the volume. The first one, by L. Arlotti and N. Bellomo, reviews the results on the initial value problem for the Boltzmann equation. It consists essentially of two parts. The first one deals with the analysis of existence and uniqueness results for small initial data. The second one with the survey of the results for large initial data. This lecture also offers suggestions on various open problems in the field. The second lecture, by M. Lachowicz, presents an asymptotic analysis of nonlinear kinetic equations in relation to the hydrodynamic limit. In addition to a rewiew of the mathematical results known in the literature, it also provides a unified treatment of this problem. The analysis applies not only to the Boltzmann equation, but also to the Enskog equation and some generalized models. The third lecture, by J. Polewczak, begins with the revised Enskog equa­ tion for moderately dense gases. The main topic is a development of the square-well kinetic theory for dense gases and its connection with the max­ imum entropy formalism. The lecture also presents mathematical problems related to advanced models of kinetic theory. The fourth lecture, by W. Walus, deals with the computational aspects related to the solution of nonlinear kinetic equations. Classical methods are reviewed: finite elements, particle methods, Monte Carlo methods. This lecture will be of interest to scientists who are involved in applications of the Boltzmann equation to fluid dynamic. A few words about limitation of scope are appropriate. Some topics have not been included. Among these are the treatment of boundary conditions, v

PREFACE

vi

the analysis of the initial-boundary value problems and the rigorous deriva­ tion of kinetic equations. However, the interested reader can find references in the bibliographies, expecially in the first and second lectures. A broader collection of references is given at the end of the book. The visits of M. Lachowicz and W. Walus were supported by Scholar­ ships of the European Community. The visits of J. Polewczak were sup­ ported by the National Research Council. The authors of the first lecture are indebted to L. Arkeryd and V. Protopopescu for useful advices in the preparation of their work.

Nicola Bellomo

CONTENTS

Preface

v

Lecture 1 by L. Arlotti and N. Bellomo On the Cauchy Problem for the Boltzmann Equation 1 The Nonlinear Boltzmann Equation

1

2 On the Initial Value Problem in the Absence of External Force Field

4

2.1 The Cauchy Problem for Small Initial Data

7

2.2 The Cauchy Problem for Large Initial Data . . . .

18

2.3 The Cauchy Problem Near Equilibrium

31

3 On the Generalized Boltzmann Equation

34

4 Open Problems

42

4.1 On the Cauchy Problem with External Force Field .

42

4.2 On the Cauchy Problem Related to Wave Phenomena 44 4.3 On the Cauchy Problem Related to Hydrodynamics

46

4.4 On the Cauchy Problem for the Discrete Boltzmann equation

47

4.5 On the Cauchy Problem for Nonlinear Continuous Models

50

5 Final Remarks

52

References

54

vii

viii

CONTENTS

Lecture 2 by M. Lachowicz A s y m p t o t i c Analysis of Nonlinear Kinetic Equations: T h e H y d r o d y n a m i c Limit 1 Introduction

65

2 Kinetic and Hydrodynamic Equations

67

2.1 Kinetic Equations

67

2.2 Dimension Analysis

74

2.3 Hydrodynamic Equations

76

2.4 Notations

79

2.5 Basic Estimates

84

3. Compressible Dynamics 3.1 Singularly Perturbed Boltzmann Equation

86 . . . .

86

3.2 Hilbert Procedure

88

3.3 Modified Hilbert Expansion

92

3.4 Initial Layer Expansion

95

3.5 Equation for the Remainder

98

3.6 Hydrodynamic Limit for the Boltzmann Equation

105

3.7 Hydrodynamic Limit for the Enskog Equation for a< e

106

3.8 Hydrodynamic Limit for the Enskog Equation for a~ e

116

3.9 Hydrodynamic Limit for the Povzner Equation . . .

120

3.10 SDE Approximating Solutions of the Hydrodynamic Equations

121

4. Incompressible Dynamics

130

4.1 Level of Smooth Solutions

130

4.2 Level of Weak Solutions

133

References

137

CONTENTS

ix

L e c t u r e 3 by J. Polewczak A n I n t r o d u c t i o n t o Kinetic T h e o r y of D e n s e Gases 1 Introduction

149

2 The Revised Enskog Equation

151

3 The Square-Well Kinetic Equation

160

References

176

L e c t u r e 4 by W . Walus C o m p u t a t i o n a l M e t h o d s for t h e Boltzmann Equation 1 Introduction

179

2 The Splitting Method in Kinetic Theory

184

3 Numerical Schemes for the Boltzmann Equation

. . . 188

4 Numerical Evaluation of the Collision Operator

. . . 194

5 Correction Techniques

205

6 Other Methods

206

References

213

General Bibliography Books and Lecture Notes - Kinetic Theory

225

Books and Lecture Notes - Fluid Dynamics

226

x

CONTENTS Books and Lecture Notes - Mathematical Methods

. . 227

Review Papers

228

Research Papers - Cauchy Problem

230

Research Papers - Derivation of Kinetic Equations Research Papers - Discrete Boltzmann Equation

. . 234 . . . 234

Research Papers - Enskog Equation

237

Research Papers - Hydrodynamics and Asymptotic Methods

238

Research Papers - Mathematical Methods

242

Research Papers - Nonlinear Models

243

Research Papers - Numerical Schemes

244

Research Papers - Problems with Boundaries Research Papers - Wave Solutions

. . . .

253 255

Lecture 1 O N T H E C A U C H Y PROBLEM FOR T H E BOLTZMANN EQUATION

L. Arlotti and N. Bellomo

1. The Nonlinear Boltzmann Equation The celebrated nonlinear Boltzmann equation is a mathematical model of the phenomenological kinetic theory of gases which describes the evolution, in time and space, of the one-particle distribution function for a simple monoatomic gas of a large number of identical particles. The mathematical model is reported in the classical literature, see [CE2], [FK1], [KOl], [RE1], and [TR1]. The evolution equation for the distribution function / = /(*,x,v):

E+ x E 3 x M3 -► K+,

(1.1)

equates the total derivative of / to the gain and loss terms due to the collisions which define the creation and destruction of particles which at the time t 6 [0,T] are in a neighbourhood of the phase point x,v, where x G D C E 3 is the space and v G I 3 is the velocity. The evolution equation, for a large class of pair-particles interaction potentials, is

^ + v • Vx + F • V v ) / = J(fJ) = MfJ) - MfJ),

1

(1.2)

2

L. Arlotti and N. Bellomo

where F is the external force field, and J\ and J2, which denote the gain and loss terms, respectively, are given by Ji(/I/)(*,x,v)=

jB(n,q)/(t,xJv/)/(*,x,w,)rfndw

j

(1.3a)

R 3 xS2_

and J2(/,/)(t,x,v) = /(t,x,v)

J

B(n,q)/(t,x,w)dndw,

(1.36)

R3xS^

where n is the unit vector in the direction of the apse-line bisecting q = w - v and q' = w' — v'. Moreover, v, w are the precollision velocities of the test and field particles, respectively, and v ' , w ' are the postcollision velocities, which are related to v and w by the relations v' = v + n(n , q) (1.4) w' = w — n(n , q) where (•, •) defines the vector inner product and S^_ is the integration do­ main of n defined as §^ = {n <E R3 :

|n| = l ,

(n,q)>0}.

(1.5)

Finally, the term B is a kernel which depends upon the intermolecular force law which, for a large class of interaction potentials, see f.i. [CE2] or [FE1], can be written as B(n,q) = A W ^ ,

(1-6)

where 6 is the azimuthal angle of v' in a spherical coordinate system at­ tached to v, with center in the point of the binary collision and the reference axis oriented as q. Moreover, a is a collision parameter such that "hard" col­ lisions correspond to a > 4, the so called "Maxwell molecules" correspond to a — 4 and "soft" interactions correspond to 2 < u < 4. In particular, the hard sphere model corresponds to the case of a tending to infinity. The derivation of the Boltzmann equation is based upon suitable sta­ tistical arguments, with somewhat heuristic features, which are reported

ON THE CAUCHY PROBLEM

3

in the classical literature and, in particular, in the books which have been cited above. In addition, we address the interested reader to the first part of the survey paper by Zweifel [ZW1], where an intuitive approach to the phenomenological derivation of the Boltzmann equation is presented. This topic is not dealt with in this paper which is mainly devoted to a survey of the mathematical results on the initial value problem for the nonlinear Boltzmann equation. In particular, we recall that the phenomenological derivation of the Boltzmann equation requires, among others, the assumptions that in the low density limit only binary collisions are taken into account that can be considered as instantaneous and local in space; and the number of pairs of particles, in the volume element dx, which participate in the collision with velocities in the ranges [v, v + dv] and [w, w +rfw],respectively, is given by f(t,x,v)dxdvf(t,x,w)dxdw.

(1.7)

It is well known that both assumptions are hard to justify and have to be regarded as a phenomenological approximation of physical reality. An alternative approach is the one of the BBGKY hierarchy, which links the Hamiltonian dynamics of the gas particles to the derivation of a sequence of evolution equations on the probability distributions of the statistical state of the particles, see [CE2]. The first equation involves the one particle and the two particles distribution function, the second equation additionally involves the three particles distribution function and so on. The Boltzmann equation can be obtained under suitable limits and closures. This lecture deals with the spatially inhomogeneous equation. Specifi­ cally, the lecture provides a survey of the mathematical results on the initial value problem in the whole space IR3 for the nonlinear Boltzmann equation [CE2] and for generalized equations [BL9]. The fundamental theorems will be reported with a sufficiently extensive sketch of the proof. There, the main ideas of the proof will be reported which are useful to the final result, whereas technical calculations will be omitted and replaced by bibliograph­ ical indications. This lecture is organized in five sections as follows: The first section is this introduction. The second section deals with the statement of the initial value problem and with a survey of the most relevant mathematical results on the solution, global in time, of the initial-value problem for the spatially inhomogeneous

4

L. Arlotti and N. Bellomo

Boltzmann equation in the absence of an external force field. This section is divided into three parts: Existence theory for small initial data in L°°, for large L 1 data, and finally for small perturbations of Maxwellian equilibria. The third section deals with the qualitative analysis of the solutions to the initial value problem for the generalized Boltzmann equation. This equation is derived on the basis of assumptions similar to the ones which lead to the Boltzmann equation. However, the simplification of the localization of the binary collision is not taken into account and the variations of the distribution function are taken into account within the action domain of the test particle. The fourth section provides a brief analysis of some research perspectives in the field. A discussion and some conclusions follow in the last section. We remark, in conclusion of this section, that this lecture is restricted to the analysis of the initial value problem in the whole space. This leaves out some classical problems such as the analysis of the boundary and the initial-boundary value problem. The reader is referred to the papers [BUI] and [MS2] to recover the bibliography on this important topic. The gen­ eral mathematical framework can be found in the paper by Beals and Protopopescu [BP1]. 2.

On the Initial Value Problem in the Absence of External Force Field

This section provides a survey of the mathematical results on the qualitative analysis, existence and asymptotic behaviour of the solutions to the initial value problem for the spatially nonhomogeneous Boltzmann equation in the absence of an external force field. The literature on the spatially homogeneous problem is reported in [TR1] and [WE2] and refers to the work essentially developed after the papers by Arkeryd [AK1-3]. As already mentioned, this type of results will not be reviewed in this paper, which is devoted to the more general case in which the space dependence is taken into account. In order to deal with the Cauchy problem, one first needs to define the problem and provide suitable definitions of solutions. Consider then, the Boltzmann equation in the whole space E 3 with given initial conditions /°(x,v) = /(0,x,v) : I

3

xE

3

4R

+

.

(2.1)

ON THE CAUCHY PROBLEM

5

Often, the problem is such that one assumes decay at infinity in space of the initial conditions either to zero or to Maxwellian distributions. If the evolution is defined in a rectangular domain D e l 3 , then one assumes periodicity conditions. The initial value problem for the Boltzmann equation can be written, in integral form, as follows /#(*,x,v)=/°(x,v) + f J#(5,x,v)ds,

(2.2)

Jo

where / # ( * , x , v ) = /(*,x + v * , v ) ,

(2.3)

J*(t, x, v) = J{t, x + v t , v ) .

(2.4)

and, analogously The evolution problem (2.2) can be written, in abstract form, and with obvious meaning of notations, as

f = Uf.

(2.5)

Classically, in order to provide a definition of solution to the initial value problem, we define a suitable Banach space, B, and state that a solution belongs to such a functional space and satisfies the initial value problem (2.2) or (2.5). In particular and without specifying, for the moment, the selection of the function space, the following definitions can be given: Definition 2.1. A function f = /(£,x, v) is defined mild solution to the initial value problem for the Boltzmann equation, if f 6 B and Eq.(2.2) is satisfied. Definition 2.2. A function f = /(£,x, v) is defined classical solution to the initial value problem for the Boltzmann equation, if f G B is continu­ ously differentiate with respect to t and x and Eq.(2.2) is satisfied in the classical sense, i.e. pointwise. Definition 2.3. A function f = /(£,x,v) is defined strong solution to the initial value problem for the Boltzmann equation, if f £ B is strongly differentiate and Eq.(2.2) is fulfilled in the norm ofB.

6

L. Arlotti and N. Bellomo

The same definitions can straightforwardly be applied to alternative definitions of the mild formulation of the initial value problem. In fact, the statement (2.2) of the problem is not unique. For instance, if J\ and J2 make sense separately, the following can also be stated

|| 9-£ +dff*L* *+f*L*{f)-Q*U>f) U)=Q*U,f)

(22<) (2.2')

(/#(0) f#(0) = /°(x,v), f°(x.v), where the following notation has been used

Mf,f) = Q(fJ), Q(fJ),

Uf,f) == fHf)fL(f)UfJ)

Moreover, formal computations show that any solution to the initial value problem for the Boltzmann equation should satisfy, for a l H > 0, the following conditions /

33 3 3

/(£,x, f(t,x,v)dxdv v) dxdv = = /

JR xR ./R xR

/I

33 3 3

33

33

JR xR xR ff H

W+-7 (fi + \ l / 4

/°(x, / ° ( x , vv)) ddxdv x d v ,,

(2.6a)

vv/°(x, / ° ( x , vv)) ddxdv x d v ,,

(2.66)

| |vv| |22//°°((xx, ,vv))ddxxddvv, ,

(2.6c) (2.6c)

3

33

3 3

3

XR ./RJRxR

vv/(£,x, / ( £ , x , vv)) ddxdv x d v = //

JR xR ./R xR

/f

3

7 RJRXXR R

||v| v | 22//(*,x,v)dxdv ( * , x , v ) d x d vlv-==- ///

3 3

3

3 XXR 7 RJR */R xR

I/

33 3 3

JO Vo JR
e{s,x,v)dsdxdv= e(s,x,v)dsdxdv=

[

33

JR xR xR «/R

3

//°° l o g / 0° d x d vv,,

(2.6d)

where H{t) H(t) = I

33 3 3

./R xR JR xR

f(t, f{t, x, v) dx dv /(*, x, v) log f(t,

(2.7) (2.7)

and e(t,x,v) e(t, x, v) = /

VR3xS2_

B ( n ,q) q )(f(t, ( / ( i , x, x ^v')/(*, B(n, w') / ) / ( * , x, x,w') \

/ ( X,V - / ( *( , x , v ) / ( £t , x , w ) J) log [7ftx,v')/(t,x,w') ' ' l ( * ' X ' W ? 1 dndw.(2.8) dnrfw.(2.8) f *: . /(*,x,v)/(t,x,w) /(*,x,v)/(£,x,w) j [ /(*.x,v)/(t,x,w) J 5

ON THE CAUCHY PROBLEM

7

It can be noticed that the following inequality //

R3

-/R 3 xR3

/(*,x, / ( t , x , vv)log/(t,x, ) l o g / ( t , x , v) v ) ddxdv x d v << 1/


/°(x, / ° ( x ,v)log/°(x, v ) l o g / ° ( x ,v) v )dx d x ddv, v,

7]R3 x R 3

(2.9) (2.9)

is a consequence of (2.6d). As we shall see, the formulations and definitions, which have been given above, can be used in order to deal with existence and uniqueness of the solutions. If the qualitative analysis is referred to existence only, then one can deal with the definition of r e n o r m a l i z e d solution, proposed by DiPerna and Lions [DPI], which will be presented later in this section. Local existence can be proven in several ways by classical methods of functional analysis: fixed point theorems [MT1], semigroup theory [BM1,2], as well as by the so called Kaniel and Shinbrot iterative scheme [KAl]. The interested reader is referred to the review paper by Greenberg, Polewczak and Zweifel [GR2], as well as to the detailed analysis by Shizuta [SHI], in order to obtain a complete information on this topic. The analysis of the Cauchy problem for large times has provided, in the last ten years, several interesting results concerning existence of solutions, uniqueness and, at least in some cases, asymptotic behaviour. The mathematical literature can be organized in three main fields: • Solutions for small initial data. This type of results refers to existence and uniqueness of mild solutions for initial conditions close to vacuum, to local Maxwellians and to the solution of the spatially homogeneous case. Classical solutions are obtained, as usual, under suitable smoothness assumptions on the initial data. • Solutions for large L 1 data. This type of results, due to DiPerna and Lions [DPI,2], exploits, as we shall see, the averaging Lemma by Golse, Lions, Perthame and Sentis [GOl] to obtain existence of weak solutions without smallness assumptions. • Solutions close to the global Maxwellian. In this case one studies existence of solutions and their asymptotic trend to equilibrium. 2.1.

T h e Cauchy P r o b l e m for Small Initial D a t a

This section deals with the Cauchy problem for small initial data: Existence and uniqueness of mild solutions for initial conditions close to vacuum, to local Maxwellians and to the solution of the spatially homogeneous case.

8

L. Arlotti and N. Bellomo

Initial data close to vacuum A survey of mathematical results on the existence theory for the Cauchy problem for small initial data which decay to zero at infinity in the phase space is proposed in this section. The initial data can be in either L 1 D L°° or L°°. In the first case, the mass of the gas is finite. In the second case, as we shall see, the mass of the gas can be infinite. This topic is now well understood, as documented in the book [BL2], as well as in papers [ILl], [HAl], [BLl] and [TOl]. Several other papers have been published about this type of results. Nevertheless, the main results are contained in the papers which have been cited above. Specifically, paper [ILl] refers to a hard sphere gas and to initial condi­ tions which tend exponentially to zero at infinity in the phase space. Paper [HAl] generalizes the result of [ILl] to the Boltzmann equation with gen­ eral interaction potential between pairs of particles. In the case of both papers, the mass of the gas is finite. On the other hand, the result of [BLl] refers to a gas with infinite mass. In fact, the initial conditions are assumed to decay in the physical space with inverse power behaviour: in this case, for sufficiently small decay the mass of the gas can be infinite. A further generalization is due to paper [TOl] where global existence is proved for ini­ tial data decaying with inverse power behaviour in the whole phase space. The two papers, [BLl] and [POl], refer both to the Boltzmann equation with general interaction potential for the internal forces between pairs of particles. In addition to the papers which have been cited above, we mention two further results due to Polewczak. In the first one [POl], the author gener­ alizes the existence theory previously limited to the case of mild solution, also to the case of classical solutions. In the second one [P02], the mathe­ matical results are further generalized to the case of initial conditions with more general decay to infinity in the phase space. In order to survey the mathematical results, we define a suitable function space. Let • V>(x, v) be a strictly positive, bounded and continuous function. • B(ip) be the space of all functions / = / ( £ , x , v ) which are measurable, with respect to a suitable measure fx, which is finite over every bounded Borel set of R3 x E 3 x [0, T] and such that |/#(*,x,v)|
(2.1.1)

ON THE CAUCHY PROBLEM

9

• The norm defined in B(ip) is the following 11*11

||/||= ||/||=

#

ess sup ess sup 3

3 te[0,T],x<ER te[0,T],xGR 3,v<ER ,veR3

/ ( * , X ,vv ) , u!/*(*»*» >l )' \ ; | .

(2.1.2)

^ ( X , V J)

The function space B(ip) is, with the norm defined in Eq.(2.1.2), a Banach space. We shall assume that local existence in 6 can be technically proved in a time interval [0,T], where T is inversely proportional to the norm of the initial datum. Technical details can be found in Chapter 2 of [BL2]. Global existence can be stated in the following Theorem 2.1.1. Consider the initial value problem for the Boltzmann equation characterized by a collision kernel such that

^

L

„0 << *(5<< !1,,

(2.1.3) (2.1.3)

o //°(x, ( x , v) v ) << aQ^ «( X x ,, vV)) ==a M |x|)0(|v|), ah(\x\)g(\v\),

(2.1.4)

sin cos 6 sin06 cosy


q

with initial conditions

where g = e _ r l v l , r > 0, and h is a bounded function tending to zero at infinity such that the L\ norm of h(z), with z = |x| is bounded. Then, there exists a smallness condition a < ac, which is inversely proportional to the L\ norm ofh(z), such that the initial value problem (2.5) has a unique global solution. In particular, admissible functions are 2 hee = MW) M M ) -= e-*M e-p|x|\ ,

P>0,

(2.1.5)

or t

L

(\-,-\\ —

ftr=WM)= hT - Air(JxlJ

-

—TT'- ,

(1 + |Xf) 2 |X| 2 )V

>1 1. p* > P* -

2 L6 ((2.1.6) - )

This theorem was proved by Illner and Shinbrot [IL1] for the hard spheres Boltzmann equation in the relatively simpler case h = he. This result was generalized by Hamdache [HA1] to the Boltzmann equation for

10

L. Arlotti and N. Bellomo

quite general interaction potentials. The proof for h — hr is due to Bellomo and Toscani [BL1], still without restrictions on the interaction potential. Sketch of the proof: The proof of Theorem 2.1.1 is based on several inequalities and can be obtained either by the use of Kaniel and Shinbrot iterative scheme [KAl], as documented in Chapter 2 of [BL2] or by classical fixed point theorems [SMI]. This second method is here followed in alternative to the proof provided in [BL2]. We recall, with reference to [SMI], that if V is a closed convex subset of B defined by

V = {feB:{feB: V

H/ll < 211/1}, 2||/°||} , ll/ll

(2.1.7) (2.1.7)

then, global existence and uniqueness of the solution to Problem (2.5) is assured if the following conditions hold V / e6 £8 : :

UfeB, W/6B,

(2.1.8)

V/eD:

UfeV, W /GD,

(2.1.9)

V/i,/2€P :

|l | W / i - W / 2 | | < | | / i - / aa | || ..

(2.1.10)

If now we consider the truncated evolution problem, i.e. Problem (2.5) where J2 — 0, then, V / i , / 2 G £>, the following inequality holds # \Uh-Uf l ^ / i - ^ M2f(t,*,v) ( * , x , v ) <<

< / <

/

+ /

/

Jo Jo

Jo

3 JR JR3XS\ XS\

B 5 ( ^ q ) / i ( 5 , x + v a , v ,, ) | / i - / 22 | ( 5 , x + vs,w ,,)dndwcfa £B(tf,q)/ ( 0 , q ) / 22((*,x « , x + v s*,lw v') dndwds w''))| |//i i -_//22||((«5 ,j xX + vs, v*,v')iiDdwcto

3

JR XS\

= = f(i f Jo Jo

| J i ( / 1, >/ l/ )1 -) ^- Jl i( (/ /2a,,//22) )| |##((5«, ,Xx,, vv))dd S» / IJl(/1 Jo

3 2 JR3XS

m q ) f / i/ i##(( ^ « ,xx B(»,q)

+ ( v - v '' )) SS ,) v '' )

JR X§\+

w') x |/i - / 2 | # ( s , x + (v - w > , w') + / 2 # ( s , x + (v - ww>> ,, w ') x | |/i / i --//22||## (( ss ,, xx + ((vv -- vv' >) s, ,vv'' ))

dndwds.

(2.1.11)

ON THE CAUCHY PROBLEM

11

ijj and using the definition of Multiplying and dividing both terms by \jj norm yields

\Ufi-Uf || (UAH ++ | |||/ / 2 2| |||)/(t, ) / ( i , Xx,,v) v) i « / i - -Uj, 2\*(t,x,v) ! | # ( t , x > v )<\\h < | | / 1 -- / /a2IKIIMI <4a||/ 1 -/ <4«||/i - Al|/(*,x, v), 2 ||/(t,x,v),

(2.1.12) (2.1 .12)

where where

"fj 1=

Jo

2 JR3xS XS+2+

B(0,(* w') ds .

Jo 7 R 3

Technical calculations show that the following bound holds

(2.1 (2.1.13) .13)

C f^iH*,"* c ^ ( x , v )011%, ||A||i, I/ <<< where Ch is is a constant and />oo1 />OC

\\h\\i == iifciii

h(z) dz , h(z)dz,

/ Jo Jo

z= — \x\. |x|.

(2.1 (2.1.14) .14)

The proof of the theorem for the truncated equation follows from inequalities (2.1.11) and (2.1.12). The proof holds true also for the whole equation. In fact it is a technical problem to show that the evolution equation (2.5) maps positive functions into positive functions. One simply has to use the formulation stated in Eq.(2.2'): the differential equation, for given Q and L is linear in / ; the formal solution of such equation shows the positivity of the map. The crucial point is then the proof of inequality (2.1.13). If /i = he then the proof is technical. In fact one has

-fJ

1I -= f I Jo Jo

x(+x+(v ( v --vv'»)s)2 2 > q ) e - p (p B{n J5(n,q)e-

JR3X§\ JR3X§*_

p ( x + ( v w > ) )V'(V e r ++ w e~-p(x+(v-w')s )

e

_

_

* - ( v " 2 w, , „"I )2 \

dndwds

dndwds.

Then, conservation of energy, standard inequalities and integration over time yields 2

S2

i»M e -- r | v | / < 2||/l||1e-*MV'M J<2||fc||ie-

// 3

JR xS\ JR3XS\

B(n,q) _ r 2 a *£iS) - ^ dndw, ee-'W dndw _ Q Q

L. Arlotti and N. Bellomo

12

and the result follows. In particular, the exponential decay is exploited in order to obtain the bound (2.1.13). ■ If h — hr the proof requires additional inequalities. In particular, the result follows from the orthogonality of the vectors (v — v') and (v — w') and from the decay defined by the function hr. This theorem has been revisited by several mathematicians who have improved and generalized it in various manners. Without claim of com­ pleteness, we list some of the aforementioned generalizations in the following remarks Remark 2.1.1. Shinbrot [SI1] dealt with the hard spheres Boltzmann equation, in the case ofh = he for a gas in the presence of a convex obstacle with deterministically reflecting walls. In particular, he was able to show that inequality (2.1.13) holds true also in the presence of the obstacle with specular reflection law on the wall. Remark 2.1.2. Toscani [TOl] proved the same theorem when the decay in the velocity space behaves as 9 = 9{\v\) = — - 7 , (1 + |v| 2 )2

k>3.

The crucial point of Toscani's proof still refers to the manipulation of in­ equality (2.1.13) to obtain the same result in the case of decay in the velocity space as defined in this remark. Remark 2.1.3. Two generalizations are due to Polewczak, who was able to show that the solutions of Theorem 2.1.1 are classical ones under suitable regularity assumptions on the initial data, and prove asymptotic conver­ gence, in time, of the solution [POl]. The first part of the proof, (a similar one can also be found in [BL3]), substantially consists in deriving an evolu­ tion equation for the space derivative of the solution and then in applying to such equation an analysis of existence of solution similar to the one applied to the original problem. The second one [P02] refers to a generalization of the function ip. Polewczak was able to show that the result holds also if the decay in time and space are related and not factorized and if the initial datum is unbounded. Remark 2.1.4. The main difference in the choice of the functions he and hr is that if h = he the inital datum refers to a gas with finite mass f° G

ON THE CAUCHY PROBLEM

13

L n L°°. On the other hand, if h = hr, then the mass of the gas can be infinity, if p* < 3, namely f° € L°°. This point is quite important in order to compare the result presented in this section with the ones of the next sections. Initial data close to a local Maxwellian An additional result on global existence of solutions to the Cauchy problem refers to the case of initial conditions near a local Maxwellian. We refer to papers [PAl] and [TOl] to state the main result: T h e o r e m 2.1.2. Consider the initial value problem for the Boltzmann equation, characterized by a collision kernel such that condition (2.1.3) holds, with initial conditions near a steady Maxwellian UJ

( l + ee)w(x,v), )o;(x,v), (1 e)w(x,v) < //°°((xx, vv)) < (1 ( 1 ---e)w(x,v)

(2.1.15)

aa;(x,v) ; ( x , v ) = ccee--QQ| x| x| 2| 2ee--/ 3/ 3l |vv| |22..

(2.1.16)

where

Then there exists a sufficiently small €Q such that, if e < EQ, then the initial value problem (2.5) has a unique global in time solution which belongs to a Banach space B(u). Sketch of t h e proof: This theorem can be proved applying the classical Kaniel and Shinbrot iterative scheme [KA1]. The scheme, as is known, uses the formulation (2.2) and can be written as follows

[' dl^+l#L#(u d» +l#L#(u n-1) n <1

d

dt

= n 1)-Q#(l

lAn l) Q#(l n-Uln-l)

d +W -^+uiL*{ln#_l)

n^

0 « - l )== Q*(un-uu O # K - l 5n-«l)n - l )

(2.1.17) (2 L17) '

Jn(0)=un(0)=/°(X,V). To begin the iteration one needs a pair of functions lo,u0. Then, following Kaniel and Shinbrot [KA1], we say that such a pair satisfies the b e g i n n i n g conditions if

0
(2.1.18)

14

L. Arlotti and N. Bellomo

Still according to [KAl], if the initial condition is positive and if the pair IQ,UO satisfies the beginning conditions, then the sequence of iterative solutions ln and un are such that ln is increasing and un is decreasing and both converge, in the L1 sense, to the mild solution of the initial value problem. This scheme is used to prove the theorem. The first step consists in putting f( u*(t,x,v) 4(tyx,v) = (l + e(t))Lu(x,v) e(t))u;(x,v) {1 (l#(t,X,v)=(l-£WMxI v).

(2.1.19) (2.1.19)

JJ.

Then to prove that the beginning condition is verified one has to show that (*

4(t)-f°(t) -

f*

f

Jo

J

<

'o#Wf/ ° --Cw-

/

Jo ./o

>0 4(s)L*(u u*(s)L*(u o-lo)(s)ds>0 0 -lo)(s)ds ■l0)(s)ds > 0. l*(s)L*(u lo*(s)L#(u 0-lo)(s)ds>0. 0-

Both inequalities are true if

e + d [ e(s){l e{s)(l + + e{s))G(s,x,v)ds e(s))G(s, x, v) ds < e{t), e(t),

(2.1.20)

^ Jo0

where d is a constant, e(t) < 1 and G ( t , x , vv)) G(t,x,

2

-a|x- -/?|w| £ ( n ,q)e(" q)e< - a l- qx *- |q2 )' el(V - 0 W)a dndw. )dndw. B(n

== / f

3 2 JR JR3XS X +s 2

+

The proof follows from a detailed analysis of the properties of the func­ tion G. Then condition (2.1.20) is verified under a suitable smallness as­ sumption on the constant defined in Theorem 2.1.2. The proof is technical if the collision kernel does not include rigid spheres s = oo. For rigid spheres, one can take, as shown in [PAl] a sequence of collision kernels Bn which tend to the rigid spheres kernel for n -> oo. The existence theorem is technically proved for the solution fn corresponding to Bn. The qualitative analysis of the asymptotics, for n -4 oo, completes the proof. Since the solution satisfies the inequality

to(t) * > ( * <)
B(UJ).

■

ON THE CAUCHY PROBLEM

15

Initial data close to the spatially homogeneous case Another result on the Cauchy problem refers to small perturbation of the spatially homogeneous problem and is due to Arkeryd, Esposito, and Pulvirenti [AK5]. This paper deals with the problem with initial data close to the solution of the spatially homogeneous problem and refers to a gas of particles assumed to be confined in a three dimensional domain with spec­ ular reflection on the boundary, for particles interacting by a hard cut-off potential. Consider the following Banach spaces: • Ba is, for s > 1, the space of all real valued and continuous functions / = /(v) such that | | / | | . = sup(l + < ; 2 r | / ( v ) | < o o , veK3

(2.1.21)

where || • ||s represents the norm in Bs. • He is, for — oo < I < oo, the space of all L^ functions / : D —> E such that |/|?=

(l + A : 2 ) | / W | 2 < o o ,

£

(2.1.22)

Jfc€27rZ3

where / denotes the Fourier transform of / . • Bt s is, for s > 1 and — oo < I < oo, the space of all measurable functions / : D x E 3 -> E such that /(-,v) <E He, Vv <E E 3 and |/(-,v)|, e Bs. Such a space is endowed with the norm

11/11,,.= Sup(l + v2y\f(;y)\t.

(2.1.23)

v€R3

After these preliminaries the main result can be stated: Theorem 2.1.2. Consider the initial value problem for the Boltzmann equation (for particles interacting with potential harder than the one of Maxwellian molecules) such that the initial condition f° is positive and continuous on D x E 3 and satisfies the following conditions fl>(v)= / / 0 ( x , v ) d x e f l , + 1 , JD

(2.1.24)

L. Arlotti and N. Bellomo

16

and iio(x, v) - / ° ( x , v) - p 0 (v) 6 Blt8.

(2.1.25)

There exist suitable so > 1, to > § and a constant at)S such that, if 5 > so, £ > £o and if

IKIks < a£,s,

(2.1.26)

then the initial value problem has a unique, positive strong solution / € B£_1_fi,_i(1+e).

(2.1.27)

In addition / G C

1

^ ^ ) , ^ ^ ^ ) )

,

(2.1.28)

and / ( t ) —> u in Bt,s, as t —> oo, where u is the Maxweih'an with the same mass, momentum, and energy as the initial condition. Sketch of the proof: The proof of the theorem which is stated above is in three steps: Step 1: The first step consists in showing that fixed to > 0, and fixed So and £Q sufficiently large, then Vs > SQ , and £ > £Q, it is possible to deter­ mine three constants bgtS, b'£ s and j£^s such that, if i*o € Bis is a positive continuous distribution having the same mass, momentum and energy as

/°, | | F o - w | | / l 5
(2.1.29)

then there exists a unique positive function /€Cl([t0,oo),

fl£_1_ffi,_j(1+c))

,

that is strong solution of the Boltzmann equation for t > to and satisfies the condition fto = F0. Moreover ft G BijS and ll/t-"lka<&i,,exp(-7o0-

(2.1.30)

S t e p 2: The second step deals with the initial value problem for the spa­ tially homogeneous Boltzmann equation, say jr.9 = Ji {9,9) + 9Hg),

9(0, v) = 0 O (v),

(2.1.31)

ON THE CAUCHY PROBLEM

17

where g0 e Bs+i is delivered as stated in Eq.(2.1.24). One can show that for sufficiently large SQ, if s > so, indeed there exists a unique positive solution g to problem (2.1.31) such that gt € Bs+i ,

Vt>0,

and

lim \\gt - LJ\\S = 0 . t—>oo

This result is a known one due to Carleman for rigid spheres further gen­ eralized by Maslova [MSI]. It implies that there exists a time t\ such that \\gt-u\\s<\bt,s,

Vt>tls.

(2.1.32)

Step 3: The third final step consists in showing that, given two positive constants bi)S and t*t s, then an additional constant a^)S can be defined in such a way that the initial value problem for the Boltzmann equation has a unique strong solution in Bts in the time interval [0, tj J , which satisfies the inequality ll/t-*IL<5**,.,

V0 < * < * ? , , ,

(2.1.33)

providing that the smallness assumption defined in (2.1.26) is satisfied. Finally, according to the results of Step 2 and Step 3 and if inequality (2.1.26) is satisfied, then the following inequality ll/t: a -u\\t,s

< &/,*,

holds. The final result follows by application of Step 1 with t0 = t*is and F0 = ft* , which completes the proof of the theorem. ■ Remark 2.1.5. The constants which are defined in Theorem 2.1.2 can be chosen as locally independent on u and in such a way that b'ls -* 0,

as betS -> 0.

Therefore, according to this property it follows that the solutions defined in the theorem are stable in the space Be)S. Namely, if f° is the initial condition defined in the theorem, then for every given e > 0, there exists a 6 > 0 such that f°eB,s,

\\f° - f°\\t.. < 6 => \\ft - fth.

< «•

(2-1-34)

18

L. Arlotti and N. Bellomo

Remark 2.1.6. The analysis described above has been further developed by Wennberg [WE2], who was able to generalize Theorem 2.1.2 to the case of pseudo Maxwellian molecules. In addition, it is shown that, for Maxwellian molecules and for particles interacting with pair forces harder than the Maxwellian, the above analysis can be developed in spaces more general than Bt,s. In these spaces it is proved that the solution to the initial value problem strongly converges to the Maxwellian distribution with the same mass, momentum and energy. 2.2.

The Cauchy Problem for Large Initial D a t a

The analysis of the initial value problem for the Boltzmann equation for large initial data in L 1 is due to DiPerna and Lions [DP 1,2], who proved a theorem which can certainly be classified as the most important one in the analysis of the Cauchy problem in kinetic theory. Their result refers, as we shall see, to the existence of weak solutions according to a definition of renormalized solution which will be given later on. Considering the relevance of such a result, the proof will be reported in several details. This survey refers mainly to the original result, although several technical developments, generally due to other authors, followed this fundamental paper. On the other hand, still in the line of [DPI], a fundamental contribution was recently given by Lions [LN1,2], who discovered several important properties of the collisional operator and was able to show convergence to equilibrium. This result will be reviewed later in this section. As in the case of the existence theory for small initial conditions, one needs suitable assumptions on the collision kernel B(n, q) defined with reference to Eq.(1.13). In particular, two assumptions are needed. The first one is A s s u m p t i o n 2.2.1. The collision Kernel B(n,q) condition

satisfies the following

1 3 ] B(n,q) B(n, 6 L°° DL3; (E ; L L( §l(S2 )2)), q) eL^DL^R ),

which is equivalent to A G L°° n L ^ E 3 ) , where

, q) dn. A(q) A ( q ) == / B(n. B(n,q)cfn. 2 7s The second one, which is known as the mild g r o w t h c o n d i t i o n is a generalization of Assumption 2.1 and can be stated as follows

ON THE CAUCHY PROBLEM

19

Assumption 2.2.2. The collision kernel B (n, q) satisBes the following condition 1 j-——^r 2 / A(q) A(q) dq dq —> -- > o0,, (1 + v ). / | qq --v|
->• oo , as v -¥

Vi? Vfl<< oo oo..

In addition, the definitions of renormalized solution and mild solution are needed for the analysis of the Cauchy problem. In more details let 0
vVx9 2 ++T - V =„ -iT7 J 7(/ J
{2Xl) (2.2.1)

in the sense of distributions. Definition 2.2.2. / = /(*,x,v) is a mild solution of Eq.(1.2) if for almost all (x, v) # 1 J i ( / , / ) #((*t , x l v ) € L [ 0 , r ] Il

1i = l,2, 1,2,

VVTT<
(2.2.2)

and # , v ) --/ / #*( (s», ,XX, ,vV) ) == x,v) // * (( *t ,, X

J(f,f)*(*,

I J(f,f)*((T,X,v)d(Tx, v) JJ Ss

da

< t < oo. V 0 < ; >V0<S
The following lemmas are directly related to the definitions which have been given above, where the second one is similar to the one we have already given previously in this section:

20

L. Arlotti and N. Bellomo

L e m m a 2.2.1. Let / > 0 and

= 1,2, 1,2, ij =

Mf, / ) € 6 L£lM MfJ) U [ 0 , oo) o o )xxEE33x x E 33 ) ,

then / is a solution of (1.2) in the sense of distributions if and only if f is a renormalized solution of (1.2). Moreover, if f is a renormalized solution of (1.2), then V(3 G C1[0,cx)) such that \/3'{t)\(l +t) < c for some constant c, the composition {3(f) is a solution in the sense of distributions of

d0 dt

==p(J)J(f,f), ^+W+xpvV x /3 = P(f)J(fJ),

(2.2.4)

where the right hand side term is well defined *in" L} oc. In fact — " i . — ^~.^„ oc

PU)J(f,f)PU)J{f, f) = = {/?'(/)(! + /)}{(i / ) } { ( ! + /)■■V(/,/)}, fr'JUJ)}, with /?'(/)(i + / ) e L ° ° . L e m m a 2.2.2. Let f > 0 and

^MfJ) ( / , / ) €€ L ;L}o oc [ 0t,oo) o o )xx E 33 xxEE33)),, c (([Q

iI = =1 1,2, ,2,

then f is solution of (1.2) in the sense of distributions if and only if it is a mild solution. Moreover, i) f is a renormalized solution of (1.2) if and only if it is a mild solution and

MfJ) MfJ) i +/

c

rl

e lioc.

i• _

-. o = 1,2.

ii) Let # F #((tt,,X x , vV ) := f '(Lf)*(Lf)*(*,x,Y)da ' ( • , x, v) da

Jo

and assume that LWELH^TIXBRXBR), LWeL'dO^xBRxBn),

VR,T
ON THE CAUCHY PROBLEM

21

where BR is the ball with radius R. Then f is a mild solution to the initial value problem for Eq.(1.2) if and only if the following exponential form is satisfied

/#(t,x,v)-/#(S,x,v)exp[-(F#(t)-F#(S))] = / Ji(/,/)#(a,x,v)exp[-(F#W-F#(a))] da, Jo V0<s<£
and for a.e. x , v .

(2.2.5)

The analysis developed in what follows uses the equalities (2.6a) and (2.6d) stated in Section 2. Formal computations show that any solution to the initial value problem for the Boltzmann equation should also satisfy, for all t > 0, the additional equality /(*,x,v)|x-v*|2dxdv = /

/ 7R3xR3

/°(x,v)|x|2dxdv.

(2.2.6)

7lR3xR3

All this implies that if / is a non-negative solution of (1.2), then the following inequality holds / ( t , x , v ) (1 + | x - vt\2 + v2 + |log/(t,x,v)|) dxdv < oo .

sup / t>0 7]R3XR3

(2.2.7) After these preliminaries, the main result given by DiPerna and Lions can be stated in the following two theorems Theorem 2.2.1. Suppose that Assumption 2.2.1 holds and that there ex­ ists a sequence fn of non-negative distributional solutions to the initial value problem for the Boltzmann equation such that fneLl

([0,T]xIR 3 x E 3 ) ,

Ji(/n,/n) € Lf oc ([0,oo) x R 3 x R 3 ) ,

VT>0,

(2.2.8)

for i = 1,2,

(2.2.9)

fn satisfies the entropy identity and for some constant C independent of n / n ( * , X , v ) ( l + | x - V * | 2 + V 2 + | l 0 g / n ( * , X , v ) | ) dxdv

SUP / 3


3

t>0JU xR

(2.2.10)

22

L. Arlotti and N. Bellomo

Assuming, Assuming, without without loss of generality, that fn weakly converges in L1 ([0,T] x iE33 xx lE33 ) , , to to i) i) ii)

VT r < oo,

a function f and that / ° = = fn(t = 0) weakly weaWy converges converges in Ll1 (E 3 x E 3 ) a function f°, the following properties can be proved: 3 3 f/ €eCHO^oo^L^R C ( [ 0 , o o ) ; X 1 ( ^ 3 xx ^l 3 )) )) f satisfies the initial condition f(t = 0) = f° and VT < oo l^jMfJ) l^rjJi(fJ) e€ LL11 ([O.rjjL^R ([o.rjjL^R33 xx EE33)))) ,,

(2.2.11)

■^rjMfJ) ([0,00); Z ^ R 3 xX E E 33 )))) , r n 4 ( / , / ) e L°° ([O.ooJjZ^R

(2.2.12)

iiij / is mild solution, or equivalently renormalized solution, to the initial value problem for the Boltzmann equation such that L(/) L°° ([0,00);infill ([O.ooJjL^R83 x E 3 )) , L(f) € L°° and, in addition, condition (2.2.7) and the Entropy 1

H(t}+-■ f - / H(t)+4

/

4J

JO Vo

V RJR 3 X3xR R3

e(s,x, v)

< lim iinf nf/ n-s-oo n-yoo

(2.2.13) inequality

dsdxdv

//°J llog Q gfl/ J dx d x dv dv,

7R3

X R

(2.2.14)

3

where e(t,x,v) e(t, x, v) = /

3

./R V R 3xS XS

2

B(n,q) B(n, q) (/(*, x, v')/(*, x, w') ^/(*,x,v')/(*,x,w') V

7(*,x,v')/(t,x,w')l dndw(2.2.15) - / (/ *( t, ,xx,,vv) )/ /( (*t,,xx,,ww))J) log l l ^ Q ^ h l Q . ] dndw (2.2.15) _ / ( t , x , v ) / ( t , x , w ) ) [ /(t,x,v)/(t,x,w) are satisfied. Theorem 2.2.2. Suppose that Assumption 2.2 and the requirements of Theorem 2.2.1 hold. The result of Theorem 2.2.1 holds replacing conditions

ON THE CAUCHY PROBLEM

23

(2.2.11-2.2.13) with the following ones: 1 . 1 11 l l3 Y^jMfJ) L ([0,T];L (Rp? x B Y^jJiUJ) L ([Q BRR)))) ,, 9T\;L J \UJ)ZL€ {[Q,T\;L\. 1+ /

(2.2.11')

X 1 33 ^e L°° ([Q.oo^L ([0,oo);L (E^ xy B B*)) L°° ([o.ooj^^R K)) ,

(2.2.12')

oo L(/) L ( / ) €GLL°° ( [(0[ O , o. O o )C;OLJ1L(1E^ 33 xxBB*)) jR)) ,

(2.2.13')

~MfJ) YjTjMfJ) YjrjMfJ) and

VR,T < oo. We shall give a sketch of the proof of Theorem 2.2.1. The proof of Theorem 2.2.2 can be obtained with simple technical modifications. Sketch of the proof: The proof can be given in several steps. A relevant aspect of the proof itself refers to the properties of the sequence fn. Step 1: One shows that VT < oo there exists a positive constant CT, which does not depend upon n such that sup SUP

0
/

JRS-xR*

/ n ((«,x,v) «,X,v)

(1 + | x - vt\22 +v22 + |log/„(*,x,v)|) dxdv < CT, (1 + |x - vt\ + v + | log/„(t,x, v)|) dxdv < CT ,

(2.2.16) (2.2.16)

and I/ Jo

JO

T

/ en(t,x,v)dtdxdv IJR*XR 3 en{t,x,v)dtdxdv

V

«/R 3 xR 3

< < CCT T ,,
(2.2.17) (2.2.17)

where the expression of en is obtained from (2.2.15) by replacing / with fn. r-\

1

Step 2: The second step deals with the analysis of the sequence L(fn) and, for every positive constant S, the sequences 1 1 +0fn

Z ! T7~Ji\Jni Ji\Jn-> Jn) >t

and 1

7 (f

f ~\

,MfnJn) <Ji\Jn-) Jn) •

l + 6ffn TTJ]7^

'

24

L. Arlotti and N. Bellomo It is proven that these sequences are weakly compact in L Ll 1 ([0,T] xxEl3 3 xx ll 3 )

and that the two sequences L ( / „n ) ,

and

r~TT7 Mfmfn) Mfnjn) 1+ ofn +
are also bounded in L°°([0,T]; L ^ E 3 x l 3 ) ) , First the sequence L(fn)

VT
is taken into account, then considering that

11 f/n 11 n 11 II Zt = -,-, .. rr rr L(f ) < -rI/(/ ) , Ztn JtUnifn) ^a(/n*/n) = 1 + 0f 1 + 0fn ^ ( /nn ) < 0jL(fn) n , and

l + Sjf dv rr^n^ n

j2(/n /n) =

M/n fn)

'

L{fn)

+ Sjfnddv ' " li+sff

LUn)

n V

mL

fn

- s ~w Uh

~ s ~M

fn

•'

the properties of the sequences referred to J2 are proved. Then, taking into account Arkeryd's inequality [AK3] JiUn, /n) fn) <
n

,

(2.2.18)

V/c > 1, i,l = 1,2, i ^ ^, the entropy inequality (2.2.17), the properties of J2 and also the properties related to the terms Ji are proven. S t e p 3: This step consists in showing that VT < 00, and VipeL00

([0,T] x E 3 x E 3 ) ,

one has /

JR3

/ n (£,x,v)(*,x,v)dv-> (*,x,v)^(t,x,v) / JR3

/(t,x,v)^(t,x,v)dv

(2.2.19)

ON T H E CAUCHY PROBLEM

25

in ( L P j L 1 ^ 3 ) ) , Vp 0 the non-negative sequence g*nn = --ll oogg((ll + 6fn)
(2.2.20) (2.2.20)

which, according to Step 1, is weakly compact and let g6 be its weak limit. Then, thanks to Lemma 2.2.1 one has (d \dt + v

vx

>\

1

X

)gi =

1 + a/n

J(fn ,/n)

(2.2.21)

and consequently, taking into account Step 2, and denoting by T0 the transport operator 3 d dt + v ' V x , one verifies that Tog6n is also weakly compact. The compactness lemma by Golse et al. [GOl] allows to state that (2.2.19) is satisfied by gi^. It is enough to take the limit for 6 -> 0 + in order to transfer the result to fn. S t e p 4: The following convergence is proved L(/») -> L(f) L(fn)->L(f)

in

3 3 L^1 (([0,T] M x lxt (M f x3 x RK )))) .

(2.2.22)

As in the preceding case the convergence is proved for the sequence
/

MfnJnWdv-* Ji(fnJn)ll>dv->

[f

7R3 JR3

,2, 1i = l1,2,

(2.2.23)

7J K R3 3

in measure on [0, T] x .£?# .BR , 1 l1l Ji{fJ){x)eL (Rl) (Ml) JJJi(fJ){x)eL (f,f)(x)&L (Rl) (Ml) li(f,f)(x)eL

Jiifjtydv, Mf,Mdv,

V.R < oo , and, in particular

a.e. for for a.e.

ti >> 000,,, i*

xG € EE GR E333,,

= 1,2. i:, 2 . 11,2. f2i =

(2.2.24)

26

L. Arlotti and N. Bellomo

We point out that the property (2.2.23), see Remark 1, p.333, of [DPI], is an unexpected continuity property of the collision operator under weak L1 convergence of the solutions of the Boltzmann equation and plays a crucial role in passing to the limit in several formulations of the initial value problem. Step 6: This step consists in showing, VT < oo that the following properties hold

JlifJ) MfJ) llitJl 1+ /

MfJ) MfJ) ^llJl 1+ /

l G LLI (

[([0,T] E 33 xX E 3 )) )) , x ( (E 0,T]X

,

(2.2.25) (2.2.25)

° o ( ([O^oo); [ 0 j O O ) . LL\R l ( f f i 3 xxE 3E) )) j }, ee £L°° 3

3

(2.2.26) (2.2.26)

c

and L(f) G L{f) e L°° L ^ d([0,oo);L O ^ J j Ll(M? ^ E 3 x EE33)))) .

(2.2.27)

The proof of (2.2.26) and (2.2.27) is immediate. Moreover, taking into account the inequalities (2.2.17) and (2.2.18), together with the compactness properties of the sequences J Jt{fnJnh TTTTT^ *(fn,fn), 1l + 6 J fn6ff dv ndv

i =-= l 1,2, ,2,

(£5>> 0 ,

One can prove that / satisfies the inequality Ji{fJ)
VA * :>>11 ,

i1 == l1,2, ,2,

i±l. i±l.

(2.2.28) (2.2.28)

Here, I denotes a non-negative element of L1 ([0,T] x (E 3 x E 3 ) ) . above inequality and (2.2.6) yield (2.1.25).

The

Step 7: This step consists in showing that / G CC([0, ( [ 0 , o oo); o ) ; LL1J((E E 33 fe

xX EE33)))) .

This is proved first, by showing continuity for the sequence g^f. Then the property is shown for / # under suitable limits. The continuity of / is obtained by weak limits of f*.

ON THE CAUCHY PROBLEM

27

Step 8: / is super-solution to the initial value problem, i.e. V0<s<£
and almost everywhere for

x,v6l3xl3 :

/#(i,x,v)-/#(s,x,v)exp[-(F#(t)-F#(s))] > / ^ i ( / , / ) # ( ^ x , v ) e x p [ - ( F # W - F # ( ( 7 ) ) ] da.

(2.2.29)

Js

Indeed, by virtue of Lemma 2.2.2 one has

/#(t,x,v)-/#(s,x,v)exp[-(F#(t)-F#(5))] = / Ji(/„,/n) # ((7,x,v)exp[-(F#(<)-F#(<7))]d<7.

(2.2.30)

JS

Then, according to what is shown in Steps 4 and 6, one can show that the left hand side of (2.2.30) is weakly convergent and that its weak limit coincides with the left hand side of (2.2.29). On the other hand, the right hand side of (2.2.30) is not less than I*Mh*,/£)*(
[-(F*(t)-F*(
da,

Js

where h* = inf{/ n , R} , VR < oo. However, the integral which is defined above and h„ are weakly conver­ gent as n —> oo and / = lim (w — lim h^\ . Then, (2.2.29) follows. Moreover, we point out that (2.2.29) implies J\ (/, / ) G Ll [0, T],

almost everywhere for

x,vGl3xE3.

This property and inequality (2.2.18) imply also the following h ( / , / ) € Ll [0, T],

almost everywhere for

x,v6i3xl3.

Step 9: / is sub-solution to the initial value problem, i.e. V0<s<£
and almost everywhere for

/ # ( t , x , v ) - / # ( 5 , x , v ) < f J(fJ)#(
x,vGi3x!3 : (2.2.31)

28

L. Arlotti and N. Bellomo

Indeed the weak limit g5 of the sequence g„ defined in (2.2.20) satisfies the equation

^(t,x l v)-^*(» f x l v)=y [J(^#(*,x,v)-^#(s,x,v)=

/

>-h -h5Lff(a,x,v)da, Lf]*(a,x,v)da, [J*-h*Lff(a,x,v)da,

(2.2.32)

where Jf and hs represent the weak limits of Jl(fn,fn) Jlifntfn) l1 + Sfnn and fn 1 + SfSf n n' respectively. Then, the result (2.2.31) follows from the fact that J? < Jx , V£ > 0, which is a consequence of Step 5, and from the fact that both g6 and h5 tend to / ad S -> 0 + . Step 10: / i s sub-solution to the initial value problem, and satisfies V 00 <<; sJ << i t<
x x,, vv G 6 lI 33 x EE3 :

and almost everywhere everywhe re for and

# /f#(t,x,v) ( i , x , v ) - / f#(s,x,v)exp[-(F#(t) # ( s , x , v ) e x p [--(F*(t)- - -**(.))] F#(s))]

, x, v) exp [--(F*(t)< [ MfJ)*(° Ji(fJ)#(<J,x,v)exp[-(F#(t)-F#(a))}

-F*(a))]

da da..

(2.2.33) 2.33) (2-

JJss

Again, g6 satisfies the equation g6#(t,x,v) #(t,x,v)

- g6#(s,x,v)exp[-{F#(t) #(s,x,v)exp[-{F#(t)

-

F#(s))] F*(s))]

-J: ■i: = Jj

# Jf(f,f)#(a,x, J?(/,/) Ji(fJ)#(
/ ] # ( a , x , v ) e x p [-( [-{F*(t) - F#(a))] F*{a))] da, + / [[j/SLLff(a,x,v)exp[-(F#(t)-F#(a))] da, F#(t) Js

where j 6 represents the weak limit of the sequence at

9n 9n

fn 1 ! ++ Sf */»" n

(2.2.34) (2.2.34

ON THE CAUCHY PROBLEM

29

Then, the result (2.2.33) follows from the fact that J( < Jly V6 > 0 and from the fact that the last integral in (2.2.34) tends to zero as S -> 0 + . S t e p 11: / i s super-solution to the initial value problem, i.e. V0<s<£
and almost everywhere for

x,v6l3xE

/ # ( * , x , v ) - / # ( s , x , v ) > f J!(/,/)#(cr,x,v)da.

3

: (2.2.35)

Js

Inequality (2.2.35) follows from the fact that for almost every x , v the function t -> ( e x p ( F # ) / # ) is absolutely continuous and satisfies, because of (2.2.30), the inequality

j f (exp(F#)/#) > 71(/,/)#exp(F#). Theorem 2.2.1 is then obtained following the sequential steps we have summarized above with exception of the entropy inequality. This last step, which is left to the readers who wish to read the paper by DiPerna and Lions [DP2], requires some additional analysis based upon suitable convexity and weak limits related to the entropy functional. The survey of the proof of the theorem is then complete. ■ Theorems 2.2.1 and 2.2.2 can be regarded as two weak stability theo­ rems. In particular, they state that the weak limit of a sequence of classical solutions is a renormalized (or, equivalently, a mild) solution to the Boltzmann equation. Prom each of them an existence theorem can be stated. In particular T h e o r e m 2.2.3. Suppose that Assumption 2.2.1 holds and that f° satisfies f° > 0 almost everywhere in M? x R3 and that / ° ( l + | x | 2 + i ; 2 + | l o g / 0 | ) dxdv
f 3

7R XR

(2.2.36)

3

Then, there exists an f, which satisfies items i-iii) of Theorem

2.2.1.

T h e o r e m 2.2.4. Suppose that Assumption 2.2.2holds and that f° satisfies the conditions stated in Theorem 2.2.3. Then the result of Theorem 2.2.3

30

L. Arlotti and N. Bellomo

holds provided that Conditions (2.2.11-2.2.13) are replaced by conditions (2.2.11 '-2.2.13'). Sketch of the proof of Theorems 2.2.3 and 2.2.4: The proof is carried out by considering the sequence of initial value problems with regularized initial condition {/£} and regularized kernels Bn. Then the approximate solution {fn} is shown to satisfy the same a priori bounds as the original problem and to maintain the same stability properties of the exact solution. The weak L1 limit of the sequence of approximate solution is then used to obtain the renormalized (or mild) solution of the Boltzmann equation. ■

As we have seen a large space of this survey was dedicated to the theory developed by DiPerna and Lions. Their existence theorem is a fundamental chapter in the mathematical analysis of the Boltzmann equation which has already captured the interest of several applied mathematicians, who have technically developed the original theory in various directions. In fact, as already shown in [DP3] this type of analysis can be further generalized to other types of kinetic equations. An additional important development is the one of [LN4], where the author studies the initial value problem for large L 1 perturbations of global Maxwellians and obtains global existence of weak renormalized solutions. This result is an important one not only because it refers to an interesting thermodynamical system, but also as it can open a path, as we shall further discuss in the next sections, to the solution of the shock wave problem. A further relevant contribution to the analysis of renormalized solution is due to P.L. Lions and proposed in a sequence of papers [LN1,2,3], where a remarkable compactness property of the so called "gain-term" of the Boltz­ mann equation is proved. Moreover, it is shown that the existence theorem by DiPerna and Lions [DPI] holds true also if the term |x| 2 of Eq.(2.2.36) is replaced by a suitable function h = h(|x|) that satisfies h>0,

(l + /i)£

is Lipschitz on

E3 ,

exp(-/i) E L ^ R 3 ) .

(2.2.37)

Therefore let / ° be the initial condition of the sequence of renormalized solution / n , then bound sup /

^ ( x , v ) ( l + /i(|x|) + |v| 2 + | l o g / ^ ( x , v ) | ) d x d v < o o , (2.2.38)

ON THE CAUCHY PROBLEM

31

must be satisfied in order to obtain the existence result. Further, analyzing the gain term only, the author is able to prove con­ vergence in measure of J i ( / n , / „ ) to J i ( / , / ) and its boundedness in suit­ able L -space. This compactness result has several important implications. Some of them are indicated by P.L. Lions in the afore-mentioned papers. We point out the following ones: • Convergence, for all bounded times, in C ([0,t];L 1 (K 3 x E 3 )) , of the sequence / n , if /£ converges in L1(IR3 x l 3 ) . • Strong L\ convergence in time of a solution of the Boltzmann equation up to the extraction of subsequences to a pure Maxwellian equilibrium for the problem in a periodic box. This result was previously obtained by Arkeryd [AK7] with the methods of nonstandard analysis. Further, quite natural, developments are reported in [LN3] where, among various results, the existence theory is applied to the equation in the pres­ ence of Vlasov-like forces. Additional developments are due to Wennberg [WE3], who was able to include in the Lions proof also the hard-sphere interactions, which were not dealt with in the original proof. The generalization of the existence theory developed by diPerna and Lions to the initial-boundary value problem is due to Hamdache [HA3] for the case of walls at the same temperature and by Arkeryd and Cercignani [AK8] for walls at different temperature. 2.3.

T h e Cauchy P r o b l e m Near Equilibrium

As is known, see [TR1], the equation J(/,/) = 0

(2.3.1)

is a functional equation which admits the so-called Maxwellians equilibrium solutions which can be written as follows ufa x , v ) = a(t, x) exp { - [ V " g(y)]2 } ,

(2.3.2)

where the terms a(t, x), b(t, x) and a(t, x) can be related to the macroscopic observables. It can also be shown that the //-functional defined in Eq.(2.2.5) is monotone decreasing in time — <0,

(2.3.3)

32

L. Arlotti and N. Bellomo

with the equality sign at equilibrium. The analysis of the initial value problem near equilibrium is generally developed for global Maxwellian, i.e. the distribution (2.3.2) is taken with constant parameters "(v) = - V e x p { - „ 2 } .

(2.3.2')

(27T)2

The analysis of the initial value problem near equilibrium refers to the existence of solutions and analysis of trend to equilibrium for the Cauchy problem with initial conditions of the type (2.3.4)

/ °° ((Xx,,Vv ))==CoJ; ( X £F0(X,V), x , vV ) ++ eFo(x,v),

where e is a small dimensionless parameter and where the solution is sought in the form f/ == u+uj^F. LJ + w 2 F .

(2.3.5) (2.3.5)

The evolution equation for the perturbation is ^^-FF = = - vv V • Vx XFF ++ LLFF ++ ivT(F, /r(F,F F), ),

(2.3.6) (2.3.6)

where BiF BLF = - v - • VVXXFF ++ L FF,, LF = uT* LJ-% (j{uj,uj^F) (J(U,LJ?F)

+ J(u/*F,tj)) J(UJIF,U)))

(2.3.7)

and vT{F,F)

=u~^J (ALJ^F,UJ^F)\ =L>-$j((u%F,(j%F)\

.

(2.3.S (2.3.8)

The initial value problem is then defined by Eq.(2.3.6) with initial condition F 0 (x,v) :

IR3 3 X xM R 33 44 R1 ++ . .

(2.3.9)

Since Problem (2.3.6, 2.3.9) is well documented in the literature and that some good surveys already exist, our review contribution will be limited to some bibliographical indication and to the historical reconstruction of the most relevant mathematical results, which will be described in a very concise form.

ON THE CAUCHY PROBLEM

33

We recall that general aspects of the problem are reviewed in Chapter 3 of [BL2], in Chapter 3 of [MS3] as well as in the second and third part of [WE2]. The spectral theory of the linear operator is reported in the book [KP1]. The book by Greenberg, Protopopescu and van der Mee [GR1] deals with the general theory of linear and linearized operators. The essential aspects can also be found in the Appendix of [BL2]. Various aspects of the problem are mathematically dealt with in the papers (among others) of Caflisch [CF1,2], Elmroth [EMI], Petterson [PS1], Maslova and co-workers [MSI,5,6], and by the Japanese school Asano, Nishida, Imai, Shizuta, Ukai. The papers of this group will be referred more precisely in what follows. The first study of such a problem is due to Grad [GDI,2] and refers to lo­ cal solutions of perturbation of Maxwellian for the equation with hard cutoff potential. His study was developed in a bounded space domain D = [0, l ] 3 with specularly reflecting walls. As is known, such a problem is equiva­ lent to look for periodic (in the space variable) solutions in the whole space. Grad's proof shows that the linear problem possesses a unique solution with linear growth in time. This result is then used to show local existence for the nonlinear problem with small distance, in norm, between the initial condition and the Maxwellian. Later Ukai [UK1] was able to prove that the solutions to the linearized problem dealt with by Grad decays to zero in time as t tends to infinity. This estimate was sufficient to provide the first global existence proof for the solutions to the initial value problem for the Boltzmann equation. In that case, of course, near equilibrium. Both studies by Grad and Ukai were developed in a special Banach space with the property of a Banach algebra, such that if / belongs to such a space, then also the nonlinear term belongs to the same space. Tech­ nical details are given in Sections 3.2 and 3.3 of [BL3]. Ukai's result was subsequently developed by several authors, mainly based upon the semi­ group theory [BM1,2], [MT1] and, in particular, Shizuta [SHI] proved the existence of classical solutions. The results which we have surveyed above refer to the Boltzmann equa­ tion with hard cutoff potential. The problem for the equation with soft cutoff potential was dealt with by Caflisch [CF1,2], who obtained existence and decay to equilibrium in spaces of the types L2 and L°°. The real Cauchy problem, i.e. the problem with initial conditions in the whole R3 space without periodicity conditions involves additional techni-

34

L. Arlotti and N. Bellomo

cal difficulties. Local existence was studied by several authors. Then two independent groups one in the formerly Soviet Union, Maslova and Firsov [MS5], and one in Japan, Nishida and Imai [Nil], obtained global existence for the equation with hard cutoff. Such a problem was also examined by Ukai [UK3]. Later, essentially in the same time as Caflisch, an important improvement is due to Asano and Ukai [UK4], who proved global existence for the equation with soft cutoff potential in a domain 2}, which can coin­ cide either with the whole space E 3 or with a bounded box with specularly reflecting walls. The proof line of the papers which have been cited above does not sub­ stantially differs from the one followed by Ukai [UK1] in his paper. The initial value problem is dealt with in a suitable Banach space X by show­ ing that in such a space the term BL generates a Co-semigroup, say etBt. Then, suitable decay estimates as t —Y oo need to be obtained for such a semigroup. To this aim the spectrum and the resolvent of BL need to be carefully studied near the imaginary axis. It turns out that the problem for the equation with hard potentials is relatively simpler than the one for the equation with soft potential. In fact, in such a case, the spectrum, near the imaginary axis, consists in a finite number of discrete eigenvalues. In the other case, it contains a left neighbourhood of the imaginary axis. We conclude this section by pointing out that the existence theory near equilibrium also contains stability results and trend to equilibrium. The same problem for the spatially homogeneous equation can be studied ex­ ploiting suitable Liapunov functionals, see the survey by Wennberg [WE2]. It would be of great interest to develop these results to the spatially inhomogeneous case as obtained by Desvillettes for the BGK model [DEI] or, as we shall see, for the generalized Boltzmann equation. 3.

On the Generalized Boltzmann Equation

The classical Boltzmann equation is derived under the assumption, among others, that the two particles that are about to collide are uncorrelated and that the variations of / are negligible at distances of the order of the radius R of the effective cross sectional area. The latter assumption implies that the spatial arguments in the one particle distribution functions are taken at x = X! = x 2 , where Xi and x 2 are the centers of the two colliding particles.

(3.1)

ON THE CAUCHY PROBLEM

35

In order to obtain a more detailed description of physical reality one should compute the trajectories of the particles during the interaction and estimate the evolution of the distribution function along the trajectories. This type of modelling leads to the generalized Boltzmann equation, which involves somewhat complicated calculations and can be regarded as a well understood model only in the spatially homogeneous case. An indirect way to model the kinetic equation avoiding the simplification stated in Eq.(4.1) consists in averaging the distribution function of the field particles in the action domain of the test particles. An example of this type of modelling is the one reported in [BL6]. It is well understood, after the papers of Morgenstern [MGl] and Povzner [PV1], that if the distribution function is suitably smeared in a domain of the phase space, then the evolution operator U gets all the good properties needed to develop a nice existence and uniqueness theory for the solutions to the initial value problem. The crucial point is to define if the averaging has some physical consistency or can be regarded as an artificial manipulation purposely done in order to simplify the mathematical problem. Various authors have worked on the generalized Boltzmann equation, see among others [LA3], [LA4] and [BL9]. The common background is, as already mentioned, the attempt to develop a hopefully reliable modelling of the kinetic equations and the related mathematical analysis. It is worth providing, after these preliminaries, a specific example of generalized Boltz­ mann equation and showing some of the results which can be obtained in the mathematical analysis of such a model. In particular, we refer to the specific example reported [BL9], where the distribution function of the field particles is replaced by its mean value in the action domain of the test particle averaged by a suitable probability density P ( r ; n) such that P(r;n) = P ( r ; - n ) .

(3.2)

This yields the following model of generalized B o l t z m a n n equation

{it+Vx)f where

= r{fJ) = r+ifJ)

~ J:(/'/}'

(3,3)

36

L. Arlotti and N. Bellomo

j;=

f

H(n,q)/(t,x,v')/(*,x + rn,w/)P(r;n)drdndw,

{ A 22xxK K3 ^S

JO '0

•7?

J I = /(*,x,v) / 7o

/

tf(n,q)/(£,x

2

JS xR

3

+ r n , w ) P ( r ; n ) d r dndw.(3.4)

Remark 3.1. The generalized Boltzmann equation, we have just seen, is an example of how one can particularize the equation with molliher introduced by Morgenstern [MG1] and Povzner [PV1]. Both models were proposed for mathematical purposes, i.e. to obtain existence proofs for the initial value problem for large initial data. In order to give some details of the models defined in Remark 3.1, let 0 < V(x) be a smooth function such that / V(x) dx = 1,

and V{-x) = V(x),

Vx G D .

(3.5)

JD

Then one can consider the generalized equation with P(r, n) replaced by V(x). The following alternative expressions of the collision operators are then obtained Jv+ ( / » / ) = /

B(n,d)f(t,xy)f(t,y,W)V(x-y)dydndw,

JDx§2xR3

Jv. (/. / ) = /(*. x, v) /

B(n, q ) / ( t , y, w)7>(x - y) dy dn dw .

JDxS2xR3

/o g\

Equation (3.3) exhibits several properties which are similar to the cor­ responding properties of the Boltzmann equation. In particular, mass mo­ mentum and energy are collision invariants. Moreover, referring to a gas in a periodic box D, one can easily show that, for non-negative solutions to Eqs.(3.3-3.4), the functional H*(t) defined by

H*(t)=

f JDxR

3

/(t,x,v)log/(t,x,v)dvdx,

(3.7)

ON THE CAUCHY PROBLEM

37

is such that dH* <0. dt

(3.8)

The equality sign in (3.8) holds if and only if 3/2

/(i,x,v) = p ( t , x ) ( ^ )

exp(-^(v-u(0)2.

(3.9)

where p(t,x) is local density, u(t) is fluid velocity and (3(t) = m/kBT(t) with T(t) fluid kinetic temperature and m mass of the gas particles. The proof that the functional form given in (3.9) provides the only pos­ sible stationary points of H* (t) is based on the following Hypothesis 3.1. In the class of measurable functions on D x R3, the only solutions of the functional equation V>(x, v) + ip{x. + rn, w) = -0(x, v') + ip(x + rn, w'),

(3.10)

a.e. in (x, v, w, n) £ D xR3 x E 3 x § 2 , are given in the form V>(x,v) = A(x)+Bv

+ Cv2.

(3.11)

Here, r > 0 is given, A is a measurable function in x 6 D, B G R3 is a constant vector, and C a constant. The first formal proof of Hypothesis 3.1 was given by Resibois [RE2] in the case of a periodic box. The proof becomes rigorous if / is smooth and log / is integrable in x and v. A rigorous proof has been recently given by Arlotti [AR1] under quite general conditions. This assumption is the starting point for the analysis of the initial value problem near equilibrium. Moreover, in order to deal with such a problem, some further assumptions on the structure of the equation and the definition of a suitable function space are necessary. Moreover one needs the following Hypothesis 3.2. Let x G D, then referring to the term B we assume the following 5(n,v-w) = /iW|v-w|/3,

0
(3.12)

38

L. Arlotti and N. Bellomo

Hypothesis 3.3. Suitable constants c\ and c2 exist such that ci sin C\ sin<90 cos cos 00 << h(B) h{9) < < cc22 sin sin 99cos cos 90 ,

(3.13) (3.13)

where 9 is the azimuthal angle ofv' in a polar spherical coordinate system attached to v with the reference axes oriented in the same direction as w — v. Then, we consider, for k > 0, the following function space X ( D xxM Xk, = (| / eGLL1l(D R3): I V

/

/ n viu3 3 JDxR

(1 + | v | 2 ) ^t //((tt,,xx,, v ) d x d v < cooi ol . )

(3.14) (3.14)

In In addition, addition, we we define define Xk,Q ^Gfe,a

= \< / f(t, v) : : measurable measurableon on = ( ^ xx, , v) [0,oo) x D x E R3 ,,

supexp(at)||/(t)||ife < oo 1I . supexp(a*)||/MIU t>o oo J)

(3.15)

Here, the norm in Xk is denoted by || • H* and the norm in Xk,a by || • |U,Q. For h(t) G X0, define /i # (*,x,v) = / i ( t , x + *v,v),

a.e. in

(x, v) <E £> x R 3 .

(3.16)

Referring to the initial value problem, the following results hold Theorem 3.1. Existence Let 0 < f° € X2 and / ° l o g / ° G X0 then there exists a solution f/ :: R+ X22 K+-> ->X to the initial value problem, with initial conditions f°, such that /I

JDxR JDxR33

//

33

JDxR JDxR

/(*,x,v)dxdv / ( * , x , v ) d x d v = fI

//°(x,v)dxdv, °(x,v)dxdv, JDXR3 JDXR

vv/(£, / ( t , xx,v)dxdv , v ) d x d v = //

3

JDxR JDxR

vv/°(x,v)dxdv, /°(x,v)dxdv,

(3.17a)

(3.176)

39

ON THE CAUCHY PROBLEM

/

v2/(*,x,v)dxdv < /

3

JDxR

3

i;2/°(x,v)dxdv,

(3.17c)

JDxR

and I

JDxR3

/(*,x,v)log/(r,x,v)dxdv<

/0(x,v)log/°(x,v)dxdv.

f JDXR3

(3.17d) In addition, iff € Xk,for k > 2, then the solution f is such that f G Xk,o and the inequality (3.17c) holds with the equality sign. 0

T h e o r e m 3.2. Uniqueness Let k > 2+ (3 and / ° , h° are initial conditions given in the framework of Theorem 3.1. Then, the following inequality holds fO

LO

H/W " M*)IU < ll/° " h»\\keMMk+(3t),

(3.18)

where f(t) and h(t) are the solutions to the initial value problem with initial conditions f° and h°, respectively, and Mk+p > 0 is a constant which depends on f° and h° not exceeding k + (3. In particular, the solution is unique. Moreover, work in progress is announced on the following T h e o r e m 3.3. Convergence to equilibrium Let us denote by UQ the unique MaxwelHan defined by the conditions f

f°(x,v)dxdv

= Meas.D I

JDXR3

f

v / ° ( x , v ) d x d v = Meas.D / 3

cj 0 (v)dv,

(3.19a)

JR3

vu0(v)dv,

(3.196)

3

JDXR

JR

and v2f(t,x,v)dxdv

f 3

JDxR

= Meas.D I v2u0{v)dv. ^R

(3.19c)

3

In addition, if f° is given in the framework of Theorem 3.1 and f° € Xk , with k > 2 , one has f(i) -> UJQ, as t -> oo inXk> ,

0 < k' < k , in the norm of Xk> .

(3.20)

40

L. Arlotti and N. Bellomo

Theorem 3.4. Exponential convergence to equilibrium Let f° and UJQ be given in the framework of Theorem 3.3, then there exist suitable k0 and a 0 with k0 > 2 + (3 and ct0 > 0 such that, whenever f° e Xk*, with k* > ko, then | | / ( 0 - "oil*-,* < C(fe,ar)||/° - uo\\k. ,

(3.21)

for all k < k*, 0 < a < OLQ and where C > 0 depends only on k, a and f°. The results reported in Theorems 3.1-3.4 hold also for the Povzner equa­ tion. However, the additional assumption that P is bounded from below by some positive constant is necessary. The reader is referred to [BL6] and to [BL9] for all details on the proofs of the theorems which have been reported above. The proofs of Theorems 3.1 and 3.2 follow a line similar to the one already known for the original Boltzmann equation in the spatially independent case. On the other hand, the proof of Theorem 3.3 uses the weak compactness argument in L 1 , re­ viewed in Section 2, to obtain weak L1 -convergence to the solution to the unique absolute Maxwellian. The main results are in Theorems 3.3 and 3.4. The first one, which deals with strong compactness is based on the recent compactness result by P.L. Lions [LN1] related to the gain term of the Boltzmann equation. This result is developed in [BL9] in order to be applied to the generalized Boltzmann operator defined in this section. Once this project is completed, exponential convergence can be is obtained by the properties of the linearized exponential generalized Boltzmann operator. An additiona open problems still still is the detailed definition of P used to average the distribution function. Still, it can be observed that, if P(r) is assumed to be a delta function of the type P(r) = S(r), then the model yields the Boltzmann equation. On the other hand, if P(r) = S(r — R), then the model yields the so-called Boltzmann-Enskog model. This model is such that the gas particles behave as rigid balls with radius R. Moreover, if the pair correlation function is inserted in the integral operator, then the full Enskog equation is obtained [RE1]. As is known [BL5], this model can be written as

idi+

Vx 7 =

= I JS2+xR3

)

^ ' / ; / ) = £+(/>/;/) - E-(fJ-J)

Y+(f,R)(n,ci)f(t,x,v')f{t,x

+

Rn,w,)dndw

ON THE CAUCHY PROBLEM

-/(t,x,v) /

41

y - ( / , f l ) ( n , q ) / ( * , x + i?n,w)dn
(3.22)

where Y is a functional of / related to the pair correlation function and R is the radius of the hard sphere, see [BL5] for more information on the original model and on its developments. When P(r) is a delta function, the model involves the same difficulties as that with Boltzmann equation. In fact, the smoothing effect of the generalized equation is lost. In addition, the generalization of the existence theory developed by DiPerna and Lions cannot be regarded as a simple technical development for the full Enskog equation. The first crucial aspect is the construction of a suitable Liapunov functional to play, in the Enskog equation, the same role of the if-functional in the Boltzmann equation. This problem was solved by Polewczak [P03], who was able to give an explicit form to the functional proposed by Resibois [RE2]. Polewczak's functional reads as follows r ( / ) (*) = H (f){t)-P'

(/)(*),

(3.23)

where H(f)(t)=

f

(/log/)(t,x,v)dxdv 3

(3.24)

3

JR xU

and P * ( / ) ( * ) = / I(f)(s)ds, Jo

(3.25)

with I(f){t)=

[

M(/)(*,x,v,w,n)dxdvdwdn 3

(3.26)

3

7R3XR XR XS^.

and M(/)(t,x,v,w,n) = (n,w-v)/(s,x,v)|y+(/,JR)(S,x,n) x f{s,x

+ ifri, w) - y - ( / , R)(s, x, n ) / ( s , x - Rn, w) j .

(3.27)

Once the crucial problem which was outlined above is solved, various aspects of the existence theory for large L\ data can be developed as doc­ umented in the book [BL5], where the bibliography on the mathematical analysis of the Enskog equation is reported.

42

4.

L. Arlotti and N. Bellomo

O p e n Problems

A survey of the mathematical results on the initial value problem for the space dependent Boltzmann equation in unbounded domains was given in the preceding sections. This section deals with a presentation of some math­ ematical problems which still remain open after the various contributions which are reported in the literature and which are cited in this lecture. Certainly, one can regard as an open problem the possibility of improv­ ing the mathematical results which have been reviewed. Technical improve­ ments are always possible and sometimes can be classified as important contributions in applied mathematics. However, the reader can identify without much difficulties the technical possibility of improving the math­ ematical theory existing in the literature. On the other hand, we prefer (being aware of the fact that this is a matter of personal taste) to provide a brief discussion on some open problems which have not been dealt with with sufficient details. In details, this last section will report about the following problems: • The Boltzmann equation in the presence of an external force field. • Analysis of the initial value problem related to wave phenomena. • The singularly rescaled initial value problem. • The analysis of the initial value problem for the discrete Boltzmann equa­ tion. • Analysis of some nonlinear continuous models of the Boltzmann equation. Each of the five topics which have been indicated above is dealt with at a qualitative level without technical details. However, detailed bibliograph­ ical indications will be given. The choice of the topics which have been mentioned above is certainly not unique and must be regarded as a matter of personal taste. 4.1.

On the Cauchy Problem with External Force Field

The mathematical literature on the qualitative analysis of the Cauchy prob­ lem for the Boltzmann equation in the presence of an external force field, i.e. for the full Eq.(2.1), is rather limited compared with the literature we have presented in Section 2. On the other hand, real physical systems of particles are often subject to some external field. In some cases, for in­ stance: charged particles, the field is determined by the trajectories of the particles and depends upon the distribution function. Therefore, it seems interesting to develop this topic further.

ON THE CAUCHY PROBLEM

43

Consider first, the spatially inhomogeneous equation for neutral particles subject to a given force field F = F(t, x, v ) , then, the paper by Asano [AS1] needs to be cited. Local existence is proven, in this paper, for quite general initial conditions. Asano's formulation of the problem is the starting point of all studies developed after his paper. Further studies of the problem can be found in papers [AS3], [GUI], [HA2], [BL4]. In particular, the papers by Asano [AS3] and Grunfeld [GUI] provide a global existence proof for the solution with initial conditions close to equilibrium and a conservative force field driving the system towards equilibrium. Hamdache [HA2] deals with initial data decaying exponentially to zero at infinity in the phase space and trajectories prescribed by an oscillating field. Quite general is the result of [BL4], where global existence is proven for decaying data, in the framework of Theorem 2.1.1 and for a general force field acting in a time interval [0, T] with T large, but finite. As we have seen, the results which have been mentioned above can be regarded as a generalization of the ones already reported in Sections 2.1 and 2.3 for the case F = 0. In fact, once the mathematical problem is properly stated, such a generalization is purely technical although several sharp estimates are needed for the existence proof. Due to these reasons, our review will be limited to state the formulation of the problem and to indicate the main lines of the development of the existence analysis. The starting point, as already mentioned, is the mild formulation essentially due to Asano [AS1]. Considering that the gas par­ ticles are subject to a force field, one has to state the initial value problem for the dynamical system

dt

= v* (4.1.1)

l

dv = F(*,xt,vt) dt with initial conditions x = x(t = 0 ) ,

v = v(* = 0 ) .

(4.1.2)

The mathematical proofs are based upon the assumption that the force field is such that the solution (x*(x,v), v*(x, v)) to Problem (4.1.1-4.1.2) is unique.

L. Arlotti and N. Bellomo

44

Under this assumption the operator S(t) S(-^/(*,x,v) = /(i,x<(x,v),vf(x,v)) ,

(4.1.3)

which acts upon / , can be defined so that the integral formulation of the initial value problem is /(t,xJv)=5(t)/°(x,v)+ / S(t-s)J(s,x,v)ds,

(4.1.4)

Jo

or, in compact form f = UFf,

(4.1.5)

where the operator UF follows from the expression of the evolution problem stated in Eq.(4.1.4). The analysis developed in the literature which was cited above is ad­ dressed to estimate the semigroup and contraction properties of the oper­ ator UF jointly related to the qualitative analysis of the solutions to the dynamical system (4.1.1) under suitable assumptions on the force field. Detailed results are not reported here as the technical aspects of the problems do not substantially differ from the ones obtained for small per­ turbation of vacuum and equilibrium. On the other hand, it is worth ad­ dressing the attention of the reader to the fact that the existence theory due to DiPerna and Lions [DPI] for large Ll data was only recently [LN3] extended to the Cauchy problem in the presence of a force field. The impor­ tant aspect of the result in [LN3] is that it refers to force fields depending on the distribution function. That is entirely different from the case described above, which is relatively easier and, somehow, less interesting problem. Due to this reason, it seems important to point out this topic and the anal­ ysis of [LN3] as an interesting field certainly worth future studies. 4.2.

On the Cauchy Problem Related to Wave P h e n o m e n a

The mathematical results on the initial value problem which was dealt with in the preceding sections do not explicitly refer either to fluid dynamic applications or to the numerical solution of the Boltzmann equation. Several statements of mathematical problems with interest in fluid dynamics can be found in the classical literature [KOl], see Chapters 4-6. An example of the first class of problems refers to the analysis of the shock wave problem. Such a problem consists, in one space dimension, in

ON THE CAUCHY PROBLEM

45

finding a smooth profile / = f(x,v) and the asymptotic conditions lim

x—*±oo

which satisfies the Boltzmann equation

/(x,v)=o;±(v),

(4.2.1)

where C J ± ( V ) defines two prescribed Maxwellians. The solution of such a problem may be related to the solution of the Riemann problem, i.e. the initial value problem with initial conditions ' x<0 < \x>0:

:

/0(x,v)=w-(v) +

f°(x,v)=LJ (v).

(4.2.2)

The mathematical background for the analysis of the shock wave prob­ lem is the paper by Caflisch and Nicolaenko [CF1], which refers to weak shock wave solutions, that is when the distance between the asymptotic Maxwellians is small. On the other hand, very little is known on strong shock wave despite the fact that careful numerical simulation, such as the one performed by Ohwada [OH1], have shown existence of strong shock waves. The mathematical analysis of strong shock wave phenomena is still awaiting a satisfactory treatment. It may be promising to attempt its solu­ tion along the lines developed in [LN4]. Similarly, one may deal with rarefaction waves such that the initial con­ ditions satisfy the asymptotic conditions defined in Eq.(4.2.1). Once more, a preliminary answer to this problem is given in [LN4]. Somewhat similar comments can be developed for the analysis of exis­ tence theorems related to the direct solution of the Boltzmann equation. Such a problem is now starting to be reliably solved, by use of different techniques, see among others [BT1], [NE1], [PE2]. In particular, we point out that the important, and difficult, problem of linking kinetic equations with the ones of the macroscopic hydrodynamics is carefully studied in sev­ eral papers, see amongst others [COl], [GS1], [OH1], [PE2]. In general, it would be useful to obtain existence theorems which are directly related to the applications of numerical schemes. An example of what we mean can be found in paper [LA4]. The discus­ sion is limited, however, to the discrete Boltzmann equation. In the paper a comparison is carried out between the solution (obtained by a collocationinterpolation method for nonlinear equations) to the initial value problem

46

L. Arlotti and N. Bellomo

for the full equation compared with the one with the gain term only. It is indeed shown that the numerical solution for the full equation remains bounded for large initial data. On the other hand, the one for the truncated equation blows up for large initial data and shows stability only for small data. This experiment may be related to the analysis developed in Section 3. In fact, the existence theory for small initial data essentially refers to the truncated equation, which generates the blow up, remarked in [IL2], for large initial data. On the other hand, the analysis of the problem by the method by DiPerna and Lions takes into account the contribution of the loss term. Developing a similar analysis for the full equation would certainly be a challanging, although difficult, research topic. In particular, it would be interesting to develop further investigations on the blow up of solutions remarked in [IL2] related to the compactness result of [LN1]. 4.3.

On the Cauchy Problem Related to Hydrodynamics

The derivation of the equations of hydrodynamics from the Boltzmann equa­ tion in suitable scaling and asymptotic limits is one of the classical topics of the mathematical kinetic theory of fluids. Several important research works can be quoted, see among others, Asano and Ukai [AS4], [UK2,4], Bardos and Golse [BAl], Bardos and Ukai [BA2], Lachowicz [LAI,3], Bardos, Golse and, Levermore [BA3,4]. This topic is dealt with exhaustively in the lecture [LA6] of this book. Therefore, we simply mention some aspects of this topic related to the Cauchy problem. The reader is referred to the lecture [LA6] for a deeper insight into this matter. The initial value singular perturbation problem is the following

|

+ v • Vx) / = i J(/, / ) ,

/ ° ( x , v) = / ( 0 , x, v ) ,

(4.3.1)

where the dimensionless parameter e is proportional to the mean free path. When e tends to zero the model tends to the description of the continuous fluids. The singular initial value problem is studied as a suitable perturba­ tion of the Maxwellian solution. A different scaling was proposed in some recent papers [DM1], [BA3,4]. This scaling is obtained by interpreting the parameter e as ratio between the Mach number and the Reynolds number. As observed in [BA4], fluid

ON THE CAUCHY PROBLEM

47

dynamics is obtained in limits where the mean free path becomes smaller compared with the macroscopic length scales. At fixed Reynolds number, the Mach number must also vanish. The flow will be incompressible if its kinetic energy in the acoustic modes varies on a faster timescale than the rotational modes. Since the acoustic modes vary on a faster timescale than the rotational modes, they may be suppressed by assuming that the solution is consistent with motion on a slow timescale; the separation scale may be measured with the Knudsen number. This physical idea can be mathematically expressed by scaling time of the order of 1/e and by setting the distance from the absolute Maxwellian to be of the order of em for some m > 1. This procedure leads to the incompressible Navier-Stokes equation through the analysis of the singular perturbation problem

£ ~ / = - v • V x / + i J(/, / ) ,

/ ° ( x , v) = / ( 0 , x, v ) .

(4.3.2)

Such a problem is certainly of great mathematical interest. Referring to the existing literature, we address the reader to the book by Maslova [MS3], Chapter 8, where some uniform estimates are provided for short and long time behaviours, and to Bardos, Golse and Levermore [BA4], where the analysis is developed for renormalized DiPerna-Lions solutions. We mention, as a research perspective, that a further development of this type of analysis can certainly be developed by using the compactness results by Lions [LN1,2], which have been reviewed in Section 2.2. 4.4. On the Cauchy P r o b l e m for the Discrete B o l t z m a n n Equa­ tion As is known, see Gatignol [GA1] and Monaco and Preziosi [MOl], the dis­ crete Boltzmann equation is a model of the phenomenological kinetic theory of gases which defines the evolution of a finite number of densities

Ni = Ni{t,x),

i = l,...,n

(4.4.1)

of a gas of particles which can move only by a finite number of velocities v . If one takes into account the case of binary and triple collisions only,

L. Arlotti and N. Bellomo

48

the structure of the equation is

j t + ViVx) =J<2>[JV] + J^[N] = \ J2 4 * lN»N> '

NN

' A

jhk

+^

E

A ^ [NhNkNm

- NiNtNj]

,

(4.4.2)

see [GA1]. The transition rates A depend on the relative velocity of the collid­ ing particles, on the velocity discretization and on the collision mechanics. These terms satisfy the reversibility and indistinguishability properties Ahk _

A

ij

~~

Ahk _

3* ~

Akh _

ij

~

Akh _

ji

~

Aij

_

AJi

hk — • • • — s±kh >

and similarly for triple collision transition rates. The discrete Boltzmann equation is based on the rather crude simpli­ fication of the velocity discretization. This simplification involves several inconsistencies such as the difficulty in using second and higher order mo­ ments to recover macroscopic observables. However, an interesting feature is that it can include triple or even multiple collisions with probability the same order as that of the binary collisions. On the other hand, in the Boltzmann equation the probability of triple collisions is negligible with respect to the binary collisions. It is interesting to point out the role of triple (or multiple) collisions in the existence theory. In some cases, i.e. perturbation of equilibrium, multiple collisions have a stabilizing effect which contributes to the trend towards equilibrium, see [BL7]. Referring now to the mathematical results on the initial value problem we anticipate that we will restrict our attention to general models skipping all those results, however important, which are strictly related to some special model and are obtained exploiting the particular geometry of the model itself. Analogously to the case of the Boltzmann equation, the mathematical literature can be grouped for special classes of problems: i) Problems for small initial data decaying to zero at infinity. ii) Problems for small initial data in L 1 . iii) Solutions near equilibrium.

ON THE CAUCHY PROBLEM

49

iv) Solutions in one space dimension for large initial data. On the other hand, no result is available for large L1 data analogous to the ones obtained for the full Boltzmann equation. This feature is essentially due to the fact that the averaging lemma by Golse et al. [GOl] cannot be applied for a discretized distribution function. In fact, the compactness result, which is the basis of the theory by DiPerna and Lions is obtained by averaging over the velocity space, which is discretized for this type of equation. Nevertheless, in the one-dimensional case, global existence and unique­ ness can be obtained for large initial data. This result is obtained by exploit­ ing conservation of mass and energy jointly with the decreasing behaviour of iif-functional or analogous Liapunov functionals. This type of results are due to Beale [BE1,2], Bony [BOl] and Cabannes and Kawashima [CAl]. Similar methods can be developed, as shown by Cabannes [CA2] and Brzezniak, Flandoli and Preziosi [BR1], to obtain existence of solutions to the initial value problem for models with binary and triple collisions. Unfortunately suitable smallness assumptions need to be applied in this case. In fact, in this case, conservation of mass and momentum cannot be used independently. The generalization of the results which have been mentioned above to the analysis of the Cauchy problem in several space dimensions does not seem, at present, possible. In fact, conservation of mass is related to geometrical arguments which hold true only in one space dimension. The mathematical theory for general discrete velocity models with small 1 L data in several space dimensions was essentially developed by Bony [B02], who was able to exploit conservation of mass suitably linked to the properties of the diffusion term. Bony's important paper can be taken as the starting point to develop an existence theory for large initial data in several space conditions. The analysis of perturbation of equilibrium and trend to Maxwellian states was developed by Kawashima [KW1]. Suitable stability criteria to assure, for "stable" models, trend to equilibrium for the solutions of the initial value problem with data close to a global Maxwellian were defined in [KW1]. This type of analysis requires the preliminary proof of existence and uniqueness of Maxwellian states, which is possible only for some models in discrete kinetic theory. This topic is systematically studied by Gatignol and co-workers [GA2,3].

50

L. Arlotti and N. Bellomo

Gatignol's analysis is also the starting point for the analysis of the Cauchy problem for models with multiple collisions. In particular, it is shown in [BL7] that an existence theory can also be developed for models with multiple collisions. Such a theory is the natural development of the one proposed in [KWl] and already cited in this section. Moreover, it is shown that some models which do not satisfy, with binary collisions only, the sta­ bility conditions, towards Maxwellian equilibrium, satisfy such conditions when higher collisions are included. The problem with small initial data decaying to zero at infinity appears to be very interesting. In fact, the analysis of this problem leads to re­ sults analogous to the ones, obtained with the same technique, for the full Boltzmann equation. The interested reader can reach a deeper knowledge on this topic by the survey papers [BL8] and [PL1], where all relevant results, besides the very recent one, are reported. Additional useful information can be found in [CB2]. The starting point of the analysis of the Cauchy problem for this model is, as is known, the pioneering paper by Nishida and Mimura [NM1]. We conclude this section by commenting, with similar arguments of the ones proposed in [CB2], that an interesting aspect of the existence theory for the discrete Boltzmann equation (may be the most interesting) is the analysis of models with multiple collisions referred to lattice gas dynamics. As a matter of fact, in lattice gas dynamics the probabilities for binary and multiple collisions are of the same order of magnitude, whereas the Boltzmann equation refers to a dilute gas with binary collisions only. 4.5.

On the Cauchy Problem for Nonlinear Continuous Models

Several other models are known in the phenomenological kinetic theory of gases. Such models are useful for some specific applications, or allow for existence results stronger than those obtained for standard models. A first example is the one of the Boltzmann equation on a lattice [CE2], global existence was proved for large initial data. Another example of this type of analysis is given by Arkeryd [AK5], who studied the initial value problem for the Boltzmann equation in one space dimension i G [0,1] with periodicity conditions at x = 0 and x = 1. The technical assumption by which he modifies the original equation consists in a particular restriction for small relative velocities in the rc-direction. In more details, it is assumed that the collision kernel B is a bounded and measurable positive function with the usual symmetries and satisfies

ON THE CAUCHY PROBLEM

51

the further restriction that the terms 8 ( n , v - w) -: 1— , Pi - wi\

and

Bin. v - w) —— —^ \v[ -w[\

are both bounded for small values of vx -w\ and v[ -w[, where the subscript refers to the direction of the x-axis. Let L\ and L\+ be, respectively, the Banach space of the measurable functions / on [0,1] x E 3 and the corresponding positive cone such that {l + \v\*)f{x,v)

£Ll([Q,l]xR3).

Then under the afore-mentioned assumptions, the following existence theo­ rem is proved, in [AK5], for large initial data T h e o r e m 4 . 5 . 1 . If f° G L]+ and / ° l o g / ° <E L j , then the Boltzmann equation has a unique mild x-periodic solution in Co(M+,L 1 + ). This solu­ tion has a decreasing H-function, preserves mass and first moments in v. Moreover the energy is bounded by its initial value. Two further research lines will be mentioned amongst others. The first one refers to the so-called BGK model which is such that the collision opera­ tor is replaced by weighted difference between the distribution function and the Maxwellian distribution with the same moments. The Cauchy problem for large initial data was first studied by Perthame [PE1], who was able to generalize the DiPerna and Lions theory towards the analysis of this model. Then a further development of the mathematical analysis of the same model generated two interesting results: convergence to equilibrium proved in [DEI] and a uniqueness result based upon a suitable application of fixed point theorem linked to bounds partiallly recovered in [PE3]. It may be interesting revisiting these results in a more general framework and, in particular, under the light of the compactness result [LN1] we have already surveyed and discussed. Somewhat different is the analysis developed in [ES1,2], which refers to models of particles with finite size undergoing non-elastic collisions. In this case the model retains all features of the Boltzmann or Enskog-Boltzmann equation. However, dealing with dissipative collisions can bring to some interesting descriptions of some properties of Saturno's rings. The analysis of papers [ES1,2] leads to the existence for large L1 data. An open problem is the analysis of existence of solutions in the presence

52

_ L. Arlotti and N. Bellomo

of a force field. Some of the results of [LN3] need to be exploited in order to include the presence of a force field.

5. Final Remarks We aim, in this final section, to provide some free speculative thoughts with­ out the constraints of the mathematical formalizations which characterized the preceding sections. Once more, these speculations will not refer to the two important topics left out in this survey: existence of steady solutions and boundary value problems. The books by Cercignani [CE2] and Maslova [MS3] can provide the interested reader with the information needed to ap­ proach these interesting and challenging topics. Referring then to the Cauchy problem in unbounded domains, we need to mention that one of the challenging open problems (may be the most im­ portant one) is the proof of uniqueness of the solutions for large initial data. This problem will probably require a great effort to be solved and, maybe, an approach qualitatively different from the one up to now developed. As a matter of fact, the existence theory developed by DiPerna and Lions was based on entirely new ideas. We remark, having in mind the afore-mentioned problem, that the analysis of [LN1] already provides a pre­ liminary answer to such a problem. In fact the equivalence between weak and strong solutions, when the latter exist, suggests to exploit the bounds recovered in the analysis of weak solutions to enlarge the set of admissible initial conditions suitable to generate strong solutions. We have seen that uniqueness can be obtained for the generalized equa­ tions where the distribution function is averaged, in some way, over the action domain of the field particles. This averaging weakens, in some sense, the concept of solution, but leads to uniqueness. It may be interesting to study this topic more deeply of what it has been done until now. In this line, one should understand more deeply also on the ground of physics, the difference between the Boltzmann equation and the generalized model. Another important topic, somehow related to the uniqueness problem, is the analysis of the Cauchy problem in regimes spanning from contin­ uum to rarefied. Namely from regimes where the models of continuum fluid-dynamics are valid to the ones where the Boltzmann equation must substitute the afore-mentioned models. Then, the physical space has to be sub-divided into various subdomains, each related to a certain model. In the simplest case, the decomposition is

ON THE CAUCHY PROBLEM

53

settled into a domain where the Boltzmann equation is valid, in a second domain where the transition from continuum to rarefied occurs and in a third domain where the models of continuum fluid-dynamics are valid. Even the statement of this problem has to be regarded as an open one. In fact, it is plain that the moments of the distribution function provide the macroscopic observables, i.e. the transition from Boltzmann to continuum models is possible. On the other hand, a few moments are not sufficient to identify the distribution function for the Boltzmann equation. Therefore, it is important to study the transition in a certain domain exploiting the gradients in the contiguous domains. Solving this problem in a satisfactory way would mean providing a fun­ damental contribution to the mathematical theory of the application of the Boltzmann equation to fluid-dynamics.

L. Arlotti and N. Bellomo

54

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L., Asymptotic behaviour of the Boltzmann equation with infinite range force, Commun. Math. Phys., 86 (1982), 475-484.

[AK2]

L., On the Boltzmann equation in unbounded space far from equilibrium, Commun. Math. Phys., 105 (1982), 205-219.

[AK3]

ARKERYD

[AK4]

L., E S P O S I T O M., and PULVIRENTI M., The Boltzmann equation for weakly inhomogeneous data, Commun. Math. Phys., I l l (1987), 393-407.

[AK5]

L., Existence theorems for certain kinetic equations and initial data, Commun. Math. Phys., 103 (1988), 139-150.

[AK6]

L., in N o n e q u i l i b r i u m p r o b l e m s in m a n y - p a r t i c l e s s y s t e m s , Springer Lect. Notes in Math. n. 1551, Springer (1993), 14-57.

[AK7]

L., On the strong L 1 trend to equilibrium for the Boltz­ mann equation, Stud. Appl. Math., 87 (1992), 282-293.

[AK8]

L. and CERCIGNANI C , A global existence theorem for the initial-boundary-value problem for the Boltzmann equation when boundaries are not isothermal, Arch. Rat. Mech. Anal., 125 (1993), 271-287.

[AR1]

L., On a functional equation in mathematical kinetic the­ ory, to appear.

ARKERYD

ARKERYD

L., Loeb solutions of the Boltzmann equation, Arch. Rat. Mech. Anal, 86 (1984), 85-97. ARKERYD

ARKERYD

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ARLOTTI

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ON THE CAUCHY PROBLEM

55

[AS4] ASANO K. and UKAI S., On the fluid dynamical limit of the Boltzmann equation, Lecture Notes in Num. Appl. Anal, 6 (1983), 1-19. [BAl] BARDOS C. and GOLSE F., Different aspects de la notion d'entropie au niveau de l'equation de Boltzmann et de Navier-Stokes, Comp. Acad. Sci. Paris, 299 (1984), 225-228. [BA2] BARDOS C. and UKAI S., The classical incompressible Navier-Stokes limit of the Boltzmann equation, Math. Models Methods Appl. Sci., 1 (1991), 235-257. [BA3] BARDOS C , GOLSE F., and LEVERMORE D., Fluid dynamics lim­ its of kinetic equations I: Formal derivation, J. Statistic. Phys., 63 (1991), 323-344. [BA4] BARDOS C., GOLSE F., and LEVERMORE D., Fluid dynamics limits of kinetic equations II, Comm. Pure Appl. Math. (1993). [BE1] BEALE T., Large-time behaviour of the Broadwell model of a discrete velocity gas, Commun. Math. Phys., 102 (1985), 217-235. [BE2] BEALE T., Large-time behaviour of discrete velocity Boltzmann equ­ ation, Commun. Math. Phys., 106 (1985), 659-678. [BM1] BELLENI MORANTE A., Applied Semigroup Theory, Oxford University Press (1978). [BM2] BELLENI MORANTE A., A Concise Guide to Semigroup Appli­ cation, Advances in Mathematics for Applied Sciences n. 19, World Sci. (1994). [BL1] BELLOMO N. and TOSCANI G., On the Cauchy problem for the non­ linear Boltzmann equation: Global existence, uniqueness and asymp­ totic behaviour, J. Math. Phys., 26 (1985), 334-338. [BL2] BELLOMO N., PALCZEWSKI A., and TOSCANI G., Mathematical Topics in Nonlinear Kinetic Theory, World Sci. (1988). [BL3] BELLOMO N. and LACHOWICZ M., On the asymptotic equivalence between the Enskog and the Boltzmann equations, J. Statist. Phys., 51 (1988), 233-247. [BL4]

BELLOMO N., LACHOWICZ M., PALCZEWSKI A., and TOSCANI

G.,

On the initial value problem for the Boltzmann equation with force term, Tranp. Theory Statist. Phys., 18 (1989), 87-102.

L. Arlotti and N. Bellomo

56

[BL5]

BELLOMO N., LACHOWICZ M.,

POLEWCZAK J., and T O S C A N I

G.,

Mathematical Topics in Nonlinear Kinetic Theory II: The Enskog Equation, Advances in Mathematics for Applied Sciences n. 1, World Sci. (1991). [BL6]

N., POLEWCZAK J., and P R E Z I O S I L., Liapunovfunction­ a l for the Enskog Equation, in Advances in Partial Differential equations and Application in Mathematical Physics, Buttazzo G., Galdi G.P., and Zanghirati L. Eds., Plenum Press (1992), 1-13.

[BL7]

N. and KAWASHIMA S., The discrete Boltzmann equation with multiple collisions: Global existence and stability for the initial value problem, J. Math. Phys., 31 (1990), 245-253.

[BL8]

N. and GUSTAFSSON T., On the initial and initial-boun­ dary value problem for the discrete Boltzmann equation, Review. Math. Phys., 3 (1992), 137-162.

[BL9]

N. and POLEWCZAK J., On the Generalized Boltzmann Equation: Existence, uniqueness and exponential convergence to equilibrium, Comp. Rend. Acad. Sciences Paris, 319 (1994), 893898.

[BOl]

J., Solutions globales bornees pour le modeles discrets de l'equation de Boltzmann, Equations aux derivees partielles, Ecole Polytecnique, Palaiseau (1987).

[B02]

J., Existence globale a donnees de Cauchy petites pour le modeles discrets de l'equation de Boltzmann, Comm. Partial Diff. Equations, 16 (1991), 533-545.

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R. and P R O T O P O P E S C U V., Abstract time dependent ansport equations, J. Math. Anal. Appl, 121 (1987), 370-405.

[BTl]

J . F . , L E TALLEC P . , T I D R I R I D., and Qiu Y., Numerical Coupling of nonconservative or kinetic models with the conservative compressible Navier-Stokes equations, INRIA Report n. 1426 (1991).

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Z., FLANDOLI F., and P R E Z I O S I L., On the initialboundary value problem for the discrete Boltzmann equation with multiple collisions, Stab, and Appl. Anal, in Cont. Media, 2 (1992), 153-182.

BELLOMO

BELLOMO

BELLOMO

BELLOMO

BONY

BONY

BEALS

tr­

BOURGAT

BRZEZNIAK

ON THE CAUCHY PROBLEM

57

[BUI BUSONI G. and PALCZEWSKI A., Stationary Boltzmann equation near equilibrium, Math. Models Methods. Appl. Sci., 3 (1993), 595639. [CBl CABANNES H. and KAWASHIMA S., Le probleme aux valeurs initiales en theorie cinetiqe discrete, Comp. Rend. Acad. Sciences Paris, 307, I (1988), 507-501. [CB2 CABANNES H., Global solution of the discrete Boltzmann equation with multiple collisions, Eur. J. Mech. B/Fluids, 11 (1992), 415-437. [CFl

R.E., The Boltzmann equation with a soft potential: Part I, Commun. Math. Phys., 74 (1980), 71-95. CAFLISCH

[CF2 CAFLISCH R.E., The Boltzmann equation with a soft potential: Part II, Commun. Math. Phys., 74 (1980), 97-59. [CF3 CAFLISCH R.E., The fluid dynamic limit for the nonlinear Boltz­ mann equation, Comm. Pure Appl. Math., 33 (1980), 651-666. [CF4 CAFLISCH R.E. and NIKOLAENKO B., Shock wave solution of the Boltzmann equation, Commun. Math. Phys., 74 (1980), 71-95. [CE1 CERCIGNANI C , GREENBERG W., and ZWEIFEL P., Global solu­ tions of the Boltzmann equation on a lattice, J. Statist. Phys., 20 (1979), 449-462. [CE2 CERCIGNANI C , Theory and Application of the Boltzmann Equation, Springer (1988). [CHI CHOUVAT P . and GATIGNOL R., Macroscopic variables in discrete kinetic theory, in Discrete Models in Fluid Dynamics, Advances in Mathematics for Applied Sciences n. 4, Alves A. Ed., World Sci. (1991). [CH2 CHOUVAT P . and GATIGNOL R., On a class of discrete models of gases with different moduli, Nonlinear Kinetic Theory and Ma­ thematical Aspects of Hyperbolic Systems, Advances in Math­ ematics for Applied Sciences n. 9, World Sci. (1992). [COl CORON F. and PERTHAME B., Numerical passage from kinetic to fluid equations, SIAM J. Num. Anal, 28 (1991), 26-42. [DM1 D E M A S I A., ESPOSITO R., and LEBOWITZ J., Incompressible Navier-Stokes and Euler limits of the Boltzmann equation, J. Statist. Phys., 42 (1989), 1189-1214.

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L., Convergence to equilibrium in large time for Boltzmann and BGK equations, Arch. Rat. Mech. Anal., 110 (1990), 73-91.

[DE2]

L., Convergence to equilibrium in various situations for the solution of the Boltzmann equation, in M a t h e m a t i c a l As­ p e c t s of K i n e t i c T h e o r y , Bom V. et al. Eds., World Sci. (1992), 101-114.

[DPI]

R. and LIONS P.L., On the Cauchy problem for the Boltz­ mann equation: Global existence and weak stability results, Annals Math., 130 (1990), 1189-1214.

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R. and LIONS P.L., Global solutions of the Boltzmann and the entropy inequality, Arch. Rat. Mech. Anal., 114 (1991), 4755.

[DP3]

R. and LIONS P.L., On the Fokker-Planck-Boltzmann equation, Commun. Math. Phys., 120 (1988), 1-23.

[ELI]

R.S. and P I N S K Y M.A., The first and second fluid approxi­ mation of the linarized Boltzmann equation, J. Math. Pures Appl., 54 (1975), 125-156.

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T., Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range, Arch. Rat. Mech. Anal, 82 (1983), 1-12.

[EN1]

D., Kinetiske Theorie, Svenka Akad., 63 (1921). English Translation in K i n e t i c T h e o r y , Brush S. Ed., Vol.3, Pergamon Press (1972).

[ES1]

M. and P E R T H A M E B., Solutions globale de 1'equation d'Enskog modifiee avec collisions elastique ou inelastique, Comp. Rend. Acad. Sci. Paris, 309 (1989), 897-902.

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M. and P E R T H A M E B., On the modified Enskog equation for elastic and inelastic collisions. Models with Spin., Ann. Inst. Poincare, 8 (1991), 289-308.

DESVILLETTES

DESVILLETTES

DIPERNA

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DIPERNA

ELLIS

ELMROTH

ENSKOG

ESTEBAN

ESTEBAN

[FE1] FERZIGER J. and K A P E R H., M a t h e m a t i c a l T h e o r y of T r a n ­ s p o r t P r o c e s s e s in Gases, North Holland (1972). [GA1] GATIGNOL R., T h e o r i e C i n e t i q u e des Gaz a R e p a r t i t i o n Di­ s c r e t e d e Vitesses, Lect. Notes in Phys. n. 36, Springer (1975).

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[GA2]

R. and COULOUVRAT F., Description hydrodynamique d'un gaz en theorie cinetique discrete, Comp. Rend. Acad. Sci. Paris, 306 (1988), 169-174.

[GA3]

R. and COULOUVRAT F., Constitutive laws for discrete velocity models, in Discrete Kinetic Theory, Lattice Gas D y ­ namics and Foundation of Gas Dynamics, Monaco R. Ed., World Sci. (1988), 121-145.

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F . , LIONS P . L . , P E R T H A M E B . , and SENTIS R., The reg­ ularity of the moments of the solution of a transport equation, J. Fund. Anal., 76 (1988), 110-125.

[GDI]

H., Asymptotic equivalence of the Navier-Stokes and the non­ linear Boltzmann equation, in Proc. S y m p . Appl. Math., Amer. Math. Soc. Publ., 17 (1965), 154-183.

[GD2]

H., Asymptotic theory of the Boltzmann equation, in Rar­ efied Gas D y n a m i c s , Laurmann J.A. Ed., Acad. Press (1963), 2959.

[GR1]

W., VAN DER M E E C.V.M., and P R O T O P O P E S C U V., B o u n d a r y Value P r o b l e m s in Abstract Kinetic Theory, Birkhauser (1987).

[GR2]

W., Z W E I F E L P . , and P O L E W C Z A K J., Global exis­ tence proofs for the Boltzmann equation, in Nonlinear P h e n o m ­ ena: T h e B o l t z m a n n Equation, Lebowitz J. and Montroll E. Eds., North Holland (1983), 21-49.

[GS1]

GROPENGIESSER F.,

GATIGNOL

GATIGNOL

GOLSE

GRAD

GRAD

GREENBERG

GREENBERG

NEUNZERT H.,

STRUCKMEIER J.,

and

WIE-

SEN B., Hypersonic flow around a 3D-Delta wing at low Knudsen numbers, in Rarefied G a s Dynamics, Beylich A. Ed., VCH Press (1990), 332-336. [GUI]

C.P., On the nonlinear Boltzmann equation with force term, Transp. Theory Statist. Phys., 14 (1985), 291-322.

[KA1]

S. and SHINBROT M., The Boltzmann equation: uniqueness and local existence, Commun. Math. Phys., 95 (1984).

[KP1]

H.G., L E K K E R K E R K E R C.G., and H E J T M A N E K J., Spec­ tral M e t h o d s in Linear Transport Theory, Birkhauser (1982).

GRUNFELD

KANIEL

KAPER

60

L. Arlotti and N. Bellomo

[KW1] KAWASHIMA S., Global existence and stability of the solution for dis­ crete velocity models of the Boltzmann equation, Led. Notes Num. Appl. Anal, 6 (1983), 59-96. [KW2]KAWASHIMA S., The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320. [KOl] KOGAN M., Rarefied G a s D y n a m i c s , Plenum Press (1968). [HA1]

K., Existence in the large and asymptotic behaviour for the Boltzmann equation, Japan J. Appl. Math., 2 (1985), 1-15.

[HA2]

HAMDACHE

[HA3]

K., Initial boundary value problems for the Boltzmann equation. Global existence of weak solutions, Arch. Rat. Mech. Anal., 119 (1992), 309-353.

[IL1]

R. and SHINBROT M., The Boltzmann equation: Global existence for a rare gas in an infinite vacuum, Commun. Math. Phys., 95 (1984), 117-126.

[IL2]

R. and SHINBROT M., The blow up of solutions of the gainterm-only Boltzmann equation, Math. Methods Appl. Sci., 9 (1987), 251-259.

[IL3]

R. and NEUNZERT H., On simulation methods for the Boltz­ mann equation, Transp. Theory Statist. Phys., 16 (1987), 141-154.

[LAI]

M., On the initial layer and the existence theorem for the nonlinear Boltzmann equation, Math. Methods Appl. Sci., 9 (1987), 342-366.

[LA2]

M. and PULVIRENTI M., A stochastic system of par­ ticles modelling the Euler equations, Arch. Rat. Math. Anal., 109 (1990), 81-93.

[LA3]

M., On the asymptotic behaviour of nonlinear kinetic equations, Ann. Mat. Pura et Appl, CLX (1991), 41-62.

[LA4]

M. and M O N A C O R., Existence and quantitative analy­ sis of the solutions to the initial value problem for the discrete Boltz­ mann equation in all space, SIAM J. Appl. Math., 49 (1989), 12311241.

HAMDACHE

K., These, Univ. "Pierre et Marie Curie", Paris (1988).

HAMDACHE

ILLNER

ILLNER

ILLNER

LACHOWICZ

LACHOWICZ

LACHOWICZ

LACHOWICZ

ON THE CAUCHY PROBLEM

61

[LA5] LACHOWICZ M., On a system of stochastic differential equations modelling the Euler and the Navier-Stokes hydrodynamic equations, Japan J. Ind. and Appl. Math., 10 (1993), 109-131. [LA6] LACHOWICZ M., On the hydrodynamics of the nonlinear Boltz­ mann equation, in Lecture Notes on the Mathematical Theory of the Nonlinear Boltzmann Equation (1994). [LN1] LIONS P.L., Compactness in Boltzmann equation via Fourier inte­ grals operators and applications I, J. Math. Kyoto Univ., 34 (1994), 391-427. [LN2] LIONS P.L., Compactness in Boltzmann equation via Fourier inte­ grals operator and applications II, J. Math. Kyoto Univ., 34 (1994), 429-460. [LN3] LIONS P.L., Compactness in Boltzmann equation via Fourier inte­ grals operator and applications III, J. Math. Kyoto Univ., 34 (1994), 539-584. [LN4] LIONS P.L., Conditions at infinity for Boltzmann equation, Comm. Partial Diff. Equations, 19 (1994), 335-367. [MT1] MARTIN R.H., Nonlinear Operators and Differential Equa­ tion in Banach Spaces, Wiley (1976). [MSI] MASLOVA N., Existence and uniqueness theorems for the Boltz­ mann equation, in Dynamical Systems II, Sinai Ya.G. Ed., Sprin­ ger (1992), 254-279. [MS2] MASLOVA N., Mathematical methods of studying the Boltzmann equation, St. Petersburg Math. J., 1 (1992), 41-80. [MS3] MASLOVA N., Nonlinear Evolution Equations, Advances in Mathematics for Applied Sciences, Vol. 10, World Sci. (1993). [MS4] MASLOVA N. and FIRSOV A.N., Solutions of the Cauchy Problem for the Boltzmann equation, (in Russian), Vest. Leningrad, 19 (1975), 83-88. [MS5] MASLOVA N. and TCHUBENKO R.P., On the solutions of the nonstationary Boltzmann equation, (in Russian), Dokl. Akad. Nauka USSR, 202 (1972), 800-803.

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[MS6]

N. and TCHUBENKO R . P . , Asymptotic properties of the solutions of the Boltzmann equation, (in Russian), Vest. Leningrad, 1 (1973), 100-105.

[MOl]

R. and P R E Z I O S I L., Fluid Dynamic Applications of the Discrete Boltzmann Equation, Advances in Mathematics for Applied Sciences, Vol. 3, World Sci. (1991).

[MG1]

D., Analytic studies related to the Maxwell-Boltzmann equation, Arch. Rat. Mech. Anal, 4 (1955), 533-555.

[Nil]

T. and IMAI K., Global solutions to the initial value prob­ lem for the nonlinear Boltzmann equation, Publ. RIMS Kyoto Univ., 12 (1976), 229-239.

[NM1]

T. and MIMURA M., On the Broadwell model for a simple discrete velocity gas, Proc. Japan A cad., 50 (1974), 812-817.

[OH1]

T., Structure of normal shock waves: Direct numerical analysis of the Boltzmann equation for hard-sphere molecules, Phys. Fluids A, 5 (1991), 217-234.

[PA1]

A. and TOSCANI G., Global solution to the Boltzmann equation for rigid spheres and initial data close to a local Maxwellian, J. Math. Phys., 30 (1989), 2445-2450.

[PE1]

PERTHAME

[PE2]

B., Boltzmann type schemes for gasdynamics and en­ tropy properties, SIAM J. Num. Anal., 27 (1990), 1405-1421.

[PE3]

B. and PULVIRENTI M., Weighted L°° bounds and uni­ queness for the Boltzmann BGK model, reprint.

[PS1]

R., Existence theorems for the nonlinear space inhomogeneous Boltzmann equation, IMA J. Appl. Math., 30 (1983), 81-105.

[PLl]

T. and ILLNER R., Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory, SIAM Review, 30 (1988), 213-255.

[POl]

J., Classical solution of the nonlinear Boltzmann equa­ tion in all space: Asymptotic behaviour of the solutions, J. Statist. Phys., 50 (1988), 611-632.

MASLOVA

MONACO

MORGENSTERN

NISHIDA

NISHIDA

OHWADA

PALCZEWSKI

B., Global existence to the BGK model for the Boltz­ mann equation, J. Diff. Equations, 82 (1991), 191-200.

PERTHAME

PERTHAME

PETTERSON

PLATKOWSKI

POLEWCZAK

ON THE CAUCHY PROBLEM

63

[P02]

J., New estimates of the nonlinear Boltzmann operator and their application to existence theorem, Transp. Theory Statist. Phys., 18 (1988), 235-247.

[P03]

POLEWCZAK

[PVl]

A.Y., The Boltzmann equation in the kinetic theory of gases, Amer. Math. Soc. Transl. Ser. 2, 47 (1962), 193-216.

POLEWCZAK

J., Global existence in L1 for the modified nonlinear Enskog equation in M3, J. Statist. Phys., 56 (1989), 159-173.

POVZNER

[RE1] RESIBOIS P . and D E LEENER M., Classical Kinetic Theory of Fluids, Wiley (1977). [RE2]

P . , H-theorem for the (modified) nonlinear Enskog equa­ tion, J. Statist. Phys. 19 (1978), 593-609. RESIBOIS

[SHI] SHIZUTA Y., On the classical solution of the Boltzmann equation Comm. Pure Appl. Math., 36 (1983), 705-754. [SI1]

SHINBROT M., Flow of a cloud past a body, Transp. Theory Phys., 15 (1986), 317-332.

Statist.

[SMI] SMART J., Fixed Point Theorems, Clarendon (1974). [TOl] TOSCANI G., On the Boltzmann equation in unbounded domains, Arch. Rat. Mech. Anal, 95 (1986), 37-49. [T02] TOSCANI G., Global solution of the initial value problem for the Boltzmann equation near a local Maxwellian, Arch. Rat. Mech. Anal., 102 (1988), 231-241. [T03] TOSCANI G., H-theorem and asymptotic trend of the solution for a rarefied gas in the vacuum, Arch. Rat. Mech. Anal., 100 (1987), 1-12. [TR1] TRUESDELL C. and MUNCASTER R., Fundamentals of Maxwell's Kinetic Theory of a Simple Monoatomic Gas, Acad. Press (1980). [UK1]

S., On the existence of global solutions of mixed problem for the nonlinear Boltzmann equation, Proc. Jap. Acad., 50 (1974), 179184.

[UK2]

S., The incompressible limit and the initial layer of the com­ pressible Euler equation, Math. Kyoto Univ., 26 (1986), 323-331.

UKAI

UKAI

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[UK3]

L. Arlotti and N. Bellomo

S., Les solutions globales de l'equation non-lineaire de Boltz­ mann dans l'espace tout entier et le demispace, Comp. Rend. Acad. Sci. Paris, A282 (1976), 317-320. UKAI

[UK4] UKAI S. and ASANO K., On Cauchy problem for the nonlinear Boltz­ mann equation with soft potentials, Publ. RIMS Kyoto Univ., 18 (1982), 477-519. [WEI]

B., On an entropy inequality for the Boltzmann equa­ tion, Comp. Rend. Acad. Sci. Paris, 315 (1992), 1441-1446.

[WE2]

B., Stability and E x p o n e n t i a l C o n v e r g e n c e for t h e B o l t z m a n n E q u a t i o n , Thesis Dept. Math. Goteborg Univ. and Chalmers Univ. of Techn. (1993).

[WE3]

B., Regularity estimates for the Boltzmann equation, Internal report No. 1994 - 02, Dept. Math. Univ. Goteborg (1994).

[ZW1]

P . F . , The Boltzmann equation and its properties, in Ki­ netic T h e o r i e s and the B o l t z m a n n E q u a t i o n , Springer Lect. Notes in Math. n. 1048, Cercignani C. Ed., Springer (1984), 111175.

WENNBERG

WENNBERG

WENNBERG

ZWEIFEL

Lecture 2 ASYMPTOTIC ANALYSIS OF NONLINEAR KINETIC EQUATIONS: THE HYDRODYNAMIC LIMIT M. Lachowicz

1.

Introduction

Asymptotic analysis plays a significant role in the qualitative analysis of equations modeling natural phenomena. Mathematical models of phenom­ ena which take place in nature contain a certain number of parameters representing the physical quantities, pertinent to the phenomena in ques­ tion. These parameters, defined (as a rule) as dimensionless quantities, are elements either of N or of K+, according to their physical sense. The essence of the asymptotic analysis is to examine the behavior of solutions of model equations when some suitably selected parameters tend to their critical values. Prom the point of view of physics, such a limit may signify the transition from the application range of one physical theory to another one. Three different aspects (at least) of the asymptotic analysis can be iden­ tified: • It enables to establish the general links between two theories and to "derive" model equations of one theory from those of the other theory.

65

66

M. Lachowicz

• It is possible to obtain existence and uniqueness theorems for an equation in one theory from the corresponding theorem in another one. Thus, the existence theorems for relatively more difficult equations can be reduced to theorems for relatively simpler equations. • It allows to define the numerical algorithms for the sophisticated model equations. This needs the methods selected for evaluation of the solution which allow to estimate the error of approximations. In most cases, the methods of asymptotic analysis lead to singularly perturbed problems, e.g. [EC1], [LOl], [MI1,2], [OM1] and [VA1,2]. The study of boundary layer or initial layer effects appears to be then an essential part of the qualitative analysis, e.g. [VAl]. Classical Newtonian mechanics, being a set of general laws, on the be­ haviour of physical systems, expressed quantitatively, is the fundamental physical theory to deal with the wide spectrum of phenomena. In this theory, every body (e.g. fluid area) can be considered as a set of micro­ scopic objects (particles). The state of such a system is given by positions X i , . . . , XAT and velocities v i , . . . , v^v of the particles constituting the sys­ tem. The evolution of the system is given by the solution of a set of ordinary differential equations. Such a system is totally deterministic. However, assuming a random initial distribution of the particles, one could introduce randomness into the system. This procedure leads to the Liouville equation, e.g. [CE1], [LN1] and [PE1]. A statistic description, in the framework of kinetic theory, is used for systems composed of a great number of particles. Evolution of such a system is described by probability density function of one particle (the so called oneparticle distribution function), regarded sometimes in non-normalized form, satisfying a kinetic equation. The Boltzmann equation is the most widely known and used kinetic equation (cf. [CC1], [CE1] and [FE1]). Other models of the kinetic theory, can be taken into account. For in­ stance the Enskog equation, proposed in 1922 by Enskog [EN1], and dealt with thereafter in many modified forms [BE5]. The purpose of introduc­ ing this equation was to construct a consistent theory for moderately dense gases (the Boltzmann equation has a physical sense for very rarefied gases). The Enskog equation appeared to be a very exciting mathematical model from both physical and mathematical points of view. On the other hand, other kinetic equations, such as the so-called Povzner equation [POl], was introduced, contrary to the Enskog model, because of purely mathematical reasons. The problem was to overcome difficulties of proving the global ex-

THE HYDRODYNAMIC LIMIT

67

istence and uniqueness of the solutions to the initial value problem. In fact, the Povzner equation has better mathematical features than the Boltzmann model. In continuum theory the microscopic material structure is not taken into account and every body (fluid area) is considered as a continuous nondisturbable area. Therefore, the state of the system in this theory cannot be described by using the positions and velocities of the particles (as in case of Newtonian mechanics) but by means of the function which describes the space distribution of physical quantities (e.g. local density, velocity and temperature). Such quantities do not relate to the fluid particles but to the space points (the so-called Euler variables) at any moment of time. The basic equation in continuum theory describing the motions of fluid without viscosity and heat conduction is the system of Euler equations for com­ pressible fluid. The other model in the continuum theory which allows to describe fluid motions, involving its viscosity and heat conduction, is the system of Navier-Stokes equations. The systems of Euler or Navier-Stokes equations are referred to as the hydrodynamic equations. This lecture is devoted to a unified description of asymptotic relations between the solutions of various nonlinear models of the kinetic theory and those of the continuum theory. This content is developed in Sections 3 and 4. Referring to the mathematical theory of linearized models, the interested reader is addressed to papers [AS1], [EL 1-3], [GK1], [PI1,2] (and references therein). Referring to the relations between particle systems and the kinetic theory equations as well as the hydrodynamic equations, such a problem is dealt with in Section 3.10. The essential bibliography is cited along the lecture. The interested reader will also find additional informations in the references at the end of the lecture itself. 2.

2.1.

Kinetic and Hydrodynamic Equations

Kinetic Equations

The B o l t z m a n n equation, well-known and studied for more than 100 years, has a quite vast literature. The Reader is referred to the Lecture 1 as well as to [BE3], [CE1], [CC1], [FE1], [GR1], [MS2] and [TR1] for a

M. Lachowicz

68

review of physical and mathematical results. It is worth pointing out that referring to the analysis of the existence of solutions, a fundamental result was obtained by DiPerna and Lions [DL1]. The Boltzmann equation describes the time evolution of the one-particle distribution function / = / ( £ , x , v ) , where t, x and v are, respectively, the time, the space and the velocity variables. Throughout this Lecture, functions on a three-dimensional torus T 3 , with respect to the space variable x, are considered. The torus T is always looked on as a rectangular parallelepiped [0,1]3 C M3 with the opposite faces identified. If necessary, a function on T can be interpreted as a periodic function on R 3 , by a simple extension. Obviously, t > 0 and v 6 i 3 . The Boltzmann equation reads Df = J(fJ),

(2.1.1)

where D is the free-streaming operator d_

D = — + v • Vx dt and J is a bilinear, symmetric collision operator

J(flj2) = \{j+(flj2)

+ J+(f2,fl)-J-Uuf2)-J-(f2jl))

,

J+(/i,/2)(x,v) = y,y/1(x,w')/2(x,v')B(n,w-v)dndw, R3 § 2

J-(flj2)=

K/ 2 )/l,

i/(/)(x,v) = / / /(x,w)B(n,w - v)dndw. »3 s 2

Standard notations are used: A particle with the center at x and the velocity v collides (interacts) with a particle with the center at x and the velocity w, and after the collision (interaction) the velocities are v' and w', respectively. S2 = { n E R 3 :

|n| = 1} .

THE HYDRODYNAMIC LIMIT

69

v' and w' are functions of v and w as well as of a vector n G § 2 according to v' = v + ((w - v) • n ) n ,

(2.1.2a)

w' = w - ((w - v) • n ) n .

(2.1.26)

B is the collision kernel; it depends upon the particle interaction poten­ tial (cf. [CEl], [CRl], [GR3,4] and [TRl]). Throughout this Lecture, B corresponding only to Grad's cutoff hard potentials is considered Assumption 2.1.1. B is a non-negative measurable function on S2 x E 3 such that i.)

B(n,v) < c o n s t ^ r - p (l + |v| A ) , where A G [0,1],

|v| 1+ v ~

ii.);

f J

———? < const / B(n, v ) d n . §2

Some examples can be given Example 2.1.1. The hard sphere model B(n,v)

=nvV0,

(2.1.3)

where C\ V C2 means the maximum of c\ and c>i. Example 2.1.2. The model of the inverse s-th power hard potential with a cutoff (cf. [GR1]) B(n,v) = | v | £ r 4 B , ( a ) ,

(2.1.4)

Tl ■ V

where s > 4, a = —rand Bs is a bounded function such that Ba(a) = 0 v

ll

for a < 0. By microscopic conservation laws one has / ^ ( v ) J ( / i , / 2 ) ( v ) d v = 0, R3

(2.1.5a)

70

M. Lachowicz

for i = 0 , . . . , 4 and each / , for which the integrals make sense, where * s l ,

Wv)=«i

4(v) = |v| 2

(i = 1,2,3),

(2.1.56)

are the collision invariants and Vi is the i-tb. component of the vector v. The following moments of the distribution function / , corresponding with fluid dynamics, are called the fluid-dynamic parameters of / : x Qf(tfx)= *,x,v)dv £/(*, ) = / / (/(*,x,v)dv

(2.1.6a)

R33

is the local density; u/(t,x) == f\i/(t,x) l ) — U Z X

1

f,|

Qf(t,i

\

f | v1 /V>7(*,x, (Af C, x> X, vJ V^ ) ,d/U.V va V V

(2.1.66)

XT' 1 *H 17"

3 R3

is the macroscopic velocity vector and T (t,x)fc\ — = Tx(f I ft x

f\ i )

ZQ

—«

l

/.

22 Uv t , x ) Y| u^ /l fi it^, xf )Y| M2 av v- g / fn.rf/ x ) y ' |/ vI|vv l / fft/ 7,r xY,x vv)vd

\ / l l A > > J

(2.1.6c) (2.1.6c)

2

k?/^,x;|uy^r,xj|

R33

is the macroscopic temperature. A unique class of solutions of the equation J(fJ) = J(fJ) = 0o

(2.1.7)

is that of Maxwellians, which are denoted in this Lecture by M |v-u(*,x)|2\ M(t,x, (t, x) (2*T(t,x))" exp f-\L£&*2L\ , M(t,x,v)v) = = eg(t,x) {2
2T(t,x)

7 '

where g, u and T are the fluid-dynamic parameters of the Maxwellian M. In order to stress the dependence of the Maxwellian M of the parameters g, u, T, the following notation is also used: M == M[£,u,T]. M[^,u .21-

THE HYDRODYNAMIC LIMIT

71

A Maxwellian with constant fluid-dynamic parameters is called a global Maxwellian. The global Maxwellian M[Q0,Q,T0], where g0 = const and To = const, is denoted by u U) LJ

= =

M[QO,0,T ]. M[g 0 ,0,T 0 0].

The Boltzmann model assumes that the overall dimensions of the particles can be neglected. In the case of dense gases, one has to replace this mass-point model by models which can take into account the overall dimensions of the particles. One such attempt leads to the Enskog equation model ([BE1,5], [EN1], [FE1]) in which each particle is regarded as a hard sphere with a nonzero diameter a. The Reader is referred to the book [BE5] for a detailed analysis of the mathematical problems and results of various versions of the Enskog equation. The Enskog equation reads Df = Ea(f-JJ), Ea(f;fJ),

(2.1.8)

where Ea is the following operator E K(/i;/2,/3)=i^(/i;/2,/3) + E+(h-f ^(/i;/3,/2) 3j2) c itfk; 1/2J3) --E-(h;f ^ a " ( 2/ ,f3)-Ei ; / 2 j 3a ) - ^ - (- /( i/ ;i /; /33,, // 22 ) )

and and ^(/i;/2,/3)(x,v)=||^(/i;x,x x,x + + an) an) h ,/ 3 )(x, v) E+(h\ 3 R S2 1R kS

ndw,, -1- an, an ,w')(n w')(n • (w (w -- v) v) V 00)) ddndw xx// 2 (x, ( x , vv')/ ' ) / 33((x x + ^ a ((hy. , X— - aan) n) = J yJy a / i ; Xx,x

^ a -"(/i; ( / i,h ; / 2,/J3 /)(x, 3 ) ( v) x,v)=y E«

,

R3 § 2

x/2(x,v)/3(x — - an,w)(nw -- v) V 0 ) d n d w . an,w)(n- ((w Standard notations are used: A particle with the center at x and the velocity v collides with a particle with the center at x - an and the velocity w. The

72

M. Lachowicz

collision kernel corresponds to the hard sphere model (see Example 2.1.1). ya represents the pair correlation function. The different ways in which one models the pair correlation function give rise to the different types of the kinetic Enskog equation found in the literature (cf. [BE5]). Here very general assumptions on the factor ya are needed: It has to be locally Lipschitz-continuous with respect to f\ and has to be regular with respect to a and such that ya(f)

= l

for

a = 0

or

/ = 0.

(2.1.9)

The precise formulation of the assumptions about the factor ya is proposed in Section 3.7. A simplified model can be obtained simply by putting ya = 1. This model is called the Boltzmann-Enskog model. The Povzner equation was introduced, as opposed to the Enskog equa­ tion, by purely mathematical purposes. The Povzner equation can be re­ garded as a modification of the Boltzmann equation which differs from the classical version because it allows for a spatial "smearing" of the process of collisions. Such an idea was first used by Morgenstern [MOl] and next, more generally, by Povzner [POl]. In spite of the fact that the latter pro­ vided some consistent justification of the model, the Povzner equation is rather often ignored by physicists. The mathematical relevance of the Povzner equation lies in that it di­ rectly corresponds to stochastic systems of particles (cf. Theorem 3.10.1). The various mathematical aspects of various versions of the Povzner equation can be found in the papers [AS3], [BE6], [BM1], [CE2], [LA5-8], [MOl], [POl] and [VOl]. Consider the following version of the Povzner equation

Df = P(fJ),

(2.1.10)

where P = P+ - P ~ , and

P

+

(/l,/2)(x,v)=y

>

|/1(x,v'(x1-x))

K3 T 3

x/2(xi,w'(xi - x ) ) # ( x i --x,w --v)dxi dw,

THE HYDRODYNAMIC LIMIT

73

P^ _ ■( /( /i1, /,/22))(x, ( x , vv)) = =J / / /// il (( xx,,v)/ v ) / i2((xi,w) x1,w) 3

3

R3 T R T3

xx5(x 5 ( x— - xXi, i , w — v)dxi dw.. v) dxi dw The post-collision velocities v' and w' are functions of the pre-collision velocities v and w as well as of the distance between the two particles xi — x. 6 is some function. Assuming (cf. [LA5,8]) that the smeared collision is possible only if the distance between two particles is less than a constant 0,2 and greater than a constant ai (where 0 < a\ < a2 < 1) one can put ,/ x . / ( XiX l- X V -- x) X) = = v V + + ((w 1 (W -- v) V) •• :, v'( X(Xi ~ l

and

Xi

-X

|xi |xi -- xx || yy |xi |xi -- xx ||

VV

('

,

\

M , X l - X, ,,

Xi - X

\

Xi

-X

(2.1.116) (2.1.116)

(oi < |x| B (I]XT, v )] xx(ai jx| < ao22),),

(2.1.12)

Xl X> X l X w'(xi x) = w f (w v) ■ ~ | ~ , w (xi x) = w (w v) ■ V |xi V |xi -x\J -x\J |xi |xi -- xx ||

5(x,v) #(x, v) -=

TT^—B V Vn , fin ai,a 2

(2.1.11a)

\\ lIxXl

//

where 47rVai)a2 is the volume of the set {x E T 3 : a\ < |x| < 02}; -B is as in Assumption 2.1.1 and x(truth) = 1, x(^se) = 0- After the change of the variables Xi —> x 4- an; where x G T 3 is fixed and a € [01,02]> n € § 2 ; one obtains a2 a2

P++Ul: ( / i ,■/ M , v=) = ^ — y | o : 2 | / 1 ( x , v ' ) / 2 ( x -1+ a n ,,W) w') 2 ) (Xx,v) P R33 ai ai R

2 §§ 2

S(n, w — v) dn da dw x .B(n, a-2

a2 Mx v)Mx

-vLII J '

i 5 " ( / i /)2/)2( X),(Vx) :, v ) = ^ - | J a 2 | / i ( x , v ) / 2 ( x - Q- nan, ) ) ww) P-(fu R 33 0.1 R 0.1

§2 §2

x B(n, £ ( n ,w w -—vv) ) ddn n dda a ddw w ,,

(2.1 (2.1.13) 1.13)

where v' and w' are given by (2.1.2). Throughout this Lecture when the Povzner equation is quoted - it means that it refers to Eq. (2.1.10) with (2.1.13).

74

M. Lachowicz

2.2.

Dimension Analysis

This Section is devoted to a short and rather formal introduction of the dimensionless forms of the Boltzmann equation. The Reader is referred to the books [CEl], [KOl] as well as [ZIl] for a physical context of the presentation. The symbols used here are specific only for this Section and they can appear with different meanings in the rest of the Lecture. Consider the following physical parameters: d - typical macroscopic length scale, r - typical macroscopic time scale, U - typical mass velocity, c - the sound speed, / - the mean free path, V - the typical particle velocity, 0 =

the mean free time, v

\i - the viscosity coefficient, Q - the typical mass density scale Moreover, let us introduce the following dimensionless parameters (cf. [ZIl]): the Knudsen number Kn = - , a the analog of the Strouhal number Sh = — , UT

the Mach number Ma = — and c the Reynolds number Re = . Small Knudsen numbers correspond to a condensated gas whereas large ones - to a flow where particles have a small number of interactions with each other. Therefore, the concept of the continuum matter in the flow field is valid only in the limit Kn-+0.

(2.2.1)

Kn —> oo

(2.2.2)

On the other hand, the limit

THE HYDRODYNAMIC LIMIT

75

corresponds to a free molecular flow (the particles have no interactions with each other). The Strouhal number is frequently assumed to be equal to 1 5/i = l .

(2.2.3)

Von Karman [VKl] pointed out that there is a relation between the Mach number, the Reynolds number and the Knudsen number Ma Kn = - — . Re

(2.2.4)

In this Lecture the limit (2.2.1) is considered in two different cases. I.) U~V~c. (2.2.5) Assuming that both Ma = 1 and (2.2.3) hold and referring all the vari­ ables to their respective typical parameters, one obtains the following dimensionless form of the Boltzmann equation

Df =

Jif f)

(22 6)

ik ' '

-

where all the variables are now assumed to be dimensionless (each symbol is, however, preserved). According to (2.2.1) the Knudsen number is a small parameter and is denoted by e. The asymptotic behavior of Eq. (2.2.6) in the limit (2.2.1) is studied in Chapter 3. II.) V ~ c and U < V . (2.2.7) Assuming that (2.2.3) holds and again referring all the variables to their respective typical parameters, one obtains the following dimen­ sionless form of the Boltzmann equation

Ma

+ v Vx/ =

% '

J(/ /}

ife ' '

(2-2 8)

-

Here both the Mach as well as the Knudsen numbers are small pa­ rameters and one can assume that Ma = Kn = e -> 0 .

(2.2.9)

The asymptotic behavior of Eq. (2.2.8) in the limit (2.2.9) is studied in Chapter 4.

76

M. Lachowicz Note that the first case corresponds to Re = - , whereas the second one £

to Re = 1. 2.3.

Hydro dynamic Equations

In several problems of gas dynamics the gas is allowed to be treated as a continuous mass of fluid (cf. [CW1]). The continuum assumption enables one to describe a fluid in terms of macroscopic, instead of microscopic parameters. As it has been pointed out in Section 2.2, if the Knudsen number is very small, the continuum assumption can be considered as a valid assumption. This Section is devoted to a presentation of some models of continuum theory - the hydrodynamic equations (historically, this branch of mechanics was concerned with water). First, consider the models which are "hydrodynamic limits" in the framework of Point I of Section 2.2. They refer to the compressible fluids. As a first model, consider the system of Euler equations for a compressible fluid (in Lecture referred as the CEE)

£« + £ £ < « * > = °.

(2.3.1a) (2.3.1a)

1= 1

3

n

d

-(Qu + J^—(euiu(gui g-t(SUj)
l

o,

jJ = 1,2,3, 1,2,3,

(2.3.16)

■*

d 1 + + T+ +euiT = (231c) T + guiTj = (2.3.1c) = 0, (.(Jr dt k (• 0+ H) t £ (^(l 6+ 2 H ) °'

a

7

7

z=l

dxi

(QUI

U

-)

where Ui is the z-th component of the vector u, i = 1,2,3. g, u and T are the macroscopic fluid parameters: g is the local density, u is the velocity and T is the temperature of the fluid. The mathematical theory for the CEE can be found in the book by Majda [MAI]. One can expect only local existence of smooth solutions to the CEE because it is known (see [SI1]) that the solutions of the CEE become singular after a finite time. This corresponds to a physical phenomenon of appearing of shocks in a fluid. The second model is the system of Navier-Stokes equations for a com-

THE HYDRODYNAMIC LIMIT

_ _ _ 77

pressible fluid (in Lecture referred as the CN-SE) 3

<9

f)

di9 + ^dx~{6Ui) l

= >o,

i=3

fa + ± &<*«*) + £ w-it 3

0

|(W)-

i=i

d

dxj

ox%

3 dx j

MT)

d

fduj (T) lw - ■{dxi g £ (MH \dxi

ieT):

"I^(Hbe; |j£j)}' )}• 2 d

2 3 2fl ((2.3.2a) - ' )

TUi) == 0

j=i 23 j = 1,2,3,

(2(2.3 326.2b) )

'''

3 +£ E

(g+£)) du. dx. : ) )

--

1(^4 Is;(«"(l -G 'G H - ) IG(^ + H) g^(-G 33

'))♦

=£

3

f)

(

=1 1=1 t=i

du, 3uj \

U

i i

, _„, dT 1

+

(T

!}■ -3 ^J ^ W/'

r+ Tr+ + 11?l

U (

dxi \

(duj UJ ; ( " j \dXi

g^r•mi:

2

9

22

++ ffUiT r j

' )

duA dxj) (2.- 12c) (232C)

where \iy and ^jjj are the coefficients of viscosity and that of heat conduc­ tion, respectively (cf. [CE1], [FE1] and [KA1]); e is the small parameter introduced in I of Section 2.2. For the mathematical theory for the C N - S E see [KA1], [MN1,2], [VZ1] and the references therein. The classical Chapman-Enskog procedure (cf. [CE1], [CCl] and [FE1]), at the formal level, gives the possibility of a refinement of the C N - S E by adding terms with a higher power of e. This, however, leads to the Burnett and super-Burnett equations, which are found to be less accurate than the CN-SE both from the physical [FI1] and the mathematical [BOl] points of view. For explicit formulation of the Burnett equations see [BT1] and

[FOl]. Consider now models for incompressible fluids which are "hydrodynamic limits" in the framework of Point II of Section 2.2.

78

.

M. Lachowicz

The following model, in Lecture referred as IEE, is the Euler equation for an incompressible fluid 3

Q0

(2.3.3a)

5^77— ii» = 0, T** 2 = 1 OXill i=l

dr\

A3

dr\

dr\ dxj

1 2 3 j j== 1,2,3,

^ + £ " ^ = -^7>2= 1 2= 1

'''

J

(2(2.3.36) 3 36)

--

where p corresponds to the pressure and is considered as an unknown function determined from (2.3.3). The Navier-Stokes equation, in the Lecture referred as the IN-SE, reads d V J2£u 0, i =n 2=1

d u + J-* d Ui Uj

=

ai ' ^ d^

*

- a ^ p + ,i'y^mUs' J J

2= 1 2= 1

(2.3.4a) (2.3.4a)

2=1 2=1

J= 3 46 i =1 2l , 23 , 3 , 2 (2.3.46)

' ' ' (-- )

l l

where \i*v is the coefficient of kinematic viscosity and is a given constant (cf. [DM1], [BA2]). For smooth initial data there exists a smooth unique local solution (u(£,x),p(£,x)) of the IN-SE, whereas for small initial data or in two dimensions - the solution is global. Moreover, there exists a global weak solution (for the existence theory see [LEI] and [LD1]). The IN-SE corresponds to a fluid with a constant density go > 0 and a constant temperature To > 0. However the IN-SE can be completed by the Boussinesq relation

( + r)

4 * -°'

— (e + T) = 0,

j3 = 1,2,3, 1,2,3,

(2.3.5a)

as well as the equation

fr + t^-Atfy. liT

+

t^

2=1

= A±^T,

2= 1

l

'

(2.3.56) (2.3.56)

THE HYDRODYNAMIC LIMIT

79

for the density and the temperature fluctuations (around the constant states £o and To): g and T, respectively; where /i*H is a given constant parameter (see [BA2]). Finally consider the system of the Stokes equation, referred as the SE 3

Q

u

X ^ * = °> zi = il

d dtu>

, i, ^ da22

dd

=

J •> ■»

a T_

3 = 1,2,3,

l »

ii = ==lll 9

,« VA a^2

T

(2.3.66)

(2.3.^ (2.3.6c)

xl

i=l 1=1

2.4.

(2.3.6a) (2.3.6a)

%

Notations

Throughout the paper, ma is the following function Q 2 |v|2 )r/ m (v) = (i (l + |v| / , mQa {v)

2

2

1 aaeR € R. l .

(2.4.1)

L P (E 3 ) is the Lebesgue space of measurable, real-valued functions whose p-th power is integrable on R3; 1 < p < -l-oo. The norm in L P (E 3 ) is denoted by || -)LP(R3)\\. The inner product in L 2 (E 3 ) is denoted by ('.O^m 3 )Let W^3, Wj3 WJ*XB* be strictly positive, smooth functions on K , T and T 3 x I 3 , respectively. B ^ W ^ a ) denotes the space of continuous, realvalued functions on E 3 with the norm i/;Boo(WRR»)|| ||/;Bc»(w .)| = s u p | W RRs. / | . 3 R m Let B ^ = Boc(mQ) and B^(W R 3 ) = B O O K ^ R S ) . C fc (T 3 , WTs) is the space of the functions which are continuous together with all their derivatives of orders |^y| < k and equipped with the norm 3 ||/;C*(T ;WvT) 3)|| f\Ck{Y*\W

=

sup

o<|-r|
xeT

Q\l\f W^^" 1 dx^ OX'

80

M. Lachowicz

Naturally, C*(T 3 ) = C*(T 3 ; 1). L P (T 3 ) is (for 1 < p < oo) the space measurable, real valued function whose p-th power is integrable on T 3 and with the norm | •; L P (T 3 )||. In addition, consider the following spaces consisting of functions on 3 T x E3: 5£oo oo equipped with the norm fc fc •;C (T33 )||);BS,| )||);B^ , , =11(11 I ■• ||a0'0ifoo = || (li •;e'er ,00 jl V ||

^oo,g equipped with the norm T3 );BS,| (| •; £,(1^)11 II i ~aoo,q , , = IKIh-M )|);B~ll.,

Xpi^Q equipped with the norm 3 3 3 3 || • | | pp,oo ; ^ = ||(||-;C"=(T )||);i (R LPp(R )\\)||,, ( -;C*(T )||);

XPiq equipped with the norm 3 3 I ■||p,q (R3)\\ )||.. )|); LP(R ( •;i,(T )|);L P., =~ ||(|-;£,(T

Consider the following assumption Assumption 2.4.1. Let Q, U, and T he the smooth Q ': Q

functions

3 [0, tt00]] xxTT 3- -> }I!11, , [0,

3 3 u : [0,«o] [0,t 0 ] x TT 3 -->> E 3,,

and 3 1 T :: [0,t [0,t0o]] x TT3 -->+ EE 1, ,

which can also depend on the parameter e e]0,e o ]; for some t0 e]0, +oo[ and some EQ €]0, +OO[; and are such that i) Q, and T satisfy g(t,x) 0 ( t , x )>>cci i >>O0,,

r(t,x) r ( i , x ) >c > c 2 >>0o, ,

Vt€[0,tb]i Vt€[0,tb]i

where c\ and C2 are constants (independent of a), and

VVxxeeTT33,,

THE HYDRODYNAMIC LIMIT

81

ii) Q, u, T together with their derivatives (up to some - sufficiently high order) are bounded independently of e G]0,£o]Now let M = M[Q, U , T ] , where Q, U and T are as in Assumption 2.4.1. Moreover, let M 0 = M\t=Q = M M t = 0 , u | t = 0 , T | t = 0 ] . A simple consequence of Assumption 2.4.1 is the existence, for each a G M1, of positive constants c~, c+, c_ and c+ - independent of e and such that V V *tG € [[0,t ( M0o],] ,Vx VxG € T 3 ,, Vv VvG GRE33, , (2.4.2) where u- and a;+ are the global Maxwellians C C"CJ_(V) Q CJ_(V)

<mma a( (vv))M A (f (*i , x , vv) < c+w+(v), < )
u- = = M[1, M [ l , 00,c_], ,c_], u;_ 0,c+]. u u+ =M[1, M[l,0,c + = +], Let YQ,A: and Y",A: be the spaces equipped with the norms: a S C ^ M - ^0 -*)|)jBS I D J I C I) ||,, I •.||J-* = ||(|. C*(T ;M i ?* =I ( 3 t III-• | | +| ^'* — = IIII||(||-;C*(T )||);B^(a.; i )||, v ■JC*(I»)|);B£,(«;*)| , Q

—

respectively. For fixed t G [0, t0] and x G T 3 consider the space L2{M] equipped with the norm - M - * ; L 22 (K ( E 3 )|| )|| | / ; Li 22 { M M}}| | == | | //•M-^;L and with the inner product = ((A / l ••Af" M} = ^ _ 1 ' / ?2/ )2L) L2 (2 (K R 33 )) • ( / iI ,»/ /22))LL 22{{M}

Analogously, for M 0 instead of M, for fixed x G T 3 , define the function space L2{Mo}. Now define the "hydrodynamic" and "nonhydrodynamic" subsets in L2{M} N , . •. .- , 4 } , M = \m{M*Pi lin{MV>i : i%==00,.

M. Lachowicz

82 where i/'o, • • •, 4>A

are

the collision invariants (cf. (2.1.56)), and

U = ML = {f e L2{M}

: (/,M4>i) Li{M} = 0; i = 0 , . . . , 4 } .

Analogously, define the "hydrodynamic" JVo and "nonhydrodynamic" 7?o subsets in Z^iM)}Define in L2{M} the projection operators V and "P 1 = 1 — V onto TV and 71, respectively. Moreover, in L2{M} - the projection operators VQ and VQ- = \ -VQ onto No and 7£0, respectively. Consider the following operator £ = 2J(M,«)

in

L2{M}.

It is well known ([CE1], [FE1] and [TR1]) that ker£ = ker£* =N.

(2.4.3)

The operator £ can be split into "regular" and "singular" parts [GR3] Cf = )Cf-vf,

(2.4.4)

where v = v{M) is the same as it was defined in Section 2.1. Consider also the operator L L/ = 2M-2j(M,M5/)

L2(R3).

in

Then kerL = kerL* = lin { M ^ : i = 0 , . . . , 4 } C L 2 (E 3 )

(2.4.5)

Lf = Kf-uf.

(2.4.6)

and

Moreover, introduce the following ^-depending two-parameter family of op­ erators t

(£4(Mi)/)(x,v) =/(x-(i-t 1 )v,v)exp(-iy i/(*2,x-(*-t2)v,v)dt2) (2.4.7)

THE HYDRODYNAMIC LIMIT

83

and the operator t

(V£ef)(t) mt) = ^Ju±Ju )f(t11)f(t )dt11)dt . 1. e(t,tel(t,t

(2.4.8) (2.4.8)

0o

Analogously, let CQ be the following operator (2.4.9)

Caa/f = = ^o/ £af-Va-f, £ - ^a * / , where

an,w')/(x, (JC f f {M(x ++ ,an, w')/(x, v') v')/(x + an, ,/Ca .a/f)(x, <m, w') v') + + M(x,V)/(x ) ( « , v) = //{M(x R3 § 2

w)} (n • (w - MM( (xx,,vv ) / ((xx - an, an,w)} (w-- v) v) VV0)dndw 0) dndw and - an,w)(n an, w)(n • ((w i/ v) VV00) )ddnnddww. . w --- v) x , V ) == / / MM(x «*cia((x,v) ( X / / R3 § 2

Let LLa/ = = Kaf-v f -a-f,va af

■

/

(2.4. 10) (2.4.10)

,

where ;re i

,

1

Kaaff ==--M'M-*lCa(M*f). - * * . i(itf*/)Note that for a = 0 the operators Co, /Co, ^o, LQ and K0 coincide with £, K,, v, L and K, respectively, with B given by (2.1.3). Let (Z4, e (t,*l)/)(X, (Mi)/)(x,v) x - (t - *i)v, *!)v,v) (Wa, V) = / ((X V) /

\

t

xxexp exp l - i- / zi// aa (*2)X( t 2 , x - ((t*- -* 2*2)v,v)dt ) v , v ) d t 22 I

(2.4.11)

and t

(VOlC (V /)(0 = = ~Ju j / z 4a,,£(t,t *(M i )1/)dt ( *1i. ) d * i . fl,e/)W 1)f(t 0

(2.4.12)

84

M. Lachowicz

Finally, write / ## ((£t , x ,,vv)) == // ( * , x + **v,v). v,v).

(2.4.13)

2.5 Basic Estimates Consider the following extensions of Grad's estimations [GR3,4]. Lemma 2.5.1. ([LA7] - Lemma 5.1)

< -Co(/,/)L -co(/,I)L22{Af> {M } >, <

(/,£I)L (f,£f)L 2{ M} 2{M}

v/eft, v/ e ft,

(2.5.1)

| / ; L 2 {M}|| { M } | | ,, V/Gft, ICr -1^f;L j ^2{M}\\ l M } ! <0;

of t G [0, to]j

(2.5.2) x

6 T 3 and

I C-ifiBZol™-*)] r- 1 /,!*^™-*)! << cc*a | | /,B»(Af-*)| / , l i ( M - ^ ) | | ,, V V/GRflBS,, /GRflBS,, for each a > 2, where cQ is a constant independent oftE £ >0.

[0, to] >

(2.5.3) x

€ T 3 and

The following inequality holds (cf. [BE3] - (A.ll)) c_niA c_m.\ < v < cc+m\ + m>

(2.5.4) (2.5.4

where A G [0,1] depends on a particle interaction potential (see Assumption 2.1.1) and c_, c + are positive (> 0) constants independent of t G [0,£o], x € T 3 and £ > 0. The operator L is non-positive in L 2 (R 3 ), cf. [CE1] ( // >^£/ /))LL 22 ( R R33)) <<005 ,VV/ /G G L 2L(2I(IR R 3 3))..

(2.5.5)

The operator K has the form ([PA2] - (2.16) and [BE3] [BE3] -- (A.21)) (Kf)(tt x, v) = (Kf)(t, = f/ k{t, x, v, w w)/(t, ) / ( t , x, w) dw ,

(2.5.6a)

R3

where |Jc(t,x,v,w)| < const/c |Jc(£,x,v,w)| const ]q_ (v,w) + ++(v,w)

(2.5.66)

THE HYDRODYNAMIC LIMIT

85

and *++(v,w) = | v -- w |

x

(

d v | 2 _ | W | 22 )A22\\

c(|vp-|w|

exp | —c|v - c | v --wp w| - c ^ 2

^ i -2 i |v — wl / z |v — w\ )

(2.5.6c)

for some constant c > 0. The following inequalities hold (cf. [PA2] and [LA2]):

| /KfiB^M'i) C / ; 1 ^ ( M - ^ ) | | < const||/;L const /;L 22{M}|| {M}| ,

(2.5.7)

Kf\\QJt00< const] f 2 £ , l*/|K~<«»wtI/h£,

(2.5.8) (2.5.8)

|/C/,B£,(Af-*)|| /C/,B£,(AT*) < cconst o n s t | //,-BB«^(JM I f--^*)) || ,

(2.5.9)

| |/C/ £ / | | +£** < const | / /| | | «'•*, *''\

(2.5.10)

| |Kf ^ / | ^Sfoo o o <
(2.5.11)

for a = 0 , 1 , 2 , . . . and k = 0,1,2,...; where a' = =00 for a = 0 a ' == a — — 11 for for Qa== 1,2,.. 1,2,..., a' • )

—

dx^y

<< const m m22\|y7\M, |M, <

(2.5.12)

0|7lj

——r < const m\ . (2.5.13) 7 <9^ arc In general, the operator J has a singular character with respect to the velocity variable v, but this singularity "is not too large" \lJU1J2) II Afuh)\\\

KkKk < < const||| const h AI o ^ i| |l A ro~ M l «•*, o"* .

|H|I| J(h,h)l%~ W i , / 2 ) | rKkA ' * < c
(2.5.14) (2-5.14) (2.5.15) (2.5.15)

k = 0,1,2,...; where A is as in Assumption 2.1.1. 2.1.1. for a -== 0,1,2,...; fc

M. Lachowicz

86

The function k++, defined by (2.5.6c), satisfies [CA3] the following esti­ mate / |v - w| a ]q_ + (v, w)dw < constm_i(v),

(2.5.16)

R3

for a > - 2 . The following classical inequalities ([HA1] - Theorems 200 - 202) hold:

(/(E/iWW $ * £ (Jff(x)dx)\ 3

(£(//,(*)dx)

(2.5.17)

3

P

)'
3

(2.5.18)

3

( / ( / f{x,y)&y)Pte)P < I (j

f*(x,y)dxydy

(2.5.19)

for non-negative functions and p > 1.

3.

3.1.

Compressible Dynamics

Singularly Perturbed Boltzmann Equation

Denote the Knudsen number Kn by e and assume e « l . Consider the singularly perturbed problem (cf. [LOl], [OM1] and [VA1,2]) for the Boltz­ mann equation: Df = -J(fJ),

(3.1.1a)

£

= F.

(3.1.16)

t=o In general, the initial data F may depend on e, but as long as such a dependence is regular, it does not cause any difficulties and will not be considered in this Chapter. Grad [GR3] considered the singularly perturbed problem for the "semilinear" Boltzmann equation locally in time. For the full nonlinear Boltz­ mann equation Nishida [Nil] as well as Ukai and Asano [UK2] proved that for the initial data F - sufficiently close (in the sense of norm) to a global

THE HYDRODYNAMIC LIMIT

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Maxwellian u and analytic in the space variable x, a strong (even classical) solution of the Boltzmann equation exists and is unique in a time interval [0, t0] (where t0 is independent of e) and the solution converges, as e \, 0, to the local Maxwellian M whose fluid-dynamical parameters solve the CEE. Obviously the strong solution / converges to the initial data F as t± 0: / tio

e4.o

► M Uo

Wo On the other hand, if there exists M0 - a strong limit of M as t \. 0, then it has to be a Maxwellian. Therefore, in general, because of the initial data F may not be a Maxwellian, the convergence for e \. 0 is not uniform in the variable t near t = 0. This singular behavior corresponds to the initial layer effect (cf. [VA1]). Ukai and Asano [UK2] proved that the necessary and sufficient condition for a uniform convergence is that the initial data function F is itself a Maxwellian. The structure of singularity resulting from the initial layer was studied in [UK3]. The conceptually reverse result was proposed by Caflisch [CA2] (see also [CA4]). Following the previous paper [CA1], concerned with the Broadwell model of the Boltzmann equation, he deduced an existence and asymptotic behavior theorem for the Boltzmann equation (3.1.1) from the hypothesis on the existence of smooth solutions to the CEE. He obtained the existence and asymptotic relation theorem for the one-dimensional problem (3.1.1) with initial data F in the form of a Maxwellian. The solution exists and the asymptotic relation holds as far as a smooth solution of the CEE exists. The form of initial data (a Maxwellian) naturally eliminate the initial layer. In paper [LA3] the initial layer was included and the many-dimensional problem was regarded. The main theorem in [LA3] contains the following result (cf. Section 3.6): for e and the nonhydrodynamic part of the initial data F - sufficiently small, a solution of the problem (3.1.1) exists on a macroscopic time - interval as long as a smooth solution of the C E E exists and this Boltzmann solution is well approximated by the CEE solution. As in Caflisch's case the Boltzmann solution is unique if the CEE solution is unique. Moreover the solution is continuously differentiable. The analysis of

M. Lachowicz

88

the smoothness of the solution is based on the paper [LA2]. The result of the paper [LA3] was generalized in [LAI], where the generalization concerns the condition on the initial data: the spatial derivatives of the nonhydrodynamic part of initial data were not assumed to be small in [LAI]. The similar problem was studied in [MSI]. In [LA6] the analysis of the asymptotic behavior of solution to the prob­ lem (3.1.1) was continued and, in contrast to the previous studies in the Sobolev spaces (with respect to the space variable x), the analysis was car­ ried out in the C° - space setting (see Section 3.5). The basis for the papers [LAI,3,6] and [MSI] was the Hilbert procedure (cf. Section 3.2) which results in the CEE. However, instead of the Hilbert procedure, the modified procedure (cf. Section 3.3), proposed by Caflisch [CA5], can be used. This modified procedure starts with the C N - S E . This was the basis for the successive step in the mathematical description of relations between the kinetic theory and the continuous fluid dynamic on the level of the C N - S E ([LA7]). Caflisch's attempt can be modified in such a way that it results in the Burnett-type equations or even in abstract "hydrodynamic" equations. This "kinetic nonuniqueness of the CN-SE" is discussed in Section 3.3. The asymptotic theory for various kinetic equations (the Enskog equa­ tion, the Povzner equation, the BBGKY-type hierarchy of equation) was initiated in the paper [LA4] and continuated in the papers [LA5,6] (at the level of the C E E ) , [LA7] (at the level of the C N - S E ) as well as [LA8,9,10] (both) - see Sections 3.7 - 3.10. The asymptotic relationships of the solution of the problem (3.1.1) and the solution of the C S - S E in the limits t -> +co, for fixed e > 0, was in­ vestigated in the paper [KA1], for the initial data F - sufficiently close to the global Maxwellian u>. 3.2.

Hilbert Procedure

Although the problem (3.1.1), for £ | 0, is a singularly perturbed problem, Hilbert [HI] assumed a solution in the regular form of a (formal) power series

/m=xy/ ( i ) w

(3.2.i)

j=0

(cf. [CA1-4], [CE1], [CC1], [FE1], [GR1,2], [LA3], [TR1]). Substituting (3.2.1) into the Boltzmann equation (3.1.1a), expressing each side of the resulting equation as a power series in e, one obtains the

THE HYDRODYNAMIC LIMIT

89

Hilbert procedure equations: J(/<°\/<°>) = 0,

(3.2.2a)

2J(/(1),/(0))=£>/(0),

(3.2.26)

i-i

2J(/W,/(°))=D/t*-i)-^J(/»|/Ci-0)

for

j = 2 , 3 , . . . . (3.2.2c)

FVom (3.2.2a) it follows (cf. [CRl] and [TRl - Chap. VIII]) that /<°) is a Maxwellian /<°> = A % u , T ] .

(3.2.3)

Assume that the fluid-dynamical parameters g, u, T are known and are such as in Assumption 2.4.1. C = J(M, •) is a linear integral operator and thus Eq. (3.2.26) is a linear integral equation for the function / W . Analogously, for given functions / ( ° ) , . . . , / ^ _ 1 \ j = 2 , 3 , . . . the right-hand side of (3.2.2c) is a known function and (3.2.2c) is a linear integral equation for /(■". Therefore the problem of determining the Hilbert expansion terms f(J) (j = 1,2,...) is one of solving an iterative system of linear integral equations. In virtue of Predholm's Theory, the integral equations (3.2.26) and (3.2.2c) can be solved, provided that the right-hand sides are orthogonal to Af (in L2{M}). The solvability condition for (3.2.26) (with (3.2.3)) i.e. VDM = 0

(3.2.4)

is the C E E for the fluid-dynamical parameters g, u, T of the Maxwellian M. Split now every / k ' ) , j = 1,2,3 . . . , into two parts fU) - gU) + hU) t

(3.2.5)

where gW and h^ are the nonhydrodynamic and the hydrodynamic com­ ponents of the j-th term of the Hilbert expansion, i.e. gw

e n,

h{j) e N.

90

M. Lachowicz

Prom (3.2.26), (3.2.3) and (3.2.4) one has g(i)

1 1 DM. =c~ DM. =c~

(3.2.6a)

Then the solvability condition for (3.2.2c), j = 2 is {l) VDh{l) VDh

{1) {l) = -VDg . . -VDg

(3.2.7a)

Analogously p / O ' - U _-£./(/«>,/«-")) £ J(/ ,/°'- )) »«) =£-» /(op-* (i)

gU)=£-i

i)

(3.2.66)

and VDh VDh^{j)

= -VDg -VDgU){j),

(3.2.76)

,

f o r ;i = 2 , 3 , . . . . for System (3.2.7a) is the linear nonhomogeneous symmetric hyperbolic sys­ tem for ( ^ W J U ^ J T W ) - the linearized C E E - linearized around the solu­ tion of the C E E (a, u, T). The nonhomogeneous term {l) -VDgM -VDg

involves second derivatives conductive terms from the the linearized C E E (again system) for (g^2\u^ ,T^) -VDgW -VDgW

l l = -VDC~ -VDC~ DM DM

of a, u, T and is precisely the viscous and heatC N - S E (cf. [CA5]). Next (3.2.76) for j - 2 is a linear nonhomogeneous symmetric hyperbolic with the nonhomogeneous term

l1 ±l 1 l± 1x 2± 2 2 = - VD{C-VD(£~VPD)V D)MD)MM -VDCV^Dh^ --VDC- VDC~ V^Dh{l) V Dh^ 1 {1)(1)l +VDC~lJ{g^+h( +VDCJ(g(9 \gM+h^), gM+hV + hP} ) , + *«", /> (1 \,g™ ■ » ) ,

where the first term on the right-hand side is the third order term in g, u, T of the Burnett system, the second term is the linearized Navier-Stokes term in g^\ uS1', T^ and the third term is the nonlinearity leaving out of the linearized CEE for a (1) , u^, T^ (cf. [CA5]). The next equation (3.2.76), j = 3 , 4 , . . . , has an analogous structure. The initial data for System (3.2.7), j = 1,2,3..., will be specified in Section 3.4 by the conditions of the decay of the initial layer solutions.

THE HYDRODYNAMIC LIMIT

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In all that follows the truncated (up to the order n) Hilbert expansion is considered. Nevertheless, the solvability condition for /( n + 1 ) (i.e. (3.2.76) with j = n) is preserved. By the general theory of symmetric hyperbolic systems and classical Grad's estimates [GR3] one has Lemma 3.2.1. ([LA3]) Assume that there exists a solution (g,u,T) of the CEE such that Assumption 2.4.1 is satisfied for some t0 > 0. Then the Hilbert expansion terms / ( 0 ) , / ( 1 \ . . . , f^ exist and are smooth with respect to both t G [0, t0] and x G T 3 variables. Moreover, one has d*+W dWd^f

mot

,n

i

< constM 2 , <™nstM*,

(3.2.8)

for all t G [0, t0], x G T 3 , v G IR3 and for each a >>00, ,

k>0,

|M > 00 ,, 7 |>

jjeG{{o0,,.. . . , n }} ,,

where the constant denoted by "const" depends only on a, k, \j\ and j . The arguments presented in Section 3.1 show that the Hilbert procedure cannot give an accurate approximation of the solution of the Boltzmann equation in the initial layer. On the other hand, the Hilbert procedure is not accurate either for the description of shocks, boundary layer or long time behavior (cf. the discussion in [CA5]). The asymptotic relations of the Broadwell model of the Boltzmann equation with its corresponding C E E in the presence of shocks were studied by Xin [XI]. He showed that the hydrodynamic approximation is valid even if there are shocks, although there are thin shock layers in which the approximation does not hold. More precisely, by assuming that the hydrodynamic solution is piecewise smooth with a finite number of noninteracting shocks and suitably small oscillations, he proved that there exist solutions to the Broadwell equations such that the Broadwell solutions converge to the fluid dynamical solutions away from the shocks at a rate of order e, as e tends to zero.

92

3.3.

M. Lachowicz

Modified Hilbert Expansion

Some of the limitations of the Hilbert procedure, mentioned in the end of Section 3.2, caused by the fact that it yields the CEE, could be removed by the approximation by the C N - S E . The CEE can also be obtained by the Chapman-Enskog procedure (cf. [CA5], [CE1J, [CC1], [GR1,5], [FE1]) at leading order, but at next (first) order the C N - S E is yielded. Then the systems of Burnett (at the second order) and of super-Burnett equations result. Resulting at higher order in the systems of Burnett equation can be considered as a disadvantage of the Chapman-Enskog procedure because these systems are found to be less ac­ curate than the C N - S E (cf. [FI1], and [BOl]). On the other hand, up to now, the Chapman-Enskog procedure has no adequate rigorous mathemat­ ical foundation, which the Hilbert procedure has, except for the Carleman model of the Boltzmann equation [PA1]. The modified Hilbert expansion proposed by Caflisch [CA5] is more adopted to the mathematical purposes. It is obtained from the Hilbert expansion by a rearrangement of terms and thus its nature is different from the Chapman-Enskog procedure. The equations resulting from the modified expansion are first the C N - S E and thereafter the linearized C N - S E . Similarly as in the Hilbert procedure, the solution of the problem (3.1.1) is assumed in the form of a power series with respect to e

/(0 = $ > 7 ( i ) ( 0 However now, in contrast to the Hilbert expansion, the terms f^ assumed to be e-independent. Decompose each f^ in the same way as in (3.2.5) fti) - gU) + hU) >

(3.3.1a) are not

(3.3.16)

where gW G 11 and h® E M. Naturally / ( 0 ) has to satisfy Eq. (3.2.2a) and therefore /(°)=M[0,u,r] with fluid-dynamical parameters g, u, T.

(3.3.2)

THE HYDRODYNAMIC LIMIT

93

Assume that (g, u,T) is known and such that Assumption 2.4.1 is satisfied. Inserting now (3.3.1) with (3.3.2) into (3.1.1) and decomposing the term g^1' into gM = g' + p", one can obtain the following set of equations: ± 2J(M,p') = V±VDM 2J(M,g') DM, }

(3.3.3a)

VDM = -eVDg', -eVDg',

(3.3.4a)

L 2J(M,g") =eVDh^ 2J(M, g") = eV^Dh^

,,

(3.3.36)

VDh{1) = -VDg" -VDg",,

(3.3.46)

and ± Dh^

±

2J(M,gW) 2J(M =sVsV i9W)-- =

± DgU-V

1_

j-i

( 0 {ijU-i)) Dh^ +V + V Dg^- ^ -£ J2j ( /JU \l{J~}i]) > ±

U) {j) {j) VDhiJ) = -VDg -VDg

(3.3.3c)

(3.3.4c)

for j = 2 , 3 , . . . . In virtue of the Fredholm theory the integral equations (3.3.3) can be solved in I/ 2 {M}: ' = £-l(V £-±1DM), (VJLDM)J gg'

(3.3.5a)

C~1l [eVL(sV DhL{l)Dh^) ^j g" = C'

(3.3.56)

and

and

g& =C-11 ( e±P ± I > ^ + 7 >L - L 2 ? ^ - 1 > - y ) j ( / ( i ) l / t i - < ) ) ] g& = C" UV DhW V

+V DgV-V

-f^JifWjV-*)) i=i

)

(3.3.5c) (3.3.5c)

for . 7 = 2 , 3 , . . . . for jNow = 2(3.3.4a) , 3 , . . . . leads to the equation Now (3.3.4a) leads to the equation l ±

l ± = -eVDC~ -eVD£-lV -eVD£V V^DM DM, DM, VDM = VDM

,

(3.3.6)

which is precisely the C N - S E for g, u, T (cf. [CA5], [CE1], [FE1], [KA1]).

94 .

.

M. Lachowicz

Equations (3.3.46 - c) lead to l) 1 ± l {1) VDh{l)l) = -eVDjCVDhS -eVDC~ V DhV^DhS , ,

(3.3.7a)

and {j) VDhiJ) VDh = - eVDCeVDC-11VV±LDh^ Dh<<j)

VDCVDC-11VV±±DgiJu~-1)

-

i + VD£-1( fcj(f( + VD£~ £ j ( / \fWy\ ,/ '-°) l

( 0

0

,

] ,

(3.3.76) (3-3.76)

j = 2,3,... which are linearized C N - S E . The nonhomogeneous terms can be recognized as Burnett and superBurnett terms as well as higher order nonlinear terms. Prom the mathematical point of view the present procedure needs more caution than the Hilbert one for e-dependence of the expression terms. However, because of Lemma 2.5.1, one conclude that none of the operations described in this Section causes any singularity with respect to e. Therefore, one has: Lemma 3.3.1. ([LA7]) Assume that there exists a solution (Q,U,T) of the CN—SE such that Assumption 2.4.1 is satisfied for some to > 0. Then there exist /<°> as in (3.3.2) and solutions /<*>,..., f^ (/(*) = g& + J#>, j ' G { l , . . . , n } ) of Eqs (3.3.3), (3.3.4) sufficiently smooth with respect to t G [0, t0] and x G T 3 and such that

0*+M

ma m _.

Q

i

p3> o n s t M 2 ,, f^ <
(3.3.8) (3.3.8)

for all t G [0,*0], x G T 3 , v G R3, e G]0,eo] and for each

a >>00, ,

kfc>0, > 0,

|W 7 l>>o0 ,

jJ€G{{00, ,ll,,.. . . , n } ,,

where the constant denoted hy "const" depends only on a, k, \j\ and j . The idea of rearrangement of terms, which is the starting point of the procedure presented in this Section, leads to the conclusion that a system

THE HYDRODYNAMIC LIMIT

95

different from those of the CEE and C N - S E can be used to establish hydrodynamic approximation of the Boltzmann equation. Using the expansion equations ± 2 J(M, g') = V^DM + VV±T[M] T[M]

(3.3.9a)

and 2 J(M, g") = eV±±Dh 2J(M,g") Dh^{1)

- VV±±T[M] Jr[M]

(3.3.96)

instead of Eqs. (3.3.3a) and (3.3.36) the asymptotic theory can be formulated, starting rather from the following "hydrodynamic" equation than (3.3.6) VDM = -eVDC'1VDM V±LDM

1 ± 1 -- £VDC~ V T[M], eVDC^V -T[M],

where T is an operator. For example, the case T[M] — -DM mation. Whereas

(3.3.10)

goes back to the CEE approxi-

j DM --eJ{C~ 1 ± T[M] == eDC~1- Vi p±DM T[M]-- eJ(C~lV^DM, V DM,C

1 1± ± C~ DM) V VDM)

gives the Burnett system approximation. On the other hand particular choices of T result in the possibility of starting from abstract systems of equations in which added terms are of the same order (with respect to e) as the viscous and heat-conductive terms in the C N - S E . Therefore, from the point of view of the kinetic theory, the CN—SE can be considered as only one of many possible refinements of the CEE. This "kinetic nonuniqueness of the CN-SE" was described by Ellis and Pinsky [EL3] in the linearized case. 3.4. Initial Layer Expansion It is very well known, see [VA1], that a bulk expansion, like those presented in Sections 3.2 and 3.3, has to be completed by the initial layer expansion as in Section 4 of the paper [LA3]

/ « = £/ W ) w+ £/<%*>)■ 3=0

j=0 .7=0

(3.4.1a)

96

M. Lachowicz

where d = - is the "stretched" time variable, fW are the bulk expansion £

terms defined in Section 3.2 or Section 3.3 and fw) are the initial layer terms. The latter are decomposed fU) =gU) i « )?| /0) = gU) ++ ft(J)

(3.4.16) (3.4.16)

where jtf) <= 7 ^ (j = 0,1,...), &<*> G Af0 (j = 1,2,...) andfc(°)= 0. The bulk expansion terms are expanded in a power series in tf fc-i fc-l (j) j i) A W,fc) /fW(e#) ( ^ ) = ^ ee7 (V " ' ^ (W v) + + ee*A We). V."* (#,«),

(3.4.1c)

i=0

where

'^""<* r) ' ^-1 ( Tc L

t=oJ

and

•#•*•>'"■"<•'« -£5((T¥ L ) . ) ,

\

" -

■ » - " /

C€]0,1[. Then the representation (3. 4.1) leads to the initial layer • equations (cf. [LA3]) Then the representation (3.4.1) leads to the initial layer equations (cf. [LA3]) 8/to) = j ( /(o)/(o) ) + 2 / ( M o ] / - ( o ) ) ) (3.4.2a) ^ where G = F --Mo, where G = F - M 0 ,

= J(f{0)J{0)) /(0)

+ 2J(Jlf 0 ,/ ( 0 ) ),

(3.4.26)

tf=0

= G,

(3.4.26)

dhW M(i) 1 - _ = -■ « pn0t(vv'.V Vxx //^^- 1)))) ,, /fc« i(i)

0=0

(3.4.2a)

(3.4.3a) (3.4.3a)

=ffW,

(3.4.36) (3.4.36)

(i -J r ==2J(M J ( / W , ^ ) ) ++ £ = 2J(M00 + + /(°),5 / W , p ^>)) + +2 2J(/(°),/iO)) J2 J(/(0,/W) J(f{i)J{k)) i,fc>i

THE HYDRODYNAMIC LIMIT

+ £

97

2J(/( i ),/( fc '^)-^ d L (v.V x /^- 1 )),

(3.4.4a)

t,fc,fc'>0

i + k+k'=j

gV) tf=0

=

(3.4.46)

-9U>0)

for j = 1,2,..., where H^ has to be specified. It is determined by the solutions of Eqs. (3.2.6) (for the approximation on the level of the CEE) or (3.3.3) (on the level of the CN-SE) at t = 0 as in (3.4.46). Note that in order to reduce singularities with respect to e the initial layer terms must decay rapidly (exponentially) with d —> oo. The zeroth order equation (3.4.2a) is the spatially uniform nonlinear Boltzmann equa­ tion (the space variable x is only a parameter). Note that all its terms as well as the initial data (3.4.26) are independent of e. Despite the fact that large initial data results are available ([AR4], [DE1-2]), here an exponen­ tially decaying solution is constructed for the initial data G satisfying the smallness condition G

4,0 < K 0 -

(3.4.5)

where K is a critical constant independent of e (cf. [LA1,3,6]). Next, the initial data H^ in (3.4.36); j = 1,2,... ,n; have to be specified such that the initial layer terms h^\ j = 1,2,..., n; vanish exponentially. The following (obvious) lemma can be used: Lemma 3.4.1. Let (3.4.6)

suplUe^HWlllo^ < const i9>0

for some a > 0, a > 0 and k>0.

Then the solution of the problem dh

= -n

(3.4.7a)

+oo

h tf=o

= In*)dti

(3.4.76)

J

satisfies sup|||e a1? /i(^)|||o' fc < const i?>0

(3.4.8)

98 _ _ ^

M. Lachowicz

Hence the initial data H^; j = 1, 2 , . . . , n; are chosen as follows +oo

j p 0 ( v ■ V x / ^ - 1 ' ) dtf.

JffCi) -

(3.4.9) (3.4.9)

o 0 This specifies the initial data for Systems (3.2.7) (on the level of the CEE) or (3.3.4) (on the level of the C N - S E )

kfcO) «

=-h{j)

i?=0 1?=0

t=0

=-H{j),

ij = ll , 2 , . . .., ,nn..

(3.4.10)

Therefore, the following lemma can be formulated Lemma 3.4.2. ([LA3]) Let G satisfy the smallness condition (3.4.5) and G G 7£o DYQ' with a and k being large enough. Then there exist unique so­ lutions / ( 0 ) , . . . , / ( n ) of the initial layer equations (3.4.2), (3.4.3) and (3.4.4) with (3.4.9) and numbers otj, hj, o3 > 0 such that (i) € C 11 ( [[00ff++oooo[[;;YY??F i' * i' ) /W

(3.4.11)

( i ) (ti)l\ £'•*' < const sup I e't* sup e a ' Vft) Will ?'**'' < const

(3.4.12)

and tf>0

for j j= for = 00,.. , . . .., ,n. n. 3.5.

Equation for the Remainder

Let / ( 0 ) = M[Q,U,T], f(l\ . . . , /( n ) be the bulk expansion terms given by Lemma 3.2.1 (for (a, u, T) being a solution of the C E E ) or by Lemma 3.3.1 (for ( a , u , T ) being a solution of the C N - S E ) . Let / ( 0 ) , . . . , / ( n ) be the initial layer expansion terms given by Lemma 3.4.2. Insert

/w=E^w+iy/w (-) +^w ;j=o =0

jj=o =0

\\

£ £

^'

(3.5.i) (3.5.1)

into the Boltzmann equation (3.1.1a) and note that the remainder r satisfies the following nonlinear equation {j) ( j j Dr Dr == + --J(/( J ( / ( 00)),r) , r ) + 22V£ ^ T £3J~-l1J{f + ( ft\r) + eZl~-11J{r + £ nn-'2l, = -Cr -£r J(/ V/ '\r)+£ J(r,r)+£ -^, ir) £ £ *—* ■

I

7=1

(3.5.2a)

THE HYDRODYNAMIC LIMIT

99

with the initial data r

t=o

= 0,

(3.5.26)

where 21 is a complicated term depending on / ( ° ) , . . . , /
'

f^,

The nonlinear J(r, r) and the nonhomogeneous 21 terms are multiplied by numbers el~l and en~l, respectively. Therefore, for n and / being properly chosen, Eq. (3.5.2a) is weakly nonlinear. Its classical approach is based on the analysis of its linearized form. To deal with the linearized equation (3.5.2a) Grad's idea [GR3,4] can be used and the operator C can be symmetrized to get the non-positive in L2(M ) (with respect to the velocity variable v) operator L (cf. (2.5.5)). In this case the -L-term does not cause any growth. Since M depends on the t and x variables, Grad's symmetrization leads to a new term (M~*DMz)r which is like m^r and has an uncontrolled sign. To overcome this difficulty, Caflisch [CA2] proposed the decomposition of the remainder r into "low velocity" and "high velocity" parts using the two different Maxwellians: M and u+ as in (2.4.2). In the case when the initial layer is included, the linear (but unbounded) with uncontrolled sign and singular (with respect to e) operator - J ( / ^ ° \ •) appears. For this reason, in the paper [LA3] (see also [LAI]) the decomposition of the high velocity parts into two terms was proposed and the decay of /(°) (cf. 3.4.12) together with the GronwallBellman inequality were used. The norm of the space had to be appropriate in order to deal with both the linear (the L2-norm) and the nonlinear (the Loo-norm) terms. This has led in [LA 1,2,3] to Sobolev spaces (with respect to the space variable x). However, to control x-derivatives, one had to use a large number n of terms in the truncated power series in e and therefore this approach needed the high smoothness of the solution of the C E E (or C N - S E ) . On the other hand, using the Sobolev spaces does not seem to be suitable for the rescaling of the space variable. The paper [LA6] dealt with the approach in the C°space (with respect to the x-variable) setting, by applying an idea suggested by Heintz [HE1,2]. The C°-space setting seems to be natural to study the asymptotic behavior (see discussion in [LA6]).

100

M. Lachowicz Keeping this in mind, decompose the remainder r as follows r ( * I x , v ) = A f * ( t , x , v ) r 0 ( * , x , v ) + « | ( v ) ] T !■*(*, x , v ) ,

(3.5.3)

i=l,3

where the first term in the right-hand side of (3.5.3) is the "small velocity part component", whereas the second one - the "high velocity part compo­ nent" . In this way the following system of three equations for ro, r\, r2 is obtained: Dr00 = -Lr £>r -Lr0 0 + 4- ~- YY e

*-^ e£ *-^

^M'^KifAn), ^M^KiAn),

(3.5.4a)

i=l,2 DT DnX

1 r + -(1 - ^)wT*/C(o;iri) + - v •ri n == - uu>I*(DM*)r + ' ( M ' ) r0 0 +

i£ ( l - $ > + ^ ( 4 0

£

+ - a ;)^; iJJ(/(°',4'-i) ( / ( ) , u ; | r 1 ) ++^ (^r(or ,or,1r, ir,2r)a )++2i, Sl +-u £ 0

5.46) (3.5.46) (3

£

1 l! Dr + -vr )w7*/C(wiri) + - wa +^ *J J( (/ /<( 00) ,\<44rraa + M *^ r o0 ) , 2) L>r22 + -C i /2 ■r2 == i^C ( l --♦W)^K(u\r £ £ c£ 5.4c) (3 (3.5.4c) with the initial data r0 ro

t=0 i=0

= rn i =

= = r r22 t=0

t=o *=0

= 00..

(3.5.4d)

^77 _ $»?Qv|j j s a continuous, non-negative function such that ^**(M) ( | v | ) = 1 if |v| <*?,
if ||v| v | >>7] ? 7 + <$, 8, •n

where r\ has to be properly chosen and 0 < S < - . The operator A is given by nn

^ ( ^0,ri,r , r i l2f) ^ ) === 2 w , J2| rf 0l r+o u+ |; l^, ri 1r + ; irr 28 ) . A{r 1 +c( ji ^ a ;; ;*^^£eJ '--1VJ ( / W ^ +)/+( j/ )W ,M i=i

The term 21 corresponds to the sum of the nonlinear and nonhomogeneous terms

*;V

"'21), 921 = w + ^ e ' "-1x J(r, J ( r , rr)) + £ nn""'2l),

THE HYDRODYNAMIC LIMIT

101

and is assumed, at this stage, to be known. Using the estimations of Section 2.5, by the same method as in the paper [LA3] (see Lemma 8.1 in [LA6]) one has Lemma 3.5.1. Choose r\ > 0 sufficiently large (but independent of e). If e > 0 is sufficiently small then the solution {r0(t),ri(t),r2(t)) of the linear system (3.5.4) satisfies the following (a priori) estimate sup ro [o,t]

a,fc oo,oo

0,fc

< const sup || r 0 || 2'So + cv 5 2 sup [o,t] feiilM

sup r\ Sfoo < conste | [0,*]

T-o 2;So

[o,t]

a—A,A: 00,00

5

(3.5.5a)

(3.5.56)

[0,1]

r

2(*)||Sjfoo
+ S U P 21

00,00 '

£

/ [0,1]

ro 2;oo + c ^ s u P

Of II - A , f c '**|| 0 0 , 0 0

[0,1]

(3.5.5c) for all t e [0,to] and each a > 0, k > 0; where A is as in Assumption 2.1.1, /c as in (3.4.5), 0 < a* < cr0, cr0 is as in (3.4.12). The following result, which allows to combine C°-estimates with the Z,2-properties of the linearization of J, is crucial in the paper [LA6] Lemma 3.5.2. sup || (^V£K)4^f\\

2 ;^

[0,1]

< Cr ,£-i sup I / | | 2,2

(3.5.6)

[0,1]

for a = 0,1,2; e > 0, t £ [0;t0]. Outline of the proof: Write Zf = {WV£K)2Wf. one has

By (2.5.4) and (2.5.6)

t ti

l^x,v)|<^////exp(-^- 4 2 )) 0

0 M3 R 3

x x ( | v |
x | / f e , x - (t - *i)v - (h - t 2 )vi, w)| dvi dwdt 2 d*i,

102

M. Lachowicz

where c > 0 is a constant and iq_+ is given by (2.5.6c). Changing the variables Vi -> Xi = x — (t — *i)v - (ti - t2)vi one gets t

ti

3 \Zf(t x v)\<^fJJJe p(~(t-t ))'(t -t )\Zf(t,x, i v)h 1 X 2- « » ) ) 1( < l -2- t ) 2

0

3

0 1»3 ] R 31D.3 M3

v w ) x ( | w | <
X x

x i dw |/(t2,xi,w)I dwd^2 d^2 dt, X l , w ) | ddxi l/(*2,

where now Vi = (xi — x + (t — £i)v)/(^2 - ^i) and cv is a positive constant depending on r\ and toThen t

ti

- 33 \Zf(t,x,v)\
0 0

x T? + + ^)^++(v,vi)x(|v ( 5 ) I C ( V V I ) X ( | V1I| | w) w) ++

11? 3 ]R IK.

J

3 TP3 T 11

\P p / ( pP - lI ))

xx xx((||w w ||
\N (( pp -- l ) / p

ddxij xi)

(T |||/(t | / ( * 22 ,.,w);L , -,w) ; i p p (T

33

dti. ) | dwd^2 )|| dwdfedti.

Now, changing the variables xi —>• vi = (xi — x + (t — £i)v)/(£2 - £i) one obtains tt ti ti

3/p |2/(t,x,v)| --(t- \Zf(t,x, v)| < < J / /exp (-|(t -3/p - « 2t,)) > ) ( *(ti 1 - - ^*22))-

0 0

x J ( I X(M < < r) v+QHflr + ( S ) ^ " 1 l5) (( vv,, vi)x(|vi < ry++ ^ W ^5)k*^-D x/(/x(|v| - ^ C v i (vijW , w)) V l )x(|viI ^Ty UJ33 M IK

3 IK'S 1R KL

1

x x ( | w |
P

P

3 , - , v i ) ; Lp (T , ,(T3 )|| ||/(t )|| ddw wdd t 22 d *t ii.| / ( *22,.,vi);L

Then, assuming that p > 3, by straightforward calculations one has ssup u p\Zf | | 2 / 2| ;oo< s uS UpP| | // l| a2 ,*p -. 2 ^ < -F£^ 3/ lvP [o,t] [CM]

^

[o,t] [o,t]

/ p

(3.5.7)

THE HYDRODYNAMIC LIMIT

103

An analogous procedure can be followed to estimate | Zf(t)\\ Young's theorem one can prove that SUP sup [0,1] [o,t]

2)P .

|| Zf\\ Z / 2 ,i pp < < Cc^e-^-V/W , e - 3 ^ - 2 ) / ( 2 " ) SUP sup || // | | 22,,22 ,, [0,1] [o,t]

Using

(3.5.8) (3.5.8)

provided that 3 < p < 6. Combining (3.5.7) and (3.5.8) yields (3.5.6).I Lemmas 3.5.1 and 3.5.2 imply the following Lemma 3.5.3. ([LA6]) Choose r\ > 0 sufficiently large (but independent of e). If e > 0 is sufficiently small then the solution (r 0 (t),ri(t),r 2 (t)) of System (3.5.4) satisfies the following (a priori) estimate: const a 00 ^ ) 0tOO .% 0 << -3TH /J

sup sup rI0r 0 2]2 | 2,2++const conste esup sup 21 || 2l||0 0^ ^ , ,

(3.5.9a) (3.5.9a)

■ . 0 const 1 sup II rin II£^) 0 0 << —^ sup 2,2 +const conste sup|| ||52l|| e sup | S^, ^°0 », ,_ sup ||r 0r 0 ||2,2+

(3.5.9ft) (3.5.9ft)

sup I r 0 || [o,i] [o,t]

[o,t] [0,1]

i,z

£

V ^£

[0,1]

[o,t] [o,t]

[o,t] [0,1]

[0,1]

_. const _, nn COnst // tt \ I =-II _ \ n 2+ + c oconst n s t s £ sup | 2t|-A,0 sup 2,2 r 22 ^00 ^ , o c< -££^3j 7// 22T «K ee xxPp ^ —€ gac^ * l) SU P rro ° 2> ^ u p 21 0 0^, 0 0, i [0,1] [0,1] \ J' [o,t] [o,t] (3.5.9c) for £ € [0,to] and a = 0,1,2,...; where the constants denoted by "const" may depend on 77. Now the main lemma can be stated. for some a > 2. If GE Loo(0,*b;X££°) Lemma 3.5.4. ([LA6]) Let 2l # < Loo(0,*o;X££ 0 ) for ri,r2:j)) of the integral eE > > 0 is sufficiently small then a unique solution (r 00,,ri,r version of the linear system (3.5.4) exists in Loo Loo(0, £0;A o o , c

CdO.toliX^ 0 ), 4rf e6C°([0,
at at estimate holds for i = 1,2,3; and the following for i = 1,2,3; and the foJiowing estimate holds w a 0 SUP £%^<<^5 P s»Pl 13||~2,-A . suP Mn S " A ° > £ £ [0;(o] [0;10]

'

(O.'o) [O.lol

3 5 10 ((3.5.10) -- )

104

M. Lachowicz

for i = 1,2,3; where the constants denoted by "const" depend on to. Proof: By Lemma 3.5.3, the statement (3.5.10) will be proved once one find the appropriate estimations of || r*o || 2,2- Following [LA3] (see (5.33-5.36)), by (3.5.4a), (2.5.5) and (3.5.5b,c) one has

sup r 0 2,2 < const / I 1 + £- e x p ( - — 0+) ] sup r0\ 2,2 £ [o,t] J L\ J [o,t] + sup sup|2f I S | "*£ " A £ dti [o,t] [o,t] JJ

(3.5.11)

fort€[0,«o]. Applying the Gronwall-Bellman inequality (see [LL1] and [MM1]) one has

W|a|s£,

sup |I r 0 2,2 < exp ((const const (t 0 H + — JJ ) sup I 211 ^ A £ , [0,t] \\ \V
(3.5.12) (3.5.12)

for te [0,t0]. Combining (3.5.12) with (3.5.9) yields the estimation (3.5.10). The existence and smoothness of the unique solution of the integral version of the linear system (3.5.4) follow from the same methods as in [LA2].i Differentiating formally, with respect to the space variable x, both sides of System (3.5.4) one obtains the system for the x-derivatives of (7*0, ^1,^2). Analysis such as that in the proof of Lemma 6 in [LA3] shows that with each differentiation one power of e is lost. Thus one has Lemma 3.5.5. Let 2l # G L o o ( 0 , t 0 ; X ^ * ) for some a > 2 and k > 0. If e > 0 is sufficiently small then a unique solution (ro,ri,r2) of the integral version of the linear system (3.5.4) exists in (L00(Oito\'K£>kOQ))3 and rrff 6eC°([0,to];X££*), ^([0,*b];X££ f c ),

jf?

™-a-2,fc\ -2,fcx 7/

6 LQO Loo(0,^o5 (0,^o; A o o , ,o0o0

€loo(0,t,iXE£»)

THE HYDRODYNAMIC LIMIT

105

fori = 1,2,3, i 1 n k ^ const —n x t SUP || r, I « £ . < g £ sup || a|| £» . sup|r,|^oo<£e,+3/2sup 21 ££*;*.

[O.'o]

'

(3.5.13)

[0,t0]

If in addition k > 2 and a G C7°([0,*tt];X^^fc"1) then

K,

l l , 2a -2,fer. r« eECC°(l0,*to];X^i^ '([0M )nC l ([0 f
-l,fc-

3.6. Hydro dynamic Limit for the Boltzmann Equation Keeping in mind the result of Lemma 3.5.5 one can fix n and / (in (3.5.1)) sufficiently large and use the successive approximation method to obtain Theorem 3.6.1. ([LA6]) Let to €]0, +oo[ be such that on the time interval [0, t0] there exists a solution (g, u, T) of the CEE (or of the CN-SE) such that Assumption 2.4.1 is satisfied. Let the initial data (3.1.16) be such that M0 + F = Mo + G, G, where Mo = M[g\ (3.4.5) and

,u t_0,T\

(3.6.1)

] and G satisfies the smallness condition

k Gellon ,, G <E Tlo HY^ Yj'*

(3.6.2)

with a and k being large enough. If 0 < e < £o, where s0 = £o(M is a fco critical value, then a solution f of Eq. (3.1.1) exists in L^ ((Uo;Yj°' ); for some cto > 0, ko > 0; and

sup

o o A ), /*6C°([o,(o];r; /#eC ([0,io];n ''°)>

3 6 3<J ((3.6.3a) -- )

h 4-J*eLoo(0,to;Yl°' °), 4-J* e L»0>.<w «?•*•). dt

(3.6.36) (3-6-36)

at

|| /(«) - M(t) - fW Q

•?•*» <

0
Qo£>

k sup 11 f(t) M(t) /<°» (-) I %°' ° < ctoe, where Ct0 is a constant (depending on to). EMoreover,

0
\

(3.6.4) (3.6.4)

J

where cto is a constant (depending ont ). Moreover, , >fco 1 0 1

>fco- 1)nC 1([0,tto];V?, *00"aa), /€C°([O ^J;Yy l f e C°([O l ^J;Yy - )nC ([0,tto];V? * " ),

(3.6.5) (3-6.5)

M. Lachowicz

106

provided that ko > 2. Theorem 3.6.1 states that under suitable smallness condition (3.4.5) on the nonhydrodynamic part G of the initial datum F there exists a solution to the singularly perturbed problem for the Boltzmann equation (3.1.1). The solution exists macroscopically as long as a smooth solution of the hydrodynamic equation (CEE or C N - S E ) does and then the asymptotic relation (3.6.4) holds. The method of the proof shows that the Boltzmann solution is unique provided that the hydrodynamic solution is unique. Fi­ nally, the Boltzmann solution is continuously difFerentiable (ko depends on the smoothness of the hydrodynamic solution and on k). Note that the smallness condition (3.4.5) does refer only to the function G itself, but not to its x-derivatives. This permits to cover a class of physical problems (phenomenon of oscillation with a high frequency). The function /(°) describes the initial layer effect. Note that, if the initial datum is purely hydrodynamical (i.e. G = 0), one has sup \\\f(t)-M(t)\\\l°'ko
(3.6.6)

0
and therefore in this case, the convergence to the bulk solution M is uniform on the interval [0, to] (and the initial layer effect disappears). The estimation of the error (in (3.6.4)) for the C N - S E approximation of the Boltzmann solution is of the same order 0(el) as for the CEE one. This is because the spirit of the method used here (Section 3.3) differs from that of the Chapman-Enskog procedure: in the latter the second order 0(e2) estimation can be expected (cf. [PA1] and [Mil,2]).

3.7.

Hydrodynamic Limit for the Enskog Equation for a < e

Consider the Enskog equation (2.1.8) in the dimensionless form: Df=±Ea,£

(/;/,/),

(3.7.1a)

with the initial data f\t=o = F'

(3.7.16)

where a and e are two dimensionless parameters of the model: a represents the scale of the hard-sphere diameter and e is as in Section 3.1.

THE HYDRODYNAMIC LIMIT

107

In the same manner as that in case of the Boltzmann equation (3.1.1a), Eq. (3.7.1a) becomes a singularly perturbed equation in the hydrodynamic limit a I 0, e I 0. In addition, the factor

yfAf) = ya

-/;x,x±anl

has a singular behavior in the limit e I 0. This singular behavior, however, can be compensated by a good behavior with respect to the limit a I 0 provided that suitable relations connecting both the parameters a and e are fulfilled (cf. (3.7.4)). The following assumption will be needed throughout this Section Assumption 3.7.1. The factor y^e£ is is such that (i) 3£,(0) ' y±e(o) = = 1,

(3.7.2)

(ii) - yt(h)\ < const - 2/ 2 | | °o,0 const 2 ^ j | | flh-f J°'°,] ( (3.7.3) 3 73) (n) \yt(fi) \y£,e(h)-y£Ah)\ V/i, }i G #2 > £0, for some /?i pi > > 0, fa > 0 and a«o0 > > 0,0, ai£,, Va,e: a < e < £o, V/1,/2 eDDa>£ Y"°'° iis is a subset of the physical consistency of the Enwhere Daa,££ C Yj°'° fLA4l): skog description (cf. Section 5 in [LA4]): fi D , = {feY%° a e 0 . , = { / 6 * ? * ::| /f |\ ?T" * <<4da,e] M } -■

Hydrodynamic limit for the Enskog equation can be investigated in two cases: I.) a is small with respect to e: £\ R0 AS a 7.4) 0 < 1, (3. '' 00 < a < £ < a < e < 1, (3.7.4) for some /3 > 1; II.) a is of the order of e, say: . 0 <
(3.7.5)

In the former case one can expect the same kind of the hydrodynamic limit as that for the Boltzmann equation, at least for large 0. This problem was considered in [LA4,6,7,9]. Note that the condition (3.7.4) corresponds to the case of Na2+i


(3.7.6)

M. Lachowicz

108

in the limit a I 0, TV —> +00, where N is a number of particles. In the case II one can expect something different from the classical Boltzmann hydrodynamic limit. Discussing this problem is put off to Section 3.8. Consider then the case I. Note that if (3 is sufficiently large (depending on Pi and fa) then all the bulk expansion solutions as well as those of initial layer can be exactly the same as for the Boltzmann equations (cf. [LA4,6,7,9]). The difference is only in the equation for the remainder. On the contrary to the operator L, the operator La defined by (2.4.10) has not, for a > 0 the properties similar to that of (2.5.5). However, the operator La can be split as follows (LA9]): La = L + Aa + aA'a ,

(3.7.7a)

where (Aa/)(x,v) = y*y,{M^(x,v')(/(x + a n , w ' ) - / ( x , w / ) ) - M ^ ( x , v ) S2 E 3 S'

x (/(x - an, w) - / ( x , w ) ) } M ^ ( x , w)(n-(w - v) V 0) d n d w

(3.7.76)

and the operator A'a has a "good" behavior with respect to both the small parameters a and e. The crucial ingredient is an estimation of the term -Aa showing that it e does not cause any growth either. Such an estimation needs some sources of regularity (with respect to the x-variable). By Golse et al. [GOl], one can expect that the velocity (and time) averaging can bring some regularity in the x-dependence. It is indeed so (this result was proposed, without a proof, in the paper [LA9]) Lemma 3.7.1. I f 0 < a < e < l

then

sup || (VEAa)2f\\ %i2 < c o n s t , / f sup II ' i £ & ° i [<M]

( 3 - 7 - 8a )

V e [ 0|t ]

sup || V£KV£Aaf\\ £, 2 < constW - sup [o,t] V e [0,t]

/ a+2,0.

(3.7.86)

sup VeAaVeKf\2o2

f\ a+2,0. 00,00 '

(3.7.8c)

[o,t]

< const*/-sup V e

M

00,00 >

THE HYDRODYNAMIC LIMIT

109

for all t 6]0,t 0 ], a > 0. Outline of the proof: (A detailed proof will appear in a forthcoming paper). Let Aa = A{al) +A{a\ where

(4 1 )/)(x,v)=y , |M^(x,v')M^(x,w){/(x + an,w')-/(x,w')} R3 S 2

x (n ■ (w - v) V 0)dndw and (42)/)(x,v)=||M^(x,v)M^(x,w){/(x-an,w)-/(x,w)} R3 S2

x (n • (w — v) V 0)dndw . One has V«Ai1>V«JC/| So>2 < const (I Vf\\ %<2 + a\\ f\\ S,i2) , where £>/(*, x,v) ti

t =

t

2- I [ f I /" j exp ( - i /" K*3,x - (i - t a )v + an, v)dt 3 ) 0 R3 § 2

0 R3

ti

x A f * ( t l l x - ( t - t i ) v + an,v') x Af *(*i,x - (t - ti)v + an, w)(n • (w - v) V 0) x exp ( - - f v{t3,x-

(t - ti)v - (ti - h)w' + an,w')d* 3 )

*2

X Jc(*2> x - (* - ^ ) V - (*1 " * 2 ) W ' + ° n ' W ' ' ® x / ( t 2 , x - (t - ti)v - (*i - t 2 )w' + an,fl

110

M. Lachowicz tt

\r,v)dt 3 J 3,X- (t-tsY — // v(t z/(*3,x-(£-t -- eexxpp (f — 3 )v,v)d£3) ti t\

t i ) v ,,v')M v ' ) M ^k(ti,x ( t i , x--((tt -- *t ii )) vv ,, w ) V 0) w )) (( nn -- (( w w -- vv) V0) x MM^2( (t ti l,5xx--((tt---ti)v ti ; ti)v -- ( *(tii - *- 3£)3w )w', x eexpx p( (- --j y/ \i>(*3,x'(*3>X- (t(f-- ■ti)v ' , ww')d* ) d t 33)J

to ti <2

X x lt(£ k(t22,x,X - (t(t- *i)v tx)v -- ( *{ti! - - *i22)w' ) w ,W,0 ',w',0 X x /f(t( 2i ,x( * - *i)v 2 , X - (t-ti)v-(ti

- ( * ! -

• t 2 )w' -t2)w',Z)}dZdt ,f)}dfd* 2 dndwdti. 2dndwdti.

Using the Bessel-Parseval equality (with respect to the space variable x) and then integration by parts (with respect to the time variable t\) lead to | !>/(*,-,v) ; ^2((T ||P/(rv,v);L < ^aj2( T 33))||| | <

mi' •l| // // M (i(vv - ww') i ( vv ))))"£ W O -■mm+ j+^i M (E E S m e Z 33

3

meZ

11

2

E3 §2

1 22 25

xx (n ( n •((w w - -vv) ) V0) V0) // // e exxpp((--iixx--mm){ ) { ......}}ddxxdd££ddnnddw w|| )J ,, E33 T33 E

where z = (—l)1/2 and {...} substitutes a complicated expression. Classical changing of variables (see [CR1] and [GR3]) yields v' -H- W' and thus i

| P / ( V , v ) ; L 22((T T 33))|| || ||D/(t,-,v);L i

ME ^ T H

E

mGZ 3

meZ

m_i(v I H ( / / [ l ( i ( v - v ')' )•' m m ++ ^- m(i y( v)) ) - V , { v --vv') ') H(//[|(i( e V — \

r

1

R 3 E3 §2 §2

m xx m_ m_ 33 // 22(w' v)| •• |n |n •• (w (w' ■-- v)| (w -- v)|v)| •l|mi(v - v')m (w' --v v) ) i(v — v')m 33/ /22(w'

x / J/ exp(exp(-zx m){.. .}dxd£| d wJj J J 2 -zx •• m ) { . . .}d> :df|].idnndw 3 E33 T E T3 i

H ( / / ( l ( v - v-V) ' ) - m•m\| S?(I E H(//(l(v

22

^ (

m G Z 33

m£Z

33 RII?

§2 &2

l r2 _1 ++ — m (v)J J m 22( v ) ) m_ m _22(v( v--V v '))

THE HYDRODYNAMIC LIMIT

111

x m_ 3 (w' - v)|n- (w-- v)|dndw) ( / / m 2 ( v - - v ' ) m 3 ( w ' - - v ) R3 § 2 2

x |n- (w - v)| / / e x p ( --zx i • m){.. .}dxd£

IndwJ 1

R3 T 3

Let now (see [GR3] - (37)) v^ and then

= n(n ■ (w - v)) and w ^ = w - v - v^1)

dndw = - ^ — d w ( 1 ) d v ( 1 ) . |v' ' The sequence of integration is the following: at first (for fixed v^) w^ is integrated over the plain perpendicular to v^1) and then v^1) is integrated over E 3 . Thus (cf. [GOl]) l m l / / (l( v - v ' ) • m ! 2 + ^ m 2 ( v ) ) _ 1 m _ 2 ( v - v')m_ 3 (w' - v) R3 § 2

x |n • (w - v)|dndw < const|m| / ( | v ^ • m| 2 + —m 2 (v)) _ 1 R3 x m - a O ^ ^ l v ^ r M v ^ < constem_i(v). Moreover, m2

Yl meZ3

( v " v ' ) m 3 ( w ' - v)|n - (w - v)| • | / / exp(-zx • m)

R 3 §2

K3 T 3

x {.. .}dxdf| 2 dndw) < const£ 2 m 2 (v) / / m 4 (v (1) ) x m 3 (w( 1 >)m 2 (v + vM)\vM\m-B(-v

+ w^Jw+fv + w<*>)

xt^v + v^+w^) x ( / , l c + + ( v + v( 1 ) I Osup|/(t,x,fl|df) 2 |v( 1 )|- 2 dw( 1 )dv( 1 )

w < const e2m2(v)

f f mz(w^)m2(y

+

VW)JILJ(V

+ w<x>)

M. Lachowicz

112 x exp ( - c l v ^ l 2 ) exp ( - c\v + ^

x m _ ( 2 Q + 6 ) ( v + v^( i )')

sup [o,t]

+ w^l2) Iv^l"1

/ a+2,0 CX),00

which is clear from (2.5.16). Combining these inequalities yields ma(v)||P/(V,v);L2(T3' sup || / | | ^ 0 ( m 2 a + 4 ( v ) J j m.2a-A{v

< const J" 1

r(i)

2

x exp(-c|v( )| ) exp(-c|v + V

+ v*1*)

+ wWftlvWpMw™ dv^)1/2

Since (2.5.16), one has

sup Vf

%j 2 < c o n s t * / - s u p

[o,t]

V £ [o,t]

/ a+2,0 00,00

The other terms can be estimated in the same way. I In the same manner as in (3.5.3) the remainder r can be split into the "small velocity" and "high velocity" components. However, now one should also split the "small velocity" component: ro = r 0 i + r02 •

(3.7.9)

Then the following linear system of 4 equations for r 0 i , r 0 2, n , r2 can be considered: 1

Dr01 = -Lr01 £

1

1

+ -Kr02 £

+ - J^ £

^M^M^+n),

(3.7.10a)

fc=l,2

1 1 , -t>r02 + - i / - r 0 2 = - A a r 0 , e e

Dri + -i/ a -ri=-w + ^ f M ^ ] 7-0

(3.7.106)

THE HYDRODYNAMIC LIMIT

113

+ V ' l M ; r 0 + i(l-**)w+*/C«04ri) 9

1 5

+ - ^ ; ^ a , e ( 0 ; / W , a ; | r i ) + Q(roi,r 0 2,ri,r 2 ) + 0 ,

(3.7.10c)

Dr 2 + - i / ■ r a = - ( l - ^ K ^ / C a ( 4 ^ 2 ) + |w;*^a,e(0; / ( 0 ) , < 4 r 2 + M * r 0 ) , (3.7. lOd) with the initial data roi

«=o

= 7*02

<=0

= 7*1

f=0

=

^2

= 0 t=0

(3.7.10e)

The operators L, K and the function i/ are as in Section 2.4 with B given by (2.1.3); Ka and z^a are given by (2.4.9); Aa and A^ are given by (3.7.7); $v was introduced in Section 3.5; (f;f,r) - | v K , e ( 0 ; / ( 0 ) + / < * V ) , (3.7.11) where r = M 2 r o + w | ( r i + r 2 ), r 0 is given by (3.7.9) and Q(roi,r 0 2,ri,r 2 ) = | v £

a

,

e

n

£J U)+ {j) f=j2 v f ) 3=0

(3.7.12)

Moreover /(°) = M[^,u,T], f ^ \ . . . , / 1 V

ft + 1 one A

can now proceed analogously to the proof of

Lemma 3.5.1 to obtain estimations of ro = roi + r 0 2, »*i and r-i analogous to those of (3.5.5). Then using Lemma 3.7.1 yields the following lemma (analogous to Lemma 3.5.4). Lemma 3.7.2. Let Condition (3.7.4) be satisfied for /3 > 6 V — - — . Let 0 # eLoo(O,to;X££°)

M. Lachowicz

114

for some a > 4. If e > 0 is sufficiently small, then a unique solution r = (roi,r 0 2,ri,r 2 ) of the integral version of the linear system (3.7.10) exists in the space (WO^oiX^))4, r#e(C°([0,*o],X^°))\ ^^(^(O^oiX^ dt

0

))

4

and the following estimations are satisfied sup|N|£%<^sup||e||^°, £

[O.to]

'

(3.7.13)

[O.to]

for i = 01,02,1,2 where the constants denoted by "const" depend on to. Keeping in mind Lemma 3.7.2, one can look for a solution to the problem (3.7.1) in the form (3.5.1) with n and / being properly chosen. As in Section 3.6 the successive approximation can be applied for 0 given by:

< 4 * = J7TT (^«,e(f + ^ +^^(f;

f + e?r> f + e V ) - ^ ( f i f + ^

f + <^))

r,r) + -jL- (£a,e(f; f, f) - £ a , e (0; f, f))

+ ^ r (Ea,e(0; f, f) - J(f, f)) + e n - ^ ,

(3.7.14)

where f and r are such as in (3.7.11) and (3.7.12) and 21 as in Section 5.3. a01 The first term of the right-hand side of (3.7.14) is like (cf. Assumption 3.7.1), the second one like el 1, the third one like - . , , . . Because of the regularity of f (cf. Lemmas 3.2.1, 3.3.1 and 3.4.2), the fourth term is like -. Finally, the fifth term is like en~l. £i+

Now one can properly choose I and n (cf. [LA6]: / = 3, n = 5) and assume that /3 in (3.7.4) is sufficiently large to obtain the main theorem.

THE HYDRODYNAMIC LIMIT

115

Theorem 3.7.1. ([LA9]) Let the parameters of the model be such that (3.7.4) is satisfied with £>6V^i5.

(3.7.15)

Let t0 e]0, +oo[ be such that on the time interval [0,£0] there exists a solu­ tion (e,u,T) of the CEE (or of the CN-SE) such that Assumption 2.4.1 is satisfied. Let the initial data (3.7.16) be such that F = MQ + G, where M0 = M [g\t=0,u\t=Q,T\t=0] (3.4.5) and

(3.7.16)

and G satisfies the smallness condition

G<Eft0nY£'\

(3.7.17)

with a and k being large enough. If 0 < e < Co, where e 0 is a critical value (depending on t0), then a solution f of the problem (3.7.1) exists in Loc(0,t0] Yj°'°), for a0 as in (3.7.3), and /#€C°([0,*o];Yj0'0),

^ /

#

e Loo(CMo;Y«°'0),

sup II f(t) - M(t) - /<°> (-) [o.to]

\

£

I r ° < ct0e,

(3.7.18)

(3.7.19)

(3.7.20)

;

where Q 0 is a constant (depending onto). In the same way as in Theorem 3.6.1 the existence in Yj 0 ' °, for A:0 > 1, as well as the time-regularity (3.6.5) of the solution / can be proved under condition that the little more restrictive assumption than Assumption 3.7.1 is supposed allowing to control of derivatives up to the order k0 of the factor ya,e and /?>(6 + 2fco)V/?2+!

+

*°.

(3.7.21)

Pi

Note that using the bulk expansion adopted to the Enskog equation (see Section 3.8) instead of the bulk expansion of the Boltzmann equation

M. Lachowicz

116

(as in this Section) one obtains the hydrodynamic equations with an addi­ tional "Enskog" term which is 0(e^~l). For such a "Enskog" hydrodynamic equation the theorem analogous to Theorem 3.7.1 (i.e. for large (3) can be formulated. However, this does not seem to be an interesting case, because a highest flexibility can be attained (for j3 > 2) by using the modified bulk procedure as in Section 3.3. 3.8.

Hydrodynamic Limit for t h e Enskog Equation for a ~ e

Condition (3.7.15) was not optimized. Some largeness assumption on /3 is anyhow essential for the proof of Theorem 3.7.1. Therefore the method of Section 3.7 is not suitable for the case (3.7.5). For the Enskog equation in the case (3.7.5) one can expect different hydrodynamics from those of the CEE or of the C N - S E . In fact, the formal consideration below shows that even for the simplified model - the Boltzmann-Enskog equation (i.e. for ya,£ = 1) new terms appear in the hydrodynamic equations for the case (3.7.5). Define Ja(/l,/2) = £a,e(0;/l,/2) and consider a solution of the problem Df=\jaUJ),

f\t=0

=F

(3.8.1a)

(3.8.16)

in the form

/ = Xy/(j), fU) = 9{j) + hP , g{j)eTi

j > 1,

(3.8.2a) (3.8.26)

and h® <E N.

Inserting (3.8.2) into (3.8.1) one has 4)(/(0),/(0))=0, 2Jo(f{0\9W)

= V^DfM

_ V±41)(f(Q\f«»),

(3.8.3) (3.8.4a)

THE HYDRODYNAMIC LIMIT

117

l 1 7>D/(o) _ ( /if(V» 0 ) ,)/ ( 0 ) )=a 0to, PDfW-p4 \fV>)

(3.8.5,,) (3.SJ.5o)

± 0 )i2))=V 2J ( ^fi0(\9 Df^- L D/( 1 >-7o(/ 2Jo(/ ,9( 2 >)=PW ( 1( 1) ),/,/((11 ) )

1) ) ) / ( o ) ) _ 2Jp-4 - i2-jW -2P ± JJ-4 (/( 11\fW)-2V ( /(o) > / (o) ) j (3.8.46) -2-p \f(°)jlo)), ( S Hf( (3.15.46) 1 1] {1) ( 0 ) « U«>\ ( / ( 0 ) j / /«>>), (0))| VDh -VDgM+ +2v4 2v4 (f , / ) + + 2W4 P J 2) PDhW = -PDgM \fM,fW) (3.8.56) 156) (3.i (l)

• •• >

where Jo J0 = =
) )i)_ --44)-(flj2)-4 ~(fuf2) - 4-(f2jl)\ (/2,/i))

,

(3.8.6a)

^ i ) + (/i,/2)(x,v)=|y (/i,/2)(x,v)=|y,,((n.V ((n.Vxx)7iCx,wO )7i(x,wO R3 § 2 ; x /2(x, /2(x,vv') ) J f n -•((w w --vv) ) VOJ V 0 Jdndw, dndw,

(3.8.66) (3.8.66)

fJ -«(-/(1/, 1/ 2) /)2()x(,xv,)v = ) =|||| (( ( - n - V x ) 7 7ii((xx))w w)) R3 S 2

/ 2 (x,v) jJ (f n - ( w - v ) VOJ ddnnddw w..

(3.8.6c)

Since (3.8.3), /^°^ = M[g, u, T] for some fluid-dynamic parameters a, U , T . The hydrodynamic equation of the "0-th order" corresponding to the Boltzmann-Enskog equation (3.8.1) for (3.7.5) is defined by Eq. (3.8.5a). Comparing it with Eq. (3.2.4) one can see that a new term PJ 0 (1) (/(°>, /<°>) appears. The same effect is present for the Navier-Stokes-type approximation (the "1-st order"). In fact the modified expansion (see Section 3.3)

118

M. Lachowicz

results in the following "1-st order" hydrodynamic equation: l 1L L VDM - V4 VDM V41]1]{M,M) (M, M) = e(-VDC~ E(-VD£V VDM DM 1) ) + VDC(M,M) VDC-11VVLLJ4{Q (M,M)

+

2V41)1)(C{C-11VV±±DM,M) DM,M) 2V4

X X 1 L1 1)L 1) 2) 2) \CV 4V (M,M),M) 2V4 (M,M)). --2V4 2V4 \C~ 4 (M, M), M) + 2V4 (M, M)).

(3.8.7)

Consequently the Enskog hydrodynamic limit differs from that of the Boltzmann equation due to the presence of a term of the Oth order in e (as well as some terms of higher orders). In the case of (3.7.4) this Enskog term was O(e0~l) and for this reason was neglected at the hydrodynamic level. From the mathematical point of view the asymptotic behavior of solutions of the Enskog equation in the hydrodynamic limit a = e \. 0 has to be considered as an open problem. The reason is that a "good" behavior of the term -La was proved only in the case (3.7.4) for large (5: (3.7.15). In £

the case of (3.7.5) it is still an unsolved problem. In view of the lack of a suitable mathematical theory allowing to derive the hydrodynamic limit for the Enskog equation for (3.7.4) it is important to consider a modified Enskog equation. In the paper [LA4] the Enskog equation in the form introduced by Arkeryd [AR2] was used (actually Arkeryd considered the modified Boltzmann-Enskog equation). In order to prove the (usual) H- Theorem Arkeryd altered the collision operator by applying a wider range of integration with respect to the collision parameter n. In the physical case there are collisions only if n • (w — v) > 0, while Arkeryd used the whole § as the range of integration with the same velocities after collision for both n and - n . This modification leads to the collision operator Eo )£ with |n • (w - v)| instead of (n • (w - v) V 0). Called it - Arkeryd's modification. Note that after Arkeryd's modification, one can change n into —n without changing the collision kernel. Note that for Arkeryd's modification the term VJQ (M,M) in the hydrodynamic equation defined by Eq. (3.8.5a) vanishes and therefore the 0-th order hydrodynamics is classical one, i.e. the C E E (provided that the term yajE does not originate an essential hydrodynamic contribution). On the other hand, the 1-st order hydrodynamics is defined by the following equation ± ± 2) VDM e(-VDC-lVlV DM M)). VDM = e(-VDCDM + 2'PJ 2Vj£\M, ($ (M, M)).

(3.8.8)

THE HYDRODYNAMIC LIMIT

119

However, the modified expansion can also result in the C N - S E , i.e. Eq. (3.8.8) without the term 2eVJ^2)(M,M). Keeping this in mind, consider the following lemma which is the basis for the asymptotic analysis of the Enskog equation with Arkeryd's modification: L e m m a 3.8.1. ([LA4]) The operator La with Arkeryd's modification can he split as follows (3.8.9)

La = La + aAa , where La is such that (/i^a/)La(R3)<0,

V/GL2(E3)

VaeE1

(3.8.10)

and M*Aaf

a,k

" ,K < const

/

a,k

+

(3.8.11)

One can now state the following version of Theorem 3.7.1 for Arkeryd's modification. Proposition 3.8.1. Let the condition (3.7.5) and Assumption

A >A +3

3.7.1 with (3.8.12)

be satisfied. Let to G]0, +OO[ be such that on the time interval [0, to] there exists a solution (#, u , T ) of the C E E (or of the C N - S E ) such that the Assumption 2.4.1 is satisfied. Let the initial data F be such that the con­ ditions of Theorem 3.7.1 are satisfied. If 0 < e < eo then a solution f of the problem (3.7.1) with Arkeryd's modification exists in L ^ O , £o; Y+°'°), for a0 as in (3.7.3), and (3.7.18) - (3.7.20) hold. Proposition 3.8.1 is certainly valid for the Boltzmann-Enskog equation (3.8.1) with Arkeryd's modification. The rather restrictive condition (3.8.12) seems to be essential for the proof: it allows avoiding singularities in the nonlinear term. Whether the result remains valid for a weaker condition is still an open problem. It is worth pointing out that for (3.7.5) and ft = ft + 1 a new 0(1) order term should appear in the 0-th order hydrodynamic equation. It is worth bringing to the attention of the Reader the analysis of the pa­ per [KMl] which shows, at the formal level, that different relations, between

M. Lachowicz

120

e and a suitable parameter characterizing the deviation from equilibrium, yield to different versions of hydrodynamic equations. 3.9.

Hydrodynamic Limit for the Povzner Equation

Consider the Povzner equation (2.1.10-2.1.13) in the dimensionless form Df=-P(fJ),

(3.9.1a)

£

with the initial data

f\t=o = F>

( 3 - 9 - lfe )

with dimensionless parameters e, ai, a 2 . Similar considerations as those of Section 3.7 yield the following theorem Theorem 3.9.1. Let the parameters of the model be such that 0 < ai < a 2 < ep < 1,

(3.9.2)

for (3 > 6 + 2/c0 and some k0 € { 0 , 1 , 2 , . . . } . Let t0 G]0, +OO[ be such that on the time interval [0, to] there exists a solution (g, u, T) of the CEE (or the C N - S E ) such that Assumption 2.4.1 is satisfied. Let the initial data (3.9.16) be such that F = Mo + G, where M0 — Af [ g | t = 0 , u (3.4.5) and

t s = 0 ,T t _ 0 ]

(3.9.3)

and G satisfies the smallness condition

GelloH Y^k ,

(3.9.4)

with a and k being large enough. If 0 < e < £0, where £Q is a critical value (depending on t0), then a solution f of Eq. (3.9.1) exists in the space Loo(0, £0; Y+°'*°); for some a0 > 0; and /#GC°([0,to];Y^'fco),

^ /

#

€ M0,to;Y£°'fco),

sup II /(*) - M(t) - /«» f -) I "-fc0 < Q o £ , 0
\£ /

(3.9.5a)

(3.9.56)

(3.9.6)

THE HYDRODYNAMIC LIMIT where cto is a constant (depending onto).

feC°

121 Moreover

([CMoliY*0-*0-1) n c 1 ([(Mo];Yao'fc°-2) ,

(3.9.7)

provided that ko > 2. Theorem 3.9.1 is a basis for Theorem 3.10.2. Hydrodynamic limit of the Povzner equation for the case 0 < ai < a 2 = e

(3.9.8)

can be discussed similarly as the same type of a limit for the BoltzmannEnskog equation in Section 3.8. Note that the hydrodynamic equations corresponding to the Povzner equation are exactly the same as those for the Boltzmann-Enskog equation. In particular for the Povzner equation with Arkeryd's modification (actually it was an equation originally considered by Povzner [POl]) the analogue of Proposition 3.8.1 can be stated P r o p o s i t i o n 3.9.1. Let Condition (3.9.8) be satisfied. Let to £]0,+oo[ be such that on the time interval [0,to] there exists a solution (g,u,T) of the C E E (or of the C N - S E ) such that Assumption 2.4.1 is satisfied. Let the initial data F be such that the conditions of Theorem 3.9.1 are satisfied. If 0 < e < £o t h e n a solution of Problem (3.9.1) with Arkeryd's modification exists in Loo(0,*o; Y+ 0 '* 0 ), for some a 0 > 0 and k0 > 0, and (3.9.5)-(3.9.7) hold. 3.10. S D E Approximating Solutions of the Hydrodynamic Equa­ tions One of the most important but still unsolved problems of the applied math­ ematics is derivation of the kinetic equations and hydrodynamic equations from the Newtonian laws of motion for the particle system (more precisely: from the Liouville equation). The basic question is how to derive the Boltz­ mann equation from the system of particles which are hard spheres. Grad [GR1], having considered the matter formally, described the role played by the Boltzmann hypothesis ("Stosszahlansatz") about the molecular chaos and has proved that the limit transition should be done so as to maintain the product of the number of particles and the square of the particle diam­ eter constant and greater than zero. This kind of limit, when the number

122

M. Lachowicz

of particle approaches infinity, is called the Boltzmann-Grad limit. The inverse of this product is the scale of the mean free path which determines the mean path of the particle between the collisions. The transition from the Liouville equation to the Boltzmann one is done through a system of equations (for sequence of correlation functions) called BBGKY hierarchy (Bogolubov, Born, Green, Kirkwood, Yvon). The problem of the existence of the Boltzmann-Grad limit for the BBGKY hierarchy was considered by Lanford III [LN1], although he sketched only roughly the proofs of the formulated theorems. Nevertheless, the Lanford paper is regarded as the fundamental and revolutionary contribution in solving the problem. Shinbrot [SHI], Uchiyama [UC1] as well as Gerasimenko and Petrina [GP1] have proved that series of iterations of the BBGKY hierarchy is convergent locally (in time) for initial data from the space of sequences of bounded functions with respect to the position variables and decaying exponentially (at infinity) with respect to the velocity variables. Illner and Pulvirenti [IL1] have obtained an analogical result globally in time, for small initial data, decaying exponentially with respect to both the velocity and the position variables. However, it should be emphasized that all the aforementioned papers are extremely complicated mathematically. On the other hand, the results discussed above concern very specific (simplified) conditions: the lo­ cal theorems related to the time intervals of the order of the mean free path and the global theorems - to initial data with the norm being smaller than a critical constant of the mean free path order. Thus, none of the above theorems is suitable for transition from the Liouville equation to the hydrodynamic equations (the mean free path tends to 0). When the simplifying conditions are omitted, the analysis becomes so difficult that the presently observed no-result state does not seem to be overcome in the near future (cf. [CD1], [GP1J, [MU1], [OE1] and [UC1]). In view of the lack of a suitable mathematical theory allowing to derive first the kinetic and then the hydrodynamic equations from (deterministic) particle systems, the very important problem is a construction of a cor­ responding theory starting from stochastic particle systems. In the case of the Liouville equation - the randomness enters into the system only by the initial distribution of the particles in the phase space. However, one can consider systems with stochastic interactions between particles. Such stochastic systems, by their random nature, can make the analysis of long time behavior easier and can permit to exploit the ergodic properties (cf. [CD1]). On the other hand, some important features of the determinis-

THE HYDRODYNAMIC LIMIT

123

tic model can remain in a stochastic description. Moreover, the stochastic modeling of complex physical systems, even if they are deterministic, can be important in that a deterministic model itself cannot take into account all factors involved in the evolution of the system (see discussion in [BE2] and [SKI]). Such elements as complicated geometrical forms of the particles, quantum effects and internal degrees of freedom may have an influence on the evolution. Last but not least, the stochastic modeling can provide a useful nu­ merical algorithm for the evolution both at the kinetic theory and at the hydrodynamic levels. The major part of the numerical schemes is based on the simulation of the huge particle systems that can approximate the evolution of the distribution function given by a kinetic equation (cf. [IVI], [KU1], [LU1,2], [WA1,2] and references therein). Therefore, the stochastic particle systems play a significant role in developing the numerical analy­ sis for both the kinetic and the continuum theories, and the mathematical results on convergence as well as estimation of the error of approximations are of great importance. In the spatially homogeneous case the idea of derivation of a simplified Boltzmann-type equation (for the so-called caricature of a Maxwellian gas) from a stochastic model was proposed by Kac [KC1,2]. Similar idea had appeared earlier in the paper by Leontovich [LT1] but had rather been ignored by the mathematicians. Kac's model was studied by McKean [MC1]. Nowadays there is a huge literature on various stochastic approaches in the kinetic theory - see for example [AS2,3], [HOI], [LA5,8,10], [SKI], [SZ1], [TA1], [UC1], and [WA1,2]. In the paper [LA5] the starting point is a particle system similar to that of soft spheres considered by Cercignani [CE2]. It is a system of identical point particles that can interact in pairs, when the distance between two of them is not greater than some number (which is one of the parameters of the model). As a result of the interaction there are jumps of the velocities of two interacting particles. The main theorem of the paper [LA5] is related to approximations (in L\) of the Maxwellian (with the fluid dynamic parameters given by a solution of the C E E ) by the solution of the prospective equation (i.e. the modified Liouville equation) of the model. Approximations are realized if the parameters of the model are suitably chosen. All estimations of the paper [LA5] allow to control the rates of approximations. The paper [LA8] was a continuation of the paper [LA5]. In a sense,

M. Lachowicz

124

Skorohod's approach [SKI] - which allows to construct the processes in complex phase spaces by the stochastic differential equations (SDE) - was combined, in the paper [LA8], with derivation of the hydrodynamic equa­ tions from the kinetic equation. In [L10] approximations at the kinetic theory level were compared to those at the hydrodynamic one and the role of various parameters of the model was described. In the system considered by Skorohod, as a result of stochastic inter­ actions there are jumps of the velocities of two interacting particles, where moment of time and size of jumps are described by the Poisson random measures (cf. [IK1]). Skorohod's attempt was carried out by Arsen'ev [AS2], who obtained in the limit N -> oo (N is the number of particles) the kinetic Povzner equation for the hard sphere interaction potential. The common disadvantage for both these results is that the convergence is in a weak sense and they do not allow to control the rate of approximation. A dif­ ferent approach was proposed by Arsen'ev [AS2] in the space homogeneous case, in order to construct the appropriate numerical algorithm for the space homogeneous Boltzmann equation. The general strategy of the paper [LA8] is similar to that of the paper [LA5]. Analogously to Skorohod [SKI] and Arsen'ev [AS3], the starting point is a system of S D E of the jump type: interactions between different pairs of particles are independent and as a result of an interaction there are jumps of the velocities of two interacting particles. Consider a system composed of TV identical point particles with positions inT3. The evolution in time of the state is given by N stochastic processes Zi = ( X i ? V 0 ,

t= 1

JV,

(3.10.1)

where Xi(t) and Vi(t) describe the position and the velocity, respectively, of i-tb. particle at time t > 0. The conditional probability that the jump of velocity of i-th. particle in the interval of time At is in a Borel set W c l 3 can be written as P{Avi <E Wl*,*,-) = pL(WiZiiZj)At, where //(., Z{, Zj) is a measure on E . Consider: the probability space (fi,3,-P) and the measurable space (0,9Jt).

(3.10.2)

THE HYDRODYNAMIC LIMIT

125

Let b be a measurable function and q - a measure on 0 such that q({0 G 0 : 6(z i3 Zj,0) G W}) = / x ( W , * , Z j ) .

(3.10.3)

The existence of such a function b for an arbitrary measure \i was discussed by Skorohod [SKI]. Definition 3.10.1. We say that on [0, to]xQ a Poisson random m e a s u r e ^3 is defined if, for each Borel set I C [0, to] and each U G VJl, the random variable ^3(7 x U) (on the probability space (H,3>P)) is such that i) some measurability and independence conditions are satisfied (e.g. [IKl]). ii) ^3(1 x U) is Poisson distributed and £ < P ( / x U ) = |/|q(U),

(3.10.4)

where |.| is the Lebesgue measure on Mr, q is a real-valued function on Wl and E is the expectation. For a given a-finite measure q on (0,9K) one can always construct (see [IKl]) a Poisson random measure ^3 such as in Def. 3.10.1. Moreover, the probability law of ^3 is uniquely determined by the measure q. q(d0) is called the characteristic measure and dtq(dd) - the inten­ sity measure of the Poisson random measure *J3. Now let 0 be the finite set as follows 0 = { 1 , . . . , /} where I is an integer. On 0 define the following measure q(U) = (sNVa^aJ)-1^

■ £ ( 2 0 - 1) eeV

(3.10.5)

for U C 0 , where £, N, Oi, a 2 , / and 7?* are (positive) parameters of the model, and 47rV^lf0a is the volume of the set {x G T : ax < |x| < a 2 } . Now, let {tyi j}i
x d0) = dtq(dO),

(3.10.6a)

for 1 < i < j < N. Moreover, let Vij = Wji

for

1 < i < *<

N

■

(3.10.66)

126

M. Lachowicz

Now, assume that on the time interval [t, t + dt] the jump of the velocity of i-th particle, due to the interaction with j-th one, is defined by the following random variable

Jb(x Jb(Xi(t),V d0), i(t),y t(t)]x 3(t),v 3(ty,e)^ l(t);X 3(t),V J(t) ]e)¥lJ(dt lJ(dtxx de), e where ^b(Xi,V»Xj*Vj,6) (X^V^XJ.VJ,^)

= | X i - Xj\)^{\Vi =$$a Xj\)9*{\Vi ■~- vVj\) ,|) a i iltaa2 2((|Xi

x ^e(zi,Zj)(0 V (VJ (VJ --v{v,-) )- ■nij)nij x $0(zi, Zj)(0 V nij)nid . . In the above formula the following notations are used Ilfj| 1, nitj — = ^Xj (xi - XjJ|Xf Xj)\xi - Xj Xj'l"

for for Xj x» / Xj Xj ,

$v and ^e are smooth, non-negative functions such that ay
*oi,a2a(2/) (!/) == 0 °

if 2/ < ai -— < <55 or r > a a,22 + 6 S ,,

*"(y) =1 **(y) = l if 2/ y < 77, ^7, n y > TJ + -1- 68 $*(y) * (y) = 0 if y>r}

and

^ ( x ^ iV = ^(xj,vi;Xi ,vi), 90(xi yii;xj xiiy^j)V j ) =

^e(xi}
if (x*, Vjjx^v^) € V
^(xl-,vi;xijvJ)=0

33 if ( x i ^ i j x j . v j ) <E (T3 x R )\V*, xR )\Ve,

where V* = {{(Xi.vtfxj,^) ( x ^ j x ^ V j ) G (T3 x M3)22 : |\VJ\ vj| < ?*,
x, Xi # ^ XXjj ,

||vi| 77* , , V i | < rj*

-1 2 ( ^ - l l)r)+r ) 7 ? , rl1 < < {VJ (v,- -- Vi) 2(0 v^) ■ - nm ^ jtj <<2077*/ 20T]J-}1}

(3.10.7) (3.10.7)

THE HYDRODYNAMIC LIMIT

127

and Ve C Vg is such that

s 7'

meas(V*\V,)< meas(V < j , e\Ve) where meas (•) is the Lebesgue measure in (T3 x R3)2. The evolution of the system is defined by the following system of SDE N

r

dVi(t) = JJ2j / 60C|(«),V 60t«W.Vii(t);3C,-(t),V (d* x d0), (t);3C,-(t),Vi(t);tf)!pwv(d* d0), ddXi(i) Xi(i)= =Vi(t)dt V , ( t ) dt t )

(3.10.8a) (3.10.8a)

(3.10.86)

for % i =—1,...,JV. 1,...,JV. Smoothness of the function 6 guarantees both the existence and uniqueness of a solution to System (3.10.8). Denote this solution by (Zj (*),..., Z J V M ) , where Z4 = (Xi,V*). Introduce the following "statistical distribution function" (a random measure) of the system of particles described by (3.10.8) 1

JV N 1

^f(U) = - N- N-Y,J2x(Zi(t)eV), x(Mt) G U), rf(V)

(3.10.9) (3.10.9)

i=l

for the Borel set U c T 3 x l 3 ; where x(truth) = 1 and x(false) = 0. The random measure ^ can approximate solutions of the kinetic as well as the hydrodynamic equations. First consider the Povzner equation (2.1.10) for the hard sphere model (2.1.3). The following theorem can be stated ([LA8-10]) Theorem 3.10.1. (Approximation at the level of the Povzner equation). Let the initial distribution of the particles be given by a function F e Xj^oo,- a > 2; such that m3F G X^i and m2F G Xj j00 , where mQ is

128

-

M. Lachowicz

given by (2.4.1). Let e > 0 and 0 < ai < a 2 < 1 be fixed parameters. note

De­

<&„ = — V ^ - V t f V — . N / 77*

(3.10.10a)

0 < 6* < 60 ,

(3.10.106)

If

where 80 > 0 is a critical constant, t h e n in Xi f i there exists a unique (mild) global solution f of the Povzner equation (2.1.10) {with (2.1.13) and (2.1.3)) with the initial data F, and the following estimation holds sup | | £ / 4 v - / t | | < A i < 5 i / A l + A 2 - , o
(3.10.11)

for each fixed t\ E]0, +OO[, where ft is the measure with the density f(t), Ai and A2 are positive quantities depending on the parameters of the model: Ai depends only on n, ti, e, a>\% a>2, whereas A2 does on t\} e, 01, a2; l l l l Is the following norm \\m\\=

sup I*|LIP<1

/

/

0(x,v)m(dx, dv)

ima rm?

and I • I Lip is the norm in the space of the Lipschitz-continuous

functions.

Theorem 3.10.1 shows that the solution of the Povzner equation for the hard spheres model can be approximated by the system of S D E , provided that the parameters of the system are suitably chosen. For a given value of the permissible error, one can choose firstly 77 - sufficiently large dependently on e, 01, ei2 and then 8* - small dependently on e, a\, a2 and 77 to make the error less than the given permissible value. R e m a r k 3.10.1. The result of Theorem 3.10.1 gives the possibility of ap­ proximations of the solutions of the Boltzmann equation for the hard sphere model as well as of the Boltzmann-Enskog equation. (i) The approximation of the solution of the Boltzmann equation is real­ ized (see [LA5]) if e is fixed (e ~ 1), a\ <& e, a2 <^ £, n » a\ V a2 V e and 5* <3C a\ Va,2 V77 Ve, on the time interval [0, to], where to e]0, +00[, on which o

the smooth (in the space variable x G T ) solution to the Boltzmann equa­ tion exists. Note that the smooth solution are known to exist for example for the case close to equilibrium, see [ST1] and [UK1].

THE HYDRODYNAMIC LIMIT

129

(ii) The approximation of the solution of the Boltzmann-Enskog equation is realized if e is fixed (e ~ 1), a\ = a > 0 is fixed and 0 2 - a « a V e , 77 » a V a 2 V e and 8*
(3.10.12a)

0
(3.10.126)

0 < 0"* < 50 ,

(3.10.12c)

where £Q and So are critical constants, 6* is defined by (3.10.10a), t h e n sup | | ^ - M f | | < A 1 ^ o
sup I WoEfi? 0
V sup

-Qt\v

/ A l

+A2-+c *?

sup I WiEtf

t 0

£,

(3.10.13)

- [guil

0
|W4^r-W3^+|u|2)];|
*****

(3.10.14)

t = 1,2,3; _ where Ai, A 2 , 6*, | | - | | are rein Theorem 3.10.1; Mti Qt, [gutfti [Q&T + \u\2)]t are the (signed) measures with the densities M(t) = M[g,u,T](t), g(t), 2 g(t)ui(t) and g(t)(ST(t) + |u(t)| ), respectively; i = 1,2,3; Wjtn(dx) = / Vj(v)m(dx, d v ) ,

130

M. Lachowicz

ipj are given by (2.1.56), j = 0 , . . . , 4; and

|n|=

/n(dxMx) sup / n(dx)0(x) .. |0|Lip
Theorem 3.10.2 shows that the fluid-dynamic quantities Q, U, T given by a solution of the hydrodynamic equation ( C E E or C N - S E ) can be approximated by the moments of the expectation of the random measure ("statistic distribution function"), describing the number of particles in domains of the phase space and corresponding to the system of SDE, provided that the parameters of the stochastic system are suitably chosen. For a given value of the permissible error, one can choose firstly e - sufficiently small, then a\ and Q2 small dependently on £, rj - large dependently on £, oi, &2 and finally S* - small dependently o n e , oi, 02, 1 to make the error less than the given permissible value. 4. 4.1.

Incompressible Dynamics Level of S m o o t h Solutions

Consider the singularly perturbed problem for the Boltzmann equation in the form

e^ + v+ vV vx / / == i jJ( / , // ) ,

(4.1.1a)

f\t=0 = FF.4=0=

(4.1.16)

W - "

? t/' >'

and note that the case of q = 1 corresponds to (2.2.8) type of singularly perturbed problem was investigated and [MS2] (see also [SO 1,2]). First, consider the case of q — 1. Analogously as solution to Eq. (4.1) can be searched in the form of the nn

ei Ci)

/(*) = E / = 00 jj =

with (2.2.9). This in [BAl-5] [DM1] in Section 3.5 the following sum:

nn

W+^c'/u)W+filr(*)l j=0 j=0

where now d is the following "stretched" time variable fl = —. e2

(4.1.2)

THE HYDRODYNAMIC LIMIT

131

As previously, /(°) satisfies the following equation J(/(°),/(°))=0

(4.1.3)

and therefore /(°) has to be a Maxwellian fW=M[6iu,T].

(4.1.4)

Assume that Q = Q0 = const,

T = T0 = const

(4.1.5)

and u are known and such that Assumption 2.4.1 is satisfied. Decompose now fW and fW as in Sections 3.2 and 3.4 fU)

= gU)

+ hU) f

(4.1.6)

where gW € K, /i ( i ) € N and / 0 ) = §W +h{j),

(4.1.7)

where g(J> G 7 ^ , /i ( j ) € M), for j = 1,2,3,4,... . The bulk approximation equations (corresponding to those of Sections 3.2 and 3.3) are the following: 2J(g^\M)

2J(gU\M)

= VL(v ■ V X M),

= V^iy ■ V x / ^ - 1 ) + 5!±—- - ^

at

(4.1.8a)

J ( / W , / « - * ) ) ) , (4.1.86)

for j = 2 , 3 , 4 , . . . , dM 1 p £ £ i + -V{x ■ V X M) 4- Mv • V x p = - P ( v • V x o ( 1 ) ), at e

(4.1.9a)

( v • V , ^ 1 ) ) - Mv • V x p) = -7>(v • VxP<2)) , J at e \

(4.1.96)

V

^ L at

+ I T ? ( V • Vxh&) = -V(v ■ V x (/' + 1 >), e

(4.1.9c)

132

M. Lachowicz

for j = 2 , 3 , . . . . Now Eq. (4.1.9a) together with (4.1.8a) result in the following equation for (u,p):

(4.1.10a) (4.1.10*)

£g-0, 1=1

duj

1 Y^

^ + iE".^ t1 = 1i

duj

= --dx^ + ^Ea^. • ' ^LtdiA 1=1 J J

1=1

1 2 i> = =1,2,3. . ,3.

(4.1.106) (4-1.106)

%

*

This hydrodynamic equation contains the singular terms (with respect o e). In order to avoid this difficulty, one can assume that U u = e£ u '' ,,

(4.1.11a)

ep' p = ep'

(4.1.116) (4.1.116)

and (u',p') satisfies the I N - S E (2.3.4). Such a type of assumption was proposed in [DM1]. Note that the initial layer equations are similar to those of Section 3.4. Moreover, the equation for the remainder has the analogous form and can be treated exactly in the same way as that in Section 3.5. Thus one has: T h e o r e m 4.1.1. Let to G]0,+oo[ be such that on the time interval [0,to] There exists a solution (u',p') to the I N - S E (2.3.4) and (0, u',T) satisfies Assumption 2.4.1, where g = g0 = const andT = T0 = const. Let the initial data (4.1.16) be such that

F =M MQ0 + G, G,

(4.1.12)

where r Mo0 = Af M[0o,eu [go, €u' , |t=o' t = o ,Tb] °l

and G satisfies the smallness condition (3.4.5) and G G Ho D Y^k with a and k being large enough. If 0 < e < e 0 , where e 0 = £o(to) is a critical value, then a solution f of Eq. (4.1.1) with q=l exists in L ^ O , ^ ; Y+°'fc°), for some e*o > 0, k0 > 0 and sup 1| f(t) M[eo,eu',T «/w (£\ 0](t) 0](t)--- /<°> / ( * ) --- M[eo,eW,T fm ( 1 ) -- «/W

€[0,to]

(?) U/

(?) Ve 2 y

2 ,:o < I ^°' »•■*• Cto cetoe2,

(4.1.13)

THE HYDRODYNAMIC LIMIT where cto is a constant (depending ont0).

133 Moreover,

f e C o ([0,
(4.1-14)

Estimate (4.1.13) for G = 0 (and so that for /W) = 0 for j = 0,1,...) was obtained in [DM1]. Consider now the case of q = 2 , 3 , . . . in Eq. (4.1.1). Then the following result can be formulated (cf. [DM1]): Remark 4.1.1. If (u',p') is a smooth solution of the IEE (2.3.3) then the theorem analogous to Theorem 4.1.1 can be formulated for Eq. (4.1.1) with q = 2,3,... . Theorem 4.1.1 leads to the conclusion that putting aside the initial layer the solution to Eq. (4.1.1) for q = 1 can be represented as /(*) = u + ef'(t),

(4.1.15)

where u is a global Maxwellian and / ' is an unknown function (nonsingular with respect to e). Therefore, one can consider the problem (4.1.1) with the initial data F = LJ + EF',

(4.1.16)

where F' is assumed to be independent of e. In the paper [BA3] the connection between the classical solution for the Boltzmann equation in the whole space 1R3 (due to Ukai [UK4]) for F' sufficiently small but independent of e with the classical solution of the I N SE for small initial data was studied and the strong convergence of / ' as e I 0 was proved. Some useful estimates regarding the dependence of the solution of Eq. (4.1.1) which is represented according to formula (4.1.15) on the parameter e can be found in the book by Maslova [MS2]. 4.2. Level of Weak Solutions The existence of a (time) global weak solution to the IN-SE (with large initial data, but corresponding to finite kinetic energy) was proved by Leray [LEI]. On the other hand, a (time) global solution (the renormalized solu­ tion) of the Boltzmann equation (with large initial data, but corresponding to finite mass, total energy and entropy) was obtained by Di Perna and

M. Lachowicz

134

Lions [DL1]. Both theorems have a similar nature as (cf. the discussion in [G02]) they allow certain global quantities (kinetic energy in the case of the IN—SE and total energy in the case of the Boltzmann equation) not to increase (in time) as well as to have a dissipation rate higher than expected. Both theorems do assert neither regularity nor uniqueness of the solutions. The first steps towards the mathematical description of the relation be­ tween the Di Perna and Lions renormalized solutions and Leray weak solu­ tions were done in [BA 1,2,5] - see also [GO2]. Consider the more general representation than that of (4.1.15) f(t)=u

+ e«'f'(t),

(4.2.1)

for q' > 1. The following (formal) result was formulated in [BA2] (see also [BA1,5] and [G02]) Theorem 4.2.1. Let f be a non-negative solution of Eq. (4.1.1) such that, when it is written according to formula (4.2.1), / ' converges in the sense of distribution and almost everywhere to a function h! as e tends to zero. Furthermore, assume that / /'(*, x, v) dv ,

/ Vif'(t, x, v) dv,

y « < |v| 2 /'(*,x l v)dv 1

/ ViVjf'(t} x, v) dv,

/'ay{v)»*/'(* 1 x f v)dv l

/nw(v)J(/',/')(*, x,v)dv, y,Si(v)^/'(i,x,v)dv,

ySi(v)J(/'J//)(t5x,v)dv

converge in the sense of distributions to / /i'(*,x,v)dv,

/ Viti(t,x,v)dv,

/

ViVjti{t,x,v)dv,

/ ^|v| 2 /i'(£,x,v)dv, /'ny(v)i; Jfc fc'(t,x l v)dv J

/^(v)J(/i',/i')(*,x,v)dv,

THE HYDRODYNAMIC LIMIT

y 3,(v)vift'(tlx,v)dv>

135

/ H i ( v ) J(/i',/*')(', x , v ) d v ;

for i,j, k = 1,2,3; where flij and Sj are £/ie unique solutions to the following equations: J(Lj,ni:j)(v)

= (viVj - -\v\A u(v)

(4.2.2a)

and J(w,3<)(v) = ( i|v| 3 ti, - 5 V i )

W(V).

(4.2.26)

Moreover, assume that all formally small terms in e vanish. Then the lim­ iting h' has the form

/i'(i,x,v)=a;(v)^,x)+v.u(t,x)+Q|v| 2 -0r(t,x)) , (4.2.3) where the density and temperature fluctuations g and T satisfy the Boussinesq relation (2.3.5a) and (g,u,T) is a weak function of: (i) the I N - S E (2.3.4) together with (2.3.5) for q = 1 and q' = 1; (ii) the SE (2.3.6) for q = 1 and q' > 1; (iii) the I E E (2.3.3) together with (2.3.56) with 0 instead of fiy for q > 1 and a7 = 1; (iv) £/ie trivial equations (2.3.6) tu^/i 0 instead of both ^iv and fi*H for q > 1 and q' > 1. The program by Bardos, Golse Levermore ([BA1,2,5] and [G02]) of find­ ing the connection between the Leray solutions of the I N - S E (the case (i)) or of weak solutions of the SE (the case (ii)) and the renormalized Di Perna and Lions solutions of the Boltzmann equation is not yet completed in the rigorous way. The hydrodynamic limits are based on the fundamental prin­ ciple of dynamics and then are strictly related to the local (in the space variable x) conservation laws. However, the only conservation law known to be satisfied by the renormalized solutions is that of mass and the ques­ tion whether the renormalized solutions satisfy local momentum and energy conservation is still open (cf. [DL1] and [BA5]). In [BA5] - Theorems 2.5 and 6.2 it is shown that if the family of renormalized solutions satisfies the local momentum conservation then the limit at the level of SE (the case (ii)) holds. At the level of the I N - S E (the case (i)) additional assumption are re­ quired to gain some weak time regularity for the moments f Vif'(t, x, v) dv

136

M. Lachowicz

(i = 1,2,3) as well as to avoid the concentration phenomena as e I 0 (see [BA1] - Theoreme 2, [BA2] - Section 5 as well as [G02] and [BA5] - As­ sumption (H2)). With these assumptions one can show (cf. [BA2,5]) the convergence at the level of I N - S E (the case (i)). One can hope (cf. Con­ clusions in [BA5]) that local conservation of momentum and energy for the renormalized solutions of the Boltzmann equation are only needed in the limit e —► 0. In such a case the assumption on conservation of momentum could be dropped and the Bardos, Golse and Levermore program would be completed at the level of the SE (the case (ii)).

Acknowledgments The content of this lecture was prepeared mostly when I enjoyed the hospi­ tality of the Politecnico di Torino (Italy) with the support derived from the Commission of the European Communities, Grant ERB-CIPA-CT-92/2245 p. 3623. The final stages of the research were partially supported by the Polish Research Council under Grant No. 2P 301 027 06.

THE HYDRODYNAMIC LIMIT

137

References

[ARl] ARKERYD L., On the Boltzmann equation. Part I: Existence. Part II: The full initial value problem, Arch. Rational Mech. Anal, 45 (1972), 1-34. [AR2] ARKERYD L., On the Enskog equation in two space variables, Transp. Theory Statist. Phys., 15 (1986), 673-691. [AR3] ARKERYD L., On the Boltzmann equation in unbounded space far from equilibrium, and the limit of zero mean free path, Comm. Math. Phys., 105 (1986), 205-219. [AR4] ARKERYD L., Stability in L 1 for the spatially homogeneous Boltz­ mann equation, Arch. Rational Mech. Anal., 103 (1988), 151-167. [AR5] ARKERYD L., CAPRINO S., and IANIRO N., The homogeneous Boltz­ mann hierarchy and statistical solutions to the homogeneous Boltz­ mann equation, J. Statist. Phys., 63 (1991), 345-358. [AR6]

ARKERYD L.,

LIONS P.L.,

MARKOWICH P.A.,

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[LAI

Yu.N., On a statistical approach to solving of the Boltzmann equation, Zh. Vychisl. Mat. i Mat. Fiz., 26 (1986), 15271534, in Russian. KONDYOURIN

LACHOWICZ M., Initial layer and existence of a solution of the nonlin­ ear Boltzmann equation, Proc. XV Internat. Symp. on Rarefied Gas Dynamics, Grado (1986), Eds. V. Boffi and C. Cercignani, Teubner (1986), 150-159.

[LA2 LACHOWICZ M., Differentiability of the solution of a system of linear equations, Arch. Mech., 38 (1986), 127-141. [LA3 LACHOWICZ M., On the initial layer and the existence theorem for the nonlinear Boltzmann equation, Math. Methods Appl. Sci., 9 (1987), 342-366. [LA4 LACHOWICZ M., On the limit of the nonlinear Enskog equation cor­ responding with fluid dynamics, Arch. Rational Mech. Anal., 101, n. 2 (1988), 179-194. [LA5 LACHOWICZ M. and PULVIRENTI M., A stochastic particle system modeling the Euler equation, Arch. Rational Mech. Anal, 109, n. 1 (1990), 81-93. [LA6 LACHOWICZ M., On the asymptotic behavior of solutions of nonlin­ ear kinetic equation, Ann. Mat. Pura Appl., 160 (1991), 41-62. LACHOWICZ M., Solutions of nonlinear kinetic equations at the level [LA7 of the Navier-Stokes dynamics, J. Math. Kyoto Univ., 32 (1992), 31-43. LACHOWICZ M., A system of stochastic differential equations model­ [LA8 ing the Euler and the Navier-Stokes hydrodynamic equations, Japan J. Industr. Appl. Math., 10 (1993), 109-131. M., On the Enskog equation and its hydrodynamic [LA9 limit, Kinetic Theory and Hyperbolic Systems, Eds. V. Boffi, LACHOWICZ

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[LL1] LAKSHMITKANTHAN V. and LEELA S., Differential and Integral Inequalities, Vol. 1, Academic Press (1969). [LN1]

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[LOl] LOMOV S.A., Introduction to General Theory of Singular Perturbations, AMS (1992). [LTl]

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A.V. and SMIRNOV S.N., An efficient stochastic algorithm for solving the Boltzmann equation, Zh. Vychisl. Mat. i Mat. Fiz., 29 (1989), 118-124, in Russian.

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[LX1] LAX P . D . , Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math., 10 (1957), 537-566. [LX2] LAX P.D., Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964), 611-613. [MAI] MAJDA A., Compressible Fluid Flow and Systems of Conser­ vation Laws in Several Space Variables, Springer (1984). [MC1]

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Lecture 3 AN INTRODUCTION TO KINETIC THEORY OF DENSE GASES

J. Polewczak

1.

Introduction

The Boltzmann equation describes well temporal behavior of a gas in a dilute gas regime. It can be formally derived from the BBGKY-hierarchy equations by taking the low density limit (the Boltzmann-Grad limit, e.g., [LN1], [CE1]) when the initial ensemble of the system possesses certain fac­ torization property. One of the most important and attractive properties of the kinetic theory based on the Boltzmann equation has been its Hfunction, and the corresponding ^-theorem. Monotonicity in time of H(t) together with its property of being equal to the equilibrium entropy when the ensemble is in equilibrium, makes Boltzmann's H-function an excellent candidate for the non-equilibrium entropy, at least in the case of the ideal gas. These facts establish an important bridge between macroscopic irre­ versible processes and microscopic dynamics. When we pass to moderately dense gases, the Enskog equation [EN1], [FE1] plays an important role in describing temporal behavior of a gas consisting of hard spheres. The ad­ vantage of considering only hard sphere system resides in two facts: the collisions are instantaneous and influence of multiple collisions (i.e. simul­ taneous encounters of more than two spheres) is negligible. In moderately dense gases the molecular diameter is no longer small compared with the 149

KINETIC THEORY OF DENSE GASES

151

conservation of energy. Furthermore, in contrast to the Boltzmann and the Enskog equations the square-well kinetic theory in [KRl] consists of two coupled equations for the one-particle distribution function fi(t, xi, vi) and the potential energy density up(t,xi). The function up(t,xi) is given in terms of the two-particle distribution function / 2 by *fp(*,Xl) = -

/

^ S W ( | X i - X 2 | ) / 2 ( « , X i , V i , X 2 , V 2 ) d V i d v 2 d X 2 . (1.2)

fixR3xR3

In this lecture, I want to present the above-mentioned square-well kinetic theory. Although this theory is well-established in statistical physics, it is relatively unknown in the mathematical community, even by mathemati­ cal physicists working on rigorous results for the Boltzmann and Enskog equations. Section 2 reviews the revised Enskog equation and will serve as a com­ parison for the more advanced kinetic theory developed in Section 3. In ad­ dition to many physical concepts underlying the theory, I will also present several properties of the square-well kinetic theory that seems to be im­ portant in obtaining existence and stability results for the corresponding system of equations. As the two preceding lectures show, several mathe­ matical theorems have been obtained for the Boltzmann equation and some simplifications of the Enskog equations. Among these are stability and global existence results and various hydrodynamical limits for these equa­ tions. None of these theorems has yet been proven for the square-well kinetic equation. In addition, the method of maximization of ensemble entropy that is used in derivation of the square-well kinetic theory is interesting in itself and, in principle, could be used to obtain still more advanced models of kinetic theory. The author hopes that the material presented here will raise interest in these directions. Parts of this lecture are based on the works of Polewczak [POl] and Polewczak and Stell [P02]. 2. The Revised Enskog Equation Consider a fluid consisting of hard spheres of diameter a and, for simplicity, mass m = 1. The state of the fluid is described by the one-particle distribu­ tion function / i ( t , x i , Vi) which changes in time due to free streaming and collisions. The function / i ( t , x i , v x ) repesents at time t the number density of particles at point xi with velocity vx. When two particles, with positions

J. Polewczak

152

at xi and x 2 , collide, their velocities Vi, v 2 take postcollisional values v[ = v i - n(n, vi - v 2 ) ,

v 2 = v 2 + n(n, vi - v 2 ) .

(2.1)

Here, (•, •) is the inner product in R 3 , and n is a vector along the line passing through the centers of the spheres at the moment of impact, i.e. n G §2+ = {n E M3 : |n| = 1, (vi - v 2 , n) > 0} . The exact rate of change of the distribution /i(t,Xi,Vi) is given by the equation

"gr

+ V l

^r

=

/

rfx

2^v2^12/2(*,Xi,Vi,X2,V2),

(2.2a)

where ^12/2 =a2 / [ / 2 ( t , X i , v i , x 2 , v 2 ) J ( x ! - x

2

-an)

-/2(i,xi,vi,x2,v2)(5(xi - x 2 +an)](n,vi - v2)dn.

(2.26)

The density of pairs of particles in collisional configurations is described by the two-particle distribution function / 2 . Expressions for the fn are given and discussed in terms of closed systems in section 3. We note that in the case of the gas considered in a spatial domain Q, ^ R3, equation (2.2a) with (2.2b) needs to be supplemented with suitable boundary conditions. Equation (2.2a) with (2.2b) is one way of writing the exact first BBGKY hierarchy equation for a hard sphere system, for which the matching condi­ tion /2(^,xi,v / 1 ,x 2 ,v 2 ) = / 2 ( t , x i , v i , x 2 , v 2 )

(2.3)

is satisfied for all v i , v 2 , and x i , x 2 with |xi - x 2 | = a + , where v i and v 2 are post-collisional velocities. The matching condition is typically lost when one introduces an approximate / 2 into (2.2a) and (2.2b), which be­ come irreversible. The usual way of introducing an approximate / 2 is by expressing it as a functional of f\ that is assumed to be independent of

J. Polewczak

150

mean free path between collisions. As the collision frequency is increased by the presence of other spheres, Enskog introduced an additional factor in the collision operator that is responsible for this increase. An impor­ tant consequence of this is that the transport of momentum and energy during collisions (negligible in the Boltzmann-Grad limit, and consequently in the Boltzmann equation) takes place over distances comparable to the separation of the molecules. In spite of its success in well predicting the transport coefficients (correct within 5% error for densities up to 3/4 of the close-packing density, see, for example, [BJ3]), in its original version, the standard Enskog theory (SET) suffers from several drawbacks. It is not known whether the kinetic theory based on the SET exhibits an if-function (see, however, [HU1] and [GM1] for partial results in this direction). Also, the SET does not drive a system with an external stationary force to the correct equilibrium distribution, and finally, in the case of mixtures, Onsager's reciprocity relations are not satisfied. The revised Enskog theory (RET) (see, [BJ1] and [RE1]) rectifies these drawbacks. In addition to the one-particle distribution function / i , description of dense gases and liquids, even on the hydrodynamical scale, requires the knowledge of the two-particle distribution function ji- Indeed, the expres­ sion for the potential energy density involves fa. (In the case of hard spheres system the potential energy density is zero.) This together with the fact that real molecules are not hard spheres has led many in search of a gener­ alization of the RET to the kinetic equations with more realistic potentials. Since smooth potentials can be approximated by a sequence of step func­ tions, the square-well potential 0 and 0 < a < R, <j)SW is given by ' 00 ,

for r < a

SWW = < - 7 , foTa
(l.l)

for r > R .

Recently, for the potential sw in (1.1), Stell et al. [ST2] (see also [KR1]) developed the square-well kinetic theory using the tool of maximization of entropy to close the hierarchy equation. The resulting theory improves greatly on the older version of the square-well kinetic theory of Davis, et al [DAI]. It exhibits an H-functional and has built in mechanism of the

KINETIC THEORY OF DENSE GASES

153

boundary conditions. For our purposes it is convenient to do this by first writing f2 in the form /2(*,xi,vi,x2,v2) = G(i,xi,v1,x2,v2)/i(Mi,Vi)/i(*,X2,v2).

(2.4)

Next, let us observe that f2 and hence G enters the kinetic equation only for arguments that characterize the precoUisional conditions of binary hard sphere impact, i.e., for x12 = |xi - x 2 | = a+ and (vx - v 2 ,n> > 0. This is obvious for the second / 2 in (2.2b). The first / 2 is evaluated for primed velocities but at x 2 = xi — on, which makes it "precoUisional" also. I shall denote G for such precoUisional variables as Y. The way in which one approximates the exact two-particle correlation function G at impact gives rise to the different kinetic equations found in the literature. The Boltzmann equation is obtained by assuming that Y = 1 and that the change of /i(£,Xf,Vi), i = 1,2, over a length a for arbitrary t and Vi is negligible, so that /i(£,x;, v^) w fi(t,x.i + an,Vi). This choice for Y is adequate in the low density limit. For more general Y, the dilutegas limit is achieved when Y —> 1. In the homogeneous case (/i does not depend on a position) Y —>■ 1 is implied by pa3 —> 0. In nonhomogeneous situations, however, Y —> 1 is formally implied by the local mass estimate /

pit, y) dy -> 0

JB(x:,a)

for x 6 fl, where /o(t,xi)= /

/i(*,xi,vi)dvi,

J5(x,o) = {y € ft: |y - x| < a} ,

and Q C IR3 is a spatial domain where the fluid is confined. We observe that in the homogeneous case the local mass estimate is equivalent to pa3 — ► 0. An interesting generalization of the Boltzmann equation (in principle it is adequate to describe the dilute-gas limit no matter how rapid is the spatial change in / i , and hence in p, on the length scale of particle diameter) follows from making only the assumption Y = 1. In the literature this model is known under the name of the Boltzmann-Enskog equation. In the standard Enskog theory (SET) (see, [EN1] and [FE1]) Y is given by

YSET(t, X!, x2) = g2 ( V I p(t, 5 ± £ ) ) ,

(2.5)

154

J. Polewczak

where p(t, x) is the local density, and #2(^12 | p) is the pair correlation function at particle separation x\2 in an uniform equilibrium state at density p. In the revised Enskog theory (RET) (see, [BJ1] and [RE1]) Y is taken to be the "contact value" of the pair correlation function g2 for a nonuniform system at equilibrium with local density p(x) in which the correlations depend upon p(x) and the excluded volume of the spheres. In particular, there are no correlations between velocities in the system. In this case one can write RET YRET (t,x = 92(Xl, g2x(x2 l,x \p(tr•)) )) (t, uxX2)l , x 2 ) = I 2p(t,

•

(2.6) (2.6)

x12=a+ xi2=a+

The term "revised" points to the fact that in the revised Enskog equation g2 describes correlations of a nonuniform rather than a uniform equilibrium state, so that p2(xi,X2 | /?(•)) is a functional of p(-) rather than simply a function of the uniform density p. In terms of the formal Mayer cluster expansion, g2 has the form [ST1], HS 02(xi,x g2(xux2\p)2 \p) = eexp x p ((-P - / f y i f 5 ( {\xi | x i ---xx22||))))xx (( xx ii , x 22 | p ) ,

(2.7) (2.7)

where 00

1

f

■■Jdxdx p(3) . .... p(k) X(xi,x = l1 + +^ J T^ — dx 3 ■• V(12 | 3 — Jb). p(ib)V(12|3...ik). k k p(3) X ( x 1 , X22 |\p) p) = _ 21 ) —! yy dx 3 • k =3 3 =

Q

Q

Here, p(k) = p(t>xk), and j3 = l/kBT, V(12 | 3 . . . k) is the sum of all graphs of k labeled points which are biconnected when the Mayer factor 5 /12 / 1 2 = eexxpp[[--^^5 ((|| xX 1l -- xx 2 | )) ]]--- l1

is added. The function (pHS is the hard spheres interaction potential given by HS

(r) =

{l o ,

( • 00 0 0 ,,

0,

for for r < a for r > a .. for

(2.8)

In the case of the hard sphere system, the Mayer factor / 1 2 = 0 1 2 — 1, where 0 i 2 = @(|xi - x 2 | - a), and 0 is the Heaviside step function. As an

KINETIC THEORY OF DENSE GASES

155

example, I provide below the expressions of 1/(12 | 3 ... k) for k = 3,4 V(12|3) V(1213) =(l)/i — ( l ) / l83/28, /23,

(2.9a)

y(12|34) = ( 2 ) /1l 33//3344/24 /24 + V(U | 34) =(2)/ + (2)/i; (2)/l3/34/24/23 + (2)/i 3 //34/l4/24 $/3< l / 2 4 / 2 3 3
(2.9b) (2.%)

The numbers in parentheses represent the corresponding symmetry factors. This is precisely the above algebraic structure of g2 that plays a fundamental role in obtaining various a priori estimations leading to existence and stability results for the revised Enskog equation (RET). With the help of (2.4), (2.6) and (2.7) equation (2.2) assumes the form r»f f\to revised r o t r i c o H Enskog TT'.nclrrkcr equation omiafion of the ^+^=E(f) %+v!x~=E{f)

= E+(f)-E-(f), -E+{f) - E~{f)'

(2 io) (2.10) -

with

E+(f) = = a2 E+(f)=a2

/

R 3 xS2_ I

2 E'(f) == a

/

E-(f)=a2

f

Y ' i ? £ ; T / ( ^ x , v ' ) / ( t , x - a n , w , ) ( n , v - w ) d n d w , (2.10a) YRETf(t,x,v')f(t,x-an,w')(n,v-w)dndw,

(2.10a)

y ^ r / ( t , x , v ) / ( t , x + a n , w ) ( n , v - w ) d n d w , (2.106) YRETf(t,x,v)f(t,x

+ an,w)(n,v-w)dndw,

3

(2.106)

ET R x S 2 .RET where Y±RET = Y (t,-x.,x ± an). Here, for simplicity, the indices were where Y = YRET(t,x,x ± an). Here, for simplicity, the indices were dropped in /1 and in the independent variables, as well as V2 was replaced dropped in f\ and in the independent variables, as well as V2 was replaced by w. by w. Note that equation (2.10) becomes the Boltzmann equation with the Note that equation (2.10) becomes the Boltzmann equation with the hard spheres potential when YRET is replaced by one and the expressions hard spheres potential when YRET is replaced by one and the expressions x qF an, in (2.10a) and (2.10b), are replaced by x. x qF an, in (2.10a) and (2.10b), are replaced by x. The revised Enskog collision operator E(f) exhibits two important propThe revised Enskog collision operator E(f) exhibits two important properties. The first one is an analog of the corresponding collision invariants for erties. The first one is an analog of the corresponding collision invariants for the Boltzmann collision operator. The second states that the RET possess the Boltzmann collision operator. The second states that the RET possess an //"-function that formally drives the system to equilibrium. an //"-function that formally drives the system to equilibrium. 3 3 3 and f(t) G C0(n x M3), for t € [0,T], one For ip measurable on fl x E For ip measurable on fl x E and f(t) G C (n x M ), for t G [0,T], one 0

J. Polewczak

156 has the identity [P03], [P04]

j

= ±a2

i;(x,v)E(f)dvdx

QxR3

J

V>(x, v') + (x + an, w')

QxR 3 xR 3 x§2_

■0(x, v) - ?/>(x + an, w) / ( £ , x , v ) / ( £ , x + a n , w ) x yKiii(i,x,x + an)(n,v- w)dndwdvdx,

(2.11)

where, at this stage, integrabihty conditions were ignored (e.g., it is enough to assume that YRET is bounded). Observe that for ip = 1, ip = v, and ip = |v| 2 , the left-hand side of identity (2.11) vanishes; this property corre­ sponds to conservation of the mass, momentum, and energy. Furthermore, in contrast to the case of the Boltzmann collision operator, for identity (2.11) to be true we had to integrate over position domain f l C R 3 . For / , a non-negative solution of (2.10), and for HRET(t) defined by [MAI] HRET(t)=

J

/(*,x,v)log/(t,x,v)dvdx

fixR3

" E ^ / k 2

~

d x i

■•• /<*KkP(2)...p(*)V(l...*) > (2.12)

Q

Q

where V(l...fc) is the sum of all irreducible Mayer graphs which doubly connect k particles, we have dHRET dt

<0.

As an illustration, I provide the expressions of V(1...

(2.13) k) for k = 2,3,4

ni2)=(l)/i2, V(123) = ( l ) / i 2 / 2 3 / i 3 ,

(2.14a) (2.146)

V(1234) = ( 3 ) / 1 2 / 2 3 / 3 4 / l 4 + ( 6 ) / i 2 / 2 3 / 3 4 / l 4 / l 3 + (l)/l2/23/34/l4/l3/24 •

(2.14c)

KINETIC THEORY OF DENSE GASES

157

As before, / y = 0 ( | x ; - Xj\ - a) are the Mayer factors and the numbers in parentheses represent the corresponding symmetry factors. In the case of the Boltzmann equation the corresponding Boltzmann //-function is given by the first term on the right-hand side of (2.12). Next, equality in (2.13) holds if and only if [RE1] /(*, x, v')/(*, x + an, w') = /(*, x, v)/(*, x + an, w ) ,

(2.15)

for all x, v, w and n such that (n, v - w) > 0. As pointed out by Resibois in [RE1], the last condition can be relaxed. Following the analysis in [RE1] together with the requirement that / ( £ , x , v ) be integrable with respect to v one obtains /(t,x,v)=p(^,x)(^)3/2exP(-^(v-uW)2),

(2.16)

where u ( t ) is fluid velocity and (5{t) = l/kBT(i) with T(t) fluid kinetic temperature. We observe that the form of / in (2.16) is very different from local equilibrium solutions of the Boltzmann equation, for which u and j5 can be functions of positions. In the case of the Boltzmann equation the if-function is constant in time if and only if the solution has the form /(«,x,v)=p(«,x)(^)3/2exp(-^(v-u(t,x))2).

(2.17)

Furthermore, due to the non-local character of the revised Enskog collision operator E(f)y the / which satisfies (2.13) with equality sign (i.e., / given in (2.16)) does not make the collision operator E(f) vanish. This fact is in contrast to the Boltzmann theory, where / given in (2.17) makes the Boltz­ mann collision operator vanish. Thus, relaxation processes in a dense fluid, as described by the RET, are more complicated as compared to the relax­ ation processes in dilute gases. Finally, after substituting / from (2.16) into (2.10), the stationary solutions (i.e., with df/dt = 0) satisfy the following equation [BJ2] Vlogp(x)= /

-£MX(x,x2|p)dx2,

(2.18)

where x i s given in (2.7) and the term / i 2 = 0 ( | x - x 2 | - a) is the Mayer factor. Equation (2.18) is the first member of the BBGKY equilibrium

J. Polewczak

158

hierarchy for the system of hard spheres. Since g i > 3, 0 2 ,i(xi,x 2 |p(*,-)) =exp{-/3(/>HS(\x1

1+

-x2|))

E ( F ^ / d X 3 - ' ' / d X f c p ( 3 ) " ' ' p w v r ( 1 2 | 3 " ' ' A : ) '(2,19) k = 3

Q

Q

are non-negative for / > 0. Indeed, non-negativity of <72,i|x12=a+ is crucial in obtaining the H-theovem for the corresponding kinetic equation [P04], [BL1]. While it is well known that #2 given in (2.7) is non-negative, at present except for i = 3, it is not known whether non-negativity of / implies 92,i\x =a+ > 0, for i > 4. In [BL1], this problem (i > 4) was formally resolved by removing from (2.19) all the terms containing odd numbers of fij (i.e., all non-positive terms). Due to lack of physical importance, this procedure is unsatisfactory. Equation (2.10), with p2)3 replacing the original pair correlation function #2, exhibits the following if-function H(t)=

j

/(t,x,v)log/(t,x,v)dvdx

f2xR 3

2 / / /i2p(*,xi)p(t,x 2 )dxidx ; Q

n

KINETIC THEORY OF DENSE GASES

-g / /

fi2f23fi3p(tyXi)p(tix2)p(t1X3)dxidx2dx3.

159

(2.20)

n Q Q

Let us note that whenever g2ti\x = a + > 0, for i > 4, the corresponding ^-function is expressed by the expression (2.12), with the infinite series truncated at k = i. The arguments developed in [DPI], [AK1], and [BL1], yield the follow­ ing global existence theorem for equation (2.10), with YRET replaced by 22'3*12=a+-

T h e o r e m 2.1. For vRET _ _ 1

and an initial value /(0,x,v) = / o ( x , v ) > 0 satisfying I

(l + |v| 2 + |x| 2 + | l o g / o ( x , v ) | ) / 0 ( x , v ) d v d x < o o ,

(2.21)

fixR 3

there exists a non-negative renormalized solution f(t,x,v) [0,T],T > 0, with the property

of (2.10) on

sup / (l + |v| 2 + |x| 2 + | l o g / ( i , x , v ) | ) / ( ^ , x , v ) d v r f x < C ( T ) . (2.22) te[o,T] J nxR3 Here, Q is the whole M3, or a three-dimensional torus. Further details on renormalized solutions can be found in [DPI] and [BL1]. I end this section with a remark regarding the physical interpretation of p2,i given by (2.19). The function #2,1 is the truncation (at k = i) of the Mayer cluster expansion (2.7), which at equilibrium is of order pl in the number density p, and thus yields the equation of state correct to order pi+1. We recall that the equation of state (written in terms of the virial coefficients Bk (T, p)), 00

pkBT

= l + YlPkBk+i(T,p),

(2.23)

160

J. Polewczak

:onsidered for hard spheres of diameter a contains virial coefficients inde­ pendent of the temperature T. For example, the second virial coefficient [k = 1) is equal to 2/37ra3, and the corresponding equation of the state is a;iven by

= l + ?7ra3/9,

(2.24)

pksT 3 which is the equilibrium equation of state of the fluid governed by equation (2.10) with YRET = 1. Successive approximations of the equilibrium equa­ tion of state, correct to order pi+1, are obtained by considering the equation (2.10) with YRET = g2,* i i 2 = a + 3.

The Square-Well Kinetic Equation

In this section I consider the kinetic equation with the square-well potential

(f>sw(r) r

oo,

(j)sw(r) = I - 7 , ,0,

for r < a for a < r < R

(3.1)

for r > R.

Here, as in section 2, particles have mass m = 1. The "collisions" are understood to be instantaneous and take place at the points where <j)SW is discontinuous. There are four different types of collisions that can be distinguished in the case of the square-well potential: (1) a collision at the hard core (at r = a + ) , (2) a collision entering the square-well (at r = i?~), (3) a collision leaving the square-well (at r = -R + ), and (4) a collision rebounding at the inner side of the square-well (at r = R~). The last type of collision occurs when the radial relative velocity is too small for escape from the well, i.e., is smaller than |v e s c | = y/^y. Observe (see, [BJ3]) that the collision here was defined as a single passage through the square-well edge, or a rebound from the hard core, or the inside of the square-well edge. At low densities a full two-body collision consists of a sequence of either entering, hard core and leaving collision, or an entering and a leaving collision. Furthermore, one has bound states consisting of either type (1) and type (4) collisions, or a sequence of type (4) collisions. Therefore, the partial collisions of one full collision are correlated. This is not the case for high densities. Indeed, at high densities the mean free path is small as compared to the square-well width R, and thus, between

KINETIC THEORY OF DENSE GASES

161

two successive partial collisions of any pair of particles the same pair of particles undergoes a large number of collisions with other particles. In the square-well kinetic theory, presented below, partial collisions will be assumed to be not correlated. Consequently, the regime of applicability of the theory is restricted to high density. As in the case of hard sphere systems one can derive an evolution equation, similar to the pseudo-Liouville equation, that governs the exact dynamics of the system interacting with the square-well potential (f)SW, as well as the corresponding analog of the BBGKY hierarchy for hard sphere systems [BJ3], [ER1], and [KR2]. The first hierarchy equation for the oneparticle distribution function / has the form ++ vv == QSW % t £ % § £ QSW(M(M

Qi +Qs Q4 /2) (3 (3.2) 2) == Q1U2) + Q»(h) + 0 (/ ) + Q4U2), 3 2 ^ + *•(/*) ^ + < - -

where Qi, for i = 1,2,3,4, are the collisions operators corresponding, respectively, to the four types of collisions mentioned above. The collision operators Qi(f2) = Qtih) ~ Q t 7 (/2), for i = 1,2,3,4, are given as follows 22 + w')(n, w)) ddn, Qi~(/2)=a ' ) ( n ,vv - w n , ddw w ,, Qtih) ■■= a // / 2 ( i£, ,xx, v, v, x, x -- aa+ n , w

(3.3a)

E 2 + Qiif2) c xx,) vv,, xx + n , v --ww) ) < idn n ddw w ), = a / / f22f (t, -1- a n ,, w w)) ((n, Qi(f2) ■■=

(3.36)

E 2 2 _ Q+(f2)=R ) d) dn nddww, , Qtih) ■■= R I/ / 2 ( £* ,, xx,,vv' t, ,xx- - i ?r nn, ,wwt' ) ( nn,,vv -- w w

(3.4a)

E 2 2 Q2(f2)=R ? ++ nn,,ww) )( n( n, v, v-- ww) )ddnnddww,, Q2U2) ■-= R / / 2/ 2((*£,,xx,,vv , x + i. R

(3.46) (3.46)

E 22

Q + ( / 2■■ ) ==J RR Qtih)

/ / /2 2((**,,xx,, v tJ , x + . R +fl+n,wt)(n,v-w)0 n , w i ) ( n , v - w )3dndw, 0 3 d n d w :, (3.5a) (3.5a) E

2 Q~(f = R2 / / /22((*£ , x ,,vv, ,xx -- i irtnT,nw, w) )( (nn,,vv - -w )wG) 30d3 dn nd dww, , 2)=R Q3U2) ■■

(3.56)

E

gQt(h) + ( / 2 ) ■==J RR22 / / 2/ 2((t*, x, x, ,vv,',,xx + i J?R- n" n, w , w' )' () n( n, v, v- -ww) )004 4ddnnddww,, (3.6a) E

J. Polewczak

162 Q-{f2)=R2

//2(*,x,v,x-irn,w)(n,v-w)04dndw.

(3.66)

E

Here, £ = R3 x S i , f2 is the two-particle distribution function, 63 = 0 ( ( n , v - w ) - | v e s c | ) , and 04 = 0 ( | v e s c | - ( n , v - w ) ) . The prime velocities in Q^ and Q% are given by v' = v — n(n, v - w ) ,

w' = w + n(n, v - w ) ,

(3.7)

while the dagger and double dagger velocities in Q^ and Q~£ are defined by v 1i = v

1 n (n, v — w) -a+ 2

+

1 (n, v —w) — a+ , (3.8) w = w + -n 1

with a+ = y ((n, v - w ) ) 2 + |v e s c | 2 and by

t = vV -

n (n, v — w) — a'

V1" =

t

1 (n, v — w) — a , (3.9) w = w + -n +

with

= V «n'

a

wr\y _ | v

12

The notation a+ and i? + in $1 above indicates that the function has to be evaluated just outside the hard core and the square-well, respectively; this evaluation is usually obtained by taking a suitable one-sided limit. The notation R~ indicates the evaluation taken just inside the square-well. The first important distinction, as compared to the hard sphere system, is the fact that, although the pairs of velocities in (3.8) and (3.9) obey conservation of the momentum, they do not obey conservation of the kinetic energy. A part of the kinetic energy is exchanged with the potential energy Df square-well. Indeed, we have vf2

+

w

f2

=

v2 +

w

2

+

2^

v$2 +

w$2 =

v2 +

w

2 _

2 7

^

^.1Q)

where |v e s c | = \flq. Equation (3.2) for / involves the unknown f2. As indicated in section 1, the closure on the one-particle distribution level is

KINETIC THEORY OF DENSE GASES

163

not viable due to the fact that / alone cannot describe the potential energy density of the system with a square-well potential. Since the closure on the level of the two-particle distribution function / 2 is too complex, instead, one closes equation (3.2) using an additional equation for the potential energy density up(t,x) (expressed in terms of / 2 , see, (1.2)), and then one tries to express / 2 as a functional of / and up. This last step is achieved through a maximization of the ensemble entropy subject to the constraints that at given t, the one particle distribution function /(£,x,v) and up(t,x) are reproduced correctly. First, from the second hierarchy equation (see, for example, [BJ3]) the equation for up(t,x) has the form —u p (t,x) + - —

/

05M/(|x-x2|)/2(i,x,v,x2,w)dwdvdx2

fixR3xR3

jR2

/

/2(*,x,v,x-i?

n,w)(n,v-w)

R3xR3xS*.

x 0((n, v - w) —

7#

\vesc\)dndwdv

/ 2 (£,x, v , x + R+n, w)(n, v - w)dndwdv.

/

(3.11)

R 3 xR 3 xS2_

Now, the maximum entropy procedure is used to make equations (3.2) and (3.11) the closed system of equations for / ( t , x , v) and u p (t,x). Sim­ ilarly to the case of the revised Enskog equation [RE1], one constructs an (appproximate) ensemble fff that maximizes the ensemble entropy (defined here up to an additive constant) S(fN)

= -kB

/ fN^i,.

■. ,zN)\og[N\fN(zu

... ,zN)] dz1---dzN,

(3.12)

under two constraints. The first one expresses the fact that fjjf reproduces correctly the one-particle distribution function / , i.e., f(t,z)=

f

N

226(z-Zi)fjjf(zu...,zN)dz1"-dxN.

(3.13)

164

J. Polewczak

Here, z = (x, v), z^ = (x^, v;), for i = 1 , . . . , JV, where JV denotes the number of particles, and fw is the normalized TV-particle distribution function, i.e., ffsfdz\ •• ••••dz;v dzpi == 11.. // /iV^Zl I point out that in the case of the hard sphere potential (2.8), constraint (3.13) alone generates RET's f$f as a functional of / , so RET's closure relation is attained at the level of the first hierarchy equation. Here, in order to simplify notation /w was considered only in a canonical ensemble. The second constraint expresses the fact that fj^f reproduces correctly the potential energy density up(t,x), i.e.,

r N \6(x-

/*

uupp(t,x) (t,x) = = Y Y, J

I

N

-

1

SW

S(X-^)-J2^

(\^-^J\)

-x,|)

i=l

^ z i • •dzAT. (3.14) fffdzi-'-dzN. (3.14)

/JV

Next, one maximizes S(/N) (see, for example, [KR1] and [BJ3]) under constraints (3.13) and (3.14). In doing so, we use the method of undetermined Lagrange multipliers. This results in the following variation with respect to IN

r /• " log /N+ A(t, z) 2_^ $fN |\ // /IN fN + / X(t, Y <5(z
JV N

-

+y*/3(*,x) + /W.xJ^^x-xOr^^dxi-Xj-Ddx J

1ml

i 1

dzi
j*i

.

(3.15)

}

where A(£, z) and /3(x, t) are Lagrange multipliers corresponding to the constraints for / ( £ , x , v ) and up(t,x), respectively. Equation (3.15) yields

A(£, z, ) / */*{log/jv + l + ]£ A(t,zO

J

^ ^

^

1i

»=i »=i N N

+ ^(t,x )^^(|x + 2 ^ ( * ' X * )i ] C ^ S W ( I X *i "-XxJ,I| ))

n Idzi • = 0. f d z 1 - - -•d•zdz;v A r = 0.

(3.16) (3.16)

KINETIC THEORY OF DENSE GASES

165

The solution of (3.16) has the form fM

JN —

-.

1 E(t)N\ ° X P

N

N

N

-£A(t,z,-)--EE^ S H / (|xi- -Xil)

.

(3.17)

Here,

fly i9(t,x<*)] + 0{t,: \[m*i) 4) + i )] N == |[)8(*,x and E(£) is the normalization constant depending on A(£,x;,v;) and (3^. The field &#/?(£, x) can be interpreted as the inverse of the local potential energy temperature that in equilibrium coincides with the kinetic energy temperature. Furthermore, in equilibrium, A = |/?|v| 2 and E(t) (modulo a multiplicative constant) reduces to the canonical partition function. As in the case of the corresponding approximate ensemble for the RET [RE1], the velocity dependence in (3.17) appears only in the factors exp(—A(i,Xi, v^)). This means that velocity correlations are neglected (partial collisions at the square-well edge are assumed to be uncorrelated). As indicated in the beginning of this section, this approximation is good for high densities. The expression (3.17) for fjif yields the following formulas for

//,,.^( t, ,xx, v, ,vj -) =p(*,x)exp[-A(*,x,v)] ^ x ) e ^f;(-t xx y) ' x - v ) ' ,

(3.18) (3.i8)

where £(t,x) is the local fugacity, C(t, x) = / exp[-A(t, C(t,x)= exp[-A(*,x,v)]dv, x, v)] dv,

(3.19)

and / 22 (i,z ( i , z 1i ,,zz 22 ) = / ( i , x i1 , v 11 ) / ( i ,,xx22,,vv22))pp22((xxi1,,xx22 | p (\p{t,-),(3(t,-)). f,-),^,-))-

(3.20) (3-20)

Both, the local fugacity ( and the pair correlation function g2l are functional of p(t, x) and /3(t, x), which may be expressed in terms of the Mayer expansion [ST1], where each vertex is weighted by the density field p(t,x) and the Mayer factors assume the form w f/i:j„ = exp =exp{-0 {-0n^ij (\xisw(\xi--- Xx,|)} , | ) } - 11..

(3.21)

J. Polewczak

166

In particular,

^2(x1,x2|/9,^)=exp(-^5iy(|x1-x2|))x(x1,x2|p,/3),

(3.22)

where X= l +

f^j^-^jdx3--*=3

Q

JdxkP(3)...p(k)V(12\S...k). Q

Here, as in expansion (2.7), p(k) = p(t, xk), K(12 | 3 . . . k) is the sum of all graphs of k labeled points which are biconnected when the Mayer factor /12=exp[-/W5^(|xi-x2|)]-l is added. Except for a different expression of the Mayer factors fij, the above expansion has the same algebraic structure as the Mayer expansion (2.7) of g2 in the RET. A noteworthy property of p 2 in (3.22) is its discontinuity at the squarewell edge: g2(xi,x2+R~x12\p,/3)

= exp(/3i 2 7)p 2 (xi,x 2 -I- R+xi2

| p,(3),

(3.23)

where x i 2 = (xi — x 2 ) / | x i — x 2 |. Discontinuities at a and R are the sources of collisional transfer of momentum and energy that in turn generate strong exchange between kinetic and potential energy. Now, from (3.14) (see also (1.2)), up(t, x) is also a functional of p(t,x) and /3(t,x), and by formally inverting this relation, one can express j3(t,x) as a functional of up(t,x) and p(t,x). Thus, inserting (3.20) and (3.22) into (3.2) and (3.11) yields the square-well kinetic theory consisting of the coupled equations for / and up. Finally, I point out that instead of f(t, x, v) and up(t,x), one has an option of switching to / ( t , x , v) and /3(£,x) as the unknown functions appearing in equations (3.2) and (3.11). One of the important features of the square-well kinetic equation based on (3.2) and (3.11) is its if-theorem. For the ensemble (3.17), with (xi, v x ) replaced by (x, v), the entropy (3.12) has the form ^Wj5f)(t)=log(EWAr!) + | / ( t , x 1 , v 1 ) A ( t , x l j v 1 ) d v 1 d x 1

KINETIC THEORY OF DENSE GASES + J0(t / 0(t,xi)u p(t,x.i)dx.i 1x1)up(t,x 1)dxi.

167 (3.24)

.

Using the fact that the normalization of fff yields --log(EJV!) log(SiV!) + yf /f-± / pu^dx! dxx = 0, — ddw d xl i ++ fu p^ V ldx

(3.25)

one obtains

i1 dS(fff) rfS(/ff) r df dv\dx.i = x dt dt kB i;^^ j -mdVidXi +

-I'

r ,dudu p T, dx\ . J -drdtd*1-

*{, 0

(3.26) (326)

Next, using equations (3.2) and (3.11) together with the property (3.23) (for more details, see [KR1] and [BW1]), one has 1 dS(fff) dS(fM) k dt dt kB

X

=

1 4 f f w\ An, 2 ^ \ l ^ ( x , xx + ADin n | p\p,0l , / ? ) //(*,x, ( t , x ) vv)/(t,S ) / ( t , x :++ D W1 i n,w) i=l i=\

v

p

*-

//(*,x,v)/(t, ( * , x , v ) / ( t , xx + AAnn,,ww) ) .j (i))/(t,x + D ( i ) ) exp(-£; e x p ( - J Ei ;[/3(i,x) ^,x + An)]/2) ~°&S / /((tt, ,xx,,vv(0)/(*,x A inn,, w w(0) p(t,x i [ / 3 ( ^ x+) +

-- / (/ ft c, xx, , vv)/(t, ) / ( t ) xx + ) + ( t , xx,) vv^))/(t, W ) / ( t ) xx + D 1nn,) wW) +A A nn ,, ww) + //(t, +A w
r = nn>x RR33 xx ER33 xxSs2 2+ +, , " R++,, --R-, i ? - , - i 2-RA = a+, a+, R and Ei ===0,0,--- 77,, 77, 0 £<

(3.27) (3.27)

J. Polewczak

168

for i — 1,2,3,4, respectively. Here, v (i)

_ v'5

v

t jv t ?

w ( l ) = w', w t ,

v',

w

t , w',

and 0 ^ ( r ) = a 2 0 ( r ) , R2Q(r),

R20(r

- |v e s c |),

fl20(|vesc|

- r),

for i = 1,2,3,4, respectively. Since g2 > 0 for / > 0, the inequality zflogrc - logy) > x - y, for x,y > 0, applied to the terms within the square brackets of (3.27) for i = 1,2,3,4 implies dS(fff)/dt>0.

(3.28)

In other words, the kinetic theory based on the coupled equations (3.2) and (3.11) possesses an H-functional (equal to -S(fj!f)). Next, applying the fact that x(\ogy — logx) —y - x only when x = y to the terms within the square brackets of (3.27), for i — 1,2,3,4, we get the functional form of stationary points of S(fj!f)(t) /.,(t,x,v) = p ( ( , x ) ( ^ ) 3 / 2 e x p ( - ^ ( v - u ( t ) ) 2 )

(3.29)

and

Mt x)=

(3 30)

' k£®-

-

Thus, the macroscopic parameters p, T and u completely determine the ensemble fjlf which satisfies dS(f}f)/dt = 0. As in the case of the RET, a stationary solution of (3.2) and (3.11) for which dS(fff)/dt = 0 yields the following equation for p(x.)

Vlogp(x) = j \j^J n

X(x,x 2 \p,{3eq)dx2

,

(3.31)

where x is given in (3.22). Here, the Mayer factor has the form /12 = exp [-Peqsw(\x - x 2 |)] - 1,

(3.32)

KINETIC THEORY OF DENSE GASES

169

and - ^

= {S(x - x 2 - a+) + [1 - exp(/3 e ,7)] <*(* - x 2 - R+} x ,

(3.33)

where x = (x - x 2 ) / | x - x 2 |. Equation (3.31) is the first member of the BBGKY equilibrium hierarchy for the system with the square-well potential (3.1). Hence, at equilibrium, fjlf reduces to the usual canonical ensemble. Inequality (3.28) shows that one can construct a kinetic theory for strongly interacting particles that is nonlocal in character and has built in mechanisms of conservation of the total energy, as well as trend to equi­ librium. Furthermore, its equilibrium state is in full agreement with the equilibrium statistical mechanics of the system interacting with square-well potential (3.1). Similarly to the revised Enskog theory, the nonlocality is manifested in the fact that dS(fj{f)/dt = 0 is a necessary, but not sufficient condition for Qsw(f) = 0. The above kinetic is the third in succession version of the kinetic variational theory introduced in [ST2]. It is usually referred to as KVTIII. Next, I present some additional properties of the KVTIII that are important in obtaining existence and stability results for the coupled equations (3.2) and (3.11). I start with an analog of Boltzmann's collision invariants. P r o p o s i t i o n 3 . 1 . Let f € C0(Q x R 3 ) and f2 E C0{Q x E 3 x 17 x R3). Then, for all tft measurable functions o n f i x l 3 ,

J 0(x,v)Qi(/)dvdk

(3.34)

fixR3 1 2 a

= 2

I

V>(x, v') -I- ^ ( x + an, w') - ip(x, v) - ^ ( x + an, w)

QxR 3 xR 3 xS^_

x /2(£,x,v,x + a+n, w)(n,v - w ) d n d w d v d x ,

/ fixR 3

^ ( x , v ) Q2(f) + Q*U) dvdx

(3.35)

J. Polewczak

170 -R2

(x, v t ) + ^ ( x + i t a , w t ) - ^ ( x , v) - V ( x +

j

Rn w

, )

fixR3xR3xS2_

x /2(t,x,v,x + #+n,w)(n,v - w)dndwdvdx

+ -R2

/

^(x.v-t) + V > ( x - # n , w t ) - - 0 ( x , v ) - ^ ( x - i ? n , w )

nxR3xR3xS2.

x / 2 (*,x, v , x - i? n , w ) ( n , v - w)@((n, v - w) - |v e s c |) d n d w d v d x , and |

V>(x,v)Q 4 (/)dvdx

(3.36)

QxR 3

ifl2

/

L ( x , v') + ^ ( x - Rn, w') - 0(x, v) - ^ ( x - Rn, w)

fixR3xR3xS2_

x / 2 (£,x, v , x — i2~n, w)(n, v — w ) 0 ( | v e s c | — (n, v — w)) d n d w d v d x , where / 2 is provided in (3.20) with g2 given by (3.22). In order to avoid any problems with convergence of the integrals in (3.34), (3.35), and (3.36), the functions / and / 2 were assumed to be con­ tinuous and with compact supports. The proposition also holds under more general conditions on / and / 2 that make the above integrals well defined. Proof: The proof of identities (3.34) and (3.36) is very similar to the proof of identity (2.11) for the RET (see, for example, [REl] or [BLl]). In order to show (3.35), observe that the Jacobians of the transformations (v, w) \-¥ ( v t , w t ) and (v,w) (->• (v+,w+) are (n,v - w ) / a + , and ( n , v - w)/o: _ , respectively (with a+ and a - defined in (3.8) and (3.9)). Furthermore, it is easy to see that (n,vt - w t ) = a+ and (n, v+ — w+) = a~. Next, the change of variables of integration, (v,w) -» ( v t , w t ) , in Q j ( / ) together with the fact that in this process v (as a function of v t and w t ) becomes

KINETIC THEORY OF DENSE GASES

171

v+, yields dvdx = Jj 0(x, iP(x,v)Q+(f)dvdx =■i» ±R v)<# 2

j

(/)

fixR33

?/>(x, V ( x ,Vv+t))//22( (t ,tX, x, X, x- - iJJT- nn,w) ,w)

/

3 j

3

n xx iSS>j . Q

x (n,v - w ) 0 ( ( n , v - w) - |v csc |) dndwdvdx.

(3.37)

Similarly, after the change of variables of integration, (v, w) -¥ (v+,w+), in Qt(f) ( m this process v becomes v ' ) one obtains /

(3.38)

T/;(x,v)g+(/)dvdx iP(x,v)Q$(f)dvdx

ftxR3 fixR

= -R = - i l22

//

^(x, w)(n, v — - w) w)dn dndwdvdx dw dv dx . . ^ ( x , vv 't ))/2(i,x, / 2 ( t , x , vv ,, xx + + iln, i2n,w)(n,v-

3 3 32 O x R3xR x RxSx+S 2 QxU

Finally, after the change of variables of integration, v ^ w and n —>■ —n, in the sum of (3.37) and (3.38) one obtains (3.35). ■ An immediate consequence of Proposition 3.1 is Corollary 3.2. Suppose f G C0(ft x E 3 ) and f2 e C0(Q x E 3 x O x R 3 ). Then, for ip — 1 and ip = v, SW /f ipQ i>QSW (f) dv dv dx dx = = 0, (f)

(3.39)

QxU3

and for ip = |v| 2 , /

+ sw \\v\ n,w) i | v2|Q2 SW Q (f)dvdx=^J\f '(/)
QxR Q x R3 3

(3.40)

r

dv dx , n , v --- w) dn dw dndwdvdx, - / 2/ 2((*t ,>xx, ,vv, ,xx - i 2R~n,n , w)0((n, w ) 0 ( ( n ,vv --- w) -—|v esc csc|) ((n, where T = Q x M3 x M3 x S 2 .

172

J. Polewczak

In contrast to the RET, identity (3.40) implies that, for solutions of (3.2) and (3.11), the kinetic energy

1

1, 2 2 f(t,x,v)dvdx v) dv dx -l |-\v\ v | /(*,x, 3 JQxR 3 ./Ov R 2

/

is not conserved. For solutions of equations (3.2) and (3.11), identity (3.40) yields the following conservation of the total energy

d 2 / i-\v\ = 00. . | v | 2f(t,x,v)dvdx+ / ( * , x , v ) ( i v d x + / u p (utp,(t,x)dx x)dx = dt at [ J 3 2 J QXR n

—

fixR3

(3.41)

Q

Let us note that when the depth 7 of the square-well potential (3.10) approaches zero and R -¥ a (i.e., when <j>sw {r) -> 4>HS(r)), formally at least, fnup(t,x)dx —>• 0, and equation (3.41) reduces to the conservation of kinetic energy for the RET. This is in agreement with the fact that for systems interacting with hard spheres potential (2.8), the potential energy density up(t,x) (see (1.2)) is zero. Although inequality (3.28) shows that KVTIII possesses a monotonic functional, the form of S}f is not very operative. One would like to express an if-function directly in terms of / and up. Consider the functional sw Hsw (f,f3)(t) x,v)log/(£,x,v) ivdx H (f,P)(t)== J /(t,x,v)log/(«,x,v)dvdx J Ht,

(3.42)

QxR 3

dXi

-^h.J

~ Yl if / k=2

Q

dxi

-hit,

x)u (t,: p(«, K) dx, dxkp(l)...p(k)V(l. . . * ) - k) - f fi(t '■' •• •/ / fokpil) ■ ■ ■ p(k) V(1... x) dx, t px)ix n n

where V ( l . . . k) is the sum of all irreducible Mayer graphs, given in (3.21), which doubly connect k particles. The following result holds [P02] Proposition 3.3. For / ( t , x , v ) > 0 and up(t,x), equations (3.2) and (3.11),

jjtt[H [Hswsw u,mt)
solutions of the coupled

(3.43) (3.43)

Furthermore, ft[Hsw (f,(3))(i) = 0 if and only iff is given by (3.28) (for any p(t,x), u(t), and T(t)) and P(t,x) = l/kBT(t).

KINETIC THEORY OF DENSE GASES

173

Note that the first two terms of Hsw have the same algebraic form as the ^-functional for the RET. Here, however, the Mayer factors j { j appearing in V(l... k) correspond to the square-well potential <j>sw. Due to the lack of convergence properties for the infinity sum in (3.42) the above result is rather formal. Similarly to the case of the RET, one can consider a truncation of the infinite series for g2 in (3.22) at k = i. As before, one can show that as long as the contact values of the truncated g2 are non-negative (for / > 0), the corresponding coupled equations (3.2) and (3.11) (i.e., with Qi replaced by the truncated series at k = i) exhibit an if-function given by (3.42), with the infinite sum truncated at k = i > 3. Next, by recognizing that the second term of (3.42) evaluated at feq is equal to PeqAeXcess, where Aexcess is the (non-uniform) equilibrium nonideal part of free energy density of the system with the potential <j>sw, one can write Hj?qw in the form Hsw e «

= — kBT

— kBT'

(3 44) {6AV

where A is the Helmholtz free energy and U = / w p (t,x)o?x is the internal energy. The classical equilibrium thermodynamics identity

A=

U-TS,

where S is the equilibrium entropy, shows that, modulo an additive constant, the right hand side of (3.44) is equal to -S/ksI also point out that the expression for —Hsw coincides, up to an ad­ ditive constant, with the formula for the entropy given in [BW1] as well as with the expression for Sj§. In this section I have considered the square-well kinetic equation. Its success depends in part on how well it describes various transport properties (and in particular, the density and temperature dependence of the transport coefficients they predict, see [BJ3] and [KR2]) and also in part on possibility of proving existence and stability theorems for them. I end this section with the estimation that is crucial in proving existence and stability results. Due to enormous mathematical complexity of the kinetic equations presented

J. Polewczak

174

here, I consider a simple model of the KVTIII that arises when one retains only the first term of gi expansion given in (3.22). Consider equations (3.2) and (3.11) with gi given by

fO,

for Ixi — X2I < a

g2 = e x p ( - / W ^ ( | x 1 2 | ) ) = < exp[7& 2 ], for a < |xi - x 2 | < R (3.45)

U,

for Ixi — X2I > R,

where x i 2 = xi — X2 and 1

& 2 = -[/?(*, xx)+/?(*, x2)] The corresponding if-function has the form: Hsw(f,{3)(t)=

J

/(t,x,v)log/(t,x,v)dvdx

fixR 3

p(t, Xi)p(t, x 2 ) dx 2 dxi

/

(3.46)

Jix(nn{o<|xi-x2|
[70i2 - l]p(*,xi)p(«,x 2 )exp[7/3i2]dx 2 rfxi .

/

fix(nn{a<|xi-x2| 0 and (30 satisfy

j

(l + |v| 2 + | l o g / o ( x , v ) | ) / 0 ( x , v ) d v d x < C 0 < o o

(3.47)

nxR3 and [\up(0,x)\dx
(3.48)

KINETIC THEORY OF DENSE GASES respectively, then, smooth solutions f(t,x,v)

175 > 0 and /?(*,x) satisfy

f (l + |v| 2 + | l o g / ( * , x , v ) | ) / ( * , x , v ) d v d x < C

(3.49)

JlxR 3

and / |up(t,x)|dx
(3.50)

Q

uniformly in t e [0, oo). Here, Q = [0, L{\ x [0, L2] x [0, L3] with the periodic boundary conditions. Theorem 3.4 provides the estimations that are essential in obtaining global existence theorems for (3.2) and (3.11). Work on these problems is in progress. An important area of research not mentioned in this lecture is study of the hydrodynamic limit for the KVTIII. Formally (see, [BJ3] and [KR2]), the hydrodynamic equations are derived by applying the Chapman-Enskog procedure to the system of kinetic equations (3.2) and (3.11). In this case, the procedure needs some modifications in order to accommodate nonvanishing potential energy density. As the result of this limiting process, one obtains not only macroscopic equations for the fluxes of mass, momentum, and energy, but in addition, explicit expressions for the transport coeffi­ cients such as viscosity, conductivity, and diffusivity. Although the trans­ port coefficients appear explicitly in hydrodynamics equations, they cannot be computed on the basis of the macroscopic description alone. That is why the hydrodynamic limits of kinetic equations play an important role in understanding the gap between the atomic structure of matter and its continuum-like behavior at the macroscopic level.

J. Polewczak

176

References [AKl]

L. and CERCIGNANI C , Global existence in L1 for the Enskog equation and convergence of solutions to solutions of the Boltzmann equation, J. Statist. Phys., 59 (1990), 845-867.

[BJ1]

H. and E R N S T M.H., The modified Enskog equa­ tion, Physica, 68 (1973), 437-456.

[BJ2]

H., Equilibrium distribution of hard sphere systems and revised Enskog theory, Phys. Rev. Lett, 51 (1983), 1503-1505.

[BJ3]

H., Kinetic theory of dense gases and liquids, in Fun­ d a m e n t a l P r o b l e m s in Statistical M e c h a n i c s V I I , H. van Bei­ jren Ed., Elsevier (1990), 357-380.

[BL1]

BELLOMO N., LACHOWICZ M., POLEWCZAK J., and TOSCANI

ARKERYD

VAN B E I J E R E N

VAN B E I J E R E N

VAN B E I J R E N

G.,

M a t h e m a t i c a l Topics in N o n l i n e a r K i n e t i c T h e o r y I I : T h e Enskog E q u a t i o n , Advances in Mathematics for Applied Sciences n. 1, World Sci. (1991). [BW1]

J. and STELL G., Local //"-theorem for a kinetic variational theory, J. Statist. Phys., 56 (1989), 821-840.

[CE1]

CERCIGNANI C., T h e o r y a n d A p p l i c a t i o n of t h e B o l t z m a n n E q u a t i o n , Springer (1988).

[DAI]

H.T., R I C E S.A., and SENGERS J.V., On the kinetic theory of dense fluids. IX. The fluid of rigid spheres with a square-well attraction, J. Chem. Phys., 35 (1961), 2210-2233.

[DPI]

R. and LlONS P.L., On the Cauchy problem for the Boltzmann Equation: Global existence and weak stability results, Annals of Math., 130 (1990), 1189-1214.

[EN1]

ENSKOG D., Kinetische Theorie, Kungl. Svenska Vetenskaps Akademiens Handl., 63, No. 4 (1921), English transl. in S. Brush, Kinetic T h e o r y , Vol. 3, Pergamon Press (1972).

[ER1]

E R N S T M.H.,

BLAWZDZIEWICZ

DAVIS

DIPERNA

DORFMAN J.R.,

H O E G Y W.,

and

VAN LEEUWEN

J.M.J., Hard sphere dynamics and binary-collision operators, Physica, 45 (1969), 127-146. [FEl]

J.H. and K A P E R H.G., M a t h e m a t i c a l T h e o r y of T r a n s p o r t P r o c e s s e s , North-Holland (1972). FERZIGER

KINETIC THEORY OF DENSE GASES

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[GMl] GRMELA M. and GARCIA-COLIN L.S., Compatibility of the Enskog kinetic theory with thermodynamics. I, Phys. Rev. A, 22 (1980), 1295-1304. [HU1] HUBERT D., tf-theorem for the Enskog kinetic equation, Phys. Chem. Liq., 6 (1977), 71-98. [KR1]

KARKHECK J.,

VAN BEIJEREN H.,

DE SCHEPPER I., and

STELL

G., Kinetic theory and H theorem for a dense square-well fluid, Phys. Rev. A, 32 (1985), 2517-2520. [KR2] KARKHECK J., Irreversible processes, kinetic equations, and ensem­ bles of maximum entropy, in Flow, Diffusion, and R a t e P r o ­ cesses, Advances in Thermodynamics Vol. 6, S. Sieniuticz and P. Salamon Eds., Taylor & Francis (1992), 23-57. [LN1]

O.E., Time evolution of large classical systems, in Dy­ namical Systems, Theory and Applications, J. Moser Ed., Lecture Notes in Physics, Vol. 38, Springer (1975), 1-111. LANFORD

[MAI] MARESCHAL M., BLAWZDZIEWICZ J., and PIASECKI J., Local en­ tropy production from the revised Enskog equation: General for­ mulation for inhomogeneous fluids, Phys. Rev. Lett., 52 (1984), 1169-1172. [POl]

J., Global existence in L1 for the generalized Enskog equation, J. Stat. Phys., 59 (1990), 461-500.

[P02]

POLEWCZAK

[P03]

J., Global Existence in L1 for the modified Nonlinear Enskog Equation in M?, J. Stat. Phys., 56 (1989), 159-173.

[P04]

POLEWCZAK

[RE1]

P., //"-theorem for the (modified) nonlinear Enskog equa­ tion, J. Statist. Phys., 19 (1978), 593-609.

[ST1]

G., Cluster expansions for classical systems in equilibrium, in T h e Equilibrium Theory of Classical Fluids, H. Frisch and J. Lebowitz Eds., Benjamin (1964).

[ST2]

G., KARKHECK J., and H. VAN BEIJEREN, Kinetic mean field theories: Results of energy constraint in maximizing entropy, J. Chem. Phys., 79 (1983), 3166-3167.

POLEWCZAK

J. and STELL G., On some mathematical and physical aspects of the if-theorems in various kinetic theories, to appear. POLEWCZAK

J. and STELL G., New properties of a class of gener­ alized kinetic equations, J. Statist. Phys., 64 (1991), 437-464.

RESIBOIS

STELL

STELL

Lecture 4 COMPUTATIONAL METHODS FOR THE BOLTZMANN EQUATION

W. Walus

1.

Introduction

The Boltzmann equation describes an evolution of a rarefied substance whose particles during a flow undergo binary interactions g + v V

x

/ = J(/,/),

(1.1)

where f(t, x, v) is the distribution function of the particles. The distribution function depends on the time t € IR+, the position x € IR and the velocity v eIR 3 . The mathematical analysis of the Boltzmann equation (1.1) for a wide range of important models is well developed at present (e.g. existence and uniqueness, analysis of asymptotic expansions, correlation with fluid dy­ namics) as presented in [BE1], [AB1], and [LAI]. The equation (1.1) de­ scribes physical phenomena which are often of great engineering and tech­ nological importance (in aerospace industry, or currently in semiconductor design). For that reason, analytical and computational methods of solving (1.1) have been studied extensively since the first computer hardware mak­ ing these calculations feasible. In general, the computational techniques for the Boltzmann equation can be divided into two categories: deterministic 179

180

W. Walus

methods and stochastic methods. In this lecture we shall concentrate on the deterministic methods giving fairly detailed description of the methods and in some cases relevant mathematical framework. Most of the deterministic and stochastic methods for nonstationary flows use the so-called splitting method (see e.g. [MK1] for general formulation and applications to problems in mathematical physics). The splitting method in kinetic theory, as a procedure to approximate solutions of the Boltzmann equation, was suggested by Grad in Principles of the kinetic theory of gases in 1958 ([GR1], pp. 246-247). In this method during each time step two consecutive stages are carried out: one corresponding to the free flow prob­ lem and the other one corresponding to the spatially uniform relaxation. In the context of simulation methods the method is phrased as the decoupling principle of physical processes, while used in numerical schemes to solve the Boltzmann equation it leads to the actual splitting of the equation. In Sec­ tion 2 we describe the splitting method applied to the Boltzmann equation and we present the convergence result due to Bogomolov [BOl]. The deterministic methods combine finite difference, finite element or finite volume approximations of the free flow equation with an appropriate evaluation method for the collision operator. Relevant numerical schemes for the Boltzmann equation are described in Section 3. In these schemes the substantive difficulty is the evaluation of the collision operator. These evaluation methods can be divided into two groups: statistical quadratures and regular quadratures. In the first group the Monte Carlo quadratures are applied to evaluate the collision operator. This approach was initiated by Nordsieck in 1955 as described in [NH1], and then generalized and improved by e.g. Hicks and Yen [HY1], Yen and Lee [YL1], and by Tcheremissine in [TR1-5]. In the second group the collision operator is evaluated analytically or numerically (using regular quadratures) for particular discretizations of the distribution function, as was done e.g. by Aristov [AR1-3] and Tan et al. [TA1]. Recently, more progress in this direction was achieved by: Bobylev et al. [BP1-3], Buet [BT1], Michel and Schneider [MSI] and Ohwada [OH1]. The evaluation methods for the collision operator are presented in Section 4. A correction technique that enforces the fulfillment of the (discretized) conservation laws for the numerical solutions is presented in Section 5. Fi­ nally, in Section 6 we describe briefly a few other deterministic methods for the Boltzmann equation that do not accord with our previous classification or apply in special cases.

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The stochastic methods constitute the other important area within com­ putational techniques for solving the Boltzmann equation. Flow simulation methods, known as Direct Simulation Monte Carlo (DSMC) methods, were initiated by Bird [BR1]. Since then his method was undergoing subsequent improvements and modifications, see e.g. [BR2-3], [KR1], [KR2], [DN1-2], and [ER1]. A few years later similar simulation algorithms, based on the Kac master equation, were given by Belotserkovskiy and Yanitskiy [BY1,2], Yanitskiy [YA1,2], Ivanov and Rogasinsky [IR1-3], and by Deshpande [DEI]. The simulation method derived from the Boltzmann equation was presented by Nanbu in the series of his papers [NB1-4]. The essential modifications of the Nanbu method, that reduced its high computational complexity and that made the algorithm applicable, were done by Babovsky [BA1]. Further developments of simulation schemes were given in [BA3-5], [BS1], [LC1-2], [WL1]. Comparative discussion of the schemes was presented e.g. in [BR5-8], [GN1], [IN1], [IR1-2], [NB7], [NG1], [PS1], [SSI], while theoretical analysis can be found in [IR1-3], [IR4], [NB5-6], [PR1], [YA3-5]. The new edition of the monograph by Bird [BR9] gives an excellent exposition of DSMC methods (see also [BR4]). In addition, it provides demonstration simula­ tion programs. The DSMC methods proved to be very successful in solving complex flows of rarefied gases, e.g. in computations for the space shuttle re-entry. Wide spectrum of applications of the DSMC methods is given, for instance, in the Proceedings of the Rarefied Gas Dynamics Symposia. The validity of the DSMC methods, i.e. the convergence of the solutions obtained via the relevant algorithms to the solutions of the Boltzmann equa­ tion was an open problem for a long time. The first convergence proof was given by Babovsky and Illner [BA2], [BI1] for the Nanbu method, and only recently by Wagner [WG1] for the Bird method. Another stochastic approach to solve the Boltzmann equation was in­ troduced by Arsen'ev [AS1,2]. The simulation method follows from refor­ mulation of the Boltzmann equation into the form of the Ito's stochastic differential equation. This approach was further developed by Lukshin and Smirnov [LS1,2], Smirnov [SMI], and Voronina [V01,2]. We conclude this introduction by giving some basic expressions and for­ mulas to which we shall refer in the course of this lecture. The differential part on the left-hand side of (1.1) is usually called the free flow (or stream­ ing) operator, while the right-hand side «/(/, / ) is the collision operator that corresponds to the particle interaction (the collisions). In the rarefied gas dynamics the collision operator has the following form

182

W. Walus

J ( / , / ) ( v ) = [/ J(fJ)W=

3

/

B ( f f , c o B «B(q,cosO)(f(V)f(w')-f(v)f(w))dndw. )(/(v')/(w,)-/(v)/(w))Aidw.

Jm. Js

2

(1.2)

+

In the above formula v' and w' are the post-collision velocities and w' w' == w w -- n(n n ( n ••qq) ), , v' = v + -1- n(n • q) and

(1.3) (1.3)

where n is the unit vector in the direction of the apse-line bisecting q = w - v and v' - w'. Moreover, q = |q| and qcosO = n • q. The symbol dn denotes the integration over the hemi-sphere S^ {nG (w- -v )v) >> 0 0} }. . S^. = {n eE 1R33 : |n| = 1, n -■( w If 6 e [0,7r/2] is used as the azimuth angle and e G [0, 2TT] as the polar angle on the hemi-sphere, then dn = sin 0d9de . The kernel of the collision operator B(q, z) > 0 is determined by the intermolecular potential. It is often assumed to be expressed in the form B(q, co*e)=qyj3(0),

(1-4)

with a smooth function ft and 0 < 7 < 1. For example, for the power potential 0(r) ~ r~K B(q,z) = q1-^"/3(z), (1.5) where the function (5{z) is continuous on [0,1] and C\ < /3(z)z~1 < c2 for some positive constants C\ and c^. In the formal limit « -> 00 one obtains the hard sphere model for which B takes particularly simple form B(q,z) = d2qz

(1.6)

where d is the diameter of the spheres. The hard sphere model, when taken as the intermolecular potential, lessens technical difficulties inherent in the Boltzmann equation but at the same time leads to gas parameters that disagree with these observed in real gases (e.g. the viscosity coefficient). In order to overcome this deficiency of the hard sphere model while retaining its advantages, the variable hard sphere (VHS) model was introduced by Bird [BR6] for which B(q,z) CKql-i'«z, B(q,z)=C=Kq1-*/"z, (17) (1.7)

COMPUTATIONAL METHODS

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where K is the exponent in the potential and CK = 3\/2(2 - s ) _ s / r ( 4 _ s) with 5 = 2/AC. There are also other, alternative to (1.2), expressions for the collision operator. Using the impact parameter b and the polar angle e to describe the particle interactions, we have J(/./)(v) = /

/ * / """ ( Z ( v ' ) / ( W ) -f(y)f(w))qbdb
JJR3 JO

(1.8)

JO

for the intermolecular potential with finite range bmax. In the case of the hard sphere model bmax = d. For finite range intermolecular potentials the collision operator J ( / , / ) can be decomposed

JU,f) = JiVJ)-Mf,f)

(1.9)

into the gain term J\(f,f) and the loss term J 2 ( / , / ) = / K / ) > D 0 t n terms having the obvious definitions which follow from (1.8). For infinite range potentials (e.g. the power potential) the integration with respect to the impact parameter b in (1.8) extends to oo. It is well known that this causes mathematical difficulties. For instance, the total cross-section is infinite and in general the decomposition (1.9) of the integral (1.8) is not possible as J i ( / , / ) and v(f) diverge. Therefore, one often introduces a cut-off that limits the impact parameter, say by bmax. Then for potentials with such a cut-off, the expression (1.8) is valid and the splitting (1.9) holds as well. Note, that if the collision operator is written in the form (1.2), a cut-off removes the singularity of the kernel B at 9 = J . In what follows we shall assume that either the potential has finite range or a cut-off has been introduced. In (1.8) the post-collision velocities are computed using the formulas v' = v c + - q ' and w ' = v c - - q ' ,

(1.10)

where v c = | ( v + w) and q' stands for the relative velocity after the colli­ sion. If x denotes the scattering angle, i.e. the angle between q and q', the components of the vector q' in the original co-ordinate system are , q'x = q x c o s x ,

q' = q2 cosx

9i93 s i.n <7l2 9293

X c o s e "I

.

sinxcose


. 929 s i n.
Xsin e >

sinxsine,

<7l2

184

W. Walus

where #12 === y/qf + q%. #2- The Theangle anglexx depends dependson onq,q, andthe the intermolecular intermolecular , bband A2+ potential 4>(r) <j>(r)

_ xX =*

n

p bdu Jo -^(j){uJo y/\-b v"1 2-u2b2u2 - l4<)lmq £( u2"-l)/mq2

'

where m is the mass of the molecule and UQ is the smallest positive root of 1l-b-2ub2 2u2 -A^u-^/mq - ±(u-2l)/mq2 = 0. 0. Moreover, note that \ - IT - 20, where 0 is the angle used to describe the vector n in (1.3). The reader unfamiliar with kinetic theory is advised to consult the texts by Cercignani [CE1] or by Truesdell and Muncaster [TM1] for the physical background and mathematical treatment of the Boltzmann equation. 2.

The Splitting M e t h o d in Kinetic Theory

The major approaches to obtain approximate solutions of the Boltzmann equation for nonstationary problems are based on the splitting method. In the case of the deterministic methods the splitting procedure decouples the Boltzmann equation on each time step into two equations: the spatially homogeneous Boltzmann equation and the collisionless transport equation, which are then solved numerically by the appropriate schemes. In the case of the DSMC methods this procedure is presented as the decoupling of the physical processes undergoing in the course of the evolution of the gas. During each time step the gas molecules first undergo collisions in each spatial cell obtaining new velocities, and then the molecules are moved freely over the computational domain according to their instantaneous velocities. The simulation of the collision process is compatible with the spatially homogeneous Boltzmann equation. In both cases the approximate solution of the Boltzmann equation is constructed in the following routine. Suppose that the time interval [0,T] has been divided into TV equal subintervals of length r = T/N, and suppose that the (constant) value fk of the approximate distribution function fT on the time interval ((fc-l)r, kr] has been computed where k = 1 , . . . , N - 1 (/° = f0 [s the initial value of the distribution function). Then fk+1, the value of the distribution function on the next time interval (kr, {k + l)r), is obtained in two steps. In the first

COMPUTATIONAL METHODS

185

step, one solves the spatially uniform relaxation problem %-

= J(f',n

on

(fer, (* + l ) r ] ,

r(kr) = f.

(2.1)

The solution of (2.1) evaluated at the endpoint (k + l ) r serves then. as as the initial value for the second step which corresponds to the free flow problem roblem ^ + v V

x

/ " = 0

on

(*r, (* + 1)T] ,

r*(kr) = } ' ((* + 1)T).

(2.2)

Finally, using the solution of (2.2) one sets / * + i = /••((* + i ) r ) . Hence, the approximation fT on the time interval (kr, (k + l)r] is defined by fr{t)=fk+l for t e ( f c r , ( H l ) r ] . In the routine just described the word solve has to be understood properly depending on the context. In the deterministic methods it means further discretization of the position and velocity variables and solving the resulting algebraic systems for the values of the distribution function at the grid points. In DSMC methods it means to simulate a stochastic process based on (2.1) with finite number of molecules distributed over the cells covering the spatial domain. In the case of an initial boundary value problem the appropriate bound­ ary conditions have to be taken into account when solving the free flow stage (2.2) of the splitting procedure. Let us also note that the order in which the two steps of the splitting procedure are preformed can be reversed, as was done for example in the Aristov and Tcheremissine algorithms [AT1-6]. When suggesting the splitting method Grad [GR1] conjectured that the approximate distribution function fT converges to the solution of the Boltzmann equation as r —► 0. The convergence was established by Bogomolov [BOl]. Also, recently, a convergence of the splitting procedure towards the DiPerna-Lions solutions of the Boltzmann has been investigated by Desvillettes and Mischeler [DMlJ.

W. Walus

186

Below we shall present Bogomolov's convergence result. The result is limited to the initial value problem for the Boltzmann equation in the case of the hard power potentials with the cut-off (cf. Introduction for terminology and notation). For mathematical convenience, the convergence is established for the perturbation h of the distribution function / which is defined by the relation / = u) + CJ2 h

where UJ is the Maxwellian distribution u{v) = ( 2 7 r ) " e x p ( - - i ; 2 ) . Then, the perturbation function h satisfies the following equation

g+v.v,& = <j(M),

(23)

/i(0,x,v) = / i 0 ( x , v ) , where Q{h,h) = -Lh +

vr{h,h),

with L and vT defined by vY(h,h)

=

u~2j(u)2h,u%h),

Lh= -2u~$J(uj,Lj*h)

=vh-

Kh.

The properties of these operators are well documented, for example, in [BE1]. Bogomolov considered the splitting procedure for (2.3) in the reversed order which was presented in the beginning of this section: Let h° be the value of the perturbation of the initial distribution function, and assume that the constant value hT(t) = hk of the distribution function on ((k l)r,kr] has been computed where k = 1,...,N -1. Then, one solves — + v ■ V x /i* = 0 on dt h*(kr) = hk,

(kr, (k + l ) Jr ] ,

(2.4)

and dh** = Q(h*\h**) dt h**{kr) =fc*((fc + l ) r ) ,

on

fcr,fe

+ l)r], (2.5)

COMPUTATIONAL METHODS

187

which gives hT on the next time interval (kr, (k + l)r] hT(t) = hk+1 =h**((k + l)r)

for

te(kr,{k+l)r].

(2.6)

The convergence is established in the function space F? = {/ such that sup(l + v2)^ X,V

Y^ l^x/(x, v)| < 00}

(a > 0)

|fe|<m

with the norm | | / l k * = SUp(l+*, 2 )f Y, X V

'

I^x/(X,V)|.

|fc|<m

First we quote the existence and uniqueness result (cf. Theorem 6 in [MAI]) that provides solutions of (2.3) in the space F™ for the hard power potentials (1.5) (i.e. with K > 4): Theorem 2.1. Let a > 2. If ho € F™, then there exists T > 0 such that the initial value problem (2.3) has a unique solution h £ L oo ([0,T];F ( J x ). The convergence of the splitting procedure is asserted in the following the­ orem. Theorem 2.2. Suppose that ho € F% with Qi > a + 3 where a > 2. Let h be the unique solution of the initial value problem (2.3) on [0,T] (where T is as in Theorem 1^.1.) and let hT be the function defined in the splitting procedure (2.4)-(2.6). Then \\h - hT\\o,a =

0(r2),

uniformly on [0,T]. Outline of the proof: Let hk+1 — ST(hk) denote the solution of the one step of the splitting procedure with hk as the input. Then hT(t) = hn = S?(/io)

for

t € ((n - l)r, TIT] .

Moreover, let h(t) = Vt(ho) be the solution of the initial value problem (2.3). For t = nr we write as follows Vt(ho) - S?(h0) =V?~lVT(ho) - V?-lST(ho) +V?-2VT(hl) - VTn-2ST(hl) + • • • +V/- 1 V r (/i n - j ) - Vi-lST{hn-') +VT{hn~l) -

ST(hn~l).

+ •••

188 .

W. Walus

Each difference term above has the following form 1 nJ n n n 3 J (SUT-1)T (h(S ->)). vJVrlVAh vT-(h) -') -- vr'sAh"-') vrlST(hn-J) = =v V{3 -,1)r (vT(/i"- )) )) ---VU-1)TV )T(Vr(h r(h»-i)). 0_

Therefore, to estimate h - hr = Vt(h0) - S?{h0) one needs two arguments: (i) Lipschitz continuity of Vt with respect to the initial data: \\V - V Vt(ho) Vt(go)\\a ||ho0 --- gP0o\\lQi o, t(ho) -t(go)\\a < const \\h which holds for ho and go belonging to Hai, and (ii) a norm estimate of the difference between the solution of the initial value problem (2.3) on the time interval [0, r] and the function obtained in one step of length r of the splitting procedure if both problems have the same initial data: \\VTT(ho) {ho) - SSTT(ho)\\a (ho)\\a = 0 ( r 33 ) , which holds for ho satisfying the assumptions of Theorem 2.2. The full proofs of (i) and (ii) are given in [BOl]. Using these estimates, one obtains finally 3 \\V (h0) --- S?(Ao)|| Vt(ho) Sr ( M I U0 < ^ constnO(r ) =

3.

0(r2).

Numerical Schemes for the Boltzmann Equation

Numerical schemes for nonstationary problems As already mentioned in Section 2, the majority of numerical schemes for solving nonstationary problems for the Boltzmann equation

^+v-V V xx / = JJ((//,,//)),,

(3.1)

use the splitting method. The method was first applied in numerical procedures to solve (3.1) by Aristov and Tcheremissine [ATl-3,5-6]. Since the method decouples the equation (3.1) into the free flow equation and the spatially homogeneous Boltzmann equation, we shall describe the relevant numerical schemes for the two equations separately. Schemes for the free flow equation The equation of the free flow

df

§£ + v - V x / = 0

(3.2)

COMPUTATIONAL METHODS

189

can be solved numerically using the standard finite difference, finite element or finite volume methods for the hyperbolic advection equation. Finite difference

schemes

Let At be the length of the time step and denote tn = nAt and let Ax, Ay and Az be the mesh steps for the x, y and z variables, respectively. The value of the distribution function at the n-th time step tn = nAt and at the point Xijtk = (iAx,jAy,kAz) is denoted by /™ f t , i.e. fr,j,k(w)

=/(*n,Xi, i > f t ,v).

In one dimensional (ID) flows, when we can assume that the distribution function does not depend on y and z, we shall suppress the indices j and k in the above notation. Here, for simplicity, we shall present the schemes for ID flows (say, in the x direction). Initially, the first-order upwind scheme fn+l

fn — fn

_ fn A,

+«*-

At Ji

7

A^

=

A

0

lf

U

=0

if

vi<0,

l

> 0

>

Ax Ji

i+

+vi \

Ji

Ax

(33)

was used [AT1-6,TR3-5,TA,YN1,YT]. In (3.3), vl denotes the component of the velocity v in the x direction. Recently, higher order schemes have been applied. For example, in [TV1], the following centered fourth-order scheme was employed /f + 1

= a_ 2 /f_ 2 + a - i / J L i + a0f? + ai/f + 1 + a2f?+2 ,

(3.4)

where

a0=i(l-c2)(4-c2), a±l=i(-l±c)(-4

+ c2),

D

a ± 2 -^(-2±c)(l-c 2 ), and

€ = *■£.

(3.5)

The schemes (3.3) and (3.4) are solved pointwise for each velocity v belonging to some finite velocity lattice V. They require the CourantPriedrichs-Lewy condition ensuring in this case the stability max Id < 1. vev

(3.6)

W. Walus

190

The CFL condition limits the maximum allowable time step, and therefore one might be tempted to use an implicit scheme that is unconditionally stable. However, solutions at tn+1 of an implicit scheme at interior points of the computational domain depend also on the boundary values of the distribution function at tn. This means that particles from the boundary at tn would influence an interior point at tn+\ without undergoing collisions during this single time step and that is incompatible with the splitting procedure. Finite volume schemes In higher dimensions the finite difference schemes might pose technical prob­ lems during solution procedures. To overcome these problems a directional splitting might be necessary. However, this kind of splitting often has un­ pleasant side effects. Therefore, in 2D or 3D flows finite volume schemes are preferable. Moreover, they provide easier handling of complex computa­ tional domains and possess good conservation property. An introduction to finite volume methods applied in fluid dynamics is given in [HR1] (Volume 1, Chapter 6). Here, following Buet [BT1], we present finite volume scheme for 2D flows governed by the advection equation. Let M denote the partitioning of the computational domain into quadri­ laterals. Let M e M be a control cell having the vertices A, B, C, D. We denote by nAB the unit outward normal to the side AB, by \AB\ its length, by SAB the midpoint of AB, and finally by MAB the cell adjacent to the side AB. Similar notation is also used for the same concepts for each of the other sides. Moreover, \M\ stands for the area of the cell M and S M for its center. In what follows we shall suppress in the notation the explicit dependence of / on the velocity variable. Let f]fr be an approximation of

fitn x)dx

w\L ' ' and let g be a function defined on the computational domain such that its restriction QM to the cell M satisfies

f =

9M{x)dx

" w\L

-

Let qAB be the flux through the side AB defined as follows . qAB

~

f w ■ nAB 9%\AB\, \VUAB9TB\ABI

ifv-nAB>0, ifv.nA5<0,

COMPUTATIONAL METHODS

191

where 9AB =

limc

0( x ) i

g%=

lim

S(x).

Likewise, we define the fluxes through the other sides of the cell M. The finite volume scheme for the free flow equation (3.2) is defined as follows /M

= / M ~ ~\M~\(qAB + QBC + QcD + qDA)-

(3.7)

The first order scheme corresponds to the choice gM (X) = fu

on the cell

M,

and the second order scheme is obtained with PM(X) = / ^ + ( V

X

/)^(X-5

M

)

for

xGM.

In the above formula (V x /) Jf denotes an approximation of the gradient of / on the cell M such that min(fMJMxY)

< gfy < m a x ( / ^ , / J ^ x y ) ,

where XY stands for the sides AB,... ,DA of the cell. The stability condition for the schemes (3.7) follows from the condition under which the schemes preserve positivity of /jjf. In the case of the first order scheme the condition is max max —— > maxfv • rij^y, 0)\XYI < 1. V<=VM(EM \M\ *-£

v

n

'

(3.8)

_

Note that in the case of ID flows (3.8) reduces to the CFL condition (3.6). The second order scheme preserves positivity, and hence is stable, under a stronger condition max max ——• max(max(v • n ^ y , 0)|Xy|) < - . v<EV MeM \M\ XY

(3.9)

4

Finite element schemes Finite element schemes might offer yet another method to solve the free flow equation (3.2). The formulation of the schemes depends on the com­ putational domain and the boundary condition associated with (3.1) which

W. Walus

192

in the splitting procedure is considered together with the equation (3.2). Recently, a finite element scheme in the case of 2D flow in ( - c o , 0] x IR with the specular reflection on the boundary of a two dimensional model gas (i.e. v 6 IR2) was used by Rogier and Schneider [RSI]. Schemes for the spatially homogeneous B o l t z m a n n equation The equation

(3-10)

% = Af,f) is solved numerically using the standard finite difference schemes: explicit fn+l _ fn = Jn, At

(3.11)

implicit fn+l _ J

fn J

= J? ~ fn+l"n

Af

,

(3.12)

exponent fn+i

_ fn

exp(

_ A t vn}

where, recall, J(f,f) — Ji(f,f) the exact integration of

+

A . (i _

- fv{f)-

exp

( - At vn)) ,

(3.13)

The scheme (3.13) is obtained by

df on the interval [tn,tn+i] with fn as the initial data. The equations (3.11)— (3.13) are solved pointwise at each grid point Xiyj,k (or for any cell M if a finite volume method is used for the free flow equation) and at each velocity vi from the relevant velocity lattice V in IR3. Therefore, in these equations fm = f%,k,x (m = n , n + 1) and J" denotes J ( / £ . fc>/£. J ( V l ) (with the likewise notation for JJ1 and vn). Note that during computation of Jn the values of the distribution function at all the velocity lattice points are used. Methods for evaluation of the collision operator J ( / , / ) are described in Section 4. Most of the computational effort on each time step of the splitting method is spent during the relaxation stage, since this stage involves eval­ uation of the collision operator. Therefore, in order to shorten computing time, one may modify the splitting scheme in the following way: during each cycle the free flow stage is performed m-times with a time step At

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allowed by the stability condition (3.6) or (3.8), and then the relaxation stage is done once over the time step of length mAi. However, the number of repetitions m of the free flow stage should be reasonably small to retain the order of the approximation of the scheme. Finally note that the splitting procedure is particularly well suited for parallel and multiprocessor computing. Suggestions for the organization of parallel codes using deterministic schemes were given by Yen and Lee [YL1], cf. also Aristov and Tcheremissine [AT5,6]. Numerical schemes for stationary problems There are two approaches to the numerical solution of the stationary Boltzmann equation v • V x / = J(/, / ) . (3.14) One approach is based on the preposition that a stationary solution of the Boltzmann equation is obtained as the asymptotic state of the nonstationary equation for large time. Then, one employs the methods for nonstationary equation performing sufficiently large number of time steps. In the other approach an iterative scheme is used. The simplest schemes correspond to ID flows. The following explicit one was used by Nordsieck An+l)

_ An+l) 7

vih

*~i

-, =

i(j<"> + jM)

for

Vl

>0,

(3.15)

where f. stands for the n-th iteration of distribution function evaluated at (xi, v). The iterations are computed pointwise at each v from a velocity lattice. Tcheremissine [TR1,2] employed the following implicit schemes An+l) „, [i

An+l) Ji-l _

T(n)

An+l)

(n)

/o -. fi\

and r(n+l)

Sj

An+l)

-—P^^ Ax

.,

= k4f - /ln+1)^n)+Ati - ftt^tl).

(3.i7)

for vi > 0. Recently, the scheme (3.17) was also exploited by Ohwada [OH1].

W. Walus

194

Yet another iterative scheme derived from the integral form of the equa­ tion (3.14) was proposed by Tcheremissine [TR1,2]

/!-> =to-p(-g^> + # u - «rf-£*i») l

~l

\

J{1\

. Ax (n)

X e x p (

-2^"'

)+

#)

(1_eXp(

J

(

"2^

(3.18)

Ax (n)

!/

'

'^

i

for vi > 0. Any of these iterative schemes needs an initialization - a function /**) defined at all spatial and velocity nodes which satisfies the relevant bound­ ary conditons. The same applies to the time-dependent approach, where one has to specify an auxiliary initial distribution function. Further modifications of the presented schemes are possible. For exam­ ple, when solving the flows where large spatial gradients of the distribution function might occur, one should use a variable space mesh Ax^ instead of the uniform one Ax. 4.

Numerical Evaluation of the Collision Operator

In this section we describe numerical methods for evaluation of the collision operator. At an early stage of the development of numerical procedures for the Boltzmann equation the collision operator was evaluated mostly by the Monte Carlo quadratures. Recently, since the advances of computer hardware, like parallel and fast computer units that nowadays make possible large scale and complicated computations, the so-called regular methods, i.e. the ones involving only analytical and numerical integrations serve as a prospective development. In the first subsection we describe the Monte Carlo integration methods, and in the second the regular methods. Before we go on to the description of the relevant methods, we shall make some comments on problems one encounters when evaluating numerically the collision operator. The first problem arises when one replaces the velocity space IR3 in the collision operator (1.2) (or (1.8)) by a bounded domain V C IR3. The domain has to be chosen in such a way that the contribution from the inte­ gration over the complement of V is negligible. This requires the knowledge of the support of the distribution function / in advance. For simple flows,

COMPUTATIONAL METHODS

195

when the distribution function is not far from the equilibrium, one can lo­ calize the support easily. Usually one assumes that the support is within the ball of radius equal to a few mean mass velocities of the flow. The replacement of IR by V imposes an artificial bound on the velocities of the particles of the gas, which contrast with the Monte Carlo simulation methods where the particles are allowed to have any velocity. In addition, no matter how we choose V, there will be pairs of velocities in V such that after a collision at least one of them falls outside V. It has been shown in numerical experiments that the portion of such collisions may be quite sig­ nificant (16% - as reported by Tcheremissine [TR1]). Therefore, unless one is sure that these post-collision velocities go out of the support of the distri­ bution function, one should employ an extrapolation technique to take into account the contribution of the distribution function at these post-collision velocities. The treatment of the velocity variable depends on the way one handles the evaluation of the integral defining the collision operator. In most of the proposed numerical schemes for solving (3.1), esspecially those that use the Monte Carlo quadratures for the collision operator, the velocity variable is discretized. Having chosen the bounded velocity domain V, one introduces a finite lattice V = {vi : i € 1} (4.1) in V where I is a finite subset of Z 3 . The value of the distribution function at a lattice point is then denoted by /i, i.e. we have /i(t,x) = /(*,X,Vj).

The choice of the velocity lattice depends on the properties of the flow under consideration and on types of approximation applied to represent the distribution function. Typically, one uses a regular lattice with the mesh step Av Vi = (Av)i where i = (ii,t2,*3) € I C Z 3 . (4.2) In the case of stationary ID flows, where the distribution function has the axial symmetry, i.e. when it depends on (vj, \Jv\ + vf), the velocity lattice shall be chosen in a relevant subspace of IR x [0,oo), see e.g. [NH1]. An example of a particular choice of the velocity lattice for ID shock wave problem is given in [OH1]. There are a number of technical aspects of the collision operator that make the numerical evaluation difficult and time consuming. Let us point

196

W. Walus

out some of them. The collision operator is defined by the five-fold integral. Therefore, direct quadrature formulas applied to this integral may lead to inefficient computations, unless one takes additional simplifying assump­ tions. The usual remedy is to employ the Monte Carlo quadratures. In this case, however, one has to face the calculations of the post-collision veloci­ ties. Since, in general, the post-collision velocities do not lay on the velocity lattice, in addition to already mentioned extrapolation, one needs an inter­ polation method to obtain the relevant values of the distribution function. Besides, in most of the methods one considers the distribution function at all velocity lattice points, even at the points where the distribution function is negligibly small (hence carrying no essential information). Recently Tan and Varghese [TV1] proposed a self-adopting method that overcomes some of the above mentioned difficulties. In their method, only these velocity lattice points at which the distribution function is greater than a prescribed threshold are considered and, moreover, the velocity lattice varies in time thus allowing the particles to have any velocity. Monte Carlo quadratures First we recall the formulas for the Monte Carlo quadratures. The reader unfamiliar with the topic is advised to consult e.g. [BUI], [KW1] or [HB1]. Let £ be n-dimensional random variable on Q C IRn with given proba­ bility density function V. Suppose that a function / is Lebesgue integrable on Q. The statistical Monte Carlo approximation of the integral

/1== /f f(u)du f(u)du Jn is defined by

7

1

N

11

^ = ^£^-y/te),

(4.3) (4.3)

where £ i , . . . , £N are randomly sampled points from Q obtained in TV inde­ pendent trials. In particular, for & uniformly distributed over the domain H, V(u) = l/m(ft) if u G H and 0 otherwise (here m(ft) denotes the Lebesgue measure of ft). In this case, the Monte Carlo approximation formula (4.3) for / reduces to its commonly used form

IMC /MC = = - N^

£ / ( ^f(€i). &)■ Z= i=l

l

(4.4)

COMPUTATIONAL METHODS

197

For the approximate formula (4.4) one can give an error estimate. Let C = m(f2)/(£), so that the expectation value of ( is just 7", and let Z)( denote the variance. Then the error estimate I-lKc\
(4.5)

holds with probability ~ p, where C depends on p. Observe, that the error of the Monte Carlo quadrature decreases as l/y/N and this feature does not depend on the dimension of the integral. One can also approximate the integral / applying the quasi Monte Carlo quadratures. For a Riemann integrable function / on Q — [0, l ] n the quasi Monte Carlo quadrature is given by 1

N

W = iv£ /(Ui) '

(4.6)

where the nodes u i , . . . , \IN are defined by a deterministic formula in such a way that the sequence {iij} is equidistributed in [0, l] n . For functions / that have finite variation Vf on [0, l ] n (in the sense of Hardy and Krause) one has the error estimate \I-IqMc\
(4.7)

where DN is the discrepancy of the sequence {u;}. Use of the quasi Monte Carlo quadratures is advantageous if one can provide points {u^} for which the error estimate (4.7) is better than (4.5). As a matter of fact, there are examples of finite sequences for which DN — 0(N~1 logn~ N). The examples of such sequences and the full exposition of the quasi Monte Carlo quadratures is given in the review paper by Niederreiter [NR1]. The Monte Carlo technique was introduced for the evaluation of the collision operator by Nordsieck in 1955 as first described in [NH1]. Since then it has been refined by many authors, see e.g. [HY1], [TR1-5], [YH1], [YN1,2] and [FP1]. Our description follows to some extend the exposition given by Aristov and Tcheremissine in [AT5,6]. Let us assume that the velocity space H 3 has been replaced by a bounded domain V and that a lattice V = {vj}iei in V has been introduced. We shall consider an approximation of the value of the collision integral at the fixed

W. Walus

198

point v k . The evaluation method described below applies to the collision operator expressed in the form (1.8) in which additionally the change of variable b -* k(r^—)2 n a s been performed and the multiplicative constant 2, V Umax

resulting from this substitution has been ignored. Let (wi, &», Cj), i - 1 , . . . , N be random points sampled from V x [0,1] x [0,27r] accordingly to the probability distribution function V(w,b,e). Then the Monte Carlo approximation of the collision frequency term is given by the following formula (cf. (4.3)) „(/)(vk)

S

* = I f

^ - ^ I v x

- w,|/(w.),

(4.8)

- w i |/(v k )/(wi).

(4.9)

1=1

and likewise for the gain term

J l ( /,/)(v k )

K

Jlk = I f ) ^ ^ y K 1=1

In general the randomly chosen velocity Wj in the formula (4.8) and the post-collision velocities vj^, wj in (4.9) do not belong to the velocity lattice V, and therefore one has to use an interpolation to obtain the values of / ( W J ) , / ( v [ J and /(wj). Let us also note that, although we have splitted the collision operator when giving the Monte Carlo approximations (4.8) and (4.9), both terms v^ and J i k are computed using the same samples of the random points (wi,6;,€j). This provides higher accuracy of the evaluation of the collison operator and lessens the computational effort. The various Monte Carlo formulas for the evaluation of the collision integral are obtained by defining the appropriate probability distribution function V and by applying special techniques to reduce the variance. Monte Carlo quadratures for one dimensional flows The simplest quadrature formula for the collision operator is obtained with the uniformly distributed random points (WJ, &»,€») {i = 1 , . . . , N) in V x [0,1] x [0,2TT], i.e. for V = l/(2irvol(V)). This formula was used at early stage of the development of the numerical integrations of the collision op­ erator, and was applied mostly to spatially one dimensional problems. In stationary ID flows (say, in the x direction), one could assume the symme­ try of the distribution function with respect to the v\ axis of the velocity variable, thus reducing the multiplicity of the collision integral. In such setting the Monte Carlo method was first used by Nordsieck [NH1].

COMPUTATIONAL METHODS _

_

^

. 199

Despite its simplicity, this quadrature has -obvious disad¥antages. In fact5 using the uniformly distributed random points, one does not take into account the fact that the distribution function decreases rapidly for velocities apart from the mean mass Yelocity. Moreover, basing on physical arguments, one can predict that in ID lows the distribution function evolves with respect to U2 and t% velocity components to a state close to the Maxwellian distribution much faster than with respect to v\ velocity component. Since the variance of the random variable used in Monte Carlo quadrature decreases when its probability density function is close to the (normalized) integrand, one can improve the accuracy of the Monte Carlo approximation using the random points for (102, W3) that have the normal probability distribution function 1

ml + twl

^P 233 (TU2,ttl3) ( ^ , » 3 ) == ^ e2 CXP{ x p ( - ^ 2 i )] f , 2fru

2CF

with suitably chosen parameter o~ (see [AT5,6]). In addition, one can further increase the accuracy, using group sampling technique with respect to w\ variable: first one divides the w%-domain into M equal length subintervals and then in each of them performs uniform sampling. The number of points sampled in the subintervals may vary from one subinterval to another, choosing more points in the subintervals where the integrand contributes significantly to the integral. Monte Carlo quadratures for two and three dimensional flows When calculating 2D or 3D fiows the accuracy of the quadratures for the collision operator and the computational effort become important factors. Therefore, the evaluation techniques for the collision integrals should compromise these factors. This can be achieved by using the group sampling technique and by employing the so-called symmetric points in the quadrature formula, or by passing to the quasi Monte Carlo quadratures. For example, in some calculations performed by Tcheremissine, the integration domain V x [0,1] x [0,2ir] (V is a bounded parallelepiped in JR3) is divided into 8 five-dimensional subdomains in the following way. The w\interval is divided into 4 intervals of equal length, and [0,2ir] is splitted into [0, w] and [TT, 2fr]. Then, in each of these subdomains, after having generated N (uniformly distributed) random points additional N points are obtained by the symmetric reflections with respect to the center. The reflected points have the same probability distribution as the randomly generated points.

W. Walul

200

Note that the number of points in the corresponding quadrature formula is a multiple of 16, and that further divisions of the intervals of the integra­ tion variables increases the minimal number of the points in the quadrature formula. The collision integral can also be approximated using the quasi Monte Carlo quadratures. On theoretical basis they provide better accuracy than the statistical Monte Carlo quadratures, A quasi Monte Carlo quadrature with the nodes forming the pamllelepipedal net was employed by Teheremissine in [TR5]. The parallelepipedal net in [0, l] 5 is defined as follows «* = ( { ^ } . " - . { ^ } )

f° r

k=

i,...,N,

where {•} stands for the fractional part of a number, and a i , . . . , a s are so-called optimal coefficients moduio TV (see Korobov [K02]). These coeffi­ cients depend on TV (in general also on the dimensionality of the integral), and are integers relatively prime with respect to TV. The algorithm for Inding at is given in [KOl], some indications are also in [K02]. Regular quadratures Regular quadratures for the collision operator are performed for distribu­ tion functions which have been suitably approximated, e.g. by (truncated) expansions with respect to some basis functions. Often, these approxima­ tions lead to a discretized Boltzmann equation which resembles equation! for discrete-velocity models (DVM) in kinetic theory. The idea of passing from the Boltzmann equation to a system of DVM equations was first em­ ployed in the context of regular quadratures for the collision operator by Aristov [AR1-3]. Similar approach, also based on a piecewise constant ap­ proximation of the distribution function, was presented by Tan et at, [TA1]. Recently this idea was further developed by Bobylev et al [BP1-3] and Buet [BT1]. First, we indicate the method proposed by Aristov. Assume that the velocity space 1R3 has been replaced by a bounded domain V and let V = fvi = (Av)i : i € 1}

(4.10)

be a finite and regular lattice in V (whence I C Z 3 is finite). Moreover, let Ai denote the cube with edge Av centered at vi and let x\ be the characteristic function of Ai. Suppose that the distribution function is

COMPUTATIONAL METHODS ^ ^ _ ^

. 201

approximated as follows / =

jf Jr

(4.11)

_j ^J *** ^ ^ \^ > » **

!l€;X €;* *\z*

Because of the the assumed form of the distribution function, it is sufficient sufficient to evaluate evaluate the to the collision collision operator operator at at v§. v§. Eecall Eecall that that the the collision collision operator involves two involves two integrations: integrations: one one over over the the velocity velocity domain domain and and the the other other one one F t.hf* U vci S Ci .# Using %JolOM. o# JLcl^wCfriJi&yLAiJfeJL Q U Uuadr c M X X dat v Uure I f j * IUIIXXULXc* U VP c*X lilXtS over XOX IXXUXc* ACJX ACJX lIjjlXl \l 5t i AXIIbciLJLcW. AXIIbciLX cW O velocity domain velocity domain V V,, we we obtain obtain U l J t l VJLA JtA\*?%.4k

JLV#Jk Jt Ji*s

\f#JL

«**%-#

\J& A O 1 U JL JL ^ # VJL lJ *.%#*■«

JL fJLJL JL %»# t^ JL\«r JL JL *

JL %S JliO

%J> J L * C * « Ctetj?%«JJL

JJ(/> ( / , //))((vv,i)) = A t » ) s/ 5 y^t/y J u ,i — ((Au) where

(4.12)

r

J u = / £(gy,cos0)(/(v|)/(vj) B(f i J 5 cos#)(/(vI)/(v])-/i/j)cfii (4.13) J..= - fifs)dn 2 is with gy = |vi withgy |vi-Vj[. — Vj[. It remains to evaluate the integrals Jy. Let It remains to evaluate the integrals Jy. Let njj11 = {n € S 22 | vj € Ak, v| € Ai} , njj = {n € S I vj € A k , vj € Ai} , and CM.!,*-* (4.14) rij(ft) = / B(qry, cos 0)dn r u ( n ) = Ja [ B(qlh ms0)dm (4.14) Jo : for 0 C S22. Then, the part of (4.13) that corresponds to flip crairi term can for 0 C S . Then, the part of (4.13) that corresponds to the gain term can be written as be written as (4.15)

Er«(°u1)^

k,i€i

(415)

k,i€l

and and the the part part that that corresponds corresponds to to the the loss loss term term becomes becomes 2 f. f. F..fc2\ rij(S )/i/j. 1 s m fJiJs •

(4.16)

vj j is if the sphere Sy centered at |(VJ §(vi + VJ) with radius | ||vV i| --- Vj| Note that, if n V v,j tnen yllcll contained iin (4.17) r (s 2 )=5]r (nH. 1 ), (4.17) ij

ij

| C 3,113 *

V. Using whereas (4.17) holds approximately if a part of Sy lays outside V. (4.17), we combine (4.12)-(4.16) to obtain the following approximation of the collision integral

j(/,/)(v,)= J(/,/)(vi)= £ JjlC IfcJl

rH'(/k/, - /i/j), rgUA-A/j),

(4.18) (4.18)

nnn £t\J£t

W. Walus

where r | ! = (Au) 3 rij(0| l ). The approximation described aboYe leads to a discretized Boltzmann equation having the form

M + V i .y x / i = J2 r| ! (Mi-/i/j)

foriei

(4.19)

J I l i t | # %Z M-

similar to equations for discrete-velocity models (DVM) in kinetic theory. Note that although it may seem difficult to evaluate the coefficients 1 Fy , once it has been accomplished they can be stored and then used in subsequent numerical computations. Some indications on how to compute these coefficients analytically for the hard sphere potential can be found in [AR1-2]. The method proposed by Tan et al. [TAl] is similar to the one by Aristov. Again the approximation (4.11) of the distribution function is assumed, however, the domain of integration in the collision integral is kept 1R . Then the value of collision operator at v§ for the assumed form of the tJLIo u l 1 0 U ItitJIl 1 LlIliL h I O i l

OtJC^UIlltJo

7(/,/)(vi) - J2 MXkiXiU^Mi

- ^KXk)(vi)/k/i -

(4.20)

Here, in contrast with Aristov's method, the coefficients in (4.20) are com­ puted numerically using standard weighted quadratures (see [TAl]). Next we present another approximation of the collision operator devel­ oped by Bobylev et al [BPl-3]. For this approximation consistency result was also established. It differs from Aristov's method mainly in the evalu­ ation of the integrals (4.13). Suppose now that the velocity lattice V is unbounded and can be iden­ tified with (Av)Z 3 V = {vj = (Av)i : i = (t 1 ,i 2 ,* 3 ) € Z 3 }. Again, the distribution function is discretized by (4.11) with 1 = Z 3 , Using a rectangular quadrature formula for the integral over R 3 in the i^vllloiljjiji l^O'tjJ, CAIJOJL * iTf%J OLJuclilll

. / ( / , / ) ( v , ) ~ ( 2 A t ; ) 3 J2 iei(i)

J

n>

(4.21)

COMPUTATIONAL METHODS _ _ _ ^ ^ _ _ _ _ ^ ^ _ _ ^

203

where I(i) = fj | i—j £ 2Z 3 }, so that the components of the relative velocity vi - vj are even integer multiples of Au and therefore we have (2Au) 3 factor at the sum. Clearly, as before, J

u

=/

B(f U 5 cos^)(/(vi)/(v])™/i/j)dn,

(4.22)

with fy = |vi — vj|. It remains to approximate the integrals Jy. Now, recall that the post-collision velocities vj and v | constitute an antipodal pair on the sphere oy ot radius — |v§ —■ Vj| and center at ovV| + vjj tor any n € o . Therefore, the integral in (4.22) can be approximated as follows

«% ~ w E ,J

4!(A/I

-m ,

(4.23)

(k,l)

where the sum is taken over all antipodal pairs (vk,vi) on Sy which be­ long to the velocity lattice and Py is the number of such pairs. By1 = B^ijjCOSpy1)) and the angle Off corresponds to the collision (vi,Vj) -» (vk,vi). Combining (4.22) and (4.23) we obtain the following approxima­ tion of the collision operator

J(fJ)(n)^MfJ)=

E j,k»i€Z

r«(AA-A/j),

(4-24)

3

where Ty1 = — By1(2Aw)3 if i — j € 2Z 3 and (vk, vi) is an antipodal pair on the sphere Sy, and Ty1 = 0 otherwise. Below we quote the theorem due to Bobylev, Palczewski and Schneider [BP2-3] which states a consistency result for the approximation (4.24), Let us denote by S& (n > 3/2 and * < n ~~ 3/2) the subspace of the Sobolev space U R (Il 3 ) which consists of functions / having the (finite) norm

ll/lki.fcHI/lln + Bup^M 1 J^ P a /( v )l» t=0

|aj=t

where || • || n is the norm in £P(IR 3 ). Thus functions belonging 5£fc, together with their derivatives up to the orderfc,decay at infinity faster than |v| z . T h e o r e m 4 . 1 . Assume that 0 in (1.4) is a bounded C$+a function and that f g SirTHen there « * . a constant C iepenMn, on a, s and 11/11 *+a,5,3 such that for any mesh step Av

\J{f,f)W-JiV,f)\
W. Walus

204

The proof of this theorem given in [BP2-3] uses deep results from number theory. As is indicated there, the error estimate in Theorem 3.1 can be improved to 0((Av)i^e) provided a certain conjecture in number theory is true. Besides the consistency the approximation (4.24) possess another im­ portant feature. Note that F has the following symmetries r»kl _ p k l _ -plk _ p i j 1 ij — l ji — i ij — L k l '

Therefore, the approximation (4.24) retains the conservation property of the collision operator ^^i(/,/)^(vi)=0, l€Z3

where #o(v) = 1, ^ ( v ) = u» for t = 1,2,3, and ^ ( v ) = |v|. This prop­ erty is essential for the performance of numerical schemes. In fact, the algorithms based on Monte Carlo quadratures for the collision operator, which do not retain this property, need an additional correction procedure to ensure accuracy of numerical results (cf. Section 5). An approximation similar to (4.24) was also proposed by Buet [BT1], Additionally, he developed acceleration procedures for evaluation of (4.24) which reduce its 0(N2) computational complexity (N denotes the number of the velocity lattice points). However, these acceleration procedures in­ troduce some stochasticity thus breaking the deterministic character of the method. In actual computations when one uses a inite velocity lattice, the pro­ cedure that yields the approximation (4.24) shall be modified accordingly the sum in (4.21) becomes finite and the delnition of T's changes slightly. The drawback of these approximations is that they involve a large vol­ ume of data (e.g. the coefficients rg 1 ). Even if symmetries of the coefficients are used (as in [AR2], [TA1]), the volume of the data remains large. On the other hand, these coefficients can be computed once in advance and saved. Moreover, the methods based on these approximations provide accurate numerical results (see [BT1] for report on some 2D calculations). The approximations described above lead to a discretized Boltzmann equation (4.19) that resembles equationsfordiscrete-velocity models (DVM)

COMPUTATIONAL' METHODS _

_

^

_

_

_

_

_

_

^

205

in kinetic theory. Therefore, these approximations may serve as a link between the Boltzmann equation and DVM equations. Yet another method of converting the Boltzmann equation to DVM equations was proposed by Nurlibayev [NU1]. The DVM equations (4.19) with large number of the discrete velocities were solved numerically by Inamuro and Sturtevant [ISl] using a deterministic (finite difference) method. The coefficients in the bilinear form on the right-hand side of (4.19) postulated in [ISl] agree with these obtained from formulas for the T coefficients in (4,24) for the hard sphere potential. Flows of discrete-velocity gases were also investigated by Goldstein et al. [GB1], [GOl] using a Monte Carlo simulation (called Integer Direct Simulation Monte Carlo Method) in parallel computing environment. 5. Correction Techniques c Since the Boltzmann equation is based on a balance principle and since the hydrodynamic equations for the macroscopic quantities have the form of the conservation laws, one may expect that the proper numerical scheme should in some sense reflect these properties. The free low stage of the splitting method is in the divergence form and therefore can be treated by conservative schemes. However, the schemes corresponding to the second stage (3.10) do not retain the basic property of the spatially homogeneous Boltzmann equation - the conservation of mass, momentum and energy. The main source of the error comes from the numerical evaluation of the gain and the loss terms of the collision integral. Additional errors are caused during interpolation procedures for evaluating the distribution function at post-collision velocities, and by use of the implicit schemes for solving (3.10). Ita^ Jfc 4t Jfc \ - ^ >-#

The necessity of a correction that would enforce conservativeness of the numerical schemes for the spatially homogeneous Boltzmann equation was noticed already by Nordsieck and Hicks [NH1] (see also [HY1], [YH1]). They used a mean square technique to correct the collision integrals evaluated by Monte Carlo integration. A different technique was proposed by Aristov and Tcheremissine (see [AT3,5,6]). The numerical solution / n obtained in the second stage of the n-th time step of the splitting procedure is corrected by adding a term _—^ / '

> a•iWi t=0

j

206

_

_

_ W. Walus

where #o(v) = 1, ^»(v) = % for % = 1,2,3, and ^ ( v ) = |v|. The numbers at are determined by requiring that the (discretized) conservation laws

Yl ^3 (vk)/k(l k€l 1

k)) + ^ o# t (v'k))

~ w

are satisfied tor j

=

t*=0 =0

::

k / /i k

kk e il

0,«.«, 4,

This correction technique was used successfully in a number of computations significantly improving the accuracy of the results (see [AT5,6]). However, it has been noticed that the correction may yield negative values of the distribution function for large velocities. Since, as observed in computations, these negative values are relatively small in magnitude, one may set the distribution function to zero at such velocities. 6. Other Methods In this section we shall describe a few selected methods which are based on other ideas than presented in the previous sections or are applicable in particular situations. We start with the method proposed by Chorin [CHI]. In this method a truncated Hermite series is employed to represent the distribution function. Use of Hermite series in kinetic theory comes from Grad (see [GR1] or [TM1] for full exposition of the topic). Let Hk(v) be the Hermite polynomial of order k. The set {e^v2^2Hk} forms the orthogonal base in L 2 (M). In order to make formulas compact we denote

HiiWHiti^Hbto), JJi(v) = H^MHiti^Hiilvz), where iI = (i*i,i (ui,tJ2»tJ ,tJ33). The approximation of the distri(*i ,12,13) 2,*3) and v = (tJi,tJ bution function at in = nAt is sought in the form Jr(x,v) 1 5 /=

1 1 ar(x)Hi(u)e 2 /—n f————— y cij |xjiii^iiije

^T^Pf g

"" •

,

(eu) (6.1)

where « u =: |u| and the vector u has components «m «- = ^

£

„ ^

(m = 1,2,3).

(6.2) (6-2)

COMPUTATIONAL METHODS

207

cj^ and s ^ may may Yary with Xrini6 time and As the notation shows, the numbers cj^ vary witn 3JIQ may also depend on the position x. A natural choice for these numbers is to take for c^ the components of the mean velocity and for sajj, j , the the directional temperatures at t n _ i . The summation in ^ t t l X Xtj*%j» out %J U t over I = {i = in (6.1) is tcarried : e orthogonality (*i»*2»*3) : 00 < L2,0 0<
i.*" «w%*s

J *

»*M- = op(x) V

X S

S

f/

S

? 2 3 •'R

3

i

1 UrJU&j|

tlJJU

n /rfrMt..)*, (x,v)fli(u)dv.

(6.3) (6.3)

Using the change of variable (6.2), we transform the integral (6.3) to the fr Y F l T l w /#11 /a[*(x) I- 1 V 1= *"— /1 ft I Y u)e~ 11W (6.4) ff(x, du (6.4) :

VIR 3

appropriate for the weighted Gaussian quadrature. In (6.4), 2

n tt 5 u + cj, Jti (6.5) ft u) —=: v ^y/s?s%s$f ? 5 j r ( xn(x,s?ui , s?t*i "T* + t'1 cf,ji sf ? ^2 + cJ,5Jli3 5 3 + § i (x, V ^ J **v + cj)ifi(u)e cJ)iIl(ll)e ,1 .. (6.5) 1

J*%*^

NTO 1,2,3), Then, for suitably chosen J¥ ,2,3),, of can be approximated by the m (m = 1. quadrature formula formula Wi Nt Ni

iV J¥ N222 JV N33

k k) (i) (i) i q ,«#\ t t ,tiu{ '3 / ^4M l 2 3 , > E E Etf(x.«[°. 2

W

W

w

(6.6) (6.6

t == 00 jjj=0 ==00 fe=0 fc=0

where u^, u^, «g are the roots of the polynomials HNX , #JV2 > flj¥3, respectively, and C4i|l)5 w^', wf5 are the corresponding quadrature weights. Prom (6.5) and (6.6) it follows that to compute the expansion coefficients (6.3) it is enough to evaluate the distribution function fn at the points fc)

((x,c? XjCi+ +S8?vP,<$ j W i » C 2+ + 5a?vg\<$ , C 3 i+ - Ss3|W 43 2 « 2

)J..

(6.7) (6.7)

This can be achieved using the Euler scheme for the Boltzmann equation

i i r(x, ) = Ar-^v) ++ Aa(rM f f - ?^r" f - )(x, ^ x jV)),, / n ( x ,V v ) = Afn~~l(^v)

(6.8) (6.8)

provided the distribution function at t n _i is known. The value of the collision operator at points (6.7) is approximated by a weighted Gaussian quadrature. One can also use Monte Carlo quadrature to evaluate the collision operator in (6.8) (see Section 4), Note that, if / " - 1 is expressed as

208 _ _ _ _ _ _ _ _ _ _ _ _ ^ _ _

—.

_ W. Walui

in (8.1) (with n replaced by n - 1), /""H** v ) c a n b e computed for every velocity v and hence there is no need for any interpolation during evaluation of the distribution function at post-collision velocities. Obviously, one computes / " at a discrete set of position points x. Then, the linear operator A in (6.8) is such that ( / n - _4/ n _ 1 )/Ai approximates the free low operator £>/ = | £ + v- V x / . Usually, A corresponds to the first order upwind scheme. The time step At and the mesh size of the position grid Ax must satisfy the CFL condition (see Section 3), Use of Hermite series to approximate a solution of the Boltzmann equation was developed further by Sod [SD1]. Below we indicate his method. it is based on the Hilbert expansion procedure (see [CE1],[LA1],[TM1]). Assume that the distribution function has the form of a (formal) power series

i%) + eSfW + _J_ . ..... .> // === /«>> /<°) + eef/ (D

Substituting it into the Boltzmann equation (in which the factor ~ has been introduced at the collision operator), one obtains a sequence of integral equations for f(mK The first one implies that /(°) j s a local Maxwellian. The equations for m = 1,2,... have the form TO — —11

ra

m l)

2J(/< >,/<°>) = D^ ~

- J2 J(f(t)Jim"l))

>

(6.9) (6.9)

i-a

Performing suitable transformations on (6.9), one can show that they lead to Rredholm integral equations of the Irst kind w)pmm^)(w)dw (w)dw = g^ (v), / JC(v, £(v,w)/( 0<m>(v), 3 Jm Jm?

(6.10)

with an explicit formula for the kernel K (in the case of the hard sphere potential), and where g^ depends on /<0 for J = 0 , . . . ,m — 1. Note that during the transformation to (6.10) the integration with respect to the angular variables in the collision operator in (6.9) has been carried out, thus reducing the dimensionality of the integrals. As in Chorin's method, / ( m ) at tn is sought in the form of a truncated Hermite expansion (6.1). Now, the values of f^ at the points (6.7) are obtained by solving a system of algebraic linear equations that results from the equation (6.10) by applying Gaussian quadrature formulas to both sides. The matrix of this system is ill-conditioned and therefore one is forced to use special methods to solve .

m -1} it. The values of fif^171 ~^1) at Cwv time u l l l l c t n _i D 1 Q 6 of OI this uUlb -1 enter into the right-hand side ; system of Df^^K , Clearly, one takes only few system through tnrougn the tne approximation approximation 01 ut% uiearly, one takes only tew lid.AJ,JU3 %jJL wXAtS Hilbert JTAlIOCJlij "expansion J L O & l l Q l O i l . %3%J terms of the to get get an an approximation approximation of of the distribution the distribution

o

* . I # & J L J L \ ^ Id J L \ J F JLJ. ■

There are also some special methods which have been developedforthe Boltzmann equation under additional simplifying assumptions either on the distribution function or on the intermolecular potential. due to TOKolyshkin j\oiybiiKin et Below we present a method due et al al [KE1] fKEll based on on ad« specific transformation of the homogeneous Boltzmann equation in L i l t ? case v^clat; in the i n r*f* i #™ITI JiJtC^AJt^C wv%Z assume Ct»oCSUlJlJ,JLC that the distributior of distribution -fin function. Hence we that the distribution of the the isotropic isotropic distribution function |v|. Let = |v|. Let functi / depends on the velocity v by v = Af (v, a) = ((-) - ) 3 / 22exp(~o;ii exp(-Q!V22), w at wJL t %f m \*M* §

A i m (m I " 1 stands ol^cWlUcifor JLUUL the uIlcJ mass H I C M J O of %Ji. molecules, IXlOlCl^utltJo| which becomes which becomes a a Maxwellian Maxwellian if if a a = = -^ T for the temperature and kforBoltzmann*s Boltzmann*s constant). Let Let (phea, tp be a (unique) (unique) y€#ro.por3rwiJiirc* strict 1 tor

ft i n f t i o n b i l U

that ,hat /*O0

v)= = / f(tt1v) Jo

M(v, a)
(6.11) (6.11)

Hence, the relation (6.11) means that / is the Laplace transform of the function ( a ) 3 / 2 ^ ( t 5 a ) at v 2 . Then the homogeneous Boltzmann equation is converted to the following equation for

ai,a2) corresponds to the collision operator for the Maxwellians M(vtai) and M(v,ot2)

dip dip

fOO

M(v1a2)) J(M(v,ai)tM{v,at2))

= /I

Jo Jo

M{v M(v,a)A(a,a a2)da. 1a)A(at1ctiiua2)da.

The kernel A can be expressed explicitly for the hard sphere and power potentials (see [KE1] for properties of A and the references cited therein for formulas). Since the inverse of a has the meaning of the temperature, we pass to the new variable T = If a. Using this change of variable and

210

-

—

W. Walus

2 2 A(1/T,l/T ,l/T denoting #^ ( T ) = tp(l/T)/T ip(l/T)/T2 and ^4(T, A(T,TUTi,T2) T2) === A(l/T, 1/TU 11/T we 2)/T 2)/T f ,ve obtain the following equation for ip /•OO f OO fkh f°° f°° = / / AFtTuTiWfaTxWtWdTKirn. ^/ AFtTuTiWfaTxWtWdTKirn. dt = JO JO dt Jo Jo Moreover, the moments of / and i/> are correlated by the formula /•OO r°°

/ Jo JO

2 2 ui?2fck(4xv (4
(6.13) (6.13)

(9k 4- Il)!! V f°°

{(2Jfe +

kk * '" / TTt/j(t,T)dT, i/>{t,T)dT. 2fc Jo ^ io

Therefore, the macroscopic parameters of the gas can be evaluated knowing only the function %p without the need of calculating the original distribution function / . To solve (6,13) numerically, we introduce a finite grid TQ < T% < . . . < TM in (0, oo) and then we approximate ip by N TV

W,T) = '£Mt)6(T-Tt'(t)),

(6.14)

where T*(t) belongs to the interval (Ti-i,!*). Clearly, &(*) == / #i(t)

JL jCLJL # i)(t,T)dT.

ipVb^

JTi-i JTi-1

Note that this approximation means that the distribution function / has been expressed in the form JV N

^ ^(t)M(v, 1 /T" £^WM(t;,i/r;(t)). ' ( * ) )

■

Substituting (6.14) into (6.13) and integrating over [!*_!,7*], we obtain for bubst: k = 1. ,.,N *w*

*w

™— JL | *

==

N

J2 ^mWumm, Aij (t)il>i(t)il>j(t)i, -Jr E dt CM-

, .

%**§ •"*■*"* A

where

where

k

%**§ •"*■*"* A

^i*(t

1

—

#

Jifc-1

Am(t)= /


/ » J\ //

4(r,r*(t),r;(t)).

(6.15) (6.i5)

COMPUTATIONAL METHODS ^ _ _

^ _ _ _ _

211

The system (6.15) Is solved using the explicit Euler scheme AT N

V£

+1

== # + At ^

^|fc#^| ■

= i tt », ji = i

To proceed to the next time step [i n + 1 , tn+2] we need also to determine the temperatures T*(in+i). For example, we may put T*(tn+1)=T*(t AT, = r;(t + Ar, n) n) + choosing AT the same for every % = 1 , . , , , N such that the energy of the system is conserved during each time step. Note that since N .Z........^

)) = = oo,

■>k(t

U— 1

the number density is also conserved. Another way of determining T*(tn+i) which also ensures the energy conservation is given in [KE1]. Another course of solving the Boltzmann equation for the case of the hard sphere potential and the isotropic distribution function was proposed by Kugerl [KG1]. It is based on the expansion of the distribution function in terms of generalized Laguerre polynomials. The Boltzmann equation is then transformed to the (infinite) system of nonlinear ordinary differential equations for the coefficients in the assumed expansion of the distribution function. All the parameters appearing in the quadratic term that replaces the collision operator are evaluated explicitly. The case of the one-dimensional stationary Boltzmann equation, where the distribution function has the axial symmetry (i.e. it depends on vi and vr = <sjv\ +1»!)» was considered by Ohwada [OH1]. Here, a (truncated) expansion of the distribution function in terms of the Laguerre polynomials with respect to u r variable combined with quadratic finite element basis functions with respect to v± variable is used. This expansion is subsequently converted to the form which allows its efficient evaluation once the values of the distribution function at lattice points have been determined and which is suitable for the evaluation of the collision operator. The lattice is uniformly spaced for v\ variable whereas for vr variable is taken as the square roots of zeros of the appropriate Laguerre polynomial. The collision operator is reduced to a quadratic expression with coefficients which can be computed

21 o

W. Walus

in advance. These coefficients - the values of the collision operator at the lattice points for certain basis functions - are computed for the case of the hard sphere potential using partially analytical and numerical quadratures. Moreover, symmetry properties of these coefficients are utilized to reduce the volume of the data. Finally, we remark that some methods were developed for the model Boltzmann equation in which R 2 is taken as the velocity space; see [MRl], [MRS1] and [RSI]. The method initiated in [MRSl] and [RSI] was subse­ quently extended to the case of K 3 velocity space in [MSI]. Acknowledgments This work was supported by the Commission of the European Communi­ ties, Grant ERR-CIPA-CT-92-2245 p. 3622. The author acknowledges the hospitality of the Politecnico di Torino (Italy) where substantial part of the work was carried out. Final stage of this research was also supported in part by the Polish Research Council under Grant N. 2P 301 027 06.

COMPUTATIONAL METHODS

213

References [ABl] ARLOTTI L. and BELLOMO N., Lecture 1: On the Cauchy Problem for the Boltzmann Equation, in Lecture Notes on t h e Mathe­ matical Theory of t h e Boltzmann Equation, World Sei. (1995). [ARlJ ARISTOV V.V., On solution of the Boltzmann equation for discrete velocities, Dokl. AJcad. Nmk USSR, 283, no. 4 (1985), 831-834 (in ■ttussian j . [AR2] ARISTOV V.V., The solution of the Boltzmann equation by the method of integration with respect to the collision parameters, Preprint, Computing Center of USSE Academy of Sciences, Moscow (1988) (in Russian). [Artijj ARISTOV V.V., JJevelopment ot trie regular metnod of solution ot the Boltzmann equation and nonuniform relaxation problems, P r o c . 17 t h Intern. Symp. on R G D , A. BeyJieh Ed., VCH Weinheim (1991), 879—883. [ASl] ARSEN'EV A. A,, On the approximation of the solution of the Boltz­ mann equation by solutions of the Ito stochastic differential equa­ tions, USSR Compui. Math. Math, Phys., 27(2) (1987), 51-59, and Zh. VychisL Mat. Mat. Fiz., 27 (1987), 400-410 (in Russian). [AS2] ARSEN'EV A.A., On the approximation of the Boltzmann equation by the stochastic differential equations, Zh. VychisL Mai, Mat, Fiz., 28 (1988), 500-567 (in Russian). [ATI] ARISTOV V.V. and TCHEREMISSINE F.G., The splitting of the nonhomogeneous kinetic operator in the Boltzmann equation, Dokl, Akad. Nmk USSR, 231, no. 1 (1976), 49-52 (in Russian). [AT2] ARISTOV V.V. and TCHEREMISSINE F.G., Solutions of nonstationary and stationary boundary Yalue problems for the Boltzmann equa­ tion using the splitting method, in Numerical M e t h o d s in Rar­ efied Gas Dynamics, Vol. 3, Computing Center of USSR Academy of Sciences, Moscow (1977), 117-140 (in Russian). [AT3] ARISTOV V.V. and TCHEREMISSINE F.G., The conserYatiYe split­ ting method for the solving of the Boltzmann Equation, USSR Cornput. Math. Math. Phys.t 20, no. 1 (1980) 208-225 and Zh. VychisL Mat. Mat. Fmn 20, no. 1 (1980), 191-207 (in Russian). [AT4] ARISTOV V.V. and TCHEREMISSINE E.G., Solution of the Euler and Navier-Stokes equations based on the operator splitting in the kinetic

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[AT6] ARISTOV V.V. and TCHEREMISSINE F.G., Direct Numerical Solutions of the Kinetic Boltzmann Equation, Computing Center of Russian Academy of Sciences, Moscow (1992) (in Russian). [BA1] BABOVSKY H., On a simulation scheme for the Boltzmann equation, Math. Methods Appl Appl. Sci., 8 (1988), 223-233. [BA2] BABOVSKY H., A convergence proof for Nanbu's Boltzmann simulation scheme, Em. J. Mech. B Fluids, 8(1) (1989), 45-55. [BA3] BABOVSKY H., Monte Carlo simulation schemes for steady kinetic equations, Transp. Theory Stat Siat, Phys., 23(1-3) (1994), 249^264. 249-264. [BA4] [BA4] BABOVSKY H., GROPENGIESSER F., NEUNZERT H., STRUCKMEIER J., and WEISEN B., LOW discrepancy methods for the Boltzmann equation, in P r o c . l i t h Intern, Symp. on R G D , Pasadena (1988), E.P. Muntz, D.P. Weaver Weaker and D.H. Campbell Eds., in Ser. Progr. Astronautics and Aeronautics Vol. 118, 85-99. [BA5] [BA5] BABOVSKY H., GROPENGIESSER F., NEUNZERT H., STRUCKMEIER J., and WEISEN B., Application of well-distributed sequences to the numerical simulation of the Boltzmann equation, J. Comp. Oomp. Appl Appl Math., 31 (1990), 15-22. [BE1] BELLOMO N., PALCZEWSKI A., and TOSCANI G., Mathematical Topics in Nonlinear Kinetic Theory, World Sci. (1988). [Bill [BI1] BABOVSKY H. and ILLNER R., A convergence proof for Nanbu's simulation method for the full Boltzmann equation, SIAM J. Numer. Anal, 26 (1989), 45^65. [BOl] BOGOMOLOV S.V., The convergence of the splitting method for the Boltzmann equation, Zh, Zh. VycMsL Mat Mai. Fiz., 28 (1988), 119-126 (in Russian). [BP1] BOBYLEV A.V., PALCZEWSKI A., and SCHNEIDER J., Discretisation of the Boltzmann equation; numerical methods and discrete-velocity models, to appear in Proc. 19 t h Intern, Symp. on R G D .

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[BR2]

BOBYLEV

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BIRD

G.A., Aerodynamic properties of some simple bodies in the hypersonic transition regime, AlAA J., 4, no. 1 (1966), 55-60. [BR3] BIRD G.A., Direct simulation and the Boltzmann equation, Phys. Fluids, 13, IS, no. 11 (1970), 2676-2681. [BR4] BIRD G.A., Molecular Gas Dynamics, Oxford UniYersity University Press (1976). [BR5] BIRD G. A., Monte Carlo simulation of gas flows, in Annual Review of Fluid Mechanics,"1 Vol. 10 (1978), 11-31. * \ * " / i -■* CRRfil context, F Jrroc. roc. iJttOJ BIRD • G.A, 1*>th JLIJ. Ser. 0%JJL * Progr. 2™ Intern, S y m p . on R G D , Sam Sum S. 5. Fischer Ed., Ed. 1 in Astronautics and Aeronautics, 74 (1981), 239-255. methodsininrarefled rarefledgas gasdynam­ dynam­ [BR7] I J A JLIMLJ' \ « j F * j&lL mm JL %jJL %*#%*? %Jt t i ***#*■* >LI* JL* UlJLJLJiCJL *%^*Cll*methods th ics, P r o c . 16 Intern. Symp. on R G D , Pasadena 1988, E.P. Muntz, D.P. Weaver and D.H. Campbell Eds., in Ser. Progr. Astro­ — nautics and Aeronautics, vol. iio, zii £* JL J.ZJ&O. —&£0. nauiicB ana Aeronautics vol. n o , BIRD

w

tt

[BR8]

1 MmJfJL%\j !

^3* * Jt*,\*Jr'JthM* *t# \<*i

^-mr %Jwmt Jk%jF

%*$ £«JkMeS*- %Jh*m W&M& %£ .§.%?*JkJfe

■

3hm*Jh %M$£*m- %&tdb&L£*%■fcdfeift.sw' %^Jfa Ji*iS>.J**Tfc

t

G. A.,. Efficiency and discrepancy discrepancy in the direct simulation meth\J *£»~»a «-% #"*f #■*§ *¥""% A ¥*H i"n T f tf H V"! ™^ J 1% F^^TT 'ytrti *W ±. Intern. £** Syi » * * § » # * on ods, P r o c . **1JL §*fHrlh t A . Beylieh Ed., VCH d o » "W*% JtCljJJ MTJTCJIC^* I JLJIJIIi»'wJl. JLJLll#iiJM. MJLm £3 VMJLM.|L#*> VMJLM.jL#* UFJUJ, UPJUJ, J£I»*%JI"JL***'* jLl»*\ jrJL««'% **,» JLi'oVAA\*JL*. JL*t*VXJ,\^Jl* JLiUL*% V V^AJL 655-662. 1991)., 655-662. Weinheim Weinheim (1991), (1991), Tf\* A, O * G.A., Molecular Molecular JLVJLIJF JLCL* UULoyL Gas 1»J"Ci»S Dynamics and the the Direct Direct MmjMM.JL %li>%~*vSimuh*Jf JLJLJLSLMJL B II R RD D G.A., B Gas Dynamics and the Simuas Plows, Claren don Press (1994). latl BOYD I.D. and STARK J.P.W., On the use of the modified Nanbu direct simulation scheme, J. Comput. Compui. Pbys,, 80, No. 2 (1989), 374386. B U E T C., A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics, Preprint (1994). BUSLENKO N., GOLENKO GOLENKO D.I., SHREIDER Yu.A., SOBOL I.M., and SRAGOVICH V.G., T h e Monte Carlo M e t h o d (The M e t h o d of Statistical Trials) % Pergamon Press (1966). BIRD JLJPJ,Jt\*JL#

1 W%

T 1*1 £ |

"jB™1"^, a>*™^§ 'W'^fe

1

[BE9] [BR9] [BR9] [BS1] [Biblj

[BT1] fRTTIl [BUI]

##

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.

„

.

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^jU 1

[BY2] BELOTSERKOVSKIY O.M. O.M, and YANITSKIY V.E., The statistical meme thod of particles in cells for the solution of problems in rareled rarefied gas ga dynamics; 2. Computational aspects of the method, Zh. Vychisl Vychisl. Vychisl Mat. Mat. Fiz., 15 (1975), 1553-1567 (in Russian), Russian). [CE1] CEROIGNANI C , The Boltzmann Equation and its ApplicaApplications, Springer (1988). rpwii [CHI] CHORIN A.J., Numerical solution of Boltzmann's equation, Comm Comm. I \J*XX X I

Pure Appl Math., 25 (1972), 171-186.

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X

X\/#

[DEI] DESHPANDE S.M., An unbiased and consistent Monte-Carlo game game simulating the Boltzmann equation, Report 78 FM 4, Indian Institute Institute of (1978). [DM1] DESVILLETTES L. and MISCHELER S., Convergence of the splitting method for the Boltzmann equation, in preparation. [DN1] [DN1] DIENISIK S.A., MALAMA Y U . G . , LEBEDEV S.N., and POLAK D.S.. D.S., Solutions of problems in physical and chemical kinetics using MonteCarlo method, in Applications of Numerical Mathematics in Chemical and Physical Kinetics, Moscow (1969), 179-231 (in JUbUoiDlwIl 1 *

[DN2] [DN2] DIENISIK S.A., MALAMA Y U . G . , LEBEDEV S.N., and OSIPOV A.I., Application of the Monte-Carlo method to solve the problems in gas kinetics, Fiz. Gor. i Vzryva (Physics of Combustion and Detonation), 8, No. 3 (1972), 331-349 (in Russian). J^owmm *»-W ^

[ER1]

*|

ERMAKOV S.M., . Y U . , and S.M., NEKRUTIN V.V., V.V., PROSHKIN A A.Yu., and SIZOVA

A.F., On Monte Carlo solution of the nonlinear kinetic equations, Dokl Akad. Nauk. USSR, 230(2) (1976), 261-263 (in Russian). Dokl. fFPll [FP1] PREZZOTTI A, and PAVANI R., Direct numerical solution of the l J Boltzmann equation for a relaxation problem of a binary mixture of hard sphere gases, Meccanica, 24 (1989), 139-143.

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fGBl [GB1] G GOLDSTEIN 1 3 # * and CIIJUJI BROADWELL JL# K%AJALJ W JCJJUJL# J.B., MJ * J O * Investi­ O L D S T E I N D., D,,, STURTEVANT STURTEVANT B., Invest!th gth gation of the motion of discrete-velocity 16 In-velocity gas model, PP rroo cc . •t In^^t JOIL •#I^^u^ni^p^ R CR J TG D D( 1 (1988), 9 8 8.) E.P. , ^ft*€2€**IE*it*n I^^u^ni^p^wwcuntcunt E.P. Muntz, Weaver Muntz, D.P.D.P. Weaver andand D.H.D.H. Campbell Eds., in Ser. Progr. Astronautics and Aeronautics Vol. 118, 118: ""■"""H

j^Cl#i^i\«lfJ.lt

\J*JI

U*A\!#

JULAlur UJL^JJJLJ*

%JJL

1

1™"%

~w- *»~»k

%JL.M,G%^&.\J%J\J

100-117. 1 AA 11*7

rpNi [GN1] GROPENGIESSEE N., and brROPENGIESSER P., F . , NEUNZERT JNEUNZERT IN., STRUCKMEIER J., J - , 'Com­ and STRUCKMEIER tornfor tthe h p Boltzmann equation, putational methods for putational and uation, in Applied A p p l i e d and Industrial I n d u s t r i a l Mathematics, M a t h e m a t i c s , "D R. Spigler opigier Hid, Ed.,, Kluwer Academic PubAcademic Publ i s W f l Q t m 1Q-4Q rriD i

[GR1]

EL,, Principles theory of gases, in in H Handbuch of the kinetic th eory of andbuch P]rinciples of 12, Springer (1958), 205™ der P Physik, 208-294. -294. h y s i k , 12, GRAD

%&J* J[\iM%,M*if

JL JL ■

[GS1]■J GOLDSTEIN STURTEVANT B., Discrete velocity gasdynamics G O L D S T E I N D. and STURTEVANT [GS1 simulations in in parallel processing environment, AIAA AIAA Paper Paper No. 89tth h 1668 (19891: 1868 (1989); AIAA 24 ThermoDhvsics Thermophysics Conf. (1989). [HB1] [nj>1

Review, S., Numerical evaluation of multiple integrals, SIAM SIAMReview, 12, 4 (1970), 481481-524. -524. L2, No. 4

HABER ci/VJDJiiJrv

M »«

JL l ! U A X M ^ i L M,%»*€WJL

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[ i r p i HlRSCH C , N u m e r i c a l C o m p u t a t i o n of I n t e r n a l aand n d Ext< ernal ■[HR1] Numerical of Internal External Rl] I air (1991), nciem lows, Wiley » L dUl \3i*^%-£ gas flow flow [HY1] HICKS and Y YE EN NS.M., S.M., Collision integrals for for Jrarefied I C K S B.L. B . L . and 1J±¥J H PJ4 c Nocilla Ed., EdS y m p . on R G D , S. problems, P: P rroc. o c . 7§t h Intern. JjMern«, Symp. itrice Tecnico Scientica (1971), 2, 845-852. 845-852. (1971.), Vol. 2, JLJLJLXi*fcJ\^JLX

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[V02] VORONINA V.A., A stochastic method-for solving an initial-boun­ dary value problem for the Boltzmann equation, Zh. VycMsl. Mat Mat Fiz.t 32, No, 4 (1992), 578-586 (in Russian). [WGl] WAGNER W., A convergence proofforBird's Direct Simulation Mon­ te Carlo method for the Boltzmann equation, J, Stat. Phys,f §6, No. 3/4 (1992), 1011-1044. [WL1] WILMOTH R.G., Adaptive domain decompositions for Monte Carlo simulations on parallel processors, in P r o c . 17 t h Intern. Symp. on RGD, A. Beylich Ed., VCH Weinheim (1991), 709-716. [YA1] YANITSKIY V.E., Application of the stochastic Poisson process for calculation of a collision relaxation of nonequilibrium gas, Zh. VycMsl Mat Mat Fiz», 13 (1973), 506-510 (in Russian), [YA2] YANITSKIY V.E., Application of random walk processes for simula­ tion of gas free flow, Zh. Vychisl. Mai, Mat Fm,f 14, No. 1 (1974), 259-262 (in Russian). [YA3] YANITSKIY V.E., Probabilistic analysis of direct statistical simula­ tions of collision processes in a rarefied gas, appendix 1 of the Russian translation of the book by G.A. Bird, Molecular Gas Dynamics, Mir Publ. (1981) (in Russian). [YA4] YANITSKIY V.E., The stochastic models of perfect gas of Inite num­ ber of particles, Preprint, Computing Center of USSR Academy of Sciences (1988) (in Russian). [YA5] YANITSKIY V.E., Operator approach to direct Monte Carlo simula­ tion theory, in Proc, 17 t h Intern. Symp. on R G D , A. Beylich Ed., VCH Weinheim (1991), 770-777. [YH1] YEN S.M., HICKS B.L., and OSTEN R.M., Further developments of a Monte Carlo method for evaluation of Boltzmann collision integrals, in P r o c . 9 t h Intern. Symp. on R G D , M. Becker and M. Piebig Eds., DFVLR Press (1974), paper A. 12. [YL1] YEN S.M. and LEE K.D., Direct numerical solution of the Boltz­ mann equation for complex gas flow problems, in P r o c . 16 t h In­ t e r n . Symp. on R G D (1988), E.P. Muntz, D.P. Weaver and D.H. Campbell Eds., in Ser. Progr. Astronautics and Aeronautics Vol. 118, -342.

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L., LIONS P.L., MARKOWICH P.A., and VARADHAN S.R.S, Nonequilibrium P r o b l e m s in Many-Particle Systems, Lect. Notes in Mathematics 1551, Springer (1993), ARKERYD

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