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

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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

Copyright © 1995 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, Massachusetts 01923, USA.

This book is printed on acid-free paper.

Printed in Singapore by Uto-Print

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)

(221 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 )
> ,, 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 ) 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 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) * > ( * 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 to and satisfies the condition fto = F0. Moreover ft G BijS and ll/t-"lka 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\\stls.

(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