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SINGULARLY PERTURBED EVOLUTION EQUATIONS WITH APPLICATIONS TO KINETIC THEORY

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Series on Advances in Mathematics for Applied Sciences - V o l . 34

SINGULARLY PERTURBED EVOLUTION EQUATIONS WITH APPLICATIONS TO KINETIC THEORY

J. R. Mika & J. Banasiak University of Natal Durban South Africa

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

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Library of Congress Cataloging-in-Publication Data Mika, J. R. Singularly perturbed evolution equations with applications to kinetic theory / by J. R. Mika, J. Banasiak p. cm. — (Series on advances in mathematics for applied sciences ; vol. 34) Includes bibliographical references and index. ISBN 9810221258 1. Evolution equations. 2. Perturbation (Mathematics). 3. Matter, Kinetic theory of. I. Banasiak, J. II. Title. III. Series. QC20.7.E88M55 1995 515'.353-dc20 95-21986 CIP

Copyright © 1995 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in anyform 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.

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Preface This book is devoted to singularly perturbed evolution equations. One of the au thors (J R M) has studied this subject for many years and it was his idea to adapt the Chapman-Enskog asymptotic method, originally developed for the Boltzmann equation, to abstract evolution equations of the resonance type, including ordinary differential equations. The second author is a late-comer to this field but his expertise in applied functional analysis was a vital contribution to the final shape of the book and he is responsible for most of the analytical results concerned with the various types of linear evolution equations. It is not intended to be a general survey of the field but aimed at presenting the results obtained by the authors: it may therefore be a little biased. Our main objective is to present the compressed asymptotic approach and convince the reader that it may serve as the universal method for the asymptotic analysis of singularly perturbed evolution equations of the resonance type, such as, for example, linear kinetic equations. This book would never have been written if the authors had not been encouraged by Nicola Bellomo who proposed to publish it in the Series on Advances in Mathematics for Applied Sciences. The cooperation with our Italian friends (particularly with Giovanni Frosali) made possible by the generous support of the Italian Consiglio Nazionale della Ricerche represented by Vinicio Boffi, greatly contributed to the final shape of the book. We would like to acknowledge the financial support received from the South African Foundation for Research Development and the University of Natal Research Fund and the encouragement from colleagues in the Department of Mathematics and Applied Mathematics. One of authors (J B) is greatly indebted to the authorities of the Technical University of Lodz and to the Institute of Mathematics, who supported his prolonged stay at the University of Natal. Last but not least, the authors are grateful to Andrzej Palczewski, who was kind enough to read the manuscript and suggest numerous corrections and changes, as well as to Joyce Ross who looked into our English. However, neither are to be blamed for any shortcomings in this book.

v

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Contents Preface

v

1

Introduction

1

2

Mathematical preliminaries

7

1

Introduction

7

2

General definitions and notation

3

Banach and Hilbert spaces

10

4

Distributions and Sobolev spaces

19

5

Vector-valued functions

28

6

Unbounded operators

32

7

Elements of spectral theory

41

3

7

Semigroup theory 1

Introduction

61 .

61

2

Generation of semigroups .

62

3

Fractional powers of closed operators

72

4

Perturbation theorems

76

5

Asymptotic behaviour of solutions

82

6

Inhomogeneous Cauchy problem

86

7

Applications to partial differential equations

vii

.

93

CONTENTS

Vlll

4

5

6

7

8

Development of asymptotic methods for singularly perturbed evo lution equations 105 1

Introduction

105

2

Single evolution equations with a small parameter

107

3

Systems of evolution equations with a small parameter

111

Some singular-singularly perturbed evolution equations and kinetic equation 125 1

Singular-singularly perturbed evolution equations

125

2

Model system: exact solution

133

3

Model system: standard asymptotic analysis

136

4

Model system: compressed asymptotic expansion

139

5

Modified model system

143

6

Singular-singularly perturbed evolution equations: compressed approach 148

7

Singularly perturbed linear kinetic equations

151

Hilbert space theory for equations of kinetic type

161

1

Introduction

161

2

Preliminary results

162

3

Properties of terms of expansion

168

4

Estimates of error of asymptotic expansion

179

5

Remarks on non-selfadjointness of C

192

Applications to kinetic equations with bounded collision operators 197 1

Introduction

197

2

Properties of linear Boltzmann equation with unbounded velocity range 199

3

Diffusion approximation to linear Boltzmann equation .

207

Applications to equations of Fokker-Planck type

215

1

Introduction

215

2

General assumptions

219

3

Variational setting

222

4

Fokker-Planck equation of electron scattering in plasma

224

CONTENTS

9

IX

5

Fokker-Planck equation of Brownian motion

228

6

Fokker-Planck equation of vibrational relaxation of harmonic oscillator 234

Applications to spatially inhomogeneous linear Boltzmann equation239 1

Introduction

239

2

Properties of neutron transport equation in slab geometry

240

3

Asymptotic expansion

246

4

Remarks on general spatially inhomogeneous equation

254

10 Application to kinetic equation with external field

field

257

1

Introduction

257

2

Properties of collision operator

260

3

Formal asymptotic formulae

263

4

Initial layer part .

267

5

Bulk part

271

6

Error of asymptotic expansion

276

11 Miscellaneous results

279

1

Introduction

279

2

Asymptotic analysis of telegraph systems

279

3

Remarks on compressed asymptotic method in .^-setting

290

4

Carleman model .

294

Bibliography

301

Index

309

Chapter 1 Introduction Since the times of Newton, physical phenomena were being described in terms of partial differential equations with time as one of the independent variables. Some 50 years ago such equations acquired the collective name evolution equations and a comprehensive theory of semigroups of operators related to solutions of evolution equations has been developed. In mathematical physics the equations often contain small or large parameters placed in such a way that the standard perturbation approach leads to difficulties and the apparent lack of convergence the resulting asymptotic series. Such equations are usually referred to as singularly perturbed. A singularly perturbed evolution equation may exhibit a singular behaviour with respect to the time variable, in which case we have the so-called initial layer phe nomenon, or with respect to spatial variables which leads to the boundary layer phe nomenon. In this book we are solely interested in the initial layer case and dispose of the boundary layer by choosing either an infinite medium or periodic boundary con dition. Reader who is interested in boundary value phenomena, is advised to consult the articles [14, 26, 73] or the survey in [25], vol. VI. Before the advent of the semigroup theory of evolution equations, a singularly per turbed differential equation or a system of such equations were treated independently of one another by ad hoc methods. Even for ordinary differential equations a com prehensive theory of singularly perturbed systems, which are very often referred to as stiff, was started only in the 1950s. It was not long before the newly developed theories of singularly perturbed ordinary differential equations and of semigroups of operators met and the theory of singularly perturbed evolution equations started to emerge from the pioneering results reported in the monograph of Krein [48]. By now the asymptotic analysis of singularly per turbed evolution equations was well established (see e.g. [31, 32]).

1

2

Singularly perturbed evolution equations

To explain the essence of the asymptotic analysis of a singularly perturbed system, let us consider a family of operators A€ in, say, a Banach space X and a family of initial value problems dtu = Aeu,

w(0) = 7,

(1.1.1)

where 7 is a given element of X (in some instances it may also depend on e). The crucial role is played by the (supposedly) small positive parameter e. It serves to label small or large terms in the equation and it is usually identified when appropriate units are chosen and the equation is cast into a dimensionless form. The problem is that there is not always an agreement on which units have to be chosen so that an equation describing a given physical situation would have diverse forms with the small parameter appearing at various places in the equation. In some cases it is not harmful, as it simply means moving the terms from one level of approximation to another. In other cases the resulting approximate equations differ significantly from each other. We will see the latter situation happening for the kinetic equation which is studied in detail in this book (for further discussion on the subject see Section 5.7). The asymptotic analysis of Eq. ( I l l ) consists of attempting to approximate the solution by the truncated series u (n) (f) = u0(t) + e Ul (r) + t2u2{t) + ■■■ + e"un{t).

(1.1.2) (1.1.2)

The coefficients uk, k = 0,1,2, ■•-,«, are determined by inserting u' n ' into Eq. (1.1.1) and then comparing terms of the same order in e. We say that u (n) is an asymptotic solution of Eq. (1.1.1) of the order n if we have for 0 < t < T, where T > 0, \\u(t)-u^(t)\\=0(e^). 0(tn+1).

(1.1.3)

Sometimes we require a little less, namely, | | u ( i ) - u w ( * ) | | = o(e n ).

(1.1.4)

For a singularly perturbed system the above procedure fails to produce the approxi mate solution satisfying the condition (1.1.3) or (1.1.4). The reason is the existence of the initial layer in the vicinity of t: = 0 extending over the interval of the order e. To cope with this we introduce a new time variable r = t/e which serves as a sort of magnifying glass to look at the initial layer in detail. The transformed equation is treated in a similar way as the original one. As a result one obtains another truncated series, but in terms of -r. The approximate solution is taken as the sum of the two truncated series, but care must be taken in choosing the initial conditions for the

Chapter 1. Introduction

3

equations to determine the coefficients in the series so that the approximate solution satisfies the original initial condition up to terms of 0(en+1). It is often said that the singular perturbation method is more art than science. In any field of mathematical research there is almost always an initial stage when intuition plays a crucial role. After that comes the formulation of basic assumptions and then the proof based on logical deduction. It seems that in the case of the singular per turbation the intuition plays a particularly important role. Unfortunately, different researchers may, and often do, have different intuitions. This certainly contributes to the view that the singular perturbation methods make a capricious and hardly predictable part of applied mathematics. The field of the singularly perturbed evolution equations is certainly in the later stage of its development. But to some extent various types of linear (and, obviously, nonlinear) kinetic equations fall away from the general pattern. The main objective of this book is to give the reader a comprehensive and, hopefully, rigorous picture of the asymptotic properties of linear kinetic equations in the framework of evolution equations by proposing to treat them consistently with the compressed method which is a modified Chapman-Enskog method to be introduced and described in Chapter 5. To introduce the subject of kinetic equations let us remind the reader that perhaps the most famous singularly perturbed equation in mathematical physics is the celebrated Boltzmann equation of kinetic theory, formulated by Boltzmann in 1872 [20]. At that time the fluid dynamics, in which gases or liquids are treated as continuous media, was already well developed and the Euler equations for inviscid, and the Navier-Stokes equations for viscous fluids, were offering an adequate description of the physical reality. The Boltzmann equation is entirely different since it treats a fluid as a collection of particles moving around with various velocities and colliding with walls and with each other. Many physicists did not believe in the underlying molecular hypothesis of matter and it took some time before Boltzmann and his equation gained general recognition. Nevertheless, there remained an embarrassing problem: how to reconcile the two apparently different descriptions of fluids. The breakthrough came from one of the most eminent mathematicians of that time. In 1912 Hilbert published a treatise on integral equations (see the American edition [44]) in which he took the Boltzmann equation and performed its asymptotic analysis to illustrate the Fredholm Alternative. In the process, at the lowest order approx imation to the Boltzmann equation, he obtained the Euler equations showing that fluid dynamics is an asymptotically valid description of the medium if the collisions play the decisive role when a system approaches thermal equilibrium. It is to be noted, however, that although Hilbert's analysis of the resulting integral equations was perfectly rigorous, his asymptotic analysis was purely heuristic. In a few years the next step was taken, this time by physicists Chapman and Enskog,

4

Singularly perturbed evolution equations

who modified the Hilbert asymptotic expansion in such a way that, instead of the Euler equations, the procedure led to the Navier-Stokes equations. Needless to say, it was again the heuristic derivation. For many years the Hilbert and Chapman-Enskog asymptotic expansions were a part of folklore in the kinetic theory (see e.g. [86, 39]). Only in recent years did articles appear in an attempt to put the asymptotic theory of the Boltzmann equation on a sound mathematical basis. The classical Boltzmann equation describes systems of particles interacting with each other so that the equation is nonlinear. In the case of rarefied gases it is possible to apply the low-density approximation and linearize the Boltzmann equation which then supplies a fairly good description of such gases. There are, however, kinetic systems of particles for which collisions between themselves can be safely neglected. In such cases we speak about the linear Boltzmann equation which is also called the linear kinetic equation or the transport equation. The kinetic theory of such particles is often referred to as the transport theory. The most important examples of particles which are described by the transport theory are neutrons, electrons and Brownian particles. A similar situation exists in the case of photons whose behaviour is the subject of the radiative transfer theory. The asymptotic theory of the linear kinetic equation or transport equation has been developed somewhat independently of that related to the original nonlinear Boltz mann equation. The first serious interest in transport theory arose in connection with the neutron fission chain reaction in nuclear reactors (and bombs). The parti cle distribution in the transport equation depends on time, three spatial and three velocity variables. The equation itself is an integro-differential equation so that its numerical treatment is extremely difficult and time-consuming. At the same time, in the early stages of the development of the nuclear reactor theory, it was noted that for most physical situations a nuclear reactor can be adequately described by the neutron density which only depends on time and three space variables. The relevant equation is of the form of the diffusion equation which can be derived from the first principles if one assumes the validity of Fick's law relating the current with the gradient of density. For many years the neutron diffusion equation has been used to describe a nuclear reactor and the transport theory applied only for solving some particular problems. The two descriptions of neutrons in a nuclear reactor exist more or less independently, although the monographs and textbooks were quoting various heuristic derivations like, for instance, via the expansion of the neutron distribution function into spherical harmonics. There was an apparent lack of progress in all attempts to derive the diffusion equation, since the standard type asymptotic expansion, of the type employed by Hilbert, would not give the diffusion equation for neutrons and also other particles. The way out of

Chapter 1. Introduction

5

this difficulty was found by Larsen and Keller [50] who, following the earlier idea of Kurtz [49], introduced a rescaled time into the transport equation. This line of research was followed subsequently by a number of researchers for neutrons and also other particles (see [14, 13, 26, 74, 77]). At the end of 1970s Mika adapted the Chapman-Enskog asymptotic expansion proce dure to a class of linear evolution equations [58, 59]. The main feature of the adapted method was that the equation is projected onto the null space of the collision operator and onto its complement. Later the approach was extended to singularly perturbed systems of ordinary differential equations by Mika and Palczewski [66, 65]. The pro cedure is referred to in this book as the compressed asymptotic expansion. In a series of papers Banasiak and Mika [7, 8, 10, 11, 61, 62] have applied the compressed method to the linear kinetic equation for various physical systems showing how the method yields the diffusion equation without any time rescaling. The compressed method, as applied to various types of the linear kinetic or transport equation, is the main subject of the application of this book. In Chapter 2 we present the functional-analytical background used in the develop ment of the asymptotic theory of evolution equations. Chapter 3, in turn, gives the rudiments of the semigroup theory and the basic theory of the Cauchy problem for evolution equations. In Chapter 4 we describe some earlier results concerned with ei ther a single singularly perturbed evolution equation or a singularly perturbed system of evolution equations in which all but one operator are bounded. The asymptotic procedure employed in this chapter is standard or of the Hilbert type. Chapter 5 is devoted to the exposition of the compressed method as applied to evo lution equations in which an operator with a nonempty null space is multiplied by a large parameter. Such equations are referred to in the literature as singular-singularly perturbed or of the resonance type and the linear kinetic equation is the most important example. We start with an equation with bounded operators and use the projection approach combined with the standard asymptotic expansion. It is demonstrated that this approach does not lead to the diffusion type equation. To explain the essence of the compressed method we introduce a simple system of ordinary differential equations whose asymptotic properties are the same as those of the kinetic equation. The advantage of such an approach is that the exact solution can be written explicitly and compared with the asymptotic solutions obtained by the standard and compressed methods. Chapter 5 also contains the application of the compressed method to a singular-singularly perturbed evolution equation with bounded operators and it ends with the discussion of the asymptotic features of linear kinetic equations. Chapter 6 consists in the comprehensive exposition of the theory of linear kinetic equations in Hilbert spaces and is based on the results of Banasiak [7]. It serves as a basis for the developments concerned with particular kinetic equations studied in the

6

Singularly perturbed evolution equations

subsequent chapters. Chapter 7 is devoted to the kinetic equations with bounded collision operators. The most important application is the neutron transport equation. Chapter 8 deals with Fokker-Planck equations for which the collision operator is an unbounded differential operator. In Chapter 9 the restriction that the coefficients in the kinetic equation are independent of the space variables is lifted. This is also important in the case of neutrons. Chapter 10 describes the modifications of the general theory which have to be in troduced in the presence of external fields as in the case of electrons. We consider equations of the type which is important in semiconductor theory. Finally, Chapter 11 presents some miscellaneous results. In particular, the asymptotic theory developed in the book is applied to the analysis of a singularly perturbed telegraph equation. Also some results pertaining to the L\ theory are given. Finally, the application of the compressed method to the Carleman model of the nonlinear Boltzmann equation is briefly presented. Most chapters in this book are divided into sections. Only Chapters 2 and 3, which contain a great amount of condensed information, have the sections divided into subsections. Sections are enumerated within each chapter by single numbers. Smaller units like subsections, theorems, lemmas etc. are given labels consisting of the number of the running section and the number of the unit. Formulae are enumerated by three numbers: the number of the chapter, the number of the section and the number of the formula within the section. If we refer the reader to a section, subsection and so on, within the same chapter, we drop the first number which indicates the chapter. Otherwise, sections are referred to by two numbers and the other units by three. For instance, Subsection 2.3.4 denotes the fourth subsection of Section 3 in Chapter 2 and Subsection 3.4 is the same subsection when referred to in Chapter 2. Formulae, however, are always referred to by their full numbers.

Chapter 2 Mathematical preliminaries 1

Introduction

In the next two chapters of the book, which are of introductory nature, we give a survey of results from functional analysis and semigroup theory which are relevant to the theory developed in this monograph. Since this book is intended to be selfconsistent, we recapitulate a number of standard facts, but we assume that the reader is familiar with basic functional analysis. Whenever we think that the result is new or presented in a non-standard way, the proof will be given. Otherwise we refer the reader to textbooks or original articles. Not all results presented here will be applied but we think that they are necessary both to present the main theory in its proper background and for its further development. The general references for Sections 3, 5, 6 and 7 are the monographs [47, 76, 89, 90] and Section 4 is based on [1, 25, 51]. Some particular results are, however, referred to in the text.

2

General definitions and notation

2.1 The following general notation will be used throughout this book. The symbol ":=" will denote "equal by definition" The sets of all natural (including 0), integer, real and complex numbers will be denoted by N, Z, M, C, respectively. If we want to exclude 0 from these sets, we complement the appropriate symbol with the asterisk. For example, N* := N \ {0}. If A € C, we then write Re A for its real part, ImX for its imaginary part and A for its complex conjugate. The symbols: [a, 6], ]a, b[ denote closed and open intervals in M, respectively, and K + :=]0,+oo[, M+ := [0,oo[. 7

Singularly perturbed evolution equations

8

2.2 Let / be a function defined on a set Q and x £ Q. We shall use one of the following symbols to denote this function: / , x —► }{x) and /(•). The symbol f(x) is in general reserved to denote the value of / at x. For example, the symbol f(-,y) denotes the function obtained from (x, y) -* f(x, y) by fixing y. If {z^neN is a family of elements of some set, then the sequence of these elements, that is the function n —> xn, will be denoted by (z„)„ SN . The derivative operator will be denoted by d. To indicate the variable with respect to which we differentiate we write dt, dx, d\x ... If x = (xi,...,xn) € R", then dx : = ( # * ! , . . . , #»„)■

The symbol dk f is to be understood as any derivative of / of order k. The symbol A:

Y, d'f means the sum of all derivatives of / of order less than or equal to k (including the function itself). If Q, C K", then by fi and Int fi we denote, respectively, the closure and the interior of Q with respect to R" If Q is an open set, then for k G N U {00} the symbol Ck(Q) denotes the set of k times continuously differentiate functions in Q with the understanding that C(Q) := C°(Q) is the set of all continuous functions in fl and 00

C°°(Q) := f|

Ck{Q).

Functions of Ck{Q.) need not be bounded in fi. If they are required to be bounded together with their derivatives of order up to k, then the corresponding set will be denoted by Ck{U). If fi is a set with non-empty interior satisfying fi C Int fl, then Ck(Q) will denote the set of all functions which, together with their derivatives up to order k, are continuous and bounded on fi. For a function / , defined on fi, we define the support of / as supp/ = { i e $ l ; f{x) # 0}. The set of all functions with compact support in Q which have continuous derivatives of order smaller than or equal to k will be denoted by CQ(Q) with the understanding that C0(f2) : = CQ(Q) is the set of all continuous functions with compact support in Q and 00

cs°(fi) == fl cftn). k=0

Chapter 2. Mathematical preliminaries

9

2.3 In the book we shall be dealing with linear spaces over C. Let X be a linear space and x\,... ,xk G X. By Lin{x1, ...,xk} we denote the linear envelope (the set of all linear combinations) of elements x;,... ,xk. Let M be a linear subspace of a vector space X. The quotient space X/M is the set of all cosets [x] := x + M\ two cosets [x] and [y] coincide if and only if x - y € M. The codimension of M in X is defined to be the dimension of X/M, that is to say, codim M := dim X/M. The space X/M can be made a linear space with the natural definition of linear operations and in the sequel we shall use the notation X/M in that meaning. Let / be a function on a linear space A' with values in another linear space Y say that / is linear if for every Xj, x% g X and a, (3 6 C we have

We

f(ax f(axi +2)Px2) === af{xaf(x,)+(3f(x 1+0x l)+Pf{x2),2), and antilinear if f{axl+0x (3f(x2). f{axi +2)(3x2) = af(x af(Xll)) + /3f(x Linear functions will usually be called linear operators and denoted by capital letters. The value of a linear operator A at x G X is usually written as Ax. If A is a linear operator acting from X into Y, then the set N(A) := N(A): = {xeX; {x ex- Ax == 0} 0} is called the null-space, or kernel, of A and the set R(A) := {y G.Y; y = Ax for some 1 6 A'} is called its range. If M C X, then A\M denotes the restriction of A to M and AM :=A(M) :=R(A\M). Linear or antilinear functions with values in C are usually called functwnals (linear or antilinear, respectively). If g is a C - valued function defined o n l x Y which is linear with respect to each variable separately, then we say that g is a bilinear form over X x Y If g is linear with respect to the first variable and antilinear with respect to the second, we call it a sesquilinear form.

10

3

Singularly perturbed evolution equations

Banach and Hilbert spaces

In this book we shall mainly deal with Banach and Hilbert spaces over C 3.1 Let A" be a Banach space and fi C X. As in R", by fl, Int fi and dQ we denote, respectively, the closure, interior and the boundary of Q. We say that Q is compact if every sequence of elements of Q contains a subsequence convergent to an element of Q and relatively compact if Q is compact. A set fi is said to be dense in X if Q = X. We say that A" is a separable space if there is a countable, dense subset of X. 3.2 If i € A", then \\x\\x will denote the norm of x in A'. If no misunderstanding is possible, we shall drop the subscript X. Let X and Y be two Banach spaces such that Y C X (set-theoretical inclusion). If the norm in Y is stronger than the norm in A", that is, for some constant C and all y € Y we have ||y||x < C||y||yi then we say that Y is continuously imbedded in X and write

y <->• x. If, in addition, Y is a dense subset of X, then we say that Y is densely imbedded in X and denote this by

M l If the unit ball in Y is relatively compact in X, then we say that Y is compactly imbedded in X and write Y 4 X. X. If the norm || ■ ||y is stronger than || ■ \\x and || ■ \\x is stronger than || • ||y then we say that the norms || • \\y and || • ||x are equivalent. 3.3 The most important subclass of Banach spaces are Hilbert spaces. A Hilbert space is a Banach space with a norm generated by a sesquilinear form, called an inner (or scalar) product and denoted by (•,•)//> m t n e following way. If H is a Hilbert space and x € H, then

\\A\H = \/(x,x)H. As usual, in obvious cases we shall omit the subscript H. Elements x,y € H are called orthogonal if {x,y)H = 0. A set of elements {xa}aeA is said to be orthonormal if its elements are mutually orthogonal and satisfy ||.xQ||// = 1 for a G A.

Chapter 2. Mathematical preliminaries

11

If the Hilbert space H is separable, then there exists a countable orthonormal set {en}neN C H, called the orthonormal basis of H, such that for arbitrary x G H we have oo

n

x = ]T(x, e,) w e, := Jirr^ £ ( z , e,)„e, t=0

"

(2.3.1)

°° , = 0

and OO

l|x||2w = EK^e.)«| 2

(2-3-2)

3.4 Let X, Y be two Banach spaces. The set of all continuous linear operators from X to Y will be denoted by C(X,Y). We recall that A 6 C(X,Y) if and only if there is M G R + such that for any x G X we have ||Ac||y < M||i||x- The greatest lower bound of such values of M is called the norm of A and with this norm C(X, Y) becomes a Banach space. An operator A G C(X, Y) is called an isomorphism if it is one-to-one (injective) and onto (surjective). Then the Banach theorem ensures that the inverse operator satisfies A'1 G C(Y,X). If A is an isomorphism and for every i € l w e have ||A:r||y = ||a:||x> then A is said to be an isometric isomorphism between X and Y Two spaces X and Y are said to be isometrically isomorphic if there exists an isometric isomorphism between X and Y 3.5 By X' we shall denote the dual space to X defined as X' := C(X,C). X', x € X, then we shall write

If x' €

x'{x) =< x,x

(2.3.3)

>x*x',

where again the subscript will be omitted if it does not lead to any misunderstanding. The bracket < , > is often called the duality pairing between X and X' Clearly this is a bilinear form over X x X Let H be a Hilbert space. Then it is more convenient to consider the adjoint (or antidual) space to H, denoted by H*, which is defined to be the space of all antilinear functionals over H. We shall use the same notation (2.3.3) for antiduality pairing; then < , ■ > will be a sesquilinear form. The reason for introducing the concept of the adjoint space is the Riesz theorem which states that each Hilbert space is isometrically isomorphic with its adjoint. Thus it is customary to identify a Hilbert space with its adjoint; then the scalar product is the antiduality pairing between these two spaces. If, however, we have two Hilbert spaces H\ £ H satisfying Hi «-4 H, then it is impossible to identify simultaneously Hi with H[ and H with H* (see [5], p. 64). In such a case we usually put H = H* (a Hilbert space H satisfying H = H* is called a pivot space) and consequently we have

Singularly perturbed evolution equations

12

#! 4 # 4 H[.

(2.3.4)

We shall encounter such a situation when dealing with variational approach to differ ential equations in Section 6. The space A" := £(A",C) will be called the bidualoi X. Equation (2.3.3) shows that X can be in a natural way identified with a subspace of X" If each functional of X" can be identified with an element of X as in (2.3.3), then we say that X is a reflexive space. By the Riesz theorem all Hilbert spaces are reflexive. 3.6 We say that a sequence {xn)ne^ of elements of a Banach space is weakly conver gent to x 6 X if for any x' £ X' we have lim < xn,x

> = < x,x' >

n—>oo

Similarly, we call (xn)nSN a weakly fundamental sequence if (< xn, x' >n)n&N is a fun damental (Cauchy) sequence for any x' £ X' A set Q C X is weakly compact if any sequence of elements of Q contains a subsequence which is weakly fundamental. 3.7 Let A € C(X,Y). The dual operator A1 £ C(Y',X') is defined by the require ment that for all x £ X and y' £ V the following equality holds < Ax,y' >Y*Y'=<

x,A'y' >x*x',

(2.3.5)

where < -, > is the suitable duality pairing. It follows that \\A\\ = \\A'\\. As before, in Hilbert spaces we use the concept of the adjoint operator which is defined in a similar way but with the duality pairing in Eq. (2.3.5) replaced by the appropriate antiduality pairing or, if A is a pivot Hilbert space, by the scalar product. In the latter, if A £ C(X) satisfies for every x, y £ X {Ax,y)x

=

{x,Ay)x,

then A is called a self-adjoint operator. 3.8 Example. In this book we shall be concerned mostly with Banach spaces of integrable functions Lp(fl), 1 < p < oo, where Q is a measurable subset of R" Let us denote by m(Q) the Lebesgue measure of fi. In what follows we identify two functions which differ from each other on a set of the Lebesgue measure m equals to zero, therefore when speaking of a function we will mean a class of equivalence of functions. For our applications, however, this distinction between a function and a

Chapter 2. Mathematical preliminaries

13

class of functions is irrelevant so that if we say that a function u e Lp(£l) has certain property for almost all x € 0,, it means that all functions of the class determined by u have this property. For 1 dxY

(2.3.6)

It is important to note that the convergence in L p -norm does not imply pointwise (nor even almost everywhere) convergence. If, however, a sequence (/n)neN converges to some / in Lp-norm, then there is a subsequence {fnk)ken converging to / almost everywhere. It is customary to complete the scale of Lp spaces by the space Loo(fi) defined to be the space of all Lebesgue measurable functions which are bounded almost everywhere in fi, that is, bounded everywhere except possibly on a set of measure zero. The corresponding norm is defined by ll/lloo := II/HL^Q) := inf{M; m({x G Q; \f(x)\ > M}) > 0}.

(2.3.7)

The expression on the right-hand side of (2.3.7) is frequently referred to as the essen tial supremum of / over fi and denoted by ess sup |/(x)|. If m(ft) < oo, then for 1 < p < p' < oo we have LP.(Q) - » LP(Q)

(2.3.8)

II/IU = ton ll/llp,

(2-3.9)

and for / £ £«,(£})

which justifies the notation. However, D

Lp(Q) ± Loo(Q),

l
as demonstrated by the function f(x) = l n i , x 6]0,1]. If m{Q) = oo, then neither (2.3.8) nor (2.3.9) holds. If 1 = ff(x)g(x)dx,

f e LP(Q), g e L f (fl).

(2.3.10)

n In particular, this implies that the space Z/2(f2) is a Hilbert space and with the above identification it is a pivot space. The antiduality pairing coincides then with the scalar product. Another important consequence of the above result is that the spaces LP(Q) with 1 < p < oo are reflexive. If p = 1, then we have (Li(Q))* = L00(Q) with antiduality pairing (2.3.10). However, the antidual to Lx is much larger than L\(Q) and thus Li(Q) is not a reflexive space. Another important property of spaces LP(Q) with 1 < p < oo is that the set C^°(fi) is dense in Lp(£l). Hence, in particular, these spaces are separable. This is, however, not true for Z-oo(fi). We shall frequently use weighted Lp spaces for p € [l,oo[. Let w be a measurable function defined on fl, satisfying 0 < w(x) < +oo almost everywhere on Q. By Lp(Q,wdx) we denote the set of all Lebesgue measurable functions on U with finite norm p

I

||/||,,« := 11/llMaw*) := f / \f(x)\"w(x)dx)

'

(2.3.11)

)

If for some constants 0 < ci < c2 we have C] < w(x) < c2 almost everywhere on Q, then Lp(Q.,wdx) = LP(Q.) in the sense that the sets of functions are equal and the norms equivalent. In general, the space Lp(Q,wdx) is isometrically isomorphic with Lp(fi) with the isomorphism defined by Af = fw, f € Lp(Q,wdx) so that their topological properties are the same. For p — 2 the spaces L2{il,wdx) are Hilbert spaces. If for some c > 0 we have w(x) > c almost everywhere on fi, then L2(Q,wdx) °^ L2(Q). Taking L2(Q) as a pivot space we see that we can identify (L2(Q,wdx))' with L2(fl, w'ldx). The corresponding antiduality pairing will be the extension of the scalar product in L 2 (0) onto L2(Q,wdx) x L2{£l, w~~ldx): < 1,9 >= I f{x)g{x)dx, o

f € L2(Q,wdx),

g£

L2{Q.,w~ldx).

3.9 Example. Let x := (i/OfcgZ" be a multi-indexed sequence of elements of a Banach space X. We say that x € L(X) if

Chapter 2. Mathematical preliminaries

£

15

||xfc|U < ° ° -

(2.3.12)

fceZ"

The space lp(X) with norm defined as above becomes a Banach space. The space l„(X) can be defined as the linear space of bounded sequences with a supremum norm. From the definition we see that / 2 (A) is a Hilbert space, provided that A is a Hilbert space. If A is a separable Hilbert space, then by (2.3.2) A is isometrically isomorphic with fe(C) with the isomorphism defined by Ax = ((ar,e,-)x)i€H ■ We complete this section by introducing the notions of the direct sum and the tensor product of Banach spaces. 3.10 For an arbitrary X\ C A we define the annihilator of Xi in A ' by the formula: X^ := {x1 G A'; < x,x' > = 0 for all x g Xx}. X^ is always a closed linear subspace of X. If A is a pivot Hilbert space, then A; 1 (with the duality pairing replaced by the corresponding scalar product) is called the orthogonal complement of Xx. If Ai is a closed linear subspace of a Banach space A, then XjXA equipped with the norm ||[x]||= m f f j l z - m | | x ; m e l , } becomes a Banach space. If A is a Hilbert space, then X/X\ is a Hilbert space. It follows that codim Ai = dim Aj 1 ,

(2.3.13) with the norm (2.3.13)

codim X^ = dim Xx.

Let Ai and A 2 be closed linear subspaces of a Banach space X satisfying X\ n A 2 = 0 and A = A'i + X2 := {xi +x2; xi € Xx,x2 G A 2 }, so that each element x G X has a unique decomposition x = y + z, y G A'i, z e A 2 . Then we say that A is a direct sum of Xi and A 2 and denote it by

X =

Xl+X2.

Singularly perturbed evolution equations

16

The subspaces Xi and X2 are then said to be complementary and dim X2 = dim

X/X1.

If X is a Hilbert space and X2 = Xf-, then we say that X is an orthogonal sum of Xi and X2 and denote it by X = X1®X2.

(2.3.14)

Closed linear subspaces of a Banach space X are closely related to projections. An operator P £ C(X) is called a projection if P2 = P. The operator Q = I - P, where / is an identity operator, is also a projection. Linear subspaces Xx = PX and X2 = QX are closed as the null-spaces of a continuous operator and it follows that X is a direct sum of X! and X2. Conversely, the decomposition (2.3.14) of X defines two projections P and Q = / — P such that X\ = R(P), X2 = N(P) and Xx = N(Q), X2 = R(Q). If P is a projection in X and Q = I — P, then P* and Q* are projection in X* and we have P*X* = (QX)1- and Q*A* = (PX)1 The projections are bounded which follows from the Closed Graph Theorem (see Subsection 6.1). If X is a Hilbert space, then, given a closed linear subspace X\ C X, we can always find an orthogonal complementary subspace X2 so that X =

X1®X2.

The corresponding projectors P and Q, called the orthogonal projectors, are selfadjoint (see Subsection 3.7) and satisfy ||P|| = \\Q\\ = 1 (unless one of them is the null operator). If X is an arbitrary Banach space then it may happen that there is no complementary subspace for a given closed subspace of XY. However, if X C X\ satisfies either dim X\ < 00 or codim X\ < 00, then Xi always has a complementary closed subspace. 3.11 The concept of tensor products of Banach spaces is quite deep and we will not go into details since in our applications we shall need only quite basic facts. The interested reader is referred to e.g. [28], pp. 221-260, [76], pp. 49-53. To introduce the notion let us assume that X is a space of functions of two variables (2:1,2:2) -> f{xi,x2). In many such spaces (e.g. for LP(R2) spaces) each element / can be represented as a limit of functions of the form N

(xux2)

-» Yl ft(xi)gi(x2),

(2.3.15)

1=1

where ft and g, are elements of some other Banach spaces, say Xi and A'2, respectively (in the given example Xi = X2 = LP(R)). This, to a certain extent, allows us to

Chapter 2. Mathematical preliminaries

17

consider variables Xi and x2 separately which is important when operators acting in X have different properties with respect to particular variables. Since the algebraic manipulations with functions of the form (2.3.15) are analogous to those with elements of an algebraic tensor product, it is customary to call (11,12) -> f,(xi)gi(x2) the tensor product of /, and ft and denote it by /, ® ft. The completion of the space of all combinations of the form (2.3.15) in a suitable norm is called the tensor product of Xi and X2. An important example of this kind is the tensor product of spaces Xi = £2(^1) and X2 = L2(Q2), defined as a completion of functions (2.3.15) in the norm of L2(£l\ xfi 2 )- Clearly, the tensor product of X\ and X2 is equal to L2(0.l xQ2). When X! and X2 are arbitrary Banach spaces, then the construction of their tensor product is by no means obvious, as the multiplication of two elements of Banach spaces is in general not defined. In the first step we construct the tensor product of elements X\ € Xi and x2 € X2, denoted by X\ ® x2 in the same way as it is done in linear algebra. If X\ and X2 are pivot Hilbert spaces, then X\ ® x2 can be defined as a sesquilinear form which acts on ip = (4>i, ip2) 6 X\ x X2 by {x1®x2)(iP)

= {xuipl)Xl{x2,iP2)X2.

(2.3.16)

Another simple example is when X\ is a space of scalar functions and X2 is an arbitrary Banach space. Then the tensor product X\ ® x2 can be identified, in a natural way, with t —> Xi(t)x2The linear envelope of all X\ ® x2 is called the algebraic tensor product of X\ and X2 and denoted by Xi ® X2. There is a variety of norms which can be introduced in Xx ® X2. The most important of norms are defined as follows. Let u € Xi ® X2, that is N

« = E/.®5.,

(2-3.17)

for some N € N and /,• g Xu ft 6 X2 for i = 1 , . . . , N. Then we have three cases. (a)

|MU := infiO/.IUIs.lk; /. e Xu ft e x2}, i=l

where the infimum is taken over all possible representations of u of the form (2.3.17) defines the greatest cross norm ox the■protectivetensor norm in Xy®X2. (b) Let < , > denote the canonical sesquilinear form on (A^ ® X2) x (X{ ® X2), that is

Singularly perturbed evolution equations

18

JV JV

N

< / , * > < 5 . ,4>>,

®Si; (t>®lp>~11=1 =1

11==11

then \\ul:=sup{®4>>;

4>^X[, i>£X*2, U\\xi < 1 , II^Hxj < 1 }

defines the least cross norm or the injective tensor norm on Xi ® X2(c) If Xi and X2 are Hilbert spaces, then we can define a scalar product by (ui ®u2,vx

®v2) := {uuVx)xi{y-2,v%)x2>

and extend it by linearity onto X\ ® X2\ this defines also a norm on Xi ® X2 which coincides with || ||e. The completion of X\ ® X2 with respect to the norms (a) and (b) are called the ■protective and infective tensor products of Banach spaces Xi and X2, respectively, with notation X^®nX2 and Xi®(X2. When X\ and X2 are Hilbert spaces, we will be using only the completion of Xx ® X2 with respect to the norm defined in (c) and denote it by AiOAj. In most cases it will be obvious which tensor product we are using and we shall use the abbreviated symbol Xi®X2. For the most important spaces appearing in applications we have: For the most important spaces appearing in applications we have: (a) if X% = C(fi») for t = 1, 2, then (a) if X, = C(fi») for t = 1, 2, then

c(n!)®ec(n2) = c(fij x n2), (b) if A", = Li(Qi) for i = 1, 2, then L 1 (fi 1 )® T L 1 (n 2 ) = L 1 (Q 1 x f i 2 ) , (c) if X, = L2{Qt) for i = 1, 2, then L2(fii)®.£<2(fi2) = -Mfti x fia)The space X2 in (a) and (b) can be replaced by an arbitrary Banach space and in (c) by an arbitrary Hilbert space which gives rise to spaces of vector-valued functions. We shall discuss these further in the next section.

Chapter 2. Mathematical preliminaries

19

Our applications are focused on Hilbert and Lx spaces so that we will be using mainly results of points (b) and (c). Finally, we point out that if we have operators A € C(X{) and B 6 C(X2), then the mapping N

N

J2f*®9>^u .=i

= YiAf,®

Bgx,

i=\

defined on Xx ® X2, is linear and continuous in X^®X2 for any choice of the tensor product norm, hence it can be extended by continuity onto Xi®X2. This extension will be denoted by A ® B; it satisfies \\A ® B\\ = \\A\\ ■ \\B\\.

4

Distributions and Sobolev spaces

4.1 The classical notion of the derivative is very often insufficient for analyzing differential operators. To cater for such difficulties the concept of the distributional, or generalized, derivative is introduced. The basic idea comes from the generalization of the formula of integration by parts. Let f,g € LP(R). We call g a generalized derivative of / if J f{x)dx4>(x)dx = - J g{x){x)dx R

(2.4.1)

Dt

for every

-> J g{x)4>{x)dx n

(2.4.2)

Singularly perturbed evolution equations

20

is a linear functional on C£°(ft), We introduce in CJ°(0) a notion of convergence in the following way: a sequence (0n)„6N of elements of CJ°(Q) is said to converge to <j> in C£°(ft) if (a) there is a set ft satisfying f l c f i such that for all n G N supp $n C ft', and (b) for arbitrary t e N lim dkcpn = dk4> uniformly in ft. With this notion of convergence the space C^°(Q) becomes a topological space (but not a Banach space), denoted by X>(fi), in such a way that the functional (2.4.2) is continuous. The dual space (the definition is the same as for Banach spaces) is denoted by V'(Q). We call elements of Z>'(fi) distributions. We note that there are more distributions in V(Q) than those of the form (2.4.2). The duality pairing between V(Q.) and P'(fi) is denoted again by < , >. By (2.4.2) we can identify each Lp(ft) with a (dense) subspace of ©'(ft), that is we can write for every 1 (ft) 4 Lp(il) A Z?'(0).

(2.4.3)

With that both duality pairings between X>'(ft) and V(Q) and between LP(Q) and L,(ft), where l / p + 1/g = 1, are uniquely determined by their restrictions to Lp(ft) X CJ°(ft) so that we have < f,

w(fi)xP(n) = <{, >Lp(n)xLq(n),

(2.4.4)

for / G LP(Q) and 4> G C~(ft). This justifies our using of the same symbol for duality pairings in the V(Q) and LP(Q). In the L-i space context, however, it will be more convenient to use the antiduality pairing defined as (u, ) : = < u,4> > (2.4.5) (2.4.5) Then the distributional antiduality pairing (2.4.5) is uniquely determined by an ap propriate restriction of the scalar product. Perhaps the most widely used distribution which cannot be represented by a function is the so-called Dirac distribution 5, defined by

Chapter 2. Mathematical preliminaries

21

< &,4> >= 4>(0). Distributions are generated by (2.4.2) not only by functions from Lp. The largest sensible class of such functions is the space LUoc(tt) consisting of functions which are integrable over every bounded set fi' such that FF C fi and referred to as locally integrable in £1 A distributional derivative of any / e LUoc(Q.) can be defined by an extension of the formula (2.4.1). Thus we say that g 6 D'(ft) is the partial distributional derivative with respect to xt, i = 1 , . . . , n, of / € LUoc(Q) if for every 6 V(Q) we have < g,4> >= - < f,dx,4> > = - J n

f{x)dXi4>(x)dx,

shortly written as >=-

< f,dXt4>>

For example, the Dirac distribution is the distributional derivative of the step function H defined by H(x) = 1 for x > 0 and H(x) = 0 for x < 0. Clearly, this definition can be extended to derivatives of arbitrary order and also to any / g V(Q). An important property of the distributional differentiation is that this is a continuous operation in V'{Q.) in the sense that if lim < fn,

=<■■< J f,4>> n-too

for every <> / G D(fi), then for every k lim = n—foo

This property has an application in identifying certain limits. If, for example, a sequence (/ n ) n gN of elements of LP(Q) converges in Lp(ft) to certain / , dxJ„ € LP(Q) for n G N and {dxJn)nen converges to some g € LP{Q), then g = dxJ. This follows since the convergence in Lp spaces yields the convergence in the distribution space by (2.4.3) and since the differentiation is continuous in X>'(£2). 4.3 The spaces Lp(Q) for p < oo (and the space L2{Q) in particular) are much more convenient than the space C(fi). It is therefore desirable to introduce Lp-like counterparts of spaces Cm(Ti). With the notion of a generalized derivative such a construction is possible. The Sobolev spaces Wp (Q) are defined in the following way:

Singularly perturbed evolution equations

22

Wpm(Q) := {u 6 L p (ft); d'u G L p (fi), 0 < I < m)

(2.4.6)

where d'u is a distributional derivative. It follows that W£(fl) endowed with the norm

lhlU,P:=ll«II^(n):=(f;i|3l«|rMn)) '

(2.4.7)

becomes a Banach space and, if p = 2, then (2.4.7) defines a Hilbert norm with the corresponding scalar product given by

(«, v)w; *(«) : =

,9*u )ij(n)ii=0 =0

4.4 Sobolev spaces can be identified in a natural way with closed subspaces of Carte sian products of the respective Lp spaces so that they inherit most of their properties. We shall not go into details, as only selected properties of W™ spaces will be needed in the sequel. In applications we shall mainly have n = R" and the properties listed here refer to this case (but, of course, in most cases they are valid in a more general setting). The interested reader is referred to e.g. [1], Chapters III and V and [25], Vol. II, Chapter IV. First we have C0°°(Rn) A W™{Rn)

(2.4.8)

for any m and 1 '(lRn) and that the antiduality pairing between V(Rn) and T>'(R") satisfies < / , U > F ( R » ) x P ( R » ) = < f,U

>w,-"'(R")xWp'"(K")'

whenever / 6 H^" m (R n ) and u e £>(Rn). As in (2.4.4), all duality pairings here are uniquely determined by the pairing between L p (R n ) and Lq(Rn) and henceforth we shall be using the notation < , > without subscripts. If p = 2, then again it is more convenient to work with the anti-duality pairings between the spaces H/r2~m(Rn) and W ^ R " ) and between V(Rn) and V(Rn) which are determined, in the above mentioned sense, by the scalar product in L 2 (R"). The operator of differentiation in W™(f2) satisfies

Chapter 2. Mathematical preliminaries

dk e C{W™(Rn),W™-k(Rn)) and the multiplication by a function 0 is a continuous operation in Wm(Rn) that € CT{W) for any m € Z, k e N and r < m.

23

(2.4.9) provided

4.5 A major role in the theory of partial differential equations is played by the socalled Sobolev imbedding theorems which establish relations between Sobolev spaces and various standard spaces. We have the following results: (a) W™{Rn) ->■ L,(R") for 1 < p < q < np/(n — mp) if n ^ mp and for 1 < p < q < +00 otherwise. for 1 < p < q < np/(n — mp) if n ^ mp and for 1 < p < q < +oo otherwise. (b) (b)

m W/ ++m (R") -»■ C J (R") W p ' (R n ) -»■ CJ(Rn) for n < mp or p = 1 and m = n.

The last imbedding is quite interesting since Wpm(R), as asubspace of L P (R"), consists of classes of equivalence of functions. Hence, this imbedding should be understood in such a sense that each class of W p 7+m (R") contains a representative which is a classically differentiable function. This property makes functions from W™(Rn) useful in the theory of differential equations, particularly in initial and boundary value problems, enabling us to define values of functions at a given point or on a manifold of lower dimension, for instance, on the boundary of a domain. Note that for ordinary Lp function it is impossible, since the boundary of (sufficiently regular) set in R n has n-dimensional Lebesgue measure zero and an Lp function can assume arbitrary values there. Unfortunately, from (b) we see that to consider pointwise values of / we need m > n/p which is very often too restrictive. 4.6 We can relax the condition on m if we agree to consider boundary values of / in a weaker sense and here we shall briefly outline the procedure (see e.g. [80], pp. 40-43 for the presented approach). Let T be an (n - l)-dimensional smooth surface in R n We know that each / € W™(Rn) is a limit in Wpm(R") of a sequence of C°° functions. The restriction of each fn to T is a well-defined function on T, called the trace of /„• If m > 1/p, then the traces of smooth functions converging in W p m (R n ) to a given / form a sequence which converges in LP{F) and the corresponding limit is called the trace of / on T. It can be proved that this limit does not depend on the choice of the approximating sequence, and that if / is a smooth function, then its trace coincides with the pointwise restriction of / to T. This concept can be extended to higher order derivatives provided that m is sufficiently large.

Singularly perturbed evolution equations

24

4.7 An important role in the theory of differential equations is played by the Fourier transform which is originally defined for u G Li(R n ) by the formula (Fu)(y) := u{y) := — ^

/
(2.4.10)

It is a one-to-one mapping with the inverse defined by (F-lu){x)

:= — ; L _ |

c«*fi{y)dy.

(2.4.11)

The applications of the Fourier transform defined in Li are limited due to the fact that there is no explicit description of FLi(R"). Unfortunately, the extension of F to other Lp spaces is not immediate, since L P (R") <£. Zq(R n ) and for arbitrary u € Lp(Rn) the integral in (2.4.10) can diverge. The most elegant generalizations of the Fourier transform is via the distribution the ory. The main idea of the approach is similar to that of the distributional derivative. Namely, we operate not on the distribution itself but on suitable test functions. In deed, changing the order of integration we obtain for any u G Li(R n ) and <j> G V(HL")

< Fu, 4> > = / (Fu){y)4>(y)dy = I u{x){Fcj>){x)dx =< u, F >

(2.4.12) (2.4.12)

The space T>(W) is, however, not invariant under the Fourier transform and we cannot insert an arbitrary u G X>'(R") into (2.4.12). Hence we have to introduce a smaller class of distributions, called tempered distributions, defined as continuous linear functionals on a space of test functions S (see e.g. [79], pp. 182-204). The latter consists of all C°°(R") functions which, together with all their derivatives, tend, as \x\ —> oo, to zero faster than any polynomial. Clearly C~(R n ) C 5 and the inclusion is strict since, for instance, x —> e~x G <S\CJ°(R n ). The space of tempered distributions is denoted by <S' and for any p we have P(R) A 5 4 L p (R) 4 S' 4 P'(R)

(2.4.13)

where the identification of Lp elements with tempered distributions is performed in exactly the same way as with ordinary distributions. We shall also use the usual notation for the duality pairing on S x S1. The Fourier transform is an isomorphism of S onto S and this allows us to define its extension onto <S' by formula (2.4.12). In other words, we say that g G S' is the Fourier transform of u G S' if for each ip G S we have

Chapter 2. Mathematical preliminaries

25

< g,ip> =

(2.4.14)

The Fourier transform defined in such a way is a unique extension by continuity of the Fourier transform defined in Lx, therefore we shall use the same notation in both cases. In addition, F is an isomorphism from S' onto S' and the inverse transformation is given by the analogous extension of F " 1 By (2.4.13) we can use the Fourier transform for functions from any Lp. The most interesting is, however, the L2 theory, since F is an isometric isomorphism from L 2 (R n ) onto L 2 (K"), that is, for any u 6 L2{W) we have ll^llMR") = IMlMR")-

(2-4.15)

This is the famous Plancherel theorem. The importance of the Fourier transform lies in the fact that it converts differentiation into multiplication according to the formula (Fdku)(y)

= (iy)k(Fu)(y),

u e S<, t e N "

(2.4.16)

Formula (2.4.16) enables us to introduce another notion of the generalized derivative. Namely, we say that dx,u exists in Lp{Rn) if F'^y -* iyt(Fu)(y)] € Lp(Rn). In accordance with this we denote by ff"(Kn) the set of all Lv functions for which F~l[wml2F{u)} e Lp(Rn), where w(y) := 1 + \y\2 Note that the space H2(Rn) is the space of solutions of the equation u-Au

=f

(2.4.17)

in L p (E n ), since (Fu)(y) = (l +

\y\2)-1(Ff)(y).

Fortunately, for p > 1 we have r7™(Rn) = Wpm(R") which means that solutions of the Helmholtz equation with the right-hand side in Lp are twice differentiate in Lp and, due to the imbedding theorem, we may infer that if the right-hand side is smooth, so is the solution. This result, which is a rough statement of the celebrated Shift Theorem for second order elliptic differential equations, fails to hold for p = 1. Since in our considerations the Li space is one the basic spaces, we shall return to this problem later. The reader interested in the theory of H™ spaces is referred to [1], pp. 219-223 and [82], pp. 116-159.

Singularly perturbed evolution equations

26

4.8 We conclude this section with a brief description of periodic distributions and Sobolev spaces of periodic functions (see e.g. [25], Vol. 3, pp. 4-9, [33], pp. 57-60 or [88], Chapter I). Let m € N U {oo}. By C™ we denote the set of all functions u G C m (R") which, together with all their derivatives of order not exceeding m, are periodic with period 27r in each variable Xj, where x = ( i i , . . . , i „ . . . , i n ) 6 Kn This class can be identified with the class C™(fi), Q = [0,27r]n, of all functions from Cm(Q) which satisfy the periodic boundary conditions:

(d r «)U=o=(d r «)U =2 .

(2-4.18)

for all/ = 1 , . . . , n and r <m. With the standard Cm norm the space C™(£2) becomes a closed subspace of C m (ft). Clearly, 2n can be replaced by any other constant. The space C™(fi) is isomorphic with the space Cm(Tn) where Tn is an n-dimensional torus and an isomorphism is given by the following identification: C™(Q) 3 ^ f t ^ G Cm{Tn) where J>(xl,...,xn)

= 4>(e'I\...,e^).

(2.4.19)

Here i denotes the imaginary unit. This identification is extremely useful as it shows that the topological properties of spaces of periodic functions are the same as that of spaces defined on smooth compact manifolds. We shall not go into details but confine ourselves to remark that, roughly speaking, these spaces have all the advantages of the spaces of functions defined on R n without drawbacks caused by the unboundedness of R" We can define the space of distributions, V(Tn), as the topological dual of C°°(Tn) and the space of periodic distributions, X>^(Q), can be obtained similarly as in Eq. (2.4.19). Consequently, Sobolev spaces W™(Tn) can be defined as in (2.4.6) and the corresponding Sobolev spaces of periodic functions, W™ (Q), can be obtained either through (2.4.19) or equivalently as the closure of C™(fi) in W™(f2). Using the procedure of taking traces of functions from Sobolev spaces described in Subsection 4.6 we obtain an alternative characterization of W™(Q). We say that u € W™ (fi), if for all / = 1 , . . . ,n and r < k — 1, Eq. (2.4.18) holds. Here the boundary values are understood in the sense of traces (see Subsection 4.6) and equality is in Lp(dQ). Note that W^(Q)

= Lm(Sl)

= Lp(n)

for all p e [l,oo]. As in Subsection 4.4 we denote by Wr™(0), where l / p + \/q = 1, the space adjoint to FF™ (fi).

Chapter 2. Mathematical preliminaries

27

All imbedding theorems described in Subsection 4.5 are valid here. Moreover, for mi > m 2 , m i , m 2 G Z, we have

w%(ti)Awg(n). This result does not hold if Q is an arbitrary unbounded domain; here it follows from the corresponding result for Sobolev spaces on compact manifolds. There is a counterpart of the Fourier transform theory of Subsection 4.7 applicable to periodic functions. Let p = 2 and u G L2[£l), H = [0,27r]" Defining for k 6 Z n it* := / n

u(x)e~'xkdx

we have that u G L2(f2) if and only if Fu := (ut)jt£zn 6 I2 Then u(ar) = J ^ H M t e s - K * ) = E

" ^

( 2 - 4 - 20 )

fcgZ"

where, again, the convergence and equality is understood in the L2 sense. As in the free-space case, this finite Fourier transform converts differentiation into multiplication and if u G Cv(O), we then have dXlu(x) = E

(iki)uketkx

keZ"

or equivalently F[dxiu] = {(ikt)uk}kez«

(2.4.21)

for I = 1 , . . . ,n. Accordingly, for u € C™(Q) c-1 £

(1 + |fc| 2 ) m |^t 2 < / E

|9fc«(.T)|dx < c E (1 + W 2 ) m K I 2

(2-4.22)

for any m G N and we can equivalently define the Sobolev spaces W£,{(1), m G Z, by U ^ ( Q ) := {ti G l>;(ft); E (1 + l^l 2 ) m l^l 2 < <»}•

(2-4-23)

fc€Z"

As in the case of free-space Fourier transform, similar considerations can be done for p G]l, oo[ but in the sequel we shall need only p = 2.

Singularly perturbed evolution equations

28

5

Vector-valued functions

To introduce the topic we start with a simple example. 5.1 Example. Let / 6 /^(R 2 )- By Fubini's theorem we have ll/lll = J \f(x,y)\2dxdy = j U\f(x,y)\2dx) R2

R

\R

dy < oo

(2.5.1)

/

and for a. e. (almost every) y £ K the integral f\f(x,y)\2dx

is finite. Thus we can

R

think of / as a function of one variable, say y, with values in the space L2(M.)- With some abuse of notation we shall denote the function y —► / ( y , •) by the same symbol / and this convention will be used throughout the book, unless it creates confusion. Should such a situation occur, the distinction will be introduced in due course. With this understanding the relation (2.5.1) can be rephrased as

/ll/(-.2/)lli2(R)*/<°°> R

therefore we can introduce the following notation: /ei2(l,L2(l)) which can roughly be read as: y —> f(y) is a function such that for a. e. y e R, f(y) is an element of L 2 (R) and moreover y —> ||/(2/)||L 2 (R) >S a square-integrable function over K. However, to make the definition of L2(R, L2CR)) precise, we must indicate what it means to integrate a vector-valued function. We shall do it later in this section but first we introduce simpler spaces of vector-valued functions which will be commonly used throughout the book. ■ 5.2 Let A" be a Banach space and T > 0. By C(]0, T[, X) we denote the space of X-valued functions which are continuous on ]0, T[ in the following sense: for every ta e]0, T[ we have lim||/(i)-/(t0)IU=0.

(2.5.2)

If T < 00, then C(]0,T[, X) = C([0, T], X) and the latter, equipped with the norm ll/l|c([o,T],x) = sup ||/(t)||x, (€[0,T)

Chapter 2. Mathematical preliminaries

29

is a Banach space. If T = oo, then ]0, oo[ = [0, oo[ but, clearly, according to our definition C(]0,oo[, X) ^ C([0,oo[, A), as the second space contains functions which may be unbounded in infinity. Therefore we have to keep the notation C(]0, oof, X) or, in shorter form, C(R + , A). This is also a Banach space. We say that an A-valued function / is differentiate at t0 e]0, T[ if there is an element x € X such that lim|| /fa +

ft)-/(*o)_ g|| \x

= 0.

Then x is called the derivative of / at t0. If / is differentiable at every point of ]0, T[, we say that it is differentiable. Since it does not cause any confusion, we shall denote the derivative of / by the usual symbol dtf. In a similar fashion we define higher order derivatives of / . The set of all functions which are continuous on ]0,T[ together with their derivatives up to an order k, will be denoted by Ck(]0,T[, X). The comments made for C(]0,T[, X) are also applicable here. We note that the differentiation commutes with any bounded linear operator on X, that is, if A G C(X, Y) and / is differentiable with respect to t, then dtAf = Adtf.

(2.5.3)

If / is continuous on a segment [a, b], —oo < a < b < oo, we can then define its Riemann integral exactly as in the scalar case and such an integral has most of the properties of the scalar Riemann integral. In particular, we can define an improper integral in the same way as in the scalar case. We also have the following estimate: i>

6

\\Jf(t)dt\\x<J\\f(t)\\xdt a

Wxdt

a

and for any A € C(X, Y) the equality b

A f f(t)dt= a

b

I Af{t)dt

(2.5.4)

a

holds. 5.3 Let us consider a function / defined on an open domain fi C C with values in a Banach space A". We say that / is holomorphic (or analytic) in Q if it is differentiable at any point z € fi. As in the scalar case, this implies that / has derivatives of arbitrary order and each point of Q has a neighbourhood where / is the sum of its Taylor series.

Singularly perturbed evolution equations

3D

Similarly to the scalar case we can introduce an integral of an X-valued function / of a complex variable z £ ft along a rectifiable curve F lying in ft. If / is holomorphic in ft, then the Cauchy theorem remains valid and the Cauchy integral formula holds

M=hlj^ dz,

(25 5)

-

r where T is a closed rectifiable curve such that the index of z with respect to it is equal to one. 5.4 As a Banach space X, to which the values of t —> /(c) belong, we can take, in particular, a space C(Xi, X2) of continuous linear operators. Then we can define two types of continuities of / . If / is continuous in the sense of definition (2.5.2), that is, the limit is taken in the norm of X = £.(Xl,X2)} then it is said that / is uniformly continuous. It is a very strong property and for many applications it is enough to assume that for any x € Xi the function t —> f(t)x (f{t) is here treated as an operator on X{) is continuous as an AVvalued function. It is then said that / is a strongly continuous function. The same remarks refer to the other properties discussed above, like differentiability, integrability or analyticity. 5.5 Occasionally, the Riemann integral will be too restrictive and we shall resort to an extension of the Lebesgue integral to vector-valued functions. The most commonly used generalization of the Lebesgue integral is offered by the Bochner integral which will now be briefly discussed (see [89], pp. 130-136, [45], pp. 58-92). The starting point in the definition of the Lebesgue integral is the notion of measurability of a function. The standard definition used in the real function theory cannot be used here and will be replaced by the following construction. Let X be a Banach space and let { f t i , . . . , n m } be a finite collection of mutually disjoint, measurable subsets of ft € R71 and ( i i , . . . , i m } be a collection of points of X. The function / : ft —> X defined by m

/(*) = £ * * » ! * ( * ) .

(2.5.6)

k=i

where \nk is the characteristic function of Qk (that is xnk s 1 on ftfc and \ n = 0 otherwise), is called a simple function. A function g defined almost everywhere on ft is called measurable on ft if there exists a sequence (/n)„eN of simple functions such that Blim||/n(t)-/(<)||x

= 0

Chapter 2. Mathematical preliminaries

31

almost everywhere on £2. It can be shown that if the range of / is separable, then / is measurable if and only if, for every x' e X', the scalar function t ->< f{t),x' > is measurable. If / is a simple function (2.5.6), then we define its integral by m

r

f{t)dt='£xkm(nk).m(Qk). a *=' If for a given function / we can choose a sequence of simple functions (/n)ngN in such a way that

HmJ||/»W-/(0IM* = 0. n

then we say that / is (Bochner) integrable on Q and define the Bochner integral by / f(t)dt := Hm, f n n

fn(t)dt.

This definition is independent of the choice of the sequence {fn)n€tt- A measurable function / is Bochner integrable on Q if and only if t —> \\f(t)\\x is Lebesgue integrable on fi and we have

\\Jf(t)dt\\x

<J\\f(t)\\dt

It follows that, due to the fact that the definition of the integral involves only linear operations and passing to the limit, the integration commutes with bounded linear operators: for any A 6 C(X, Y) we have , I f(t)dt=

I

Af{t)dt.

(2.5.'

The notion of the Bochner integral allows us to complete the example from the be ginning of this subsection by providing a rigorous definition of LP(Q,X) spaces. For a Banach space X we define LP(Q,X) as the space of (classes of equivalence of) measurable functions with finite norm

f (JII/MM*

for

I
ll/IUP(n.x) = { e ^j S U p[[/(a;)|| x for p = 00.

Singularly perturbed evolution equations

32

The spaces LP(Q,X) are Banach spaces which have most of the basic properties of the scalar Lp spaces. With this definition it can be proved that for 1 < p < oo and fi = fii x f22 we have Lp(nx,Lp(Q2))

=

Lp(n).

In cases considered most important for applications we also have L1{Q,,X) = Li{Q)®X for any Banach space X and L2{Q,,H) = L2{Q.)®H for any Hilbert space H. In all of the three last formulae the equality sign is to be understood as the "isometrically isomorphic" 5.6 The Fourier transform can also be defined for functions from Li(R", X), where X is a Banach space, by the same formula (2.4.10). We shall need only its generalization to L2(Rn, H) (where H is a Hilbert space) which can be obtained as in the scalar case. As in the scalar case we know that F is an isometric isomorphism from L2(Rn,H) onto L2 (Mn, H) with the Plancherel formula \\FJ\\LW,H)

= II/HMR-,*)-

(2-5.8)

All formulae for differentiation, inversion and algebraic manipulations are the same as in the scalar case.

6

Unbounded operators

A differential operator d : C"(R) -> C(R) is a bounded linear operator. It is, however, inconvenient that it acts between two different spaces. Indeed, the equation du+u = 0 formally does not make sense in this setting, since on the left-hand side we have an addition of elements of two different spaces. It is therefore desirable to define the operator of differentiation in such a way that it acts from a given space into itself. This can be done in two ways. If we are determined to keep continuity, then we have to define it either on some C°° space, or on a space of distributions, but both choices have many drawbacks. First of all neither space is a Banach space which makes the

Chapter 2. Mathematical preliminaries

33

whole analysis quite complicated. Moreover, the first choice offers too small, and the second too large, spaces for most applications. Another way is to define the differentiation in C(R), not on the whole space but on a subspace, for instance on C'(R), which is then treated as a subspace of C(R). It follows that in this case we lose the continuity of the operator but the advantages of such an approach outnumber its drawbacks and in what follows we shall be dealing with such a type of definition of differential (and other) operators. 6.1 Let us consider a Banach space X and a linear operator A defined on a linear set D{A) C X with a range R(A) in a Banach space Y The set D(A) is called the domain of A and the operator A with its domain D[A) is frequently referred to as (A,D(A)). However, when no misunderstanding is possible, it will be denoted, as previously, by A. An operator B is said to be an extension of A if D(A) c D(B) and B\D(A) denote it A C B.

= A and

We say that (A,D(A)) is a closed operator if for any sequence (i n )„ e N of elements of D(A) satisfying lim xn = x 6 A' and lim Axn = f e Y we have x 6 D(A) and n—>oo

71—>oo

Ax = / . An equivalent definition is that the graph of A, g{A)-{{x,Ax); S(A): = Ux,Ax); x£ xeD(A)}, D(A)\, is a closed set in the product space X x Y Clearly, every bounded operator defined on the whole space is closed but there exist closed operators which are not bounded. An example is rendered by the operator u —»• du defined in C([0,1]) with D(d) = C'QO, 1]). However, if for a closed operator A we have D(A) = X, then A is bounded. This is a statement of the famous Closed Graph Theorem. 6.2 The set of all closed operators from X to Y will be denoted by C(X, Y) with a natural simplification C(X) if X = Y Unfortunately, C(X,Y) is, in general, not a linear space, since the sum of two unbounded closed operators may not be closed. However, if A e C(X) and B e C(X), then A + B e C(X). 6.3 Numerous differential operators, in particular in Lp setting, are not closed, but in a certain sense, they can be extended to closed operators. Making this notion precise, we say that an operator (A, D(A)) is closable if its closed extension exists. The smallest closed extension of (.4, D(A)) is called its closure and denoted by (.4, D(A)) (or in short A). If (A,D(A)) is a closed operator, S C D(A) and ( 4 | s , S ) = (A,D{A)), then we say that S is a core for .4. In other words, values of .4 are completely determined by its values on the core.

Singularly perturbed evolution equations

34

The operator A is closable if for every sequence (xn)„en of elements of D(A) such that lim xn = 0 and lim Axn = / we have / = 0. n—foo

n—foo

All differential operators with sufficiently smooth coefficients defined on CJ°(R n ) are closable in every Lp(Rn). In fact, if for (wn)neN C C^°(R") we have lirn^n = 0 and lim Aun — f in LJR"), then by (2.4.3) and the continuity of differentiation in X>'(Kn) we see that / = 0 in P'(R n ) and therefore also / = 0 in Lp(Rn). The characterization of the domain of the closure is, however, very often a complicated task. We shall describe some examples of this kind in Section 3.7. 6.4 In what follows we will often encounter operators either defined on finite-di mensional subspaces of a Banach space X, or having a finite-dimensional range. The latter are called finite rank operators. An operator with finite-dimensional domain of definition is always bounded. However, for a finite rank operator this is true only if it is closable. 6.5 If A € C(X,Y),

then D(A) equipped with the so-called graph norm \W\DW:=(h\\'( x I

2 l I * + \\Au\\ x) P

(2.6.1)

is a Banach space. If A" is a Hilbert space, then D(A) becomes a Hilbert space. Let B be an operator in X and D(A) C D(B). some a, b > 0 and all x 6 D(A) we have

We say that B is A-bounded if for

| | B z | | K < a | | x | | x + &||Ax||y.

(2.6.2)

It follows that B is A-bounded if and only if B\D{A) is bounded in the graph norm (2.6.1). If, moreover, B is a closable operator, then the inclusion D(A) C D(B) is already sufficient for A-boundedness of B (see [47], p. 191). 6.6 If an operator A is invertible and closed, then the inverse A~l is also closed. We shall very often use the following result: if for some A 6 C the operator (A - A/)" 1 e C(X), then A is closed. This follows from Subsection 6.2, boundedness of AT" and the identity A = (A - XI) + XL 6.7 The formulae (2.5.3), (2.5.4) and (2.5.7) hold to some extent also for closed operators. Let A € C(X,Y) and both / and Af are difTerentiable. Then

Adf = dAf.

(2.6.3)

Chapter 2. Mathematical preliminaries

35

If / is a Bochner integrable function over Q such that for almost all x we have f(x) € D(A), Af is measurable and

j \\Af(x)\\ydx

< CO,

then (see [45], p. 83) A J f{x)dx = j Af(x)dx. n n

(2.6.4)

6.8 For an unbounded operator we can introduce the notion of the dual operator in a similar way as for bounded operators; the construction, however, is much more complicated. Let A € C(X,Y). We define D(A') as the set of all y e Y' for which there exists an z 6 X' with the property < AX, y >yxy,=<

X, Z >XxX'

(2.6.5)

for all x £ D(A). In general, z is not uniquely defined unless we assume that A is a densely defined operator , that is, the domain D(A) is dense in X. If this is the case, we define the dual (or transposed) operator (A',D(A')) by A'y = z, where z is determined by (2.6.5). In the Hilbert space setting we shall use rather the adjoint operator, which can be defined in exactly the same way as the dual operator, with the dual space replaced by the adjoint space and the duality pairing replaced by the antiduality pairing. The operator adjoint to A is denoted by A' If an operator A is densely defined, then A' G C(Y',X'). If A is closable, then A' = (A)' If, moreover, X is reflexive, then A' is densely defined and (A1)' = A. 6.9 Let A £ C(X) be densely defined. A1'1 e £ ( X ' ) and (/!-')' = (A')" 1

The inverse A'1

6 £(A') if and only if

If an operator is not surjective but its range R{A) is closed, we can still say a great deal about its invertibility. We have the following Closed Range Theorem (see [89], p. 205). If A € C(X, Y) is densely defined, then the following properties are equivalent: (a) R(A) is closed,

Singularly perturbed evolution equations

36 (b) R(A') is closed, (c) R(A) =

N(A')\

(d) R(A') =

N(A)1

6.10 Let H be a pivot Hilbert space. An operator A satisfying (A,D(A)) C (A*,D(A*)), or equivalently [Au,v)u = (u,Av)n for all u,v € D{A), is called sym metric. If we have (A,D(A)) = (A*,D(A*)), then A is called a self-adjoint operator. A self-adjoint operator is, by definition, closed and densely defined. If, for a sym metric operator A, we have D(A) = H or R(A) = H, then A is self-adjoint. If a self-adjoint operator A has the inverse A"1, then A - 1 is also self-adjoint. If a selfadjoint operator A satisfies ( A U , K ) / | | U | | # > m for some m > 0 and all 0 / « 6 D(A), then the operator A is said to be positive and positive definite if m > 0. 6.11 If A,B e C(X), then we say that A and B commute if AB = BA. It is not easy to extend this definition to unbounded operators due to difficulties related to the domains. Usually it is partly done, when one of the operators belongs to C{X). An operator A is said to commute with B 6 £(A) if BA C AB.

(2.6.6)

Let X = A'i © A'2. We say that A is decomposed according to Xi and X2 (or reduced by Xi and X2) if P£>(A) C D(A), AXl C X : , AA 2 C X2,

(2.6.7)

where P is the projection on A'i along X%. This definition is equivalent to the condi tion that PA C AP When A is decomposed as above we can define parts A\ and A2 of A in Xi and X2. Part Ai is defined as an operator in the Banach space A'i with domain D(A{) = D(A) n Xx. Part A2 is defined similarly. If A is closed and densely defined, the same is true for both Ai and A2. The first inclusion of (2.6.7) yields in particular that PD(A) = D{A{) (see [47], pp. 171-172). 6.12 An important class of unbounded operators is the class of dissipative oper ators. If H is a Hilbert space, then the definition is quite simple. We say that a linear operator (A,D(A)) acting in H (not necessarily closed or densely defined) is dissipative if for every u 6 D(A) Re(Au,u)H

< 0.

(2.6.8)

Chapter 2. Mathematical preliminaries

37

Since we will work also in Lp spaces, we must generalize this definition to the Banach space setting but for that we shall need some prerequisites. Let X be a Banach space and X' its dual. For an arbitrary element u e X we denote by Ju an element of X' satisfying \\Ju\\x,

= \\u\\x,

<Ju,u>=\\u\\2x.

(2G.9)

The existence of Ju is guaranteed by the Hahn-Banach theorem but Ju is, in general, not uniquely defined. From now on we shall denote by Ju the set of all elements satisfying Eq. (2.6.9). In an important case of Lp spaces with 1 < p < oo and for Hilbert spaces J is a function; J is linear only for Hilbert spaces (see e.g. [31], p. 118). It is said that an operator (A, D(A}) acting in a Banach space X is dissipative if, for every u € D(A), there is u' 6 Ju such that Re < Au,u

> < 0.

(2.6.10)

Note that if X is a pivot Hilbert space, then Ju = u and (2.6.10) and (2.6.8) are equivalent. An important characterization of dissipativity is offered by the following result. A linear operator is dissipative if and only if, for all u 6 D(A), and A > 0 we have ||(A/-A)«||x>||u||jf.

(2.6.11)

This shows that a dissipative operator has a closed range. Dissipative operators satisfying R(I-A)=X

(2.6.12)

are called m-dissipative. Since R(I - A) is closed by (2.6.11), for m-dissipativity of A it is sufficient that R(I - A) is dense in A'. We shall note some properties of dissipative operators which will be useful in the sequel. If A is m-dissipative, then R(XI-A) = X for all A > 0. A dissipative operator with dense domain is always closable and the closure of any dissipative operator is also dissipative. If X is reflexive, then any m-dissipative operator is densely defined. We say that a dissipative operator A is maximal dissipative if .4 has no proper dis sipative extension. An m-dissipative operator is always a maximal dissipative. The converse is true in Hilbert spaces provided A is either densely defined or closed. If .4 is a closed or densely defined dissipative operator in a Hilbert space, then it always

38

Singularly perturbed evolution equations

admits an m-dissipative extension but its construction is usually not explicit and thus this result is of limited usage (see [31], pp. 156-158). An important subclass of dissipative operators acting in a Banach space X is the class of conservative operators. We say that A is a conservative operator, if for every u 6 D(A) there is v! G Ju such that =0, = 0,

(2.6.13)

with natural simplification of the definition when X is a Hilbert space. In the latter case one can see that A is a conservative operator if and only if B := — IA is a symmetric operator. 6.13 We conclude this section by discussing the class of unbounded operators gen erated by sesquilinear forms. The presentation here follows [25], Vol. II, pp. 367-374. Let Hi be a Hilbert space and let (x, y) —> a(x, y) be a continuous sesquilinear form on Hi x Hi, that is, satisfying o(ilW) 0 and all x, y e H\. It follows that with every such form we can associate a unique operator A e C{Hi,H{), defined by a(x,y)=<

Ax,y>H;xHl

(2.6.14)

The form a*{x,y) = a(y,x) is called the adjoint to a. If B is an operator generated by a* according to (2.6.14), then it turns out that B = A", as defined in Eq. (2.3.5). We say that the sesquilinear form a (or the associated operator A) is Hi-coercive if there is a constant a > 0 such that for every x € Hi we have Rea(x,x)>a\\x\\2Hi. M H l '

(2.6.15)

If A is the operator associated with a, then it follows that the two problems: (a) for given / G H{ find x € Hi such that Ax = /, and

(2.6.16)

Chapter 2. Mathematical preliminaries

39

(b) for given / 6 H[ find x € Hi such that for every y € Hi a{x,y)=, f,v

(2.6.17)

>>

are equivalent. Formulations (2.6.16) and (2.6.17) are often referred to as, respec tively, the operator and variational formulations of the problem. The solvability of (2.6.17) (and of (2.6.16)) is dealt with by the celebrated LaxMilgram Theorem: if a is a continuous sesquilinear coercive form on Ht x Hit then the associated operator A is an isomorphism of Hi onto H"[. The same is clearly true if A is replaced by A* In many situations we are faced with the equations which can be cast in the form Ax + \x = f,

A 6 C,

(2.6.18)

rather than in the form Ax = f. To deal with such problems it is more convenient to consider two Hilbert spaces Hi and H (with H being a pivot space) satisfying (see (2.3.4))

(2.6.19)

HXAHAH{.

With this identification the notation of (2.6.18) is justified since x can be considered as an element of //,*. The antiduality pairing < ■, > : = <

,

>H\xHl

is uniquely determined by the scalar product (■, -)H, that is, < x,y >=

(x,y)H,

whenever x 6 H and y 6 Hi. Equation (2.6.18) can be written in an equivalent variational form: for a given / e H[ and A € C find x g H\ such that a(x,y) + \(x,y)H= f , v >

(2.6.20)

for all y € Hi. We say that a sesquilinear for a is Hi-coercive with respect to H if there exists A0 G 1 such that Rea(x,x)

+ X0\\x\\2H>a\\x\\2Hi

I*,

(2.6.21)

Singularly perturbed evolution equations

40

for all x € H\. This inequality is frequently referred to as the Garding inequality. It immediately follows from the Lax-Milgram theorem that if a is // r coercive with respect to H, then the problem (2.6.20) (and equivalently (2.6.18)) has a unique solution whenever ReX > A0. The space H{ very often turns out to be too large for many applications: in the example below / G H[ may not be a function. To deal with this we shall modify the definition of A which will make A an unbounded operator acting in H. Let A be an operator associated with a according to (2.6.14). We define an unbounded operator (A,D{A)) as follows:

D(A) Ax

:= {x£ Hu Ax£H}, := Ax . 4 J for x 6 D(A).

(2.6.22)

If we assume that the generating form a is //^coercive with respect to H, then the operator A is closed, D(A) is dense in both Hi and H. The adjoint in H of A is the restriction of A* (generated by a*) to D(A') := {x 6 Hi; A*x e H] and D(A') is dense in both Hi and H. In particular, if a is symmetric, that is to say, a(x,y) = a(y,x) for all x,y e Hi, then A is self-adjoint. If, in addition, a is Hicoercive, then A is positive definite. Both A + XI and A* + XI are isomorphisms of D{A) and D(A*), respectively, onto H for any A with Re X > X0. In the sequel we shall usually use the same notation for the operators defined by Eqs. (2.6.22) and (2.6.14). 6.14 E x a m p l e . Let Hi = W2l(Rn) and let a be given by the integral expression a(u, v)=

I Yl dku{x)dkv(x)dx.

(2.6.23)

R" * = 1

Clearly a is a continuous sesquilinear form on W^M") x W^iW) but it fails to be W2(Rn)-coercive. However, a is H/21(Rn)-coercive with respect to L 2 (R n ) for any A0 > 0. Therefore, if Re X > 0 and / € W f ^ R " ) , then the variational problem: find u e W}(Rn) such that for all v 6 W}(Rn) u{x)dkv{x)dx R"

t = 1

+ X I u(x)v(x)dx R"

= J f{x)v{x)dx

(2.6.24)

R"

for all v 6 W ^ R " ) . has a unique solution. To determine the operator A associated with a we note that for v € C£°(Rn) the definition of the distributional derivative gives

Chapter 2. Mathematical preliminaries

41

a(u, v) =< —Au,v > This identity is valid also for arbitrary v G W^(Rn) by the density result (2.4.8), and by (2.4.9). Therefore the (distributional) Laplace operator is associated with a and problem (2.6.24) is equivalent to the differential equation - Au + Xu = f in W{l{W)

(2.6.25)

and this equation is uniquely solvable in W ^ R " ) for any A with Re A > 0 and

few2-l(w).

7

m

Elements of spectral theory

Let A be a linear operator with domain D(A) and range R{A) contained in a Banach space X. In applications, we are usually interested in properties of the family of operators {AA}A£C defined as Axx = (XI - A)x,

x G D(A).

Analysis of invertibility of Ax is the subject of the spectral theory and we shall now summarize its basic results which will be useful in the sequel. 7.1 We start with several definitions. Any point A for which there is a continuous inverse of A\ is said to belong to the resolvent set of A and the inverse Axl is called the resolvent of A and denoted by R(X, A). The resolvent set is denoted by p(A) and thus p{A) := {X £ Q R(X, R(X,A)e£(X)}. The resolvent set p(A) is an open subset of a complex plane and A —I R(X, A) is a holomorphic C(X)-valued function in each component of p(A). We note two properties of the resolvent of A which will be useful in the sequel. If B is a closable operator with D(A) C D(B), then BR(X,A) 6 C(X) for any A G p(A). If B G C(X) and p(A) / 0, then A commutes with B (see Subsection 6.11) if for some A G p(A) B commutes with R(X,A). It then follows that B commutes with R(X,A) for any A G p{A). The set a( A) := C\p(A) is called the spectrum of A. Some information on the location of the spectrum of A is provided by the spectral radius r(A) defined as follows

Singularly perturbed evolution equations

42

r(A) : = sup |A|.

(2.7.1)

For bounded operators we have HA) = lim

\\An\\l/n

and r(A) < \\A\\ which, in particular, implies that the resolvent set of a bounded operator is not empty. To distinguish different cases for which A\ is not invertible we subdivide a (A) into three disjoint sets: the point spectrum of A, crp(A), the continuous spectrum of A, ac(A) and the residual spectrum of A, aT(A) (see e.g. [89], p. 209). The point spectrum is of major interest to us so we shall discuss it in some detail. A complex number A0 belongs to the point spectrum of A if and only if the equation Ax = XQX has a non-trivial solution i 0 . Then A0 is called an eigenvalue of A and XQ is called the eigenvector. The null-space N(X0I - A) of A\0 is called the eigenspace of A corresponding to A0 and its dimension is called the geometric multiplicity of A0 and denoted by m9. 7.2 Let A € C(X) and let T(A) be the set of scalar functions / which are holomorphic in some neighbourhood fl of a (A) (possibly depending on / ) . If / e f(A), then we can define f{A) by the Cauchy type integral called the Dunford integral

f(A)~±-Jf(\)R(\,A)d\,

j \

(2.7.2)

r where T is a positively oriented rectifiable curve in £}\<7(J4). From the Cauchy integral theorem it follows that f(A) is independent of the choice of F in Eq. (2.7.2). The set T(A) is linear and it is also closed with respect to the superposition of functions. We have the following Spectral Mapping Theorem ([89], p. 227): if / <£ T(A), then

f(a(A)) = a(f(A)). If A is an unbounded operator, then its spectrum is, in general, unbounded and the construction (2.7.2) is not directly applicable. We shall discuss some extension of these results later in this section and in Section 3.3. 7.3 Sometimes it happens that a(A) contains an isolated point A0. Then A0 is an isolated singular point of the resolvent R{-,A) and the latter can be expanded into Laurent series

Chapter 2. Mathematical preliminaries

43

+00

R(X,A)=

J2 ( A - A o ) n A .

(2.7.3)

k=—00

with A n = (2™)-' /(A - Ao)-"_1fi(A,j4)dA, r where T is a positively oriented circle centered at A0 and of radius small enough not to enclose other points of a (A). Of particular importance is the operator P : = A_! = ( 2 ™ ) - 1 1 R(X,A)dX, r

(2.7.4)

called the spectral projection. It is indeed a projection and if we denote Xi = PX and X% — (I — P)X, then X\ and X2 reduce the operator A. Denoting by A\ and A2 the parts of A in Xx and X2, respectively, we have o(A{) = A0 and a{A2) = o{A) \ {A0}. The principal part of the series (2.7.3) may be infinite. It is finite (thus A0 is a pole of R(-,A)), however, if and only if Xi = PX is finite-dimensional ([47], p. 181). The dimension of X\ is called the algebraic multiplicity of A0 and denoted by ma. In this case A0 is an eigenvalue of A. Unfortunately, in general, the eigenspace N(XQI — A) is a proper subset of X\ and ing < ma. We have, however, the following result:

PX = N({X0I -

A)m),

Xj

=

X2

= (I - P)X = R((X0I -

A)m),

for any m > u, where v is the index of A0 defined as the order of the pole of R(-, A) at A0, and X = N((X0I - A)") © R((X0I - AY).

(2.7.5)

If ma = 1, then we call A0 a simple eigenvalue and then mg = ma. It is of great interest to determine when the eigenvalue is simple or, more generally, when m„ = mg, because then (and only then) v = 1 and the decomposition X = N(X0I -A)®

R{X0I - A)

(2.7.6)

is the spectral decomposition and spectral projections P and I — P are the projections along the range and the null-space of (X0I - -4), respectively.

44

Singularly perturbed evolution equations

It is straightforward to determine whether the eigenvalue is simple when A is a selfadjoint operator, since then the index of each eigenvalue is equal to 1 which implies ma = mg ([47], p. 273). Another case of practical interest, when v — 1, is treated in the formula (2.7.9). Let A be a densely defined operator so that A1 exists. Then p(A) and cr(A) are complex conjugates of p(A') and a (A1), respectively. Moreover, R(X,A') = R(X,A)' for A € p{A). If A0 £ a(A) is an isolated eigenvalue with finite algebraic multiplicity m a , then A0 is an isolated eigenvalue of A' with the same algebraic and geometric multiplicity (see [47], p. 184). We have the following results: AXox = / is solvable if and only if / € (N(X0I - A'})1,

(2.7.7)

and A'-Xog = y is solvable if and only if y 6 (N(X0I - A))-1

(2.7.8)

Note that in view of the Closed Range Theorem this result is equivalent to the ob servation that if A0 is an isolated eigenvalue of finite multiplicity, then A — X0I has a closed range. This allows us to give another simple condition for ma = mg, which will be useful later. It follows that N(X0I - A) JLN(X0I - A') implies ma = mg.

(2.7.9)

Indeed, let Ao have the index v > 1. Then for some x0 we have (X0I — A)"xa = 0 which implies (A 0 /-^)"- 1 2:o = y o e A r ( A o / - A ) . This requires j/o 6 (N(X0I — A'))1 from (2.7.7) which contradicts the assumption of (2.7.9). Thus v = 1 and ma = mg. Note that self-adjoint operators satisfy the assumption in (2.7.9). 7.4 For a large class of operators which are important in applications, such as com pact or power compact operators, we can give a much more complete description of the spectrum. We start with some definitions. We say that a linear operator A' acting in a Banach space A" is compact if for any bounded sequence (i n ) n e N of elements of X the sequence (Axn)n€N contains a Cauchy subsequence. The space of compact operators in A" will be denoted by K-(X). The

Chapter 2. Mathematical preliminaries

45

space K,(X) is a closed linear subspace of C(X) - hence the limit in C(X) of a sequence of compact operators is a compact operator. Any bounded finite rank operator (see subsection 6.4) is compact. Therefore, if an operator A is a limit in C(X) of finite rank operators, then A G K(X). If A G C(X) and K G K(X), then AK,KA,K' G K(X). It is said that A is a power compact operator if Ap G K.(X) for some p G N. The set of power compact operators will be denoted by VK.{X). If A G VfC(X), then a{A) is a countable set with no accumulation point different from zero. Any A G o{A) \ {0} is an eigenvalue of A of finite multiplicity. Therefore, for power compact operators, results in (2.7.7) and (2.7.8) can be improved. Indeed, any Ao / 0 is either in the resolvent set of A (and then the equation A\0x = f has a unique solution for every / G X) or it is an isolated eigenvalue of finite multiplicity, whence (2.7.7) holds. An analogous remark is true for the adjoint operator. This result is frequently referred to as the Fredholm theory in the Hilbert space setting (where, according to our convention, we use the adjoint operators) and the Riesz-Schauder theory in the Banach space setting ([90], pp. 330-344). When dealing with operators arising in the theory of differential equations, we cannot expect them to be bounded and compact. Very often, however, the resolvent of such an operator is compact, or power compact, at some Ao G p(A). Since the equation Axx = f is equivalent to ((A0 - A)" 1 / - R{X0, A))x = (Ao - A)-1i?(A0, A)f, we see that a(A) consists entirely of isolated eigenvalues of finite multiplicity with no finite accumulation points. An analogous theory can be developed for equations of the form \A + A' where A" G /C(-Y) and A is an isomorphism, since the equation {XA + K)x K)x = / is equivalent to {XI-A'1K)x

= A~1f

and A~lK £ JC(.\"). If A" is only power compact, then such a result is in general not true, as A~lK may not be a power compact operator. However, if for instance A and

Singularly perturbed evolution equations

46

A" commute, then clearly A ' A € VK.(X) and again the Riesz-Schauder theory is applicable. 7.5 Let A be a self-adjoint operator in a Hilbert space H. In this case the spectrum of A is real and any isolated point of spectrum is an eigenvalue. Let us denote by N\ the eigenspace corresponding to the eigenvalue A and by P\ the spectral projection onto N\. Each N\ is finite-dimensional and, as we mentioned earlier, geometric and algebraic multiplicities of A are equal so that (2.7.6) holds. Since for self-adjoint operators relations (2.7.7) and (2.7.8) are identical, we see that (2.7.6) is an orthogonal decomposition and therefore P\ coincides with the orthogonal projection onto N\. The spectral projections corresponding to different eigenvalues are orthogonal and consequently NxLN^ for A / \i. If, moreover, A 6 VIC(H), then a{A) \ {0} is a countable set of isolated eigenvalues. Zero is always a point of the spectrum and it is an eigenvalue if N0 := N(A) ^ {0}; dim No, however, may be infinite. If ei)// = 0 for I =fi k. Note, however, that we may have \k = A/ despite k ^ l\ this is due to the fact that some eigenspaces may have dimension greater than one and we may have different eigenvectors corresponding to the same eigenvalue. With this notation it follows that H can be represented as the following orthogonal sum (see e.g. [25], Vol. Ill, pp. 20-26): H = N0 © Lm{e n }„ e N .

(2.7.10)

For an arbitrary x £ H Eq. (2.7.10) can be rewritten in the form oo

x = x0 + ^2(x,en)Hen,

(2.7.11)

n=l

from which it follows that oo

Ax=Y^K{x,en)Hen.

(2.7.12)

n=l

Here we adopted the convention that if a0(A) is finite (which is possible) containing, say, N elements, then An = 0 for n > TV. If, however, A is injective, so that A^ = {0}

Chapter 2. Mathematical preliminaries

47

and H infinite dimensional, then {A n } n € N is infinite and { e n } n 6 N is an orthonormal basis in H. As in Subsection 7.4, we can obtain similar results if A is an unbounded operator with the compact resolvent R((M,A), n € p(A) ([25], Vol. Ill, pp. 38-40). We see that Xk € a(A) if and only if (/j, - Xk)~l € a{R(n, A) and the sets of eigenvectors coincide since zero is not an eigenvalue of R(^,A). Therefore, with the same convention referring to the arrangement of spectrum and eigenvalues, we see that the set of {|An|}nsN diverges to infinity and {en}ngm is an orthonormal basis in H. However, some care must be taken if we want to use the representation for A given in (2.7.12), as the series does not converge for all x G H due to the unboundedness of the sequence {An}nsN- However, it follows that (2.7.12) converges if and only if oo

Ux\\2 = j:\\\n\\2\(x,en)H\2
< 00,

(2.7.13)

n=l

and the domain of A can be characterized alternatively by D{A) = {x<EH; {Xn(x, eOtf}nSN 6 I2}.

(2.7.14)

This situation is frequently encountered when we work with variational problems described in Subsection 6.13. Let H be a pivot Hilbert space and Hi densely and compactly imbedded in H. We consider a continuous symmetric form on Hi x Hi (see (2.6.21)) which is /^-coercive with respect to H and the associated unbounded operator (A, D(A)) defined as in Eq. (2.6.22). Since the form is coercive, D(A) is densely and continuously imbedded in Hi, the resolvent set of A is non-empty, and A is self-adjoint. Let A 6 p{A), then R{X,A)H = D(A) ^-> Hi •-* H and, since the last imbedding is compact, R(X, A) € IC(H) and the whole theory applies. In addition {e„}neN is an orthogonal (but not orthonormal) basis in D(A) and Hx. If A is a positive compact operator, then o(A) consists of positive numbers converging to zero and it follows that the operator

Bx = Y,\/*n{x,en)Hen,

(2.7.15)

n=l

satisfies B2 = A, thus it can be denoted by A1/2- It follows that this is a unique self-adjoint and compact square root of A (that is operator B satisfying B2 = A). Other functions of A can be defined in the same way but we shall not dwell on that. Similarly, using Eq. (2.7.15) we can define the square root All2 for a positive operator with a compact resolvent. The resulting operator will again be an unbounded positive operator with a compact resolvent defined on

Singularly perturbed evolution equations

48

D{A1'2) := {x e H; {^K(x,

ek)H}nen

€ I2}.

(2.7.16)

Clearly, D(A) C £>(A1/2). In particular, for operators generated by coercive symmet ric forms we have D{A1'2) = Hi

(2.7.17)

(see Section 6.13). Another function of an unbounded operator which we frequently encounter is the exponential function, defined as oo

exp(-At)x

= £ e~A"'(z, en)Hen.

(2.7.18)

71 = 1

If a(A) is positive and t > 0, it can then be checked that exp(—At) defines a family of bounded operators with properties analogous to that of ordinary exponential function. We shall discuss a much more general definition of the exponential function of an unbounded operator in the next chapter. 7.6 To complete our discussion of spectral representation of operators we consider a generalization of the representation formula (2.7.12) to the case when A is an arbitrary self-adjoint operator. Then its spectrum is no longer a countable set of isolated eigenvalues but can contain continuous parts and the sum in (2.7.12) has to be replaced by an integral. To do that we first introduce the notion of a spectral family. Let H be a Hilbert space. We say that the family of orthogonal projectors {£A}A€R is a spectral family if for all A , / j £ l and x 6 H (a) E\Ell = E1, (b)

7 = min{A,^},

lim Exx = 0 and lim Exx = I, A—►—oo

A—>+oo

(c) lim Ex+ex = Ex i-t0+

Any power compact (or with compact resolvent) self-adjoint operator A generates a spectral family defined by EX-=

E

P*,

a{A)3\i <X

where Pxk is the orthogonal spectral projector onto eigenspace N\k.

Chapter 2. Mathematical preliminaries

49

An arbitrary self-adjoint operator generates a unique spectral family. Let x € H and / € C([a, b}). Then we can define the integral of / with respect to any such spectral family Ex as the limit in H of the Riemann sums *>

N

Jf(X)dExx

: = U r n £ / ( A w ) ( £ A ( i + 1 ) - Exlk))x,

(2.7.19)

where the limit is taken in X over all possible partitions of [a, b] of the form a = A(o) < A(i) < . . . < \{N) = b, X{k) e]A(fc),A(fc+1)] and S = maxJlA^+i) - A(fc)|}. The integral / f(X)dExx

can then be defined as the improper Riemann integral,

—oo

that is, as the limit of integrals of the form (2.7.19) as a —> - c o and b —> +00. Note that (2.7.19) can be equivalently expressed as

(f(A)x, y) = j f(X)d(Exx,

y),

x,yeH,

where, for any fixed x,y € H, the above integral is the Stieltjes integral. With this notation we can state the result known as the Spectral Theorem (see e.g. [25], Vol. Ill, p. 126, [79], p. 368). Let (A,D(A)) be a self-adjoint operator on a Hilbert space H; then there exists a unique spectral family {.EAIASR such that for any x € D(A) +00

Ax=

j XdExx.

(2.7.20)

— CO

The domain of A is characterized by +00

D(A) = {xeH;

J \X\2d{Exx,x)

< 00}.

—00

It follows (see [79], p. 368) that the integration in (2.7.20) is in fact performed over the spectrum of A so that Ax=

j

XdExx.

(2.7.21)

a(A)

This observation relates a number of properties of a self-adjoint operator .4 to the loca tion of its spectrum. First we note that {Ax,x)H > m\\x\\2H if and only if inf a (A) > m

Singularly perturbed evolution equations

50

and (Ax,x)H < M\\x\\% if and only if supa(A) < M. In particular, A is a dissipative operator if and only if a (A) C (-00,0] and A is bounded if and only if a(A) is bounded. If / is a measurable function on R (or rather on a {A)), we can define a function by the formula f(A)x = J f(X)dExx.

f(A)

(2.7.22)

The operator f(A) is a closed, densely defined operator with the domain D(f(A))

J l/(A)| 2 di \f(X)\2d(Exx,x)<^}.

= {xeH;

c(A)

Clearly, f(A) is self-adjoint if / is real-valued, and bounded if / is bounded. In particular, if A is a positive operator, then there is a unique self-adjoint square root of A defined by A1'2!

=

f Xl/2dExx.

(2.7.23)

"(A)

Again, as for bounded operators, if A is generated by coercive symmetric form (see Section 6.13), then D(A^2) = Hx (see e.g. [84], p. 29). 7.7 One more subset of spectrum turns out to be of importance in the spectral the ory. Let A € £(X) where X is a Banach space. We say that A is a Fredholm operator if both N(A) and X/R(A) are finite-dimensional. If the range of a closed operator is of finite codimension, then it is a closed subspace so that Fredholm operators are a special class of operators for which the Closed Graph Theorem holds. We call the Fredholm domain of A the set PF(A) of all A e C for which the operator XI — A is a Fredholm operator. The essential spectrum is defined as aess(A) := C \ pp{A). The Fredholm domain is an open set, hence aess(A) is closed. If A is an eigenvalue of A of finite multiplicity, then A e PF{A). Conversely, an element of an unbounded component of PF{A) either belongs to the resolvent set of A or is an eigenvalue of finite multiplicity. The essential spectrum of A is stable under compact perturbations ([29], p. 40), that is to say, if A 6 C(X) and K G IC(X), then ae3S(A) = aess(A + A').

(2.7.24)

Chapter 2. Mathematical preliminaries

51

7.8 Example. Let us consider a function / € C(M") satisfying m a x / = M and m i n / = m and the corresponding multiplication operator Mj : Lp(Rn,wdx) -$• Lp(Rn,wdx), where w is a weight function, (see Example 3.8), defined by (M}g)(x) := f(x)g(x) for almost all x G R" It is clear that p(A) C C\[m, M\ On the other hand, if A € [m, M], then /(z*) = A for some zA e R n and the inverse operator M (A _;)-i is not defined on the whole Lp(Rn,wdx), since (A - / ( z ) ) _ 1 is not essentially bounded (see [47], p. 143). Let g £ fi(M(A-/)), then any function of the form G(x) = g(x) for z in some neigh bourhood of / ~ ' ( { ^ } ) and G(x) = 0 elsewhere does not belong to R{M(X-D) either, and functions of this form span an infinite-dimensional subspace of LP(R"). Conse quently, aess(A) = [m, M\. m We conclude this section by discussing several examples relevant to the further de velopment of the theory. 7.9 Example Let H be a Hilbert space. We say that A € C-(H) is a Hilbert-Schmidt operator if there exists an orthonormal basis {en}n£N in H, such that oo

||/l||2:=£l|,4en||2
(2.7.25)

n=0

It follows that ||A|| 2 is independent of the choice of an orthonormal basis and satisfies the axioms of the norm which justifies calling ||A|| 2 the Hilbert-Schmidt norm of A. The adjoint operator A* is also a Hilbert-Schmidt operator and ||A*||2 = ||J4|| 2 . From (2.7.25) it follows that any Hilbert- Schmidt operator A is a limit of finite-rank operators of the form N

x -¥

^2{x,en)Aen, n=0

which are compact, and therefore Hilbert-Schmidt operators are compact (Section 7.4). If A is a self-adjoint operator, then A is a Hilbert-Schmidt operator if and only if o(A)\{0}

=

{Xn}n^ oo

where A„, n G N, are isolated eigenvalues and Y, \

< °°-

The most important result in the theory of Hilbert-Schmidt operators can be stated as follows (see e.g. [47], p. 262). Let H = L2(Q,wdx) be a weighted space (see Example

Singularly perturbed evolution equations

52

3.8) with the norm given by (2.3.11) where fl is an arbitrary measurable subset of R n Then K 6 £(L2(fl,wdx)) is a Hilbert-Schmidt operator if and only if there is a function k such that the function

(z,y)

\w(x) , ^\k(x,y)

G L2(fl x (l,dxdy)

(2.7.26)

and, for almost all x G fl, (Ku)(x) = j k(x,y)u(y)dy. n

(2.7.27)

Moreover, ||A'||2 = / \k(x, y)\2w(x)w(y)dxdy. n Let us consider an operator C : L2(R.n) —> L 2 (R") defined by (Cu)(x) = v{x)u{x) + j k(x, y)u(y)dy,

(2.7.28)

R»

where u € C(R") satisfies min v = m and maxv = M and fc satisfies (2.7.26). By Example 7.8 we have that a(M„) = aess{Ml/) = [m,M) and by the compactness of the integral operator aess(C) = [rn, M}. Hence any element of a{C) \ [m.,M] is an isolated eigenvalue of finite multiplicity. ■ 7.10 E x a m p l e Let us consider the integral operator defined in Eq. (2.7.28) but this time acting in the space L^(Q, wdx). This is a bounded operator provided / ess sup \k(x,y)\w~l(y)w(x)dx

< oo.

(2.7.29)

Note that according to (2.7.29) it is enough to carry out the analysis in Li(fi) with the understanding that k was replaced by km defined as kw{x,y) := k{x, k{x,y)w-l{x)w(y). We denote the corresponding operator by Kw. Unfortunately, unlike in other Lp spaces, the condition (2.7.29) does not ensure the compactness of Kw unless we impose additional assumption on k. The operator Kw is compact if, in addition to (2.7.29), we assume that

Chapter 2. Mathematical preliminaries

53

X->kw{x,-)€C(a,L00(n)).

: « ) ) ■

Indeed, for a given e > 0 we can always find by (2.7.29) a compact set fi< C Q with the property / ess sup \k(x,y)\w~1(y)w(x)dx n\n.

< t.

(2.7.30)

Let us define the operator Kw>( by the formula (Kwtu)(x) := {Kuw)(x)\nt. By (2.7.30) we see that limA^, = Kw in £(Li(fi)), hence Kw is compact provided all KWil are compact. We see that KW<1 6 /C(Li(fi), C(Qt)) because the kernel is a uniformly continuous function of x G Q ( . Therefore A*Wr, is also compact in Li(Qe) since the imbedding C(S7e) >-> Ai(fl£) is continuous. Next, the operator of extension by zero outside Ctt is a continuous operator from Ll(Q€) into Li(fi) which yields Kwl G /C(Z/i(fi)), hence Kw e /C(Li(f2)) and consequently A' £ )C(Li(Q),wdx). The assumption that fcTO is continuous is often too restrictive for applications. Fortu nately, it follows that (2.7.29) is enough to ensure that Kw is weakly compact, that is, if (/n)n6iN is & bounded sequence in Ai(f2), then the sequence (Kmfn)nefj contains a subsequence {Kwfnk)k
k(x,y)u(y)dy,

where v G (7(R n ). In L\ setting its analysis is slightly more complicated if k is not continuous because the integral part is no longer a compact operator. To obtain a counterpart of the results in L2 we proceed as follows. Let, as before, min v = m and max v = M, that is, a{Mu) = aess{Mv) = [m, M). Hence the operator XI - M„ is an isomorphism for A £ [m, M). If we fix such A, then the equation {XI-C)u

=f

(2.7.31)

is equivalent to (7 - (A/ - Mu)-lK)u

= (XI - Mv)-lj.

(2.7.32)

For arbitrary power compact operators this would not work since the composition of a power compact operator and continuous operator may not be a power compact

54

Singularly perturbed evolution equations

operator. However, for arbitrary g G L<x,(Rn) we have ((XI - M„) ^'g G L^W1) from < (XI - M„y1Kf,g>=<

Kf, ((XI - M„)-l)'g

and

>

we see that if (A'/n)nsN is a weakly fundamental sequence, then ((XI — Mv)'1 K f n ) n ^ is also a weakly fundamental sequence and consequently (XI-Mi/)'1K G VK,(Li(W)) along with K. We can apply the whole Riesz-Schauder theory to (2.7.32) to conclude that 1 is either in the resolvent set of (XI — M„)~1K or is an isolated eigenvalue of finite multiplicity (see section 7.4). Returning to the original equation (2.7.31) we see that if A ^ [m, M\ then either A € p(C) or A is an isolated eigenvalue of finite multiplicity for C, as in the case when K is compact. ■ Next we shall apply the results of the spectral theory described above to the so-called Sturm-Liouville problem which consists in finding a solution of the ordinary second order differential equation - dx(p{x)dxu(x))

+ q(x)u(x) = f(x)

in [a, 6] C R,

(2.7.33)

subject to the boundary conditions:

otiu(a) + f5xdxu(a) = 0, a2u(a) + fadxu(b) = 0,

(2.7.34)

where a;,/?,, i = 1,2 are constants. If both a and b are finite, p is strictly positive, q is bounded on [a,b] and |a,| + \P,\ > 0 for i = 1,2, then the problem (2.7.33)(2.7.34) is referred to as a nondegenerate Sturm-Liouville problem, otherwise we have a degenerate Sturm-Liouville problem. We shall discuss three examples of the latter which will appear later in applications. 7.11 E x a m p l e . In the first example we deal with the Legendre operator given by the following differential expression: (Lu)(x) := -dx [(1 - x2)dxu(x)]

, i€]-l,l[.

(2.7.35)

Here the coefficient k(x) = 1 - z 2 vanishes at the end points of the domain so that we have a degenerate problem. As we shall see below, there is no need to complement (2.7.35) by any boundary conditions as they will be incorporated into the definition of the space where the solution will be sought.

Chapter 2. Mathematical preliminaries

55

We introduce the space Hi := {u € L 2 ( [ - l , 1]); X -> x/l - x 2 a x u(x) e L 2 ( [ - l , 1])}

(2.7.36)

and a symmetric sesquilinear form l

h(u,v)

:= / [(1 - x2)dxu{x)dxv(x)

+ u{xjv(x)] dx.

(2.7.37)

-l

It follows that Hi with the scalar product (■, ■)#, := lt(-, ■) is a Hilbert space. Note that the sesquilinear form l

l(u,v) := f [(1 - x2)dxu(x)dxv(xj\

dx

(2.7.38)

-i

could not have been used for that purpose since it vanishes for constant functions and thus does not define a scalar product. Application of the Lax-Milgram theorem shows that for any given / 6 L2([— 1,1]) there exists a unique solution u e Hi of the variational problem I

h(u,v) = I f(x)v(x)dx,

for all « e Hi-

(2.7.39)

-i

Proceeding as in Section 6.13 we associate with /i the unbounded operator (Aj,, D(Atl)) so that (2.7.39) is equivalent to the operator equation Ahu

=f

(2.7.40)

and u € D(Ah). In fact, taking v € CJ°((] - 1,1[) we see that A^u = Lu + u where the differentiation in L is understood in the distributional sense. We can actually prove that

D(Ah) = {u e w}(] - i, ID; x -> (i - x2)u(x) e w2(] - i,i[)}.

(2.7.41)

Let us now turn to the spectral properties of Alx ([25], Vol. Ill, pp. 53-61). It. follows that the imbedding Hi ^-» H is compact, hence A^ is a self-adjoint operator with compact resolvent and thus it has purely point spectrum with no finite accumulation points. It can be found that a(Atl) = {A m } m e N , where Am = mirn + 1) + 1 and all eigenvalues are simple. The set {Pm}m(=N of its eigenfunctions is an orthogonal basis

Singularly perturbed evolution equations

56

in all three spaces D(Alx), Hi, and L2([-l, 1]). The eigenfunction Pm, corresponding to the eigenvalue Am, and normalized in L 2 ( [ - l , 1]), has the form

^H^)1^2-1^^0'1

(2 7 42

-- >

Functions Pm are called Legendre polynomials. We note that the Legendre polynomi als satisfy the following recurrence relation: xPm(x)

=

m, -f-1 Pm+1 (X) \J2m + A\J2m + 1

vn Pm-l(x). \J2m + IV2m — 1

(2.7.43)

The above results were obtained for the operator L +1 and not for L itself, since we were able to prove the coercivity of only the first one. However, if we denote by At the operator associated with I defined via (2.7.38), then clearly Ai = A^ — I and At coincides in L2{[— 1,1]) with L. We then have cr(Ai) = {Am - l } m 6 N = {m(m + l)} m6 N, where Am are eigenvalues of Atl. The spectrum of Ai is therefore located on the right half-axis which means, in particular, that —At is a dissipative operator by (2.7.21).■ 7.12 E x a m p l e . Let us consider the differential expression (Hu)(x) :=(-6g +x2)u(x), i £ l .

(2.7.44)

Here the degeneracy of the associated Sturm-Liouville problem is due to the unboundedness of the domain. As in the previous example there is no place here for any boundary conditions, they are replaced by the requirement that the solution of the problem belongs to L2(M.). To make the concept precise we start with a symmetric sesquilinear form on V x V defined for u, 4> e V by >:= h{u, <£) := J{dxu{x)dI(j)(x)

+ x2u(x)4>{x))dx

(2.7.45) (2.7.45;

and note that the form h is a scalar product on CJ°(K). We define Hi to be the completion of CJ°(R) with respect to the norm || ■ \\Hl := Jh(-,-) generated by h. This means that Hi is a Hilbert space and h is an i^-coercive form. It can be proved that Hi ^ W 2 W "">• -MR) a n d t n a t the imbedding Hi *-* L 2 (R) is compact. Let us define (Ah, D(Ah)) to be the operator associated with h according to the formula (2.6.22). As in the previous example, for u € D(Ah) the equality Ahu = Hu is valid in the distributional sense and consequently in L2(R)-

57

Chapter 2. Mathematical preliminaries

From the Lax-Milgram theorem and (2.6.16) and (2.6.17) we see that for every / € L 2 (K) there is a unique solution u G D(Ah) of the equation (Ahu)(x)

= (Hu){x) = f(x),

(2.7.46)

and this equality holds for almost all x 6 L 2 (R). Since h is symmetric and the imbedding Hi '-¥ H is compact ([25], Vol. Ill, pp. 64-68), Ah is a self-adjoint operator with compact resolvent so that it has a purely point spectrum {A m } m € N and the corresponding set of eigenfunctions {//m}m£N is an orthogonal basis in all D{Ah), Hx and L 2 (R). It can be found that Am = 2m + 1 for m G N, all eigenvalues are simple and the corresponding normalized in L 2 (R) eigenfunctions, called the Hermite functions, are given by the formula Hm{x) : =

( 1)m ~ ex2/2d?e-x2 yJ^K2mm\

(2.7.47)

Functions Hm obey the following recurrence formula: Vm+lHm+i{x)

- V2xHm(x)

+ yflHHm-i{x)

= 0-

(2.7.48)

We also need the relation D(A][2) = Hx

(2.7.49)

which stems from (2.7.17).

■

7.13 E x a m p l e . In the last example we discuss the Laguerre operator which is generated by the following differential expression:

(Cau)(x)

= -dx(xdx)u(x)+(-^^ /

V

Q + l + ^ + ^)u(x),

2

x€R+,

a>0.

(2.7.50)

Let H = L 2 (R + ). We associate with £ in a natural way a sesquilinear symmetric form on P(R+) x V(R+) by the formula

a{u,v)

:= j

xdxudxv(x)

/ q+ l — + - + — 1 u{x)v(x) dx. V 2

+ I

(2.7.51)

As in Example 7.11 this form is not a scalar product due to the presence of negative term - ( 1 + a ) / 2 so that we consider an augmented form

Singularly perturbed evolution equations

58

l'a(u,v) := la(u,v) + (1 + a){u,v)„.

(2.7.52)

Now we have to distinguish two cases: a = 0 and a / 0 which are substantially different since in the first case the term a 2 /Ax, unbounded at zero, is absent. We therefore define Hh0 := {u G V'(R+); x -> s/xu{x) € H, x ->■ ^/x~dxu{x) £ H)

(2.7.53)

and, for a / 0, #i,a ■= {« e P'(R+); 2 -» v^u(x) 6 W^(R+)}.

(2.7.54)

Since fl"i0 «->■ if, the norm oo

oo

oo

IMIs, o : = / Mz)| 2
0

(2.7.55)

0

is well-defined on U\fl and makes it a Hilbert space. To deal with the space / / l Q for a ■£ 0 we recall Hardy's lemma which states the following: if x —> dx{xu{x)) G L2(K+)i then u G L 2 (R+). This yields that if u G # l a , then x -*■ -^= € H,

ue H.

(2.7.56)

Therefore the norm given by

oo

114*.. := J\u(x)fdx 0 oo

oo

+ I \^x~u{x)\2dx + I 0

0

2

l

-^^-dx

oo

+ I \^x~dxii{x)\2dx

(2.7.57)

0

is well-defined on / / l Q with a / 0 and makes it a Hilbert space. In particular, we see that the imbeddings JT/IIQ «-► H are continuous for all a > 0. Let X^ and J4^ be the operators generated according to the (2.6.22) by [0 and 1^, respectively. Since H\a <-> H, their domains coincide. Now, from (2.7.55) and (2.7.57) we see that l'a is #1|C1-coercive for any a, therefore the operators A^ are self-adjoint and since they have compact resolvents ([25], Vol. Ill, pp. 69-81), the

Chapter 2. Mathematical preliminaries

59

spectral theory is applicable. It follows that o(A\^) = {m)m^ and all eigenvalues Xm = m are simple. The complete set of eigenfunctions is an orthogonal basis in L 2 (K+). Normalized in L 2 (K+) eigenfunctions of the Laguerre operator are called generalized Laguerre junctions and are given by a/2 -x/2Qa

*« (x) :=

I \

j = g - W J m ! r ( m + a + 1)

(2.7.58)

where Q ° ( i ) are Laguerre polynomials defined as Qam{x) = x-aexd?(e-xxm+a).

(2.7.59)

The Laguerre functions obey the following recurrence formula:

**no = -\^+i$r(o + (a+i)C'(a 0 ^ ( 0

=

-yfc

+ m + l)(m + l)^]+1(0

-^m(m + a)*£ii(0.

+ (2m + a + l)*L a ) (0

m > 0.

(2.7.60)

From the location of spectrum we immediately obtain that the operators — A^ dissipative.

are ■

A more detailed analysis of the operators discussed in the examples above can be found in Chapter 8.

Chapter 3 Semigroup theory 1

Introduction

In this chapter we describe a theory which enables us to solve an abstract Cauchy problem, that is, the problem of finding u which satisfies the following equation: dtu = Au,

t> 0,

(3.1.1)

together with the initial condition u(0)=u0,

(3.1.2)

where A is an operator acting in some Banach space A' and uo € X is given. If A reduces to the multiplication by a number a, Au := au, then the exponential function t —> u0eat provides a unique solution to (3.1.1), (3.1.2). More generally, if A is a bounded operator, then we can define the exponential function of tA by the formula oo

jnjn

exp(M):=£—— fc=o n-

(3.1.3)

and, as in the previous case, the function t —> exp(t.A)u0 (note that in the last ex pression we have the operator exp(L4) acting on element u 0 ) is a unique solution to (3.1.1), (3.1.2). Unfortunately, formula (3.1.3) cannot be used when A is an un bounded operator since it involves iterates A" and their common domain can shrink to {0}. Abstract Cauchy problems with unbounded operators are, however, of great importance, as they include all parabolic and hyperbolic differential equations; then 61

Singularly perturbed evolution equations

62

A is a differential operator with respect to spatial variables. In Sections 2.7.5 and 2.7.6 we saw how to construct the exponential function for self-adjoint operators and it can be checked that they provide a solution to (3.1.1), (3.1.2) but clearly the as sumption that A is self-adjoint turns out to be too restrictive for most applications. In this chapter we discuss how to construct the exponential function for a large class of operators. Most of the presented material is standard and can be found in text books on the semigroup theory, for instance, [31, 36, 69, 72, 84]. There are, however, a few results which, to our knowledge, are new. For these we provide detailed proofs.

2

Generation of semigroups

In this section we describe the construction of the exponential function for a large class of unbounded operators and discuss some of their properties. 2.1 Let X be a Banach space. A family (G(t))t>o of continuous linear operators in A is called a strongly continuous semigroup of operators in A", or shortly a C0-semigroup in A, if it satisfies the following three conditions: (a) G(t + s) = G{s)G{t) for all t, s > 0, (b) G(0) = /, (c) lim \\G(t)x - x\\x = 0. t-X)+

For any C0-semigroup (G(t))t>o there exist constants M and u> such that \\G(t)\\<Meut,

te[0,oo[.

(3.2.1)

If u> — 0 in the above formula, then the semigroup (G(t))t>o is called uniformly bounded and if w < 0, of the negative type. Conditions (a) and (c) and Eq. (3.2.1) imply that lim\\G(t)x-G(t0)x\\x t-n+

=Q

for all x 6 A and to > 0 which explains the name of a strongly continuous semigroup (see Subsection 2.5.4). 2.2 The linear operator (A,D(A))

defined by

Chapter 3. Semigroup theory

63

D{A):={xeX;

lim t~\G{t)x - x) exists}

(3.2.2)

and Ax := limt~l(G{t)x-x) for ieD(A),

(3.2.3)

is called the (infinitesimal) generator of the semigroup (G(£)),>0. If the semigroup generated by A satisfies the estimate (3.2.1) for some M and u>, then we say that A€g(M,w). In future we shall be dealing with various operators and the corresponding semigroups. To shorten the exposition we adopt the convention that {GA(t))t>o is the semigroup generated by A. However, when there is no possibility of confusion, we shall omit the subscript referring to the generator in the notation of the semigroup. If (G(t)) ( >o is a strongly continuous semigroup, then D(A) is dense in X, A is closed and for all x0 € D(A) we have G(t)x0 e D{A) for t > 0. Moreover, the function t —> G(t)x0 is differentiable for t > 0 and it satisfies on ]0, oo[ the equation dtG{t)x0 = AG{t)x0.

(3.2.4)

Taking into account the condition (b) of the definition of the semigroup we see that t —> x(t) := G(t)x0 is a solution of the Cauchy problem (3.1.1), (3.1.2) with the initial condition a:(0) = x0. Moreover, it follows that the semigroup is uniquely determined by its generator, thus G(-)x0 is a unique solution of the Cauchy problem (3.1.1), (3.1.2). Note, however, that we are restricted here to the initial values taken from the domain of A. If we replace (c) by the condition (c»)

lim \\G(t) - I\\c{X)

= 0,

(->0+

then such a semigroup is called a uniformly continuous semigroup. It is also a Cosemigroup, hence it satisfies (3.2.4). However, (G(t))t>o is a uniformly continuous semigroup if and only if its generator A G C(X). Thus uniformly continuous semi groups give solutions of Cauchy problems with bounded operators and, by the unique ness, they are completely described by exponential functions defined by (3.1.3). Since our main interest lies in solving Cauchy problems with unbounded operators, we shall focus our attention on Co-semigroups. At present we can solve the inverse problem: for a given semigroup, by Eq. (3.2.4), we can determine the Cauchy problem solved by the function t -> G(t)x0. Our main interest, however, is to find a solution to a given Cauchy problem. To this end, we must first determine the class of operators

Singularly perturbed evolution equations

G4

which can be generators of Co-semigroups. The answer to this problem is given by the famous Hille-Yoshida theorem (see e.g. (72), pp. 8-22) which asserts that A e Q(M,LJ) if and only if A is a closed, densely defined operator whose resolvent set satisfies p(A) D]w,oo[ and the resolvent operator R(-,A) for every n 6 N and A €]w, oo[ satisfies the estimate M ||fi(A,A)l<^-^-

(3.2.5)

A direct verification of (3.2.5) is in most cases very tedious, if not impossible, and therefore very often we resort to conditions which are less general but less complicated than (3.2.5). A significant simplification can be achieved if the resolvent of A satisfies ll#(A,A)|| < —!— f o r A > c j . A— w Then, clearly M = 1 and A e

(3.2.6)

Q(l,u).

The resolvent of A and the semigroup (G(t))t>o generated by A are related by the Laplace transform oo

R{\, A)x = f e-MG(t)xdt o

iorxeX,

A > w.

(3.2.7)

As we mentioned earlier, the semigroup generated by A plays a role of the exponential function exp(tyl) but there is no easy way to find it constructively. Neither the series approach nor the formula eai = lim (1 + ta/n)n, n-»oov

'

'

valid for scalar a, is applicable here, since they would involve iterates of an unbounded operator. The variation of the latter, however, makes the trick and the semigroup (G(t))t>o generated by A can be calculated according to the formula G(t)x = lim (7 - -A] 71

\~" x = lim "nn „ fn I

"-K»

A

)]" x,

for x G X,

(3.2.!

where the limit is uniform in t on any bounded interval in [0, oo[. 2.3 When the semigroup (G(i))t>o satisfies the estimate

I|G(0II<1,

i ;t>0,

(3.2.9)

Chapter 3. Semigroup theory

65

it is then called a contraction semigroup. The contraction semigroups are closely related to dissipative operators (see subsection 2.6.12). This is described by the Lummer-Phillips theorem which states that a linear, densely defined operator in X is a generator of a contraction semigroup in X if and only if it is m-dissipative ([72], p. 14). In a reflexive Banach space any m-dissipative operator is densely defined and the assumption of the density of the domain can be dropped. Let us assume now that A is a conservative operator. The conservative operators form a subset of dissipative operators so that the Lummer-Phillips theorem holds. A semigroup (G(t))t>0 with the conservative generator has, however, an additional property of being a semigroup of isometries: for any x0 € X and ( > 0 w e have l|G(r>o|| = ||z 0 ||.

(3-2.10)

Conversely, if (G(£)),>0 is a semigroup of isometries, then its generator is a conserva tive operator. Note that the property that A is the generator of a Co-semigroup in X is independent of the norm in X, as long as we take equivalent norms. The property of (G(£))(>0 being a contraction semigroup (or of A being a dissipative operator) depends, however, on the choice of the norm in X. We shall discuss now some aspects of generation of contraction semigroups which will be useful in the sequel. It follows that if A is a closed, densely defined and dissipative operator in X and A* is also dissipative, then A is m-dissipative, hence it generates a semigroup of contractions. This remark is especially noteworthy if X is a Hilbert space. Then from Eq. (2.6.8) we see that if A is dissipative and D(A) = D{A"), then A is m-dissipative. In particular, if A is self-adjoint and cr(A) c ] - oo,0[, then A is dissipative by (2.7.21). Since A = A*, it follows that A is m-dissipative. In many instances we will be dealing in this book with generators and semigroups depending on a parameter. For dissipative generators it is particularly easy to handle using the proposition stated below. Let us consider the space X := L r (ft, X), where 1 < r < oo, ft is a measurable subset of M" and X is a Banach space. In what follows we shall understand the notation p e ft as "for almost all p € ft" (see Subsection 2.3.8). Let us suppose that we are given a family of operators {(Sp, D(Sp))}pen in X. We define the operator (S, X>(5)) acting in .1' by the following formulae: V(S) := {u e X; u(p) 6 D(SP) forp 6 ft, Su <E X} and, for u €

(3.2.11)

V(S), (Su)(p) := Spu{p),

p € ft.

(3.2.12)

Singularly perturbed evolution equations

66 We have the following proposition:

Proposition 2.1 If Sp are m-dissipative operators in X for any p € fi, then the op erator S is an m-dissipative operator in X. If(Gp(t))t>0 and(Q(t))t>o are semigroups generated by Sp and S, respectively, then for every p 6 fi, t > 0 and u € X we have (g(t)u)(P) = Gp(t)u(p).

(3.2.13)

//, in particular, (Gp(t))t>o is, for p € Q, a semigroup of isometries, then (G(t))t>o also has this property. Proof. Since for p 6 Q the operator (SP,D(SP)) (2.6.11) that for u(p) G D{SP)

is dissipative in X, we have by Eq.

||(A/-5>(p)||x>AHp)||x,

A>0.

(3.2.14)

Let / € X = Lr(Q,X). Since for p e 0. we have /(p) 6 X, by m-dissipativity of Sp there is u(p) 6 D(SP) satisfying (XI - Sp)u(p) = f(p) and therefore u(p) = R(\,Sp)f(p). By (3.2.14) we get | K p ) | U = \\R(X,Sp)f(p)\\x

< A- 1 ||/(p)|| x .

(3.2.15)

If u is a function p —> w(p), then by integration we have IMU < A- 1 ||/||*. Hence u £ X. Consequently, XT - S is surjective onto X. Since by Eq. (2.6.11) S is dissipative in X, it is m-dissipative, hence it generates a semigroup of contractions, say (S(i))j> 0 , in X. Let 1Z(X,S) be the resolvent of S. From the preceding considerations it follows that for every / € X (TZ(X,S)f)(p)

= Rp(X,Sp)f(p).

(3.2.16)

By Eq. (3.2.8) we have, for an arbitrary u G X and t > 0, Q(t)u = lim

n„ in

,

n-Kx

Using the classical result from the L r -theory (see Example 2.3.8) we can extract a which converges in X almost everywhere in ft. On subsequence ( [ ^ ( ^ , 5 ) ] " ' )) which converges in X almost everywhere in fi. On

Chapter 3. Semigroup theory

67

the other hand, this subsequence converges in X to Gp(t)u(p) by (3.2.16), since Sp is the generator of (Gp(t))t>0. Therefore (3.2.14) holds. To complete the proof we note that if (Gp(t))t>o are semigroups of isometries, then (G{t))t>o has also this property by (3.2.14). ■ Remark 2.1 In some applications we shall have fi = Z n ; then the Lr space will be replaced by the space lT{X) of X-valued sequences (see example 2.3.9). Proposition 2.1 remains valid in such a case with essentially the same proof. ■ Let A be the generator of a semigroup (G(t))t>o in X and let Q be a projector onto a subspace W of X. We are very often interested as to whether the property of A being a generator is preserved when it is restricted to W Even if A is reduced by W the answer is not straightforward, as demonstrated, for instance, in [72], pp. 121-125. In many applications W is not an invariant subspace so that (A, D(A) n W) does not have a range in W In such a case it does not make sense to ask whether it generates a semigroup in W. However, we can consider the operator (QA, D(A) n W) where Q is a projection onto W This operator has its range in W but it is not clear whether XI — QAQ, where / denotes the identity operator in W, is invertible and surjective. Below we shall give a positive solution to this problem in a Hilbert space H if A is a dissipative operator and W is a subspace of finite codimension in H. Theorem 2.1 // (A,D(A)) is an m-dissipative operator in a Hilbert space H and W is a subspace of H of finite codimension, then the operator (QAQ,D(A) n W) is m-dissipative in W', Proof. Let us denote Aw = (QAQ, D(A) n W) and P = I - Q. First we observe that Aw is a dissipative operator. Indeed, for u G D(Aw) = D(A) f~\W we have Re [Awu,u)

= Re [{Au,u) - [PAu,u)\

= Re [Au,u) < 0

since u -L PH. We have to prove that Aw is m-dissipative in W. Denoting B = / - A we see that B is an isomorphism of D(B) = D{A) onto H. W is a closed subspace in H with codim W := n. Since the topology of the graph norm in D(A) is stronger than that induced from H, D(A)nW is closed in the graph norm and therefore the image B(D(A)nW) is closed in H, as B is an isomorphism. We shall prove that dim D{A)/{D{A)

n W) = n.

(3.2.17)

(is

Singularly perturbed evolution equations

Indeed, we have dim D{A)/(D(A) n l V ) < n. Otherwise, we would have n + 1 independent cosets of D(A)/(D(A) n IV), say [ e i | , . . . . | e n . , j , so that n*l

£ o * M = 0 if and only if ot = 0, k=l,...,n

+ l,

kml

which, in turn, implies that "•1

£ o * e j , e D(A)CiW

if and only if ot = 0. k = 1

n+I.

*=i

Bute* € D(A).k

= 1

n + l . s o we have

Y,nkPk G W if and only if ak = 0, * = 1

n+1.

*=i

That means <•» € D(-4), k = 1 , . . . , n + I are linearly independent modulo IV which contradicts the assumption dim Hf\V = n. On the other hand, let dim D{A)/{D{A) poser) as follows D(A) =

O IV) = * < n. Then D(A) can be decom-

(D(A)nW)®Lin{eil.^,ek),

where e i , - - - , « * € D{A). Since it is a direct sum, we have * £ o , c , e l V if and only if a* = 0,

fc«l1...,n

+ l,

•-I

thus wc can consider a direct sum H =

W®Lm{eu...,ek).

Let us suppose that H is not dense in H. Then there is A 9* 0 and h ± H, that is, ft satisfies *
for arbitrary v € IV and arbitrary scalar? o,, 1 = 1,. .. k. On the other hand, every element of D[A) can also be decomposed in such a way which implies that D(A) is

Chapter 3. Semigroup theory

09

not dense in H either. This contradiction shows that H is dense in H; consequently k > n and Eq. (3.2.17) is proved. Since H = B(D(A))

and A is an isomorphism, we get dim H/B(D{A)r\W)

=n.

Subspaces W and B(D(A)nW) are both of codimension n in H, so their intersection is of codimension at most 2n in H. Indeed, let e i , . . . , e „ and Si,..., §*, k < n, span the orthogonal complements to W and B(D[A) n W), respectively. The set £ = { e i , . . . , e n , e i , . . . ,e/t}, of dimension not exceeding 2n, is orthogonal to W D B ( D ( A ) r W ) . L * t e ± L m { e , } f = 1 © L m { e j } j e n , where j 6 n if e, / B(23(A)nW), In particular, e _L Lm{e,}* =1 , hence e € B(D(A) n W). Also e ± e., for every e, / S(D(A) n W) and if for some j k , eJk 1 B(D(A) n W), then e i t g Lm{e,}f =I , thus e 1 eJk and e e W Combining, we get e € B ( D ( J 4 ) n W) n W so that B = W n B(£>(A) n W) © Lin{e,}ti © L t n { e , } i e n , and dim H/(B{D{A)

nW)nW)<

< In,

Therefore, by (3.2.17) dimB{D{A) n W)/B((£>( A) n W) n W) = I < n, and also dimW/(B(D{A)nW)n\V) w)nw) = l.

(3.2.18)

We therefore have B(D{A) nW)

= [B(D(A)

nff) nW)nW}®Z,

where Z is the orthogonal complement to B(D(A) nW)n W in B(D(A) n IT) (with respect to the scalar product in H) with dim Z = I. Let Z = Lin{fu . . . , / ; } and assume that there is / € Z such that / 1 W ( / ^ 0). Let us consider ux and u 2 = uj + /• Both elements belong to B(D(A) n W), thus i^ = B i j , u2 = Bx 2 for some xi and i 2 in £ ) (A)nH / This shows that QBQxx = QBQx2, which is impossible, as QBQ = I - QAQ with dissipative QAQ. Thus neither / G Z is perpendicular to H^ Let us consider

Singularly perturbed evolution equations

70

QB{D(A) r\W) = [B{D(A) n W) n W] © QZ, where QZ = Lin{Qfu ■ ■ •, Q/i}, and assume that there is / € QZtlB(D(A)nW)nW. Then for some h 1 W we have g = f + h€ Z. However, as / € W, we get (3, })H = (/, / ) « # 0 (unless / = 0) which contradicts Z X A(D(A) CiW)nW So / must be equal to zero which in turn gives g = h £ Z, but this was ruled out by the above considerations. So, neither of / £ QZ belongs to B(D(A) n W) n W This implies that the set Lin{Qfi, ...,Qfi} is linearly independent modulo B(D(A) nW)nW. In particular we get dimQZ = I and by (3.2.18) W = [B(D(A) n W) n W] © QZ = QB(D(A) n W) which means that B = I—QAQ is surjective and consequently QJ4<5 is m - dissipative. As M7 is reflexive as a Hilbert space, D(QAQ) = D(A) n VV is dense in W and the application of the Lummer-Phillips theorem ends the proof. ■ 2.4 Let (G(t))(>o be a Co-semigroup on a Banach space X generated by A and let us consider the family of dual operators (G'(t))t>0. Prom the definition of the dual operator it follows that this family satisfies the semigroup property (a) so that it is natural to call it the dual semigroup of (G(t))t>o- The dual semigroup need not be a Co-semigroup, as the operation of taking dual does not necessarily preserve the strong convergence. If, however, X is a reflexive Banach space, then the dual semigroup is a Co-semigroup generated by A'. Otherwise, if X is not reflexive, then the domain of A' may not be dense in X' and consequently A' may not be the generator of a semigroup. In such a case we introduce the concept of the Phillips dual in the following way (see e.g. [31], p. 79). Let X* be the closure of D(A') in X1 The operator (A#,D(A*)) is called the Phillips dual of A if it is the part of A' in X*, that is D{A#) := {x e D(A') n X*; A1 6 X*}

(3.2.19)

A*x := Ax.

(3.2.20)

and for x e D(A*)

If A is the generator of (G(i)) t > 0 , then A* is the generator of a semigroup (G*(t))t>o in X* which is the restriction of (G'(t))t>o to X*. This result is important for the analysis of semigroups generated by differential op erators in L\. As we know, L[ — L^,, but L* is a certain subspace of continuous functions determined by the type of the differential operator and boundary conditions.

Chapter 3. Semigroup theory

71

2.5 As we mentioned earlier, the direct checking of the estimate in the Hille-Yoshida theorem is often not feasible. We have showed that for the particular case of con traction semigroups it is enough to check the estimate (3.2.5) for n = 1 only, that is for the resolvent itself. Here we shall discuss another important example when the availability of the Hille-Yoshida theorem follows from the estimates of the resolvent of A, but this time we shall need some additional assumptions concerning the properties of R(\, A) off the real line. Precisely, let the resolvent set of A satisfy

p{A) D Sz+S := {A 6 C; |argA| < - + <5}u{0} for some 0 < 5 < - ,

(3.2.21)

and let there exist C such that for every 0 / A G Sn/2+s the following estimate holds: \\R{\,A)\\<^-

(3.2.22)

Then ([72], pp. 30-32) A is the generator of a uniformly bounded semigroup (G(t))t>o (the constant M in (3.2.1) being not necessarily equal to C) and (G(i)) ( > 0 is given by (compare with (2.7.2)) G(t) = — [ extR(\,A)d\. 27TZ J r

(3.2.23)

Here T is an unbounded smooth curve in Sw/2+s traced in anticlockwise direction and coinciding with rays {A; A = te±,e,t > 0}, TT/2 < 0 < TT/2 + 6 as |A| -> oo. The integral in (3.2.23) is convergent in the operator topology uniformly with respect to t in [£o,°o[ where t0 > 0. Hence, in particular, t —> G(t) is continuous in the operator topology for t > 0 (it cannot be continuous for t > 0 unless A is a bounded operator). R e m a r k 2.2 The case when p(A) D u + S^^+s for some w > 0 and \\R(\, A)\\ < C/\\ - LJ\ for A € w + Sn/2+s can be reduced to the above one by introducing A := A — LU'I with u)' > w. ■ It turns out that the semigroup (G(t))t>o generated by A satisfying (3.2.21)-(3.2.22) is more than merely continuous. In fact, it is the restriction to the real line of a function z -> G (z) which is analytic in the sector S6 in the uniform operator topology. Moreover, G has the semigroup property in Si, that is, for any zltz2 € S6 we have G'{z\ + z2) = G'{zl)G'(z2). We shall use the same symbol (G(t))t>o for the semigroup and its analytic extension and call (G(t))t>o an analytic semigroup. If A is the generator of an analytic semigroup (G(i)) ( >o, then t —> G(t) has deriva tives of arbitrary order (in the uniform operator topology) on ]0, oo[ and G(t) €

Singularly perturbed evolution equations

72

C(X, D(An)) for any t > 0 and n g N, where £>(An) is equipped with the graph norm described in (2.6.1). Moreover, d?G(t) = AnG(t) and lim s u p r | | A r l G ( 0 | | < o o .

(3.2.24)

(-»0+

This shows, in particular, that t -> G(£)xo solves the Cauchy problem (3.1.1)-(3.1.2) for arbitrary initial data x0 £ X. This is a significant improvement upon the case of an ordinary C0-semigroup, for which x0 6 D(A) was required.

3

Fractional powers of closed operators

In Subsections 2.7.2, 2.7.5 and 2.7.6 we saw that for particular classes of operators it is possible to define functions (and, in particular, fractional powers) of operators. Here we shall describe how to define fractional powers of operators related to the generators of semigroups. Throughout this section we assume that B is a densely defined closed operator in X satisfying p(B) O £ := [ A e C; 7 < |ArgA| < TT} for some 7 > 0

(3.3.1)

and that there exists C such that for every O ^ A e S the following estimate holds: \\R(X,B)\\<^-

(3.3.2)

We have that 7 < 7r/2 if and only if A = —B is the generator of an analytic semigroup. This is the case of main interest to us. 3.1 First we discuss the definition of a fractional power of B under the additional assumption that 0 e p{B). Then, since the resolvent set of B is open, it is possible to find a path V running in p(B) from ooe~' s to ooetS, to < 9 < n, avoiding the negative real axis and the origin. Having done this, we define

B~a ~—^iJ A"a/?(A, B)d\,

(3.3.3)

r where the branch A"Q is taken to be positive for real positive values of A. It can be checked that for a = n e N the fractional power B~a defined by (3.3.3) coincides with the iterate B~n = (B~l)n

Chapter 3. Semigroup theory

73

It can be proved ([72], pp. 69-75) that, defined in such a way, B'a is a bounded one-to-one operator, hence it makes sense to define fractional powers (Ba,D(Ba)) of operators with positive exponents by the formula D{Ba) := R(B~a)

(3.3.4)

and Ba := (B-")-1

(3.3.5)

Additionally, we define B° := I. If a > 0, then Ba is an unbounded, closed and densely defined operator satisfying D{Ba) C D(Bp) for a > (5. Moreover, for arbitrary a,0 € 1 we have BaB<3x = Ba+/3x for every x e D(B1) where 7 = max{a, /?, a + P}. The interval ] — 00, 0] C p(Ba) and for 0 < a < 1 the resolvent of B" is given by the formula

R(X, Ba) = ^f Z7TI

J

r

~J^R{», B)dn, II — A

(3.3.6)

where T is the same contour as in (3.3.3). We can make this integral real by deforming the path of integration into the upper and lower sides of the negative real axis so that

Ja(X) := R(X, B«) = ^ V

'

V

7. 7T

' ^ t m

J t2a - 2Xta COS TO + A2

dtt

(3.3.7)

0

valid for all A > 0. Putting A = 0 we obtain an alternative formula for B~a oo

B-° =

sin tra si - ^[r°R(-t,B)dt IT JJ IT 00

(3.3.8)

Since .4 = -B is the generator of a semigroup (G(i)) ( > 0 we can use (3.2.7) and after some calculations we obtain the following formula relating fractional powers of -A and G: 00

(-A)-a = -LJr a G(t)dt,

(3.3.9)

Singularly perturbed evolution equations

74

where T(-) is the Euler T-function. 3.2 Let us turn our attention to the case when 0 0 p{B), that is, we assume only (3.3.1)-(3.3.2). Now we cannot define Ba by (3.3.5) since, in general, it is impossible to construct B~a It can be proved, however, that Ja(X), defined in Eq. (3.3.7), is also the resolvent of some closed operator when 0 0 p{B) ([84], pp. 39-43). It is therefore natural to define Ba as the operator having JQ(A) as its resolvent, that is B" := XI - J~l{\).

(3.3.10)

This definition is not immediately useful but it can be proved that D((B + SI)a) = D(Ba) for all 6 > 0 such that 0 g p(B + SI) and ||(B + &I)ax - B°x\\ < K5a\\x\\

(3.3.11)

for all x 6 D(Ba) and some constant K. This allows us to derive a number of useful properties of Ba In particular, all properties of B°, a > 0, mentioned above for the case 0 € p(B), are also valid if this assumption is not satisfied. We shall now state some other properties of fractional powers assuming only (3.3.1), (3.3.2) and 0 < a < 1, unless specifically stated otherwise. The presentation here follows [72], pp. 73-75. The domain of B is the core of Ba and for x € D(A) we have sin iviy r

Bax = —

7T

/ taR{-t,

J 0

B)Bxdt.

(3.3.12)

There exists a constant M > 0 such that for every x € D(B) and s > 0, we have ||BQ:r|| < M (sa\\x\\ + a—^l&rll)

(3.3.13)

||B a :r|| < 2M||z|| 1 - a ||Bx|| Q

(3.3.14)

and

These two inequalities are commonly referred to as the moment inequalities. Let C be a closed operator satisfying D(C) o D(Ba) for some 0 < a < 1, then there is a constant K such that for all s > 0 and x € D(B) we have \\Cx\\
+ s°-l\\Bx\\).

(3.3.15)

Chapter 3. Semigroup theory

75

Conversely, if C is a closed operator satisfying D(C) D D{B) and for some 7 satisfying 0 < 7 < 1 and every s > So > 0 we have \\Cx\\-1\\Bx\i)

Bill)

for x e D(B), then D{Ba) c D{C).

(3.3.16)

If A = - B is the infinitesimal generator of an analytic, semigroup (G(t))t>0 in X, then G(t) : X -> D(Ba) and G{t)Bax = BaG{t)x for every r. > 0, a > 0 and~z e D(Ba). For every £ > 0 the operator BaG(t) is bounded and \\BaG(t)\\ < Mata

(3.3.17)

for some constant Ma. If, moreover, 0 G p(-B), then there is S > 0 such that ||S Q G(i)|| < Mat-ae~u

(3.3.18)

For arbitrary t > 0, 0 < a < 1 and x 6 D(B°) we have ||G(0* - x|| < Cata\\Bax\\

(3.3.19)

for some constant Ca. 3.3 If X is a Hilbert space and B is a positive self-adjoint operator, then we have two definitions of the fractional power of B: by Eq. (2.7.22) and that given in this section. It follows that if Ba is defined either by (3.3.5) or (3.3.10), then 00

Bax=j\adExx

(3.3.20)

0

for x e D(Ba), where {E>,}xeu i s the spectral family generated by B. Thus the two definitions coincide ([84], p. 44). We note the following important result, called the Heinz-Kato theorem ([84], p. 44), which is useful in characterizing domains of fractional powers of operators. Let X\ and X2 be two Hilbert spaces, -A and —B be m-dissipative operators in A'i and A'2 respectively (so that fractional powers of A and B are defined), and T € C{XX, X2)If T maps D(A) into D(B) and there is M > 0 such that for all x e D(A) we have IIBTzll < Af|[Ax[|. Then for each a € [0,1] we have

Singularly perturbed evolution equations

76

TD(Aa)

C D{Ba).

(3.3.21)

As an immediate application of this theorem with X\ = X2 and T = I we see that if for m-dissipative operators -A and -B we have D(A) = D(B) then D(Aa) = D(Ba) for all a€ [0,1]. 3.4 The domains D(Aa), a > 0, equipped with the graph norms become Banach spaces (or Hilbert spaces when X is a Hilbert space). We call them fractional order spaces and denote by Xa with the convention that XQ := X. The norm in Xa will be denoted by || ■ ||Q. We have Xa^t X for any a > 0. We shall not go into details of the theory of fractional order spaces which is closely related to the interpolation theory (see e.g., [5], pp. 113-115, [51], Ch. 1, [24], pp. 38-57). We shall need, however, a particular result which can be stated as follows. Let {Xa}aiu and {Yp}pe\& be two families of fractional order Hilbert spaces and let T € C(Xao,YPo) n C{Xai,YPl) for some a 0 , A>,ai,/?i- If a2 = a 0 (l - 9) + axd and P2 = /50(1 -6) + ^6 for some 0 < 6 < 1, then T € C(Xa2,Y02).

4

(3.3.22)

Perturbation theorems

We are frequently faced with equations of the form dtu = Au + Bu,

(3.4.1)

where A is known to be the generator of some semigroup (G(t))t>o and B is an operator with a simpler structure. In this section we consider the question: under which conditions to be satisfied by B will the suitably defined sum A + B generate a semigroup and how this semigroup is related to (G(i)) ( > 0 . 4.1 One of the most widely used results is the Bounded Perturbation Theorem ([72], pp. 76-78). It can be stated as follows: if A,B are operators in a Banach space A' satisfying (A,D(A)) € Q(M,u) (see Subsection 2.2) and B 6 C(X), then (A + B,D(A)) £ g(M,u> + M\\B\\). Moreover, if (G(i)) ( > 0 and (T(t)) t > 0 are semigroups generated by A and A + B respectively, then (T(i))<>o is the unique solution of the Volterra equation t

T(t) = G{t) + J' G{t- s)BT(s)ds, o

(3.4.2)

Chapter 3. Semigroup theory

77

and as such it can be obtained in the form of the series

r(t) = £r n (t),

(3.4.3)

n=0

where the sequence (T n ) n 6 N can be determined iteratively by T0(t) := G(t) and t

Tn{t)=

JG(t-s)BTn.1{s)ds. ii

4.2 Unfortunately, in many applications the perturbing operator B is not bounded. In general, we have to impose on B the condition that it is "weaker" than A in the sense that D(A) C D{B). If B is closable, then this condition implies that B is A-bounded, i.e. for some a > 0 and b > 0 we have (see (2.6.2)) | | B i j | x < a | | A c | U + 6|NI. xeD(A).

(3.4.4)

Even then the sum A + B may not be a generator and we have to impose some additional conditions. We note two results of this kind ([72], pp. 80-84). Let A generate a holomorphic semigroup and B be a closed operator. There is 5 > 0 such that if B satisfies (3.4.4) with some a € [0,(5], then (A + B,D(A)) generates a holomorphic semigroup. This result is not immediately useful since the constant <5 is not given explicitly. It follows, however, that the assumptions of this theorem are satisfied if B is a bounded perturbation and also if for some a G]0,1[ we have D((-A)a)

C D(B).

(3.4.5)

Therefore in both these cases (A + B, D(A)) is the generator of a holomorphic semi group. The constant <5 can be effectively determined if we are concerned with dissipative operators. The following theorem holds: if A is an m-dissipative operator, B is dissipative with D(B) D D(A) and (3.4.4) holds with a e [0,1[ and b > 0, then (/I + B, D(A)) is also m-dissipative. When a — 1, then the statement is, in general, false but the following weaker version holds. If the assumptions of the previous

Singularly perturbed evolution equations

78

statement are satisfied with a = 1 and B' is densely defined, then (A + B,D(A)) is m-dissipative. In particular, if X is a reflexive Banach space, then the dual of a closable, densely defined operator is always densely defined and in this case the assumption on B' is automatically satisfied. 4.3 In the two preceding subsections we were concerned with a perturbing operator B which was weaker, in one sense or another, than A. Here we shall discuss a more symmetric case when both operators enter (3.4.1) on equal rights. The main technical result in this subsection is Chernoff's product formula ([36], p. 50) which can be written in the following way. Let {V(t)}t>0 be a family of contractions on X with V(0) = I (not necessarily a semigroup). Suppose the right-hand derivative d+Vx/dt\t=0 of V exists for all x from some set D and that (d+V/dt\t-0, D) generates a semigroup of contractions (G(f))(>0. Then for each x € X, lim V n-+oo

m

\x = C G{t)x,

(3.4.6)

and the limit is uniform in t on compact subsets of K + . If A generates a contraction semigroup, then Chernoff's formula with V(t) = (I tA)~l yields the exponential formula (3.2.8). We shall often use the following version of Chernoff's formula known as the Trotter k

k

J=\

}=1

product formula: suppose that A,, j = 1 , . . . ,k, and ( £ AJt fl D(A3)) generate on X semigroups of contractions (G 7 (t)) t > 0 and (G(i)) ( > 0 , respectively. Then for each X£X,

lim n—*oo

(o

G)

■■■
c = G(t)x,

(3.4.7)

and the limit is uniform for t in compact subsets of R + . Ik

k

\

The jY,] A,, The Trotter Trotter formula formula can can be be strengthen strengthen by by not not requiring requiring that that A,, f|f| D(A,) D(AA ) V=i J=I generates a semigroup but then one must adoot another assuirmtion which is sufficient/ generates a semigroup but then one must adopt another assumption which is sufficient for that. For example, if the image of I f] D[A3) 1 under the operator I XI — T, A,

'

is dense in X, then the assumptions of the Trotter formula are satisfied. This follows from the fact that the sum of dissipative operators is dissipative and the closure of a it it dissipative operator has a closed range. T h e range of (XI — £ A,, |~l D(Aj)) i=i

k

k

j=i

j=i

is thus

j=i

both closed and dense and ( £ A,, f] D{Aj)) must be m-dissipative ([72], pp. 92-93).

Chapter 3. Semigroup theory

79

4.4 Results of the type discussed above are not very convenient since in addition to some assumptions on components one has to assume certain properties of the sum of operators as a whole, and that is usually rather difficult to verify. The properties of the components determine fully the behaviour of the sum only in special cases, one of which we consider below ([69], pp. 23-24). Let X\ and X2 be two Banach spaces and X = Xi®X2, where ® denotes any tensor product discussed in Section 2.3.11. If A generates in Xx a C 0 -semigroup (G(t))i>o, then A®IX2 generates a C0-semigroup (Q(t))t>0 in X and (£(t))i>o = (G(t))e>0<8>/xv Let us now suppose that (G(t)) ( > 0 and (T(t))i>o are C0-semigroups generated in Xx and X2 by A and 23, respectively. Then (G(t) ® T(r.))<>0 is a C 0 -semigroup on X generated by (A ® IX2 + IXl ® B, D(A) ® D{B)). 4.5 Let us note then that in a number of previous results the domain of the generator was not determined explicitly but only in terms of its core. Given a semigroup (G(t))t>o it is very often quite difficult to determine the domain of its generator A in a constructive way. On the other hand, it is important to know which subspaces of D(A) determine A, and consequently (G(i)) ( > 0 , in a unique way. Certainly, any core of A is such a subspace. Moreover, it follows that if D0 C D(A) is not a core of A then there exist an infinite number of different extensions of A\Do which are generators. Thus the cores are the only subspaces of D(A) which determine A (see [69], p. 47). We note a simple condition for a subspace D0 to be a core. If D0 is a dense subspace of X satisfying D0 C D(A) and G(t)D0 C DQ for all t > 0, then D0 is a core of A. 4.6 Now we formulate and prove a result which partly addresses problems discussed in the last three subsections. Let X be a Banach space. We denote by Q, either a measurable subset of R" or the set of integer n-tuples Z" and define

X :■■

f Lr(Q,X) \ lr{X)

if if

ncRn, Q = Zn,

where 1 < r < oo. In what follows the phrase "for almost all p G Q" is to be understood as "for all p G n except possibly for some p in a set of the Lebesgue measure zero" in the first case and "for all except possibly a finite number p 6 Z° " in the second case and the notation "for p G Q." means "for almost all p G Q" We consider k families of m-dissipative operators {S,,P}P6«, j = 1, ...,fc, denoting the corresponding semigroups by {Ghp(t))t>0 and we assume that for p G Q

(Tp, D(TP)) := \X St, Q D(sJ\

(3.4.8) (3.4.8)

Singularly perturbed evolution equations

so

is the generator of a semigroup of contractions (Gp(t))t>0 in X. If Ap denotes one of the operators Tp or SjiP, then we define its extension A to X in the following way: D{A) := {x 6 X; x{p) e D{AP) for p 6 Q, p -4 Apa;(p) 6 * }

(3.4.9)

and for i e D ( i ) (^b)(p) := A p x(p).

(3.4.10)

Thus T and 5, will denote the extensions of the corresponding operators in the sense of the above definitions. We have the following result: Theorem 4.1 The operator {T,D(T)) tions in X, say (G{t))t>o, given by

is the generator of a semigroup of contrac

(G(t)x)(p) = Gp(t)x(p)

(3.4.11)

for every x € X. If there exist a set

D0 C fl H D(Sk,p)

(3.4.12)

penj=i

which, for almost all p € fi, is a core for Tp, then this semigroup is given by the Trotter formula

^w^JJa^G)-*©)"*- xex

(3 413)

-

where for j = l,...,fe the operators Q}, are extensions of GJP defined as in Eq. (3.4-11). Proof. The first statement of the theorem and (3.4.11) follows directly from Propo sition 2.1. Before we start proving (3.4.13) let us note the following general observation. For any operator (A,D(A)) acting in X, if A(D(A)) is dense in X, then the image A(D) of any core D of A is also dense in X. Indeed, let for some ft e I ' we have < Ax, h > = 0 for all x 6 D. Taking arbitrary y G A(D(A)) we have / = lim Axn for some xn € D, n—>oo

n = ! , ■ • - , and

Chapter 3. Semigroup theory

81

< / , h >=< lim Axn, h > = lim < Ax„, h >= 0. n—*oo

Since A(D{A))

n~+oo

is dense in X, we obtain from this h = 0. k

k

3=1

]= \

Let us now consider the operator ( Y, S,, f] £)(<5,)). As a sum of dissipative operators, it is itself dissipative. Let v £ X' = Lq(Q, X'), 1/p 4- 1/q = 1, be such that t

< ( I - £ S, ))z, V >*,<*,= 0 J= I it

for arbitrary a; € fl D(Sj).

Let, for the time being, S7 C R n We consider arbitrary

J=I

h £ D0 and a scalar function G Co°(Q). By (3.4.12) and (3.4.9) we have tf>h e D D{S3) and k

< {I -iZS3))4>h,v

k

>XxX,= I' 4>[p) < (I -t,SJ,p)h>v(p)

i=i

i

>***' dp.

7=1

By density of C0°°(fi) in Lp{fl) we get

*: <(I-Y,S),p)h>V(P)>XxX'=°XxX'—

0

J= l

for arbitrary h € D0 and p € £1 But the image of D0 under I — £ 5 J i P is dense 3=1

in X for any p e fi since D 0 is a core of Tp and Tp is the generator of a dissipative semigroup. Therefore u(p) = 0 for p 6 fi and consequently n = 0 in LP(Q, A'). If fi = Z", then according to our assumptions, there may exist a finite subset Zg C Zn such that Do is not a core for the operators Tp with p 6 ZJ. Hence, we cannot deal with this difficulty as in the case £2 = R" since the equality in /P(A") means equality everywhere. To overcome this difficulty we define a sequence h(p) by the formula

hip)

■ g(p) for h for !

peZ£ p^ZJ,

where g(p) £ D(TP) and h € D0 are arbitrary. The remaining part is analogous to the case X = LP(Q,X) with an obvious replacement of the space CJ° by the space of sequences which have only a finite number of elements different from zero. Therefore we see that in both cases the closure

Singularly perturbed evolution equations

82

k

D

I k

\\

\j=l

j]

Ik Ik

&'- (0M)=1

\ }1 = 1

kk

\

1=1

/

£$, ,n^(^) \j=l j=l ') \ ,= i

generates a semigroup of contractions given by the formula (3.4.13) (see Subsection 4.3). To prove that this semigroup is equal to the semigroup generated by T we con-

71

k

V

sider x G D\ £ S, . Then i = lim xn where z n G 0 D(S,) for n = 1 , . . . with n_vo V=i / ° J=1 *: lim (V) S,)xn = f in X, hence there is a subsequence such that for p G n we have J n - » O j=- j lim x(p) xnk(p) and lim ( E SJ,p)xnk(p)

= f{p) in X. But then x n t (p) 6 £>(T„)

x{p) = lim xnk{p) and lim ( £ S 3if ,)x nt (p) = /(p) in X. But then x„ 4 (p) G £>(7;) so that x(p) G D(TP) and Tvx(p) = f{p), since Tp is closed on D(TP). Consequently, x G D(T) and £ 5, C T Since T is dissipative, it is a dissipative extension of £ 5 ; j=i

j=i

and because the latter, as a generator, is a maximal dissipative operator, we have

T=ZSr J=I

Formula (3.4.13) follows directly from the equality above and also from the Trotter formula. ■

5

Asymptotic behaviour of solutions

One of the most important questions in the theory of evolution equations is to de termine the behaviour of the solution to (3.1.1), (3.1.2) for large values of t. We know that for each semigroup (G(r.))t>0 there are constants u>' and M^ such that the relation \\G(t)\\ < M „ , e u ' ' ( 6 [0,oo[ holds (see Eq. (3.2.1)). The number u> := M{ui' G R; ||G(i)|| < M^e"' 1 for all i > 0 and some M ^ }

(3.5.1) (3.5.1)

is called the growth bound of the semigroup (G(t))t>a and it can be shown that

Chapter 3. Semigroup theory

83

w= l i m r l l o g | | G ( * ) | | .

(3.5.2)

t—¥QO

From the definition of w it is clear that the large-time behaviour of the solution is determined by the value of w, that is, if w < 0, then the zero solution is (exponentially) stable. In what follows we shall concentrate on describing the relations between the growth bound w, the spectral radius of (G(t)) t > 0 defined as r(G(t)) := sup{|A|; A 6 a(G(t))},

(3.5.3)

and the spectral bound of the generator A of (G(i)) ( > 0 given as s(A) := sup{Re\;

A 6 a {A)}.

(3.5.4)

We will, in particular, seek conditions to be satisfied by the generator of a semigroup which ensure ui < 0. 5.1 If A is a finite dimensional linear operator so that (3.1.1) is a system of ordinary linear differential equations, then the solution to the above-mentioned problem is given by the famous result by Lyapunov. It states that if the real parts of eigenvalues of A are all negative (that is in our terminology s(A) < 0), then the zero solution is (exponentially) stable. It follows that this result holds true if A is an arbitrary bounded operator ([69], pp. 60-61). Then the uniformly continuous semigroup (G(i)) ( > 0 generated by A is the exponential function of tA which can be defined also by the Dunford integral, hence the Spectral Mapping Theorem (see Subsection 2.7.2) gives a(G{t)) = eia{A)

(3.5.5)

From (3.5.2) and the expression of the spectral radius (Subsection 2.7.1) it follows that eut = r(G(t)) and, consequently, LU = S(A).

(3.5.6)

Unfortunately, Eq. (3.5.6) fails to hold in the general case and there are a number of constructive examples of semigroups with negative spectral bound of the generator and positive growth bound of the corresponding semigroup. This is mainly due to the fact that the Spectral Mapping Theorem is not available for C0-semigroups and it follows that there is no one-to-one correspondence between the spectra of A and

Singularly perturbed evolution equations

34

(G(t))t>o- We shall not enter, however, into details of the spectral theory of semi groups but discuss only selected results which are relevant to the developed theory. 5.2 For arbitrary semigroups we have a partial result which can be called the Spec tral Inclusion Theorem ([69], pp. 84-85), stating that expta{A)Ca(G{t)), ')),

(3.5.7)

with an additional result for the point spectra exptap(A)=ap(G{t))\{0}.) \ { 0 } -

(3.5.8)

Note that Eq. (3.5.7) implies s(A)
w

(3.5.9)

-

If, however, we impose some restrictions on the generator, then the Spectral Map ping Theorem is valid. This happens if (G(t))t>o is an eventually norm continuous semigroup, i.e. for some t0 > 0 t —> G(t) is a norm continuous function on ]f.0,oo[. Uniformly bounded semigroups are clearly eventually norm continuous. A nontrivial example of eventually norm continuous semigroup is offered by a holomorphic semi group which is eventually norm continuous with t0 = 0. If A is the generator of an eventually norm continuous semigroup (G(t))t>o then expta(A)

=a(G(t))\{0} {0}

(3.5.10)

and consequently we have u = s(A).

(3.5.11)

5.3 In the previous subsection we discussed some ways to determine the growth bound of the semigroup from appropriate properties of its generator. Here we show how some information about the spectrum of the semigroup can be used for this purpose. Let us recall (Subsection 2.7.7) that aess(T) is defined as the set of all A 6 a (A) for which XI — T is not a Fredholm operator. If T is a bounded operator, then we can define the essential spectral radius of T as r „ . ( T ) :=sup{|A|; A g

aess{T)}.

(3.5.12)

Chapter 3. Semigroup theory

85

If (G(t))t>o is a semigroup generated by A, then its essential growth bound, ujess, defined as e"*"' := ress{G(t)),

t >0

satisfies we33 < ui and the equality occurs only when ress(G(t)) = r(G(t)) for t > 0. Now, if ujesa < ui, then there is an eigenvalue A 6 a{G(t)) satisfying |A| = r(G(t)) and by (3.5.8) there exists A! e ap(A) with ReXi = u ([69], p. 74). Hence s(A) > u which, by (3.5.9), implies s{A)=u>.

(3.5.13)

In other words, for arbitrary semigroup (G(t))t>0 with generator A we have LO = max{oj ess , s(A)}.

(3.5.14)

An important concept, which is closely related to the problems discussed presently, is that of the quasi-compact semigroup ([69], pp. 214-218). We say that a semigroup (G(t))t>o on a Banach space X is quasi-compact if limdist(G(f),/C(X)) = 0, t-*oo

where )C(X) is the space of compact operators and the distance is calculated in the norm of C{X). Any semigroup (G(£)),>0 with negative growth bound is quasi-compact since it approaches the zero operator. The following statements are equivalent: (a) (G(t.))t>o is quasi-compact. (b) u)ess < 0. (c) For some t0 > 0 and K 6 K(X) we have \\T(t0) - K\\ < 1. From (b) and (3.5.14) it follows that if (G(t))t>o is a quasi-compact semigroup and s{A) < 0, then u> < 0. An important class of quasi-compact semigroups can be obtained by perturbations, since if (G(t))t>o is a quasi-compact semigroup generated by A and K is a compact operator, then the semigroup (S(t)) t >o generated by A + K is also quasi-compact. More generally, if (G(t))t>o is an arbitrary semigroup which is generated by A, and (S(t))t>o is a semigroup generated by A + K, where K G IC(X), then the essential growth bounds of {G(t))t>o and (5(r)) ( > 0 are equal.

Singularly perturbed evolution equations

86

5.4 Finally, we give a result which relates the growth bound of a semigroup to yet another subset of spectrum. For a closed and densely defined operator A on & Banach space X we define the approximate spectrum of A, aa(A), as the set of all A e C such that either XI — A is not injective or (XI — A)(D{A)) is not closed in X. The name follows from the fact that aa{A) can be equivalently characterized as the set of all A G C for which there exists a sequence (x„)tlgN> called an approximate eigenvector, with the properties: xn £ D(A), \\xn\\ = 1 for n e N, and lim ||Ai n - Xxn\\ = 0. Let now (G(t)) t > 0 be a semigroup generated by A. The following statements are equivalent ([69], p. 108): (a) u) < 0; (b) s(A) < 0 and for some t0 > 0, sup{|A|; A e aa(G(t0))}

< 1;

(c) for every (or some) p > 1 and every x € X oo

f \\G(t)x\\pdt < +oo. 0

6

Inhomogeneous Cauchy problem

In this section we consider the inhomogeneous Cauchy problem

dtu(t) = u(0) =

Au(t) + f(t), u0,

0 < 0
(3.6.1)

for some T e]0,oo], where / is a function defined on [0,T[. Throughout this section we always assume that A is a generator of a Co-semigroup (G(t))t>0 on a Banach space X so that the corresponding homogeneous (/ = 0) initial value problem has a unique solution for every initial value u 0 G D(A). 6.1 We say that a function u is a classical solution to the problem (3.6.1) if it satisfies the following conditions: (a) w e C ° ( [ 0 , T [ , X ) n C 1 ( ] 0 , T [ , X ) , (b) u{t) €D(A)

for*€]0,T[,

(c) Eqs. (3.6.1) are satisfied.

Chapter 3. Semigroup theory

87

If u is a classical solution to (3.6.1) and / € £-i([0,T],X), then it is given by the formula

u(t) = G(t)u0+

I G(to

s)f(s)ds,

0
(3.6.2)

Consequently, if / G Lx([0,T],X), then for every u0 € X the inhomogeneous Cauchy problem (3.6.1) has at most one solution and if a solution exists, it is given by (3.6.2). The function u, defined by Eq. (3.6.2), is continuous but not necessarily differentiable and therefore it is not always a solution to (3.6.1). We call u the mild solution of the Cauchy problem (3.6.1) on [0,T], though the word "solution" is a little misleading here as the mild solution may not actually solve (3.6.1). The most important question is to identify assumptions which should be imposed on / and/or A so that (3.6.1) has a classical solution. The continuity of / is, in general, not sufficient. 6.2 We start with the general result ([72], pp. 107-108). Let / e Li([0,T],X) C°Q0,T],X) and let the function given by

n

t

v(t) := I G(to

s)f{s)ds

(3.6.3)

satisfy one of the following conditions:

(a) u eC 1 {]0,r[ 1 A'), (b) v € veC°(}0,T[,D(A)). Then the Cauchy problem (3.6.1) has a classical solution on [0, T[ for every uQ e D(A). Conversely, if (3.6.1) has a classical solution for some u0 6 D(A), then both (a) and (b) are satisfied. The direct use of this result is limited by the fact that the conditions involve not only / but also the semigroup itself. We note, however, that the above assumptions are satisfied if either / £ Cl{[0,T],X) or / 6 L,([0,T], D[A)) n C°(]0, T[, X). 6.3 The continuity assumptions imposed on / in the previous subsection are very often too restrictive and not easy to check. Therefore we shall introduce another type of solution called the strong solution adopting the following definition (see [72], pp. 109): we say that u is a strong solution to (3.6.1) if

88

Singularly perturbed evolution equations

(a) u € ueW}([0,T\,X), (b) u(t) E D(A) for almost all t E}Q,T[, (c) u(0) = u 0 and dtu(t) = Au(t) + f(t) almost everywhere on [0, T[. Note, that from (a) it follows that u is absolutely continuous and therefore the first part of (c) makes sense. Clearly, the strong solution is weaker than the classical one, but it is sufficient in the sense that, contrary to the mild solution, the strong solution can be inserted into the equation and satisfies the equation almost everywhere. For strong solutions we have counterparts of results of Subsection 6.2. Let / € Li([0,T], X) and let v be defined by (3.6.3) and satisfy one of the following conditions: (a) vE€ W,W}{[0,T\,X), (b) v e vELx([0,T\,D{A)). Then the Cauchy problem (3.6.1) has a strong solution on [0, T] for every uQ € D(A). Conversely, if (3.6.1) has a strong solution for some UQ E D(A), then both (a) and (b) are satisfied. Again, this result is used mainly through the following corollary: if either / E W{{[0,T],X) or / € Li{[0,T\,D(A)), then the problem (3.6.1) has a strong solu tion for every x E D(A). Note, that if X is a reflexive Banach space, then it is enough to assume that / is Lipschitz continuous on [0,T], 6.4 If we assume that the semigroup (G(t))t>o is holomorphic, then the requirements of the previous subsection can be substantially weakened ([72], pp. 110-115). First we recall that the homogeneous Cauchy problem has a solution for any u0 E X (see Subsection 2.5). We start, as before, discussion of the inhomogeneous case with a general technical condition. Let A be the generator of a holomorphic semigroup (G(t))t>0 and / € Li([0,T],X). If for every t G]0, T[ there is St > 0 and a function Wt E C°([0, oo[) such that

l1/(*)-/(*)ll<W t (|i-*|) and <5<

1W {r) /

0

t

W t { T )

T

Chapter 3. Semigroup theory

89

then (3.6.1) has a classical solution for every u0 € X. This result is mostly used through the corollary which we state below, but before that we must recall the concept of the Holder continuous function. Let I C R be an interval. We say that / : I —>■ X is Holder continuous on I if there are constants L > 0 and -d e]0,1] such that for any t, s € I we have \\f(t)-f(s)\\
(3.6.4)

If i? = 1, then we say that / is Lipschitz continuous. If every t € / has a neighbour hood in which / is Holder continuous, then we say that / is locally Holder continuous in I. In this case both L and i5 may depend on t. If I is compact, however, then each locally Holder continuous function is Holder continuous. Now we are ready to state the announced corollary. Under the assumptions of this subsection, if / £ £i([0, T],X) is locally Holder continuous on ]0,T[, then (3.6.1) has a classical solution for every u0 £ X. As in the previous subsection we can derive another set of conditions for the solv ability of the problem, related this time to the range of / . Let us consider the space Xa := D((-A)a) equipped with a suitable graph norm (Subsection 3.4). With this a standard result can be formulated as follows. If / 6 C(]0,T], X) and the function t —> \\f(t)\\xa is bounded on [0,T] for some a > 0, then for every u 0 € X there is a classical solution of (3.6.1). For later applications we shall need, however, a stronger result which we will prove below. Theorem 6.1 Let A be the generator a holomorphic semigroup (G(t))t>o and f € C°(]0,T],X). Let us assume that for some a > 0 the function s -¥ q(s) := ||/(s)||x„ £ Li([0,T]) and is bounded over compact subsets of]0,T]. Then for every initial value ua € X the Cauchy problem (3.6.1) has a unique classical solution. Proof. Since (G(t))t>0 that

is an analytic semigroup, it suffices to show (Subsection 6.3)

t

F(t) =

AJG{t-s)f(s)ds o

is a continuous function for t > 0. Using analyticity of (G(tf))t>o we can move A under the integral sign and obtain for a specified in the formulation of the theorem t

\\JAG(t-s)f(s)ds\\x(s)ds\\x o

Singularly perturbed evolution equations

90

< J \\(-A)l-°G{t J\\(-A)l-aG(t-S)(-A)af(S)\\xdS o <

a Kj\t-srK I \t-s\ ~' 1\\f(a)\\x.d8

o

j)\t-sriq(s)ds

= U+ <

K[{t/2)a-lQ{t/2)

+ Q-1

\

max q(s)(t/2)a) t/2<s
,

(3.6.5)

I

where Q is the primitive of q satisfying Q(0) = 0. This proves that F(t) is well-defined for t > 0. To show that F is continuous we proceed as follows. We fix t > 0 and consider arbitrary h > 0. Then we have

t+h

F{t + h)-F[t)

=

f AG(t +

h-s)f(s)ds

t t

+1QAQ o

(G(t + h-s)-G(t-s))Kt-s)) f(s)ds =h + h-

We estimate 7j and I2 separately. Since for a fixed t the function ||(—A) a /(s)||x is bounded for s £ [t,t + h] we have as in Eq. (3.6.5)

\\h\\x
I \t + h- s M K - A r / M l l i / d s < ^ha J a t

To estimate 72 we first note that for t > 0 | ! ( _ A ) i - G ( t + ft) - (-A)l-°G(t)\\cm

<

K

2ta~\

which follows from Eq. (3.3.18). Then we have ||(-4)i-«G(* + ft) - ( - A ) 1 - ° G ( t ) | | £ ( x ) < which is obtained from the following identity:

(-A)l-"G(t

+ h)x -

{-A)l-aG(t)x

K3he~2,

(3.6.6)

Chapter 3. Semigroup theory

91 t+h

= (-A)1-a

J dtG{s)xds

=

{-A)2-aG{s)xds.

J

t

t

Combining the two inequalities we obtain that I K - A ) 1 - ^ * + h) - (-Ay-aG(t)\\ciX) where fi(h, t) = ta~l min{l, h/t).

<

Ktn(h,t),

Thus we have t

\\h\\ < K4 J(t - s)a-ln{h, o

t-

s)q(s)ds

and we proceed by splitting the integral into two parts as follows. Let h < t/2, then */2

a_2

J(t-s)a-ln{h,t-s)q(s)ds
< h {-)

t/2

Jq(s)ds
(3.6.7)

and

f(t-s)a-1iJ,(h,t-s)q{s)ds s)q(s)ds t/2 t/2

< max q(s) j Ta-ln{h,T)dT ~s~ o

oo

I h

< KJJTa-ldT \o < K6ha

+

\

hfTa-2dT h J (3.6.8)

Combining (3.6.6)-(3.6.8) we obtain the continuity of F from the right. However, for semigroups this also yields the continuity from the left, hence F is continuous and (3.6.1) is classically solvable. ■ If we are interested only in strong solvability of (3.6.1), then the assumption of the previous theorem can be substantially weakened. We have the following theorem: Theorem 6.2 Let A be the generator a holomorphic semigroup (G(t))t>o and f 6 Li([0,T], X0) for some a > 0. Then for every initial value u0 G X the Cauchy problem (3.6.1) has a unique strong solution. Proof. We have to show (Subsection 6.3) that

Singularly perturbed evolution equations

92

z

F(t) := Av(t) = A I G(t-

s)f(s)ds

is integrable over [0,T]. Let us consider (/ n ) n£ N such that fn G C°([0,T], Xa) for n G N and lim / „ = / in Li([0,T],Xa). Since the norm of Xa is stronger than that 71—J-OO

of X we see from

| K ( t ) - V(t)\\x = \\JG(t

- S)(}n(s) - f(s))dS\\X

< KX\\fn -

f\\Ll(l0,T] X)

that lim vn(t) = v(t) for every t G [0,T1. Since fn are also X-continuous functions, n—i-oo

the results of the previous theorem apply so that Fn = Avn G C°(]0,T[,X) can interchange A and the integral sign getting

and we

F,;(r) = Avn{i) = J AG(t -

(3.6.!

s)fn(s)ds.

Now let us consider the sequence (F n ) n £ N in Li([0,T],X). obtain

T II

For some rn,n € N we

(

/ r jr AG(t - s)(fn(s) - fm(s))d

d£ x

T

(

< J J\\(-Ay-°G(t-s)(-A)°(fn(s)-fm( a))b SjlllA" dsdr. 0

0 T

: #.

t

J /i*-*rMi/«w-/m(a I

T

T

I

dsdt

0

ll/»(*)-/m(*)|U./ t - s ^ d r .>) ds

0

< a-^A-ll/. - /m||il{lwl,x.) <

^"sll/n - /m|Ui([0,r],X„),

(3.6.10)

so that (F n )„ 6 N is a Cauchy sequence in Li([0,T],X). Hence there is F' G Li([0,T],X) which satisfies lim F = F' in Li(fO.Tl, A'). By the classical result in the measure theory there is a subsequence of (F71)n£fj which is convergent almost everywhere to F', that is lim Avn (t) = F'(t) for almost all t G [0,T1. Since A is closed, we obtain k—too

Chapter 3. Semigroup theory

93

F'(t) = Av(t) = F(t) almost everywhere which yields F € Li([0,T], X).

7

u

Applications to partial differential equations

In this section we shall discuss several types of operators which generate semigroups and which are relevant to the applications discussed in the latter parts of the book. 7.1 Example. We consider the evolution equation in Lx (Mn) related to the following partial differential equation: n

(dtu)(t,x)

= ^2a,(x)(dx,u)(t,x)

+ ao{x)u(t,i),

((.ijel+xf

(3.7.1)

i=i

where a,, i = 0,.,.,n, condition

are bounded Lipschitz continuous functions, with the initial

u(0,a;) = u0(x),

x £ R"

(3.7.2)

We shall study this equation as an abstract evolution equation in the Banach space Li(K n ). Let us denote by A an unbounded operator defined on the domain n

D{A) := {u e Li(R n ); £ > A , u G M » n ) }

(3-7.3)

i=i

by n

(Au)(x) :=1£/at(x)(dx,u)(x)

+ a0(x}u(x),

u 6 D{A).

(3.7.4)

t=i

The partial differential equation (3.7.1) can now be written as the following abstract Cauchy problem in Li(R n ):

d,u u(0)

= Au, = u0.

t > 0, (3.7.5)

This case is relatively easy to deal with and it is possible to give an almost explicit form of the semigroup {G(t))t>0 generated by A (see e.g. [4]). We start with some

Singularly perturbed evolution equations

94

preliminaries. Let us consider the following Cauchy problem for the system of ordinary differential equations for y = (j/i, ■ ■ ■ ,yn) G R™ with an independent variable s G R: dsVn = an(y),

dsVi = a^y),

yi\s=t

=

xu

yn\s=t

-

(3.7.e

Xn,

where t G R and x := (xu... ,xn) € R" are arbitrary. It follows (see [42], Chapter V ) that if a,, i = 1 , . . . ,n, are Lipschitz continuous, then the problem (3.7.6) has a unique solution for every (£, x) G R" +1 denoted by s->-#(s,£,x) = (<j>i(s,t,x),...,n(s,t,x)),

s G R.

(3.7.7)

Moreover, if the functions a, G C m (R") for i = 1 , . . . , n, then $ G C m ( R n + 2 ) . With this notation we have the following result: if a,-, i = 0 , . . . , n, are bounded and Lipschitz continuous functions on R", then the linear operator A defined by (3.7.3)(3.7.4) is the infinitesimal generator of a strongly continuous semigroup (G(t))t>0 given by

(G(t)uQ)(x) = exp

/ a 0 ($(s,0,x))ds

u0($(t,0,x))

(3.7.8)

for u0 G L1(Rn) and t > 0. Moreover, for t > 0 we have: \\G(t)\\ <

supa {x)t e

°

7.2 The remaining part of this section will be devoted to the discussion of solvability of the diffusion-type Cauchy problems

(dtu){t,x)

=

J2 a *. (a%j(x)dXju{t,x)^J +J2a*{x)dx,u(t,x)

u(0,x)

=

u0,

+

a0(x)u{t,x), (3.7.9)

where t > 0 and x G ft C R" The problem (3.7.9) should be supplemented by some boundary conditions defined on the boundary of ft. In our applications, however, we will be concerned with two special cases:

Chapter 3. Semigroup theory

95

(a) no boundary conditions due to ft = R n , (b) ft = [0,2TT]" and periodic boundary conditions (see Subsection 2.4.8). A great advantage of both (a) and (b) is that in all variants of the formula of in tegration by parts, the boundary integral is absent. This essentially simplifies all considerations, as we shall see below. Many results given in the literature, however, deal with the case of bounded domain ft and various boundary conditions different from the periodic ones. The corresponding results for the free-space and periodic cases can be obtained along similar lines with appropriate modifications. We shall not go into details of the general theory but rather outline the main ideas and point out how the results pertaining to the cases (a) and (b) can be derived. We shall follow here the presentation of Fattorini, Chapter 4 of [31] unless stated otherwise. According to our general philosophy we shall convert (3.7.9) into an abstract Cauchy problem for an ordinary differential equation in some Banach space X which will be one of Lp spaces or spaces of continuous functions on ft = R n or ft = [0, 27r]n We start with some necessary definitions and notations. Let us define the domains D0 : = C^(R n ) (see Subsection 2.2.2) and Dn := C£([0, 2w]B) (see Subsection 2.4.8) and the operators (An,Do), (A*, D„) for u € Da by n

(Aau)(x)

n

:= J^ dXi (aij{x)dX}u{x))

+'£at(x)dXtu(x)

+ aB(x)ii(x),

(3.7.10)

.=1

«,J=1

where we adopt the convention that a is either 0 or n. To simplify the presentation we will not use any indices with ft, adopting the convention that ft := R" in the free-space case and ft := [0,2^]" in the periodic case. We assume that the coefficients of the operator defined by Eq. (3.7.10) are real and for », j — 1 , . . . , n they satisfy a%j,a, e Cjt(fi), Qo G L^Q.) in the periodic case and aij,a,i € C ^ H ) , a0 £ L^Q.) in the free-space case. Without any loss of generality we can let ai3 = aJt. A crucial assumption is that Aa is strongly_elliptic in fi, that is, for some constant A" > 0 and all £ = ( f t , . . . , £„) € R" and x € ft we have 71

E

2

h(xMi K'\tf<\

a

(3-7.11)

i,j=l

An important role is played by the formal adjoint to Aa defined on the domain D*a := Daby

(A»(s):= £ i,j=l

aXj ( a ^ i j ^ ^ x ) ) - E 9 x , («,(^)«(i)) + i o ( i ) « ( i ) . '=1

(3.7.12)

Singularly perturbed evolution equations

90

It is customary to define the formal adjoint by the integral identity /' {Aau)(x)v{x)dx a

= J u{x){Aa*v){x)dx a

(3.7.13)

valid for u, v e C0°°(n). We can easily extend Eq. (3.7.13) to hold for u,v e Da. For a = 0 this is true as C£°(Rn) is dense in C$(R n ) and for a = IT, integrating the left-hand side of Eq. (3.7.13) by parts, we can show that the boundary terms vanish due to the periodic nature of all involved functions and coefficients. 7.3 The operators (Aa, Da) are dissipative under relatively mild conditions. Through out this section we shall always assume that (3.7.11) holds. With this agreement we have the following results: if u„ := ess sup a0(x)

1

"S^d^a^x)

I < 0,

(3.7.14)

V

then (Aa,Da) is dissipative in LP(Q) for a = 0,7r and p € [l,oo]. The proof of this result for a = 0 is standard and for a = n is reduced to the standard one upon noticing that the suitable integrals over the boundary of Q vanish due to the periodicity of involved functions. For the limiting case p = oo in Eq. (3.7.14) the value of 1/p is understood to be zero. This is also the dissipativity condition in C s (0). Precisely, if esssupa 0 (:r) < 0,

(3.7.15)

then (Aa, Da) is dissipative in C7(M") for a = 0 and in C„.([0, 2n]n) for a = IT. Again, the result for a = 0 follows in the standard way and for a — TT goes along similar lines since CV([0, 27r]n) can be thought of as a space of functions defined on a torus where there is no boundary creating difficulties when dealing with C(H) for a general case. With obvious modification of Eq. (3.7.14), the operator (A*, Da) is dissipative in Lp spaces; clearly in the spaces of continuous functions no modification of the condition (3.7.15) is necessary. Unfortunately, the operators (Aa) Da) are not closed. Being dissipative they are, however, closable. Our next task is to discuss the properties of their closures. In particular, we are interested in whether they are m-dissipative and whether they generate holomorphic semigroups. This will be done in two parts. First we deal with the L2 theory which is relatively easy in both free-space and periodic cases. In the second part we shall be mainly concerned with the Li theory for the periodic case. We discuss, however, some aspects of it for the free-space case.

Chapter 3. Semigroup theory

97

7.4 Example. The L2 theory makes little use of the results of the previous subsec tion since it is more convenient to apply the variational theory described in Subsection 2.6.13. Our pivot space here will be H := L 2 (fi) and the variational spaces of Sub section 2.6.13 will be defined as Hlfl := Wj(fi) and #i | 7 r = W^Q) in the free-space and periodic cases, respectively. The sesquilinear form corresponding to —AQ is

aa(u, v) := /

Y^ a,ij{x)dXlu{x)dX]v(x)

- ^2a,(x)dXju(x)v(x)

-

a0(x)u{x)v(x) dx. (3.7.16)

Let us denote by -Aa formula

£ C(Hiia, H[a) the operator generated by aa according to the

< -Aau,v

> W ; o x// 1 , a = aa(u,v),

u,v e HUa,

and by ( — A2i0, D2ia) the corresponding variational operator unbounded in L2(Q) and defined on DXa

:= {u € Hha; AQu 6 L2(Q)}

(3.7.17)

by the formula A2,au:=Aau,

uGD2ia.

(3.7.18)

If Aa is strongly elliptic, then aa is i/i, a -coercive with respect to L2(Q) (see Eq. (2.6.21)) and therefore (XI - A2,a)D2,a

- L2{Q).

(3.7.19)

Moreover, there exists u such that Re((A2xa - w/)«,w)i,(n) < 0 so that A2,a € 6(1,w) (see Subsection 2.2). Under the same assumptions A2,Q generates a holomorphic semigroup (see e.g [80]). Again the proof for a = 0 can be found in the literature and for a = ir is similar, due to the absence of boundary integrals. To answer the question as to whether the dissipativity conditions (3.7.14) yield the dissipativity of A2iCI we need to show that Aa = A2%a (see Subsection 2.6.12). This is

Singularly perturbed evolution equations

98

again done not directly, but by more precise characterization of the domain D 2 Q. It follows that £>2,o = W22(Rn),

(3.7.20)

which can be proved by the difference quotient method in the same way as the interior regularity of solutions to general boundary value problems. Also in the periodic case we have D2,T, = WI„{[Q,2*Y)

(3.7.21)

and the proof is similar as in the free-space case. In both cases, if coefficients of Aa are smooth enough, then we have the following Shift Theorem: for / G W2*(RB) (resp. / € W£n([0, 2n]n)) the solution u to the problem ^2,QU

=

/,

where a = 0 or a = n, respectively, satisfies u € W 2 t+2 (R n ) (resp. u € W^2{[0, 2ir]n)). From the embedding theorems it follows that, in particular, if / e C£°([0, 27r]n), then u has the same property. With these results we see that the W% and graph norms are equivalent on Z?2,Q and since Cg(K") (resp. C 2 ([0, 27r]")) is dense in W*(Rn) (resp. W?itr([0,27r]n)), it is also dense in Z)2,o (resp. D2i1T)- Hence Aa = A2]a for a = 0,7r and J 4 2 Q is dissipative provided Aa has this property. Moreover M,au = Aau for u e D2,Q and A2,a = (A*a)'

(3.7.22)

where A* is the formal adjoint (3.7.12) and the asterisk denotes the L2 adjoint. This follows from the fact that the adjoint form generates the operator which is the unique extension by the density of A*a. m In the next example we shall discuss the generation of a semigroup by a suitable extension of An in LP(Q) and, particularly, in Li(Q). The reason of omitting the free-space case is that the Lp theory for bounded domains relies heavily upon the L2 theory through the inclusions

Lpl(Q) 4 L2(fi) 4 LP(Q), p' < 2, p > 2,

Chapter 3. Semigroup theory

99

which fail if 0 is unbounded. Some results concerning Li(R n ) case, which is of some importance to us, will be discussed at the end of this section. 7.5 E x a m p l e . Since the proof of that the closure of A„ in Li(Q.) generates a semigroup in Lx is considerably more involved than that in L2 we shall start stating the main result. Let us assume that An is strongly elliptic and that the coefficients ahJ G C*+](Q), a, € C*(fi) for k > n/2. We denote Ai,„ := A~^ where the closure is taken in L^fl). Then A1|7r G ^ ( l , ^ ) (see Eq. 3.7.14). The domain of A ]|lr can be characterized as follows: L>i,„ = {u G Li(fi); A,u G Li(fi)}

(3.7.23)

where again A,, is understood in the distributional sense. Moreover, Dlt. C

0

^,»(«)

(3-7.24)

9
and there is A > 0 such that for every q < n/(n - 1) there is C(q) such that N K v ( f i ) < C(q)\\(\I

- Alt7,)u\\Lm.

(3.7.25)

The proof relies upon results in other Lp spaces and will be carried out in several steps. Firstly, let us consider the operator (A^,D„) with Dw C LP(Q). Then for any p G [l,oo[ the closure Apa := Aa in LP(Q) satisfies Apa G Q(1,UJP) and for p = oo the closure A^^ := AQ in LX(Q) (or, which is equivalent here, in C„(fi)) generates a semigroup in CV(Q) (but not in L^Q^)) and satisfies A ^ ^ G £ ( 1 , ^ ) . The main points of the proof of this statement are as follows. For p e [1, 2[ we have L2(Q) ^ i p ( ^ ) a n ( i since A^ does not depend on the underlying space we have from Example 7.4

(XI - An)Dn 4 L2(Q) 4 LP(Q) for large enough A. Thus the image of D„ through the closure XI - An is also dense in Lp(fi) and, since this closure is dissipative, its range is closed. This shows that XI - A* is m-dissipative. For p > 2 we take / G C£°(f2), then by the Shift Theorem the solution u to the equation (XI - A2,Q)w = / is also a C°°-function and thus belongs to D„ C D2,K. We then have (AT" - Aa)Dn D C™(0) and the latter is dense in Lp(fi). Hence the remaining part of the proof follows as before. The case p = oo is

Singularly perturbed evolution equations

100

dealt with in the similar manner but we observe that D„ C CJ(O), hence the image (XI - Aa)D„ C CV(fi). Since on C»(Q) the norm induced from LX(Q) is equivalent to the supremum norm, the closure of (XI - Aa) will be an m-dissipative operator in CV(n) and not in L^Q). Secondly, as in the L2 case, we can prove that for p G ]1, oo[ we have Ap,a = Ta = (Aa*)' = ( 1 7 ) ' ,

(3.7.26)

where ^ a * is understood to act in Lq(ftn), l / p + l/q = 1- Moreover, for those p we have D(Ap,a) = W£„(n). (3.7.27) Again, thanks to the periodic structure of the problem, the proof of (3.7.27) follows the lines of the proof of the interior regularity of the solution to a standard boundary value problem. Unfortunately, the preceding result does not cover the case p = 1 which is of the most interest to us. This is partly due to the fact that Li(f2) is not a reflexive space and hence we cannot expect that the duals of operators will generate Co-semigroups in the dual spaces; in fact, we saw that the closure of the operator An in Loo(^) generates a semigroup in C„(Q) and not in L0O(Q). This problem can be handled, however, using the theory of the Phillips dual described in Subsection 2.4 and the characterization of Z \ „ is obtained via the corresponding result for .Doo,*- Under all assumptions introduced in this subsection we have the following result:

D(A00>n) = | u 6 0 KM

A u€ C

*

*(Q) \

( 3 - 7 - 28 )

The proof is based on the theory of Phillips dual applied to the space X = Li(Q) and the operator A\ n = /L* which, as we know from the above result, generates a Co-semigroup. The Phillips dual of X is defined as X* = D((A\J)

=

D((A*)'),

where the closure is taken in ./^(fi) and it follows, exactly as for the standard bound ary value problems, that X*=C„(Q).

(3.7.29)

The Phillips dual of A*hn is defined as the part of (A*n)' in A'* (see (3.2.19) and (3.2.20)) and in a similar manner we show that (Alw)# = (A*)*. As a by-product we obtain that u G (A*ln)# satisfies requirements of (3.7.28). The last step is to

Chapter 3. Semigroup theory

101

show that actually AXt* = (A\D* This is done by noting that A„ C {A*)# so that Ax^ C (AD* and since the resolvent sets of both operators have a nonempty common part, it follows that A,, = (AD*■ The characterization of D(AhD, given in Eqs. (3.7.23) and (3.7.25), is again obtained by using the theory of the Phillips dual but technically it is much more complex. First, we need the following general result on weak solvability of periodic boundary value problems: if we assume that 1 < p < oo, then for sufficiently large A there is a unique solution u € W^^(Q) of the problem n

(A/-/U)u = /0+ £ & , / ,

(3.7.30)

•=i

where ft € LP(Q). Moreover, for some constant C the solution u satisfies the inequal ity n

ll«llw£.(n)
(3.7.31)

where An is understood to operate on u in the distributional sense. In the standard case the proof of this result is based upon the L2 theory, Sobolev imbedding theorems, and a priori estimate of the form (3.7.31). As we saw earlier, all these are available in the periodic case. The proof of the results given in Eqs. 3.7.23-3.7.25 is based on the theory of the Phillips dual applied to the operator A^^ in C,(H) and it goes along the same lines as for the standard boundary value problems since Cn(f2) is isomorphic to the space of continuous functions on the torus (see subsection 2.4.8). The torus is a compact manifold so that the topological properties of C^(Q) are the same as those of C(Q) for arbitrary bounded Q,. The first step is the identification of (Cn(Q)f

:= D*,,« C ( 0 , ( 0 ) ) ' ,

where, by the preceding remark, the last space is isomorphic to the space of measures on the torus. Since for p > n we have W*^(Q), any measure v 6 (Cn(Q))' also defines a continuous linear functional on W^%(Q) by *(u) = /' u(x)dv.

(3.7.32)

Since Wl n(Q) is isometric with a closed subspace of the Cartesian product of n + 1 copies of Lp(£l), the isometry being u -> (u, -dxiu,..., -dXnu), by the Hahn-Banach

102

Singularly perturbed evolution equations

theorem we can extend $ onto L p (Q) n + 1 so that there are functions /, G Lq(Q), i = 0,1,..,,n, 1/p + Xjq — 1 such that for u G W^(Q.) we have

*(«) = / f/o«(i) - £ / A . » ) d l -

(3-7-33)

Let us take a measure n satisfying ft G D((A*)') so that for any u € C%(Q) /(A/ - A;)u(x)d/i = I u{x)du n n

(3.7.34)

for some measure v. Let us construct functions /, G Lq(Q), i = 0,. . . , n as in Eq. (3.7.33) and let w G W^Q) be the solution to Eq. (3.7.30) with these /,. Upon integration by parts we immediately obtain that for u G C%(Q) I {XI - A*a)u{x){dn - w{x)dx) = 0. Taking A > ux we have that (XI - A*a)Cl(Q) is dense in CW(Q) which implies dfj. = wdx and consequently D{(A*)') C W^„(Q) for all q < m/(m - 1); in particular D((Al)') C Li(fi). Since Cl(Q) C D({A*)'), the density argument shows that ( a ( f i ) ) # : = D ( ( . 4 ; ) ' ) = L 1 (n). By the definition of the Phillips dual we see that £ ( K o , « ) # ) = {« G L,(fi); A„u G A ( n } } . The identification (J4OO,TT)* = /l l 7 r is carried out as before. We have /l l 7 r = A^ C (AJj,)* and since both operators are the generators we must have Ax,* = (A,)* = ( ^ , J # . In a similar way, adapting the results for the standard boundary value problems (see [72], pp. 217-219 or [83]), we can prove that Aa,,* is the generator of a holomorphic semigroup and, again by the Phillips duality, it also follows that Ai,n generates a holomorphic semigroup. ■ In the last example we shall discuss the generation of a semigroup by a suitable extension of the Laplacian in Lj(R n ). Here we follow [43], pp. 32-40.

Chapter 3. Semigroup theory

103

7.6 E x a m p l e . Let us consider the following initial value problem in

{dtu)(t,x) u(0,x)

= =

(Au)(t,x), u0{x).

(>0,ieR

Li(W):

n

(3.7.35)

The operator A0 := A, defined on the space Cg(R n ) is dissipative (see Subsection 7.3). By means of the Fourier transform we can show (see Eq. (2.4.17)) that its distributional extension, which we denote by Ax, onto H2(Rn)

■- {u £ Li(R n ); F'^wFu]

€ Li(R")}

where w(y) := (1 4- |y| 2 ), is m-dissipative. Moreover, Ax generates a holomorphic semigroup. We equip H2(Rn)

with the norm ll«llff?(R») ~ \\F~>FW\\\LW).

It follows that C0°°(Rn) is dense in H2(Rn)

(see [1], p. 221) so that Al = A~ain

(3.7.36) Lx{Rn).

Generalizing, we define spaces H[(Rn) in the following way: //f(R n ) := {u € Li(R B ); F ~ V / 2 f > ] ] € Li(R n )}

(3.7.37)

with norms analogous to (3.7.36). It follows that Hi(W)

= £>((-,4 1 ) s / 2 ).

(3.7.38)

Unfortunately, unlike other H* spaces, the H[ spaces, in general, do not coincide with respective Sobolev spaces. For integer k we have ([82], pp. 160-161) (a) ^ ( R 1 ) = H${Rl) when k is even, (b) when n > 1 and k is even, then Wf{Rn)

C Hf (R n ) and the inclusion is strict,

(c) for any n, when k is odd, then neither W*(R n ) C fl?{R"), nor W*(R") 3

flJ(R").

On the other hand, fff spaces are "close" to Sobolev spaces. Precisely, for integer k we have (i) W?{Rn) C fff(Rn) for 5 < fc,

104

Singularly perturbed evolution equations

(ii) Hi(Rn) C Wl(&n) for k < s. These results can be applied to arbitrary second order elliptic differential operators with constant coefficients, hence they can be transformed, via linear change of vari ables, to operators having the Laplacian as their principal part. Then the zero and the first order terms can be treated as a perturbation satisfying, by Eq. (3.7.38) and (ii) above, the assumptions (3.4.5). Note, however, that in general the space Hf[Mn) itself is not stable even under a linear transformation and consequently the solution of the original (untransformed) equation will not be an element of i/j 2 (R n ).

Chapter 4 Development of asymptotic m e t h o d s for singularly p e r t u r b e d evolution equations 1

Introduction

Perturbation methods and asymptotic analysis have been used in applied mathematics for a long time but some fifty years ago it was realized that a great many systems of ordinary and partial differential equations, describing physical reality, contain small parameters which enter in such a way that a naive asymptotic analysis leads to major difficulties. Such systems are referred to as singularly perturbed or, in the case of ordinary differential equations, stiff. The first important step towards understanding the asymptotic behaviour of singu larly perturbed systems of ordinary differential equations was made by Tikhonov [85]. We will present briefly his result. Consider the system of ordinary differential equations of the form

edtx(t) = f(x,y,t), dty(t) = g(x,y,t),

(4.1.1)

where x and / are n-dimensional and y and g are m-dimensional vector functions and e is a small positive parameter. The initial conditions are x(0) = /i,

2/(0) =7/,

(4.1.2)

with fj, and r) given n-dimensional and m-dimensional vectors, respectively. A finite 105

106

Singularly perturbed evolution equations

time interval, say, [0,2"], where T > 0, is taken. In some cases the interval may be extended to [0, oo] but we will not take into account such a possibility. Setting e equal to 0 in Eqs. (4.1.1) we obtain the degenerate system

0 = dty{t) =

f(x,y,t), g(x,y,t),

(4.1.3)

where y is to satisfy the same initial condition as y in (4.1.2). The form of the degenerate system (4.1.3) shows the basic feature of the singular perturbation, when some of the differential equations become of a lower order (in the present case simply algebraic equations) so that the original initial conditions cannot be satisfied unless special initial layer solutions are introduced. To solve the system (4.1.3) we should be able to solve the equation 0 = f(x,y,t) and express uniquely x in terms of y, at least in some region in R" +m Thus we assume that there exists a function such that x = <j>{y, i) and insert it in the second equation in (4.1.3). As a result we obtain a system of equations dty(t) = g(4>(y,t),y,t)

(4.1.4)

with the initial condition y = r/. The next assumption to be made is that (4.1.4) has a unique solution on [0,T]. We also consider the associated system dTx{r) = f(£,y,t),

(4.1.5)

in which y and t are treated as parameters, and assume that its stationary point x = 4>{y,t) is stable in the Lyapunov sense in the required region. Additionally, we introduce a special case of (4.1.5) for y = rj and t; = 0 dT£{T) = f(x)r1,0)

(4.1.6)

with the initial condition x(0) = \x and make the last assumption that the solution to Eq. (4.1.6) satisfies T limz(r)

= (j)(r},0).

Finally, if the functions / and g are sufficiently smooth, then the Tikhonov theorem states that there is a constant e0 > 0 such that for all e € (0, e0] there exists on [0, T] the solution {x(t,e),y(t,e)} such that

Chapter 4. Development of asymptotic methods

limx{t,e)=x{t)

= (j>(y(t),t),

\imy(t,i)

= y(t),

107

0 < i < T, 0
Drawing from the results of Tikhonov and some others such as, for instance, Gradstein [39], the asymptotic analysis of singularly perturbed systems of ordinary differential equations rapidly developed, parallel to development of numerical methods for stiff systems, into a flourishing field of applied mathematics. The description of the field as well as the survey of the literature can be found in several monographs [70, 81, 87]. A natural extension of the singular perturbation theory for ordinary differential equa tions is to equations of evolution in Banach spaces. The first step has been made by Krein [48] who derived the zero order approximation for a single evolution equa tion with a small parameter multiplying the time derivative. In this chapter we will describe the extension of Krein's results to an arbitrary order given by Mika [54, 55]. By far more important, from a practical point of view, is the case of a system of equations, in which only one equation is singularly perturbed. We will present in this chapter the analysis of a special type of such a system following Mika [56, 57],

2

Single evolution equations with a small param eter

In this section we will study asymptotic properties of single evolution equations in which a small parameter multiplies the time derivative. We will follow closely [55]. Let X be a Banach space and the operator A with the domain D(A) dense in X be the generator of a semigroup (G(t))t>0. Then the operator \A where e is a positive parameter, is the generator of the semigroup (G(t/e)) ( > 0 . If the function q : [0,T] -> X where T > 0, is continuously differentiable, then the singularly perturbed evolution equation edtx(t) = Ax[t) + q(t)

(4.2.1)

i(0) = /x 6 D(A),

(4.2.2)

with the initial condition

has on [0, T] a unique classical solution

Singularly perturbed evolution equations

108

x(t) = G Qfi

+ ~ £ G (i~)

q(s)ds.

(4.2.3)

This result is a direct consequence of the results of Section 3.6. If e is small and the semigroup (G(t))t>0 satisfies the condition ||G(t)|| <e~ui,

LU>0,

te[0,T],

(4.2,4)

then we may try to expand the solution into powers of e. Since our equation is singularly perturbed we may expect the existence of an initial layer, as in the case of ordinary differential equations. To account for such a layer we follow the well established procedure and represent the solution as the sum of two functions. The first one denoted x(t) is called the bulk solution (in some cases it is being referred to as the outer or global solution). The second function, the initial layer solution (sometimes called the inner or local solution) is supposed to depend on the rescaled time variable r = t/e and is denoted by X(T). Thus the whole solution to (4.2.1) is written as the sum x(t) = x{t)+x{T).

(4.2.5)

Inserting this into Eq. (4.2.1) we obtain edtx(t) + dTx(r) = Ax(t) + AX(T) + q{t). We replace the above equation by two equations separating the two time variables t and r and write

tdtx{t)

=

Ax(t) + q(t),

8TX(T)

=

AX{T).

(4.2.6)

We note that the equation describing x is no longer singularly perturbed. Defining the bulk asymptotic solution of the order N, which is expected to represent the exact solution outside the initial layer, as

S

W = ^e"x!(i),

(4.2.7)

n=0

substituting it into the first equation in (4.2.6), and comparing terms of the same order in e, we obtain the system of equations

Chapter 4. Development of asymptotic methods

109

A-lq(t),

x0{t)

=

xn{t)

= A-ldtxn^(t),

n = l,2,...,N.

(4.2.8)

From these equations we see that the differentiability of p is not sufficient. Actually, we have to assume that q has a continuous derivative of order N + 1. Then from Eq. (4.2.8) we have xn = (A- 1 )" + 1 a ( "g(0,

n = 0,1,2,...,

N.

(4.2.9)

This can be shown by induction. First, observe from (4.2.8) that Eq. (4.2.9) is valid for n = 0. Next assume it to be valid for n = k. Then Eq. (4.2.8) yields

l l A~A' dtd: txk(t) l = A~ dt\A-ldt(A-l){k+l)dfq{t)

xk+l

=

=

(A-vf+Vd{tk+l)q(t).

This follows from the fact that for any differentiable function / dtA-1f(t)

A-ldtf(t)

=

(see Eq. (2.5.3)). If the initial layer solution of order N is taken as

x(W)W = E e " £ " M .

( 4 - 2 - 10 )

n=0

and the initial values for xn denoted with fin, then from (4.2.6) we have £n(r)=G(T)£n,

n = 0,l,2,...,N.

(4.2.11)

Let now *»(0)=A»

n=

l,2,...,N,

then from the original initial condition (4.2.2) we obtain the relations p n + An = M>n, where <50„ is Kronecker's symbol.

(4.2.12)

Singularly perturbed evolution equations

110

Combining all the above results we can write the asymptotic solution of order N in the following form

z<w>(i) = =

f W ( t ) + *<">(-)

= £W(t) +
(4.2.13)

is given by Eqs. (4.2.7) and (4.2.9).

The above approximate solution was obtained by performing the classical asymptotic analysis. In this case, however, a direct approach can also be used to prove the asymptotic convergence of i ^ ' to the exact solution. Let us return to the exact solution as given by Eq. (4.2.3) and integrate by parts the integral term. We have

- f

G (—)

q{s)ds

=

AA~1q(s)da

- J* G (—■) ■AG(

in e

( V

-

r)

r 0 (s)ds

i-s^

Lds \G \~r) *o(s)ds > [ ' — i ) x0(S)|< - j ■''o

G(i-i)

dsx0(s)ds

(t) - G (V) x 0 (0) - e f l-AG (j-~j

f

)

fi(*)ds.

Repeating TV times the differentiation by parts we arrive at the formula x(t) = xm{t)

- e{N+1) [' -G{s)dsxN{s)ds.

The integral term admits the estimate

1

ft — s )

Jo -G I

dsxN(s)ds

<

M [<■

I exp (-w(t - s)/e) ds

Jo

=

M{\ - exp(-wt/e)) < M,

where M = mp{\\dtxN{t)\\

:0
(4.2.14)

Chapter 4. Development of asymptotic methods which is finite since xN(t) is expressed in terms of of be continuous.

111 q{t) which were assumed to

We will now summarize the results of this section in the theorem. T h e o r e m 2.1 Let the operator A with the domain D(A) dense in X be the generator of a C'o-semigroup, the initial value fi 6 D(A) and the nonhomogeneous term q{t) continuously differentiable (N + 1) times. Then there exists an asymptotic solution x' N '(<) of order N such that | | i ( f . ) - x " v ) | | = 0( e ( / v + 1 )) uniformly on the interval [0,T], The asymptotic analysis presented in this section is still valid if the operator A is replaced by the operator valued function A [0,T] —v X for which there exists an evolution operator U(t,s) defined for all t, s € [0,T]. Additionally, the operator function A has to be continuously differentiable sufficiently many times. The formulae are almost the same with the semigroup operator G replaced by an evolution operator U. The last remark is in place concerning the relevance of the approximate solution derived in this section. The exact solution is readily available for the problem (4.2.1) but it is expressed in terms of an integral as in Eq. (4.2.3). The evolution of the integral may be both tedious and time consuming and the differentiation needed to find the asymptotic solution could be considerably easier and much faster.

3

Systems of evolution equations with a small pa rameter

In this section we will study systems of evolution equations, of the form similar to that of the singularly perturbed systems of ordinary differential equations (4.1.1). For simplicity, following closely [56], we take a system of two equations with one of them containing a small parameter e > 0. We start with an unperturbed system

dtv dtw

= Bv + Pw, = Qv + Aw,

(4.3.1)

with the initial conditions i,(0) = ft e A',

w(0) = V 6 D(A).

(4.3.2)

Singularly perturbed evolution equations

112

Theorem 3.1 Let A be the generator of a C0-sermgroup (G(t))t>o m a Banach space X, and P,Q and B bounded operators in X. The system evolution equations (4-3.1) with the initial condition (4-3.2) has a unique classical solution on any finite time interval [0,T] where T > 0. Proof. Write (4.3.1) as a single equation of evolution in X x X: dtz = Ez + Rz, 2(0) = 0, where z = (v,w),

(4.3.3)

9 = (H,TJ) and

E=

B 0 0 A

0 P Q 0

R

(4.3.4)

The matrix operator E is clearly the generator of a semigroup (K(t))t>0 given by

K(t)

H(t) 0

0 G(t)

in X x X

(4.3.5)

'

where H is a uniformly continuous semigroup generated by the bounded operator B in A'. Since R is a bounded operator in X x X, the application of the perturbation theorem (Subsection 3.4.1) shows the validity of the theorem. ■ The solution to the problem (4.3.1) satisfies the Volterra integral equation z(t) = Klt)B + I Kit Jo

a)Rz(a)da.

(4.3.6)

Writing again z{a) in the form (4.3.6) we obtain z{t) = K(t)6 + f K[t - a)R \K(O)6

Jo

I

j(t) =m{t)+

[ Jo

+ [' K{a -

Jo

s)z(s)ds da

or U{t,s)Rz(s)ds,

(4.3.7)

where U{t, s) = J K(t - a)RK{a - s)da

(4.3.8)

Chapter 4. Development of asymptotic methods

113

and (4.3.9)

m(t) = K{t)e + U{t,0)6.

Here U is a function defined and continuous in the sense of norm m X x X on the triangle {(t,s) : 0 < s < t < T} and m is continuously differentiable on [0,T]. The solution to the integral equation (4.3.7) can be obtained by the method of suc cessive approximations z0(t)

=

zk+i{t)

=

m(t), / Jo

U(t,s)zk(s)ds,

with

*(*) = £**(*)■ It is to be noted that the components v and w of z are fully separated in (4.3.7) since U(t,s)R is a matrix operator with the only nonzero terms on the main diagonal. In fact we have

U(t, s) =

t SlHii J sH{t-a)PG(a-s)da

0 fs G(t - a)QH(a - s)da

0

and

U(t,s)R

JtsH{t~a)PG(a-s)Qda

Qda

0 t

It Git-a) J sG{t~a)QH(a-s)Pda

0

We now introduce a small parameter e > 0 and replace the original system (4.3.1) by the singularly perturbed system

dtv edtw

= Bv + Pw, = Qv + Aw,

with the same initial conditions (4.3.2).

(4.3.10)

Singularly perturbed evolution equations

114

For a singularly perturbed system we need an additional assumption that the semi group (G(t))t>o satisfies the inequality M > 0,

\\G(t)\\<Me e^\

u > 0,

t > 0,

(4.3.11)

otherwise the solution to (4.3.10) would increase exponentially with time and no asymptotic expansion could be possible. The formulae derived for the unperturbed system remain the same if the operators Q and A are replaced by ~Q and \A, respectively. The semigroup generated by \A is {G(t/e))t>0 and the group (H(t))t>o remains unchanged so that H(t) 0

Kit)

0 G{t/e)) .

(4.3.12)

The operator function U(t, s) will now have the form \HH{t-a)PG{

0

U(t,s) =

llGl

s)da :*?)\QG{a ' - s)da

'-)da ?(2fi)0

(4.3.13)

0

The proof of the existence of the solution obtained for (4.3.1) will be valid for the singularly perturbed system. We now turn to the asymptotic analysis of the system (4.3.10). Following the standard asymptotic expansion procedure we express both v and w as the sums of the bulk and initial layer solutions

v(t)

=

it) (t)

=

v(t)+v(r), w(t)

+ UI(T),

(4.3.14)

where r = t/t. The bulk solution satisfies the equations

dtv tdtw

Bv + Pw, Qv + Aw,

(4.3.15)

and the initial layer solution the equations

dTv dTw

= tBv + ePw = Qv + Aw

(4.3.16)

Chapter 4. Development of asymptotic methods

115

we start with the bulk solution and set

vW(t) = £yen(t), n=0

u)W(t)

=

£e«tD„(i).

(4.3.17)

71=0

The expansions are introduced into (4.3.15) and the terms of the same order in e are compared. This results in the following system of equations:

dtvn

=

~Bvn + Pwn,

edtwn-\

=

Qv„ + Awni n = 0,l, ...,7V.

(4.3.18)

Here again we adopt the convention that vb-i = 0 . The initial conditions for the equations for wn are assumed to be vn(0) = iineX,

n = 0,1,...,7V.

(4.3.19)

The initial values p,n are to be determined with the help of the initial layer solutions.

Lemma 3.1 The system of equations (4-3.18) with the initial values (4-3.19) has a unique classical solution {VO,VU---,VN;W0,WU...,WN}.

,wN}

Proof. On account of Eq. (4.3.11) the inverse operator A'1 exists so that from Eq. (4.3.18) we obtain

wn dtvn

= A'1 (d t w n _i - Qvn), = (B - PA~1Q)vn + PA-'dtwn.u n = 0,1, ...,7V.

(4.3.20)

Now B - PA~1Q is a bounded operator and the differentiability of wn_i follows by induction. This shows that Eq. (4.3.18) with the initial conditions (4 3.19) has a solution of required properties. ■

Singularly perturbed evolution equations

116

For the initial layer solution we proceed in a similar manner. Writing

S"°(r) = £ ^ n ( r ) , wr,W (r)

n=0 N

=

£e»«>„(r), n=0

(4.3.21)

inserting this into (4.3.16), and comparing terms of the same order in e, we obtain the system of equations

dTvn dTwn

= B«„_i + Pu>„-i, = Qvn + Awn, n = 0,l,...,N,

(4.3.22)

where again V-i = 0 and u)n_i = 0. These equations have to be supplemented by the initial conditions w„(0) = fjneD(A),

n = 0,l,...,N.

(4.3.23)

Since the functions vn are supposed to decay exponentially with increasing r we must take

V0(T)

vn{r)

= 0, f°° = / [Bvn-i{s) + Pwn-^ds

(4.3.24)

n = 1,2,...,AT. FVom the second equation in (4.3.22) taken for n = 1 we see that w0{r)

= C7(T)T70,

which is differentiable since 770 is assumed to belong to the domain of A and satisfies the inequality

IKMII < ti0)e-\ r > 0, where $]

= \\f,Q\\.

Chapter 4. Development of asymptotic methods

117

Next we have from Eq. (4.3.24) VX{T) = -JT

PG(s)fj0ds,

which is differentiable and satisfies the inequality l|fii(T)|| < c l V * " , r > 0 , where a[

= \\P\\ \\rj0\\.

Proceeding with n = 1 we have U>I(T) = G{T)TIX +Q

f

Jo

G{T -

s)vAs)ds,

which again is differentiable. Taking the norms on both sides we obtain which again is differentiable. Taking the norms on both sides we obtain

lk(r)|| < e-T|^|| + ||Q||/ T e -'J ( T - S ) a i e - S d s lk(r)||

<

e-T||»h|| +

= (Pi0) +

in

fi1)r)e-

WT

\\Q\\j\-^-^aie-sds

where

P[0) = 11*11, ti1] =

\\Q\\a[0)

We will now show by induction that for arbitrary n such that 0 < n < N, the system of equations (4.3.22) with the initial condition (4.3.23) has a differentiable solution satisfying the inequalities

\\Vn{r)\\ <

E ^ r - e

—

m=l

IK(r)||

<

^ m J T m e - " r , r > 0,

£ m=l

where Q^m) and /3*,m) are some constants. Assume that for some n = k the above statements are true. Then we have V*+l( r ) = - /

[Biik{s) + Pwk{s)]ds

(4.3.25)

Singularly perturbed evolution equations

118

which is obviously differentiable. Taking the norms on both sides and using (4.3.25) we have

fc_1

\\vk+x(r)\\

.

.

fc

roo

< ||B|lE«lm)/

s^e-'ds

,

+ WPW^P^

.

roc

/

, - e/Te^da -

m=\

m=\ k

= £< i;*"^ m=l

where the constants Q H are, in an obvious way, related to the constants a^

and

( ]

pr

In a similar way we can show that wt+i is differentiable and satisfies the inequality (4.3.25). We combine the results in the lemma. L e m m a 3.2 The system of equations (4-3.22) with the initial conditions (4-3.23) has a differentiable solution {v0,v1,...,iiN;

wo,Wi,Wn}

which satisfies the inequalities (4-3.23). We see that in the bulk solution the initial values of uin are not arbitrary but follow from the second equation in (4.3.18). Similarly, the initial values of iin are calculated from (4.3.24). This shows that the values of p,n and f\n may be calculated iteratively in the following way. First we note that Wo = 0 and VQ = 0 so that Mo = M.

'7o = >7,

(4.3.26)

where, by Eq. (4.3.2), 17 e D(A). Then for any n, such that 1 < n < N, we evaluate vn(0) from Eq. (4.3.24) and put fin = -w„(0).

(4.3.27)

Similarly, we calculate wn(0) from the first equation in (4.3.20) and take fjn = -wJO).

(4.3.28)

Chapter 4. Development of asymptotic methods

119

Now Eq. (4.3.20) implies that ijn <E D(A). With the above formulae we see that if we define the asymptotic solution of order N as vW{t)

= vW(t)v^(t)+v^(r),

w^(t)

=

IBW(0+IBW(T),

(4.3.29)

where r = t/t and the formulae (4.3.26), (4.3.27) and (4.3.28) are used, then „<">(()) = M,

u|W(0) = !j,

(4.3.30)

so that the asymptotic solution satisfies the exact initial conditions. To prove the asymptotic convergence of the approximate solution as given in (4.3.29), we introduce the functions XN(t) yN(t)

N

= e- (v(t) E-N(v{t)-vW(t)), N

= e- (w(t) e-N(w(t)-vW(t)),

(4.3.31)

which satisfy the system of equations dtxN

= BxN + PyN + pN[t),

edtUN = QxN + AyN + qN{i),

(4.3.32)

with the initial condition X*r(0)=0,

j/Af(0) = 0.

(4.3.33)

The nonhomogeneous terms in (4.3.32) are given as P„(t)

=

qn{t)

=

BVN{T)

+

PwN{t),

-edtwN(t).

(4.3.34)

We will now prove the main theorem of this section. Theorem 3.2 With all the assumptions made in this section about the operators ap pearing in the system of equations (4-3.1), the approximate solution {v ( ; v l(t),io w (£)}, as given in (4.3.29), converges to the solution {v(t),w(t)} of (4-3.1) uniformly on an interval [0, T] where T > 0 as e w + 1 In other words, \\xN(t)\\=0(e), uniformly for t £ [0; T].

||y w (t)|| = 0(e)

Singularly perturbed evolution equations

120

Proof. We modify the derivation of the Volterra equation equivalent to the original system of equations (4.3.1) to account for the inhomogeneous terms pt/{t) and qN(t) appearing in (4.3.32). If rN = (PN>QN) and uN = (xN,yN), then the system (4.3.32) is equivalent to uN(t) = hN{t) + f U(t,s)RuN{s)ds, Jo

(4.3.35)

where hN(t)=

f [K(t-a) Jo

+ U(t,s)]rN(s)ds.

(4.3.36)

The matrix operator R is defined in Eq. (4.3.4) and the matrix operator function U in Eq. (4.3.13). We have already noticed that the functions xN and y^ are completely separated in (4.3.35). In particular, y# satisfies the following integral equation yN{t) = fN{t) + f L(t, s)yN{s)ds

(4.3.37)

where L{t, s) = - / ' H{t - o)PG (-—-)

Wdo

(4.3.38)

and

fN(t)

=

^G(~)g (s)ds C°
+- Jo/ M{t,s)pN(s)ds

(4.3.39)

with

M(t, S) = -JG

(*—-] QH{<7 - s)da.

(4.3.40)

From the very beginning we have considered our system of equations over a finite time interval [0, T] even if the system is autonomous. The reason is that various constants appearing in the estimates of the relevant functions in the above equations may depend on the length of the time interval, i.e. on T. Keeping this in mind we

Chapter 4. Development of asymptotic methods

121

first observe that qN is bounded so that the first integral in (4.3.39) can be estimated as follows

I

t

ft — s G\-)qN{s)ds

<

<

rnax\\qN(s)\\-(l-e-Jr1) - e -

(4.3.41)

0<s
where cx is a constant which may depend on T. This will also be true for other constants to be introduced in the course of this proof and denoted by ck with k running up to 8. The kernel M(t,s)

satisfies the inequality

\\M(t,s)\\

< HQII max \\Q\\m^T\\H(t)\\-J^e-^-^da " "

< c2,

0
(4.3.42)

so that the second integral term in Eq. (4.3.39) admits the estimate

I

/ M(t,s)pN(s)ds Jo <

C2Jo

<

ec3.

[||B|| 2

> r Q) +iiP|iE^ m ) Q e~

wt/t

dt

(4.3.43)

Combining Eqs. (4.3.42) and (4.3.43) we have \\fN{t)\\ < ec4.

(4.3.44)

The kernel L(t,s) is bounded since

\m,s)\\

<

||P||-||Q||max||//(f)||-/e-^-s»/'da

<

||P||||Q||m«r||^(t)||i(l-«"-{'-'/«)

<

cs.

(4.3.45)

We recall now that the integral equation (4.3.37) can be solved by the method of successive approximations as it was in the case of (4.3.7). Thus we have

Singularly perturbed evolution equations

122

yW(t)

=

+1

y% \t)

=

fN(t), / Jo

L(t,s)yP(s)ds,

from which it follows that

!/*(<) = El/ft'(*)•

(4.3.46)

k=0

We see from (4.3.44) and (4.3.45) that

M < «c4 teff VF(t) k\ so that (4.3.47)

\\VN{t)\\ < «c6 where o^T

(4.3.48)

c 6 = c4e

Finally, from the first equation in (4.3.32) we obtain that

xN (t)

=

/ Jo

H(t-s)PyN{s)ds

+

I H(tJo

s)pN(s)ds.

The first integral admits the estimate 1/ \\Jo

H{t-s)PyN(s)dt5

<

€C7,

(4.3.49)

where c7 = | | P | | c 6 m a x | | / / ( i ) | | . As for the second integral we have / H(t Jo

s)pN{s)diS < «c8,

(4.3.50)

Chapter 4. Development of asymptotic methods

123

using exactly the same procedure as in the case of Eq. (4.3.43) by simply replacing M{t,s) with H(t- s). Combining (4.3.49) and (4.3.50) we have

IM0ll<e(c 7 + c8),

(4.3.51)

and, finally, from (4.3.48) and (4.3.51) \\xN(t)\\

< ec, \\yN(t)\\

< «■

(4.3.52)

where c = max{c6,C7 + c8} which is exactly the estimate predicted in the statement of the theorem. ■ The original system of equations can be made more complicated if the constant oper ators are replaced by operator functions sufficiently smoothly depending on time. We can also add inhomogeneous terms, provided they are sufficiently smooth functions of time. The analysis is, however, more complicated and, instead of repeating it here, we send the reader to the original paper [56].

Chapter 5 Some singular-singularly p e r t u r b e d evolution equations and kinetic equation 1

Singular-singularly perturbed evolution equa tions

In the previous chapter we considered a single equation of evolution in which the time derivative is multiplied by a small parameter. We turn our attention to the asymptotic properties of evolution equations in which an operator is multiplied by a large parameter but at the same time it has an eigenvalue zero. Such equations are sometimes referred to as singular-singular or of the resonance type. In this section we will study a particular type of a singular-singular equation following [58]. Let C be a bounded operator in the Banach space X, such that zero is its simple isolated eigenvalue with the eigenspace V, consisting of such element y 6 X for which Cy = 0. Then X can be represented as a direct sum A' = V © W, where both V and W are invariant subspaces of the operator C and C is one-to-one from W onto itself. Let P be the spectral projection associated with the eigenvalue A = 0 (see Eq. (2.7.4)). Thus we have V = PA,

W = QX, 125

Singularly perturbed evolution equations

126 where Q = I — P

We further assume that the spectrum of C, denoted a(C), is, except the point zero, located in the left half-plane so that ReX < 0 if A 6 a{C) and A ^ 0. We will also assume that a(C) is bounded away from zero so that sup{i?eA; A e a(B),X / 0} = - w < 0. The operator C defined as an operator from W onto itself will be denoted by QCQ by virtue of the fact that PC = CP = 0. The operator QCQ is a generator of the uniformly continuous semigroup (G(t))(>0 which, thanks to the assumption on a(C) and Eq. (3.5.6), satisfies the inequality \\G{t)\\ < Me~^;

t > 0,

(5.1.1)

where M > 0 is a constant. Now if B is a bounded operator in X, the evolution equation dtu = Bu + -Cu,

(5.1.2)

u(0) = 9 6 X

(5.1.3)

with the initial condition

will have a unique classic solution in X. The well established approach to the asymptotic analysis of (5.1.2) which dates back to Hilbert, who applied it to the Boltzmann equation of the kinetic theory in 1912 (see [20]), consists in expanding the bulk solution to (5.1.2) in powers of e so that u{t) = uo(t) + eui («) + ■■■

(5.1.4)

and inserting it for u(t) into (5.1.2). Comparing terms of the same order in e we obtain the following first three equations

Ciio = 0, Cu\ = dtu0 — Bu0, Ciii = dtU\ — flfil.

flfii.

(5.1.5)

Chapter 5. Singular-singularly perturbed equations

127

From the first equation we see that u0 6 V Since Cux £ W, by applying the projection operator P to the second equation, we obtain P{dtu0 - Bu0) = 0, or dtu0 = PBu0,

(5.1.6)

which is the equation to be satisfied by the zero order bulk solution. Since X is a direct sum of V and W, the function v,\ can be represented as the sum Ui = u f + u\. The operator C is invertible on W so that u™ = -C~lQBu0.

(5.1.7)

As previously Cu2 G W which gives P{dtul - Bui) = 0 and, in turn, dtu\ = PBu\

+

PBu?

Finally, using (5.1.7), we obtain the equation satisfied by u( in the form dtu\

= PBu\

- PBC^QBUQ.

(5.1.8)

The above procedure is slightly involved and not particularly transparent. In what follows we will employ the projection method first used for singular-singularly per turbed evolution equations by Mika [58]. The projection method was later applied to the nonlinear Boltzmann equation by Caflisch [18]. The method consists in writing the evolution equation (5.1.2) as a system of evolution equations in subspaces of V" and W To that purpose we first introduce the notation

Pu = v,

Qu = w,

P6 = H,

Q0 = V,

(5.1.9)

Singularly perturbed evolution equations

128

and then apply the projectors P and Q to the equation (5.1.2). Using Eq. (5.1.9) we nhtain nf equations ermations obtain thf the fivstem system of

dtv

=

PBPv + PBQw,

dtw

= QBPv + QBQw + -QCQw,

(5.1.10)

with the initial conditions v(0) = n,

w(0) = r].

(5.1.11)

The spurious operators P and Q are written for symmetry. Clearly, the above system of equations (5.1.10) with the initial condition (5.1.11) is equivalent to the original evolution equation (5.1.2) with the initial condition (5.1.3). We see that it is slightly different from the system (4.3.10) but the standard asymp totic analysis is very similar. Thus, as in the previous case, the solution to (5.1.10) will be represented as a sum of the bulk and initial layer solutions so that

v(t) w(t)

= v(t) + v(T), = w(t)+w(r),

(5.1.12)

where r = t/e. The bulk solution satisfies the original system of equations which can be written in the form

dtv edtw

=

PBPv + PBQw,

= eQBPv + eQBQw + QCQw,

(5.1.13)

and the initial layer solution will satisfy the system

dTv

=

ePBPii + ePBQw,

dTw

=

eQBPv + eQBQw + QCQw.

(5.1.14)

The standard asymptotic procedure calls for the expansion of all the functions ap pearing in the above equations. Thus for the bulk solution we put

Chapter 5. Singular-singularly perturbed equations

129

££ttM*).

v^(t) --

n=0 N

tf)W(t) \t)

= Z^n(t),

(5.1.15)

n=0

and for the initial layer solution we take

vW(r) = £>"iin(r), W

(N)

n=0 N

(r) = £ ^ „ { r ) .

(5.1.16)

n=0

Further procedure is almost identical to that employed in the previous section. For the functions vn and wn representing the bulk solution we obtain the equations

dtvn QCQwn

= PBPvn + PBQwn, = dtWn-x-QBPVn^-QBQwn, BQWn, n=

(5.1.17)

0,l,...,N.

As before, w_i := 0 and w_i := 0. We assume that the initial conditions for v„ are vn{0) = fin € X.

(5.1.18)

We first see that, since QCQ is invertible, the second equation has a unique solution for each n. In particular, wo{t) = 0

(5.1.19)

and

wn(t)

= (QCQr^dtWn-i-QBPVn-r-QBQwn-J, B<3tD„_i), n= 1,2,..., N.

(5.1.20)

Singularly perturbed evolution equations

130

If these functions are inserted into the first equation in (5.1.17), then the resulting nonhomogeneous equation will have a unique strongly differentiable solution for ar bitrary initial functions p,n. Let us write the equations for the first two terms in »W, For v0 we obtain dtv0 = PBPv0.

(5.1.21)

Next we evaluate u>i which is given by Wi =

-(QCQ)~lQBPv0

so that dm = PBPvi - PBQ{QCQ)-xQBPva.

(5.1.22)

For the initial layer solutions we obtain the following system of equations obtained in the same way as the analogous system in the previous section

drvn dTwn

= PBPvn-.i + PBQwn-u = QBPvn-i + QBQwn^ + QCQwn, n = 0,\,...,N, i)_i = 0,ty_! = 0.

(5.1.23)

The solutions to the first equation in (5.1.23), which decay exponentially, must be taken in the form similar to that as in Section 4.3.

w0

=

0,

vn

= -j

ran

n-

(PBPvn.1{s)

+ PBQwn^l(s))ds,

(5.1.24)

1,...,N.

The second equation in (5.1.23) require the initial conditions. We denote the initial values for wn by ^n(0) = ^„,

n = 0,1,2,..., AT.

(5.1.25)

Let us calculate the first two terms in j}W and w ^ ' For w0 we have w 0 (r) = G(T)T)O,

(5.1.26)

Chapter 5. Singular-singularly perturbed equations

131

and for w^

tui(r)

=

G(T)ih - [T G(r Jo

=

G(r)fji - I' G(r - s)QBQG(s)fi0ds.

s)QBQwo(s)ds (5.1.27)

Jo

From (5.1.24) we obtain do = 0 and OO

Vl

(r)

/ =

-PBQ

PBQw0{s)ds j " G{s)fj0ds OO

-PQB

/

(QCQ)-lQCQG(s)f,0ds )ds

-PBQ(QCQ)-1

=

/ JT

— G(s)rj0ds ds

l

=

PBC{QCQy G(T)i~lods.)ds.

(5.1.28)

Since we require that

f>W(0)+5W(0) u* (A,) (0)+w (A,) (0)

= /x, = ??,

(5.1.29)

the initial values p,n and ?7n can be evaluated iteratively in exactly the same way as in the previous example. As an example we will list the initial values for all the moments up to the first order:

M°)

= Mo = M,

u 0 (0)

=

Ao = 0,

u>o{0) =

r?o = 0,

wJo(O) =

fj0 = r), Ai = -PB(-PBQ(QCQ)-lri,

MO) = MO) = h = °BQ{ PBQ(QCQ)-ln, 0 i (0) = fh = -{QCQ)-xQBPti, ^j(O) = Jh = {QCQ)-lQBPfi. V-

(5.1.30)

The asymptotic solution of order JV is now taken as the sum of bulk and initial layer solutions

Singularly perturbed evolution equations

132

vW(t)

»<"!(()

=

cW(t)+«W(T),

=

w(N\t)+ww(r),

(5.1.31)

(r),

where r = t/e. From (5.1.29) it is seen that vw{0)

= p,

wm(0)=r1.

(5.1.32)

To prove the asymptotic convergence of the procedure we have to form the functions

xN(t) yN(t)

= £">(*)-i>{A°(t)), = e-N(w(t)-WW(t)), '(*)),

(5.1.33)

and prove that xN and y^ are of the order e uniformly on an interval [0, T] where T> 0. Taking into account all the previous equations we see that i/v following system of nonhomogeneous equations dtxN{t)

= PBPxN{t)

+

dtyN{t)

= QBPxN(t)

+ QBQyN{t)

an

d VN satisfy the

PBQyN{t)+pN{t), + -QCQyN{t)

+ qN{t),

(5.1.34)

with the initial conditions arjv(0) = 0,

yN{0)=0.

(5.1.35)

The nonhomogeneous terms are given as

pN{t)

=

QN(t) =

PBPVN{T)

+

PBQWN,

-dtwN(t) +QBP{vN(t)

+ vN(t)) + QBQ{wN{t)

+ WN(T)).

(5.1.36)

The proof of convergence for the present case follows closely the proof given in Section 4.3 and will not be repeated here.

Chapter 5. Singular-singularly perturbed equations

2

133

Model system: exact solution

In the previous section we considered a particular type of singular-singularly per turbed evolution equations and applied the standard asymptotic analysis as proposed by Hilbert. We now want to introduce the reader to a different asymptotic procedure for such equations. To make our point and all considerations simple, we introduce a system of ordinary differential equations which has the basic features of singularsingularly perturbed evolution equations appearing in the linear kinetic theory. We will explain its connection with kinetic theory in Section 7. Let us consider the system of the ordinary differential equations

dtv + Av + Sw = 0, dtw + Sv + Bw + -w = 0, e

(5.2.1)

with the initial conditions v{0) = n,

w{0) = ri.

(5.2.2)

All the coefficients A, B and S are assumed to be time-independent and the initial values ix and r\ to be independent of the small parameter t. The characteristic equation for (5.2.1) has the form eA2 + [1 + e{A + B)}\ + A + eAB - eS2 = 0.

(5.2.3)

The discriminant is given as E = [l + 2t{B -A)+e2((B-

A)2 + 45 2 )] "2 ,

(5.2.4)

and the two eigenvalues are

* = -h-^A+B) A2 =

l

- --\{A

+

+ B ) - l

i (5.2.5)

The general solution of the system (5.2.1) is

v = w =

aie

Alt

+ a 2 e A2t ,

a 1 7ie A l t + Q272eA2!,

(5.2.6)

Singularly perturbed evolution equations

134 where ai and a2 are arbitrary constants and

7i

=

72

=

Aj + A

1

1

S X2 + A

S 1

2e 1

~ S

2e

S

5l»-*-i > ^) + 2e

-

(5.2.7)

Substituting t = 0 in (5.2.6) and using (5.2.2) we obtain the system of equations for Q! and a2

ai + a2

=

n,

» i 7 i + «272

=

V,

(5.2.1

which yields 1

1 \

e (B-

A

«i

(V22 + 2EJ ■2E)I1+EE V 2

Q2

( 2~2E)^

1

1 \

e /A+

~E{

2

0 B fJ. + Sr) 0 Sr,

i i

(5.2.9)

We now assume that e is very small so that the exponential factors in (5.2.6) behave in ways entirely different from each other. We expand all the quantities appearing in (5.2.4)-(5.2.9) into power series in e and we will truncate the series on first order terms except in those cases where higher order terms are needed due to cancellation of £ appearing in the denominator. Thus we have

E 1

Ps

*

l + e ( B - ^ ) + 2e252, l + e ( B - A ) + e 2 ( ( B -- A ) 2 - - 2 5 2 ) ,

E A,

Ps

- A + eS2,

A2

P3

-I-B-eS2,

7i

sa

-eS2,

72 Pi

- + B - A + eS2,

OH Pa

[i — eSrj,

a2

Pa

«i72 f= a »272

-a

eSrj, —tSfi, 77 + eS/x.

(5.2.10)

Chapter 5. Singular-singularly perturbed equations

135

Here and throughout the paper the symbol " w " denotes "approximately equal" (up to O(e) or 0(e 2 ) terms). The terms containing e A,t change slowly with time due to (5.2.10), exactly as the bulk solution in the singular perturbation theory, and consequently will be referred to as the bulk solution of our system. It is given by v{t) = a,e A l ',

w(t) = Qi7ie Al '

(5.2.11)

By (5.2.10) the exponential factor eA2' decays rapidly with time for small e and thus the terms, which contain that factor, represent what is called in the singular pertur bation theory the initial layer solution. It is given by v(t) = a 2 e A2 ',

m(t) = a2j2eX2'

(5.2.12)

Having expanded all the coefficients multiplying the exponential factor e Al ', as well as \i itself, we must decide whether we also expand the exponential function into the power series in terms of the exponent or leave it unexpanded. Accordingly, we may write the expansion of e Al ' in two ways: e A ' ( ^e-M{l

+ eS2t)

(5.2.13)

or eA" «

{ A+eS2)t e

-

(5.2.14)

Using (5.2.13) we obtain the bulk solution in the form

v(t) w{t)

{n + (sVc(S2vt-Sr,))e-Al, -(Sne-At,

« «

(5.2.15)

and from (5.2.14) we have

v(t)

«

(n-tS fa-eSriel-*-"**,

w(t)

«

-eSve{-A+tS2)t-

(5.2.16)

The second form should be a more accurate approximation to the exact solution, valid over a longer period of time, since the exponential function is unexpanded.

Singularly perturbed evolution equations

136

With the initial layer solution the situation is somewhat different. First of all we see that due to the term -t/e the whole function decays rapidly and becomes negligible after the time-interval of order e. Therefore, in accordance with a well established rule in singular perturbation theory, we first introduce the new time variable r = t/e which serves as a sort of a magnifying glass to look more closely at the initial layer. With this X2t = e\2r w - ( 1 + eB + e 2 5 2 )r, so that ex2t = e a 2 T w

g-r^ _

eBr^

(5.2.17)

if we retain only terms of order not exceeding 0(e). For the initial layer solution we do not leave the exponential function unexpanded, as in the case of the bulk solution, to obtain a better approximation. The reason for this is the fact that the physical meaning of the initial layer solution is to supply proper initial conditions for the bulk solution which describes a given physical phe nomenon for most of the time. These initial conditions do not depend on whether the exponential factor eA2' is, or is not, expanded in terms of e. Using Eqs. (5.2.10) and (5.2.17) we can write the approximate initial layer solution in the form

V(T) W(T)

« «

eSqe T, {7) + t{Sn-tBr]T))e-T,

(5.2.18)

where the terms proportional to e2 have been neglected. We use the same letter to denote the function of t and of r in Eqs. (5.2.12) and (5.2.18) but that should not lead to any misunderstanding. In the next section we show that the application of the standard asymptotic expansion method leads to the approximate solution of the problem (5.2.1) in the form given by Eqs. (5.2.15) and (5.2.18).

3

Model system: standard asymptotic analysis

We will now follow the standard asymptotic expansion procedure as introduced in Section 1. We start with the bulk solution and assume that it can be expanded into a power series in e. Retaining terms of the first order we write

Chapter 5. Singular-singularly perturbed equations

v « w «

137

v0 + evly wQ + euii.

(5.3.1)

Inserting these equations into Eq. (5.2.1) and comparing terms of the same order in € we obtain the system of equations

0(1) 0(e)

: dtv0 + Av0 = 0, w0 = 0, : a ( v! + Avx - S2v0 = 0, tDi = -SVQ.

(5.3.2) (5.3.3)

The initial conditions to be satisfied by v0 and V\ will be determined with the help of the initial layer solution. If we introduce the notation ^o(0) = Ao,

»i(0)=A,

(5.3.4)

then solving the differential equations in (5.3.2), (5.3.3) we obtain

fioe'At,

v 0 (t)

=

Bi(t)

=

(pi + S2p0t)

w0(t) wi(i)

= =

0, -SAoe - ""

e~A\ (5.3.5)

To obtain the approximate initial layer solution we introduce r = t/e into the system (5.2.1) which yields a new system of equations

drv + eAv + eSw = 0, dTw + eSv + tBw + w = 0.

(5.3.6)

As in the case of the bulk solution we expand v and w in terms of e and truncate the expansion on terms of order e

V Pa v0 + evi, u) « tio + ewi-

(5.3.7)

Singularly perturbed evolution equations

138

Inserting these into (5.3.6) and comparing terms of the same order in e we obtain

0(1) 0(e)

: aTti0 = 0, dTw0 + w0 = 0, : drvx + Av0 + Sw0 = 0, drwi + Sv0 + Bw0 + wi - 0.

(5.3.8)

It is to be remembered that all these functions represent the initial layer solution which is assumed to vanish exponentially with time. This shows that

Mr)

= 0,

Mr)

=

I

SwQ(s)ds.

(5.3.9)

Denoting the initial values for w0 and u>i by 770 and r^, respectively, we can solve the equations for w0 and u>i to obtain

wo{r) = V0e T, Wi(r) = {fJi-Bf,0r)e-T

(5.3.10)

If we denote TJO(0) = p,0 and Ci(0) = /i,, from (5.3.9) we obtain /Jo = 0,

/JI = Sfjo-

(5.3.11)

Finally, we use the fact that the approximate solution to the original problem (5.2.1) is given as a sum of bulk and initial layer solutions. Thus

So(0)+«'i(0) + i)o(0) + e7j1(0) = n, u)o(O) +ewi (0)+tZ)0(0) + ew1(0) = 77, which gives in turn

Mo + «/ii + Mo + e/ii % + ejji + % + e?7i

= £», =

77,

Chapter 5. Singular-singularly perturbed equations

139

and it follows that the initial values for Eqs. (5.3.2), (5.3.3) and (5.3.8) can be taken as

Mo = M, Mi =

Mo = 0.

~ST],

Mi =

Vo = 0 , fji =

Sr],

Vo = T),

-Sfj,,

r)\ =

Sfj..

(5.3.12)

Combining Eqs. (5.3.5), (5.3.9) and (5.3.10) with (5.3.12) we have, finally,

v(t)

= e 0 (t) + evi(<) =

w{t)

= w0(t) =

v(T) W(T)

(n + e(-Sr, +

S2iit))e-At,

+ewi(t)

-eSne-At,

= V0{T) +evi(r) = eSrje~T, = WO{T) + ewi(r) = {ri + i{Sii- Br,r))e-T

(5.3.13)

These formulae are in perfect agreement with those derived in the previous section by the direct expansion of the exact solution and given in (5.2.15) and (5.2.18). We note, however, that the standard asymptotic analysis leads to the fully expanded bulk solution. The question arises whether there is an asymptotic expansion method which would lead to the bulk solution as given in (5.2.16). The answer is affirmative: in the next section we will introduce the compressed asymptotic expansion method and show that it has the required property.

4

Model system: compressed asymptotic expan sion

We will now make a number of assumptions defining a modified asymptotic expansion which may be called compressed to account for the fact that different levels of ap proximation are compressed into a single function. Such procedure dates back to the method of Chapman and Enskog in kinetic theory. There is extensive literature on the subject (see e.g. [20]), concerned mainly with the nonlinear Boltzmann equation.

Singularly perturbed evolution equations

140

The application of the procedure to linear evolution equations was developed by Mika [58], and to ordinary differential equations by Mika and Palczewski [65, 66]. The basic features of the method are: (a) The bulk approximation to v is not expanded into powers of e so that v is con sidered as 0(1). This means that, in some sense, all orders of approximation of v are compressed into a single function. In the sequel we will use the notation p for a given order approximation to v to stress the special role that this function plays in the asymptotic expansion for singular-singularly perturbed equations. (b) The bulk approximation to w is explicitly written in terms of p and expanded into powers of e . This is clearly possible for autonomous systems. Let us now see how the above assumptions modify the asymptotic procedure employed in the previous section to find the approximate bulk solution. The assumption (b) tells us that there is a function 2D such that w(t) = W(p(t)).

(5.4.1)

Expanding 2D into powers of e we have 2D(p)=2D 0 (p)+e2D 1 (p) + . . . .

(5.4.2)

With this notation we can now write the equation for p in the form dtp + Ap + S(2D0(p) + e9Di(p) + ...) = 0.

(5.4.3)

This equation plays a pivotal role in the compressed method. We see from it that, although the function p is not expanded, its time derivative is effectively expressed as a power series in e. In fact, we have dtp = p\0 + ep\1 + ...,

(5.4.4)

where

Mo = -Ap-SWo{p), dtp\k = -S2D fc (p), k = 1,2,....

(5.4.5)

Chapter 5. Singular-singularly perturbed equations

141

This feature of the compressed method (and of the original Chapman-Enskog method) may look like a contradiction and it is usually discussed in literature in rather vague terms. With our simple model system of equations we hope to give a convincing explanation of this phenomenon. Some additional comments are given in Remark 4.1 at the end of this section. To find 2Dfc we have to rewrite the equation for w. First we notice that from Eqs. (5.4.1) and (5.4.4)

edtw

= e{dpW0 + ed.W, + ■ ■-){dtp) =

e(dpW0 + edpWi + ■ ■ ■) (dtp\0 + tdtpU + ■ ■ ■).

(5.4.6)(5.4.6)

With the above formulae from the second equation in (5.2.1) we obtain the following equation

-e{dpW0 + edpWi + ■ • -){Ap + {SW0 + eSWi + ■ ■ -)(p)) +Sp + B{W0 + eW1 + ...)(p) + *{WQ + eW1 + ---){p) = 0.

(5.4.7)

From this we can now determine recursively the functions Wk which are then inserted into Eq. (5.4.3). In particular, we have 2B0(p) = 0,

W1(p) = -Sp,

(5.4.8)

so, up to O(e) terms, we obtain the following equation for p dtp + Ap-eS2p = 0}

(5.4.9)

p(t) = p(0)e ( -^ + £ S 2 ) (

(5.4.10)

whose solution is

It remains for us to find the initial value p(0). Due to the existence of the initial layer solution we cannot expect p(0) to be independent of e. Thus, up to 0(e) terms we write p(0) = Mo + e^i,

Singularly perturbed evolution equations

142

where p,0 and /Ui have to be determined with the help of initial layer functions. As in the previous section we obtain fio = fi,

pi =

-Srj,

so that p{t) = {fi - eSr))e{-A+(S2)t

(5.4.11)

From Eq. (5.4.8) we have w(t) ~ —eSp, so that omitting the term of order e2 we obtain w{t) ~ -iSp.e{~A+tS2)t

(5.4.12)

Both p in Eq. (5.4.10) and w in Eq. (5.4.12) are identical to v and w derived directly from the exact solution without expanding the exponential factor and given by Eq. (5.2.16). We see that the compressed asymptotic expansion indeed leads to a better approxi mation of the bulk solution valid over a longer period of time. At the same time this simple model system explains what is the actual meaning of the assumption (a). It prevents us from expanding the function v into a power series in t so that it stays as a far more sophisticated (and accurate) function of e than a polynomial. The quality of the approximation supplied by the compressed expansion becomes dramatically more important if the constants S, A and B are replaced by operators and Eq. (5.2.1) becomes a system of evolution equations. Then Eq. (5.4.9) is a much more sophisticated equation than the system of equations (5.3.2), (5.3.3). The drawback of the standard asymptotic procedure is removed if the original system of equations is modified by rescaling various terms. We will consider such a modified system in the next section. Before doing this, however, we present another view of the assumptions of the compressed asymptotic method and an explanation of the so-called contradictions related to them. Remark 4.1 Let us consider once again Eq. (5.4.1) which is a symbolic version of the assumption (b). We see that it amounts to taking p as a new independent variable or, in other words, to writing the original system in the (p, iu)-phase space. Then the assumption (a) is a statement of the obvious fact that the independent variable is not expanded into powers of e. The price we have to pay is that the notions of

Chapter 5. Singular-singularly perturbed equations

143

the time derivative and the initial value of p do not make any sense in the phase space. Hence, the so-called contradiction of the Chapman-Enskog method that the time derivative and the initial value of p are expanded into powers of e despite the fact that p is unexpanded are caused by the fact that the phase space and the physical space descriptions of the system are mixed up. To clarify this confusion we note that there exist operators acting on p in the phase space such that their realization in the physical space coincide with evaluating the time derivative and the initial value of p and these operators are expanded in powers of e (see [6]). ■

5

Modified model system

To explain how the standard approach works when the system is scaled in a different way, we change the scaling in the model system (5.2.1) and write it in the form

tdtv + eAv + Sw = 0, e2dtw + eSv + eBw + w = 0.

(5.5.1)

The same initial conditions as in Eq. (5.2.2) will be adopted. The characteristic equation is e2A2 + (1 + eB + e2A)\ + A - S2 + eAB = 0, with the discriminant E = [l + 2eB + e2(B2 -2A

+ 4S2 - 2e 3 5 3 )] ^

The eigenvalues are

11 /Z11

A, X2

2

2 \e i/1

= -2{72

B B

AA

e

J

B

+

J

A\

+

E E

2e2' E

A

)~2Tr

The general solution to the system (5.5.1) is written in the form identical to that given by Eq. (5.2.6) with 7l

= - | ( A , + A);

i = l,2-

Singularly perturbed evolution equations

144

The constants of integration ct\ and a 2 are obtained from the initial conditions as in Eq. (5.2.8). We will not state here the exact values of at and a,7; but give only their expansions up to terms of order e. We have

«!

ss [i — eSr],

a2

w eSri +

e2(S^-BSr,),

and

£*i7i »

-e5^,

0272 »

r? + «M-

As before, we have to keep terms proportional to e2 due to the fact that e appears in the denominator in A2. The exponential factor e A,t can be written as gAlt K

e(S

2

-X)((1 _

S2t)

tBS2t)

(5 5

2)

or 0 A,t

(S2-A-eBS2t)

^

(5.5.3)

Using Eq. (5.5.2) we obtain the first order approximation to the bulk solution in the form

v(t)

«

(n - ((Sr! +

w(t)

«

-€Sfieis2-A)t

BS2fit))e<s'-A*, (5.5.4)

With Eq. (5.5.3) we have

{s2 A tBs2 Sv)eSV)e - - »,

v(t)

«

(ji -

w(t)

«

-eS^s2-A-tBs2^.

BS2)(

(5.5.5)

To expand the exponential factor eA2' we introduce the new time variable r = 1/f2, similarly as in Section 3, except that e is replaced by e2 Expanding the resulting exponential factor el X2T into power series we obtain

Chapter 5. Singular-singularly perturbed equations

145

e£2A2T « e"T (1 - CBT). As a result the first order approximation to v and w is

V(T)

W tSr]e T,

w(r)

»

{r) + t{Sn- Br)T))e~T

(5.5.6)

Now we repeat, almost without modification, the reasoning employed in Section 3 to derive a standard asymptotic approximation. Writing

v w v0 + evi, w ~ w0 + ew1,

(5.5.7)

we obtain

0(1) O(i)

wo = 0, wi = -Sv0, dtv0 + {A- S2)v0 = 0, : w2 = -Svi + BSv0, 8tVi + {A- S2)vi + BS2v0 = 0.

(5.5.8) (5.5.9)

Using the same notation as in Section 3 we can write the solutions to the above equations as w0(t)

= 0,

Wo(0 = «H(0

=

fioe{s2~A)\ -5/ioe(52-^,

v,{t) = (ft - BS2p.0t)e^-A^

(5.5.10)

To find flo and fix we have to turn to the initial layer solutions v and w and write the original system (5.5.1) in terms of r = t/e2

dTv + e2Av + eSw = 0, dTw + eSv + eBw + w = 0.

(5.5.11)

Singularly perturbed evolution equations

146 Then we take

v « w «

Va + tih, wg + ewi.

(5.5.12)

Inserting Eq. (5.5.12) into the system (5.5.11) and comparing the terms of the same order in e we obtain

0(1)

: dTv0 = 0,

0(e)

dTw0 + wo = 0, : dTVi + Sw0 = 0, dTuii + Sv0 + Bw0

(5.5.13) wt = 0.

(5.5.14)

As previously, we easily find the solution to the above equations to be W0(T) = ij0e~T,

V0(T) = 0,

U>I{T) = (fjl ~ BTT)0)e~T',

iii(r) = Sfj0e~\

(5.5.15)

where the initial values fjo and 771 together with p,0 and fii are to be determined by demanding that the sum of the bulk and initial layer solutions satisfies the original initial conditions with 0(e 2 ) accuracy. It is easy to see that it can be achieved by the following choice of the initial conditions for (5.5.10) and (5.5.15)

fro = p,

pi = -Sri,

Vo = v,

Vi = 5M-

(5.5.16)

With the above initial values we can now write

v(t)

= Vo{t) + evi(t)

w(t)

= (ii-e(SV + BS2T,t))eis2-A)t, = w0(t) + eu>i(t)

= V(T)

W(T)

-es^s2-A)t,

=

VO(T)+€SI(T)

=

eSr)e~T,

=

WQ(T) +

=

(r, + e(Sfi - BrjT))e-T

6WI(T)

(5.5.17)

Chapter 5. Singular-singularly perturbed equations

147

Comparing Eq. (5.5.17) with Eqs. (5.5.4) and (5.5.6) we see that, if the equation is of the form (5.5.1), then the standard asymptotic expansion method leads to a result of the same quality as the result obtained for the equation of the form (5.2.1) by the compressed method. However, the problem is that the two equations differ from each other. In some cases the equation of the form (5.5.1), rather than (5.2.1), describes the actual physical situation and the labeling of various terms in (5.5.1) corresponds to their true relative importance. In other cases the actual equation is of the form (5.2.1) and the rescaling is achieved by introducing the new time variable 9 = it so that Eq. (5.2.1) takes the form of Eq. (5.5.1). This approach, however, requires that the constant A in Eq. (5.2.1) be either zero or multiplied by e, since otherwise the standard asymptotic procedure will produce the expansion with all terms equal to zero. We shall discuss this in more detail in Section 7 in connection with the kinetic equation. Even if Eq. (5.5.1) represents the reality, we may still ask ourselves whether the compressed approach might lead to improvements. To answer this question let us return to Section 4 and adapt the compressed approach, described there, to the present case. Using the same notation as in Section 4 we obtain the equation for p dtp + Ap + S{Wl + eW2)(p)=Q

(5.5.18)

where already we have taken into account that 2D0 = 0. The functions 2Di and W? are obtained from the second equation in (5.5.1) in which the term e2dtw is 0(e 3 ) and can be neglected in the lower order approximations. We have Wl(p)

= -Sp,

W2(p) = BSp,

so that Eq. (5.5.18) becomes d,p+(A-

S2 + eBS2)p = 0,

(5.5.19)

whose solution is p(t) = p(0)e<s2-A-
(5.5.20)

The initial value p(0) can be found by considering the initial layer solution as it was shown in other cases. It is not difficult to see that p(0) = n - eSi), so that

Singularly perturbed evolution equations

148

2

p{t) = {n-eSr1)e^-A-tBS^\ -eBS )t

(5.5.21)

from which we have 2

-tBS )t w{t) « eWlP = -eStie{s2-A-(BS2)t

(5.5.22)

We see that, again in this case, the compressed approach leads to a better approxi mation in which the exponential factor remains unexpanded and only the exponent is expanded. It can also be checked that even if A is not multiplied by e, the compressed method gives a sensible asymptotic expansion, whereas the standard method fails.

6

Singular-singularly perturbed evolution equa tions: compressed approach

In Section 1 we have applied the standard asymptotic analysis to a singularly per turbed equation (5.1.2). Having seen how the application of the compressed method improves the asymptotic approximation in the case of a simple model, we can now see what happens in the general case. The bulk solution Ci'w) will now be considered as a function pN of order zero and the function u / ^ ' assumed to be of the form u>W{t) = jrtnWnPN(t),

(5.6.1)

n=l

where 20„ are time-independent bounded linear operators acting from V to W. The first equation in (5.1.13) is now replaced, not by a system of equations as in (5.1.17), but by a single equation N

dtPN = PBPPN

+ Y, tnPBQWnPN. n=0

The derivative terms in the second equations in (5.1.17) is written as N

N

edt 53 enWnPN = n—n

n

e^enWndtpN. n=0

edt J2 e MnPN = t £

enWndtPN.

On the other hand, dtpN is given by (5.6.2) so that On the other hand, dtpN is given by (5.6.2) so that

(5.6.2)

Chapter 5. Singular-singularly perturbed equations

N

N

edtY/enWnPN

149

/

N

= tYJ^wn[PBPpN

71=0

n=0

+

V

\

YJ^PBQWTAPN n=0

/

JTen(wn-iPBP+YiWkPBQWn-l-k)pN.

=

n=l

V

k=0

(5.6.3) I

In the above expression all terms of order higher than N have been omitted. Now we insert (5.6.1) and (5.6.3) into the second equation in (5.1.13). Omitting pN we obtain the following relation for the operators 2Dn N

j

n-1

£ e" Wn^PBP + £ WtPBQWn-! n=l

V

k=0 N

N

= (QBP+J2tnQBQW)n-1 + QCQY,enWn n=l

= 0.

(5.6.4)

n=0

Comparing terms of the same order in 6 we obtain from (5.6.4) the system of equations in which the first two equations have a different structure from that of the remaining ones

2D0 QCQ2Di

= 0, = -QBP, n-l

QCQWn = W^PBP-QBQWn^

+ ^WkPBQWn-Lk,

(5.6.5)

k=\

n = 2,3,...,N. The operators 2Dn can be evaluated iteratively since the operator QCQ is invertible on the subspace W To supply the initial value for pN, which is written as N

UN

= £ £»&,, n= 0

we have to use the initial layer solution V^N\T) for which the initial value is to be calculated from the integrals in Eq. (5.1.24). We take

s(w)(o) = £ * n M ° ) = !>»<""> 71 = 0

71=0

Singularly perturbed evolution equations

150 with p0 = 0. Since

p w ( 0 ) + c W ( 0 ) = /i, we must put p,n = —fin',

n = 1,2,..., TV.

From the above formulae it is seen that the initial values up to the first order will be, in the present case, exactly the same as given in Eq. (5.1.30) for the standard approach. The proof of the asymptotic convergence of the compressed method follows very closely that for the standard expansion in Section 4.3 and will not be reproduced here. To appreciate the difference between the two asymptotic procedures described in this section, let us write the resulting equations for p0 and p\. For p 0 w e have dtPo = PAPp0,

p0(0) = p,

(5.6.6)

and for pt dtfn = {PAP - ePAQ{QCQ)-lQAP)Pu

(5.6.7)

with the initial condition pi(0) = p-ePAQ(QCQ)-1r1.

(5.6.8)

In many applications the operator PAP is very simple. For instance, in the case of the Boltzmann equation for neutrons it describes the attenuation of neutrons. On the other hand, PAQ and QAP are first order differential operators so that the second term in (5.6.7) is the Laplace operator. It is clear then that (5.6.7) will give a much more sophisticated description of the behaviour of the function v than the combination of the equations (5.1.21) and (5.1.22) obtained with the standard approach. In some cases it is possible to achieve a similar result by rescaling the original equation as in the case of the simple model. It seems, however, that the compressed approach is much more straightforward and it will be applied to all specific singular-singularly perturbed evolution equations considered in this book.

Chapter 5. Singular-singularly perturbed equations

7

151

Singularly p e r t u r b e d linear kinetic equations

In previous sections of this chapter we considered the singular-singularly perturbed evolution equations with bounded operators. We also introduced the model system of ordinary differential equations to explain the significance of the compressed approach. In this section we will present to the reader linear kinetic equations, perhaps the most important example of singular-singularly perturbed evolution equations. They differ from the evolution equation considered so far in that the operators involved are, or might be, unbounded. A generic linear kinetic equation can be written in the following form [20, 40] dtu = Au + Su + Cu, (5.7.1) where u is the particle distribution. A kinetic equation describes the behaviour of particles which, for all practical pur poses, do not collide with each other but only with particles of the host medium. The operator A is, in all practical cases, bounded and describes the attenuation and/or production of particles, whereas S is always an unbounded differential oper ator related to the streaming of particles. Finally, C is the collision operator which is bounded or unbounded depending on what kinds of particles are involved. For example, in the case of neutrons the operator C is bounded and in the case of Brownian particles or electrons it is unbounded. A more detailed description of various standard collision operators and their analysis are given in Chapters 7-9 where we perform the asymptotic analysis of particular types of linear kinetic equations. We also note that in certain physical situations the operator C contains a streaming op erator in velocity space. This occurs, for instance, in the kinetic theory of charged particles in the presence of a strong external field (see Chapter 10 and [52, 74]). There are two possible ways of treating a given equation which we know is perturbed (regularly or singularly). One way is to put the equation into a nondimensional form by choosing appropriate units and look for small parameters which then have a well defined physical meaning. The difficulty with this approach is that the choice of units is, in general, rather arbitrary and in the end, the crucial role is played by physical intuition. An alternative procedure is to identify, also on physical grounds, small or large terms in the equation and label them with e or \jt respectively and treat e formally as a small positive parameter. When considering the kinetic equation in this book we will adopt the second procedure. In the case of a kinetic equation we need labels indicating what mechanisms prevail when the system tends to equilibrium (see e.g. [13, 19, 20]). In most cases of physical interest the collision operator C plays the major role in that process. Thus we put a coefficient 1/e in front of C and Eq. (5.7.1) is written in the form dtu = Au + Su + -Cu

(5.7.2)

Singularly perturbed evolution equations

152

and adopt it as the standard scaling of Eq. (5.7.1). The operator C, whether it describes only collisions or some other phenomena, will have the properties described in Section 1 so that the single equation of evolution (5.7.2) can be treated with the projection method, and written as a system of evolution equations in subspaces V and W of the original space X in the following form

dtv

= V{A + S)Vv + V(A + S)Qw,

dtw

=

Q{A + S)Vv + Q(A + S)Qw + -QCQw.

(5.7.3)

For the generic kinetic equation we use calligraphic letters to denote spaces and operators since Roman letters will be used later for the Fourier transformed equation (see Chapter 6). The function v represents the zeroth-order moment of the particle distribution func tion or the so-called hydrodynamic part of the solution and w contains all the higher moments. It is a well known fact in kinetic theory that the hydrodynamic moment (or in some cases moments) gives the basic description of the fluid so that the main objective of the asymptotic analysis should always be to supply a reliable equation describing v. For an arbitrary linear kinetic equation such a role is played by the diffusion equation. Heuristically the diffusion equation may be derived from first principles, provided a relationship between the current and density is postulated. This is referred to as Fick's or Darcy's law, depending on the physical context. The unfortunate situation in kinetic theory is that the standard analysis presented in Section 1 fails to lead to the diffusion equation. In other words, neither of the equations analogous to (5.1.21) and (5.1.22), which in the present case will have the form, dtv0 + V{A + S)Vv0 = 0

(5.7.4)

and dtvx + V(A + S)Vvx = V{A + S)Q{QCQ)~lQ{A

+ S)VvQ,

(5.7.5)

represent a diffusion equation. The same is true for higher order approximations. This situation calls for a remedy. In this book we advocate the compressed asymptotic expansion method. Another possible approach is to rescale the equation so that the small parameter e appears in a different way. From the analysis of the simple system of singularly perturbed equations (5.2.1) we see that in the compressed approach the exponential factor eAl< remains unexpanded and only \\t is represented as a truncated power series in e. In the language of op erators, as in Eq. (5.7.3), it means that the compressed approach, at each level of

Chapter 5. Singular-singularly perturbed equations

153

approximation, leads to more sophisticated semigroups, whereas in the standard ap proach, the semigroups remain fixed and the sequence of nonhomogeneous equations is obtained. This seems to be a very strong argument in favour of the compressed approach. We now show how various asymptotic procedures work for kinetic equations. As mentioned in Section 2, the simple system of differential equations discussed in this chapter has the asymptotic properties almost identical to those occurring in kinetic theory. Moreover, if we choose constants A, B and 5 in a suitable way, the spectral properties, which are relevant to the asymptotic analysis, are also similar. Thus our study hopefully helps one to understand what can be expected in the asymptotic analysis of kinetic equations, and why. Here we shall discuss the implications of our results in some detail. A generic kinetic equation has the form (5.7.1) and Eq. (5.7.2) gives the standard version of scaling which goes back to the original Hilbert approach. With such a scaling of kinetic equations, the parameter e can be interpreted as the mean free path of particles, and the term 1/e in front of the collision operator indicates that the collisions play the main role in bringing the system to equilibrium. Let us write the system (5.2.1) in the form of a single differential equation in K2 dtu = AmU + Smu + -Cmu,

(5.7.6)

where the subscript m indicates that Eq. (5.7.6) is a model equation, u = [v, w] and "^m —

' Aa 0

0" Ba _

,

*->m —

' As S

S ' Bs

, cm =

'0 0

0 ' 1

■

(5.7.7)

Here we split constants A and B of Eq. (5.2.1) to account for the attenuation and streaming operators. We see that a{Cm) contains zero so that Cm has the basic spectral property of the general collision operator C which is relevant to the asymptotic analysis. In particular, the element 1 in Cm corresponds to the operator QCQ of Eq. (5.7.3). Constants AS,BS and S can be chosen in such a way that a(Sm) is purely imaginary, as in the case of the streaming operator <S and, according to our convention, they correspond to projections of 5 in (5.7.3). Let us recall that S contains the spatial gradient, so the term S2 in Eqs. (5.3.3), (5.4.9), or in Eqs. (5.5.8), (5.5.9), corresponds to the diffusion operator. The problem with the scaling in Eq. (5.7.2) is that the standard asymptotic technique, introduced by Hilbert, does not yield, in a natural way, the diffusion equation expected to be valid by physicists. This can be seen in Eq. (5.3.3), where the term with S 2 ,

Singularly perturbed evolution equations

154

corresponding to the diffusion operator, acts on v0 instead of vx and consequently Eq. (5.3.3) plays a role of a nonhomogeneous ordinary differential equation. One way to improve this situation at this stage (see e.g. [74]) is to multiply the equation for v\ in (5.3.3) by e and add it to the equation for v0 in (5.3.2). Next, if we add and subtract the term e2S2vi, we obtain dti/M + Avw

- eS2v^

= -e2S2vu

(5.7.8)

where s' 1 ' = v0 + evi. The right-hand side is 0{t2) and as such can be discarded at the discussed level of approximation. Thus we obtain the diffusion equation, identical to (5.4.9) with v^ corresponding to p, but the method is far from being straightforward. However, the first systematic way to remedy the situation came in the early 1970s, when the idea of time rescaling was proposed and subsequently developed in [49, 53, 41, 50, 14, 26] among others. As is seen in Section 5, the standard asymptotic anal ysis of the rescaled kinetic equation (5.5.1) (with t replaced by 9 = it) leads indeed to the diffusion equation in (5.5.8) and it has been rigorously proved for a large class of transport equations that the error of the asymptotic equation is of the order e, in accordance with the level of approximation in (5.5.8). However, this method has serious drawbacks, displayed even by our simple model. First of all, we see that it requires the constant A in (5.5.1) to be zero or of order e which corresponds to the requirement that the operator P(A + S)P should have the same properties. For a large class of kinetic phenomena we have no attenuation so that .4 = 0. However, the operator PSP, being zero for some kinetic equations (e.g. neutron transport (see Chapter 9), Fokker-Planck equations of Brownian motion and electron scattering (see Chapter 8)), fails to have this property in a number of physically relevant cases, as in the case of the linear Boltzmann equation with a strong external field discussed in Chapter 10. Another example is offered by the Fokker-Planck equation for en ergy relaxation of a hard-sphere Rayleigh gas, or vibrational relaxation of harmonic oscillators in the continuous (high temperature) limit which is dealt with in Section 8.6. Several remarks concerning this observation are in place here. Let us denote by H the singularly perturbed operator in the kinetic equation. Firstly, in all examples from the kinetic theory discussed in this book, the general solution of the equation HE0 = 0 can be written in the form

E0(x,O = f(x)eo(O,

(5.7.9)

where / is an arbitrary function. In most cases the operator V, is simply the collision operator C and e0 is the Maxwellian. However, for the Boltzmann equation with a strong external field this is not the case since then TZ contains the gradient with respect to the velocity (see Eq. (10.1.4)).

Chapter 5. Singular-singularly perturbed equations

155

The operator V is the spectral projection, associated with the eigenvalue 0 of R, onto the space generated by the functions of the form (5.7.9) and, due to properties of the operator 1Z, we have

(VU)(x,0 =

eo(oJu(x,OdC',

where E is the domain of admissible velocities. If we confine ourselves to the classical streaming operator, defined by

(Su)(x,Z) = Zdxu(x,Q, then the operator VSV is given (in three dimensions) by the formula

(VSVu)(x,S)

= £c(fl /\ 1=1

i

U

(x,Oeo(m'

U'MCW jj> |

One can easily note from this that the rescaling method is available whenever the first moments of e0 vanish (for instance when e0 is symmetric with respect to each coor dinate of £). This happens for standard transport phenomena, but not, for example, for the transport of electrons in the presence of a strong electric field as discussed in Chapter 10. The second remark refers to kinetic equations of the Fokker-Planck type. As we shall see in Chapter 8, for a number of them the complete set of eigenfunctions of the collision operator is a set of orthogonal polynomials and the streaming operator (more precisely the operator of multiplication by £) satisfies the classical three-point recurrence formula [75] £* n = a n $ n + i + fcn$n + c„*„_ 1 .

(5.7.10)

It can be checked that if bn (in particular b0) are zero, then the operator PSP is also zero. This is, for instance, the case of the Fokker-Planck equation of Brownian motion (Hermite polynomials) and electron scattering and neutron transport in slab geometry (Legendre polynomials). The coefficients bn are not equal to zero for Laguerre polynomials and, as we noted earlier, the rescaling cannot be applied in that case. In the neutron transport theory the operator VSV is equal to zero, but there is attenuation/production of particles so that A / 0 and for the rescaling method to be applicable we must multiply A by e. Physically this means that the diffusion approximation can be obtained by the rescaling technique only for relatively small attenuation/production. Another point worth mentioning is that the rescaling technique in the form presented here gives a diffusion approximation of order t. If we wanted to obtain an approxima tion of order e2, we would then have to use two terms of the asymptotic expansions

156

Singularly perturbed evolution equations

(5.5.7) and (5.5.12). Proceeding in the same way as when deriving Eq. (5.7.8) we multiply the equation for V\ in (5.5.9) by e, and adding to the equation for v0 in (5.5.8), we obtain dev(x) + {A- S2)v{1) = -eBS2v0 (5.7.11) which is a non-homogeneous diffusion equation for v^ = vQ + evi. That seems to be unsatisfactory. However, if we return to the original time t = 9/e, then the righthand side of Eq. (5.7.11) will be multiplied by e2 which can be discarded on this level of approximation. Thus the rescaling technique gives the diffusion approximation of order e2, identical to Eq. (5.4.9) with li' 1 ' replacing p, but again the approach is not exactly straightforward. The problems mentioned while discussing both the standard and the rescaling meth ods, can be dealt with in a natural way by the use of the compressed method, based on the Chapman-Enskog asymptotic procedure. We saw in Section 4 that this ap proach yields a much more sophisticated asymptotic solution (5.4.11), (5.4.12) than (5.3.13), obtained by the standard method, and unlike the rescaling method, can be used for arbitrary operators PSP and A. In the following chapters we apply the compressed method and provide its rigor ous mathematical justification for a number of kinetic problems including the linear Boltzmann equation with and without the external field, Fokker-Planck equations of Brownian motion with the Laguerre collision operator and the neutron transport equation. Our analysis is carried out in an abstract setting which makes it available to a variety of other linear kinetic problems, not specifically mentioned in this book. The compressed method has also been successfully applied to the nonlinear Carleman equation [71]. We briefly describe this result in Section 11.4. To complete this survey we note that in some cases the scaling (5.5.1) (or similar, with other combinations of powers of e) is derived by a dimensional analysis [13, 27, 52] and consequently the equation corresponding to (5.5.1) is written in real time. This is particularly visible for the full nonlinear Boltzmann equation, where the coefficients in front of the streaming and the collision operators have a physical meaning and are called the Strouhal and the Knudsen numbers, respectively. As an example of the scaling (5.5.1) in a linear theory we mention the linear transport equation with a weak external field (10.1.3). However, the justification of putting the parameter 1/e in front of the streaming operator in the general (particularly linear) case seems to be doubtful, as for most phenomena the Strouhal number is either of order 1 or of the same order as the Knudsen number [20]. There is, however, an interest in investigating such scalings in the full nonlinear Boltzmann equation where the relationship between the powers of e is expected to account for various degrees of disparity between microscopic and macroscopic scales of the process, see e.g. [13, 27]. In particular, the hydrodynamic limit of the full nonlinear Boltzmann equation is

Chapter 5. Singular-singularly perturbed equations

157

the compressible Navier-Stokes equations for the standard scaling (5.7.2) and the incompressible Navier-Stokes equations for the scaling (5.5.1), see [13]. We would like to stress, however, that the justification of a particular scaling is outside the scope of any mathematical theory and, as the scaling (5.5.1) is recognized in physics, we shall also discuss basic features of the introduced asymptotic expansions in this case. From the formal point of view, the standard method applied to (5.5.1) has all the drawbacks of the rescaling methods. Moreover, to obtain an approximation of order e2 we proceed as previously, but in the present case we are not able to use the same method to dispose of the inhomogeneity in (5.7.11) since we are now working with the actual time. However, it is interesting that in all the real cases discussed earlier, where the rescaling technique is applicable (i.e. when VSV = 0), the term VSQ{QCQ)~l

QSQ(QCQ)-1

QSVv0

(5.7.12)

which corresponds to the inhomogeneity in Eq. (5.7.11) is equal to zero [62] and consequently the diffusion equation gives an approximation of order e2 Unfortunately such an occurrence is not a general rule, as can be seen in the model discussed in this chapter, where we put A = 0 and B, S / 0. We can also apply the compressed method to (5.5.1), as was done in the second part of Section 5. It is seen that the compressed method gives a notrivial expansion even if V{S + A)V is of order 1. This operator, multiplied by 1/e, appears in the result ing diffusion equation and represents the convection and/or attenuation as in Eqs. (10.3.10) or (9.3.8). Another significant difference is when we go for an approxima tion of order t2 This time, instead of the inhomogeneous equation (5.7.11), we obtain the homogeneous evolution equation (5.5.19) which is more sophisticated. In kinetic theory Eq. (5.5.19) corresponds to a third order differential equation related to the Burnett equation (the third order term is given by (5.7.12) and its advantage seems to be doubtful). However, it is interesting to note that if the operator defined by Eq. (5.7.12) is equal to zero, then both the standard and the compressed method yield the same diffusion equation which provides an approximation of order e2 We note also that in the modified asymptotic procedure of Caflisch [19] the higher order approximation does not produce a third order equation but (in the linear cases) an inhomogeneous diffusion equation. We complete this section by pointing out differences between various asymptotic procedures, which are of more mathematical character. Firstly, in linear kinetic theory the singularly perturbed equation of the type (5.5.1) was derived from the generic equation (5.2.1) by the rescaling of time t —> et. Hence, both versions describe essentially the same phenomenon and it is interesting to know whether the asymptotic expansions corresponding to these two models are equivalent. In other words, let us rescale time in Eq. (5.2.1), find the asymptotic expansion of the solution in the new time variable, and return to the original time. Do we get the

Singularly perturbed evolution equations

158

asymptotic expansion obtained directly from Eq. (5.2.1)? In mathematical language we are asking whether the operations of time rescaling and asymptotic expansion commute with each other. Clearly, since the equations for the initial layer are identical, the initial layer terms will coincide. The situation is different, however, for the bulk part terms as easily seen if we compare formulae (5.3.5) with (5.5.10) and (5.4.11), (5.4.12) with (5.5.21), (5.5.22). It immediately follows that the standard asymptotic expansion produces completely different asymptotic expansions for real and rescaled times. This fact has been observed in [77], On the other hand, the compressed asymptotic procedure provides us with the same expansions (up to the time rescaling) so that we can say that the compressed asymptotic expansion commutes with the time rescaling or is stable with respect to the time rescaling. This can easily be explained if one notices that in the compressed method the variable p plays the role of an independent variable (see Eq. (5.4.2)) and thus it is not affected by the change of time. The second point we would like to emphasize here is related to the error estimates in the time rescaling approach. If we look closely at Eqs. (5.5.8), (5.5.9), we notice that to obtain zero level terms we need the relation coming from the first level of the expansion; similarly, to get first order terms we employ the relation from the second level, etc. This fact has its consequences in the error estimate. The error of the expansion on zero level, defined by = v0, --v0, yy:=v-v

z := z:=w-W w -wo, 0,

satisfies the system edty + eAy + Sz = -Sw0 - eS2v0, e dtz + eSy + eBz + z = -eSv0 - eBw0. 2

(5.7.13)

Using the approach of Example 6.4.1 we can prove that, due to the presence of the terms with v0, the solution of this system satisfies y = 0(1) and z = 0(1). This is unsatisfactory, since we expect that the zeroth-order terms of the expansion should approximate the solution with 0(e) accuracy and the expansion of the exact solution in Section 5 shows that it is so. The reason for this discrepancy is that, as we observed earlier, to find zeroth-order terms we used relations from the first level which were absent in Eq. (5.7.13). The standard way to remedy the situation is to go to the higher order approximation. For example, if we take the first order expansion defined by Eq. (5.5.17), then we can prove that the resulting error is of order e. Hence, taking for simplicity only the bulk part of v, we have |,bj-flo + «it||

=0(e).

But the term tv\ is of order e and finally we obtain ||t; - Boll = 0(e),

Chapter 5. Singular-singularly perturbed equations

159

as required. This procedure, when applied to the full kinetic equation, has a serious drawback. To explain this we note that in the real case the operators corresponding to S and B are unbounded. From Eqs. (5.5.8) and (5.5.9) we see that the higher the level of approximation the more iterations of B and 5 are involved, and consequently, the higher regularity of the data is required. In other words, to get O(e) accuracy of the expansion by the rescaling method we need to assume the regularity of the data which would be enough to obtain the 0(e 2 ) accuracy by the compressed method without rescaling.

Chapter 6 Hilbert space theory for equations of kinetic type 1

Introduction

As we mentioned in Chapter 1, a good asymptotic analysis of a problem should consist of two major ingredients: an algorithm providing a sensible asymptotic solution of the problem and the proof that this asymptotic solution is convergent to the actual solution of the original problem. In the previous chapter we described and compared several asymptotic procedures for the abstract initial value problem related to the kinetic equation of the form

dtu u(0)

= Au + Su + -Cu, e = u.

(6.1.1)

We also asserted that if the operators involved in Eq. (6.1.1) are bounded, then the asymptotic solutions produced by these procedures converge to the solution u and determined the rate of convergence. The model with bounded operators is not realistic since the streaming operator S and, in many cases, the collision operator C, are unbounded. In this chapter we shall show that for a large class of unbounded operators S and C the compressed asymptotic method of Section 5.6 provides an asymptotic solution which converges to the solution u of Eq. (6.1.1). For simplicity, and due to relevance in applications, we confine ourselves to the first level of expansion where the asymptotic solution solves a diffusion-type equation (see Sections 5.4 and 5.6). We shall also show that the error of approximation is then of order e2, in accordance with the 161

Singularly perturbed evolution equations

162 formal approach.

Unfortunately, it is common in applied mathematics that mathematically sound gen eral theory does not cover all phenomena envisaged by the formal considerations, and the theory developed in this chapter is not an exception. However, it caters for a variety of kinetic equations with both bounded and unbounded collision operators described in details in Chapters 7 and 8. Moreover, in our opinion, it identifies the main goals which should be achieved when one performs an asymptotic analysis of a kinetic equation which does not fit into the framework of this chapter and thus provides a plan to follow. Examples of this kind are demonstrated in Chapters 9 and 10. Finally, we note that all the considerations, carried out in this chapter for the com pressed asymptotic expansion, can be repeated with minor changes for the expansion of the Hilbert type. This may be important in some cases since the terms of the bulk part of the Hilbert expansion are defined by first order differential equations (compare Eqs. (5.3.2) and (5.3.3)) and these usually are solvable under milder conditions than the diffusion equation yielded by the compressed method. However, as we pointed out in Section 5.7, the Hilbert method has numerous drawbacks so that we decided to focus our considerations entirely on the compressed method.

2

Preliminary results

In this section we introduce basic notation and assumptions which will make possible the asymptotic analysis of the abstract kinetic equation

{dtu){t,x) u(0,x)

= {Su)(t,x) = u(x),

+ (Cu)(t,x),

f>0,ie(l, (6.2.1)

where both operators may be unbounded. To simplify the analysis we have omitted the operator A which appears only in the kinetic equation for neutrons. It will be reintroduced in Chapter 9 dealing with that case. The whole asymptotic analysis will be based upon the theory of semigroups which was presented in Chapter 3. For that we shall cast (6.2.1) into the form of an abstract evolution equation. To avoid additional difficulties related to a possible occurrence of a boundary layer we confine ourselves to two particular types of problems. We assume that, with respect to x, either we are dealing with the whole space so that fi = R", or with the so-called periodic boundary conditions (see Subsections 2.4.8 and 3.7.2) and then fi = [0,27r]" With some abuse of notation, the function (t,x, £) —> u(t,x, 0 will be treated as a

Chapter 6. Hilbert space theory

163

function (t,x) -> u(t,x) with values in some Hilbert space H (a space of functions of velocity variable) and for every t the value u(t) itself will be an element of H = L2(Q)0H = Z<2(fi,iJ), where ® denotes the tensor product of Hilbert spaces (see Subsection 2.3.11). We assume that C is independent of x in the sense that C = I®C where C is an operator acting in H. The operator S will be taken in the form

(5«)(x,0 = E((5Ik®%))«)(x,0, fc=i

where S^), k = 1 , . . . ,n are operators acting in H. Then the differential equation in (6.2.1) can be written as n

dtu = -($2 dXk ® S{k))u + (I ® C)u.

(6.2.2)

Let us introduce the space W = L2(Pn,H), where P = K in the free space case and P = Z in the periodic case. Precisely, in the latter the space L2 reduces itself to the space l2 of square-summable, //-valued, multi-indexed sequences. Since this will not lead to any misunderstanding, we shall use the same labels, P and L2, for both sets of parameters and both spaces, respectively. Also, in what follows, the phrase "for every p € P n ", when referred to the first case, is to be understood as "for almost every p e l n " We have already seen (see Subsections 2.4.7, 2.5.6 and 2.4.8) that the Fourier trans formation with respect to x, u := Fu, is, for both n = R" and fi = [0,27r]n, an isometric isomorphism of L2{Q,H) onto L2(Pn,H). Hence the analysis of solvability of Eq. (6.2.2) in L2(Q, H) is equivalent to the analysis of solvability of dtu = Su + Cii, where S := E ipkSw

in

(6.2.3)

L2(Pn,H).

k=l

We shall see below that under certain assumptions the variable p in the Eq. (6.2.3) can be considered as a parameter. This will allow us to discard restrictions due to the unboundedness of the differential operators dXk. Since from now on we shall work only with the transformed equation Eq. (6.2.3), in what follows we shall drop "hat" in the notation of the solution to Eq. (6.2.3). We now introduce basic assumptions which will be used throughout this chapter. Since we shall prove that it is enough to treat p as a parameter, we will be con cerned with the family of operators indexed by p £ P n acting in the space H and our assumptions refer to this case. Let P™ equal either R n or Z n We consider an operator (C, D(C)) and a family of operators {(5 P , £>(S p ))} pe p„. Bearing in mind the

164

Singularly perturbed evolution equations

applications we assume that p -> Sp is a linear mapping of P" into the set of closed operators in H so that n i

Sp--

(6.2.4)

Z^PkS(k), k=l

where all (S(k),D(S(k)))

l,...,n.

are closed linear operators for k

We denote

Ds := ft D(SW) k=l

and equip Ds with the norm 1

2

2

'T

M D s . = (\\u\\ H + J:\\S(k)u\\ Hy

(6.2.5)

Since each operator S(k) is closed, Ds becomes a Hilbert space. We also have \\Spu\\H

< M\p\\\u\\Ds,

UGDS,

(6.2.6)

for some constant M which is independent of p. Let the operators have the following properties: P . l C is a self-adjoint operator, generating an analytic semigroup of contractions, (Gc(0)(>o> in H. Zero is a simple isolated eigenvalue of C (see Subsection 2.7.3) with the eigenfunction m and sup Re{a{C) \ {0}} = - 7 < 0.

(6.2.7)

P.2 For every p € P", Sp generates a semigroup of isometries (G5 (t))f>oP.3 D{C) n Ds is dense in H and me

DsnD(C).

(6.2.8)

It follows that the operators (C + SP, D(C)nD(Sp)) are dissipative and, having dense domain, they are closable (see Subsection 2.6.12). We define (Tp, D(TP)) = (C + Sp, D(C) n D(SP)) and postulate that Tp has the following property:

(6.2.9)

Chapter 6. Hilbert space theory

165

P.4 For each p £ P n the operator Tp generates a C0-semigroup in H, denoted by (GrP(t))t>oBy the Trotter product formula (Subsection 3.4.3) G

^u

= hm (GC (1) GSp (£j)\

in H,

(6.2.10)

and the limit is uniform in t on bounded subsets of R + . This formula implies that (GTr(t))t>o is a contraction semigroup. We also require the following: P. 5 For each p G P" D(T„) = D(C) n D(SP)

(6.2.11)

Dr:=D(C)nDs

(6.2.12)

and

is a core of Tp. It is clear that DT, equipped with the norm \W\\DT--=(\\U\\2DS

+ \\U\\1{C))II\

(6.2.13)

is a Hilbert space. We can now introduce precise definitions of operators C and S which appeared in (6.2.3) and are now written without hats. We define C = / ® C on the domain D(C) = D(I ®C) = L 2 (P") ® D(C) and (Su)(p,t;) = (Spu)(p,t;), treated as an operator in K, with domain D(S) = { u £ « ; u(p) € D(SP) for p g P", Su e H}.

(6.2.14)

By P.l and Subsection 3.4.4, (C,D{C)) generates in H a semigroup of contractions and, since C = / ® C, we conclude by Subsection 3.4.4 that C also generates a semigroup of contractions. It follows that S satisfies all assumptions of proposition 3.2.1, hence the following result is valid:

Singularly perturbed evolution equations

166

Proposition 2.1 The operator (S, D(S)) generates a semigroup of isometries in H, say (Gs(t))t>o, such that for every p € P" and u G % we have (Gsu)(p) = GSpuv in H.

(6.2.15)

Let us define the operator T by the formula (Tu) (p) = (Tpu) (p) for p 6 P n acting on the domain D(T) = { « € « ; u(p) € D(TP) for p € P", Tu € U}. Using the above proposition we see that all assumptions of Theorem 3.4.1 are satisfied with Do = DT. Thus the following result holds true: Theorem 2.1 Operator (T,D(T)) generates a semigroup of contractions in H, say (Gr{t))t>0i given by the Trotter formula G r ( 0 « = H m ( G c ( l ) G 5 (£))"«,

uen.

(6.2.16)

This semigroup satisfies (GTu)(p) = GTpu{p). for p £ Pnand every u £ H.

(6.2.17) m

With this result we can carry out the asymptotic analysis in the space H, ensuring that the estimates are in a suitable sense uniform in p. As we saw in Section 5.1, the main role in the compressed (modified Chapman-Enskog) asymptotic procedure is played by a decomposition of (6.2.1) into two subspaces. One is the eigenspace of C corresponding to the zero eigenvalue, and the other is its orthogonal complement. Since these subspaces are invariant under the Fourier transform with respect to x, the same is true for the transformed equation (6.2.3). Here we shall prove that under assumptions P.1-P.5 such a decomposition is justified and investigate some properties of the resulting system of equations. Let us consider the following evolution equation in H: dtu = Spu + Cu

(6.2.18)

for some p € P n We define P to be the operator of the orthogonal projection onto the kernel of C, PH = N{C) = V, and Q = I - P, QH = V1 = W

Chapter 6. Hilbert space theory

167

Let us fix p e P" and let u e D(TP). We then have Pu = a m for some scalar a and (6.2.8) yields Pu € D{SP) so that PD(TP) C D{TP). Moreover, Qu = u - Pu e D{TP) and since D{TV) = D{SV) n D(C), we obtain Tvu = SpPu + CQu + SpQu. We introduce the notation v = Pu,

w = Qu.

(6.2.19)

The projection operators commute with dt (see Eq. (2.5.3)), hence applying P and Q to both sides of (6.2.18) and taking into account that PC = 0 and CP = 0, we obtain the following system of equations (compare Eq. (5.1.10)):

dtv dtw

= =

PSpPv + PSpQw, QSvPv + {QCQ + QSpQ)w,

(6.2.20)

where we have left spurious notation for the projection operators for the sake of symmetry. We have the following: Proposition 2.2 A pair (v,w) G V x (D(Tp) f~l W) solves (6.2.20) if and only if u = v + w e D{TP) solves (6.2.18). Proof. Deriving (6.2.20) we actually proved that if u is a solution to (6.2.18), then (v,w) solves (6.2.20). Let (v,w) 6 V x (D{TP) n W) solve (6.2.20). Then v, treated as an element of H, satisfies v G D(TP) = D(C) n D(SP). Similarly, w e DTDW C D(C) n Ds so that u = v + w € D(C) D D$ and the assertion follows by adding both equations of (6.2.20). ■ To perform the asymptotic analysis we will need certain results on a generation of semigroups by projected operators in W Since C is self-adjoint, the projections P and Q are orthogonal and we can apply Theorem 3.2.1 to the operators Tp, C, Sp, obtaining the following corollary: Corollary 2.1 The operator QTpQ generates a semigroup of contractions in W, say (GQT a(t))t>a, which satisfies for t > 0 \\GQTpQ(t)\\w<e-

(6.2.21)

Singularly perturbed evolution equations

168 where 7 is defined in (6.2.7).

Proof. The Trotter formula gives for each u &W GcrrrQ(t)u = Urn [GQCQ

( £ ) GQSpQ ( £ ) ) " ,

(6.2.22)

where {GQCQ{t))t>o and (GQSpQ(t))t>o are the semigroups generated by QCQ and Q5pQ, respectively. Now, by (6.2.7) we have sup a(QCQ) < - 7 < 0 and, since QCQ is self-adjoint, the spectral representation (2.7.21) of (Ggco gives

HCWOII < e-*

(6.2.23)

Since GQSPQ is a semigroup of contractions, Eqs. (6.2.22) and (6.2.23) imply the estimate (6.2.21). ■ Remark 2.1 In the asymptotic analysis we shall be concerned with the operator of the form TtxP = C1C + Sp,

(6.2.24)

which generates a semigroup of contractions similarly as in Corollary 2.1. Since supi?ecr(e- 1 QCQ) < - 7 / e and QCQ is self-adjoint, the corresponding semigroup, say {GQTK Q(0)OO> satisfies \\G
3

(6.2.25)

Properties of terms of expansion

As we saw in Section 2 the asymptotic analysis of the singularly perturbed abstract kinetic equation in U = L2{0.,H) is equivalent to the asymptotic analysis of the system of equations with parameter p £ P n in the projected form:

dtv edtw

= PSpPv + PSpQw, = tQSpQw + eQSpPv + QCQw,

Chapter 6. Hilbert space theory -

.

v(0) w(0)

169 o

o

= v := Pu, = w:=Qu,

UIEH.

(6.3.1)

The full formal compressed asymptotic analysis of the problem analogous to (6.3.1) with Sp replaced by bounded operator B, was described in Section 5.6, but since the formulae for the terms of the asymptotic expansion on the second (diffusion) level of approximation are considerably simpler than the general ones, we shall recall them here. Since it will not cause any misunderstanding, we shall also omit in most cases the superscript n which in Section 5.6 indicated the level of approximation. The solution of (6.3.1) is sought in the form

v(t)

= v[t) + v(r)

w{t)

=

w(t) + w(r),

(6.3.2)

where r = t/e. We approximate w, v and w by the truncated power series in e:

w =

w0 + eilii + 0{e2),

v = i>0 + tVi + 0(e2), w = w}0 + «Z>i+O(e 2 ).

(6.3.3)

According to the compressed asymptotic procedure the bulk part of the hydrodynamic part v is not expanded. These expansions are inserted into (6.3.1) and, truncated on the e2 level, yield the following set of equations (see section 5.6): The bulk part: dtvw

=

PSvPv(l)

g (D(0)

=

v - ePSpQ(QCQ)~lw, ",

-

l SPQ\tPSpQ(QCQ)- QSvPv^\

p

(6.3.4)

and

wo = <j>, =

0 -{QCQ)-lQSpPvW. u^.

(6.3.5) (6.3.6)

Singularly perturbed evolution equations

170 The initial layer part: tJ0(r)

s

0,

WO(T)

=

GQCQ{T)W,

VX{T)

=

PSPQ{QCQ)-1WQ{T),

(6.3.7) (6.3.8) (6.3.9)

■M,

and

9ru>i

= QCQwi + QSpQwo,

wi{0)

= {QCQ)-lQSpP°v.

(6.3.10)

The initial conditions for the bulk and initial layer part have been found in the manner described in Chapter 5. The equation in (6.3.4) is the basic goal of the asymptotic procedures for the kinetic equations discussed in the book and, as shown in Proposition 3.1, always has the form of the second order parabolic equation referred to in the literature as the diffusion equation. Thus the approximation of the solution u = v + w to (6.3.1) takes the form 1

vW(t)

\° = vW{t) + wi(t/e) = v(l)(t) + tPSpQ{QCQ)-lGQCQ{t/t)w, e)w,

(6.3.11)

where v^ is the solution of the problem (6.3.4), and wM{t) = we(t/e) + e(wi(t) + Wtit/e)),

(6.3.12)

where the terms are defined in (6.3.8), (6.3.6) and (6.3.10). Below we shall prove that the terms of the asymptotic expansion are well defined by (6.3.4) and (6.3.10) and that u' 1 ' and u^ 1 ', defined by (6.3.11), (6.3.12), approximate u with an error of order e2 To do this we need to find an equation satisfied by the error which is defined as

y(t) z(t)

= w(t)-[fi(1)(i)+oii(tA)]. = w{t)-[wo(t/e) + e(ffl 1 (0+wi(t/e))].

(6-3-13) (6.3.14)

Inserting (formally) the error to the first equation of (6.3.1) and taking advantage of (6.3.4) and (6.3.9) we obtain

Chapter 6. Hilbert space theory

dty

171

= dtv - dtvW - dTv\ =

PSpPv + PSpQw - dtv{1) -

=

PSpPy + (LPSPPV} -cV(1) -

=

ePSpQw0

+ PSpQz + PSvQwa + tPSpQw, + ePSpQu>i

PSpQw0

PSpPy + PSpQz + ePSpPiJ! + PSpQwv

(6.3.15)

The equation for z, with the help of (6.3.6), (6.3.8) and (6.3.10), looks as follows

dtz

= dtw =

dTwo — tdtW\ — dTxb\ e QSpPy + QSpPv + eQSpPv! + QSpQz + QSpQw0 +tQSpQw,

+ eQSpQwi + -QCQz + -QCQw0

+ QCQw1

—d T wo — edtWi — 3Tu)i e =

QSpPy + QSpQz + +eQSpQwl

-QCQz

+ eQSpPvi + eQSpQuii - edtw1.

(6.3.16)

We shall need explicit formulae for various projections of the operator Sp. Let us recall that PH = V = Lin{m} and Q = I — P First, for any u £ H we have

Pu

=

m(u,m)//

Qu

=

u — m(u,m)//.

(6.3.17)

Using the definition of Sp we obtain for w € D(SP) and p = ( p i , . . . ,p„) £ P" the following formulae:

(PSpPu)

=

im(u,m)H^2pk{S{k)m,m)H,

{PSpQu)

= im^Pk ,Pk ( ( % ) « , m ) w - (u, m)H{S{k)m,m)H)

(6.3.18)

m)u,

,

(6.3.19)

/t = l

{QSpPu)

= i(u, m ) „ Y, Pk (SWm

~ m(5(*)m. m)w) .

(6.3.20)

fc=I

(QSpQu)

= i^Pk

(S{k)u - m(S{k)u, m)H - (u, m)m)HS{k)m

-m(Swm,m)H) m)w)

(6.3.21)

Singularly perturbed evolution equations

172

Since the operators PSPP and QSPP are defined on a one-dimensional space, they are continuous (Subsection 2.6.4). For PSPQ we have the following lemma: Lemma 3.1 The operator PSPQ, defined on D(SP), is closable and its closed exten sion is a bounded operator on H. Proof. Since Sp is conservative and from (6.2.8) we have for u € D(SP) PSpu = (Spu, m ) „ m = - ( « ,

Spm)Hm,

it is clear that PSP is closable and can be extended by continuity onto H. The state ment for PSPQ follows from the formula PSPQ = PSP - PSPP and the continuity of the latter. ■

Remark 3.1 In what follows we shall denote the continuous extension of PSPQ by the same symbol. With this convention we see that the term vi of the expansion, given by (6.3.9), is well defined. ■ In the next step we analyze the solution to (6.3.4) and in particular we prove that it is indeed a diffusion equation. Proposition 3.1 There exists a vector function n

P -> W(P) ; = S ^kPk k=l

with real coefficients w\., k = 1 , . . . , n, and a complex matrix function n

p -> Cl{p) := £

QMptPi

(6.3.22)

*:,i=l

such that the solution v^{t)

to (6.3.4) is given by the formula vt-1){t)=e^-^)tvw{0).

(6.3.23)

The function O satisfies the inequalities Re n(p) > oj.WQSpPWl > uj[\p\2 and

(6.3.24)

Chapter 6. Hilbert space theory

173

I«(P)I < \\(QCQ)-l\\H\\QSpP\\H\\PSpQ\\H

< cu2\p\2

(6.3.25)

for some positive constants wx, u>'1 and u>2, independent of p. Proof. First we note that by Remark 3.1 for each p e P n the operators PSPP and PSpQ(QCQ)~lQSpP are well-defined one-dimensional operators. From Eqs. (6.3.18)(6.3.20), and by the linearity of p -¥ Sp, we see that there exists a vector function p -> a{p) and a matrix function p -> Q,(p) such that for any v € V PSpPv = a{p)v and PS,Q{QCQ)-lQS,Pv

= Cl(p)v.

Therefore (6.3.4) is an ordinary linear differential equation with a general solution given by (6.3.23). Since Sp is conservative, taking an arbitrary v € V satisfying \\v\\n = 1 we have, Re a{p) = Re [PSpPv,v)H

= Re (Spv,v)H

= 0,

so a(p) = iw(p) = i E ^kPk with real coefficients u>kfc=i

Moreover, due to orthogonality of the projections P and Q and to the self-adjointness of C, and again by the conservativity of Sp, we obtain

ReCl(p)

= = =

(PSpQ(QCQ)-lQSpPv,v)H -Re{{QCQ)-'QSpPv,QSpP)H -Re (wp,QCQwp)H = -Re {QCQwp,wp)H

Re

> 7lKI| 2 w , (6.3.26)

where wp = (QCQ)'lQSpPv and 7 is defined by Eq. (6.2.7). In the last inequality we used the fact that for negative definite operators the spectral bound is equal to the least upper bound of the numerical range. The operator Sp is linear in p and v is fixed so that p -> QSpPv is a linear operator defined on P" with values in W and its range is a finite-dimensional subspace W of W Since (QCQ)-1 is a one-to-one operator in W, its restriction to W is an isomorphism onto its (finite-dimensional) range, therefore \\{QCQ)-lQSpPv\\H

> ii\\QSpPv\\H

Singularly perturbed evolution equations

174

for some constant fi which is independent of p. Moreover, we can find v e V, \\v\\H = 1, such that \\QSpPv\\H = \\QSPP\\H.

(6.3.27)

Thus the formulae (6.3.26) and (6.3.27) give the first inequality of (6.3.24) with wj = y/x. The second follows directly from the continuity of p —> \\QSpPv\\H and the finite dimension of P" To prove (6.3.25) we use the continuity of the operators PSPQ, (QCQ)~l and QSPP to obtain

\Q(p)\

= <

(PSpQ(QCQ)-1QSpPv,v)H \\(QCQ)-1\\H\\QSPP\\H\\PSPQ\\H

where again we used the fact that p —> Sp is a linear function. Let us go back to the equation in (6.3.4). Since V = Lin{m), be written as a product

■ 1

the function c' ' can

^ (1) (*,P.O = P(<-P)m(0, where m is the function spanning V ft will be explicitly given in concrete applica tions. With this notation the equation in (6.3.4) can be written as a scalar equation

d

tP - M 5Z P*w* ) P ~ e I H Qi.kPiPk P,

(6.3.28)

which, according to Proposition 3.1, is the Fourier-transformed diffusion equation with a convection term. For future reference we shall write this equation in the original, differential form. We shall denote the solution of this equation by the same character p which will not lead to any misunderstanding. With this convention we have: dtp = £uj k d X k p + t £ A:=l

ntikdlmp,

(6.3.29)

l,k=l

The central topic of the applied part of this book is the demonstration that the diffusion equation (6.3.29) is an asymptotic limit of a linear kinetic equation when ever the collisions are the main mechanism driving a given physical system towards equilibrium.

Chapter 6. Hilbert space theory

175

The considerations of Chapters 4 and 5 show that we must ensure that the initial layer terms decay exponentially. This is not so easy to achieve, as Eqs. (6.3.9) and (6.3.10), which define the initial layer terms v\ywo and W\, involve compositions of (usually) unbounded operators. Therefore, before we proceed any further, we need to introduce a number of technical assumptions which will ensure the required behaviour of the initial layer terms. First let us note that, thanks to the assumption P.l imposed on the collision operator C, we can define fractional powers of — C (see Section 3.3). Moreover, since the defi nition of the fractional power (3.3.3) involves only the resolvent of -C, the subspaces V and W, reducing C, also reduce (-C) Q - In particular, Wa := D({-QCQ)a)

= D((-C)a)

n W = H„ D W,

(6.3.30)

where Ha is the fractional order space (see Subsection 3.3.4). Certainly, if C is bounded, we have D((-C)a) = H. Now, let X denote one of the spaces: H, Wa or DT, and Z be an arbitrary subspace satisfying Z =-*■ X. We assume that for every

we

znW t -4 QSpQGQCQ{t)w

G C(]0, oo[, H).

(6.3.31)

The assumption of the continuity can be replaced in some cases by the measurability in H. Moreover, we assume that \\QSpQGQCQ(t)w\\n

< q^e-^lpWlMlx

(6.3.32)

\\QSpQGQCQ(t)w\\x

< q2(t)e-^\p\\\Mz,

(6.3.33)

and

where 7' > 0 and gi,g 2 a r e functions bounded over compact subsets of ]0,oo[ and belonging to Li(R+,e~1''dt). In what follows we introduce the notation 7o := min{7,7'},

(6.3.34)

with 7 defined in (6.2.7). Assumptions (6.3.31)-(6.3.33) may look quite awkward but in what follows we shall see how they are satisfied for various choices of the operator C. Remark 3.2 Let us introduce the notation

176

Singularly perturbed evolution equations

Dsi := {u e Ds; 5(t)U € Ds for k = 1 , . . . , n} and equip the space £>s2 with a graph norm. It is easy to see that if QCQ is a bounded operator (so that Ds = DT, see Eq. (6.2.12)) such that its suitable restric tions generate semigroups of negative type in both Ds and Ds^, then the assumptions (6.3.31)-(6.3.33) are satisfied with X = Ds and Z = Dsi- Such a situation occurs in the theory of the linear Boltzmann equation and will be discussed in Chapter 7. ■ Unfortunately, sometimes we will have to impose a stronger regularity of m than that required by (6.2.8). In such cases we assume that for each p € P" Spin S X,

(6.3.35)

where X is the space for which (6.3.32) and (6.3.33) hold. R e m a r k 3.3 If X = H, then it is easy to note that assumptions (6.3.32) and (6.3.33) can be replaced by a single inequality \\QSpQGQCQ(t)w\\H

< qi(t)e-^\p\\\w\\H.

(6.3.36)

Indeed, inequality (6.3.33) follows then by the obvious inequality

IIHItf < IMU, w e Z. In this case the requirement (6.3.35) is identical to (6.2.8).

■

We shall see in Chapter 8 that (6.3.31)-(6.3.33) and (6.3.35) are also satisfied for a number of unbounded collision operators defined by second order differential expres sions. We note two simple consequences of the assumptions (6.3.31)-(6.3.33). We note two simple consequences of the assumptions (6.3.31)-(6.3.33). Proposition 3.2 For some constant K we have P r o p o s i t i o n 3.2 For some constant K we have (6.3.37) \\QSpQ(QCQ)-llw\\H < K\p\\\w\\x \\QSpQ{QCQ)- w\\H < K\p\\\w\\x (6.3.37) where w is in the space X for which the assumptions (6.3.32) and (6.3.33) are valid.

Proof. Since the type of GQCQ is negative, we have by Eq. (3.2.7)

Chapter 6. Hilbert space theory

177

(QCQy'w ■■

/ GQCQ{t)wdt,

w € H.

o 0 By (6.3.32) we get for w e X

\\QSpQ(QCQrlw\\H

= \\ j QSpQGQCQ{t)wdt\\H\p\\\w\\x 0

j

qi{t)e^dt

0

< ^IPIIIHU

(6.3.38)

where moving QSPQ under the sign of integral is justified by the assumption (6.3.31), closedness of QSPQ, and (2.6.4). ■ After these preliminaries we turn to the question of existence and regularity of the solution t«i of (6.3.10). Proposition 3.3 7/ the assumptions (6.3.32) and (6.3.33) are satisfied with either X = Wa, 0 < a < 1, or X = DT, and the assumption (6.3.31) holds, then for every initial value Wi 6 Zfl W the Cauchy problem (6.3.10) has a unique classical solution which satisfies the estimate

IK(r)||* < e-^rQ,(T)Ip|||S|U,

(6.3.39)

where Q\ is an antiderivative of q\ and u is the initial value of the original problem (6.3.1). If the function in Eq. (6.3.31) is only measurable, then (6.3.10) has a unique strong solution which satisfies the same estimate (6.3.39). Proof. First, let X = Wa. We denote f{t) = QSpQGQCQ{t)w. then we consider the function q(s) =

If it is continuous,

\\QSpQGQCQ(t)w\\xa

which, by (6.3.33), satisfies assumptions of Theorem 3.6.1, hence (6.3.10) is classically solvable. To prove (6.3.39) we take the representation of the solution to (6.3.10) in the form

wi(r) = GQCQ(T){QCQ)-1QSPPV QSPPZ -+ jGQCQ{T

o

- a)QSpQGQCQ{a)wdo

(6.3.40)

178

Singularly perturbed evolution equations

and by continuity of QSPP and (6.3.32)

IKMIU T

< \\GQCQ(T)(QCQ)-1QSPPZ\\H

PV\\H

+ J \\GQCQ(T

-

a)QSpQGQCQ(a)w\\Hda

o <e-™ T (K||£||„ + Q;(r)|p|||^|U a ) <e- T0T Q.(7-)|p|||S|U- o , where Q[ denotes the antiderivative of qi satisfying Q[ (0) = 0 and 70 was defined in (6.3.34). Second, let X = DT. We note that since C generates an analytic semigroup, Ha is defined and (6.3.33) is valid not only for X = DT but also for X = Wa for arbitrary a due to Eq. (6.2.13). Hence, the function q has all the required properties. To complete the proof let us observe that if the function in Eq. (6.3.31) is only measurable, then by Eq. (6.3.33) it belongs to Li([0, oo[, Xa) and the application of Theorem 3.6.2 gives the strong solvability of the problem (6.3.10). The estimate (6.3.39) is obtained as above, since the representation formula (6.3.40) is also valid for strong solutions (see Subsection 3.6.1). ■ We end this section with two simple cases when assumptions (6.3.32) and (6.3.33) are satisfied. P r o p o s i t i o n 3.4 If (6.3.36) is satisfied, the function in Eq. (6.3.31) is measurable and w e D(QCQ), then the problem (6.3.10) has a unique classical solution satisfying \\MT)\\B

Zer^QiirMlvU,\V\\\V\\H,

(6.3.41)

where Qi is an antiderivative of q\. Proof. Since the type of GQCQ is negative, we have 00

(QCQ)-1x

=

JGQCQ{t)xdt 0

for x € W From the estimate 00

WQS.QiQCQr'wWw

= || J QSpQGQCQ(t)wdt\\w 0

<

A'i|p|||ui|| w
00

< \p\\\w\\w

9I'

jqi(t)e-^dt

0

(6.3.42)

Chapter 6. Hilbert space theory

179

where we moved QSPQ under the sign of the integral which is allowed due to the closedness of QSPQ and measurability of the function in Eq. (6.3.31), we see that the operator QSPQ{QCQ)~X is bounded. For w e D(QCQ) we have QSpQGQCQ(t)w

= QSP< QS.QiQCQr'GgcQWQCQu'

and (GQCQ{t))t>o is an analytic semigroup, hence the right-hand side of the differ ential equation in (6.3.10) is a differentiable function for t > 0, and therefore it is locally Holder continuous on ]0, oo[. By Subsection 3.6.4 we obtain that (6.3.10) is classically solvable and Eq. (6.3.41) follows as in the preceding propositions. ■

Proposition 3.5 // assumptions (6.3.31)-(6.3.33) Z = Dsi, the functions q\, q2 are bounded and

are satisfied with X = Ds and

QCQw e D j f l W, then the problem (6.3.10) has a unique classical solution satisfying \\*i(r)h<er™TQi{T))p\\\«\\Da,

(6.3.43)

where Q\ is an antiderivative of q\. If, in addition, (GQCQ(^))(>O is analytic in Ds, then the proposition holds with w e

Dsnw Proof. Since (GQCQW)<>O is a semigroup in Ds, t —> QSpQGQcc}(t)w is differ entiable on [0,oo[ which proves the proposition by Subsection 3.6.2. The estimate (6.3.43) is obtained as in Proposition 3.3. If (GQCQ(0)I>O is analytic in Ds, then t -> QSpQGQCQ{t)w is differentiable for t > 0 for any w G Ds D W and we complete the proof as for Proposition 3.4. ■

4

Estimates of error of asymptotic expansion

In this section we shall prove that the asymptotic expansion, defined by Eqs. (6.3.11) and (6.3.12), gives an approximation of the order e2 to the solution u or, in other words, that the solution y,z to (6.3.15)-(6.3.16) is of the order e2 The proof of this estimate is quite long, so we decided to start with an example, related to the model kinetic equation discussed in Section 5.2, where the error can be calculated and estimated explicitly.

Singularly perturbed evolution equations

180

Example 4.1 Let us consider the model kinetic equation (5.2.1). To further simplify the model we put A = 0 and a := -B = —5, which does not affect asymptotic properties of the solution. With this Eq. (5.2.1) takes the form

dtv

=

aw,

dtw

=

1 av + aw - -w

v(0)

=

o V,

w{0)

=

£

0

(6.4.1)

W.

If we take for the streaming operator 0 a a a

5= and for the collision operator

C =

0 0

0 -1

i

then Eq. (6.4.1) will have the same asymptotic properties as the generic kinetic equa tion (6.2.1) in the projected form given by Eq. (6.3.1). Performing the compressed asymptotic analysis of (6.4.1) we obtain the error equations (6.3.15) and (6.3.16) in the following form: dty

az + taw\,

dtz

ay + az

2/(0) z(0)

z + e(auii + av\ + awi — dtwi)

0, 0,

(6.4.2)

where Vi, Wi, Wi are defined in (5.3.10), (5.3.9) and (5.4.12), respectively. We are interested in obtaining 0(e2) estimates for y and z. First we note that the inhomogeneity in Eq. (6.4.2) consists of two parts with completely different characteristics: the part resulting from the initial layer F =(

awi a{wi +vi)

and the part resulting from the bulk solution

!

Chapter 6. Hilbert space theory

181

F =e

0 avj\ — dtWi

Let r be the error resulting from the initial layer part, that is, let f solve the initial value problem dtf

= Sf 4- -Cf + F,

f(0)

=

' 0 ' 0

(6.4.3)

We know that the solution to (6.4.3) is given by the exponential function of the operator 71 := <S + -fC according to the formula

r(t) = eT<'f(0) + I

eT'it-s)F{s)ds

(6.4.4)

Due to the spectral properties of % we have, as in the general case, |e''{H
(6.4.5)

and this is enough to obtain the required estimate of r. Indeed, taking advantage of Eq. (5.3.13), we see that for some constant M we have \\F(t)\\ < eMe-t/c\\u\\ and therefore t

\\r{t)\\

\\eT-{t-s)\\\\F(s)\\ds

< I o

i

<

e M | | u | | y V * / e d s < A/^HulJ

(6.4.6)

where M and Mi are some constants. The error due to the bulk part contribution, say f, is the solution to the problem

dtf

10)

= Sr+ -Cf + F, " 0 ' 0

€

(6.4.7)

Singularly perturbed evolution equations

182

Unfortunately, we cannot apply the same method as in estimating f, since F depends only on t and not on t/e. We will have to solve Eq. (6.4.7) explicitly. To do so we need a coordinatewise expression for eT,t- As in Chapter 5 we see that the characteristic values of Tc are given by

Ai

=

A2 =

1 a E ~2e + 2+Ye' 1 a E ~Ye + 2 ~ 2e'

(6.4.8)

where E is the square root of the discriminant .1/2

E = (l - 2ea + 5 e V )

After some calculations we arrive at the following formula:

(Ai-As)"1

aT.t

-A -A2eAl( + A ^ 2 ' A1A2CT -1

_„\tt

0-(eAl* _

,\2t') +, eA2

eA2<)

A ie Al( - A2eA2(

The solution to the problem (6.4.7) can be written in the form

f{t) = eT't

je~T'5F{s)ds.

(6.4.9)

0

To shorten the subsequent calculations we shall focus our attention on one component of F only, namely on eawi, and denote by fj the expression corresponding to (6.4.9). By (5.4.12) we see that omitting the higher order terms we can take eaw.it) = eva2eec,2t Disregarding the irrelevant factor va2 we have

/ , o

e

0 „((T2S

a (g(-Al+e
e

(I.S

A, - X

/

e Ai — A2

where we introduced the notation

2

2

)s _

^Ie(-Al+^ )S _ CT(AJ

- A a )i

AiAj - A2A2

e(-A2

+ £(T2)s\

^e(-X2+^2)s

da

Chapter 6. Hilbert space theory

183

g (-Ai+«r

A, =

J

)« _ j

-Ai + ea2

and e (-A 2 +ea

A2 =

2

)( _ j

-A 2 + e<72 '

Upon multiplication by e T,( we obtain

fi = 7

Ai — A2

CT(eAl'Ai-eA2(A2) e A l %A, -e A 2 t A 2 A 2

(6.4.10)

To estimate fj we note that by (6.4.8) we have

A! =

«r 2 + 0(e 2 ) = 0(e),

A2 =

- - + a-ea2

+ 0{e2) = O

n

©■

so that 1 Ai-Aj

l-a +

1 2ea2+0(e2)

0(e)

and

eA'(A, A,eA-'A,

0(1) O(l), j = 1,2.

Thus from Eq. (6.4.10) we obtain fl(t) = 0(e2). Similarly we can estimate the contribution to f coming from dtw. Combining these with the estimate (6.4.6) we see that indeed the compressed asymptotic expansion gives an approximation of order e2 to the solution u = [v , v] of the problem (6.4.1).

Singularly perturbed evolution equations

184

After these preliminaries we are ready to state and prove the main theorem of this section. We recall that the error of asymptotic expansion is given by formulae (6.3.13) and (6.3.14), that is

y(t)

= «(t)-[€«(*)+ «Ci(*A)].

z{t)

=

w{t)-[w0{t/e)+ew1(t)+ew1(t/e)].

Theorem 4.1 // assumptions PA P.5, (6.3.31)-(6.3.33) and (6.3.35) are satisfied for some choice of X and Z then for any T, 0 < T < oo there is a constant M independent of p such that \\y(t) + z(t)\\H < SM max3{\p\k\\u\\z}

(6.4.11)

uniformly for 0 < t < T. Proof. Let us denote by P i ( / , g) the Cauchy problem

dty

=

dtz

= QSpPy + QSpQz + -QCQz + g,

2/(0) = z(0) =

PSpPy + PSpQz + f,

0, 0.

(6.4.12)

We denote by P2C1) the equivalent Cauchy problem (see Proposition 2.2)

dtr

= Spr + -Cr + h, e r(0) = 0.

(6.4.13)

From the previous section we see that we can insert the error denned by (6.3.13) and (6.3.14) into (6.4.12). Indeed, from Propositions 3.1, 3.3 and 3.4 it follows that we can differentiate all terms of y and z. Since QSPP and PSPQ are continuous (the latter identified with its continuous extension, Lemma 3.1), terms PSpQz and QSpPy are well-defined. Also QCQz is well-defined by Proposition 2.2, Eqs. (6.3.6), (6.3.8) and Propositions 3.3-3.5. Next, the fact that QSpQw0 is well-defined follows from the assumption (6.3.32), provided w 6 X, and for QSpQwi it follows from the estimate (6.4.15) below. The last term to be checked is QSpQwi. We have by Proposition 3.2

Chapter 6. Hilbert space theory

185

\\QSpQv>i\\n = \\QSvQ{QCQ)-lQSvPv\\H

< A|p|(t>,m)„||QS p m|| x ,

which is finite by (6.3.35). Moreover, p —► QSpm is a finite-dimensional linear function with values in X so that we obtain WQSyPw^H < K2\p\2{v,m)H.

(6.4.14)

Therefore the formal derivation of equations for the error (6.3.15) and (6.3.16) is justified and the error is a classical (or strong, if the function in (6.3.31) is only measurable) solution of the problem Pi(ef,eg), where

/

=

PSpPvl

+ PSpQu,!

g = QSpQwi + QSpPvi - dtuii + QSpQwi,

or, equivalently, r = y + z is a classical solution of P2(e(/ +
+QSpQwi.

Unfortunately, we cannot mimic the proof given in the example, as it utilized special relations between characteristic values Ai and A2 and which are unavailable in the general case. To overcome the difficulties we have to split the estimates even further, as the behaviour of g is not uniform with respect to p and t for small t. In the proof

Singularly perturbed evolution equations

186

below we estimate first the short-time influence of g, denoted by f0. In the estimate (6.4.28) of f0 the additional e appears due to the length of the interval of integration. The most difficult part is to estimate the long-time influence of g which we denote by f\. To explain the difficulty, we note that though the bulk-part contribution g appears only in the second equation of (6.4.12), due to the coupling in the system it also influences the first equation which does not depend explicitly on e. Therefore the direct estimate of fj, analogous to that for f, is impossible because fi contains terms which do not depend on t/e, as can be seen in (6.4.10). To circumvent this difficulty we use Remark 2.1 to solve an auxiliary equation (6.4.30) in the subspace W By this remark, the solution h of this equation depends on t/e and it transpires that the difference fi = fi — h solves an appropriate problem P 2 with the right-hand side depending on t/e and the additional e can be extracted by integration, as in the initial layer part. This is done in (6.4.32). Hence r = f + f0 + fi is of order e2, as required, and the theorem is proved. Now we shall carry out this program in a rigorous way. Firstly, we deal with the part f(t) of the error which is due to the initial layer phenomena and therefore let f (t) be a mild solution of the problem ^ W / + <j), AO- We shall need the following estimates:

\\QSpQwx{r)\\H <

e-^QiMlppllSH*,

(6.4.15)

2 ||PS P <M(T)IU < e -^Q 2 (r)|p| ||£|U, 2 HQSj-PSiMIU < e-^Q 3 (T)M ||£|U, 70T 2 \\PS,Ph(r)\\H < e- Q4(r)|p| ||S|U,

(6.4.16)

where the functions Q; e Ia(R + , e

(6.4.17) (6.4.18)

70T

Let us prove (6.4.15). Using the representation of the solution to (6.3.10) we obtain

QSpQwi(T)

( QCQ(QCQ)-lQSpPl = QSPQ G

+

J GQCQ{T

-

a)QSvQGQCQ{a)wda7

(6.4.19)

If Sp is bounded, then the estimate follows immediately. If not, we proceed as follows. We claim that

\\QSPQGQCQ(r){QCQr'QSpPv\\H

< g 1 (r) e -^ T |l(QCQ)- 1 QS p P|p|||«|| x < mie-^qi(T)\p\2f\\H,

(6.4.20)

Chapter G. Hilbert space theory

187

for some constant m^ The first inequality in (6.4.20) follows from (6.3.32). To prove the second inequality we have to distinguish two cases. If either X = H or A' = Da, then the result is immediate as both {QCQ)-lQSpP and {-Cy(QCQ)-lQS„P are bounded in H by continuity of QSPP (and Proposition 3.2 in the second case). Let now A' = />/■; then by continuity of QSPP we have

\\(QCQ)-'QSpPv\\D{c) < kMfWn and by Eqs. (6.2.5), (6.2.13), (6.3.32) and (6.3.35) we obtain

\\(QCQ)-lQSpPv\\Ds

< \\(QCQ)-lQSpP°v\\H

Y,\\QSPll)Q(QCQ)-1QSpPi\\H

+ 1=1

< k2\\QSpPv\\DT

= (°Mii)„||S p m - (m,S p m)„m|| D 7 . <

k3\p\f\\H,

where k{ are constants and p(i) := ( 0 , . . . , 0,1, 0 , . . . , 0) with 1 on the l-th place. Thus (6.4.20) is proved. For the second term of (6.4.19) we have by (6.3.32) and (6.3.33)

T

\\SPQ J GQCQ(T

-

a)QSpQGQCQ(o)wda\\H

o T

<

Jqi(r

-

a)e-^-^\p\\\QSpQGQCQ(a)w\\xda

0 T

<

e-Hp| 2 ||u|U J"9I(T

- a)q2(a)da,

(6.4.21)

o

where in the last line || • \\z can be replaced by || \\H if A = H. Then clearly (ji = q-i s q. Let us denote T

Q = / ?I(T o

o)q2{o)do.

Using properties of q\ and q2 and 70 from (6.3.34) we obtain

PO

|e-70TQ(r)dT 0

OO

=

/

T

je-
V

ijq1(T-a)q2((j)da\dT Vo

/

Singularly perturbed evolution equations

188

I

192(0-)

U(a)Je-™Tqi(T-a)dT\da

OO

OO

/e-""g 2 (a)da

f

e"1^qi{d)dd < oo.

(6.4.22)

Combining estimates (6.4.20) and (6.4.21) we get (6.4.15) with QY = m^ by (6.4.22) it is clear that such Qx has the postulated property.

+ Q and

The proof of the remaining three inequalities is similar but easier, due to the boundedness of operators PSPP, PSPQ and QSPP Since (Gyp<(i))t>o is a semigroup of contractions, we obtain the following estimate of

wmh < K,e£ 2 + e />*../«

£Q,

(;)bH|fi|U *

+ /C-T** fe Q4(0|pf2ll«IU) t'^

<

1 K2e=22 1

<

K 3 6 2 |p| 2 ||S|| z ,

(6.4.23)

which is the required estimate of the initial layer contribution to the total error of the asymptotic expansion. Next we estimate the part of the error related to the bulk solution given by the solution to P2(e
g = -dfiy, + QSpQiD, =

(6.4.24) l

n{

nc)t

((w - Sle) - QSpQ){QCQ)- QSvPe^~ SpPe^(v "-

(Z

-

l

ePSpQ{QCQ)- w).

As we mentioned earlier, the properties of g are not uniform with respect to e and p for small t, so we split g into a sum of two functions, say g0 and gl. To define g0 we consider, for fixed e, a decreasing C°°-function 4>e such that

kit) := Now let

I

1 for t < ie 0 for t> | e .

Chapter 6. Hilbert space theory

189

ffo = 4>e0, 01 = (1 - 4>t)g-

(6.4.25)

Let f0 be a mild solution of P2(e0o)- First we note that we have to take special care of the composition QSpQ(QCQ)~l, as QSPQ is not bounded. To this end we refine the second part of the estimate (6.4.20). Since both QSPP and PSPQ are bounded (see Remark 3.1) we see that we can extend the identity (QSpPx,y)H

=

-{x,PSpQy)H,

originally valid for x,y S D(SP), onto H which shows in particular that a(p) := \\QSPP\\ = \\PSPQ\\} and, by linearity of p —► Sp, we have \a(p)\ < A\p\ for some constant A. Using this notation we have exactly as in (6.4.20) WQS.QiQCQy'QS.PHu

< Li|p| 2 ||S[[*

(6.4.26)

and

\\QSpQ{QCQ)-xQSpPPSpQ(QCQ)-lw\\H
m)«| < L3\P\2HP)\\\M\H ii

ii

(6.4.27)

for some constants Lt, i = 1,2,3. Moreover, Re f2(p) > tj 1 a 2 (p) and \Q\ < u'a2{p) < oi 2 |p| 2 (see Proposition 3.1) and we obtain as in (6.4.23)

3c/2

INI < e J

\\ga(s)\\Hds

o

i; <

M lC J e - ^ 2 ' ' " |a(p) + ea2{p) + e\p\2a{p) o + e2\p\2a2(p) + a2(p) +

ea3(p)}\\u\\Hds

Singularly perturbed evolution equations

190

3E/2

<

—l_ —1_ ( 1l + « + ^ =

eMamKibl'liailir} /

i/2eecjis,

Id.

u

<

M,max{lp|*||u||J/}e8,

(6.4.28)

where we have used the relation k - ^ —

max(gte-'£f'2)=

fc/2

,

(6.4.29)

valid for t > 0, to estimate the term e2a(p)e~UJia ' p ' which remains after including factors of the third degree in p into the initial value. Finally we estimate the mild solution fl=y1 + Si of the problem P2(e
dth

= QSpQh + -QCQ + egu e

h(0)

= 0.

(6.4.30)

Let us note that (6.4.26) shows that QSPQ(QCQ)~1QSPP is a bounded operator in H. Therefore the function t -> QSpQwi{t) = -QSpQiQCQ^QSpPv is differentiable whenever v is differentiable. Also, d ; ^ is differentiable if dtv is differentiable. However, € is a solution of an ordinary linear differential equation with constant coef ficients by Proposition 3.1 and as such it is a C°°-function. Therefore gi is sufficiently regular for h to exist and

MO = £ / Gl
s)gi{s)ds,

where Gcp is defined in Remark 2.1. We see that the function f\ = S\ + y\ — h is a solution of the problem P2(PSpQh) and we have

\\Ut)h

< N,|p|e- U » ca2{p)t{a(p) + ea2(p) + e\p\2a(p) +e2\p\2a2(P) +

a2(p)+ea3(p)}\\u\\H

N 2 maxJ| P r N |W(l

+

^

+ 7

=_),

(6.4,1)

Chapter 6. Hilbert space theory

191

where we used Proposition 3.1 and again (6.4.29) to estimate terms of the fourth and fifth order in p. With (6.4.31) we can estimate fx as follows:

t

\\h{t)\\H

< J' \\PSpQh(s)\\Hds

<eJ

<

R l £ max |p|*||S||„ / o
<

e2R2 rnax3\p\k\\u\\H

/

JJ

s' e^~^\p\\\g,(S')\\Hds' (Is

/

/e-('-s'»/((l + e 2 - ^ + . , ds' J \ eewis' eeui\S ^/2eeuj1s'

ds

(6.4.32)

for 0 < t < T, for some constant R depending on T. In a similar way we obtain \\h{t)\\H < e2D max \P\k\\u\\H

(6.4.33)

0
for some constant D. The sum of mild solutions f, fo and fj = f\ + h is a mild solution of the problem Pa(e(/ + 9)) a n d since this problem has the classical (or strong) solution r = y + z where y and z are given by (6.3.13) and (6.3.14) we have (see Subsection 3.6.1)

V = 2/+ 2/o+ 2/i, z = z + z0 + %\, where y0 = -P^o and 5o = Q^o- Combining estimates (6.4.23), (6.4.28), (6.4.32) and (6.4.33) we get the required estimate (6.4.11). ■ As an immediate consequence we obtain the estimate for the error of the approxima tion in V. (and, equivalently in L2(£l, H)) where, with some abuse of notation, we use the same notation for the error in H as in H. Theorem 4.2 If assumptions P.I for some choice of X and Z and

max

P.5, (6.3.31)-(6.3.33)

J \p\k\\u\\2zdp

and (6.3.35) are satisfied

(6.4.34)

0<*<3

IP

that is, u is a Fourier transform of a function from the Sobolev space W23(fi, Z), then for any T, 0 < T < oo there is a constant C such that

Singularly perturbed evolution equations

192

\\y(t) + z(t)\\n
(6.4.35) *

Note that in the periodic case the integral in Eq. (6.4.34) should be replaced by the summation.

5

Remarks on non-selfadjointness of C

The assumption that C is a self-adjoint operator, appearing in P.l, very often turns out to be too restrictive. Here we shall show that, under certain restrictions, the assumption P.l can be replaced by the following one: P . l ' C is the generator of an analytic semigroup of contractions, (Gc(t))t>o, in H. Zero is a simple isolated eigenvalue of C with the eigenfunction m and sup Re{a(C) \ {0}} = —7 < 0. Moreover, the spectral projections P and Q, corresponding to the eigenvalue A = 0 are orthogonal. To discuss possibilities of obtaining a counterpart of Theorem 4.1 under this as sumption, we note that the self-adjointness of C was used in two places. Firstly, in Corollary 2.1 and the subsequent Remark 2.1, to prove the existence of the semigroup (G£J,(t))(>o which was then used to solve the auxiliary equation (6.4.30). Secondly, in Proposition 3.1 to prove the positive definiteness of the matrix of diffusion coefficients {fi.j} in (6.3.26). In many cases it is possible to establish the positive definiteness of {fi.j} through other means (see, for instance, Proposition 7.2.1). If we managed to do this, then we still have to discuss solvability of Eq. (6.4.30). We know that, due to the analyticity of (GQCQ(£))I>O, the spectral bound of QCQ and the growth bound of (GQCQ(£))(>O coincide and, in particular, \\GQCQ(t)\\ < Me-*

(6.5.1)

for some constants M > 1 and [i > 0 (see Section 3.5). Unfortunately, if we cannot ensure that the constant M in Eq. (6.5.1) is equal to 1, then the results of Remark 2.1 seem not to be available. However, we can still estimate the bulk part contribution to the error, as in theorem 4.1, if we introduce additional assumptions. Namely, we will require that for every w € Z n W t -4 QSpQGQCQ{t)w is X-measurable

(6.5.2)

193

Chapter 6. Hilbert space theory and QSpm 6 Z.

(6.5.3)

With the measurability assumption we can prove, exactly as in Proposition 3.2, that \\QSpQ(QCQ)-lw\\x

< K\p\\\w\\z

(6.5.4)

for w € Z and that X for which assumptions (6.3.32) and (6.3.33) is valid. With these results we can formulate the following counterpart of Theorem 4.1: Theorem 5.1 // assumptions P.l', P.2 P.5, (6.3.32) and (6.3.33), (6.5.3) and (6.5.2) are satisfied for some choice of X and Z, and if the matrix of diffusion coef ficients {0,j} satisfies (6.3.24), then for any T, 0 < T < 00, there is a constant M independent of p such that \\y(t) + z(t)\\H<e2Mmix{\p\k\\u\\z} 11«IU}

(6.5.5)

uniformly for 0 < t < T. Proof. We use the same notation as in Theorem 4.1. The estimates for the error functions, f and the "short-range'' part of f, f0, obviously remain unchanged. To estimate fi, the "long-range" part of f, we introduce an auxiliary function h as a solution of the problem

dth

=

-QCQh + egu e = 0.

h(0)

(6.5.6)

As in the proof of Theorem 4.1 we see that the function gx is sufficiently regular for h to exist and t

h(t) = el GQcQ(t -

s)g1(s)ds.

0

This time the function ?i = Z\ + yi — h is a solution of the problem P2{SpQh). have t

\\SPQh\\„

< e J \\QSpQGQcQ{t 0

-

s)gi(s)\\Hds

We

Singularly perturbed evolution equations

194 t

■ Je-rt-V'qt oa

(y^-)\p\\\h{s)\\xds.

(6.5.7)

Next, using (6.4.24) we have by (6.5.4) and (6.5.3)

||QS P Q»ilU <

\\QSpQ(QCQ)-lQSpPv\\x

<(v,m)H\\QSpQ(QCQ)-1QSpPm\\x

< \p\(v, m ) w | | 5 p m - m\\z < M\p\2\\v\\H. (6.5.8)

Similarly to (6.4.20) we obtain

= \\(QCQ)-lQSvPdtv\\ < Ml\p\\\dtv\\H.

\\dtw\\x

(6.5.9)

Further, taking advantage of (6.5.8) and (6.5.9) we obtain for |p|||
\\h(t)h

< J \\SPQh(s)\\Hds

<

/

[e-nW'tol

/

< Ue max 0<*<3

|p|||Si(
^

/

k

\p\ fh

,j / e -o( - ')A ,

s s

?2

( ^ )

,(1

e2

+

+-

( (2s-0/2£

< e'L2ngs\p\k\\u\\nJ

J

b

?)

ds'I ds

I

e-">°q2(o)da

°

< e 2 L 3 max|p|*||«|| w

(6.5.10)

for 0 < t < T, with L 3 proportional to T. In a similar way we obtain ||ft(t)lk<e2Kmax|p|k||u||H.

(6.5.11)

U
The remaining part of the theorem is proved exactly as for Theorem 4.1.

■

Chapter 6. Hilbert space theory

195

Remark 5.1 If we are unable to prove that the matrix {QtJ} is positive definite, then still from (6.3.26) we have that - Re(QCQwp, wp)H > 0,

(6.5.12)

which shows that it is at least non-negative. That result is certainly unsatisfactory from the physical point of view: nevertheless it is sufficient to prove a counterpart of Theorem 4.1. A closer look at its proof shows that the strict positivity was used only in estimate (6.4.29). Without this estimate we are unable to balance the powers of p by the behaviour of the exponential function so that there is no advantage in splitting g into short- and long-range parts. Consequently, all powers of p must be included into the initial data. The estimates then go as the estimates in the proof of Theorem 5.1. ■ With this remark we have the theorem. Theorem 5.2 If assumptions P.l', P.2 - P.5, (6.3.32)-(6.3.33), (6.5.3) and (6.5.2) are satisfied for some choices of X and Z then for any T, 0 < T < oo there is a constant M independent of p such that \\vP(t) + zp(t)U

< e2M max {\p\k\\u\\z}

(6.5.13)

uniformly for 0 < t < T In both cases discussed in this section, Theorem 4.2 has to be modified in an obvious way.

Chapter 7 Applications t o kinetic equations with bounded collision operators 1

Introduction

In this chapter we shall consider applications of the theory (which was developed in Chapter 6) to a class of kinetic equations with bounded collision operators, that is, to various types of the linear Boltzmann equation which arise when the collision between the particles can be neglected and only the interactions with the host medium are relevant in the process. We follow the presentation of [6, 8]. Accounting for a mathematical complexity we divide this class into two subclasses. The first one will involve operators with bounded velocities; in this case the ap plication of the results of the previous chapter is almost trivial as the operator of multiplication by the velocity is bounded and most of the assumptions are automat ically satisfied. We shall not discuss this case separately here. The second subclass will include operators with unbounded velocities so that the multiplication by the velocity is no longer a bounded operator and we have to prove that the assumptions of Chapter 6 are satisfied. Let us consider the following initial value problem for a transport equation:

dtu(t,x,o = -£dxu(t1x,0-v(ZMt,x,$) + Jk(t,e)u{t,x,e)d£, *>o, J en, u(0,x,0 = u(x,0-

(7.1.1)

We assume that we deal either with the free space case and then the spatial variable satisfies x € Q := K3, or with periodic boundary conditions in which case x 6 Q := 197

Singularly perturbed evolution equations

198

[0, 27r]3 The variable £ 6 E C U3 is the velocity and (r, x, £) -> u(c, x, f) describes the time evolution of the distribution function in the position-velocity phase space. The collision frequency u and the scattering kernel k are known functions which are assumed here to be independent of x. From the physical point of view a natural space for analysis of this equation is Li(R6, dxd£), as the integral Judxd£ gives the total number of particles in the systern. However, this space has many drawbacks. Firstly, the Fourier transform is not sensibly invertible in L\ and this creates problems which were avoided in Chapter 6 by reducing the spatial variable to the role of a parameter. Thus, to make the theory of Chapter 6 available we have to use the space L 2 (K 3 ) with respect to x variable. Our choice of the space for the velocity is dictated by the mathematical simplicity. It follows from e.g. [20, 23] that the linear Boltzmann equation and ki netic equations of the Fokker-Planck type become self-adjoint if considered in the space H = L2(R3, m - 1 ^ ) ^ ) , where m is the equilibrium solution, or in other words, the normalized eigenfunction corresponding to the eigenvalue A = 0 of the collision operator C. Also, as we shall see in Section 2, many relevant formulae are formally identical in H and L\ settings. This will allow us to use the results of the Hilbert space theory for some problems posed in L\ which are discussed in Chapter 11. Taking the above into account we use, as the basic space for our considerations, the space

H = L2(BL3X,H) = L 2 ( R 6 , m - I ( 0 # d a : ) -

(7.1.2)

Considering Eq. (7.1.1) as an evolution equation in the space U and following results of Section 6.2, we apply the Fourier transform with respect to x to Eq. (7.1.1) and obtain the equivalent problem in L2{Pn,H), given by

dtu(t,P,0 --: iPdxu(t,p,o - i/(0fi(t,P,0 + / fc(£,0«(*.P.£'R', «(0,p,£)

=

«(p.O-

fi(p,C).

(7.1.3)

As in section 6.2, P n = Mn in the free space case and P n = Z" in the periodic case. Due to Theorem 6.2.1 we can analyze the problem (7.1.3) in H, treating p as a parameter. Following Section 6.2 we denote by calligraphic letters the operators acting in fi and by the corresponding roman characters their counterparts in H.

Chapter 7. Bounded collision operators

2

199

Properties of linear Boltzmann equation with unbounded velocity range

In this section we analyze the diffusion approximation of the equation (7.1.3) in the case when 3 = R3 We recall that we are dealing with the following initial value problem

dtu(t,p,£)

== ipdxu(t,p,£) --v(Ou(t,p,0+

1 HU')u(t,p,C)dti' R3

u(0,p,0

=

u(p,0.

(7.2.1)

where p £ P 3 and ,,,

i JI£33 Z

for the free space case, for the periodic boundary conditions.

The scattering kernel k is usually written in the following form * & O = m(0*(£.O.

(7.2.2)

where m is the normalized Maxwellian distribution in a given temperature 6 m ( 0 = (27r0)- 3 / 2 exp(-£ 2 /20)

(7.2.3)

and is called the scattering cross-section. We assume that A.l O<0(£,£')
(7.2.4)

3

for almost all £',£ € R and some constant Cj. Our analysis does not require ^ to be a symmetric function. However, in realistic cases, the so-called principle of detailed balance [20, 52) asserts that fc(£,Om(0

= *(£', O m ( 0

(7.2.5)

and that yields *(£,O = 0«'.fl-

(7-2.6)

Singularly perturbed evolution equations

200

Another assumption taken from the kinetic theory is that the operator C, defined as

(Cum) ■■= -"KMO+/*& eww, R3

is conservative, that is, for every u € Li(R) we have

f c W £ = 0.

(7.2.7)

R3

This equation yields an important result: since for any u e £i(R) we have This equation yields an important result: since for any u e £i(R) we have 0

= j cun = 11 -i/(e)«(^) + / k(f, t'Mt'W] di,

0 = JCudZ = f(-v(S)u(Z) 3

3

R

R

+

Jk(Z,Z')u(?)dt'\dt 3

V

/

R

= /«(o(-"(o + /fc(eUR'W R3

V

R3

/

(provided that the change of order of integration is justified), then clearly " ( 0 = / fcte', t)d? = / m(£')tfte', Odf. R3

(7.2.8)

R3

At this stage we introduce another technical assumption: namely, we postulate that A.2 0 < c2 < v(£) < c3

(7.2.9)

for every £ G R3 and some constants C2,c3. As we mentioned in the introduction, the most convenient space for the analysis of Eq. (7.2.1) is L2(Pn,H), where

H=

L2{R\m-\OdO-

Note that with m defined by Eq. (7.2.3) we have H = L2(]R3, m " 1 ^ ) ->• Li(R 3 ) so that all assumptions introduced above in the L\ setting remain valid in H. We have:

Chapter 7. Bounded collision operators

201

Proposition 2.1 If the assumptions A.l, A.2 and Eqs. (7.2.6) and (7.2.7) are sat isfied, then the operator C has the property P.l of section 6.2. If instead of (7.2.6) we have, for some constant c4,

c„>o, £,reR 3 ,

(7.2.10)

then C has the property P.l' of Section 6.5. R e m a r k 2.1 From Eq. (7.2.8) we see that the inequality (7.2.10) together with A.l implies the assumption A.2 with c2 = c^. Proof. Let us introduce the following notation:

(Ku)(0 := Jfc(£,£>(?)# = m ^ / *K- 0 ^ ' K , R3

R3

then Mvu = Ku - Cu is simply the operator of multiplication by v. First of all let us note that C G C(H). Indeed, by assumption A.2 we see that Mu is bounded. Now, for every u € H we have

\\Ku\\l

=

/(m(O|/0(^>(m'l) R3 V

R3

m(£)"X /

< //(^,f)Wf)"'(0) : / 2 |«(OHO' 1 / , )V«« R3 R3

- / (/**(«.Om(0m(O^/l«(OlMO" 1 d€')* R3

<

\R3

R3

N 2 || U || 2 „,

/

(7.2.11)

where N2 = j j>(^')m(0m(O«

)m(Od?d£ < oo.

R3R3

Therefore C = — Mv + K is bounded. In particular, this implies that the semi group (G c (t)) ( >o, generated by C, is a uniformly continuous semigroup, hence an

Singularly perturbed evolution equations

202

analytic one. Moreover, it follows that the kernel k of the integral operator K satis fies k € L 2 (K 6 ,m" 1 (^)m" 1 (^')d^d^') and therefore K is a Hilbert-Schmidt operator (see Subsection 2.7.9). We use this later to establish the necessary spectral properties of C. Due to the assumption A.2, the operator Mu is an isomorphism. Therefore we can apply the Fredholm Alternative (Subsection 2.7.4) to the operator C,, := -\iMu + K.

(7.2.12)

We obtain that \x / 0 is either an eigenvalue or a regular point of C^ and that the set of eigenvalues of C^ is either finite or countable with 0 being a single accumulation point. If we apply this result for the spectral problem for C\ := C = —M„ + K \u-Muu

+ Ku = Q

(7.2.13)

we see that A = 0, corresponding to fj, = 1 in Eq. (7.2.12), can either be a regular point or an isolated eigenvalue of — Mu + K. We shall show that it is a simple eigenvalue with a corresponding eigenfunction m. First we note that m is an eigenfunction of C corresponding to the eigenvalue 0. Indeed, by Eq. (7.2.8) we have (A-m)(0 = m ( 0 ^ ( U ' ) m ( r ) r f r = ^ ) m ( 0 = (Mum)(Q.

(7.2.14)

To prove that the eigenspace, corresponding to A = 0, is one-dimensional we calculate, for an arbitrary u £ H, the following scalar product

Re{Cu,u)H

= Ref(Cu)(t)u{Om-l(®dZ

(7.2.15)

R»

=

- / KO«(Ofi(£)m - 1 (Ode + ReJ

J

1

fc^eXCXOm- ^)^. Hz,axam

We have two ways of calculating v : Eq. (7.2.8) and Eq. (7.2.14). If we calculate the first term on the right-hand side of (7.2.15) using (7.2.8) we get

/ KOHoiwm = / / R" R"

and if we use (7.2.14) we obtain

K0|2

^ )2m ( g ' W

m(0

(7.2.16)

Chapter 7. Bounded collision operators

203

/KaiuCflpm-^ode = / /

K

^^W

m(0

i«te')iafc(£,o d^d^'

(7.2.17)

m(£')

R" R"

If we add equations obtained from (7.2.15) by substituting (7.2.16) and (7.2.17), we obtain

2Ref(Cu)(Qu(0m-1(0dt

(Ode = ■ ~fj

*(£, ? ) m ( 0 m ( O

R" R'

u(0 m(0

u(C) m(f')

d(,dE,'

(7.2.18) 2

2

2

where we use the complex identity /?e(|a| — lab + |b| ) = \a — b\ Now, if u is any eigenfunction of C, then Re(Cu,u)H

= 0.

However, due to the assumption A.l and (7.2.18) this is possible only if "(fl m(0

"(f) ra(e')

= Q

for almost every £,£' G R3 and that yields u = am for some constant a € C Since C is self-adjoint, we see by Subsection 2.7.3 that A = 0 is a simple eigenvalue. The equation (7.2.7) can be written in the following form (Cv,m)H

= J(Cu)(t)dt R

=0

3

from which we see that R(C), the range of C, is orthogonal to m in H. Since the eigenspace of C is one-dimensional, R(C) has codimension 1 and therefore W:=R(C)

=

{u;u±m}.

(7.2.19)

Singularly perturbed evolution equations

204 Putting V := Lin{m] form

we see that we have an orthogonal decomposition of H in the

H = V ®W and the corresponding spectral projections: P onto V and Q onto W are orthogonal. As a byproduct of (7.2.18) we obtain Re{Cu,u)

<0

which means that C is a dissipative operator and, since it is bounded, we have Rea{C) < 0 ([47], p. 268). Now, if Eq. (7.2.6) is satisfied, then

{Cu,u)„

= |(-KO|u(O|2 + u(Om(o/0(£,r)«(m')m-HO^

j «/(oi«(o iMOde+/ /ttt,eMz'MVdzd? = {u, cu)H R3

R3R3

so that C is a self-adjoint operator and the assumption P.l of Section 6.2 is satisfied. To estimate the constant 7 in P.l we note that —7 = sup{A; A € o{C) \ {0}}. On the other hand, the interval [—c3, — c2] coincides with the essential spectrum of M„ (see Subsection 2.7.8) and since by Eq. (2.7.24) the essential spectrum is invariant under compact perturbations we see that there is 0 < 7 < c2 such that sup{i?e A; A 6 a(C) \ {0}} < - 7 .

(7.2.20)

If, instead of (7.2.6), the assumption (7.2.10) is satisfied, we rewrite Eq. (7.2.18) in the following form

m

2ReJ(Cu)(Ou(S)m-1(C)dli R"

JJHU R" R"

<

-2c4

- 2

u(Qy/Mf)

U(?)y/M0)

yfctf)

/ V ™(^)

-

<%<%'

C

/ «'(o^/«(e)^' (eK/^ IS7 ^ M d ^dtd?' --h

<■>)

(,

-2c4 (\\u\\2H - {u,m)H(m,u)H)

i-<

,

R3

Chapter 7. Bounded collision operators

205

where we used the property ||m||# = 1. Therefore, by Remark 2.1, we obtain for u 6 W (that is ulxn) Re(Cu,u)H<-c2\\u\\l which means that C is coercive on W A.2.

(7.2.21)

The constant <■■> was defined in assumption

Since QCQ is the part of C in W, we have a(QCQ) = ff(C) \ {0} and since QCQ is continuous we finally obtain sup/?eff(CCC) = sup{i?eA;AGff(C)\{0}} < - c 2 .

(7.2,22) ■

The operator Sp is formally given by the formula 3

V ( 0 = »Pf«(f) = E *P*6«(0,

(72.23)

where f = (£1162, £3), P = (Pi>P2,P3) £ R3- The space 7J5, discussed in Section 6.2, is defined by Ds := L 2 (R 3 , (1 + | £ | 2 ) m _ 1 ( f R ) -

(7-2.24)

It can be checked that the semigroup generated by Sp is given by the formula GSp{t)u = ei*lu.

(7.2.25)

This semigroup is clearly conservative so that the assumption P.2 is satisfied. Since D(C) = H, we have DT = D{C) n Ds = Ds = L 2 (R 3 ,(1 + l ? | 2 ) m - ' ( 0 ^ ) which is dense in H, as C0°°(R3) C L 2 (R 3 , (1 + I ^ H m ^ ^ R ) and C0°°(R3) is dense in H. Moreover, CJ°(R 3 ) is a core for each Sp. This follows from Subsection 3.4.5 and Eq. (7.2.25). Clearly, m satisfies all the assumptions (6.2.8), (6.3.35) and (6.5.3). The operator Tp generates a semigroup by the Bounded Perturbation Theorem (Sub section 3.4.1). The Trotter formula (6.2.10) gives, however, a better estimate of its growth. We obtain in particular that (Gr (t))(>o ' s a contraction semigroup. The

Singularly perturbed evolution equations

206

Bounded Perturbation Theorem also shows that D[TP) = D(SP) so that DT is a core for each Tp and thus assumptions P.4 and P.5 are satisfied. Next we turn to the assumptions of Section 6.3 (or Section 6.5). The operator QCQ is defined in natural way in a subspace W C H. We denote by (QCQ)r its restriction to W C\HT, where Hr := L 2 (R 3 , (1 + \t\2Y™-\t)d{)

(7.2.26)

and r € N* Proposition 2.2 Let r e N* be arbitrary. Under assumptions of this section, the operator (QCQ)T generates a uniformly continuous semigroup in HT of (negative) type which is independent of r. Proof. The result for r = 0 follows from Proposition 2.1. Let us denote by CT,Kr,... restrictions of respective operators to HT. We start with analyzing the structure of the spectrum of CT. First we prove that each Kr is a bounded operator. To this end we estimate the norm

||tf r u|| a H ,

=

/ m ^ l / ^ - O ^ ' R f U R

<

3

+

l ^ r m -

1

^

3

M

/ m ( o ( / 0 2 ( e , e ' ) ( l + le'|2)-rm(rR' R3

VR3

x / mi 2 (i + irfrm-^odn a + KIM R3

/

< KhWk, < N?||tC,,

(7.2.27)

(7.2.27)

where

N2 = jjm(o(i + i^i2r^(e,f )(i + \e\2rm-\odw

< +00,

R3K3

due to the properties of the Maxwellian m. The kernel kr of Kr, considered as an integral operator in Hrt is given by

2 T

kr(W) = --m(ou,e)m(0(i+\e\ and clearly

r

Chapter 7. Bounded collision operators

207

kT e L 2 (R 6 , (1 + l e p y m - ^ X l + | C ' | J ) ' m - l ( O d f d O . therefore Kr is a Hilbert-Schmidt operator in HT. It is seen that MUJ is an iso morphism of Hr so that, repeating the steps in the proof of Proposition 2.1, we see that A = 0 is an isolated eigenvalue of —M„T + KT. For arbitrary A satisfying - c 2 < Re A < 0, the operator XI + M„ r is an isomorphism and, since KT is compact, that A is either a regular point or an eigenvalue of XI + MU)T - KT. However, since HT C H0 = H, no new eigenvalue is possible in the strip {A; —c2 < ReX < 0}, thus - c2 < sup{Re A; A € a(Cr) \ {0}} < sup{i?e A; A € a(C) \ {0}}.

(7.2.28) (7.2.28)

Therefore, if the assumption (7.2.6) holds, we have sup{i?eA; A e a(Cr) \ {0}} < - 7 ,

(7.2.29)

where 7 is defined by Eq. (7.2.20) and if Eq. (7.2.10) is satisfied we have by Eq. (7.2.22) sup{Re A; A € a(Cr) \ {0}} = - c 2 .

(7.2.30)

Now, (QCQ)r is a bounded operator in Hr fl W for any r This follows from Eq. (7.2.27) and from the fact that W is an invariant subspace for C. It is easy to see that subspaces Hr n W and V = Lm{m} give the spectral decomposition of Hr corresponding to the decomposition of the spectrum a(Cr) = {0} U {o{CT) \ {0}}. Therefore we have a((QCQ)r) = cr(C r )\{0} and since (QCQ)T is a bounded operator, its spectral bound is equal to the growth bound of the semigroup generated by this operator (Subsection 3.5.1). ■ Since QCQ generates semigroups in both Ds = Hi and Dsi = H2, it follows from Remark 6.3.2 that assumptions (6.3.31)-(6.3.33) are satisfied. Moreover, since m is the Maxwellian, defined by Eq. (7.2.3), we see also that the assumption (6.5.3) is satisfied (yielding of course (6.2.8) and (6.3.35)). Therefore all assumptions of Theorems 6.4.1 and 6.4.2 (or Theorem 6.5.1) are satisfied and the whole theory of Chapter 6 is available.

3

Diffusion approximation to linear Boltzmann equation

Before we apply the results of Chapter 6 and formulate the main theorem of this section, we shall return to the formulation of the kinetic equation Eq. (7.1.1) and also

Singularly perturbed evolution equations

208

translate all general formulae of Chapter 6 into the language of this section. We start with operators PSP, PSQ, QSP, QSQ which were written explicitly in equations (6.3.18)-(6.3.21). Remembering that

JU(om^~1(o^

(u,v)H = R3

we obtain

Pu

= {u,m)Hm

= m (u{£,)d£,

(7.3.1)

R3

Qu

= u - (u,m)gm

= u - m / u(£)
(7.3.2)

R3

Since in the L2(0.,H) setting we have

(Su)(t, x, 0 = fa,ti(t, x, 0 = £ £kdtku{t, x, 0 , fc=l

it follows that

(PSP)u(t,x,0 = m(0J2 / a ^ f U . C K / ^ m K ' K 4 = 1

\R3

K3

=0,

(7.3.3) (7.3.3)

/

where we have taken advantage of the equality / e W C R ' = 0,

k = 1,2,3,

(7.3.4)

R3

which follows from the symmetry of Maxwellian, as defined in Eq. (7.2.3). Using Eq. (7.3.3) we easily obtain formulae for the remaining operators

(PSQu)(t,x,0

0

= m(0i;/&S.Xt, *,?)#.

(7.3.5)

fc=lR3

(QSPu){t,x,a

0

= m(0E^/^«(M,('K',

(7-3.6)

Chapter 7. Bounded collision operators

(QSQu)(t,x,0

=

209

E K ^ ( t , x , 0 - m ( 0 / ^ 9

I k

^ , x , O <

-m(0&/^ 4 ti(t 1 i,Ode , J

(7-3.7)

In the next step we find an explicit form for the coefficients of the diffusion operator defined in Eq. (6.3.29). The coefficients are invariant under the Fourier transform (as they do not depend on x) so that we can write the operator appearing in Eq. (6.3.29) with S instead of Sp. We denote l PSQ Vu = PSQ{QCQ)~ QSPu.

(7.3.8)

From the previous considerations it follows that there is an operator D such that V = D ® / on the space L 2 (R")®Lin{m}, that is, V acts on functions of the form u(x,0 = p(z)m(0

(7.3.9)

V(pm) = {Dp)m.

(7.3.10)

in the following way

We shall list and prove some properties of D in the proposition below. P r o p o s i t i o n 3.1 Operator D is a second order elliptic differential operator

(Dp)(x) = £

dkldlkiXlp(x),

(7.3.11)

with real coefficients d« given by the formula

dki = JdktfKM,

(7.3.12)

where dk are unique solutions in W of (Cdk)(£) = &m(fl, ft = 1,2,3.

(7.3.13)

/ / operator C is self-adjoint, then D is formally self-adjoint, that is, du = dlk. If the collision frequency satisfies the following assumption: for any linear isometric mapping r of K3 we have

Singularly perturbed evolution equations

210

tf(r£,rO

f.feR5,

= 0te,r),

(7.3.14)

then dkt = dSkl

(7.3.15)

/or some d > 0, uj/iere (5^ is t/ie Kronecker delta. Proof. From equations (7.3.8) and (7,3.10) we obtain {Dp)m = PSQ(QCQ)-lQSP{pm).

(7.3.16)

To find how D operates on p we take the scalar product in H on both sides of (7.3.16) with m to obtain (PSQ{QCQ)-1QSP{pm,m)H,

Dp =

where we used ||m||// = 1. Due to Proposition 6.3.1 we know that D is an elliptic operator. To find its explicit form we use the self-adjointness of P and Q and we represent S in the form 3

3

(Su)(x,0 = £ ( ( & , ® 5 & ) u ) ( i , 0 = E ^ V f e O > k=\

k=\

where the notation is self-explanatory. Clearly, the operators Sik are self-adjoint. Moreover, P % m = 0 by Eq. (7.3.4) so that QSaPm

= Sikm,

(7.3.17)

Using also the self-adjointness of P and Q we obtain

(PSQ(QCQ)-lQSP(pm),m)H

=

£

a ^ , I , p ( x ) ( ( Q C Q ) - 1 % m , S e ,m)„

fc,i=i

=

E ^ t , * )

(7-3.18)

so that Eq. (7.3.11) is proved. To express coefficients dki in the form (7.3.12) we note that since % m € W = QH there exists a unique solution to the equation

Chapter 7. Bounded collision operators

(Cu)(0

211

=

fcmtf),

which will be denoted by dk. Thus, dk = (QCQ)~lS(k)m

dk, =

and

{dk,Sx,m)n=Jdk{?)$dt'.

Thus Eq. (7.3.12) is proved. Clearly, all the coefficients are real. This follows from the fact that the operator C is real and invertible on W If C is self-adjoint, then D is also self-adjoint. Indeed du = ( ( Q C Q ) - ! 5 & m , S ^ m ) i / = ((QCQ)-»5 & m, Sitm)H

= dlk.

Finally, let the assumption (7.3.14) be satisfied. We introduce the notation f(ij = (6.6)> £{2) = (£i,6) and £(3) = (£i,£ 2 ). We see that £ i m ( 0 is invariant under any rotation in the plane £i = 0. Also, since all coefficients of C are invariant under such rotations, the same must be true for the solution d$£) which means that there is a function of two variables, say d0, for which di(fl=4>Ki.lf(i)l). where | | denotes here the Euclidean norm in K2 Similar considerations can be carried out for the remaining indices so that we obtain dk(Z) = do{£k,\S{k)$, k= 1,2,3.

(7.3.19)

Similarly, we can show that d0 is an odd function with respect to the first variable, that is, do(-6,|£(*)|) = - d o f o . M ) -

(7-3-20)

With this result the equation (7.3.12) gives dki = 0 for k / I and if we use (7.3.19) we obtain that dkk = d for some positive constant d. m Next we shall write in (more or less) explicit form the remaining terms of the expan sion: Eqs. (6.3.11) and (6.3.12). We start with the initial value for (6.3.29) which is given by 8(0) = v -

ePSQiQCQ)-1™,

212

Singularly perturbed evolution equations o

where v(x,£) = m{^)Q{x) = m(£) J u(x,?)d? R

_ o

o

o

(see (7.3.9)) and w = Qu = u - xaQ.

3

Here u is the initial value of the original equation (7.1.1). o

If we define d to be the unique solution in W to the equation

(Cd)(z,0 = «(*,0-m(0£(z) (such a solution exists as the right hand side belongs to W), then by (7.3.6) and from o

o

Qd = d we obtain (PSQ{QCQ)-lw){x,0

= (PSQd)(x,£)

= m(fl £

j?dXkd(x,£')<,

k=lR3

and consequently we obtain the following initial value problem for p

dtp(t,x)

d

= e J2

kidlk,xlP(t,x),

k,l=l

p(0,x)

=

°e-eJ2

IadXkd{xt(,')d€! l

(7.3.21)

R3

Next, from Eqs. (6.3.6), (7.3.13) and (7.3.17) we obtain 3

Mt, *, 0 = -{{QCQ)-lQSPv){t,

x, t) = - £ dh(t)dXkp(t,

x).

(7.3.22)

The operator QCQ usually cannot be simplified but, since it is bounded, the corre sponding semigroup (T<jcQM)t>o is simply the exponential function and we can write the equations for w in the following form u>o(t/e) = exp (-QCQ)

W.

(7.3.23)

To evaluate the initial value for u>i we recall Eqs. (6.3.10) and (7.3.13) to get 3

i(0) = {QCQ)~lQSPv and by Eq. (6.3.40) we obtain

I

= £ dkdXkQ

(7.3.24)

Chapter 7. Bounded collision operators

Q

=

213

exp ^ Q C Q ) tBi(O)

(7.3.25)

T

+- e x p

( - Q C Q ) | e x p f - - Q c g ) QSQexp (-QCQ\

wds,

where tZ)j(O) and QSQ are given by formulae (7.3.24) and (7.3.7), respectively. Finally, we write down the formula for the initial layer for the hydrodynamic part of the solution, Wj. From Eq. (6.3.9) we have CI(T) = P5Q( PSQ(QCQrlw0(r)Denoting by d the unique solution in W to

(Crf) (£,*,*) == w0 (-,x,n

== expQgCQJ w(x,0

and using Eq. (7.3.5) we get

i)

1

^,x,^=m(OpQ,x)=m(OE/^td(^.e')<.

(7.3.26)

Using these results we can write down the approximation u' 1 ' = o' 1 ' + ty' 1 ' to the solution u of (7.1.1) obtained by the compressed asymptotic method in the following form V(D

_

c (i) + e S j ]

(7.3.27)

(1)

=

i5 0 +€(ti>i+tfi),

(7.3.28)

w

where s' 1 ' = pm and the remaining terms of the expansion are defined by Eqs. (7.3.21), (7.3.26), (7.3.23), (7.3.25), respectively. Alternatively, if we are interested only in the approximation of the hydrodynamic part of the solution w, defined as e(t, x) = I u(t, x, f )d£', R

then the approximation to g is given by

3

(7.3.29)

Singularly perturbed evolution equations

214

p
(7.3.30)

where p was defined in Eq. (7.3.26). Theorem 6.4.2, specified to the considered case, has the following form Theorem 3.1 // assumptions A.l, A2, (7.2.7) and either (7.2.6) or (7.2.10) are satisfied and

max3 I \p\n\\u\\2H2dp

< +oo

(7.3.31)

(or, equivalently u belongs to the Sobolev space W23(Q., H2)), then for any T, 0 < T < 00, there is a constant C such that for every t € [0, T] we have \\u(t) - (v^(t)

+ w{l)(t)\\H

< Ce2

(7.3.32)

and ll^)-p(1)WIU2(R3)
(7.3.33)

where S (1) , w(1) and p{l) were defined in (7.3.27),(7.3.28)

and (7.3.30), respectively.

Chapter 8 Applications to equations of Fokker-Planck type 1

Introduction

In the previous chapter we described the linear Boltzmann equation

(dtu){t,x,Z)

=

-(dxv.(t,x,£) + J {-k(£, Z)u(t, x, 0 + k(£, C)u(t, x, e')) d?,

O0,i€l",

where we used Eq. (7.2.8) to write Eq. (7.1.1) in a more symmetric form. This equa tion describes the time evolution of particle distribution function when the main mech anism causing the transition of the particle from one state to another are collisions. If we replace collisions by some other stochastic mechanism, then the corresponding evolution equation can be written in the following form [75]:

(dtiL)(t,x,o = -e&ufo'.o (s-i-i) f/W- ■*z)u(t,x,?)-A(t-*e)u(t,x,t))d?,o o . i e R " , 3 IR3 R

where A(£' —> 0 is the transition rate from the state f to f. referred to as the master equation of the Kramers type.

Equation (8.1.2) is

Very often this nonlocal integro-difierential equation is approximated by a local differ ential equation. This is done by expanding A into a power series, called the KramersMoyal series (see e.g. [75]), about x = x' and truncating this series at some finite 215

Singularly perturbed evolution equations

216

term. If we truncate the Kramers-Moyal expansion on the second term, then the Fokker-Planck equation of the Kramers type is obtained

, (D?J(tMt,ir . f ) ) -

dtu{t,x,£) = —£dxu(t,x,£) +

(D^(Ou(t, *.f)), 1' = = 11

t1.1=1 ,j=i

(8.1.2) where for i,j = 1,...

,n

A(1)(0 = /te-CW^O< R3

and

oSJ(o = ^/(6-fi)te-<;-M(r-4 ex, R3

where <5,_, denotes the Kronecker delta. The operator

(FnU)(o := £ diXi (Dl2J(0u(t,x,$) - £ a f t (z?(1>(?M£)) t, J = i

«=i

will be referred to in the sequel as the Fokker-Planck operator. A general analysis of n-dimensional Fokker-Planck equation is quite difficult and thus far beyond the scope for this book. The main difficulty is that the matrix {Aj}ij=i,.,n is v e r v often not positive definite; hence the Fokker-Planck operator turns out to be a degenerate elliptic operator. For that reason we shall focus in this chapter on selected examples which are of interest in applications. However, before this we shall discuss some general aspects of one dimensional Fokker-Planck operators. Let us consider the one-dimensional version of the Fokker-Planck operator

( F l U ) ( 0 := d\ ( £ ( 2 ) ( C M O ) - h ( £ ( 1 ) ( 0 « ( 0 ) , - o o < a < ( ( 2 ) (0«(0) - O ( 1 ) (0"(C) = 0 for :r = a,6.

(8.1.4)

This corresponds physically to the zero flux at boundaries. In many cases, however, either a = - o o and/or b = +00, or the coefficients vanish at the boundary points.

Chapter 8. Fokker-Planck equations

217

Then we have the degenerate problem, similar to that encountered in Section 2.7. In this case the boundary conditions (8.1.4) should be replaced by an appropriate condition for the behaviour of the solution at the singular points. Unfortunately, the Sturm-Liouville theory cannot be applied directly here since Fi is not a formally self-adjoint operator. To remedy the situation we introduce an eigenfunction of the Fokker-Planck operator corresponding to the 0 eigenvalue, that is, the function eo satisfying a£2(/?(2)(Oeo(0)-^(^(1)(Oe0(0)=0, subject to the appropriate conditions. It can be explicitly calculated that

5

*-^|-jV*l

<"->

We can use eo to write F\ in a compact form

(FlUm = d, (eoCO^HOWKMO) ■

(8-1-6)

It is now easy to see that Fi is a formally self-adjoint differential operator in the weighted space L2([a, 6], e^'d^), that is, for any v € C^°(]a,b[) we have b

m=

b 1

f{Fxu)(t;W£)e^(Z)dt a

a

«(o Ju(0(F^)(Oe; (t)dt.

In fact, it can be proved that if we take a suitable realization of F\ in L2{{a, 6], eo"1), which takes into account the boundary conditions, then the resulting operator will be self-adjoint. Note that we encountered a similar situation in Chapter 7 where the analysis of the linear Boltzmann equation was carried out in the weighted space L 2 (R, m _ 1 d£)Alternatively, to be able to take advantage of the Sturm-Liouville theory, we can introduce a new unknown function to transform the Fokker-Planck operator into a formally self-adjoint operator in the standard L2 space. To this end we define U(0 ■= eo l/2(Z)u(a

(8.1.7)

It can be calculated that such defined U satisfies the following equation corresponding to Eq. (8.1.2):

Singularly perturbed evolution equations

218

dtU(t, x, O = -^dxU(t,

x, 0 + F[U{t, x, 0 ,

(8.1.8)

(9fD

(8.1.9)

where F[U = dt (DWdfU) + \ (dtD^

- diD^

-

' ^ 7 2 f " > ) 2 ) U.

Clearly, Fx' is formally self-adjoint in L2([a,b]) and its suitable realization is a selfadjoint operator. In this chapter we shall use both these approaches. Finally, we point out some choices of Z)(1) and Z?(2) which lead to classical degenerate Sturm-Liouville problems [23]. (a) The choice a = - 1 , b = 1, Z>(2)(?) = 1 - £2 and D(1>(£) = - 2 £ leads to the Legendre operator discussed in Subsection 2.7.11. Here we have e0 = 1 so that in this case the Fokker-Planck operator is self-adjoint in L2{[—1,1]). The asymp totic analysis of the corresponding Fokker-Planck equation, which describes, for example, electron scattering in plasma, will be carried out in Section 4. (b) The choice a = - c o , b = - o o , L>(2)(£) = 1 and D ( 1 ) ( 0 = ~H, k > 0 gives the Hermite operator. Here e0(£) = e" 2 and the Fokker-Planck operator is self-adjoint in L 2 (R,e 2 ) The transformed operator F[ is now related to the harmonic oscillator operator of Subsection 2.7.12. In fact, it is exactly the harmonic oscillator operator, defined by Eq. (2.7.44), if k = 2. Otherwise, F{ can be transformed into the harmonic oscillator operator by rescaling of the velocity variable. This will be done in Section 5, where we perform the asymptotic analysis of the corresponding Fokker-Planck equation. The FokkerPlanck operator of the Hermite type describes the Ornstein-Uhlenbeck process on an infinite line and, in particular, Brownian motion [75, 23], (c) Another interesting choice is when a = 0, b — 00, Z)(2'(£) = x and -D' 1 '^) = a+1—£,a > —1. Then e 0 (O = e~(-xa and the modified Fokker-Planck operator F[ is the generalized Laguerre operator discussed in Subsection 2.7.13. The corresponding Fokker-Planck equation describes, for example, the vibrational relaxation of the harmonic oscillator in the continuous (high temperature) limit and the energy relaxation of a hard-sphere Rayleigh gas: this will be dealt with in Section 6. Before starting to analyze concrete examples we shall reformulate the assumptions of Chapter 6 to make them more appropriate for collision operators of Fokker-Planck type. The asymptotic theory presented in this chapter is due to [7].

Chapter 8. Fokker-Planck equations

2

219

General assumptions

We assume here that the operators involved satisfy the assumptions P.1-P.4 of Section 6.2. Since, however, the collision operators appearing here are second order differential operators, their properties differ strongly from those of the operators discussed in the previous chapter. In particular, the collision operators are usually "stronger" than the operator of multiplication by the velocity. Thus, it is possible to replace assumptions P.5 of Section 6.2 and (6.3.31)-(6.3.33) by the following simpler one: P . 5 ' For p e P" D(C) C Ds

(8.2.1)

and SPD{{I - C)1+a) C D((I-C)a)

(8.2.2)

for some a > 0. Remark 2.1 If Sp G C(H) for p € P", then there is no need to assume inclusions (8.2.1) and (8.2.2). Indeed, in this case P.4 is satisfied automatically by the Bounded Perturbation Theorem (see Subsection 3.4.1) and, as we shall see in Proposition 2.1, the assumption (8.2.2) is not necessary. ■ Remark 2.2 It follows that we can replace Eq. (8.2.2) by a slightly stronger, but easier to check assumption: for p € P", SP(D(C2))

C D(C).

(8.2.3)

Indeed, we have D((I - C)') = D(C') for i = 0,1,2 by the results of Subsection 3.3.2. By e.g. holomorphic interpolation (see Subsection 3.3.4) we see that the domains D((I — C)1+a) and D((I — C)a) are interpolation spaces of the same order between the pairs [D((I - C)2),D(I - C)} and [D(I - C),H], respectively. From the Eqs. (8.2.1) and (8.2.3) it follows that Sp € C(D((I - C f ) , D{I - C)) n C[D{J - C), H) and by Eq. (3.3.22) we have Sp g £(£>((/ - C*) 1+Q ),D((/ - C)a)) with

||Sp||£(£>((/-C) 1 +'>),D((/-C)°))

^

C m a X

<

Slpl,

(\\Sp\\cmC*),D(C))i

\\Sp\\c(D(C),H))

(8.2.4)

Singularly perturbed evolution equations

220

where the constants c and S are independent of p. The last inequality follows from the fact that the mapping p -> Sp, being finite-dimensional, is continuous from P n to both £(£>((/ - C) 2 ), D(I - O) and C(D(I - C), H). m Remark 2.3 Assumptions P.4, Eqs. (8.2.1) and (8.2.2) (or (8.2.3)) can be replaced by another requirement which is often easier to check: Z)((-C) Q ) C Ds

(8.2.5)

for some 0 < a < I. Indeed, P.4 follows immediately by Subsection 3.4.2 and we shall see in Proposition 2.1 that the assumption (8.2.5) is sufficient for the availability of (6.3.31)-(6.3.33). We shall use Eq. (8.2.5) in the analysis of the Fokker-Planck equation of the Brownian motion in Section 5. It fails to hold, however, in the case of the Laguerre operator, discussed in Section 6, since the multiplication operator is "of the same order" as the Laguerre differential operator. ■ Let us recall the notation Ha = D((—C)a) where the latter space is equipped with the graph norm. We shall prove that if Sp and C satisfy the assumptions listed above, then (6.3.31)-(6.3.33) are satisfied. Proposition 2.1 If C and Sp satisfy D(C) C Ds and SpD({-C)l+a)

C D((-C)a),

(8.2.6)

or D((-C)a)

C Ds

(8.2.7)

for some a > 0, or Sp G C(H),

(8.2.8)

then (6.3.31)-(6.3.33) are satisfied with X = Ha and Z = D{C) in the first case and X = Z — H in the last two cases. Proof. First we note that since {GQCQ(t))t>o three cases D(C) C Ds, then the function t ->•

is an analytic semigroup and in all

QSpQGQCQ(t)w

(see Eq. (6.3.31)) is continuous on ]0, +oo[ for any w e W

Chapter 8. Fokker-Planck equations

221

Now let the assumption (8.2.6) be satisfied and let, for the time being, SP}0, denote the part of Sp in Ha, that is, S p Q := Sp\Dsa where Ds,a ■= {ueHa\

Spu€

Ha}.

Since the norm of Ha is stronger than that of H and Sp is closed in H, it follows that SPia is a closed operator in Ha. Therefore by Eq. (8.2.6) we have Spa € £(Wi+ Q , Ha) and by linearity of p —> Sp we get \\SP\\c{Hi+a,Ha) < KM-

(8-2.9)

This inequality holds also for a = 0. Now, since W = QH reduces C, Eqs. (8.2.6) and (8.2.9) hold also in W (see the discussion leading to Eq. (6.3.30)). So, if Eq. (8.2.6) is satisfied, we then have by Eq. (8.2.9) with a = 0 and Eq. (3.3.18)

\\QSpQGQCQ(t)w\\H

< \\QSVQ{QCQY \\QSpQ{QCQ)-\-QCQ)l-aGQCQ(t)(-QCQ)aw\\H _1 a 1 i < M i t aM t - e^ \\\p\w\\Ha 1e

for some 7' < 7. Next, by Eq. (8.2.9) with a > 0 we have (-QCQ)aQSpQ(-QCQ)-l-a

,-l-a

G C(H)

which yields by Eq. (3.3.18)

\\(-QCQrQSpQGQcQ(t)w\\H)*\\H < <

a 1 a a \\(-QCQ)°( \\(-QCQ) QSpQ(-QCQ)- - (-QCQ) GQcQ(t)(-QCQ)w\\H

M2t-ae^''\\\p\w\\D(C),

so that in the first case the assumptions (6.3.31)-(6.3.33) are satisfied. Now let Eq. (8.2.7) be satisfied. Then, similarly as in Eq. (8.2.9), we obtain \\Sp\\C(Ha,H) < K2\P\ and

\\QSPQGQCQ(t)w\\H

< \\QSPQ(\\QSPQ(-QCQ)-°(-QCQrGQCQ(t)w\\H <

M3rQe-y'|p|||w|U,

Singularly perturbed evolution equations

222 and the thesis follows by Remark 6.3.3.

Finally, if Eq. (8.2.8) is satisfied, then we clearly have

\\QSPQGQCQ{t)w\\H <

UMWMIH

and the proposition follows again by Remark 6.3.3.

■

It follows from this proposition that all terms of the compressed asymptotic expansion, discussed in Section 6.3, are well defined under any of the assumptions introduced above so that the whole asymptotic analysis of Section 6.4 is valid. Checking, however, the availability of these assumptions in particular cases is very tedious, hence in the next section we present another look at them from the point of view of the variational theory, developed in Subsection 2.6.13. Note that we encountered a similar situation when we were dealing with solvability of the diffusion equation in Example 3.7.4.

3

Variational setting

Fokker-Planck type collision operators are given by second order differential expres sions of elliptic type, hence it is natural to consider them in the variational setting discussed in Subsection 2.6.13. Unfortunately, the assumptions P.4 of Section 6.2 and (6.2.11) have to be cast in a different form so that it is easier to use them in this setting. Let us suppose that Hi is another Hilbert space densely embedded in H and let c be a continuous sesquilinear, symmetric form on Hi x Hi such that —c satisfies the Garding inequality (Eq. (2.6.21)), that is, - Rec{u,u)

+ X0\\u\\2H>a\\u\\2Hi,

u £ Hu

(8.3.1)

for some constants a > 0 and A0. We assume that the collision operator C is asso ciated with - c in the sense of Eq. (2.6.22) and satisfies the remaining assumptions of P.l in Section 6.2, that is, C is dissipative and 0 is its isolated simple eigenvalue. Note that by the remark after Eq. (2.7.23) we have Hi =

D((-C)1'2)

and D(C) is dense in H. Let us turn our attention to the operators S p , satisfying P.2 of Section 6.2. In addition to Eq. (8.2.1) we assume that for some constant M and all u, v E Hi n Ds the following inequality holds

Chapter 8. Fokker-Planck equations

223

\(Spu,v)H\<M\\u\\Hl\\v\\Hl.

(8.3.2)

It follows that D(C) C Ds n Hi and it is dense in Hx, thus Eq. (8.3.2) yields that (SPU,V)H can be extended to a continuous, sesquilinear form sp : H\ x Hx —> C. For p e P n we denote by Tp the operator generated by i p = c + s p with the domain D{%) = { « 6 # i ; f„u € / / } . We assume that D(fp) = D(C)

(8.3.3)

for every p € P" By virtue of Eq. (8.2.1) this assumption may appear to be the same as the assumption (6.2.11). It concerns, however, the domains of variational operators which are usually much easier to determine, as they do not involve passing to limits as in the definition of Tp in Eq. (3.4.8). Moreover, we shall prove that under the introduced assumptions the operators Tp and Tp coincide. T h e o r e m 3.1 For every p € P the operator Tp generates a strongly continuous semigroup of contractions, say Gf , given by the formula G

rP«- = i™(Gc(£)G

5 p

(£))\

ueH

(8.3.4)

and the limit is uniform with respect to t on bounded intervals. In particular, fp = Tp.

(8.3.5)

Proof. First we prove that Tp is dissipative. Let u G D{TP). Then by Eq. (8.3.3) we have u € D(C) and Eq. (8.2.1) yields u € Ds, thus Re tp(u, u) = Re (Cu, u)H + Re (Spu, u)H < 0 by dissipativity of C and conservativity of Sp (see P.l and P.2 of Section 6.2). Since D{C) is dense in Hi and tp is continuous on Hi x Hi we get Retp{v,v)

<0

for every v e Hx and consequently Re(7>,u)w < 0

Singularly perturbed evolution equations

224

for u e D(fp). By Eq. (8.3.1) and by the conservativity of Sp we see that tp satisfies the Garding inequality and thus fp is an m-dissipative operator. Hence, it generates a semigroup of contractions. By Eqs. (8.2.1) and (8.3.3) we have %u = Cu + Spu for u £ D{C), therefore (fp,D(fp)) = {C + SP,D(C)) = (C + SP,D{C)), where the last equality follows from the fact that % is closed as an m-dissipative operator. Equation (3.4.8) implies that fp = Tp and Eq. (8.3.4) is a consequence of the Trotter formula (6.2.10). ■

4

Fokker-Planck equation of electron scattering in plasma

Let us consider an infinite or periodic medium filled with plasma consisting of heavy ions and electrons. The electrons are supposed to have the same constant kinetic energy. We assume that their velocity distribution is invariant under rotations of a fixed coordinate axis Ox. Therefore, the distribution function u of electrons depends only on their position x and the cosine of the angle between the axis Ox and their velocity. This cosine will be denoted by fi. The evolution is due to the combined effects of the streaming modelled by (Su)(x,n)

=

ndxu(x,n)

and the Coulomb-type scattering against heavy ions which, in the Fokker-Planck theory, is modelled by the Legendre differential expression (see Subsection 2.7.11) {-Lu){x, n) = d„((l - i/^uix,

/u)),

where x 6 K or x G [0, 27r] in the free-space and periodic cases, respectively, and fie [-1,1]. After the application of the Fourier transformation the corresponding initial value problem for the Fokker-Planck equation has the following form

dtu(t,p,)i) u(0,p,fj,)

= -ipnu(t,p,fj.) =

u{p,n),

+ dIJ((l-

M2)9Mu((,p,/u)), (8.4.1)

where, as in Chapter 6, we dropped the hat in the notation of the Fourier-transformed function u. The basic facts about the Legendre differential operator were given in Section 2.7.11. Here we recall that the variational space Hx (see Eq. (2.7.36)) is given by

Chapter 8. Fokker-Planck equations

225

Hx := {u € L 2 ( [ - l , 1]); p -> y/l - p^u{p)

6 L 2 ( [ - l , 1])}.

The collision operator, which somewhat inconsequently, but in accordance with the notation of the "applied" part of the book will be denoted here by C (C = At in the notation of Subsection 2.7.11), is generated by the symmetric, sesquilinear form defined by I

l{u,v) '■= ~

[(1 - ^2)d^u(p)dtlv(p)^

dp, u,v 6 Hi.

-l

We know that the unbounded operator (C, D{C)), where by Eq. (2.7.41) we have

D(C) := {« e Hi; Cu € H) = {u e Wj'd - 1,1[); A* -4 ( l - p X r i G Wa2(] - 1,1[)}, is self-adjoint, dissipative, its resolvent is compact and the spectrum is given by {—m(m+ l)}m€N- Therefore we see that the assumption P.l of Section 6.2 is satisfied with 7 = —2. Since p 6 [—1,1], the operator Sp defined for p 6 Pn by {Spu){p,p) :=

-ippu(p,fi)

is bounded and by Remark 2.1 all the remaining assumptions are satisfied. Therefore Theorem 6.4.1 with H = L2{[-1,1]) and Z = / / is valid. We complete this section by deriving the explicit form of the initial value problem for the diffusion equation (6.3.29), the solution of which gives the diffusion approximation to the hydrodynamic part of the solution of the initial value problem (8.4.1). Since the normalized eigenfunctions of C, that is, the Legendre polynomials (see Eq. (2.7.42)), form an orthonormal basis in L2([—1,1]), for any u € L2([— 1,1]) we have oo

u= Y, umPm, m=0

where I

v-m ■= /

and (u m ) m6 N € h- Clearly

u{p)Pm{p)dp

(8.4.2)

Singularly perturbed evolution equations

226

Cu = - J2 m(m + t)umPm m=0

and D(C) can be alternatively characterized (see Eq. (2.7.14)) by D(C) = {u € L2([-l,

1]); (m(m + l ) « J m e N € l2}.

Since for each p G P" the operator Sp is bounded, by Eq. (2.7.43) we have for any «€la([-l,l])

OO

(SPU) = -ipY^UrnllPm

(8-4.3)

m=0

_

"

V P

(

m + l

ra

P

/

m

i

P

"l

m+1 +

^ 0 > v ^r+3v 2^TT _. ^ f m+l Jp " ^0\x^zT3/2^Tl"m+1

y ^ T T v ^ T ^ T m_1) m 1 / m 1 x^fr+Tv 2l^^T" - J m'

where we used the fact that the multiplication by fi is a bounded operator and that the sequences m+l \ ^ 2 m + 3V2m + i ; m e N

m

and

0.

\ \ / 2 m + l ^ m - I, ,

are bounded. Note that in Eq. (8.4.4) we used the convention u_j := 0. Let us return to the original problem (8.4.1), that is, we replace p by x and the multiplication by ip by the differentiation with respect to x. Similarly as in Eq. (7.3.29), we use the following notation: l

e{t,x) := fu(t,x,ii)dfi. d/i.

(8.4.4)

-l

The stationary solution e0 is, by Eq. (2.7.42), given by e0(fJ.) = P0(fj) = ~ , and by Eq. (8.4.4) we easily obtain

(8.4.5)

Chapter 8. Fokker-Planck equations

227

(PSPu)

= 0,

(PSQu)

=

-i=aiUlP0,

(QSPu)

=

—y=d x v^P v v3

(8.4.6)

Consequently, the diffusion operator of Eq. (6.3.4) will take the form PSQ(QCQ)-lQSPu

= iflguoPo

(8-4.7)

b

and for the initial value corrector we have the following formula PSQ(QCQ)'1w o

o

po

o

= 4=ft«ii,o, v3

(8-4.8)

o

where w = Qu = Y, umPm and u is a sufficiently smooth initial value of the problem m=l

(8.4.1). As in Section 7.3 we denote v{l){t, x, ft) = v{l](t, x, fi) + evi(t, x, n) = p(t, x)P0(n) + ep(t, x)P0(p.).

(8.4.9)

With this notation the initial value problem for the diffusion equation will have the form

dtp = p(0)

=

-g5xP. uQ-—=dxu1.

The explicit form of the initial layer corrector iii = pP0, defined by Eq. (6.3.9), is given by

>G) =

\e-2^dxuv

Thus Theorem 6.4.1, specified for this example, states that if u 6 H'|(R, L2([—l, 1])) in the free space case or u € W23iff([0, 27r], L 2 ( [ - l . 1])) in the periodic case (see Sub section 3.7.4), then

Singularly perturbed evolution equations

228

Q{t)-p(t)-epfy

= 0(e2) L 2 (nx[-i,i])

uniformly for t in bounded intervals of [0, oo[. Here, Q = R in the free space case and fl = [0, 2n] in the periodic case.

5

Fokker-Planck equation of Brownian motion

In the case of n-dimensional Brownian motion the collision operator corresponds to the three-dimensional differential operator

(Cu)(p,o = ^(e + ae)u(p,o

(8.5.1)

and the (Fourier transformed) streaming operator is, as usual, of the form

(5u)(p,0 = »Pf«(p,0.

(8-5-2)

where £,p £ Rn Here u is the particle distribution function in the phase space, x denotes the position and £ the velocity of the particle. First we shall transform the Fokker-Planck operator to the harmonic oscillator oper ator whose one dimensional form was discussed in Subsection 2.7.12. As in Section 1, we use the following transformation of the unknown function u(£)- For £ = \/2x £ Rn we define y(x) = {Anu)(x) := (V2)te^"«(-s/2a;).

(8.5.3)

mi This is an isometry of the space L 2 (R n , e i d£) onto L2(M.n,dx) which transforms the Fokker-Planck collision operator C into

Cy = ^ y ^

e

' ^

[dlv - M2V + ny) ■

(8.5.4)

Dropping the normalizing factor we arrive at the harmonic oscillator operator in L2{Rn), denoted hereafter by H, (Hy)(x) = d2xy(x) - \x\2y(x) + ny{x).

(8.5.5)

Note that the scaling factor \/2 in the transformation was necessary to arrive at the standard form of the harmonic oscillator operator defined by Eq. (8.5.5).

Chapter 8. Fokker-Planck equations

229

To analyze this operator we introduce the sesquilinear form

H4>, 4>) = J (dx4>dxi> + \x\24>ii + } dx,

(8.5.6) (8.5.6)

R"

defined originally on C£°(W) and the Hilbert space Hi defined as the closure of C$°(W) with respect to the norm ||0||#, = \fh(,<j>). We have the following lemma: Lemma 5.1 The imbedding Hi —> L 2 (K n ) is compact. Proof. It is clear that Hi c W 2 l (R n ) with continuous injection. Let B(x0,r) denote the open ball centered in x0 with radius r. We take a function G C,^°(Rn) with the following properties: 0 < <j> < 1 and for some r > 0, 0 = 1 on B(0,r) and supp 4> C B(0, 2r). Let (un)n€?i, un G / / i , be a sequence which satisfies for every n € N IKIk < K

(8.5.7)

for some K > 0. The sequence (||un||n/i(R")) is also bounded so that by [89], p. 126 we can extract a subsequence (u ni )„ 6 N which converges weakly to some u in both Hi and VJ^R 71 ). If so, the sequence (i>„Jn€N, where vnk = L 2 (5(0,2r)) is compact, from each subsequence of (^nJngN we can select a subsequence which converges in the norm of L2{B(0,2r)). Since each subsequence has a weak limit, which is equal to v, it must also be a strong limit of each subsequence in L2(B(0,2r)) and therefore the whole (I>„J„ 6 N must converge to v in L2(-B(0, 2r)). Now we see that j \unk - u\2dx <2 J (t>2\unk - u\2dx + 2 / (1 - 4>)2\unk - u\2dx. R»

R"

(8.5.8)

R"

The second integral can be estimated as follows:

| ( 1 - r

\unk - u\2dx

<

^

J

M 2 K - u\2dx < - j ,

|l|>r

where K' = AK2 by Eq. (8.5.7) and the definition of the norm in H\. For any e > 0 we can choose r in such a way that 2K'/r 2 < e 2 /4 and for this r we can take k0 (note that the sequence (i>„t)neN is independent of r but for the multiplier <j>) such that

Singularly perturbed evolution equations

230

\vnk - v\2dx <

I

f

-

4

\X\<2T

which gives the estimate of the remaining term in Eq. (8.5.8).

■

Let Ah denote the operator associated with h according to Eq. (2.6.22) (see also Subsection 2.7.12). It follows that h satisfies the Carding inequality, thus it is a self-adjoint operator with compact resolvent by Lemma 5.1. This means that the spectrum of Ah consists only of eigenvalues and the operator itself can be expressed in terms of the series of its eigenfunctions. Using the separation of variables and the one-dimensional theory of the harmonic oscillator (Subsection 2.7.12) we obtain the following expression for the eigenfunctions of Ah'.

H£n)M = ^ J W C I K ^ 1 * ' 2 = n*£}fe).

(".9)

where x g R", a = (ait... ,a„) is a multi-index and H^J are normalized onedimensional Hermite functions corresponding to the eigenvalue A = —a, defined by Eq. (2.7.47). Let C denote the Fokker-Planck collision operator obtained from Ah by the inverse transformation (8.5.3), and thus corresponding to the differential expression (8.5.1). For k = 1 , . . . , n and the multi-index 0 = (Pi,. . . J3k) we define

*?> - A , - 1 ^ ' that is, $L )(?)

"

= (o^Tn3*''^

mi 2

= ft <'te).

Since A t is an isometric isomorphism, the family ( $ ' n ) l |;|2 basis in L2(Rn,e * d£). We therefore have

(8-5.10)

forms an orthonormal Q€N"

oo

u =

(8.5.11) |a|=0 ld=0

in L2(Rn) and

Cu=-

Y, |a|«Q<E>in), |a| = l

(8.5.12)

Chapter 8. Fokker-Planck equations

231

so that it is clear that C is dissipative and satisfies all the assumptions in Section 3. Moreover, it is easy to see that Ds = {u € L2(Rn);

|e|wGL 2 (R")}

so that from Eq. (8.5.6) we obtain that Hx = £>((-^n'. Let us adopt the following convention a(i, ±1) = ( e n , . . . , a, ± 1 , . . . , On). From the one-dimensional recurrence formula (2.7.48) for the Hermite functions we derive the following relation for $*>n'. Let i — 1 , . . . , n, then fc*M = v V f T * f r { l , + 1 ) ( 0 + v / ^ U ) ^ ) -

(8-5.13)

If some Q, = 0, then naturally the second summand vanishes. By Eq. (8.5.13) we obtain formally

oo

/

n

\

Spu = -i £ «« $ > (x/^TT*L(!,+1) + V^*5U>) |a|=0 n

= ~l E P* k=l

VJb=l /

/

oo

\

(V®~kUa(k,-i) + Vak + lutt(t,+i)) *Ln)

£ \|a|=0

(8.5.14)

(8.5.14)

/

To prove the last formula for u € Z?(C) we note that by Eq. (8.5.12) for such u we have oo

Y, |a| 2 |u Q | 2 < oo. M=o For uN =

N

E « a we obtain |c|=0

(8.5.15)

Singularly perturbed evolution equations

232

ZkUN =

£

^ (v'at + l*L"l,+i) + \/o* $ L"Li)

|a|=0 N

J2 (\/a~kUa(k-i) + Vak + l«0(*,+i)) *Ln) + # N ,

=

|o|=0

where RN = £

>/5T+T(u 0 *$ i+1)

-ua{k,+l)^).

By Eq. (8.5.15) it is clear that lim \\RN\\n = lim Y, (tt* + X) (M2 + lu«(*,+i)l2) = ° and also the first summand converges so that Eq. (8.5.14) is established. We can now identify operators appearing in Eq. (6.3.29). Let, as in the previous section, v = p$o ' and V\ = p$o Introducing the notation 0(*;!) = ( 0 , . . . , l , . . . , 0 ) and O(i,j;/c,/) = ( 0 , . . . , f c , . . . , Z , . . . , 0 ) , where I (resp. (k,l)) are in the i-th (resp. z-th and j-th) place, we get

^

= ->EP4WIP

and further SpQ(QHnQ)~'QSpPv = - £ M , i=i

=

(f>
-Ep*(EMi*oJt;i,)p

= -x>(i h=\

Projecting this onto $o

we

/

\vi=i«¥*

p/
g e t the diffusion operator in the form

Chapter 8. Fokker-Planck equations

233

PSpQ{QCQYxQSvPv

-\p\2v.

=

Similarly for the corrector of the initial value we obtain

PS.QiQCQr'QS.Pw

=i ^

£Pt«o(*.i) fc=i

and the initial layer corrector ii will have the form 't\ -i-(n)' vi ( - ) = ie •$i, n ' ^PkUo(k,\), tJ

k=i

where uQ and Mo(t;i) are the zeroth and first moments of the Fourier transformed initial value for u. To formulate the final result of this section we return to the original quantities, that is, the multiplication by ipk will be replaced by dXk. We also recall the notation g{t,x) := (u{t,x,-),${0n]

{■))„= J E

u(t,x,Odt,

where u is the solution of the initial value problem for the Fokker-Planck equation of Brownian motion. The following version of Theorem 6.4.1 is valid. Let u e 3 1

(R",L 2 (K n ,e 1 ^dO), then

g(t) - p(t) - ep Q )

mi

llL2(R"xRn,e4-(/idO

=0(e2)

(8.5.16)

uniformly for t in bounded intervals of [0, oo[. Here p is the solution of the following initial value problem

dtp

=

{dip,

p(0)

=

u 0 -e£d.r t uo(jfc;i),

n k=l

and the function p in the initial layer corrector vi = p$ 0 is given by

P(Z)

=e

t/e

T,d**uo(kV-

Singularly perturbed evolution equations

234

6

Fokker-Planck equation of vibrational relaxation of harmonic oscillator

In the next example we shall consider the collision operator which appears for example in connection with the vibrational relaxation of harmonic oscillators in the continuous (high temperature) limit [78, 23], and the energy relaxation of a hard-sphere Rayleigh gas [2, 23]. For simplicity, we shall be dealing directly with the modified Fokker-Planck operator (8.1.9) which in the present case is given by the modified Laguerre operator

(^)(p,0

ha

£

Q2

\

= d&deu{p,8)+(^-£-^\u(p,Q, ( £ l + , a > 0 . (8.6.1) ~ ' 4 " 4t)J«(P,0.

This operator was discussed in Subsection 2.7.13. As we saw, there are two slightly different cases corresponding to a = 0 and a > 0. We shall consider only a > 0 here and drop the subscript a in the notation. The case a = 0 can be treated in a similar manner. Let us recall that the basic space is H := L 2 (R+) and the corresponding variational space Hi can be defined by Hi : = { « G H; f - > ^ w G W}{R+)}.

(8.6.2)

This definition, simpler than that given in Eq. (2.7.54), follows from Eq. (2.7.56). From Subsection 2.7.13 we see that the sesquilinear form c := [ (see Eq. (2.7.51)) generates the collision operator C := Ah defined on the domain D{C) := {ue Hi- Cue

H},

which satisfies the assumption P.l of Section 6.2. Let us now consider the streaming operator on the domain Ds = {ue H; |f|ti € H}. It follows from Eq. (8.6.2) that (Spu,v)H can be extended onto Hi x Hi so that the assumption (8.3.2) is satisfied and we can define on Hi x Hi the form tp = c + sp which generates the operator fp with the domain D{fp) = {u e Hi;

TpueH}.

In other words, u e D(fp) if and only if for some / € H we have

Chapter 8. Fokker-Planck equations

235

tp(u,v)

= {f,v)„

(8.6.3)

for every v G H\. To check assumptions (8.2.1) and (8.3.3) we need more precise information about D(TP). To this end we have the following lemma: L e m m a 6.1 If u G D(TP), then £ —> i / ? u e H\. Proof. The form tv is given by 00

tp(u, v) = -J

°° /

1

2\

Zd(ud(vdZ - j I f - + tpj £ + ^7 j ««&;•

Let us consider for R > 0 a continuous function f 1 for 0 < £ < fl 0R(£) = J linear for R < £ < 2R [ 0 for f > 2fl. We define »R

(e) = v^Mewo

and

«A(0 = V^«(OMO = tf*(fl«(0. It can be checked that S/j G Z/2(R+), lim uR = J^u

in L 2 (R+)

(8.6.4)

and vR € Hi. Thus we may take v = t/# in Eq. (8.6.3) to get oo

tp(u,vR) = - Ud(ud(vRdi tp{u, vR) = - J Zd(ud(VRd£

oo ,

2\

- 1 f - + ip) £ + — \ uRuRdZ, - J f f - + z'pj £ + — j uRUR~dZ,

where we used the identity UVR = uRuR. The first term can be rearranged as follows where we used the identity uv« = URUR. The first term can be rearranged as follows / Zdzud<:vRd£ o oo

oo

oo

0

0

0

= / ? M 2 - / £[df(^0*)] 2 M 2 # - 2t7m / ^ ( V ^ « ) ( ^ ) ^ C

Singularly perturbed evolution equations

236 Using Eq. (8.6.3) we obtain

oo

t„{uR,uR)

oo

= j eRffi,uRd£,-

f

0

mw m{ftfR))M dt, 2

0 oo

-2ilm

J{ds(^6R)(^8R)d(uud$,)d^uud^, o

(8.6.5)

so that oo

oo

/W^-- Jt[

Re tp(uR,uR)

J^d((^9R)}2\u\2^-

= ReJeRffiuRd£ o

Next, the family of functions V & ( V ^ K ) = 2W

+

ZWR

is bounded with respect to R since |£9{0R| < 2R/R = 2. Thus oo

oo

Jtldd\ft(>R)?M2dt<4j\u\2dZ. 0

0

Applying the Cauchy inequality to the first term in Eq. (8.6.5) we obtain for any t > 0 OO

OO

OO

I / 0Rfy/ZuRdZ\ < - f | / | 2 ^ + \ o o o

\{\uR?di.

and by the coercivity of tp we obtain oo

II«X, *?/

0

oo

\u\2di +

25tJ

l/| 2 ^ + ^llfi-llir,, 0

where 6 is the coercivity constant for tp. Taking sufficiently small e we see that the family {||u fl ||// 1 } fl>0 is bounded and therefore we can extract a subsequence {uRk)keN which converges weakly (see Subsection 2.3.6) to some w in H\. By Eq. (8.6.4) we must have w = %/?"• ■ With this lemma we see that

Chapter 8. Fokker-Planck equations

237

1. If u G D(fp), then ( u £ fl so that the domain of Tp is independent of p and, subsequently, D(fp) = D(C), 2. D{C) C Ds. Therefore, by Theorem 3.1, the operator Tp coincides with that given by Eq. (6.2,9), and the assumptions P.3 and (8.2.1) are satisfied. The last assumption to lie checked is (8.2.2). We shall prove that the stronger assumption (8.2.3) is satisfied. Let us take u G D(C2) and, dropping p in the streaming operator, we immediately obtain

CSu = dciZdttfu)) + f ^ - j p - \ - ^)

(ft) = SCu + u + £d(U.

The first two terms on the right-hand side are in H. From Lemma 6.1 we infer that £ —> ft € //i(K + ) so that the third term is also in H. Therefore, the assumptions (8.2.1) and (8.2.3) are satisfied and by Proposition 2.1 we obtain that the assumptions (6.3.31)-(6.3.33) hold. Hence, all assumptions of Section 6.2 are satisfied and Theorem 6.4.1 is available. To complete our considerations we write down the explicit form of the diffusion equa tion (6.3.29). To this end we recall (see Subsection 2.7.13) that the complete set of orthonormal eigenfunctions { $ ° } n 6 N of C is given by {cm,Q£a/2e~'e/2<3m)(0}>nsN, where Q^ are the Laguerre polynomials given by Eq. (2.7.59) and cmiQ are the nor malizing constants defined in Eq. (2.7.58). Let u = Y. "m*m . then Cu = — J2 mum$)£' m=0

Returning to the x variable and

m=l

using the same notation as in previous sections we obtain the following initial value problem for the diffusion equation (6.3.29):

dtp v(0)

= -{a + l)dzp + e(a + l)d2xp, = uQ - e\/a + ldxUi,

where u0 and U\ are zeroth and first moments of the initial value u. Contrary to the two previous cases, here we have to assume that u 6 iy 2 3 (R + , D(C)) (see Proposition 2.1).

Chapter 9 Applications to spatially inhomogeneous linear Boltzmann equation 1

Introduction

In this chapter we discuss some aspects of the asymptotic analysis of the linear Boltz mann equation with coefficients depending on the spatial variable x, that is, we will be concerned with the following initial value problem:

dtu(t,x, £) =

-£dxii(t, x, £) -v{x, Qu{t, x, £) + I k(x, C, g)u(t, x, £')d£,

u(Q,x,0

= &(*,£),

t > 0, x € ft, (9.1.1)

where we are dealing either with the free space case and then ft = R n , or the periodic case with ft = [0,1]" The occurrence of the spatial inhomogeneity in Eq. (9.1.1) makes the direct applica tion of the theory of Chapter 6 impossible since the Fourier transform is not available here. Additional difficulty is created by the fact that the gradient operator does not commute with the collision operator. We shall see, however, that the formal compressed asymptotic expansion remains valid and that the results of Chapter 6 constitute, in some sense, signposts which should be followed throughout the proof of the counterpart of Theorem 6.4.1 for the space inhomogeneous case. For transparency we shall discuss in detail a simplified model related to the neutron 239

Singularly perturbed evolution equations

240

transport in the slab geometry. Some comments referring to the general case of Eq. (9.1.1) are given in Section 4. The results this chapter were first proved in [11] where, in the absence of general theory, the authors made extensive use of the Legendre polynomial representation of the operators involved.

2

Properties of neutron transport equation in slab geometry

We consider the one-velocity neutron transport equation in one spatial dimension with anisotropic scattering. In this case we have [46]

dtu u(0)

= Au + Su + -Cu, =

t > 0,

u,

(9.2.1)

and the appearing operators are suitable realizations of the following differential ex pressions

(Su)(x,ft) (Au)(x,n)

= -^idxu(x,ii), = -aa(x)u(x,/j,),

(Cu)(x,n)

=

-o,{x)

(u(x,n)

(9.2.2) - £

bm(x)Pm(n)

J u(x,fi!)Pm(n')dA

.

Here x g £1 C E (fi is equal to either R in the free-space case or to [0,1] in the periodic case), /i e [— 1,1], aa and as are macroscopic absorption and scattering cross-sections, respectively, with as(x) > c > 0 for x G 0 and {bm)meN is a bounded sequence of functions which describe the anisotropy of scattering. In particular, bo = 1, bx is the average cosine of the scattering angle and inf

(1 - bm(x)) = b > 0.

(9.2.3)

The functions Pm in Eq. (9.2.2) are the normalized Legendre polynomials defined by the formula (2.7.42). The basic spaces in our considerations are the Hilbert spaces H =

L2(\-1,1])

Chapter 9. Spatially inhomogeneous equation

241

and ■H:=L2{Qx

[-1,1]) = L 2 (Q,//).

From Eq. (9.2.2) we see that C is a bounded operator in L2(Q x [—1,1]) and that 0 is an isolated eigenvalue of C with e0(x,/j,) := m(x)P0 = m(x)/\/2, where m is any function satisfying ||m||i,(n) = 1> being a corresponding normalized eigenvector. Thus the kernel of C can be written as N(C) = L2(0.) ® Lm{P0}. This corresponds to the assumption P.l in Chapter 6 where, for the collision operator of the form C = / © C, we required that the kernel of C be one-dimensional in H. Let us define for u 6 T-L

um{x) = 1

u{x,^,')Pm(n')d)i'.

-l

Now the initial value problem (9.1.1) takes the following form:

8tU

=

-OaU - fldxU + -
u(0)

=

u.

(9.2.4)

In the periodic case the equation (9.2.4) are to be supplemented by the boundary conditions u(0,|t) = u(l,/i). (9.2.5) As the first step we shall establish the existence of a solution to the problem (9.2.4). In what follows we shall use the same notation for the operators in Hilbert space and the corresponding differential expressions which, however, will not lead to any misunderstanding. If we consider operators defined by Eq. (9.2.2) in the "H-setting we see that both A and C are bounded whereas S, defined on

D(S) = {u e H; Su e U) in the free space case and on D(S) = {ueU\

Sue

H,u(0,n)

= u{l,n)

fora.e. p, G [-1,1]}

in the periodic case, is a closed and densely defined operator (see e.g. [3]). It is worthwhile to note that if Su G K then for almost every /j, we have U(-,/J.) € U'^ft), so that the boundary conditions in the definition of V(S) make sense. Let us introduce the transport operator Tt defined on the domain

242

Singularly perturbed evolution equations

D{%) = D{S)

(9.2.6)

Teu = Au + Su + -Cu.

(9.2.7)

by the formula

We have the following theorem: T h e o r e m 2.1 For each e > 0 the operator Tt is an infinitesimal generator of a C0semigroup (5r<( r ))oo ln H which for every t > 0 satisfies the estimate

< e~°\

WGTMW

(9.2.8)

where a = inf aa(x).

(9.2.9)

x£fi

Proof. Since the free space case is well known (see e.g. [46]), we shall deal here with the periodic case only. First we shall investigate the properties of the operator S. It follows that S is a conservative operator. Indeed, for u 6 D(S) we have

/

/ fi.dxuudx \ dn

=

=

/ yU | u(l) | 2 - | w(0) I2 - / udxudx

- /I

f ndxuudx\

d[i

d/x,

(9.2.10)

thus Re (Su, u) = 0 and the statement is proved. Now, the solution of the boundary value problem lidxu + Au = u(0,/i)

=

/, u M ,

where / € "H and A > 0 is given by

u(x, n) = =

;

11 A

{ 1 eil-')f(a, rids++e e-t -I]f(s n)ds

n(i( l—- ee -h M 0 Vo

.

X

[e^f(s,, ~' C '/( S , /u)ds

•)

Chapter 9. Spatially inhomogeneous equation

243

Estimation of the norm of the term in the brackets gives for almost all n, x

1

Je»{B-x)f(s,ii)ds

+ e~->

fei(s-x)f(s,ti)ds tj([0,l])

< \\n-,ri\\i*mU'-'I'te + '-'i Je~t'dsj =

y(l-f'-")||/(-,M)|k2((o,,]),

where we used the Young inequality for convolutions (see e.g. [31], p. 23) with the understanding that the function / is extended by 0 outside the interval [0,1] and the exponential factor is extended by 0 outside [0,1] and [—1,0] in the first and second summands, respectively. Finally we obtain

Hl« < f ll/lk which shows that the operator AX — S is a surjection for every A > 0. Since 5 is conservative, the assumptions of the Lummer-Phillips theorem are satisfied (see Subsection 3.2.3), hence S generates a semigroup of contractions. Next, both operators C and A are bounded. From the Bounded Perturbation Theorem (see Subsection 3.4.1) it follows that 71 generates a semigroup in H but the estimate of its growth offered by that theorem is unsatisfactory for our purposes since, according to it, we would have 1/e multiplying the positive exponent in the estimate of the semigroup. To improve the estimate we shall use the Trotter formula (Subsection 3.4.3) that for every v € H we have ^At)v=x^{G^)gA{^gs{i))\

(9.2.11)

and the limit is uniform on bounded intervals. Thus, the estimate (9.2.8) follows from the properties of each semigroup and the equation above. ■ From the properties of solutions to evolution problems we know that if u G D(Tt), then u, as a solution to Eq. (9.2.4), satisfies u(t) G D(T,) for every t > 0. In other words, we can differentiate u with respect to x for almost every n, but the resulting derivative may not be square-integrable over Q x [0,1] unless multiplied by JJL. Such a result is insufficient for our purpose. In the next lemma we shall show that the required regularity of the solution can be achieved by imposing additional assumptions on the initial datum and the coefficients in .4 and C. Let us recall the convention of Subsection 3.7.2 that H'2fcQ(f2) denotes either the stan dard Sobolev space with Q = R for o■ = 0 or the Sobolev space of periodic functions

Singularly perturbed evolution equations

244

(see Subsection 2.4.8) with Q. = [0,1] for a = w. Analogous convention applies to the spaces of continuous functions C* with the additional assumption that for a = 0 we have C*(fi) = Ch(R) (see Subsection 2.2.2). Throughout the remaining part of this section we assume that <7a,
m = 0,l,...,

(9.2.12)

and, denoting by 9"C, dxA the strong derivatives of C and A with respect to x (see Subsection 2.5.4), we require that dkxA, dkxC e C(H),

k-

0,1,2.

(9.2.13)

We also introduce the following notation K ■= Wla(n),

(9.2.14)

and ft£:=W&(n)®M[-l.l]). ([-1.1]),

(9.2.15)

where, if no misunderstanding is possible, we omit the subscript a. We have the following lemma. Lemma 2.1 If u e %\ then the solution u to Eq. (9.2.4) for every t > 0 satisfies u(t) £ Hi.

(9.2.16)

Additionally t

{dxu){t) = gTt{t)dzu

+1QTXt o

- s)dx (-C + A ) uds.

(9.2.17)

Proof. Let a = 0 first. We shall denote -C + A by B. Since S is bounded, the bounded perturbation theorem gives us the following representation of the semigroup

(SrAt))t>o oo

Sr.W = E 5 " W . where S0(t) = Qs(t) and, for every w

£H,

(9-2.18)

Chapter 9. Spatially inhomogeneous equation

Sn{t)w = j Qs{to

245

s)BSn.-[wds

n = l,2,...

(9.2.19)

The convergence in Eq. (9.2.18) is in the uniform operator topology. 00

o

First we prove that the series £ dxSn(t)u

is convergent. This is done by induction.

n=0

We conjecture that for n = 1, 2 , . . . , the following inequality holds ||0xS„(Ou||„ < -. {\\dxB\\ + \\B\\r HUIIMI n!

(9.2.20)

where here, and in the remaining part of the proof, || || denotes the norm in C(H). Since differentiation commutes with (<7s(0)t>o, we see that Eq. (9.2.20) holds for n = 0. From Eq. (9.2.19) we obtain t

dxSn{t)u

= f Qs(t - s) (dxBSn-i{s)u o

+ BdxSn-i{s)v)

ds

(9.2.21)

and, since 5 is a conservative operator, t

t

\\dxSn(t)u\\n

< \\dxB\\ I \\Sn^(s)u\\ndS o

+ \\B\\ I \\dxSn^(s)u\\Hds. o

(9.2.22)

In the Bounded Perturbation Theorem we have the following estimate (see [72], p. 77)

wsn-iWiy
||a^„(t)«ll« < — {ll^BHiieir-1l + ||B||(||^B|| + ||B|irl}||fi||H4 1 \\dxSn(t)u\\H

< £ {lia^HIIBir- + ||B||(||4,0|| + H B U r } ||n|| w i .

The term in brackets can be estimated as follows The term in brackets can be estimated as follows

II^BIHieil"-1 + ||B||((M + ||B||)-i = \\dIB\\\\B\\n-1 + 'lpo{n-kl)\\d^\\'!\\B\rk < (\\dxB\\+\\B\\rkr°

246

Singularly perturbed evolution equations

where we used relations ("71) + 1 = (") and (""') < (*). Thus inequality (9.2.20) holds and consequently u G Til,. In a similar way we can prove that the series of second derivatives of Sn is convergent. The inequality (9.2.20) in this case is replaced by \\d2xSn(t)u\\n < £ (ll^SH + 2\\dtB\\ + \\B\\)n \\u\\nl.

(9.2.23)

Next we prove that dxu is differentiable with respect to t. To this end we investigate again Eq. (9.2.19). Since u e U2a implies dxu e D(S), we see that dxS0(t)u = Gs(t)dxu is differentiable for t > 0 and an easy induction also shows that each dxSn{t)u has this property. Thus dtdxSn{t)u

= n\gs(t

- s) (d2xBSn^{s)u

+ 2dxBdISn.1(s)u

+ B ^ S ^ s J u ) ds

+ a r BS„_ 1 (r.)u + £d I S n _ 1 (t)w. 00

Now, combining inequalities (9.2.20) and (9.2.23) we see that the series £)

n=0

o

dtdxSn(t)u

is convergent and thus dxu is differentiable with respect to t. These results allow us to differentiate Eq. (3.2.14) with respect to x to obtain that dxu is a classical solution of the following Cauchy problem

dtdxu dxu(0)

= %dxu + dx (-C + A) U, =

dxu}

from which Eq. (9.2.17) follows. The statement for periodic solutions follows from the fact that under the assumptions (9.2.12) and (9.2.13) neither operation in the proof affects the periodicity in x of the involved functions. ■

3

Asymptotic expansion

As in Chapter 6 we denote by P and Q the projectors of H (here H = L2{[—1,1])) onto V = PH := Lin{P0} and W := QH = Lin{P m } m >,, respectively. Accordingly, V := / 0 P is the orthogonal projector onto V := L2{£1) ® V and Q := I ® Q, where / denotes the identity operator on L2(Q). The distribution function u can be decomposed as u = Vu + Qu = v + w

Chapter 9. Spatially inhomogeneous equation

247

and the projected equation (6.3.1) has the following form:

dtv

= V(A + SyPi> + VSQw,

dtw

=

Q(A + S)Qw + QSV» + QCQit>,

v(0)

= v := Vu,

-u,(0)

=

w := Qu,

(9.3.1)

where we have taken advantage of the fact that the attenuation operator .4 is diag onal. The equivalence of problems (9.2.1) and (9.3.1) can be proved exactly as in Proposition 6.2.2. To derive the explicit forms of the operators appearing in Eq. (9.3.1) we expand it € H into the series of Lcgendre polynomials oo

oo

U

U = Y.

">P'" = "° P 0 + E

m=0

U

mPm-

(9-3.2)

m=l

We can proceed as in the derivation of Eq. (8.4.4), the only difference and difficulty being the presence of the differentiation operator dj. in S instead of the pointwise multiplication by p in Sp. If, however, u is the solution to the problem (9.2.4) with the initial value u satisfying the assumption of Lemma 9.2.2, then for each t > 0 we have (.r, (i) —► u(t, .r, //.) £ \\'%rt{Q) 0 La([—1,1]) In particular, for m > 0 we have

um e w'rjn), which implies GO

9"u=

£d*uraPm,

A-= 1,2.

m=0

Using this equation we obtain

T(A + S)Vu

=

PSQu

=

-
73 QSPv

=

QCQu

=

--j=dxu0Pi.

- J2 <^»(1 -

bm)umPm,

(9.3.3)

Singularly perturbed evolution equations

248

Q(A + S)Qu

{■

= -Y,\a°u™

+ dxUm

+

+

/2m + 3\/2rn + l

, + (1 -

tflm)>/2m+r>/2m_iaxUm-l

}

,> P m ,

(9-3.4

where 5 l m denotes the Kronecker delta. As the next step we shall write down the explicit forms of Eqs. (6.3.29), (6.3.6), (6.3.10) which define coefficients of the asymptotic expansion. Recalling the notation introduced in Eq. (9.2.14) we assume that u € Til

(9.3.5)

and denote dm = os{l-bm)

for m = l , 2 , . . .

(9.3.6)

Introducing the notation of Eqs. (8.4.4)-(8.4.9), after simple formal calculations we see that the initial value problem for the diffusion equation (6.3.29) in the case of neutron transport theory is of the form

dtp = -oap +e-dx[jdxp\ P(0) = u0--j=dx(^-j

,

(9.3.7) (9-3-8)

From Eq. (6.3.6) we immediately obtain

m(t)

~ -7u,d*m

(9.3.9)

In order to have p and tSj properly defined we have to investigate solvability and properties of the solution to the problem (9.3.8). Let (At|Q, D(A,,,))

be the variational operator defined by the differential expression At,a = -aa(p + -dx {-j-dx<,

on the domain D(A(,a) = Wln(n)

= Hl

Chapter 9. Spatially inhomogeneous equation

249

Lemma 3.1 (a) Let l/dl € C%(n) and aa € Loo(fi). Then Ae0. (b) Let (-ACiC,)K denotes the fractional power of(-Ae%a). natural number, we have D((-A(,a)l) = HZ,

Then, if K IS an arbitrary (9.3.10)

provided \/dl e C£(Q) and aa 6 CK~l(Q). o

o

(c) //, additionally, € W2*Q(fi), then <j> = GA satisfies the inequality

urn,|

Hr+-

< (et)-ie-^^U\\H!n W\H

(9.3.11)

for every m, K 6 N and £ > 0. Proof. Point (a) and Eq. (9.3.10) are straightforward consequences of the theory developed in Section 3.3 and Subsection 3.7.4. Equation (9.3.10) also gives the equiv alence of the norms which, however, may not be uniform with respect to e. Thus the necessity of the estimate (9.3.11) which we now prove. By the variational theory we have D((-Ai,a)1/2) = H„ with equivalence of norms. Since (—A ta ) l l 2 is a positively defined operator, the norm || ■ || £ 0 in £)((-A <EQ ) 1/ ' 2 ) is generated by the sesquilinear form associated with (—A^Q) 1 / 2 , that is, \Hl*

:= /
±-\dxu(x)\2dx.

Hence, for sufficiently small t > 0 we obtain yfihhl

< \W\kc < H | j / « .

(9.3.12)

Since A[
\Im\{

The careful analysis of the proof of the theorem on the generation of analytic semi groups in [72], p. 62, shows that then the constant M in the estimate of the resolvent ||/?(A£,Q,A)||<^

(9.3.13)

Singularly perturbed evolution equations

250

for A 6 SIL+S (see Eq. (3.2.21)) is independent of e. Next we recall Eq. (3.3.18) which in our case will have the form

U-A):aGA.jt)\\

< MJ-^-^"",

(9.3.14)

where again it can be proved that the constant MK is independent of t (see [72], pp. 63 and 75). Combining Eqs. (9.3.12) and (9.3.14) we get

\m)Ui

< -r\\(-A)%GA.Jt)4>\\„ M l

^ :

1

= e

- ( ^ ) t

M f l

y/Te"

and inequality (9.3.11) follows by induction.

■

This lemma settles questions concerning properties of the bulk part of the asymptotic expansion. In particular, for the solution of the problem (9.3.8) with its particular form of the initial data, the formula (9.3.11) yields the following inequalities for

\U L.,+1

1 M H < < e~" (l + yf) ||«||Hj

I e~at {js +1) H«lk

p = 0,1,2,

for p = 3, for

(9.3.15)

p =4-

As far as the initial layer part is concerned we start with solving the equation (6.3.! for wo which in our case gives

Mr)

■= GQCQ(T)W = ]T

e-d"TumPm.

(9.3.16)

From Eqs. (6.3.7), (9.2.3) and (9.3.4) we obtain

V\{T)

=dx

/e-"' T Pi■) P0. \ dx -0

(9.3.17)

The last term of the initial layer expansion, IUI, is also well defined as follows from the next lemma. L e m m a 3.2 If (9.3.5) is satisfied, dm € C*(Q) and

Chapter 9. Spatially inhomogeneous equation

sup

251

II5X.II < oo

(9.3.18)

mgN.xSR

for k = 0,1,2, then the Cauchy problem

dTwx *i(0)

= QCQwl + Q(A + S)Qw0, =

—^-dJioPu

has a unique classical solution which for every r > 0 satisfies Mr)

e Wjjn).

(9.3.19)

Proof. To prove the first part of the lemma it is enough to show that T -> Q(A + S)QW0(T)

(9.3.20)

is a continuous function, since QCQ is a bounded operator. It is clear that r —► QAQw0(r) is continuous. We can write S = dx ® S^, where Sp is the operator of multiplication by \i. By Eqs. (9.2.13) and (9.3.16) we see that r —¥ dxw0(T) is also a continuous function. Since dx commutes with Q and S^ is a continuous operator we obtain the continuity of T —t QSQW0(T), hence the function (9.3.20) is continuous. Using Eq. (3.6.2) we see that the solution w^ is given by the formula

U>I(T) = -jfzcQ

(O.UQPI)

+ JGQCQ{T

- v)Q(A + S)QgQCQ(v)wdv.

(9.3.21)

It can be directly checked that if the assumptions of the lemma are satisfied then Eq. (9.3.19) holds. ■ Lemma 3.2 corresponds to Propositions 6.3.3-6.3.5. Note that in this case, due to the continuity of the operators QCQ and 5^, the result is almost trivial. We are now ready to prove the counterpart of Theorem 6.4.1. Theorem 3.1 Let u be the solution of the problem (9.2.4) with the initial value u satisfying Eq. (9.3.5). Assume moreover that \j&\ € C'(Q), a 0 G C\ and that (9.3.18) holds. If we define the error of the asymptotic expansion by

Singularly perturbed evolution equations

252

y(t)

=

z(t)

= w(t) - [w0(t/e) + ftZ)i(t) + ew^t/e)],

e(t)P0-[p(t)Po

+ tPi(t/e)Po), (9.3.22)

where we used the notation of Eqs. (8.4-4)~(8-4-9), then for any fixed T, 0 < T < oo, \\y(t) + z(t)\\n = 0(e2),

(9.3.23)

uniformly for 0 < t < T. Proof. The proof is analogous to that of Theorem 6.4.1 except for a few technical differences so that we only outline it using the same notation as before. Results of Lemmas 3.1 and 3.2 and Eqs. (9.3.16) and (9.3.17) show that the error, defined by Eq. (9.3.22), is a classical solution of the problem P i ( / , g) where

/ == e(V{A + 9 =

S)Vvi+VSQ)wu

e(Q{A + S)Qu>! + QSVvx -dtwx

+ Q(A +

S)Qwl).

Firstly we estimate the part of the error f which is due to the initial layer phe nomena. To this end we consider the problem P2(—e(VSQvi + VSQiLi + Q(A + S)Qu>i + Q(A + S)Vvi)). Tedious, but straightforward, calculations involving Eqs. (9.3.4), (9.3.16), (9.3.17), (9.3.19) and (9.3.21) show that the following counterparts of estimates (6.4.15)-(6.4.18) hold

||G(./4 + S)GtiSj(r)||«

<

e-cbTM3(T)\\u\\n,

\\V(A + S)Qw1(r)h

<

e-c6TM3(r)||u|k,

WQiA + SWvtWU \\V(A + S)Vv1(r)\\n

<

e-^M2(T)\\u\\ni, ebr

< e- M1(r)\\u\\Hi,

(9.3.24)

where Mk, k = 1,2,3 denote polynomials in r of order k and c and b are defined in the text following Eq. (9.2.2). Using Eq. (9.3.24) similarly as in Eq. (6.4.23) we obtain \\f(t)\\n = ^K\\kni

(9.3.25)

for some constant K. In the next step we estimate the part of the error related to the bulk solution. Chang ing the order of differentiation in Eq. (9.3.9) we have by Eq. (9.3.8)

Chapter 9. Spatially inhomogeneous equation

W ) =

253

-73^M-^+H;H} A

(9 126)

"

and by Eq. (9.3.4) we obtain

Q(A + S)Qwl(t) = aa^J—-dxpPl V3ajPo

+ ^=dx (±-dJ) P2. 3V5

Vdi

(9.3.27)

/

Let g= -dtwx

+ Q(A + S)Qwi.

As in Eq. (6.4.25) we write g = go + gi- The mild solution f0 of the problem P2(—ego) is estimated as in Eq. (6.4.28) but we use Eqs. (9.3.15) instead of Eq. (6.4.29) getting ||ai||«<£8KI||u||M..

(9.3.28)

The estimate of the mild solution fj of the problem P2(—e
< e- t(c6< "' + ' 7) .

(9.3.29)

The rest of the proof follows almost exactly the corresponding parts of the proof of Theorem 6.4.1 with the sole difference that the auxiliary function h, defined in Eq. (6.4.30), is dependent on x and to prove that it is sufficiently regular with respect to x we use Lemma 2.1 specified for (Q%(t))t>oTherefore we have the estimate ||fi(i)||*<£ 2 K 2 ||£||„3

(9.3.30)

and combining estimates (9.3.25), (9.3.28), (9.3.30) we obtain ||y(t)+*(f)|JK<e2K4ll"lk from which we obtain Eq. (9.3.23).

■

Singularly perturbed evolution equations

254

4

Remarks on general spatially inhomogeneous equation

In the previous section we discussed a simplified version of the linear Boltzmann equation with rr-dependent coefficients

dtu(t,x,£)

=

: -SdMt,x,Z)-nx,ZMt,z,Z) + f k(x,S,?)u(t,x,?)d?

t > 0, x € «.

(9.4.1)

The particular form of that equation allowed us to obtain in a relatively easy way two crucial results: (a) That the operator QCQ is negative definite on W. (b) That all the initial layer terms decay exponentially as t —> oo and, in particular, that w\ is well defined. This point was made even easier since the velocities were bounded. Without going into details we merely sketch here the proof that the results (a) and (b) are valid for the general form of the linear Boltzmann equation (9.4.1). For the spatially homogeneous linear Boltzmann equation the fact that QCQ is pos itive definite, and has the other properties required by the assumption P.l of Section 6.2 or P.l' of Section 6.5 was established in Proposition 7.2.1. A closer look at the proof of this proposition shows that it can be carried out with x-dependent coefficients provided the estimates in the assumptions A.l and A.2 are in an appropriate sense uniform with respect to x. Once this is done it follows, in particular, that the matrix of diffusion coefficients { J \ 7 } is positive definite. The proof of this statement is sim ilar to that of Proposition 3.1 and even simpler since QCQ is a bounded operator. Then the estimates (9.3.15) follows as in the proof of Lemma 3.1. The necessary regularity with respect to x of the solution u, initial value u and their projections is established by Lemma 2.1 which does not depend on the particular shape of operators, provided the assumption (9.2.13) is satisfied. Moreover, the property (a) yields the exponential decay in W of the semigroup (^Qca(t))t>o- To establish the estimates (9.3.24) we need a counterpart of Propo sition 7.2.2. If the velocity space is bounded, as we had in the example, then (9.3.24) follows as in Theorem 3.1. If the velocity space is unbounded, we must require that all the operators <9£(?QCQ(0> k = 0,1,2, t > 0, which are well-defined by the assumption

Chapter 9. Spatially inhomogeneous equation

255

(9.2.13), satisfy the inequalities \\dkxGacQ(t)\\n, < M k e - * ' for some constants M k , where %r = L 2 (Q)®// r for r = 0,1,2. This follows exactly as in Proposition 2.2 provided 0 (and possibly m which may depend on x) are sufficiently smooth. The remaining points can be dealt with as in Theorem 3.1.

C h a p t e r 10 Application t o kinetic equation with external field 1

Introduction

In this chapter we are interested in the analysis of the initial value problem for the onedimensional linear kinetic equation which describes the time evolution of the spatially dependent electron distribution function (t,x, £) —> u(t, x,£) in a weakly ionized host medium under the influence of a spatially uniform time independent electric field. The one-dimensional linear Boltzmann equation for that problem takes the form

dtu(t, x, £) + £dxu(t, x, £) -I- ad^u(t, x, £) +00

= -i/(0«(t,i,0+ / k(Z,t')u(t,x,S')d?,

t>0,

(10.1.1)

—00

with the initial condition u(0,x,£)=sv.{x,t),

(10.1.2)

where x, £ G R are the space and velocity variables, respectively, t is the time, a = qE/m > 0 is the constant electrostatic acceleration, with —q the electron charge, E the electric field and m the electron mass, i/(£) is the collision frequency and /;(£,£') is the scattering kernel. Under the hypothesis that the ionization and recombination effects balance each other, &(£,£') is a non-negative measurable function such that for any £ € R, +00

/ *(f,£'K == KO —00 — oo

257

Singularly perturbed evolution equations

258 (see Eq. (7.2.8)).

At least two different scalings of this equation exist in the literature (see also Section 5.7). These are

ftu(*.*.0 = - - ( f M U { ) + a ^ ( « . * , 0 ) + i f-i/(0«(t,i,0+ / *&€>(*, * . 0 # )

(10.1.3)

and

dtu(t,x,£) = —^dxu(t, x, f) + 1 f-aStu(t,a;,f)-i/(Ou(e 1 a; > OH- j k(£, £ > ( * . * , 0 < * f )

(10.1.4)

In Eq. (10.1.3) we have dominant collisions with a relatively small external field and in Eq. (10.1.4) the collisions are still dominant but the external field is of the same order of magnitude. Note that in Eq. (10.1.4) the singularly perturbed operator consists not only of the collision operator, as in all the examples discussed earlier, but also of the gradient with respect to the velocity. We discussed scalings of this type in Section 5.7. Here we shall reiterate that although it is argued that the scaling of Eq. (10.1.3) is physically justified, our belief is that it was introduced since the time rescaling technique, resulting in such a form of equa tion, proved to be so successful in deriving the diffusion equation for various kinetic equations. However, as was pointed out in Section 5.7, in the case of a strong external field the scaling with 1/e multiplying the spatial gradient and 1/e2 multiplying the remaining two terms on the right-hand side, produces a trivial asymptotic expansion and thus the need to return to the generic scaling resulting in Eq. (10.1.4). We note also that the compressed asymptotic expansion gives the diffusion equation on the first level of approximation also for generic scaling of equation with a weak external field, that is, for the equation

dtu(t,x,£)

=

-£dxu(t,x,£)-ad(u(t,x,(;)

+ J I -v(£)u(t, * . £ ) + / *K. CHt, x, 0 dA

(10.1.5) (10.1.5)

However, we shall not discuss this point but will focus our attention on the equation (10.1.4) whose analysis is more interesting.

Chapter 10. Equations with external field

259

The main difficulties arising here are related to the specific form of the singularly perturbed operator in Eq. (10.1.4) which has the form £ + C where £ is formally defined by the differential expression {£u)(x,Q:=-adtu(x,t)

(10.1.6)

and C is the collision operator defined as in Chapter 7 by +oo

J k{a,e)u(x,s')de

CU(X,O-=-V(OU(T,O+

(10.1.7)

—00

We shall discuss the relevant mathematical properties of £ and C in Section 2. Here we note that the integro-differential operator £ + C is not self-adjoint. Moreover, we do not know the explicit form of its eigenfunction e0 corresponding to the eigenvalue A = 0. Therefore the introduction of the space of the form % = L2{R, m _ 1 d£) which was crucial in Chapters 7 and 8 seems to be not feasible. In particular, we cannot prove that the diffusion coefficient is positive. It follows that the only sensible space with which we can work is Li(R) and this creates many difficulties, as we saw in Chapters 2 and 3. In this space even the weakened assumptions P.l' of Section 6.5 is not satisfied and the results of Chapter 6 are to a large extent useless. However, as we pointed out in Section 9.1, all the formal results remain valid and the analysis performed in Chapter 6 identifies main goals which should be achieved in particular cases, even if they are beyond the scope of general theory. Due to these difficulties we present a rigorous justification of the formal results for a very simple model of BGK equation with a constant collision frequency u0 [35]. In this model the collision term is described in terms of the Maxwellian m ( 0 = (27r(9)-1/2exp(-<e2/20),

(10.1.8)

normalized to unity with +oo

9 :=< e >= j e^iodi. —oo

and Eq. (10.1.4) takes the form ftuftx,*)

= -£&«(*.*.£)

+ - j -ad(u(t,

(10.1.9)

x, 0 - uQu{t, x, 0 + n , m ( 0 J u[t, x, £') d£'J , t > 0.

Singularly perturbed evolution equations

260

The results of this chapter were first proved in [9]. Some other aspects of this problem are discussed in [74] and [34].

2

Properties of collision operator

Throughout the chapter we will be using a number of different spaces so that it is useful to list them in one place and introduce suitable notations. Let Xr be the Banach space Li(R2, (1 + |£|) r dxd£) with the usual norm + oo

I H U = /|«(x,OI(l + KI)rdxdr — oo

for r e N f l . This space can be represented in a natural way (see [28]) as XT :=XX®XU

: = L i ( R , d i ) ® L 1 ( K , ( l + |C|) r dO-

(10-2.1)

We shall need also the following spaces of the Sobolev type Hfr) := W{(R, dx) ® XLr = U (H, (1 + |<e|M, W*(R, ate)) where Wf(R,dx) denotes the usual Sobolev space of functions Li(K, dfi, Z) denotes the space of Z-valued functions which are with respect to the measure pi. In case Z = R we simplify the ing Li(R,d/j,). For r = 0, when we have the standard L\ spaces, subscript 0 in the notation and write X', X^ and Wk

(10.2.2)

of x-variable and integrable over K notation by writ we shall drop the

We shall need a subspace X° of X( = L\ (R, d£) determined by the condition +oo

u € X° if and only if

f u(£)d£ = 0.

(10.2.3)

—oo

We adopt the following convention: if Z is any space defined in Eqs. (10.2.1) or (10.2.2), then Z° = ZnX°. Let us define the operators (Su){x, 0 := - a « ( i , i),

D{S) = {u G X- Su G X) ,

(10.2.4)

Chapter 10. Equations with external field (£u)(x,0

:= -ad(u{x,£),

261 D(£) = {u e X; £u 6 X)

The collision operator C is defined by Cu := —Mvu + KLu, where the operator K, is given by

[Ku){x,S)::= j k(U'Mx,S')d?. -co

The general theory of the problem defined by Eqs. (10.1.1)-(10.1.2) was developed in e.g. [35, 4, 34, 74], However, the detailed presentation is beyond the scope of this book and we confine ourselves to the survey of some relevant results. We assume that the collision frequency v satisfies the following assumption. We assume that the collision frequency v satisfies the following assumption. (a) v is bounded on R, positive and (a) v is bounded on R, positive and +co

[ v{£)d£ = oo.

(10.2.5)

—oo

This is the necessary condition for the process governed by Eq. (10.1.1) to relax towards a steady-state solution [35] or, using more analytic terminology, for e0 to exist. To discuss the spectral properties of the operator £ + C we note that all operators <S, £, C, K, and Mv can be written, as in Chapter 6, in the form of the tensor products / ® S, I ® E, I ® C, I ® K, I ® M„ where the operators denoted by the Roman characters act in X(. Let us consider the stationary equation in X^, corresponding to Eq. (10.1.1), that is, Etp + C

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