Singularly perturbed evolution equations with applications to kinetic theory

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

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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.

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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*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 < p < oo we define LP(Q) to be the set of Lebesgue measurable functions with finite norm

ll/ll P := H/llMn)

i

I

J\f\>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(x)dx = - J g{x){x)dx R

(2.4.1)

Dt

for every
(2.4.8)

for any m and 1 < p < oo. The space adjoint to Wpm(R") is denoted by W-m{Rn) where 1/p + \/q = 1. From (2.4.8) it follows that it can be identified with a dense subspace of X>'(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) := — ^

/ 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)-/(< 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 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,=
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)=
H;xHl

: =
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 £}\ 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