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Second Order Linear Differential Equations in Banach Spaces
 0444876987, 9780444876980, 9780080872193

Table of contents :
Content:
Editor
Page ii

Edited by
Page iii

Copyright page
Page iv

Preface
Pages v-vii
Buenos Aires

List of Symbols
Page xiii

Chapter I The Cauchy Problem for First Order Equations. Semigroup Theory
Pages 1-23

Chapter II The Cauchy Problem for Second Order Equations Cosine Function Theory
Pages 24-42

Chapter III Reduction of a Second Order Equation to a A First Order System. Phase Spaces.
Pages 43-99

Chapter IV Applications to Partial Differential Equations
Pages 100-125

Chapter V Uniformly Bounded Groups and Cosine Functions in Hilbert Space
Pages 126-164

Chapter VI The Parabolic Singular Perturbation Problem
Pages 165-237

Chapter VII Other Singular Perturbation Problems
Pages 238-269

Chapter VIII The Complete Second Order Equation
Pages 270-302

Bibliography Review Article
Pages 303-314

Citation preview

SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS IN BANACH SPACES

NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (99)

Editor: Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro and University of Rochester

108

SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS IN BANACH SPACES

H. 0.FAlTORINI University of California at Los Angeles USA.

1985

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD

OElsevier Science Publishers B.V., 1985 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 87698 7

Publishers:

ELSEVIER SClENCE PUBLISHERS B.V. P.O. BOX 1991 1000 BZ AMSTERDAM THE NETHERLANDS

Sole distributors for the U.S.A. and Canada:

ELSEVIER SCIENCE PUBLISHING COMPANY, INC 52 VAN DERB ILT AVE NUE NEW YORK, N.Y. 10017 U.S.A.

Library of Congress Cataloging in Publication Data

Fattorini, H. 0. (Hector O.), 1938Second order l i n e a r d i f f e r e n t i a l equations i n W a c h spaces. (North-Holland mrthematics s t u d i e s ; 108) (Notas de m a t d t i c a ; 99) Bibliography: p. 1. Differential equations, P a r t i a l . 2. Differential equations, Linear. 3. Banach spaces. I. Title. 11. Series. 111. Series: lVDtas de m a t d t i c a (Amsterdam, Netherlands) ; 99. w . 1 8 6 no. 108 510 s ~515.3'541 84-28658

wav1

ISBn 0-444-87698-7

PRINTED IN THE NETHERLANDS

V

PREFACE

An initial value or initial-boundary value problem u t = Au, u

=

uo

for

(1)

t = 0

A

is a partial differential operator in the space variables x l ' . . . can be recast in the form of an ordinary differential initial value

where

x rn problem

~ ( 0 =) u0'

u'(t) = Au(t), where

A

(2)

is thought of as an operator in a function space E

and the

boundary conditions, if any, are included in the definition of the space E or of the domain of A . I f

E is suitably chosen, solutions of (2) will

exist for sufficiently many initial data on

uo

in the norm of

uo

and will depend continuously

E. This yay of looking at (1) was initiated by Hille

and Yosida in the forties and resulted in the creation and development of semigroup theory, now an integral part of most advanced treatments of parabolic and hyperbolic partial differential equations. A second order initial value or initial-boundary value problem

utt = Au, u = uo, u

=

u1

for

(3)

t = 0

can be reduced in the same way to an ordinary differential initial value problem ~"(t) = Au(t), where

A

~(0) = u0, u'(0)

=

u

(4)

1

is defined as in (2). This, however, can often be avoided reducing

(3) to a first order system following the "take the derivative as a new function" rule one learns in elementary theory of partial differential equations. That this trick always works, at least if one measures the derivative in a new norm, is in fact one of the results in Chapter I11 of this book. Moreover, the choice of this norm is usually natural and has physical meaning. However, reduction to first order is of no particular help in a problem as elementary as the growth of solutions of u"(t)

=

(A

+

cI)u(t)

PREFACE

vi

in terms of the growth of the solutions of

u"(t)

=

Au(t).

In other

problems, such as singular perturbation, direct consideration of second order equations leads to simpler and more inclusive theories. Finally, the formalism associated with ( 4 ) has proven useful in other fields, such as the control theory of hyperbolic equations. These and other reasons give motivation to the development of a theory of second order differential equations in Banach spaces. This work presents a few facts on that theory and some applications.

NO claim of completeness is made, either in the text or in the references; many important results have been left out and many important papers are not mentioned. Chapter I expounds semigroup theory; Chapter I1 presents cosine function theory, which stands in relation to the second order equation ( 4 ) as semigroup theory stands in relation to the first order equation (2). Chapter I11 deals with the reduction of ( 4 ) to a first order system mentioned above and other related topics. The next four chapters are on applications; in Chapter IV we treat the initial-boundary value problem (3) with

A a second order uniformly elliptic partial differential operator in a domain of m-dimensional Euclidean space, with either the Dirichlet boundary condition or a variational boundary condition. Chapter V treats the second order equation ( 4 ) in Hilbert spaces, where many special results are available; there are applications to equations with almost periodic and periodic solutions. Chapters VI and VII are on singular perturbation problems, with applications to diverse physical situations. Finally, in Chapter VIII we touch upon the theory of the "ctmplete" second order equation u"(t)

t Bu'(t)

+

Au(t)

=

0

(5)

without going too far into it; mostly, we search for the correct definition of correctly posed initial value problem for (5). Some shortcuts through the book are possible, and we do not bother to indicate them explicitly; for instance, Chapter 1 1 1 is only briefly needed in Chapters IV and V and not used at all in Chapters VI and VII. Some effort has been made to make this book as self-contained as possible; nothing isneededexcept the elementary theory of Banach and Hilbert spaces and some acquaintance with parabolic partial differential equations. The functional calculus for self adjoint operators is only used in Chapters IV and V and in exercises in other chapters. The exercises throughout the book cover parts of the theory not in the text or related facts of interest; references are included for the less

PREFACE

vii

immediate.

I am glad to acknowledge my thanks to many colleagues who read parts of the book and suggested improvements and to the Instituto Argentino de

MatemAtica, Consejo Nacional de Investigaciones Cientificas y TQcnicas, Argentina, for its hospitality during March I983 and August 1984, at which time the actual writing was concluded. Finally, and most important of a l l , the undertaking of this project would have been impossible without the understanding support of the National Science Foundation, which support extended during the entire time it t o o k to complete it. As always, my wife Natalia was encouraging, patient and understanding and to her go my very special thanks.

Buenos Aires, August 1984

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ix

CONTENTS

PREFACE. LIST OF SYMBOLS. CHAPTER I.

............................................. ..........................................

V

xiii

THE CAUCHY PROBLEM FOR FIRST ORDER EQUATIONS. SEMIGROUP THEORY

....... .................... ..........................

51.1

The Cauchy problem for first order equations.

1

51.2

The Cauchy problem in

6

(-m,m).

51.3 The Hille-Yosida theorem. 51.4 51.5 51.6

.................................. The inhomogeneous equation. ........................ Miscellaneous comments. ............................ Semigroup theory.

7 13

15 18

CHAPTER 11. THE CAUCHY PROBLEM FOR SECOND ORDER EQUATIONS. COSINE FUNCTION THEORY

311.1

The Cauchy problem for second order equations.

511.2 The generation theorem.

511.3 Cosine function theory.

........................... ...........................

....................... .............. ...........................

511.4

The inhomogeneous equation.

511.5

Estimations by hyperbolic functions.

511.6

Miscellaneous comments.

CHAPTER 111.

....

24 28 32 35

37 38

REDUCTION OF A SECOND ORDER EQUATION TO A

FIRST ORDER SYSTEM. PHASE SPACES. 5111.1 Phase spaces.

....................................

5111.4

........... Resolvents of fractional powers. ................. Translation of generators of cosine functions. ...

5111.5

The principal value square root reduction.

43

5111.2 Fractional powers of closed operators.

50

5111.3

56

5111.6 9111.7 5111.8

....... Second order equations in Lp spaces. ............. Analyticity properties of bb(t). ................. Other square root reductions. ....................

59 62 71

80 86

CONTENTS

X

5111.9

Miscellaneous comments.

95

. * . . . . . . . . . . . . . . . . . I . . . .

CHAPTER IV. APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS 5IV.l Wave equations: the Dirichlet boundary condition. sIV.2 5IV.3

5IV.4 5IV.5 sIV.6 sIV.7

SIV.8 sIV.9

CHAPTER V.

. The phase space. ................................ The Cauchy problem. ............................. Wave equations: other boundary conditions. ...... The phase space. ................................ The Cauchy problem. ............................. Higher order equations. ......................... Higher order equations (continuation). .......... Miscellaneous comments. .........................

100

104 109 112 116 117

118

120 124

UNIFORMLY BOUNDED GROUPS AND COSINE FUNCTIONS

IN HILBERT SPACE

...... ... Uniformly bounded groups in Hilbert space. ...... Almost periodic functions. ...................... Almost periodic groups in Hilbert space. ........ Banach integrals. ...............................

133

5V.6 Uniformly bounded cosine functions in Hilbert space.

145

sv. 1 g.2

5v.3 sv.4

5v.5 sv.7

The Hahn-Banach theorem: Banach limits.

Almost periodic cosine functions in Hilbert space...

.........................

SV.8 Miscellaneous comments.

126 128

138 142 153 158

CHAPTER VI. THE PARABOLIC SINGULAR PERTURBATION PROBLEM sVI.1

Vibrations of a membrane in a viscous medium.

sVI.2

Singular perturbation. Explicit solution of the perturbed equation.

5VI.3 sVI.4 sVI.5 5VI.6

............................

The homogeneous equation: convergence of u(t;E) Convergence of u'(t;E)

..

165 166

....

and higher derivatives...

...

The homogeneous equation. Rates of convergence.....

171 180 192

Singular integrals of liilbert space valued functions and applications to inhomogeneous first order equations.

.....................................

202

5VI.7 The inhomogeneous equation: convergence of u(t;E) and u'(t;E).

...................................

210

sVI.8 Correctors at the initial layer. Asymptotic series. 5VI.9 Elliptic differential operators.

218

sVI.10 Miscellaneous comments.

233

............... ........................

228

xi

CONTENTS CHAPTER VII. OTHER SINGULAR PERTURBATION PROBLEMS sVIT.1

A singular perturbation problem in quantum

mechanics.

.....................................

sVII.2 The Schrodinger singular perturbation problem..

sVII.3 sVTI.4 sVII.5 sVII.6 sVII.7 sVII.8

Assumptions on the initial value problem.

....

238 239

......

24 1

......

245

The homogeneous equation: convergence results

................ Elliptic differential operators. ............... The inhomogeneous equation. .................... Miscellaneous comments. ........................ Verification of the hypotheses.

250 258 262 265

CHAPTER VIIT.

THE COMPLETE SECOND ORDER EQUATION sVIII.1 The Cauchy problem. ........................... sVIII.2

Growth of solutions and existence of phase spaces.

270 271

sVIII.3 Exponential growth of solutions and existence of

................................. ................. sVIII.5 Miscellaneous comments. ....................... BIBLIOGRAPHY. ................................................. phase spaces.

§VIII.4 Construction of phase spaces.

276 289 298 303

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xiii

LIST OF SYMBOLS

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1

CHAPTER I

THE CAUCHY PROBLEM FOR FIRST ORDER EQUATIONS. SEMIGROUP THEORY

$1.1 The Cauchy problem for f i r s t order equations.

E

We denote by with domain

D(A)

be complex and

a general Banach space and by

E

in

D(A)

Unless otherwise s t a t e d

w i l l be dense i n

E.

We shall use i n t h e s e q u e l t h e

t o i n d i c a t e functions

t h e i r i n d i v i d u a l values a t with values i n t h e i i m i t as

-t

t

are

u(t), f ( t ) ,

0

t h e l i m i t i s one-sided a t [0,

-t

m).

t = 0.

t

will

E

u f t ) , t + f ( t ) , ...;

etc.

of t h e quotient of increments

e x i s t s and i s a continuous function of other than

t

i s continuously d i f f e r e n t i a b l e i n

E

h

a l i n e a r operator

E.

...

symbols u(^t),,);('f

A

and range i n

u(t)

A function

t 2 0

h-l(u(t

i n t h e norm of

i f and only if

+ E;

h)

-

u(t))

of course,

Similar d e f i n i t i o n s a r e used i n i n t e r v a l s

A (stron_g or genuine) s o l u t i o n of t h e a b s t r a c t d i f f e r -

e n t i a l equation ~ ' ( t =) Au(t) in

[0,

that

i s a continuously d i f f e r e n t i a b l e E-valued f u n c t i o n u ( t )

m)

u(t)

(1.1)

E

D(A)

and (1.1) i s s a t i s f i e d f o r

t >_ 0.

such

Solutions a r e

correspondingly defined i n other i n t e r v a l s . The Cauchy o r i n i t i a l value problem f o r (1.1)i n

t >_ 0

is that of

f i n d i n g s o l u t i o n s s a t i s f y i n g t h e i n i t i a l condition u ( 0 ) = uo

.

The Cauchy problem f o r (1.1)i s w e l l posed ( o r properly posed) i n

t? 0

i f and only i f t h e following two assumptions hold:

(a) f o r any

-fying

(Existence). uo

(1.2).

E

D

There e x i s t s a dense subspace

t h e r e e x i s t s a solution

u(t^)

of

D

(1.1)

of

E

such t h a t

t r_ 0 s a t i s -

2

FIRST ORDER EQUATIONS

(b)

There e x i s t s a nonnegative, f i n i t e

(Continuous dependence).

function

t

defined i n

C(t^)

10

such t h a t

Ilu(t)': 5 c(t)llu(o)ll f o r any s o l u t i o n of (1.1).

(1.3) i s assumed t o hold f o r

Note that o r not

u(0)

Also, s i n c e t h e equation must hold i n p a r t i c u l a r f o r

D.

E

s o l u t i o n of (l.l), whether

t = 0 we must have D

.

5 D(A)

(1.4)

F i n a l l y , t h e Cauchy problem f o r (1.1)i s uniformly w e l l posed (or

t 5 0

uniformly properly posed) i n

t5 0

C(f)

i s nondecreasing i n

C($)

i s bounded on compacts of

E(t)

=

sup {C(s); 0

(1.7) i s obvious.

and consider t h e f u n c t i o n

s o l u t i o n of (1.1)with

un +

t2 0

The f i r s t e q u a l i t y

D

E

( s , t >_ 0)

such t h a t

i s s t r o n g l y continuous i n

Proof.

t

sfs)s(t)

such that, f o r every

w

i f t h e Cauchy problem f o r

pick

=

( i f t h e Cauchy problem f o r (1.1) i s uniformly

6

w e l l posed t h e r e e x i s t s

t)

-+

(1.lo)

S(t)u()

S(^t)uo i s s t r o n g l y measurable t h e r e .

Accordingly,

i s almost s e p a r a b l y valued, t h a t i s , t h e r e e x i s t s a n u l l s u b s e t m)

such that

Xo = { S ( t ) u o ; t d, do)

i s s e p a r a b l e ( f o r t h i s and

o t h e r r e s u l t s on measurable v e c t o r valued f u n c t i o n s s e e for i n s t a n c e HILLE-WILLIE generated by

[1957:1,Ch. 1111. ru01 IJ Xo

It follows t h a t t h e c l o s e d subspace

is s e p a r a b l e and t h e r e e x i s t s a sequence

f t n ;n 2 l’j contained i n t h e complement of do such that t h e s e t (uol (J ( S ( t n ) u O ; n 2 11 i s fundamental i n Eo ( f i n i t e l i n e a r

Yo =

Eo

FIRST ORDER EQUATIONS

4

Yo a r e dense i n E 0 ). L e t now t k do ( n = 1,2, ...); then S ( t ) u o and S ( t ) S ( t n ) u O =

combinations of elements of

t + tn )! do

that

S ( t + tn)uO belong t o eo

Eo.

-

eo = do U (do

Define

t l ) U (do

-

such

t2) U

...;

i s a n u l l s e t and it follows from t h e preceding arguments that

c_

S(t)Eo

0

denote by

Em;

since

IIS(:)vnil

with

t

Given

i s t h e supremum of t h e sequence

m(Z)

a dense sequence i n t h e unit sphere of Moreover,

S ( t ) E m E Em for

t h e norm of t h e r e s t r i c t i o n of

i s separable,

with

i s t h e closed subspace generated by t h e

i s separable and

complement of t h e n u l l s e t

En

and a separable subspace

hence i f Em

n’

en

to

El, {vn]

Em and is i t s e l f measurable.

em we have

m(s

+ t)

=

u E Em, hdl 5 11 5 SUP f lb(s>vli; v E Em, b l l < _ m ( t > l Accordingly, a c o n t r a d i c t i o n r e s u l t s from:

1/s(s)S(t)uli;

< - m(s)m(t).

LEW 1.2.

t

defined i n

Let

1 0

m( 0, t ,d e ) ,

m(s)m(t) (0,

c a< B
_ we have eU d U ( a - d) 2 [ O , a ] s o t h a t Id] + ] a - dl >_ a , where 1.1 indic a t e s Lebesgue measure. But Id1 = la - d l , hence Id1 >_ a/2. Assume now t h a t m($) is unbounded i n [ a , f31, so that t h e r e e x i s t s Proof.

Let

a sequence

[anl

a > 0.

If

t h e r e with

s

= a, t

m(an) +

m.

.

Applying t h e argument above we

5

FIRST ORDER EQUATIONS

deduce t h e existence of a measurable set

dn

[0, p ]

in

with

.Jm(an)

i n dn, which c o n t r a d i c t s t h e a/2 and m ( t ) 2 /dnl 1 an/2 f a c t t h a t m(t) i s everywhere f i n i t e . This completes t h e proof of

Lemma 1.2. End of proof of Theorem 1.1. Let 0

I h / 0,

E,

E

0 < r < to,

We obtain from ( 1 . 7 ) t h e e q u a l i t y

r.

-

S ( t O ) ) u = S ( t ) ( S ( t O+ h

t 0

the function

S(i)u

i s continuous, hence bounded

in

6 5 t = ~ ( 1 / 2 ) " ~ ( t n/2)

t h e l a r g e s t i n t e g e r with

n/2

-

llS(t)ll ,

@(;)

(1.1) be well posed i n

(1.1)i s u n i f o r d y well posed

i s s t r o n g l y continuous i n

@(;)

(-,m).(ii)

s a t i s f i e s t h e cosine f u n c t i o n a l equations C(0) hi?)

=

I, C(S

+

t) + C(~-t)[email protected](s)@(t)

There e x i s t constants

C ,m 0

(-m


ho).

(2.a)

i s t h e s o l u t i o n o p e r a t o r of

The proof i m i t a t e s t h a t o f Theorem

REMARK 2.4.

8(u)

E

eht@(t)u d t

F

05

and such t h a t

0

Assume t h a t , f o r each

A

if

ll5

Then

W

-a

i s t h e union of a l l

8 ( W )

be a n o p e r a t o r such t h a t

A

hot @(:)

Reh

2 K (CO,w)

to

g2 i s t h e union of all t h e

and

2 Q ( 0 ) i s empty f o r

(note t h a t

0

The c l a s s

(1.10).

21

C0

belows

(1.1) i s (uniformly) w e l l posed i n

t h e Cauchy problem for and

A

A c l o s e d , d e n s e l y defined o p e r a t o r

(1.1).

1.3.4 and we omit it.

I n e q u a l i t i e s ( 2 . 1 ) follow from t h e i r r e a l c o u n t e r p a r t s

( s e e Remark 1.3.5).

$11.3 A

Cosine f u n c t l o n t h e o r y . (E)-valued f u n c t i o n

C(i)

defined i n

-m

g

with

i s replaced by

t

continuously d i f f e r e n t t a b l e and

when

W

= 0).

u

0

=

0

then

U( 0,

(5.1)

implies t h e following e s t i m a t e f o r

(5.1),

d(i): (5.2)

If

= 0,

t h e i n e q u a l i t y i s (1.15).

We can e a s i l y o b t a i n a g e n e r a t i o n theorem based on

(5.1) r a t h e r t h a n

although t h e c o u n t e r p a r t s of i n e q u a l i t i e s ( 2 . 1 ) a r e l e s s simple.

on ( l . l O ) ,

THEOREM 5.1.

Let

A

be c l o s e d .

The Cauchy problem for (1.1)2

uniformly well posed i n

(-m,m)

i f and only i f

e x i s t s i n t h e h a l f space

R(h2;A)

[l(hR(h2;A))(n)ll

5

with propagator

@(:)

satisfying

Reh >

(5.1)

(1)

C o ( - l ) nn!(Reh((Reh)*-U 2 )-1) ( n )

where t h e i n d i c a t e d d e r i v a t i v e s on t h e r i g h t hand s i d e a r e t a k e n with respect t o t h e variable

Proof.

ReX.

Combining t h e b a s i c formula (2.11) (which i s obtained

e x a c t l y as i n Theorem 2.1) with i n e q u a l i t y

(5.1) w e

obtain

38

SECOND XDER EQUATIONS

W e use t h e n again (2.11), t h i s time f o r t h e s c a l a r cosine function cosh

wt

(whose i n f i n i t e s i m a l generator i s

sequence o f i n e q u a l i t i e s

To prove t h e converse, we only need t o

t h e r e s u l t i s the

make a few minor changes i n

Observe f i r s t t h a t t h e f i r s t i n e q u a l i t y (5.3)

t h e proof of Theorem 2.1.

implies t h e f i r s t i n e q u a l i t y ( 2 . 1 ) .

l(t;u)

W2);

(5.3).

Thus t h e c o n s t r u c t i o n of t h e function

i n (2.12) and t h e p r o o f of i t s p r o p e r t i e s proceeds i n e x a c t l y t h e

same way.

However, t h e e s t i m a t i o n ( 2 . 1 5 ) i s s l i g h t l y d i f f e r e n t .

W e use

again t h e Post i n v e r s i o n formula (1.3.14) obtaining

where w e use on t h e right s i d e Laplace transform i s

h( h2

(I.3.u)for t h e function cosh

- w2)-l).

lilt

(whose

The r e s t of t h e proof i s j u s t l i k e

t h a t of Theorem 2.1 and we omit t h e d e t a i l s .

J u s t a s ( 2 . 1 ) , i n e q u a l i t i e s ( 5 . 5 ) follow from t h e i r

REMARK 5.2. r e a l counterparts

( s e e Remarks

$11.6

1.3.5 and 2 . 4 ) .

This can be again proved using Taylor s e r i e s .

Miscellaneous comments. Strongly continuous cosine f u n c t i o n s were introduced by SOV‘A [1966:1],

who defined t h e i n f i n i t e s i m a l generator and proved t h e generation theorem 2.1.

Other p r o o f s of Theorem 2.1 were given by DA PRATO-GIUSTI

and t h e author

[1969:3 1 i n c e r t a i n l o c a l l y convex spaces.

proof i s t h e one we have employed here. t h e norm of ( E )

[1967:17

This l a s t

Cosine functions continuous i n

were considered e a r l i e r by KUREF’A [1962:1] who t r e a t e d

a s well t h e case where t h e cosine f u n c t i o n t a k e s values i n a Banach algebra; t h e end r e s u l t of t h i s v e r s i o n of t h e theory i s t h e r e p r e s e n t a t i o n

C(t)

= cos(tA1I2)

( s e e Exercise

2 below).

The d e f i n i t i o n of properly

posed Cauchy problems f o r higher order equations (of which (1.1))i s a p a r t i c u l a r case) i s due t o t h e author [1969:21, a s well a s t h e r e l a t i o n between s t r o n g l y continuous cosine functions and s o l u t i o n operators of second order equations.

Theorem 1.1 i s due t o t h e author [1969:21; a

39

SECOND ORDER EQUATIONS

r e s u l t of t h e same "measurability implies c o n t i n u i t y " type was proved

KUREPA

e a r l i e r by

[1962:1], where both measurability and c o n t i n u i t y a r e

understood i n t h e norm of ( E ) ( o r , more g e n e r a l l y , i n t h e norm of a Eanach a l g e b r a ) .

Theorem 2.3 i s due t o t h e author [1969:2].

EXERCISE 1. Let

A

be a bounded operator i n a Esnach space E.

Show

n

that

A

E

t h a t i s , t h a t t h e Cauchy problem f o r

Ed,

i s uniformly well posed i n generated by

A

-00

0.

a + B = 1. Let

u E D(A)

y - K u Y

a + p < 1.

and c o n s i d e r t h e f u n c t i o n (2.24)

56 in

PHASE SPACES

5

0 < y

C o n t i n u i t y of

1.

y

t h e d e f i n i t i o n and c o n t i n u i t y a t

u

Accordingly, i f

-A

The p r o o f t h a t

E

2

D(A )

2 (-A)

0 < y < 1 i s obvious from

in

(2.24)

1 h a s b e e n proved i n Lemma 2 . 2 .

=

we have

o! (-A)@

i s e x a c t l y t h e same a s i n t h e p r e v i o u s

c a s e ; t h e o p p o s i t e i n c l u s i o n depends on t h e f a c t t h a t i f r e s t r i c t i o n of

A

to

D(A2)

argument employed above for

$111.3

Ti'

then

K;/

.

=

A

which i s shown u s i n g t h e

We omit t h e d e t a i l s .

R e s o l v e n t s o f f r a c t i o n a l powers.

A s s e e n i n t h e next result, c o n d i t i o n of

R(A;A)

The s e c t o r

c (cp-)

Obviously,

A

=

0

t h e s e t of a l l

h

implies existence

cp',

Proof:

0
0)

(4.11)

A2

Accordingly,

Ab

belongs t o t h e

62

PHASE SPACES

3(C )

class

2

= (b I - A )

k

defined i n 4111.2 and t h e f r a c t i o n a l powers




V = ess.

Assume t h a t t h e s c a l a r product

sup c

e l l i p t i c i t y condition

a s i n (1.10).

( U , V ) ~

i s chosen

Then, t a k i n g t h e uniform

(1.6) i n t o account we o b t a i n from (4.4) and ( 4 . 5 )

that

(4.6) t h u s it i s obvious t h a t , i f

o/

i s s u f f i c i e n t l y l a r g e , t h e f i r s t of t h e

two i n e q u a l i t i e s

w i l l hold f o r

u

E

8 ; that t h e second i s as w e l l t r u e follows from

(4.6) with no p a r t i c u l a r requirements on CY beyond cy > v . The f a c t 8 i s dense i n $(n) (Theorem 4.2) and t h e Schwartz i n e q u a l i t y

that

(u,~),

imply t h a t argument

for

can be defined, using a n obvious approximation

arbitrary

u

E

$(a).

Since t h e norm defined by ( 4 . 2 )

d(R), we

i s e q u i v a l e n t t o t h e o r i g i n a l norm of follows t h a t

$(n)

s h a l l assume i n what

Il-IIcy.

i s endowed w i t h

From t h i s p o i n t on, t h e c o n s t r u c t i o n of t h e o p e r a t o r corresponding t o t h e s e l f a d j o i n t p a r t (1.8) of condition

B

A

Ao(B) and t o t h e boundary

i n (4.1) proceeds e x a c t l y i n t h e same way as i n t h e c a s e

of t h e D i r i c h l e t boundary c o n d i t i o n :

u

E

D(A,(@))

(w

E

$(n))

i f and o n l y i f t h e

l i n e a r fbnctional

w

-

i s continuous i n t h e norm of

(u,v),

L2(R);

AO(B)u where

v

i s t h e unique element of

=

(4.9)

we d e f i n e cuu

L

2

- V, (n) s a t i s f y i n g

(4.10)

116

PARTIAL DIFFERENTIAL EQUATIONS

(4.11) A s i n sIV.1, coefficients

motivation f o r t h i s stems from t h e f a c t t h a t i f t h e a

jk function such t h a t

and t h e boundary

cm - A 0u

= v

a r e smzoth and

in

and

0

D"u = y,

i s a smooth

u

t h e n (4.11)

follows f o r any smooth w. Operating a s i n 6 I V . l we show t h a t

( U - AO(B))D(A,,(B)) t h i s time for any

h > a,

CY

=

so l a r g e t h a t (4.7) holds.

estimate of the type of (1.19) and prove t h a t

A,

i n t h e same range o f Finally,

Ao(B)

Ao(p)

W e o b t a i n an

U-AO(B) i s one-to-one

(AI-A0(p))-'

so t h a t

h >

exists i n

cy.

i s symmetric s o t h a t , using Lemma 1.1 we show t h a t

i s s e l f a d j o i n t and bounded above by

depending not only on t h e cosine f u n c t i o n

OIV.5

(4.12)

L2W,

a, where

v but also on t h e c o e f f i c i e n t

@,(t)

generated by

Ao(@)

ff

y.

i s a constant Accordingly,

i s t h i s time given by

The phase space.

The arguments i n s I V . 2 have an obvious counterpart h e r e . c o n s t r u c t i o n of t h e square r o o t

B of

Ao(B)

The

proceeds i n t h e same way,

as does t h e proof of THEORFM 5.1 D(E) =

d(n).

(5.1)

The phase space f o r t h e equation

u " ( t ) = Ao(B)u(t) i s now

(5.3)

El

=

$(".

(5.4)

117

PARTIAL DIFFEREWIAL EQUATIONS

Again, t h e phase space The group

Go(;)

( 5 . 3 ) i s t h e same one provided by Theorem 111.1.3.

propagating t h e s o l u t i o n s of ( 5 . 2 ) i s given by (2.11)

with i n f i n i t e s i m a l g e n e r a t o r D(210(f3)) = D ( A O ( f 3 ) )

(2.12), i t s domain being i d e n t i f i e d by

x €$(D).

To t a k e c a r e of t h e f i r s t order terms we

use Theorem 2.3 a p p l i e d t o t h e bounded p e r t u r b a t i o n o p e r a t o r (2.13).

I n t h i s way we o b t a i n ;

Let

THEOREM 5.1.

r,

A

0

be a domain of c l a s s

t h e operator ( l . 2 ) ,

(3

measurable and bounded on

I-.

with domain

D(A(f3))

=

CiLi

t h e boundary c o n d i t i o n

w i t h bounded boundary

( 1 . 4 ) with y

Let -

D(AO(f3)).

d(n)

Then t h e space

X L'(0)

is a

phase space f o r t h e e q u a t i o n

u"(t)

Q1v.6

=

.

A(B)u(t)

(5.6)

The Cauchy problem.

A l l t h e r e s u l t s i n S e c t i o n IV.3 have a n immediate c o u n t e r p a r t h e r e ; we

d e f i n e t h e semigroup B O ( i ) given by ( 3 . 1 ) i n t h e product space 2 2 = L (0) X L ( a ) ; again, depends on t h e p a r t i c u l a r square r o o t

z0(t)

of

Ao(B)

chosen.

B

However, we need

This can be achieved by r e p l a c i n g l a r g e i n t h e d e f i n i t i o n of

Ao(f3);

t o have a bounded i n v e r s e .

c ( x ) by

m Pu = C b . ( x ) D J u + j =1 J TmOREM boundary

y

2

6.1.

r, A

fi

t h e o p e r a t o r (1.2), B

D(A(B))

i s w e l l posed i n

-m

= D(Ao(f3)).

< t
’ ;X

a(x)) < m,

(7.8)

N

a(x), t h e spectrum of

where

E

i s e a s i l y i d e n t i f i e d as

A,

As proved i n E x e r c i s e 11.5, (7.8) i s e q u i v a l e n t t o t h e f a c t t h a t

~(x)

i s contained i n a r e g i o n of t h e form Re h < - w2 - ( I m h ) > / 4 3 . LEMMA 7.2.

-

A

2

i f and o n l y i f

E

p

(a)

(7.9) i s even ,(b)

a

is P -

r e a l with (-l)p’zap (c) j

is r e a l i f

aj

i s odd,

j

>

j

i s even

>

p/z,

,

(7.10)

(d) aj i s imaginary i f

p/2.

Assume t h a t ( a ) , ( b ) , ( c ) and ( d ) hold.

Proof:

P(t) = where

j

p,

Since

a n i n e q u a l i t y of t h e type of

51.

not hold i n t h i s c a s e f o r l a r g e

This ends t h e proof

of Lemma 7.1. We note t h e c u r i o u s consequences o f Lemma

belongs t o

2

,

d

-

A =

+

-

=

(-$I8

6

+(-&)5 (d/dx) 5

does not, i n s p i t e of t h e f a c t t h a t (d/dx)8

although t h e o p e r a t o r

t h e operator A

of

(dx)

7.1:

than

(d/dx)

6

i s a “tamer” p e r t u r b a t i o n

.

I n t h e following s e c t i o n we s h a l l attempt a t h e o r y of t h e equation

(7.l),

b u t only i n t h e c a s e where

t h e D i r i c h l e t boundary c o n d i t i o n . c o e f f i c i e n t s of

O1v.8

A

of o r d e r > p/2

B

i s t h e h i g h e r order v e r s i o n of

Lemma

7.1 i n d i c a t e s

that the

w i l l have t o be s u i t a b l y r e s t r i c t e d .

Higher o r d e r e q u a t i o n s ( c o n t i n u a t i o n )

We study here t h e e q u a t i o n

(7.1) w i t h

an operator

A

of t h e form

121

PARTIAL DIFFERFNTIAL EQUATIONS

c

7

Au =

(-l)Ial-'D"(a+(x)D

Bu ) +

I4 5 k

la1 5 k The c o e f f i c i e n t s

101 T

am, ba

k

a r e r e a l and defined i n a bounded domain

Rm.

in-dimensional Euclidean space

of

W e s h a l l assume t h a t t h e c o e f f i c i e n t s

of t h e p r i n c i p a l p a r t of t h e operator

a

(8.1)

bo/(x)Dau.

A,

OB

c

(-l)ial-lDw(a~D')

,

(8.2)

Ictl=k [BI=k

a r e continuous i n

-

n;

t h e r e s t of t h e

simply measurable and bounded i n r e s t r i c t i o n of

A

R.

a

*'

a s well as t h e

A(B)

The operator

obtained by imposition a t t h e boundary

b,

are

denotes t h e

r

of t h e

D i ric h l et b ound a ry cond it ion

... =

u = D"u =

(Dw)k-l~ = 0

(x

E

r)

(8.3)

(8.3) w i l l be s a t i s f i e d only i n a generalized sense t o be

(although

c l a r i f i e d l a t e r ) . We assume t h a t

and t h a t

A

f o r some

K

i s u n i f o r d y e l l i p t i c , which i n t h i s case means t h a t

> 0.

The following r e s u l t (Ggrding's i n e q u a l i t y ) w i l l be b a s i c .

To s t a t e

it w e introduce t h e Sobolev spaces wk'p(fi) (1 5 p < m ) c o n s i s t i n g of u defined i n fl and having p a r t i a l d e r i v a t i v e s of

all f u n c t i o n s

5k

order

For

p

(understood i n t h e sense of d i s t r i b u t i o n s ) i n

LP(R);

the

w k ~ p ( n >i s

norm of

=

2

wky2(n) =

( t h e only case of i n t e r e s t t o u s ) we s h a l l w r i t e

$(Q).

( i n t h e norm of

The space

Hk(n)).

$(n)

The statement t h a t

v e r s i o n o f t h e boundary conditions THEORail

8.1 L e t

L

i s t h e c l o s u r e of u E %(a)

(8.3).

be a d i f f e r e n t i a l operator;

d)(n)

in

$(n).

i s t h e weak

122

PARTIAL DIFFERENTIAL EQUATIONS

i n a bounded domain

7

c

bI5k

lPl5k

Q

5 Rm.

(-l)lN(-lDTY(aOIT;Dpu)

Assume t h a t a l l t h e c o e f f i c i e n t s

a r e measurable and bounded and t h a t

ICY~ =

=

k.

i s continuous i n

a$

Then t h e r e e x i s t constants

x

14 5 k Is I I

0

when

such t h a t

C,CY

JaaM(x)D?DBu

-

dx

2

k

For a proof see FRIEDMAN [1969:1,p.321. W e proceed t o t h e c o n s t r u c t i o n of a phase space f o r t h e equation

where

A.

i s t h e s e l f a d j o i n t p a r t of A, A~ =

7 Ao(B)

The d e f i n i t i o n of renorm t h e space

where

CY

$(n)

Y

(8.8)

(-i)lml-lDm(aOIT;Dpu).

IBl5k

l+k

follows t h a t f o r t h e second order case.

W e

by means of t h e s c a l a r product

i s t h e constant i n (8.6).

We have

(8.10) The second i n e q u a l i t y follows f r o m t h e boundedness of t h e c o e f f i c i e n t s

of A ; t h e first i s a consequence of Theorem 8.1.

u

E

4(.".)

belongs t o

D(AO(B))

i f and only i f t h e l i n e a r f u n c t i o n a l

w -, ( u , ~ ) , i s continuous i n t h e norm of element of

2

L (a)

L

2

(Q),

A,(@).

being t h e orily

that satisfies

W e show i n t h e same way as i n t h e case a d j o i n t and t h a t

An element

Ao(B)

k = 2

i s bounded above

g e n e r a t e s t h e cosine f u n c t i o n

(by

that o!),

Ao(B) so that

i s self Ao(B)

123

PARTIAL DIFFERENTIAL EQUATIONS

C(t) and a square r o o t

cash t A o ( B ) 1 / 2

=

=

(8.12)

can be defined as i n gIV.2: we have

B = A,(@)'/' D(B)

,

#(n)

=

D((U h >

t h e l a s t i n e q u a l i t y holding for

- A ~ ( B ) 1) /2 ),

(8.13)

Theorem 111.5.4, combined with

cy.

(8.13) i m p l i e s t h a t Q =

i s a state space for (8.7).

(8-14)

H p ) x L2(Q) To show t h a t

Gf

i s as well a s t a t e space

f o r the f u l l equation

we i n c o r p o r a t e t h e lower order terms i n

(8.1) through p e r t u r b a t i o n

(Theorem 2.3) d e f i n i n g

(8.16) and

:]

? = [ : We o b t a i n i n t h i s way: THEOREM 8.2.

Let A

(8.1), @

be t h e operator

the Dirichlet

boundary c o n d i t i o n (8.3), and l e t

(8.18)

A(B) = Ao(B) + P w i t h domain

Then t h e space

D(A(@)) = D(Ao(@)).

a phase space f o r t h e equation

$(Q)

x L2(n)

(8.15).

The t r e a t m e n t of t h e Cauchy problem f o r (8.15) f o l l o w s word by word t h a t f o r second order e q u a t i o n s i n pIV.3; THEOREM

8.3.

L A A

we only s t a t e t h e f i n a l r e s u l t .

be t h e operator (8.1),

boundary c o n d i t i o n ( 8 . 3 ) , and l e t

B the Dirichlet

124

PARTIAL DIFFERFNTIAL EQUATIONS

A ( B ) = Ao(B) with domain

D(A(B))

( 8 -19)

P

Then t h e Cauchy problem f o r t h e

= D(AO(B)).

equation (8.15) i s well posed i n

9IV.g

+-

in

0.

such t h a t

(AI-A)u = 0

eigenvalue

#

that is, there exists

L*(O,~)).

U(5)d5

(9.4)

125

PARTIAL DIFFERENTIAL EQUATIONS

EXERCISE 4 . that

f o r all R(p;A)

Let

(Hint:

p E p(A) =

be a n o p e r a t o r i n a Eanach space

A

i s compact f o r some

R(h;A)

h

E

Then

@(A).

E

such

i s compact

R(p;A)

use t h e second r e s o l v e n t e q u a t i o n

R(A;A) + (A-p)R(p;A)R(A;A)

and t h e f a c t t h a t t h e sum o f two

compact o p e r a t o r s and t h e product of a compact o p e r a t o r and a bounded o p e r a t o r a r e compact; s e e KATO EXERCISE 5.

Let

A

[1976:11).

be as i n Exercise

empty o r c o n s i s t s of a sequence that

+

h

then

m

A

EXERCISE

6.

o(A)

is

and t h e space

Show t h a t i f

A

E

a(A)

EB(h) of g e n e r a l i z e d

-

enjoys t h e p r o p e r t i e s described i n E x e r c i s e 2. Show t h a t t h e r e e x i s t s a n o p e r a t o r as i n Exercise

(Hint:

a(A) =

A

Show t h a t

of complex numbers such

i f t h e sequence i s i n f i n i t e .

i s a n eigenvalue of

e i g e n v e c t o r s of

with

\, h2,. ..

4.

t r y t h e inverse of the Volterra operator

4

(9.4)).

n be a bounded domain i n m-dimensional into Ehclidean space Rm, B a l i n e a r bounded o p e r a t o r from L2(n) EXERCISE

7.

Let

$(".

Show t h a t

L2(Q),

i s compact (See MIHAILOV

E,

thought of a s a n o p e r a t o r from

L2(")

into

[1976:1]).

7 show t h a t t h e second o r d e r o p e r a t o r s i n (3.15) and (6.2) and t h e h i g h e r o r d e r o p e r a t o r s i n (8.19) enjoy t h e s p e c t r a l p r o p e r t i e s i n Exercise 5 ( H i n t : show t h a t R(h;AO(p)) i s compact u s i n g Exercise 7 and t h e n apply Ekercise 5 ) . EXERCISE 8.

Using Exercise

126

CHAPTER V UNIFORMLY BOUNDED GROUPS AND CmINE FUNCTIONS IN HILBERT SPACE

4 v.l

The Hahn-Baaach theorem: Let

E

Banach l i m i t s .

be a n a r b i t r a r y r e d l i n e a r space.

A functional

p :E

-W

i s c a l l e d sublinear i f

for

u, v

E

E

arbitrary.

THEOFEM 1.1. (Hahn-Banach). a linear functional.

Let F

be a subspace of

E, cp : F

J

R

Assume t h a t

Then t h e r e e x i s t s a l i n e a r f u n c t i o n a l

0 :E

-.

R

such t h a t

For a proof see BANACH [1932: 1, p . 281.

With t h e h e l p of Theorem

1.1 we can c o i s t r u c t a n i n t r i g u i n g

extension of t h e notion of l i m i t .

Let

bounded complex functions defined i n t

l i m i t in

B

i s a functional

p e r t i e s , where

f(t), g ( i )

E

B = B[O,m)

1. 0 .

be t h e space of d l

A Banach l i m i t or peneralized

LIM : B -t 6: enjoying t h e following pros +m B and C U , ~ a r e complex numbers.

127

I N HILBERT SPACE

5

lim i n f f ( s )

(d)

S-

(f)

LIM f ( s )

r e a l valued:

Banach l i m i t s i n

LIM f ( s )

s+

arbitrary.

B

for

f

S-

m

m

$( be t h e subspace of

Let

B

c o n s i s t i n g of

Define

t h e i n f i m u m taken over all p o s s i b l e f i n i t e sequences

BR,

of nonnegative numbers. Using Theorem 1.1 f o r a linear functional

We check i n s t a n t l y t h a t rp = 0

and

F = (0)

Q :BR

LIM = Q.

Obviously,

satisfies

p

(5,)

(1.1).

we deduce t h e existence of

R such t h a t

-+

(f

Q(f) < P ( f ) Set

LIM f ( s ) s +-m

LIM R e f ( s ) + i LIM I m f ( s )

=

r e a l valued f u n c t i o n s .

E

B exist.

once t h i s done we simply s e t

S-'W

f

is real.

f

m

Obviously, it i s enough t o construct

Pro3f:

for

if

if t h e l a t t e r e x i s t s .

l i m f(s)

=

s-

THEOFZM 1 . 2 .

E

sup f ( s )

S-m

S-m

m

L3-m

f

zlim

@z

~ L I Nf ( s ) [ I l i a sup]f(s)(.

(el

for

LIM f ( s ) s-

S-+W

(a)

holds.

E

%I.

(1.6)

Replacing

by

f

(1.6)

in

-f

S-.m

we o b t a i n

This y i e l d s (d)

(c).

-

p ( f ) < U r n sup f ( t ) and - p ( - f ) z

Since

l i m inf f ( t ) ,

(1.6) and (1.7); obviously, ( d ) implies we take 5 1 = h, 5 2 -- 2h, ...,cn = nh i n (1.5)

follows from

To check

(b) p(f(c

+

h)-f(i))

5:

(f). so t h a t

l i m sup ( f ( s + n h ) - f ( t ) ) . S+'X

Since

n

way t h a t that €3

is arbitrary,

p(f(i)

p(;'(t

- f ( f , + h ) ) 5 0,

Q(f(: + h ) ) = Q(f(;)).

be such t h a t

+

eie LIM f(s)

h)

- f ( ; ) ) 5 0.

t h u s it follows from

Finally, we show =

We deduce i n t h e same

ILIM f ( s ) l .

(e)

Then

(1.6)

and

a s follows.

(1.7) Let

128

I N HILBERT SPACE

This concludes t h e prsof of Theorem 1 . 2 . A Banach l i m i t

5

sequences

(co,Cl,

=

i n t h e space

LIM

n-

... )

is

p r o p e r t i e s corresponding t o

limits i n

l i m inf n- m

cn 5

I LIM

(el)

n(f')

LM n- m

Proof:

LIM n- m

5lim n-

m

5,

=

COROLLARY 1.3.

and

i n the d e f i n i t i o n of Eanach

(f)

lim n- rn

LIM

cn I'_ l i m

sup

cn

{Cn]

if

is real.

n-m

SUP IS,^. m

5,

if t h e l a t t e r e x i s t s .

am

Ba.nach l i m i t s i n

exist.

Define

LIM n- m s+

-

(a)

4" enjoying t h e

functional i n

R

B:

(d')

where

of complex bounded

Rm

m

cn

=

LIM f ( S ) , s-

m

i s one of t h e Banach l i m i t s constructed i n Theorem 1 . 2

m

f(s) =

5,

in

n

5

s < n + 1.

Uniformly bounded gro-ips i n Hilbert space.

8V.2

Throughout t h e r e s t of t h i s chapter (except i n Section V .3) we

shall assume t h a t Let

B

E

= H

i s a complex H i l b e r t space.

be a s e l f a d j o i n t operator i n

Then it follows e a s i l y

H.

from t h e f u n c t i o n a l c a l c u l u s f o r self ad j o i n t operators t h a t

U(;),

where U ( t ) = exp(itB)

(-a
0

o+, [1958:1, p . 5671),

a s claimed.

This ends t h e proof

147

I N HILBERT SPACE

of Theorem

6.1.

The next r e s u l t i s a n exact c o u n t e r p a r t of Theoyem 2.2 for c o s i n e However, t h e method of proof i s somewhat d i f f e r e n t .

functions.

THEORZM 6.2. @(s

+ t) +

@(s

Let c(

=

e(E,T,u)

0, E

=

5

Ct;o 5 t 5

t h e c h a r a c t e r i s t i c f u n c t i o n of

t

Proof: Set equation

=

s

= u/2

cr

e

E

E

H,

(6 -12)

ll@(t)uII < ~ l l u 1 1 3 .

e.

i n t h e (second) cosine f u n c t i o n a l

0/2

=

@(u) + I

-

we have

Accordingly, i f

so t h a t

hence

(11.1.9). m e r e s u l t i s 2qu/2)2

Hence, i f

T,

(v);

using

(6.11).

+ l), u

1/(2C

m

&

f! e.

5 1/(2C + 1)

we deduce t h a t

It follows t h a t i f

shows t h a t t h e f u n c t i o n s

x (i)

and

u

x

E

f2;)

e

then

2a

#

e,

which

have d i s j o i n t support.

Hence

by

(i)

and

t i o n we o b t a i n

(v).

Taking t h e change-of-variable property i n considera-

(6.13),

thus ending t h e proof of Lemma 6.4.

149

I N HILBERT SPACE

Proof of Theorem 6.2.

m e operator

i s t h i s time defined by

P

ds

E y v i r t u e of Lemma

6.4

with

E =

1/(2C m

t h u s it f o l l o w s from t h e d e f i n i t i o n of

+ 1)

P

.

(6.14)

we have

that

On t h e o t h e r hand, it i s obvious t h a t

Accordirgly, i f inequalities Let now

i s t h e p o s i t i v e , s e l f a d j o i n t square r o o t of P,

Q

(6.8) h o l d . t

be a real number,

u,v

elements of

H.

Using t h e

c o s i n e f u n c t i o n a l e q u a t i o n s and Theorem 5.2 we deduce t h a t ( P @ ( t ) u , v ) = LIM

L T ( @ ( s ) C ( t ) u , @ ( s ) v d) s

T-‘M T L =

1LIM 2 T-m

$lo -T

( @ ( s + t)u,C(s)v) ds

$k

T

+ 1_

*

LIM T-m

+ 1_

LIM T-m

(@(s - t)u,@(s)v) d s

PT



for

u,v

E

H.

$ j o ( @ ( s ) u , @ ( s+ t ) v ) d s

Accordingly,

(6.18) P r e - and p o s t - m u l t i p l y i n g by QC(t)Q-l

Q

-1

= Q-’@(t)*Q

we obta.in =

(Q@(t)Q-’)*.

(6.19)

150

I N HILBERT SPACE

This completes t h e proof of Theorem

6.2.

The coriiments following Theorem 2.2 apply h e r e a s w e l l :

replacing

t h e o r i g i n a l s c a l a r product by ttie ( t o p o l o g i c a l l y e q u i v a l e d c ) s c a l a r product

(2.12)

-

@ ( t )s e l f a d j o i n t .

r e n d e r s each

COROLLARY 6.5.

Assume i n a d d i t i o n t h a t

C(;)

Q

B

u

B

2

t

E.

Then t h e r e e x i s t s a s e l f ad-

tnd a bounded s e l f a d j o i n t o p e r a t o r

0

(6.8)

s a t i s f y i n g i n e q u a l i t i e s of t h e form

@ ( t=) Q - l c o s (tB1/')Q @(;)

Conversely, e v e r y

(-a

E 2 EI f o r some

(6.7).

and such t h a t

0.

E

The following d i s c r e t e v e r s i o n of Theorem

6.2 corresponds t o

Theorem 2.4.

THEOREM tors in

2C C mn

H

Let {cn;-m < n c

6.6.

m)

be a sequence of bo,mded opera-

s a t i s f y i n g t h e " d i s c r e t e c o s i n e equations"

+ Cm-,

= @,+,

m,n.

f o r all

=

I

'

Assume t h a t

Then t h e r e e x i s t s a bounded, s e l f a d j o i n t o p e r a t o r

2-l(2c +

0

l)-'hIl2 5

(Qu,.)

Q

5 Cllul12

satisf'ying

(6.22)

and such t h a t

fin

i s s e l f ad,joint f o r a l l operator

U

n.

=

QCnQ

-1

Equivalently, t h e r e e x i s t s a u n i t a r g

such t h a t

@n

=

2

Q-l(V" +

U-n)Q

.

(6.24)

The proof i s r a t h e r s i m i l a r t o t h a t f o r t h e continuous v e r s i o n . The o;lerator

P

i s now defined by

151

I N HILBEHT SPACE

(Pu,v) where

1.3.

Il@2muII 2

then

P r o c e e d i n g as i n t h e p r o o f o f Lemma 6.4 we c a n show

IICmuII < Ellull

t h a t if

-

LM n- m

i s one o f t h e Banach l i m i t s o f sequences c o n s t r u c t e d i n

LIM

Corollary

=

11

f o r which

f o r an arbitrary integer

EIIU// i s

PmuII 5

a t least e q u a l t o

5

indicates the largest integer

[s]

m

and

5

E

hence t h e n m b e r o f i n t e g e r s between

Ellu/l;

s.

[ ( n - 5)/41,

Taking

E = 1/(2C

(2C +l)-l

0

and

n

where

+ 1) we

obtain

and it i s o b v i o u s t h a t

Q,

thus

(6.22).

t h e p g s i t i v e s e lf a d j o i n t s q u a r e r o o t of

hence e a c h

(6.23)

in

=

C*P, n

(6.27)

is self adjoint.

Consider now t h e sequence of o p e r a t o r s {AS,) m = n

satisfies

(6.17) shows t h a t

A computation e n t i r e l y s i m i l a r t o

P@n

P

{fin;-m

1; s i n c e u(fin) must b e

I A1

w e deduce t h a t

E = arc cos

a1, where

i n the interval &

a r c cos A

i s the function

[ - ~ / 2 , ~ / 2 ] . Let

n = c o s (nB)

.

(6.28)

152

I N HILBERT SPACE

Then it follows from t h e f b n c t i o n ? l c a l c u l - u s f o r s e l f a d j o i n t o p e r a t o r s

{&

that

s a t i s f i e s a s w e l l t h e d i s c r e t e c o s i n e f u n c t i o n a l equation;

n

i n particular,

&

which shows i n d u c t i v e l y t h a t

(6.23)

with

n

(6.28)

We only have t o combine

(6.24),

t o obtain

8 f o r all n s i n c e & = 8 1 1' n 1 ( i n t h e f*orm & = f e x p ( i n E ) + e x p ( -inB)}) n 2 where LJ = e x p ( i E ) .

1

-

Theorem 6.1 i s obviously e q u i v a l e n t to t h e following r e s u l t for second order a b s t r a c t d i f f e r e n t i a l e q u a t i o n s .

Let

THEOREM 6.7.

be a c l o s e d , densely defined o p e r a t o r i n t h e

A

H such t h a t t h e Cnuchy problem f o r

H i l b e r t space

u"(t)

- 0

we have

The argument below is standard i n approximation theory.

In

0

5

s

Ilc(s)u

- uII

taking

0

Once

d i v i d e t h e i n t e r v a l of i n t e g r a t i o n i n (2.21) a t

given, we

5

w e use ( 2 . 1 8 ) , t a k i n g

p

C &/2

there; i n

w2)

by t h e f i r s t e q u d i t y (2.11).

S(f ) A.

Re

We apply t h e n Theorem 2.3 t o deduce t h a t

i s a s t r o n g l y continuous semigroup with i n f i n i t e s i m d g e n e r a t o r That S(i) can be extended t o a ( E ) - v a l u e d f u n c t i o n a n a l y t i c i n

5 > 0,

G(m)

(2.23)

a s well as t h e e s t i m a t e s corresponding t o t h e c l a s s e s

can be proved r e p l a c i n g

we have

t

by

6

i n (2.23).

If

Re

6>

0

171

PARABOLIC SINGULAR PERTUREATION

ds

(2.24)

which implies t h e d e s i r e d e s t i m a t e s .

BVI.3

The homogeneous equation:

convergence of

W e i n v e s t i g a t e here t h e convergence of *

E

-+

0

we o b t a i n uniform bounds on that

cp,$

2

/lu(t;E)/I

where

f(t)

i n t h e homogeneous case

C0

0

u(

CY. Essentially

t h e same manipu-

reveal that t h e e s t i m a t e

where

does not depend on

E,t.

I n e q u a l i t y (3.14)

l e a d s t o t h e estimates below.

holds, where

i s a constant t h a t does not depend on

C

s,t,E.

We make use of (3.14)-(3.15) observing t h a t 2 1/2 2 (Es/t) ) 5 1 ( E s / t ) /2 i n t h e exponent and t h a t (Es/t) 2 ) 1/2 >_ (1 (Es(E)/t)')1/2 = 2~ i n t h e denominator of

Proof.

(1 (1

-

-

(2.14);

t h e term

-

2 4E /t

i n s i d e t h e p a r e n t h e s i s i s p o s i t i v e and can

be dropped.

where t h e c o n s t a n t

C

does not depend of

Proof. We use a g a i n (3.14)-(:,.15)

hand s i d e of t h e i n e q u a l i t y

s,t,E.

keeping i n mind t h a t t h e r i g h t

(3.13) is a n increasing f i n c t i o n of

x.

Accordingly, we can e s t i m a t e t h e r i g h t hand s i d e of (3.14) by t h e value obtained i n s e r t i n g t h e highest p o s s i b l e value of (which i s t h e summand

(1

-

2

(Es(E)/t)2)1/2

4E /t

= 2q).

(1

-

(Es/t

2 1/2

)

Once t h i s is done we d i s c a r d

i n t h e o u t e r p a r e n t h e s i s o f (3.15).

The r e s u l t

i s (3.17). A s a n immediate consequence of (3.17) and of the estimation (3.8)

PARABOLIC SINGULAR PERTURBATION f o r the length of the interval

s(E)

t h e r e and i n o t h e r i n e q u a l i t i e s

C

s ,t ,&

0 . We s a y t h a t a family of f'unctions converges uniformly i n t > - t ( E ) t o a f u n c t i o n g(:) if and t(E)

> 0

f o r each

E

only if sup

1 h

-

llg(t;E)

g(t)lI = 0 .

Ed0 t)t(E)

I f t h e supremum i s t a k e n i n s a y that

t

2

t ( E ) 0

for

a r b i t r a r y we

uniformly on compacts of

g(:)

t(E). We prove below that f o r every

on compacts of

t ,t(E)

s u b s e t s of

as long as

E,

u

E

E, R(t;E)u-r S(t)u

uniformly w i t h r e s p e c t t o

t(E)/E2

4

(E

m

.+ 0 )

uniformly

u on bounded

.

(3.20)

I n f a c t , assume t h i s i s f a l s e . Then t h e r e e x i s t s a bounded sequence

[u,]

sequence

{t,]

For each

n

C

E,

a sequence 2

such that

we choose

tn/En

'n

-21-

{En]

*

with and

m

End

0

and a bounded

lIR(tn;En)un

- S(t,)unll

2 6 > 0.

such that 'in

-

WE

n

n --

(3.21)

n (note t h a t zero,

'n

-

n 1/2

< 1/2: moreover, s i n c e both as

n

+ m).

En

and

E n t n-1/2

tend t o

We d i v i d e t h e i n t e r v a l of i n t e g r a t i o n i n

(2.14) according t o t h e e q u a l i t y (3.7) with

q = q n'

We have

176

PARABOLIC SINGUMR PERTURBATION

The f i r s t i n t e g r a l tends t o zero as convergence theorem: that

(note that

1

-

(Es/t)2

uniform estimate

due t o the dominated

m

+

9

-

cp(t,,s;E,)

n

i n f a c t , t e asymptotic r e l a t i o n (3.10) shows e-'

/4tn-+ 0

2 211, -,

1. hence

(3.16).

as

-

n-,

Es/t

m

0)

for

s

fixed

and we have t h e

The second i n t e g r a l tends t o zero by (3.18).

A s f o r t h e t h i r d it i s e a s i l y seen t o telescope making the change of variable

ti1/'s

=

(J

tn a r e bounded. I n f a c t ,

and r e c a l l i n g t h a t t h e

Now, it follows from (3.8) and (3.20) that S(E

n

)

=

tn 2 1/2 (1 - 4q ) n n

t

2

s(En) >_ 2 t 1/4&-1/2 n

Thus

8n

-

m

(1

-

,3/4

2'in)l/2 2 2

as

n

-

m

n -

-

(3.24) El/2 n s o t h a t (3.23) tends

t o zero (we note t h a t i f

w = 0 t h e i n t e g r a l (3.23) tends t o zero 2 under the s o l e assumption that t,,/En -,m , where t h e tn may be

unbounded; t h i s f a c t bears on a resuLt below). a contradiction and j u s t i f i e d

OUT

claim about

We have then obtained sf.

We prove next t h e corresponding statement f o r

5(t;E).

The

estimates a r e obtained i n a s i m i l a r fashion, thus we only s t a t e the final results.

Formula (3,10)-(3.11) has t h e following counterpart:

with X(t,S;E) = (77t)-1/2(1 The estimate is uniform i n The inequality

holds i n

0 5 s 5 t, where

-

2 (y))-1/4(l +

0 5 s 5 s(E).

.

(I(%))(3.26)

PARABOLIC SINGULAR PERTURBATION

177

p(t,S;E) = t and t h e constant

(3.28)

does not depend on

C

we use t h e asymptotic formula (3.5) f o r

m = 0

E,t.

To o b t a i n (3.25)-(3.26)

m = 1; t h e same formula with

yields the inequality

Using t h e i n e q u a l i t y (3.26)-(3.27) we e a s i l y o b t a i n t h e following counterparts of L e m 3.2 and Lemma 3.3 :

holds, where the constant

where t h e constant Using

C

does not depend on

does not depend on

C

s,t,E.

s,t,E.

(3.31) and (3.8) we obtain

We prove t h a t

6(t;E)u uniformly i n of

t >_

S( t ) u

uniformly with respect t o

t(E)

i n e x a c t l y t h e same way used f o r

E

(3.33)

R;

u

i n bounded s e t s

d e t a i l s a r e omitted.

A f t e r a n elementary estimation of t h e f i r s t term i n (2.5) t h e proof of t h e following r e s u l t i s complete:

Let

THEOREM 3.6. UO(E)

and l e t

u(t;E)

-b

v,

uO(E),ul(E) 2

E UJE)

E

E

-. uo

be such that

- v

(E

+

0)

be the generalized s o l u t i o n of ( 3 . l ) ,

number such t h a t (3.20) holds.

,

(3.34) t(E) > 0

Then

U(itjE) -. u(Z)

(3.35)

178

PARABOLIC S INGUMR PERTURBATION

unifwmly i n compacts of s o l u t i o n of ( 3 . 2 ) w i t h respect t o

uo,v

REMARK 3.7.

if.

2

t

t(E)

u(%)

where

u(0) = uo.

i s t h e generalized

The convergence i s uniform with

(Iuo(I, /(v(I a r e bounded.

does not converge t o

uO(E)

thus t h e r e i s a "boundary l a y e r " near zero where approximation t o

t >_ 0 u

Obviously, uniform convergence i n

expected s i n c e i n g e n e r a l

u(^t)

cannot be

as

0

-

E

0,

i s not a good

u(;;&)

[1981:1] f o r a thorough

( s e e KEVORKIAN-COLE

treatment of t h e one dimensional case). Note a l s o t h a t m i f o r m -opt -w2t e u(t;E) t o e u ( t ) i n t >_ t ( E ) cannot be

convergence of

assured even i n t h e s c a l a r case.

To s e e t h i s , l e t

s o l u t i o n of t h e i n i t i a l value problem

-w

< w2

since ?(&),A*(&) e-',2tew2t = 1,

we have

e

L3t -2 1 with

0 l e t t ( E ) be such that 2 -, a , where 0 < a < m . Then we seE from (3.3) that

follows. 0 , 0 c E c 1/&)

independent of

u)

> 0

and of

5

116"(t;E)(E-1R(&-1;A))ll

C(w4

f

2 w 2 / t + l/t 2 ) eu) t

The proof i s straightforward b u t tedious. that

U E

D(A),

B"(t;E)u

s o that

.

(4.26)

Assume f o r t h e moment

e x i s t s ; a n e x p l i c i t formula f o r it

can be obtained from (4.1): -t/2&2 G,"(t;E)u =

E3

-t/2E2 C'(t/E)u

-

4

-t/2E C(t/E)U

2E

+

te

2 C(t/E)U

8E6

C(s)u ds

te

2 -t/2E f

t e 4E7

C(s)u ds 2

-t/E

Jo

Ii(((t/E)2

-

2 1/2

s )

/2&) C(s)u ds

Il(((t/E)2

-

S2)l/'/2E) C(s)u ds

PARABOLIC SINGULAR PERTURBATION

187

e 4E5

where terms a r e grouped t o g e t h e r as t h e y appear i n d i f f e r e n t i a t i n g

(4.1).

Note a l s o that t h e t h i r d and f o u r t h i n t e g r a l s a r e i n d i v i d u a l l y

divergent and must be combined i n t o one.

We t a k e a look f i r s t a t t h e

terms that l a y o u t s i d e of i n t e g r a l s . For t h e second we have

and t h e same estimate o b t a i n s for t h e t h i r d and t h e f o u r t h ,

SO

that

t h e y s a t i s f y (4.26) even without t h e i n t e r c e s s i o n of t h e mollifying operator then

E-1R(E-1;A).(4)

For t h e f i r s t term we note that i f

C(^t)v i s continuously d i f f e r e n t i a b l e w i t h

hence

C' ( t ) v = d(t)Av

Since

w

7

v

E

D(A)

C " ( t ) v = C(t)Av,

and we have

0, l/S(t)ii 5 C e x p ( ~ t ) ( ~ and ) t h e r i g h t hand s i d e of (4.29)

can be estimated i n t h e same way as (4.28). To e s t i m a t e t h e six i n t e g r a l s i n (4.27) we d i v i d e t h e domain of integration a t specified l a t e r .

s = s(E)

given by

(3.7), with q < 1/2 t o be

For t h e f i r s t o u t e r i n t e g r a l we t a k e advantage of

(3.17) f o r cp(t,s;E), divided by t E 2 ; f o r t h e i n t e r v a l of i n t e g r a t i o n we use (3.8). The r e s u l t is a bound of t h e form t h e estimate

The s e c o n d , f i f i h and s i x t h i n t e g r a l s a r e t r e a t e d i n t h e sane way: i n a l l c a s e s , due t o t h e a d d i t i o n a l f a c t o r e s t i m a t e of t h e form

t/E2

we end up with a n

188

PARABOLIC SINGULAR PERTURBATION

A s pointed out a f t e r (4.27) the t h i r d and f o u r t h i n t e g r a l s must be

combined i n t o one t o a m i d divergence a t

s = t/E

w r i t t e n s e p a r a t e l y only f o r typographical reasons).

( i n f a c t , they a r e The basis of the

r e s u l t i n g estimation w i l l be t h e asymptotic s e r i e s f o r the h n c t i o n obtained from (3.6): Q(x) = X-~(X-~I~(X))'

we deduce from it that

The combined integrand of the f o u r t h and f i f t h i n t e g r a l (including f a c t o r s outside of the i n t e g r a l ) i s

-

2 -t/2&2 t e Q ( ((t/E)2

16E~

-

~~)'/~/2E)C(s)u.

I n view of (3.30) we have

where p(t,S;E) = t

(3

(4.34)

is increasing w e can bound the r i g h t hand side of (4.34) by i t s value a t s = s ( E ) subsequently deleting the f a c t o r 6$/t from t h e outer parenthesis. The r e s u l t i s an upper bound f o r

Since

-t

x)-5/2ex

t h e combined integrand of the form

Therefore, the i n t e g r a l can be bounded by the following expression:

This completes the consideration of the outer i n t e g r a l s .

We look a t t h e inner i n t e g r a l s . t h e i n t e g r a l belaw:

We begin by grouping them i n t o

PARABOLIC SINGUL4R PERTURBATION

189

O(t,s;E)C(s)u ds.

Using t h e asymptotic developments (3.5) f o r

Io, I1 and Ii of

m = 1 i n t h e f i r s t and f o u r t h i n t e g r a l s and of order

order

i n t h e r e s t we o b t a i n f o r

B

m = 2

a n expression of t h e form

a l i n e a r combination of terms of t h e form

with X

with

(4.36)

j = 2,1,0,C

f o r each t e r m

expression f o r

J

>

We then use T a y l o r ' s formula of order 2 ) 'j, ending up w i t h t h e following

U . , ~ .0.

J

(1 - ( E s / t ) X:

o(&)) 2

where each

i s independent of

X,(t,s)

t > 0

cosine f u n c t i o n

and apply formula C(^s) = cos og,

(in fact,

E

(4.27) i n

where

t h e space

(4.39)

is a f i n i t e

X,

>_ 0 ) . We

s@t-* w i t h ct:B

l i n e a r c o m b i d t i o n of terms of t h e form then f i x

2j

E = C

t o the

i s a r e a l parameter.

0

Naturally, t h e r e s u l t must be t h e second d e r i v a t i v e of t h e s o l u t i o n of E 2t " ( t j E )

4-

< ' ( t ; E ) = w 2< ( t j E )

,

(4.40)

w i t h i n i t i a l conditions _ t ( E ) uniformly w i t h r e s p e c t t o i s any bounded s u b s e t i n E.

uniformly on compacts of E

(u E D(A))

We have t2(6''(t;E)U

u

independent of

C

< 1/40) such t h a t

E

where CB

The homogeneous equation.

Rates of convergence.

We show i n t h i s s e c t i o n t h a t i f t h e r e i s no ''crossover" of i n i t i a l conditions ( i . e . i f we have uo,

uO(&)

r a t h e r than (3.34)) then

t

5

2 E

u1(&) -.

E

respect t o

u

D(A)

D(A)

(5.1) u(%)

u0

E

uniformly i n

D(A)

or t o c e r t a i n

I n contrast with the not be uniform w i t h

(Iu/(is bounded.

be t h e o p e r a t o r a c t i n g on t h e i n i t i a l c o n d i t i o n

i n (2.14), i . e . &(t;E) = e 4 / 2 2 C(t/E) E

and E.

3vI.4, convergence w i l l

even i f

Let Q(t;E)

o

-t

converges t o

u(^t;E)

subspaces intermediate between

u

as

w i t h p r e c i s e r a t e s of convergence i f

0,

r e s u l t s of 3Vr.3 and

If

o

both

R(:;E)u

d i f f e r e n t i a b l e , t h u s s o is

+ 5 1R ( t ; E ) + 2 1& ( t j E ) .

(5.2)

a r e twice continuously

and B(t;;E)u u(;;E)

uO(E)

= Q(;;&)U.

The d e r i v a t i v e

r)

v(t;E) = u ' ( t ; E ) i s a generalized s o l u t i o n of (2.1) with i n i t i a l conditions v(0;E) = u ' ( 0 j E ) = 0 and v ' ( 0 ; E ) = u"(0;E) = E -2Au. Hence, by uniqueness, we must have ( t ; E ) = & ' (t;E)U = 6(t;E)Au.

(5.3) 2 On t h e other hand, we may w r i t e (4.1) i n the form W ( t ; E ) = E - Q ( t ; & ) -2 E 6 ( t ; E ) , hence U'

G(tjE)U = &(tjE)U

-

2 E

G'(tjE)U

.

Applying t h i s e q u a l i t y t o a n element of t h e form Au we o b t a i n

(5.4) and using (5.3)

-

PARABOLIC SIPU'GUWI PERTURBATION Q ' ( t j E ) u = AiS(t;E)U

-

193

.

2 6'(t;E)Au

E

(5.5)

s o t h a t Q ( i ; & ) u i s a genuine s o l u t i o n of t h e nonhomogeneous first order equatior? (2.2). Consequently, t h e variation-of-constants

(1.5.3)applies and we have

formula

-

Q(t;E)u

lb(t;E)ll w)

,m

L(h)u

E

Adi(A)u - 1 C(A)U -u + 1 2 h2 &*A

+ L m e - A t 6 f ' ( t ; E ) u d t = &'A2

( h > w).

E

so t h a t

2 h2 I f A

(E

- A)L(A)u =

u and we deduce using denseness of

D(A) that ( A > w2 ).

X(A) = R(E2 A2 + A;A)

(5.9)

Accordingly,

W e use now (11.2.11):

=k m

hR(A2;A)u

e-At@(t)u d t

( A > w, u

f

E).

(5.11)

Making use o f (5.11) and of the cosine f u n c t i o n a l equation (11.3.1) f o r C(t)

we obtain

2 2pv R ( ~ ~ ; A ) ;A)U R(~ =dmkae-(Ps*t)(C(~

+

t ) + C(S

-

t ) ) u dsdt

(p,v

> w).

(5.12)

194

PARABOLIC SINGUIAR PERTURBATION

Taking advantage of t h e convolutiun theorem i n t h e d e f i n i t i o n of

i n (5.6) we deduce, making use o f (5.8) and (5.10) t h a t

Q(^t;E)

2 2 m(h;A)R(E A

+

A; A ) u =

( A > w2,

dt

E).

(5.13)

( A > w 2 ),

(5.14)

u

E

By v i r t u e of (5.12) we m y a l s o w r i t e

AR(A;A)R(E

2 2

A

+

A; A)U =

[mlmh(tys,h;E)(C(s

t ) + C(S

-t

-

t ) ) u dsdt

with

Consider t h e s c a l a r cosine f u n c t i o n A

C ( t ) = cosh w t

( 4

< t


2

= e

t

( t 2 0)

I

and G(t;E) = YW(t;E), Yw

as defined i n (3.3); accordingly it follows from (5.7) t h a t $(t;E)

Applying formulas

= Ow(t;E)

.

(5.13) and (5.14) we obtain

rm

Let now

u be a n a r b i t r a r y element of

of t h e d u a l space

E*

with

* IIu /[

=:

function

According t o t h e previous arguments,

*

E, u

a n a r b i t r a r y element

l[ull = 1, and consider t h e s c a l a r

195

PARABOLIC SINGULAR PERTURBATION

Lme-Atr(

t;E)

+

dt =L F ( t , s , h ; E ) ( k ( s

t)

-

-

k(s

(5.18)

t ) ) dsdt

where

Obviously , k(s)

Let

0

(-m


- o

(in

(A?

0,

t 1 0)

if

n = 0,1,...)

We define correspondingly a l t e r n a t i n g functions i n

.

(5.21)

t >_ a.

It i s obvious that the swn of two a l t e r n a t i n g f u n c t i o n s and t h e product of an a l t e r n a t i n g f u n c t i o n by a nonnegative c o n s t a n t i s alternating.

More g e n e r a l l y , i t follows from L e i b n i z ' s formula that

t h e product of two a l t e r n a t i n g f i n c t i o n s i s a l t e r n a t i n g .

LEMMA. 5.2. alternating.

m(i)

be a f i n c t i o n such that

rn'(%)

Then 6

R ( A ) = e -m(^A)

(5.22)

i s alternating. Proof:

Obviously, it is enough t o show that each summand i n

t h e d e r i v a t i v e of order

with

j,.

.., p

*

n 2 1 of

R ( A)

i s of t h e form

1 and n k+(j-1)+ ...+(p- 1) (-1) = (-1)

n = 1; assuming it is t r u e for n,

This statement is obvious f o r validity for

its

n i 1 follows from L e i b n i z ' s formula.

LEMMA 5.3. Let

E

> 0,

*

m(A) = ( E A Then m' (A)

(5.24)

+

A)

q2

(A,O)

is alternating.

The proof i s l e f t t o t h e reader (Exercise 1).

.

(5.25)

196

PARABOLIC SINGULAR PERTURBATION LEMMA 5.4.

t

+

Aza.

Let

f ( % ) be continuous i n

Assume t h e Laplace transform

m.

t >_ 0 , f ( t )

=

O(exp a t )

Pf(^A) i s a l t e r n a t i n g i n

Then f ( t ) >_ 0

( t >_ 0 ) .

(5.26)

The proof is an immediate consequence of Lemma 1.3.2 ( s e e (1.3.14)). End of proof of Theorem 5.1. We go back t o (5.18). The d e f i n i t i o n (5.15) of t h e flmction h ( t , s , A ; t ) , Lemma 5.3, Lemma 5.2 and the is comments preceding it show t h a t h, a s a function of A, a l t e r n a t i n g f o r any s , t 2 0 , E > 0. Since t h e f i n c t i o n k ( s ) defined i n (5.19) i s nonnegative, it follows from (5.18) t h a t the Laplace i s a l t e r n a t i n g . Thus, by Lemma 5.4, transform of r(ht;E) r ( t ; E ) >_ 0 ( t 2 0 , E > 0 ) . Taking i n t o account t h e a r b i t r a r i n e s s of

*

u and u

,

(5.7) follows, completing t h e proof of Theorem 5.1.

I n a l l of t h e r e s u l t s t h a t follow u(:;&)

u(z))

(resp.

is the

s o l u t i o n of t h e homogeneous i n i t i a l value problem (2.1) (resp. ( 2 . 2 ) ) .

and applying (5.6) and (5.7) to t h e f i r s t term on t h e r i g h t hand s i d e t o estimate t h e other summands we use (3.4) which implies

of (5.28): (taking u

0

(E)

= 0

or

Ilc(t;E)ll _ 0,

E

’0)

*

(5.29)

W e obtain a simpler but l e s s p r e c i s e bound noting t h a t @,(t;&), u?t (Lemma 3.1) and i n t e g r a t i n g ( 5 . 7 ) by p a r t s ; it r e s u l t s YW(t,E) 5 e t h a t Ow(t;E) 5 (1+ w2t)eat so t h a t (5.27) becomes 2 lju(t;E) u(t>li 5 c O2 (1 ~ + w 2t ) e w2t / l ~ u +o ~coew ~ t ~ l u o ( ~ )~ , I I +

-

-

197

PARABOLIC SINGULAR PERTURBATION 2 wLt e I/ul(~)I(

( t 2 0,

+ c0E

Theorem 5.5 implies t h a t when

t

D(A)

E

0

-

/lU(t;E)

uniformly on compacts of

u

u(t)ll

E

> 0)

.

(5.30)

we have 2

= O(E

1

(5.31)

if

0

I1uO(~)- uoIl = O(E

2

and

llu,(~>Il =

o(1).

(5.32)

Estimates of t h e same s o r t can be e a s i l y obtained f o r t h e d e r i v a t i v e u'(t;E)

if

u

0

E

2

D(A )

and

uO(E)

In f a c t ,

D(A).

E

v(%;E) = u ' ( % j E )

i s t h e s o l u t i o n of t h e i n i t i a l value problem (2.1) w i t h v(O;E)=U'(O;E)=U~(E),

~ ( 2 =)

On t h e o t h e r hand,

= u"(0;E) = E

v'(O,E)

-2

(AuO(E)

-

ul(E)).

(5.33)

i s t h e s o l u t i o n of (2.2) w i t h

u'(t)

(5.34)

~ ( 0= ) ~ ' ( 0 =) AuO. Accordingly, we have

THEOREM 5.6.

Assume t h a t

u

2

0

E

and

D(A )

uO(E) E

2 2 l ( u ' ( t ; E ) - u ' ( t ) l l 5 COE @ w ( t ; E ) l ( A uolI + CO+w(t;E)lIU1(E)

-

+ COYW(t;E)/lU1(E)

2

D(A).

-

Then

AuolI

AU,(.)ll

2

2 w t 2 (1 + w t ) e IIA uolI

5

COE

+

c0 ew t(I/U1(E)

2

- AUoll

.

( t 2 0) It follows from t h i s r e s u l t that i f

- Auo(E)/I)

IlU,(E)

+

uo

(5.35) 2

D(A )

E

uO(E) E D(A)

and

then llu'(t;E) uniformly on compacts of Ilu,(E)

-

t 2 0 2

- u'(t)l/ =

2 O(E

(5.36)

)

if

- Au~(E)II =

2

)

and

IIu~(E)

= Of& )

and

/ I A u O ( ~ ) AuoII = O ( E ).

AuoI/ =

O(E

O(E

> , (5.37)

or,e q u i v a l e n t l y , i f Ilu,(E)

- Auoll

Theorems 5.5 and

2

-

2

(5.38)

5.6 a r e e a s i l y s e e n t o i n p l y convergence r e s u l t s

v a l i d f o r a r b i t r a r y i n i t i a l conditions.

PARABOLIC SINGULAR PZRTURBATION

198

Let

5.7.

THEOREM

(resp. ( 2 . 2 ) w i t h u

t i o n of (2.1)

E E

0

uO(E)

~ ( 2 ) ) be

(resp.

u(^tjE)

-. uo,

arbitrary).

-. 0 &s

E'u1(E)

t h e generalized solu-

E

+

Assume t h a t

(5.39)

0.

Then U(tjE)

uniformly on compacts o f Proof.

u.

U(0) =

6> 0

Pick

r(^t)be

Let

u(t)

+

E

(5.40)

0

+

t >_ 0 .

u

and choose

E

D(A)

with

;1

- uoI/ 5 & .

t h e s o l u t i o n of t h e i n i t i a l value problem (2.2) with

Applying Theorem 5.5 and i n e q u a l i t i e s

(5.29), ( 5 . 3 0 ) , we

obtain

-

IIU(tF)

5

COE

5

U(t>ll

2 (1+

2 (u

-

IIU(tF)

+

t ) e w t/lAiiI

5

COE

2

2 w t

e

l l ~ ~ l +l

i ~-

uOl/

2

coew t l l u o ( ~ ) 2 w t

11u1(@)11+ C06 e

-

uo1l

.

(5.41)

> 0

s u f f i c i e n t l y small we c a n obviously make t h e r i g h t hand 2 2C0 & ew a i n 0 5 t 5 a , a > 0. This ends the proof.

E

side e wt

+ Taking

c0E 2ew

U(t)lI

-

C o e " tlluo(E)

2

t

-

II3t)

:(t>lI +

2

2

Concerning d e r i v a t i v e s , we have

Let

THEOREM 5.8.

that

u(t)

u(tjE),

u~,u~(E E )D(A) AuO(E) a Au, u l ( f ~ )

-

b e its i n Theorem 5.7.

Au0 -as

E

+

Assume

0.

(5.42)

-

Then

U'(tjE) uniformly on compacts of

U'(t)

4

as

E

(5.43)

0

+

t >_ 0 .

The proof follows t h e l i n e s of t h a t of t h e previous r e s u l t . 6 > 0,

,(^t)

and choose

u

E

2

D(A )

such that

llAu

- AuoI/ 5 6 .

Let

Then, i f

is a g a i n t h e s o l u t i o n of t h e i n i t i a l value problem (2.2) w i t h

U(0) = uo we a p p l y (5.29) and (5.35),

obtaining

199

PARABOLIC SINGULAR PERTURBATION

2

+ c 08 e w t

(t

0,

E

(5.44)

> 0).

T h i s completes t h e proof.

5.9.

Convergence i n (5.31) and (5.36) is uniform i n t ? 0 ( r a t h e r t h a n j u s t uniform on compacts of t ? 0 ) i f w = 0 . O f REMARK

course, t h e same observation applies t o a l l t h e other r e s u l t s i n t h i s section. For easy reference l a t e r w e c o l l e c t t h e s e p a r t i c u l a r cases of Theorems 5.5, 5.6 5.7 and 5.8 under a single heading. THEOREM 5.10.

cosine m c t i o n

Assume that A

c(Z>

with I M t ) II 5

Let

u(ht;E)

(2.1)

,

u(%)

generates a s t r o n g l y continuous

co

e w Let

u

f

Ha.

(t

2

0)

.

(5.53)

Define

IuI,

= ~~u~~ + sup ( w 2

+

l/t)a-le-W

2 tl/AS(t)ul/.

(5.54)

t>_o Obviously,

I *ICYi s

a norm i n

31

CY

(which, i n c i d e n t a l l y , makes

Ha

a

Banach space). The following two r e s u l t s a r e formal counterparts of Theorem 5.5 and 5.6.

The proof of both r e s u l t s i s based on a d i f f e r e n t estimation of t h e operator

a.

commute with A

Assume f i r s t t h a t

u

E

D(A);

since

S

and

6'

and with each other we can w r i t e

=L t

B(t;&)u

It follows from (4.1) that

V(t

-

s;E)AS(S)U d s .

(5.57)

202

PARABOLIC SINGULAR PERTURBATION

-

Take the

(1 cu)-th power of both s i d e s , take the

a-th

power of both

sides of (4.20) and multiply the i n e q u a l i t i e s thus obtained term by term.

The r e s u l t is ll5'(tF)Il

Hence, i f

u

(u)

2

+

2

l/tlffew

(t

2 01.

(5.59)

Za,

E

/b(tF)UII 5

w

t

2

(w

(t;E) 0

7.7.

such t h a t

t h e constant i n (4.20) and (4.26)) and define

=

o

for

t c 0.

There e x i s t constants

E

0'

B > 0

independent of

E

215

PARABOLIC SINGULAR PERTURBATION

-

(7 19)

Proof:

Write 6"(sjE)U

for

u

E

D(h),

of (4.27) and

where

5

0

=

X0 ( S ; E ) U +

is t h e f i r s t term on t h e r i g h t hand side

Xl i s t h e sum of t h e r e s t .

r a t h e r , Theorem 4.7)

(7.20)

Zl(S;E)U

Using Theorem 4.4 ( o r ,

we deduce 2

~ ~ ~ ( s ;5&c(w2 ) ~ -t ~ w2s-'

kt

t > 0, u

Kp(S;E)U-Kp(S

By Theorem

E

D(A).

-

tjE)U

+

s-2)eW

5

C's-2e ps

( s > 0)

.

(7.21)

We have =

4.1 we have

Putting together (7.21) and (7.23) we can estimate t h e integrand of t h e

f i r s t i n t e g r a l i n (7.22) by

Cb-2, thus t h e i n t e g r a l i t s e l f is

bounded by a constant times

1 s - t

- -s1 '

(7.24)

The second i n t e g r a l i n (7.22), a f t e r i n t e g r a t i o n by p a r t s , becomes

(7.25) A look at t h e integrand i n (7.25) makes p l a i n that it can be estimated

by a constant times

216

PARABOLIC SINGULAR PERTURBATION

thus t h e i n t e g r a l c o n t r i b u t e s another serving of (7.24).

Putting

t o g e t h e r a l l estimations and taking advantage of t h e f a c t t h a t

D(A)

is dense i n E we deduce t h a t

-p

c

+

~

2 -t),-( S-t)/2E eW( S-t)/E e

(5

-

,

E

(7 27)

t h e last two summands o r i g i n a t i n g from estimation of t h e boundary terms On t h i s basis, we proceed t o estimate t h e i n t e g r a l

i n (7.25).

(7.28) The i n t e g r a l of t h e f i r s t t e r m i n (7.27) i s computed as i n Lemma 6.3. The i n t e g r a l of t h e second term i n (7.27) i s

To compute t h e i n t e g r a l of t h e last term we make t h e change of variables

s

-

t =

t h e domain of i n t e g r a t i o n i s then

0;

< s +

(P

- 4

- b))1/2 - w ) I E 2 ( p -id2i-

A/

1x1

V/ 11-11

=

-

+ l)1’2(P/lUl)1’2)

(Re((U

2

UlAl-1’2}\U

0 < Re 1-1

i s t h e unique multiple of

h

- kdl

by

i s bounded away from zero i n t h e s t r i p t l y small, where

(p

(7.32) 1/2

]A\-’, setting u = E A we see t h a t it i s enough t o show t h a t

Multiplying numerator and denominator noting t h a t

217

(7.33)

+ 111’2 2 with

1.

sufficien-

E~

111 = P.

on t h e l i n e

p

and

We check e a s i l y t h a t (7.33) never vanishes, thus we only have t o show t h a t it i s bounded away from zero for 1 ~ +1 m. Note t h a t , f o r 1U1 = r a t t a i n s i t s minimum a t Re((u + 1)1’2(~/1~])1’2)

u

=

+ir, thus

(7.34) On t h e o t h e r

I

EOlUl -

1x1

hand,

q u + 111’2,

>

lhl-1/21U

so t h a t

Ei21~1

thus our claim holds f o r

Proof of Theorem 7.6.

T h a t the kernel

independent of

< 1, 1/2w.

K (t;E) P

satisfies

(a)

in

was shown i n Lemma 7.7, while (6.3), with B

E

likewise independent of

was t h e s u b j e c t of Lemma 7.8.

E

t h e operator

-Lt

f(;) i s bounded i n

-
_ 0 .

The function w(;;E)

= u(;jE)

- vo(G;E)

i s a s o l u t i o n of t h e i n i t i a l value problem ~

+

E2w"(tjE)

~ ( 0 ; s )=

1

w = w

U (E)

0

-

w'(t;E) = A w ( t ; E )

+ vo,

2 e-t/E Avo,

~ ' ( 0 j E ) = Ill(&)

- E -2

(8.10) To.

i s t h e s o l u t i o n of t h e homogeneous equation with t h e assigned i n i t i a l conditions and w2 is t h e s o l u t i o n

Write

-k

w2

where

w1

of the inhomogeneous equation with zero i n i t i a l conditions and 2 f ( t ; E ) = -e-t/E

We apply t o w1 while

w2

(8.W

AV0 '

Theorem 5.5 (with t h e simplified estimate (5.30)),

i s handled by means of Theorem 7.2 ( s p e c i f i c a l l y , t h e f i r s t

i n e q u a l i t y (7.4)). The final estimate i s

with t h e obvious modification i n t h e last term i f

w = 0.

This ends

t h e proof. For

N = 1 an a d d i t i o n a l c o r r e c t o r must be used, namely

2 vl(tjE) = -e -t/@

THEOREM 8.2.

1'

Assume t h a t ( 8 . 3 ) holds f o r

N = 1, t h a t i s

221

PARABOLIC SINGULAR PERTURBATION

u

0

u

=

(E)

= E - 2v 0

1 u , u 1, vo, v1

-~ and t h a t

!I

U(tjE) = u ( t ) +

D(A

U

2

0

t

u n i f o r m l y on compacts o f

O(E ),

f

+ d v l i-

o(1)

(8-131

1. Then

V0(tj&) f

0

2

t Eul

(E)

E(ul(t) +

(resp.

uo(i)

where

0,

+ O(E 2

Vl(tjE))

1

(8.14)

&

u,(t))

the s o l u t i o n o f (8.2) w i t h

u0 ( 0 ) = uo If -

i.

vo

(resp.

w = 0 , (8.14) holds uniformly i n

u,(o)

= u1 +

(8.15)

t >_ 0 .

We c o n s i d e r t h i s t i m e t h e f u n c t i o n

Proof.

w(^tjE) = U(ntjE)

-

-

vo(i;E)

= Aw(tjE)

E2w"(t;E) i w ' ( t ; E ) =

U

As i n Theorem

0

(E)

+

vo

C EV1,

2

-

W'(0,E) =

(8.16)

Evl(t;E)

t h a t s o l v e s t h e i n i t i a l v a l u e problem

w(O,E)

q.

e-t/E

U (E)

1

-

Av

0

-

2 @e-t/E Avl,

E -2V

(8.17)

-1

- E

y

1'

0

8.1, we write w as t h e sum o f a s o l u t i o n w1

(8.18)

of t h e

homogeneous e q u a t i o n t a k i n g t h e a s s i g n e d i n i t i a l c o n d i t i o n s and a solution

w2

of t h e nonhomogeneous e q u a t i o n w i t h z e r o i n i t i a l c o n d i t i o n s .

We a p p l y a g a i n t h e s e c o n d i n e q u a l i t y (7.4) t o

and

w2

(5.30) t o

W1,

obtaining

Obviously, a d i f f e r e n t t a c k must b e a d o p t e d f o r N >_ 2 , s i n c e t h e f i r s t term o n t h e r i g h t hand s i d e s of ( 8 . 1 2 ) and (8.19) c a n n o t b e squeezed smaller t h a n

level.

O(E2).

We p r o c e e d at first o n a p u r e l y f o r m a l

The a p p r o x i m a t i n g h n c t i o n w i l l b e of t h e form

u

N

(tjE)

= u (t) + EUl(t)

0

N

f

*-.

-t E UJt)

,

(8.20)

222

PAMBOLIC S I N G L U R PERTURBATION

a r e defined a s before and t h e where u0 (t), u1(^t) s a t i s f y t h e d i f f e r e n t i a l equations un' ( t ) = Aun(t)

-

~:-~(t)

u

( t >_ 0 )

(t), n 2 2, .

(8.21)

Noting t h a t t h e c o r r e c t o r s 2 vo, v1

used i n t h e cases N = 0,l a r e of 2 t h e form v O ( t j E ) = v (t/E ), vl(t;E) = v (t/E ), we s h a l l use a 0 1 combination of c o r r e c t o r s of t h e form II ( t ; E ) = v (t/E N 0

The

v

n'

n >_ 2

2

)

+

tVl(t/E

2

+

) +

N 2 VN(t/E ).

E

(8.22)

w i l l s a t i s f y t h e d i f f e r e n t i a l equations vn" ( t )

+

vA(t) = Avn-,(t)

(t

2

0)

,

(8.23)

and t h e decay condition vn(t)

-, o

as

t

4

m

and

(8.24)

n = 1,2

Note that t h e equation s a t i s f i e d by , ) : ( u U'n (

. is

(8.25)

t ) = Aun(t),

vn, n = 1 , 2 s a t i s f i e s v p )

+ vA(t)

= 0

.

(8.26)

Consider now t h e f'unction

N

I1=0 p )

= ( E 2U

N

+ &-2

N

c &"Vi(t/E2) + E-2 c E"V;l(t/E2)

+ E-2

17;O

+ up)) +

E ( E 2 u;l(t)

c E"(yll(t/&2) t VA(t/E2)) -2

+ up))

223

PARABOLIC SINGULAR PERTURBATION

+

N-2

c

EnAvn(t/E2)

n=0 N

=

c

EnA(un(t)

+

2

vn(t/E ) )

ti=O

The i n i t i a l conditions on u

Il’

u0 ( 0 ) = uo

-

E N - 1 AvNml(t/E2)

n = 0,l

+ vo,

ENAvN(t/E 2 ).

(8.28)

a r e those i n Theorem 8.1: = u1

u,(o)

+ v1

.

(8.29)

On t h e other hand, t h e i n i t i a l conditions on v n = 0 , 1 must be n’ 2 2 those t h a t insure t h a t v (t;E) = v (t/E ), v (t;E) = v,(t/E ), 0 0 1

v0 ( t ; E ) , v1(t;E)

where

a r e t h e correctors used i n Theorem 8.2.

Accordingly, $0)

= vo, vi(0) = vl,

hence, taking (8.26) and (8.24) i n t o account,

v ( t ) = -e 0

For

n

2

2,

-t vo, v,(t)

t h e i n i t i a l conditions f o r

= -e

un(i)

-t

v

1’

and

vn(t)

are,

respectively un(o) = un

thus for

un(E) tU(t;E)

-

must be constructed a f t e r

(8.32)

vn(o)

vn(t).

The i n i t i a l conditions

are obtained from (8.29) (8.30) ( 8 . 3 2 ) and (8.33):

PARABOLIC SINGULAR PERTUBBATION

224 lo ( 0 ; E ) = N

cN Enun(O) + cN Envn(o) = =O

Il=O

= u

0

- v

+

v

-

0

-

-

vn(o>>

G 2

+

EV

cN Envn(o) = cN n=2

0

=

c

+ N E n(un

“(9 + vl)

f

E

nun

(8.34)

=O

N

N

n=0

n=0

c Enu’n( 0 ) 4- c Enm2vn

N-2

c

E=O

EnU’(0) = n

c

n - 2 v n + E N-1 U&l(O)

E

I1;O

N

f

E

up)

(8.35)

Hence, i n view of (8.3), IlU(0F)

- mN(o;E)I/

= O(EPst1)

-

=

(8.36)

and IlU’(0;E)

lo$OjE)I/

O(EN-l)

.

(8.37)

We face now t h e problem of making a l l t h e s e computations valid. Roughly speaking, t h i s amounts t o :

(a) showing t h a t every d e r i v a t i v e w r i t t e n ( a s i n (8.171, (8.21), (8.24), e t c . ) a c t u a l l y e x i s t s . (b)

etc.),

showing t h a t every time we w r i t e

(as i n (8.17), (8.21),

Au

u a c t u a l l y belongs t o t h e domain of

A.

This w i l l be done by r e q u i r i n g “smoothness” conditions of varying degree on t h e c o e f f i c i e n t s

un’ vn

u o ( t > = S(t>(U0 + v&

i n (8.3). u,W

We begin with

= S(t)(U1+

a r e made e x p l i c i t i n (8.31).

“J

(8.38)

while

vo(t), vl(t)

v,(t),

v3(t) we solve (8.23) with t h e i n i t i a l condition (8.33) at and t h e decay condition (8.24) as t -. m:

t = 0

To construct

225

PARABOLIC S TNGULAR PERTURBATION

v;(o)

v2

=

-

v"(t) 3 v ~ ( o= ) v3

3

v2(t) -,o

u$o), f

-

vl(t) 3

= -e -tA v ~

-, o

u~(o), v 3 ( t )

t

as

=,

4

( t 2 0),

-

t

as

(8.40) m

.

Solving e x p l i c i t l y these e q u a t i o n s , v2(t)

= te

-t Avo

-

-t v ( t ) = t e Av

3

-t

(v2

- e -t (v3

1

- Au0 -

2Avo)

9

(8.41)

- AU1

avo)

9

(8.42)

-

A(uo + v,), U i ( 0 ) = A(U1 + vl). u s i n g t h e equation (8.21) and t h e

where w e have used t h e fact t h a t We compute next

e

u 2 ( t ) , u3(t)

U'

0

(0) =

i n i t i a l c o n d i t i o n (8.32) :

u;(t)

2

AU2(t) - S(t)A (u0 + v0)

=

( t >_ 0 ) , (8.43)

u2(o) =

?(t)

Au3(t)

=

-

U2

+ To

7

2

( t 2 0),

S(t)A (ul + vl)

(8.44) u ( 0 ) = u + v1 3 3

9

2 ) where we have used t h e f a c t s t h a t u " ( t ) = S(t)A (uo t v,), ~ " ( t = 0 1 2 (see (8.29) and (8.37)). = S(t)A (ul -t v,), v;)(O) = vo, v i ( 0 ) = v1

Hence

U,(t)

=

S(t)(U2 + v0)

=

S(t)(U2

f

v0)

-

u ( t ) = S(t)(u3

3

With

S(t

Lt

-

s)S(S)A

2

(u0 + v0) dS

tS(t)A 2 (u0 + v0) ,

-k

2

vl) - tS(t)A (ul

(8.45) -k

v~).

(8.46)

up(%),u 3 ( t ) ,

see that

v4(i),

y2(i), v3(t) already manufactured, we can e a s i l y v 5 ( t ) w i l l have t h e form v4(t)

=

e-tP4(t), v5 ( t )

=

e-tP5(t)

,

(8.47)

PARABOLIC SINGULAR PERTURBATION

226

where

is a polynomial of degree 2 whose c o e f f i c i e n t s a r e l i n e a r

P4(%)

combinations of

AJu

0’

AJvo ( j 5 j), Au2

and

Av2

uo, vo 7u2yv2 replaced by On t h e other hand, we have

t h e same polynomial with respectively.

U4(t)

- P4(0))

2tS(t)A3(uo

u (t) = S(t)(u

5

+

u

(i)

(resp. u5($))

-

vo)

- P5(0))

5

42

is

2

-2tS(t)A3(ul + vl) thus

P (t)

+ tS(t)A (u2 + v0) -

= S(t)(~4

-

and

5 u1,v1,u3,v3

t2 S(t)A4 (uo + v,),

+ tS(t)A2 (u3 i- vl)

(8.48)

-

- t2 S(t)A4(ul + v,),

(8.49)

can be constructed i f uo, vo E D(A 4 ), 4 2 D(A ), u3 E D(A ). However, i f we wish (8.47)

u E D(A ) (resp. ul,vl E 2 t o be a genuine solutions of (8.23) we a c t u a l l y need that

vo E D(A 5 ) and u E D(A 3 ), u4, P4(0) E D ( A ) ; i n view of our 0’ 2 previous comments about P4, it is s u f f i c i e n t f o r t h i s t h a t 4 3 2 uo, vo E D(A ), u2 E D(A ) v2 E D(A ) and u,, E D(A). Likewise, i f u

we wish (8.48) t o be a genuine s o l u t i o n of (8.21) we must a s k t h a t

3

2

E D(A5), u3 E D(A ) v3 E D(A ), u5 E D(A). It w i l l be of ul’ “1 i n t e r e s t l a t e r t o a s c e r t a i n t h a t u 4 ( t ) , u ( t ) a r e twice continuously 5 6 d i f f e r e n t i a b l e . This w i l l be t h e case i f u0’ V0Y U1’ v1 E D(A 1, u2, u3 E D(A 4), v2, v3 E D(A 3) and u4”-15 E D(A). From t h e s e observations we surmise t h e following r u l e s , v a l i d f o r

arbitrary m

2

1. I n t h e f i r s t place, we have

v,(t> where

(resp. Pml(%))

P,(t)

a r e l i n e a r combinations of (j

5

2m.-

A L ~A , J

3),

= e -tP*,(t)

= e-tP&),v;w,(t)

-

(8.50)

i s a polynomial whose c o e f f i c i e n t s

AJu ,AJ, 0

... Ajua-4,A3uh-4.(j

(j ~ 5~an

,

0

(j Y ~-2’~-ZjYvN-2’vN-3 u ~ , u ~ , vE ~ D(A , ~ ~).

THEOm8.3. Odd’

-4-

D(A )

3

N

,...,

N

N

n=O

n=O

(8.55) uniformly on compacts of

t 2

o

(uniformly i n

t

o

if

U)

= 0).

g

PARABOLIC SINGULAR PERTURBATION

228

i s even t h e same r e s u l t obtains under t h e assumption t h a t 2 Nt2

N>_ 2

uN>vN E D(A

1,

Proof. N = 2m

+

%-17%-29vN-19v~q-2

f

We consider f i r s t t h e case

1 and apply r u l e ( a ) .

N

odd

Avo(t),

...,AvN(t)

>_ 3;

we s e t here

Since conditions (8.51) a r e s a t i s f i e d

...

(with something t o spare) we deduce t h a t with

1.

D ( A ~ ) , . * * > U ~ > U O0, V E ~D(A >V

continuous.

vo(t), , v N ( t ) E D(A) Taking (8.50) i n t o account we

deduce t h a t

(8.56) This w i l l be used t o estimate the l a s t two terms on t h e r i g h t hand s i d e of (8.28):

f o r t h e f i r s t two terms we simply use t h e f a c t ,

u~-~(t)and

assured by (b), t h a t differentiable.

%(t)

a r e twice continuously

Using t h e f i r s t inequality (7.10) i n (8.28) we

obtain

where, i n v i e w of (8.56), the contribution of t h e l a s t two terms i s O(EW1),

This ends the proof.

The case

N

even >_ 2

i s handled

much i n t h e same way and we omit t h e d e t a i l s .

@?I. 9 E l l i p t i c d i f f e r e n t i a l equations. We apply the theory i n t h e lust eight sections t o t h e d i f f e r e n t i a l operator

m

m

i n a bounded domain R

.

m

of m-dimensional space w i t h boundary

T; here

229

PARABOLIC SINGULAR PERTURBATION

A(p)

d e n o t e s t h e r e s t r i c t i o n of

o b t a i n e d by means of t h e D i r i c h l e t

A

boundary c o n d i t i o n

o

=

U(X)

r),

(X E

(9.2)

or b y means of t h e v a r i a t i o n a l boundary c o n d i t i o n N

D ~ ~ ( X= )

The c o n s t r u c t i o n of

(x

y(x)u(x)

E

r).

(9.3)

w a s c a r r i e d o u t i n Chapter IV i n c o n s i d e r a b l e

A(@)

9IV.3 ( f o r t h e D i r i c h l e t boundary c o n d i t i o n ) and i n SN.6 (for t h e boundary c o n d i t i o n ( 9 . 3 ) ) t h a t A ( B )

d e t a i l ; i n p a r t i c u l a r , it w a s shown i n

g e n e r a t e s a s t r o n g l y continuous c o s i n e f u n c t i o n , t h u s a l l t h e r e s u l t s i n t h i s chapter apply automatically.

SVI. and u 0

5 , for 5.13.

i n s t a n c e t h e “ i n t e r m e d i a t e ” e s t i m a t i o n s i n Theorems 5.12 Combining Theorem

D ( ( b 2 1 - A(@))‘)

E

u n i f o r m l y on compacts of D((b21

-

5.u w i t h Lemma 5.15 w e deduce t h a t

-

u ( t ) I I = 0(E2’)

t > - 0.

p

Hl(n).

E

-o

(9.4)

The most i n t e r e s t i n g c a s e i s

D((b21 - A ( B ) ) if

as

can b e identified.

A(f3))‘)

if

l a r g e enough) t h e n

(b

jlu(t;E)

where

Of s p e c i a l i n t e r e s t a r e t h o s e i n

c1 =

1/2,

I n f a c t , w e s h a l l show t h a t

(9.5)

= Hi(Cl)

i s t h e D i r i c h l e t boundary c o n d i t i o n and

bl,

...,bm

belong t o

To show ( 9 . 5 ) we n o t e t h a t it h a s a l r e a d y b e e n proved t h a t

D ( ( b 2 1 - AO(@))1/2) especially

=

(see ( N . 2 . k ) ) a n d r e c a l l Theorem IV.2.2, HO(R) 1

(IV.2.6)). Thus, D((b21

We s k e t c h t h e p r o o f of

w e o n l y h a v e t o shod t h a t

- Ao(p))lb2) (9.6).

cosine function generated by used t o c o n s t r u c t

Let

- A(p))”I2)

.

(9.6)

C ( t ) = c o s h t Ao(@)1’2

Ao(p).

b e the 0 It f o l l o w s from t h e p e r t u r b a t i o n

e0(t) (or

6(t) from

cosine function generated by

= D((b21

A(p)lb

d i r e c t l y ) that

C(t),

the

c a n b e e x p r e s s e d b y means of t h e

perturbation series C ( t ) U = C0(t)U where domain

+

gTJF*Co(t)u

+ qTJF*qTJF*Co(t)u +

d e n o t e s t h e ( o n l y ) bounded e x t e n s i o n o f

O1 H (Q))

t o a l l of

* * *

,

So(t)P

(9.7) (with

L2( Q ) ; t h a t t h i s e x t e n s i o n e x i s t s follows

230

PARABOLIC SINGULAR PER’IURBATION

S ( t ) P i s bounded ( i n t h e norm o f 0

from t h e f a c t t h a t

1

L2(n))

in

~ ~ ( n )s,i n c e s o ( t ) P = (sinh t Ao(B)1/2)Ac(B)-1/2P, and

Using (1.5) and t h e “ r e c i p r o c a l ” series Co(t)u = we show t h a t

@(t)U

-

qqF*C(t)u

+

f40P*rn*C(t)u +

(9.8)

@ ( t ) u i s continuou.;ly d i f f e r e n t i a b l e i f and o n l y i f (9.6) follows f r o m Theorem

@,(ti) i s c o n t i n u o u s l y d i f f e r e n t i i b l e , t h u s

111.6.4. However, i n t h e p r e s e n t s i t u i t i o n , estimates on rates o f convergence l i k e (1.4) c a n b e o b t a i n e d under weaker assumptions b y more e l e m e n t a r y methods.

We s k e t c h below t h i s theeory i n a s u i t a b l y ” a b s t r a c t ” v e r s i o n .

Let

E = H

be a H i l b e r t s p a - e and

A.

a s e l f adjoint operator

such t h a t

with

K

>

0.

We c o n s i d e r t h e o p e r a t o r A = A.

where

P

+ P,

(9.10)

i s such that

m-l

(9.11)

i s bounded, where B = ( - A )1/2 d e f i n e d as i n srV.3. Using essentially 0 t h e same methods i n srV.4 we show t h a t A g e n e r a t e s a s t r o n g l y

c o n t i n u o u s c o s i n e f u n c t i o n , t h u s ill r e s u l t s i n t h i s c h a p t e r apply, i n particular those i n

sVI.5. We e x p l o i t t h e s e below.

Using t h e f u n c t i o n a l c a l c u l u s f o r s e l f a d j o i n t o p e r a t o r s we can d e f i n e f r a c t i o n a l powers

( -Ac)‘

of

-Ao

where

E = C1, A u = ku w i t h

A

E

= H

i s a H i l b e r t space and

It f o l l o w s from Fxercise 11.5 -that A

E

A

8(C,w)

g e n e r a t e s a s t r o n g l y continuous c o s i n e f u n c t i o n

3.1,

is a

(that

C(t)

satisfying

Ilc(t)ll 5 i f and only i f

cr(A),

C e U J l tI

t h e spectrum of

(-m

A,

-

W(E),

i s a s o l u t i o n of t h e e q u a t i o n a m - -

- w(E)2

-

E

5 (4a)-1/2

b2 , 4 4E)2

4E2 if

a ( & )= a

The h a l f - s t r i p corresponding t o

w

where

(5.3).

we have

and t h e f i r s t e s t i m a t e (3.8) i s v e r i f i e d , s a y for

i s a c t u a l l y s a t i s f i e d f o r all

(5.6) away from z e r o ) . S(t;A

E

2

0,

0

_
l l sc b > E a ( i + I t l ) a / 2 e " l t l ~ ~ ( - ~ o ) 1 ~ u o ~ ~ (5.38)

258

OTHER PROELEMS

The proof c o n s i s t s i n applying Theorem t h e proof of Theorem

u0

4.2).

u0 (E)

D((-AO)lw),

E

E

llul(E)

-

iAuOII

~ ' ( t )( s e e

u'(t;E),

A s a consequence, we o b t a i n e a s i l y t h a t , i f D(A)

then

- u'(t)ll

IlUYtjE)

uniformly on compacts of

5.5 t o

0

is linear i n

i s a parameter t o be f i x e d below. v,

conjugate l i n e a r i n

u

3'

and w e

This w i l l be achieved by

s l i g h t m o d i f i c a t i o n s of t h e arguments i n Chapter I V

E

satisfies b

(6.6) t h a t we o u t l i n e

below, beginning with t h e D i r i c h l e t boundary c o n d i t i o n . u,v

and t h a t

B ,

bounded.

(6.3)

C(l)

For

Obviously,

[u,v],

and we check e a s i l y t h a t

260

OTHER PROBLEMS

[v,u],

Using t h e uniform e l l i p t i c i t y assumption ( 6 . 4 ) , t h e

= [u,v&.

5

IDJ,),[

inequality

+

( ~ / 2 ID%[* )

and i t s counterpart

(1/2E) IvI2

[ Z J v I we e a s i l y show t h a t i f

for

i s l a r g e enough, t h e f i r s t

CY

ineq u a l i t y 2 c (u,.)

5

c > 0,

holds for some

5c

[U,Ul,

where

2

(u

(u,u>

#(W

E

(6.8)

i s t h e o r i g i n a l s c a l a r product of

(u,v)

t h e second i n e q u a l i t y ( 6 . P ) i s a consequence of t h e assumptions

#(O);

on t h e c o e f f i c i e n t s .

We s h a l l from now on assume

AO(R)

The operator

((@I

-

=

[u,wl,

(w

c o n s i s t i n g of all u

E

$w,

$(n)

E

right-hand s i d e of (6.9) continuous i n t h e norm of of t h e theory of (u,v),

has:

(6.9)

which make t h e

L2(Q).

The r e s t

unfolds e x a c t l y as i n Chapter I V , s i n c e it i s

Ao(B)

[u,v],

based on t h e p r o p e r t i e s of t h e s c a l a r product same

[U,U~/‘.

i s defined by

AO(B))U,W)

A (5) 0

the domain of

endowed with

$(Q)

(lullCY=

t h e s c a l a r product (6.7) and i t s associated norm

which a r e t h e

Ao(B)

we check i n t h e same way t h a t

i s symmetric

and densely defined, t h a t i t s c o n s t r u c t i o n does not depend on

(11 - AO(B))D(AO(B))

-

h

E

A1

-

if

p(AO(D))

A?

3 a> ,

h 2 a.

Ao( fi) i s one-to-one f o r byproduct of (6.8) t h a t ( A 1 A0(3))-’ and t h a t

(A

E

=

that

(6.10)

We a l s o o b t a i n as a

i s bounded, so t h a t

This i s known t o imply t h a t

(Y.

Q‘,

is

Ao(i3)

s e l f ad j o i n t ( see Lemma IV.l.1). The f u l l operator Bu =

2

( b .Dju J

The assumptions on t h e operator.

We define

A(f3)

b

i s constructed by p e r t u r b a t i o n .

+ cu

D’(bju))

3

and on

A(5)

=

c

=

imply t h a t

no(fi)

and i t follows from Theorem 5.1 t h a t

- 2 7 (Djbj)u

for

u,v

E

$(O).

cu

.

A(@)

(6.11)

i s a bounded

(6.12)

+ B

The case where t h e boundary c o n d i t i o n a g a i n a s i n Chapter I V .

B

+

Let

s a t i s f i e s Assumption 3.1.

(6.3) i s used

i s handled

This time, however, t h e f u n c t i o n d i s

The d e f i n i t i o n of t h e operator

A0(6)

is

261

OTHER PROBLEMS

-

((@I The o p e r a t o r

A ~ ( Q ) ) ~ , W ) = [U,~I:,

(6.11).

B

t h e f u l l operator

3.1.

A(B)

satisfies

Summarizing :

Let R

THEORFM 6.2.

be a domain

& Rm,

A

(6.1) with

t h e operator

dJrn(C2),

b.

a

8)

A(

i s t h e bounded o p e r a t o r defined

It f o l l o w s a g a i n from Theorem 5.1 t h a t

Assumption

(6.14)

$,(Q)).

E

Ao( 0) i s a g a i n s e l f a d j o i n t :

i s obtained by formula (6.12), where by

(w

c E Lm(Q), E Assume, moreover t h a t t h e a are real jk’ J jk and s a t i s f y t h e uniform e l l i p t i c i t y assumption (6.4) and t h a t t h e b j a r e p u r e l y imaginary. If 8 i s t h e D i r i c h l e t boundary c o n d i t i o n (6.2)

I

A ( B ) defined by (6.7) and (6.9) s a t i s f i e s Assumption R i s bounded and of c l a s s and R i s t h e boundary measurable and bounded i n r t h e n t h e c o n d i t i o n (6.3) w i t h o p e r a t o r A ( 8 ) defined by (6.13) (6.9) s a t i s f i e s Assumption 3.1.

t h e operator

7’)

If

3.1.

6.3. Theorem 5.5 h a s an i n t e r e s t i n g a p p l i c a t i o n i s not e a s i l y i d e n t i f i a b l e even f o r D( ( -Ao( B))cy)

REMARK Although

here. @

=

1

under t h e p r e s e n t smoothness assumptions, we have show i n Theorem I v . 2 . 2 and Theorem

Iv.5.l

that

D((-Ao(8))1’2) when

8 is

=

5.5

8

E

HbQ)

%(a)

(6.3).

Using Theorem

E)

-

u(t)ll

(6.17)

= O ( E1/2)

and

Ilu0(d

-

uoII =

o(E1/2),

IlU,(E)II

u

E

$(a).

0 c o n d i t i o n (5.24) holds.

(6.18)

= O(E - 3 F ) .

The same r e s u l t h o l d s for boundary c o n d i t i o n s we assume t h a t

(6.16)

B i s t h e D i r i c h l e t boundary c o n d i t i o n ,

we deduce t h a t i f

uo

=

i s t h e v a r i a t i o n a l boundary c o n d i t i o n

Ilu(t; if

(6.15)

t h e D i r i c h l e t boundary c o n d i t i o n ( 6 . 2 ) , and

D((-*o(B))1/2) when

$(a)

B of type (6.3) where

However, we c a n only guarantee

(6.17)

This i s e a s i l y seen t o be t h e case i f

r Djbj,

c

E

$’“(n).

if

262

OTHER PROBLEMS

Sv11.7

The inhomogeneous e q u a t i o n .

A s pointed o u t i n SVII.2, t h e e x p l i c i t ( g e n e r a l i z e d ) s o l u t i o n of t h e i n i t i a l value problem ( 2 . 1 ) with n u l l i n i t i a l c o n d i t i o n s

is

UJE)

U0(E),

=k t

-

Gi(t

u(t;E)

s;&)f(s;&)ds.

We have a l r e a d y noted ( i n Example 4.6) t h a t s t r o n g l y convergent a s

(7.1) of

E

+

0.

(7-1) i s n o t even

ei(t;E)

However (and somewhat s u r p r i s i n g l y )

t u r n s out t o t r a n s l a t e convergence of

i n t o convergence

f(t;E)

a t l e a s t f o r a c l a s s of o p e r a t o r s c o n t a i n i n g t h e d i f f e r e n t i a l

u(t;E)

o p e r a t o r s i n Sv11.6.

THEORFM

7.1. Let

E = H

be a H i l b e r t space,

(7.2)

A = A0 + E , where

A.

operator.

L ~ -T,T;H) (

i s a s e l f a d j o i n t o p e r a t o r bounded above, Let

such t h a t f(s;E)

in

o

T > 0, k ( s ; E);

1

L (-T,T;H).

-

E

f(s)

Finally, l e t

5

as

u(t;E)

E

F

~

-

)a

a bounded

B

f a m i l y of f u n c t i o n s i n

(7.1)

0

be t h e (weak) s o l u t i o n of t h e

i n i t i a l value problem 2

EU"(t;E)

-

iU'(t;E)

+ f(t;E)

= AU(t;E)

( I t ]5 T ) ,

-

(7 4) U(0;E)

uniformly i n

It I 5 T,

= 0, l ~ ' ( 0 ; E ) = 0 .

u(t;

where

E)

i s t h e weak s o l u t i o n of

u ' ( t ) = i A u ( t ) + i f ( t ) (It1 5 T)

, (7.6)

u(0) = 0 .

Proof:

We c a n obviously assume t h a t

incorporate i n t o

B

t h e " p a r t of

s h a l l show f i r s t Theorem 7.1 f o r E,

c o n s i d e r i n g f i r s t t h e case

P(dp)

A.

a(Ao)

5 ( 0 , ~ )( i f

with spectrum i n

IJ.

2

not we 0").

We

and t h e n mix t h e p e r t u r b a t i o n A. f ( t ; E ) = f ( t ) independent of E . Let

be t h e r e s o l u t i o n of t h e i d e n t i t y f o r

A.

and

Ei(t;E;Ao)

the

263

OTHER PROBLEM2

(7.3) w i t h E = 0 . The same c o n s i d e r a t i o n s l e a d i n g t o Examples 4.5 and 4.6 show t h a t

(second) propagator of t h e e q u a t i o n

PO

for

u

are

t h e r o o t s o f t h e c h a r a c t e r i s t i c polynomid

Let

0

E

5

E,

t

where

5

T.

We can w r i t e

t

u(t;E)=

[ ei(t - s ; € ; A 0 ) f ( s )

c, d s = [:P(dp)L[

e(t -

s ; & ; p ) f ( s ) ds

(7.10)

" 0

a f t e r an e a s i l y j u s t i f i e d interchange i n t h e order of i n t e g r a t i o n . note next t h a t

hence

On t h e o t h e r hand,

Since

we deduce t h a t , f o r

p

fixed,

12(t;p;

E)

-

ieipt -/ote'Psf(s)

ds

W e

264

OTHER PROBLEMS

uniformly i n

0

5 t 5 T.

To handle t h e f i r s t i n t e g r a l we note t h a t

-

+

h (p;~)

as

i m

-o

E

and use t h e following uniform v e r s i o n of t h e Riemann-Lebesgue lemma: if -

g(t)

i s a ( s c a l a r or v e c t o r - v a l u e d ) f u n c t i o n i n

lim

J'''eiosg(s)

the

L1

0

5

t

5 T;

t h e proof i s achieved approximating

g

in

Applying (7.15) t o t h e f i r s t

norm by smooth f u n c t i o n s .

integral i n

(7.15)

ds = 0

0

a-'m

uniformly i n

then

L1( 0 , T )

(7.13) we o b t a i n T(t;p;E)

uniformly i n

0

5

Assume t h a t

t

as

0

F

-

(7.16)

0

5 T.

u(t;&)f. u ( t )

itn],

e x i s t s a sequence

-+

0

uniformly i n

5 tn 5

T

0

5t 5

and a sequence

T. {En],

Then t h e r e E~

-+

0

such t h a t

llu(tn;tn) Making use of

(7.14), (7.16)

-

Il(t,)II

of t h e range

-T

0

5t 5

5t 5

0.

(7.17)

6 > 0.

and a v a r i a n t of t h e dominated convergence

theorem we o b t a i n a c o n t r a d i c t i o n with h o l d s uniformly i n

2

T.

(7.17).

This shows t h a t

(7.5)

An e n t i r e l y s i m i l a r argument t a k e s c a r e

The case where

f

depends on

E

i s handled

writing

+ktGi(t

- S;E;A)f(S)

ds

(7.1.8)

and making use of t h e uniform bound (7.11). We i n c o r p o r a t e f i n a l l y t h e p e r t u r b a t i o n

B.

It results from (3.7)

and from t h e p e r t u r b a t i o n formula (5.20) ( o r d i r e c t l y ) t h a t we have

+ 6. ( t * & ' A ) x BGi(t;&;A0)u 1 " O

265

OTHER PROBLEMS

hence U(t

;E)

ei(t;€ ; A O )

=

Y

+ S ( t ; &;Ao)

f (t ;&)

+ ei(t;&;Ao) * EEi(t;E;A0) Now, using

Y

E e i ( t ;€ ; A O ) x f ( t ; E)

*Eei(t;&;AO)

...

*f(t;&)+

(7.20)

(3.1) we show t h a t t h e n-th term of t h e s e r i e s ( 7 . 2 0 ) i s

bounded i n norm by

On t h e o t h e r hand, using r e p e a t e d l y t h e p r e v i o u s l y proved r e s u l t on convergence of that

e.(t;E-A

3 0

1

) * f ( t ; & ) i n each term of (7.20) we deduce

ei(t;&;Ao)*E6.(t;&;AO)*f(t;E), 1

E6i(t;E;Ao)*f(t;c),...

qt;E;Ao)

*Bei(t;&;Ao)x

all converge uniformly i n

It/

5

T;

the

l i m i t of t h e n-th term of ( 7 . 2 0 ) i s

. .. x E i S ( t ^ ; i A o ) * f ( t ) S(i;iAo) * iES(i;Ao) . .. * i B S ( f ; A o ) * i f ( ; ) * BiS(t";iAo) x

iS(;;iAo) =

t h u s t h e sum of t h e s e r i e s converges uniformly, as S(_ 0 .

f o r any s o l u t i o n of (1.1) I f the function

t >_ 0

(1.3)

C(t)(llu(O)lI + l l U ~ ( 0 ) l l )

C(t)

in

(1.3) can be chosen nondecreasing i n

( o r , more g e n e r a l l y , bounded on compacts of

t

0)

then we

say t h a t t h e Cauchy problem f o r (1.1) i s uniformly w e l l posed ( o r

t >_ 0 .

uniformly properly posed) i n

The propagators o r s o l u t i o n operators of (1.1)a r e defined by

u(2)

where

(resp.

u ( 0 ) = u, u ' ( 0 ) = 0

C ( t ) (resp. of

D

Since both C(t)

and

0

a(t)

v(%))

i s t h e s o l u t i o n of (1.1)with v(0) = 0, v ' ( 0 ) = u).

(resp.

The d e f i n i t i o n of

s(t)) makes and

sense f o r u E D ( r e s p . f o r u E D1). 0 Dl a r e dense i n E we can extend (using (1.3))

t o all of

E

a s bounded operators; t h e s e operator-

valued functions r e s u l t s t r o n g l y continuous i n Iic(t)li

5 C ( t > , ils(t>iI 5 C ( t >

Moreover, by d e f i n i t i o n ,

C ( 0 ) = I,

S ( 0 ) = 0.

t >_ 0

U(t) =

c(t)u(o) +

and s a t i s f y

( t >_ 0 ) .

(1.5)

F i n a l l y , we prove e a s i l y

u(%) i s a n a r b i t r a r y s o l u t i o n of (1.1)i n

t h a t if

(1-4)

S ( t ) u = v(t),

@ ( t ) u =u ( t ) ,

t 2 0

then

S(t)Ul(O).

(1.6)

The proof i s t h e same a s t h a t of (11.1.6). We s h a U assume from now on that t h e operators

A

and

B

are

closed. $vIII.2

Growth of s o l u t i o n s and existence of phase spaces.

The d e f i n i t i o n of phase space i s , except f o r small modifications, t h e same i n $111.1. A phase space i n

t

0

f o r t h e equation (1.1)

equipped with any of i t s product n o r m , @ = Eo x E 1' El a r e Banach spaces s a t i s f y i n g t h e following assumptions:

i s a product space where

E 0 (a)

(-.

D1

dense i n (b)

and

E ,E

6 E with bounded inclusion; moreover, 1Do Eo El) is dense i n E i n t h e topology of Eo ( r e s p . i s

0

n

0

El i n t h e topology of

El).

There e x i s t s a s t r o n g l y continuous semigroup G ( t )

272

THE COMPLETE EQUATION

E = E

in

0

X

t 2

E

o

1

such that

with

for any s o l u t i o n u ( i )

u(0)

E

E ~ u , l(0)

E

E ~ .

The comments a f t e r t h e d e f i n i t i o n of phase space i n $111.1apply here:

we omit the d e t a i l s . We examine i n the rest of t h e s e c t i o n t h e r e l a t i o n of t h i s notion

w i t h t h a t of w e l l posed Cauchy problem i n the case where

E = Q2 is

t h e set of a l l sequences with

2 ~ ~ { u n= ] ~ Iu ~n

c

l2

u = [un jn >_ 1)= {un] of complex numbers and A , B a r e the operators c

ACunI = lanun),

B{ un] = Fbnun],

(2.2)

rb ) sequences of complex numbers t o be determined l a t e r : n t h e domain of A c o n s i s t s of a l l {u ) E E w i t h {a,.,) E E. The n domain of B i s s i m i l a r l y defined; observe that both A and B a r e

{an]

and

normal operators commuting w i t h each other.

u(%) = [un(%)] i s a s o l u t i o n of (1.1)then each u (%) s a t i s f i e s the s c a l a r equation n u''(t> + b n u ' ( t ) + a u ( t > = 0. On t h i s b a s i s , we deduce that the n?? propagators C(%), b ( t ) of (1.1) must be given by

where

+

hn,A,

If

a r e the r o o t s of t h e c h a r a c t e r i s t i c equation h2 + b A + a n = O ,

(2.5)

h = h- (a case t h a t we w i l l n n Obviously, a necessary condition f o r t h e Cauchy problem

w i t h the modifications de rigueur when

avoid here).

f

f o r (1.1) t o be w e l l posed i n

a(t) =

Ilc(t)II

t

2

= SUP

n>_l

0

i s that

-

A+ e n

n

A$

THE COMPLETE EQUATION

273

and

be bounded on compacts of

0

5 t
1 for a l l

Also, a,

m(n> >_

t >_ 0 ,

i s increasing i n

m(^t)

accordingly t h e f u n c t i o n on compacts.

n

thus bounded

n = wn

'n

(n

2 1)

.

(2.17)

Define yn = l o g I n view of t h e i n e q a l i t y

Cyn

+ Lwl/". n n

= log w;/n

log x i x

5

2-1/2ex,

valid for

(2.18) x

5.

0, we

h.ave yn We s e l e c t now

a n,bn i n

n

-

n'

(2.19)

(2.5) i n such a way that (2.20)

We have i

( h I = p > e . n n -

(2.21)

On t h e o t h e r hand, i n view of (2.19),

2

(8, thus the sequence

A =

{A'-]n /A/

-

2 1/2

VJ

>_ Yn>

i s contained i n t h e region e,

Accordingly t h e r e e x i s t c o n s t a n t s

0 C_

Re h

0> 8

7

5 I m A. 0

independent of

fl

such that (2.22)

275

THE COMPLETE EQUATION

I n view of t h e d e f i n i t i o n (2.14) of

m(t)

we obtain from (2.23) and

(2.24) that

e(m(t> -et> 5 a(t>5

5

e(m(t) - e t > in

t 2 0.

T(t)

o(m(t>

+ e

t

1,

(2.25)

5 o ( m ( t > + et >,

(2.26)

The i n e q u a l i t i e s on t h e right-hand s i d e s of (2.25), (2.26)

imply that t h e Cauchy problem f o r (1.1) i s w e l l posed i n only remains t o choose t h e sequence i n e q u a l i t i e s (2.10) a r e s a t i s f i e d .

0

To do t h i s , we assume ( a s we

Keeping i n mind that t h e

constant i n (2.22) i s independent of t h e choice of

Both conditions (2.U) a r e obvious. the greatest integer

5 t.

It

0.

i n such a way t h a t t h e

u(%) i s nondecreasing.

obviously may) t h a t

t

we s e t

fl,

On the other hand l e t

t

2

1, n = [ t ] ,

Then, taking (2.16) and (2.27) i n t o account,

we o b t a i n

whence t h e f i r s t i n e q u a l i t y (2.10) r e s u l t s from (2.25); t h e second follows i n a similar way from (2.26).

EXAMPLE 2.2.

Let

a > 0.

Then t h e r e e x i s t A , B of t h e form

(2.2)such t h a t t h e Cauchy problem f o r (1.1)i s w e l l posed i n 0

b u t not w e l l posed i n any i n - t e r v a l 0

2

0

0.

SUP

s20

e

-us

IFr( s ) u l l ,

(4.5)

Eo

=

(4.6)

D ( A ) _C E o .

i s a Banach space i s much t h e same as t h a t f o r t h e

(111.2.1) and w e omit i t .

THEORE3l 4.2.

t

2

W e obviously have Do

equation

c o n s i s t s of all

so l a r g e t h a t ( 3 . 2 ) , t h e f i r s t i n e q u a l i t y (3.57)

W'

and ( 4 . 2 ) h o l d .

The proof t h a t

Eo

is

Eo

W',

The space

i s continuously d i f f e r e n t i a b l e i n t

l I ~ 1 1=~ IIuII + where

(4.4)

Let t h e Cauchy problem f o r

and l e t Assumption 3.1 be s a t i s f i e d .

f o r (4.1). Proof:

We must show t h a t

( 4 . 1 ) be well posed i n Then

Em i s a phase space

29 1

THE COMPLETE EQUATION

We prove f i r s t t h a t each Em. q t ) i s a bounded operator i n Em. I n order t o do t h i s we t a k e u E D0 and f i x t > 0 . Due t o time invariance of ( 4 . 1 ) t h e f u n c t i o n i s a s t r o n g l y continuous semigroup i n

u(i)

=

C ( t + g)u

i s a s o l u t i o n of ( 4 . 1 ) t h u s it follows from formula

(1.5) that C(S

+

= C(s)C(t)u

t)U

This e q u a l i t y i s extended t o

+ S(s)C'(t)u

u

aJ_1

E

Eo

(s,t

as follows:

2

(4.8)

0).

integrate i n

0 5rzt,

-rote(. +

T)u dT = c ( s )

Lt

C(7)U dT

+

and extend (4.9) t o a r b i t r a r y

u E E by denseness of we d i f f e r e n t i a t e and o b t a i n (4.8). The analogue of ( 4 . 8 ) f o r S ( t ) i s S(S

+ t)u

= C(s)S(t)u

+

u(s) = 8(s

+

Do;

u

for

E

2

0),

(4.10)

t)u, u

E

D1;

since all

u

operators i n (4.10) are bounded we can extend t h e e q u a l i t y t o all We note i n passing t h a t (4.8) i t s e l f can be extended t o all

a( s ) C ' ( t )

modified form observing t h a t

Eo

(s,t

S(s)S'(t)u *I

and i s shown by applying (1.5) t o

(4.9)

S(s)(c(t)u-u),

u

f

E

E

E.

in a

must have a bounded extension.

We s h a l l not make use of t h i s i n what follows. We prove t h a t each

qt)

i s a bounded o p e r a t o r i n

Em. To do

t h i s , we m u s t show t h a t t h e o p e r a t o r s C ( t ) :Eo

-

c ' ( t ) : Eo

E 0 E

are bounded i n t h e spaces i n d i c a t e d .

8 ( t ) :E

-

8 ' ( t ) :E

Eo

-.

(4.11) E

This i s r a t h e r obvious f o r

c'(t)

E ) and f o r a t ( t ) (from Assumption 3.1). 0 Note a l s o t h a t it follows from Corollary 3.7 and Lemma 4.1 t h a t

(from t h e d e f i n i t i o n of

292

THE COMPLETE EQUATION

( h e r e and i n o t h e r i n e q u a l i t i e s

C

denotes an a r b i t r a r y constant, not

n e c e s s a r i l y t h e same i n d i f f e r e n t p l a c e s ) . Continuity of

C(t)

C(s)C(t)u = C ( s + t ) u - S ( s ) C ' ( t ) u j

form

and d i f f e r e n t i a t e with respect t o

apply t o an element

u

+ t ) u - 8'(s)CI(t)u.

i s a bounded operator from

C(t)

i n the of

E~

We obtain

s.

Cl(s)C(t)u = C l ( s If follows t h a t

Write ( 4 . 8 )

i s proved a s follows.

Eo

(4.13) into

and

Eo

(4.14) f o r some constant

Finally, boundedness of

C.

Write (4.10) i n t h e form

It follows t h a t

' 0

i s continuous i n

+ t ) u - 8t(s)Sl(t)u.

)

5

wt

(t

Ce

(4.15)

L

(4.16)

0)

W e have t h e n completed t h e proof t h a t each

C.

Em: moreover, t h e r e e x i s t s a constant

wt

l l ~ t I l l ~ 5~ Cme ) f o r some constant

W e then

The r e s u l t i s

i s a bounded operator and

8(t)

lls(t)ll(E.E f o r some constant

i s shown a s follows.

+ t)u-S(s)S'(t)u.

C(s>S(t)u = 8 ( s

d i f f e r e n t i a t e t h i s e q u a l i t y term by term. C f ( s ) S ( t ) u= S l ( s

S(t)

C,

t h e constant

(t

C

10 )

q t )

such t h a t (4.17)

being t h e same i n Corollary 3.7

w

and Lemma 4.1. The semigroup equation

follows from (4.8) and (4.10) and t h e i r d i f f e r e n t i a t e d versions (4.13) and (4.15). n

The next step i s t o show t h a t

q t )

i s s t r o n g l y continuous.

It

i s enough t o prove t h a t Ilqh)u as

h

qtk

-

O+.

- uII( Em)

+

(4.19)

0

However, we s h a l l skip t h i s step since we show below t h a t

has a derivative at t h e o r i g i n ( i n t h e norm of

s)

for u

in

293

THE COMPLETE EQUATION

Gm; t h i s , combined w i t h t h e uniform bound (4.28)

a dense subset of obviously i m plies

THEOREM 4.3.

(4.19),

q;)

since

i s a s t r o n g l y c o n t i n u o u s semigroup w i t h

8 given by

infinitesimal generator

8=

=

c l o s u r e of

71,

(4.20)

where

w i t h domain D(%) = D ( A )

The f u n c t i o n

i s a s o l u t i o n of

u(;)

u(t)

(4.22)

( D ( A ) fI D ( E ) ) .

X

(4.1) o n l y i f

= [u(t),u'(t)l

(4.2?)

i s a s o l u t i o n of

u'(t)

=

%u(t).

Proof: We b e g i n b y showing t h a t t o p o l o g y of

Eo.

i s dense i n

D(A)

To d o t h i s we s e l e c t a

(4.24)

"6-sequence"

Eo

{@,I

i n the

of scalar

f u n c t i o n s l i k e that used i n t h e proof o f C o r o l l a r y 3.5 ( b ) , and show that

u as

n

-

( f o r any

f o r each

m

u

E

E)

(4.13) we s e e that

u

n

= J$ , ( t ) c ( t ) u

E

E

0' i s obvious.

dt

That ( 4 . 2 6 )

-

(4.26)

u

h o l d s i n t h e t o p o l o g y of

Assume now t h a t

u

E

EO.

Then, using

E

294

THE COMPLETE EQUATION

and we check e a s i l y t h a t

e-WSC'(s)un converges uniformly i n

to

un

emWSCt(s)u, s o t h a t

-

u

in

20

t

EO'

W e show next t h a t

u

Em f o r each

in

l i m i t r e l a t i o n s as

E

h

-

D(3).

This i s equivalent t o t h e following f o u r

0+:

-

h'l(C(h)u for

for

u

u

u

for

E

E

E

u

E

h-lS(h)u

-

u

(4.28)

h-lC'(h)u

-

-Au

in

Eo

(4.29)

D(A) n D(B),

D(A),

D(A)

n D(B).

- u)

+

--. -Bu

in

0

(4.31)

E

To prove (14.28) we use (4.13) i n t h e form

s,

i s bounded i n norm by a constant

t h e constant described a f t e r

e W I S , m'

(4.32) as h

(4.30)

E

in

and

This expression, as a f u n c t i o n of

times

in Eo

D(A),

h-'(S1(h)u for

u ) -, 0

(4.5).

The l i m i t of

is

C"(~)U-S'(S)C"((~)U= C"(S)U + S'(S)AU = 0 after (3.11). C1(s)(h"S(h)u

To show (4.29) we w r i t e (4.10) in t h e form

- u ) = h''(St(s

+ h)u

- S t ( s ) u ) - g'(s)h''(8'(h)u - u ) - c ' ( s ) u , (4.33)

which i s bounded i n norm as well by a constant times

as

h

-

O+

eWts; i t s l i m i t

is

S"(S)U-S'(S)S"(O)-C'(S)U

=

S"(S)U + S'(S)BU

+

S(S)AU = 0

(4.34)

295

THE COMPLETE EQUATION

i n view of Corollary 3.5(d).

F i n a l l y , t h e two l i m i t r e l a t i o n s (4.30)

and (4.31) a r e obvious, since and

u

E

h-l(W'(h)u

- u) n D(E)

-D(A) 1

D

-

-

h-lC'(h)u

( s e e Corollary

3.5

%

=

=

-AC(O)u- EC'(0)u

= -Au

-Eu f o r

(c)).

q;)

Having proved (4.27), we know t h a t semigroup and t h a t , i f

C"(0)u

- BS'(0)u

S"(0)u = -A8(0)u

i s a s t r o n g l y continuous

i s i t s i n f i n i t e s i m a l generator, t h e n

(4.35)

U C B . T o improve (4.35) t o (4.20) it w i l l be s u f f i c i e n t t o prove t h a t Uh

f o r all u

E

D(9J)

s e l e c t a sequence

In f a c t , i f

{un]

(t)u d t

= kJhE

5 D(Z)

E

D(U)

(4.36)

(4.36) i s t r u e and

-

un

with

u

in

u Qm

E

(g, we may

(that

D(8)

is

dense i n ( u )h + 11

follows from (4.26) and f r o u Corollary 3.5 ( b ) ) . Then whereas u(un) h = 8 ( u n ) h = h -1(F(h)un- un) -+ h -1( S ( h ) u - u);

uhn€ D(E)

f o r any

u

E

Gh = Assume that tends t o 8 u

u

that

E

h > 0

Qm and any

and

-

h-'(5(h)u

.

u)

(4.37)

u E D(%). Taking i n t o account t h a t t h e r i g h t s i d e of (4.37) as h+O+ it follows from t h e fact t h a t i s closed

D(@

u

and

uu

= %u, which completes t h e proof of (4.20).

The i n c l u s i o n r e l a t i o n (4.36) c a n be reduced t o t h e f o u r r e l a t i o n s

(4.38)

(4.39) / g h C l ( t ) u d t = C(h)u

L h S t ( t ) u d t = S(h)u-u If

u

E

D(A)

E

-

u

D(A)

we have

E

D ( A ) fl D ( B )

n D(B)

(U E

(u

D(A)

=L

E

D(A)),

(4.40) (4.41)

D(E)).

h

AJOhC(t)u d t

so t h a t (4.38) h o l d s .

8( a + p l o g (1 + I h l ) I n Exercises ASSUMPTION

u

for all

of

E

{u

(5.8)

5 t o 11 we r e q u i r e p a r t ( a ) of Assumption 3.1, t h a t i s

5.1.

i s continuously d i f f e r e n t i a b l e i n t

S(i)u

E

5.

U s i n g Exercise 2 show t h a t t h e operator

0

Do Tl D(B); Bu

D1]

E

8(t)B

(with

h a s a bounded extension

t o all

given by = C(t)

EXERCISE 6.

Define

with

m(F)

-

Sf(t)

.

(5.9)

R(h;c~) as i n (3.19),

R( h;fn)u

Jbwe-htm(t)s(t)u

=

,

dt

a t e s t f u n c t i o n i d e n t i c d l y equal t o

(5.10)

1 near zero.

(a s l i g h t l y extended v e r s i o n o f ) (5.6) show t h a t if u such t h a t

Au, Bu

E

where

N(t;rp) = 2 v f ( t ) 3 ' ( t ) EXERCISE

Show t h a t for

7.

E

Do

Using

n D(B)

is

D1 t h e n (3.37) holds, that i s ,

+ $h;a)u

R(h;a)P(h)u = u

Define

-t

p"(t)s(t)

R(h)

as i n

+

,

(5.11)

a ' ( t ) W .

(3.41)

for

Reh

3 W,

w l a r g e enough.

u as i n Exercise 6 we have R(h)P(h)u = u

EXERCISE 8.

h(:;rp)

2

E E.

EXERCISE domain

.

For Reh

given by (3.46).

R( A). EXERCISE 9 .

Define

2

w,

W

.

l a r g e enough, d e f i n e

S(;)

Prove that t h e Laplace transform of

(5.12)

- (3.48),

by

8 equals

301

THE COMPLETE EQUATION

jta-'/r(a) ya ( c o n v o l u t i o n by

;lo

0)

(t < 0 )

Ya produces t h e " a n t i d e r i v a t i v e of order a " ) . Show

t h a t , m u l t i p l y i n g (5.12) by o b t a i n , u s i n g Exercise

for

2

(t

and i n v e r t i n g Laplace t r a n s f o r m s we

8,

as i n Exercise 6.

u

EXERCISE 10. Assume t h a t t h e s e t of d1 u

Au, Eu

E

D1 i s dense i n t h e space X

Do

E

n D(B)

= D(A)

n D(B)

such t h a t

endowed with t h e

norm

EXERCISE 11. Snow t h a t , i f

(Y1

€3 I

+ Y2

Combining (5.13) and

@ E

u

E

D1,

+ Y @A) * 3

SU = Y

€3 u

3

(t

2

(5.14)

0).

( 5 . 1 4 ) , prove t h a t , under t h e c o n d i t i o n s of

Exercise 10,

qt) = E(t). EXERCISE 12.

Under t h e c o n d i t i o n s of Exercise 1 0 , show t h a t

(3.53) ( r e s p . (3.54)) h o l d s for st(;)

formula

t h a t there exist constants

EXERCISE

u

E

E

SL(t)B).

c(t)u

Show

i s continuously d i f f e r e n t i a b l e i n

if

t 10

(4.3) h o l d s , so t h a t Ilcl(t)UII

C

(resp. f o r

such t h a t

C,W

13. Under t h e c o n d i t i o n s of Exercise 8 show t h a t ,

i s such t h a t

t h e n formula

with

(5.15)

( b u t not

w)

5

Ce

wt

may depend on

under t h e p r e s e n t hypotheses.

(t

u.

L

0)

-

(5.17)

Show t h a t Theorem 4 . 2 i s v a l i d

THE COMPLETE EQUATION

302

EXERCISE w e l l posed i n

14. 0

5t 5

E

D(A)

n D(B)

i s well posed i n

2

0

u E.

i s dense i n

t

t h a t Assumption

a (a > 0),

t h e r e and t h a t t h e s e t of all

Bu

(5.3) i s 3.1 i s s a t i s f i e d

We suppose h e r e t h a t t h e Cauchy problem f o r

f

D ( A ) I- D(E)

such t h a t

Then t h e Cauchy problem for

(5.3)

and Assumption 3.1 i s s a t i s f i e d . Note t h a t 611

t h e assumptions i n t h i s Exercise a r e s a t i s f i e d f o r t h e incomplete equation

u"(t) + Au(t) = 0

(5.18)

under t h e only assumption t h a t t h e Cauchy problem for

(5.18) i s w e l l

posed; of course, t h e r e s u l t for (5.18) can be proved i n a more elementary way by ad hoc methods. FOGTNOTES TO CHAPTER VIII

(1) We note t h e i n c o n s i s t e n c y of n o t a t i o n involved i n w r i t i n g t h e incomplete e q u a t i o n

u" + Au

=

0,

and not

u"

=

Au

as i n Chapters I1

and 111. (2)

Although t h e argument could be completed using (3.25), t h e " l e f t -

(3.41) s i m p l i f i e s some of t h e arguments. ( 3 ) We might s e t h e r e W1 = min(U,wl): f o r i f W ' < U, R(A) c a n be ana'Lytically continued t o Reh > W ' by means of Q ( h ) . ( 4 ) Convolution by Y i s employzd h e r e t o avoid using convolution of 3

handed" r e p r e s e n t a t i o n

distributions.

303

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