Asymptotic Analysis of Singular Perturbations

Asymptotic Analysis of Singular Perturbations

STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUME 9

Editors: J. L. LIONS, Paris G. PAPANICOLAOU, New York R. T. ROCKAFELLAR, Seattle

NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM

'

NEW YORK ' OXFORD

ASYMPTOTIC ANALYSIS OF SINGULAR PERTURBATIONS

WIKTOR ECKHAUS Mathematisch Instituut Rijksuniversiteit Utrecht

1979 NORTH-HOLLAND PUBLISHING COMPANY -AMSTERDAM ' N E W YORK ' OXFORD

@ North-Holland Publishing Company, I979

All rights reserved. No purt of this publicution muy be reproduced, stored in u retrievul system or trunsmitted, in uny form or by uny meuns, electronic, mechunicul, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN 0444 853065

Publishers:

NORTH. ,AND PUBLISHING COh P NY AMSTERDAM ’ NEW YORK ‘ O X F O R D Sole distributors for the U.S.A. und Cunudu: ELSEVIER NORTH-HOLLAND, INC. 52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

Library of Congress Cataloging in Publication Data Eckhaus, Wiktor. Asymptotic analysis of singular perturbations. (Studies in mathematics and its applications; v. 9 ) Bibliography: p. Includes index. 1. Differential equations - Asymptotic theory. 2. Perturbation (Mathematics) 3. Boundary layer. I. Title. 11. Series. QA371.E26 515’.35 79-1 1324 ISBN G 4 4 4 8 5 3 0 c - 5

PRINTED IN T H E NETHERLANDS

To Beutrice

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PREFACE The theory of singular perturbations is a fascinating mixture of rigorous analysis, heuristic reasoning and induction from experience. The aim of this book is to give these aspects a coherent presentation. The scope of the book is limited to problems exhibiting the so-called boundary layer behaviour. Singular perturbations as a discipline has emerged from problems confronted by physicists, engineers and applied mathematicians. A wealth of techniques and results can be found described in the books of Van Dyke (1964), Wasow (1965), Cole (1968), Nayfeh (1973), Lions (1973) and O’Malley (1974), which also contain further reference to hundreds of papers in the periodical literature. I have become interested in singular perturbations some fourteen years ago, and ever since, in a succession of lecture notes, and in various papers, it was my aim to develop a line of thinking which would permit a deductive presentation of the theory. Some aspects of this line of thinking have been described in an earlier short monograph (Eckhaus (1973)),which to some extent can be considered as an introduction to the present study. A companion volume to that monograph, announced in the monograph, never appeared, because a first draft convinced me that I should develop parts of the earlier material in greater length and depth, and confront the various aspects, rather than separate them. The final result is the present volume. This book is an inquiry into the mathematical structure of the theory of singular perturbation. The book can be studied in various ways, depending on the readers interests and motivation. For example, the practitioner of heuristic analysis who is mainly interested in applications may read superficially the first three chapters, study mainly Chapters 4 and 5 and dismiss the rest. However, if he would care to do some reading in Chapter 6, he may discover ways of turning the heuristic analysis of some specific problem into a rigorous theory. On the other hand, the mathematician interested in the techniques of proof of the validity of formal approximations may concentrate on Chapter 6, while a student of elliptic p.d.e.’s can content himself with parts of Chapter 6 and study Chapter 7. Let me further mention that the first three chapters are rather selfcontained, and should be of interest to those who are puzzled and fascinated (as I have been for many years) by the somewhat bizarre foundations of the method of matched expansions. Along similar lines, differently flavoured courses, or series of seminars, can be based on the contents of the book. However, if one wants to gain understanding of the interplay of the various ingredients in the theory of singular perturbations, vii

...

Vlll

PREFACE

then most of the material presented in this book should be covered. Parts of the manuscript of this book, at various stages of its development, were read by Eduard de Jager, Bob O’Malley, Jan Besjes, Harry Moet, Will de Ruyter, Jan Sijbrand and Ferdinand Verhulst. I want to thank them for their constructive criticism and many useful suggestions. I am particularly grateful to Aart van Harten, who read almost the entire manuscript and helped me clarify various delicate points of the analysis. Finally, I want to acknowledge my gratitude to the Department of Mathematics of the University of Utrecht, for giving me freedom from most of my duties in the academic year ’77-’78, and making it thus possible for me to complete the manuscript of this book. Amstelveen, September 1978

CONTENTS Preface

vii

Contents

ix

Chapter 1. Asymptotic definitions and properties 1.1. 1.2. 1.3. 1.4. 1.5. 1.6.

Order symbols, sharp estimates and order functions Asymptotic sequences and asymptotic series Orders of magnitude Asymptotic approximations and asymptotic expansions Regular approximations and regular expansions Gauge functions, gauge sets and the uniqueness of regular expansions

Chapter 2. Functions with singularities on subsets of lower dimension (Boundary layers)

1

1 5 8 11 16 19

21

2.1. Qualitative description of singular behaviour 2.2. Regularity in subdomains. Extension theorems 2.3. Local analysis of continuous functions; local limit functions and local expansions 2.4. The formalism of expansion operators

22 23

Chapter 3. Matching relations and composite expansions

38

3.1. Significant approximations and boundary layer variables 3.2. Further applications of the extension theorems; the overlap hypothesis 3.3. Matching in the intermediate variables and uniform approximations on the basis of the overlap hypothesis 3.4. Overlap hypothesis and intermediate matching in the case more general local expansions

40

ix

29 33

42

50 53

X

CONTENTS

3.5. Correction layers and composite expansions; an asymptotic matching principle 3.6. Composite expansions and an asymptotic matching principle from the hypothesis of regularizing layer 3.7. Asymptotic matching principles and composite expansions from the overlap hypothesis 3.8. Validity of asymptotic matching principle without overlap

61 66

Appendix: Proof of Theorem 3.7.1.

68

Chapter 4. Heuristic analysis of singular perturbations. Linear problems

76

4.1. Degenerations of linear differential operators 4.2. The differential equations for the first term of the regular and the local expansions 4.3. Recurrence relations for regular and local expansions 4.4. The correspondence principle 4.5. Further development of the heuristic analysis: some onedimensional problems 4.6. Heuristic analysis continued: some two-dimensional problems 4.7. The concept of formal approximations 4.8. Expansions with a regularizing factor: the WKB approximation 4.9. Expansion by the method of multiple scales

77

102 116 120 133 139

Chapter 5. Heuristic analysis continued. Non-linear problems

145

5.1. Degenerations of non-linear operators 5.2. The differential equations for the first terms of the regular and the local expansions 5.3. The differential equations for the higher order terms of the expansions 5.4. Analysis of some one-dimensional problems 5.5. Significant degenerations and the correspondence principle reconsidered 5.6. Some one-dimensional problems exhibiting strong non-linear effects 5.7. Some elliptic second order problems in R2 5.8. Remarks on the formal approximations in non-linear problems

146

55 59

87 91 98

148 150 153 163 165 181 105

Chapter 6. Foundations for a rigorous theory of singular perturbations

188

6.1. General introductory considerations 6.1.1. Classical perturbation analysis in a Banach space

188 189

CONTENTS

6.1.2. Regular problems 6.1.3. Singular problems: a classification 6.1.4. The problem of validity of formal approximations 6.2. Estimates for linear problems 6.2.1. The maximum principle for elliptic operators and its applications 6.2.2. Estimates in Hilbert spaces from positivity of the bilinear form 6.2.3. Estimates in Holder norms. Elliptic equations of higher order 6.2.4. Estimates for initial value problems 6.3. Non-linear problems 6.3.1. Non-linear applications of the maximum principle 6.3.2. Upper and lower solutions 6.3.3. Initial value problems. Tichonov’s theorem 6.3.4. Estimates for the remainder term based on contraction mapping in a Banach space

XI

195 197 198 200 20 1 206 212 220 225 226 230 234 236

Chapter 7. Elliptic singular perturbations

244

7.1. Linear operators of second order. Elementary boundary layers 7.1.1, Zeroth order degenerations 7.1.2. First order degenerations. Subdomains with ordinary boundary layers 7.1.3. First order degenerations continued. Parabolic boundary layers 7.2. Linear operators of second order continued. Refined analysis of boundary layers 7.2.1. Birth of boundary layers 7.2.2. Free layers and other non-uniformities 7.3. Non-linear operators of second order 7.3.1. Zeroth order degenerations 7.3.2. First order degenerations 7.4. Linear operators of higher order 7.4.1. Elliptic degenerations 7.4.2. First order degenerations

245 245

Bibliography

280

Subject Index

287

248 255 258 258 262 264 264 267 270 270 275

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CHAPTER 1

ASYMPTOTIC DEFINITIONS AND PROPERTIES In this chapter we introduce and study the basic tools of asymptotic analysis: the concepts of orders of magnitude, of asymptotic approximations and asymptotic expansions. These concepts, although elementary, require a careful analysis when one wishes to study (as we do in this book) functions @(x,E),x E D c R”,E E ( O , E ~ for ] E 10. Such functions will be considered as families of elements (parametrized by E ) of some normed space of functions, and the asymptotic properties of @, as E J 0, will be defined through the asymptotic properties of some suitably defined norm of @. We shall always assume continuity of Q(x,E) with respect to E on some interval E E (0, zO]. This assumption simplifies the analysis somewhat and nevertheless provides a sufficiently large setting for applications. Most of the chapter is devoted to definitions, and the discussion of their motivations. The chapter contains further a lemma on the sufficiency of order functions (Sections 1.1 and 1,3), and the Du Bois-Reymond theorem, which provides interesting insight into the behaviour of asymptotic sequences (Section 1.2). In Section 1.5 we study the so-called regular asymptotic approximations and expansions. The use of the overworked adjective ‘regular’ requires some justification. In the context of this study the motivation for this terminology is two-fold: firstly, a function possessing a regular asymptotic approximation indeed has properties that are usually associated with the term . the ‘regular’, i.e. the existence (in a certain sense) of a limit as ~ 1 0Secondly, regular asymptotic expansions are expansions of the simplest possible structure; they are sufficient for the solution of perturbation problems that are commonly called regular. We commence this chapter with two sections on functions that only depend on the parameter E . 1.1. Order symbols, sharp estimates and order functions

Consider any pair of real continuous functions f(~) and g(E), E E (0, E ~ ] .The behaviour of these functions, as E tends to zero, can be compared by using the classical Landau order symbols 0 and 0,which are defined as follows: Definition 1.1.1. (i)f= O ( g ) for E ~ ifO there exist positive constants k and C such that I f ( & ) [ < klg(E)I for 0 < E < C . (ii)f= o(g) if limE+o{f(E)/g(E)}exists and equals zero.

ASYMPTOTIC DEFINITIONS AND PROPERTIES

2

Sometimes it is convenient to use, instead of the symbols 0 and due to Hardy:

CH. 1, 91 0,a

notation

Definition 1.1.2. (i)f< g o f = O(g), (ii) f < g-f = o(g). Let d be any set of functions that are mappings of (O,E,,] into the real numbers. we can introduce an asymptotic ordering of the elements Using the relation of d.For that purpose we first define an equivalence relation.

<

Definition 1.1.3.fx g iff< g and g <J

<

One easily verifies that the binary relation has the propertiesf
g

< h *f<

Clearly, f and g are not comparable by the binary relation defined by the symbol A further, and rather fundamental difference between the asymptotic ordering defined by the symbol and the familiar order properties of the real line can be expressed as follows:

<.

<

Property 1. The statement

tf<

g a n d f + g’ does not implyf< g.

To prove the property it is sufficient to consider for examplef(&)= E sin 1/&, g(E) = E.

On the other hand, and in analogy to the order properties of the real line, we have:

Property 2. Iff x g, then f # o(g) and g # o ( f ) . Proof of this property is rather elementary and is left as an exercise for the reader. We finally introduce a symbol for sharp order of magnitude estimates:

Definition 1.1.4.f= O,(g) if (f=O(g) a n d f # o(g)). From the preceding analysis we know t h a t f x g impliesf= O,(g).However, f x g is a sufficient, but not a necessary condition to have f= O,(g). In other

CH. 1. R 1 ORDER SYMBOLS, SHARP ESTIMATES AND ORDER FUNCTIONS

3

words, and this a consequence of Property 1, there exist pairs of functions satisfyingf= O,(g),f# g. It is for this reason that it is useful to have a separate symbol 0,, defining sharp orders of magnitude estimates. We shall now define a convenient set of 'simpler' comparison functions, to be called order functions, which we shall show to be sufficient to describe sharp orders of magnitude of any continuous functionf(E). Order functions will be denoted by the symbol 6, provided (if necessary) with subscripts and/or superscripts.

Definition 1.1.5. 6 is an element ofthe set oforderfunctions d if a(&), E E ( O , E ~ ]is ) real, positive, continuous and monotonic, and if limeLo6 ( ~exists. Order functions as defined above can be used to describe the behaviour of bounded functionsf(&),and in particular to analyse the way a functionf(8) tends to zero as E J O . Well known examples of sequences of order functions are 6,(~= ) E,, or a,,(&) = exp ( - n / ~ ) n, = O,l, ... . On the other hand, functionsf(&) which grow without bound as E 1 0 may be analysed by means of functions $(E) = 1 / 6 ( ~where ) 6 E 8.This leads to the extended set of order functions 8,defined by:

Definition 1.1.6. 6 is an element of the extended set of order functions 8 if either 6 ~ orb We shall now show that the set d is sufficient for the study of any continuous function f ( e ) (Eckhaus (1973)).

Lemma l.l.l.Letf(E), E E ( O , E ~ be ] a real continuousfunction, not identically equal to zero on each interval (0,73, t E ( O , E ~ ] Then . there exists an order function 6 E 8 such t h a t f = 0,(6). Proof. (1) Suppose first that f ( ~ )is not bounded as E 10. f ( ~ is ) bounded on any compact subinterval contained in (O,EJ, and If(&)/ has a maximum in any such compact subinterval. Let {E,,}? be any monotonic sequence with limn+3o E, = 0. On any interval I,, = ( E 1 E E [E,, e O ] } the maximum of If(e)I is given by

If(4I = IfWI; max & E I, We observe that for each n If(~,*)l

E,* E 1,.

2 /f(~:1) and furthermore

= 0. limn*m [lf(E,*)lI-' Now consider the function E + 6 ( ~ defined ) by

6 ( 4 = max 9

E [G E01

If(v)I.

It is obvious that 6 is a positive, continuous monotonic function satisfying

4

ASYMPTOTIC DEFINITIONS AND PROPERTIES

6 ( ~2 ) lf(~)I

for E E (0,&,I and

CH. 1, $1

@,) = If(&,*)I.

Hence, clearly, f = 0 ( 6 ) , and furthermore we cannot have f = 0(6), because for = 1, and { E , * } ; is a sequence with limn+- E,* = 0. all n, I ~ ( E , * ) J / ~ ( E , * ) It remains to be shown that 6 E 8.For that purpose we observe that 6(E) 2

If(&,*)\

for E E (O,E,].

Hence

Since the right-hand side of the inequality tends to zero as n -, a,we obtain lirnE+,(l/d(E))= 0, that is 6 E 6. (2) Suppose now thatf(e) is bounded as E 10. Iff(&)does not tend to zero as 8 1 0 , then the function 6 ( ~ = ) 1 satisfies all conditions of the lemma. We therefore consider the remaining case lim,,,f(E) = 0. We extendf(s) to E E [O,E,] by defining f ( 0 ) = 0. Let {E,}; again be a monotonic sequence with limn+ E, = 0 and consider the intervals I, = { E / E E [O,E,]). We have = If(~,*)l; E,* max If(~)l E E

Then V n

E I,.

I,

If(&,*+

,)I d lf(~,*)l and

If(&,*)/

= 0.

We define a positive continuous function 6 ( ~ by )

44 = vmax If(rl)I. [O,El E

Obviously 6 ( ~2 )

if(&)(, E E [O,E,],

and

a(&,*) = If(E,*)I from which it follows t h a t f = 0,(6). Furthermore B(E) < lf(~,*)I for

Therefore lim,

E E

[O,E,].

,a(&) = 0 and 6

E 8.

Remark. It is possible to prove Lemma 1.1.1under somewhat weaker conditions. It is sufficient to assume, instead of continuity, boundedness of f ( e ) on any compact subinterval of (0,E , ] . The proof is only slightly more complicated. However, from the point of view of applications, functions which are not continuous in any neighbourhood of the origin do not seem to be of much interest. We conclude this section with some elementary properties of order functions.

CH. 1, 82

ASYMPTOTIC SEQUENCES AND ASYMPTOTIC SERIES

5

(i) Let 6, E 8 and 6, E 8,while c( and /3 are two real positive constants. Then 26, gs, E 8. (ii) Let 6, E & and 6, E 8.Then 6, '6,E 8.This property is not preserved in 8,as is shown by the pair of functions 6, = el", 6, = eC1" (2 + sin l / ~ ) . However, the property does hold in the complement of & in 8.We then have: (iii) Consider 6, E 8, 6, # & and 6, E 8,6, # b. Then 6, = 6, '6,E 8,6, $ b.

+

Remarks. It is easy to verify (and furthermore useful in applications), that if for some given function f(E), E E (O,E,], and some order function 6 E 8,one has an estimate f = O(6) or f = o(6) or f = 0,(6),

then one also has, respectively

1.2. Asymptotic sequences and asymptotic series Let us consider now pairs of order functions 6,, 6,, satisfying 6, 4 6,, with 6, # 0,(6,). Then by definition, 6, < 6,. We observe that the binary relation defined by the symbol 4 is transitive, that is, if 6, <6, and 6,<6, then 6, < 6,. Therefore, any denumerable set of order functions, in which every pair 6,, 6, satisfies either 6, < 6, or 6, < 6,, and which contains an element 6, such that 6, < do, Vn, can conveniently be arranged by the symbol < into a so called asymptotic sequence. Definition 1.2.1. A sequence of order functions 6,,n = 0,1, ...., N is an asymptotic sequence if for all n = 1,....,N , 6,< 6,-

,.

In the above definition N may be finite or infinite. We remark that any asymptotic sequence of order functions is a totally ordered subset 'of the set of order functions. Definition 1.2.1 is easily extended to sequences of arbitrary functions f,(~),as follows: Definition 1.2.2. A sequence of functionsf, (E) is an asymptotic sequence if for all n = O , l , ..., N one hasf, = 0,(6,), 6, E 6 (or 6, E 8), and (6,); is an asymptotic sequence. Asymptotic sequences generate asymptotic series by the usual linear operations. We obtain

cr=o

Definition 1.2.3. A sum anfn(c) where a, are constants, is an asymptotic series if {f,(~)); is an asymptotic sequence.

6

ASYMPTOTIC DEFINITIONS AND PROPERTIES

CH. 1, 52

An asymptotic series which is defined for all m can formally be interpreted as an infinite asymptotic series. Of course, nothing is implied about the convergence of such series. Furthermore, the question of convergence of an infinite asymptotic series is of no particular interest in the asymptotic theory. Instead, the concepts of asymptotic approximations and asymptotic expansions are used. These concepts will be studied in the following sections. We conclude this section with a result on the behaviour of order functions which in its original version is known as the Du Bois-Reymond theorem (Hardy (1910)).To introduce this result we remark that a relation of the type 6, < 6, is usually interpreted to mean that '6, is asymptotically smaller than 6;. However, and this is the contents of the Du Bois-Reymond theorem, there exists no asymptotic sequence {d,,}; of which the terms, for sufficiently large n, would become 'arbitrarily small' in the asymptotic sense. More precisely we have:

Theorem 1.2.1. Giuen any asymptotic sequence {6,}:, with 6,+ there exist order functions 6 ' such that

6 '

< 6,

< 6 , for

all n,

for all n

Proof. The theorem is trivial if the sequence of order function is such that for all n we have lime+' l j 6 , ( ~ )= 0. In that case it is sufficient to take 6 ' = constant. We therefore consider the case that for sufficiently large n, 6, = o(1). We observe that 6,+ < 6, implies that there exist constants &*(n),such that

a,,+

(E)

< 6 , ( ~ ) for 0 < E < c*(n).

Hence there exists a monotonic decreasing sequence such that

6,+

(E)

{E,}:,

with limn-=

E, =

0,

< b,(&j for E E (o,E,).

We now define a function a'(&), E E ( O , E ~ ) by , the following construction: we ' positive, monotone and continuous, and satisfying take 6 So(&,) = 6 n +

1 (En),

0 < So(&) < 6, + 1(~)for E It is quite obvious that we have 0 < So(&) < a,+ And consequently

for

E [E,

E E

+

1, E,].

(o,~,].

CH. 1, 92

ASYMPTOTIC SEQUENCES AND ASYMPTOTIC SERIES

7

Now, the right-hand side, in the above inequality, tends to zero as &LO. Therefore lim

do(&) ~

E-0

a,(&)

= 0,

Furthermore, since 6,

Vn. = o(l), we

must have

lirn So(&)= 0. E-0

This concludes the proof of the theorem.

As an example consider the sequence { E " ) ; . Then one may take 6' = e-l". Functions of the order of magnitude 6 ' are often called 'transcendentally small' (with respect to E"). One can deduce from the above theorem various additional interesting results on the behaviour of sequences of order functions. The corollary that follows now is in fact the formulation of the Du Bois-Reymond theorem as given in Hardy (1910). Corollary 1. Let id,}; be a sequence of order functions such that for each n, a,, > G,_and 1irnc+' 1/6,(~)= 0. Then there exists an order function 6 O , with limE+o1/6'(~)= 0, such that

Proof. Write l/$, = 6, and use the preceeding theorem. We finally state another result that is sometimes useful in applications (cf. for example Eckhaus (1977.A)).

Corollary 2. Let {8,}?be a sequence of order functions such that 8,+ furthermore, for each n, s", = o(1). Then there exists an order function such that 6' > 6, Vn.

> 8, and

8 '

= o(l),

Proof. The non-trivial part of the corollary is the assertion that go= o(1). For the proof we shall need the inverse function. Let thus 6 + i,(S) be the inverse of E + 6,(~).One easily sees that in(6)= 0. Furthermore, it is not difficult to deduce that the condition

implies lim E;, + ~

6- 0

- 0.

in@)

ASYMPTOTIC DEFINITIONS AND PROPERTIES

8

CH. 1, $3

We may now interpret the sequence of functions Ln(6)as a sequence of order functions, with variable 6, and apply our preceding theorem. It follows that there ) , lim6+o~ ~ (=6 0, ) exists a positive continuous monotonic function 6 + ~ ~ ( dwith such that lim

6 ..+ 0

Q6 % =o, C"(6)

Vn

Let now E + S"(E)be the inverse of 6 -+ ~~(6). Then J0 = o(1). Furthermore, < Ln for 6 1 0 implies that go> &, for E 10. This concludes the proof.

E^O

1.3. Orders of magnitude We now commence the study of functions @(x,E), of more than one variable. The scalar variable denoted by E, E E (O,eO], will be considered as a (small) parameter, while the variable x will be interpreted as an 'independent' (in general vector) variable in the domain D c R".We shall study functions @ which are mappings of D x ( O , E ~ ]into R" and investigate their behaviour as E 10, for any value x in D. Definitions 1.1.1 and 1.1.4 can be applied when considering any arbitrary but $xed value x = xo E D . We then obtain the pointwise order of magnitude estimates. Definition 1.3.1. Let 6 be an order function, belonging to 6 or 8. (i) @ = O(6) at x = xo if there exist positive constants k and c such that I@(x~,E)J < k6(E) for 0 < E < c. (ii) 0 = o(6) at x = xo if limE+o@ ( x ~ , E ) / ~ (= E )0. (iii) 0 = 0,(6) at x = x o if, at x = xo, @ = O(6)and @ #o(6). In the older part of asymptotic analysis, devoted mainly to the study of integrals containing parameters, there was not much need to develop further the definitions of the orders of magnitude. It was sufficient, for most purposes of the analysis, to add the following remark (Erdelyi (1956), de Bruijn (1958), Lauwerier (1974)): Definition 1.3.2. Let 6 be an order function belonging to I or 8. (i) @ = O(6) uniformly in a subset Do c D if there exist positive constants k and c independent of x such that for all x E D6 I@(X,E)\< k6(E) for 0 < E < c. (ii) @ = o(6) uniformly in a subset Do c D if lime..+o@(x,E)/~(E)= 0 uniformly in Do. In modern developments of asymptotic analysis, motivated largely by the needs of perturbation theories for differential equations, it is useful to introduce

CH. 1, $3

ORDERS OF MAGNITUDE

9

generalizations of the classical concepts. For this purpose it is convenient to , are mappings of consider any function @(x,E)as afamily of functions ~ J x )which D into R" and are parametrised by E. We then have the identity @ ( x , E )= ~ J x ) , 4&:D + R", $&E P(D), where P(D) is some set of functions which is further specified according to the needs and goals of the analysis. If P(D)is a linear space on which a norm ( 1 * [ID is defined, then this norm is a natural instrument to use as a measure of the order of magnitude of functions. We thus arrive at the following definition.

Definition 1.3.3. Consider the functions @(x,E)= 4&(x)where, V E E ( O , E ~ ] , $&:D -, R". Let the restrictions of $& to a subset Do c D be elements of a normed linear space P(Do), with a given norm ll.IIDo and let 6 be an order function belonging to € or 6. (i) @ = O(6) in Do c D if I( llDa = 0(6), (ii) @ = o(6) in Do c D if 11 4&[IDo = 0(6), (iii) @ = 0 , ( 6 ) in Do c D if @ = O(6) and @ # o(6) in Do c D. Remarks. It is quite obvious that uniform behaviour by Definition 1.3.2 is a special case of Definition 1.3.3 with

Naturally, Definition 1.3.3 also permits one to investigate orders of magnitude of @ in D by letting Do coincide with D. In applications it is often difficult, to analyse a given function in the whole domain of definition D. In such cases one analyses the function in suitable subdomains, with the purpose of deriving ultimately, from the combined results of the analysis, estimates valid in the whole domain D. It should be clear that the analysis can only be successful if the norms defined for the restrictions to various subdomains are suitably related to the norm 11 * ( I D . In what follows we shall use in subdomains norms which are essentially the same as the norm chosen in V(D),except for the obvious change of the domain of definition of functions. Thus, if we work in the supremum norm, then for any subset D, c D the norm will be defined by

Similarly, if we work in ,!,,-norm, then in any subdomain

Such definitions are easily extended to norms associated with the supremumnorm or the ,!,,-norm and involving derivatives of functions, or to weighted

ASYMPTOTIC DEFINITIONS AND PROPERTIES

10

CH. 1, $3

norms etc. The useful properties that follow from such choices of norms in subdomains can be used to define acceptable norms. This leads to: Definition 1.3.4. Consider $& E P(D) with norm l l * l l D . The norms l l . l l D , of the restriction of 4&to subdomains D , c D are consistently defined if the following condition is satisfied: D2

D2

D3

imp1ies that

+

I I ~ E ~ ~ DI ~, ~ E I I D I /I@EIID,'

We remark that (by taking for D 3 the empty set) we also have the useful property: D 2 11 4 6 11 D1' D l * 11 4~11 D Z From a somewhat more abstract point of view one can observe that the norms (1 * /ID, on the restrictions of q5&to D , are in fact a family of seminorms on V ( D ) . Thus the total structure underlying our analysis consists of a linear space V ( D ) provided with a family of seminorms 11 / I D ?. The seminorms are such that I/ * /ID is a norm on P(D); for an arbitrary but fixed D , c D, l l - l l D v is a norm of the restriction of $&to D,, and the family of seminorms satisfies the condition given in Definition 1.3.4. In applications we shall often abbreviate and simplify the notations by writing

It should be clear that, when studying a given function O(X,E),the result of the analysis of the order of magnitude may depend on the choice of the norm 11 * II and that furthermore, the function can be of different orders of magnitude in different subdomains. This is illustrated by the following example. Consider @(x,E)= e-x/', D = {x I x E [O,l]). We first study orders of magnitude in D, for various choices of the definition of norm. (a) Let ( 1 CD / / = supxED101,then 0 = 0,(1) in D. (b) Let 11 CD 11

=

supxsD/@I

+ supxsDlCDxl, then CD = O,(E-') in D.

(c) Let I / CD 11 = { j A[CD(X,E)]~~X}'/~, then CD = O,(JE)in D. (d) Let I/ CD 11 = [ 1A{[@(x,E)]' + [CD,.(x,~)]~)dx]''~, then 0 = O , ( E - ~in' ~D.) Next let Do c D be any compact subinterval of D , not containing the origin. Then it is not difficult to verify that the orders of magnitude of CD in Do are given by order functions which describe the orders of magnitude in D , multiplied by the order function ,-PIa, where p is a positive constant. Concluding the discussion of the orders of magnitude let us note that in applications need may arise for a further generalization of the concepts. Thus, in perturbation problems for differential equations it is sometimes necessary to

CH. 1, $4 ASYMPTOTIC APPROXIMATIONS AND ASYMPTOTIC EXPANSIONS

11

introduce and study expressions of the type

One loosely speaks of 'e-dependendant norms', which to some extent contradicts the familiar definition of norm (as a mapping of a set of functions into the real numbers). We shall not attempt to remove this contradiction by introducing new terminology. In what follows we shall admit as a norm any mapping which, for every arbitrary but fixed value of E , is a norm on B(D), provided that for any function x + $(x) E V(D),the expression 11411, is a continuous function of E for E E (O,eO]. We now state a result that will be used very frequently in the analysis that follows. Lemma 1.3.1. Let there be given a non-triuial function @(x,E),x E D c R", E E ( O , E ~ ]and , a norm Ij*ll, such that 11CDI(is a continuousfunction of E , for E E (O,E~]. Then there exists an order function 6 E d such that

CD

=

0,(6).

Proof. We write l \ C D [ l = f(E) and use Lemma 1.1.1. Corollary: Let @ satisfy the conditions of Lemma 1.3.1. Then there exists an order function 6 E 8 such that 1 G=: @ = O,(1). b

1.4. Asymptotic approximations and asymptotic expansions In what follows we study functions O(x,e), x E D,E E (O,eO], in arbitrary subsets Do c D. Naturally, all results and definitions also hold in the whole domain D, if one lets Do coincide with D. The concept of asymptotic approximation is most easily defined for functions which are Os(l).We then have: Definition 1.4.1. Let @(x,E) be a function such that G = 0,(1) in Do c D. A function Gas(x,e)is an asymptotic approximation of @(x,E)in Do if

G - Gas= o(1) in Do. By Lemma 1.3.1 and its corollary we know that any function, for which the norm is continuous in E , can be rescaled to a function that is Os(l).Thus

ASYMPTOTIC DEFINITIONS AND PROPERTIES

12

CH. 1, Q4

Definition 1.4.1 can immediately be generalized as follows: Definition 1.4.2. Let O(X,E) be such that O = 0,(6,) in Do c D, 6, E 6.A function Oasis a non-trivial asymptotic approximation of O in Do if 1

- {O - Oa,>= o(1)

6,

in Do.

The explicit statement ‘non-trivial’ in the above definition is made to allow later the possibility that one is not interested in computing Oasbecause the order of magnitude 6, is very small. We shall discuss this case at the end of this section. In applications the function O is usually not explicitly given, but only defined as a solution of some problem. In such cases Definition 1.4.1 may seem ‘not operational’, because the order function do, defining the sharp order of magnitude of O, may not be known a priori. We observe however that if a function Oasis an asymptotic approximation of O in the sense of Definition 1.4.2, then necessarily Oas= 0,(6,). We may thus modify Definition 1.4.2 into: Definition 1.4.2*. A function Oasis a non-trivial asymptotic approximation of @ in Do c D if

O - Q a s = o(6,) in Do where 6,

E

8’is such that

Oas= 0,(6,)

in Do.

Definitions 1.4.2and 1.4.2* are entirely equivalent. In fact, the functions @ and Oascan be considered as each others asymptotic approximations. Given as asymptotic approximation Oasof @, one may attempt to construct a ‘better’ asymptotic approximation. This can be accomplished by repeated application of Definition 1.4.2 and Lemma 1.3.1 as follows: Let O = 0,(6,) and let 4, be an asymptotic approximation of @. We define 1 6,= [@

6,

-

401

and assume that 6lis a non-trivial function. By Definition 1.4.2 we have 6,= o(1) and from Lemma 1.3.1 it follows that there exists an order function = o(1) such that 1 a -- s, Let now

=

O5(1).

41be an asymptotic approximation 62= Ol- $1= o(1).

of

Then

CH. I , $4 ASYMPTOTIC A P I ’ K O X I M A T I O N S A N D ASYMPTOTIC‘ t X I ’ A N S I O N S

We rescale again, (assuming

13

a2to be non-trivial)

1 a2= : = O,(l), 8*= O(1) 62

and proceed to define 42as an asymptotic approximation of a2.Continuing the procedure we find:

c 6”(44,(X,&)+ 0(6,(&)) m

@(X,E) =

n=O

-

where 4, = O,( l), 6, = 6 , -*- 8 ,. Since hi = o(l), i = 1, ..., n, we have 6, = o(6,- l). An expression for @(x,E)as given above is called an usymptotic expunsion of 0 to m + 1 terms. We shall now formalize this concept and summarize some of the preceding results. As a preliminary we generalize Definitions 1.2.2 and 1.2.3.

-

Definition 1.4.3. A sequence of functions {@,(x,E)):,x E Do c D is un usymptotic .wuence if {Il@,J): is an asymptotic sequence, that is if ~l@,ll = 0,(6,), 6 , = o(6, - 1 ). A series C ; = o @ , ( ~is, ~an) usymptotic series if {@,): is an asymptotic sequence.

Remurk. Any asymptotic series can be written in the rescaled form m

2

6,(44,(x,a

n=O

with 4, = 0,(1) and 6, = o(6, - l). It is in this form that we define asymptotic expansions.

Definition 1.4.4. Let @;T’(X,E) = ~ , ( E ) ~ , ( x ,4, c )= , O,(l),be an asymptotic series in x E D o c D. @::)is an usymptotic expunsion to m + 1 terms of @ in D o if @ = a:+ ’ o(6,) in Do. It should be clear from the preceding discussion that an asymptotic expansion can always be studied as a repeated process of asymptotic approximations. If, in Definition 1.4.4, m can be chosen arbitrarily large, then an infinite asymptotic expansion is obtained. Uniqueness of an asymptotic expansion, for any given function @(x,c), is not implied, even in the case of an infinite expansion. One reason for non-uniqueness arises as a consequence of the library that one may have when defining the elements of the sequence {6,). For example consider the function: (D(X,&)

{ ;

= 1-EX}-1; x

€

C0,ll.

14

ASYMPTOTIC DEFINITIONS AND PROPERTIES

CH. 1, 84

One can easily construct two different asymptotic expansions, given by the following formulas:

CD

=

1

m

c

+

&"X(X

- 1)"-

+ O(Ern),

Vm.

n= 1

A second, and somewhat more fundamental reason for non-uniqueness of asymptotic expansions arises as a consequence of the Du Bois-Reymond theorem. and can be demonstrated as follows. Suppose that for a given function @(x,E)= 0,(1) we have for all m an expansion, m

1 6n(&)4n(xjEI + o(6rn).

Q(x,E) =

n=O

Then for any sequence 6, there exist order functions 6' = 0(1),such that 6' for all n. Consider now the asymptotic series

< 6,

m

where $,, are arbitrary functions satisfying @ n - $, = O(S?).Any such asymptotic series is again an asymptotic expansion of the function 0. The lack of uniqueness of expansions needs not be a disadvantage. On the contrary, in applications one can use this property to modify an expansion to obtain some special property. For example, suppose one studies a function CD(x,&),0 < x < 1, which satisfies boundary conditions (D(0,E) = CD(1,E) = 0.

Suppose that, by some procedure, an asymptotic expansion m

(D(X,E) '

C

=

E"@,(X,E)

+qEm

+

1)

n=O

is obtained, with the property @,(O,E) = 0; @,,(l,c)= O,(e-'/&).

It is now possible to modify the expansion, so that CD;;)(x,&) will satisfy the boundary conditions imposed on 0.To achieve this, one may take an arbitrary bounded function x + O(x) satisfying O(0) = 0;

O(1) = 1

CH. 1 , 9 4 ASYMPTOTIC APPROXIMATIONS AKD ASYMPTOTIC EXPANSIONS

15

and define a new expansion by m

c

=

@%E)

Efl{&(X,E)

- @(x)4n(L&)}.

n=O

The above example is characteristic of a situation which often arises in perturbation problems. An infinite asymptotic expansion by an infinite asymptotic series may either converge or diverge. The question of convergence is of no particular importance in the asymptotic theory. We mention, in this connection, the following wellknown phenomenon: Suppose that @ has an asymptotic expansion in the sense of Definition 1.4.4, valid for all m, and that m

lim m-rm n

16,$, =

exists,

~

then it may well be that m

@(X,E)

# lim

m-3) n

=

~n(&)$n(x4 ~

A nearly trivial example of this behaviour arises as follows: Let @(x,E) + @,(x,E) where Q1 is a function possessing a convergent expansion

= O1(x,&)

c 7)

@l(X>E) =

En$n(X)

n=O

while Q2(x,c)= O,(e-'")). Then obviously we have, for all m, the expansion

c m

@(X,E) =

Eflf#Jn(X)

+ O(E"

+

1)

n=O

but m

@(X,E)

#

1 En$n(X).

n=O

The preceding example also illustrates a remark made earlier, that in certain cases one may not be interested in computing the expansion of a function because the terms involved are smaller than a preassigned sequence of orders of accuracy. Thus, in the example considered above, we computed the expansion of cD, by simply putting @, equal to zero. We have implicitly admitted the trivial function as an approximation of 0 2 It , will be useful for the further analysis to formalize this procedure. Definition 1.4.5. Let @ = 0,(6,) in Do c D and 6, = o(1). Then zero is an asymptotic approximation of cD up to the order of magnitude of do, in Do c D.

ASYMPTOTIC DEFINITIONS A N D PROPERTIES

16

CH. I , k5

1.5. Regular approximations and regular expansions Definition 1.5.1. Suppose there exists an order function 6, function $ o ( x ) independent of E , such that 1 - {0- 6,4,) = o(1) in Do c D,

(E)

and a non-trivial

60

then G0(c)4,(x) is a non-trivial regulur usymptotic upproximution of @(x,E) in D o c D. Definition 1.5.2. Suppose there exists an asymptotic sequence { S , , ( E ) ) ~ and a sequence of non-trivial functions $,,(x) independent of E , such that the series

c w)4n(4 m

Q):yw =

n=O

is an asymptotic expansion of ~ ( x , Ein) D o c D . Then 0:y)is a regular usymptotic expansion of @ in D o c D. Remurks. Obviously, regular asymptotic approximations and regular asymptotic expansions are special cases of the approximations and expansions studied in Section 1.4. Regular expansions are sometimes called Poincareexpansions, while expansions in the sense of Definition 1.4.4 are called generalized expansions. The special properties of regular approximations appear more clearly if we further analyse Definition 1.5.1 as follows: ) a family of functions 4&(x),where We identify again the function O ( X , Ewith $ & : D o+ R", 4&E B(Do) and v ( D o ) is a linear space with a given norm 11 *[I. If ~ ( x , Ehas ) a regular~approximationin the sense of Definition 1.5.1, then we have

This simply means that the rescaled family of functions ( l/hO(&))4,(x) converges in norm to 4o (x) as E 10. Or, in yet other words, there exists a function 40(x), which is a limit of (l/h0) 4&as E 10, with convergence in the given norm 11 * 11. We shall now continue the analysis of regular expansions for the special and important case in which the norm used is the supremum-norm. We also stipulate that the subdomain Do in which the regular expansion is studied, is independent of E. Given that

'

CH. I , b5

R E G U L A R A P P R O X I M A T I O N S AND R E G U L A R E X P A N S I O N S

17

it follows that for any arbitrary but fixed x E Do

I

- 4 0 ( x ) = 0;

hence 1 4 , ( x ) = lim -@(x,E). r:10 do(&)

Repeating the reasoning we obtain, for any x E Do

Furthermore, from a fundamental property of uniform convergence it follows , $,,(x), that if @ ( x , E )is a bounded continuous function in Do for all E E ( O , E ~ ]then for all n, are continuous in Do. We now study the question of uniqueness of regular approximations. Lemma 1.5.1. Consider @ ( x , E ) and let there exist two non-trivial regular approximations, that is @(X,E)

= dO(E)4O(X)

@(X,E)

= s;

+ o(d,),

( E ) 4 ; ( x ) + o(6;).

Then limE+o6;(e)/d0(~) furthermore

= c,

where c is a constant unequal to zero, and

40(4 = c 4 m . Proof. We write

and we use the fact that

Reformulating somewhat we obtain

where r 1 and r2 are functions which, for any x E Do, tend to zero as E 10. We now consider arbitrary but fixed points x E Do, chosen such that 4,*(x) # 0. Then, for sufficiently small E , we also have $g(x) + rl(,x,E)# 0 and we may hence write

18

ASYMPTOTIC DEFINITIONS AND PROPERTIES

CH. I , b5

It follows that the limit of ~ ; ( E ) / C ~ ~ (as E ) E 1 0 exists. Now, the limit, if it exists, is unique, so we write

Returning to an earlier formula we can now state that, for any x E Do,

Finally, it is obvious that c # 0, for otherwise 40(x) would be a trivial function. Thus, the non-uniqueness of asymptotic expansions persists in the case of regular expansions, although Lemma 1.5.1 does establish a relation between the different regular approximations of any function. In Section 1.6 we shall show that non-uniqueness of regular expansions can be removed by imposing further suitable restrictions on the order functions. Let us now consider anew, from the point of view of the preceding analysis, the question of admitting approximations by the trivial function. If in Definition 1.5.1 we drop the requirement that C#Jo be non-trivial, then the approximation by zero appears as a (trivial) regular asymptotic approximation. This leads to the modification of Definition 1.4.5, to: Definition 1.5.3. Let O = O,(d0)in Doc D and 6, = o( 1). Then zero is a regular asymptotic approximation of 0 up to the order of magnitude of do, in D o c D .

Regular approximations and regular expansions are important tools of the asymptotic analysis, for two reasons: Firstly, when studying a given function O(x,c), which depends on the variables x and E in some complicated way, not much is gained if we represent this function by an asymptotic approximation Oar(x,e) which is an equally complicated function. The aim of the asymptotic analysis is to represent O(X,E)in terms of simpler functions. Now simplicity, as well as complexity, are difficult to describe and define, however, it should be clear that the terms of a regular expansion $,(x) are 'simpler' than the terms $ , ( x , E ) of generalized expansions because $,(x) is independent of E . Secondly, and this is even more important, if we know that a regular expansion exists then, in the case of supremum norm and for &-independent domains we also know that the terms of the expansion can be defined by the simple limit processes described in this section. In the case of general expansions as defined in Definition 1.4.4, we have as yet no information about the

CH. 1, 56

G A U G E FUNCTIONS, A N D G A U G E SETS

19

procedures which would permit us to construct the terms of the expansions. We finally remark that a function @(x,E)may have a regular expansion to a strictly finite number of terms, and an infinite asymptotic expansion of a more general structure. The question of regularity must in fact be investigated anew in every step of the construction of the terms of the expansion. This is illustr,ated by the following example: Consider @(x,E)= @ , ( x , E )+ E ~ ~ - ~ / ~ @ , ( X ,xE E) , [O,l] where p is some positive integer, while @, and D2 have convergent series representations X

@,(x,E) =

1~'4:') (x)

and

Q2(x,c) n=O

n=O

Then, for m < p we have a regular expansion

c m

@(X,E)=

+

En+:l'(X)

O(Em)

n=O

while for m 2 p we have the expansion

c m

@(X,E)

=

xr

m -

En4k1)(X)

-t

n

(x)e-X/&,

En + ~ 4 k 2 )

n=O

n = O

This last expansion is of course not regular.

1.6. Gauge functions, gauge sets and the uniqueness of regular expansions In applications the order functions occurring in expansions are chosen as simple as possible. They usually are elements of one-parameter families of functions such as E P , (In 1 / ~ ) 4 , exp( - s / f ) , or products of such functions, where p , q and s can be any real number and CJ any positive number. These functions are called (elementary) gauge functions. Depending on the needs of the analysis one sometimes uses somewhat modified gauge functions, such as for example ( a + In l/s)q, q < O , where a is some constant. We shall now define sets of gauge functions. to be called gauge sets, through their useful properties. Definition 1.6.1. A gauge set 6, is a subset of 8 such that: (i) For any two elements 6,,6, E 6, either 6, < 6, or 6, < 6, holds. (ii) Given any 6 E z0,there exist elements 6,,6, E 6, 'such that

6,

< 6 < 6,.

(iii) Given any pair 6,,6, that

6,

< 6i < 6,.

E

go,with 6, .< 6,, there exist elements 6,E 6, such

ASYMPTOTIC DEFINITIONS AND PROPERTIES

20

CH. I , 46

A gauge set is a ‘measuring rod’ of the asymptotic analysis. If a function corresponds, by the sharp order of magnitude symbol, with an element of some given gauge set, then the correspondence is unique within the given gauge set. By this property the non-uniqueness of regular expansions can be removed. We can state this result as follows:

Lemma 1.6.1. Let Vo(D,,6,) he the set of functions of the structure 6,,(~)6~(x),x E Do, 6,

( x , ~-+)

E

6,,

n

where 6, is some guuge set. If .some given function @ hus u regular expunsion belonging to V,(D,,R,), then this expunsion i s unique in V,(D,,k,).

Proof of the lemma is almost trivial. We use Lemma 1.5.1 and observe that in 6, one cannot have lime-, iig/S, = c # 0. Hence, the regular asymptotic approximation S , ( E ) ~ , ( X )is uniquely defined. Since the regular as’ymptotic expansion is obtained by a repeated process of constructing regular approximations, the uniqueness of the expansion is assured. Lemma 1.6.1 also permits another interpretation of the process of construction of regular expansions. Let V(D,,d,) be a set of functions which have regular expansions to m 1 terms, that are elements of V,(D,,d,). This means that to every element @ E V(D,,d,) there is assigned (by construction of the regular expansion) a unique element of V,(D,,d,). Hence a mapping of V(Do,60) into Yl(D,,do) is defined. Let us write, for every @ E V(Do,60), @(x,c)= hn(i;)$,(x) o(hm)and abbreviate

+

c;=

+

c S , ( C ) ~ , ( X ) Vo(Do,do). m

Em@

=

E

n =

0

We may now consider E m as a mapping: E m : V(D,,dO) + V,(D,,d,).

E m will be called a (regular) expunsion operator. We shall develop further the concept of expansion operators in the next chapter (Section 2.4), and use expansion operators extensively in Chapter 3. These operators will provide us with a convenient shorthand notation for operations on expansions, which would otherwise be very cumbersome.

CHAPTER 2

FUNCTIONS WITH SINGULARITIES ON SUBSETS OF LOWER DIMENSION (BOUNDARY LAYERS) Consider functions O(X,E), x E D c R",E E (O,co] such that no regular approximation in D exists. Then, roughly speaking, two main types of singular behaviour can be distinguished. In the first case the function is regular (i.e. possesses a regular approximation) everywhere in D with the exception of 'small' neighbourhoods of certain manifolds of lower dimension. This behaviour is usually called boundary layer behaviour. In the second case there are n-dimensional subsets D o c D,with non-empty interior, (which may coincide with D) such that the set of xo E D o for which the limit of the rescaled function @(x,,,E) = O,( l), as E 10, does not exist, is dense in Do. This behaviour can be called oscillatory. We shall not study, in this book, functions with oscillatory behaviour. The technique of analysis for such functions is in general quite different from the analysis of functions with boundary layer behaviour. For an orientation on the formal techniques that are useful in the oscillatory case the reader can consult for example Nayfeh (1973). We further mention, as sources of information, Roseau (1966 and 1976). In what follows we deal exclusively with functions exhibiting boundary layer behaviour, and no oscillations. We start with precise qualitative description of the singular behaviour due to boundary layers, and then consider regular approximations in subdomains of D. This leads to the so-called extension theorems. Next (Section 2.3) we study the singular behaviour in the boundary layers by local asymptotic analysis. This leads to the concepts of local limits, and locally regular expansions. We shall finally formalize the apparatus of regular expansions in subdomains and local expansions, through the introduction of expansion operators (Section 2.4). These operators provide a convenient notation for manipulations with various expansions and simplify the analysis of relations between different expansions. We further introduce, in Section 2.4, expansions that are truncated to a prescribed order of accuracy (instead of a prescribed number of terms). Such expansions are again symbolized by suitably defined expansion operators. The singular behaviour of functions due to boundary layers is most easily recognized and understood when working with the supremum norm as a measure of order of magnitude of functions. This is why most of the analysis of this chapter will be done in the setting of uniform convergence. However, we

22

FUNCTIONS WITH SINGULARITIES

CH. 2, $1

shall also, occasionally, state and demonstrate results in which any other definition of norm may be used. Present chapter is, in a sense, introductory to Chapter 3 in which, starting from local expansions and regular expansions in subdomains, matched asymptotic expansions of CD, valid in the whole domain D, will be deduced and studied. Regular expansions in subdomains and local expansions in boundary layer regions, of the type studied in this chapter, are usually called ‘outer’ and ‘inner’ expansions in the applied mathematical literature. The terminology has its origins in problems of fluid dynamics, and in-particular problems in infinite domains (see for example Van Dyke (1964)). We do not adhere to this terminology (in spite of its wide use) because it becomes devoid of its intuitive meaning, and therefore confusing, in many problems that will be of interest to us. However, the reader who prefers the adjectives ‘inner’ and ‘outer’ may at any time employ the substitutions: ‘local expansion’ = ‘inner expansion’ and ‘regular expansions in a subdomain’ = ‘outer expansion’.

2.1. Qualitative description of singular behaviour If a function CD(x,&),x E D c R”,E E (O,cO], has a regular approximation in some subset D o c D, then we shall say, for brevity, that 0 is regular in Do. We consider functions 0 such that the union of all subsets in which CD is regular does not coincide with the domain D. However, the complement in D of the union of all subsets in which CD is regular is a ‘small’ neighbourhood of a subset of lower dimension 3. 3 will generally be taken a union of manifolds of dimension lower than n. Thus, if n = 1 then Scan be a,collection of isolated points, if n = 2 then Scan be a union of isolated points and lines, etc. The boundary layer behaviour can be described as follows: Definition 2.1.1. Let a function (D(x,E), x E D c R”,E E (O,cO] be such that there exists a manifold of lower dimension S c D with the following properties: (i) CD is not regular in any n-dimensional, &-independent,subset of D that contains points belonging to 5. (ii) Any point xo E 0-5belongs to some n-dimensional, &-independent,subset of D in which CD is a regular. Then CD has its singularities on 3, and we say that CD exhibits boundary layers along 9.

CH. 2,52

REGULARITY IN SUBDOMAINS. EXTENSION THEOREMS

23

For example, consider @(x,E)= e-x’a- 1, x E [O,l], and let 11 - 1 1 be the supremum norm. It is obvious that 0 is not regular in [O,l]. However, let xo be an arbitrary point belonging to (0,1]. Then there exists some p > 0, such that xo E [p,l] and @ is regular in [p,l]. Hence 9 consists of the point x = 0 and 0 has a boundary layer at x = 0. The terminology of ‘boundary layers’ has its physical roots in the fluid dynamics, where it was introduced by Prandtl(l905) in describing the motion of viscous fluids near solid boundaries. Originally a ‘boundary layer’ meant a thin layer of fluid in which due to viscosity rapid variation of fluid-velocities takes place. Later developments in the mechanics of continuous media brought to evidence frequent occurrence of ‘boundary layer behaviour’, caused by different physical small parameters. The terminology has further widely been used in the singular perturbation theory, as a loose description of the behaviour defined by the properties (i) and (ii) formulated above. In applications boundary layers most often occur along parts of the boundary of D. However, one can also have ‘interior’ or free boundary layers. An elementary example of this behaviour is provided by the function @(x,E) = tanh

X

-,

x E [ - 1,1].

&

Let the norm of the restriction of @ to any subinterval I again be defined by = SUPxe,l@l. The function 0 is not regular in D. However, for any arbitrary number p , with 0 < p < 1 we have the regular approximations

Il@ll

@(x,E)=

1 + o(1) for x E [p,l],

@(x,E)= - 1

+ o(1)

for x E [ - 1, - p ]

At x = 0 the function @ has a boundary layer. A less elementary example of the existence of an interior boundary layer is given by the following formula, which arises in the theory of heat conduction.

where x 1 E [ - 1,1], x2 E [0,1] andf(() is an arbitrary continuous function with f ( 0 ) # 0. It is left to the reader to verify that (in the norm of uniform convergence) @ has a boundary layer along x1 = 0.

2.2. Regularity in subdomains. Extension theorems Consider @(x,&),x E D c R”,with D a bounded set, and let @ have boundary layers on a union of manifolds of lower dimension 9. Then @ has regular

FUNCTIONS WITH SINGULARITIES

24

CH. 2,42

approximations in &-independent compact subsets of D that do not contain points of $. We shall show that, in a certain sense, the validity of the regular approximation can be extended up to s. This result, a so-called extension theorem, plays an important role in the foundations of the method of matched asymptotic expansions, to be studied in Chapter 3. For the further development of that method we shall also need other extension theorems, concerned with cases in which the domain D is unbounded, or grows without bound as E 10. In what follows we shall derive extension theorems under rather general conditions, for functions defined in D c R" of arbitrary dimension, and without specific choices for the norm of the function. This can easily be done, because, as we shall show, the extension theorems can be considered as consequences of two rather elementary lemmas on monotonic functions. We commence with the onedimensional situation, in the setting of uniform convergence. To define the ideas, and introduce the subject, we start with a classical result due to S . Kaplun (1957, 1967).

Theorem 2.2.1. Let @(x,&),x E [O,l], E E (O,E~],be a continuous function on [O,l] x (O,cO] and let there exist a continuousfunction 40(x),x E (O,l], such that, for any d > 0, uniformly in d < x < 1, lim(@(x,&)- $,(x)} &+O

= 0.

Zhen there exists an orderfunction B(E) = o(1) such that lim{@(x,&) - 4,(x)} = 0

&+O

unformly in 6 ( ~ < ) x

< 1.

Remark. The assertion of the uniform convergence of the limit in Theorem 2.2.1 should be understood in the following sense: Let I, denote the interval I, = (xl6(&)< x ,< l}, then lim{supl@(x,c)- 40(x)I} = 0. 8-0

xcr,

As an explicit elementary example consider @((x,E) = e-x'E, x E [OJ]. For any d > 0 we Rave, uniformly in x E [d,l], @(x,E)= 0. Furthermore, zero remains a regular approximation in an extended domain defined by a(&) < x < 1, where B(E) may be chosen any order function satisfying E = o(6). We shall prove Theorem 2.2.1 as an application of a somewhat more fundamental result, given in Lemma 2.2.1, which in spirit is related to Theorem 2.2.1, but has a wider applicability.

Lemma 2.2.1. Let g(c,d),

E E (O,cO], d E (O,dO]c

R,, be a real positive function,

CH. 2,52

REGULARITY IN SUBDOMAINS. EXTENSION THEOREMS

25

monotone decreasing with respect to d , that is, such that d ' > d implies g(E,d') < g(E,d). Suppose that for any d E (O,d,] lim&+og(e,d) = 0. 7hen there exists an orderfunction B(E) = o(l), such that lirn&-, g(E,B(E))= 0.

Proof. We are given that for any fixed d , and any p > 0, there exists q(p,d) such that g(E,d) < p

for 0 < E

< q(p,d).

Let us take any two monotonic sequences { p , } ; and {d,,};, decreasing to zero as n + co. We define furthermore a monotonic sequence {q,}:, decreasing to zero, by the formula

Then g(E,d,) < p n for 0 < E

< 4,.

Furthermore, because of monotonic behaviour of g(E,d) with respect to d , we have, g(c,d) < g(&) < pn,

for 0 < E

< q,,

d 2 d;

We now define, by any convenient construction, a monotonic continuous function E + B(E), satisfying the condition B(4,) = d,- 1 . It is clear that B(E) = o(1). Consider now the function g(E,B(E)).Let p be any positive number. There exists an integer m, such that p , < p . Consider the interval 0 < E < qm. For any value of E in that interval one can find an integer n 2 rn, such that E E [qn+l,qn]. If E is in such subinterval, then d,

< a(&) < d,-

1

and hence

&,W) < g(E9dn)G P, < P,

< P.

Thus g(E,b(E))< p

for 0 < E

< 4,,,.

This proves the lemma. We obtain now proof of Theorem 2.2.1 by a straightforward application of Lemma 1.2.1. We write g ( 4 )=

SUP xc[d, 11

I@(x,4I.

FUNCTIONS WITH SINGULARITIES

26

CH.2,92

Clearly, the function g(E,d),thus defined, satisfies the conditions of Lemma 2.2.1. We next deduce a result closely related to Lemma 2.2.1.

Lemma 2.2.2. Let g(c,d), E E ( O , E ~ ] ,d E [do,co), be a real positive function, monotone increasing with respect to d , that is such that d’ > d implies g(E,d’) 2 g ( 4 . Suppose that, for any d E [do,=), limc+og(E,d) = 0. Then there exists an orderfunction 6 ( ~= ) o(l), such that &+O

Proof. We write d = l / a a n d g(e,l/a) = g(e,d). The function g(e,d),thus obtained, satisfies conditions of Lemma 2.2.1, which proves Lemma 2.2.2. Lemma 2.2.2 permits to prove the following counterpart of Theorem 2.2.1.

Theorem 2.2.2. Let (D(x,E), x E [O,co), E E ( O , E ~ ] be a continuous function, and suppose there exists a function x -+ $o(x), x E [O,co), such that, for any d > 0, uniformly on x E [O,d] lim{@(x,&)- 4,(x)} = 0. &+O

?hen there exists an orderfunction 8 ( ~=) o(1) such that lim {sup 1 @(x,E)- &(x) 1 } E+O

=0

xel,

where I, = {x 1 x E [O,l/d(~)]}.

Proof. Write sup,,~o,dl~ ( D ( X , E) 40(x)l = g(E,d) and apply Lemma 2.2.2. In applications one often has to deal with functions which, instead of being defined on R, as in Theorem 2.2.2 have a domain which increases without bound as E 10. This requires a modification of Lemma 2.2.2.

Lemma 2.2.2 bis. Let g(E,d) be a real positive function, of which the domain is given where $(E) is a positive monotonic function, growing by E E ( O , E ~ ] ,d E [do, $(&)I, without bound as E 10. g(E,d) is monotonic increasing in d, i.e. for any pair of numbers (d,,d,) E [do,\l/(~)],d‘ > d implies g(E,d’) 2 g(&,d). Suppose that for any arbitrary, but fixed value of d , g(E,d) = 0. Then there exists an order function 6 ( ~= ) o(1) such that

CH.2,42

REGULARITY IN SUBDOMAINS. EXTENSION THEOREMS

Proof. We define a function $(&,d),E E (O,E~],d

$W)=

{:!;&&)

E

27

[ d o , 00) as follows:

for d E [d09;$(&)19 for d > $I)(&).

i

Applying Lemma 2.2.2 we have the existence of an order function a(&)= o(1) such that lim, $(&,$(&)- I ) = 0. Let now B(E) be an order function defined by B(E)- = min($((e)- $ $ (8)).

Then B(E) = o(1) and limZ1 g(e,B(&)-')= 0. We can now generalize Theorem 2.2.2 and obtain

Theorem 2.2.2 bis. The Theorem 2.2.2 holds if the function Q,(x,E) under consideration has a domain given by x E [ O , $ ( E ) ] , E E (O,E~],where $(&) is a positive monotonic function, increasing without bound as E J 0. Remarks. Proof of Theorem 2.2.2 bis is of course obtained by a straightforward application of Lemma 2.2.2 bis. On the other hand, Lemma 2.2.2 bis also permits some further generalizations of Lemma 2.2.1 and Theorem 2.2.1. Thus, in Lemma 2.2.1 the domain of g(c,d) can be replaced by E E ( O , E ~ ] x, E [&E), 11, where e"@) is a monotonic function satisfying e"= o(l), and similarly, in Theorem 2.2.1, the domain of Q,(x,E) can be replaced by E E (O,E~],x E [&(~),1], with e"(&) a monotonic function, & = o(1). The proof of these statements is left as an exercise to the reader. We shall now show that Lemma's 2.2.1, 2.2.2 and 2.2.2 bis, permit to deduce results on extended domains of validity of a regular approximation, in a much more general setting then Theorems 2.2.1, 2.2.2 and 2.2.2 bis. There is in fact no need to restrict oneself to the one dimensional situation, nor is there any need to put specific emphasis on the norm of uniform convergence. We consider Q,(x,&),x E D c R" and assume that a definition of norms is given in accordance to Section 1.3. We assume that the norms have consistently been defined as described in Definition 1.3.4. In particular we shall need the following.

Property. If D, and D, are any two subdomains and l l * l l D 1 , Il'lls are the norms on the restrictions of functions to those subdomains, then D, c D, implies ~ ~ Q , ~6~ \I'll&* D l We shall first deduce a generalization of Theorem 2.2.1. Theorem 2.2.3 states that, if Q, is a regular in any compact set not containing g, then, under certain conditions on f,Q, is regular in a subdomain D,, of which the boundary r, contains a part that approaches S arbitrarily close as E 10. More precisely, we have

FUNCTIONS WITH SINGULARITIES

28

CH. 2, $2

Theorem 2.2.3. Let S c $be a manifold of dimension k < n, along which @ has a boundary layer. Consider any one-parameter family of compact sets D, c D, satisfying the following conditions: (i) D, are ordered by inclusion, i.e. d" < d ' =. D,,c D,,,. (ii) The parameter d is defined by d = sup { inf [di~t(x,,x,~)]} XSS

X E r d

where r, is the boundary of D,, x, and x r d denote points on S, respectively r,, and dist(x, y ) is the Euclidian distance. (iii) D, is defined for any d E (O,do]. Suppose @(x,E) is regular in D,, for any d E (O,d,]. Then @(x,E) is regular in a where D, is Vor any E E (O,eO]),a compact set with boundary re, domain D, c 0-5, and lim sup { inf [ d i ~ t ( ~ , , ~ , ~=)0. ]} E - O X E S

X

E

~

~

Proof. We are given that there exists a function @,(x) and an order function JO(&) such that, for any d E (O,d,]

1

- $Jo(x)

= 0.

IIDd

Write now

The function g(&,d)satisfies conditions of Lemma 2.2.1. Hence there exists an order function h ( ~=) o(1) such that g(E,d(E))+ 0 as E -,0. Taking, for any E E (O,E,], d = B(E) we obtain from D, a family of compacts D, in which 0 is regular. The assertion stated in the last line of the theorem follows from the definition of the parameter d in condition (ii) of the theorem. In a similar way we can also obtain a generalization of Theorems 2.2.2 and 2.2.2 bis.

Theorem 2.2.4. Let D be an .+independent unbounded domain in B" and let D,, d E [O,co), be a one-parameter family of bounded domains, with d ' > d = Ddf =I D,, D, = D, such thatfor any compact K c D, there exists d > 0 for which K c D,. Let @ be defined in D,,,,, where $ ( E ) is a positive monotonicfunction, that grows without bound as E 1 0, and let @ be regular in any .+independent compact K c DYy(,,. Then there exists an order function J ( E ) = o( 1) such that @ is regular in

u;=,

D6(e)-'.

LOCAL ANALYSIS OF CONTINUOUS FUNCTIONS

CH. 2,53

29

Proof. Write

and apply Lemma 2.2.2 bis.

2.3. Local analysis of continuous functions; local limit functions and local expansions In this section we develop some basic tools for the analysis of functions @ near

g, that is, in the boundary layer region. We consider functions c R",

@(x,&), x E D

in both variables, and define the norm of the restriction of @ to any subset Do c D by E

E (O,eO], continuous

ll@ll

=

SUP

I@(X,&)I.

x E Do

If in some subset Do a regular approximation 6 0 ( ~ ) ~ of 0 (@~( x), E ) exists, then, by a basic property of uniform convergence, 40(x) is continuous in Do. For simplicity of exposition we first consider the one-dimensional situation. We take D = { x I x E [O,l]) and assume that @ is regular, except for a singularity at the origin. Hence there exists a function such that for any number d E (0, I], uniformly in d < x 6 1,

By the extension theorems of the preceding section, the above limit is also zero for x E [6(~),11 where 6(e) is some order function with 6 = o(1). There remains to be studied a small neighbourhood of the origin, and it seems natural to introduce, for the purpose of the analysis, a mathematical equivalent of a magnifying glass, given by the transformation

The variable 5, for any choice of d,, will be called a local variable. (< is also often called a stretched variable or an inner variable, (Nayfeh (1973), Van Dyke (1964, 1975)). This denomination will not be used in the sequel.) Performing the transformation we shall write

@(6,5,E) = a)*(<,&) We define now, as a tool for further analysis, a new kind of limit of the function @ ( x , E ) ,to be called a local limit.

CH. 2,§3

FUNCTIONS WITH SINGULARITIES

30

Definition 2.3.1. Let there be given a transformation to local variable 6, = o(1). For any function F ( x , E the ) local limit is given by lim F ( X , E ) 5

5 = x/S,,

lim F * ( ~ , E ) E-0

where F * ( ~ , E=)F(S,(E)<,E). A local limit, if it exists, defines a function of the variable introduce local limit function of the function @(x,E ) .

t, This leads us to

Definition 2.3.2. Suppose @(Ss[,&) = @ * ( ( , E ) = OS(Sg)for 5 E [A,B], where A and B are some constants. A function $o(() is a local (non-trivial)limit function of @ if, uniformly for 5 E [ A , B ] , 1 lim -0 5

4

=

i,b0(().

If a local limit function exists, then this function defines an asymptotic approximation of the function @ in a subdomain near the point where @ is singular. In fact we have 1 -{@* 6:

- sgq0) = o(1)

for A < 5 < B. If, in the above, it is allowed to take A = 0, then an asymptotic approximation in a full neighbourhood of the origin is obtained. It follows that the function O * ( ~ , is E regular ) (i.e. has a regular approximation) in the sense of Section 1.5, in a domain defined by 5 E [ A , B ] . Hence, if for some local variable ( a local limit function exists in some domain 5 E Do, then the function @ can be considered locally regular in that domain and the transformation to the local variable a regularizing transformation. Local regularity, in some local variable, is a property that functions may or may not possess. It is not difficult to give examples of functions which, in all local variables, have no local limits. An elementary example is the function 1

@(x,E) = [l-e-x’E]sin -, X

x E (0,l-j.

The reader will easily verify that @ given above, is a regular in x E [d, 11, Vd > 0, and that in all local variables 5 = x/S,, 6, = o(l), local limits do not exist. In applications one usually deals with functions which, contrary to the example given above, have local limits for large families of local variables. In such situations one attempts, using the extension theorems, to construct approximations in the whole boundary region by local limit functions corresponding to certain ‘important’ local variables. This procedure is successful in

CH. 2,53

LOCAL ANALYSIS OF CONTINUOUS FUNCTIONS

31

large classes of problems, and is one of the main features of the method of matched asymptotic expansions, to be studied in Chapter 3. We shall now illustrate the use of local limits on an elementary example. Consider

-1+((;)’+i+ y;

@ ( w ) =E

XE[O,l].

The function is regular in x G [d, 13, d > 0, and the regular approximation can be obtained through simple limit calculation: C $ ~ ( X= )

1 lim @ ( x , E )= -. X

&-+O

We introduce a one-parameter family of local variables 4, = x / E ’ , v > 0. The corresponding local limit functions exist for all values of v. In fact, simple calculations show that, if 0 < v < 3,then 1 1 $tJ(tv)=lim-@=5”

6,

t v

where one must choose 6, = E - ” , and the limit is uniform in and B arbitrary positive constants.

t, E [A,B],

with A

If v = i, then $ p z ’ ( t l , z j = lim E ’ ” O =

-tl,2 + (t?,z + 2)’”

1R

for If v >

E

[ A , B ] , A and B arbitrary, positive.

3,then +g)(t,,)= lim E ~ =/ J2 ~ o 5,

for t, E [A,B], A and B arbitrary, positive. Returning now to the case v = $ one can show by straightforward analysis that the domain of validity of the limit can considerably be extended. In fact one finds E-0

uniformly in tl,zE [O,E-’~~]. We thus obtain the surprising result that the local limit function $bllz)(tj z ) produces an asymptotic approximation of (3 uniformly valid on the full interval x E [O,l]. This result is certainly not typical for most applications. As we shall see later on, approximations by local limit functions in the boundary layer region must usually be combined with regular approximations valid outside that region.

FUNCTIONS WITH SINGULARITIES

32

CH. 2,43

We conclude this section with generalizations to multi-dimensional situations of the concepts so far introduced. We recall that we study functions @(x,E),x E D c R",E E (O,eO], which have singularities, in the sense specified in Section 2.1, on n-p dimensional subsets 9, with p 2 1. We shall denote by S any connected subset of 5 The components of the vector x will be denoted by X I , ..., x".

Definition 2.3.3. Let S c D be a manifold of dimension n-p, p > 0, and let there be given a transformation x + X which is one to one and continuous in some Eindependent subset of D containing S and whickis such that S is represented by X' = **-xP= 0. 5 is a local variable along S if the components of 5 are defined by

5" -x i-4 s h(4) with

8F) = o(1) for q = 1,..., p , 1 for q = p + 1, ..., n. We shall indicate, for brevity, the transformation from x to local variable 5 by x -, 4. Introducing into @(x,E) the change of variables x + 5 produces a function of ( and E, to be denoted by @*(<,E). We shall say that x + < induces @(x,E) --i a*(<,&). The domain of @* will be specified when needed.

Definition 2.3.4. Let F(x,E)be a function defined in a neighbourhood of S, 4 a local variable along S , and let the transformation to local variable x + 5 induce F(x,E)-,F*((,E).A local limit of F(x,E)along S is given by lim F ( X , E ) %lim F*(<,E). 5

el 0

Definition 2.3.5. Let x + 5 be a transformation to local variables that induces @ ( x , E ) + @ * ( ~ , E ) , and let DX be the image of some subset of D by the transformation x + 5. Suppose that @*((,E) = 0,(6,*) in D,*.A function $o(<) is a non-trivial local limit function of @ if lim<@/h: = t+b0(<), uniformly for 5 E D,*. If a local limit function .1)~(5) exists for 5 E D o then So$o(5) is a local approximation of @ in D,*.

We remarked earlier that the existence of a local limit function can be interpreted as local regularity of @. We can proceed to define local expansions (of regular type).as in Section 1.5.

Definition 2.3.6. An asymptotic series

CH.2,

THE FORMALISM OF EXPANSION OPERATORS

33

is a local asymptotic expansion of @ for 4 E D,*if @* - a,*,'")= o(6:)

for 4 E D,*.

The local asymptotic expansion defined above should perhaps more properly be called 'locally regular expansion' or 'local limit process expansion'. In general we shall abbreviate and simply speak of local expansions. Finally, in analogy to Section 1.5, local approximations by trivial functions can be defined as follows:

Definition 2.3.7. Let x + g be a transformation to local variables that induces O(X,E)-,@*(<,E), and let ( D * ( ~ , E )=

Os(S,*) for

5 ED,*

where D,*is the image of some subset of D. If a,* = o(1) then the local approximation of @ in D,*is zero u p t o the order of magnitude of 6:.

2.4. The formalism of expansion operators

In the further development of the analysis of functions with boundary layers it will be necessary to study relations between expansions in different variables and to perform, for that purpose, various manipulations on these expansions. For example, a regular expansion, and a local expansion of a function may have extended domains of validity with a non-empty intersection. In such a case one would wish to study both expansions in their common domain of validity. A similar situation may arise when studying two local expansions of a function, in two different local variables. In this section we shall develop the formalism of expansion operators, to provide us with a convenient language for manipulations with expansions. We have already introduced, in Section 1.6, the concepts of regular expansion operators. We have also remarked that local expansions as defined in Section 2.3 are of an essentially regular structure, and are in fact regular expansions of a transformed function. In order to bring out the relations between these concepts more clearly, it is necessary to deemphasize the preceding special interpretations of the variables x and 4. This is why, in the three basic definitions that follow, we use other symbols for variables, and, to avoid any possible confusion with preceding definitions, other symbols for the functions being expanded. Throughout this section the setting for all expansions is some fixed gauge set of order functions, so that uniqueness of regular expansions is assured (compare Section 1.6).

FUNCTIONS WITH SINGULARITIES

34

CH. 2 , g

Definition 2.4.1. Let F : D x (O,eo] + RP,D c R4,be a function that possesses a regular expansion in Do c D; i.e., for the restriction of F to Do, m

F=

C n=O

+ o(dm)

where {6,} is an ordered sequence of which all elements belong to a gauge set 8,;fn: Do -+ R4 are not identically equal to zero. We shall write

n=O

Em will be called a regular expansion operator.

Definition 2.4.2. Consider a mapping ( t , ~-+) F(t,E),

t

E

D@),

E E

(O,E~]

and let there be given another mapping (transformation of variables) Q :D(') x ( O , E ~ -+ ] D(t),

Then

T,F

F*(T,E)= F(Q(z,E),E), z E D'".

Definition 2.4.3. Let F be such that T,F is regular in some subdomain 08)c D('). Then EYF

gfE ~ F 7, E 08).

Remarks and applications. Definition 2.4.1 provides a coordinate-free description of regular expansions, Definition 2.4.2 describes the effect of transformation of variables, while Definition 2.4.3 combines the effect of transformation and expansion. Given a function (x,E)+ @(x,E), x E D , E E (O,cO], Definition 2.4.1 directly applies in case of regularity in subdomains, and we have (in keeping with previous notations) m

Em@(x,&)=

C ~ , , ( E ) ~ , , ( X ) ,x E

Do c D.

n=O

<

Let be a local variable, then in Definition 2.4.2 we can identify x with t, with z and F with @. By Definition 2.4.3 we obtain E;@

= E m q @ = Em@*(<,&), 5

ED:

5

CH.2, @I

THE FORMALISM OF EXPANSION OPERATORS

35

and from Definition 2.4.1 we have the representation m

EYQ =

1 8n*(&)+n(t)*

n=O

Returning to the regular expansion of O(x,c), x E Do c D we notice that this expansion can also be seen as an application of Definition 2.4.3, with T, being the identity transformation. Thus, we may also write m

EzQ =

C

an(&)$n(x)*

n=O

We shall generally prefer the notation E;O, because of the symmetry with EFO, and the clear way it presents variables. We remark that Definitions 2.4.1, 2.4.2 and 2.4.3 also permit us to define expansions in much more general situations than that just described. Thus, in Definition 2.4.2, t and o may be identified with 5 and x, respectively. Then Definitions 2.4.3 and 2.4.1 lead to ) F ( ~ , Eand ) 5 is a local regular expansion E;F, where F is a mapping ( 5 , ~+ variable. Similarly, t and o may be identified with two local variables, say t, and t2.Frequent use of such applications will be made in Chapter 3. In definitions 2.4.1 and 2.4.3 expansions are truncated to a prescribed number of terms. In applications, and in particular when one studies and compares different expansions, more important than the number of terms in the expansion is the accuracy of the truncated expansion as an approximation. We shall therefore separately define expansions truncated to a prescribed order of accuracy. We shall also allow for approximations by the trivial function. We first give the definition in a coordinate-free form.

Definition 2.4.4. Consider an expansion of F in Do in the sense of Definition 2.4.1, i.e. E'F = Let {d:)} be some ordered sequence of order functions, which may be different from the sequence We define the truncation of E'F to a preassigned order of accuracy B$)(E) as an expansion E(")F such that

C;=o&f,. {a,}.

F - E(")F = o(6:)) in Do.

Furthermore, (i) E(")F = 0 if F = o(6:)) in Do, (ii) E("'f= ~ ; = o if~ Fn #~ o(6:)) in Do where v is an integer such that for all n = O,l,..., v(m), 8, # 0(6$));f, is as in definition 2.4.1.

Remarks. Given the choice of a;), the domain of E(") is of course restricted to functions for which E(")F exists (as an expansion to a finite number of terms). On the other hand, if F is regular in Do c D,then different choices of 6;) are

FUNCTIONS WITH SINGULARITIES

36

CH. 2,V

possible for which E(")F exists. For example, let Z,,(E)in Definition 2.4.1 be the sequence with elements:

A non-trivial application of Definition 2.4.4 is obtained by taking 6 ) = E". : From Definition 2.4.3 we obtain an operator E:m) by simply writing def

Ej")F = E'")T,F.

For future convenience we shall write out the definition in detail for the expansions Ekrn)CDand EY'CD. Definition 2.4.4.* Let 6 : ) ( ~be ) any element of a pre-assigned ordered sequence of order functions. (1) Suppose O(X,E)has a regular expansion for x E Do. ELrn)@is defined by (D

- ELrn)@= o(6:)) for x E Do.

(i) ELrn)@= 0 if CD = o(6:)) in Do. (ii) EY)CD = ~ ; = , 8 , ( ~ ) 4 , if( ~CD) # o(6:)) in Do, where 0 is an integer such that for n = O , l , ..., ~ ( m )6,, # o(6:)); 4,, are non-trivial functions. (2) Suppose, in some local variable 5 , CD has a local expansion for 5 E DZ. EP)(D is defined by 7 p - E Y ) @= o(6:))

for 5 E D,*.

(i) E r ) @ = 0 if T,@ = o(6:)) in D,*. (ii) EY'Q = ~ ~ = 0 6 ~ ( ~ ) \ I if/ , (Ts(D S ) # o(6:)) in Dz, where p is an integer such that for n = O , l , ...,p( m), 6: # o(6:)); \I/, are non-trivial functions. Let us consider now more closely the functions obtained by application of the expansion operators. In the definitions we have kept the domains D o (or D;) fixed. In applications however, for functions with boundary layers, regular expansions usually exist in any compact c-independent subset of some open set D c R".Under such circumstances one can define, by extension, a function V

(x,&)

+

C bn(E)4n(X) n=O

with domain X E D , such that the restriction of this function to any Eindependent compact set D, is regular expansion E:") CD. We shall denote such a function by the symbol ELrn)@. Similarly, local expansions usually exist in any compact c-independent subset

CH. 2,94

THE FORMALISM OF EXPANSION OPERATORS

31

of some open set D* c R".One can then define by extension a function P

( 5 9 4

+

5 ED* 1 ~m$n(5),

n=O

such that the restriction of this function to any .+independent compact D,* E D* is a local expansion E y ) @ .We shall denote this function by the symbol I??)@. In our later analysis (in particular Chapter 3), frequent use of the extensions Eim)@, E y ) @will be made. They are formalized through the following definition.

Definition 2.4.5. Let @ have regular expansions in any compact &-independent subset Do of an open set D. Then V

E y @ = 1 dn(&)4n(x),

XED

n=O

is a function such that the restriction of Eim)@ to any Do is the regular expansion EF)@ (in the sense of Definition 2.4.4). Let @ have local expansions in any &-independentcompact subset D,*of an open set D*. Then

EF)@ =

5 ED*

d;(~)i+h,(t), n=O

is a function such that the restriction of E y ) @to any D,*is the local expansion E F ) @ (in the sense of Definition 2.4.4). Introducing the extensions Ex@ and Eta makes it possible to analyse 'expansions of expansions', for example, expressions of the form EF)Eim)@.Let us illustrate what we mean by the one-dimensional example. Let @(x,E), x E [0,1] be regular for x E [d, 11, Vd > 0. Then, by extension, a function Eim)@, x E (O,l] may be defined and, in any local variable t, the expansion EF)Eim)@ may be studied (if such an expansion exists). Suppose further, that in some local variable 5, we have E:'")@, ( E [O,A], V A > 0. Then, by extension a function Eim)@,5 E [O,co)can be defined. We can next study the regular expansion E y y ) @= E'"'T,E:"'@, In Chapter 3 we shall show that, for a class of functions, and with special provisions for the sequence d;), one has the relation Ep)E(m)E(m)@= E ( m ) E ( m ) @ . x

5

5

x

This relation is the so-called asymptotic matching principle, and is fundamental to the method of matched asymptotic expansions. It would be very difficult, and cumbersome, to express such relations without the formalism of expansion operators.

CHAPTER 3

MATCHING RELATIONS AND COMPOSITE EXPANSIONS The process of matching of expansions is an essential tool in the analysis of perturbation problems, where it permits to determine unknown constants and functions occurring in various expansions. There are two schools of matching in the applied mathematical literature: One, originated by Kaplun and Lagerstrom (1957) employs the so-called intermediate variables, and can be justified by the overlap hypothesis, that is, by the assumption that the extended domains of validity of two expansions (for example a regular and a local expansion) have a non-empty intersection. Extensive application of this method can be found for example in Cole (1968). A second method is based on what is called, after Van Dyke (1964), an asymptotic matching principle. This method leads to efficient computations, and has therefore been popular in applications. A justification of asymptotic matching principles can be obtained from suitable hypotheses on the structure of uniformly valid approximations in the whole domain of definition of the function under consideration. An extensive analysis along this line has been presented by Fraenkel(l969). In Eckhaus (1977) it has been demonstrated that an asymptotic matching principle can also be deduced from the overlap hypothesis. Closely related to the problem of matching is the problem of constructing the so-called composite expansions. These are approximations uniformly valid in the domain of definition of the function under consideration, and build up with the aid of the regular approximations in subdomains, and certain ‘important’ local approximations. We begin our analysis in this chapter by formalizing the notion of ‘important’ approximations, which will be called significant. Studying some consequences of the extension theorems, we then make the overlap hypothesis plausible and proceed to develop the theory of matching in the intermediate variables. This development i s side-tracked in Sections 3.5 and 3.6 where asymptotic matching principles are introduced. We deduce the asymptotic matching principles from the concepts of correction layers and regularizing layers. It should be emphasized that these concepts have an importance of their own, and are useful in applications. They also immediately imply the existence of composite expansions. In Section 3.7 we go back to the overlap hypothesis and show that, under certain conditions, it implies the validity of an asymptotic matching principle and the existence of composite expansions. The proof of the main result is 38

CH. 3

MATCHING RELATIONS A N D COMPOSITE EXPANSIONS

39

technically rather involved and lengthy, and is presented separately in the appendix to this chapter. Finally, in Section 3.8, we show by an example, that the overlap hypothesis is a sufficient but not a necessary condition for the validity of an asymptotic matching principle. We use extensively in this chapter the formalism of expansion operators, described and defined in Section 2.4. For a good understanding of the chapter the reader should at least be familiar with the contents of Definition 2.4.4*, which defines regular and local expansions truncated to a prescribed order of accuracy. To facilitate further the study of the chapter we shall now briefly explain again the most frequently occurring operations, for the simple onedimensional case, and assuming non-trivial expansions. Consider functions @(x,&), x E [OJ], E E (O,cO], and the local variables

where 6, are elements of some gauge set of order functions. A regular expansion is given by

c 6fl(&)4fl(x),

a(m)

ELrn)@=

n=O

local expansions are given by

c d:(&)$;)(l,).

r(m)

E(m)@ 5" =

fl=O

The integers p(m), o(m) are defined in Definition 2.4.4*. Thefunction ELm)@is defined by extension of ELm)@to x E (OJ]; similarly, the function E c ) @ is defined by extension of E c ) @ to (, E (0,co). The operators T,, qvdescribe the effect of transformation of variables on functions, with the subscript indicating the new variable. One thus has

Extensions and transformations of variables lead to 'expansions of expansions'. Thus, E?)EI,"j@ is a local expansion in the <,-variable of the function ELm)@,Eg)E:4,'@"can be read as a local expansion in the <,-variable of the function T,EE)@,while Eim)Et)@is a regular expansion of the function TxEtv)@.

40

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

CH. 3, $1

3.1. Significant approximations and boundary layer variables Let @ be a function that exhibits boundary layer behaviour along some manifold S . Performing a local analysis, as described in Section 2.3, one can introduce along S various different local variables (corresponding to different choices of the order functions 6kq) in Definition 2.3.3) and study local approximations in those different local variables. The question then naturally arises: must one study 'all possible' local variables along S ? In the example given in Section 2.3 we have seen that some local variables are 'more important' than others. We shall now formalize the notions of 'more important' and 'less important' variables. In what follows we consider two local variables t1 and t2 to be different if in Definition 2.3.3, for some components q, the order functions 6): in the transformation x + t1 are not of the same (sharp) order of magnitude as the corresponding order functions in the transformation x -+ t2.For example, in the one-dimensional situation, the local variables

cl=--, X

t 2 = &X

6, are considered to be different if 6,# 0,(6,), & # 0,(6,). All expansions will be symbolized by expansion operators described in Section 2.4. When studying some given function we shall always suppose that some (fixed) sequence of order functions {Sq)) has been chosen to define the accuracy of truncated expansions, so that one has

cD - EL")@ = 0(6:'),

Do, cD* - EY'cD = 0(6;'), t E D*. x

E

The expansions EL!")Oand EY)cD are further explicitly described in Definition 2.4.4.* With these preliminaries we have: Definition 3.1.1. Let and t2 be two different local variables along S and suppose that,,for some values of q and p

t1E D f , Eg)cD exists for t2E 0:. E ! f @ exists for

We shall say that E$)cD is contained in E:)cD if, for some values of p ,

Ep:@

= E'q)E'P'@ 51 c2 '

4 1 E D*. 1

One may readily generalize Definition 3.1.1, to allow for cases in which a regular expansion is contained in some local expansion, or vice-versa. For that

CH. 3, $1 SIGNIFICANT APPROXIMATIONS (AND BOUNDARY LAYER VARIABLES) 41

purpose it is sufficient to replace in Definition 3.1.1 either

or t2by the symbol

x:

Definition 3.1.2. Suppose a regular approximation ELm)@, x E Do exists, and furthermore, in some local variable {,, local approximation EFj@ exists for 4 , E 0:. Then

It is clear that an approximation that is contained in some other approximation can be considered 'less important'. We are thus led to search for and to define 'important approximations', which will be called significant. Definition 3.1.3. A regular approximation ELm)@ is significant if ELrn)@is not contained in any local approximation. A local approximation E g ) @is significant if E'@ is not contained in ELm)@, for all m, and furthermore if there exists no 1ocal.ipproximation E f i @ , with 4, different from {,, such that EK)@ would be contained in EFL@. Remarks. It should be clear that significant approximations are important elements for the analysis of @(x,c), x E D. In applications, given a function 0,one usually finds that the number of significant approximations is finite, and often even small, One then proceeds to construct an approximation of @ valid in the whole domain D, using the significant approximations as building blocks. The procedures leading to such a construction will occupy us through the remainder of this chapter. We remark however that one can easily define nice (continuous) functions @(x,E)for which the number of significant approximations is not finite, and even not denumerable. Examples of such pathological behaviour, which fortunately does not often occur in applications, can be found in Eckhaus (1973).

The local variables for which the local expansions are significant, are naturally of particular importance. This is expressed in the following: Definition 3.1.4. If a local approximation is a significant approximation, then the corresponding local variable is called a boundary layer variable. We now illustrate the definitions given above by some examples. In Section 2.3 we have studied the function

42

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

@(X,E) =

-5& +

CH. 3,52

((;)' + ;+ y2

One can verify that there is only one boundary layer variable, 51,2 = x/Jc. The corresponding significant approximation, to the first order of accuracy, is given by E ( rOt )a @

= E- ""-5,,2

+ ( 5 ? ! 2 + 2)' 21.

There is no other significant approximation. Next consider the function E2

@(x,E)= -e-*/&*

, O<x
X+E2

We choose as the sequence defining the accuracy of expansions { E " } . The regular expansion in 0 < d < x < 1 is then zero to all orders of accuracy. One may verify that there are two significant approximations, in the boundary layer variables =;

X

and

X t2 =?.

The significant approximations are given by

where A and B are arbitrary positive constants. The regular approximation is contained in E E ) @ .

3.2. Further applications of the extension theorems; the overlap hypothesis Approximations that are significant in the sense of Definition 3.1.3 are certainly 'important', because they cannot be deduced from some other approximation in other variables. However, the definition does not imply that, given a function @ ( x , E ) x E D, the set of the significant approximations of @ is sufficient to construct an approximation of @ uniformly valid in D. In this section we shall formulate an additional and fundamental condition, called the overlap hypothesis, which in the sequel will assure the possibility of such a construction. In order to bring out matters clearly, we consider the one-dimensional situation, and suppose that @(x,E),x E [0,1] has a boundary layer at x = 0. For

CH. 3, $2

FURTHER APPLICATIONS OF THE EXTENSION THEOREMS

simplicity we assume first that there is just one boundary layer variable defined by

43

to,

Generalizations of this simple case will be discussed at the end of this section. We further assume that the regular expansion Eim)@is a significant approximation valid in any compact interval contained in ( O , l ] , while the local Explicitly: expansion E K ) @is valid in any compact interval contained in [0, a). @ - ELm)@ = o(6:))

Tco@- E E ) @ = o(6:))

for x

E

for

[ d , 11, V d > 0,

toE [O,A],

V A > 0.

The meaning of the symbols used above is given in Definition 2.4.4* We further recall that the operator 7& denotes a transformation of variables, i.e.

To@= @(dS(&)to,&) = @*(to,&). We shall now investigate, using the extension theoreins of Chapter 2, various ways of defining extended domains of validity of Eim)@ and E K ) @ as approximations of @. Lemma3.2.1.Suppose that the regular approximation Eim)@ of @ is such that - Ekm)@= 0(6:’) for x E [ d , l ] , V d > 0. Then for any 1 = O,l, ...,m, there exist order functions b; = o(1) such that

@

@ = ELm)@= o(~:)-J

The order functions

for x E [&I].

& satisfy

Proof. Consider, for any 1 = O,l, ...,m, the function d, defined by

We have

%=o ( 8 )

= o(1)

for x E [ d , l ] , V d > 0.

m-1

Hence, the regular approximation of % is zero up to the order of magnitude of 6:)/6:)-1 in x E [d,l], V d > 0. We now apply to the function d, the Theorem 2.2.1 and obtain the existence of an order function & = o(1) such that lim E+O

d, = o

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

44

CH. 3,82

uniformly in x E [& 13. Thus & = o(l), and consequently @ - EL")@ = o(d;)Lf), for x E [6;,1], which proves the main assertion of the lemma. The possibility of choosing & such that 4- is a simple consequence of the fact that

6<

@

- EL")@ = o(d;)-l+l)

@

- EL")@ = o(c5;)LJ.

implies

a, = 4Comments. The choice a, = 4- is of course trivial. The essence of Lemma 3.2.1

Hence one can just take

lies in the possibility of enlarging the extended domain of validity of E;")@ as approximation of @, at the expense of the accuracy of the approximation. This can be accomplished if it is possible to choose the order functions such that

6

&-&I

while for the accuracy of the approximation one has @ - Ekrn)@= O ( ~ ; ) - J # 0(6:)-~+~) in x E [&,I].

We illustrate the procedure by the following elementary example: Consider 1

@=A x E [O,l]. X+E'

We have

and 0 - ELm)@ = o(E")for x E [ d , 11, Vd > 0. In order to investigate the possible extensions of the domain of validity of EL")@ as an approximation of 0,we analyse explicitly the expression

It is not difficult to deduce that for x 2 c&lirn , where c is an arbitrary constant, one has +

R , = o ( E " - ' ) , 1 = 0,1, ...,m

Hence, in application of Lemma 3.2.1 to this example, one can take $ - &lim + 2 1 -

CH. 3,§2

45

FURTHER APPLICATIONS O F THE EXTENSION THEOREMS

We now formulate a result analogous to Lemma 3.2.1, for local approximations E g ) @ : Lemma 3.2.2. Suppose that the local approximation EL)@ of @ is such that Tso@- E c ) @ = o(Sp))

for toE [O,A], V A > 0.

Then for any p = O,l, ...,n, there exist order functions

The order functions

Sp = o(1) such

that

Zp satisfy

Proof and comments. The proof is very analogous to the proof of Lemma 3.2.1, using now Theorem 2.3.2, with modification, 2.3.2 bis. Consider for that purpose, for each p = O,l,.. ,, n, @** =

1 ~

SpL

[Tco@- E(snd@].

The reader may verify that @**satisfies conditions of Theorem 2.3.2,2.3.2 bis, when the variable x in these theorems is replaced by to,and $o put equal to zero. The proof can then be completed following the reasoning of the proof of Lemma 3.2.1. Again, the essence of Lemma 3.2.2 lies in the possibility of enlarging the extended domain of validity of EE)@ as an approximation of 0,at the expense of the accuracy of the approximation. Such case arises when it is possible to choose the order function & such that while for the accuracy of the approximations one has

With Lemma 3.2.1 and 3.2.2 at our disposal we can envisage the possibility of existence of a subset of x E [0,1] on which both ELrn)@and ,??El@are valid as approximations of 0. Let us rewrite the definition of the extended domain of validity of E g ) @ , as given in Lemma 3.2.2, in terms of the variable x . We have the relation to= x/6,. Therefore,

[ iPl [ ip1.

t o €0,-

* X E

0,L

46

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

CH. 3,62

Comparing the results of Lemma 3.2.1 and 3.2.2 it should be clear that under can have certain conditions the extended domains of validity of ELrn)@and Egd@ a non-empty intersection. The assumption that this is the case is called the overlap hypothesis. We shall give this hypothesis a formulation that will provide a convenient basis for the further development of the theory.

Definition 3.2.1. The extended domains of validity of ELrn)@and EE)@ otierlap strongly if for each m = O , l , ..., M there exist order functions s’, and To (as defined in Lemma 3.2.1 and 3.2.2) satisfying

Remarks. In the case of strong overlap there is no need to extend the domains of validity of ELrn)@and Eg)@at the expense of the accuracy of the approximations in order to achieve a non-empty intersection. This then obviously is the ‘nicest’ case possible. If there is no strong overlap, then one may attempt to achieve a non-empty intersection at the costs of accuracy, on the basis of Lemma 3.2.1 and 3.2.2. Since enlarging the extended domains of validity diminishes the accuracy, one is led to search for ‘smallest possible’ non-empty intersections (which still have some useful properties), thus retaining ‘highest possible’ accuracy. These considerations motivate the following definition.

Definition 3.2.2. The extended domains of validity of ELrn)@and Egd@overlap if for any k = O,l, ..., K there exist integers m and n, and furthermore order functions 6,$which satisfy 6< $, such that

The above definition contains the case of strong overlap (Definition 3.2.1) if one can choose m = n = k. It is sometimes convenient to express the overlap hypothesis in a yet different way, using a so-called ‘intermediate variable in the overlap region’. Such a variable is defined by

ti = X / h i where di = o(1) is an order function satisfying

$ai<& From Definition 3.2.2 we then obtain Corollary. If the domains of ualidity of

and E$,@ otierlap, then for any

CH. 3,§2

FURTHER APPLICATIONS O F THE EXTENSION THEOREMS

47

k = O,l, ..., K there exist integers rn und n, and furthermore order functions di satisfying 6, < Si< 1, such that @ - ELrn)@= o(6f)) for x

CD

-

T,EK)CD

=

o(6f))

The estimates being ualid for any constants.

E

[tiS,,l],

for x E [O,SiSi].

tiE [Ai,Bi],where Ai > 0, Bi > 0 are arbitrary

The above corollary not only is a consequence of Definition 3.2.2, but can also be shown to be equivalent to Definition 3.2.2. This can be accomplished by starting with the corollary and using the extension theorems to obtain an overlap region as given in Definition 3.2.2. The exercise is left to the reader. We repeat that the definitions of significant approximations as given in Section 3.1 do not imply that the conditions of overlap will be satisfied. The overlap hypothesis is an additional element of the theory. From the analysis of this chapter it will appear that imposing the overlap hypothesis on significant approximations provides a sufficient condition for the development of a theory of matching and the construction of uniform approximations in the domain of definition of CD. We now return briefly to the starting point of the analysis of this section. We have assumed that there is just one boundary layer variable to.This simplifying assumption can be dropped and the analysis extended to the case where there is a denumerable number of boundary layer variables t,, v = O,l,,,., defined by

5,

= x/66”’

with 650) > 6:” t 6j2’> . . ..

In that case, for any significant approximation EE’CD one has

qvCD- E C ) @= o(6:))

for 4, E [A,,B,], V A , > 0, VBv > A , > 0.

The reader should have no difficulty in rephrazing Lemma 3.2.1 to obtain extensions ‘to the left’ of the domain of validity of EE)CD,while Lemma 3.2.2 will provide extensions ‘to the right’. Overlap hypothesis between Ec)CD and E:“,‘+, can then be formulated as in Definitions 3.2.1, 3.2.2, and the corollary of Definition 3.2.2. In this way a basis for treatment of boundary layers with a more complex structure is obtained. Remark. In applications one sometimes encounters problems in which @(x,E) is defined for x E [6(&),1],6(&)= o(l), and exhibits a boundary layer behaviour near x = a(&). Such problems can of course be dealt with by a simple transformation

48

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

CH. 3, $2

of k variables, however, the transformation is not even needed, if 6 ( ~is) identical with the order function defining a boundary layer variable (and this is often the case in problems mentioned above). For example consider @(x,E),x E [~.1] and suppose that there is just one boundary layer variable defined by

to= X/E. One easily verifies that the analysis given in this section is directly applicable in such a case. The only modification needed is an obvious change of the domain of the local variable into toE [1,A], A > 1. We conclude this section with two elementary examples of functions for which the significant approximations satisfy the overlap hypothesis. The examples are elementary in the sense that only elementary functions are involved. However, the second example will already show that determining of the overlap region, the intermediate variable, and the accuracy of approximations may be a tedious procedure. This provides already some motivation for the development of the theory in a later part of this chapter, where the overlap hypothesis will be used in the foundation of the analysis, but the final results will not contain any explicit reference to the overlap domain. Example 1. 0 = e-X/E+Ax), x E [O,l], whereflx) has a convergent power series representation cc

1 aflxfl.

f(x) =

fl=O

We take as measure for the accuracy of the approximations the sequence

6;) = E

~ ,

m = O,l, ...

and obtain by straightforward procedure

ELm)@=f(x),

.

x E [&I],

V d > 0.

Furthermore with the boundary layer variable E g ' @ = e-
+

m

d"',,tz,

to = X/E:

toE [O,A],

VA > 0.

n=O

For the investigation of the extended domains of validity it is convenient to use in this example elements of the set of order functions 6, = E", v E (0,l). We find V v E @,I): @ - ELm)@= o(cm)for x E [~",1],

V p E (0,l): @

- TxEgj@= OS(.dm+'jfl) for x E [ o , E ~ ] .

We now choose l>p>rn/m+J

CH. 3, $2

FURTHER APPLICATIONS OF THE EXTENSION THEOREMS

49

so that Os(dm+ l)P) = o(cm). The overlap is achieved for any v > p, while the specific condition imposed on p assures that the overlap is strong. Example 2. @ = (E/(x+ 6)) +f(x), x E [O,l], wheref(x) is as in Example 1, and again

6;) = Ern,

to= XJE.

We find:

=f(x)

qm)@

+

c ( - 1)P(:)"",

m-l p=o

-

x E [d,l],

Vd > 0,

n

1

We analyse now the extended domains of validity as in Example 1, and obtain CD = Os(&(1-V)(m+1)), x E [EV,l], v v E (O,l), J

y

@

0 - T,E:"d@= OS(&"("+ I)),

x E [O,EP],

v p E (OJ].

In order to achieve overlap one must be able to determine, for any integer k , integers m and n such that (1-v)(m+l) > k

and p ( n + l ) > k

while furthermore one must always satisfy the condition v > p.

We find that the overlap is not strong for k > 0. To establish this it is sufficient to consider k = m = n = 1. Then the first two conditions yield v <+,

p >

+

so that it is impossible to satisfy v > p. In order to have overlap with k = 1 one can choose n = 1 and m overlap domain is then defined by

3>v

=

2. The

> p > +.

Considering now k = 2 one can satisfy the conditions with n = 3, m = 4 and obtain an overlap domain defined by

3 > v > p > +. For any k > 2 the analysis must similarly be repeated. One finds that a rapidly increasing number of terms in the expansions ELm)@and E';)@ is needed in order to achieve the desired accuracy in the overlap domain.

50

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

CH. 3,43

3.3. Matching in the intermediate variables and uniform approximations on the basis of the overlap hypothesis The overlap hypothesis leads directly to matching relations in an intermediate variable, that is, to certain identity relations between expansions of functions derived from the significant approximations. In fact, from the corollary of Definition 3.2.2 one immediately obtains the estimate

q , E y D - T<,EE)@= O ( S [ ) ) ,

5, E [ A , , B , ] ,

VA,, V B ,

where 5, is an intermediate variable in the overlap region. We recall again that the symbol T?, describes the effect of transformation of variables on functions. Explicitly in the present case fl

7pLrn)@=

1 q4+p(4ti),

p = o

with integers p and v as defined in Definition 2.4.41" The above estimate already is, in a sense, a matching relation. To obtain a more elegant formulation we assume that the functions ELm)@and E E @ posses local expansions in the intermediate variable ti.This leads to Lemma 3.3.1. Let the extended domains of calidity of Eirn)@and Ef:@ ouerlap, and let thesefunctions posses local expansions in the intermediate cariahle 5,. Then for any k = 0, ..., K there exist integers m and n such that

Proof of the lemma follows immediately from the corollary of Definition 3.2.2. To illustrate the procedure of matching in intermediate variable we shall analyse now a problem, which in certain aspects already is representative for the use of matching relations in applications. The problem to be considered is derived from Example 2 of the preceding section. Suppose that the exact representation of a function @(x,E),x E [O,l] is unknown. However, we are given that Ekrn)@= f ( x )

+ 2

( - l)p

m p =- lo

@-

Ey)@=o(E~)

;

!i)"+l?

for x

E

[d,l], Vd > 0

where f(x) in some given function which has a Taylor expansion in the vicinity of x = 0. Furthermore, in the boundary layer variable to = X / E

CH.3,53

MATCHING IN THE INTERMEDIATE VARIABLES 1

51

n

TeoO- E t ) 0 = o(E")

for toE [O,A], V A > 0

where ap, p = O , l , . . . , n are constants that are unknown. We assume that the conditions of Lemma 3.3.1 are satisfied, with the intermediate variable given by

ti= X/EU where o is some number, (T E (0,l). In order to analyse the matching relation given in Lemma 3.3.1 we first write out explicitly

+

Te,E!yD=f(&"ti)

c (-

m-1

p=o

1)P

uJ+

-

We now compute the local expansions in the variable expansion operator E g ) . According to our definitions

ti by application of the

where p ( k ) is an integer such that

and E(' - " ) ( p # O(E~). It should be clear that for any integer k , and any value of (T E (0,l)one can find an integer m such that +

Similarly by Taylor expansion:

f(&"&) - E p f =

O(&k)

and P' # o ( E ~ ) . Therefore, for any integer k and any value of ~7E (0,l)one can find an integer n

52

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

CH. 3,53

such that n

p=o

p=o

provided that the coefficients up satisfy

Thus, application of matching relations fully determines the unknown coefficients a, of the local expansions E';)@. Overlooking the computations on the present example one discovers the following surprising aspect: in order to apply the matching relation in the intermediate variable of the overlap region one does not need to know the extent of that region. In fact, the matching relations are satisfied for any cs E (0,l). This may lead to the conclusion that there is overlap for 'any cs E (0,l), yet, from the computation of Example 2, Section 3.2, we know that the conclusion is false, for any finite m and n. On the other hand, one may begin to suspect that the hypothesis of existence of an overlap region, without specific information about that region, may for certain classes of problems be a sufficient basis for the theory. This idea will be pursued further in later sections of this chapter. We conclude this section with a demonstration that significant approximations which satisfy the overlap hypothesis permit to construct an approximation of @ uniformly valid in the domain of that function. Our demonstration is constructive, but is mainly of theoretical interest, because the uniform approximation that will be given has the drawback of containing explicitly an intermediate variable in the overlap region, and is therefore not yet very suitable for applications. On the other hand we do show that, with the overlap hypothesis, the significant approximations contain all information that is needed to define uniform approximations.

Lemma 3.3.2. Let ELm)@ and EL)@ be significant approximations satisfying the overlap hypothesis, i.e. for some integer k, m and n @

- E p @ = 0(6y), x

@

- T,E'r"d@= O ( c q ) ) , x

with 6< b. Consider

E

[&,I, E

[O,b]

CH.3,§4

and

53

MORE GENERAL LOCAL EXPANSIONS

~ ( 4 ,is) an arbitrary continuous function %(ti) =

0 1

for

tiE [O,cci],

cli

satisfying

> 0,

for ti E [ P i , a),Pi > mi 7hen R = o(dp))forx E [O,l]. Proof. Consider first the restriction of R to x

E

[PiSi,l]. Then R

=@ -

ELm)@

= O(Sp).

Consider next the restriction of R to x E [riSi,PiSi3(which corresponds to We write:

tiE [cli,fli]).

In the interval under consideration both E:)@ and mations. Therefore

Eg)@ are valid as approxi-

R = ~(drl). Consider finally the restriction of R to x E [O,sr,S,]. Then R = 0 - T,E:",'@ = O(d[)).

The union of the intervals considered above covers the interval [O,l], which proves the assertion of the lemma.

3.4. Overlap hypothesis and intermediate matching in the case of more general local expansions It is useful, at this stage, to retrace the main steps of our analysis, and their motivation. We have been working with local expansions of the structure defined by the operator E r ' , (that is Poincare-type expansions in terms of a local variable), because the expansion .Eirn)Ois uniquely defined (with a given choice of the gauge-function 6:)' and a constructive procedure for the terms of the expansion is available. We have introduced the concept of significant approximations as an operational criterium permitting to define important local variables (i.e. the boundary layer variables). Then, analysing the consequences of the extension theorems, we have given a precise formulation of the overlap hypothesis, which in turn permits to derive matching relations. Significant approximations satisfying the overlap hypothesis are shown to be sufficient for the construction of uniform approximations of the function under consideration. However, in application the program may fail at different stages: A significant approximation of the structure given by E g ) @may not exist, or not satisfy the overlap hypothesis, Also, in certain problems, working with local expansions of a more general type (i.e. not Poincare-expansions) may be 'more natural' or, for

54

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

CH. 3, @

some reasons, more advantageous in performing the computations. Lagerstrom (1976) gives examples of such situations. Working with generalized local expansions one looses unicity and constructive procedures defined by the operator Ek;). However, overlap hypothesis and matching relations can still be formulated. This is the purpose of the present section. Adapting earlier definition we have

Definition 3.4.1. Consider the asymptotic series

@%,4

c ll/p(5,&) Ir

=

p=o

<

with ll/p = 0,(6,*) for E DO*, 6: = 0(6,*+ 6: # o(6;)) where 6;) is an element of a pre-assigned sequence of order functions. 62)(<,&) is a generalized asymptotic expansion of cD* for 5 E D& with accuracy 6;), if

cD* - e2)= o(6;)) for 4 ED:. Using the extension theorems as in Section 3.2 we can derife

Lemma 3.4.1. Suppose that the generalized local expansion e:)(50,&) that

qo@ - eg(to,&) = o(6;))

for toE [o,A], V A > 0.

Then for any 1 = O,l,..., n there exist order functions

7he order functions

of cD is such

Sp = o(1) such

that

S, satisfy

s, < 6-,. We now write the overlap hypothesis in a form analogous to the corollary of Definition 3.2.2.

Definition 3.4.2. The extended domains of validity of ELrn)@ and B2)overlap if for any k = O,l, ...K , there exist integers m and n, and furthermore order functions 6, satisfying 6, < hi< 1, such that

0 - ELrn)@= o(6f)) for x E [<,6,,1], @ - TyL) = o(6f)) for x E [0,
The estimates being valid for any tiE [A&], where A i> 0, Bi> 0 are arbitrary constants.

CH. 3,55

CORRECTION LAYERS AND COMPOSITE EXPANSIONS

55

This leads to matching relations as in Section 3.3. Lemma 3.4.2. Let the extended domains of oalidity of E1;”)@and 9:)ooerlap and let these functions possess local expansions in the intermediate oariable ti.Then for any k = O,l, ...K there exist integers m and n such that EWEON@= E(k)@N= t,

x


~t)@.

dS

The necessity to use generalized local expansions is in applications an exception rather than a rule. This is why, in the sequel, we shall continue the analysis in the framework of local expansions defined by the operator E?’.

3.5. Correction layers and composite expansions; an asymptotic matching principle In this section, and in the next one, we introduce the concepts of ‘composite expansions’ and the so-called ‘asymptotic matching principles’ by taking as starting point of the analysis certain hypotheses about the structure of the expansions. The hypotheses to be used do not invoke the overlap hypothesis. The interrelations between the various concepts will be studied in the later parts of this chapter. In the present section we consider the case in which the function

c 6p(44p(X! P

E5;“’@=

p=o

can be extended as a continuous function to x E D (i.e. to the whole domain in which @ is defined). One can then define and study

6 = @ - ELm’@,

x

E

D.

To be more specific we consider the one-dimensional situation and assume the existence of the regular and the local expansions of 6,i.e.

& - ELm)6= o(6:)) Tco&- E K ) 6 = o(6:))

for x

E

[d,O], V d > 0,

for toE [ O , A ] , V A > 0.

It is trivial to verify that

ELm)$ = 0. Suppose now that one can construct the local expansion E E ) & such that for any integer k, there exists an integer m, for which

E$)EE)& = 0.

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

56

CH. 3, $5

Then the local expansion E g ) & contains the regular expansion ES;")&,and one could expect that E g ) & in fact is valid as an approximation of 8 for x E [0,1]. We are thus led to the following: Definition 3.5.1. Let @ ( x , E ) ,x 1

@ =@

E

[0,1],

E E (O,E~] be

such that for each m,

- E?)@

exists as a continuous function for x E [O,l]. The local expansion E g ) & is a correction layer, if for each integer k one can find an integer m such that

d, - TxEg% = o(Sf)) for x

E

[0,1].

Corollary 1. Suppose that E g ) @ and EE)ELm)@exist. I f E g ) & is a correction layer, then @ = Eim)@

+ TxEg)@- T X E g ' E y @+ O ( S p ) ,

x

E

[O, 11.

Corollary 2. A necessary condition for E g ) & to be a correction layer is

EOE(rn)@ 50 = 0,

x E [d,l],

Vd > 0.

This implies

The expression for 0 in Corollary 1 is called a composite expansion. It is obtained from the definition of the correction layer by the substitution 6 = @ - Eim'@. The relation given in Corollary 2 is called an asymptotic matching principle. It is obtained from the composite expansion by application of the operator Eim). One can deduce the following result from corollary 2, for the case m = k = 0. Corollary 3. Suppose corollary 2 holds for m = k = 0 and ELo'@ = $ o ( ~ ) , Suppose further that lim5,

lim 5-1X

, E r ) @ = 1,!/~(5). 3o

exists. Then

= lim 4 0 ( x ) . x-10

The result given in Corollary 3 is the simplest, and probably oldest asymptotic matching principle in use in the applied mathematical literature. As a very simple illustration of the preceding concepts we consider an example derived from Example 1, Section 3.2. Suppose that the exact representation of the function @(x,E),x E [O,l], is

CORRECTION LAYERS AND COMPOSITE EXPANSIONS

CH. 3, $5

51

unknown. However, we are given that (D - ES,"'@ = o ( E ~ ) , x E [d,l],

Ekm)@ =f(x),

Vd > 0.

Heref(x) is a given function, which has a Taylor expansion in some neighbourhood of x = 0. Suppose further that in the boundary layer variable 50

= XJE,

E K ) @ = e-ro

+

n

c EPapG,

p=o

Tt0@- E k ) @ = o(E")

VA > 0

for toE [O,A],

where up, p = O , l , ..., n are as yet unknown constants. Assume now that E(m) @ - Ep)@} to

is a correction layer. By a straightforward computation one finds

One can now impose the matching condition of Corollary 2, with k = rn, and obtain

u p = - (1- ) dPf p ! dxP

=o.

The unknown constants a p are thus determined by matching. The validity of the composite expansion given in Corollary 1 is trivially verified in the present case. In fact, one finds

EF)@ + TxEg)@- T,Eg)Ej,")@=f(x)

+

One thus recovers the exact representation of the function 0,as given in example 1 of Section 3.2. In applications, the matching conditions of Corollary 2 are often replaced by simpler conditions, which are obtained as follows: Consider

We are given, by Corollary 2, that

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

58

CH. 3,55

This suggests that the functions qp(t0) are small, for large values of the argument to,and leads one to expect that the matching condition of Corollary 2 could be replaced by the condition lim CO-

i J P ( ~ ,=) 0;

~p = 0, ...,p.

m

We emphasize that, although the reasoning given above often leads to correct results, the justification can only be based on a special structure of the functions iJp(t0) under consideration. In general, the condition EF),TE)5= 0 does not necessarily imply that every term of the expansion I?:)$ should tend to zero as to+ co. This is shown by the following counter-example (which concludes our discussion of correction layers). Let @(x,E),x E [0,1], E E (O,E~]be given by

We consider expansions with the accuracy defined by

It is not very difficult to deduce that, for all m, ELm’@ = 0,

x E [d,l],

Vd > 0.

To demonstrate this result we rewrite the function @ as follows: ln{l+ ln(x + Ee) @(X,E)

= -

In In 1/E

X+E

This formula permits to deduce: @ = O((In !)-l(ln

In

!)-I)

x E [d,l]

V d > 0.

Hence @ is ‘transcendentally small’, as compared to 6:), for all m. The local expansion, in terms of the boundary variable to = x / e is obtained by straightforward expansion, and one finds, when m is odd

E

For rn is even one obtains

CH. 3,96

EXPANSIONS AND MATCHING PRINCIPLE

59

One can show next that Corollary 2 holds in the following sense: For k is even

Epg@

= 0.

For k is odd E(k)E(k+l ) @ = 0. x

€0

This can be demonstrated by rewriting the formula for T x E g ) @in a form analogous to the one used in the analysis of EL!)@. Let us now write

For p is odd we have +p(50)

=

(&)L

In (50+e)

and although Corollary 2 holds, t+bp(t0) grows without bound as

50

+

a.

3.6. Composite expansions and an asymptotic matching principle from the hypothesis of regularizing layer The analysis of Section 3.5. does not apply if one cannot 'subtract the regular approximation', because ELm)@does not define a continuous function for x E [O,l]. However, one can develop a reasoning that is a counterpart to the reasoning of Section 3.5, by subtracting the local expansion and assuming that @ - E g ) @defines a function that is regular (up to a certain order of accuracy) for x E [O,l]. We are thus led to Definition 3.5.1. E E ) @is a regularizing layer if for each integer k there exists an integer m such that @ - T,Eg)@ -

EL!)(@ - E g ) @ )= o(6:))

for x E [O,l],

and if furthermore Ejl")(@- E(")@)= EL")@ - E L m ) E ( m ) @ €0

€0

is a continuous function for x E [O,l]. Corollary 1. I f E g ) @ is a regularizing layer then

+ T,Eg)@ - Eim)Eg)@+ o(6f))

@ = ELm)@

Corollary 2. I f E g ) @ is a regularizing layer then

for x E [O,l].

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

60

CH. 3,§6

The composite expansion given in Corollary 1 is obtained by rewriting the definition of the regularizing layer. The matching principle of Corollary 2 is obtained by applying to the composite expansion the operator Eg)@. As a simple illustration of the preceding results we consider anew an example studied in Section 3.3. We are given that the function @(x,E),x E [OJ] has a regular expansion

p=o

@

- ELm)@ = o ( E ~ ) for x E [d,l],

V d > 0:

where f(x) is some given function which has a Taylor expansion in some neighbourhood of the origin, Furthermore, in the local variable X

40

=El

Tc0@- E g ) @ = o(E")

for

toE [O,A], V A > 0

where ap, p = O,l, ..., m are unknown constants. Assume now that E g ) @ is a regularizing layer. Simple computation shows that m

Ekm)@- Eim)EE)@=f(X) -

1 aPxp. p=o

Clearly, the function given above can be extended as a continuous function to x E [O,l]. Next we apply matching relations as given in Corollary 2. It is again a matter of very simple computation to establish that, for any k d m

Hence, imposing the matching condition, we find

It appears that, at least in the present example, the determination of the constants ap by the method of this section requires much less labor than match-

CH. 3,§7

ASYMPTOTIC MATCHING PRINCIPLES AND EXPANSIONS

61

ing in the intermediate variable on the basis of overlap hypothesis, as performed in Section 3.3. We finally compute the composite expansion given in Corollary 1, and find:

We thus recover the exact representation of the function CD in Example 2, Section 3.2, which was used to generate the present example.

3.7. Asymptotic matching principle and composite expansion from the overlap hypothesis The assumptions in Sections 3.5 and 3.6 lead to elegant results, comprised in Corollaries 1 and 2. The question arises whether such results can also be established on the basis of overlap hypothesis. The answer is affirmative, provided that certain (mild) restrictive conditions have been imposed, and provided furthermore that a rather detailed formulation and description of various expansions has been given. One thus obtains a theory which is based on the hypothesis of the existence of an overlap region, but does not contain any explicit reference to that region in the final results. The development given in this section is not only of theoretical interest. As a result of the analysis we also obtain a careful formulation of the asymptotic matching principle. In particular, a precise rule for truncating the expansions will be given. The rule is essential to avoid erroneous results in certain applications. In what follows the setting for the theory is provided by conditions to be imposed on various expansions, the main result is stated in a theorem, and various essential assumptions are accompanied with comments and further interpretation. The proof of the main theorem is rather technical and somewhat lengthy, and is therefore presented separately from the main text, in the appendix to this chapter. Condition 1. There exist regular, local and intermediate expansions Q,

- Eim)CD= 0(6:)),

T O O- Eg'CD = ~(d:'),

q2Q, -E

y D = O(dL)),

x E [d,l],

V d > 0,

l oE [O,BO], VBo > 0, tiE [ A , , B , ] , V B , > A i> 0

where (1.1) the boundary layer variable is defined by X 50

=Ep

p is some positive number.

62

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

CH. 3,§7

(1.2) the intermediate variables under consideration are given by Y

& .

ti = EL'

-

1 E (0,p).

(1.3) the sequence defining the accuracy of the truncated expansions is chosen to be 8':

m

= crn-Y,

=

l,2,...

with y an arbitrarily small positive number.

Comments. The choice of the variables in (1.1) and (1.2) is quite usual, and most frequently occurring in applications. By the statement that there exist expansions (with accuracy prescribed by (1.3)) we always mean expansions with finite number of terms. The special choice made in (1.3) thus excludes the so-called 'purely logarithmic case'. In such case one has for example

and no finite expansion achieves the accuracy prescribed by 6'"')= c1 - Y . We shall briefly comment further on that case later on. The special choice of the sequence defining the accuracy of truncated expansions in (1.3) has the following essential significance. Consider functionsf(t,e), (where t may be identified with either x,or ti,or to), that are finite sums of the structure b k=-a

Let p be some real number, a and b arbitrary integers, or zero. Compare this function with the order function

6;)

=.p-Y

where m is some integer, and y an arbitrary small positive number. Then, if p 2 m, all terms off(t,E) are 0(6:)), while if p < m, no term off(t,E) is o(6:)). In other words: truncating expansions to the order of accuracy prescribed by 1.3 does not break up groups of terms of the structure given byf(t,&).We thus have a provision that makes it impossible to 'cut between logarithms' when truncating expansions. That the provision is a necessary one was clearly recognized in Fraenkel (1969);.see also Van Dyke (1975).

Condition 2. Consider the functions

ELrn)@=

8p(~)q5p(x)and I?:)@ p=o

S:(~)l(l~(<~).

= p=o

CH. 3,57

ASYMPTOTIC MATCHING PRINCIPLES AND EXPANSIONS

63

(2.1) Each of the functions 4,(x) has asymptotic expansions for x l 0

c 4bp'(x) + Pl(4

k(l)

4,(x) =

q=O

with limXI0

4ri

l(X)

lim-PIC4 = 0

= 0,

4h"'CX)

XlO

XI

where I can be chosen any positive integer. The functions all have the structure

~Y)(X)

4:)(x)

= c,,x'(ln

x)"

where z can be any real number, and D any positive integer, or zero. (2.2) Each of the functions $,(to)has asymptotic expansions for to-, co k(l)

$,(to)

=

c $:'(to)

q=o

+ Pl(t0)

with

where I can be chosen any positive integer. The functions $:)(to) all have the structure

$ytO) where

T

= &t'o(ln t o ) "

can be any real number and

D

any positive integer, or zero.

Comments. The essential part of Condition 2 is the restriction imposed on the structure of the functions +bp)(x), $:)(to). Excluded by the condition are functions like (In to)-",r~ > 0, or ln(1n to).On the other hand, behaviour admitted by Condition 2 is most widely encountered in applications. The essential useful properties of the functions +:)(x), $:)(to) concern the effect of the transformation of variables. Consider for example +:)(x) and the transformation X

t = ?, E'.

i. E (0,pl.

(This includes the intermediate variables, and the boundary layer variable.) Then

64

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

CH. 3,#7

Because CJ is a positive integer (or zero), the expression automatically is an asymptotic expansion (with a finite number of terms). More precisely, there exists an integer m, (that depends on A and z) such that EF)C#I:) = 0 for m

< m,,

EF)C#I:) = 7&5hp) for m > m,. Similar properties hold for the functions $:)(to). These properties play an important role in the proof of the theorem, of which the formulation follows now:

Theorem 3.7.1. Consider @(x,E),x E [O,l], E E (O,E,] and suppose that Conditions 1 and 2 are satisfied. Suppose further that there exists an overlap domain, such that for any integer s one canfind an integer m and an intermediate variable tisatisfying Condition 1.1, for which

EK)ELrn)@= EK)E(rn)@ 50 =Ef)@, 7hen for any integer 1 > 0 (l)E(l)@ =

E(')E(I)E(1)@,

E,O x to Furthermore, for x E [O,l]

x

50

+ E p D - E y E g ) @+ 0(6!').

@ = TxEg'@

Comments. We stress that the validity of the asymptotic matching principle, given in the theorem, is only assured if truncation of expansions is done according to prescribed accuracy by Condition 1.3. Note that the theorem admits weak overlap. It is somewhat surprising at a first glance that a possibly weak overlap does not affect the structure of the asymptotic matching principle. Thus we can apply the principle with 1 = 1, while Eil)@and E g ) @ may not overlap. This seemingly paradoxical result is well-known from examples in the literature (Fraenkel (1969)). However, there is no true paradox. The essence of the proof of the theorem, given in the appendix to this chapter, consists of computing the left-hand side of the identity

Following the computations one can clearly see that the repeated process of transformation of variables and re-expansion has the effect that certain terms which are important for the validity of overlap, simply disappear from the final result. Weak overlap does manifest itself in the accuracy of the uniform approximation for x E [OJ]. As stated in the theorem, the accuracy of the composite expansion may be lower than the accuracy of Eirn)@and E E ) @in the domains of validity of these expansions.

CH. 3,57

ASYMPTOTIC MATCHING PRINCIPLES AND EXPANSIONS

65

We noted earlier, that Theorem 3.7.1 leaves out of consideration the so-called , are of purely logarithmic case. In such case all order functions Bp(&), B ~ ( E ) B:)(E) the structure (In l/c)-P. From Fraenkel (1969) it is known that only a weaker version of the asymptotic matching principle can hold under such conditions. One then has the existence of pairs of integers (p,q) for which E(q)E(P)E(q)@ = E(4)E(P)@. to x to to x

The admissible pairs (p,q) can be determined on the basis of further explicit information on the behaviour of Ep)cD for x 1.0 and Eg)cD for lo+ 00. The interested reader should consult Fraenkel (1969) for details. We conclude this section with an example showing that correct truncation of expansions can be essential in application of the asymptotic matching principle. The example that follows is a simplified version of an example given in Fraenkel (1969), and discussed from the point of view of application of Theorem 3.7.1. in Eckhaus (1977). We consider @(X,E)

=f(x)/m,

x E c411

wheref(z) = In z + z2(ln z + 1). It is elementary to derive that, for x E [d,l], Vd > 0 qX,E)

=

1 In E

E2

E2

- -f(x) In

- -j-(x) (In E)'

E

+

0(&4).

Next we consider local expansion, with local variable defined by

to= XI&. Straightforward computation yields, for In t o In E

toE [O,A], V A > 0 E2

@*= 1 +-+~~(t~-1)+-{t(#1<,+1)

In

"2

- In t 0 - 1 } - -L.-

(In E)'

E

+

In to O(c4).

Consider now, in an elementary approach neglecting Condition 1 of Theorem 3.7.1, expansions truncated to the accuracy given by the sequence

Checking the validity of the asymptotic matching principle one finds

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

66

4

CH. 3,58

( o ) E ( o ) @# E(O)E(O)E(O)@, 0

To

x

x

co

E(1)E(1)@= E(l)E(l)E(1)@ co

To

x

x

co

5

# E$iE(L)E(I)@, 1 = 2,3, E (1)E:)O 50 x to (4)E(4)@ = E(4)E(4)E(4)@,

Eco x To x co The reader may verify that the two cases in which the principle holds correspond to truncated expansions in exact agreement with Condition 1 of Theorem 3.7.1, while in all cases in which the principle fails, the condition has been violated.

3.8. Validity of asymptotic matching principle without overlap In Sections 3.5 and 3.6 we have derived asymptotic matching principles from hypothesis on the structure of the expansions without invoking the overlap hypothesis. Next, in Section 3.7, we have shown that, for a somewhat restricted class of functions, the overlap hypothesis also leads to the validity of an asymptotic matching principle and the existence of a composite expansion. However, it should be recognized that the overlap hypothesis was used as a sufficient condition to establish the result. That the overlap is not necessary for the validity of an asymptotic matching principle will be shown by an example that follows now. Consider, for x E [EJ], the function @(x,E)=

1 In x

~

+ In1E ~

+ In x -In1 E + 1

’

We adopt, as sequence defining the accuracy of truncated expansion

6;) = (In We shall study local expansion in the boundary layer variable

21 and furthermore expansions in intermediate variable 50

= X/E,

ti= X/El,

50

VA E (0,l).

Straightforward computation yields

CH. 3,58

VALIDITY OF ASYMPTOTIC MATCHING PRINCIPLE

1

Ct

m-1

1 (In ti)” I In E n1 =O ( - l ) ” m - ( l - l i ) l n E

=-

61

(lnti+1)” c [(1-~)lnE]n‘

m-l

n=O

We now analyse the possibility of existence of an overlap domain. For that purpose we compute

We claim that E g ) @ # E:)ELm)@.This can already be seen by considering s = 1. In that case one has

Similar computation produces

Again E g ) @ # E g ) E g ) @ ,as can be seen from the case s = 1. In that case

It follows that in all intervals x E [AiE93i&q

where I is an arbitrary number satisfying l E (0, l), A i and Biarbitrary constants, neither EL!”)@nor E g ) @is a valid approximation. Hence there is no overlap. We next compute formally

68

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

CH. 3, APP.

Comparison shows that for any integer m we have the identity

EE)Ei")@ = EE)Eim)Eg)@. It is amusing to compute finally a composite expansion. Using preceding results one finds T,EpD

+ E ! p D - Jqp= 0.

Hence the composite expansions is, in a trivial way, a uniformly valid approximation, because the expansion is identical with the function that has been expanded. Appendix. Proof of Theorem 3.7.1 What follows can be considered an extensive exercise in the analysis of the effects of repeated application of expansion operators, on functions satisfying Conditions 1 and 2 of Section 3.7. The proof of the validity of the asymptotic matching principle is obtained in essence by straightforward computation. The proof of the validity of the composite expansion is a demonstration that, under the conditions stated in the theorem, E E ) 0 is a regularizing layer in the sense of Section 3.6. A.l. In the first step we derive some explicit formulas for various expansions occurring in the theorem. Consider EL")@. The behaviour of this function for x i 0 is given by Condition 2 of Section 3.7. We have

Ep0=

y dp(&){ y

p=o

$bp'(x)

q=o

+

o(x')j.

Using Condition 1 of Section 3.7, and with special attention to the truncating properties of 8;) = P - 7 (where y is an arbitrary small positive number), we deduce that

where

fPq(ti) =o fPq(ti)= 1

if

q,Bp(&)4bp)(x) = o(E~-Y),

if q,dp(&)4bp)(x)# o ( E ~ - Y ) . The number of combination of indexes p , 4 for which follows from

p=o

C4'0

I

fpq(si)# 0 is finite. This

CH. 3, APP.

PROOF OF THEOREM 3.7.1.

Hence, for any integer s we can find an integer k ( l ) such that By an entirely analogous reasoning we find

69 8''

= O(cS).

where

A.2. We summarise some useful simple properties of expansions gi1 It should be clear that

From this we compute

70

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

CH. 3, APP.

and for any m 2 s

E(s)E(s) p O 5, 50

@ = E ( S E) ( s ) E ( s ) @ , .

50

50

5,

50

A.3. Consider any of the functions $f)(x), as defined in Condition2 of Section 3.7, and the functions obtained from that function by the transformation of variables q,and q,.It is elementary to deduce that the orders of magnitude in the sequence 4f),T',,$?), Tt04f)either increase or decrease monotonically, as we go from the variable x, over ti, to toand consider successively, in each variable, restrictions to closed intervals independent of E. Similarly, +:)(to),T5,+f)(t0),Tx+f)(to) either increase or decrease monotonically in orders of magnitude as we go from to,over ti,to x .

A.5. We shall demonstrate that = EtiE!)@,

E(S)E(S)E(S)@ 50 x

L

The expansion on the left-hand side can be computed from

with ip4(ti) as defined in A.l. In fact, explicitly we have 50

E ( S ) E ( ti S ) E (xS ) @

= TCQ

1

1

d p ( ~ )

fpq(Ci) F p q C t o ) 4 f ) C x )

with f p q ( C 0 )again defined in A.1. Hence, using the formulae for Et:Et)@ from A.l, we obtain

CH. 3, APP.

PROOF O F THEOREM 3.7.1.

71

Assume now that the expression on the right-hand side is non-zero. Then there must exist pairs of indeces p , q such that ipq(t0)

= 1,

ipq(ti)

= 0.

This would mean that

qodp(E)4b""4# O(ES- 7, TC,hP(E)+b")(X)

= O(ES-')

and by the monotonicity results of A.3 = O(ES-').

dp(E)(fy(X)

However, by the definition of the expansion operators, p(s) is an integer such that dP(&)# O ( E ~ - * ) . We thus arrive at a contradiction, and conclude that E(S)E(S)E(S)@ - p y @ = 0. 50

51

x

A.6. Next we analyse the right-hand side of the identity obtained in A.4. As a preliminary, by reasoning analogous to A S and using A.l, we write

Et),p)EE'W@ - ,vt),Eg)(~ = 5,

50

The non-zero terms on the right-hand side occur for indices p,q such that xpq(x) =

1,

Xpq(ti)

= 0.

This means that vp*(E)$y(tO)

# O(ES- y ) ,

Ty,hp*(&)$b")(
Such terms can occur, because m 2 s. However, contribution of these nonzero terms to the expansion E(S) 50

{

E(s)'z)Eg)Q - Et)Eg)@} x

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

12

CH. 3, APP.

must of course be zero. Hence we have shown that

(s)E(s)p)E(W@ =

E,Q

5,

X

(S)E(S)E(m)@. x eQ

E
TO

A.7. By reasoning entirely analogous to A.2, for an m 2 s we have E(S)E ( S ) 1';") @ = E ( S )E ( S ) E ( S ) @ ) . TO

0

X


50

X

Combining now results of A.4, A.5 and A.6 we find E ( S ) E ( S ) @= p "'E'"'E:"d@. x

50

X


The relation holds for any integer s and hence yields proof of the asymptotic matching principle.

A.8. We now study the validity of the composite expansion, that is we investigate, for x E [O,l], the expression

R,

=@ -

TxEg)@ - Eirn)@+ Eim)E(m)@ TO

where, for any integer s, the integer m is chosen such that

Et)Eim)@= E i ) E g ) @= Eg,)@. First we show that the expression is meaningful, i.e. that R , indeed is a continuous function for x E [O,l]. For that purpose consider

EL,)@ - E (xm ) E (to m )@ =

a(,)

c

dpCE)&(X).

p=o

Explicit formulae for 4,,(x) can be deduced from results of A.l, however, it is sufficient to remark that $,,(x) satisfy the Condition 2 of Section 3.7. This means that

where

4:) (x) are functions defined in Condition 2.1 of Section'3.7, and ipq(to) =o

if d p ~ o @ ) ( x=) o(E,-Y),

j p q ( t o= ) 1 if d,?,c@)(x)

#

o(E,-Y).

Now, by the asymptotic matching principle of A.7 E(m)IE(m)@ 50

I

x

- E(m)Eg)@}

= 0.

X

Given the structure of the functions $F)(x), it is not difficult to deduce that the condition can only be satisfied if for all pairs, p , q one can

ipq(to)= 0. Furthermore, because 6, # o ( E ~ - ~$F)(x) ), must be decreasing functions for

PROOF OF THEOREM 3.7.1

CH. 3, APP.

x

13

10, and from this it follows that

A.9. We observe that

R, = 6 - ELm)6,

6 = @ - TxEg'@.

We claim that 6 has the same properties with respect to the overlap domain, as the function @. To prove the assertion we consider first (fj - TXE(W6 = @ - T,E'"'@ - TXE("){@ - E'"'@} = @ - T E'"'@ CO

CO

CO

CO

x

Co

It is hence trivial that

6 - ~ ~ E g=)o(6r)) 6

for x E [0,6,ti],V A , > 0.

Next we consider

6 - Ei")&= @ - EL")@- T x { p " ) @- E;m'E'"'@}, CO

50

We are given that @ - ELm)@= o(6r)) for x E [6i
In the following step of our analysis we shall prove separately that T,E(")@ to - Ekm)Eg)@= o ( 6 f ) ) for x E [6,ti,l].

The expression under consideration is only non-zero if for some pairs p,q one has

MATCHING RELATIONS AND COMPOSITE EXPANSIONS

74

Xpq(5i)

CH. 3, APP.

x,*(x) = 0.

= 1,

Suppose this to be the case, then the expansion term

qgCE,$bp)(50) #

E t ) E g ) @would contain a

O(ES-Y)

such that T,6p*(E)$bp)(S0) = O(Em-Y).

However, by the overlap between

Eim)@and E g ) @ ,we have

E ( s ) E ( m ) @ = E(s),F;m)@

€t

and therefore also

T . E ~ ) E ~=’ @ T,EE’E;m’@. The right-hand side (see A.l) cannot contain any ferm that is O ( E ~ - ? )and , therefore such terms are excluded in the left-hand side as well. It follows that, = 1, &(x) = 1, and the expression we study is hence identically whenever xPq(ti) zero. A.ll. We now summarize A.9 and A.lO.

R, = 5 - ELrn)&,

5 - T , E ~ ) $= o(6:))

for x

6 - ELrn)&= o(6:))

E

[0,6,5,],

for x E [6,5i,1].

Hence we have

ELrn)&-


for x = 6,ri, tiE [Ai,Bil.

= o(6:))

However, E g ) 6 = 0, because E ‘ m ) d , = EbW{@- E ( m ) @ > - 0. €Q



.f

-

It follows that

ELrn)&= o(6:))

for x = ai5,, t i E [ A , , B i ] .

This in term implies that

ELm)&= o(6:))

for x E [O,t,S,].

The above conclusion is obtained using the properties of the expansion

E;!%

c

p(m)

=

p=o

derived in A.8.

Bp(E)&(X),

x E [0,1]

CH. 3, APP.

PROOF OF THEOREM 3.7.1.

We can now assert that

& - ELm)&= o(6r)) for

xE

[O,tiSi].

The demonstration is trivial, because

& = o(6f)) for x E [O,tiSi]. Combining the results we have

d - ELrn%= 0(6!))

for x E [0,1].

This concludes the proof of the theorem.

15

CHAPTER 4

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS LINEAR PROBLEMS In this chapter we commence the study of functions which are implicitly defined as solutions of differential equations satisfying some supplementary conditions, such as boundary conditions or initial conditions. The operators defining the problem contain a small parameter E. Our goal is to determine an asymptotic approximation, or asymptotic expansion, of the solution for E 10, without the knowledge of an explicit representation of the solution. Our approach in this chapter is to assume that the solution is a function of the type studied in Chapter 2 and 3. We thus assume. the existence of regular expansions in subdomains and local expansions in boundary layer regions. However, the extent of the domain of validity of regular expansions and the location of boundary layers is not known a priori. We further assume the validity of the overlap hypothesis when dealing with significant approximations. The first section is preparatory. It deals with various formal limits of differential operators and introduces concepts analogous to Section 3.1. Then, in Section 4.2 and 4.3, we deduce rigorously the differential equations for the terms of the regular and the local expansions. The analysis of Sections 4.1 to 4.3 is illustrated by various explicit examples. The next question is: if along some manifold the solution exhibits boundary layer behaviour, what are the boundary layer variables (i.e. variables corresponding to a significant approximation)? At this point we introduce and discuss the correspondence principle, as a guide in search for boundary layer variables (Section 4.4). Sections 4.5 and 4.6 serve to show that the results of the preceding sections permit to develop a full heuristic analysis of perturbation problems, that is, the construction of all expansions, including the determination of the location of the boundary layers. We analyse in these sections some one- and two-dimensional problems for differential equations of second order. The material included is rather extensive, and serves also the purpose of providing exercise in the use of the concepts and the techniques. Naturally, all attempts are made to render the heuristic analysis as convincing as possible, which may lead to the impression that a proof of validity of the results, as an asymptotic approximation of the solution, is unnecessary. To remove this impression we interject, in Section 4.5, a counter-example showing that heuristic reasoning may lead to wrong results, without any warning. In Section 4.7 we give a new interpretation of the results of the preceding 76

CH. 4, $ 1

D E G E N E R A T I O N S OF L I N E A R D I F F E R E N T I A L O P E R A T O R S

77

analysis, from the point of view of the concept of formal approximations. A formal approximation is a function which, in a certain sense, satisfies approximately the differential equation and other conditions of the problem. Constructing a formal approximation one actually solves a ‘neighbouring problem’, and the expected validity of a formal approximation as an approximation of the solution is connected with the question whether neighbouring problems have neighbouring solutions. Section 4.7 contains an introductory discussion of these aspects, which will be studied in a more general setting, and in much more detail, in Chapter 6. In the last two sections of this chapter we discuss two alternatives for the construction of formal expansions. The alternatives are based on certain detailed and specific assumptions on the structure of composite expansions. They are known as the WKB-approximation, and the method of multiple scales.

4.1. Degenerations of linear differential operators Let a function @(x,E),x E D , E E ( O , E ~ ] ,satisfy a differential equation L,@ = E L 1 @

+ Lo@= 0

where L , and Lo are linear and &-independent. Suppose that there exists a regular approximation of @ in a subdomain D o , given by @(X,E)

= &(x)

+ o(l),

x E Do.

It is reasonable to expect that the function q ! ~ ~will satisfy

LO(bO= 0. In Section 4.2 we shall prove that this indeed is the case. Consider next a more general differential equation L&@= 0 in which L, depends on E in a more complicated way. Again, if a regular approximation &(x) exists, then one would expect that, under certain conditions, 40(x)will satisfy a differential equation in which the differential operator is a formal limit of L, as 810. In this section we shall define and study such formal limits of L,. We shall also consider formal limits of L, when transformed to a local variable. Such limits will be needed to define in turn the differential equations satisfied by the functions which occur in the local expansions. Finally, in analogy to Section 3.1 where the concept of a significant approximation has been defined, we shall introduce an ordering of the formal limits of L, in different variables, leading to the important concept of a significant degeneration. The abstract definitions given in this section will be illustrated by a series of examples.

78

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

CH. 4,61

As an introduction we consider first the one-dimensional case. A linear ordinary differential operator can be represented by an expression of the structure n

L, =

dP C u ~ ( x , E-,dxP ) p=o

x ED

where a,(x,e), ...,a,(x,~)are given functions. Suppose that there exist functions aE(x),...,uf(x) which are not all identically zero, and which are such that p = 0,...,n; x E D.

lim a,(x,r) = apO(x), E-0

Then the operator n

Lo =

1 a,(x)= dP 0

p=o

can be considered a formal limit of L, as &LO. We shall call such a formal limit a degeneration of L, (for &LO). If a;(x) = 0, p = 0, 1,...,n, then, generalizing somewhat, we can associate to L, a non-trivial degeneration, given again by the formula

where a;(x), p

= 0, ...,n

are now defined by

a ;,(Ex ) lim ~ ( : ( E ) ( L ~=( X E-0

and $G d is some order function such that not all a;(x), p = O , l , ...,n are identically zero. Consider next the effect of transformation to a local variable

where xo is arbitrary but fixed. The transformation induces, in an obvious way, a transformed operator 9,, explicitly given by

We can now look for formal limits of 9,. Suppose that for some 8~ &, there exist numbers ap(xo),p = 0, ...,n, not all equal to zero, such that

+

lim ~ ( ( E ) [ ~ ( E ) ] - ~ U ~ J( (XE )~< , E )= rp(xo), E-0

< G D*

CH. 4, $1

DEGENERATIONS O F LINEAR DIFFERENTIAL OPERATORS

19

where D* is the image of D after transformation from x to the local variable 5. We can then define the non-trivial operator

is naturally associated to L,, we shall call Y othe degeneration of Because Y E L, in the (-variable. It is obvious that one can introduce in the same explicit way degenerations of linear partial differential operators. However, the formulas become complicated and generalization to non-linear operators, later on, rather cumbersome. We therefore proceed now with the general definitions in a different way, without making use of the explicit representation of L,.

Definition 4.1.1. Let L,, E E ( O , E ~ be ] a family of linear differential operators in which the derivatives have been taken with respect to the components of the variable x E D. The degeneration of L, in the x-variable is a differential operator Lo, not identically zero, such that for all functions u(x), x E D, independent of E , for which L,u exists and is not identically zero, and for some order function $(E) lim SL,u = L,U. E-.O

Next we describe the effect of transformation to local variables x -+ ( on the operator L,. We recall (Section 2.4) that the effect of this transformation, or its inverse, on functions, can be described by the operators Tt or T,, i.e. T<@(X,E) = a*(<,&), T,@*(S,E)= @(X,&).

With this preliminary we have

Definition 4.1.2. Let L, be as in Definition 4.1.1 and let there be given a transformation to local coordinates x + ( which is bijective for x E D o c D , 4 ED,*, and is sufficiently differentiable. The transformed operator Y Eis a differential operator such that for all functions %((), ( E 08,for which LETx% exists one has L?,% = T,L,T,%.

Definition 4.1.3. Let L, be as in Definition 4.1.1 and consider a transformation to local variables x -+ 5 which induces a transformed operator Y Eby Definition 4.1.2. The degeneration of L, in the {-variable is a differential operator Yo,not identically zero, such that for all functions %((), ( E Dg independent of E , for

80

H E U R I S T I C ANALYSIS OF S I N G U L A R P E R T U R B A T I O N S

C H . 4, 81

which Y&eexists and is not identically zero, and for some order function $(E), lim $2,~ =Y ~ Q , &*O

We shall now compare degenerations in different local variables and order them according to concepts analogous to Section 3.1. Definition 4.1.4. Consider an operator L, and some given manifold S along which two different local variables t 1 and t2 have been defined. Let Yb')and Yb2)be the corresponding degenerations, and let the transformation t1+ t2 induce Ybl)+ ~ ? b ' . ~ ) .We shall say that Yb2)is contained in Yb" if for all exists and is not identically functions $(t2)independent of.&for which Yb1.2)$ zero, and for some order function &((E),one has lim 8

~ b ~= , ~9 b)2 )$$ .

E-0

Remurks. In the definition given above two local variables are considered different if they are different in the sense described in Section 3.1. The notion )Yb'.2) is analogous to what has that the transformation t1+ t2 induces Yb'+ been defined concerning the effect of transformation x + 5 on L,. Thus 2b'.2' is a differential operator in which derivatives are taken with respect to components of the variables t2, and which is such that for any sufficiently differentiable function $(t2)one has

The property described in Definition 4.1.4 naturally leads to Definition 4.1.5. Given L,, and some fixed manifold S along which local variables are defined, a degeneration Y oin a local variable 4 is called signijicunt if there exists no local variable different from 5 such that the corresponding degeneration would contain Yo. Significant degenerations are also sometimes called 'distinguished limits' of the equations (Cole (1968)). We shall now, as an illustration, apply the concepts introduced in this section to some explicit perturbation problems. Example 1. Let the domain D be an interval of the real line, and consider

L, = EL] LI =

+ L,,

d2 a2(x) --y dx

d + q ( x ) -& + ao(x),

CH. 4, $1

81

DEGENERATIONS O F LINEAR DIFFERENTIAL OPERATORS

All coefficients are continuous functions in D, and u2(x)# 0, Vx E D. We can define local coordinates in the vicinity of any arbitrary but fixed point xo of D,through the transformation x-xo

5=--

, 6(&)= o(1). 6 (4 For simplicity of presentation we shall restrict the analysis to a family of local variables given by x-xo

(=-

,

1'

EV

> 0.

The transformation L, + Y E is now easily performed and one finds

+ E"5)- ddt22 + E~-"u1(xo+ &"()-+d5d

Y E= &1-2"u2(xo

+ E-"bb,(X, +

E"5)-

d d5

&Uo(Xo

+E"5)

+ bo(xo+ E V < ) .

In the further analysis a number of cases must be distinguished.

E.1.1. Suppose.b, = 0, Vx E D. Computing the degenerations from the formula we find for 9, 3 0

for v E (O,+),

= bo(x0)

Y o= U

~

d2 X d5 (

+ ~bo(xo) ) ~ for v = 9,

The reader may verify, using the Definition 4.1.4 and 4.1.5 that there is only one significant degeneration, for v = 3,and that the significant degeneration contains all other degenerations. Hence, the totality of degenerations of L,, for any xo E D,is very simply described in this case, by the formula giving the significant degeneration.

E.1.2. Assume now b , # 0, Vx E D. Straightforward computation gives as degenerations 2 0

= h(X0)-

d d5

for v E (0,l),

82

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

dZ Y o= az(xo)7 dt

+ bl(xo&d

CH. 4, $1

for v = 1, for v > 1

Again, there is only one significant degeneration, for v = 1, which contains all other degenerations. A somewhat hidden aspect of the result given above, comes to light if we compare the significant degenerations for xo at the two end-points of a closed interval D.To be specific consider x E [O,l]. Then for xo = 0, 4 , = X/E,to2 0 and the significant degeneration is given by

However, for xo = 1, t1 =

x-1

and 4 , d 0.

Hence, for a better comparison of the degeneration, we modify the definition of the local variable into -41

= r1=

1-x f , 2 0. 7,

This leads to the significant degeneration

Since the coefficients az(x) and bl(x) do not change sign in the interval, the significant degenerations at the two end-points have structures that differ considerably, because of the change of sign in the first derivative term. This difference will be of importance in applications (see Section 4.5). E.1.3. We briefly consider the case in which b,(x) is zero at isolated points of the interval. This is a so-called turning-point problem. Of course, for x, such that bl(xo) # 0 the preceding result holds. Studying the vicinity of the turning-point one must specify the structure of b1(x). As an example consider with limx,xo 6,(x) = 0, b, constant unequal to zero. Substituting into the general expression for one finds again only one significant degeneration, for v = given by

4,

CH. 4,$1

DEGENERATIONS O F LINEAR DIFFERENTIAL OPERATORS

83

The significant degeneration contains all other degenerations. More complicated turning-point structures can be treated similarly. Example 2. Let D .be a domain in R 2 and consider L&= EL,

+ Lo

where L , is a linear elliptic &-independentoperator of second order, while Lo is of zeroth order, i.e. Lo = g(4.

We shall assume that g ( x ) # 0 in D,We study the case that S (Definition 4.1.4) is a smooth curve r which lies either in D or along (a part of) a boundary of D.In order to define local variables we must first introduce in a neighbourhood of r a new system of coordinates p,8, which is such that for any point ,P(p,O), p = 0 implies P E r. One can take p to be the distance measured along normals on r, and 8 the distance along r. One can also take a more ‘oblique’ system. We shall have the opportunity to give explicit examples when studying later more explicit geometries of S . Since L , is a general elliptic operator of second order, a transformation of variables can produce nothing else but another general elliptic operator of second order. Hence, written in the new variables, one must have

By an essential property of elliptic operators a and y cannot be zero in the domain under consideration, and are furthermore of the same sign. Next we introduce local variables along S , by the formula

5 = PIEY, v > 0. Assuming all coefficients to be continuous functions one finds without difficulty that 9 0

= -g(O,Q)

for v E (O,+),

a2

Y,, = cc(o,e)---I-+ g(o,e) for v = +,

at

yo= .(o,

e)ata 2

for v > +.

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

84

CH. 4, $1

Further inspection shows that for v = 3 we have a significant degeneration, which contains all other degenerations. It is remarkable that in spite of the rather general formulation of our problem, all degenerations, for all choices of the curve S , can be represented by just one formula. It should further be noted that 04po is an ordinary differential operator, with coefficients containing the second variable as a parameter. We finally consider the case that S is a point in 0.Without loss of generality the case can be studied in the p,6 coordinate system, by introducing local variables

t 2 = (O-6,)jsF,

= PIE',

1'

> 0, p > 0.

The reader should have no difficulty in verifying that there is one significant degeneration, for v = p = containing all other degenerations. The corresponding Y ois a partial differential operator of elliptic type with constant coefficients and is explicitly given by

4,

yo= U(o,e,)---

z2 + 2j(o,e,)

zt?

E2 ~

+ "A0,Qo):

c't,E52

z2 + g(0360).

zt 2

Example 3. Let D be a domain in R 2 and consider

+

L, = EL1 Lo where L , is as in Example 2 and Lo is a linear differential operator of first order, i.e.

We shall assume that the characteristics of the operator Lo do not intersect in

6.

We proceed now as in Example 2, define a smooth curve r in 6 and a new coordinate system p,6 in a neighbourhood of r, the new coordinates being such that for any point P(p,O), p = O implies P E r. This leads to a transformation of the operator L, into

If now S, that is the set along which local variables have been defined, coincides with (a part of) r, then local variables along S are again given by

4

= PIEV,

1'

> 0.

CH. 4, $1

DEGENERATIONS OF LINEAR DIFFERENTIAL OPERATORS

85

As before, because of ellipticity of L , , r and j , are non-zero in D, However with respect to the coefficient p various cases must be distinguished. E.3.1. p(0,O) # 0 on S. Geometrically this means that the curve S does not coincide with, and is nowhere tangent to a characteristic of the operator Lo. Verification of the above statement is a simple exercise in the analysis of first order partial differential operators. Introducing the transformation to the local variables one finds one significant degeneration, for v = 1, which contains all other degenerations. The significant degeneration is given by

As in Example 2, the significant degeneration defines an ordinary differential operator. E.3.2. Let now S coincide with a characteristic of L o , and let us furthermore specify the coordinate system p,O in such a way that any curve p = const. is a characteristic of Lo. Then p ( p , 0 ) = 0. One finds again one significant degeneration, for v = *, which contains all other degenerations, and is given by

We see that Y onow is a parabolic differential operator. E.3.3. Suppose that the curve is tangent to a characteristic of Lo at some 0 = (lo, and is such that, in some neighbourhood of (lo, one has PC(Pl(lJ = ((l-0o)CPo

+ ,L~(P>~)I

with p(Ja constant not equal to zero, and lim p(p,O) = 0.

0 -or,

Then the result of E.3.1 holds if S is an open subset of I- not containing 0 = O o , and the vicinity of 0 = 0 , must be investigated separately. The set 'along' which local variables are defined, say So, now consists of the point p = 0, 0 = do, and local variables are given by

t, =

t2 = (0&0")/>/F,

v > 0, p > 0.

Assuming u(O,O,) # 0 one finds, by a careful analysis, two significant degenerations. Forp=v=l

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

86

CH. 4,§1

For v = 213, p = 1/3

Any other degeneration is contained in one of the two significant degenerations.

E.3.4. Consider finally the situation described in E.3.2, that is acoordinate system that p(p,e) = 0. If we now study degenerations in the local variables 51

= PIEV,

iJ2

= (Q-Qo)l&p,

v > 0, P > 0,

then there is one significant degeneration, containing all others and occurring for v = p = 1:

E.3.5. We illustrate all preceding cases by an explicit example. Consider

a and a domain D defined by (see Fig. 4.1)

x:

+ x: < 1,

XI

2 0.

We study the degenerations along the bounury r of D. The characteristics of Lo are lines x1 = const. The part of the boundary x1 = 0, / x 2 -= / 1 is a characteristic boundary, and the result of E.3.2 applies.

Fig. 4.1.

CH. 4,$2

T H E DIFFERENTIAL EQUATIONS FOR T H E FIRST TERM

87

In order to analyse the remaining part of the boundary we introduce a new coordinate system x1 = ( l - p ) c o s O

and x2 =(l-p)sinO.

Then the result of E.3.1 holds, except for the vicinity of 8 = 0, where r is tangent to a characteristic. In that neighbourhood E.3.3 can be used. Finally, the vicinity of the points where the boundary has a discontinuous tangent, x1 = 0, lx21 = 1, can be analysed by E.3.4.

4.2. The differential equations for the first term of the regular and the local expansion Let @(x,E),E D,be a function that satisfies the differential equation

Lea= Lo@+ L,@ = F where Lo is the degeneration of L, in the x-variable. The definition of degeneration implies that L, is an operator such that for all functions U(x) independent of E, for which LEU exists, one has lim L,U = 0. E-0

We further assume the existence of lim F(x,E)= F,(x). E-0

We shall be concerned with the following basic question: Suppose that @(x,F) has a regular approximation +,(x) for x E Do c D.Does the function $o satisfy the differential equation L&, = F,? To make the question more precise one must specify in what sense (i.e. norm) the regular approximation exists, and furthermore, in what sense the differential operation is interpreted. In our analysis we consider differential equations in the classical sense, so that the statement that @ ( x , E )satisfies Le@ = F for x E D means L,@(x,E) = F(x,E), VX E D.

Furthermore the asymptotic behaviour is studied in the norm of uniform convergence. The reader interested in the variational formulation of perturbation problems for differential equations, and the asymptotic analysis in the sense of norms of the corresponding Hilbert spaces, will find abundant information in Lions (1973).

88

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

CH. 4, $2

Let us show by an example, that the question posed above is not a trivial one. Consider @(x,E), x E [O,l], defined as the solution of the initial value problem

g)x=o

@(O,&) = 1,

= O.

One may verify by inspection, that the solution reads

3

@(X,&) = 1 +& 1- cosy

[

.

The degeneration in the present example, can simply be computed: lim L,O(x) = Lo@)

= -O(x)

E+O

while furthermore lim F ( x , E= ) F,

= 0.

&+O

On the other hand lim @(x,E) = q50 = 1. &+O

It is clear that Lo+, # F,. It will be convenient, in our analysis to use the so-called ‘test functions’ in Do. These are &-independent, infinitely differentiable functions with a compact support in Do (i.e. functions that are identically zero outside some compact subset of Do), With the aid of test functions we define adjoint operators as follows: Given an operator L, the adjoint z i s an operator such that for any pair of test functions O1 and 8, one has

j o,Le,dx Do

=

j e,Zo,dx. Do

These preliminaries permit to formulate the following result (Eckhaus (1977)):

Theorem 4.2.1. Let @(x,&),x E Do satisfy the equation

L&@= Lo@+ Lp@= F , x E Do where Lo is the degeneration of L, in the x-variable, F is a continuous function in Do and

CH.4, $2

THE DIFFERENTIAL EQUATIONS FOR THE FIRST TERM

89

lim F(x,E)= Fo(x) &+O

uniformly in 6,. Lo and L, are linear operators and the adjoint 2, satisJies further the condition that, for any test function 0, 1imE+,L,O = 0 uniformly in Do. Suppose that there exists a function 40(x) such that 1imE+, @(x,E)= $,(x) uniformly in Do, and L&o exists as a continuous function. 7hen L 0 4 , = F , in Do.

Proof. Consider, for any test function 8, the identity

S Do

BLo@dx+

OL,cDdx = Do

Introducing the adjoint operators

S @ZoOdx+ S cDZ,Odx Do

S

BL,@dx

Lo and L,,

we have

OF dx.

=

Do

OFdx.

= Do

Do

Do

We can compute the limit of these expansions as E 0, using the property of uniform convergence, which permits to interchange the process of integration and taking the limit. We find

S Do

4oLoOdx =

S OFodx. Do

And hence

s e [ ~ , e , - ~ , l d x = 0.

Do

Now, F , is a continuous function (by uniform convergence of F -,F,) and L04, is a continuous function by a condition of the theorem. Hence we can use the classical Lemma of Lagrange (also called the fundamental Lemma of the calculus of variation) and conclude that

Remarks. The reader interested in the study of differential equations in the framework of functional analysis will have recognised that half-way our proof we have demonstrated that 4, satisfies in the weak sense the equation Lo$, = F,. Starting with this observation one can modify somewhat the theorem, and replace the a priori requirement of differentiability of 4, (existence of L04,,) by the statement that Lo is hypo-ellyptic, or of first order with continuous coefficients. Using results of Hormander (1964), one can again demonstrate the main assertion of the theorem, by showing that weak solutions are in this case classical solutions.

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

90

CH. 4, $2

ep,

On the other hand, the condition on the adjoint stated in the theorem, is essential. This can be seen by considering anew the example given earlier in this section. In that example Lo$o # F o e Computing the adjoint one finds

e,

Clearly the condition on

zpof the theorem is not satisfied.

We now turn to the local analysis of @(x,c).Given a transformation to local variables x -+ 5, such that T@(x,E)= @*(t,c), 5 E D*, the differential equation L,@ = F transforms to

ED*. Suppose that for 5 E D,*c D* a local approximation S,*$o(<) of @* exists. Again we ask the question: what differential equation does Go satisfy ? The Ye@* = F*,

answer follows by a straight-forward paraphrasing of Theorem 4.2.1.In order to avoid any ambiguity in further applications we write out this result in full detail.

Theorem 4.2.1." Let @(x,E)satisfy Le@= F and consider a transformation to local satisJies variables x -+ 4 such that @*((,E) = T<@(x,E) Lzo@* + Ye@* =F where Y oand Y pare linear operators, Y ois the degeneration of L, in the tvariable, and 9, is such that for any test function 8* in Dg the adjoint 2, satisfies uniformly in D,* lim Pee*= 0. E+O

Suppose that there exists a limir-function $o(t) such that,for some order function d,*(&),uniformly in D,*,

and Lfot,bo is a continuous function. Suppose further that, uniformly in 0;.

Then Y o * o = F,*

CH. 4,§3 RECURRENCE RELATIONS FOR REGULAR AND LOCAL EXPANSIONS 91

4.3. Recurrence relations for the regular and the local expansions The results of the preceding section can be extended to regular and local expansions in arbitrary number of terms, because the pFocess of constructing the expansions is a repeated process of constructing approximations (Section 1.4). We start with an elementary case. Consider

+

LEO= EL,@ Lo@= F

where L, and Lo are linear and &-independent, and F has an asymptotic expansion m

F ( X , & )=

1

&"fn(X)

+ O(Em),

x E Do.

n=O

Suppose that there exist in Do a regular expansion of the structure m

ET0

=

1

E"$,(X).

n=O

The usual heuristic procedure consists of 'substituting the expansion into the equation and putting coefficients of powers of E equal to zero' This leads, in a straightforward way, to the set of differential equations: LO$n

= f n - &$n-1,

n 2 1.

Of course, the procedure is only meaningful, if the functions +,, have sufficient n = 1,..., m, and Lo& exist. differentiability properties, so that L14n-l, We can justify the procedure, using the Theorem 4.2.1. Consider for that purpose the function 0,defined by

(0- 4 0 ) / & . Substituting in the differential equation we obtain 1 L,0, = EL,@, + Lo@,= ;[F-fo] - L,(h0. 01 =

exists. Assuming the existence Given is that a regular approximation $1 of of the adjoint 2, of L , we can apply the theorem and obtain Lo41

=f1 - L140.

One can now continue the analysis considering

0 3 =

etc.

1 $0 - $ 0

-4 1

- E2$21,

H E U R I S T I C ANALYSIS OF S I N G U L A R P E R T U R B A T I O N S

92

C H . 4,93

We observe that the degeneration LOplays a predominant role in defining the differential equations for all $,,, n = 0,..., m. In fact, for any n, we have a nonhomogeneous equation Lo4n

=fn

where the right-hand side f n is a function that can be computed at each step. We finally observe (putting together the preceding results) that the sequence of functions @ ( q ) , defined by 4

@(4)

E"$,(x), q = 0,1, ...,m

=EfQ = n=O

satisfies a recurrence relation Lo@(4)= E f f - L 1 W-l), q = 1,..., m.

In the siniple case studied above not much is gained by this interpretation. More in general, however, defining expansions by suitable recurrence relations permits a concise notation for cumbersome expressions. The general result that follows now is formulated in this way. All expansions occurring in the theorem are in the norm of uniform convergence. Theorem 4.3.1. Let (D(x,c),x E Do satisfy the equation LO@+ L,O = F , where Lo and L, are linear operators defined in Theorem 4.2.1. Suppose there exist a regular expansion m

E:@

=

1 dn(E)$n(x),

x

E Do*

n=O

Let ELm) be a regular expansion operator with pre-assigned order of accuracy defined by the sequence {d,,}. Then for each q = 1,...,m L o E 4x Q

= Eiq'F -

E$'L,E:-'@.

The above result holds under the natural condition that L o E f @ exists as a continuousfiinction, and the expansions E!$F and E$)L,E:-'Q exist for x E Do.

Proof. One can demonstrate the theorem by studying consecutively

=

(@ - do$o)/do, Q2 = (0- 60+o - dl+l)/61,etc., applying each time Theorem 4.2.1.

This leads to rather cumbersome computations. We present here a direct proof, which is along the lines of the proof of Theorem 4.2.1. Consider, for any test function 8, the identity

J BLO(@-E:@)dx Do

+J Do

+J

Do

8Lp(O-Ef@)dx +

J BL,(EfCD-Ef-'@,)dx Do

8{LoE:@ + L,E:-'@ - Ei4'F}dx = J B(F - ELq'F)dx. Do

CH. 4,&3 RECURRENCE RELATIONS FOR REGULAR AND LOCAL EXPANSIONS 93

We introduce in the first two terms the adjoint operators multiply the identity by (6,)- This yields

'.

1

+ f T{LOE:@ + L,Ef-'@ Do

4

=

1 f -(F

- ES,4'FjOdx

Lo and L,,

+ f L,+,Odx

and

=

DO

-

E$'F)Odx.

D O 'Y

Now we observe that 1

lim -(@ - E $ @ ) = 0 and c-0

6,

Similarly, considering expansions of L,EY,.

1

lim - ( F - E!$F) = 0. c - 0 6,

' one has

1

lim-{ELq)LpEz-'O - L,E:-'@J.= 0. E-0 6, Using this information, we compute the limit of the identity for cJ0. We thus obtain lim 6-0

1

f --.(L,E:O + Ej,Y)L,E:-'O - E$'F)Odx Do

= 0.

6q

By the definition of the expansion operators, the expression between brackets has the following structure: Y

&E:@

+ ELq'L,,E$-'@ - ELq'F = 2

J,(E)W,(X)

n=O

where w,(x), n = 0,...,4, are continuous functions. Hence we have

Since 6,+ = 0(6,), Vn, the limit above cannot be zero, unless

f w,(x)O(x)dx = 0,

n = 0,1, ..., 4 .

Do

This in turn implies that w,(x)=O,

XED,,, n = 0 , 1 , ..., q

which yields the proof of the theorem.

94

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

CH. 4, $3

It should be clear that an entirely analogous result holds for local expansions. We write it out in full detail. for future convenience.

Theorem 4.3.1?' Let @(x,E)satisfy L8@= F and consider a transformation to local variables x + 4 such that a*((,&)= TS@(x,e)satisfies Yo@* Yp@* = F, 4 ED,*

+

where Y oand Y Pare linear operators defined in Theorem 4.2.1. Suppose there exists a local expansion m

ET@ =

c 6,*(&)@,(0*

fl=O

Let EY' be a local expansion operator with pre-assigned order of accuracy defined by the sequence {6,*}. Then for each q = 1,...,n 9 E 4 @ = E ( 4 ) F - E ( g ) Y E4-'@. o r t: < P S The above result holds under the natural condition that Y o E j @ exists as a continuous function and the expansions EF'F and EF'YpEE-'@ exist for 5 E D,*. We note that in the preceding results, the structure of the expansions E : @ , or ET@, (that is, the nature of the sequences of order functions (6,) and {6:}), was considered as given. In applications the structure of E ; @ and E y @ is not known a priori, and must be deduced in the course of the analysis. Quite often the study of the differential equations and other data of the problem suggest in a straightforward way the correct structure for the expansions. However, in certain cases, 'unexpected' terms in the expansions can arise. In order to show that the structure of the differential equation can only suggest, but does not determine, the structure of the expansions we consider anew the elementary case studied at the beginning of this section. Let @ satisfy cLl@ Lo@= F , where L , and Lo are linear, &-independent operators. Suppose for simplicity that F is &-independent,and that any other given constraint on the function @ (such as the boundary conditions) also is Eindependent. Then the structure of the problem indeed strongly suggests a regular expansion of the structure

+

c m

E;@ =

E"+,(x).

n=O

However, the possibility of a different struciure for the regular expansion cannot be excluded. Let us write m

E;@ =

c 6n(c)+n(X)

n=O

CH. 4, $3 RECURRENCE RELATIONS FOR REGULAR AND LOCAL EXPANSIONS 95

and assume 6, = 1. There may exist non-trivial terms of the expansion such that for say n = l , , , . , , p ,E = o(6,). The corresponding functions q5, must then satisfy the homogeneous equation Lo& = 0, n = 1,...,p .

Similarly one can have a group of terms in the expansion, for which the order functions satisfy c2 4 6, 4 E , etc. In applications one usually deduces the order of magnitude of these ‘unexpected’ terms from the requirements arising in the matching procedures. In the second one of the two examples that follow now a simple problem exhibiting expansions of an unexpected structure will be discussed. We commence however with a more elementary example, illustrating the analysis of this section. Example 1. Let the domain be the interval x E [O,l], and consider the boundary value problem LEO= F ,

@(O,E) = O ( ~ , E =)0

where: L,

= EL,

Lo

=

+ L,,

b,(x),

a,(x) # 0, b,(x) # 0, vx E [O,ll. For simplicity we assume here that F is independent of E . Suppose that there exists a regular expansion in some subinterval of [O,l]. It is not difficult to deduce that the structure of the expansion must in this case be given by m

E y=

c

c“,(x).

n=O

The functions q5Jx) are defined by

It should be clear that the subdomain in which the regular expansion of (B exists in general cannot contain the endpoints x = 0 and x = 1, because E:O does not satisfy the boundary conditions.

96

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

CH. 4, $3

Consider next local expansions in the vicinity of some arbitrary but fixed point x = x,. (The preceding result on the regular expansion suggests that xo = 0 and x, = 1 should be studied.) We study local expansions in the local variable that define the significant degeneration, which, in Section 4.1, Example E.l.l was found to be

5 = (x -X0)/&1’2. We rewrite the differential equation in the local variable, separating off the degeneration, and obtain Yo@*

+ Y P O *= F ,

+

Y p= {a 2 ( x

581’2)

d2 d5

- a,(x,)} 7 d

+ &1’2a1(xo+ W2)+ b,(x, + ( E l ’ , ) d5

+ &ao(xo+

- b,(x,)

5&1/2).

If the coefficients a,, a,, a, and b, have convergent Taylor series expansions in some neighbourhood of x = xo, then m

Where 9;) are operators independent of E. Similarly, if F has a convergent Taylor series expansion in the vicinity of x = x, then we can. write m

q=o

The structure of the differential equation in the local variable suggests the local expansion m

EFa

=

c .“’2+“(‘z).

n=O

Our later analysis will show that this structure in the present case is indeed correct. For the first term of the local expansion we have the differential equation

CH. 4,53 RECURRENCE RELATIONS FOR REGULAR AND LOCAL EXPANSIONS 97

For n > 0, the functions

$n

satisfy the inhomogeneous equations

= Fn.

The right-hand side of these equations can be computed using Theorem 4.3.1*. For n = 1 one has 90$1

=fl - 2 r ) $ 0 7

The explicit expression for Fn becomes increasingly complicated, as n increases. Example 2. Consider now, on the interval x E [0,117 the boundary value problem LE@= &L1@+ Lo@= F ,

@(O,E) = @ ( 1 , ~ )= 0

where d2 L1=2, dx Lo = x

d2 z dx

+ -,dxd

m

F=

C

~"f.

p=o

The representation for F is a convergent series in which the f n ' s are constants. Suppose that there exists a regular expansion in some subinterval of [O,l]. The differential equation suggests for the expansion the structure

c Enq5"(X). m

EF@ =

n=O

Using Theorem 4.3.1 we obtain for the functions q5,, the differential equations '040

=fO7

Lo4n = f n - Llq5n- 1' Consider next local expansions. The neighbourhood of x = 0 appears particularly interesting, because x = 0 is a singular point for the operator Lo. Analysis of degenerations, along the lines of Section 4.1, shows that there is one significant dgeneration, occurring for

5 = XI&.

98

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

CH. 4, $4

Y ois in this case, in essence, the full differential operator L,, transformed to the local variable. The differential equation for @*(<,E) = T,@(x,E)reads: 90@*

= (1+5)-

d2@* d@* dt2

+x=EF*

The differential equation suggests for the local expansion the structure m

Jy@ = 1 E"ijn(4). n=O Using Theorem 4.3.1 we find Yoijo = 0 and

=fn-l.

However, in the present problem, the structures of the regular and the local expansion, as suggested by the differential equation, are both incorrect. This can be seen from the exact solution which reads ln(x + E ) - In E ln(1 + E ) - In E Analysing this function we find in x E [d,l],V d > 0 a regular expansion @(x,E) =fo(x - 1)

1 + -fob x + O,(E). In E

Similarly, in the local variable expansion

@*(Ts)

1 In E

= -foln(l

5 = X / E , we find for 5 E [O,A],V A > 0,a local

+ 5 ) + O,(E).

4.4. The correspondence principle Consider a function @(x,E),x E D, E E (O,E~]which satisfies a differential equation L,@ = F and suppose that @ is a function of the type described and studied in Chapter 3. More precisely assume that there exist significant approximations of @ in a finite number of different local variables, and that any approximation of @ is contained in some significant approximation. The problem now is to determine the boundary layer variables. Our preceding analysis, and terminology, suggests a relation between boundary layer variables and the variables that define significant degenerations. In fact, if for some fixed subset S , to is a variable that defines a significant degeneration 9Lo)', and tisome different variable for which the degeneration is then the function E:,E:,,@ satisfies the differential equation contained in Ybo),

CH. 4, #4

THE CORRESPONDENCE PRINCIPLE

99

that E:,@ must satisfy. This can be demonstrated using results and techniques of Section 4.2, but it does not yet prove that E:,@ is contained in Ej,,@. There is, at the present date, no theory which would predict a priori that (for certain classes of operators L,) the boundary layer variables correspond to local variables defining significant degenerations. But there is an overwhelming a posteriori evidence: in great many problems the assumption of correspondence has permitted the construction of the asymptotic approximation. This leads to the following: Heuristic principle. Consider a set of dgerent local variables along some given manifold S . I f there exists a significant approximation, then the degeneration in the boundary layer variable is a significant degeneration. The principle given above should be carefully interpreted. It does not state that if, in some local variable, the degeneration is significant, then the corresponding local approximation is significant. That such inversion is in general incorrect will be shown in Example E.1.2 below. We remark that in explicit applications, the analysis also often shows that a significant degeneration is the only possible candidate to produce significant approximations. This, and some .other aspects of the preceding discussion, will be illustrated now by some examples. Example 1. We consider, as in Section 4.1, the differential operator d2 a,(x)a

}

d

+ a, (x)-d x + a,(x) + b,(x)-d x + b,(x)

and the boundary value problem for a function @(x,E),x Le@= 0;

@(O,E)

= CI

# 0;

E

[OJ]

@(l,&) = p # 0.

E.l.l. Let b , = 0, Vx ED, and assume a2(x)> 0, b,(x) < 0, x E D. This last assumption is the classical condition for the existence and unicity of solutions, for E arbitrarily small. We study local approximations in the neighbourhood of x = 0. Extending slightly the analysis of Section 4.1, we find, for the local variables defined by t d =X / W

the following degenerations: 9

0

=boo

for d2

+

9 0= a 2 ( x 0 ) 7 b,(x,) d ts

> JE,

for 6 ( ~= ) JE,

H E U R I S T I C ANALYSIS OF S I N G U L A R P E R T U R B A T I O N S

I00

C H . 4, #4

Clearly, the significant degeneration arises for 6 ( ~= ) JE. A local approximation $o(&) is a solution of the differential equation = 0.

It is easy to compute all functions that satisfy these equations. We find

$O(L)

=0

$o(<,)

= Ae-w(0)t6 Bew(O)eS for

$0(56)

= A,

for

+

+ Bat,

for

B(E)

> JE,

J ( E ) = JE,

4 JE

~ ( E I

with wo = ( - bo(0)/uo(O))l’z. Clearly no solution for 6(&)> JE, or J ( E ) 4 JE, can contain a solution for B(E)=JE, unless A = B = A , = B , = O . But in this case there are no significant approximations. Hence, the only possible candidate for a significant approximation is a solution of the equation for the significant degeneration. We now pursue the analysis, using the extension theorem of Section 2.2. There ) exists an order function d1(&) such that for all order functions 6 ( ~satisfying 6, > 6 > JEthe corresponding local approximations are contained in $o(56) for 6 = J EThis . immediately leads to the conclusion that B = 0. Similarly, there exists an order function a1(&)such that for all order functions 6 ( ~ satisfying ) d1(&)< 6 ( ~< ) JE, the corresponding local approximations are contained in $(<,) for 6 = JE. This leads to B, = 0.

We thus arrive at the conclusion that, under the hypothesis of the existence of local approximations by local limit functions, there is only one significant approximation which corresponds to the significant degeneration, and which contains all other local approximations. The significant approximation is given by

Because the extended domain of validity of this approximation contains arbitrarily small neighbourhoods of x = 0, one may put A = a. A similar analysis, near x = I, produces the significant approximation

E.1.2. Let now b l ( x )# 0, Vx E D. Taking up, and generalizing slightly, the results of Section 4.1, we find for the local variables

CH. 4,$4

101

T H E CORRESPONDENCE PRINCIPLE

for 6 ( ~> ) E,

for 6 ( ~< ) E. The local approximations $o(ta), which are solutions of YOlLO

=0

are therefore of the structure for B(E) > E ,

$o(ta) = A ,

+ &-(bl(o)ia2(0))t6

$o(
for 6 = E , for S(E) < E ,

+ Bat,

Reasoning as on E.l.lone easily finds that the only candidate for a significant degeneration is a local approximation corresponding to 6 = E, i.e. to a significant degeneration. Furthermore using the extension theorem, it follows that, if b 1(O)la,(O)

’0,

then for a(&) > E, Ab = A + B, while for 6(&)< E, A , = A and B , = 0. However, if bl(O)/a,(O) < 0,

then B=O, and there are no significant approximations. In this last case, repeating the analysis in the neighbourhood of x = 1, one finds a significant approximation $o = A + Be(bl(l)/az(l))(l-x/&). The reader is invited to verify the details of the analysis outlined above. Example 2. Let D be a bounded domain in R2,with boundary L&= EL2

r, and

+ g(x)

where L , is a linear elliptic differential operator of second order. Consider the boundary value problem L,@=O;

We assume that neighbourhood of

@ = $ on

r.

r is a smooth curve and study local approximations in the r. We can take up the results of Section 4.1. Assuming

102

H E U R I S T I C ANALYSIS OF S I N G U L A R P E R T U R B A T I O N S

C H . 4,$5

a(0,O)> 0 and g(0,O)< 0 one finds results entirely analogous to Example E.l.l. The reader is again invited to perform the analysis in detail. Example 3. We shall now consider a problem with a more complex boundary layer structure. Let @(x,E), x 0, satisfy

and the initial condition @(O,E) = 1.

We introduce local variables

5, = X/EY. Straightforward analysis shows that there are two significant degenerations, which contain all other degenerations. The significant degenerations are given by 1,

1. It is easy, in this case to check explicitly the correspondence between significant degenerations and significant approximations. The exact solution of the problem is given by EL

@(x,E)= -e-'/'.

X+EZ

Analysing this function one finds two significant approximations:

One can further verify that the significant approximations contain the approximations in all other variables. 4.5. Further development of the heuristic analysis: some one-dimensional problems

In this section, and in the next one, we show how the concepts and the results of the preceding sections, and those of Chapter 3, can be used as tools of analysis

C H . 4, $5

F U R T H E R D E V E L O P M E N T OF T H E H E U R I S T I C ANALYSIS

103

for the full investigation of perturbation problems. We shall study linear boundary value problems and initial value problems of second order, investigated already to some extent in the examples given in Sections 4.1, 4.3 and 4.4. Our aim will be to determine completely all expansions, and the location of boundary layers. Example 1. We consider d2 Let @(x,E),x E [O,l],

+ u1(x)-dx + a,(x) + b1(x)-dxd + b,(x).

}

E E (O,E,]

be defined as a solution of

Lg@= F

satisfying the boundary conditions @(O,E)

= c(

# 0;

CD(1,E) = fi

# 0.

For simplicity we assume that F is independent of E . Furthermore, F(x) and all coefficients of the differential operator are infinitely differentiable. Let now b, = 0, Vx E D,and assume that a,(x) > 0 and bo(x) < 0. If a regular expansion of @ exists in some subinterval, then, following Section 4.3. m

EF@

=

1

Efl$n(X),

n=O

where L, = a

d2 dx

2 7

+ a,-dxd + a,.

It should be clear that in general

Hence we must expect boundary layer behaviour in the vicinity of x = 0 and x = 1. Assuming the validity of the correspondence principle we have, from Section 4.4, near x = 0 the b,oundary layer variable

H E U R l S T I C ANALYSIS OF S I N G U L A R P E R T U R B A T I O N S

104

C H . 4, a5

and the significant degeneration A2

U

+

b,(O). d50 Similarly, near x = 1, the boundary layer variable is given by 2 0

51

=a 2 ( 0 ) 7

= (1- X ) / J E

and the significant degeneration is d2

9,= u 2 ( 1 ) 7 + bO(1). d5 1

We can now proceed as in Section 4.4 and develop the differential equations for the terms of the (significant)local expansions in the two boundary layers. We perform the analysis in some detail for the neighbourhood of x = 0. Assuming, as in Section 4.3, the local expansion m

We find, by straightforward application of Theorem 4.2.1,

9,t+bp= F(0). The general solution of this equation reads

t+bbo)((o) = A,e-w(o)~o+ B 0ew(o)e;o + F(0) where w(0) = (bo(0)/ao(O))1’2. Using the overlap hypothesis it is quite simple to deduce that

B, = 0. This follows from the fact that the regular expansion E;O is bounded for &LO, while t+bbo)((,), considered in an intermediate variable, is unbounded, unless B, = 0. Next we observe that, since there is only one significant degeneration, the correspondence principle permits to deduce (see E.1.1 of Section 4.4) that the significant local approximation EToO is valid on toE [O,A], VA > 0. Hence we may impose the boundary condition at x = 0, and obtain A , = a - F(0).

Proceeding to higher terms of the expansion one can develop the differential II > 0, from the result of Theorem 4.3.1*, i.e. equations for t+b!,’)( ,), 9,E;o(D

=

Egy-

where, in the present case,

Eg)9pE;o-1(D,

CH. 4,$5

FURTHER DEVELOPMENT OF T H E HEURISTIC ANALYSIS

d2

+ dt0

9p= [ a 2 ( t o & ” 2) a2(0)],

105

d &1’2a1(to&1’2)-

dt0

+ & U 0 ( t 0 & ” 2 ) + bo(to&’/2)- b,(O). Explicit results for n = 1,2, ... can be obtained by introducing for the coefficients of the differential equation, Taylors expansions in the vicinity of x = 0. The reader may consider it an exercise to convince himself that for n = 1,2,... one has

$!,‘)(to)= An(
*!,yo) = 0 and matching according to any of the principles developed in Chapter 3, yields n = 1,2.... unique determination of the functions $Lo)(to), The computational effort in the calculations outlined above, and in particular the application of matching conditions, can be simplified and made more systematic, by ‘subtracting the regular approximation’, as outlined in Section 3.5. Using that approach we start out with the function

a, = 0 - E ! p . We then have the problem

L z 5 =&m+lpm, pm= -

with boundary conditions m

5(0,&)= a -

c En4,(O);

n=O

m

a,(l,&) =p -

c &nc$n(l).

n=O

Obviously, the regular approximation of d, is zero, up to the order of magnitude of Local analysis of the boundary layers can be performed as before. Let us write 2m

E;:0

=

1 &“’*$Ip’((o). n=O

Then again, the general solution of the differential equations for the terms of the expansions is of the structure

$Lo)(t0) = An(to)e-w(o)Co + Bn(to)ew(o)~o + f,,(t,).

106

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

CH. 4 , # 5

The boundary conditions now are

4&o)'co) = - 4 0 0 ,

$;"do)

=

- &(O),

p = 1,2,.. .,

$Ip)(O) = 0, for n odd. Matching with the regular expansion (which is zero) reduces to the very simple set of conditions = 0, V n . lim $:')(lo) 50- Jt

Of course, entirely analogous analysis can be performed for the neighbourhood of x = 1. Putting together the results one obtains a function which, if all heuristic hypothesis that have been made are true, is an asymptotic approximation of @(x,E)on x E [0,1]. The explicit result for the first approximation reads F(x) @ ~ ~ ) ( x=, & ) + [ a - F(o)]e- w ( O l x g E + [P-F(1)le-"""' - x ) / g , where w(x) = ( - bo(x)/ao(x))"2. We conclude with a remark which, although rather obvious, may be useful in avoiding confusion when computing higher order terms of local expansions. Because of the polynomials occurring in I)!,~)(;",), the expansion EFo@ will contain terms of the structure gn(&,tO) = & " i 2 5 ne-w(0)co. 0 Transforming into x-variable one obtains

T,g,(&,to)= x"e - "(O) One may now begin to wonder whether the term under consideration is not really of the order of magnitude of unity. This of course is not true, and one has, by elementary analysis, the strict estimate

Example 2. We consider again, for x E [O,l], the problem L,@ = F ,

@(O,E) = CZ, cD(1,~)= /3

where L, = EL1

+ Lo,

CH. 4,45

FURTHER DEVELOPMENT OF THE HEURISTIC ANALYSIS

107

We now study the case

b,(x) < 0, x

E

[O,l].

As before we take a 2 ( x )> 0, F independent of E , and furthermore F ( x ) and all coefficients of the differential operator infinitely differentiable for x E [0,1]. Assume that a regular expansion of 0 exists in some subinterval. Then, following Section 4.3, m

E:Q, =

C

En+n(X),

n=O

Lo40 = F,

Lo+,, = -L1+,,-,,

n = 1,2,....

Lo is a differential operator of the first order and therefore, contrary to Example 1, the regular expansion is not uniquely determined by the relations given above. In order to determine E T 0 one needs either boundary conditions or matching conditions. The correct choice of the conditions to be imposed on the regular can be made only if one has the knowledge of the location of the boundary layers. Consider the neighbourhood of x = 0 and assume the validity of the correspondence principle. Then, according to the analysis in E.1.2 of Section 4.4, there is no significant approximation in any local variable and hence no boundary layer behaviour. This is simply a consequence of the structure of the significant degeneration. According to our earlier analysis the significant degeneration arises for

to= x/e and is given by d2 d Y o = a 2 ( 0 ) 7 b,(O)-. dt0 dt0 In the present example we have

+

b,(O)/a,(O)< 0. It follows that all non-constant solutions of Yo*o = 0 grow exponentially for increasing conditions.

to,and must be excluded by the matching

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

108

CH. 4,aS

Since there is no boundary layer at x = 0, we conclude that the domain of validity of the regular expansion includes the origin, and we impose the boundary condition Solving the differential equations for the functions $,,(x) we obtain explicitly

X

where q(x) = J(bo(f)/bl(f))dX 0

Now in general (E,"@)x=l# P and we must therefore expect a boundary layer at x = 1. To simplify the computation we now reformulate the problem by subtracting the regular expansion. We thus write (5 = @ - E!pD

and consider the problem LEO= &m+lpm, Pm = -L14m with boundary conditions m

&(O,E)

= 0;

6 ( l , & )= p -

c Eyb"(1).

n=O

The regular approximation of (5 is of course zero up to the order of magnitude of Ern+l.

Assuming the validity of the correspondence principle we have, from Section 4.1, near x = 1 the boundary layer variable 51

= (1-X)/&

and the significant degeneration

Using Theorems 4.2.1* and 4.3.1*, and assuming local expansion of the structure

FURTHER DEVELOPMENT OF T H E HEURISTIC ANALYSIS

CH. 4, $5

109

m

EF16 =

C

~~$i~)(t~).

n=O

We find

9 0 $ p= 0. The differential equations for

$hl),

n > 0 can be derived from

9oE!16 = -EK' LZPE!F16 where in the present case d2 d Y P= [ U 2 ( l -&tl)- U 2 ( 1 ) ] 7 - & U , ( l -&tl)di"1 dtl

For n

=0

+ &2Uo(l

-E<1)

we have explicitly

From the overlap hypothesis, and recalling that the regular approximation of up to the order cm", we deduce

6 is zero

-

Bo = 0. Since there is only one significant degeneration, the correspondence principle implies by the analysis outlined in E.1.2 of Section 4.4 that the significant local approximation E;,O is valid on t1E [O,A],VA > 0. Hence, we may impose the boundary condition at x = 1 and obtain

A0

=

P - $o(l).

The reader should again consider it an exercise to pursue the analysis to higher orders of approximation. Analogous to Example 1, one can establish that the functions $!,"(i",) are of the structure

$il)(tl)= A n ( t 1 F P t 1+ where An(tl)and $,,(t,)are polynomials of degree n. The functions be uniquely determined by imposing the boundary conditions

$L1)(tl)can

&(O) = - On(1), n = 1,... and matching with the regular expansion. Given the structure of $hl)(tl), and the fact that the regular approximation of 5 is zero up to the order of E ~ " , the matching relations simply reduce to lim t1-K

gn(t1) = 0,

~ n .

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

110

CH. 4,55

Combining the results obtained so far we find that, if all heuristic hypothesis that have been made are true, an asymptotic approximation of @ for x E [OJ] is given by the function

+

@i:)(x,c) = ~ $ ~ ( x )[ f i - C$,(I)]~-”(’-~)’~

where p = - bl(l)/al(l). We conclude by remarking that if instead of the case b,(x)/a,(x) < 0

one considers the case b,(x)/az(x) > 0,

then an entirely analogous analysis can be performed. In that case the regular expansion is valid on x E [d,l], Vd>O, and a boundary layer at x=O occurs.

Example 3. Consider, as in Example 2, for x E [O,l] the differential equation L,Q

=F

with L,

= EL1

+ Lo,

We shall now study the initial value problem defined by the initial conditions @(O,E)

= a,

&)x=o

= p*

We exclude turning-points by assuming b,(x) # 0, and impose furthermore the condition b,(x)/az(x)

=- 0,

vx E [ O J l

This condition is essential for our analysis, as will be shown at the end of this section. Assuming the existence of a regular expansion m

ErQ

=

C n=O

we find again

Lo40

= F,

E’$,(X)

CH. 4,$S

FURTHER DEVELOPMENT OF THE HEURISTIC ANALYSIS

111

n = 1,2,....

Lo$,, = -L14n-l,

Since Lo is a first order differential operator, the regular expansion cannot satisfy in general the initial conditions at x = 0, and a boundary layer behaviour must be expected. On the other hand, it is not clear yet what boundary condition should be imposed in order to determine uniquely the expansion EF@. Before examining the boundary layer in detail we first transform according to Section 3.5, by introducing

8 = 0 - E;0. The transformation can formaly be performed although the function I?:@ is not yet uniquely determined. In fact we shall see that the analysis of 5 also leads, in an efficient way, to conditions defining EF@. Performing the transformation we obtain

LE8=

-&m+l

LI 4 m 3

The regular approximation of

8 is again zero up to the order of magnitude

of

& m t l

We now introduce the local variable that produces the significant degeneration, i.e.

t o = XI& The differential equation transforms to To$* + 9P 5*=

-&m+2

~ O L4 Im

where

The initial conditions become m

(5*)so=o =M

-

1 En4,(0);

n=O

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

112

CH. 4,95

In the next step we assume the existence of a local expansion m

q 0 5=

c

Efl$fl(to).

fl=O

Then

etc. Reasoning as in the preceding examples we deduce that the domain of validity of Erod, must certainly contain the origin, and impose therefore the conditions

qO(o)= u - 40(~);

$,,(o) =

- 4,(0)

for n = I, ...,m.

Furthermore, we claim that, in the present problem, the existence of the local expansion implies, that the terms of the expansion satisfy initial conditions derived from the initial conditions for (d@*/dt,),, = o, as follows:

To prove the assertion we rewrite the problem for %* as an equivalent integral equation

+ with ,u = b,(O)/a,(O).

rz] dto

50-0

50

f e-”‘d(

+ &*(O,E)

0

Now, given that LfP satisfies the conditions of Theorem 4.2.1*, given furthermore that 5*and its derivative is bounded at to= 0, and that $* tends to uniformly as E J O , it is not difficult to show (using essentially integration by parts) that the first term on the right-hand side in the above identity tends to zero as E 10. Hence we find

lim 5*(t0c) = i+To(t0) = lim &+O

c-0

[*]:

50

__ 0

From this it follows by differentiation, that

f e-”{dt


+ 5*(0,0).

CH. 4,95

FURTHER DEVELOPMENT OF THE HEURISTIC ANALYSIS

113

This proves our assertion for n = 0. The reader should convince himself that, extending this technique, the initial conditions for n > 0 can also be deduced. n = O,...,m, together with the given initial The differential equations for qfl, conditions, uniquely determine these functions. If the coefficients of L, have Taylor expansions near x = 0, then all functions Ffl, n = 0, ...,m, are of the structure J n ~ t o= ) an(i"o)e-peo+ SnCto)

where &(to)and $,(to)are polynomials of degree not higher than n. We finally assume the validity of the overlap hypothesis and examine the .effect of matching with the regular expansion for 8 (which is zero up to the order of magnitude of t r n + l ) . It follows that

Qt0)= 0, Vn. We now apply the results so far obtained for explicit computation of the terms of the expansion. Starting with n = 0 we have

tJo(t0) = AIoe-Nc0 + So. Imposing the initial conditions we find CO(C0)

=

- 40(0)*

The matching condition then requires $o(to) = 0;

40(0) = a.

We have thus found an initial condition which determines uniquely the function

40W. Proceeding to n = 1, we have

ql(t0)= Ale-pco + Sl where

2, and S, are constants. Imposing the initial condition yields

9,= -(b1(0) - A",. Imposing the matching condition we obtain

41(o)= a1= -IJ1(O).

pl(t0)now is fully determined and we have also obtained an initial condition

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

114

CH. 4, $5

which uniquely determines the function 41(x).Proceeding to n = l,...,m one can determine in this way all the functions occurring in the expansions. Combining the results one finds that, if all heuristic hypothesis that have been made are verified, an asymptotic expansion of @(x,E)for x E [0,1] is given by the expansion m

@XX94 =

c

m

&"4n(X)

n=O

+

c Oz(t0).

n= 1

The conditions to be imposed on the functions occurring in the expansions can be summarized as follows: The functions 4,,(x), n = 0,...,m satisfy

do@)= The functions lim to+

c(

= @((I,&);

4,(0) = $,,(o)

for n

=

I,. . .,m.

$,,(to),n = 1,...,m satisfy

IC/n(50)

=0

5

and furthermore

x=o

f o r n = 2 ,...,m.

We now turn to the case P = b,(O)/a2(0) < 0. The preceding analysis is not affected by conditions on the sign of p, until the stage in which matching conditions are imposed. Taking up at that point, for p < 0, one would reach the conclusion that

An((0)= 0,

$,,(to)= 0,

Vn.

But then one cannot satisfy in general the initial conditions. The conclusion is that some of the hypotheses made in the course of the analysis are not satisfied. It is not too difficult to discover that the hypothesis that is not correct in the present case is the existence of a regular expansion of 0 on some non-degenerate subinterval of [O,l]. This can already be seen from the simple example of the initial value problem E-

d20 dx2

+ p-d@ = 0, dx

CH. 4,45

FURTHER DEVELOPMENT OF THE HEURISTIC ANALYSIS

115

The exact solution reads

+ (cc-fl).

@(x,E)= fle-Px'&

It is clear that, when p < 0, the solution for any x > 0, grows exponentially as 10, and there is no regular approximation. One can show (see Gee1 (1978))that this behaviour occurs in general when The case p < 0 is also instructive from another point of view, because it produces examples in which the heuristic reasoning leads to wrong conclusions. Consider, for x E [0,1], the problem d2CD

dCD

&---=f

dx2

de- l i x f=,x,

>

dx

Integrating once, and taking into account the initial conditions we find that CD must satisfy

Assume now that there exists a regular expansion of CD in some subinterval. Then by the standard procedure, we obtain the function m

@%&)

=

1 &n4fl(x),

fl=O

C$~(X)

= -e-l'",

CDrS(x,&) defines an infinitely differentiable function for x E [0,1] and satisfies all initial conditions imposed on the function CD. Furthermore, m can be chosen arbitrarily large. There is hence no obvious reason to investigate the boundary layer behaviour, and one may be tempted into conclusion that CDrS(x,&) is an asymptotic expansion of CD for x E [0,1]. However, the exact solution of the problem reads 1

X

&

0

@(x,&)= ?e(l/&)X e-(l/&)x'e-l/X'dxj

Consider x > xo > 0, where xo is an arbitrary but fixed number. Then, by elementary estimates

H E U R I S T I C ANALYSIS OF S I N G U L A R P E R T U R B A T I O N S

116

CH. 4, %

It follows that the solution @(x,E)grows without bound as E 0. The function @rS(x,&) is bounded for E 1 0 and can therefore not be an approximation of the solution. The example treated above shows that heuristic reasoning does not guarantee the correctness of the results, even if the construction is free of contradictions. In other words, a proof of validity of the presumed approximation is indispensable as the closing link of the analysis. We shall discuss briefly some basic aspects of the proof in Section 4.7. Full attention to the problems of proof will be given in Chapter 6.

4.6. Heuristic analysis continued: some two-dimensional problems We shall analyse in this section, along lines similar to Section 4.5, some elliptic Dirichlet problems in R2. We restrict ourselves to simple differential operators and simple geometries so as to obtain simple explicit results. This will permit us to develop the reasoning unobstructed by the technical complexity of the calculations. We emphasize however that the simplifying assumptions used in this section (differential operators with constant coefficients, elementary geometries) are entirely unessential for the main steps of the analysis. This will appear clearly in Chapter 7. In what follows our goal will be to obtain global insight in the structure of the approximations, that is, the location of the boundary layers, the differential equations defining the various expansions and the (expected) domains of their validity. The objective of this section is, as in Section 4.4, to show by examples how the concepts and results so far obtained can be used for the investigation of perturbation problems.

Example 1. Let D c R 2 be given by D

=

{x,JI

x2

+

< 1).

We consider the differential equation

&A@- @ = F where A is the Laplace operator. On the boundary r, defined by

r = {x,J(x 2 + y 2 = 1)

C H . 4, $6

H E U R I S T I C ANALYSIS C O N T I N U E D

117

we impose the condition @ = 8,

r.

(X,y)E

F and 8 are given functions, which we assume to be infinitely differentiable. Here, and in the sequel of this section we assume for simplicity that F and 8 are independent of E . The analysis that follows is very analogous to Example 1 of Section 4.5. Assume that in some subdomain a regular expansion of @ exists. Following Section 4.3 we write

c m

E;@ =

En4,(X,Y)

n=O

and obtain

4 n = A 4 n - l , n = 1,2,....

q50= - F ,

Clearly, in general

E ; @ # 8, for (x,Y) E r hence we must expect a boundary layer along r. We now subtract the function E;@ and obtain

6=@-E p , & A 6- 6 = E ~ 6=u-

c

+ ' A ~ ~ ,

m

[;nan,

x,y

E

r.

n=0

Obviously, the regular expansion of

6 is zero up to the order

of magnitude of

[;m+ 1

In order to be able to define local variables along r, we must first introduce a new system of coordinates p,v which is such that for any point P(p,v), p = 0 implies P E I-. In the present case it seems quite logical to use for this purpose the polar coordinates x = (1 -p)cos v,

y

= (1 -p)sin

v.

Our problem then transforms to

6=

m

C

E~U,,(V)

for p

= 0.

n=O

It is now straightforward to verify (see Example 2 of Section 4.1), that there is one significant degeneration that arises for

CH. 4,$6

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

118

and is given by

a2

9 -7-1. O - a g

In this local variable the differential equation reads

Yo5*+ YP5*= E

~

~

~

A

~

~

where

Assuming the validity of the correspondence principle and using results of Sections 4.2, 4.3 we write

n=O

and obtain

etc. We observe that the equations defining the functions (I, are ordinary differential equations in which the variable v occurs as a parameter. For this reason the boundary layer along r is called an ordinary boundary layer. Using the overlap hypothesis we can impose on the expansion E:m& matching conditions, which again take a simple form, because the regular approximation is zero (up to the order c m + l ) . We can furthermore impose the boundary conditions for p = 0, which take the form $zp(0,v) = q v ) ,

P

= OJ...,

$,,(O,v)= 0 for n is odd. There is no difficulty in determining now the terms of the expansions. We find, for the first two terms $ O ( t J ) = 0o(v)e-C,

@,((,v)

= -+3,(v)<e-C.

CH. 4, 66

HEURISTIC ANALYSIS C O N T I N U E D

1 I9

Putting together the results obtained so far, one obtains a function which, if all heuristic hypotheses made are true, is an asymptotic approximation of (D for x,y E D. Explicitly the result reads

When interpreting the result given above the reader should keep in mind the remark made at the end of Example 1, Section 4.5, concerning the order of magnitude of terms of the structure ("e-'.

Example 2. Consider now the differential equation 8Q) EAQ)- / A T = F , /A > 0. G'Y

The domain is as in Example 1, i.e.

D

= { x , y / x 2 + y 2< l ) ,

r = ( x , ~ ~ J x ~=+1;.L . ~ Furthermore Q, =

0 for

(XJ) E

r.

We shall find that the analysis is, in certain aspects, analogous to Example 2 of Section 4.5. However, new phenomena will also appear. Assume the existence of a regular expansion in some subdomain. Then

c c"&,(x,y), m

EYQ, =

n = fJ

In order to determine the regular expansion we need boundary conditions. We therefore analyse the possible location of the boundary layers. We introduce for that purpose the polar coordinates x = (1 -p)cos v, 4' = (1 -p)sin v. The problem then transforms to

-"I@

1 i i2 + 1 E[zF - ~1- p i p (1

-p)2

2

Q)

= tl

for

p =

0.

iv2

cos V i Q ) + ,usin v iQ) y + ~- F, G p 1 - p iv

120

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

CH. 4, $6

The significant degenerations can now be determined, following Example 3 of Section 4.1. We find that, if the neighbourhoods of v = 0 and v = x are excluded, then there is only one significant degeneration which arises for the local variable

5 =PIE and is given by

-Yo = p

+ p sin v--.85a

Excluding neighbourhoods of v = 0 and v .= n: means excluding neighbourhoods of the extremal points A and B of D. This leads naturally to a subdivision of the boundary r into a ‘upper part’ r +and a ‘lower part’ r- (see Fig. 4.2). Consider r-,i.e. v E (n:,2x).By a reasoning analogous to Example 2 of Section 4.5 (see also E.1.2 of Section 4.4) we deduce that there is no significant approximation. This can simply be seen from the fact that along r- any nonconstant solution of -Yoqo = 0

grows exponentially with increasing p, which makes matching with regular approximation impossible. There is thus no boundary layer along r-,and consequently r- must belong to the domain of validity of the regular expansions. Let us describe the boundary condition along r- as follows @(x,Y,E)= d-(x)

for y = - (1-x2)’/’.

We are now able to determine the terms of the regular expansion and obtain

Fig. 4.2.

HEURISTIC ANALYSIS CONTINUED

CH. 4, $6

121

The result given above is somewhat deceptively nice. The reader should have no difficulty in discovering that, for n = 1,2,,.., the functions $,(x,y) in general possess singularities at x = T 1. This corresponds to the points A and B, which we have already been forced to exclude while considering the significant degenerations. Excluding neighbourhoods of A and B we pursue the analysis in a reduced domain D,, defined by

D , = { x , y ) x ’ + ~ . ~ < l-;l + d <

x

<

l-d,d>O}.

There is no difficulty to follow now the procedure used in the earlier examples. We define

5=@-Ep which leads to

m

@ =

e - nC= O E n $ n ( X , y ) ,

0 = 0,

(x,Y)E r,, (X,Y)E

r-.

Transforming to polar coordinates, and introducing the boudary layer variable

5 = PIE one obtains

Yo8*+ Yp6*= E ~ ” A $ ~ where

This leads to the expansion m

122

HEURISTIC ANALYSIS OF S I N G U L A R P E R T U R B A T I O N S

CH. 4 , # 6

etc. As in Example 1, the differential equations defining the terms of the local expansion are ordinary differential equations, in which the variable v occurs as a parameter. The boundary layer along r +is therefore called again an ordinary boundary layer. We rewrite the boundary condition for the function 6 on T + as follows:

6=

m

1 &ne,'(v)

for p = 0, v E(O,TC).

n=O

Imposing this condition on $,(O,V)

E T 6 yields

= e,'(V).

Finally, imposing matching with the regular approximation of 6 (which is zero up to the order & " + l ) again leads to the simple set of conditions lim $,,(<,v)

= 0,

Vn.

5-a

There is no difficulty in determining now the functions result is

$,,(t,v).

For n = 0 the

i 0 ( 5 ,4 = 0; (4exp { - p < sin v} Let us now examine briefly the nature of the difficulties that arise in the neighbourhoods of the extremal points A and B, excluded in the preceding analysis. These neighbourhoods are regions of transition from the boundary r along which there is no boundary layer, to the boundary r + along which boundary layer occurs. It is for this reason that the problem of investigating the solution near A and B has been called the problem of birth of boundary layers (Grasman 1971). From the point of view of the investigation of degenerations, A and B are the two points at which the boundary r is tangent to a characteristic of Lo. In E.3.3 of Section 4.1 we have established that there are then two significant degenerations. One can now use the correspondence principle in order to analyse CD in the neighbourhood of these points, and thus complete the analysis of CD in D. We shall describe this analysis in detail in Chapter 7. We finally remark that if p < 0, then, as in Example 2 of Section 4.5, the situation with respect to the location of the boundary layers is reversed, and one has a boundary layer along r- and no boundary layer along r+.

Example 3. We consider as in Example 2

CH. 4, $6

HEURISTIC ANALYSIS CONTINUED

123

Fig. 4.3.

@=8

for (x,Y)E r.

However, r now is an arbitrary smooth convex curve which is cut twice by any line x = c, c E ( x o , x l ) .The situation is sketched in Fig. 4.3. From the analysis of Example 2 we may expect a boundary layer along r+ and no boundary layer along r-. We should also expect difficulties in the neighbourhoods of the extremal points A and B where problems of boundary layer birth will arise. We can study the boundary layer along r+by introducing, as in Section 4.1, a new system of coordinates s,p, which is such that s measures the distance along r+while p measures the distance along normals to r+.Using such a coordinate system one must keep in mind that the transformation from x,y to s,p may be only defined in a strip along I-+, because normals to r+may intersect in D. This leads to certain technical difficulties, which can be overcome, as will be shown in Chapter 7. The purpose of the present analysis is to show that if one restricts the investigation to a domain in which neighbourhoods of A and B are excluded, then one can use another system of coordinates which avoids the difficulties described above and leads to simple results. This will also permit us to make some rather fundamental observations about the behaviour of solutions of the problem under consideration. From another point of view, the analysis that follows also is an illustration for a remark made in Section 4.1, concerning a certain freedom of choice in the definition of local variables in R2. It will be convenient to describe the boundaries r+and r-,and the boundary condition on r, as follows

r+= ( X , Y I Y = r + ( x ) ) , r- = { X , Y I Y = Y - ( X ) ) , @(X,Y) =

8 + ( x ) for Y

@(x,y)= & ( x ) for y

=

r+(4

= ?-(X).

The reduced domain D,, for any d > 0 is defined by

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

I24

D, = {X,Y I Y - ( x ) 6 y 6 l/+(x),xo+d d x 6

CH. 4, $6

-d, d > 0 ) .

Assuming the existence of a regular expansion in a subset of D, we deduce, analogously to Example 2, m

KP

=

c &m4n(X,y),

n=O

1

~ O ( X )=

y

S

-p

1

4 , ( ~=) - -

F(x,y‘)dy’

+ O-(x),

Y-(X)

1y

A$n-l(x,y’)dy’, n = 1,2,...,m.

Y-(X)

Next we subtract the regular expansion by introducing

6 = 0 - E!pD and obtain

m

%Y)

= O+(x) -

Ef14,(X,Y)

for y

= ?+(XI,

n=O

for y = y - (x).

6(x,y) = 0

We now introduce a new system of coordinates by the transformation

x = x;

y = y+(x)-y.

We shall write

a(x,Y+(x)-y,E) = a(x,y,&). We further abbreviate

Performing the transformation one finds

+

a2a [l+W2(2)]7

ay

+ 2W(X)- axay

aa = E ~ + ’ A ~ ~ . + o(%)-aa +ay ay The domain is defined by

0 < y 6 y + ( 3 - Y-(X),

CH. 4,96

HEURISTIC ANALYSIS CONTINUED

125

xO+d<x<x,-d.

The boundary conditions take the form

c m

G(X,O) =

&ne,,(q

n=O

where O O ( 4 = O + ( 8 - +o(%Y+(a> O,(X) = - +,,(T,y + (z)))), n = 1,. . . , m .

Furthermore = y+(g)- ?-(X).

G ( x , j )= 0 for j

Analysis of the degenerations in local variables along j = 0 again yields one significant degeneration which arises for the local variable '1 = YIE.

Transforming to this local variable we find

-

U

(32

Y o=

[1+

w2(2)]

i

+ 7, 6'1

c'1

(32

Y p= & 2w(X);: uxa'1

+

This leads to the expansion

c &ntJn(X,'1), m

qif,=

fl=O

YOtJO

= 0,

etc. Reasoning as in the preceding examples we impose on the terms of the expansion the boundary conditions

tJfl(x,o)= e,(x). Furthermore, matching with the regular expansion of @ (which is zero up to the order cm+ ') yields the simple condition lim $,,(x,?)= 0, n q - T~

=

o,.. .,m.

126

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

CH. 4,Q6

Explicitly, the first term of the expansion reads

Further terms of the expansion can of course be obtained by straightforward computation. Summarizing and writing the result in terms of the original variables we find that, if all heuristic hypotheses that have been made are correct, an asymptotic approximation of @ in D, is given by

where

Let us assume that the result given above is indeed correct (we shall prove that this is the case in Chapter 7). We can then draw certain conclusions concerning the behaviour of solution under perturbation of boundary data and geometry. Our result holds in any domain D,, defined by the restriction

x,+d<x<x,-d,

d>0.

These are subdomains of D bounded by any two characteristics of Lo which cut r twice. The approximation of the solution in D, only depends on data in D,. Consider now two problems for which the geometry and other data only differ outside some fixed domain D,. Then the asymptotic approximation for both problems in D, is the same. In other words: within the accuracy of asymptotic approximation, geometry and other data perturbation outside D, are not felt inside D,. We shall make this statement more precise in the analysis of elliptic problems in Chapter 7.

Example 4. We consider the prototype problem

CD = 8 for (x,Y)E r and study the case in which the boundary r contains a segment that coincides with a characteristic of Lo,i.e. a line x = const. The simplest case is a rectangular domain, sketched in Fig. 4.4.

CH.4,$6

HEURISTIC ANALYSIS CONTINUED

127

Fig. 4.4.

Let us again commence by subtracting the regular expansion which satisfies the boundary condition along r- (i.e. along y = yo). We then have Z) = @ - E:@,

6=0

for y

Z) = g'O)(y,E) for

= yo.

x = xo,

6 = g")(y,E) for x = x l , 6 = ~ ( x , E ) for y = j1 where g'"(y,E) = [@ - ET@]x=x,l,

g"'(y,E) = [@

- Ey@]xFxl,

f ( ~= )[@

E:@],=,,.

-

The regular expansion of @ is zero up to the order E * & ' , while along y = y1 a boundary layer of the type studied in Example 3 will occur. We must further expect new boundary layers along x = xn and x = xl. Consider the neighbourhood of x = xn. Studying local variables we find (see Section 4.1) one significant degeneration, that arises for the local variable 50 =

(x- xo)/&.

The differential equation takes the form

Yo&* + gP6*= E ~ + ' A $ ~ ,

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

128

CH. 4, $6

This suggests a local expansion m

q0& = 1E"Up(SO,y) n=O

with CJo u p = 0, dpoujp)=

a2 ---U'O' dY2

n-l,

n = 1,...,m.

The functions defining the boundary layer are now solutions of partial differential equations of parabolic type. The boundary layer along x = xo is therefore called a parabolic boundary layer. In order to determine the boundary conditions for the terms of the local expansion we must determine what is the domain of the variables. From the correspondence principle we can deduce, as before, that

toE [O,A], V A > 0. This leads to the boundary conditions Ub0)(O,Y)= (@LXO - ($o)x=xo, Ulp'(0,y) =

-($n)x=xo,

n = 1,...,m.

Considering the extent of the domain of validity of the local expansion in the variable y we can only make a hypothesis. Let us assume that the domain of validity is given by Y E

CY0,Yll.

Then UIp)(t,y)= 0, Vn.

The functions ULo)(t,y)can now be determined if we impose matching with the regular expansion of 5, which again reduces to the simple condition lim U$"?(to,y)= 0, n = 0,...,m. 50-

m

It is a simple exercise in the solution of the heat-equation to establish that

CH. 4,57

T H E C O N C E P T OF F O R M A L APPROXIMATIONS

129

where One can formally proceed to determine VIp)(<,y),n = 1,...,m. However, explicit computation shows that in general these functions possess singularities for 5 = 0, y = yo. Hence at the point A difficulties arise, which again can be characterized as a problem of birth of the (parabolic) boundary layer. The neighbourhood of that point must be studied separately, using the significant degeneration derived in Section 4.1. It should be clear that, for the boundary layer along x = x 1 one can perform an entirely analogous analysis, using as local variable 5 1 = (XI -X)IJ&.

Furthermore, along y = y , one can construct an ordinary boundary layer, as in Example 3. One is then left with the problems of boundary-layer birth at points A and B, and one should also expect some difficulties at points C and D where two different boundary layers interact. Further analysis of these aspects of the problem will be given in Chapter 7. 4.7. The concept of formal approximations The analysis of the examples in Section 4.5 and 4.6 can be given a new interpretation by introducing a concept which, although implicit in the preceding constructions, has not yet been explicitly defined and discussed. We commence therefore with some definitions. In what follows the symbol x denotes again in general a vector. Definition 4.7.1. A function Qas is a formal approximate solution of the differential equation L,@ = F , x E D if LED,,,= F + p , p = o(1) in D. Formal approximate solutions will be called formal approximations.

Definition 4.7.2. A formal approximate solution )@ ::

of

L E @ =F , X E D

is a formal asymptotic expansion (of some solution of that differential equation) if

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

130

CH. 4,57

6,

with S n + l = 0(6,), = Os(l). (ii) For any q = O,l, ...,in there exists an order function 6:) such that

We claim that in the examples of Sections 4.5 and 4.6, the construction, when carried out to a sufficient number of terms, produces formal approximations and formal expansions. This can be verified by inspection, and is furthermore already implicit to some extent in the Theorems 4.3.1 and 4.3.1*, which define the differential equations for the terms of various expansions. However, performing the verification a suitable grouping of terms may be needed in order to obtain formal expansions. For instance, in Example 2 of Section 4.5 a formal asymptotic expansion is given by m+ 1

m

@g)(X,&)=

1

&“(b,(X)

n=O

+ n1 &“$p((o);6’: =O

= &Q+l .

In other words, one needs, for consistency, one term more in the local expansion, then in the regular expansion. Similar phenomena arise in Example 2 and 3 of Section 4.6, where one obtains formal expansions in the reduced domain D,. In the Example 4 of Section 4.6 one obtains formal expansions in a subdomain in which neighbourhoods of the four corner points are excluded. The reader should consider it a useful exercise to demonstrate explicitly that the analysis of Section 4.5 and 4.6 indeed leads to formal expansions. He may also consult, in particular for Section 4.6, Eckhaus and De Jager (1966). The functions constructed in the examples of Section 4.5 and 4.6 not only are formal expansions in the sense of Definition 4.7.2, they also satisfy, to a certain degree of accuracy, the boundary conditions or initial conditions imposed on the problem. Let us, to be more specific, consider Dirichlet problems for second order elliptic equations (or two-point boundary value problems for ordinary differential equations). Then the problem we wish to solve is defined by L,@=F,

XED;

@=e,

X E r

where r denotes the boundary of D. Suppose that the formal expansion that has been constructed satisfies L,@iy)= F )@ ;:

=

+ p,,

e + I,,

p m = O(Sg)), x E D,

im = o(~g)), x E r.

0;;) is thus a solution of a problem that is, in a sense neighbouring’ the problem we wish to solve. @i;)is’an asymptotic approximation of @ if for the problems under consideration, it is true that ‘neighbouring’ problems have suitably ‘neighbouring’

CH. 4,$7

THE CONCEPT OF FORMAL APPROXIMATIONS

131

solutions. Let us make this statement more precise. We formulate the problem for the remainder term R,, which, due to linearity of L, and the boundary conditions, takes the following form:

CD = CDiT) + R,, p, = 0(6:)), x E D,

L,R, = - p m , R,

i, = o(a;)), x E r.

= -(,,

Taking again advantage of linearity, we rescale R,

=

6;)Rm,

c, = -6(*)C

p, = 6;'p,,

m

m'

Then L,R,

= P,,

x

ED;

R, =

c,-,

xE

r.

The problem for R, is of the same general nature as the original problem for

CD, only the data on the right-hand side of the equations are changed. However,

we do not have to solve for R,, all we need is an estimate of R,. In other words, we need some general a priori estimates for a class of problems, to which the problem for CD belongs. We thus arrive at

Lemma 4.7.1. Let 4 and W be two sets of functions, and suppose that for each F E 4,BE W ,the solution 5 of

&=e",

X E r

E ~ ;

@=e,

X E r

L,$)::) = F __ 6(')p m rn,

XED,

a$):

r.

L,G=F,

XED;

exists and is bounded as E 10. Let CD satisfy

L,CD=F, X and let

satisfy

=8 -

6;)Cm, x E

Cm E W ,then CD - CDZ)= o(6;)).

If P, E 4 and

The proof of the lemma is contained in the reasoning preceding the lemma. The problem of obtaining the a priori information concerning boundness of solutions, given in the first part of the lemma, will be studied in detail in Chapter 6 . We consider now formal expansions, in the sense of Definiiion 4.7.2, and apply the result stated in Lemma 4.7.1.

132

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

CH. 4, $7

Then

c 6fl(&)qjfl(x,&)+ o(6;)). m

@(X,&) =

n=O

In applications one usually has

8;’

= h,,,.

Then the result given above constitutes the proof that the formal expansion indeed is an asymptotic expansion of the solution 0.For somewhat more exists for all m, and 8;) general sequences { 8;))one also obtains the proof, if is such that for m 2 m, ’6;

+

Sp(m)

where p(m), m = m,, m, + l,..,is an increasing sequence of integers. The boundedness of solutions, required in the first part of Lemma 4.7.1 is not a necessary condition for the validity of a formal expansion as an approximation of the solution, and can be replaced by a somewhat milder requirement.

P E F,&E a, 6 be the solution of L&F, x E ~ ; 6=8, xdsuppose that, if F and & are bounded as 10, 6 satisfies the estimate I1 6 II G

Lemma 4.7.2. Let, for each and

w.

Let @ satisfy L,@=F,

and let):g@

XED;

a==,

+ 6;)Pm,

x E D,

X E r

satisfy

L,@i;) = F

mi:)

=

e + 6;ym,

E

r.

rm 9% rm= Os(l),and furthermore that there

Suppose that jim E 9, jim = Os(l), exists an integer p < m, such that

E

A(&)db‘)(&)= o(1).

Then @ - 0::)= 0(6;’/6b*’).

The proof of the lemma is obvious (and proceeds along the lines of the proof of Lemma 4.7.1). The lemma clearly admits the possibility that A(&) grows without bound as &LO, provided that the growth is limited by the condition stated in the lemma. It should be clear that results analogous to Lemma 4.7.1

CH. 4, @

THE W K B APPROXIMATION

133

and 4.7.2 can be formulated with boundary conditions replaced by initial conditions. The crucial point of the preceding considerations is the a priori estimate for the order of magnitude of the solutions. The problem of finding such estimates will be studied in Chapter 6, where the question of validity of formal approximation as approximate solutions will further be investigated, in a more general setting.

4.8. Expansions with a regularizing factor; the WKB approximations Our preceding analysis was based on expansions in terms of limit functions. In this section and in the next one, we shall discuss two well-known alternative methods for constructing expansions. Our main goal is to compare the methods, and study their relative merits. In the method that we study in this section, one attempts to represent a function @(x,E),x E D by a linear combination of a finite number of functions of the structure F

(g;) __

H(x,E)

where H ( x , E )is a function that has a regular expansion for x E D . Thus, the ) ) . course, all singular behaviour of @(x,E)is ‘removed’ by the factor F ( q ( x ) / d ( ~ Of functions F,q,H have as yet to be determined. Often there is some reason to expect exponential behaviour, and one makes a priori the choice for F to be the exponential function. Approximations that are constructed in this way are commonly called WKB approximations, because they have extensively been used in the well-known WKB method of mathematical physics for the analysis of the transition from oscillatory to non-oscillatory motion through a so-called classical turning point (see Morse and Feshbach ( 1953)). When F is an exponential function, then an alternative formulation can also be used, in which one attempts to represent the function under consideration by a linear combination of functions of the structure

where Q(x,E)has a regular expansion for x E D. The choice of the formulation to be used is essentially a question of computational efficiency. To be more specific about the method, we start with the following simple prototype problem: Let @(x,E),x E [O,l] be the solution of

H E U R I S T I C ANALYSIS OF SINGULAR P E R T U R B A T I O N S

134

E

C H . 4, $8

d2@ y - w(x)@ = 0, w(x) > 0, dx

@(O,E)

= a;

@(l,&) = p.

We look for functions

r6:E)

U(X,E)= exp --Q(x,E)

I

which produce formal approximations of solutions of the differential equation by regular expansions of Q(x,E).A rather obvious choice is

a(&) = J E . Then, by substitution, we find that Q(x,E)must satisfy the equation

Assume now that Q has a regular expansion m

Q(x,E)=

1 EnlZqn(x)+

O(E~”).

n=O

By standard procedure we obtain, for the terms of the expansions, the equations

etc. The solutions are X

qo(x)= T J Jw(x ’)dx ’

+ const,

ql(x) = -In [ ~ ( x ) ] ” “ .

We thus find two formal approximate solutions, which can be written as follows:

CH.4, 58

T H E W K B APPROXIMATION

135

where A and B are arbitrary constants. Consider next

oas= ugg, + ubf! Assuming that Das is an asymptotic approximation of D we impose the boundary conditions and find

+ O(e-n.'e)), B = P[w(~)]'/~+ O(eCn'Je)

A

= CC[W(O)]~

where 1

R = S w(x ')dx '. 0

Let us now compare this result with the result that we have obtained earlier in Section 4.5. The reader should have no difficulty in verifying, by constructing for Ubg) + Uif) local expansions and proving their validity for x E [0,1], that Das = ae-"!JE

+ pe-(l-x)fd'e

+ o(l),

x E [O,l].

Hence, the WKB approximation constructed here is in the first approximation equivalent to our earlier, much simpler result. Is there then any advantage to use the much more complicated WKB formulas? To answer the question we investigate the accuracy of the results. The differential equation considered here is a simple case of the SturmLiouville equation for which many classical results are available. From Erdelyi (1956), Chapter 4, we can use a result which, in the notation of this section, can be stated as follows: There exist two linearly independent solutions U(l)and U " ) of the differential equation, such that, uniformly for x E [O,l], U(1' = u"' as + o(JE)I, U ( 2 )= U'2' as

~1 + o(JE)I.

An important point in the above result is that the error is relative (to the order of magnitude of Ugi) and Ugf) at any point x E [O,l]). Written out in full detail our results thus read

H E U R I S T I C ANALYSIS OF S I N G U L A R P E R T U R B A T I O N S

136

C H . 4, $8

On the other hand, the much simpler result of Section 4.5 has only an absolute accuracy, i.e. CD(~= , &ae-x/v/E ) + pe-"

- X) ~J E

+ o(JE),x E [0,1].

The advantage of the WKB approximation thus appears to be one of high precision, for which one pays the price of complexity of the expressions. Of course, one can pursue further the expansions to higher order, by computing q 2 , q3 etc. It is important to observe at this stage that using WKB approximations one can achieve consistent high precision in special problems only. As an illustration of this remark we consider the problem of constructing approximations to @(x,E),x E [O,l] defined as solution of d2@ dx

&2 - w(x)CD = F(x), w > 0,

CD(l,&) =p

@(O,&) = a ;

where F ( x ) is a given function, not identically zero. We commence by subtracting the regular approximation m

E:Q =

C cn4n(x)5 n=O

We thus define

@ = CD - E ! p and obtain d2@ dz4m & 2 - w(x)@= E m + dx dx2 ' m

@(O,&)

=a -

1 &"@,(O);

n=O

m

@(1,&)= p -

C

&"&(l).

n=O

We consider, for simplicity, the first approximation, i.e. m = 0, and decompose further

@=&+r3 E

d2& 2 - w(x)6 = 0, dx

CH. 4, S;S

THE W K B APPROXIMATION

d2r dx

&y - w(x)r = &-

137

d240 dx2’

r(1,E) = 0.

r ( 0 , ~=) 0;

Assume now that the problem for r satisfies the conditions of Lemma 4.7.1. (That this is the case will be shown in Chapter 6.) Then r = O(E). The problem for 8 can be solved by WKB approximation as in the preceding case, using now the modified boundary conditions. The final result is @(x,s)= &(x)

+ U:;’(X,&)[l +O(JE)l + U::)(X,&)[l +O(JE)I + O(E).

We observe that for x E [d,l -4, V d > 0,

u::)= O(e-d/J&),

(2) uas

- O(,-d/J&),

Thus, the contributions of the WKB approximations in this interval are much smaller than the error produced by the regular approximation. In other words, the relative high accuracy of the WKB approximation becomes irrelevant due to the error produced by the regular approximation and one may as well use the much simpler formulas obtained in Section 4.5. One can of course attempt to obtain better results by proceeding in the classical way, that is by defining the Green’s function for the problem in terms of two independent solutions of the homogeneous equation, introducing WKB approximations for these two independent solutions and then expanding the particular integral. The complexity of the analysis becomes then rather prohibitive. As a further illustration of the use of the WKB approximations we discuss briefly the following problem: O(X,&),x E [O,l] is defined as the solution of E

d2@ dO x a(x)- - w(x)@ = 0, a(x) > 0, dx dx

@(O,&)

+

=r;

@(1,&)= p.

We look for solutions of the differential equation in the form

[a 1

U(X,I) = exp -Q(x,E) Q must then satisfy

.

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

138

C H . 4. $8

Next we suppose that m

Q(x,E)=

&”q,(X) n=O

+O(E~).

Introducing this expansion one finds, by standard procedure, two formal approximations, which can be written as follows:

Consider now

mas = ug:,+ ug:,. Imposing the boundary conditions we find

A

= a(0)“

-

ug:)(o)].

U!$ thus determined is identical with the first term of the regular expansion, when constructed by the method of Section 4.5. Ugk’ is a WKB approximation. Concerning the accuracy of the result one can make observations similar to those of the preceding example. If /? = 0, then a consistent approximation for 0 with a high precision is obtained. If /? 0, then the relative accuracy of the WKB approximation is spoiled in the final result by the error of the regular approximation and one may as well use the simpler results of Section 4.5. WKB approximations for problems in ordinary differential equations give more complicated formulas than expansions in terms of limit functions, but the results are still tractable. Generally speaking use of these approximations can be considered a question of taste, in problems exhibiting boundary layer behaviour. However, exceptional situations can occur where use of WKB approximations, because of their precision, becomes essential, and the analysis cannot be performed in terms of limit functions. This can be seen in Cook and Eckhaus (1973). WKB approximations can also be used (and have been used) in problems for partial differential equations. Levinson (1950), who was the first one to study rigorously elliptic problems of the type described in Section 4.6, used essentially WKB approximations to define the ordinary boundary layers. The method has further been employed in elliptic problems by O’Malley (1967). However, the method has very serious disadvantages in partial differential equations, as compared to expansions in terms of limit functions. Using WKB approxi-

+

CH.4, $9

EXPANSIONS BY T H E M E T H O D OF MULTIPLE SCALES

139

mations one can define the terms of the expansions as solutions of first order non-linear partial differential equations. Solving the equations is not a trivial matter, and for explicit results each geometry must separately be studied. The interested reader should consult the original publications for details and compare with Section 4.6 and Chapter 7. We finally remark that WKB approximations cannot describe parabolic boundary layers. The reason for this is, that the functions describing a parabolic boundary layer are of a much more complicated structure than the exponential function.

4.9. Expansions by the method of multiple scales

In Section 4.8 we have studied expansions of the structure m

fa

c ~n(&)4n(X),i

= q(X)/m.

n=O

Generalizing further we may attempt to represent a function @(x,&),x E D c R’, by a linear combination of functions which, for x E D, have expansions of the structure m

c 6,(E)&(X,O,

i= q ( x ) / W *

n=O

This is the starting point of the method of multiple scales. The development of the formalism is attributed to Cochran (1962), Mahony (1962) and Cole and Kevorkian (1963), and was popularized further in Kevorkian (1966). The method has extensively been used in a great variety of problems and is further described in Van Dyke (1964), Cole (1968), and Nayfeh (1973), where many examples of applications can be found. One can go still further in generalizing the form of the expansions by assuming, for x E D, m

1 4L&)6n(xL)9i= Q(x,E)/~(E) where now Q(x,E)= IS,=, JP(&)qp(x). n=O

Finally, one can introduce several ‘new variables’ of the type of the variable i, by considering expansions m

c

p=o

CH. 4,$9

H E U R I S T I C ANALYSIS OF S I N G U L A R P E R T U R B A T I O N S

140

In applications, the structure of the expansion chosen as the starting point of the analysis usually reflects some a priori insight in the expected behaviour of solutions. The essential rule that defines the terms of the expansions in the method of multiple scales is the so-called non-sedarity condition, which is commonly stated as follows:

Condition. For a / / n $n+

.,m - 1 and x E D

= O,l,..

1/6n

must be bounded as E 1 0. Furthermore, for a / / n

= O,l,.

. .,m - 1, x E D,p

=

l,...,po

'26n

/a;

?[

must be bounded as consideration.

~ 1 0 p.o is

the order of the dgerential equation under

When interpreting the above condition one must distinguish between problems with oscillatory behaviour, and non-oscillatory (i.e. boundary layer) behaviour. In cases of oscillatory behaviour (and this is where the method of multiple scales has most frequently been used) the condition is entirely natural and assures that the expansion indeed is an asymptotic series and that furthermore the derivatives of the expansion, up to the order needed for substitution in the differential equation, still are asymptotic series. To see this consider for x E [O,l] the simple case X

[ = -; E

6, = E n .

Suppose that

Suppose furthermore that, violating the non-secularity condition

then

n=O

n=O

In other words: we do not have an asymptotic series for x E [OJ]. In problems with boundary layer behaviour the interpretation of the nonsecularity condition is quite different. The condition is not needed to assure that

EXPANSIONS BY T H E METHOD OF MULTIPLE SCALES

CH. 4, $9

one has an asymptotic expansion. To see this, consider for x

E

141

[O,l] the series

m

C

&"a,(x)i"e-'; i = X / E .

fl=O

This is an asymptotic series (see remark at the end of Example 1 of Section 4.5) but the expansion is ruled out by the non-secularity condition. In fact the condition imposes approximations in a sense similar to those obtained by the WKB method. This can be seen by considering the function m

@'m)

C

=

dn(&)$n(x>i).

n=O

Consider further an approximation of this function obtained by taking only the first term of the series. Then

+

= dn(&)C#lfl(X,i) [l O(l)].

The non-secularity condition on the derivatives of $, imposes similar behaviour for the derivatives of the expansion. In view of the interpretation given above, one should expect that in application to problems which can be dealt with by the method of Section 4.8, the method of multiple scales will give similar results. As an illustration we consider a problem briefly discussed in Section 4.8. @(x,E),x E [O,l], is solution of d2@ d@ a(x)- - w(x)@= 0, a(x) > 0, dx dx

E - 2

+

@(O,&) = 3,

@(1,&)

=

p.

As in Section 4.5 we define a regular expansion m

E!p=

1 &"C#l,(X) n=O

Next we define and obtain E

d2G y dx

+ u(x)-dG - w(x)G = dx

Em+'-

d2+m dx2 '

HEURISTIC ANALYSIS OF SINGULAR PERTURBATIONS

142

CH. 4, $9

m

@(O,&) = x - $bO(O) -

c E"$bn(O);

@(l,E) = 0.

n= 1

The boundary layer near x = 0 will now be studied by the method of multiple scales. We introduce

i = q(X)jE, q(x) is as yet uriknown, but will be supposed to be positive and monotone, with q(0) = 0.

We write ii,(X,&)= 6(X,i,&), Differentiation yields dii, a6 dx ax

1

86

+ -E q ' ( x c) 7i ,

a26 -1q " ( x 56 2 a26 d% ) 7 + -q'(x)dx2 dx2 E GL & 8x35

+

1 a 26 + +& l ' ( x ) 1 2 a- Ti .

In the above primes denote differentiation with respect to x. Substituting in the differential equation we find

+ &2';iacx26= 0. Next we introduce the formal expansion m

n= 0

This leads to

-1

etc. Solving for 40(x,[) we find

EXPANSIONS BY T H E M E T H O D OF MULTIPLE SCALES

CH. 4,89

143

We abbreviate

and compute the right-hand side of the equation for

dA0 (2q‘R-q)-dx

+ WB,

4,. We obtain:

+ 2q“A0-dR + q ’ A , + wAo dx

dB dx

- a>.

One may verify that, in order to suppress secular terms one must impose dR AoR- = 0 dx ’ (2q’R-Q)-

-wB,

dA, dx

+ 2q’Ao-ddRx + q”A0 + wAO = 0,

+ u-dB0 = 0. dx

We hence find that R is a constant. One can further verify that without loss of generality one can put

n = 1. This means that X

q(x) = J a(x’)dx’. 0

The variable iis thus determined. Proceeding to the second non-secularity condition, we have now a-

dA0 dx

+ (u’ + w)Ao = 0.

The solution is

144

H E U R I S T I C ANALYSIS OF S I N G U L A R P E R T U R B A T I O N S

CH. 4. $9

Finally, from the last non-secularity condition

where c is a constant. The function &Jx,() can now fully be determined by imposing the boundary conditions. We find c = 0,

AO(0)=

- 40(0).

Collecting our results we discover that 6, is identical with the WKB approximation given in Section 4.8. Application of the method of multiple scales to elliptic boundary value problems has been studied by Bouthier (1977), who also found the results to be equivalent with the WKB approximation. Bouthier (1977) has further shown that the method of multiple scales is incapable to deal with the problem of a parabolic boundary layer. On the other hand it should be emphasized that the method of multiple scales is a rather general and flexible procedure, capable of producing formal approximations in problems for which no WKB approximations are known (higher order differential equations, systems of differential equations). On the basis of the discussion of this section one may expect that in applications to problems with a boundary layer behaviour one will obtain, as a consequence of the non-secularity condition, approximations of which the accuracy has properties similar to the WKB approximation.

CHAPTER 5

HEURISTIC ANALYSIS CONTINUED NON-LINEAR PROBLEMS Our aim in this chapter is to develop further the heuristic analysis of Chapter 4 while studying problems in which the differential equation contains non-linear terms, and to investigate the effects of non-linear phenomena on the constructive procedure. The first three sections are preparatory: the concept of degeneration is adapted for non-linear operators and the equations for the terms of the regular and the local expansions are deduced by a technique analogous to Sections 4.2 and 4.3. An important (and well-known) result is that in general only the first term of any expansion is a solution of a non-linear equation, while the equations for higher terms are linear. In Section 5.4 we show by some selected examples of ordinary differential equations, that for certain classes of problems the effects of non-linearity are very mild. In fact, the analysis is entirely parallel to Chapter 4 and no new phenomena occur. The last example of Section 5.4 introduces some more spectacular non-linear effects. This motivates a renewed analysis of the concept of significant degeneration and the correspondence principle, which we give a formulation more adapted to non-linear problems, in Section 5.5. In Section 5.6 we analyse some selected examples of ordinary differential equations which exhibit strong non-linear effects. Of interest to us are the effects on the constructive procedure. The following fundamental phenomena are shown to arise: (i) The location of the boundary layers depends on the boundary data and cannot be determined a priori form considerations of the structure of the differential equation. (ii) The order function defining a boundary layer variable cannot always be determined from considerations of significant degenerations and are found by imposing matching conditions. (iii) The correspondence principle may fail. (iv) Interior boundary layers may occur. The existence and location of these layers depend on the boundary data and can not be predicted a priori from considerations of the structure of the differential equations. In spite of these difficulties we achieve a consistent construction by heuristic analysis. We also show that introducing suitably defined generalized local expansions (instead of expansions in limit functions) one can remove some of the difficulties, and even remove the failure of the correspondence principle. The use 145

146

H E U R I S T I C ANALYSIS C O N T I N U E D . ( N O N - L I N E A R P R O B L E M S ) CH. 5, C1

of generalized local expansions in certain non-linear problems has strongly been advocated by Lagerstrom (1976). In Section 5.7 we turn to partial differential equations and show that in elliptic problems analogous to Section 5.4 the construction is again as in the linear case. Finally in Section 5.8, we briefly discuss the results from the point of view of the concept of formal approximations. Singular perturbations in connection with ordinary non-linear differential equations, to which parts of this chapter are devoted, have attracted considerable attention in the recent years. We emphasize that problems of this type treated in this chapter are only selected examples, meant as a vehicle to convey the concepts and the techniques. Our interest is in the method of analysis and we did not attempt here, nor elsewhere in this book, to give a survey of results for ordinary differential equations. The reader interested in this field should consult Dorr, Parter and Shampine (1973), O'Malley (1974), Howes (1977) and van Harten (1975, 1978), and will find there abundant further references to the relevant literature.

5.1. Degenerations of non-linear operators The analysis of degenerations, analogous to Section 4.1, retains its usefulness when dealing with non-linear problems, however various complications appear. The first difficulty that one faces comes from the fact that in non-linear problems, the relative order of magnitude of various terms of a differential equation may depend on the order of magnitude of the function under consideration. For example, let L, be defined by L,cD = E

d2cD 2 dx

+ (d@ I-, dx

When we restrict ourselves to functions that are OJl), then we may associate to L, a degeneration Lo given by du Lou = u--. dx However, if cD = 8 6 , if, = 05(1),then

A degeneration in this case is defined by

d2u du Lou = __ + udx2 dx

CH. 5 , $1

DEGENERATIONS OF NON-LINEAR OPERATORS

147

We learn from the example that in non-linear problems degenerations must be defined with respect to functions of some given order of magnitude. This leads to certain modifications of definitions of Section, 4.1. In what follows we consider, without further detailed specification, an operator as a mapping of some space of functions into some other space of functions. We shall say that an operator is non-trivial if there exist non-empty subsets in its domain of which the image by the operator is not identically zero. This is a generalization of the statement that an operator is not identically zero, in the linear case. The operator L, will naturally always be assumed to be nontrivial, and furthermore such that (D = 0 implies L,(D = 0. Definition 5.1.1. The degeneration of L, in the x-variable, with respect to functions that are OS(6(&)), is a non-trivial operator Lo such that for all functions u(x), x E D, independent of E , for which L,6u exists and is not identically zero, and for some order function 5(&) lim drL,6u = Lou. &-+O

Definition 5.1.2. Let there be given a transformation to local variable x -+ 4, which induces a transformation of the operator L, into 9,. The degeneration of L, in the (-variable, with respect to functions that are OS(6(&)), is a non-trivial operator 9,such that for all functions v(<), 4 E D o , independent of E , for which 9,6v exists and is not identically zero, and for some order function 4 ~ ) lim $3,6v = 9, v. E-0

It should be clear that the dependence of the degenerations on the order of magnitude of the functions under consideration can lead to difficulties in applications. When studying some given problem one must either have a priori information about the order of magnitude of the solution, or deduce the order of magnitude in the course of the analysis. In the linear case, the study of degenerations has led us to the concept of significant degenerations and to the correspondence principle. Similar concepts can be introduced in the non-linear case. We shall find that, to a certain extent, one can use without any modification the definition of significant degenerations given in Definition 4.1.4 and 4.1.5, and the formulation of the correspondence principle given in Section 4.4. However, the dependence of the degeneration on the order of magnitude of functions makes it necessary, for certain classes of problems, to analyse anew these concepts. This will be done in a later section of this chapter.

148

H E U R I S T I C ANALYSIS C O N T I N U E D . ( N O N - L I N E A R P R O B L E M S ) C H . 5. $2

5.2. The differential equations for the first terms of the regular and the local expansions In this section we shall establish results analogous to Section 4.2. Roughly speaking we shall show that, under certain conditions, the limit of the solution for E LO, satisfies the formal limit of the differential equation. To establish such results in the non-linear case it is of course necessary to prescribe certain features of the operator. Our approach will be to impose sufficient conditions such that the simple technique of proof used in Section 4.2 can be employed again with almost no modification. In the analysis that follows it makes no difference whether one considers regular or local approximations. We therefore use in this section neutral symbols, and consider functions u(t,E),f ( t , ~ )t, E Do c R",E E ( O , E ~ ] .One can identify in the final results t with x, or with some local variable 5. We further suppose u(t,E) to be a rescaled function, i.e. u = 0,(1) in D o . This does not affect the generality of the results. It simply means that in the differential equation a transformation has been performed, based on a priori estimate of the order of magnitude of the solution in the domain under consideration. With these preliminaries we can formulate the following result: Theorem 5.2.1. Let u(t,E),t E D o , E E ( O , E ~ ]satisfy the equation You

+ Y p u =f

and suppose that

c Yg'gl(u(t),t) S

0)

YOU(t) =

1=1

where Yg),1 = 1,...,s, are linear operators, independent of E , and g,, 1 = 1,...,s are (sufficiently differentiable) mappings of R x D o into R. 4

(ii)

Y p u ( t )=

1Y g ) h , ( u ( t ) , t ) I=1

where Yg),1 = 1,...,q, are linear operators such that,for any test function c ( t ) in Do, the adjoints L?g) satisfy, uniformly in D o ,

lim L?g)c = 0. El0

f i e functions h,, 1 = 1,...,q are (sufficiently differentiable) muppings of R x Do into R. (iii) f i e r e exists a function uo(t)such that

lim u ( ~ , E=) u o ( t ) EL0

T H E DIFFERENTIAL EQUATIONS FOR THE FIRST TERMS

CH. 5 , $2

149

uniformly in Do, and 2’ouo is a continuous function. (iv) ‘There exists a function fo(t) such that

uniformly in Do. Under these conditions uo sutisfies you0

=fo,

Proof. As in section 4.2 we consider, for any test-function v, the identity

1 v Y 0 u d t + 1 v Y P u d t = 1 vfdt. D0

DCJ

I)IJ

Using data given in conditions (i) and (ii), and introducing the adjoint operators Pg’ and we find

a:’,

S

{

g,(u,t)s?g)u d t

I = 1 Do

4

+

1 h l ( u , t ) ~ ~d Jt v= 1 ufdt.

I = 1 Do

D0

We now take the limit for & L O . The uniform convergence specified in conditions (ii), (iii) and (iv), and the continuity of the functions g, and h,, permit to establish that S

1

f g , ( u o , t ) ~ g J v d=t

1 ufodt

Do

Do

and hence

f u[LZ,u,-f,]dt

= 0.

Do

Because Y o u o-fo You0

is a continuous function for t E D o we conclude that

=fo.

Comments. The applicability of the theorem is limited to a certain extent by the conditions on the structure of the differential equations, imposed in (i) and (ii). To see the nature of the limitations we consider second order equations. An example which fulfills the structural conditions of the theorem is given by

where Y Eis some linear operator. The conditions are in general not fulfilled if the differential equation contains quadratic terms in the first derivatives, or, if the coefficients of second derivatives depend on the function u and its first derivatives.

150

HEURISTIC ANALYSIS CONTINUED, (NON-LINEAR PROBLEMS) CH. 5,53

One can generalize the theorem to cover such situations, while retaining the simple techniques of proof, as follows: (i) Replace the function g, by g",(u,u,,t), where u , symbolises the vector of which the components are the derivatives duldt,, i = 1, ...,n. (ii) Replace similarly h, by h,(u,u,,t). (iii) Require that, not only lim u ( ~ , E=) u,(t)

&+O

but also

. a u au, lim- = -;

at,

&+Oat,

i = 1, ...,n.

One can treat analogously higher order equations.

5.3. The differential equations for the higher order terms of the expansions We continue the analysis of the function u ( ~ , E t) ,E D o ,E E (O,E,], which satisfies

Y e u= Y o u + Y p u= J In applications one can again identify in the final results the variable t with some local variable t, or with the variable x. Y ois the degeneration with respect to functions which are OJl). We suppose now that there exists an expansion m

EYu =

C

6n(E)Un(t)

n=O

with 6, = 1, and 90%

=fo.

We shall derive the differential equation for the next term of the expansion and indicate the procedure for higher terms. We shall find that in general the functions u,,,n 2 1 satisfy linear non-homogeneous differential equations. We recall that, by the definition of degenerations, Y pis an operator such that for all functions u ( t ) independent of E for which Y p u exists, one has lim Y p u = 0. &+O

We assume that it is possible to rescale the operator by writing S p U

= 6(E)PpU

where 6 ( ~= ) o(1) and Ppis an operator such that for all functions u ( t ) independent of E for which Ppuexists, the function P , u is bounded as ~ 1 0 . As a second preliminary we introduce the Frechet derivative 3:of Y eat uo.

CH. 5,53 THE DIFFERENTIAL EQUATIONS FOR THE HIGHER ORDER TERMS

151

2;is a linear mapping such that 2&(U,

+ w) = z ; w + P ( w ) + Y e u o

and .P(iW) lim -= 0. i.

-

0

1,

We assume that the limit is zero uniformly in E E ( O , E ~ ]E, ~Do. Because of the given decomposition of the operator Y Ewe , also have

9; = 9; + 69; where 2;is the Frechet derivative of Y oat u o , and of PPat uo. We now write U(t,E) =

u,(t)

9; is the Frechet derivative

+ 6 , (&)GI(t,t;).

The function U1 satisfies the differential equation

P & U l = f - f o - d9',u, where L?&U, = 9 z ( u , + 6 , U l ) - Y c u , = Y,,'dlU1

+ W d , U,).

We claim that ifthe Frechet derivative Y,!, of Y oat uo is not identically zero, then the degeneration 8, of$?, with respect tofunctions that are 0,(1) can be identified with the linear operator YC;. The proof is nearly trivial. For any function u ( t ) independent of 9 1 , u= Yl;dlu

E

we have

+ Y(6, u ) = d , { Y ' b u + d P ; u ) + 9 Y 6 , u ) .

Hence 1 Iim _P8u = Y A U

&-PO

u) + ~-i m6 8 i u + lim- Y(6, = 6, I

0

r-0

We proceed further on the assumption that the linear operator 2;is not identically zero. Rewriting the differential equation for 6,we obtain

9;u,

1 + Pb"U1= q{f-f"

- 69pu,J.

1

where 1 q%,= 69;U, + T.Y(6,U,). dl

If

2;)is an operator of the structure considered in Section 5.2, then one can

152

HEURISTIC ANALYSIS CONTINUED. (NON-LINEAR PROBLEMS) CH. 5, $3

apply the theory given there, to establish that

Obviously, 6, must be such that the limit on the right-hand side of the equation exists. ) be analysed in a similar way. We Higher terms of the expansion u ( ~ , Ecan write

Substituting in the differential equation for U,,we obtain a differential equation for Uz which can be written as follows: 1 + Pf’u,= -{f-fo

Yi”;,U,

-

61 s 9 p u o }- -[Yi”;,ul + 8’Y’uJ

62

62

where

82)

is an operator of the structure considered in Section 5.2, then Again, if one can apply the theory given there to establish that

One can now proceed to establish the differential equations for u,, n > 2. If the structure of the differential equations permits repeated application of the theory of Section 5.2, then the functions u,, n = 1,2,,.,, satisfy linear nonhomogeneous equations, in which the differential operator is the Frechet derivative of the degeneration at uo. is zero. Going Consider now the special case that the Frechet derivative 9; back to the equation for U,we can define the degeneration 8,of PEwith respect to functions that are O,( 1) and introduce the resulting decomposition of 8,. One then has

P0U1 + 8 p u 1= S{f-f,

-6

9p0}

where & is an order function occurring in the definition of the degeneration (Definition 5.1.1, and 5.1.2). If 8, and gPare operators of the structure considered in Section 5.2, then one can conclude again

p0ul

= lim S { f - f , - 6 ~ ? ~ u , } , &+O

provided of course, that the limit on the right-hand side exists. The degeneration 8, will in general be a non-linear operator.

CH. 5 , $4

ANALYSIS OF S O M E ONE-DIMENSIONAL PROBLEMS

153

5.4. Analysis of some one-dimensional problems

In this section we shall study some selected examples, as an illustration of the preceding analysis of this chapter. Our aim is further to develop an heuristic reasoning, analogous to Chapter 4, and to gain some understanding in the nature of the difficulties that may arise due to the non-linearity of the problems under consideration. We shall find that, to a large extent, the effect of nonlinearities is not spectacular, and that an analysis similar to Chapter 4 can be performed without any essential difficulties. However, for certain problems, entirely new phenomena occur. The last example given in this section introduces such typically non-linear effects. Example 1. Let @(x,E),x E [O,l] be a solution of

+

&L1@ Lo@= 0,

@(O,E)

= CI,

@(l,&)= p

For simplicity we assume that the functions a,(x), a,(x), g(x,z) are infinitely differentiable for x E [0,1] and z E ( - c0,co).The function g further satisfies g(x,o) = 0,

ag/az 2 d > 0.

Suppose that in some subinterval a regular approximation exists. One easily establishes that the regular approximation is zero. We therefore turn to the analysis of boundary layers. Consider the neighbourhood of x = 0 and introduce the local variable

5 = X/6(&). The equation transforms into

We suppose CI # 0 and independent of E. This suggests that, at least in some local variables, @* = O,( 1). We consider corresponding degenerations of the operator Y EThe . analysis of the degenerations is now entirely as in the linear case. Following Definition 4.1.4 and 4.1.5 we find that there is only one significant degeneration which occurs for

6 = JE and is given by

154

H E U R I S T I C ANALYSIS C O N T I N U E D . ( N O N - L I N E A R P R O B L E M S ) C H . 5,94

The significant degeneration contains all other degenerations of 9& with respect to functions that are Os(l). Assuming the validity of the correspondence principle we obtain, in the boundary layer variable 5 = x / & ' / the ~ equation

If now

then, using Section 5.2,

where we have abbreviated g(O,$,) = go($o). The equation for $o can be solved in an implicit way. In the first step we get

We want the solution t j 0 to match with the regular expansion (which is zero). An elementary analysis shows that this requires the choice of the minus sign in the right-hand side of the equation. With this choice, for small values of $o, the solution behaves like exPC-tJgb(o)l. The implicitly given solution reads

We now turn to higher terms of the expansion for @*((,&). Let us assume that the coefficients al(x), ao(x) and the function g(x,z), have Taylor expansions m

q(x)=

1 ahl)x", n=O

a,(x) =

1 upxn,

n=O

CH. 5, $4

ANALYSIS OF S O M E ONE-DIMENSIONAL PROBLEMS

155

Then the structure of the differentia1 equation for @* suggests an expansion of the structure

@*(5,4

=

*o(t) + e1’2*1(5) + o(E”2).

The Frechet derivative of Z0at

rl/o is easily computed:

We thus obtain, for t+h1the equation

This linear non-homogeneous equation can be fully solved, in terms of starting with the observation that d$o/d5 is a solution of the homogeneous equation, i.e.

One can proceed in similar way to higher terms of the expansion for @*((,E). Furthermore, for the neighbourhood of x = 1 an entirely analogous analysis can be performed.

Example 2. The function @(x,E),x E [O,l] is a solution of d@ &L1@ p(x,@)- - g ( x , @ )= 0 dx

+

with

d2 d LI = 7 + %(XIdx dx

+ ao(x)

and

@(O,E) = a,

@(I,&) = p.

Again we assume all functions occurring in the equation to be infinitely differentiable. We further assume P(X,Z) 2 Po > 0. Suppose that in some subinterval there exists a regular approximation @(X,&)= cbo(x) + o(1). Then $o must satisfy

As a final hypothesis on the structure of the equation we assume the function g(x,z) to be such that the equation

HEURISTIC ANALYSIS CONTINUED. (NON-LINEAR PROBLEMS) CH. 5, 84

156

Lou = 0

for any given u ( x o )= uo, xo E [O,l], has a unique solution u(x) for x E [OJ]. We consider briefly higher terms of the regular expansion. The structure of the differential equation suggests the expansion =4

@(X,E)

O W + E41(X)

+ 4E).

Computing the Frechet derivative of Lo at linear non-homogeneous equation

4owe find that 41must satisfy the

where

In order to determine the functions +o and d1 one needs a boundary condition, and therefore an insight into the possible location of the boundary layers. Consider the neighbourhood of x=O and introduce the local variable

5 = X/8(E). The equation for @ transforms to

- g(6(,@*) = 0. We suppose a # 0 and independent of E. This suggests that, at least in some local variables, @* = OJ). Studying the corresponding degenerations of the operator we find one significant degeneration, which occurs for

a=& and is given by d2u

The first term of the local expansion t,b0(5) is thus a solution of d2*0

+ p(O,*o)- d*O d5

-

d5

= 0.

CH.5, $4

ANALYSIS OF S O M E O N E - D I M E N S I O N A L P R O B L E M S

157

It will be convenient to write = c0

$O(O

+ O(5)

where c0 is an, as yet, arbitrary constant, and 0(5) satisfies

Because p 2 p O > 0, one can deduce that O ( 0 , for 5 > 0, decays at least exponentially, and hence exhibits boundary layer behaviour. One can perform an entirely analogous analysis in the neighbourhood of x = 1, with local variables

r

1

= (1 - x ) / m .

The significant degeneration again occurs for 6 = E. However one finds that all non-constant solutions of the equation defining the local approximation grow at least exponentially for t l > 0, and must be dismissed by the requirement of matching with the regular approximation. We conclude that there is no boundary layer in the neighbourhood of x = 1, and impose therefore, on the regular approximation the boundary conditions &(l) = p

and

&l(l) = 0.

This defines the functions +,(x) and $,(x). Going back to the boundary layer at x = 0 we can determine the constant co by matching with the regular approximation qbO(x)and obtain CO

=

40(0).

The function O(() can now be determined by imposing the boundary condition $"(O) = &(0)

+ O(0) = x .

The implicit formulas for O ( ( ) reads

We now proceed to the next term of the local expansion in the boundary layer variable. The equation for @* suggests an expansion of the structure

@*(t,E)

= Go(()

+

E$1(5)

+

Ok).

Again it will be convenient to assume the Taylor expansion

c '/,

P(X4 =

n=O

P,(Z)X".

158

HEURISTIC ANALYSIS CONTINUED, (NON-LINEAR PROBLEMS) CH. 5, $4

Proceeding as in Section 5.3, we find for

$1

the equation

This linear inhomogeneous equation can be fully solved, in terms of I ) ~ ,starting with the observation that the function d$,/dS is a solution of the homogeneous equation, i.e.

i+hl can then be determined by imposing the boundary condition $1(0) = 0, and furthermore matching with the regular approximation. One can ' finally combine all results in a composite expansion.

Example 3. We consider the initial value problem for the differential equation of Example 2. Thus @(x,e),x E [OJ], is the solution of EL1 @

+ p(x,@)-d@ - g(x,@) = 0, dx

We shall use now a variant of the constructive procedure, in which the regular expansion is subtracted and a correction layer is constructed. The procedure could of course also have been followed in the analysis of Example 2. We commence by writing @(X,4 = @(X,E)- 4 0 ( 4 - E 4 1 ( X ) where, as in Example 2,

4o and 41 satisfy

Lo40= 0 and Lb$, = -L1& Boundary conditions for the functions 4o and q51 have as yet to be determined. For the function @(x,E)we have the following problem:

where

CH. 5, 44

ANALYSIS OF S O M E O N E - D I M E N S I O N A L P R O B L E M S

159

R(x,E) is a function uniformly bounded as 610, for x E [0,1]. The initial conditions are

ww =

- 40(0) - &41(0),

E)

=B-($)

x=o

d4 x=o

-&&)

x=o

We introduce the local variable ( = X/&

and assume that there exists a local expansion

G * ( M = $0")

+ & $ I ( < ) + O(&).

The degeneration of the differential operator is easily deterrriqed. We find that must satisfy the differential equation

The initial conditions are

It will again be convenient to write

I)~(() satisfies all conditions of the problem if one takes co = $ O ( O )

and impose furthermore O(0) = 0.

Because fi 2 p o > 0, the function O(C) decays at least exponentially for ( > 0. We recall now that the regular expansion of G)(x,E) is zero up to the order of . overlap, and matching the first terms of the regular magnitude of E ~ Assuming and the local expansions we thus find co = 0.

However this means that $O(O)

= 0.

We hence must conclude that $o(() is identically zero. The result is not entirely trivial, because it implies an initial condition for the function Cpo(x),i.e.

160

HEURISTIC ANALYSIS CONTINUED. (NON-LINEAR PROBLEMS) CH. S,94

40(0) = a.

Proceeding with the analysis of the boundary layer, we must now determine the degeneration of the operator in the boundary layer variable with respect to The computation is straightforward and one obtains for functions that are O,(E). the differential equation the function +h1(t)

d2rl/,+ P(O,O,O)---d*l dt

dt

= 0.

The initial conditions are

The solution is

where p0 = P(O,O,O) = ~ ( 0 , 4 ~ ( 0 ) ) . Finally, imposing matching with the regular expansion (which is zero up to the order of magnitude of E ~ ) we , find

This condition determines the function q51(x). One can proceed of course in this way to higher terms of the expansion.

Example 4. Let @(x,E),x E [OJ], 2

) +& d2@

E

E ( O , E ~ ] be ,

a solution of

=o,

dx @(O,&) = a;

@(l,&) = p.

We shall study first the exact solution, which is given by @(x,E)= a - ~ l n ( 1 - x +xe('-~)lE}. If a < p, then @(x,E)exhibits boundary layer behaviour in the neighbourhood of x = 1. Introducing the local variable
1-x ; 6, '

ds=exp(q)

we find the local expansion

E:,@ = p - & l n ( t l+ 1).

CH. $64

ANALYSIS OF SOME ONE-DIMENSIONAL PROBLEMS

It is not difficult to see that E i l @is a significant approximation, and hence a boundary layer variable. In fact, the function

(

T x E i , @= x - cln 1 -x+exp

161

is

~

is a uniformly valid approximation of @ ( x , E )for x E [O,l]. Consider now the case x > p. Then @(x,E)exhibits boundary layer behaviour in the neighbourhood of x = 0. Introducing the local variable 50

=

z, 6, X

S

= exp

( ).

we find the significant approximation Eio@ = CI - Eln(1 +to).

The significant approximation yields again a uniformly valid approximation of @(x,E)for x E [O,l]. The location of the boundary layer is thus determined by the sign of CI - p, and not by the structure of the differential equation (as was the case in all the problems that we have studied until now). It follows that, when attempting an heuristic analysis analogous to the preceding examples, we cannot hope to be able to determine a priori the location of the boundary layer. One can only assume the location of the boundary layer and try to perform a coherent construction. If this does not succeed one must reverse the assumption on the location of the layer. The second novelty in this example is the rather unusual order function which occurs in the definition of the boundary layer variables. We now proceed to an heuristic analysis of the problem. Assume that there is a boundary layer in the neighbourhood of x = 0, and that there exists a regular expansion

Jw= 4OW + 61(441(X) valid in x E [d,l], d > 0. We find that

Hence, imposing the boundary condition, we get 4o(x) = B.

The Frechet derivative of Lo at +o is zero, so that the equation governing depends on the order function 6,,which is unknown at this stage. If 6, = o(E), then 41is a constant which can be put equal to zero, because of the boundary condition.

162

HEURISTIC ANALYSIS CONTINUED. (NON-LINEAR PROBLEMS) CH. 5,§4

If dl(&) = E , then

41satisfies

Imposing $,(1) = 0 we find

(2;)

41(x) = -In -

where A is an unknown constant. We turn to the analysis of the boundary layer, and introduce X

The differential equation transforms to

The surprising aspect of the above equation is that the degenerations are independent of the order function 6, which determines the stretching in the local variable 5. Because of the boundary condition at 5 = 0, it is reasonable to assume

Jq@ = $o(t) + 6T(E)$l(t). We find $o(t) = CI.And, if

Solutions satisfying $,(O) $1(t)=

= E,

=0

are given by

-w + B t )

where B is an unknown constant. We shall now attempt to match the regular and the local expansion in an intermediate variable

Transforming, and reexpending the local expansion we find

E i i E i @ = CI + ~ l 6n- E In hi - E In B - E In ti. Considering ,T:@ one finds that the expansion of this function in the intermediate variable does not contain a term proportional to In ti,unless A = 0.

CORRESPONDENCE PRINCIPLE RECONSIDERED

CH. 5, $5

Imposing A

=0

163

we find

E i t E t O = p - E In 6, - E In

ti.

Assuming the validity of the overlap hypothesis we obtain

B = 1 and 6 = e-(@--P)ie. It follows that 5 is indeed a local variable if t( > p. The reader should have no difficulty in verifying that we have indeed determined correctly the expansion E i oO,which was deduced earlier from the exact solution. Furthermore, if p > LY, then a similar analysis can be performed, starting with the assumption that the boundary layer is at x = 1. A striking aspect of the present example is, that the study of degenerations does not permit to determine the boundary layer variable. We have found the order function defining the local variable from the conditions of matching.

5.5. Significant degenerations and the correspondence principle reconsidered The concept of significant degenerations, and the correspondence principle, were formulated in Chapter 4 for linear problems. They were to some extent successfully used in the examples of non-linear problems studied in Section 5.4. However, the last example of Section 5.4 indicates a need for further analysis of these concepts. It will be useful to introduce an operational notation for the process of computing degenerations, as defined in Section 5.1. We shall write Y o = Y(L,;5,6).

Y is an operator which maps L,, for some given local variable 5 and order of magnitude 6, into its degeneration Y o (Definition 5.1.2). Generalizing is contained in Y(L,;('2),6(2)) if Section 4.1, we shall say that Y(L,;("),6(1)) Y(Y(L,;t'2',6'2');t"',6"') = Y ( L & ; p , 6 ' " ) .

Two degenerations Y(L,;t,6)and Y(L,;s'$') will be called equivalent if they are contained in each other. With these preliminaries we can formulate a generalization of the definition of significant degenerations as follows: Definition 5.5.1. Given L,, and some fixed manifold along which local variables are defined, a degeneration -Yo = Y(L,;<,G)is.sign$cant if for all degenerations Y(L,;5',6') which are not equivalent to Y(L,;5,6)one has Y (Y(L,;t ' 3 6 '1; 5,s) # Y(L,;5&.

In linear problems the degenerations are independent of the choice of 6. In

164

H E U R I S T I C ANALYSIS C O N T I N U E D . ( N O N - L I N E A R P R O B L E M S ) C H . 5,45

non-linear problems the degenerations may be independent of the choice of local variables. This was the case in Example 4 of the preceding section. The reader should verify that when studying in Example 4 the local expansion the choice that has been made for a?(&), is the one which, in accordance with Definition 5.5.1, produces a significant degeneration. We now turn to the correspondence principle, as formulated in Section 4.4. We can state the principle in a somewhat different way, by removing the emphasis on the role of the local variables in the concept of significant degenerations. This leads to the following formulation: Modified heuristic principle. Consider some fixed manifold along which local variables are defined. lf there exists a significant approximation then the dijjierential operator in the governing equation is a significant degeneration. For linear problems the formulation given above is equivalent to the formulation given in Section 4.4, and leads to the determination of the boundary layer variables. In non-linear problems, using for significant degenerations the Definition 5.5.1, one may attempt to determine by the correspondence principle both the boundary layer variables and the order of magnitude of the function under consideration. In some cases this is indeed possible, as will appear from the examples to be treated in Section 5.6. In other cases, application of the correspondence principle may lead to only partial answers. This is shown already by the example 4 of the preceding section, where the choice of the order function 6: is in accordance with the correspondence principle. However, in this example the principle yields no information on the boundary layer variable (which was determined by the conditions of matching). We recall from Section 4.4 that there is no general theoretical foundation for the validity of the correspondence principle, but only ‘experimental’ evidence that it holds in great many problems. In Section 5.6 we shall show that in nonlinear problems the principle may fail in a certain sense. In order to analyse the nature of the failure we must analyse in some detail the contents of the principle. The correspondence principle states in fact that, in order to determine the significant approximations, it is necessary and sufficient to analyse approximations which correspond with significant degenerations. The statement ‘sufficient’ in the above interpretation is based on the expectation that significant approximations corresponding to significant degenerations have extended domains of validity that are large enough to contain approximations in other local variables. In the selected, but representative, examples to be treated in Section 5.6 we shall find that the construction of approximations which correspond with significant degenerations is always necessary. However, in certain problems it is not sufficient. Additional significant approximations, corresponding to de-

CH. 5, $6 S O M E PROBLEMS EXHIBITING STRONG NON-LINEAR EFFECTS

165

generations that are not significant, may arise, because the extended domains of validity of approximations corresponding to significant degenerations are not large enough. We shall also find that this failure of the correspondence principle can be removed by using, instead of expansions in terms of limit functions, suitably defined generalized expansions.

5.6. Some one-dimensional problems exhibiting strong non-linear effects In the examples that follow we continue the exploration of the effects of nonlinearities on the heuristic construction. The analysis will also provide an illustration of the discussion given in Section 5.5. Example 1. Let @(x,E),x E [0,1],

(:)i

E L , @ - WZ(X) @(O,E) = a,

E E

=

( O , E ~ ]be a solution of

-f2(x);

w > 0, f > 0,

@(l,E) = p

where d2 d L2 = 2 d x + a1(x)-d x

+ ao(x).

The problem stated above is a generalisation of Example 4,Section 5.4. From that example we have already learned that one cannot determine a priori the location of the boundary layers from an analysis of the structure of the differential equation. One must assume the location of the boundary layers and attempt to perform a coherent construction. We assume that the function @(x,E)has a boundary layer at x = 0, and exhibits no boundary layer behaviour at x = 1. We commence with the regular expansion. Consider

Jw) 4 d X ) + =

E41(X).

Then

The sign at the first term on the right-hand side is unknown at this stage. Proceeding in the usual way we further find that the function 41(x) is the solution of the linear differential equation

166

HEURISTIC ANALYSIS CONTINUED. (NON-LINEAR PROBLEMS) CH. 5,96

with = 0.

We now turn to the analysis of the boundary layer. Introducing local variables

5 =X we obtain, for E

/ W

@(a<,&)= @*((,E),

d2@*

the differential equation

1

d@*

Consider a local expansion

@*(5,4

=

$o(t) + 6 T ( E ) $ l ( O + O(JT).

One finds for any 6 = o(l),

$o(t) = $o

= constant.

The value of $o cannot yet be determined, because nothing is known at this stage about the domains of validity of the local expansions in various local variables (i.e. for various choices of the order function 6(E)j. We reformulate the problem, by writing

@*(W- $ 0 = JT(&)$)*(S,E), 6*(5,&) = OS(l). The order function 6: is again unknown. The function differential equation

6* satisfies

the

- f 2 ( 6 O + E6Tao(65)$0. Using Section 5.5 we deduce that there is one significant degeneration of the which arises for ST = E , and is given by operator Y&, =

The significant degeneration is independent of 6. The correspondence principle implies in the present case that for any significant approximation one must have

6:

= E.

CH. 5,96 SOME PROBLEMS EXHIBITING S T R O N G NON-LINEAR EFFECTS

167

However, the significant degeneration arises for any choice of 6 and we therefore have at this stage the possibility of significant approximations in any local variable. It is quite easily seen that any choice of 6 > E can be dismissed. The reader should have no difficulty in verifying that in that case, the choice 6: = E (leading to the significant degeneration) is inconsistent with the assumption of the = )$1(5) + o(1). existence of an approximation G * ( ~ , E We are thus led to consider, as a first candidate to produce significant approximations, the local variable 51

=.;

X

Then $1(51) is a solution of the differential equation ~w2(o)($) 2 = -f2(0). d5; The equation is not difficult to solve, and one finds

where co and c are constants, c < 1. The domain of validity of the local approximation $o+~$1(51) is as yet unknown. However, in any local variable such that 6 > E there is no significant approximation, and it is therefore reasonable to assume that the extended 1 ( ~ ) One can now domains of validity of and 4 0 ( ~ ) + ~ 4 overlap. investigate matching according to any one of the methods given in Chapter 3. Using the asymptotic matching principle we compute

qlw

= (PO(0) + E(41(0) + 4;,(0)51). One easily verifies that the matching relation can only be satisfied if for the function q40(x) one makes the choice

Matching further yields $0

= 40(0);

co = 4Jl(O).

The regular expansion is thus fully determined while for the local expansion in the variable t1we have

168

HEURISTIC ANALYSIS CONTINUED. (NON-LINEAR PROBLEMS) CH. 5 , $6

The constant c is still unknown. Suppose that the local expansion has a domain of validity that contains the origin. In that case one would have

In order to satisfy the relation one must take

Hence c is no more a constant (but depends on E ) and the expansion is no more an expansion in limit function. We shall return later on to the discussion of the function which arises when the value of c as determined above has been substituted in the local expansion. Pursuing the analysis in the framework of expansions in limit functions we must conclude that E j , @ cannot be valid up to the origin and consider the possibility of significant approximations (by limit functions) in local variables ( = XI6

with 6 < E. We now write @(5:,E)

= 1Go(4)

+ 61(&)$1(5:)+ o(61).

We have already found that, in any local variable $,(()

=

3, = constant.

Furthermore, in order to have a significant approximation we must take

6; = E. The function

G1(()then satisfies

Imposing the boundary condition at the origin one finds the expansion &

E j @ = x --ln(1

w 2(o)

+ B()

where B is an unknown constant. Furthermore, the order function 6 in the

C H . 5, $6 S O M E P R O B L E M S EXHIBITING S T R O N G N O N - L I N E A R E F F E C T S

169

definition of the local variable is also still unknown. Assume now that the extended domains of validity of E i CD and E i , 0 overlap. We investigate matching in an intermediate variable

ti= x/6,,

6 < 6,< E .

Transforming into the intermediate variable, and reexpanding we find

T t , E l @= CI

&

- -[ln

B

w 2(o)

+ In hi - In 6 + In ti]+ o(E).

On the other hand

Comparing the two expansions one finds that matching is impossible, unless c = 1.

Imposing this condition we have &

TttEil@= $o(0) - -[In w 2(o)

2f(0)w(0) - $ 1 ( ~ ) w 2 ( ~ )

+ In 6, + In 5 , - In E ] + o(E). Matching yields now

B = 2f(O)w(O)exp { - 41(O)W2(O)).

All unknown constant and order functions have thus been determined. Of course, 5 must be a local variable, and therefore 6 = o(1). It follows that the construction is consistent if. CL

- $O(O) > 0.

Combining the results we find that the boundary layer has a double structure. In the local variable 51

= X/&,

we have the expansion

51 E

CA,BI, '4 > 0

170

HEURISTIC ANALYSIS CONTINUED. (NON-LINEAR PROBLEMS) CH. 5 , $6

In the local variable

we have the expansion

We now return to an earlier formal result, obtained by imposing on the local expansion in the <,-variable the boundary condition at the origin. This procedure yields a generalized expansion, to be denoted by @"!( 4 , ,&), which is explicitly given by @('!((,,E) =

$o(0)

+E

exp{2f(O)w(0)5,} - 1

It is an easy exercise to show that @(l!((l,~) contains both E:l@ and E j @ , i.e.

E ~ , O " ' ( < l ,=~ )E;l@,

Ei0"'(5:1,~)= E l @ . We thus discover that, in problems with strong non-linear effects of the type studied here, it may be advantageous to use generalized expansions. Working with expansions in terms of limit functions we did achieve a consistent construction. However, using instead generalized expansions is computationally more efficient, because one does not have to construct then separately the sublayer E ; @ . We finally remark that, if X - $ ~ ( O ) < 0, then one can perform an entirely analogous analysis, starting with the assumption of a boundary layer at x = 1.

Example 2. Let @(x,&), x E [O,l],

@(O,E) = a ;

E

@(l,&) =p

E

(O,cO] be a solution of

CH. 5, $6 SOME PROBLEMS EXHIBITING S T R O N G NON-LINEAR EFFECTS

171

where L, =

d2 d + u1(x)dx2 dx ~

+ uo(x).

As in Example 1, we shall proceed from a priori assumptions on the structure of the solution, and attempt to perform a coherent construction. The assumption will suitably be modified if a coherent construction cannot be achieved. It will appear in the course of the analysis, that the problem under consideration here, exhibits various new and subtle phenomena. To simplify somewhat the analysis we shall assume throughout that

f(4-g(x) # 0,

vx E C0,ll.

Reasons for imposing this restriction will become clear shortly. E.2.1. We start on the assumption that @(x,E)has a regular expansion for x E [d,l], d > 0, and exhibits a boundary layer behaviour at the origin. Consider the expansion

E i @ = 4o(x) + E 4 1 ( X ) . It should be clear that for the function C$~(X) two choices are possible. We investigate first the consequences of the choice X

4o(x) = . j f ( x ’ ) d x ’ + P. 1

Proceeding in the usual way we find that 41(x) is the solution of

41(1) = 0.

The condition f-g # 0 assures unique solvability of the equation for 41. Violation of this condition introduces, as an additional phenomenon, turning point behaviour, which will not be considered in the present analysis. We shall now consider the boundary layer. It will be convenient to use a modified procedure, and subtract the regular expansion. We write W , E ) = @(X,E)

- 4o(x) - E 4 1 ( X )

and obtain for G(x,E)the following problem:

G(0,E)= CI - (bO(O) - qbl(0);

ii)(l,E) = 0.

172

H E U R I S T I C ANALYSIS C O N T I N U E D . ( N O N - L I N E A R P R O B L E M S ) CH.5,86

Introduce local variables

d@* + &ao@* = + -6 a 1 - 2 - d$l __ d x l dt:

&2[

-L2$1

+

($)’I.

Consider a local expansion

@*(5>4= $ o ( O

+ 6T(&)$l(O + O(6T).

Then, as in Example 1, for any choice of

a(&),

$o(<) = $o = constant.

We finally reformulate the problem by writing

@*(5,4

-

$0

= 6T(E)&*(t,E),

a)*((,&)= OS(l). One easily establishes that there is only one significant degeneration, which occurs for

S t ( & ) = &,

d(E) = E .

Thus, applications of the correspondence principle yields in the present problem not only the order of magnitude S r , but also the determination of the local variable. We shall write

t1= XI&. The function $l(t:l) is a solution of

where we have abbreviated

w =f(O)

- g(0).

The equation is easily solved, and one finds $l(t:,j = -ln(l-cewfl)

+ cl.

CH. 5,46 SOME PROBLEMS EXHIBITING STRONG NON-LINEAR EFFECTS

Hence, the local expansion, up to the order of magnitude of

E g ) G = $o - E[ln(l -cewcl)

E,

173

reads

+ c,].

We now impose matching of the local expansion with the regular expansion of

a, which is zero up to the order of magnitude of E. The matching relation can only be satisfied if

w =f(O)

- g(0) < 0.

Imposing this condition we find c1 = 0.

$o = 0,

It should be clear that the expansion in terms of limit functions

EL:)@ =

-E

In( 1- cewcl)

for any value of c (independent of condition

E),

cannot in general satisfy the boundary

G(0,E)= M - $hO(O) - E&(O). Pursueing the analysis in the framework of expansions in terms of limit functions we must conclude that E\t)il, cannot be valid up to the origin, and we must investigate the possibility of a sublayer. Consider local variables

5 =X/W) with 6 ( ~< ) E and the expansion

G*(5,4 = $ o w Again

-

I,F~(~) = $o For

+ E$1(5) + 44.

= constant.

one finds

The operator Y ois not a significant degeneration. Solving one finds

$l(t)= - In(l+ B < ) + c,. Finally, imposing the boundary condition at 5 = 0 one obtains $0

= a - 40(0),

c1 =

-41m

We have thus constructed a local expansion

114

HEURISTIC ANALYSIS CONTINUED. (NON-LINEAR PROBLEMS) CH. 5 , 5 6

E y ) @ = ct - 40(0)- c[ln(l + B 5 )

+ cbl(O)l,

6 < E.

5 = X/6(&),

The constant B, and the order function 6 are as yet undetermined. The reader should consider it an exercise to show that by matching E y ) @and E\:)G one finds c = 1,

B = (-w)eXP{-41(0)}9

Thus, all constants and order functions occurring in the expansions are determined. Of course, 5 must be a local variable, and therefore 6 = o(1). It follows that the construction is consistent, if c!

-

40(o)> 0.

Combining all results we find that the boundary layer has a double structure, given by two significant approximations (in terms of limit functions). In the local variable 51

=

4%

51 E

['4,BI,

'4 > 0

we have Eit)@= -&ln(l -ewti), w

=f(o)

- g(0).

In the local variable

The significant approximation E y ) @ does not correspond to a significant degeneration, and we are hence confronted with a failure of the correspondence principle. The nature of the failure has been discussed in Section 5.5: the extended domain of validity of the significant approximation Ekt)@ is not sufficiently large, so that another significant approximation, which does not correspond to a significant degeneration, must occur. As in Example 1, we can improve the results by introducing generalized expansions. We return for that purpose to the analysis of local approximations in the variable Cl, and write

CH. 5 , $6 SOME PROBLEMS EXHIBITING STRONG NON-LINEAR EFFECTS

175

Matching with regular expansion of G (which is zero up to the order of magnitude of E ) we have already found = 0,

c1 = 0.

Imposing now the boundary condition @l)(O,&)

= cc

- &(O)

-

&(bl(0)

one finds

It is an easy exercise to show, that fY')(<,,E) thus determined contains Eki'G and E p ) @ , i.e. E(1)0(1)= ~ ( 1 ) @ Er'Q(1)= E ( 1 ) @ .
C(

- $o(O)

= CI -

p + S f(x)dx > 0. 0

One can verify that the consistent construction is unique, i.e. under the conditions given above, other choices of the regular expansion and the location of the boundary layer do not lead to a consistent construction. In what follows we state results relative to cases where the conditions given above are violated. It is a useful exercise to verify the statements by performing explicitly the construction. (E.2.2. Suppose that f(x) -g(x) < 0. However 1

C(

- /? + J f(x)dx < 0. 0

Assuming that the boundary layer is at x = 1, and that the first term of the regular expansion is given by X

$ o ( ~ )= S g(x ')dx '

+

CI

0

one achieves a consistent construction under the supplementary condition I

/? - CI - J g(x ')dx ' > 0. 0

176

HEURISTIC ANALYSIS CONTINUED. (NON-LINEAR PROBLEMS) CH. 5 , $6

E.2.3. Suppose now f(x) -g(x) > 0. The analysis can be transcribed from E.2.1 and E.2.2 by interchanging the symbols f and g. E.2.4. We shall now study the problem formulated in Example 2, under the condition f ( x ) - g(x) < 0 in the case that 1

1

/i’ Sg(x)dx < a < p - Sf(x)dx. 0

0

Combining the conditions of validity of E.2.1 and E.2.2 one easily sees that this case is excluded in the preceding results. Furthermore, from the conditionf-g < 0 it follows that for any value of p there exists a non-empty interval of values of a satisfying the above inequality. As an introduction we consider a simple, exactly solvable case, given by &d xe 2- PdIx P d+x l ] = o ,

@(O,&) a and

= a;

@(l,&) = p.

fi are such that p < a < p+l.

The exact solution of the problem reads

We define xo = 1 - (a-fi),

xo E (0,l).

One easily deduces that

+ O(E) (D(x,E)= fi + 1 - x + O(E)

(D(X,E)= a

for x E [O,xo-d],

d >0

for x E [xo+d,l],

d > 0.

In the neighbourhood of x = xo the function (D(x,E)exhibits a boundary layer behaviour. This free layer can be described by the local expansion

E y ) @ = a - &In(l+e$

5

= (x-x~)/E.

Thus, in contrast to the preceding cases, @(x,E)does not have a boundary layer at some end-point of the interval, but a free boundary layer in the neighbourhood of an internal point, of which the location depends on the values of a and fi.

CH. 5,56 SOME PROBLEMS EXHIBITING STRONG NON-LINEAR EFFECTS

I77

We now turn to the more general problem formulated in Example 2. Suppose that there exists an internal point xo E (0,l) such that the function @(x,E)has regular expansions

+ E~( : ) ( x)+ o(E), @(x,E) = $g’(x) + E ~ : * ) ( x+) o(E), @(x,E)= 4!)(~)

x E [O,x0-d],

d > 0,

x E [x0 +d,l],

d > 0.

Concerning the functions occurring in the expansions, different choices are possible at this stage. Before deciding upon these choices, we shall study the structure of the internal layer. We introduce the local variables x-xo (=-

w

and a local expansion

@*(5,&)= $ O ( O + G Y E ) I C / l ( S ) + O(6T). By the standard procedure one finds

1)~(5)= $o

= constant.

Furthermore, the significant degeneration occurs again for d ( E ) = E,

cy(E)

= E.

The function $1(() satisfies the equation

Solving the equation we obtain

$1(4) =f(xo)t

+ c1

- ln(1 +cewor)

where wo =f(x,) - g(xo) < 0. We now study the matching of the local expansion with the regular expansions. Consider x > xo, which corresponds with g > 0. Using the asymptotic matching principle up to the order E, one finds

Similarly, for x < xo (and hence 5 < 0), the asymptotic matching principle yields q0 Ef(xo)S- &wO( - E In c &cl =

+

+

178

HEURISTIC ANALYSIS CONTINUED. (NON-LINEAR PROBLEMS)

CH. 5,96

The matching relations thus require that

c1 = ~ Y ) ( x ) ,

c = exp[$y)(xo) - 4:“(xo)].

These results permit a motivated choice for the regular expansions. We take

+df)(~)

X

=x

+ S g(x‘)dx’, 0 1

$J~’(x) = P - Jf(x’)dx‘. X

The functions +y)(x) and +y)(x) are now uniquely defined in the usual way. It remains to be demonstrated that there exists a unique internal point xo, for which $t’(X,,’ = 4df’(xo). We recall that we study the case 1

1

P - J g(x)dx < x < P - Sf(x)dx. 0

0

Consider the function X

F

(XI =

-

J [f(x) - g(x)ldx. 0

F(x) is continuous and monotonical increasing (because f-g < 0):

1

1

F(1)= x - J [ f ( x ) - g ( ~ ) ] d ~> J/’ - Jf(x)dx. 0

0

Hence there exists a point xo such that 1

F(x0) = P - J f W d x . 0

Thus 1

CH. 5, $6 SOME PROBLEMS EXHIBITING STRONG NON-LINEAR EFFECTS

179

and finally C(

xo

1

0

xo

+ J g(x)dx = (3 - J f(x)dx.

All expansions, and the location of the free layer, have now been determined. Reviewing the analysis of Example 2 a somewhat curious feature appears: problems with a free layer are in a sense easier to analyse than problems with boundary layers at the end points, because in the case of a free layer there is no need to introduce generalized local expansions. Example 3. We shall now consider some further generalizations of the preceding examples, and discuss in particular certain cases of failure of the constructive procedure. Let @(x,E),x E [O,l], be a solution of

@(l,E) = (3

O(O,&) = a;

where, as before d d L2 = 2 dx q ( x ) d- x

+

+ a,(x).

E.3.1. Consider the case that p o =O,f=O @(x,E)has a regular expansion @(X,E)

= 4o(x)

and suppose that, in some subinterval

+ E41(X) + O ( 4 .

Then 4o = constant, while for the function $l(x) one obtains the differential equation

One must recognize at this stage that, from the practical point of view, the analysis cannot in general be pursued any further, because the equation for q51 will not be exactly solvable (except for some special simple cases). The difficulty appears most clearly in the case that p1 = 0, p2 = constant. Then the equation for 41is the full differential equation of the problem, and if one is able to solve that equation, then there is no need to construct approximate solutions. E.3.2. Consider the boundary layer in the neighbourhood of some point x = xo. Introducing local variables

4

= (x - X d l

and a local expansion

o)

180

HEURISTIC ANALYSIS CONTINUED. (NON-LINEAR PROBLEMS) CH. 5,46

one finds again that

1)~(5)= $o

= constant.

A significant degeneration occurs for h(E) = E ,

and for the function

hT(E) = G

one has the equation

The equation is of the type already encountered in Example 2, and on the basis of experience with that example there may seem to be no reason to expect any difficulties. Yet in certain cases a rather fundamental failure of the constructive procedure occurs. Consider, as an example, the equation

We assume that a regular expansion is valid in x E [d,l], d>O, and that at

x =0 a boundary layer occurs. Introducing

5 =X / W ,

@*(W= $ O ( S )

we find

$o(5) = $o

+ hT(&)$l(t)

= constant.

Consideration of significant degenerations permits to determine

The significant degeneration occurs for any choice of J(E). The function is a solution of

Solving, and imposing the boundary condition, we obtain the local expansion 1

a*((,&) = Q - &-ln(l+

+

B5) o(E). Pz(4 By a similar construction one finds the regular expansion

SOME ELLIPTIC SECOND ORDER PROBLEMS I N R 2

CH. 5, 67

181

The constants A and B, and the order function 6 ( ~ are ) undetermined at this stage. One may hope to determine these quantities from considerations of matching. However, attempting to match the regular and the local expansion one finds that this is impossible if P2(4

# P2(B).

An explanation of the reason for the failure (Van Harten (1975)),comes from the consideration of the exact solution (which can be given in an implicit way). The function @(x,E)has a continuum of significant ccpproximations in the local variables

5 = xe"'& where

D

is any number satisfying z

1

0 < 0 < p2(z)dz. B

It should be clear that our method of analysis, which supposes a finite number of significant approximations, cannot cope with problems in which the solutions exhibits such behaviour.

5.7. Some elliptic second order problems in R 2 This section is in a sense parallel to Section 5.4. We shall find that for a class of quasi linear elliptic problems analogous to Section 5.4, the effect of nonlinearities is again not spectacular, and that an analysis similar to the linear case can be performed without any essential difficulties. We shall not purme the analysis of elliptic problems to cases analogous to Section 5.6. The interested reader can find some remarks on such problems in Van Harten (1975).

Example 1. Let D be an open bounded domain in R2, with a smooth boundary r and let @ ( x , E ) ,x E D be a solution of &L2@ + Lo@= 0, x Q,=o,

where

xEr

E

D,

182

HEURISTIC ANALYSIS CONTINUED. (NON-LINEAR PROBLEMS) CH. 5 , 57

ag/a@2 d > 0.

g(x,O) = 0,

Suppose that in some subdomain a regular approximation exists. One easily establishes that the regular approximation is zero, up to the order of magnitude of cm, Vm.We therefore turn to the analysis of the boundary layers. Let there be given a transformation of variables (x,,x,) -,(p,v) such that, for any point P , p = 0 P E r. The differential equation takes the form

with cr(p,v) > 0. We introduce the transformation to local variables

t =P / W and find, in the usual way, that there is one significant degeneration, which occurs for B(E)

= JE

and is given by aZu

9 o u = .(O,v);-5

- g(O,v,u).

OP

The structure of the equation suggests a local expansion

@(t>Vd The function

=

Iclo(t,v)

I)~((,V)

+ JE$l(S,V) + O ( J 4

satisfies

where go(v,$o) = g(o,v,Iclo)/~(o>v). The variable v occurs in the equation as a parameter, and the local expansion thus describes an ordinary boundary layer. The analysis of i,bo is further entirely analogous to Example 1 of Section 5.4. If the local variable is defined in such W - t h a t t -= 0-in D, then $o is implicitly given by e

5

t

= $0 J {2J0 g,,(v,z)dz}-’”dt.

Turning to higher terms of the expansion one finds

SOME ELLIPTIC SECOND ORDER PROBLEMS I N R Z

CH. 5, $7

183

The right-hand side can be computed explicitly. The Frechet derivative gb(v,$,) is defined by

The linear non-homogeneous equation for $ l can be fully solved, in terms of $, starting with the observation that ? ~ ) ~ / d [is a solution of the homogeneous equation, i.e.

Example 2. We now consider

= 0,

(xl,x2)E r

where A is the Laplace operator and

2 Po > 0.

P(X,,X2,@)

In order to obtain simple explicit results we take r to be the unit circle. We start with the analysis of the boundary layer structure along Introducing the transformation of variables x1 = (1-p)cos v,

E

r.

x2 = (1 -p)sin v,

= P/6(&)

we find, for the function @*( (.v,E), the differential equation E

i2@*

-~

6 2 Z(2

+ &

1 (1-6C')Z

+ ~ ( ( 1 -6()

Z2@* ZV2

1 c"@* sl-sg 6 ( &

cos v,(1 - 6 0sin v,@*)

1 c"@* - cosv c"@* -sin v [6 c< 1-65 cv

Excluding the neighbourhoods of v = 0 and v degeneration for

6 = E, given by

= T[

]=o.

we find a significant

184

HEURISTIC ANALYSIS CONTINUED. (NON-LINEAR PROBLEMS) CH. 5 , 3 7

where S sin v, u). p(v,u) = ~ ( C O v,

The first term of a local expansion $o(t,v) is thus a solution of

& + P(V,$~)sin v W O = 0. at

at

Analogous to Example 2, Section 5.4, we can write $O(t,V)

where

= c(v)

+ $O(tP)

9,is a function satisfying -

6Q

-- - -sin v f p(v,c+z)dz

at

0

because p>po>O, we deduce that tJ0((,v) for ( > O and v E ( 0 , ~decays ) at least ) <-dependent solutions grow at least exponentially. However, for v E ( x , ~ x all exponentially and must therefore be dismissed. We conclude that, as in the linear case, there is a boundary layer along the ) ) no boundary layer along the upper part of the boundary (i.e. for v E ( 0 , ~ and lower part of the boundary (i.e. for v E (q27t)).In the neighbourhood of the two extremal points of r, given by v = 0 and v = n, one has the problem of birth of the boundary layer, which we shall investigate in some detail in Chapter 7. We now interrupt the analysis of the boundary layer in order to construct the regular expansion. For the boundary condition along the lower part of the boundary one can write for x2 = -(I - x : ) ~ ’ ~ .

@(x1,x2)= e-(x,)

The standard construction yields (w,,x2) = e-(x,)

where

$ 1is

+ &41(XI,X2) +

a function satisfying

841 -8x2

41= 0

AeP(X,,XZ,Q-)’

for x2 = -(I -x;)ll2.

Returning to the analysis of the boundary layer we determine the function c(v) by matching with the regular expansion, and obtain c(v) = e - ( - cos v),

V E (0,~).

For the boundary condition along the upper part of the boundary one can write (D(O,~,&)= e+(cosv), E (o,n).

CH. 5, $8

FORMAL APPROXIMATIONS I N NON-LINEAR PROBLEMS

The function $o(

t,v)

185

must then satisfy the boundary condition

$o(o,v)= e+(cosv) - c ( ~ = ) w(v),

wr){i

E

(0,~).

This leads to the implicit formula for the solution

5

1 sin v *lo-

= __

P(v,c(v)+ z)dz

0

One can now proceed to construct higher order terms of the local expansion. Writing @*(S,V,E)

one finds that

= $0(5,v)

+ E $ l ( S , V ) + O(E)

$1satisfies

W O $l u;Ij1=-a 2 * 1 + p(v,$,) sin v -+ P ’ ( V , $ ~ ) sin v at at at a*1

= Fl(t,V>$O),

where p’(v,$,) = (8/dtj0)p(v,$,), while the function F , can be computed explicitly. can be fully solved, in terms of The linear non-homogeneous equation for tjo,starting with the observation that a$o/ag is a solution of the homogeneous equation, i.e.

5.8. Remarks on the formal approximations in non-linear problems In the examples of Sections 5.4, 5.6 and 5.7, when the construction is carried out to a sufficient number of terms, and the results are combined into a composite expansion, one obtains a function @:?)(x,E) which is, as in the linear case, a formal approximation. This is already implicit in the results of Sections 5.2 and 5.3, and can further be verified by inspection. Using Definition 4.7.1 (which is not restricted to linear operators) we have

LZe@= F ; X E D , Ye@:?) =F

+ pm;

pm = ~ ( l ) x , E D.

Furthermore, @p) also satisfies, to a certain degree of accuracy, the boundary conditions or initial conditions imposed on 0.For example, in second-order boundary value problems, we have @=

e

on

r,

@g)= e + c,;

[, = o(1) on

r.

186

HEURISTIC ANALYSIS CONTINUED. (NON-LINEAR PROBLEMS) CH. 5,58

Thus )@ :! is a solution of a problem that is, in a sense, ‘neighbouring’ the problem that we wish to solve and again we hope that these neighbouring problems will have sufficiently neighbouring solutions. In linear problems the question of validity of a formal approximation as an approximate solution can be reduced to the question of boundedness of solutions of L,$=F,

$=e’,

XED,

XET

for some suitably defined sets of functions F“ E 9, e “ a~ (see Lemma 4.7.1,4.7.2). In non-linear problems such reduction is no more possible. In fact, introducing the remainder term R, through the formula @ = CDgy)

+ R,

we obtain for R, the problem

+

L,(@Ly’ R,) - LE(@iT)) = - p,, R m = - im, X E r .

x

E

D,

One can now abbreviate L,R,

= L,(@iy’

+ R ) - L,@L:)

and have x

L,R, = -p,,

E

D.

However, L, is different from L,, and depends on CDL:). This observation should make it already clear that proving validity of formal approximations in non-linear problems is a considerably more difficult undertaking, than in the linear case. The basic problem, involved in proving the validity of formal approximations is somewhat camouflaged in the linear case by the simplifications that can be introduced due to linearity. To show this we shall now sketch a reasoning analogous to Lemma 4.7.1, 4.7.2, but without recourse to linearity. For simplicity of exposition we consider the case that the boundary condition is given by @=O

on

r.

We further assume that 0;:)has been constructed such that =0

on

r.

In applications this can be achieved by a slight modification of the constructive procedure described already in Section 1.4. In Chapter 7 a somewhat different by analogous modification will frequently be used, which consists of multiplying the boundary layer .terms by a smoothing function, i.e. a function

CH. 5 , 4 8

FORMAL APPROXIMATIONS IN NON-LINEAR PROBLEMS

187

that equals one in the boundary layer region, and zero at some distance of that region. With this preliminaries we now reformulate our problem as follows: Let V be the set of functions that are twice continuously differentiable in D and V, c Va subset of which the elements take the value zero at r. F is the set of continuous functions in D. We search for an element 0 E V, such that, for some given F LED= F . On the other hand, we have constructed an element @ ): L&O;:)= F

+ p,,

E

V, such that

pm = ~ ( l ) .

Suppose now that for each that

E

8 there exists a unique element

6 E V, such

L&5= F. This means that there exists an inverse operator L&-’ : 9 -,V, such that If now F

5=L

y .

+ p,

8,then we can write

a:)

E

- @ = L&- (F

+ p,)

- L&- F .

For the validity of the formal approximation we want @ ); - 0 = o(l), while we are given that p m = o( 1). The validity of the formal approximation can hence be demonstrated if the ‘solution operator’ L; has suitable continuity properties. This point of view will frequently be used in Chapter 6 .

CHAPTER 6

FOUNDATIONS FOR A RIGOROUS THEORY OF SINGULAR PERTURBATIONS For problems of the type investigated in Chapters 4 and 5 , a rigorous theory of singular perturbations is achieved if one succeeds in proving that a formal approximation is an asymptotic approximation of a solution of the problem under consideration. In this chapter we have collected the analytical methods and the results which, for various classes of problems, permit indeed to demonstrate the validity of formal approximations (as approximate solution). The chapter is divided into three parts. Part 6.1 presents, in a sense, background material. It is concerned with classical perturbation analysis, in an abstract setting, and contains further some basic considerations on the nature of regular and singular perturbation problems. Part 6.2 is devoted to estimates of solutions of linear singular perturbation problems. The results permit proof of validity of a formal approximation in large classes of problems. Part 6.3 is concerned with non-linear problems. It contains results that are of interest for certain special classes of problems, and furthermore a general method for proving validity of formal approximations in non-linear problems, which in spirit is related to the classical perturbation analysis described in part 6.1.

6.1. General introductory considerations In this part of the chapter we discuss in a somewhat abstract setting certain general aspects of perturbation analysis, which have some relation to the problem of proving the validity of formal approximations. For simplicity of the presentation we shall use the following formulation. and mappings L, : V+ 9, We consider linear spaces of functions Vand 9, where L, generally is a differential operator. Any boundary conditions, or initial conditions, to be imposed on a solution CD of the problem

L e o = F,

@'EX F E F

are supposed to be incorporated in the definition of the linear space T/: Clearly, this is only possible if the boundary conditions, or initial conditions, are linear and homogeneous. For example, considering Dirichlet problems for second 188

CH. 6, $ 1

GENERAL INTRODUCTORY CONSIDERATIONS

189

order elliptic operators L,, we can impose in this setting the boundary condition @ = 0 on the boundary r of the domain D. The formulation can be extended to more general boundary conditions, by generalizing the definition of the operator to a vector with components defining the boundary conditions. We do not pursue full generality of formulation here, in order to avoid the technical complexity.

6.1.1. Classical perturbation analysis in a Banach space In the perturbation theory the following procedure can be considered as classical: Given a problem

@EX F

L,@=F,

E

~

one constructs a decomposition of the operator L, into two parts:

+

L, = d, 9&,

.d, is such that the inverse .dc- : .F+ Vexists, at least locally. Y,is considered as a perturbation. Next, one writes @ = . d , - ' ( F - .YE@),

@ E r/;

F E .F.

It can now be expected that an approximation of @ will be given by

'

Q4, = .AFF.

In order to prove the validity of the approximation one must study @-

'

= n; (F - .YE@) - .dE-F .

It should be clear that if the perturbation .YE@ is sufficiently small in some suitably defined sense, and if the inverse d,' is a continuous operator with sufficiently nice properties, then one may indeed be able to demonstrate that is an approximation of 0.To pursue the analysis one must investigate the properties of S,-and 9,. We note that both operators are explicitly given by construction. The classical perturbation analysis thus combines the construction and the proof of validity of the approximation into one line of reasoning, without requiring any additional a priori information about the problem. The elegant classical approach is not directly applicable to problems of the type studied in Chapters 4 and 5. In those problems a decomposition of L,, as required by the classical analysis, can in general not be given, and @'as is constructed as a formal approximation by an heuristic analysis. It is nevertheless useful to devote here some attention to the classical approach. Although of little direct applicability to singular perturbations, classical perturbation analysis can be a powerful tool for proving validity of

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

190

CH. 6,pl

formal approximations, in particular in non-linear problems. In this section we develop some rather general perturbation results, which have an interest of their own, and which provide a convenient basis for further applications (Section 6.3.4). We return to the reformulated problem 0=de-'(F-9,@),

@EX FEF.

In an explicit approach one can attempt to demonstrate the convergence of an iterative procedure based on the above formula, starting with Oas= d , - ' F

as a first approximation. In a more abstract setting this can be accomplished by application of a fixed point theorem. We shall follow here the abstract approach. Let 2. be a Banach space, i.e. a complete normed space, with norm 11 11. We recall now a fundamental result, which is stated below in a form convenient for our analysis.

-

Banach's fixed point theorem. Let T b e a mapping of a Banach space Pinto itself and consider balls B(o) = {0E PI 11 0 11

< o}.

Suppose that, for some o = 0,one has ( i ) T is a strict contraction, i.e. VQ1 ,m2 E B(6) ]IT01- T@ZII < k l ] @ 1 - @ 2 1 ] ,

k < 1.

(ii) T maps B(6)into itself, i.e. V 0 E B(8)

llT@ll< 0. Then there exists a unique element 0 E B(8),such that 0 = TO.

Corollary. Suppose that the conditions of the fixed point theorem are satisfied for each 6 E [oO,ol] with oo < o l e.Then the solution 0 o f 0 = T 0 lies in B(oo),and is unique in B(o,). In order to apply the theorem we must reformulate our problem so that it can be interpreted as search for a fixed point for some operator Tin a Banach space. This can be accomplished as follows: Let be a Banach space with norm 11 )I which is such that V c and let B b e a normed linear space with norm 11*11*. Supposethat can be extended to a mapping of a neighbourhood of F in B into P a n d 8, can be extended to a Then mapping of a neighbourhood of d,- F in into P.

r

-

r

CH. 6. 51

GENERAL INTRODUCTORY CONSIDERATIONS

191

7-a)= d&-[ F - 9&@] is a mapping of a neighbourhood of de-'F in pinto f? For example, consider the boundary value problem for a second order ordinary differential operator L,. Then Vis the linear space of twice continuously differentiable functions, which take the value zero at the end points of the interval, and .F is the linear space of continuous functions. We can make V into a Banach space, by defining the norm as follows:

where c1 and c2 are some positive constants. (We remark already here that a special, &-dependentchoice of the constants in the definition of norms will appear to be very profitable in the further course of the analysis of this chapter.) One can also make a different choice of the Banach space. If .YEis a first order operator, then pcan be chosen the linear space of continuously differentiable functions, equipped with the norm

ll@ll = supl@

121

+ c,sup -.

We now turn to the conditions of the fixed point theorem. Condition (i) implies Lipschitz continuity of the operator 7: This in turn implies conditions for and P,,. For the simplicity of presentation we shall first treat the operators .dF; the case that .d;'is linear. The following conditions will be sufficient to prove a perturbation theorem. Continuity condition 1. Let .Fand Pbe normed linear spaces, with norms l]*ll*, respectively I[ -11. .A,;: .F-+ Psatisfies llLdt-'F / l < j . ( c ) ~ ~ F ~ ~ * .

Continuity condition 2. For each pair E B(G) =

and each

cj-

E (O,r?),

{a I ll@ll < 0).

5 > 0, one has

ll.YE@l - .Y,@,,II*6 p(E,cj-)llo1- 0 2 1 1 where

is a positive continuous function, with

~(E,G)

p(t;pl)

< p(~,cj-,) for 'il < 0 2 .

With these preliminaries we can formulate

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

192

CH. 6, §l

Theorem 6.1.1.1. Consider the problem

L,@=d&@+P,@=F, @ E Y F E F and suppose that the following conditions are satisfied: (i) The inverse dE-is densely defined and can be extended to a mapping of 9 into a Banach space Psatisfying continuity condition 1. Y ecan be extended to a mapping of Pinto 3 and satisfies continuity condition 2. (ii) For each E E (O,tO], andfor some y E (O,l), there exist numbers B ' > 0 such that %(E)p(E,c')

< 1 - y.

(iii) For eachfixed Bl

= sup B '

E E ( O , E ~ ]one

has

1 > -A(&) IIFII*.

Y Then there exists a solution @, unique in the ball B(o,), which satisfies

Furthermore, the function Oas,given by

mas= d,-' F satisfies

Comments on the theorem. The theorem can be used in various situations. In a classical context, the term Ps@ of the equation is a perturbation for E small. In that case it is logical to expect that p ( ~ , c ) = o ( l ) for , E J O . If, for example, A(&) = O(l), then condition (ii) of the theorem, for E sufficiently small, is satisfied is bounded. It follows that for all finite B ' , and condition (iii) is satisfied if JJF*ll Qas indeed is an asymptotic approximation of @. Similar results can be achieved even if I-(&) grows without bound as E J 0, provided that ~ ( E , c T ) tends to zero sufficiently fast. On the other hand, conditions (ii) and (iii) of the theorem can also be satisfied in a different context, when IIF(I* is small and p(&,a)is such that p(&,O)= 0. In Section 6.3.4 we shall obtain in this way a theorem due to Van Harten (1975, 1978) which is a powerful tool in proving validity of formal approximations in non-linear problems.

193

GENERAL INTRODUCTORY CONSIDERATIONS

CH. 6, $1

Hence, Tis a strict contraction on all balls B(o’), where o’ is a number as ,specified in condition (ii) of the theorem. Next we consider the mapping TQ,for CD E B(o),o E (O,o,]. I/TQ,II= Ild,’(F

d iL(E)llFlI*

- P&Q,)ll d WllFll* + I I ~ & Q , I I * )

+

fi(&)P(&,(T)(T

+

d I.(~)llFli* (1 -Y)o. It follows that llT@lld

0

if CT

1 2 00 = - A ( & ) ~ ~ F ~ ~ * .

Y

Thus, for o E [oO,ol]the conditions of the fixed point theorem are satisfied, while condition (iii) of Theorem 6.3.1.1 assures that the interval [oO,ol]is nonempty. Existence, uniqueness and an estimate for Q, follow from the fixed point theorem and its corollary. Finally, considering 0 - Q,’,,, we write

IIQ, - Q,)asll

= Ild,’(F

- Y E @-) d E - ’FI I d ~ w ( ~ ) l l ~ e Q , l l *

This concludes the proof of the theorem. We now return to the original formulation of our problem:

LEQ,=d,Q,+9’ECD=F,

Q , E ~F

EF.

r2

T/: One can The setting of Theorem 6.1.1.1 is a Banach space such that derive an analogous result in a different setting, by taking 9 to be a Banach space. The reasoning goes as follows:

: .F is bijective. Then for any CB E Vone can write Suppose that d EV+ @=&&-If,

f E 9 .

Substituting into the equation we get f = F - Pcde‘J:

Write now

Tf=F-Ped02,’f,

VfEF.

Then Tis mapping of F into itself and the problem is again reduced to the

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

194

CH. 6, $1

search for a fixed point of Tin a Banach space. Using Continuity conditions 1 and 2, one obtains in this way results entirely analogous to Theorem 6.1.1.2. We shall finally discuss an extension of the results to the case that d,is nonlinear. We replace the Continuity condition 1 by a condition of local Lipschitzcontinuity, as follows: Continuity condition l*.Let 9and ?be normed linear spaces, with norms 11*11* respectively I[*//. For each pair F,,F2 with IIF,ll* 0 one has

jl.02;’F, - d J ’ F 2 l I

< j-(&,d)llF1-F2(I*

where j.(E,d) is a positive continuous function with i(E,dl) < i(c,d2) for d , > d,.

One can now derive results analogous to Theorem 6.1.1.1, paying to the nonlinearity the tribute of a slightly more complicated formulation.

Theorem 6.1.1.2. Consider the problem L&CD= dECD+ B&a) = F,

CD E y

F

E

9

and suppose that the following conditions are satisfied: (i) The inverse atE-’is densely defined and can be extended to a mapping of 9 into Banach space Bsatisfying continuity condition l*. .PEcan be extended to a mapping of pinto 9 satisfying continuity condition 2. (ii) For each E E ( O , E ~ ]and for some y E (O,l), there exist numbers c’> 0 such that E”(E,T)y(E,a’)

d 1- y

where r=JIFlI*+,u(~,a’)a’. (iii)

(T,

= sup (T’

z1

= IIFII*

1

> -A(E,~,)\~FI~*,

Y

+ P(Wl)o,.

Then there exists a solution 0,unique in the ball B(o,), satisfying

Furthermore, the function a),

satisfies

= d&-’F

a,,,

given by

CH. 6 , $1

GENERAL INTRODUCTORY CONSIDERATIONS

195

Proof of the theorem is entirely analogous to the proof of Theorem 6.1.1.1. 6.1.2. Regular problem

Suppose that in a problem that can be treated by classical perturbation analysis, one can choose d,= do,independent of E and identify dowith the degeneration Lo of L,. Suppose further, for simplicity, that F = F , is independent of E. Then @as satisfies

Lo@.,, = Fo Because the problem for Qas does not contain E , we can write aas(x,e)= Cpo(x). Thus, in such problems, there exists a regular approximation 40(x) of O(x,c), which is a solution of a formal limit problem Lo40 = Fo. We shall call such problems regular in the first approximation. We shall now define, more in general, regularity of a perturbation problem as the property of existence of a regular approximation which can be obtained as a solution of a formal limit problem. From this point of view, the applicability of classical perturbation analysis is not essential for the regularity. We thus arrive at Definition 6.1.2.1. A perturbation problem Lz@= F , @ E K F E 9, for @(x,E), x E D,E E (O,E,], is regular (in the first approximation) if (i) There exists a formal limit problem Lou = F,, where Lo is the degeneration of L, in the x-variables and Fo(x)is a regular approximation of F(x,E) for x E D. (ii) There exists a regular approximation 4,(x) of @(x,E) for x E D. (iii) Lo@,is well defined and furthermore $,(x) satisfies L04, = F,.

Note that the definition does not require that 4, be an element of T/: This is a consequence of the fact that in any given problem, the answer of the question of regularity depends on the choice of a norm on T/: For example, consider the function @(x,E) = x

+ e(e-X’e- I),

x E [ @ A ] , A > 0.

If l\@ll = sup 101, then there exists a regular approximation q5,(x) = x

for x E [O,A], A > 0.

However, if 11@(1 = sup 101+ sup Id@/dxl,then there exists no regular approximation of @(x,E) for x E [O,A], A > 0.

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

196

CH. 6 , $1

The function ~ ( x , E as ) , given above, is the solution of LEO= E

d20 y dx

d0 += 1, dx

0(O,&) = 0, @x=o

x E [O,A], A > 0,

= O*

In the formulation of Definition 6.1.2.1 we can take Vto be the space of twice differentiable functions, satisfying initial conditions prescribed for the function 0.If )I*((= supI*I, then the problem is regular (in the first approximation), however C$~(X) does not satisfy all initial conditions imposed on 0. Suppose now that for some given problem condition (i) of Definition 6.1.2.1is satisfied, i.e. there exists a formal limit problem. Then one may attempt to equip Vwith a norm such that (ii) and (iii) are satisfied, so that the problem becomes regular, in that topology. This procedure is often followed in the analysis of perturbation problems in Hilbert space. Many general results of this type can be found in Lions (1973) and further in Huet (1976). On the other hand, in applications the choice of norm in r! often is dictated by some additional considerations, which imply that only approximations in the sense of certain norms are of interest. For example, when sending a spaceship from earth to moon and bringing it back to earth, an approximation of the trajectary in the sense of L,-norm will not be very useful. One then wants to know the position and the velocity of the spaceship at any moment, which requires approximations in the sense of the supremum norm for the function and its derivative. However, if one’s main inteiest is some integrated quantity (such as energy), of the type

J 0’dx D

then computing approximations in the sense of supremum norm may not be necessary for a precision that is required, and working in the ,!,,-norm may be sufficient. Suppose therefore that the choice of the norm on Vhas been made. Then the conditions of regularity of Definition 6.1.2.1 can be violated in various ways. This will lead us to a classification of singular behaviour, in the next section. As a preliminary we give some further consideration to the boundary conditions that must be satisfied by a regular approximation (if such an approximation exists). We have already remarked that C$o does not have to be an element of T/: On the other hand, given a definition of norm, elementary considerations usually permit to delimit in a sense the set of admissible functions, of which c # ~must ~ be an element. For example in an initial value problem, with initial condition @(O,E) = 0, (d@,/dx),=,, = 0, and with ] 10,1 =1sup 1 01,an approximation mas of 0 must

CH. 6, $1

GENERAL INTRODUCTORY CONSIDERATIONS

197

satisfy aas(O,&) = o(1). This defines the set of functions that are admissible as approximations. More in general we have Definition 6.1.2.2. Let P b e a normed linear space, V c P a subset defined by some auxiliary conditions (such as boundary conditions or initial conditions), and let l\*ll be a norm on T? An element & E Pwill be called admissible if

inf /l@--&/l

a€V

=

0.

A set V, c Pis admissible if each element of V , is admissible. Introduction of the concept of admissible subsets is useful in applications, because it can permit conclusions concerning regularity in an early stage of the analysis. For example, suppose that in some given problem, condition'@)of Definition 6.1.2.1 is satisfied, i.e. a formal limit problem exists. If now all solutions u of Lou = F , are not admissible then (iii) cannot be satisfied, and the problem is not regular. Only if Lou = F , has some solution u, in some admissible set V,, can one hope to be able to prove that Q, - u, = o(l), i.e. that the problem is regular. 6.1.3. Singular problems: a classification

We now study various possibilities of singular behaviour for some arbitrary but fixed choice of norm in T? Definition 6.1.3.1. A perturbation problem LEQ,= F , 0 E r! F E 9 is singular in the given topology of Pif not all conditions of regularity given in Definition 6.1.2.1are satisfied.

Investigating all possible negations of Definition 6.1.2.1one easily arrives at a classification into four main classes, as follows (Eckhaus (1977)): S.1. A formal limit problem either does not exist, or has no admissible 'solutions

In this case there are two further possibilities:

(S.l.1)A regular approximation does not exist. (S.1.2)A regular approximation does exist. S.2. There exists a formal limit problem with admissible solutions Again there are two possibilities: (S.2.1)A regular approximation does not exist. (S.2.2)A regular approximation 4, does exist, however 4, is not a solution of the formal limit problem.

We shall now discuss in some detail these classes of singular problems, in relation to the problems that have been studied in Chapters 4 and 5.

198

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

CH. 6, $1

The singular class S.1. In all problems studied in Chapters 4 and 5 a formal limit problem Lou = F , exists. We remark however that non-existence of a formal limit problem does not imply non-existence of a regular approximation. This is shown by the theory of non-linear oscillations, discussed from the point of view of singular perturbations in Eckhaus (1977). If a formal limit problem does exist, then a perturbation problem is in the singular class S.l if all solutions of Lou = F , are in some inadmissible set. This is the case in most problems studied in Chapters 4 and 5. For example consider the Dirichlet problem for a second order elliptic equation, and let the norm be the supremum norm. Then any approximation must satisfy the boundary condition imposed on @ on the boundary r of D. This is in general impossible for the solutions of Lou = F,, because Lo is of first order, or of zeroth order. Most problems studied in Chapters 4 and 5 belong further to the class (S.1.1). However, the class (S.1.2) is non-empty, as shown by the example discussed in Section 4.2. The singular class S.2. These problems are disturbing, but fortunately not frequent in applications. Confronted with a problem in the class S.2 one can construct a formal approximation as a solution of the formal limit problem in an admissible subset, without being warned in the course of the procedure that one does not construct an asymptotic approximation. Example 3 of Section 4.5 illustrates the class (S.2.1). Examples in the class (S.2.2)are unknown to this author. In Eckhaus (1977) it has further been shown that, if a problem is properly posed in the sense of Hadamard, and satisfies certain mild additional conditions, then the behaviour described in class (S.2.2) cannot occur. 6.1.4. The problem of validity of formal approximations

In this section we take up the reasoning of Sections 4.7 and 5.8, and outline the general framework in which most of the analysis of this chapter will be placed. Assume again that a formal approximation 0::)has been constructed such satisfies the boundary conditions, or the initial conditions, imposed on that 0;;) @. We then have L,@=F, L,@L?'

=F

@EK FEF,

+ p,,

E

p, E 9

If we know that the inverse L;' : 9 + Vexists, then we can write

CH. 6 , 81

GENERAL INTRODUCTORY CONSIDERATIONS

199

The study of the validity of OaSas an approximation of @ is thus reduced to the study of the continuity properties of the operator L-&’. In the linear case, the continuity of Le-’ is simply expressed by llL;1Fl] G

A(&)llF/l*, V F E 9.

We shall find in part 6.2 that for large classes of problems a precise estimate of ).(E) can be obtained without any explicit knowledge of the operator In the actual course of the analysis one does not even have to suppose at the outset that L;’ exists, and results can be derived in the form of so-called a priori estimates. These are statements of the following type: Consider the problem

L,&=F,

& E X

FEF.

If, for some FEB a solution & exists, then

1 1 q G E”(E)llFII*. Part 6.2 is devoted to such estimates for linear problems. We remark that in the linear case an a priori estimate immediately implies uniqueness of solutions. To establish the existence of solutions one can use the available global theories for various classes of problems. In the non-linear case, an a priori estimate of solutions of the original problem is not very useful if one wants to study the difference between two solutions @ and mas.It seems then natural to study the problem for the remainder term R,, given by R, = @ - @(),as 3

@,Rm= - p m ;

R,

E

V, pm

E

.9

where @&Rm = L,(R,

+ @):

- L,@::).

The modified operator @,, which depends on @!$, will in general be quite complicated. The aim of the analysis is again to obtain estimates of the structure IlRmll G

J4E)IIPmII*.

We shall find that only for limited classes of problems such estimates can directly be obtained, on the basis of some global properties of the operator @, (Section 6.3.1). More in general, if one wishes to obtain an estimate for R,, the operator @, must be studied in some detail. However, we expect that R , will be small. This suggests the use of classical perturbation analysis of Section 6.1.1, with a decomposition @&

+ 9&

= d,

200

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

CH. 6 , $2

where d,is the linearization of L, at R , = 0. Existence and continuity properties of the inverse dE- can then be established following the methods of part 6.2. The procedure will be described in Section 6.3.4. Both in the linear and in the non-linear case, we thus attempt to establish that, if IIL&@l- L & W *

is sufficiently small, then IIQ1

- @2II

is small. This can be viewed-as the property of continuous dependence of solutions on the data, or in yet other words, the continuity of the inverse Le-' (if one has uniqueness of solutions, so that L;' can indeed be defined). The approach to the problem of proving validity of formal approximations, outlined above has been frequently and successfully used in applications to both linear and non-linear problems. It should however be noted that this approach is not the only conceivable one. For example, for certain classes of problems, interesting results have been obtained by the method of so-called upper and lower solutions (Section 6.3.2). We also mention in this context a quite different alternative approach proposed in Eckhaus (1977).

6.2. Estimates for linear problems In this part we derive estimates of solutions for various classes of linear singular perturbation problems. Our aim is to obtain good insight into the .+dependence of the estimates. The starting point of the analysis is usually some global result about the class of problems under consideration (such as for example the maximum principle). The amount of labour, and the degree of sophistication needed in the analysis in order to establish the estimate, depends on the class of problems, and furthermore, on the choice of norms. In what follows we do not attempt to enumerate all known results concerning estimates of solutions of linear singular perturbation problems. We concentrate instead on the basic aspects and the underlying ideas of some methods of analysis which have been fruitful in applications, and give an indication of the relevant publications for further reading. We start, in Section 6.2.1 with estimates in the supremum norm for solutions of elliptic Dirichlet problems of second order. This section contains also some discussion, and reference to the literature, concerning the more difficult problems with so-called turning points. Next we describe a technique for obtaining estimates in certain Hilbert spaces which are more or less 'naturally' associated to a differential equation under consideration. This also provides an introduction into the use of &-dependent

CH. 6 , $2

ESTIMATES FOR LINEAR PROBLEMS

20 1

norms, which yield estimates of the function and its derivative. Estimates of this type are very important. They permit to demonstrate that the derivatives of the formal approximation are approximations of the corresponding derivatives of the solution. In Section 6.2.3 we pursue the study of estimates of solutions, and of higher order derivatives of solutions, using 'stronger' norms (i.e. Holder norms). This leads to a theory for elliptic Dirichlet problems of higher order (for which in general there is no maximum principle). Furthermore, for elliptic problems of second order, very detailed results are obtained. These results are an important tool in the analysis of non-linear problems. Finally, in Section 6.2.4 some estimates for initial value problems for ordinary differential equations are presented, and an indication of the literature containing recent results for hyperbolic singular perturbation problems is given. 6.2.1 The maximum principle for elliptic operators and its applications

Many problems associated with differential equation have so-called maximum principles. An excellent source book is Protter and Weinberger (1967). Eckhaus and De Jager (1966) used systematically the maximum principle for elliptic boundary value problems in order to derive estimation theorems, and Van Harten (1975) contributed some important improvements in this area. We describe in this section the main results. We consider second order elliptic operators and assume for simplicity that all coefficients are infinitely differentiable. In what follows we denote by C k ( D ) the space of k-times continuously differentiable functions in an open bounded domain D c R". A convenient form of the maximum principle, for our purpose, is the following one (Protter and Weinberger (1967) Theorem 6): Lemma 6.2.1.1. Let D c R" be an open bounded domain with boundary consider, for E > 0, the differential operator

r, and

Suuuose that Then any function u E C 2 ( D )n C(d) that satis$es

Leu 2 0 in D,

u 6 0 on

r

also satisfies u60

in D.

A serious limitation in Lemma 6.2.1.1 is condition (ii). A powerful tool to

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extend the applicability of the maximum principle is the following result, given essentially in Protter and Weinberger, theorem 10:

Lemma 6.2.1.2. Lemma 6.2.1.1 remains valid if condition (ii) is replaced by (ii)* There exists afunction W E C2(D) n C(d) such that in D,

L,W
in B.

W>O

Proof. Let u be a function satisfying L,u 2 0 in D,

u

<0

on

r

and consider u* = u/W where W is a function satisfying condition (ii)*. Then Leu = L,u*W=

L,u^2 0 in D

and hence t,u^>O

Now

in D,

u*
on

r

t,is an operator of the structure

where 1 $=-L,W
inD.

It follows that Lemma 6.2.1.1 holds for the operator Finally

L,,

and the function 6 .

implies u

<0

in

D.

An interesting application of Lemma 6.2.1.2, which gives a considerable extension to the validity of Lemma 6.2.1.1, is given by

Lemma 6.2.1.3. Lemma 6.2.1.1 remains valid if E is sufficientlj small, the coefficients aij(x,E)are uniformly bounded for x E 6,E 10, and condition (ii) is replaced by (ii)** There exist constants el,...,On,and 6 > 0 such that

C e,picx,E) + "/'x,E)< -6 < 0 L

in

6.

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203

Proof. Take in Lemma 6.2.1.2

1

w= exp C e,xi (i:l

Lemma 6.2.1.2 can further be used in various difficult situations by constructing the function Was a formal approximation in a suitably defined problem (Van Harten 1978). This is formalized in

Lemma 6.2.1.4. Lemma 6.2.1.1 remains valid i f &is sufficiently small and condition (ii) is replaced by (ii)*** There exists a function ~ ( x , < E d) < 0, x E D, and a function W ( X , E )2 d, d> 0, xer, with d and d &-independent,such that a formal approximation Wasof the solution W o f in D,

L,W=f

W = w on

r,

is positive in D ,

We now proceed to derive bounds for solutions of boundary value problems.

Theorem 6.2.1.1. Let CD be a solution of L,CD=F,

XED

with some prescribed boundary values on r. Suppose that the conditions of Lemma 6.2.1.1, or 6.2.1.2 are satisfied, and furthermore, that there exist two functions a,, and CDi, elements of C z ( D )n C ( b ) , with the following properties:

< F < LEOiin D , Of< O < CD, on r. L,O,

Then Oi < CD

< @,

in D.

Proof. Apply Lemma 6.2.1.1, or 6.2.1.2, to O - 0,and CDl - 0. The functions a, and CDf are called barrier functions. A suitable choice of barrier functions permits to establish asymptotic estimates of solutions. A quite general result can easily be established under conditions of Lemma 6.2.1.3and is given by

Theorem 6.2.1.2. Let 0 be a solution of L,CD=F, X E D ,

O=O,

x ~ r

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CH. 6, $2

where

Suppose that (i) 'The coefficients aij(x,&)are uniformly bounded for E 10, x exists a positive constant A , independent of E such that

c aij(x,E)titj>, A c t;

ED

and that there

in D.

i

ij

(ii) ?here exist constants 01,...,8,,,6,

independent of

C Pi(x,E)ei+ ^J(x,E)< -6 < o

in

E

such that

D.

I

?fien,for sufficiently small

where

E

[@lo G cCF10 [*lo= supxoDI-I,and

c is a constant independent of

E.

Proof. Write QU

= -0, = c[F],exp

with

1

(

c = -max exp 6 a

1 eixi)i.

i I 1

One easily establishes that 0,,0,indeed are barrier functions, when conditions (i) and (ii) are satisfied. The proof then follows by application of Theorem 6.2.1.1. The contents of Theorem 6.2.1.2 is an a priori estimate, which can immediately be put to use in proving validity of formal approximations. Furthermore, in the present case, the estimate also leads to a complete theory for the boundary value problem. We sketch here the reasoning: assume F E C c c ( D )and , r smooth, then from Agmon, Douglas and Nirenberg (1959) one has 0 E C x ( D ) . Because -of linearity, the estimate guarantees uniqueness of solutions, and uniqueness in turn implies solvability (Ladyzenskaja and Uralceva (1968)). We thus obtain the existence of LE-',for E sufficiently small, and the continuity estimate

with

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205

We note that Theorem 6.2.1.2 holds when the domain D varies with E (but remains bounded for E 10). This is evident when writing out the technical details of the proof. The observation is of importance, because it permits to apply the theorem to certain problems with a ‘free boundary’ (Eckhaus and Moet (1978)). We further note that in Theorem 6.2.1.2 only mild conditions on the Edependence of the coefficients of L, are imposed. However, condition (ii) does imply an important limitation on the applicability of the theorem. This is most clearly seen in the one-dimensional case. The theorem then requires the existence of constants 8,6 such that p(x,E)e

+ ~ ( x , E<) -6

< 0 in 6.

If y is not strictly negative, but is bounded for E 10, then the condition is satisfied if P(x,E) has no zeros in 6. The theorem is hence not applicable if B(x,E)has one or more zeros in 6 and y is not strictly negative. Points x* E 6 such that P(x*,e) = 0 are called turning points. Similarly, when the dimension of D is higher than one, then condition (ii) states that either y be negative, or that y be bounded for E 1 0 and at least one coefficient pi be definite in 6. Boundary value problems for operators with turning points, violating condition (ii) of Theorem 6.2.1.2, have only very recently become subject of rigorous investigation, by rather specialized methods, which fall outside the scope of this chapter, We indicate here some relevant literature for the interested reader. The study of ordinary differential equations with turning points has a long history. A particularly fruitful method has been originated by Langer (1931, 1949) and developed by various authors, In that method one uses a ‘comparison equation’, (that can be solved explicitly in terms of special functions) which is sufficiently ‘close’ to the original equation so that a perturbation analysis is possible. For an account of these developments the reader can consult Wasow (1965) and Olver (1974). Boundary value problems for equations with turning points present the additional difficulty that, when Theorem 6.2.1.2 does not apply, one must search for other methods to prove the validity of formal approximations. Interest in these problems has been stimulated by the work of Ackerberg and OMalley (1970) who discovered, by formal analysis, a peculiar behaviour of solutions, which they called resonance. A rigorous theory of boundary value problems with turning points has at the present date been achieved only for certain specific classes of problems. For ordinary differential equations the analysis of Olver (1976, 1978) is exemplary for the use of special functions. An entirely different approach is due to De Groen (1976,1977), who studied spectral properties of L,. De Groen’s analysis provides a theory for a class of ordinary differential

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CH. 6, $2

operators, and a class of elliptic operators which are such that the characteristics of the degeneration all intersect at one internal point of D. Let us finally remark that in problems with turning points, when condition (ii) of Theorem 6.2.1.2 is violated, one should not too hastily conclude that the maximum principle is not applicable, or of no use. In fact, Theorem 6.2.1.1 may still hold, and one can then attempt to construct suitable barrier functions for the problem. That this approach can be fruitful is demonstrated by Grasman (1977), who’studied a class of elliptic operators with y = 0 and pi, i = 1,...,n vanishing at an interior surface of D. 6.2.2. Estimates in Hilbert spaces from positivity of the bilinear form

It is technically rather easy to derive estimates in certain Hilbert spaces which are more or less ‘naturally’ associated to the differential equation under consideration. Many results in this area can be found in Lions (1973). One obtains by these procedures estimates for the function and its derivatives. However, in order to obtain sharp estimates one must use ‘&-dependent’norms. Let us first show this by the following simple introductory example. Let @(x,E),x E B t R” be a solution of

+

&A@ Y(x)@= F, x E D,

@=o,

XGI-

where A is the Laplace operator and y(x) satisfies y(x) < yo < 0, x E 6. We multiply the differential equation by the function 0 and integrate over the domain. We thus obtain the identity

+ES D

cDA@dx + S y@’dx = J F@dx. D

D

Integrating the first term by parts we find

Considering the left-hand side we can write

where ct = min[-y,,l]. On the other hand, considering the right-hand side of the identity and using the Cauchy-Schwartz inequality, we get

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207

and hence also

Combining these results we obtain the following estimate:

The left-hand side in the above estimate can be considered as the norm of Q, in a Hilbert space in which the inner product ( u p ) is defined by

(u,v) = S uudx D

au au + EJD 1 --dx. axi axi {=I

The inner product, and the norm, are &-dependent,and it is this very fact that gives us rather sharp estimates. Explicitly we find, for the function Q,

and for the derivatives

We now turn to more general problems. The possibility of obtaining estimates in suitably defined Hilbert spaces can easily be seen to be an almost direct consequence of a condition for unique solvability of the problem under consideration, as expressed in the Lax-Milgram Lemma. We show this by considering second order differential operators. It will be convenient to write L, =

a &I-aij(x,E)axj ij

a axj

+ L,

where L , is a first order operator. Let V be the closed subspace HA(D) of functions which are elements of the usual Sobolev space H’(D) and which vanish on the boundary r of D. We associate to L, a bilinear form on I!defined by

We further define a linear form P(u) = - J UFdx. D

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CH. 6 , 5 2

The 'variational formulation' of the problem L,@ = F , @ E Vnow reads (Lions (1 973)): Determine @ E V such that for all u E V B(u,@)= P(u).

If B(u,u) and P(u) are continuous and bounded, then according to the LaxMilgram Lemma (Yosida (1974)), the above problem has a unique solution if v u E v B(u,u) 2 allull2,

c(

> 0.

This condition of positivity of the bilinear form is also called the condition of coercivity, or condition of V-ellipticity (D. Huet (1976)). When the condition is satisfied we have

If furthermore

{i

@2dx}1'2 6 ll@ll,

then, using Cauchy-Schwartz inequality, we obtain the estimate

In applications one must therefore attempt to equip the space Vwith an inner product such that the positivity of the bilinear form can indeed be demonstrated. Going back to the operator L, we observe that one can associate to L, more general 'weighted' bilinear forms B J , ,.), defined by B&,v) = B(rc/u,u) where rc/ is some suitably chosen function. This opens more room for manoeuvring, when proving estimation theorems. We shall use this technique to demonstrate Theorem 6.2.2.1. Let @ satisfy where

a

a

a +

L, = E C -aij(x,E), + C &(x,E)Y(x,E). i axi i j axj axi Suppose that (i) All coefficients are C'(D) functions; the coefficients aij(x,E) together with their first derivatives, are uniformly bounded for E J 0, x E D.

ESTIMATES FOR LINEAR PROBLEMS

CH. 6, $2

209

Furthermore

c aij(x,E)titj2 A x t f ,

A > 0,

x

E

6.

i

ij

(ii) There exist constants Ol,,..,O,, and 6 such that

j - J & p i - i z -aPi +yG i i axi Then @ satisfies, for

E

-6
in6.

sufficiently small, the estimate

where c is a constant, independent of

E.

Proof. Following Van Harten (1975) we consider the bilinear form BIL(.,.), with

II/ = exp

z

Gixi.

i= 1

The constants #will be determined in the course of the analysis. Explicitly we have

Straightforward computation, involving some integration by parts, yields

-

EXi j aij8,8,ju2dx.

Take now 4= -20i, where Oi are constants given in condition (ii) of the theorem. Let further 6* be some number such that

0<6*<6 where 6 is a constant given again in condition (ii). Then for E sufficiently small:

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A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

dx

CH.6 , 5 2

+ 6*mJ u’dx

D i

D

where m = minxei$. Fin a 11y

where a = m.min[A,G*]. Now, from the differential equation:

where m* = max,,D $, Combining this Tesult with the positivity estimate of B,(@,Q,) we obtain the proof of the theorem. Similar results can, by similar techniques, be derived for parabolic problems. This is illustrated by

Theorem 6.2.2.2. Let @(x,t,E),x E D c R”,t E [O,T], satisfy

6@

--

at

+ L,@ = F ,

Q,

= 0 for x E I-,

= 0.

Suppose that L, is as specijed in Theorem 6.2.2.1. Then @ satis$es, for sufficiently small E , the estimate

[i{J@’dx O D

+ ~JZ(g)’dx}dt]’’~ D i

where c is a constant independent of

d c[[{:

F2d~}dt]“’.

E.

Proof. We multiply the differential equation by $0,where $: LJ -+ R is the function used in the proof of Theorem 6.2.2.1, and integrate over x. This yields

CH. 6, 92

ESTIMATES FOR LINEAR PROBLEMS

21 1

Next, integrating over t, and taking into account the initial condition, we obtain

$[!

T

T

0

O D

+ 1B,(@,@)dt = - 11F$@dxdt.

$02dx] t=T

Because of positivity of B,(.,.) established in the proof of Theorem 6.2.2.1, we have

The right-hand side can be estimated, using the Cauchy-Schwartz inequality twice, as follows

The proof can now be completed using steps analogous to the proof of Theorem 6.2.2.1. The Theorems 6.2.2.1 and 6.2.2.2 present quite general results for differential operators of second order, relating explicitly the condition of positivity of the associated bilinear form to conditions imposed on the coefficients of the differential operator. Considering differential operators of higher order one can achieve similar results if one succeeds in establishing conditions on the coefficients of the differential operator which assure positivity of an associated bilinear form. A simple illustration is provided by the problem &A2@- (A@-?@) = F, x E D,

aia

-= 0,

ani

i = 0,1, for x E r

where A2 is the biharmonic operator and n denotes the normal to should have no difficulty in demonstrating that, if

r. The reader

y Q yo < 0 in D, then @ satisfies the estimate

where c is a constant independent of E. Estimates for the derivatives are of great importance in applications when

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CH. 6, $2

studying properties of formal approximations. We shall show this by considering the class of problems dealt with in Theorem 6.2.2.1. has been constructed Suppose that a formal expansion m

c W4,(X,E),

=

@i?(X,E)

n=O

L&CD;;’= F

+ pm,

q;)= 0,

x

E

x

E

D,

r.

Suppose further, to be explicit, that p, = Os(Ern - P),

x

E

D

where p is some fixed number. We consider

CD - (Das = a, - -

-pm,

L&CD= F

=

a=o,

X E r .

x E D,

If L, satisfies the conditions of Theorem 6.2.2.1, then we can apply the theorem to the problem for 6.We shall write

It follows that Furthermore, for any component i = 1,...,n

Now

aa

ECD

axi - B X ,

c W $2 4*

n=O

It follows that, if m is sufficiently high (i.e. > p + ) ) , then, in the sense of the L,norm, not only CDgy) is an approximation of CD, but the derivative of the formal approximation with respect to any component of x is an approximution of the corresponding deriuative of the function CD. 6.2.3. Estimates in Holder norms: elliptic equations of higher order In this section we continue the study of estimates of solutions of boundary value problems, and of the derivatives of the solutions, using however ‘stronger’

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213

norms than those of Section 6.2.2. Our basic tool will be a priori estimates, also called Schauder's estimates, which are well-known for various classes of problems. An essential difficulty is to obtain explicitly the dependence on E in the estimates. The difficulty can be overcome by a method due to J. Besjes (1975). In order to introduce the subject on an elementary level we consider first a boundary value problem for an ordinary differential operator, i.e.

@(O,&) = @( 1,&) = 0. It will be convenient to use the following notation

We shall further extensively use an interpolation inequality between seminorms [flk, which holds for any sufficiently differentiable function (Miranda (1970)):

Lemma 6.2.3.1. For anj' p satisfjing 0 < p < q

where D i s a constant that only depends on p,q and the domain, while p be chosen arbitrurily.

E(

0 , ~can ~)

For the ordinary differential operator under consideration it is possible to derive in an elementary way an a priori estimate exhibiting explicitly the E-dependence. We rewrite the problem into d2@ If a solution CD E C 2 ( D )exists, then we immediately obtain

We next eliminate [@I, using the interpolation inequality with p = 1 and q=2. It is not difficult to see that a good choice for that purpose is

11 1

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

214

CH. 6, $2

We thus use

and obtain

This gives in turn

The two estimates for [@I1and [@Izgiven above are the a priori estimates for the solution. The analysis must be modified in the special case [PI0 = 0. Then there is no need to eliminate [@I1in order to obtain an estimate for [@I2. An estimate for [@I1 follows from Lemma 6.2.3.1 with p = 1 and q = 2 . The best choice for p is easily seen to be p = J E . We can pursue further the analysis if conditions for the maximum principle are satisfied. In that case we also have the estimate of Theorem 6.2.1.2, i.e.

[@lo < C C F l O . Combining the results one easily establishes

Theorem 6.2.3.1. Let 0 be the solution of

d2@

+

do dx

'2p-

dx

@(O,E)

+ ?@ = F ;

x E (O,l),

= @(l,&) = 0.

Suppose that p and y are uniformly bounded as C / O for furthermore, that there exist constants 9 and 6 such that

Then we have the estimates (a) For fi = 0,(1)

X E

[O,l] and

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215

We now turn to elliptic boundary value problems. We shall consider elliptic operators of arbitrary order. A priori estimates of solutions are known from the work of Agmon, Doughs and Nirenberg (1959). The setting of these estimates are Holder continuous functions. We shall therefore need suitable notations. We consider functionsf(x), x E D c R",and use for the length of the vector x the Cartesian norm

Let s be a n-tuple (,sl, ...,sn), where s, is a natural number or zero. Partial derivates offwith respect to the components of x will be denoted by Otf=

0-

zx;l ...ax:

c:=

with Is1 = 5,. The supremum norm of the derivatives is defined by

Let now a be a number satisfying 0 < x < 1. We say that Dmf(x) is Holder continuous with exponent x (or %-Holder continuous), if

where the supremum is taken over all x E D, y E D,x # y. f~ Cm+'(D)meansf€ Cm(D)and [fI,+, < x.. Holder norms are defined by IfIm+T

=

lflm + Cflm+z

where

The interpolation inequality given in Lemma 6.2.3.1 holds for semi-norms [f],+,, i.e. when q is a natural number, and p = m + x , rn = 0,1, ...,q - 1. A linear differential operator L, of order 1, is an expression of the structure

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A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

CH. 6, $2

It is well known that a real elliptic operator necessarily is of even order. Let now L be of order 2m, m a natural number. Lis uniformly strongly elliptic in D if for all real n-tuples 5 ,...,(,

where ts denotes the product ( y . 5 y ..... 5:. With these preliminaries we can formulate the following result, due to Agmon, Douglis and Nirenberg (1959).

Theorem 6.2.3.2.Let D be bounded, with a smooth boundary rand let L be a linear, uniformly strongly elliptic operator of order 2m in D,of which all coefficients are (D), with 12 2m. elements of C' - 2 m +' Consider the Dirichlet problem LO = F

in D ,

where dldn denotes normal deriljative on Assume that @ E C2m+a(D). Then (i) @ E C"a(D), (ii) I @ I f + a < c(IFIL-Zm+a + [@lo)*

r.

We now consider singular perturbation problems with L= EL,,

+Lk

where L,, is of order 2m and L, is of order k < 2m. Suppose that L satisfies the conditions of Theorem 6.2.3.2. Then we have the estimate (ii) which however is useless yet, because we do not know how c depends on E. In order to obtain an a priori estimate in which the &-dependenceis made explicit, Besjes (1975) follows a method of which the essence has been explained at the beginning of this section, when considering ordinary differential operators. Let L,, and Lk be independent of E . One can rewrite the problem into

Then, applying estimate (ii) of Theorem 6.2.3.2 one obtains I@'ll+i

it

< c -IF

- Lk@Il-Zm+a

+ [@lo}.

Working out the right-hand side of the inequality one gets Holder norms of @ with subscript lower than 1+ CI. Careful analysis of this expression with repeated

ESTIMATES FOR LINEAR PROBLEMS

CH. 6,42

217

judicious use of Lemma 6.2.3.1 (which involves in particular optimal choices of p as function of E ) permits Besjes to establish

Theorem 6.2.3.3. Let L in Theorem 6.2.3.2 be given by

+

L= ELZm L,

where L,, and L, are independent of 'Then,for any j = 0,1, ...,1,

E

and of order 2m, resp. k < 2m.

CQljd ~ 1 ~ 7 F I 1 - 2 m++ zCZE'[QIO> where cl,cZ are independent of (a) for k < 2m- 1

E,

o = l-j-2m+ k + 1 , 2m-k-1 (b)for k

=

and o,t are as follows:

t = -

j

2m-k-1'

2m- 1 1 +z 1-cc '

o = I-j-

t=

-- j

1 -!x'

Theorem 6.2.3.2 provides a sharp a priori estimate for singularly perturbed elliptic Dirichlet problems of any order. However, to make the estimate useful in proofs of validity of formal approximations, we still need an estimate for [@lo. For second order problems (i.e. m = 1 ) one can now use the maximum principle and the theory of Section 6.2.1. We shall return to this case at the end of this section and describe a further generalization of Theorem 6.2.3.3. For higher order problems ( m > 1 ) there is no maximum principle, and a serious difficulty still persists at this stage. One can of course attempt to derive estimates in suitable Hilbert spaces, following the method of Section 6.2.2. An estimate for [@I0 would then follow if one could apply the well known Sobolev's imbedding theorem. This procedure, if successful, always imposes a limitation on the dimension n of the domain D. Besjes resolves the difficulty by showing that, when 0 is a solution of the probiem defined in Theorem 6.2.3.3, then [@lo can be estimated with the aid of elementary Hilbert space results, even though Sobolev's theorem does not apply. An essential tool for the purpose is given by

Lemma 6.2.3.2.Let D have the ordinary cone property (Friedman (1969, p. 22)). Then for any f E C'(D) the following inequality holds:

[fI0

G clW-'I2{L f zdx}1'2

+ c2w[fll

where w E (O,wo)can be chosen arbitrarily.

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CH. 6, $2

Using Theorem 6.2.3.2 and Lemma 6.2.3.2 with a suitable choice for w one can in terms of This leads to the following result: now estimate [@lo

Theorem 6.2.3.4. Let L=&LZm+Lkbe as in Theorem 6.2.3.2. Then, for any j = O,l,. ..,1

+

[@Ij < c

~ E ~ I F / ~c2cT - ~ f ~Q2dx + ~ {D

where c1 and c2 are constants independent of j n/2 5 = for k < 2m- 1, 2m-k-1

!I2 E, 0

is as in Theorem 6.2.3.3 and

+

T =

j

+

n/2 for k 1-a

--

= 2m-1.

The final step in the analysis is an estimate for ll@IlL2. Here one can use methods of Section 6.2.2 in a rather elementary way. However, the structure of the operator L, must now be specified:

Theorem 6.2.3.5. Let L = &LZm + Lk be as in Theorem 6.2.3.3, and let furthermore Lk be a positive operator, in the sense that, Vu E Ck(D)

f ULkUdX D

C j D

U2dX,

C

>0

then

where c is a constant independent of

E.

Proof. We consider the identity E j

QL2,Qdx

D

+ Ds @Lk@dX= Df QFdx.

From Garding’s inequality (see for example Friedman (1969))one has

f QL2,Qdx 2 ~1 D

C

f IDs@12dx- ~2 f @’dX.

Is1 < m D

D

Hence, using positivity of L,, one finds (C

-~

~ 2 @’dX ) s

D

G f QFdx. D

Finally, using Cauchy-Schwartz inequality in the standard way it follows that

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CH. 6, 52

219

{j @2dx'1'2 < E{jF2dx)li2 D

f

where E is some constant independent of

E.

The theory is thus completed, under the condition that Lk is a positive operator. It is not difficult to see that, for elliptic operators Lk, positivity of Lk is closely related to the conditions for unique solvability of the Dirichlet problem associated with Lk, as expressed in the Lax-Milgram Lemma. If Lk is an operator of first order, then the positivity can be assured by imposing suitable conditions on the coefficients. Besjes (1974) extended the theory to parabolic problems described by

where L,, and Lk are linear uniformly strongly elliptic operators independent of the variable t, and 2m > k . It would lead us too far to review here the results, which in many ways are analogous to those for elliptic problems. We finally return to elliptic problems of second order, where the maximum principle can be used. Van Harten (1978.A) obtained very precise results for the more general case, in which the coefficients depend on E. We repeat first the formulation of the problem. We consider L,Q=F,

x E ~ ,

@=o,

X E T

where

C aij(x,c)titj2 AXi ti',

A > 0.

1J

We further introduce the following notations: A0 =

Z:IaijIo;

Aa =

ij

Yo = M o ;

YZ

1 IaijL; ij

=

I"/(,.

With these preliminaries we can formulate

Theorem 6.2.3.6. Let L, be as in Lemma 6.2.1.4 (maximum principle) and take E J 0. Then the solution 0 satisfies the following estimates: (a) ZfA, = O ( E - ~B,) ,= O ( E - ~ya) ,= O ( E - ' - ~y)o, = O(E-'),then a E (0,l). Suppose that A , and Bo are bounded for

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

220

CH. 6 , $2

The proof of the theorem is given in Van Harten (1978.A). The theorem is very useful when studying operators L, that arise as linearizations of the problem for the remainder term. 6.2.4.Estimates for initial value problems

It is possible to derive a priori estimates for various initial value problems by a technique which in spirit is somewhat similar to the construction of Liapunov functions. We commence with Theorem 6.2.4.1. Let O(X,E),x E [O,T] be the solution of

O(O,&) = 0,

g)x=o = O.

Assume that a, E Co([O,T]), a, E C'([O,T]), al,a2 and da,/dx are uniformly bounded for E 4 0. Assume further that a2(x,&)2 a > 0, x E [O,T], with x independent of E. Then, for E suficiently small, the following inequality holds for x E [O,T]:

where c is a constant independent of

E.

Proof. It will be convenient to rewrite the problem into one for a system of first order equation. Thus we put

du2 dx

E-=

- a2u2 - alul

+ F.

We now follow essentially Gee1 (1978). Multiplying the differential equation

CH.6 , 52

ESTIMATES FOR LINEAR PROBLEMS

22 1

for u2 by u2 we get

One can estimate the derivative of u2 by an expression not containing u 2 , using the following elementary inequality, which holds for any pair of numbers i“1,52:

2<,5, d

t: + 5:.

We thus write

and obtain

Next, we multiply the differential equation for u2 by the function u1 and, taking into account the differential equation for u l , we get

$I-- du:

dx

+

du2 dx

EU~-

= U,U:

+ FU

1‘

Using again the elementary inequality we find

We finally add the two estimates and obtain d

-[u,fu:+EU:) dx

+ 2EU1U2] 6

Because of the hypothesis on the coefficients, there exists a constant co independent of E such that d dx

-[u2(4

+ E U : ) + 2eulu2] < c , [ u : + F u ~ ]+ 2 F 2 .

We now integrate, taking into account the initial conditions. This yields

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

222

a,(ut

CH. 6 , $2

+ 2 ~ ~ ~ 2<4co, s [u: + E U ; ] ~ X+’ 2s F 2 d x ‘

+EM:)

X

x

0

0

On the left-hand side we use -2EUIU2 = (&1’4U1)(&3’4U2)

<

&1’22(:

+

&3’22(i

which implies 2&UtU2

Hence, for

E

a,(u:

2

-&1’2U;

- E3’2 u2

sufficiently small

+

EUt)

+ 2 E U 1 U 2 2 (a2 -

El’2)U:

+ &(a2-

&1’2)u;

2 a(u: + E M ; ) where CI > 0 is a constant independent of We finally obtain in this way u:

E.

c x 2= + EM; < 2 J [u:+~u:]dx’ + J F2dx‘. -

M O

a0

The assertion of the theorem is now demonstrated by a straight-forward application of Gronwall’s Lemma, which states that, for a positive continuous function $(x), and two positive constants k o , k , , the inequality X

+(x) d k1 J $(x‘)dx’ + ko 0

implies $(x)

< koeklx.

For proof of this lemma see for example Roseau (1966). Remarks. We have written out the proof of the theorem in some detail, to show the essential steps. This will permit in the sequel an almost immediate generalization to differential equations of higher order, with E multiplying the highest order derivative.

The positivity condition for a, is crucial for the theorem, as we have already seen in Example 3 of Section 4.6. Similar conditions will appear in a generalization of the theorem that follows now,

Theorem 6.2.4.2. Consider the n-tuple of functions u~(x,E), ..., u,(x,E),satisfying du1_ dun - 1 _ - 242, ... - u,,

dx

’ dx

CH.6 , 42

223

ESTIMATES FOR LINEAR PROBLEMS

ui(O,&)= 0, i

=

1,...,n.

are Co([O,T]) functions, unformly Assume that the coefficients a , ,...,a,bounded for E 10. a, E C ‘([O,T]), and is, together with its derivative, uniformly bounded for E 10. Furthermore

U,(X,E) 2 a > 0, x E [0,7-1, with a independent of E . ‘Then,for E sufficiently small, n- 1

T

+

1

[ U ~ ( X , E ) ]E[U,(X,E)]~ ~ 6 c F2(x‘,&)dx’, x i= 1

E

[O,T]

0

where c is u constant independent of

E.

Proof. Multiplying the differential equation for u, by a,u, we get d dx

n- 1

~1 E - U ~ U , ’

= -u,’u,‘ -

1 u,,u,,u,u, + Fu,u, + -$-ti*.du dx

p= 1

We get rid of the first term on the right-hand side by writing apupunund

1 +z u; u; + 3 .); u,’.,”, /P

1

F a n u , < i T F 2 + i?iu,’u,’ 70

where ?o,...,y,-l

is a set of numbers such that

p=o

This yields an estimate d dx

E-u

c

n- 1

u2 < c, “

p= 1

u,’

du + c n F 2+ $C:-U,’ dx

where c, is some constant independent of E . Next, multiplying the differential equation for u, by u,-

we obtain

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

224

CH. 6 , 5 2

Using again the elementary inequality, and the boundness of the coefficients u p , we find

where c n - is some constant independent of E . For u p , p = n- 2, ...,1 we have

Adding all the estimates one obtains n-2

d dx

p= 1

The proof can now be completed following steps entirely parallel to the proof of Theorem 6.2.4.1. Remarks. Written in terms of a single differential equation, we have an estimate for the solution of the initial value problem

= 0, p = 0,...,n-1.

One cannot expect similar results to hold in the case of a differential operator L, of order n, of which the degeneration is of order lower than n - 1. In that case the solutions of the initial value problem must be expected to be either unbounded, or exhibiting rapid oscillations for E 10. This can be seen from the construction of formal approximations. Furthermore, indication of such behaviour are implicit in the analysis of boundary value problems for higher order equations, as given in O'Malley (1974). We now turn to partial differential equations, in particular to second order equations of hyperbolic type. In this area, the so called energy integral method has been known and used for a long time as a powerful tool to obtain a priori estimate of solutions (see for example Friedrichs (1954)). We briefly describe the main steps of the method. Consider the problem

CH. 6 , $3

NON-LINEAR PROBLEMS

225

LEO= F, x E D c R”,

L,

= EL2

+ L,

L, is a second order operator of hyperbolic type, L , is a first order operator. On the boundary r of D the function @ and/or its first derivatives are prescribed. Related to the problem above are integral identities of the type

where Do is a subdomain (usually containing part of the boundary r),the coefficient x(x), &(x), i = 1,...,n are, as yet free. In the energy integral method one chooses Do, CI,p,, i = 1,..., n, such that on the left-hand side, applying Green’s theorems, one obtains integrals over positive definite quadratic forms (called energy integrals). Judicious manipulations of the resulting formula lead to the a priori estimate. An additional difficulty in singular perturbations is to obtain useful estimates of the &-dependence.Such estimates, in the supremum norm, have been obtained by De Jager (1975), for problems with domains in RZ.Further results are given in Gee1 (1978). An essential restriction on the theory is, that the characteristics of L , must be ‘time-like’ with respect to characteristics of L 2 .(For the notion of ‘time-like’ and ‘space-like’ characteristics one can consult Cole (1968).) With the same restriction, Genet and Madoune (1977) derived estimates in norms of Hilbert spaces, for hyperbolic problems with domain in R”. 6.3. Non-linear problems

The contents of this part is somewhat heterogeneous: it is a collection of various available rigorous results and procedures that are of some generality in the domain of nonlinear singular perturbations. In Section 6.3.1 we study, in the non linear case, the maximum principle for elliptic equations. The results, when applicable, permit immediate proof of validity of a formal approximation. However, the applicability is severely restricted to rather special problems. Next, in Section 6.3.2, we describe the method of upper and lower solutions. The method has mainly been used in boundary value problems for ordinary second order differential equations, where the basic tool is Nagumo’s theorem and its generalizations. In applications one obtains (rather easily) the proof of validity or regular approximations in subdomains. Section 6.3.3 is devoted to Tichonov’s theorem for initial value problems for systems of singularly perturbed ordinary differential equations. We have included this theorem here, because it is a fundamental result on the behaviour

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

226

CH. 6,53

of the solutions of such systems. However, Tichonov’s theorem cannot be considered ‘a tool of analysis’, i.e. it does not lead to further results. The final Section 6.3.4 is the most substantial one. The problem for the remainder term is studied in the setting of a perturbation theory in a Banach space. We describe a systematical procedure due to Van Harten. The method is very versatile and is not limited to any special classes of problems. The method does require advanced use of the techniques for obtaining estimates for linear problems described in part 6.2. With this prerequisite Van Harten’s procedure can be considered as a very general method for proving the validity of formal approximations in non linear problems. 6.3.1. Non-linear applications of the maximum principle

We shall

isider non-linear operators L, given by

L,@ = B ( x , @ , D ~ @ , D ~ @ , E ) where 9is to be interpreted as a function of n2 + 2n + 2 variables. Of particular importance will be the partial derivative of 9 with respect to 0,which shall be denoted by 9@. B will be of the structure .F(x,@,D~@,D~@,E) = E Xaijij axiaxj

The coefficients aij are functions of x,@,d@/8xi, i = 1,..., n and this section we assume ellipticity, i.e.

E.

Throughout

for all x E D, E E (O,eO], and all @ and D1@. With these preliminaries we can give the following formulation of the maximum principle (Protter and Weinberger (1967) section 2.16): Lemma 6.3.1.1. Let

D

c R” be an open bounded domain with boundary

Consider L,@ = B(x,@,D~@,D~@,E)

and impose the condition that

B ~ ( x , @ , D 1 @ , D z<~ 0, ~ ) for x E D, E E ( O , E ~ ]and all values 0 f @ , D 1 @ , D 2 @ . Then any function u E C 2 ( D )n C(D) that satisfies L,u 3 0 in D,

r.

CH. 6, $3

NON-LINEAR PROBLEMS

~

o0n r


in D.

u

227

also satisfies u

It should immediately be recognized that the conditions Fa<0, is a very serious restriction on the applicability of the lemma. To see this, consider for example

Then

It should be clear that, in order to assure 5*< 0 for all values of @ and d@/axi, one must require

aPi _ -0

a@

ay and - G O ,

a@

More in general, 5 must be of the structure 9(x,@,D1@,D2@,&) = @(x,D1@,D2@,&) + y(x,@,&). One can then satisfy the condition by requiring

a y / m < 0. Problems in which the condition Fa< 0 is not satisfied can sometimes be transformed into a form in which the condition is satisfied. This can, for example, be attempted by proceeding in analogy to Lemma 6.2.1.2, that is, by considering a transformed problem for the function 6, defined by

6 = @/w, with w a suitably chosen function. In what follows we develop some further consequences of the maximum principle assuming always the condition of Lemma 6.3.1.1 to be satisfied. Theorem 6.3.1.1. Let @ be a function that satisfies

L,@ = F , X E D . Suppose L, to be such that Lemma 6.3.1.1 holds. Suppose further that there exist two functions a,, and a,, elements of Cz(D),with the following properties: LEOu< F

< LEO,

in D,

(DL< @< 0” on r.

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

228

CH. 6,53

Proof. Consider w = @ - @,, and define

t , w = L,(w

+ mu)- L,w.

We have t , w 2 0 in D,

w

<0

on

r.

Now

L,w = F ( x , w+@,,,D’(w

+ @,,),D’(w +@,),E)

- F(X,@,,,D’@.,,D~@.,,E)

= @(x,w, D~w,D’w,E).

The condition gQ < 0 for all @,,D’@,D’@implies that gW< 0, for all w, D’w, D’w. Hence, Lemma 6.3.1.1 holds for the function w,and the operator L,. This proves the assertion of the theorem concerning a,,. Analogous analysis for W = Of- 0 will complete the proof of the theorem. A suitable choice of barrier functions @,, and can now lead to asymptotic estimates of solutions of boundary value problems. A particularly simple result is given by Qf

Theorem 6.3.1.2. Let @ be a solution of Lc@= F(X,@,D’@.,D’@,E)= F , x

@=e,

E

D,

x c r

Assume F(x,u,D~u,D’u,E)to be such that

F,,(x,u,D~u,D~u,E) < -6 < o for x E 6,E E (O,E,,], and all values of u,D’u,D’u, with 6 a constant independent of E. %en SUP XED

PI < +upXED IF1 + SUP PI}

where c is independent of

XEr

E.

Proof. We can use constant barrier functions

with a suitable choice of M . One easily establishes that, under the condition of the theorem, one has

CH. 6, $3

NON-LINEAR P R O B L E M S

L & a u< The choice M

229

--mu.

= 6-' assures

now that

a,, and CDl indeed are barrier functions.

Theorem 6.3.1.2 gives an a priori estimate for solutions of non-linear elliptic boundary value problems, Unlike the linear case, the estimate does not imply uniqueness and existence of solutions. However, assuming existence, the estimate can be used for proofs of validity of formal approximations. One thus obtains Theorem 6.3.1.3. Suppose there exists a function CD satisfying L,@ = F , X E D , @=07

XEI-

where

L&@= .F(X,CD,D'CD.,D~CD,E), and F satisfies the conditions of Theorem 6.3.1.2. Let there be given a function CDas such that

LEDa,= F =

e,

+ p,

x E D,

XEI-

then SUP

IQ-QasI

XED

< C S U P IPI XEB

where c is independent of

E.

Proof. Define R = 0 -aasand consider the problem Z,R = L,(R +aas) - LEOas = - p, x R

E D,

=o, x ~ r .

We have

Z, R

=

+ a,,, D'( R + a,,), DZ(R+CDas),&) -9 ,D '@ as, D ,E)

F(x,R

(x7mas

= @.'x,R,D

@as

R,D~R,E).

As in the proof of Theorem 6.3.1.1, from Fa< 0 (for all values of the arguments) we obtain that gR< 0 (for all values of the arguments). One can thus apply Theorem 6.3.1.2 to the operator t,and the function R, which yields proof of Theorem 6.3.1.3.

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

230

CH. 6, $3

Some applications of the maximum principle in proofs of validity of formal approximations in non-linear elliptic problems can be found in Van Harten (1975). Dorr, Parter and Shampine (1973) extensively used the maximum principle in their analysis of singular perturbations in ordinary non-linear differential operators. 6.3.2. Upper 'und lower solutions

In the special case of ordinary differential equations of second order an interesting tool for the analysis of singular gerturbation problems is provided by a theorem due to Nagumo (1937), which can be interpreted as a generalization of results obtained from the maximum principle. We discuss in this section Nagumo's theorem and its application, and conclude with some extension of the results to partial differential equations. We consider first, without special reference to singular perturbations, differential operators Lof the structure given by

+ G (x , Q , , ~ )

d2Q, LQ,= dX2

where x E [O,l], G is a continuous function on [O,l] x R2. Some essential concepts for the analysis are given by the two following definitions: Definition 6.3.2.1. Let OU(x),Q f ( x )be two continuous functions for x E [O,l], with Q,,(X)

G @Ax).

The function G(x,Q,,dQ,/dx)satisfies a Nagumo condition on [O,l], with respect to the pair Of, mu, if there exists a function $(s), s E (0,oo) such that lG(xm;! G @(lsl) for all x E [O,l], Qf G

Q,

d

a,,, /sI < 30.

Furthermore

Definition 6.3.2.2. Consider a differential equation

LQ = 0.

A function

Qf

E

C2[0,1] is a lower solution, if

La+ 2 0, x E (0,l).

A function

QU

E

C2[0,1] is an upper solution, if

CH. 6, $3

NON-LINEAR P R O B L E M S

23 1

LCDu d 0, x E (0,l). Remarks. An essential implication of the Nagumo condition is, that if G is a polynomial in dCD/dx, then the degree of the polynomial must be not higher than two.

The concept of upper and lower solution is related to the concepts of barrier functions. This becomes explicit in Nagumo’s theorem, that follows now: Theorem 6.3.2.1. Consider the boundary tjalue problem

CD(0) = A ,

CD(1) = B.

Let CDJ be a lower solution, and CDl

e a),,

a,, an upper solution, with

ole A e CD,,,

CDJ

e B d mu

and let G satisfy a Nagumo condition with respect to the pair CDJ, a,,. Then the boundary value problem has a solution CD(x), which satisfies CDJ(X)

6 @(x)d @,,(x), x

E

C0,ll.

Proof of the theorem is given in Nagumo (1937). One can also consult Jacson (1968). A generalization of the theorem has been given by Habets and Laloy (1974). Theorem 6.3.2.2. Theorem 6.3.2.1 remains true replaced by

if the boundary conditions are

and if furthermore the function CD, and CD,, are generalized lower and upper solutions, in the following sense: (i) CD,(x) is continuous, and is a piecewise C 2 function for x E [O,l], i.e. there exists a partition of [O,l] into [ti,ti+1 ] , i = 0,1, ...,n, 0 = to < t , < .** < t,+ = 1, szich that in each [ti,ti+ CDJ is twice continuously dgerentiable, with derivatives on tilti+ to be understood as right-hand, respectively left-hand derivatives. Furthermore, CD,(x) satisfies vx E (0,l) D-CDJ(x)< D+CDJ(X)

232

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

CH. 6, 53

where D-O,denotes the left-hand derivative and D+O,the right-hand derivative. (ii) Ou(x) is continuous, and is a piecewise Cz function for x E [O,l] which satisfies

vx E (0,l) D-O,(X) 2 D+O.,(X).

LO,2 0, LOud 0.

(iii) Vi, Vx E [t,,ti+

Theorems 6.3.2.1 and 6.3.2.2 can be used in the analysis of singular perturbation problems of the structure dZ# They lead to a direct proof of validity of the first term of the regular expansion in subdomains which do not contain boundary layers. We illustrate the procedure by the following simple example (Harris (1976)): Let @(A&), x E [O,l], E E ( O , E ~ ]be a solution of

O(1,c) = B.

@(O,E) = A ,

We consider the case A 2 B - 1 > 0.

Heuristic analysis leads to a regular approximation O.,(X,E) = x

+B -1

with an expected domain of validity x E [d,l], d > 0. At x = 0 there is a boundary layer, with a boundary layer variable given by

5 = XI&. We observe now that (D,=x+B-l

is a lower solution. Furthermore (D,, = x

+ B - 1 + (A-B+l)exp

[

3

(1-B)--

CH. 6, $3

NON-LINEAR PROBLEMS

233

is an upper solution. Applying Theorem 6.3.2.1 we then find that there exists a solution O ( X , Eof) the boundary problem, which is such that =x lim O(X,E)

+ B - 1.

Vx

E

d > 0.

[d,l],

E-0

It should be clear that in more advanced applications one needs a considerable insight into the expected structure of the solutions (which can be obtained from heuristic analysis) and considerable ingenuity, in order to construct suitable upper and lower solutions. First results along these lines were obtained by Bris (1954). Several interesting results are given in Habets and Laloy (1974). Systematic and extensive use of the technique is contained in the work of Howes (1976). We remark that Habet's and Laloy's generalization of Nagumo's theorem (stated here as Theorem 6.3.2.2) is of particular interest for problems in which a solution exhibits 'angular limiting behaviour', i.e. has 'free' boundary layers at internal points of the domain (see Howes (1976)). We now turn to partial differential equations. We shall state here a theorem which in spirit is related to Theorem 6.3.2.1, but requires more stringent restrictions on the structure of the equation. We consider, in a bounded domain D c R", with boundary r, the problem LO=O,

X E ~ ,

~ = e x,

~

r

where LO

=

Ci j a

2%

oxicxj +

, j ( ~ ) r

SO

i

bi(~)2Xi

+ G(x,@),

For technical reasons, the following assumptions will be made: r is smooth, aij E C"(D),bi E C""(D), 0 is a restriction to of a functions of the class Cz'"(D); G : R" x R + R is C"(D) with respect to the first argument, uniformly in the second argument, and is continuously differentiable with respect to the second argument. Finally zaij(i(j>AC(!, ij

A>0,

XED.

i

As a last preliminary, a generalization of Definition 6.3.2.2 will be given: Definition 6.3.2.3. Consider the differential equation LO A function O., E C 2 ( D )n C(D)is a lower solution if

LO., 20, XED. A function 0,E C z ( D )n C(D) is an upper solution if

LO, GO, X E D . We can now formulate

= 0,

x E D.

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

234

CH. 6,53

Theorem 6.3.2.3. Consider the boundary value problem L@=o,

x E ~ ,

@=o,

x E r

with

where L i s uniformly strongly elliptic in 6, and all data are sufficiently Holder continuous. Suppose there exists an upper solution Q U , and a lower solution 01, which satisfy

~ q ,x,E

~ ,

@l

G

o G Q ~ ,X E r .

Then the boundary value problem has a solution @(x),which satisfies @,(x)d @(x)d @,(x), x

E

6.

Proof of the theorem can be obtained by the method of monotone iteration (Sattinger (1972, 1973)),and is described in some detail in Diekman and Temme (1973). Application of the method of monotone iteration and the concepts of upperand lower-solutions to elliptic singular perturbation problems (including some cases not covered by Theorem 6.3.2.3) are given in Habets (1978), where use is made of the work of Nagumo (1954) and Amann (1976). 6.3.3. Initial value problems; Tichonov's theorem

In this section we discuss another rather special result, concerning the limit behaviour (for E 10) of singularly perturbed systems of ordinary differential equations of first order. We consider for x 2 0, the system

where v : R, -,R4, u : R, -+ RP. H and G satisfy the usual conditions for existence and unicity of solutions. We associate to the above problem, the formal limit problem: 0

= H(x,u',v,(x,u3,0),

A theorem due to Tichonov (1952) states that, under certain conditions, one has for E 1 0

CH. 6,93

NON-LINEAR PROBLEMS

u - ti = o(l),

235

v - v, = o(1)

uniformly for x E [d, q, where d is an arbitrarily small positive number, and T some positive number. A set of important conditions for the validity of Tichonov’s theorem concerns the behaviour of solutions of yet another system of equations associated to the original one. This system is given by dv* = H(X,U,V*,O). dz

~

In the above equations x and u are considered as parameters. Let x and u be restricted to some domain S, given by S

=

{XI x E [O,T,]} x

I

{u u E S, c

RP}.

Following Hoppensteadt (1966, 1967), one must require that: (i) There exists a continuous and isolated root u,(x,u) of H(x,u,v) = 0, x,u E s.

(ii) For each (x,u) E S, v,(x,u) is an asymptotically stable critical point of the equation dv* = H(x,u,v*,O). d.r

~

(iii) v,(x,u) has a domain of attraction that is uniform with respect to (x,u) E S . This statement is to be understood as follows: Let v*(z;c,,x,u) denote the solution of dv* dz

~

= H(x,u,v*,O);

[v*],=,

= uo.

There exists a number K , independent of x,u, such that, V(x,u) E S

lool

< K*

lim v*(s;v,,x,u) = v,(x,u). ,-+a

One can accomplish now the proof of Tichonov’s theorem if some further technical conditions are imposed, in particular the continuity of H and G with respect to E for E E [O,E,], uniform with respect to other variables. Imposing more stringent conditions (differentiability of H,G and 0,) stronger results can be established, i.e. the validity of zip, as approximation of u,v on any compact subset of x E (0,co)The . interested reader should consult Hoppensteadt (1966) and, of course, Tichonov (1952). The proof by Hoppenstead (1966) is based on construction of Liapunov functions and is quite different from Tichonov’s (1952) original proof. In this connection it should be noted that Tichonov used in his proof uniform behaviour according to condition (iii) given above, without

236

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

CH. 6,53

stating the condition as such. Full discussion of this matter can be found in Hoppenstaed (1967). Let us return now to the initial value problem for the pair of vector functions u(x,E),u(x,E),x E [O,T], E E ( O , E ~ ] ,stated at the beginning of this section. In the further development of the asymptotic analysis one would wish to construct approximations valid for x E [O,T]. This has been accomplished by O’Malley (1971) (see also O’Malley (1974)), following earlier work of Vasil’eva (1963). As could be expected from Tichonov’s theorem, the approximation consists of a regular expansion for x E [d,7-J1, d > 0, corrected by a boundary layer near the origin. Tichonov’s theorem, and the various methods for its proof, do not contribute much to the further development of the theory. In O’Malley (1974), the approximation is constructed as a formal approximation, the problem for the remainder term is formulated and then converted to an integral equation. Finally, a successive approximation scheme for the integral equation is shown to converge. This procedure for proving validity of the formal approximation can be considered as a special and explicit application of a.quite general method, to be described in the next section. 6.3.4. Estimates for the remainder term based on contraction mapping in a Banach space

It is quite common in non-linear analysis to use implicit function theorems or, what is closely related, methods based on the fixed point theorem for contraction mappings in a Banach space. In singular perturbations this approach was used by Berger and Fraenkel(l970) and Fife (1973,1974) in connection with certain specific problems, Berger and Fraenkel (1971) presented some more general results, and Van Harten (1975, 1978) developed the approach into a systematic procedure, which is applicable to large classes of problems. In this section we describe and discuss Van Harten’s procedure. We shall use the abstract formulation and the results of classical perturbation analysis, given in Section 6.1.1. Let Vbe a linear space, p a Banach space with norm ll*\l, 9 another linear We consider the problem space, with norm ll*/l*, and L, a mapping of Vinto 9. of determining @ E Vsuch that, for some given F E 9 L,@ = F . O n the other hand, a formal approximation Oa5E Vis given such that LEOa,= F

+ p,

pE

.F.

We formulate the problem for the remainder term R: R = @ - mas E

L&R= L,(R + (DaJ - LEO.,,=

-p .

CH. 6, 93

NON-LINEAR PROBLEMS

237

Following the method of classical perturbation analysis we define the decomposition L,R

=

A,R

+ PER.

Because we expect R to be small, we take for A , the linearization of R = 0, i.e. a linear map that is tangent to Le at R = 0. P, then satisfies

L,

at

This decomposition is uniquely defined. We assume that the inverse A,-' exists. As in Section 6.1.1, continuity conditions on A,-' and P, must be imposed. In the present context these conditions will be as follows: Continuity condition 1. Let .F and P b e normed linear spaces, with norm 11*11*, respectively 11.11. A , - ' : 9 + Psatisfies llA,-'Fli < ~ ( E ) / / F I I * . Continuity condition 2. For each pair u l , v Z E &(a), with

B(G)= { v E

G E

(O,O),where

PI IIc(( < a},

one has IlP&(Ul)- P,(v2)11* < P(W)Ilt.l -v2/I

where

is a positive continuous function, monotonic in a, and such that

~(E,G)

) 0. lim p ( ~ , o= 0-0

Condition 1 is unaltered with respect to Section 6.1.1. Condition 2 is modified ) G 4 0. One can expect that by the requirement of the limit behaviour of ~ ( E , Gas this behaviour will indeed occur in large classes of problems because of the purely non-linear structure of P,. We shall return to this matter shortly. We now formulate the main perturbation theorem:

Theorem 6.3.4.1. Consider the problem L,R = A,R

+ P E R= - p ,

RE

F

E

9.

Assume that the inverse A,-' is densely defined and can be extended to a mapping of B into a Banach space satisfying Continuity condition 1. Assume satisfying Continuity further that P, can be extended to a mapping of Pinto 9, condition 2. Let &',for each E E ( O , E ~ ]and , for some E (0,l)be the largest number from (0,6) such that i(E)P(W)

< 1-7,

f~

E (0,611

238

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

CH. 6, $3

Suppose further that p satisfies IlPll* < ?$1(4/44

Then there exists a solution R, unique in the ball B(e1), which satisjes 1

IIRll < - J.(4//Pll*.

Y

Proof of the theorem is obtained by straightforward application of Theorem 6.1.1.1. We note that the existence of 8' is assured by the behaviour of p ( w ) as specified in Continuity condition 2. Let us investigate what is needed if one wants to apply the theorem to any specific problem. The following steps can then be distinguished: Step 1. Decision on the norms / / - / / and ll-\l* to be used. Step 2. Estimate of the continuity constant i ( c ) for A; ',in these norms (where one can use the methods of part 6.2 of this chapter). Step 3. Verification of Continuity condition 2, and a good estimate of p ( w ) . Step 4. Computation of If one succeeds in all steps, and the formal approximation has been constructed such that p satisfies the condition of the theorem, then one achieves the proof of validity of Qas as approximation of CD. Van Harten (1975, 1978) has systematically treated by this procedure various classes of non-linear boundary value problems for ordinary differential equations, and elliptic differential equations of second order. Geel (1978) successfully used the procedure when studying non-linear hyperbolic problems. Geel based his analysis on a version of Van Harten's theorem in which 9 is a Banach space (compare Section 6.1.1). We now return to further discussion of the main steps of Van Harten's procedure. In applications Step 1 is crucial: the choice of norms ll*ll and 11.11* must be made such that the continuity estimates in conditions 1 and 2 can be ) tends to zero as o 1 0 and get verified. One further must show that ~ ( E , oindeed an estimate of the behaviour of p(&,cr).At this stage of the analysis a very helpful tool is a formula for P,(vl)- P,(v2), which we shall derive shortly, and which is analogous to a classical formula for the remainder term in Taylor's expansion. To establish the formula we shall need some convenient notation. Let t,v be given by

Lev = H(x,v,D'ti ,...,Drnv,&) where D', i = O,l, ...,m denote, as usual, the derivatives. We suppose that H is twice continuously differentiable with respect to v,D1v,. ..,Drnv (considered as independent variables). Let J? : 9be the mapping corresponding to H . Partial derivatives will be denoted as follows:

v+

CH.6, 53

239

NON-LINEAR PROBLEMS

2

0

BH

= -,

av

dH z1 =BD Y' etc.

We now have 1

P,vl - P,V2 =

1

di., k

1 0

di.2 &+ki[l-2{Gz 0

x ((l-l.l)Diu2

+ %,(t.,

-t ; ~ ) } ]

+ l . l D 1 ~ l ) ( D -k ~Dku2). l

The formula is an identity, and hence the order of highest derivatives occurring at the right-hand side and at the left-hand side is the same. One can derive the formula as follows: P&Vl- P&V2= 2 ( u 1 ) - P ( U 2 ) - A,(til

-Y2)

-D k v 2 )

x

((1- l . l ) D ' u 2 + ~ l D 1 ~ l ) (-Dku2). Dk~~

We can now show that, if H satisfies the differentiability requirement of existence of X k lthen , P, always satisfies Continuity condition 2 with respect to certain 'natural' norms, and that furthermore p ( w ) behaves linearly with r~ as r~ 10. This result is formulated in

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

240

Lemma 6.3.4.1. Let

CH. 6,§3

tv be given by

Lev = H(x,v,D1v,...,Dmv,&) where H is twice continuously differentiable with respect to the variables v,D'v ,...,Dmv. Consider any norm of the structure m'

1141 =

c Cs(4CDS~I0

s=o

where [*lo is the supremum norm, c,(E) > 0 are continuous functions f o r c,(E)= 0(1), m' any number satisfying

E E

(O,E~],

p<m'<m and p is the order of the highest derivative occurring in P,. T h e n f o r any pair ul,vz, llvlll Go, Ilc2ll Go, o E (0,0] [PEGl- P&VZ-J0 6 oK(E)llvl -2j211

where K(E),E E ( O , E ~ ] ,is some positive function independent of

G.

Proof. We use the formula for PEul- P,uZ. Because X k lexist as continuous functions, for llvl/l Q 0,//v211< 0,x E d one has Ixkl[E'2{ul

+ ' * l ( v l - G 2 ) } l l < czkl(E),

CJE) continuous functions for

C"kl(E)

> O.

E E (O,E~].

This yields the estimate

< c k [ (E)CDLU1 +D'v210[Dku1 Q 0,Ilv21( < G, 0 E (031, then

[P&vI -p,u2]0

k

If now Ilulll

-Dkv210'

I

2 [D'v, +D'u,], Q -0. CI(E)

We thus find [P,u, -P,v2]o

<0

c c:(E)[Dkv, - D k v J 0 m'

k=O

with

Finally, given any two sets of coefficients c ~ ( E>) 0 and c ~ ( E>) 0, one can always find K(E)such that

CH. 6, $3

NON-LINEAR PROBLEMS

24 1

c Ck*(E)[Dkr;, - - D k u , ] , 6 K(&)1 C k ( & ) [ D k U 1 - D k u 2 ] , . m’

m’

k=O

k=O

This proves the lemma.

As an explicit illustration of the procedure described in this section we shall now briefly treat the following problem, analysed in full detail in Van Harten (1978). Let @(x,E),x E [O,l], E E ( O , e O ] be a solution of L&@= E

d2@ d@ F - p(x,@)dx dx

+ q(x,@)= F(x):

@(O,E) = @(1,&)= 0.

We assume that p and q are such that, the equation d UO -p(x,Uo)dx

+ q(x,Uo) = F(x)

with boundary condition Uo(0)= 0 admits a solution Uo(x) on the whole interval x E [O,l]. We further assume p(x, a )

s 6 > 0, v u E R.

A formal approximation )@ ;: can be constructed to any order of accuracy, according to methods described in Chapter 5 (compare Example 3, Section 5.4). We assume the construction to be such that, for m = O,l, ... L,@::’ = F

+ pm,

[ P m l o = Em+1, @ I : (0,E) = a ):

(1,E) = 0.

We define the problem for the remainder R,: R,

= @ - @(m) as 3

+ q(x,Rm +a::))

- q(X,@i:’)

R,(O,E) = R,(~,E)= 0. The linearization of t a t R, = 0 is given by

=

-P m ,

A RIGOROUS THEORY OF SINGULAR PERTURBATIONS

242

CH. 6, $3

where po and qo denote the partial derivatives of the functions p and q with respect to the second variable. P, is then defined through the decomposition LRm = A,Rm

+ PERm.

Inspection of the operator A, shows that the standard conditions for applicability of the maximum principle, as given in Lemma 6.2.1.1 and 6.2.1.3, are in general not satisfied. This is caused by the strong influence of the term

in the boundary layer region. However, one can use methods based on Lemma 6.2.1.4. Combining this technique with Theorem 6.2.3.6 (which holds for a = 0, in the one dimensional case), one can establish that the solution U ( X , E ) of A,u =f,

u(O,E) = u ( ~ , E = ) 0

satisfies

where

114 = C U I O +

E[3

+&

2 [ 3 0

and 1, is independent of E. We now define the Banach space V as the space of twice continuously differentiable functions which take the value zero at the boundary, equipped with the norm ll*/l given above. Naturally, .F is the space of continuous functions equipped with the supremum norm. Next continuity properties of P , must be established. Using the formula for PEul - PEu2,and working along the lines of Lemma 6.3.4.1, one finds that, if Ilt.lll
where ko is a constant. Theorem 6.3.4.1 can now be applied. We find A

01

1-7 iko

=-

E.

The problem for the remainder term is uniquely solvable within the ball B ( 6 , ) provided that

CH. 6, S3

NON-LINEAR PROBLEMS

This condition is satisfied when m 2 1. One then has the estimate

This proves the validity of the formal approximation. The final result is

+

[Rml0

&[%lo+

c 2 [ -d2Rm ]

= 0 ( c m + ' ) for m 2 1. 0

243

CHAPTER 7

ELLIPTIC SINGULAR PERTURBATIONS This chapter is, to some extent, a survey of what has been accomplished in the domain of elliptic singular perturbations. Many investigations, scattered through the literature, have contributed to a coherent body of knowledge in this domain. We shall, however, not attempt to enumerate all results. Our aim is to show how, by combining the heuristic analysis of Chapters 4 and 5 with the methods of Chapter 6, one can indeed establish a complete and rigorous theory for certain classes of problems. We have chosen for the demonstration the elliptic singular perturbations, because the well-developed theory in this domain permits to follow in applications the methods and procedures of Chapters 4, 5 and 6 through progressively higher stages of difficulty. ' From the point of view of formal constructions, this chapter offers an extension of earlier results of Chapters 4 and 5 to more general differential operators and geometries of the domain, and more intricate boundary layer phenomena. The rigorous theory is based on Chapter 6. However, having the material of Chapter 6 at ones disposal does not mean that the proof of validity of a formal approximation always is a trivial matter. As we shall see, in some problems the proof does follow by a straightforward application of a theorem of Chapter 6, in other problems, the proof may require a considerable amount of ingenuity and labour. The literature on elliptic perturbations is quite extensive. In reviewing the literature a tribute should be paid to the pioneering work of Levinson (1950) and Visik and Lyusternik (1957), which provided much of the stimulus for the further developments. In what follows we deal with a theory in which the asymptotic order of magnitude of functions and their derivatives are measured in the norm of uniform convergence. The reader interested in the theory of elliptic singular perturbations in the framework of Hilbert spaces should consult Lions (1973), where an extensive list of references can be found. Among more recent publications in this domain we mention Huet (1976, 1977). Almost any original publication on the theory of singular perturbations features long strings of complicated formulas and intricate strings of estimates. A reasonably detailed account of the theory would give this chapter a prohib'itive length. We have chosen for a descriptive presentation, giving full details whenever possible, but restricting ourselves to the main line of reasoning when technical details become cumbersome. A consequence of this presentation 244

LINEAR OPERATORS OF SECOND ORDER

CH. 7, $1

245

is that, for any problem treated in this chapter, the relevant original publications should be consulted if one wishes to acquire a more profound understanding Qf the theory.

7.1. Linear operators of second order; elementary boundary layers

In this section we study problems which are generalizations of Examples 1 to

4 of Section 4.6. We admit equations with variable coefficients and domains with a quite arbitrary geometry. This does not introduce any fundamental difficulties. It is, in fact, an essential feature of the asymptotic method, that explicit formula for the asymptotic approximation of the solution can be derived in problems of a considerable complexity. 7.1.1. Zeroth order degenerations

Let D c R nbe an open bounded domain, with boundary problem L,O = F , in D, CP = 8, on

r. We consider the

r

where

L, = EL2

+ Lo.

For simplicity of presentation we take L2 and Lo, and the functions F and 8, to be independent of E. We thus consider

a

a2

Lz = C aij(x)+ C bi(x)--, ij axiaxj axi

ij

i

g ( x ) < go < 0, in

D.

g ( x ) is taken strictly negative, in order to be able to define without difficulty the regular expansion. We introduce now in a standard way m

EpD

=

1

n=O

ELLIPTIC SINGULAR PERTURBATIONS

246

CH. 7 , 01

n2 1. Assuming sufficient differentiability of coefficients, so that all expressions make sense, and E;@ can be defined by extension in D,we write

a =@ -Epq LEG= -

E

~

~ in~ D, L

~

~

~

,

@ = 8 - E p , on r. We now intend to construct a correction layer along r. We take r to be smooth manifold and introduce in a neighbourhood of r a new system of coordinates p,u (where u has components u l , ..., 0,- 1 ) which is such that for any point P(p,u), p = 0 implies P E r. p can be taken to be the distance measured along normals on r. We assume the transformation to the new coordinates to be one-to-one, in an &-independentneighbourhood of r. (At some distance off the normals can of course intersect.) Transformed into new coordinates the differential operator looks like

with a(p,v) > 0, y(p,u) < 0. We next introduce local variables along

r, by the formula

( = PIEV, v > 0.

One easily finds that there is one significant degeneration, which occurs for v = L2

and is given by

where cro(u) = u(O,u), yo(u) = y(0,u). Let us further assume that the coefficients cr(p,u), ui(p,u),etc. all have a power series expansion in p, convergent in some interval [O,po]. We can then write a formal series expansion for L, L, =

1P Y n n

where Y nare differential operators in terms of the variables containing E .

5,ul, ..., u,-

1,

not

LINEAR O P E R A T O R S OF S E C O N D O R D E R

C H . 7, $1

241

The correction layer can now be defined by

c

2m

EpD

=

&"'2$"(5,U).

n=O

Using methods of Chapter 4 we find = 0, YO$l

=

r: > 0;

-Y1$0,

+o = e - 40/r,

5 > 0;

$1

= 0,

5

- 92$0,> 0;

=

5 = 0;

5 = 0; $2

=

-wr, t = 0;

etc. Furthermore, imposing matching with the regular expansion for zero up to the order E ~ yields ) lim I),, 5-

= 0,

n

(which is

= 0,..., 2m.

1-

The functions $ n , n = 0,..., 2m can all be explicitly constructed (they are solutions of ordinary differential equations with constant coefficients, in which the variables ul,..., o n - occur as parameters). Furthermore, all functions I),, are uniquely defined, and decay exponentially for increasing 5. We now face a small difficulty, if we want to define by extension a function Epl@in 6, because the transformation to local coordinates is only defined in a neighbourhood of r. However, the functions

are exponentially small for p >, pl, p1 an arbitrary positive number. Without violating the accuracy of all expansions, we can define the continuations of EpJ@ to be zero for p pl, p1 > 0. This is formally accomplished by the introduction of a C ' smoothing function H ( p ) : H'p'=

o

{I

for P E [ O d , P > 0, for p >/ pl, p1 > p.

We now write '@ ::

=

ELm)@+ H(p)Ekm'@,

xE

D.

One can verify that m = O,l,,.,, is a formal approximation. Furthermore, Theorem 6.2.1.2 can directly be applied. This yields, in the supremum norm @ = )@ ::

-I- O(t,m+ 1'2),

in 6.

One thus obtains a complete rigorous theory for the problem under consideration, subject to the condition of sufficient smoothness of all data.

ELLIPTIC SINGULAR PERTURBATIONS

248

CH. 7, $1

The method and the results can further be extended to the case in which the coefficients of the operators L1 and Lo, and the functions F and 8, have power series expansions in E. 7.1.2.First order degenerations; Subdomains with ordinary boundary layers

We now consider, for an open bounded domain D c R 2 with boundary problem

L&(D= F , in D,

(D =

r, the

0, on

where

L, = EL2

+ Lo,

L2 is as in Section 7.1.1, and

a

+

Lo = - - g ( x , J2). 8x2

We take again, for simplicity, all coefficients and the functions F and 8 to be independent of E. In Example 2 of Section 4.6 we have already seen that difficulties are to be expected, even in nice convex domains, in the neighbourhood of extremal points A and B, where a characteristic of the operator Lo (i.e. a line x1 = constant) is tangent to the boundary r (see Fig. 7.1). In the present section we consider, for arbitrary domains D,subdomains such that I-, restricted to the subdomain, is nowhere tangent to a characteristic of Lo. We remark that the differential operator considered in this section, although of a somewhat special form, represents a large class of operators. To see this, consider

L&= EL2 + 2, where L2 is an elliptic operator of second order and Lo an arbitrary operator of first order. If the characteristics of Eo do not intersect in b,then one can always

x2

Ii

r+

Fig. 7.1.

LINEAR OPERATORS OF SECOND ORDER

CH. 7, 51

249

devise a transformation of coordinates such that the operator takes the form considered in this section. Let now D, be a subdomain of D defined by the restriction x1 E C T O J l l .

D: is a somewhat larger subdomain, defined by E [ T O - A , T +~ A ] , A > 0. D: is such that r, restricted to D;, is nowhere tangent to a characteristic x 1 =constant (see Fig. 7.2). We intend to construct the formal approximation in D; and prove its validity in Dr. Our only assumption on the domain D will be that the solution of the problem in D exists and is bounded (in the supremum norm) as 810. For sufficiently smooth data this is assured by Theorem 6.2.1.2. The construction of the formal approximation follows the standard procedure. On the basis of Chapter 4,we expect the boundary layer to occur along r+.We define therefore the regular expansion by m

ELrn'@=

1 E"@,(X),

n=O

Lo40 = F , L o 4 n= - L 2 4 n - 1 , n 2 1 with the boundary condition

[~y0-~,..~-

= e(X).

Next we define if) = @ - E y @ , LEG=

-Ern

-

+

Q =8-E

a=o,

X

L24,,

~ D x, E r+, ._ E

x2

Fig. 7.2.

ELLIPTIC SINGULAR PERTURBATIONS

250

CH. 7, $1

In order to construct a correction !ayer, new coordinates u,p in a neighbourhood of Ti must be introduced, such that, for any point P(p,u), p = 0 implies P E r + . In the new coordinates, the differential operator L, will look like

To define the new coordinates we can take again u measured along Ti and p along normals to I-+. We next introduce local variables along r + by , the formula P <=-

,

EY

v>o.

There is one significant degeneration, which occurs for v = l

and is given by

where cr,(u) = a(0,u);po(u) = p(0,u). It is important to observe that

> 0, po > 0. Assume finally, as in Section 7.1.1, that all coefficients a(p,u), &u), convergent power series expansions in p for p E [O,po], po > 0. We can now define the correction layer by CIO

m

EF'O =

1 E"$,,((,u). n=O

The first term satisfies L Y ~= + ~0,

rl/o = e - 40/r+, E r+.

Higher order terms satisfy y o $ n =fn($n-l,***,

$01,

$n

=

-4n/r+, x E r+

wheref,, n = 42, ..., are functions which can explicitly be computed. Furthermore, from the requirements of matching, one has lim t,hn = 0.

e-

m

etc. have

LINEAR OPERATORS OF SECOND ORDER

CH. 7 , yjl

25 1

The construction is described in full detail in Eckhaus and De Jager (1966). Although stated in general terms, the construction is such that, given any shape of the boundaries r- and I-+, all terms of the expansion can explicitly be computed. We now write )@ ::

m

m+ 1

n=O

n=O

1 E n 6 n ( X ) + H(P)2

=

En$n(t,o)

where H(p) is the smoothing function defined in Section 7.1.1. Careful analysis given in Eckhaus and De Jager (1966) shows that, under suitable differentiability conditions on the data of the problem, one has pm = O(E,+') for x in D:

LEO!$ = - p m ,

The order estimate is in the supremum norm. We turn to the proof of the validity of as an approximation of 0.The problem for the remainder term R, can be formulated as follows: R,

=@ -

as

L,R, = ppm, R, = 0

xE

5

for x

s:,

pm = 0 ( c m +'), E

I-+ and x E r-,

R, = 8, for x 1 = z1 I

R, = d l

x E D:,

+ A,

for x 1 = T~ - A.

The functions 8, and gl are of course unknown. From the fact that @ is bounded are for X E D and 0:;)is bounded for X E D : we can deduce that 8, and bounded (for &LO). We intend to demonstrate that R, = O(ernt1)in the somewhat smaller domain, defined by the restriction

4

This cannot be accomplished by a standard application of the estimates of Chapter 6. In fact, the proof is non-trivial, and is quite amusing. We shall need an auxiliary result, that can be stated as follows: Lemma 7.1.2.1. Let Dd c D be a subdomain such that the boundary restricted to d s is nowhere tangent to x 1 = const; i.e. (cf: Fig. 7.3.) D, = D n {(x1,x2) I x 1 E [ro-d,.rl + d ] } ,

d > 0,

Consider the problem

d, = 0, x E r+and x E rL,& = ~ f , x E Dd, d , = ~ , , x = T ~ + ~ ,@ = w I , x = z O - d I

r

of D,

ELLIPTIC SINGULAR PERTURBATIONS

252

CH.I , $1

Fig. 7.3.

where S,w,,w, are bounded for

E

J 0,

a

L, = EL2 - -+ g ( x ) . 8x2 L, is an elliptic operator. Then

6 = O(E) unformly for x E Ddf,where D,, is a subdomain of D,, defined by the restriction XI E

[ro-+d,z1 + $ d ] .

Proof. Let x(xl) be a non-negative C" function satisfying X(XJ =

I

0 for x1 E [z,-+d,z, ++d], 1 for x1 2 z1+ i d and x1 < zo - i d .

Let further M be a number such that Iw,I

<M,

IwjI

< M.

We consider a pair of functions 0"and CD,, given by

-CD!

= CD, =

{CE

+ MX(xl)}ekx2,

k > 0, c > 0.

It is elementary to verify that on the boundary a l l , of D, 0"2 6, Furthermore

CD, < 0.

L,@, = ( - k + g ) [ c e

+ MX(xl)ekx2+ tG(x1,x2,~)]

where G(xl,xZ,E)is uniformly bounded in 6,. It is hence possible to choose the constants k > 0 and c > 0 such that

Leou < Ef

in D,.

CH.7, $1

LINEAR OPERATORS OF SECOND ORDER

253

One then also has

Lea,2 Ef in D,. 0,and Dl are thus barrier functions, in the sense of Theorem 6.2.1.1, and it follows that

< 5 < @,

x

E

6,.

In the restricted domain X IE

we have

[ T O - i d , ? , ++d]

x = 0 and hence Du= O(E) and

Ol = O(E).

This proves the lemma. We now return to the problem for the remainder term. We apply Lemma 7.1.2.1, with d = A. This yields

R, = O(E) in a restricted domain, defined by E

[TO-~A,T, ++A].

We renormalize in the restricted domain: R, = E R ~ )

and obtain

L,Rg' = p g ) , p g ) = O(P), Rg)=O

for x E r f

and x E r - ,

Rg ) = 82) for x1 = z1 + +A, R g ) = 4') for x1 = T~

- *A

where and @z) are bounded. Hence, Lemma 7.1.2.1 can again be applied with d = +A, and we obtain RC) = O(E)

in a restricted domain defined by XI E

[TO-*A,T, +;A].

Again we renormalize, and apply Lemma 7.1.2.1.The procedure can be repeated m-times, each time in a somewhat smaller domain. The final result is

R, = O(ern+')

ELLIPTIC SINGULAR PERTURBATIONS

254

CH. 7, $1

in a restricted domain which is somewhat larger than the restriction x1 E C ~ O J l l .

In Eckhaus and De Jager (1966) the procedure of proof described above has been used to demonstrate that for problems with a smooth convex boundary, as sketched in Fig. 7.1, one has @ = Og’+O(Ern+l )

uniformly in D with the exception of arbitrary small neighbourhoods of the extremal points A and B. Using rather sophisticated barrier functions, Frankena (1968) was able to prove that, under certain auxiliary technical conditions, the first approximation (i.e. rn = 0) is valid uniformly in D. This result was further extended and improved (with respect to the estimate of the remainder term) by Mauss (1969). On the other hand, in the discussion of this section we have emphasized, that the construction of the formal approximation in a subdomain as sketched in Fig. 7.2, and the proof of the validity of the formal construction, is largely independent of what happens outside the subdomain 0;. This insensitivity of approximations in a subdomain to variation of the geometry of the problem outside the subdomain has already been anticipated in the discussion of Example 3 of Section 4.6. As a consequence, the analysis of this section also provides partial results for problems with a rather complicated geometry. Consider for example a domain as sketched in Fig. 7.4. In the subdomains D:’), 05*)and D53) one can construct the asymptotic approximation of the solution following the analysis of this section. There remains then to be studied

x2

I

Fig. 1.4.

CH. 7, $1

LINEAR OPERATORS OF SECOND ORDER

255

(a) The neighbourhoods of the points A, B and B,. One is confronted there with the problem of birth of a boundary layer, to be studied in Section 7.2.1. (b) The neighbourhood of x1 = x?). The problem is connected with the occurrence of free boundary layers, to be discussed in Section 7.2.2. We have treated in this section problems with domain D c R",for it = 2, because in this case the geometry is easily visualized. However, one can, without difficulty, extend the analysis, and obtain analogous results, for n > 2. For the operator L, of the structure

one can achieve the generalization by an analysis entirely analogous to this section, with Di defined by the restriction of the variables xl,xz, ...,x, to a suitably chosen cylinder. 7.1.3.First order degeneration continued; parabolic boundary layers The problem studied in this section is a generalization of Example 4, Section 4.6. We consider a boundary value problem as defined in Section 7.1.2,however we admit now, in the boundary r of D, segments that coincide with a characteristic x1 = const. A typical situation is sketched in Fig. 7.5. The boundary contains two characteristic segments AB and CD. We assume that the segments BC and A D are nowhere tangent to a line x1 = const. Let the segment BC be given by x2 = Y +(XA

and the segment A D by XZ

= Y -(Xl).

We assume y+(x,) > y-(x,).

Fig. 7.5.

ELLIPTIC SINGULAR PERTURBATIONS

256

CH. 7, $1

One can define now a transformation of variables which maps the domain into a rectangle. Such a transformation is given by

In what follows we assume that the transformation has been performed, and we drop the bar on the variables. We thus consider the problem

a@

LEO= EL2@- -+ gCD = F, x1 E [O,l], x2 E [OJ] ax 2 with CD prescribed along the boundary. L, is a second order elliptic operator. The construction of the formal approximation is entirely analogous to Example 4 of Section 4.6. The more general differential operator considered here brings only a slight modification in the formula for the parabolic layer. We describe briefly the procedure. First a regular expansion is constructed satisfying the boundary condition along x2 = 0. Next, one introduces = CD - ELm)@

and obtains

LE&= O(&m+ I), with zero boundary condition along x2 = 0. We consider the parabolic layer along x1 = 0. Transforming to the boundary layer variable

t = X1/&1/2 we obtain the differential operator

The degeneration is given by

where a0(x2)= a(0,x2),g0(x2)= g(O,x2). The first term of the local expansion in boundary layer variable is a function

Ubo)(&x,) which satisfies

CH. 7, $1

LINEAR OPERATORS OF SECOND ORDER

257

The equation can be reduced to the heat equation by the following transformation:

XI =

7

a,(t)dt.

0

Thus, Ubo)(c,x2)can explicitly be given in terms of the formulas derived already in Example 4 of section 4.6. One can perform an analogous construction for the parabolic layer along x 1= 1, and construct finally, by a standard procedure an ordinary boundary layer along x2 = 1. The final result is a formal approximation of the following structure:

where U'p) is the first term of the local expansion along x1 = 1, and Go the first term of the local expansion along x2 = 1. The proof of validity of the formal approximation is quite difficult, because of the occurrence of singularities at the four corner points of the domain. In Eckhaus and De Jager (1966) the singularities were suppressed by rather special techniques, and proof was given by application of the maximum principle. The estimate of the remainder term depends on the values of the coefficient p of the equation. The result, uniformly for x1 E [O,l], x 2 E [O,l], is as follows:

@=

+ R,

O ( E ' / ~if) p = 0, R = { O ( E ' ~ if~ fi) # 0.

In the case of constant coefficients Mauss (1971) improved the estimates and obtained R = { O(E) if O ( E " ~ )if

= 0,

fi # 0.

In order to pursue the analysis to higher approximations one must investigate in detail the behaviour of the solution in the neighbourhoods of the four corner points. We shall discuss these matters in Section 7.2. One can extend the analysis, without serious modifications, to problems with domains D c R",n > 2. However, for certain geometries, one finds a new and

258

ELLIPTIC SINGULAR PERTURBATIONS

CH. 7, 42

somewhat different type of parabolic boundary layer. The simplest example occurs in R3, when one considers

with the domain D a unit cube x1 E [OJ], x2 E [O,l], x3 E [O,l]. Along the four ribs which are parallel to the x,-axis one finds a parabolic layer of a threedimensional structure (which cannot occur in R’). The construction of the formal approximation is described in detail in Van Harten (1975).

7.2. Linear operators of second order continued; refined analysis of boundary layers This section is devoted to various aspects of boundary layer behaviour which cannot be described by elementary boundary layers studied in Section 7.1. Our treatment will not be very detailed, because of the inherent technical complixity of the analysis. Our aim is to describe the methods and outline the results. 7.2.1. Birth of boundary layers

We consider the problem studied in Section 7.1.2, i.e.

a@ +

LE@= EL,@- - g o

= F,

in D,

ax2

CD = 8, on

r

where L, is a second order elliptic operator independent of E , and D is an open bounded convex domain in R 2 with a smooth boundary r (see Fig. 7.6). In Section 7.1.2 we have constructed an asymptotic approximation of the solution in a subdomain not containing the neighbourhoods of the extremal

x2

I

I

r’

Fig. 7.6.

CH. 7 , $2

LINEAR OPERATORS OF SECOND ORDER CONTINUED

259

points A and B, where the boundary is tangent to a line x1 = const. We now intend to analyse these neighbourhoods. To simplify the presentation we consider a prototype problem, studied already in Example 2 of Section 4.6:

a@

&A@---=

@ = 8, on

F , in D,

2X2

r

where A is the Laplace operator

D

=

{(x,Y)Ix~+Y~

< 1).

Introducing polar coordinates x1 = (1 -p)cos u,

x2 = (1 - p ) sin u

we have

a2

1

a

1

cosva@ + sin v-a@ + ~-= F , ap I - av ~

@ = 8 for p = 0.

The points A and B of r are given by p = 0, v = n, resp. v = 0. The neighbourhoods of these points are the regions of transition from r- along which there is no boundary layer, to r +along which an ordinary boundary layer occurs. In order to analyse the neighbourhood of p = 0, v = 0, we introduce a twoparameter family of local coordinates, defined by

One finds, by a straightforward analysis, that there are two significant degenerations, given by

In Fig. 7.7 the results are summarized in a degeneration-diagram, including the significant degeneration for p = 0 (which defines the ordinary boundary layer), and the degenerations along the lines joining the points of significant degenerations. One can also show that for more general local variables, defined by

ELLIPTIC SINGULAR PERTURBATIONS

260

CH.7 , $2

Fig. 7.7.

no new significant degenerations occur. Adapting the correspondence principle (Section 4.4) one can expect: An intermediate boundary layer for v = 3, p = The corresponding boundary layer variables will be denoted by

4.

52/31

V1/3-

An internal boundary layer for v = 1, p layer variables will be denoted by 51,

=

1. The corresponding boundary

Vl*

The construction of the formal approximation can be performed in a standard manner. For the internal boundary layer we introduce a formal expansion rn

U(51,VllE)

=

c

Enbq51,Vl).

n=O

Expanding the boundary data in a Taylor series 00

e(u) =

cInun n=O

one finds

a'+,

at;

+-T---

avl

- 0,

$o = a,,

for t1 = O .

CH. 7. ~2

LINEAR OPERATORS OF S E C O N D ORDER C O N T I N U E D

26 1

Although the differential equation for $o is the full equation of the problem, the boundary value problem for $, is quite simple (it is a problem for a half-plane) and can be solved explicitly. The problems for t+hn, n > 0, can analogously be defined. For the intermediate boundary layer one can similarly define a formal approximation of the structure U*(t2/33q1/3sE)

=

i

n=O

ni3

*

@n (52/39qI/3)*

The explicit construction of all expansions is given in Grasman (1971), using matching in intermediate variables which, in the degeneration diagram, correspond to the lines joining the points of significant degenerations. A similar analysis can be repeated for the neighbourhood of the point A. Grasman (1971) combines all results into a composite formal expansion, which is free of singularities in the whole domain D.The proof of the validity of the formal approximation in d follows then by a simple application of Theorem 6.2.1.2. One can solve analogously the problem of the birth of the ordinary boundary layer for the more general differential operator L,,and an arbitrary convex domain with a smooth boundary. The details of the investigation can be found in Grasman (1971). The phenomenon of birth of an ordinary boundary layer appears thus to be characterized by the occurrence of an intermediate and an internal boundary layer. However, Van Harten (1975) has shown that, in the case of the simple prototype problem with a circular domain, considered in this section, the internal layer is contained in the intermediate layer (for sufficiently high values of p ) . This does not contradict the correspondence principle (compare the discussion in Section 4.4).It does mean that in certain problems, the analysis of the birth of the boundary layer can be somewhat simplified. We now turn to the problem studied in Section 7.1.3. In the simplest case we have

with 0 prescribed along the boundary of the rectangular domain x1 E [O,l], x2 E [O,l] (see Fig. 7.8). Along the segment AD there is no boundary layer, while along AB and DC, parabolic boundary layers occur. The singularities at the points A and D which arise in the formal approximation by the parabolic layer, are again due to the phenomenon of birth of the layer. In Grasman (1971) the problem has been studied by the method outlined in this section. For the simple prototype problem stated above, the birth of the

262

ELLIPTIC SINGULAR PERTURBATIONS

CH. 7, $2

Fig. 7.8.

parabolic layer appears to be characterized by an internal layer. Taking the internal layer into account Grasman (1971) has constructed a formal approximation which is free of singularities at points A and D. The result confirms earlier result of Grasman (1968), where the parabolic layer was studied in a quarter infinite region, and an asymptotic approximation was derived from the explicit solution. We further mention in this context Cook and Ludford (1973), who studied the prototype differential equation on a semi-infinite strip and analysed the asymptotic approximations from exact representation of the solution, using Fourier analysis. For the more general differential operator, the problem of the birth of a parabolic boundary layer has not been studied explicitly in detail, at the present date. A brief analysis of degenerations in that case can be found in Eckhaus (1972). We finally mention an extensive analysis of the parabolic layers (and their birth) in problems with almost characteristic boundaries, given in Grasman (1974). 7.2.2.Free layers and other non-uniformities

An important class of boundary layer phenomena, which has received some attention in the literature, is the occurrence of free boundary layers. For problems of the type studied in Sections 7.1.2 and 7.1.3, the reasons why a free layer may occur are of two kinds: Firstly, the data of the problem may not possess sufficient regularity for the construction of the approximation as described in Sections 7.1.2 and 7.1.3. For example consider the geometry as given in Fig. 7.6. If the curve defining r-, or the function defining the values of 0 on r-, do not have sufficient differentiability properties at some point x1 = x:, then the construction of the regular approximation fails at x1 = x:. This leads to the conclusion that the results obtained in Section 7.1.2 hold in subdomains not containing a neighbourhood of x1 = x:, and that along the line x1 = x: a new boundary layer phenomenon is to be expected.

LINEAR OPERATORS OF SECOND ORDER CONTINUED

CH. 7, $2

x21

x2

263

B"

B'

Fig. 7.9a.

Fig. 7.9b.

A second reason for the occurrence of free layers is non-convexity of the domain. In a situation sketched in the Fig. 7.9a, any attempt to construct an asymptotic approximation for the solution of the boundary value problem for the prototype equation

will lead to the conclusion that, along the line CD, a free layer must occur. A similar conclusion holds for the geometry sketched in Fig. 7.9b, where the free layer along CD is, in some sense, a continuation of the usual parabolic layer along BC. The heuristic construction of the formal approximations for free layers can be performed along the line of reasoning that we have used in most of our preceding analysis, and will not be reproduced here. For explicit results the reader is referred to the work of Mauss (1969, 1971). Free layers have also been studied in Cook and Ludford (1971), for the prototype equation on an infinite strip, using Fourier analysis on the exact representation of the solution. A quite different type of free layers occurs in problems with turning point behaviour, i.e. when the differential operator L, = ELZ

+ Lo

is such that the characteristics of Lo intersect in D. Some results for such problems have been obtained by De Jager (1972) and by Barton (1976). For the case that all characteristics of Lo intersect in one point, an extensive analysis has been given by De Groen (1976). Returning to problem studied in Sections 7.1.2, 7.1.3 (i.e. when Lo has nonintersecting characteristics) we remark that non-uniformities of a type different from free layers can occur at points of the 'upper' boundary. For example, in Fig.

264

ELLIPTIC SINGULAR PERTURBATIONS

CH. 7, $3

7.9a, localized non-uniformities are to be expected in the neighbourhoods of the points of the upper boundary AB' where the data of the problem do not possess sufficient differentiability properties, and furthermore in the neighbourhood of the point D where the free layer along CD interacts with the ordinary layer on AB'. Similarly, in Fig. 7.9b, interaction of parabolic layers and ordinary boundary layer occurs at points A ' and B". For a prototype problem, an extensive explicit study of these phenomena has been given in Cook and Ludford (1971, 1973).

7.3. Non-linear operators of second order Problems of singular perturbations for quasi-linear elliptic differential equations of second order have been investigated by Berger and Fraenkel(1970), Fife (1973, 1974) and Van Harten (1975, 1978). In this section we shall consider some relatively simple, but representative problems, and show how the heuristic analysis of Chapter 5, combined with methods of Chapter 6, leads to a complete theory. More general results can be found in the references quoted above, in particular in Van Harten (1975, 1978). 7.3.1. Zeroth order degenerations

We consider a slight generalization of the problem studied in Example 1, Section 5.7. D is an open bounded domain in R", with a smooth boundary r. CD(x,&),x E D, E E ( O , E ~ ]satisfies ,

LeCD = &L2@+ Lo@= F , in D,

CD = 8, on

where

LOO = -g(x,CD),

For simplicity we assume all coefficients, and the functions F and 0, to be infinitely differentiable, F and 8 independent of E. A regular expansion is defined by m

Ep@=

C n=O

&"I$,(X).

NON-LINEAR OPERATORS OF SECOND ORDER

CH. I , $3

265

The function +o(x) satisfies the non-linear equation g(x,+o) = - F .

The functions +"(x), n > 0 are solutions of linear equations (compare Section 5.2). The regular expansion will in general not satisfy the boundary condition for on I-. We now write (f) = Q,

- EWQ,

and obtain &L2(f) - g(x,ff),~) =fm(x,e), in D,

ii, = 8 -

on

r

where

f,

=

O ( E , + ~ ) in ,

$(X,ii,,E)

= g(x,ii,

D,

+ Eirn'Q,)- g(x,ELrn'O,).

g satisfies

as

-->,d>O. aii,

In order to construct the correction layer we introduce, as in Section 7.1.1, in a neighbourhood of r a new system of coordinates p,u (where u has components u 1 ,...,u,which is such that for any point P(p,u), p = 0 implies P E I-. We assume the transformation to the new coordinates to be one-to-one in an Eindependent neighbourhood of r. The differential equation takes now the form

Next, we introduce the transformation to local variables

and find, in the usual way, that there is one significant degeneration, which occurs for

S(&)= E l i 2 and is given by

266

ELLIPTIC SINGULAR PERTURBATIONS

Lz0U = a(O,u),

a2u

dP

CH. 7 , $3

- g(o,u,U,o).

The analysis of the boundary layer is further completely analogous to Example 1 of Section 5.7. A local expansion is defined by 2m

Er)G=

C

E"/~$,((,U).

n=O

The function b,t0 satisfies

where

A solution that tends to zero (exponentially) for

r+

00, is

implicitly given by

Higher order terms of the local expansion satisfy

$, = 4Jl- for I),,

=0

5 = 0, n even, for 5 = 0, n odd.

The Frechet derivative g'(ty,b0)is defined by

The linear non-homogeneous equations for $,, n = 1,2,. ..,can be fully solved, in terms of $o, starting with the observation that a$o/a( is a solution of the homogeneous equation, i.e.

It is furthermore not too difficult to verify that there exist solutions for $, n = 1,2,... which tend to zero as <+a. We now write

CH. 7 , k3

NON-LINEAR OPERATORS OF SECOND ORDER

0 = )@ ::

267

+ R,

where = .E;w

+ H(p)Ey%.

H ( p ) is the usual smoothing function, introduced in Section 7.1.1. The construction of )@ :: is such that for m = O,l, ..., the function formal approximation in the following sense:

Lc@iT)= O(crn+li2),in

)@ ;:

is a

D.

For the problem considered in this section, the proof of the validity of @!$ as an approximation of @ can be obtained from the maximum principle. In fact, Theorem 6.3.1.3 is directly applicable, and yields @ - )@ ::

= O(em+ l/,),

in 6.

The estimate is in the supremum norm. We recall from Section 6.3.1. that, in the non-linear case, the maximum principle does not imply uniqueness and existence of solutions of the boundary value problem. A theory which guarantees uniqueness and existence of solution can be developed, for the problem studied in this section, using the method based on contraction mapping in a Banach space, described in Section 6.3.4. ~ also a more general form of the This has been done in Van Harten ( 1 9 7 where operator L,, was admitted. We further mention the work of Berger and Fraenkel (1970), who studied a rather particular problem in which the condition Cg/’L@2 d > 0 is not satisfied.

7.3.2. First order degenerution

We consider a generalization of Example 2, Section 5.7. D is a bounded convex domain in R 2 , with a smooth boundary r. @ satisfies

@ = 0,

on

r.

As in Section 7.3.1, L , is a linear elliptic operator of second order. The coefficients of the highest order derivatives satisfy

1 uij(x)titj2 A 1 ,?;, ij

We consider the case p(x,@,) 2 p o > 0.

I

A > 0, x E 6.

ELLIPTIC SINGULAR PERTURBATIONS

268

x2

CH. 7, $3

I

Fig. 7.10.

We further assume that the problem

4o = 0 for x2 E rhas a unique C" solution in any subdomain of D not containing neighbourhoods of the points A and B (see Fig. 7.10). With these hypothesis one can construct a formal regular expansion satisfying the boundary condition for @ along r-, and furthermore a boundary layer along r+.The analysis parallels Example 2 of Section 5.7, using further the techniques of Section 7.1.2. The exercise is left to the reader (who may also consult Van Harten (1975, 1978)). The formal expansion thus obtained contains singularities at points A and B, reflecting the problem of birth of the boundary layer. We shall show that one can solve the problem for the neighbourhoods of the points A and B in exactly the same way as in the linear case. For the simplicity of the presentation we take r again to be a circle and L , the Laplace operator. Introducing polar coordinates x 1 = (1 - p ) cos u,

x2 = (1 - p ) sin u

we obtain 1

a@

1

coSua@

- p [ ( 1 - p ) cos u,( 1 - p ) sin u,@] sin u I + ~up 1-p a u

+ g [ ( 1 - p ) sin u,@] = F , @ = O(u) for p = 0.

CH. 7 , $3

NON-LINEAR OPERATORS OF S E C O N D ORDER

269

We now analyse the neighbourhood of p = 0, u = 0, by introducing the local coordinates

Assuming O(0) # 0, and hence CD = 0(1) in the local domain, we investigate the degenerations of the operator with respect to functions that are O(1). One easily establishes that, as in the linear case, there are two significant degenerations, given by

where = P(l,O,U).

The problem of constructing the corresponding local approximations looks rather intractable, because of the non-linear structure of the equations. However, by a suitable a priori choice of the structure of the expansions a surprising simplification of the problem can be achieved. Consider a formal approximation for the intermediate layer P

U,*(t2/3,71/3,E)

=

c

*

ni3 $n ( < 2 / 3 3 v 1 / 3 )

n=O

where tZl3, q1,3 are the local variables corresponding to v = 3, resp. p = 3. Let the function defining the boundary value of @ be expanded in a Taylor series

n=O

We impose on the terms of the intermediate layer the boundary condition

*0*(0,11113)

= ao,

A solution for $: is given by

*o*

= ao.

270

CH. I, 64

ELLIPTIC SINGULAR PERTURBATIONS

Starting with this solution we find, for $,*, n > 0, the linear inhomogeneous problems

*,*(O,v1,3)

= 47v;,3>

n > 0.

Van Harten (1975, 1978) has shown that one can define (and construct) solutions $,*, n = 1,2,... such that U;(t2!3,vl,3,c)is free of singularities in the domain 5213 E [O,T],~ 1 E ~[ -3 771, V T > 0, Furthermore U: matches properly with the regular expansion, and with the ordinary boundary layer, for sufficiently large p. In other words, the construction of the internal layer, corresponding to v = p = 1, is not necessary to remove the singularities at the point B, provided that the intermediate layer U ; has suitably been defined. Naturally in the neighbourhood of the point A, a similar analysis can be performed. Combining all results one can define a formal approximation 0:;) in D,such that

L,(Piy) = O(2) in D,

@:y) = 0 on r.

s is positive if all expansions have been pursued to a sufficiently high order. The proof of validity of (P;i) as an approximation of @, following the method described in Section 6.3.4, can be found in Van Harten (1978), which also contains various generalizations of the problem considered in this section. 7.4. Linear operators of higher order Singular perturbations of Dirichlet problems for higher order elliptic equations have been first studied in Visik and Lyusternik (1957). Some results in the setting of Hilbert spaces are contained in Lions (1973). A systematic analysis including proofs of validity in Holder norms, has been given by Besjes (1975), on whose work this section is based. 7.4.1. Elliptic degenerations

Let D c R" be bounded, with a smooth boundary following problem: L,@ = E L , ~ ( + P L2k@= F

as@

-=

ans

O,,

r. We

consider the

in D,

s = 0,1, ...,m-1 on

where a/& denotes the normal derivative on r. L Z mand L,, are uniformly strongly elliptic operators, of the order 2m, respectively 2k, with m > k. For

CH. 7, $4

LINEAR OPERATORS OF HIGHER ORDER

271

simplicity, all coefficients, and the functions F , O,, are assumed infinitely differentiable and independent of 8. We expect the approximation of 0 to consist of a regular expansion and a boundary layer along r. However, it is not clear at this stage what will be the structure of the expansions, and what boundary conditions must be imposed on various terms of the expansions. We therefore investigate first briefly the structure of the boundary layer. Let p,u be a new system of coordinates, with p measured along normals to r, such that p = 0 defines r. The transformation to new coordinates is one-to-one in some &-independentneighbourhood of r. In the new coordinates, the differential operator will look as follows:

In the above formula only the highest derivatives of L2, and L2k with respect to p have been retained. Introducing local coordinates by

5

=P

/W,

@) = o(l),

one finds one significant degeneration, which occurs for 6 = &1/(2(m-k)) and is given by

We assume, and this fundamental for the construction, that the differential operator L, is such that ( - l)"a, (0,v) >0

in 6,

( - l)kao(O,u)> O

in 6.

The assumption has the following consequences: Consider solutions of the homogeneous equation

Y o u = 0. Y ois an ordinary differential operator in 5, in which the variables u occur as parameters. The characteristic equation associated with 9,reads 3,2k(a,(O,u)3,2'"-k'

+ a,(O,u)} = 0.

The roots J. cannot be purely imaginary. This can be seen by writing 3, = i1 and using the conditions imposed on a, and a,. Write next A2 = Q. There are m - k non-zero different roots for Q, none of which is negative real. It follows that for 3,

ELLIPTIC SINGULAR PERTURBATIONS

212

CH. I , $4

there are exactly m - k roots with negative real part. The final conclusion is that there are exactly m - k linearly independent solutions of z 0 u = 0 which decay exponentially for 5 -,00. We now undertake the construction of the formal expansion. In order to avoid fractional powers of E we introduce a new small parameter by = &1/(2(m-Wa

Furthermore, we write a formal expansion of the differential operator in the local variable

where Y pare operators in the variable 5, u, independent of E . Omitting here any heuristic motivation we introduce, following Besjes, the expansion M

@(xi&)=

N

C P'+~<x)+ H ( p )j C ~ j + ~ $ j ( S , u+ ) RMN j=O = 0

where H ( p ) is the usual smoothing function, which is zero outside a strip near the boundary. The differential equations for the functions occurring in the expansions are derived in the familiar way, and one obtains L2k40

= F,

L2k4j

= 0, j = 1,...,2 m - 2 k - 1 ,

L2k4j

= - L2m40, j = 2m - 2k,

etc., Y o * o = 0,

,xY i $ j - l , j

Yo$j= -

j = 1,....

1=1

We next investigate the boundary conditions. Noting that on the boundary

we obtain, for s = 0,1, ...,m - 1

The relations are decomposed into two blocks: For s = O,l, ...,k - 1 we impose

CH.7, $4

LINEAR OPERATORS OF HIGHER ORDER

l

o

213

for j + s - k < 0.

For s = k , . . .,m - 1 we impose

for k - s + j > M . We note that the boundary conditions for the functions q 5 j , j = 0, ... are natural Dirichlet conditions for the differential equations defining these functions. Concerning the function t,bj, j = 0,... we remark that for each function m - k initial conditions are imposed. From the properties of the linearly independent solutions of the equation Y o u = 0, discussed earlier in this section, one can deduce that each function t,bj, j = O,l, ,.., is uniquely defined, if we require that t,bj vanishes as 5 + co. The whole system is solvable in the following way: One first determines q50. This provides the initial conditions for t,bo, which in turn defines the boundary conditions for 4 1 ,etc. We now turn to the problem for the remainder term R M N . Careful analysis shows (Besjes (1975)), that when choosing N properly, i.e.

+

N = max (M k,M + m - k - 1) one has L,RM, = F M ,

in D,

where

The definition of norms used above has been given in Section 6.2.3. The proof of validity of the formal approximation can now be achieved in two steps:

ELLIPTIC SINGULAR PERTURBATIONS

274

CH. 7, $4

One first defines a suitably smooth function U ( x ) x E d which satisfy the . one defines boundary conditions imposed on R M NNext, -

R M N= R M N- U

and applies the theory of Section 6.2.3. The final result is [ R M N I<j c ,uMM+'-j,

O<j<M

with [ * I j defined in Section 6.2.3. We conclude with a simple explicit example of the construction: Let D c R2 be a domain bounded by the unit circle. @(x,E) is solution of

a@

@ = O,,

&A2@- A @ = F , in D;

- = O,,

an

on

r

where A is the Laplace operator, and A2 the biharmonic operator. The first term of the regular expansion +,(x) satisfies

~ 4 =, - F ,

X E D ,

4,

=

e,,

XE

r.

The boundary layer variable is given by x 1 = (1 - p ) cos u,

x2 = (1 - p ) sin u,

The first term of the local expansion is J&$O(S,U).

The function $,((,u)

satisfies

There is a unique solution that tends to zero for $0(5,u)

= -Bl(u)e-r

where

The next term of the regular expansion is JE41W.

5 -+ co,and is given by

CH. 1, $4

LINEAR O P E R A T O R S OF H I G H E R O R D E R

215

The function @l(x)satisfies

=o,

XED,

=

-g1,

xEr.

The solution provides an initial condition for the next term of the local expansion, etc. 7.4.2.First order degenerations

In order to be able to visualize the geometry, we consider D c R2. @(x,E) satisfies EL,,@ -

a@ + g@ = F ,

in D,

0x2

We shall start with a situation as sketched in Fig. 7.11a. It will not be surprising that in the vicinity of extremal points A and B, where r is tangent to a characteristic x1 = const., difficulties will arise. We shall then deal, in analogy to Section 7.1.2 with subdomains as sketched in Fig 7.11b. We commence with a brief analysis of the boundary layer structure. As usual, we introduce new coordinates p,v in a strip along the boundary. In the transformed coordinates the differential operator looks as follows:

To define the situation we take (1 -)"a(p,v) > 0, p(p,v)

> 0 on I-+,

p(p,v) < 0

on

Fig. 7.11a.

r-.

Fig. 7.11b.

ELLIPTIC SINGULAR PERTURBATIONS

216

CH. 7 , $4

Introducing the local variables

4 =P / W ,

= O(1)

we find (excluding points A and B) one significant degeneration, which occurs for 6(&)= & 1 / ( 2 m - 1 ) and is given by

zzm

3 0

= ao(u)@&

a

- PO(4 -

at

where a0(u) = r(O,u), po(u) = p(0,u). We investigate the solutions of the homogeneous equation LzOU = 0.

The associated characteristic polynomial is given by c r o ( u ) P - po(u)E, = 0.

Elementary analysis shows that there are m roots with negative real parts when po > 0 and m - 1 roots with negative real parts when po < 0. Hence, for the description of the boundary layer along r- there are m- 1 linearly independent solutions which decay (exponentially) for 5 -i co,while there are m such solutions along I?+. We now undertake to construct the expansion. To avoid fractional powers we introduce a new small parameter = &1/(2m- 1)

Next, we write @(X,&)

=

M

N-

j=O

j=O

c vJ’q5j(x)+ H ( p ) c vJ+’$,:(4,u)

+@

where H ( p ) is the usual smoothing function, which is zero outside a strip along r-. The differential equations for the functions d j and $:, are derived in the standard way,’ and one has

The boundary conditions along r - are distributed by a procedure analogous to Section 7.4.1. We thus require

LINEAR OPERATORS OF HIGHER ORDER

CH. 7, $4

277

340 e; = el - 7/r-. on

Furthermore, $ ; ( ( , v ) must tend to zero as (+a. In view of the properties of the independent solutions of y o u = 0, the function $0 is then uniquely defined. In the next step we require = -t,b;(O,v)

for

X E r ,

for s = 1,

5 = 0,

for s = 2, ( = 0, for 2 < s

< m-1,

( = 0.

Again, $; must tend to zero as (-+a. The procedure can be pursued in this way to arbitrarily high terms of the expansions. In the final step we construct the boundary layer along r+.We thus write N +

@ = H(p)

1

v’$;((,t.)

+ R,

j=O

where H ( p ) is a smoothing function, which is zero outside a strip along yo*;

=o,

The boundary conditions along

r +are satisfied in the following way:

l<s<m-l, etc.

t=O,

r+;

ELLIPTIC SINGULAR PERTURBATIONS

278

CH. 7, 94

I);,

Furthermore, j = O,l,,., are required to vanish for 5 + co. We turn now to the problem for the remainder term R,: L,R,

F,,

=

x

E

D,

In a configuration as sketched in Fig. 7.11a, singularities of F , occur at the points A and B. We therefore restrict ourselves to subdomains as sketched in ~ A > 0, D, is Fig. 7.11b. 0: is defined by the restriction x 1 E [ T ~ - A , T+A], ] . restriction of the boundary somewhat smaller subdomain with x1 E [ T ~ , T ~ The r to D: is nowhere tangent to a line x1 = const. If in the construction of the expansions N + and N - are suitably chosen, then lFMll < c!JM+l-I in D:, [GO,M]O < CV, Gs,M= 0,

on

r'

and

s = 1,...,m- 1,

r-, on

r +and r-

One would wish now to establish a result analogous to Section 7.1.2, i.e. to demonstrate that R , is small in 0;. In Section 7.1.2 this has been done with the aid of barrier functions, derived on the basis of the maximum principle. However, for the problem studied here, there is no maximum principle and seemingly no tools to perform the analysis. The difficulties have nevertheless been overcome by Besjes (1975). The analysis of Besjes is quite involved, and cannot be described here in any detail. We shall only briefly indicate the main line of reasoning. The problem for the remainder term can of course be reduced to a problem with homogeneous boundary conditions along r +and r- (restricted to 0;).We therefore consider

Let

i(xl)be a C" function such that 1 inR',

O<<<

<=1

for x1 E [ T ~ , T , ] ,

i= 0 for x 1 > T~ Introduce next

+ A and x 1 < T~

- A.

CH. 7, 54

LINEAR OPERATORS OF HIGHER ORDER

219

R = (R. Then

aR

+

L,k? = EL,,,$ - - g R 8x2

=

[ F i- EL*R, x E D:,

where 8D;is the full boundary of 0;. The problem for R" is hence a Dirichlet problem of the type studied in Section 6.2.3, except of course that the term L*R is unknown. However, L* is a differential operator of the order 2m - 1, and hence of lower order than L,. Besjes (1975) analyses the problem for l? by methods analogous to Section 6.2.3, with judicious use of the interpolation inequalities between norms. In a final stage one needs an estimate for the quantity

{ IRI'dx. D;

This is accomplished by a method suggested in Visik and Lynsternik (1957). Proceeding along these lines, Besjes (1975) establishes that for the problem considered in this section, the remainder term satisfie$

[ R M ]6~'-+'VC

,

0 <j

<M,

in D,.

with [*Ij defined in Section 6.2.3. One thus obtains, for higher order elliptic problems, a theory analogous to Section 7.1.2. The theory is further an improvement of Section 7.1.2, because estimates of the derivatives of the remainder term are also obtained.

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A. ERDELYI Asymptotic expansions, Dover Publications, New York (1956).

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SUBJECT INDEX Asymptotic-approximation, 11, 12, 15 -expansion, 13 -series, 5, 7, 13 -sequence, 5, 13 Banach’s Theorem, 190 Barrier functions, 203, 228 Boundary layer 22 birth of-, 122, 258, 261 free-, 176, 262 ordinary-, 118, 182, 248 parabolic-, 128, 255 -variable, 41 Composite expansions, 38, 56, 59, 60 Correction layers, 38, 56 Correspondence Principle, 99, 122, 164 Degenerations of operators, 78, 79, 147 Du Bois-Reymond Theorem, 6 Expansion operators, 33-37 Extension Theorems 24-28 applications of-, 43, 45 Formal -asymptotic expansion, 129, 130 -approximations, 129, 185-187, 198-200 -limit problem, 195 Gauge functions, 19 Generalized expansions, 54, 170, 174, 175 Holder norms,

215

Inner -expansion, 22 -variable, 29 Intermediate variable, 38 Local -approximation, 32, 33 -expansion, 32, 33, 36 -limit, 30, 32

-limit function, 30, 32 -variable, 29, 32 Matching 38 asymptotic-principle, 38, 56, 59, 60, 6 1-64 -in interm. variables, 50, 55 Maximum Principle, 201-203, 226227, 230 Multiple Scales, 139-144 Nagumo’s -condition, 230 -Theorem, 23 1 Order symbols, 1, 2 Order Functions, 3 Orders of magnitude, 8, 9 Outer expansion, 22 Overlap Hypothesis, 38, 46, 54 Perturbation classical-analysis, 189 -Theorems, 192, 194, 237-238 regular-problem, 195 singular-problem, 197, 198 Regular -approximation, 16, 18 -expansion, 16, 36 -expansion operator, 34 Regularizing layer, 38, 59 Significant approximation, 38, 41 Significant degeneration, 80, 163 Stretched variable. 29 Tichonov’s Theorem, 234-236 Truncation of expansions, 35 Turning Point problems, 82, 205, 206 Upper and lower solutions, 230, 233 WKB approximations, 133-139 287

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