Asymptotic Analysis of Singular Perturbations
STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUME 9
Editors: J. L. LI...
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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.
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)*( 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 @*( 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 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
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 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- 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
+
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 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
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~( 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 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( 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 $( ) 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( 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( 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
$hl),
n > 0 can be derived from
9oE!16 = -EK' LZPE!F16 where in the present case d2 d Y P= [ U 2 ( l -&tl)- U 2 ( 1 ) ] 7 - & U , ( l -&tl)di"1 dtl
For n
=0
+ &2Uo(l
-E 0,
then an entirely analogous analysis can be performed. In that case the regular expansion is valid on x E [d,l], Vd>O, and a boundary layer at x=O occurs.
Example 3. Consider, as in Example 2, for x E [O,l] the differential equation L,Q
=F
with L,
= EL1
+ Lo,
We shall now study the initial value problem defined by the initial conditions @(O,E)
= a,
&)x=o
= p*
We exclude turning-points by assuming b,(x) # 0, and impose furthermore the condition b,(x)/az(x)
=- 0,
vx E [ O J l
This condition is essential for our analysis, as will be shown at the end of this section. Assuming the existence of a regular expansion m
ErQ
=
C n=O
we find again
Lo40
= F,
E’$,(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
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
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(, 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
CH. 6, 92
ESTIMATES F O R LINEAR P R O B L E M S
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 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 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, 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, $1
In order to construct a correction !ayer, new coordinates u,p in a neighbourhood of Ti must be introduced, such that, for any point P(p,u), p = 0 implies P E r + . In the new coordinates, the differential operator L, will look like
To define the new coordinates we can take again u measured along Ti and p along normals to I-+. We next introduce local variables along r + by , the formula P o.
There is one significant degeneration, which occurs for v = l
and is given by
where cr,(u) = a(0,u);po(u) = p(0,u). It is important to observe that
> 0, po > 0. Assume finally, as in Section 7.1.1, that all coefficients a(p,u), &u), convergent power series expansions in p for p E [O,po], po > 0. We can now define the correction layer by CIO
m
EF'O =
1 E"$,,((,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 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