NORTH-HOLLAND
MATHEMATICS STUDIES Editor: Leopoldo NACHBIN
Non-Linear Partial Differential Equations An Algebraic View of Generalized Solutions
E.E. ROSINGER
NORTH-HOLLAND
NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS A N ALGEBRAIC VIEW OF GENERALIZED SOLUTIONS
NORTH-HOLLAND MATHEMATICS STUDIES 164 (Continuation of the Notas de Matematica)
Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University of Rochester New York, U.S.A.
NORTH-HOLLAND - AMSTERDAM
' NEW YORK OXFORD TOKYO
NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS AN ALGEBRAIC VIEW OF GENERALIZED SOLUTIONS Elemer E. ROSINGER Department of Mathematics University of Pretoria Pretoria, South Africa
1990 NORTH-HOLLAND - AMSTERDAM
NEW YORK
OXFORD TOKYO
ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 21 1,1000 AE Amsterdam, The Netherlands Distributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, N.Y. 10010, U.S.A.
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R o s i n g e r . E l e m e r E. Non-llnear partial d i f f e r e n t i a l e q u a t i o n s : a n a l g e b r a i c view of generalized solutions 1 Elemer E . Rosinger. p. cm. -- (North-Holland m a t h e m a t i c s s t u d i e s ; 164) Includes btbliographical references. I S B N 0-444-88700-8 1. D i f f e r e n t i a l e q u a t i o n s . P a r t i a l . 2. Differential equations. I. Title. 11. Series. Nonlinear. OA377. R 6 8 1990 515'.353--dc20 90-47848
CIP ISBN: 0 444 88700 8
O ELSEVIER SCIENCE PUBLISHERS B.V., 1990 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V. IPhysical Sciences and Engineering Division, P.O. Box 103, 1000 AC Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed by the publisher for any injury andlor damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Printed in the Netherlands
DEDICATED TO MY DAUGHTER MYRA- SHARON
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A massive transition of interest from solving linear partial differential equation to solving nonlinear ones has taken place during the last two or three decades. The availability of better digital computers often made numerical experimentations progress faster than the theoretical understanding of nonlinear partial differential equations. The three most important nonlinear phenomena observed so far both experimentally and numerically, and studied theoretically in connection with such equations have been the solitons, shock waves and turbulence or chaotical processes. In many ways, these phenomena have presented increasin diff iculties in the mentioned order. In particular, the latter two pf;enomena necessarily lead to n o n c l a s s i c a l or generalized solutions for nonlinear partial differential equations. While L. Schwartz's 1950 linear theory of distributions or generalized functions has proved to be of significant value in the theoretical understanding of linear, especially constant coefficient partial differential equations, sufficiently general and comprehensive nonlinear theories of eneralized functions which may conveniently handle shock waves or turbuyence have been late to appear. Curiously, the insufficiency of L. Schwartz's linear theory and therefore the need for going beyond it was pointed out quite earl . Indeed, in 1957, H. Lewy showed that most simple linear, variable coef icient, first order partial differential equations cannot have solutions within the L. Schwartz distributions. Unfortunately, that early warning has been disregarded for quite a while. One of the more important reasons for that seems to be the misunderstanding of L. Schwarz's so called impossibility result of 1954, which has often been wrongly interpreted as proving that no convenient nonlinear theory of generalized functions could be possible, Hormander [I].
!
Nevertheless, various ad-hoc weak solution methods have been used in order to obtain nonclassical, generalized solutions for certain classes of nonlinear partial differential equations, such as for instance presented in Lions [1,2], without however developing any systematic and wide ranging non 1 i n e a r theory of generalized functions. As a consequence, the attempts in extending weak solution methods from the linear to the nonlinear case, have often overlooked essentially nonlinear phenomena. In this way, the resulting weak solution methods used in the case of nonlinear partial differential equations proved to be insufficiently founded. Indeed, in the case of weak solutions obtained by various compactness arguments for instance, one remains open to nonlinear stability paradoxes, such as the existence of lueak and s t r o n g solutions for the nonlinear system u= 0 u2 = 1
E. E. Ros inger
which would of course mean that we have somehow managed t o prove the equal it y 02 = 1 within the real numbers Appendix 6.
R,
see f o r details Chapter 1 , Section 8 and
Lately, there appears t o be an awareness about the fact that nonlinear operat ions - such as those involved in nonlinear partial differential equations - often f a i l t o be weakly continuous, Dacorogna. As a consequence, certain particular and limited solution method have been developed, such as for instance those based on compensated compactness and the Young measure associated with weakly convergent sequences of functions subjected t o differential constraints on algebraic manifolds, Ball, Murat, Tartar, D i Perna, Rauch k Reed, Slemrod. The fact remains however that with these methods only special, f o r instance conservation type nonlinear partial differential equations can be dealt with, since the basic philosophy of these methods i s t o get around the nonlinear failures of weak convergence by imposing further restrictions both on the nonlinearities and the weakly convergent sequences considered, Dacorogna. Theref ore, with such methods there i s no attempt t o develop a comprehensive nonlinear theory of generalized functions, which may be capable of handling l a r e enough classes of nonlinear partial differential equations. These metho s only t r y t o avoid the difficulties by part icvlarizing the problems considered. Thus very l i t t l e is done in order t o better understand the deeper nature of these difficulties, nature which, as shown in t h i s volume, see also Rosinger [1,2,3], i s rather algebraic then topological.
f
The conceptual difficulties which so often a r i s e when trying t o extend linear methods t o essentially nonlinear situations cannot and should not be overlooked or disregarded. However, as the above kind of nonlinear stabil i t y paradoxes show it, the transition from linear t o nonlinear methods is not always done i n a proper way. For a better glimpse into some of the more important such transit ions, one can consult for instance the excellent historical survey i n Zabuski. For the sake of completeness, and in order t o further stress the c r i t i c a l importance of the care for rigour when trying t o extend linear ideas and methods t o essentially nonlinear situations, one should perhaps mention as well the following. The transition of methods and concepts from linear t o nonlinear partial differential equations has in fact produced two sets of paradoxes. The one above i s connected with exact solutions. A second one concerns the numerical convergence paradox implied by the Lax equivalence result, and it i s presented in detail in Rosinger [4,5,6]. Since the l a t e seventies, two systematic attempts have been made in order t o remed the mentioned inadequate situation concerning weak and generalized so utions of sufficiently large classes of nonlinear partial diffeuations. The main publications have been Rosin er [I ,2] , 1,2] and Rosinger [3], a f i r s t presentation of some o the baslc ved being given earller in Rosinger [7,8].
I
I
Foreword
Colombeau ' s nonlinear theory of generalized functions, although developed in the early eighties, has started in a rather independent manner. However, it proves to be a particular case of the more general nonlinear theory of generalized functions in Rosinger [I ,2] , see for details Rosinger [3,pp . 300-3061 . In fact, the two theories in Rosinger [1,2,3] and Colombeau [1,2] have so far been somewhat complementary to each other, as they approach the field of generalized solutions for nonlinear partial differential equations from rather opposite points of view. Indeed, both theories aim to construct d i f f e r e n t i a l a l g e b r a s A of generalized functions which extend the L. Schwartz distriburtions, that is, admit embeddings V'(Q)
c A, with Q c IRn open.
Given then linear or nonlinear partial differential operators T(x,D) on Q, one can extend them easily, so that they may act for instance as mappings
In that case the respective linear or nonlinear partial differential equations
with f E A given, may have generalized solutions U E A, customary conditions may prove to be unique, regular, etc.
which under
And in view of H. Lewy's mentioned impossibility result, extensions of the L. Schwartz distributions given by embeddings
3'(Q) c A of the above or similar type prove to be n e c e s s a r y even when solving linear variable coefficient partial differential equations. Now, Colombeau's nonlinear theor develops what appears to be the most n a t u r a l and c e n t r a l class of dif erential algebras A which contain the distributions, see for details Chapter 8, as well as Rosinger [3,pp. V'(Q) 115-1231. The power of that approach is quite impressive as it leads to existence, uniqueness and regularity results concerning solutions of large classes of linear and nonlinear partial differential equations, equations which earlier were not solved, or were even proved to be unsolvable within the distributions or hyperfunctions, see for details Rosinger [3, pp. 145-1921 . In addition, Colombeau's nonlinear theory has important applications in the numerical solution of nonlinear and nonconservative shocks for instance, see for details Biagioni 121.
l
On the other hand, the earlier and more eneral nonlinear theory in Rosinger [1,2,3], has started with the ClariBication of the a l g e b r a i c and d i f f e r e n t zal foundations of what may conveniently be considered as a1 1 p o s s i b 1e nonlinear theories of generalized functions. That approach leads
E.E. Rosinger
t o the characterization and construction of a very large class of different i a l algebras A which contain the 9' distributions, and which can be used in order t o give the solution of most general nonlinear partial differential equations. In that context, i n addition t o the usual problems of existence, uniqueness and regularity of solutions, a f i r s t and fundamental role i s played by the problems of s t a b i l i t y , generality and exactness of such solutions, see for details Chapter 1 , Sections 8- 12. This general approach yields several results which are a f i r s t i n the literature. For instance, one obtains global generalized solutions for a2 1 anal y l ic nonlinear partial differential equations. These solutions are ana 1y t ic on the whole of the domain of analyticity of the respective equations, except for closed, nowhere dense subsets, which can be chosen t o have zero Lebesque measure, see Chapter 2 and Rosinger [3].
A second result gives an algebraic characterization for the existence of generalized solutions for a l l polynomial nonlinear partial differential equations with continuous coefficients, see Chapter 3 and Rosinger [3] . This algebraic characterization happens t o be given by a version of the so called neutrix or off diogonality condition, see (1.6.11) i n Chapter 1. A third type of results concerns the characterization of a very large class of differential algebras containing the distributions. One of t h i s characterizations i s given by the mentioned neutrix or off diagonality condition on differential algebras of generalized functions constructed as quotient algebras
where A = ( ~ ( 9 ) ) ' and Z i s an ideal i n A, see Chapter 6, as well as Rosinger [1,2,3]. Within a more general framework of quotient algebras
d i s an ideal i n A, a where A i s a subalgebra i n (C?(~))\nd further characterization of the structure of these algebras i s given. Indeed, it i s shown that the algebraic type neutrix or off diagonality condition i s equivalent t o a topological type condition of dense vanishing, see Chapter 3. The above three results use the f u l l generality of the nonlinear theory developed i n Rosin er [1,2,3], and it i s an open question whether similar results may be o tainable w i t h i n the particular nonlinear theory in Colombeau [ I , 21 .
f
Several other results which so f a r could only be obtained within the framework of the nonlinear theory i n Rosinger [1,2,3 are presented shortly i n Chapters 6 and 7. More detailed accounts, inc uding additional such res u l t s can be found in Rosinger [1,2,3].
1
Foreword
A t t h i s stage it may be important t o point out the utility of considerin the problem of generalized solution for nonlinear partial differentia equations within sufficiently large frameworks. Indeed, as H. Lewy ' s 1957 example shows it, the framework p'(lRn) of the L. Schwartz distribution i s too restrictive even for linear, variable coefficient partial differential equations. Colombeau's particular nonlinear theory, owing t o i t s natural, central position proves t o be unusually powerful, both i n generalized and numerical solutions for wide classes of linear and nonlinear partial differential equations. However, results such as i n Chapters 2 and 3 for instance, find t h e i r natural framework w i t h i n the general nonlinear theory introduced in Rosinger [I ,2,3] , and so f a r could not be reproduced w i t h i n the framework in Colombeau [I ,2] .
!?
What t o us seems however less than surprising i s that t h i s i s not yet the end of the story. Indeed, as seen in the results i n Chapter 4, contributed recently by M. Obergug enberger, further extensions of the general framework i n Rosinger [ 1 , 2 , 4 are particularly useful. A l l t h i s development seems t o create the feeling t h a t , inspite of the rather extended framework presented in t h i s volume, the nonlinear theories of generalized functions may s t i l l be a t t h e i r beginnings.
And now a few words about the point of view and approach pursued i n this volume. A t least since Sobolev [1,2], the main, i n fact nearly exclusive approach i n the stud of weak and generalized solutions for linear and nonlinear partial difrerential equations has been that of functional analysis, used most often i n infinite dimensional vector spaces. That includes as well the way Colombeau ' s nonlinear theory of generalized functions was started i n Colombeau [I] .
The difficulties in such a functional analytic approach i n the case of solving nonlinear partial differential equations are well known. And they come mainly from the fact that the strength of present day functional analysis i s rather i n the linear than the nonlinear realm. In addition, an exag erated preference for a functional analytic point of view can have the un ortunate tendency t o f a i l t o see simple but fundament a l facts for what they really are, and instead, t o notice them only through some of t h e i r more sophisticated consequences, as they may emerge when translated into the functional analytic language.
P
The effect may be an unnecessary obfuscation, and hence, misunderstanding, as happened for instance w i t h L. Schwartz's so called impossibility result. Indeed, the point t h i s result t r i e s t o emphasize i n i t s original formulation i s that in a differential algebra which contains just a few continuous functions, the multiplication of these functions cannot be the usual function multiplication, unless we are ready t o accept certain apparently unpleasant consequences, see for details Proposition 1 in Chapter 1, Section 2 . However, those few continuous functions are not P-smooth. I n fact, their f i r s t or a t most second order derivatives happen t o be discon-
E.E. Rosinger
tinuous. In t h i s way, a t a deeper level, the difficulty which the so called Schwartz impossibility result i s trying t o t e l l us inspite of a l l misunderstandin s , i s that there exists a certain conf 1 ict between discontinuity, multip ication and differentiation.
P
And as seen i n Chapter 1, Section 1 and Appendix 1, t h i s conflict i s of a most simple algebraic nature, which already happens t o occur within a rock- bottom, very general framework, f a r from being in any way restricted or specific t o the L. Schwartz distributions. And then, what can be done? Well, we can remember that one way t o see modern mathematics i s as being a multilayered theory in which successive layers are built upon and include earlier, more fundamental ones. For instance, one may l i s t some of them as follows, according t o the way successive layers depend on previous ones: -
set theory
-
topology functional analysis etc.
- binary relations, order - algebra -
In t h i s way, it may appear useful to t r y t o identify the roots of a problem or difficulty a t the deeper relevant layers. Such an approach w i l l bring the so called L. Schwartz impossibility result, and i n eneral, the problem of distribution multiplication to the algebra level o? the basic conflici between discontinuity, multiplication and differentiation, mentioned above. This i s then i n short the essence and the novelty of the 'algebra f i r s t ' approach pursued i n the present volume. The reader who may wonder about the possible effectiveness of such a desesc ~ lion ~ t from involved and sophisticated functional analysis t o basic mathematical structures, may perhaps f i r s t - and equally - wonder about the rather incisive insight of the celebrated seventeenth century Dutch philosopher Spinoza, according t o whom the ultimate aim of science i s t o reduce the whole world t o a tautology. I n mathematics, a good part of this d namics i s expressed i n the well known adage that, old theorems never die: t ey just become definitions!
X
Indeed, it i s obvious that a lot of knowledge, understanding, experience and hopefully simplification i s needed in order t o set up an appropriate mathematical structure, in particular axioms or definitions. I n t h i s way, the knowledge in 'old theorems' becomes ezp 1 ic i t in the very mathematical structure i t s e l f . A good illustration for that, in particular for simplification, i s the transition from Colombeau [I] t o Colombeau [2]. Certainly, in a deduction A + B , B cannot be more than A , that i s , i t cannot contain more information than A , and the nearer B i s to A , the more the information which was gotten through the deduction i s near t o 100%. Of course, we are not interested in a theory which mainly has 100%
Foreword
efficient deductions A + A. So, we should keep somewhat awa from tautology. On the other hand, the more the amount of near tauto ogical deductions in a theory, the greater our understanding of what i s after a l l , the best analogy i s a tautolo y and the best e z p l i c i l &On%e!i~ i s an analogy. I s n ' t it that a proper ey i s better than a skeleton key precisely t o the extent that it i s more analo ous with the lock, containing more explicit knowledge i n i t s very structure.
K
f
9
In t h i s respect the p r e s e n c e of 'hard theorems' - which are hard owing to their f a r from tautological roofs - i s a sign of insufficient insight on the level of the structure o the theory as a mhole. Let us just remember how the so called 'Fundamental Theorem of Algebra' according t o which an algebraic equation has a t least one complex root, lost i t s 'hard' status from the time of D'Alembert to the time of Cauchy, owing t o the emergence of complex function theory.
!
Now, as if to give some much desired comfort to those who ma nevertheless feel that, within a good mathematical theory one should, a t east here and there, have some 'hard theorems ' , the mentioned desescalat ion proves to leave room for such theorems. Indeed, l e t us mention just some of them, present also within the part of the general theory contained in this volume. In Sections 4 and 6 in Chapter 2 , one uses a transfinite inductive exhaustion process for open s e t s in Euclidean spaces and, respectively, a rather involved topological and measure theoretical argument in Euclidean spaces. In Section 4 of Chapter 3, a similarly involved, twice iterated use of the Baire category argument i n Euclidean spaces i s employed. Further, i n Section 5 of Chapter 3, a deep property of up er semicontinuous functions i s used i n a c r i t i c a l manner. In Section 4 !o Chapter 4, functional analytic methods are employed. And we can also mention the cardinality arguments on sets of continuous functions on Euclidean spaces, which are fundamental in the results presented in Chapter 6.
K
It should be mentioned that the presentations in Rosin e r [1,2,3] and Colombeau [2] , have also pursued an 'algebra f i r s t approacf , although in a different, more obvious manner, which i s directly inspired by the classical weak solution method. Indeed, it i s well known that (?'(a) for instance, i s weakly sequentially dense i n 3'(Q). Therefore, i n some sense to be defined precisely, we have an ' inclusion'
Moreover
i s obviously an associative and commutative differential algebra, when considered with the usual term wise operations on functions. I n t h i s way, a l l what a nonlinear theor of eneralized functions needs t o do i s to give a precise meaning t o the a ove inclusion' .
t k
E.E. Rosinger
XIV
In the present volume, t h i s second a1 ebraic approach follows a f t e r the earlier mentioned one, namely, that !ealing with the conflict between discontinuity, multiplication and differentiation. The outcome of such an approach i s that functional analytic methods need not be brought into play for quite a long time. In f a c t , just as in Rosinger [1,2,3] and Colombeau [2], such methods are not used a t a l l , The mathematics which i s used consists of basic except for Chapter 4. algebra of rings of functions, calculus and some topology, a l l i n Euclidean spaces alone. The connect ion with partial differential equations i s made throu h certain asymptotic interpretations in the s p i r i t of the 'neutrix calcu us' in Van der Corput, see Chapter 1, Section 6 and Appendix 4.
P
The fact that functional analysis i s not needed from the very beginning should not come as a surprise. Indeed, i n the customary approaches t o partial differential equations there are t h r e e reasons for the use of functional analysis, namely, f i r s t , i n order t o define partial derivatives f o r eneralized functions, then, in order t o approximate generalized solutions regular functions, and finally, in order t o define the generalized functions as elements i n the completion of certain spaces of regular functions. B u t , by constructing embeddings
$
into quotient algebras
one can avoid functional analysis t o a good extent. Indeed, the partial derivatives of generalized functions T E A = A l l , can be reduced to the usual partial derivatives of the smooth functions in the sequences representing T. Further, an algebraic study of exact solutions by generalized functions need not involve approximations from t e very beginning. Finally, the respective algebras A of generalized functions used as 'reservoirs' of solutions - can easily be kept large enough by simply using suitably chosen subalgebras A and ideals Z i n the construction of the quotient algebras A = A/Z.
fiven
The ease i n such an approach and the extent t o which it works can be seen i n this volume, as well as in the cited main publications Rosinger J 1 , 2 , 3 ] and Colombeau [1,2]. Further developments have been contribute i n a number of papers and two research monographs, published or due t o appear, by M. Adamczewski, J . Aragona, H . A . Biagioni, J . J . Cauret , J . F. Colombeau, J.E. ~ a l k ,F. Lafon, M. Langlais, A.Y. Le Roux, A . Noussair, M. Oberguggenberger, B. Perot and T.D. Todorov, see the References. As before, a close contact w i t h J.F. Colombeau and M. Oberguggenberger has offered the author a particularly useful help, not least owing t o the exchange of different views on what appears t o be a f a s t emerging nonlinear theory of generalized functions.
Foreword
A special mention is due to I. Oberguggenberger's recent cycle of research on Semilinear Wave Equations with rough initial data, which is shortly presented in Chapter 4. In addition to its obvious importance as a powerful contribution to a clearer understanding of the propogation of singularities in nonlinear wave phenomena, its impact on the emerging nonlinear theory of generalized functions is uniquely important at this stage. Indeed, the method used in Oberguggenberger's research is so simple and powerful precisely due to the fact that it is sufficiently eneral, in fact so general that it goes beyond the framework of Rosinger E,2,3] as well. A detailed presentation of recent results on the propagation of singularities in nonlinear wave phenomena, as well as a thorough analysis of the connection between the emerging nonlinear theory of generalized functions and earlier, partial attempts at distribution multiplication are presented in the research monograph Oberguggenberger [15]
.
Another s ecial mention is due to H.A. Biagioni's research monograph, Biagioni f2h, which presents a most important development of Colombeau's nonlinear t eory, namely, in the field of numerical solutions for nonlinear, nonconservat ive partial differential equations. This line of research is still very much at the beginning of reaching its full potential, and owing to the extensive space it takes to present the results already obtained, it could not be included in this volume. The author owes a deep and warm gratitude to Professor J. Swart, the head of the mathematics department, and to Professor N. Sauer, the dean of the science faculty at the University of Pretoria for the unique academic and research conditions created and for the friendly and kind support over the years. As on so many previous occasions, the outstanding work of careful typing of the manuscript was done by Mrs. A.E. Van Rensburg. It is hard to ever truly appreciate the contribution of such help. As on earlier occasions, the author is particularly grateful to Drs. A Sevenster of the Elsevier Science Publishers for a truly supportive approach. This is the author's third research monograph in ten years in Prof. L. Nachbin's series of the North-Holland Mathematics Studies. In our era, when 'Big Science' so often tries to dwarf us into negligible and disposable entities, subjecting us to the 'Big Industry', conveyor belt type management by 'Publish or Perish!, one should perhaps better not think about how the world may look without editors like Prof. L. Nachbin, who are still ready to offer us a most outstanding encouragement and support. And what may in fact be wrong with 'Big Science'? Well, was it Henry Ford, of the 'History is bunk' fame, who found it necessary to insist that: 'Big Organizations can never be humane'?
XVI
E.E. Rosinger
Yet, after WW 11, to the more traditional Big Organizations of Army, Priesthood, Bureaucracy and Industry, we have been so busy adding that of Big Science ...
E.E. Rosinger Pretoria, May 1990
TABLE OF CONTENTS
CEAPTER 1
CONFLICT BETWEEN DISCONTINUITY. MUTLIPLICATION AND DIFFERENTIATION .................................
.......................................... .............................................
$1
A basic conflict
$2
A few remarks
$3 An algebra setting for generalized functions
..............
................................... Construction of algebras containing the distributions ..... The Neutrix condition .....................................
$4 Limits to compatibility $5
$6
$7 Representation versus interpretation
......................
$8 Nonlinear stability paradoxes. or how to prove that 0 2 = 1 in R ............................................. $9 Extending nonlinear partial differential operators to generalized functions .....................................
........................... Nonlinear stability. generality and exactness ............. Algebraic solution to the nonlinear stability paradoxes ...
$10 Notions of generalized solution $11 $12
.......... Systems of nonlinear partial differential equations .......
$13 General nonlinear partial differential equations $14
....
Appendix 1
On Heaviside functions and their derivatives
Appendix 2
The Cauchy-Bolzano quotient algebra construction of the real numbers ................................
Appendix 3
How 'wild' should we allow the worst generalized functions to be? ................................
Appendix 4
Neutrix calculus and negligible sequences of functions .......................................
Appendix 5
Review of certain important representations and interpretations connected with partial differential equations .......................................
XVIII
Appendix 6
Appendix 7 Appendix 8
CHAPTER 2
E.E. Rosinger
Details on nonlinear stability paradoxes. and on the existence and uniqueness of solutions for nonlinear partial differential equations ...................
88
The deficiency of distribution theory from the point of view of exactness .............................
94
Inexistence of largest off diagonal vector subspaces orideals ........................................
98
GLOBAL VERSION OF THE CAUCHY KOVALEVSKAIA THEOREM ON ANALYTIC NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS ........................................
101
............................................... The nowhere dense ideals ...................................
$1 Introduction
101
$2
104
53 Nonlinear partial differential operators on spaces of
...................................... Basic Lemma ................................................ Global generalized solutions ...............................
generalized functions
$4 55
$6 Closed nowhere dense singularities with zero Lebesque
measure
....................................................
109 113 117
........
122
Too many equations and solutions?! ............... Universal ordinary differential equations ........ Universal partial differential equations ......... Final Remark .....................................
122 123 124 124
.....
126
$7 Strange phenomena in partial differential equations $7.1 $7.2 $7.3 $7.4
108
Appendix 1
On the structure of the nowhere dense ideals
CHAPTER 3
ALGEBRAIC CHARACTERIZATION FOR THE SOLVABILITY OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS .......... 131
................................................ The notion of generalized solution ..........................
$1 Introduction
131
52
132
$3 The problem of solvability of nonlinear partial differential
equations
...................................................
135
Table of Contents
XIX
$4 Neutrix characterization for the solvability of nonlinear partial differential equations .............................. 136 $5 The neutrix condition as a densely vanishing condition on
...................................................... Dense vanishing in the case of smooth ideals ................ The case of normal ideals ................................... Conclusions ................................................ ideals
$6
$7 $8
.......... ...............................
152 159 164 165
Appendix 1
On the sharpness of Lemma 1 in Section 4
168
Appendix 2
Sheaves of sections
170
CHAPTER 4
GENERALIZED SOLUTIONS OF SEMILINEAR WAVE EQUATIONS WITH ROUGH INITIAL VALUES ......................... 173
$1 Introduction
................................................
$2 The general existence and uniqueness result
.................
173 174
........................................ 185 ............................................... 194
$ 4 The delta wave space $5
A few remarks
CHAPTER 5
DISCONTINUOUS. SHOCK. WEAK AND GENERALIZED SOLUTIONS OF BASIC NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS . . 197
$1 The need for nonclassical solutions: the example of the
.............................. Integral versus partial differential equations .............. Concepts of generalized solutions ........................... Why use distributions? ...................................... The Lewy inexistence result ................................. nonlinear shock wave equations
$2 $3 $4 55
Appendix 1
Yultiplication. localization and regularization of distributions .....................................
197 202 208 209 212
215
E.E. Rosinger
XX
CHAINS OF ALGEBRAS OF GENERALIZED FUNCTIONS
CHAPTER 6
..... .
$1 Restrictions on embeddings of the distributions into quotient algebras ..........................................
. . . .. . . ................. ...... .... . . .... . ... Neutrix characterization of regular ideals .... . ... . . ... . . ..
$2 Regularizations $3
$4 The utility of chains of algebras of generalized functions
..
$5 Nonlinear partial differential operators in chains of
. . ............... . . ...... . ....... . . .... ..... . ... . . .
algebras
$6 Limitations on the embedding of smooth functions into chains of algebras . . . . . . .. ................ . . .... . . . ... . . . . .. . . . .. . CHAPTER 7
RESOLUTION OF SINGULARITIES OF WEAK SOLUTIONS FOR POLYNOMIAL NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
$1 Introduction
...............................................
$2 Simple polynomial nonlinear PDEs and resolution of
.............................................. Resolution of singularities of nonlinear shock waves . .. . . . . . singularities
$3
$4 Resolution of singularities of the Klein-Gordon type nonlinear
waves
.......................................................
$5 Junction conditions and resolution of singularities of weak solutions for the equations of magnetohydrodynamics and general relativity .......................................... $6 Resoluble systems of polynomial nonlinear partial differential
................................................... Computation of the junction conditions .. . ... . . ... . . .. . . . ... . equations
57
58 Examples of resoluble systems of polynomial nonlinear partial
............ . . . . ...... . ... . . ... . . . .. . . Global version of the Cauchy-Kovalevskaia theorem in chains of algebras of generalized functions . . . ..... . . . .... . . ... . . .... .
differential equations
$9
294
XXI
Table of Contents
THE PARTICULAR CASE OF COLOMBEAU'S ALGEBRAS
CHbPTEB8
.......
................... P(P) ...............
301
$1 Smooth approximations and representations
301
$2 Properties of the differential algebra
307
$3 Colonbeau's algebra 54 $5
~ ( p as ) a collapsed case of
chains of
.................................................... 312 Integrals of generalized functions .......................... 319 Coupled calculus in ~ ( e .................................. ) 325 algebras
56 Generalized solutions of nonlinear wave equations in quantum
field interaction
..........................................
333
57 Generalized solutions for linear partial differential
equations
Appendix 1
..................................................
335
The natural character of Colombeau' s differential algebra ..........................................
345
Appendix 2
Asymptotics without a topology
...................
354
Appendix 3
Connections with previous attempts in distribution multiplication ...................................
357
Appendix 4
Final Remarks References
An intuitive illustration of the structure of Colonbeau' s algebras ............................. 361
.................................................
367
......................................................
371
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CHAPTER 1 CONFLICT BETWEEN DISCONTINUITY, MLJLTIPLICATION AND DIFFERENTIATION
$1. A BASIC CONFLICT
There exist basic algebraic - i n particular, ring theoretic - aspects involved i n the problem of finding generalized solutions for nonlinear part i a l differential equations.
Why generalized solutions? The answer i s well known and a short, f i r s t account of it i s given i n Chapter 5, as well as i n the literature mentioned there.
B u t then equally, if not even more so, one may ask: why algebra, and why precisely i n the realms of nonlinear partial differential equations? Fortunately, the answer t o t h i s second question i s much simpler, and it can be presented here, without the need for any special introduction. In f a c t , it i s our aim t o show for the f i r s t time i n the known l i t e r a t u r e , that the issue of generalized solutions for nonlinear partial differential equations can be approached i n a relevant and useful way by f i r s t considering the algebraic problems involved in the basic trio of: -
discontinuity multiplication differentiation.
The interests i n such an 'algebra f i r s t ' approach can be multiple. First of a l l , the ring theoretic type of algebra involved belongs to a more fundamental kind of mathematics than the usual calculus, functional anal y t i c or topological methods which are customary in the study of partial differential equations. And by using such more fundamental aathemat ics, one can hope for a better and deeper understanding of the issues involved, as well as for easier solutions. Fortunately, such advantages happen to materialize t o a good extent. Another reason for pursuing 'algebra f i r s t ' i s i n trying t o draw the attention of mathematicians working i n fields f a r removed from analysis or functional analysis - such as for instance algebra, or rings of functions upon the possibility of significant applications of t h e i r methods and results i n the f i e l d of solving nonlinear partial differential equations.
r
Finally, there i s an interest i n showing t o many anal s t s and functional analysts working in the f i e l d of nonlinear partial dif erential equations, that the road can lead not only from theories and methods which are already quite complicated towards other ones, yet more complicated. On the contrary, and a t least as a temporary detour, the 'algebra f i r s t ' road can
E .E. Rosinger
lead to quite a few simplifications and clarifications. However, in view of the results already obtained along that road, Rosinger [1,2,3], Colombeau [1,2], one may as well see it as being much more than a temporary affair. Indeed, one of such results - a first in the literature - is a globalized version of the classical Cauchy-Kovalevskaia theorem concerning the existence of solutions for arbitrary analytic nonlinear partial differential equations, Rosinger [3, pp. 259-2661 . By using algebraic, ring theoretic methods, one can prove the existence of generalized solutions on the whole of the domain of analyticity of the respective nonlinear equations. Furthermore, these generalized solutions are analytic on the whole of the respective domains, except for subsets of zero Lebesque measure. Details are presented in Chapter 2. Similar ring theoretic methods can lead to another first in the literature, Rosinger [3, pp. 233-2471 , namely, an algebraic characterization for the solvability of a very large class of nonlinear partial differential equations, reviewed in Chapter 3. Important first results in the literature are obtained in the particular theory in Colombeau [1,2] as well. For instance, solutions are found for large classes of nonlinear partial differential equations which earlier were unsolved or proved to be unsolvable in distributions. In addition, solutions are constructed for the first time for arbitrary systems of linear partial differential equations with smooth coefficients, thus going beyond the celebrated impossibility result of Lewy, as well as its various extensions, Hormander. Let us now turn to the motivation of the basic algebraic settin presented later in its basic form in Section 3. For that purpose we shal give here a most simple example of the kind of conflicts we can expect when dealing with the mentioned trio of discontinuity, multiplication and differentiation. Fortunately, an attentive study of the conflict involved in this simple, one dimensional example can lead a long way towards the clarification and solution of most important problems concerning the solution of wide classes of important nonlinear partial differential equations.
P
As is known, see also Chapter 5, the classical solutions of linear or nonlinear partial differential equations are given by sufficiently smooth functions
where fl c IRn is a certain domain. For our purposes, we can restrict the setting to the simplest, one dimensional case, when n = 1 and D = R . As also seen in Chapter 5, nonlinear partial differential equations have important nonclassical, that is, generalized solutions. In particular, such nonclassical solutions may be given by nonsmooth or discontinuous functions U in (1.1.1) For our purposes, it will be sufficient at first to consider a most simple discontinuous function, such as the well known Heaviside function (1.1.2) defined by
H:R 4 R
Algebraic conflict
Now, when appearing in generalized solutions of nonlinear partial differential equations, a discontinuous function such as H , w i l l be involved in multiplication and differentiation. Therefore, it appears that the basic setting we are looking for should be given by a ring of functions (1.1.4)
AclR-+R
such that
and A (1.1.6)
has a derivative operator
D : A-+A
that i s , an operator D which i s linear and also s a t i s f i e s the Leibnitz rule of product derivative (1.1.7)
D(f.g) = (Df)-g + f.(Dg), f , g
E
A
Unfortunately, the problem already s t a r t s right here. Indeed, no matter how intuitive and natural i s the above setting as an extension of the classical, smooth case, the relations (1.1.2)- (1.1.7) have rather inconvenient consequences ! For t h a t , f i r s t we note that (1.1.4), (1.1.5) yield the relations
Further, i n view of (1.1.4), A i s associative and commutative. (1.1.8), (1.1.7) give the relations (1.1.9)
mH.(DH) = DH,
NOW,if p,q E DI,
m E
IN,
m
p,q 2 2 and p # q,
>2 then (1.1.9) implies
Hence
E.E. Rosinger
which results in
However, as seen in Chapter 5, there exist particularly strong reasons to expect that the derivative operator D in (1.1.6), (1.1.7) is such that
where 6 is the well known Dirac delta function. In this case, the relations (1.1. lo), (1.1.11) would imply
which is false, since the Dirac delta function is not the identically zero function. To recapitulate, the discontinuous function H has the multiplication property (1.1.8), which by differentiation gives the relation (1.1.10). Then, assuming the natural relation in (1.1.11) , one obtains the incorrect result in (1.1.12). The way out is obviously by trying to relax some of the assumptions involved. What we have to try to hold to, in view of reasons such as those presented in Chapter 5, is the discontinuous function H and the relation DH = 6 describing its derivative. But we are more free in two other respects, namely, in choosing the algebra A and the derivative operator D. Indeed, while A should contain functions such as H : R -+ [R, it need not be an algebra of functions from R to R. In other words, A may as well contain more general elements, and the multiplication in A need not be so closely related to the multiplication of functions. In particular, (1.1.8) need not necessarily hold. Concerning the derivative operator D, a most important point to note is the particularly restrictive nature of the assumption in (1.1.6). Indeed, this assumption implies that the elements of A are indefinitely derivable, that is (1.1.13)
~~a exists, for all a
E
A and m
E N,
in 2 1
This would of course happen if we had
which is however no2 possible in view of (1.1.5). It follows, that we should keep open the possibility when the derivative operator D is defined as follows
Algebraic conflict
where A is another algebra of generalized functions. In this case, in order to preserve the Leibnitz rule of product derivative, we can assume the existence of an algebra homomorphism (1.1.16)
A a-a
b
A
and then rewrite (1.1.7) as follows
It will be shown in Chapters 2 and 3, that the above two kind of relaxations, namely, on the algebra A and the derivative operator D, are more then sufficient in order to find generalized solutions for wide classes of nonlinear partial differential equations. In particular one can obtain the mentioned globalized version for the Cauchy-Kovalevskaia theorem, in which one can prove the existence of generalized solutions, on the whole of the domain of analyticity , of arbitrary analytic nonlinear partial differential equation. Furthermore, one can obtain an algebraic characterization for the solvability of a very large class of nonlinear partial differential equations.
52.
A FEW RENARKS
It is particularly importarat to note that the argument leading to (1.1.10) does a02 use calculus and it is purely algebraic, more precisely it only uses the a1 ebra structure of A and the fact that D is linear and it satisfies tfe Leibnitz rule of product derivative. Two, more abstract variants of this argument are presented in Appendix 1. The further results in this chapter on the conflict within the trio of discontinuity, multiplication and differentiation will also be of a similar purely algebraic nature. This is precisely the reason why the 'algebra first' approach is useful, and should be systematically pursued. Now, let us recall the essential role played by the property, see (1.1.8)
in obtaining the undesirable relation (1.1.10). We note that the infinite family of equations
E. E. Rosinger
o n l y h a s two s o l u t i o n s , namely
Therefore, t h e r e l a t i o n 1.2.1) determines uniquely t h e Heaviside f u n c t i o n H amon a l l f u n c t i o n s rom R t o R, which a r e d i s c o n t i n u o u s o n l y at x = 0 E and a r e nondecreasing.
$
\,
Let us d e n o t e by 0 and 1 t h e f u n c t i o n s d e f i n e d on R which t a k e everywhere t h e v a l u e 0, r e s p e c t i v e l y 1. Then, w i t h i n t h e framework of (1.1.4)(1.1.7), we obviously have 0 E A . Let u s f u r t h e r assume t h a t
t h e n from ( 1 . 1 . 7 ) , we o b t a i n e a s i l y t h e r e l a t i o n s
which a r e t o be expected, s i n c e t h e f u n c t i o n s 0 and 1 a r e c o n s t a n t . The unex e c t e d f a c t i s t h a t , although H is n o t a c o n s t a n t f u n c t i o n , (1.1.4)- (I. 1 . 7 ) w i l l n e v e r t h e l e s s imply, s e e (1.1.10)
i
I n view of 1 . 2 . 3 ) , t h e f u n c t i o n s 0 and 1 a r e t h e o n l y continuous f u n c t i o n s which s a t i s y (1.2.1). It f o l l o w s t h a t t h e framework i n (1.1.4)- (1.1.7) and (1.2.4) is t o o c o a r s e i n o r d e r t o d i s t i n g u i s h between t h e d e r i v a t i v e s of continuous and d i s c o n t i n u o u s f u n c t i o n s . The above c o n f l i c t between d i s c o n t i n u i t y , m u l t i p l i c a t i o n and d e r i v a t i v e app e a r s a l r e a d y at t h a t r a t h e r simple and fundamental l e v e l . Needless t o s a y , it h a s many f u r t h e r , more involved i m p l i c a t i o n s , such as f o r i n s t a n c e , t h e long misunderstood, s o c a l l e d I,. Schwartz i m p o s s i b i l i t y r e s u l t , s e e below. However, a proper t r e a t m e n t of t h a t c o n f l i c t can o n l y b e n e f i t from i t s i d e n t i f i c a t i o n at its most b a s i c and a l s o s i m p l e l e v e l s . Dtherwise, t h e complications involved may l e a d t o misunderstandings, as happened with t h e mentioned r e s u l t of L. Schwartz. S u f f i c e it h e r e t o say t h a t one of t h e most important consequences of t h e c o n f l i c t between d i s c o n t i n u i t y , m u l t i p l i c a t i o n and d e r i v a t i v e is t h e f o l lowing. Above a c e r t a i n l e v e l of i r r e u l a r , d i s c o n t i n u o u s o r nonsmooth f u n c t i o n s , m u l t i p l i c a t i o n can no l o n g e r e made i n a unique, c a n o n i c a l o r b e s t way. T h i s is t h e p r i c e we have t o pay if we n e v e r t h e l e s s want t o b r i n g t o g e t h e r i n t o one mathematical s t r u c t u r e both d i s c o n t i n u i t y and mult i p l i c a t l o n , as well as d e r i v a t i v e . I n t h i s way, t h e message of L. Schwartz's mentioned r e s u l t is not t h a t 'one cannot m u l t i p l y d i s t r i b u t i o n s ' , Hormander [p. 9 , but on t h e c o n t r a r y , t h a t one can, and i n e v i t a b l y has t o f a c e t h e f a c t t a t t h e y can be m u l t i p l i e d i n many d i f f e r e n t ways.
%
B
Algebraic conf1ict
7
By the way, for the sake of rigour, let us specify that the citation from Hormander mentioned above reads as follows: 'It has been proved by Schwartz ... that an associative multiplication of two arbitrary distributions cannot be defined'. The reader less familiar with the linear theory of the L. Schwartz distributions can omit the rest of this section and go straight to the next one, that is, to Section 3. A few useful details on distribution theory are presented in Appendix 1 to Chapter 5. After consulting them, as well as the basic issues concerning generalized solutions for nonlinear partial differential equations presented in the mentioned chapter, the reader may return to the rest of this section. The second remark concerns the longstanding misunderstandin connected with the so called impossibility result of L. Schwartz, estabfished in 1954, Schwartz [2]. This result has often been overstated by claiming that it proves the impossibility of conveniently multiplying distributions. In fact, L. Schwartz's mentioned result only shows that - similar to the situation following from (1.1.4 - 1.1.7) above - an insufficiently careful choice of an algebraic framewor or generalized functions can lead to undesirable consequences. In addition, its set up is more complicated than the one in (1.1.4)- (1.1.7), therefore, to an extent it hides the simplicity of the fundamental conflict in the trio of discontinuity, multiplication and differentiation. For convenience, we recall here L. Schwartz ' s mentioned result, a detailed proof of which can be found for instance in Rosinger [3, pp. 27-30].
11
Pro~osition 1 (Schwartz [2] ) Suppose A is an associative algebra with a derivative operator : A -+ A, that is, a linear mapping which satisfies the Leibnitz rule of product derivative , see (1.1.7) .
D
Further suppose that (1.2.7)
the following four C"- smooth functions 1, x, n!(x and x2 (Cn xl - 1) belong to A, where for x latter two unctions are assumed to vanish,
1
(1.2.8)
the function 1 is the unit in A
(1.2.9)
the multiplication in A is such that (x).(x(!n 1x1 - 1)) = x2(Cn 1x1 - 1)
1x1 - 1) = 0, the
E .E . Rosinger
the derivative operator D : A -+ A following three C1- smooth functions 1, X, x2(Ln 1x1 - 1) is the usual derivative of functions. Then, there exists no 6 E A, 6 # 0
E
applied to the
A, such that
The usual interpretation of Proposition 1 goes along the following lines. If 6 is the Dirac delta function, or more precisely distribution, then it is well known that
where VJ(R) is the set of the Schwartz distributions on R. Furthermore, we can multiply each distribution T E P1(IR) with each function $ E P(R) and obtain $T E 9'(IR) , see Appendix 1 to Chapter 5. In particular, we have
therefore
It should be recalled here that
in particular
Now, if we want to multiply two arbitrary distributions T and S from V'(IR), the above lnentioned procedure where the multiplication of any T E P'(IR) with any $ E P(R) gives $T E VJ(IR) will in general not work, since both distributions T and S may fail to belong to P(IR). For instance, in view of (1.2.16), we cannot compute b2 = 66 by the above procedure. One way to multiply arbitrary distributions from T(R) embedding (1.2.17)
2'(R) c A
is by finding an
Algebraic conf1ict
9
where A is an algebra, and then performing the multiplication in A. Now, one notes that, in view of the well known inclusion CO(IR) $ T1(IR), the four functions in (1.2.7) must belong to A. It follows that the conditions (1.2.8)- (1.2.10) on the algebra A required in L. Schwartz's above result are rather natrral and minimal. Nevertheless, they lead to , which is in conflict with the customary distributional properties of the irac delta funtion presented in the relations (1.2.12), (1.2.14). (1.2.11e And here, the usual misinterpretation occurs, according to which it is stated that there cannot exist convenient algebras A such as in (1.2.17) in which to embed the distributions. Sometime it is also concluded that in view of (1.2.11) any algebra A which satisfies the natural and minimal conditions in (1.2.7)- (1.2.10) , cannot contain the Dirac delta function. One can as well encounter the general conclusion that the multiplication of arbitrary distributions is not possible, Hormander. Let us now return to Proposition 1 above and try to assess its correct meaning. First we note that in view of (1.2.12), we shall necessarily have
for any algebra A containing the distributions, such as in (1.2.17). Now, an algebra A as in Proposition 1, may or may not contain the Dirac delta function 6. But if A is large enough, such as for instance in (1.2.17), then, in view of (1.2.18), it will contain 6 as a nonzero element. What is then the message in (1.2.ll)? Well, it is simply the following. The relation (1.2.14) is valid for the usual multiplication between C?-smooth functions and distributions, see Appendix 1 to Chapter 5. In other words, x and 6 are zero divisors in that multiplication. Or, from an analysis point of view, the singularity of 6 at the point 0 E IR is of lower order than that of the function 1/x. In the same time, in view of (1.2.11), the multiplication in any algebra A such as in Proposition 1, and which contains 6, will by necessity give Hence, x and 6 are no longer zero divisors in A. Which in the language of analysis means that, when seen in A, the singularity of 6 at the point 0 E IR is not of lower order than that of the function l/x.
E.E. Rosinger
10
53.
AN ALGEBRA SETTING FOR GENERALIZED FUNCTIONS
As mentioned, it will be sufficient to deal alone with the one dimensional case of generalized functions on W, since extensions to arbitrary higher dimensions follow easily. Further, we have seen that we may have to go as far as to construct embeddings 9' (R) c A
(1.3.1)
where A morphism (1.3.2)
-
A
A are associative algebras, with a given algebra homo-
and
A-A a-B
and a derivative operator (1.3.3)
D : A-A
which is a linear mapping such that that Leibnitz rule of product derivative is satisfied
d
Now, the first problem in setting up a framework such as in (1.3.1 - ( 1 . 3 . 4 ) arises from the fact that unrestricted differentiation on, an certain multiplications with elements of P ( W ) can be performed within the classical linear theory of distributions, see Appendix 1 Chapter 5. Let us recall a few relevant details. We have the well known inclusions
where the three spaces, except for P' ( R ) , are algebras of functions. Further, we have the multiplication, see in particular (1.2.13)
which for T E q o c ( R ) reduces to the multiplication in Finally, we have the distributional derivative (1.3.7)
D
:
9' (R)
4
qo,(R).
V 1(R)
which coincides with the classical derivative of functions, when restricted , and satisfies the following version of the Leibnitz rule of i:odu$(?erivat ive
Algebraic conflict
It follows that the first problem i s t o see t o what extent the structures of algebra and differentiation on A can be compatible with the respective classical structures in (1.3.5)- ( 1 . 3 . 8 ) .
In t h i s respect, we have already noted a few facts, which within ( 1 . 3 . 5 ) ( 1 . 3 . 8 ) , can be formulated as follows
then, w i t h the multiplication in qoc(lR) we have (1.3.10)
H ~ = H , EM,
m)1
while the multiplication in ( 1 . 3 . 6 ) gives (1.3.11)
x6 = 0
E
V1(R)
and final 1y (1.3.12)
DH = S
Let u s see t o what extent the above few basic properties in (1.3.10)(1.3.12) can be preserved within the algebra and differential structure of A. An argument similar t o that used i n proving (1.1.10) shows that ( 1 . 3 . 1 ) ( 1 . 3 . 3 ) together w i t h (1.3.10) and (1.3.12) yield
I n case
and ( 1 . 3 . 2 ) i s the identity mapping, (1.3.13) contradicts the fact that S f 0 i n A. I t follows that the algebra and differential structure on A can only t o a limited extent be compat ible with the classical multiplication and differentiation on 9' (W) , summarized i n (1.3.5)- ( 1 . 3 . 8 ) .
E .E. Rosinger
12
$4. LIMITS TO COMPATIBILITY It will be useful to further investi ate the limitations vpon the compatibility between the a1 ebra and dif erential structure of the extensions (1.3.1)-(1.3.4), calle for short
f
B
EAD , and on the other hand, the classical structures of multiplication and differentiation in (1.3.5)- (1.3.8), denoted by CMD . For convenience, we shall denote by
the particular case of EAD, when (1.3.14) holds.
L. Schwartz's so called impossibility result in Proposition 1, Section 2, is one example of such a limitation, and in Chapter 6, we shall present several related results. First however, a few simpler and more basic results on such limitations on the compatibility between EAD and CMD will be mentioned. These results concern a yet more general trio, namely that of: -
insufficient smoothness multiplication differentiation.
Let us define the continuous functions
Then, within the algebra of functions CO(R),
while obviously
we have the relations
Algebraic conflict
and with the classical derivative of functions, we have
Further, with the distributional derivative (1.3.7), we obtain
We shall s t a r t with some rather general differential algebras A, which need not contain a l l the distributions i n P'(IR). The interest in such results i s that they show the mentioned kind of limits on compatibility, even i f the algebras A contain only a fea of the classical functions or distributions. W e shall name by
any associative and commutative algebra A , together with a derivative operator D : A -+ A which i s a linear mapping and it s a t i s f i e s the Leibnitz rule of product derivative (1.1.7) . Suppose now that our DA
s a t i s f i e s the conditions
(1.4.8)
~ , x , x + , x E- A
(1.4.9)
1 i s the u n i t element in A
\
further, that the relations (1.4.3), 1.4.4) hold in A, derivative D on A satisfies the re ations in (1.4.6). Pro~osition2
(Rosinger [3] , p. 318)
The following relations hold i n A (1.4.10)
XDX+ =
X+
, XDX
= X-
and f i n a l l y , the
E .E. Ros inger
Remark 1
d
(1) The relations (1.4.7) and 1.4.16) show t h a t in case A s a t i s f i e s the minimal compatibility con i t i o n s (1.4.3), (1.4.4), (1.4.6), (1.4.8) and (1.4.9), the derivative D on A cannot be compatible with t h e distributional derivative, even f o r the continuous function x+, since s # 0. (2) The i n t e r e s t in Propositon 2 comes from t h e f a c t t h a t none of the conditions on the algebra A involve discontinuous functions. Furthermore, both t h e r e s u l t s and a s seen next, t h e i r proof a r e purely algebraic, in the sense mentioned a t the beginning of Section 2 . Proof of Proposition 2. For convenience, l e t us denote
In view of (1.4.4) we have
n owing t o (1.4.6) we obtain hence by d e r i ~ a t ~ i oand
which yield (1.4.10). Now, a derivation of (1.4.10) gives through (1.4.6) Da + x.D2a = Da, and thus (1.4.13).
Db + x.D2b = Db
A derivation of (1.4.18) gives d i r e c t l y
which in view of (1.4.10) yields (1.4.11).
In view of (1.4.3) we have
the l a s t equality being implied by (1.4.10) and (1.4.11).
and (1.4.12) i s proved.
Similarly
From (1.4.11) by derivation, we obtain
Algebraic conflict
Da.Db
t
a.D2b = 0,
Da-Db t b.D2a = 0
hence (1.4.19)
Da.Db = -a.D2b = -b.D2a
But (1.4.3), (1.4.13) yield (1.4.20)
a.D2b = (x - b).D2b = xD2b - b.D2b = -b.D2b b.D2a = (x - a)-D2a = x.D2a - a.D2a = -a.D2a
while (1.4.3), (1.4.6) give by derivation (1.4.21)
Da + Db = 1
hence (1.4.22)
Da-Db = Da.(l - Da) = Da - (Da)2 Da-Db
=
(1 - Db) .Db = Db - (Db)2
Now (1.4.19)- (1.4.22) give
for a certain c E A . (1.4.24)
We shall show that
c=O
Indeed applying twice the derivative t o (1.4.12) we obtain successively
and multiplying the last relation by x, relation (1.4.25)
we obtain in view of (1.4.13) the
a. (x.D3b) = 0
But a derivation of (1.4.13) yields D2b + x.D3b = 0 which if multiplied by a ,
gives together with (1.4.25) the relation
E.E. Rosinger
16
hence in view of (1.4.23) we obtain (1.4.24). (1.4.24) imply (1.4.14), a s well a s (1.4.15).
Now obviously (1.4.23) and
In view of (1.4.14) it follows t h a t ( ~ a =) ~~ a , ( ~ =b Db, ) ~ p E IN,
p 2 1
hence by derivation p.(~a)p-'~2a=D2a, P E W ,
p > 2
and then again by (1.4.15), we have
or 1 -.D2a = Da-D2a, p P
E
IN,
p
>
2
hence
which obviously y i e l d s
Since we can s i m i l a r l y obtain
t h e proof of (1.4.16) i s completed. We turn now t o the more p a r t i c u l a r t r i o o f : -
singular functions, such a s multiplication differentiation.
6
Suppose given a DA such t h a t
Before we go f u r t h e r , l e t us r e c a l l , see (1.2.14), t h a t with t h e multip l i c a t i o n in (1.3.6) a v a i l a b l e in V' (R) , we have t h e r e l a t i o n (1.4.27)
x6 = 0
E
V'(R)
Algebraic conflict
17
which means that at the point 0 E R, 6 has a singularity of an order less than that of the function l/x. On the other hand, as seen at the end of Section 2, in particular in (1.2.19), the order of singularity of 6 at the point 0 E R may be quite high, if not even infinite, if the multiplication is considered in a DA which satisfies (1 -4.26).
In Propositions 3 and 4 next, results detailing these opposite possibilities are presented. Suppose that in addition to (1.4.27), the algebra following conditions
A
satisfies the
(1.4.29)
the multiplication in A induces on the monomials in (1 -4.28) the usual multiplication
(1.4.30)
1 is the unit element in A
(1.4.31)
D
applied to monomials in (1.4.28) coincides with the classical derivative of functions
Pro~osition3 (Rosinger [3] , p. 35) The following relations hold within the algebra A (1.4.32)
xP.~qb= 0 E A, p,q E IN,
(1.4.35)
(6)'
=
p > q
S+D6= 0 E A
Proof In view of (1.4.28)-(1.4.31) (1.4.36)
6 + x.D6
=
0E
D applied to (1.4.27) yields A
which multiplied by x, and in view of (1.4.29) and (1.4.27), yields
If we apply D to the latter relation and then multiply by x, we have in the same way
E .E . Rosinger
hence, by repeating t h i s procedure, (1.4.32) i s obtained. In view of (1.4.30) and (1.4.31), a repeated application of D t o (1.4.36) w i l l yield (1.4.31). Further, if we multiply (1.83) by (1.4.31) , we obtain
xP,
then,
Multipl ing t h i s l a t t e r relation by (1.4.327, we obtain (1.4.34). Finally, for p = 0 and
and applying (1.4.35).
D
q = 2,
in view of
(1.4.28)-
and taking into account
(1.4.34) yields
t o t h i s l a t t e r relation, i n view of (1.4.30), we obtain
Remark 2 The above degeneracy result in (1.4.35) i s not i n agreement w i t h various other results encountered and used i n the l i t e r a t u r e , see for instance Mikusinski [2] , Braunss & Liese. In particular, i n various distribution multiplication theories, as those for instance given by differential algebras containing the distributions, it follows that ( ~ 5 ) $~ a', hence (612 / 0, see for instance Rosinger [ I , p. 111 , Rosinger [2, p. 661 , Co ombeau [ I , p. 691 , Colombeau [2, p. 381 . I t follows that, i n case we embed P'(IR), or some of i t s subsets into a single differential algebra A , certain products involving the Dirac delta distribution or i t s derivatives, may vanish as in Proposition 3 above. However, i n view of (1.2.19) , in such algebras, we may have t o expect that
The next result in Proposition 4, shows that, under rather general conditions, we necessarily have in such algebras A the relations
which means that with the multiplication in such algebras, the Dirac delta distribution has an infinite order singularity a t the point 0 E W.
Algebraic conf1ict
19
It is p a r t i c u l a r l y important to mention that the above fact - which can be seen in p v r e l y a l g e b r a i c terms as well - conditions much of the way the L. Schwartz distributions can be embedded in differential algebras. Details on the most general forms of possible embeddings are presented in Chapter 6. Suppose given a DA which satisfies (1.4.26), as well as (1.4.39)
xmcA, M E N
(1.4.40)
1 is the unit element in A
Pro~osition4 (Rosinger [3] , p. 323) If S,S2,S3,... # 0
(1.4.42)
then we have for m
E
E
A
M
xm. S # O E A
(1.4.43) Proof
Assume that for a certain m xm+l.6 = 0
(1.4.44)
E
E
I, we have
A
Then we shall have (1.4.45)
xm.p = xm.63 =
Indeed, if p
E
I, p
xm+'.#
> 2,
=
0
E
A
then (1.4.44) yields
= 0 E A
hence by differentiation (m + l).xm.SP + p.xm+l.S~-l-~S = 0E A but p- 1
>
1, thus (1.4.44) yields (m + l).xm-Sp = 0 E A
and the proof of (1.4.45) is completed.
E.E. Rosinger
Starting with
obtained in (1.4.45), in a similar way we obtain
Continuing the argument, we end up with
which contradicts (1.4.42). The conclusion which emerges from Propositions 1-4 and Remarks 1-2 abovc is the following. The EAD setting in (1.3.1)- (1.3.4) contains a large variety of rather different, if not even conflicting possiblities concerning the ways to settle the purely a1 ebraic conflict between insufficient smoothness, multiplication and di ferentiation. And the outcomes of some of these ways may be unacceptable in certain circumstances. Indeed, under eneral conditions we can have the degeneracy property (1.4.35) of rather 6, name y
P
f
On the other hand, under similarly general conditions 6 may prove to be infinitely singular, namely
as follows for instance from (1.4.43). It is therefore desirable if we can find a sufficiently natural and general way the mentioned conflict can be settled. In Section 5 next, we present such a way, which is inspired by most elementary considerations concernin rings of functions. And we are led to these rin s of functions in a most irect way whenever we consider the concept of weai solution introduced in, and massively used since Sobolev [I ,2] .
%
The framework in Section 5 is the basis for the general nonlinear theory in Rosinger [1,2,3], as well as for its particular case in Colombeau [1,2]. The insistence on the 'algebra first ' approach proves to lead to a rather powerful tool, although at present we seem to be at the beginning of its fuller use and understanding, in spite of results such as those in Chapters 2 and 3.
Algebraic conf1ict
55.
CONSTRETION OF ALGEBRAS CONTAINING TEE DISTRIBUTIONS
So far, one of the major interest in sufficiently systematic nonlinear theories of generalized functions has come from the fact that, as seen in Chapter 5, important classes of nonlinear partial differential equations have solutions of applicative interest which are no longer given by suff iciently smooth functions, therefore, they fail to be classical solutions. To the extent that L. Schwartzfs linear theory of distributions or generalized functions does not allow uithin itself for the unrestricted performance of nonlinear operations, in particular multiplication, see Appendix 1 to Chapter 5, one should try to embed the distributions in larger algebras of generalized functions. Fortunately, a wide range of nonlinear operations - beyond multiplication - will equally be available within these algebras. For algebraic convenience, let us start with the class of polynomial non1inear partial differential equations, having the form
where Q c R" while
is nonvoid, open, pij E Oln
and ci,f E CO(Q)
are given,
is the unknown function. The order of the equation (1.5.1) is by definition
assuming that none of the ci by
vanishes on the whole of Q .
Let us denote
the partial differential operator generated by the left hand term in (1.5.1). Then obviously we have the mapping (1.5.5)
T(D)
:
Cm(a) -, Co (Q)
and a classical solution (1.5.2) of (1.5.3) is any smooth function
such that, for the mapping in (1.5.5), we have
E. E. Ros inger
22
Unfortunately, the important case of nonclassical solutions of (1.5. I ) , where
cannot be obtained through (1.5.5)- (1.5.7) . Therefore, the mapping T(D) in (1.5.5) has t o be extended beyond Cm(n), t o suitable spaces of generalized functions. The various ways the mapping T(D) in (1.5.5) has usually been extended beyond P ( Q ) , have been biased by an improper balance i n the handling of our often conflicting interests concerning representation and interpretation in mathematical models and theories, for further details, see Section 7, as well as Appendices 2 and 5 a t the end of t h i s chapter. Indeed, especially since Sobolev [1,2], there has been a natural intuitive tendency t o conceive of the needed extensions of P ( Q ) as being given by an embedding
where E i s a suitable topological vector space of generalized functions, obtained solely based on an interpretat ion of approximat ion. That i s , each generalized function
i s supposed t o be some kind of limit (1.5.11)
U = l i m $v V+m
of classical functions $v
E
cm(n), with v
E
IN.
In t h i s way, the excessive and early - in f a c t , a priori and exclusive stress on the mentioned kind of approximation type interpretation, has inevitably led t o the situation where topology alone i s supposed t o give us extensions E, such as i n (1.5.9). However, as seen in particularly useful approximation inter 'topology f i r s t ' , i
P
Rosinger t o avoid retation, not even
[I ,2,3] and Colombeau [I, 21 , it proves t o be such an early and exclusive stress on any stress which can easily lead t o the usual 'topology only' approach.
The alternative i s based on the realization that in mathematics, it may often be useful t o allow representation to g o farther than interpretation.
Algebraic conf 1i c t
This may also mean that we should s t a r t w i t h a rather general represent a t ion, and then l a t e r , become concerned w i t h interpretation. Fortunately from the point of view of simplicity and clarity, as seen next, such a course w i l l lead us t o an 'algebra f i r s t ' approach. In order t o have a better understanding of the kind of eneral representation we may need i n order t o construct E in (1.5.9), Pet us f i r s t have a look a t the usual way generalized solutions (1.5.8 are constructed for equations (1.5.7). This way, which can be called t e sequential method, has i t s systematic origin i n Sobolev [1,2], and proceeds as follows.
1
Based on specific features of the partial differential equation (1.5.7) an infinite sequence of so called approximating equations or systems of equat ions
are constructed, w i t h TV(D) being ordinary or partial differential operators. The essential point in the construction of the approximating equations (1.5.12) i s that they have sufficiently smooth classical solutions
where
i s a suitable topological vector space of funtions. Now, from 1.5.13) one can often extract a Cauchy sequence i n the given uniform top0 ogy on 3, by using compactness, monotonicity, fixed points or other arguments. Thus one obtains a Cauchy sequence of sufficiently smooth function
\
labeled for convenience with the same indices as i n (1.5.13) And now one i s supposed t o take a 'leap of f a i t h ' and declare the element, which often i s not a usual function ( 1 . 5 . 2 ) , but a generalized function (1.5.16)
U = lim $, E 7 U-b
t o be the generalized solution of the partial differential equation (1.5.7), that i s , one declares that U in (1.5.16) does solve the equation
where 7 i s the completion of
3 i n i t s given uniform topology.
24
E. E. Ros inger
In t h i s way, we are suggested t o choose in (1.5.9)
Let us now review from a more abstract point of view the representational structure above, and do so f o r the time being without too early and limiting attempts a t interpretation. As i s well known in general topology, in case the uniform topology on i s metrizable, i t s completion 7 can be obtained as follows
7
where
and S i s the s e t of a l l Cauchy sequences i n T, while V is the set of a l l sequences convergent t o zero in T, see also Appendices 2 and 5. I t follows that the general form of representation f o r spaces of generalized functions E in (1.5.9) can be expected t o be given by quotient spaces of the form
where
Here A
i s a given infinite index s e t , while
are suitable vector subspaces. Then, with the term wise operation on sequences, 76 i s i n a natural way a vector space.. Finally, V and S are appropriately chosen vector subspaces in 9. Obviously, within (1.5.21)- (1.5.23), E w i l l again be a vector space. In the case of solving nonlinear p a r t i a l d i f f e r e n t i a l equations (1.5.1), we shall be interested in extensions (1.5.9) with E r e laced by an a1 ebra of generalized functions. The above construct ion in 8.5.21)- (1.5.23f can easily be particularized for that purpose. Indeed, we can choose a suitable subalgebra of smooth functions
Algebraic conflict
where
A is a subalgebra in $, while
Z
is an ideal in
A. Then the
quotient algebra
can offer the representat ion for our algebra of eneralized functions. We should note that 2 in (1.5.24) is automatical y an associative and commutative algebra. Therefore, with the term wise operation on sequences, $ is in a natural way an associative and commutative algebra. It follows that the algebra A of generalized functions in (1.5.26) will also be associative and commutative.
P
At first sight, it may appear that there exists too much arbitrariness with constructions of algebras of eneralized functions such as in (1.5.24)(1.5.26). Such concerns will e addressed and clarified to a good extent in several stages in the sequel.
%
First, the so called neutrix condition will particularize the above framework in (1.5.24)- (1.5.26). This purely algebraic condition, dealt with in the next section, characterizes the natural requirement that
yet it proves to be surprisingly powerful.
A second kind of particularization will come from the obvious requirement that the nonlinear partial differential operators (1.5.5) should be extendable to the algebras of generalized functions A. This issue is dealt with in Section 9. Finally, a third necessity for particularization, presented in Section 10, concerns the important concepts of stabi 1 it y, general it y and exactness, which are naturally associated with the concept of generalized solutions of nonlinear partial differential equations. For further comment on the necessary degree of generality in (1.5.21)(1.5.23) , or in particular in (1.5.24)- (1.5.26) , see Appendices 3, 5, 6 and
7.
s6. THE NEUTRIX CONDITION Given in general a vector space 3 of classical functions $ : a + R, we have taken vector subspaces V c S c 9 and have defined E = S/V as a space whose elements U E E generalize the classical functions $ E 3. It follows that we should have a vector space embedding
E. E. Ros inger
defined by the 1 inear injection
where
i s the constant sequence with the terms
$A = $,
for A
We reformulate (1.6.1)- (1.6.3) in a more convenient form. 0 the null vector subspace in and l e t
E
A.
Let us denote by
9
(1.6.4)
= f"($)
be the vector subspace in of a l l the e o ~ s t a n t sequences of functions Then it i s in 3 , that is, the dia onal in the cartesian product easy t o see that (1.6.1)- 6 . 6 . 3 ) are equivalent with the inclasion diagram
3".
together with the off diagonality condition (1.6.6)
V fl
=
0
which we s h a l l c a l l in the sequel the neutrix co~dition. The name is suggested by similar ideas introduced e a r l i e r in Van der Corput, within a so called 'neutrix calculus' developed in order t o simplify and systematize methods in asymptotic analysis, see Appendix 4. With the terminolo y in Van der Corput, the sequences of functions v = ( ~ ~E (A 71 E Y are called lV-negligible'. In t h i s sense, given two functions o,B E 3,B t h e i r difference a-/3 E ? i s 'V-neglibible' if and only i f u(a) - u(B = ~(cr-/3)E V, which in view of (1.6.6) i s equivalent t o cr = /3. In ot e r words, (1.6.6) means t h a t the quotient structure E = S/Y does distinguish between classical functions in 3. Let us denote by (1.6.7)
VS?,~
Algebraic conf 1i c t
the set of a l l the quotient vector spaces and (1.6.6).
27
E = S/V which satisfy (1.6.5)
Since we are interested in generalized solutions for nonlinear partial differential equations, it i s useful t o consider the particular cases of the above quotient structures given by quotient algebras. For that purpose we proceed as follows. Suppose X is an algebra of classical functions $:fl X can be a subalgebra i n the algebra (1.6.8)
-4
W,
for instance
A@)
of a l l the measurable and a.e. f i n i t e functions $:fl + 03. with the tern wise operations on sequences of functions, only a vector space but also an algebra. Let us denote by
#
In that case, w i l l be not
#
where A c is a the s e t of a l l the qvotierat algebras A = AjZ, subalgebra and Z i s an ideal i n A such that the inclusion diagram
s a t i s f i e s the neutrix or off diagonality condition (1.6.11)
I n &T,A= 0
Obviously, if
3 i n (1.6.7) i s an algebra, then
(1.6.12)
AL3,~ "'?,A
I
I n general, i t follows that similar t o 1.6.1), (1.6.2), for every quotient algebra A = A / Z E A L ~ w e h a v e t h e a g e b r a embedding ,A
given by the injective algebra homomorphism (1.6.14)
2 3 $HU($) + T E A
Again, the conditions (1.6.10) and (1.6.11) are necessary and sufficient for (1.6.13) and (1.6.14). I n t h i s way, one obtains an answer t o (1.5.7). Here we should like t o draw the attention upon a most important f a c t .
In
E. E. Rosinger
the case of quot i e n t a l g e b r a s A = A/Z E ALXaA, the purely algebraic n e u t r i z or o f f diagoraalitg c o n d i t i o n (1.6.11) i i - p a r t i c u l a r l y p o ~ e r f u l ,although it appears t o be simple and t r i v i a l . Indeed, variants of the neut r i x condition characterize the e z i s t e n c e and s t r u c t u r e of a large class of c h a i n s o f d i f f e r e n t i a l a1 ebras of generalized functions, as seen i n Rosinger [2,3]. In particu a r , the neutrix condition determines t o a good extent the s t r u c t u r e of ideals Z which play a crucial role i n the stabil i t y , generalit and exactness properties of the algebras of generalized functions, see Jection 11 as well as Chapters 3 and 6.
1
5 7. REPRESENTATION VERSUS INTERPRETATION Let us s t a r t w i t h an example which can i l l u s t r a t e the fact that in mathematical modelling it may be useful t o allow representation t o go f u r t h e r than interpretation. Indeed, in the case of the infinite set I of natural numbers, the Peano axioms give a rigorous r e p r e s e n t a t i o n for a l l n E I. However, when it comes t o interpret various n E H , we cannot go too f a r . For instance, if we take
it i s hard to find for it an interpretation which may be satisfactory t o the extent t h a t , l e t us say, it could distinguish meaningfully between n above, and n + 1. And yet, n in (1.7.1) can be rigorously represented by using s i x digits only!
On the other hand, an integer n E I extent be i t s own interpretation.
which i s not too large can t o a good
The point t o note here i s that we f i r s t represent r i orously a l l and then, l a t e r , we only interpret some of the n E I .
P
n E I,
Certainly, ri orous computations can only be done based on rigorous representations. fnd representations which are f a r reaching and convenient w i l l allow for generality and ease i n computations. Let us now consider shortly a second example which i s nearer t o the quot ient algebras of generalized functions considered i n Sections 5 and 6, and used extensively I n the sequel. This example concerns one of the ways nonstandard numbers can be constructed, and as such, it i s an extension of the classical Cauchy-Bolzano construction of the real numbers, presented in Appendix 2 . For us, the interest in t h i s second example i s i n the fact that the way the representation of nonstandard numbers i s constructed i s t o a good extent f r e e of numerical, approximation, or in general, metric topological interpretat ions. Yet, as i s well known, the nonstandard numbers thus obtained prove t o be particularly useful.
Algebraic c o n f l i c t
The c o n s t r u c t i o n o f t h e nonstandard s e t t h e quotient f i e l d
*
where A ='Q usual r a t i o n a l [pp. 7-91. It i n t h e s e n s e of
*
*
Q
of r e a l numbers is g i v e n by
*
and Z is an i d e a l i n A, w h i l e Q is t h e s e t of t h e numbers, s e e Schmieden & Laugwitz, o r S t r o y a n & Luxemburg * follows e a s i l y t h a t Q i n (1.7.2) is a q u o t i e n t a l g e b r a ( 1 . 6 . 9 ) , t h a t is
Thus i n p a r t i c u l a r , we have s a t i s f i e d t h e n e u t r i z condi2ion, s e e (1.6.11)
and we have t h e embedding of a l g e b r a s , i n f a c t f i e l d s , g i v e n by
The important f a c t t o n o t e concerning (1.7.2)- (1.7.5) is t h a t , u n l i k e with t h e q u o t i e n t a l g e b r a c o n s t r u c t ion i n Appendix 2 , t h e q u o t i e n t represen* t a t i o n of elements of Q
cannot have m e t r i c , i n p a r t i c u l a r approximation o r numerical i n t e r p r e t a * tions. Indeed, Q c o n t a i n s nonzero inf i n i t e s i m a l s , * whose m e t r i c i n t e r p r e t a t i o n would of course have t o be z e r o . Then Q c o n t a i n s many d i f f e r e n t i n f i n i t e numbers, whose m e t r i c i n t e r p r e t a t i o n could o n l y be t m,
*
s i n c e Q is ordered. d e f i n e a mapping
What can however be done is t h e f o l l o w i n g :
*
*
one can
*
on t h e s t r i c t s u b s e t Qo of Q, such t h a t * s t ( x ) E [R w i l l be t h e s t a n d a r d p a r t of t h e nonstandard number *x E Qo. But t h e r e will be * * nonstandard numbers *x E Q\ Qo which do not have s t a n d a r d p a r t . F u r t h e r examples and comments on t h e p o s s i b l e r e l a t i o n s h i p between repres e n t a t i o n and i n t e r p r e t a t i o n can be seen i n Appendix 5.
E .E . Rosinger
30
By the way, it should be pointed out that the nonlinear theory of generalized functions as developed in Rosinger [I ,2,3] and Colombeau [I ,2] , has strong connections with nonstandard methods, although the a1 ebras of generalized functions constructed by the mentioned nonlinear t eory are much more large than the fields of nonstandard numbers. For details see Oberguggenberger [6] and the literature cited there.
%
We may perhaps conclude that, since Cantor's set theory, one of the most important powers of mathematics has come from its capability to rigorously represent. And to the extent that such representations could be so general and far reaching, one of our weaknesses as mathematicians has been in interpreting the powerful representations available. This however need not lead to a situation where our limitations in interpretation would come to dictate the way we may use the representational power of mathematics. Mathematical research is for the researcher a learning process as well. And one way to learn may as well come from trying to deal with the gap between representation and interpretation, without collapsing the former within the given limits of the latter.
$8. NONLINEAR STABILITY PARADOXES, OR BOW TO PROVE THAT 0' = 1 IN R
The nonlinear stability paradoxes mentioned in this section point to long existent basic conceptaal deficiencies of the customary weak solution method for nonlinear partial differential equations, as developed for instance in Lions [I ,2], and used in a wide range of later applications. Indeed, as shown next, according to that customary method, one can prove the existence of weak - and strong - solutions U for the nonlinear system of equations lJ = O
therefore admittedly proving that in IR, we have
For that, let us now present in some detail the mentioned customary weak solution method, sketched shortly in Section 5, see (1.5.12)-(1.5.18). Suppose given a nonlinear partial differential equation, see (1.5.1), (1.5.7)
Then, one constructs an infinite sequence of approximating equations (1.8.4)
TV(D)$V(~)= 0, x E O,
v
E
#
Algebraic conflict
which admit classical solutions
in other words 7 i s a vector space of sufficiently smooth functions, such as for instance in (1.5.14). Now, with a suitable choice of (1.8.4 , (1.8.5) and of a metrizable topology on 3, one can often find a Cauc y subsequence in (1.8.5), that i s
b
which for convenience was labeled w i t h the same indices as i n (1.8.5). Usually, t h i s subsequence s or fixed point argument.
i s obtained from a compactness, monotonicity
The final and most objectionable step comes now, when the limit (1.8.7)
U = l i m $u E 7 U+w
is declared t o be a solution of (1.8.3), where 7 i s the completion of in i t s given topology.
3
It should further be pointed out that, owing t o the usual d i f f i c u l t i e s involved in the steps (1.8.4)- (1.8.6), especially when i n i t i a l and/or boundary value problems are associated with the nonlinear partial differential equation (1.8.3), one ends up with one single or w i t h very few Cauchy sequences s i n (1.8.6). What happened can be recapitulated as follows: We have a sequence s in (1.8.6) such that s converges t o U,
and
T(D)s converges t o f Then, based alone on that, we feel entitled t o define U as being a generalized solution of the nonlinear partial differential equation
Now l e t us go back t o the framework i n (1.8.3)- (1.8.7) and see what the 'leap of f a i t h ' in declaring (1.8.7) t o be a solution of (1.8.3) amounts to.
It i s obvious that the above i s equivalent t o saying that the nonlinear mapping (1.5.5) has been extended t o a nonlinear mapping
E .E . Rosinger
32
i n such a way that t h i s extension s a t i s f i e s
where K i s a suitable topological vector space of functions or generalized functions on 0 . In t h i s way, the above customary method for finding generalized solutions t o nonlinear partial differential equations amounts t o nothing else but ad- hoc point- wise extensions of the type (1.8.8), (1.8.9) for nonlinear mappings (1.5.5). Now the deficiency of t h i s solution method i s obvious. Indeed, the nonlinear extension (1.8.8), (1.8.9) .was made based not upon a1 1 the Cauchy sequences t in 7 which converge t o the same given U E 7, but only upon the very few, quite often one single sequence s i n (1.8.6). Not t o mention that the sequences s in (1.8.6) and thus U i n (1.8.7), are usually obtained by a rather arbitrary subsequence selection from (1.8.5), such as for instance, compactness arguments. However one critical point i s often overlooked. Namely, that i n general, the nonlinear mapping (1.5.5) i s not compatible w i t h the vector space topologies on 3 and 1, which means that for sequences t E we have in general
?
(1 -8.10)
Cauchy in 3 #=> T(D)t Cauchy i n 2
t
And worse yet: Cauchy in 2, nevertheless
it may happen that, even if T(D)tl for two given Cauchy sequences t l l i m tl = l i m
7
t2
7
and and
T(D)t2 are tz i n 3,
f=> l i m T(D)tl = l i m T(D)t2
X
'X
A most simple example i n t h i s connection i s given by the zero order non 1 inear partial differential operator
connected w i t h the s t a b i l i t y paradoxes (1.8. I ) , (1.8.2). Indeed, if we 3 = P(R) with the topology induced by V' (R) , and we take take 'X = V ' ) then the sequence (1-8.13)
v = (xVlv E IN),
i s Cauchy in 3 and
&(x) =
8 cos
VX,
x
E
E,
v E IN
Algebraic conflict
b o t h lueakly and s t r o n g l y i n 7 = V(iR). (1.8.15)
Yet, we shall also have
lim Tv = l i m v2 = 1 X X
b o t h w e a k l y and s t r o n l y i n X = V (R) . Therefore, according t o the customary method (1.8.3-(1.8.7), the sequence v i n (1.8.13) defines b o l h a weak and s t r o n g s o u t i o n for the system
1
For the sake of clarity l e t us show the more g e n e r a l effects s t a b i l i t y paradoxes such as i n (1.8.16) can have upon the customary sequential approach (1.8.3)- (1.8.7). Suppose T(D) in (1.5.5) contains 112 as the only nonlinear term, and 112 has the coefficient 1, that i s
where L(D) i s a linear partial differential operator. Further, suppose that the system (1.8.16) has a solution i n 3, that i s , there exist a Cauchy sequence v E % such that (1.8.18)
l i m v = 0,
3
l i m v2 = 1 X
Let us take s in (1.8.6) which i s supposed t o define a solution U E 7 of (1.8.3) through (1.8.7). Then, for a given but arbitrary A E W l e t us define
Then obviously, have
t
i s a Cauchy sequence i n
7
and i n view of (1.8.7) we
lim t = lim s = U E ?
7
7
hence t defines the same generalized function U as given i n (1.8.7) by s . If now, as i s usual, the generalized meaning of (1.8.9) i s taken t o be (1.8.21) i t follows that
lim T(D)s = f X
E .E. Ros inger
hence (1.8.18), (1.8.21) yield l i m T(D)t = f X
(1.8.23)
t
A2
t
2A l i m sv
7
provided that (1.8.24)
sv E
9 i s Cauchy
and that also, as it happens w i t h many of topologies on 3 and 1, we have satisfied
the usually encountered
l i m L(D)v = 0 X
(1.8.25)
which i s for instance the case of the topology on coefficients in L(D) are f?- smooth. Since X
E
P(R"),
if the
R i s arbitrary, (1.8.23) yields
if in (1.8.20) we use the representation (1.8.27)
U = limt € 7 3
In that way, the very same U E 7 , once happens t o be the generalized solution of the equation T(D)U = f , as i n (1.8.9), while another time, if U E 7 i s given as i n (1.8.27), then i n view of (1.8.26), it happens t o be no longer a solution. It i s obvious from (1.8.23) that i n case (1.8.23) does not hold, the above situation in (1.8.26) remains the same. In case T(D) contains nonlinear terms i n U other than in (1.8.171, the overall situation remains the same, since s t a b i l i t y paradoxes simi a r t o (1.8.16) can easily be obtained for the respective nonlinear terms. Further examples and clarifications can be found i n Appendix 6. I t follows that in the case of n o n l i n e a r partial differential equations, the extension of the concept of classical solution t o that of the concept of generalized solution along the lines (1.8.3)- (1.8.7) i s an i m p r o p e r generalization of various classical extensions, such as f o r instance the extension Q c R of the rational numbers into the real numbers. Indeed, since the usual topology on Q i s compatible w i t h multiplication, nonrational solutions u E R of equations such as for instance
Algebraic conflict
can be defined by the sequential method
u = lim s
(1.8.29) where s
R
E 'Q
(1.8.30) Indeed, if
is any Cauchy sequence i n Q ,
such that
l i m s2 = 2 R
v E Q'
i s such that
l i m v = 0,
the Cauchy sequence
0
t = s + v E Q* w i l l again satisfy both (1.8.29) and (1.8.30). Unfortunately however, as seen above, the usual topologies encoutered on the spaces 3 and X' are not compatible with multiplication. One can wonder about the reasons the customary sequential approach t o generalized solutions for nonlinear partial differential equations has managed t o overlook the above deficiency. One likely reason i s that in the case of linear partial differential operators with sufficiently smooth coefficients, the above customary method 1.8.3)- (1.8.7) hap ens t o be correct because of what i s called i n genera the phenomenon o automatic coatinuity of certain classes of linear operators. Indeed, l e t us suppose that T(D) in (1.5.4) has the following particular linear form
\
with ci E assume that
e"(Q).
P
Suppose given a Cauchy sequence
s E
311
and l e t us
lim s = U E ?
7
(1.8.33)
l i m L(D)s = f 'X
Then, even if (1.8.32) and (1.8.33) have been obtained for one single sequence s , these two relations can nevertheless be interpreted as giving a generalized solution U E 7 of the equation (1.8.34)
L(D)U = f
i n 'X
Indeed, if we take any Cauchy sequence v E
such that
E .E. Rosinger
(1.8.35)
lim v = 0 7
and define the Cauchy sequence in 7 given by
then (1.8.35), (1.8.36) yield limt = U E ? ? while the linearity of L(D) (1.8.38)
L(D)t
=
gives
L(D)s + L(D)v
Now in view of the smoothness of the coefficients in (1.8.31), we usually have the automatic continuity property (1.8.39)
lim v = 0 => lim L(D)v 7 I
= 0
In this way (1.8.33), (1.8.35), (1.8.38) and (1.8.39) yield (1.8.40)
li~nL(D)t I
=
f
I
The relations (1.8.38)- 1.8.40) prove the validity of the interpretation in (1.8.34) irrespective o the sequences s in (1.8.32) and (1.8.33). It is now obvious that, if the linear partial differential operator L(D) in (1.8.31) is replaced by a nonlinear partial differential operator as in (1.5.4), then both crucial steps in (1.8.38) and (1.8.39) will in general break down. It is precisely this double break doun which has usually been overlooked when going from solution methods for linear partial differential equations to solution methods for nonlinear ones. In general, the extension of linear methods to nonlinear ones can often involve difficulties which require essentially new ways of thinking. A particularly illuminatin survey of several such well known extensions and of the mentioned kind o difficulties can be found in Zabuski.
!?
Let us now have a better look at the above nonlinear stability paradoxes and do it in a way which avoids the usual exclusive use of approximation or topological interpretations. In particular, let us consider these paradoxes in the terms of the representations in Sections 5 and 6, terms which are purely algebraic. For convenience, let us consider the nonlinear partial differential equation (1.5.1) in the following particular case
Algebraic conflict
with
ci,f E
e)(P), and
l e t us denote by
the corresponding nonlinear p a r t i a l d i f f e r e n t i a l operator. denote by, see Appendix 1, Chapter 5,
Further, l e t u s
which converge the s e t of a l l sequences of Coo-smooth functions on IRn weakly in D'(IRn) t o a distribution, respectively t o zero. Then Y" and s" are vector subspaces in ( P ( Rn) )N and the mapping
where, f o r s = ( $ v l ~E N) E
we f',
(1.8.45)
gV(x)o(x)dx, o E qIRn)
U(o) = l i m
define
V+m RIl
i s a v e c t o r space isomorphism. Now, the mentioned customary method f o r constructing weak solutions f o r nonlinear p a r t i a l d i f f e r e n t i a l equations can often be reformulated a s follows. One t r i e s t o find a solution U E P (Rn) of the equation (1.8.41) by constructing a sequence s = ($,I v E N) E 500 such that
while, in the sense of the mapping (1.8.44), we obtain in the same time (1.8.47)
s+p++U
In view of (1.8.42), it i s obvious that (1.8.48)
P(D) : (r(IRn))""
+
P(D)
n N ( P ( R 1)
generates a mapping
E.E. Rosinger
if applied term by term t o sequences of c-smooth functions. For a moment, l e t us assume that the form
P(D)
in (1.8.42) is l i n e a r , i . e . , it has
then it follows easily, see Appendix 1 , Chapter 5, t h a t compatible with the quotient structure P/P, that is
P(D)
is
thus, one can define t h e linear mapping
I t follows that in the linear case (1.8.49), it sufficies t o construct one single sequence s E 500 with the property (1.8.46), since f o r any other sequence t E 500 with t - s E P, the l i n e a r i t y of P(D) and t h e relations (1.8.46), (1.8.50) w i l l yield
However, t h e general nonlinear mapping (1.8.42) i s not necessarily compatible with the quotient structure p/P, since instead of (1.8.50), we may have
and also
Indeed, as seen with the weak and strong solution (1.8.13)- (1.8.15) of the nonlinear system (1.8.16), it follows that
theref ore
Algebraic conflict
in particular
In this way, (1.8.55) follows from (1.8.58); o r d e r nonl anear d i f f e r e n t i a l o p e r a t o r
even in the case of the z e r o
The relation (1.8.54) can be obtained in a similar way, using the argument that a2 $ V' , see Rosinger [I, p. 111, or Rosinger [2, p. 661 . Now, in view of (1.8.54) and (1.8.55), it is obvious that we cannot always have (1.8.53) in the case of a general nonlinear operator P(D) in (1.8.42). In this way, the customary weak solution method is invalidated in the general nonlinear case. As the above simple and eneral algebraic argument shows it, the mentioned nonlinear stability parafoxes can reach rather deep in the study of weak solutions of nonlinear partial differential equations. Therefore these paradoxes deserve as general and complete a study as possible. It is easy to see that one of the b a s i c r e a s o n s for the above nonlinear stability paradoxes are the relations in (1.8.57) and (1.8.58). Obviously, in order to reestablish the inclusion ' c ' in these two relations, we can follow one of the following t h r e e possibilities: First, to replace with a s m a l l e r vector subspace
Secondly, to replace V" with a l a r g e r vector subspace
Or thirdly, to replace VOO with any other suitable vector subspace
As seen in Chapter 8, Colombeau's method does in a way replace VOO with one particular, sma 1 l e r Z c V". On the other hand, the method in Rosinger [I ,2,3], presented shortly in Chapters 2, 3, 5 and 6 is based on a study of the more general third possibility mentioned above in (1.8.62). One of the first results of that study is that V" is t o o l a r g e to be suitable for n o n l i n e a r theories of generalized functions. This is also suggested by the following simple result.
E .E . Rosinger
Lemma 1
a,,
Suppose g i v e n any sequence of p o s i t i v e numbers t h a t au -+ m , when v -+ m . Then, t h e r e e x i s t sequences v = (,yVlu E IN) E
for
u 2 p,
u E IN,
with
p
E #
with
v
E
IN,
such
p, such t h a t
s u i t a b l y chosen, p o s s i b l y dependent on
a.
Proof Let us t a k e
/3'
E g(IR),
For A E R , A > 0,
t h e n obviously
>
y
0
>
0,
such t h a t
l e t us define
y ~ P(R)
by
0 and i n view of (1.8.64) we have
we have
t h u s we can assume t h a t with s u i t a b l y chosen A , (1.8.67)
I
y2 (x) dx > 1
IR Let us t a k e p E ( 0 , l ) cv, w i t h u E IN, by
and d e f i n e t h e sequences of p o s i t i v e numbers
F i n a l l y , l e t u s d e f i n e v = (xu 1 u E IN) E (C"(IR))
DI
b,,
Algebraic c o n f l i c t
If
a
E
P(R) t h e n obviously
hence, i n view of (1.8.66) and (1.8.68), t h e i n t e g r a l i n t h e l e f t hand term of (1.8.70) t e n d s t o z e r o , when v -I a. It f o l l o w s t h a t
Assume now a E P(R) and a
>
0 , t h e n obviously
hence (1.8.63) f o l l o w s i n view o f (1.8.67) and (1.8.68). The r e s u l t i n (1.8.63) shows t h a t t h e square v2 of a sequence v which converges weakly t o z e r o , can d i v e r g e weakly t o i n f i n i t y arbitrarily f a s t . I n connect ion with t h e henomenon of n o n l i n e a r s t a b i l i t y paradoxes it is u s e f u l t o c o n s i d e r t h e ollowing milder v e r s i o n of Lemma 1 above. Let u s d e f i n e t h e sequence of P - s m o o t h f u n c t i o n s on IR
!
(1.8.72)
w = (uvlu E #)
by (1.8.73)
uU(x) =
fi ~ ( v x ) ,x
E IR,
v E IN
where we have chosen (1.8.74)
o
E
P(lR), with o 2 0 and J$(x)dx
= 1
IR Then a g a i n , an easy d i r e c t computation will g i v e (1.8.75)
uU -+ 0 and ui
-+
6, when v
-+
oo
i n t h e sense of both t h e weak and strong topology i n P (R).
E.E. Rosinger
42
It follows that (1.8.72)- (1.8.75) gives a weak and s t r o n g solution for the nonlinear system
thus admittedly proves that, in addition t o the relation (1.8.2), we can have as well
One of the interesting differences between the above solutions of (1.8.16) and (1 .8.76) i s the following. The weakly and strongly convergent sequence v in (1 8 3 which solves the system (1.8.16), has highly oscillatory terms xu, when u -+ W. On the contrary, the weakly and strongly convergent sequence w in (1.8.73) has terms w,, which do not oscillate more and more frequently, when u + W . The same i s true for the sequence v in Lemma 1. As a final remark, we shoiild note that recently, there has been a certain limited awareness in a few specific instances of nonlinear partial differential equations of the possibility of the above kind of nonlinear stability paradoxes associated w i t h weak solution methods, see Ball, Murat, Tartar, Dacorogna, D i Perna, Rauch & Reed, Slemrod. However, the respective methods developed i n order t o avoid nonlinear s t a b i l i t y para,doxes of weak solutions present several important limitations. Indeed, f i r s t of a l l , these methods are developed within the traditional functional analytic view point, where alone t o p o l o g i c a l i n t e r p r e t a t ion i s used, with the consequent exclusion of the vast possibilities offered by more fundamental a l g e b r a i c r e p r e s e n t a t i o n s . The limiting vision implied by the usual unquestioned and automatic topological interpretation i s clearly expressed for instance in Dacorogna, [p. 41 . There, the sequence
u E (x) = sin X-E
,x
E
(0,2r),
6
> 0
and the nonlinear continuous function
are considered, for which one obviously has the weak convergence properties
0. That example, similar t o the one i n (1.8.13)- (1.8.16) above, when E i s claimed t o lead t o the conclusion: ' . . .Therefore in order t o obtain weak continuity one has t o impose some
...
Algebraic conflict
restrictions on the sequence {u'}
and on the nonlinear function f . . . ' .
In t h i s way, only certain particular types of nonlinear partial different i a l equations and sequential solutions can be dealt with. Not t o mention that there i s no attempt t o develop a comprehensive nonlinear theory of generalized functions, capable of hand1ing large classes of nonlinear partial differential equations. The effect of such particular and limited approaches based on topological interpretat ion - for instance, the Tartar- Murat compensated compactness and the Youn measure associated with weakly conver ent sequences of functions subjecte% t o differential constraints on an a1 e raic manifold - has been a distancin from the basic algebraic reasons un the nonlinear stabie clouding of what i n l i t y para axes. That distancing has fact prove t o be rather simple ring theoretic phenomena.
!
$9. EXTENDING NONLINEAR PARTIAL DIFFERENTIAL OPERATORS TO GENERALIZED
FUNCTIONS I t has been noted that the basic and rather elementary algebraic conflict between discontinuity, multiplication and differentiation presented i n Section 1 leads t o the setting i n (1.3.1)- (1.3.4), where A and 1 are algebras of generalized functions extending the distributions. The way such algebras can be constructed i s shown in (1.5.24)- (1.5.27), and equivalently, i n (1.6.9)-(1.6.11). As mentioned a t the end of Section 5, the generality of the above constructions comes t o be subjected t o several natural particularizations. The f i r s t of them, dealt with in the section, i s imposed by the way polynomial nonlinear partial differential operators T(D) in (1.5.5) can be defined as acting between such spaces of generalized unctions. In order not t o miss on any of the possibly relevant phenomena involved, we shall approach the problem of the extension of T(D) t o spaces of generalized functions within the most general framework of the respective spaces. For t h a t , it i s easy to observe that one can obtain an extension
and A = ,411 E ALXYA , see (1.6.7), for suitable E = S/V E VS 34 (1.6.9 , provided that one can define the following extensions of the usual part i a derivatives
1
(1.9.2)
D~ : E
where m
i s the order of
-+
A,
p
E
@In, Ipl 5 m
T(D) , see (1.5.3)
Indeed, in such a case, for given
U E E,
the nonlinear, that i s poly-
E .E . Ros inger
44
nomial operations involved in T(D)U w i l l take place not in the domain E but in the range A of T(D) in (1.9.1). It follows that the most general framework f o r the extension of T(D) can involve different spaces f o r the domain and the range, and only the range has to be an algebra. It should be noted however that further extensions of the framework in (1.9.1) are s t i l l possible and useful, see Chapter 4. Let us now see the way extensions (1.9.2) can be obtained. do that i f we make the following natural assumptions
DPv c 2, DPsc A,
(1.9.4)
p EM".
We can easily
IpJ 5 m
Indeed, in t h i s case (1.9.2) can be defined by
DPu= DPs +
(1.9.5)
2 E A = A/2,
p E bin,
Ipl 5 m
f o r every (1.9.6)
U=S+VEE=S/V, S E S
where for every
we define
However, the above method f o r extension by reduction t o representants can be used in the followin less restrictive manner. Let us suppose that the vector subspace 3 c and the subalgebra T c Y(O) are such that
~(4
We note that (1.9.9) holds whenever X I
CO (A),
see (1.9.3).
Suppose now that
o
op(v n ( ~ ~ ( n )c) 2, ~ ) DP(S n ( ~ ( n ) ) " c A, p E wn,
IPI 5 m
We also note that (1.9.10) hold whenever (1.9.4) is s a t i s f i e d . It i s obvious that in the conditions (1.9.9) and (1.9.10), we can replace with any I E N = I U {m). In t h i s general case, it w i l l be convenient
m
Algebraic conflict
45
in the sequel t o denote (1.9.9) and (1.9.10) together under the simpler form
Finally, l e t us denote by
the set of a l l the quotient vector spaces E = S/V E VS3
We note that (1.9.13) holds whenever 3 c c'(Q),
,A
such that
see (1.9.3).
e Similar t o (1.9.12), we can also define ALTyA. Now the above definition (1.9.5)- (1.9.8) of the p a r t i a l derivatives (1.9.2) can further be extended as follows. Suppose given
such that
Then we can define the partial derivatives of generalized functions as being given by the linear operators (1.9.16)
D~ : E --+ A, p E lIn,
(pl 5 m
as follows: f o r given (1.9.17) we define
where
U=S+VEE=S/V, S E S
E.E. Rosinger
46
It is easy t o see that the above definition (1.9.16)- (1.9.19) of p a r t i a l derivatives of generalized functions i s correct and it contains as a part icular case the previous definition (1.9.5) . the Furthermore, when restricted t o classical functions in 7 n "(ill, p a r t i a l derivatives in (1.9.16) coincide with the usual ones. Fina l y , the partial derivatives (1.9.16) are 1 inear mappings, and if 7 is a subalgebra and E = S/V i s a quotient algebra, then they s a t i s f y in A the Leibnz tz rule of product derivatives. Now we can return t o the problem of defining an extension (1.9.1) The framework w i l l of course be the same with the above used in defining p a r t i a l derivatives f o r generalized functions. Namely, we suppose given
such that
where m i s the order of T(D). Further, we shall also assume that the coefficients in T(D) , see (1.5.4), satisfy (1.9.22)
cis%, 1 < i < h
which obviously holds whenever
CO (R) c
X,
see (1.9.3).
Now we can define the ex2ension (1.9.23)
T(D)
:
E
-4
A
as follows: f o r given (1.9.24)
U=S+VEE=S/V,s s S
we define (1.9.25)
T(D)U = T(D)t + Z E A = A/Z
where
Let us show that the above definition (1.9.23)- (1.9.26) i s correct. First we note that owing t o i t s polynomial nonlinearity, T(D) s a t i s f i e s the following relation f o r every z,w E (Cm(iI))'
Algebraic conf 1i c t
. while pa where z are products of ci, D pi jz and possibly DP.lJw, are some of the p i j In view of (1.9.27) we obtain the following succession of implications: i f (1.9.28)
E
sn
( ~ ( n ) ) ~w , E
v n (P(n))"
then
theref ore (1.9.30)
T(D)(z + w) - T(D)z
E
I
Now l e t us take s~ E S, s l - s E V, tl E S n ( ~ " ' ( a ) ) ~ ,t l - s , alternative representations in (1.9.24)- (1.9.26). Then obviously z = t E S n (cm(n))hnd w =
tl
- t =
E
V
for
( t l - s1) - ( t - s ) + (sl - s ) E V
hence (1.9.30) yields T(D)tl - T(D)t E 1, which proves that the definition in (1.9.24)- (1.9.26) does not depend on the representants s or t. I t i s easy t o see that the restriction of the extended T(D) in (1.9.23) t o classical functions in 3 fl cm(a) a c t s in the same way with the usual nonlinear p a r t i a l differential operator in (1.5.5). 1 0 NOTIONS OF GENERALIZED SOLUION Given the above constructed extension t o generalized functions
of the polynomial nonlinear p a r t i a l d i f f e r e n t i a l operator T(D) in (1.5.5), it is now a rather simple matter t o define a notion of generalized s o l u ~ i o nf o r the equation
as being any (1.10.1).
U
E
E
which will satisfy (1.10.2) with
T(D)
defined in
E.E. Rosinger
However, we should not miss the fact that there are less simple phenomena involved here. Indeed, if we are given a nonlinear partial differential equation, such as for instance in (1.5.1), that equation is prior to, and therefore independent of the various possible generalized function spaces involved in extensions such as those in 1.10.1). And obviously, there can be a large variety of such generalized unction spaces which could appear in these extensions.
i
The utility of considering an equation (1.5.1) within different extensions (1.10.1) will become obvious in Section 11 in connection with the nonlinear stability, generality and exactness properties of generalized solutions. Here however we should like to recall the third element which, in addition to the partial differential equations and their possible eneralized solutions, does in a natural way belon to the picture, and w ich is constituted from the various specific so ution methods. Such specific solution methods, which quite often encompass a wealth of mathematical, physical and other insight and information, usually lead to sequences or in general, families
B
E
C Q -+ R, with X E A, which are of sufficiently smooth functions supposed to define in certain ways - often, by approximation - classical or generalized solutions U, see for instance (1.8.3)- (1.8.7). It should be noted that the nonlinear stability paradoxes point out the questionable way generalized solutions U are associated with families s in the customary sequential approach for solving nonlinear partial differential equations. What the families s and the methods which lead to them are concerned, they may have their own merits, depending on the particulars of the situation involved. In view of the above, we shall define now a solution concept which focuses on such families s in (1.10.3). The interest in such a solution concept is in the fact that it eliminates the problem of nonlinear stability paradoxes, since the association s -+ U of a single family s with a generalized function U takes place in the framework of an extension (1.10.1 , see details in Section 12. Moreover, it allows a deeper study of t e stability, generality and exactness properties of generalized solutions for nonlinear partial differential equations, see Sect Ion 11.
1
Suppose T(D)
in (1.5.5) has order m and we are given the equation
in which, for the sake of generality, we can assume this time that f E M(0). Further, suppose given a family
ci,
Algebraic conflict
of functions
dA E M(O), with A
E A.
Then s i s called a sequential solution f o r (1.10.4), i f and only i f there exists a vector subspace 7 and a subalgebra I in A@), as well as m E = S/V E VS. with E 5 A , such that and A = ,411 E ALI,A, ?,A (1.10.6)
C I , .. . ,ch, f E I
and f o r T(D)
in (1.10.1) we have
where
When it i s useful t o mention the spaces E and A of generalized functions in the above definition, we shall say that s i s an E --+ A sequential solution for (1.10.4). In view of (1.9.24)-(1.9.26 , it i s easy t o see that (1.10.7) and (1.10.8) are equivalent with the con ition
d
which i s further equivalent with the following two conditions (1.10.10)
SES
and
Remark 3 Usually, it i s considered convenient t o solve p a r t i a l d i f f e r e n t i a l equations with the respective p a r t i a l differential operators actin within one single space of generalized functions. With the notation in $1.10.1) and in the case of a nonlinear p a r t i a l differential operator T(D), that would mean the particular situation when
E.E . Rosinger
As seen in Chapter 8, Colombeau's nonlinear theory of generalized functions leads to such a situation, where the linear or nonlinear partial differential operators are acting within the same space of generalized functions
It should nevertheless be mentioned that, even in the case of linear partial differential operators, the utility of different domain and range vector spaces of generalized functions is well documented in the literature. Hormander, Treves [I,2,3] . However, it is important to point out that, as shown in Rosinger 61,2,3] and presented in the sequel, see in particular Chapters 2, 6 an 7, a proper handling of such difficulties as the nonlinear stability paradoxes and the so called Schwartz impossibility result is facilitated if we consider that the nonlinear partial derivative operators T(D) act within the following particular case of (1.10.I), given by
where
and
which is more general than (1.10.2) or (1.10.14). It should be mentioned in that the utility of considering different algebras A and A' (1.10.15) does ultimately lead to the consideration of infinite chains of such algebras of generalized functions, see Chapter 6. § 11 . NONLINEAR STABILITY, GENERALITY AND EXAmNESS
Now we come to the three basic pro erties which lead to the necessary structure of any nonlinear theory o generalized functions based on the sequential approach initiated earlier in Section 5. The associated not ions of stability, generality and exactness have first been introduced in Rosinger [2], where further details can be found. These three properties which relate to generalized solutions as well as the respective spaces of generalized functions are essent ial for a proper handling of the problems which arise from the nonlinear stability paradoxes and the so called Schwartz impossibility result.
!
Algebraic conflict
Suppose given the framework in (1.9.20)- (1.9.22), in which case we can define, see (1.9.23), the mapping
Then the generalized solutions of the nonlinear partial differential equation
have the form
Let us have a better look at the relationship between U and s in (1.11.3). In view of (1.10.11), for the same U kept fixed, we can replace s with any t which satisfies the conditions
Therefore, it is obvious that the maximal stability of U means (1.11.5)
maximal V
Remark 4 Here it is most important to point out that owing to the nevtrix condition (1.6.6) which has to be satisfied by the spaces of generalized functions, it is obvious that one cannot speak about the largest V, since the off diagonality condition (1.6.6) means that V has to be contained in some vetor subspace which is complementary to UFSA. Similarly, one cannot speak about the largest ideals 1 which satisfi the respective neutrix, or off diagonality condition (1.6.11), see for details Appendix 8. This fact alone is sufficient reason to expect that a proper approach to eneralized solutions of nonlinear partial differential equations requires t e consideration of variovs spaces of generalized functions, since a canonical space of generalized functions does not appear in a natural way.
k
Now, if we go further, we note that the neutrix condition (1.6.6) appears in connection with the embedding (1.6.1) which expresses the requirement that classical functions should be particular cases of eneralized functions. Or in other words, eneralized functions should e general enough in order to include classica functions.
B
f
Owing to the well established role of the L. Schwartz distributions D ' ' in the study of generalized functions, we could ask the following stronger version of the embedding condition (1 - 6 .I), namely
E.E. Rosinger
52
which is also satisfied by Colombeau's generalized functions, see details in Chapter 8. It should be pointed out that, owing to the inexistence of solutions of certain partial differential equations within particular spaces of gene, see Rosinger [3, Part 1, ralized functions, such as for instance there exists an Chapter 3, Section I, or Part 2, Chapter 2, interest in large spaces of eneralized functioSrtp: :)3 which can offer a satisfactory Ireservoirl for the existence of generalized solutions U. In this way we are led to a second quality of the spaces of generalized functions E = S/V called in the sequel general it y, and meaning (1.11.7)
large E
=
S/V
or equivalently (1.11.8)
large S and small V
Remark 5 (1) In view of (1.11.5) and (1.11.8), it is obvious that stability and generality are conflicting. It follows in particular that there is no interest in maximal generality, unless one is ready to sacrifice stability. (2) In order to obtain a generality for a quotient space E = S/V which is not less than that in (1.11.6), it is obvious that A in (1.6.5) cannot be finite even if 3 = U(Q). However, since even such a small space as P(Q) is sequentially dense in Vf(Q), with the usual topology on the latter, one can obtain (1.11.6) for any infinite index set A, whenever for instance p(Q) c 3. Finally, we come to the third quality of spaces of generalized functions. if and E Cm(Q), We note that equation (1.11.2) has a classical solution only if there exists t E (c~(Q))~ such that $J
Further, a family of classical functions s = (tA 1A sequential solution of (1.11.2), if and only if
E
A) E (,U(Q))l
is a
Algebraic conflict
for certain spaces of generalized functions Section 10. Since 1 i s an ideal i n A,
53
E = S/V and
A = A/2
as i n
condition (1.11.10) w i l l obviously yield
I n other words, the error i n solving the equation (1.11.2), and which i s given by
satisfies the explicit algebraic tests
We note that i n the terms of the neutrix calculus, see Section 6 and Appendix 4, condition (1.11.13) means that the error wt i s 2- negligible, while condition (1.11.14) means that each 'projection' zswt of the error wt, with z E A, i s also 2-negligible. Obviously, if 1 E 'X then u(1) E A, according t o (1.6.10). Thus (1.11.14) w i l l imply (1.11.13). We shall c a l l the above algebraic test on error i n (1.11.13) and (1.11.14) the exaciness property of the sequential solution s . Obviously, better exactness means (1.11.15)
large A and small 2
Remark 6 As seen i n (1.11.9), classical solutions have the best exactness property which corresponds t o the smallest 2, that i s Z = 0, and thus t o the largest A given by A = #. That situation can no longer occur w i t h nonclassical, that i s , generalized solutions. Indeed, i n view of the inclusions
m i n (1.9.10) which appears i n connection w i t h the condition E < A between the spaces of generalized functions in (1.11.1), it i s obvious that stability and exactness are conflicting.
E .E . Rosinger
54
We can conclude that the above mentioned conf 1 ict between stability on the one hand and generality and exactness on the other, sets up a rather sophisticated inferplay between these three properties, see Fig 1 below. The way that interplay is handled can depend to a large extent on the particulars of the situations involved in connection with the nonlinear artial differential equations under consideration, see details in Rosinger !2,3] . maximal stability = = maximal V
T(D
and minimal V
=
maximal
A
and minimal Z
where E
=
S/V
E VS;,~,
A
=
#I
E
ALy,A
m and E 5 A Figure 1
We should also note the following. Both stability and generality refer exclusively to the given space E = S/V in (1.11.1) and are independent of the linear or nonlinear partial differential operator T(D) and any of its eneralized solutions U E E = S/V. On the other hand, the conditions 1.11.13) and (1.11.14) defining exactness, involve both spaces E and A, as well as the linear or nonlinear partial differential operator T(D) and its sequential solution s.
7
It is easy to see that, if we replace the framework (1.11.1) with the more particular one in (1.10.12), the conflict between stability, generality and exactness does not become easier, and on the contrary, their interplay has more constraints. It should be remembered that, no matter how useful particular spaces of generalized functions may be, the primary interest is with the linear or nonlinear partial differential equations and their classical or generalized solutions which model physical and other processes. The variety of spaces of generalized functions as well as solution methods are only the means constructed to handle the above primary interest.
Algebraic c o n f l i c t
ALGEBRAIC SOLJTION TO THE NONLIMAR STbBILITY PARADOXES
12.
Here we show, based on a purely algebraic argument, t h a t i f we solve nonl i n e a r p a r t i a l d i f f e r e n t i a l equations within frameworks such a s i n Sect ion 10, then t h e mentioned kind of s t a b i l i t y paradoxes cannot occur any longer. Suppose iven the m- t h order polynomial nonlinear p a r t i a l d i f f e r e n t i a l oper a t o r T$D) i n (1.5.4) and any of i t s extensions
m where E = S/V E VS;,~, A = d/I E ALX and E A , while ,A a vector subspace and X c M(n) i s a subalgebra, such t h a t
<
C ~ E Xf o r l < i < h
(1.12.2)
In order t o avoid t r i v i a l cases, we assume i n (1.5.4) t h a t (1.12.3)
k i > l
for l < i < h
Now we show t h a t we always have (1.12.4)
U=OEE+T(D)U=OEA
Indeed, l e t us assume t h a t U=S+VEE=S/V, S E S
(1.12.5)
then i n view of (1.9.25)
, (1.9.26) ,
we obtain
T(D)U = T(D)t + Z E A = d/Z
(1.12.6) where
t E
(1.12.7) But
U = 0
(1.12.8)
E
sn
( ~ ( n ) ) ~t ,- s E
E and (1.12.5) yield S E V
hence (1.12.7) implies
v
3
c M(n) i s
.
E .E Rosinger
m Now we recall that E
< A,
~~t E I, p
(1.12.10)
hence the relations (1.9 .lo) , (1.12.9) yield E MI',Ipl
<_ m
Finally, (1.5.41, (1.12.2) and (1.12.10) will give (1.12.11)
T(D)t
E
Z
which completes the proof of (1.12.4) From (1.12.4) it follows in particular that systems such as (1.8.1) or (1.8.76) cannot have solutions within any frmework (1.12.1) . It is easy to see that similar results will also hold for the general nonlinear partial differential operators in Section 13 next. Within the particular nonlinear theory of generalized functions in Colombeau [1,2], nonlinear stability paradoxes of the kind of those in (1.8.1) or (1.8.76) still can happen, see Chapter 8. However, their unwelcome effects can be satisfactorily handled with the help of a special equivalence relation, called association, defined for a large subset of Colombeau's generalized functions. 3
GENERAL NONLINEAR PARTIAL DIFFERENTIAL EqUATIONS
The particular polynomial form of nonlinearity of the partial differential in (1.5.4) was assumed because it made it easier to set up Operators natural ramework in Sections 5, 6, 8-10. the
TiD)
However, with minor modifications , that framework can accommodate much more general nonlinear partial differential operators, which are defined now. An m-th order continuous nonlinear partial differential equation is by definition of the form
where with
f
E
CO(ll)
is given, while
p
E
Oln,
lpl
<m
and
T
E
CO(fl
x
R'),
The left hand term in (1-13.1) generates the nonlinear partial differential operator
defined by
Algebraic conflict
Obviously, the class of a l l T(D) above contain as a particular case those defined in (1.5.4). Our aim i s t o define f o r (1.9.23).
T(D)
above extensions similar t o those in
Given the spaces of generalized functions and we denote
i f and only if the following three conditions are s a t i s f i e d
and
An example f o r the way t h i s l a t e r condition (1.13.8) i s s a t i s f i e d in applications can be seen in Chapter 2. T(D) It i s easy t o see that in case E 5 A, one can define the extension
where
Once the extension (1.13.9) was defined, one can easily extend the notion of sequential solution defined in Section 10 t o the case of T(D) given in (1.13.3). The connection between the framework above and that in Sections 9, 10 i s obvious. Indeed, suppose iven the m- t h order polynomio E nonlinear differential operator in (1.5.4) and the extension in (1. S . 2 3 r t ial
T(DP
E .E . Rosinger
58
where E
=
m
,
SJVE VS;,~
A
A
and E
A
< A.
In that case we obviously have
therefore (1.13.126 is an extension in the sense of (1.39) as well. Indeed, (1.9.9) an (1.5.4) yield (1.13.6), provided that
Further, (1.9.10), (1.5.4) and (1.13.14) yield (1.13.7). follows from (1.9.10) .
Finally, (1.13.8)
Further details as well as applications can be found in Rosinger [2] , where it is also shown the way variable transforms are treated within the general framework presented in Sections 9-12.
4
SYSTEMS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
For convenience, we shall only consider systems of polynomial nonlinear partial differential equations. The extension of arbitrary continuous nonlinear systems of partial differential equations follows easily by a direct application of the method in Section 13. Suppose we are given a polynomial nonlinear system (1.14.1)
where U = fg E tion
.X cBi(x) T T D l
(4, .. .Ua)
:
P(n)are given.
n
-
~
"pi j
~(x) = ~fp(~),~
ut n, l g < b
lRa are unknown functions, while cgi,
The order of the system in (1.14.1) is by defini-
provided that none of the coefficients cBi Let us denote
vanishes everywhere on
a.
Algebraic c o n f l i c t
(1.14.3)
Tp(D)U(x) =
T 'pi(') l
TT
1ijSkpi
D
~ 'ai j
(x) ~ , ~x
€
0~
u
o r , i n a more compact form (1.14.4)
T(D) = (TI ( D l ,
,Tb(D))
which obviously generates a mapping (1.14.5)
T(D) : (C'"(Q))~
(m-01~
It follows t h a t a classical solution of (1.14.1) is any family of functions
f o r which t h e mapping (1.14.5) s a t i s f i e s (1.14.7)
T(D)$ = f
on Q
where (1.14.8)
f = (fl
,. .. , f b )
In analogy with (1.10.5), t h e sequential solutions of (1.14.1) will by d e f i n i t i o n be given by families of functions (1.14.9)
s =
($/\I/\
E
A)
with h a given infinite s e t , and
In t h a t way
The precise d e f i n i t i o n can be obtained in analogy t o t h e one in Section 10 a s follows. F i r s t , we rearrange t h e functions in s , transforming t h e family of funct i o n s i n (1.14.11) i n t o a family of functions
E. E. Ros inger
60
Suppose now given 7 and X respectively a vector subspace and a subalgebra in M(Q), such that (1.9.9) i s s a t i s f i e d , and the coefficients cai a s well a s the right- hand terms f in the system (1.14. I ) , belong t o
a
I . Suppose further given E = S/V E VS;,~ and A = A / I E AL such t h a t X,A m E 5 A . Then, the mapping (1.14.5) can be extended t o a mapping (1.14.14)
T(D) : Ea --+ A b
defined by (1.14.15)
T(D) (st +V,
. . . ,sa+Y)
= (TI (D) t+Z,
.. . ,Tb(D)t+Z),
sl
, . . . ,sa€ S
where
. ..,ta)
(1.14.16)
t = (tl ,
and
in (1.14.19) is the term by term application of the mapping
Tp(D)ta
generated by (1.14.3) t o the sequence of functions t,
E
(P(Q))'.
A family of functions
i s called a E 4 A seqvenlial solvtion of the system in (1.14.1), i f and only if the mapping (1.14.15) maps
into
where
Algebraic conf1ict
It is easy to see that the above condition is equivalent to
which is the same as the following condition
With the above definition of a sequential solution for a system of polynomial nonlinear partial differential equations, it is obvious that the study of such solutions can actually be reduced to the simultaneous study of a finite number - in this case given by a - of sequential solutions in the sense of Section 10, which satisfy a finite number - in this case given by b - of polynomial nonlinear partial differential equations.
.
E .E Ros i n g e r
APPENDIX 1 ON HEAVISIDE FUNCTIONS AND TEEIR DERIVATIVES Suppose X is a set with at l e a s t two elements, w h i l e A is an a s s o c i a t i v e and commutative a l g e b r a on IR, w i t h u n i t 1, and with 1 # 0 E A. Let
be a subalgebra o f f u n c t i o n s f:X -+ A, such t h a t t h e c o n s t a n t f u n c t i o n s 0 , l : X -+ A belong t o 3. A f u n c t i o n H : X -+ A w i l l b e c a l l e d of Heaviside type, i f a n o n l y i f
where (Xo , X ) is a p a r t i t i o n o f X i n two nonvoid s e t s . Obviously, t h e 1 c o n s t a n t f u n c t i o n s 0 , 1 and t h e Heaviside t y p e f u n c t i o n s a r e s o l u t i o n s of t h e i n f i n i t e system of e q u a t i o n s i n f E 'A
We s h a l l now r e q u i r e t h a t
Finally, l e t
(l.Al.5)
D : 3+3
be a d e r i v a t i v e , t h a t is, a l i n e a r maping which s a t i s f i e s t h e L e i b n i t z r u l e of product d e r i v a t i v e
Then it f o l l o w s t h a t we always have
(l.Al.7)
D O = D I = D H = O E ~
Therefore, t h e d e r i v a t i v e D f a i l s t o t a k e n o t i c e of t h e nonconstancy and ' d i s c o n t i n u i t y ' of H. Indeed, t h e f a c t t h a t DO = 0 E 3 f o l l o w s e a s i l y from t h e l i n e a r i t y of D : 3 + 3. F u r t h e r , D l = 0 E 3 f o l l o w s from (1.A1.6) by t a k i n g f = g = 1 E 3 i n t h a t r e l a t i o n .
Algebraic c o n f l i c t
63
The r e l a t i o n s DO = D l = 0 E 3 a r e t o be expected, s i n c e a r e c o n s t a n t f u n c t i o n s . On t h e o t h e r hand, t h e r e l a t i o n
0,l : X
-r
A
r e s u l t s i n a similar way t o t h a t used i n proving (1.1.10). The conclusion is t h a t a d i f f e r e n t i a l framemork ( 1 .A1.9)
(X,A,~YD)
such as i n (l.A1.l)- (1.A1.6) is too c o a r s e i n o r d e r t o be a b l e t o t a k e n o t e of t h e nonconstancy o r ' d i s c o n t i n u i t y ' of Heaviside t y p e f u n c t i o n s , and t h e r e f o r e it cannot d i s c r i m i n a t e between t h e c o n s t a n t and nonconstant s o l u t i o n s of (1. A1.3) .
It is i n t e r e s t i n g t o n o t e t h a t t h e a l g e b r a i c r o o t s o f t h e f a i l u r e i n 1.A1.7) go s t i l l deeper. Indeed, let u s r e p l a c e t h e d i f f e r e n t i a l ramework i n (1.A1.9) with t h e f o l l o w i n g more g e n e r a l one
I
(1. A1.lO)
(39 D)
where 3 is an a s s o c i a t i v e and commutative a l g e b r a on R, w i t h u n i t 1, and w i t h 1 # 0 E 3. Thus 3 need not be a n a l g e b r a o f f u n c t i o n s as i n ( l A l . 1 ) . F u r t h e r , we assume t h a t D : 3 4 3 is l i n e a r and s a t i s f i e s t h e L e i b n i t z r u l e ( 1 .A1.6). Now, i n t h e proof of (1.1. l o ) , we can n o t e t h a t one o n l y needs t h e r e l a t i o n
f o r c e r t a i n p,q
EN,
2
<
q.
Therefore, we c a l l a n element H E 3 t o be o f extended Heaviside type, i f and o n l y i f it s a t i s f i e s ( l . A l . 1 1 ) . It f o l l o w s t h a t w i t h i n t h e framework i n (1. A 1 . l o ) , t h e r e l a t i o n (1.A1.12)
DO=DI=DH=OE?
w i l l hold f o r every extended Heaviside t y p e element H
E
?.
F i n a l l y , we i n d i c a t e t h e n a t u r e o f t h e p r o p e r t y (1.Al.11) which d e f i n e s t h e extended Heaviside t y p e elements i n 3. I f f o r g i v e n H E 3, we have
t h e n obviously
E.E. Rosinger
hence ( l . A l . l l ) i s satisfied. Further, i f for given (1 .A1.13) does not hold, but we have (1.A1.15)
H
E
7
the relation
lip = H
for a certain minimal p
E
I, p
> 3,
then obviously
and in general (1 .A1.16)
H ~ ( P - ' ) + '= H , . E M ,
m > 1
thus (1 .A1.12) i s again satisfied. I t follows that (1.A1.13) or (1.A1.15) are the only cases when (1 .A1.12) can hold.
Algebraic conflict
APPENDIX 2 THE CAUCBY-BOLZANO QUOTIENT ALGEBRA CONSTRUCTION OF THE REAL NUMBERS A classical example of quotient algebra construction i s given by the well known Cauchy-Bolzano method, van Rootselaar, for constructing the set of real numbers [R from the set of rational numbers Q . Let us denote by
A the subalgebra i n QM of a l l the Cauchy sequences r = (ro , r ,~. . . , r v , .. .) of rational numbers and l e t us denote by
Z the ideal in A of a l l the sequences z = (zO, z , , . . . ,zy, . . .) numbers which converge t o zero.
of rational
We shall denote by
the subalgebra i n A of a l l the constant sequences. Q the null ideal i n QM.
Finally, we denote by
Then the following inclusion diagram i s valid
and it s a t i s f i e s the condition (1.A2.2) Moreover, according t o Cauchy- Bolzano, we have that (1.A2.3)
W and A
=
A/Z are isomorphic fields.
As seen i n Appendix 4 , the condition (1.A2.2) above means that Z i s a aeutrix i n Q' and the corresponding neutrix 1 imit i s identical w i t h the usual limit for rational numbers, i . e . , the relation holds
E.E. Rosinger
66
Z - l i m r, = l i m r,
(1.A2.4)
Y+m
whenever exists.
r = ( rr
The i d e a l Z not i c e t h a t (1.A2.5)
V*
,. .. r
...
E
and one of t h e limits i n (1.62.4)
Q
h a s s e v e r a l important p r o p e r t i e s p r e s e n t e d now.
F i r s t we
Z is a maximal i d e a l i n A
s i n c e A = A/Z is a f i e l d . Furthermore, 1 is subseq~rence i n v a r i a n t , t h a t i s , a subsequence o f a sequence i n Z w i l l a l s o belong t o Z Let u s now denote by B t h e subalgebra i n Then obviously (1.A2.6)
QN of a l l t h e bounded sequences of r a t i o n a l numbers.
A i s a subalgebra i n B
The s p e c i a l r e l a t i o n between Z and B p o s i t ions.
is p r e s e n t e d i n t h e next two pro-
P r o ~ o s iito n 5 Z is a maximal svbseqvence i n v a r i a n t i d e a l i n 8.
Proof Assume t h a t it is f a l s e and such t h a t
3 is a subsequence i n v a r i a n t i d e a l i n
B,
Let u s t a k e t h e n
S i n c e 9 C B, t h e r e l a t i o n ( 1 .A2.8) g i v e s a p o i n t sequence z ' = (z; , z i , . . . ,z;, . . .), such t h a t (1.A2.9)
l i m z;/ = U+m
But
E W\{O)
and a sub-
Algebraic conflict
since J is subsequence invariant. Moreover, in view of (1 .A2.9) we can obviously assume that
Iz'
L lt1/2 > 0,
Therefore, defining z" = (zg,z;,
Y
E
1
. . . ,zL,.. .) E 'Q
by
we obtain (1.A2.12)
Z" = V
(z;,z;,
. . . ,z;, . . .)
E
B
Now, the relations (1.A2.10)- (1 .A2.12) will yield 1
=
z".z"
E
3.B c 1
hence (1.A2.13)
b = B
as 9 is an ideal B. Since (1.A2.7) and (1.A2.13) contradict each other, the proof is completed. Pro~osition 6
B is maximal among all subalgebras in 'Q
in which 1 is an ideal
Proof Assume that it is false and C is a subalgebra in 'Q
$
(1.A2.14)
B
(1.A2.15)
1 is an ideal in C
such that
C
Let us take then z=
(ZO
,ZI ,. . . ,zV,.. .)
E
C\B
It follows that there exists a subsequence zt = (zV ,zu ,. . . ,zV
z = (zO,ZI ,. . . ,zU,.. .) , such that (1 .A2.16)
lim IzV I P* P
= m
0
1
P
,. . . )
in
E.E. Rosinger
68
Obviously, we can assume t h a t (1.A2.17)
v0 < v1 <
... <
v < P
...
and (1.A2.18)
zV # O , P
Then, i n view of ( 1 .A2.17) and ( 1 .A2.18), we can d e f i n e r = (1-0 ,rl ,...,rV,...) E 'Q by
But, t h e r e l a t i o n s (1. A2.19) and (1. A2.16) y i e l d (1.A2.20)
r = ( r O , r l ,...,r v,...) E Z
t h e r e f o r e , i n view of (1.A2.15) we o b t a i n (1.A2.21)
r-z
E
2.C c Z
However, i n view of ( 1 .A2.19) we o b t a i n
and t h e r e l a t i o n s (1 .A2.21), ( 1 .A2.22) obviously c o n t r a d i c t each o t h e r . o Therefore (1 .A2.14) cannot hold. Corollarv 1
B is t h e largest s u b a l g e b r a i n 'Q
i n which i?
is a n i d e a l .
Proof With a s l i g h t and obvious e x t e n s i o n of t h e n o t a t i o n s i n Appendix 7 i n t h e s e q u e l , it f o l l o w s from P r o p o s i t i o n 6 above t h a t
Algebraic conflict
APPENDIX 3
BOW 'WILD' SHOULD WE ALLOW TBE WORST GENERALIZED FUNCTIONS TO BE? The quotient algebra structures
h
which define various algebras of generalized functions, see 1.10.1) and (1.5.24)- (1.5.26), may a t f i r s t seem t o be too general from t e point of view of convenient use or interpretation. In order t o better grasp what may in the above situation be indeed the 'minimal necessary level of generality', l e t us f i r s t r e c a l l how 'wild' some of the worst nonstandard r e a l s must be in order t o allow us the con* struction of f i e l d s R of nonstandard r e a l s in a way t h a t makes them convenient extensions of the classical f i e l d W of r e a l numbers. One of t h e ways t o obtain nonstandard f i e l d s power type construction. F i r s t , we exten[ R
*
W
i s the following ultra-
into the vastly larger W N , by the embedding
and note that Rm is i n a natural way a commutative algebra over R, with respect t o the term wise operations on sequences. In t h i s way u in (1.A3.2) i s an injective a1 ebra homomorphism and u(0) = (0,0,0,. . . , 0 ? .. .) and ( 1 = ( l , , , . . 1 , . . . a r e the zero and unit elements respectively in RN . Unfortunately, with these natural term wise operations, W [N f a i 1s t o be a f i e l d . Indeed, the multiplication in IRDI has many zero divisors, f o r instance
It follows in particular that the algebra field.
iRN
cannot be embedded into a
Therefore, a s a second step, we s h a l l redace the algebra lKN by factoring it with any suitable ideal N c RN. The point in t h i s procedure i s obvious.
E.E. Rosinger
Indeed, as i s well known RN/Y
P.
Therefore, w i l l be a field, if and only if N i s a mazinal idrol in such that we end up w i t h our problem i s t o find maximal ideals Y i n , strict inclusions
given by the embedding !R
(1.A3.4) x
-,
u
canonical
u(x) + N
u(x)
In order t o help the problem of finding suitable maximal ideals 1, we note that a l l proper ideals in RN are generated by filters on N in the following way. Given a f i l t e r 7 on I, l e t us define (1.A3.5)
K,
N
= {a = (a,lu E I ) E !R
where for a = (auJu E I ) E (1 .A3.6)
#
I
((a)
E
3)
we denoted
((a) = {V E NI aV = 0)
It follows easily that
%
i s a proper ideal i n
@
Conversely, given a proper ideal K in RDI , l e t us define
I t follows easily that relation (1.143.8)
TI
i s a f i l t e r on
N.
Furthermore, we have the
#=K
F~
Indeed, the inclusion ' c ' l e t us take
For the converse inclusion 'I',
i s immediate. a = (aVlv E IN)
E
N 3J
Algebraic conflict
Then (1.A3.5) yields
for a certain
((a)
hence i n view of (1.A3.7) we obtain
E
b = (bYlv E N)
E
1. Let us define now c = (cYlv E N)
E
RN
by if
Y E
N\((b)
if
v
((b)
E
which i s a correct definition, owing to (1.A3.6). (1. A3.9) , we have
Then i n view of
thus
and the proof of (1.A3.8) i s completed.
3 on
IN,
The consequence of (1 .A3.8), (1.A3.10) i s that a l l maximal ideals in are of the form
RN
The crucial property for us i s the following. then (1 .A3.10)
N3
maximal ideal
Given a f i l t e r
w 3 ultrafilter
as seen by a direct application of (1 .A3.5) and (1 .A3.7).
(1.A3.11)
N3
,
with 3 u l t r a f i l t e r on IN
Now, on IN, and i n general on any infinite s e t , there exist only two kind of u l t r a f i l t e r s . The fixed u l t r a f i l t e r s on N are of the form (1.A3.12)
3,= {F c MI,
E
F)
where o E N i s arbitrary but fixed. The f r e e u l t r a f i l t e r s on N are a l l the other, non fixed u l t r a f i l t e r s . They can be characterized i n the following way. Let us denote
3% = {F c N ( N\F
i s a f i n i t e subset)
then 3% i s a f i l t e r on IN, and it i s called the ~ r i c h e tf i l t e r . i s easy to see that given an ultraf i l t e r 3 on N , we have
Now i t
E. E. Rosinger
(1.A3.13)
Further, (1.A3.4)
7 free
B
TR c 7
F infinite
we note that, given any f i l t e r
7
i s indeed an embedding of algebras, since for x (1 .A3.15)
DI,
on
E
the mapping, see
IR we have
U(X)E N3 w x = 0
as follows easily from (1.A3.5) and the fact that
/,,
f 3.
A t t h i s stage a rather surprising thing happens. Namely, for a l l fixed ultraf i l t e r s 3 on IN, the embedding (1 .A3.14) gives an isommorphism
Indeed, assume that 3 = Fa, for a certain fixed a E IN, see (1.A3.12). Then the mapping (1.A3.14) i s onto. For that, l e t us take any I a = (ay l v E IN) E R I , and then denote x = a@. It i s immediate then that
hence the mapping (1.A3.14) maps x
E
IR onto a + N3
E
IRIN /NF.
It follows that i n order to obtain proper extensions (1.A3.31, we have t o use free ultraf i l t e r s 7 on DI. Fortunately that always works. Indeed, l e t 3 be any free u l t r a f i l t e r on iN and l e t us take
then in view of (1 .A3.13), for every x
E
IR,
we obviously have
therefore the mapping (1.A3.14) i s not onto, that i s , we have a proper embedding
Algebraic conflict
The conclusion i s that we encounter the following dichotomy :
-
either 7 i s a fixed u l t r a f i l t e r , and then R = RININF or 3 i s a free u l t r a f i l t e r , and t h k n
'5 #/n;
In this way we can take
for any free u l t r a f i l t e r 7 on IN. A t t h i s point we can see how 'wild' some of the worst nonstandard numbers must inevitably be. For that, l e t us take any
(1.A3.20)
a = (a,lv
E
IN)
E
IN
IR
then the equivalence class * a E IRIN /I7 defined by of a l l b = (bylu E IN) E RIN , such that (1.A3.21)
is given by the set
a
((b - a) = {v E Nibv = ay) E 7
Now we note that in view of (1 .A3.13), the sets of indices ((b - a ) i n (1 .A3.21) are infinite subsets in IN. Therefore, every sequence * b = (b,lv E IN) in the class a has infinitely many identical terms w i t h the sequence a = (a,lu E IN). In t h i s way, the completely arbitrary behaviour of the terms in the sequence a does not disappear by the factorization @/+ since it i s reproduced infinitely many times by the terms of every sequence b = (but, E IN)
i n the class
*
a.
Comin back t o generalized functions we note that even in the case of the L. Sc wartz distributions on R , we can have such 'wild' elements as for instance
k,
where a, E IR and p, E M can be chosen a r b i t r a r i l y , delta distribution.
6 being the Dirac
E.E. Rosinger
74
In the case of the algebras of generalized functions in (1.5.24)- (1.5.26), the construction proceeds i n tuo steps similar t o those in (1.A3.2) and (1.A3.3). Namely, f i r s t we extend the algebra of smooth functions into the much larger algebra $, with a given, suitably chosen c infinite index set A . Then, for appropriate subalgebras A c '8 and ideals 1 in A, we construct the quotient algebra A = All, with the aim of obtaining convenient extensions
~(h)
*
Just as in the case of dealing with nonstandard numbers in IR in (1.A3.19), or distributions i n V'(Q), i n most of the situations when we use generalized functions which are elements of algebras A = A i l i n , we shall not have t o deal with t h e i r most 'wild' instances. .A3.23) in order t o be able to construct a sufficiently simple and clear owever, eneral theory, we have t o allow the possibility that such 'wild' generafired functions are present.
L1
Such a situation repeats itself often in mathematics. For instance, we define the whole set Z, although we use only moderately l a r e integers. Then we extend Z into Q , although we use only fractions w i t moderately large numerators and denominators. We further extend Q into IR, although we shall actually use only a few irrational numbers. B u t perhaps by f a r the most significant example i s iven by Cantor's set theory, where unfathomably large sets are allowed, a though those in common mathematical use happen t o be of a rather limited size.
a
P
One could therefore talk about a kind of mathematical principle accordin t o which definitions of mathematical e n t i t i e s should be allowed t o go muc farther than the actual use of the respective e n t i t i e s . This principle has i t s parallel i n that mentioned i n Appendix 5, according t o which representation should be allowed t o go much farther than interpretation, see also Section 7.
f
Algebraic conflict
APPENDIX 4 NEUI'RIX CALCULUS AND NEGLIGIBLE SEUUENCES OF FUNCTIONS
Connected with the study of various asymptotic expansions, a 'neutrix calculus' was developed in Van der Corput, aimed to deal in a general and unified way with 'negligible' quantities. The basic idea of his method presented in the sequel - proved to be of a wider interest, being for instance useful in the theory of distributions, Fisher [1,2]. In the conditions 1.6.6) and (1.6.11) in Section 6, defining the q u o t i e n t s p a c e s respectiveb a l g e b r a s of generalized functions, we have also made use of the notion o 'negligible' sequences of functions. An other example can be seen in Appendix 2, where R is defined as a q u o t i e n t a l g e b r a of classes of sequences of rational numbers, accordin to the Cauchy-Bolzano method. In that case the 'negligible' sequences o rational numbers will coincide with the sequences of rational numbers convergent to zero.
I
And now, the definition of a n e u t r i x . Suppose iven an arbitrary non-void set X and an Abelian group G. The object of our study will be the functions f : X-rG in other words, the elements of the Cartesian product
which is in a natural way also an Abelian group. The problem is to define in a suitable and general manner the notion of 'negligible' function f E GX.
A given subgroup
will be called a n e u t r i x , if and only if
in which case the functions f E N will be called N - n e g l i g i b l e . An interest in the above notion comes from the fact that in case X has a directed partial order 5 , one can define a n e u t r i x l i m i t for functions in GX as follows. Suppose given a neutrix N c cX, a function f E 'G and 7 E G. Then we define
E.E. Rosinger
(1.A4.4)
N - lim f(x)
= 7
X*
if and only if the function g
E
GX defined by
In view of (1.A4.3) it is easy to see that the limit (1.A4.4) is unique, whenever it exists. The condition (1.A4.3) defining a neutrix can be given the following a l g e b r a i c characterization. Let us define the group monomorphism: u : C - t GX
and denote
Then Uc is the subgroup of c o n s t a n t functions in GX. Let us denote by
Q the null subgroup in GX. The following characterization follows easily.
A subgroup N c 'G
is a neutrix if and only if the inclusion diagram
satisfies the condition
or equivalently, the mapping ii defined by
Algebraic conf1ict
is a group monomorphism, where '6 is the canonical quotient epimorphism. n It is worth noticin that the condition (1 .A4.7) is the opposite of the condition that the c ain of group homomorphisms
f
is exact. Indeed, (1.A4.7) is equivalent to (1.A4.6), while (1.A4.8) is equivalent to
In view of (1.A4.7) and (1.A4.4), in case X is a directed set, we can interpret G X / 1 as a kind of sequential completion of G , obtained by using sequences in G with indices in the set X . If we take now
X
= !N
and G = U(Q)
then the inclusion diagrams (1.6.5) and (1.6.10) are particular cases of , while the conditions (1.6.6) and (1.6.11) are identical with Moreover the sequent ia 1 so 1 ut ions of polynomial nonlinear partial differential equations defined in Section 10, can be seen as neutrix limits in the sense of (1.A4.4 above. Indeed, suppose given the m- th order polynomial nonlinear partia differential equations, see (1.5.1)
?
with continuous coefficients and right hand term, and let us consider
where
,A= E = S/V E V S ~ cm(Q) ,#
All
E
ALP
Obviously, we can also consider the mapping (1.A4.11)
T(D)
:
cm(Q)
4
C" (0)
(Q)
7 1
m and E 5 A.
E. E. Rosinger
DI in which case f o r each given sequence of functions s = ($,lv r IN)~(p(fl)) it makes sense t o ask whether o r not the n e u t r i x l i m i t e x i s t s
(1.A4.12)
Z - l i m T(D)$, = ? u*
And in case the neutrix limit in (1.A4.12) e x i s t s , it w i l l obviously be a function in C?'(fl). P r o ~ o s i t i o n8 Suppose given a sequence of functions s = ($ulu E IN) E S. E -+ A sequential solution of (1 .A4. l o ) , i f and only i f (1.A4.13)
Then
s
is a
Z - l i m T(D)$y = f U-tao
Proof By definition, the relation (1 .A4.13) i s equivalent t o T(D)s - u(f) E Z which in view of (1.10.7)- (1.10.11) completes the proof. Remark 7 In Van der Corput's 'neutrix calculus1 the power of the n e u l r i z c o n d i t i o n (1 .A4.6) i s rather explanatory than operative, t h a t i s , it can unify known results rather than deliver new ones. The reason f o r that i s the insuff i c i e n t a l g e b r a i c - that i s , only group - structure on G, G X and N. However, as soon as G i s an a l g e b r a and /(I is an i d e a l in a subalgebra the mentioned neutrix condition acquires a surprizing power, as of G', seen in Rosinger [1,2,3], as well as Chapters 2, 3, 6 and 7 in the sequel.
Algebraic conflict
APPENDIX 5 REVIEW OF CERTAIN IYPORTANT REPRESENTATIONS AND INTERPRETATIONS CONNECTED WITH PARTIAL DIFFERENTIAL EQUATIONS. Since the emergence of the established theory of eneralized functions, starting with the Schwartz linear theory of distritutions, the study of generalized solutions of linear - and later also of certain nonlinear partial differential equations has become a subdomain of 1 inear metric topology on functions spaces, in particular, on Hilbert, Sobolev, Banach or in general, locally convex topological spaces. Here, for the sake of terminological simplicity, we call linear metric topology a topology on a vector space, compatible with it, and which can be defined by a family of semimetrics, that is, a uniform topology, and which may in particular be metrizable, if it can be defined by one single metric. The apparent advantage of such a course of events seemed obvious, since the respective linear metric topolo y methods involved in proving the existence of generalized solutions could %e expected to lead to approzimation methods for these solutions, useful in their effective numerical computation. Such expectations have more or less been fulfilled: more in the case of fixed point and monotonicity methods for finding generalized solutions, and hardly at all in the case of method provin the existence of generalized solutions based on compactness arguments in unction spaces.
!
Yet, the idea of basing the generalized solution methods on linear metric topology has remained most powerful, to the extent of being nearly exclusive. This situation has most probably been originated and supported by the historical fact that at the dawn of modern mathematics about a century ago, the power of linear metric topological methods on function spaces became quite impressive. Indeed, even in findin classical solutions for nonlinear ordinary differential equations, one o the best methods so far has remained that of Picard's successive approximations, not to mention the elegance and efficiency of the emer ing functional analytic methods of those times, used for instance by Fred olm.
!
f,
As a kind of counterpoint to that, one should however note that no linear metric topology on any function space is used in the proof of the CauchyKovalevskaia theorem, which so far, is one of the most - if not the most powerful and general local existence, uniqueness and regularit result for solutions of rather arbitrary nonlinear partial differentiaf equations. But is there anything wrong with linear metric topologies on function spaces, when trying to find generalized solutions for partial differential equations? In the linear case, one serious warning came with Lewy's 1957 inexistence result, strengthened by Shapira in 1967. It showed that certain rather simple linear variable coefficient partial differential equations cannot have generalized solutions in the space of the Schwartz distributions, or even in more large spaces of hyperfunctions.
E. E. Rosinger
80
I s it that such equations do nevertheless have solutions, but their solutions carry more informat ion than distributions or hyperfunctions can handle? While the above kind of limitations in the case of linear partial differential equations were noted and studied in some d e t a i l , a major difficulty i n the case of nonlinear partial differential equations, presented by nonlinear stability paradoxes, has usually been overlooked, except for some recent studies - see Section 8 and the literature cited there - which concentrating on certain special cases of the mentioned difficulty, t r y remedies within the same linear metric topological approach. And what i s in fact the essence of that linear metric topological approach? Well, from the point of view of generalized solutions, it comes down t o the following. Given a vector space X of classical, real or complex valued functions over a domain Q c IR" of independent variables, l e t us f i r s t assume that X i s endowed w i t h a metrizable topology defined by a metric
which i s compatible with the vector space structure of X. I n t h i s case, the space of generalized functions X i s the completion of X in the given metrizable topology and can be obtained as a quoteint vector space
where S i s the set of Cauchy sequences in sequences convergent t o zero i n X, hence
This means that a generalized function represent at ion F = s + V E X = S/V,
(1.A5.4)
with
s
X,
F
E
E
S
X
while
V
i s the set of
will have a quot ienl
The approzimat ion effect involved in such a situation i s obvious: a given generalized function F in (1.A5.4) i s the limit of any of the sequences s i n (1.A5.4), sequences which in view of (1.A4.3) are of the form (1.A5.5) where, for u (1.A5.6)
s = (f,(u E IN) E S E
N,
f, E X are classical functions, and
l i m d(f,f,) = 0 u+m
In case the vector space topology on X by a family
i s not metrizable and it i s given
Algebraic conf 1i c t
of semimetrics, the quotient vector space structure (1.A5.2) of the generalized function space X s t i l l remains valid. Moreover, in many important cases, X w i l l s t i l l be sequential1 dense in X , t h u s (1 .A5.3)(1.A5.5) remain valid, while (1 .A5.6) will i e replaced w i t h (1.A5.8)
l i m di(F,fu) = 0, with
i E I
u*
Usually, such an approximation property as i n (1 .A5.6) or (1 .A5.8) can also have a numerical meaning in the sense that, although (1.A5.9)
F(x),
with x
E
a,
i s usually not defined
since F i s a generalized and not nevertheless, the sequences of numbers (1.A5.10)
(fU(x)Iu E IN),
with
x
a classical
function on
Q,
E fl
allow the extraction of some numerical information by suitable limit processes. A standard example giving the L. Schwartz 3' (Q) distributions i s obtained as follows. Suppose given a vector space X of classical functions on Q , such that
is endowed w i t h the weak topology induced by Pf(il) and suppose X through the obvious inclusion X c V'(Q) implied by (1.A5.11). Then the sequences of numbers in (1 .A5.10) give for each $ E P(Q), the number denoted by F($) and defined by
Moreover, F($) does not depend on s saying that (A5.3)
i n (1 .A5.4), which i s equivalent to
,
l i m ~ g v ( x ) $ ( x ) d x = O ,with v = ( g u l u ~ ~ ) t Yy i,~ P ( i 2 ) U+m
Now, the numerical meaning of (1.A5.12) i s quite obvious: given x E D and $ E q?, then F($) contains an information on the 'value' a t and around x of t e generalized function F, t o the extent that (1.A5.14)
supp $ i s contained i n a neighbourhood of
x
E.E. Rosinger
in which case F($) neighbourhood of x.
can be seen as a kind of average value of A l l that i s simply described by
Let us shortly recall the essence of the above: A space X of classical functions is extended into a space ralized functions (1.A5.17) The space X construct ion (1.A5.18)
X
F
in a
of gene-
X C X of generalized functions i s obtained through a quotient X=S/V, V c S c x D I
The embedding (1.A5.17) i s defined by (1.A5.19)
X 3 f ~ F = u ( f +) V E X
where u ) = ( f , . . . f , . . generalized function (1. A5.20)
..
Finally, a quotient representation of a
F = S + V E X , s = ( f v l v ~ D I E) S
has a certain approzimation and even numerical interpretation in terms of specific limit processes involving numerical values of the classical functions f v , with v E M. Now, it i s particularly important t o note that there are useful and established quot ient construct ions in Analysis which, although go along lines similar t o that in (1.A5.17)- (1.A5.20), are nevertheless l e s s restrictive and simp1 ist ic in that the construction of the spaces X is not based on numerical, approximat ion, or in general, topological interpretat ions. Such interpretations can often be associated l a t e r , and usually, only t o certain strict subsets
Yet, no matter how useful such interpretations of elements of Xn may be, these interpretations need not be employed from the very beginning, when the spaces X are defined. One such well known example is given by Nonstandard Analysis, see Stroyan & Luxembur a s well as the particular nonstandard construction presented in (1.7.2)-fi.7.7), o r in Appendix 3.
Algebraic conflict
Prompted by the above, we can conclude as follows. In certain mathematical models, the representation of the e n t i t i e s involved can by i t s e l f give one or several interpretations leading back t o basic relationships between the mathematical model and the system which is modelled. A typical example i s the arithmetic of integers, where every not too large number m E Z i s t o a good extent i t s own interpretation.
A f i r s t departure from that situation happens within usual distribution theory, see (1 .A5.1)- (1 .A5.20), where a generalized function F E P (S2) with a quotient representation F = s + V E P ( Q )= S/V, s E S, s t i l l has a usual approximation interpretation (1.A5.8) i n the function space X i n (1.A5.11), but it can no longer have a usual numerical interpretation within the real or complex numbers, except for the much more relaxed version given by the weak form (1 .A5.12)- (1 .A5.15). In t h i s way we note that re resentation happens t o go farther than interpretat ion, see also (1.15.21f and (1.7.7). With Nonstandard Analysis a further departure occurs in so f a r that a construction such as in (1.7.2) and the resulting representations (1.7.6), are made from the very beginnin u~ithout even an approximation interpreIough such, as well as a numerical intertation, see also Appendix 3, a l t! pretation can l a t e r be associated w i t h part of the structure, see (1.7.7).
However, the fact that i n the above mentioned theories representations are developed farther than interpretations i s b no means a drawback, but on the contrary, it i s one of the main reasons or the power and applicability of these theories. Indeed, the rigorous computations in these theories are and can only be done on the level of representations. Therefore, one has an advantage if representat ions are developed conveniently and f a r enough.
P
Coming t o Colombeauts version of the nonlinear theory of generalized functions presented shortly i n Chapter 8, the issue concerning approximat ion and numerical interpretations i s as follows. The main instrument i s the relation of association 11which associates a distribution T E D '' to every generalized function F E G o , where Go i s a strict subset of of generalized functions. This relation of Colombeau s algebra association
has a numerical interpretation given by (1.15.23)
j $ F dx lRn
1-
#T dx,
$
E
D
lRn
which i s similar t o (1.A5.12). Concerning possible approxima t ion int erpretat ions associated with Colombeauts generalized functions, one can consult Rosinger [3, Appendices 1 and 2 i n Part 2, Chapter I ] .
84
E.E. Rosinger
It follows that with respect to the relationship between representations and interpretations, Colombeau's nonlinear theory of generalized functions is one step further then the usual linear theory of the V distributions, since the interpretations used in the former reduce it to the latter, Nevertheless, this can allow for according to (1 .A5.22), (1 .A5.23). particularly powerful results even in the numerical analysis of nonlinear nonconservat ive partial differential equations. Here, we shortly mention one such recent result. We consider the coupling between the velocity u and the stress a in a one dimensional homogeneous medium of constant density, modelled by the nonlinear nonconservat ive system Ut
+ U'U X = aX
(1.A5.24)
t>O, X E R at + u.0X
=
k2uX
where k > 0 depends on the medium. As usual, we associate with (1 .A5.24) the initial value problem
While the first equation in (1 .A5.24) being conservative, it could be replaced by the weak equation
However, the second equation in (1 .A5.24) does not offer such a possibility since it has a nonconservative form. In order to obtain a numerical solution for (1 .A5.24), (1.A5.25) a usual two time step, split numerical scheme is used. The first step discretizes the nonlinear convection terms in (1 .A5.24) by a usual Lagrangian, Eulerian and antidiffusion correction method. The second step discretizes the wave propagation terms in (1.A5.24). Each step is made through a decentered, Donnor cell scheme. In this way, for given initial values (1 .A5.25), we obtain two families of numbers (1.A5.27)
(ulln
E
N, i E Z), (alln E N, i
E
Z)
with the intended approximation property
where At, Ax > 0 are the time and space increments respectively.
Algebraic conflict
It i s further assumed that (1.A5.29)
u o , uo have bounded variation on R
and (1.A5.30)
Iuo(x)l
< kuo(x),
x E
which means that we deal w i t h positive and high velocities, a particularly useful case in certain applications. Then we denote
and assume that (1. A5.32)
r _( l/max{M,k)
where r = At/Ax > 0 i s the Courant-Friedrichs-Lewy number of the time and space discretization used. Under the above conditions (1 .A5.29)- (1. A5.32), the numerical method gives norm, numerical solutions (l.A5.27), (1.A5.28) which are stable in the '?f i n the t o t a l variation i n space and i n the Tonnelli-Cesari type total variation in time. Assuming r > 0 given, for every constant functions
Ax > 0
we can define the piece wise
Then, a Helly type compactness argument yields for each T > 0 a sequence (1 .A5.35)
Axv>O, U E N ,
limAx=O
with
V+rn
and a pair of functions with bounded variation
such that (1.A5.37)
l i m uAx = u, v
V+rn
i n the sense of
Cioc([O,T]
l i m uAx = u V+m
x
R).
V
We note that i n view of (1 .A5.35)-
E.E. Rosinger
(1.A5.37) we have
Now, a well known problem with the above so called numerical sol~rtion (1.A5.35)-(1.A5.38) of (l.A5.24), (1.A5.25) is that the functions u, u obtained are not smooth enough in order to be classical solutions of (1.A5.24), (1 .A5.25) . Moreover, since the second equation in (1.A5.24) is nonconservative, we cannot use a weak equation similar to (1.A5.26) in order to check whether indeed u and u are at least ueak solutions of (l.A5.24), (1.A5.25). In other words, we simply cannot be sure in which ways u and u may relate to our initial problem (l.A5.24), (l.A5.25), except for the fact that they have been obtained by a compactness argument from the stable numerical solutions (1.A5.27) , (1.A5.28) . And as mentioned earlier in connection with nonlinear stability paradoxes, it is in particular with compactness ar uments used for obtaining generalized solutions of nonlinear partial di ferential equations that one has to be specially careful. It is therefore precisely here that Colombeau's nonlinear theory of generalized functions proves to be particularly powerful and useful. Indeed, any family
7
in (1.A5.37) can be associated with two generalized functioras
having the grot ient representat ion
where (f ,g) are defined by the family (l.A5.39), such that (1.A5.42)
UXU, C X U
and
where the equivalence relation (l.A5.22), (1.A5.23) by (1.A5.44)
F xG
* F-G 11-
x
on
G([O,m)
0, F,G E P([O,m)
x
W) x
is defined through
W)
In this way u and u do have a pointwise numerical interpretation through ( 1 .A5.37), although they do not satisfy the equal ions (1 .A5.24), (1 .A5.25) classically or in a weak sense.
Algebraic conflict
On the other hand, U and X do not have a pointwise numerical interpretation, but they satisfy the equations (1.A5.24) in the modified form (1.A5.43). The link between u, u and U, C is given in (l.A5.42), which as mentioned, has the same averaging numerical interpretation with (1.A5.12).
A few important properties should also be mentioned. The relations (1.A5.42) determine u and r uniquely. Further, as seen in Chapter 8, the equivalence relation r is not compatible with the multiplication of generalized functions in 0. Nevertheless, in the above case of association in (l.A5.42), we also have the following stronger association property
for every two variable constant coefficient polynomial P .
l
In view of the above, it is obvious that the information contained in 1.A5.39 and transmitted to the quotient representat ion (1 .A5.40), 1.A5.41 , is only partly contained in (u,a) as obtained by the limit in .15. 37/. In other words, the quotient representation (1 .A5.40), 1.A5.41 contains more informat ion about (1 .A5.39) than the limit (u,u ) in (1.A5.37).
?
Finally, the association property (1.A5.42 has also the following advantage. The functions u, a cannot be rep aced in the equations (1.A5.24) since that would involve multiplications between nonsmooth functions and their distributional derivatives, which is not possible within the linear theory of distributions. The functions u, u cannot be replaced in weak forms of the equations (1.A5.24), since that system is nonconservative. But the functions u, r can easily be used in (1.A5.42) which is the simplest possible linear system in these two functions. Concerning the interpretations of the nonlinear theory of generalized functions in Rosinger [1,2,3 , it suffices to mention that, although developed somewhat earlier, t at theory is a kind of encompassing roof theory for a large class of possible nonlinear theories of generalized functions, which among others, contains Colombeau's nonlinear theory as a particular case. Therefore, with respect to the relationship between representations and interpretations discussed above, the theory in Rosinger 1,2,3] is yet one more step further than Colombeau's nonlinear theory, see or details Chapters 2, 3, 6 and 7, as well as Chapter 4, which in the case of a particular but important class of nonlinear hyperbolic partial differential equations goes beyond even the framework in Rosinger [I ,2,3] .
h
I
E. E. Ros i n g e r
APPENDIX 6 DETAILS ON NONLINEAR STABILITY PARADOXES, AND ON TEE EXISTENCE AND UNIQUENESS OF SOLmIONS FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Let u s t u r n t o t h e e n e r a l n-dimensional c a s e of t h e nonlinear s t a b i l i t y paradoxes i l l u s t r a t e i n S e c t i o n 8.
%
For t h a t , let us c o n s i d e r t h e nonlinear system
and show t h a t it h a s s o l u t i o n s g i v e n by sequences of on iRn
which a r e both weakly and s t r o n g l y convergent i n take f o r instance ( 1 .A6.3)
xu(*) =
JZ
cos
UXI
,
x = ( X I , . . . ,xn)
P-smooth functions
Indeed, i f we
V (Rn). E
Rn,
u E IN
we s h a l l obviously have (1.A6.4) hence v
l i m X, = 0, u+w
lim u*
= I
i n Vf(IRn)
i n (1 .A6.2) is a weak and s t r o n g s o l u t i o n of ( 1 .A6.1) i n 2' (iRn) .
Let u s c o n s i d e r now t h e f o l l o w i n g n o n l i n e a r system which is n o n t r i v i a l l y partial differential
where as u s u a l l y ,
Uxl
denotes t h e p a r t i a l d e r i v a t i v e of
t o X I , w h i l e o E (?('Kt) is a n a r b i t r a r y , given f u n c t i o n . sequence of P - smooth f u n c t i o n s
n IN
w = (wulu E IN) E ( P ( R ) )
U
with r e s p e c t We d e f i n e t h e
Algebraic c o n f l i c t
Then it follows e a s i l y t h a t
Therefore w
is both a weak and strong solution of ( 1 .A6.5) in V' (an).
It is obvious t h a t in a similar way, one can obtain both weak and strong solutions v = (xVlu E W) E (e)(lRn))' in V'(IRn) f o r a l a r g e v a r i e t y of nonlinear partial differential systems U(x) = 0,
x E lR
where m E IN, q . E Wn and 7 E f'(lRn) , 7 not i d e n t i c a l l y zero. J t h i s follows e a s i l y f o r m even, if we take (1.A6.10) f o r x = (xl
xV(x)
,. . . ,xn)
=
@(x) cos
E lRn,
VXI
v E #,
Indeed,
+ @(x) s i n vxl
and with suitably chosen
It follows t h a t both weak and strong solutions in sequences of functions
@,B E P ( R ~ ) .
V'(lRn),
given by
can be found f o r a large c l a s s of nonlinear partial differential systems of the form
where l , m i E !N,qij E ~ l " , d ~E ,p(lRn), ~ where
7 is not i d e n t i c a l l y zero.
E.E. Rosinger
90
Indeed, as follows from (1.A6.9), t h i s can happen i f a t l e a s t one of the mi i s even. Let us show now that (1.A6.12) can have both meak and strong solutions i n l I t ( P ) even i n the case when all mi are odd. For t h a t , it suffices t o show that (1.A6.9 can have such solutions f o r m odd. This happens indeed i f instead o (1.A6.10), we take
1
xv(x) = 1 t cos vx - 2 cos2vx
(1.A6.13) f o r x = (X
1
(1 .A6.14)
1
1
,. ..,xn) lim
x,
V-hn
Thus with (l.A6.13),
E
p, v = 0,
E IN.
lim
Then a direct computation gives
xi = - 314
i n 2. ( P )
V-m
it follows that
i s a sequence which i s both a weak and strong solution of the nonlinear sytem
Now the existence of weak and strong solutions f o r nonlinear systems such as in (1.A6.1), (l.A6.5), (1.A6.9) o r (1.A6.12) does have critical effects on the ezistence and uniqueness of weak solutions f o r nonlinear p a r t i a l d i f f e r e n t i a l equations, when they a r e obtained by the usual methods mentioned i n Section 8 of t h i s Chapter. Nevertheless, as noted e a r l i e r , these c r i t i c a l effects are often overlooked i n the l i t e r a t u r e . In order t o i l l u s t r a t e such effects, it suffices t o extend the argument i n 1.8.17)- (1.8.27) t o more general nonlinear p a r t i a l d i f f e r e n t i a l operators 1.5.4), using the solutions of systems (1 .A6.12). Indeed, suppose that the equation
I
i s claimed t o have a weak solution
given by a sequence
Algebraic conf1ict
for which
As usual, this means the existence of the simultaneous limits
Suppose that the nonlinear partial differential operator (1.A6.17) has the form
T(D)
in
is a linear partial differential operator with C@'-smooth where L(D) coefficients, while Q(D) is a nonlinear partial differential operator such as those in the left hand term of the second equation in (1.A6.12). Then for any sequence (1.A6.23)
s+ v
with v in (1.A6.11), which satisfies (l.A6.12), we shall have the limit (1.A6.24)
s+ v
+
U in P'(lRn)
However, we show that in general, we shall no longer have the limit
In other words, the so called weak solution (1.A6.21) does not extend to sequences (1 .A6.24), where nevertheless, we have v + 0 in V' (lRn) , this being a severe limitation on the stability of the weak solution (1 .A6.21) and therefore, on the very meaning of such a weak solution concept. Indeed, (1.A6.22) gives
while obviously
where R(D) is a nonlinear partial differential operator in two arguments. It follows that
E .E. Rosinger
92
But (l.A6.21), (1.A6.11) and (1.A6.12) yield the limits
If we assume now the existence of the limit (1.A6.30)
R(D) (s ,v)
4
g in P' (lRn)
then in view of (1 .A6.28), (1 .A6.29), we obtain the limit
which obviously need not be the same with (l.A6.25), unless
It follows that (1 .A6.25) will indeed fail, whenever (1 .A6.32) or (1 .A6.30) fail. It is now obvious that the lack of stability illustrated in (1.A6.24), (1.A6.25) above can lead to the questioning of the usual weak solution methods for nonlinear partial differential equations. Indeed, concerning the usual constructive proofs for the existence of weak solutions, it is obvious that the construction of particular sequences s as in (1.A6.21) can prove to be irrelevant, as lon as their stability properties in (1 .A6.24), (1.A6.25) are not establishef. Otherwise, ve can land in situations when
but
and
which according to the usual interpretation of weak solutions would mean that U E P' (lRn) is a simultaneous weak solution for (1 .A6.36) and
T(D)U
=
f
Algebraic conflict
93
I t i s now also obvious that with difficulties such as in (1.A6.33)(l.A6.37), the concept of the vniqueness of the usual weak solutions becomes even more problematic .
r
One wa to deal w i t h the above situation i s the one presented i n Sections 9-12 o this Chapter. See also Chapters 2-4 and 6-8.
E .E . Rosinger
APPENDIX 7 TEE DEFICIENCY OF DISTRIBWION THEORY FROH THE POINT OF VIEW OF EXACTNESS We s h a l l show t h a t from t h e point of view of exactness, it i s not conveo r in particular 3' -, 9' sequential n i e n t t o d e a l with E + P', s o l u t i o n s , even when we solve l i n e a r p a r t i a l d i f f e r e n t i a l equations. For t h a t purpose we need a few n o t a t i o n s . Suppose given an i n f i n i t e index s e t A and a subalgebra 1 c M(Q) as a vector subspace I c f . Then we denote by (1 .A7.1)
a s well
$,A(1)
t h e l a r g e s t subalgebra B c
f
such t h a t 1.8 c I.
Obviously, t h e following t h r e e p r o p e r t i e s a r e equivalent:
c
(1.A7.2)
+ *,(I)
(1.A7.3)
Z subalgebra i n
(1.A7.4)
I
f
i d e a l i n RXYA(I)
Let us denote by (1.~7.5)
IDr,n
t h e s e t of a l l t h e v e c t o r subspaces I c (1.~7.6)
z n u,,,
=
#
such t h a t , see (1.6.11)
o
and, s e e (1.6.10) (1.~7.7)
z @ ur,n
-%,A(')
I n view of (l.A7.7), (1.A7.2) and (1.A7.4) it follows t h a t
F i n a l l y , i n view of (1.A7.81, algebra
for
I
E
IDX,A
we can d e f i n e t h e q u o t i e n t
Algebraic conflict
Then (1.A7.6), (1.6.10), (1.6.11) yield
And also c o n v e r s e l y , it follows easily that
for which there In other words, contains exactly a l l I c f exists A = A/I E ALy,A. Moreover, it i s easy to see that
which means that each quotient algebra A =
41 E
ALXYA i s a s a b a l g e b r a i n
A2,J(". In view of (1.A7.6), (1.A7.7) it i s easy to see that we have the following simple c h a r a c t e r i m t ion of IDzyA as being the set of a l l I c ?h such that (1.A7.13)
I subalgebra i n
f
Suppose now that we are dealing w i t h a linear partial differential equation w i t h p-smooth coefficients (1.A7.16)
L (D) U(x) = f (x) , x
E
D
where D c !Rn i s nonvoid, open, while L(D) i s given i n (1.8.31). Further, suppose given a sequence of functions
E .E . Ros inger
96
It is easy t o see that in the case of a linear p a r t i a l d i f f e r e n t i a l equation (1 .A7.16), the definition of E + A sequential solutions given in (1.10.6)- (1.10.8 can be extended t o the case when A is also a quotient vector space an not necessarily a quotient algebra, see also Chapter 4. I t i s in t h i s sense that we assume that
d
(1.A7.18)
s i s a E -+ T1(Q) sequential solution of (1.A7.16)
where
and, see (1.8.43)
In view of (1.10.10), (1.10.11), the relation (1.A7.18) means the following two conditions (1.A7.21)
S E S
and -
-
t-s E
v =+ wt
E
P(Q)
where (1.A7.23)
wt = L(D)t - u ( f )
i s the error sequence corresponding t o t
E
S.
Now, the deficiency of the approach in (1.A7.18) from the point of view of exactness becomes apparent. Indeed, when we t r y t o strengthen the condit ion
in (1 .A7.22) on the error sequence wt by finding conditions of exactness similar t o those in (1.11.13), (1.11.14), we have t o face the f a c t that
where the notation in (1 .A7.1) was used. (1 .A7.25) follows easily from, see (1.8.57)
We note that the relation
Algebraic conflict
In view of (1.11.15), the relation (1.A7.25) means that we cannot obtain a satisfactory exactness property for the sequential solution in (1 .A7.18).
E .E . Rosinger
APPENDIX 8
INEXISTENCE OF LARGEST OFF DIAGONAL VECTOR SUBSPACES OR IDEALS We give a (F (R))' tained in be defined
very simple example of two vector subspaces V and V' in which are off diagonal, see (1.6.6), but which cannot be canan off diagonal vector subspace. Indeed, l e t w , w f E (e(IR))' by
(1.A8.1)
wv(x) = 1 + sin vx, wb(x) = s i n vx,
f o r v E N , x E IR
and l e t V and V' be the vector subspaces generated by respectively. Then it follows that
and
w
since f o r any $ E C" (IR , the relation w = u($) that $ vanishes on a ense subset of R.
or
Now l e t V" be a vector subspace in V ' . Then obviously
which contains
d
therefore V"
(co(R))'
w' = u ( ~ ) implies
and
V
w i l l not satisfy the off diagonality condition (1.6.6).
The above i s valid a s well f o r ideals i n w , w f E ( P (IR))' be defined by (1.A8.4)
w'
(P(R))'.
wv(x) = 1 + sin vx, wh(x) = 1 + cos vx,
for v
Indeed, E
IN, x
let E
IR
and l e t
be the ideals in (P (R))' generated by as above, it is easy t o prove that (1.A8.6)
w
and w'
respectively.
Then,
znzi = I ' nzi = 0 P ( R ) ,IN (R) ,N
thus the ideals (1.6.11).
2
and
2'
s a t i s f y the off diagonality condition
Assume now t h a t 2' ' is an ideal in Then obviously
( P (IR))'
which contains Z and 1'.
Algebraic conflict
(1.A8.7)
W"
= w + W' E
I + I' c 2"
But (1 .A8.4) yields (1.A8.8)
w"(x) V
= 2 + sin v x t cos v x > 0, for VEIN, X E I R
therefore
which means that the ideal 2" does not satisfy the off diagonality condition (1.6.11). In fact (1 .A8.9) implies that
thus, there is no proper ideal in
I and 2'.
(e(lR))INwhich may
contain the ideals
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CHAPTER 2 GLOBAL VERSION OF THE CAUCEY-KOVALEVSUIA THEOEEU ON ANALYTIC NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
1
.
INTRODUCTION
The aim of this Chapter is to show that all analytic nonlinear partial differential equations have generalized solutions on the who1 e of their domain of analyticity. These generalized solutions are defined in the sense of Chapter 1, Section 10, and are analytic on the whole of the domain of analyticity of the respective equations, except perhaps for closed, nowhere dense subsets, which may be chosen to have zero Lebesque measure. This result is nontrivial because of at least two reasons. First, there is little understanding of the structure of singularities of analytic functions of several complex variables. Secondly, analytic functions can tend very fast to infinity near to their singularities. These two reasons have so far been sufficient in order to prevent the theory of distributions of L. Schwartz from finding global distributional solutions for arbitrary analytic nonlinear partial differential equations. It follows that one may as well look for more eneral concepts of solutions. Fortunately, the theory in Chapter 1 is su ficient for that purpose.
f
In this way one obtains for the first time, see Rosinger [3], the following result : The analyticity of solutions of arbitrary analytic nonlinear partial differential equations is a strongly generic property of these equations. We recall that a property of a system is called strongly generic, if and only if it holds on an open and dense subset of the domain of definition of that system, see Richtmyer. The above result may present an interest from several points of view. Firstly, in case the closed nowhere dense subsets on which the generalized solutions may fail to be analytic have zero Lebesque measure, such solutions can describe a large variety of shocks. Secondly, it is well known that closed, nowhere dense subsets in Euclidean spaces can have arbitrary positive Lebesque measure, Oxtoby. In this case one may expect that the respective generalized solutions may, among others, model turbulence or other chaotical processes as well. Closed, nowhere dense sets of arbitrary positive Lebesque measure can in particular be various Cantor type sets, encountered as attractors in recent studies of asymptotic fluid behaviour, Temam. Finally, the open problem arises whether the rather large class of generalized solutions constructed in this Chapter may exhaust all, or most of the
E. E. Rosinger
s o l u t i o n s corresponding t o v a r i o u s s o l u t i o n c o n c e p t s used s o far i n t h e l i t e r a t u r e , when a p p l i e d t o s o l v i n g a n a l y t i c n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s . A p a r t i a l answer is p r e s e n t e d i n Chapter 7 , where t h e method f o r d e a l i n w i t h c l o s e d , nowhere dense s i n u l a r i t i e s d e v e l o ed i n t h i s Chapter w i 1 be a p p l i e d t o t h e r e s o l u t i o n o s i n g u l a r i t i e s o weak solut i o n s of a very l a r g e c l a s s of polynomial n o n l i n e a r p a r t i a l d i f f e r e n t i a l equations.
P
4
9
It may be i d e a l t o f i n d at once a unique and r e g u l a r s o l u t i o n f o r a g i v e n n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n , corresponding t o a c e r t a i n i n i t i a l and/or boundary v a l u e problem. However, as is well known, t h a t kind of i d e a l s i t u a t i o n seldom happens. I n f a c t , it d o e s n o t happen even i n t h e c a s e of one of t h e most simple n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s , namely, t h e shock wave equation
with t h e i n i t i a l v a l u e problem
i n which c a s e t h e Rankine- Hugoniot c o n d i t i o n t o g e t h e r w i t h t h e e n t r o p c o n d i t i o n may be needed i n o r d e r t o s e l e c t a unique s o l u t i o n , whicK t y p i c a l l y w i l l f a i l t o be r e g u l a r , due t o t h e presence of shocks, Smoller. The above example of t h e shock wave equation may be u s e f u l i n understanding t h e meaning of t h e r e s u l t s i n t h i s Chapter. Indeed, i f we do n o t a s k t h e Rankine-Hugoniot and entropy c o n d i t i o n s , t h e shock wave e q u a t i o n , f o r a g i v e n i n i t i a l v a l u e problem, may have i n f i n i t e l y many s o l u t i o n s which a r e well d e f i n e d o u t s i d e o f t h e shocks. These solut i o n s can be o b t a i n e d by p a t c h i n g up i n t h e h a l f p l a n e
s o l u t i o n s d e f i n e d by t h e method of c h a r a c t e r i s t i c s a p p l i e d on p a r t s of t h e space domain corresponding t o t h e i n i t i a l moment t = O ,
XER
It o n l y happens t h a t t h e mentioned Rankine-Hugoniot and e n t r o p y c o n d i t i o n s can o f t e n s e l e c t o u t one s i n g l e s o l u t i o n from a n i n f i n i t y of such patched up s o l u t i o n s .
It f o l l o w s t h a t , at f i r s t , we have t o f a c e a m u l t i t u d e of patched up solut i o n s . And t h e n , by u s i n g c e r t a i n a d d i t i o n a l l o b a l p r i n c i p l e s , such as f o r i n s t a n c e t h e Rankine-Hugoniot o r entropy con i t i o n , be a b l e t o s e l e c t a unique s o l u t i o n .
%
I n t h e terms of t h e above example, t h e aim of t h i s one g l o b a l and u n i v e r s a 1 p r i n c i p l e which can d e f i n e n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s s e t s of T h i s p r i n c i p l e , namely, t o be a g e n e r a l i z e d s o l u t i o n
Chapter is t o p r e s e n t for arbitrary analytic patched up s o l u t i o n s . in t h e sense specified
Global Cauchy- Kovalevskaia
103
below, does not necessarily lead t o the uniqueness of such solutions for given analytic and noncharacterist i c Cauchy problems. I t follows therefore that the uniqueness of solutions has t o result from the imposition of f u r t h e r g l o b a l o r l o c a l c o n d i t i o n s . To recapitulate, the aim of t h i s Chapter i s t o answer the following quest i o n : Can one find s u f f i c i e n t l y r e g u l a r g l o b a l solutions f o r a l l a n a l y t i c noa1i n e a r partial differential equations ?
This question i s nontrivial since classical or distributional global solut ions are not available. As mentioned, the answer t o the above question i s affirmative. Moreover, for each given analytic nonlinear partial differential equation, one can construct a l a r g e c l a s s of such global solutions. I n t h i s way, one i s led t o the second q u e s t i o n : To what extent does t h i s class of global solutions exhaust the various solutions generated by the customary weak or generalized solution concepts used so f a r in the literature, when applied t o analytic nonlinear partial differential equations ? The answer t o t h i s second question remains open. Several further comments are as follows. The proof techniques i n t h i s Chapter do not employ functional analysis. They only use basic constructions i n rings of continuous functions on Euclidean spaces, as well as classical calculus and elements of topolo y i n these spaces, added of course t o the classical proof of the original, Focal Cauchy- Kovalevskaia theorem. In particular , the whole construct ion centers around the so called nowhere dense i d e a l s
introduced i n Rosinger [1,2,3], which are used for the construction of suitable a1 ebras of eneralized functions in the sense defined in (1.6.9)(1.6.11). !ee (2.2.48 - (2.2.6) for the detailed definition. This lack of need for the use of functional analysis should not come as a surprise. Indeed, the classical proof of the Cauchy-Kovalevskaia theorem itself i s only using calculus, functions of complex variables and inequal i t y estimates, although that theorem remains one of the most general and powerful nonlinear local existence, uniqueness and regularity results. I t i s perhaps the time t o become aware of the b l i n d s p o t - if not in fact r e l a t i v e f a i l u r e - of several decades of functional analytic exclusivisrn i n solving nonlinear partial differential equations. Indeed, on the one hand, the classical Cauchy-Kovalevskaia theorem had been proved long before the advent of functional analytic approaches and i t s proof only uses Abel type
E.E. Rosinger
104
estimates of power s e r i e s , i . e . , it i s based on elementary properties of the geometric s e r i e s . On t h e other hand, t h e subsequent functional analyt i c methods, despite t h e i r numerous important contributions, have not produced one s i n g l e comparably general type independent and powerful nonlinear l o c a l existence, uniqueness and regularity r e s u l t . Curiously, t h e e x i s t e n t functional a n a l y t i c methods a r e not able t o improve on t h e c l a s s i c a l Cauchy- Kovalevskaia r e s u l t i n i t s given general terms. It appears indeed t h a t , especially i n t h e case of nonlinear p a r t i a l d i f f e r e n t i a l equations, the power of functional analysis i s r a t h e r limited t o t h e d e t a i l e d study of various p a r t i c u l a r , s p e c i f i c classes of equations. Lately, a c e r t a i n awareness of t h a t l i m i t a t i o n seems t o emerge i n a t e n t a t i v e way. For instance i n Evans, methods i n measure theory a r e presented a s supplementing t h e limited power of functional analysis. However, such ideas a r e s t i l l f a r from identifying t h e roots of t h e issues involved i n t h e way t h a t i d e n t i f i c a t i o n becomes possible through t h e 'algebra f i r s t ' approach presented i n t h i s volume, a s well a s i n Rosinger [I ,2,3] and Colombeau [I721
-
The proof of t h e f a c t t h a t t h e global generalized solutions constructed i n t h i s Chapter a r e a n a l y t i c , except perhaps on closed, nowhere dense subsets, follows q u i t e straightforward from a t r a n s f i n i t e induction argument, once the nowhere dense i d e a l s i n (2.1 .l) a r e brought i n t o t h e p i c t u r e . The respective developments a r e presented i n Sections 2-5, and culminate i n Theorem 1 i n Section 5. The f u r t h e r refinement i n Theorem 2, Section 6 , shows t h a t the mentioned closed, nowhere dense s i n g u l a r i t i e s can be constructed i n such a way a s t o have zero Lebesque measure. This refinement i s based on Proposition 1 i n Section 6 , contributed by M. Oberguggenberger i n a p r i v a t e communication. Curiously, t h i s stronger r e s u l t i n Theorem 2 i s constructive and it does not use t r a n s f i n i t e induction. I n order t o obtain Theorems 1 and 2 i n t h i s Chapter, t h e f u l l generality of the concept of solution introduced i n Rosinger [1,2,3] and presented i n Chapter 1, Section 10 was used.
The problem remains open whether t h e mentioned r e s u l t s may as well be obtained within t h e p a r t i c u l a r theory of generalized functions introduced i n Colombeau [I ,2]
.
§ 2.
THE NOWHERE DENSE IDEALS
The presentation i n t h i s Chapter, owing t o a simplified notation, i s r a t h e r selfcontained, although it follows, a s a p a r t i c u l a r case, t h e general cons t r u c t i o n i n Chapter I . Given l a domain i n lRn dix 1, Chapter 5 ,
and
t h e s e t of a l l the sequences
L
E
IA
=
W
U (m),
we denote by, .see Appen-
Global Cauchy- Kovalevskaia
e
of C -smooth f u n c t i o n s which converge weakly i n V'(fl) t o a d i s t r i b u t i o n , r e s p e c t i v e l y , t o z e r o . Then, one o b t a i n s t h e v e c t o r s p a c e isomorphism (2.2.2)
-
sl(n)lv'(n)
PI(n)
d e f i n e d by
ve (n) t- s,
s +
s E se(n)
where
The b a s i c concept, namely, t h e nowhere dense i d e a l on now. We denote by (2.2.3)
fl
is introduced
lnd (Q)
t h e s e t of a l l t h e sequences of continuous f u n c t i o n s
which s a t i s f y t h e condition
3
v (2.2.4)
r
c fl c l o s e d , nowhere dense E
n\r
:
:
3 /LEN:
v
u ? p : wu(x) = 0 U E N ,
It f o l l o w s t h a t Ind(Q) is a n i d e a l i n ( ~ ( n ) ) ' , s e e P r o p o s i t i o n 2 i n Appendix 1. Moreover, a s shown i n Appendix 1 based on a B a i r e c a t e g o r y t h e c o n d i t i o n (2.2.4) i n t h e d e f i n i t i o n of lnd(fl) can argument i n R", be r e p l a c e d by t h e f o l l o w i n g e q u i v a l e n t one
E. E. Ros inger
(2.2.5)
3 I' c fl closed, nowhere dense : v x E n\r : 3 p E IN, V c fl\r neighbourhood of v VEIN, u ) p , y E V :
x :
w,(Y> = 0 I t i s important t o note that, for both these properties, the continuity of the functions w,, as well as the condition on r being closed and nowhere dense, requested in (2.2.4), are essential, see Appendix 1. We shall c a l l Znd(fl) the nowhere d e n s e i d e a l on fl.
An easy consequence of (2.2.5) i s the relation
where !E N = IN, and D~ i s the usual p- th order partial derivative, applied term wise t o sequences of smooth functions. Let us take now a subalgebra two conditions, see (1.6.10) (2.2.7)
A c ( ~ ( f l ) ) ' which s a t i s f i e s the following
Ind(fl> c A
and (2.2.8) where II(fl)
up) c A i s the subalgebra of a l l the sequences with identical terms ~ ( $ 1= ($,$,$,
- - ,$, - - 1
corresponding t o arbitrary continuous functions $
E
C?' (a), see (1.6.4).
Obviously (2.2.9)
A
= A/Znd(n)
i s an associative and commutative algebra, with the unit element (2.2.10)
1, = u(1) + lnd(fl) E A
Moreover, the mapping
Global Cauchy- Kovalevskaia
107
is a n embedding of a1 e b r a s , s i n c e we obviously have s a t i s f i e d t h e neut rix condition, see (1.6.lfi
(2.2.12)
I n d ( a ) fi
a("
= u
as shown i n Appendix 1. For
!.
E
Ill l e t us d e f i n e t h e q u o t i e n t a l g e b r a s
(2.2.23)
h l = (A fl (ce(n) )@')/(~~~(fi) ,-(ce(fi) I )IN)
Embeddings of t h e d i s t r i b u t i o n s i n t o a l g e b r a s of g e n e r a l i z e d f u n c t i o n s , given by
a r e c o n s t r u c t e d i n Rosinger [I-31, as well as i n Colombeau [1,2], s e e a l s o Chapter 6. F i n a l l y we n o t e t h a t one can e a s i l y f i n d a subalgebra A c (c"(Q))' s a t i s f ie, (2 . 2 . 7 ) , ( 2 . 2 . 8 ) , by t a k i n g f o r i n s t a n c e A = (C"(Q))'.
vhicll
For t h e s a k e of completeness, l e t us mention t h e way t h e p a r t i a l d e r i v a t i v e o p e r a t o r s can be extended t o t h e q u o t i e n t a l g e b r a s Ae. Let u s suppose t h a t f o r a given l is s a t i s f i e d by t h e subalgebra A
E
IA,
t h e following additional condition
T h i s can be obviously secured, i f f o r i n s t a n c e , we t a k e A = (P'(S2)) Given now mapping
for s
E
A
k
n
E
IA,
(c'(n))IN.
p
E
INn,
Ipl + k
<
l,
.
we can obviously d e f i n e t h e
108
E.E. Rosinger
$3.
NONLINEAR PARTIAL DIFFEBENT1A.L OPEUTOBS ON SPACES OF GENERALIZED FDNCTIONS
Let us specify m E IN, the order of the nonlinear p a r t i a l differential operators considered i n the sequel. Suppose given an arbitrary continuous function (2.3.1)
F
E
CO(fl x
fi
where
ii
= car(p E D("
-
I
lpl
<
m)
Then we define the m-th order nonlinear p a r t i a l d i f f e r e n t i a l operator (2.3.2)
T(D) : Cm(fl)
CO(fl)
by (2.3.3)
(T(D)U) (x) = F(x,U(x)
Suppose given
t!
(2.3.4)
2
a subalgebra in
E
I,e 2 m.
,..., D ~ U ( X.).,.) , PEP, ( p( <m,
X E ~ ,u€cm(fi)
Let us take
.
(Co(ll))IN containing Ind(ll) U T(D) ( A n (Ce(fl))IN)
Finally, l e t us define the quotient algebra (2.3.5)
A
= '/Ind(n)
We show now that the mapping (2.3.2) can be extended t o a mapping (2.3.6)
T(D) : A'
-
by the definition (2.3.7)
e
for a l l
s = (#,(v E IN) E A fl (C (0))
(2.3.8)
T(D)s = (T(D)#,l
E 1)
or ,
where we denoted
Global Cauchy- Kovalevskaia
109
Indeed, i n order t o see t h a t (2.3.6) - (2.3.8) give a correct d e f i n i t i o n , it suffices t o note t h a t we have t h e property
which follows e a s i l y from (2.2.5) and (2.2.6). The i n t e r e s t i n t h e extensions (2.3.6) is i n t h e f a c t t h a t t h e global version of t h e Cauchy-Kovalevskaia theorem proved i n t h e sequel w i l l give generalized solutions which are elements i n A e
.
Remark 1 I n Sections 2 and 3 above, we have obviously followed t h e general cons t r u c t i o n i n Sections 6 , 9 and 10 i n Chapter 1, without however using in f u l l d e t a i l t h e respective notations. The simplified notation i n t h i s Chapter may o f f e r the reader a presentation which t o a good extent i s selfcontained. For t h e sake of completeness we note t h a t , with the notations i n Chapter 1, we have the following. Relation (1.6.4) gives
while (1.6.9)-(1.6.11) (2.2.16) give
together
with
(2.2.7),
(2.2.8),
(2.2.12)
and
Further (1.13.5) together with (2.3.6)- (2.3.9) give
§4.
BASIC LEMMA
Given t h e domain 0 i n I R ~ and on i t , t h e m-th order analytic nonlinear p a r t i a l d i f f e r e n t i a l equation
>
x = ( t , y ) E n , t E R, y E lRn-l, m q E INn-', with 1, 0 5 p < m, p + 191 5 m. Further, given f o r (2.4.1) the noncharacteristic analytic hypersurf ace
E. E. Ros inger
110
and t h e a n a l y t i c Cauchy d a t a
Lemma 1 I' c Q closed, nowhere dense and an a n a l y t i c function which is a s o l u t i o n of (2.4.1) on Q \ r and s a t i s f i e s
There e x i s t s @ : Q\r + C (2.4.3).
Proof For every x = ( t n ,y) a nonvoid open s e t
E
S,
t h e Cauchy-Kovalevskaia theorem, Walter, y i e l d s
Qx c Q, with x
(2.4.4)
E
Qx
and an a n l y t i c function which is a s o l u t i o n of (2.4.1), (2.4.3)
on Qx.
It follows by a n a l y t i c continuation t h a t we o b t a i n an a n a l y t i c s o l u t i o n of (2.4. I ) , (2.4.3) on t h e nonvoid open s e t
Obviously, we can only have t h e following two s i t u a t i o n s . Case 1.
If
il'
is dense i n Q , then t h e proof is completed by t a k i n g
Case 2.
If
9'
is not dense i n Q , then we o b t a i n t h e p a r t i t i o n
(2.4.7)
a = n,
u
rl u ni
where (2.4.8)
91 = i n t e r i o r of
is nonvoid open, while I'l
is c l o s e d , nowhere dense.
Now, we can t a k e (2.4.9)
xi = ( t i , y i )
(Q\Q1)
E Qi
Global Cauchy- Kovalevskaia
Then, with tl
E
R given by
XI
,
we define
which i s a noncharacteristic analytic hypersurface for (2.4.1). can consider any given analytic Cauchy data
On S1
we
In t h i s way, we reduced the original problem of proving the Lemma 1 for R and S, t o the problem of proving it for fll and S1 . This reduction obviously sets up an iterative process which can only lead to one of the following two situations. Alternative 1. After a f i n i t e number of iterations, we reach Case 1. More precisely, for some h > 1, we obtain the f i n i t e sequence of part it ions
where $+, = Q,Ql , . . . ,Qh, $2' are closed, howhere dense.
,...,nn"
are nonvoid open, while PI , . . . ,Ph+l
In that case, we obviously have an analytic solution of (2.4.1), (2.4.3) on the nonvoid open set
while RJ
i s dense in R
therefore, we can take
and the proof i s completed. Alternative 2. We never reach Case 1 , a f t e r any f i n i t e number of iterations. Then, for an ordinal number a 2 1, we obtain an open set
E. E. Rosinger
112
i n t h e followinf and t h e nonvoi construct (2.4.14)
way. I f open s e t
Ra c RP,
according t o (2.4.5).
9
o = ,8 + 1 f o r a s u i t a b l e o r d i n a l number 8, Rb c R h a s a l r e a d y been o b t a i n e d , t h e n we
nonvoid, open
F u r t h e r , we d e f i n e t h e nonvoid open s e t
and t a k e (2.4.16)
Ro = i n t e r i o r of
(R\A,)
Otherwise, i f a # 1 + 1 f o r any o r d i n a l number we d e f i n e t h e nonvoid open s e t
P,
i n s t e a d of ( 2 . 4 . 1 5 ) ,
and t h e n a g a i n , t a k e Ro as i n (2.4.16). T h i s p r o c e s s can be continued u n t i l we reach Ro = 4 i n ( 2 . 4 . 1 6 ) . I n t h a t c a s e , we obviously have an a n a l y t i c s o l u t i o n o f ( 2 . 4 . 1 ) , (2.4.3) on t h e nonvoid open s e t
and R'
is dense i n R
which means t h a t by t a k i n g (2.4.18)
r = n\n/
t h e proof is completed.
Global Cauchy-Kovalevskaia
8 5. GLOBAL GENERALIZED SOLUTIONS Given the m-th order analytic nonlinear partial differential equation
t f R , ~ER"-', mz1, O < p < m , with x=(t,y)sfl, p + l q l < m, and with the analytic Cauchy data
qr01"-',
on the noncharacteristic analytic hypersurface
s=
(2.5.3)
(x = (t,y)
E
nit = to) # 4
Let
be an analytic solution of (2.5.1), (2.5.2) as given by the Lemma 1 in Section 4. Then (2.5.5)
c fl is closed and nowhere dense
hence, since 'I
is closed, it follows that
Suppose given a p- smooth function a:IR
-
[0,1] , such that
a(x) = 0, for 1x1 < a (2.5.7) a(x) = 1, for for certain 0 < a < b
1x1
> b
Let us define the sequence of p-smooth functions
where for
u E DI,
we have
E. E. Rosinger
Then obviously, t h i s sequence s of f'-smooth functions has the property
v
x
E
n\r
:
3 p E Dl, V c Q\I' neighbourhood of V VEW, v ) p , y ~ V : d,(y) = $(Y)
(2.5.10)
Now, similar t o (2.3.2), differential operator
T (D) : Cm(Q)
(2.5.11)
(2.3.3),
-
x :
l e t us define the nonlinear partial
(? (0)
corresponding t o (2.5.1) by
with x = ( t , y ) E Q, t E R , y E Rn-l, 0 5 p t m, q E Dln-', p + lq/ 5 ?. With the above notation, the equation (2.5.1) can be written i n the equivalent form
Then (2.5.4)
,
(2.5.5) and (2.5.10) obviously imply
With the above preparations, we can now construct the algebras of generalized functions w i t h i n which the analytic nonlinear partial differential equation (2.5.1) has a g loba 1 eneralized solution for the noncharacterist i c analytic Cauchy data (2.5.27 , (2.5.3). Let us take a subalgebra
which s a t i s f i e s the conditions (2.5.16)
S E A
and, see (2.2.7), (2.2.8), (2.2.15), (2.2.18)
The possibility of such a choice is obvious, since instance, w i l l satisfy (2.5.16) and (2.5.17).
4
= (P(Q))'
for
Global Cauchy-Kovalevskaia
115
Now, with the construction in (2.2.16), (2.3.5) and (2.3.6) we obtain the following g l o b a l existence result for the equations (2.5.I ) , or equivalently (2.5.13). Theorem 1 For l E I1, l > m, the m-th order analytic nonlinear partial differential equation in (2.5.13) (2.5.18)
T(D)u(~,Y)
=
0,
=
(t,~)E
n
with the noncharacteristic analytic Cauchy data
and (2.5.20)
S = {x = (t,y) E Rlt = to) #
4
has generalized solutions (2.5.21)
U
E
A'
defined on the whole of Q.
-
These solutions U are analytic functions (2.5.22)
+ :~ \ r
c
when restricted to suitable open, dense subsets Q\r, where (2.5.23)
r c il
closed, nowhere dense.
Proof Using the construction in (2.2.16), (2.3.5) and (2.3.6) we can extend (2.5.11) to a mapping (2.5.24) where (2.5.25) while
T(D)
:
-
A' I
E.E. Rosinger
w i t h 1 being a subalgebra in Ind(Q) U T(D) (2 n (ce(n) 1')
(co(o))' containing
-
Let us now define
which i s possible i n view of (2.5.8) and (2.5.16). It only remains t o show that with the mapping in (2.5.24), we have
But i n view of (2.5.27), (2.3.7) and (2.3.8) we have
thus (2.5.28) follows from (2.5.14). Remark 2 The result in Lemma 1 i n Section 4 , and therefore in Theorem 1 above, i s an existence result. From the proof of Lemma 1, in particular from the freedom of choice in and (2.4.11), it i s obvious that in general, many solutions , (2.5.5) can be obtained. A f i r s t open problem then i s whether the class of solutions (2.5.4), (2.5.5) exhausts a l l the solutions correspondin t o various solution concepts which can reasonably be associated with ana ytic nonlinear partial differential equations. Here one can refer for instance t o situations where solution concepts are supposed t o accommodate phenomena such as turbulence, strange attractors, e t c . , see Richtmeyer, Temam and the literature cited there. Within the solution concept used above further open problems concern the uniqueness and regularity of the global solutions obtained. From the above construction i t i s obvious that these problems are connected w i t h the appropriate uses of the freedom in the choice of the subalgebras A and 2 . Indeed, the smaller these subalgebras, within the required conditions, the better the uniqueness and regularity properties of the corresponding flobal, generalized solutions in (2.5.21). This problem i s close1 connecte w l t h t e s t a b i l i t y , generality and exactness of generalized so utions for linear and nonlinear partial differential equat Ions, introduced and specifically dealt with i n Section 11, Chapter 1. I n view of (2.3.12) and (2.5.24)-(2.5.28), the global eneralized solution U in (2.5.21) has indeed the meaning defined in tection 13, Chapter I .
k
I
Global Cauchy-Kovalevskaia
117
$6. CLOSED NOWHERE DENSE SINGULAEITIES WITH ZEBO LEBESqUE I[EASWE In this Section, in Theorem 2, we improve on the result in Theorem 1, Section 5, by showing that the closed, nowhere dense singularity 'l in , (2.5.23) , outside of which the global generalized solution U in is analytic, can be chosen in such a way that it has zero Lebesque measure, i.e. mes I' = 0
(2.6.1)
The surprising fact about this strengthening of Theorem 1 in Section 5 is that it can be obtained in a constructive way, without the use of transfinite induction. The essential instrument in obtaining this strengthened result in (2.6.1) is presented next in Proposition 1, which was offered by I. Oberguggenberger in a private communication, Oberguggenberger [5]. This result has as well an obvious interest in itself, since it is, according to our best knowledge, the first result which gives the rather sharp kind of information in (2.6.1) on the size of the subsets on which the classical Cauchy-Kovalevskaia theorem may fail to yield analytic solutions. Pro~osit ion 1 (Oberguggenberger [5] ) Given the analytic nonlinear partial differential equation (2.5. I), there exists I' c fl with (2.6.2)
'l
(2.6.3)
mes 'l
and W
: fl\l'-
closed, nowhere dense in fl = 0
Q: an analytic solution of (2.5.1) on fl\I'.
Proof Assume given (t,y) E n. If we choose some initial values on an analytic hypersurface passing through (t,y) , then the Cauchy-Kovalevskaia theorem yields
and an analytic solution W
:
I
4
Q: of the equation (2.5.1) on I.
Assume given K c fl open, such that its closure K is compact and Then applying the above to points (t,y) E K, we obtain
Kc
fl.
E. E. Rosinger
118
and analytic solutions Wj : I j with 1 < j J , such that
<
Kc Assume given 1
<
u
-
C of the equation (2.5.1) on each
Ij
l<jCJ i 5 n and l e t
be the set of pairwise different elements i n taken in increasing order.
ali,.
. . ,oJi,Bli,. . . ,BJi
Let us denote by
i?
the set of a l l
where 1 < j i
a l l the I
E
<
I j,
j(i),
with
1
n.
Further, l e t us denote by
1 such that
Then obviously
But i n view of (2.6.4),
Q1
,. . . ,QH
are pairwise disjoint and
and then
i s an analytic solution of equation (2.5.1) on Qh.
Global Cauchy- Kovalevskaia
Let u s now denote f o r 1 Rh = Qh
"
which a r e obviously open and p a i r w i s e d i s j o i n t .
Indeed, f i r s t we n o t e t h a t t h e i n c l u s i o n t h e i n c l u s i o n 3 . Obviously
'c'
We s h a l l show t h a t
is immediate.
Now we prove
I f we denote
and apply Lemma 2 below t o ( 2 . 6 . 5 ) , we o b t a i n
and t h e proof of ( 2 . 6 . 7 ) is completed. S i n c e RI ,. . .,RH a r e p a i r w i s e d i s j o i n t , we o b t a i n from ( 2 . 6 . 6 ) a n a n a l y t i c solution
of t h e equation ( 2 . 5 . 1 ) on an open, dense s u b s e t K t
of
K,
From ( 2 . 6 . 4 ) it is obvious t h a t (2.6.10)
mes (K\K') = 0
It o n l y remains now t o extend ( 2 . 6 . 8 ) - ( 2 . 6 . 1 0 ) from K t o Q. Let u s t a k e a sequence KO,K1
,.. .Ky, . .
of nonvoid open s u b s e t s i n Q,
such t h a t
where
E. E. Rosinger
K, c Kv+l, Kv compact, v
E
W
E
W,
and
We apply (2.6.8)- (2.6.10)
successively to
K = KO and K = Kv\Kv-l, u
v
>
1
and o b t a i n
V E N
Wu:K;-C,
a n a l y t i c s o l u t i o n s of ( 2 . 5 . 1 ) on K;,
S i n c e K;, show t h a t
with
v
E
N
with
a r e obviously p a i r w i s e d i s j o i n t , we o n l y have t o
For t h a t we n o t e t h e f o l l o w i n g obvious r e l a t i o n s
We a l s o n o t e t h a t i n view of (2.6.10), we have mes(Kv\K;)
= 0,
v
which t o g e t h e r with (2.6.11) y i e l d
E
N
Global Cauchy-Kovalevskaia
Lemma 2 Let A, B be subsets in a topological space. If B is open and B c K , then also
BcAng Proof Then x E B yields
Take x E B \ A .
V Q o p e n : x ~ Q e B n Q # #
while x f
A implies 3 Qo open
:
x E Qo and A
n B n Qo
=
4
But in view of the above Q1
and obviously since A r7 Q1
=BnQo#4
91
=
4.
is open. If we take now y E QI c B, we obtain y f K , Hence B\A 4, which is absurd
+
An now we come to the s t r e n g t h e n e d form of the globalized version of the classical Cauchy-Kovalevskaia theorem. Theorem 2 Given the m- th order analytic nonlinear partial differential equation
x = ( t , y ) ~ Q , ~ E R ,~ E R " " , m21, O(p<m, ~ E H ~ - ' , with p + Iq( m, and with the noncharacteristic analytic Cauchy data
where (2.6.14) Then for
S
= {x = (t,y) E Q J t = to) #
4
I E IA, I 2 m, there exist generalized solutions
E.E. Rosinger
defined on the whole of fl. These solutions U are analytic functions
when restricted to suitable open, dense subsets R\I', (2.6.17)
l'
c
fl
where
closed, nowhere dense
and
Proof It follows easily from Proposition 1 above and the proof of Theorem Section 5 .
1
in
8 7. STRANGE PHENOMENA IN PARTIAL DIFFERENTIAL EqUATIONS 87.1
Too Hany Eqnat ions and Solutions?!
The results in Theorems 1 and 2 in this Chapter may at first seem strange owing to the apparent ezcessive wealth of the patched up type generalized solutions which they provide. Let us further enquire into that phenomenon of 'excessive wealth' of solutions of differential equations, and cite two recent results, one concerning ordinary, and the other partial differential equations.
A rather related fact which comes to attention when observing the various linear and nonlinear ordinary or partial differential equations which appear in the customary mathematical models is their low order, as well as relatively simple nonlinearity. For instance, the Navier-Stokes equations or those of general relativity are of second order and have quadratic nonlinearities, etc. In view of that, one may as well ask whether indeed we need general nonlinear theories capable of handing partial differential equations of arbitrary order and nonlinearity. The question becomes only aggravated when we learn about several recent results which seem to indicate that partial differential equations of high order and nonlinearity could hardly be models of physical theories, since they present strange properties concerning their sets of solutions. In the next two subsections we shall present two such recent results. They seem to indicate the existence of a kind of upper bound on the complexity of partial differential equations which may be suitable for modelling physical phenomena.
Global Cauchy- Kovalevskaia
$7.2
Universal Ordinary Differential Equations
As i f t o make things worse, t h e mentioned kind of upper bound seems t o e x i s t even i n t h e p a r t i c u l a r world of nonlinear ordinary d i f f e r e n t i a l equat i o n s of polynomial type. We mention i n t h i s respect t h e following recent r e s u l t , Rubel. There e x i s t s a nontrivial fourth order d i f f e r e n t i a l equation
vhere P i s a polynomial i n four variables with integer c o e f f i c i e n t s , such t h a t f o r every f E CO(R) and w E C0 (R), with w(x) > 0, f o r x E IR, there e x i s t s a solution u E p ( R ) of (2.7.1) with t h e property
I n other words, (2.7.1) has s o many p - smooth solutions u t h a t these solution can approximate every continuous function f on IR i n t h e rather strong sense of (2.7.2). An additional disturbing f e a t u r e of t h e above r e s u l t i s i n t h e simplicity and directness of i t s proof, whose main steps we reproduce now. Assume given f and w a s above. Obviously we can f i n d a pricewise a f f i n e conIf (x) - g(x) 1 < w(x)/2, with tinuous function g E C0 (R) , such t h a t x E R. Therefore, it suffices t o f i n d a solution u E p ( R Of (2.7-11y such t h a t Ig(x) - u(x)l < w(x)/2, with x E R. This can e done easi y as follows. Let us define $ , x E p ( R ) by
I,
Given a,B,A,B E R ,
l e t us define
x
E
Cm(R) by
Then, by an ingenious sequence of d i f f e r e n t iaions and eliminations, aimed a t a , p, A and B , one obtains (2.7.1) a s t h e equation which admits (2.7.3) a s solution, f o r a l l t h e values of a , 8, A and B.
k
Assume now given a f i n i t e interval I = a,b] c R on which g i s a f f i n e . Obviously, one can choose a,B E R , suc t h a t x has t h e constant value A f o r a x a+€ and has t h e constant value B f o r b- c x b, where c > 0 i s suitably chosen. Of course, we can choose A = g ( a ) and
< <
< <
E.E. Rosinger
124
B.= g(b) and then g and x are both monotonously increasing or decreasing on I = [a,b] . Now, by choosing I sufficiently small, the continuity of g and x will yield
Finally, the solution u E P(R) of (2.7.1) is obtained by the concatenation of all X . Then (2.7.2) follows from (2.7.4). $7.3
Universal Partial Differential E q ~ions t
After the result in subsection 7.3 above, the following similar property of nonlinear partial differential equations, Buck, may no longer seem surprising. This time however the proof is rather hard, as it uses Kolmogorov's solution of Hilbert's Thirteenth Problem. In view of this, we shall omit the proof and only present the result which reads as follows. For every n E IN, n 2 2, there exists a nontrivial polynomial nonlinear partial differential equation
where P is a polynomial with real coefficients and degree at most d(n), with the order of (2.7.5) at most m(n), i.e., p E Pin, Ip( 5 m(n) in (2.7.5), and such that for every f E CO ( [0,1]n, and r > 0, there exists a solution U E C~([O,~]") of (2.7.5), with the property
In this way, similar to (2.7.1) and (2.7.2), the algebraic partial smooth solutions U that differential equation in (2.7.5) has so many Pthey can uniformly approzimate every continuous function f on [O,lln Unlike in subsection 7.2 above, a somewhat comforting thing with the result in (2.7.5) and (2.7.6 is that - as known so far - the values of the order m(n) and degree d(n] are quite large even for small values of n, such as for instance n = 2, 3 or 4. 57.4
Final Remark
The results in the above subsection 7.2 and 7.3 show the existence of polynomial ordinary or partial differential equations which are universa 1 in the sense that their P-smooth solutions are sufficiently many in order to approximate uniformly continuous function. A difficulty arising from that situation is the following. A given ordinary or partial differential equation which appears in the modelling of a physical phenomenon will define by its solutions the possible states of this phenomenon. Therefore
Global Cauchy- Kovalevskaia
the set of these solutions should have a similar size with that of the set of states. Now, the set of possible states of a given physical phenomenon i s usually much more restricted than the set of a l l continuous functions, for instance. In view of this, i t may appear that equations such as ( 2 . 7 . 1 ) or ( 2 . 7 . 5 ) may f a i l to model any usual physical phenomena.
E. E. Ros inger
APPENDIX 1 ON TEE STRUCTURE OF TEE NOWHERE DENSE IDEALS
P r o ~ o s iion t 2 Ind(Q) is an ideal in
(P(9))'.
Proof In view of (2.2.4) it is obvious t h a t
I t only remains t o show t h a t (2.Al.l)
Ind(')
+
Ind(')
Ind(')
Let w,w' E lad@), then (2.2.4) yields
r,r' c
(2.A1.2)
fl closed, nowhere dense
such t h a t
Let us denote
then obviously I'" view of 2.A1.3), (2.Al.l) olds.
\
c Q i s again closed and nowhere dense. w"
and
I"'
will also s a t i s f y (2.2.4).
P r o ~ o s iion t 3 The nowhere dense ideal s a t i s f i e s the neu t r i x c o n d i t i o n (2.A1.4)
Ind(n) fl
a("
= 0
Moreover, i n Therefore,
Global Cauchy- Kovalevskaia
Proof Take $ E C0 (62)
such t h a t
Then (2.2.4) y i e l d s (2.A1.5)
I' c 52 c l o s e d , nowhere dense
such t h a t
But t h e c o n t i n u i t y of obviously y i e l d
$ on 62 and t h e r e l a t i o n s (2.A1.5),
An important p r o p e r t y of t h e nowhere dense i d e a l Jnd(fl)
(2.A1.6) w i l l
is p r e s e n t e d now.
P r o ~ o s ii to n 4 Every sequence w E Znd(62) of continuous f u n c t i o n s on f o l l o w i n g stronger v e r s i o n of (2.2.4)
3
r
c R c l o s e d , nowhere d e n s e
8
satisfies the
:
v (2.A1.7)
x E n\r : 3 p E M, V c R \ r neighbourhood of v uEM,u)p,yEV:
x :
w,(Y> = 0
Proof It f o l l o w s e a s i l y from Lemma 3 below. Indeed, assume w E Znd(62) and I' c 62 as given by (2.2.4) . Then a' = Q\I' is open and dense i n Q. Further
v
X E V :
3 pEM:
Let
I c 62'
be nonvoid, c l o s e d .
Then Lemma 3 y i e l d s a nonvoid, r e l a t i v e l y
E .E . Ros i n g e r
128
open AI c I large. Let
Q"
such t h a t
w,
v a n i s h e s on
for
A
v E IN
be t h e union of a l l t h e nonvoid, open s u b s e t s v E IN s u f f i c i e n t l y l a r g e .
wv v a n i s h e s f o r
sufficiently
A c Q'
on which
But Q" is dense i n Q' s i n c e i n t h e argument above, one can choose I as ranging o v e r a l l t h e c l o s e d b a l l s i n Q . Hence Q" is dense i n Q , s i n c e Q' is dense i n Q. Now I?' = Q\Q"
will s a t i s f y (2.A1.7)
o
Lemma 3
-
Suppose E is a complete m e t r i c s p a c e , E' is a t o p o l o g i c a l space and we E' , a r e g i v e n t h e continuous f u n c t i o n s f : E + E' , and f,, : E with v E IN, such t h a t V X E E : 3 VEIN: v vEIN,v>p f Jx>
= f (XI
Then f o r e v e r y nonvoid, c l o s e d s u b s e t I c E, t h e r e exists a nonvoid, r e l a t i v e l y open s u b s e t A c I and v E IN, such t h a t
Proof
It f o l l o w s e a s i l y from a B a i r e c a t e g o r y argument Indeed, f o r I c E nonvoid, c l o s e d and f o r v
E
IN,
l e t u s denote
Then t h e h y p o t h e s i s obviously y i e l d s
is c l o s e d i n But f o r p E IN, I P nuous. And I being c l o s e d i n
I, E,
s i n c e f and a l l f v a r e c o n t i it is i n i t s e l f a complete m e t r i c
Global Cauchy- Kovalevskaia
space. p E
,
129
Thus the Daire category argument implies that for at least one the interior of I relative to I i s not void 0 F
The u t i l i t y of Proposition 4 i s i n the next result. Corollarv 1 For C E [A, (2.A1.8)
we have the relations gP(%d(n)
fl (cC(~))')
Proof
It follows a t once from (2.A1.7)
Ind(n), P
' 9
(PI
5
This Page Intentionally Left Blank
CBAPTER 3 ALGEBRAIC CBARACTERIZATION FOR THE SOLVABILITY OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
I n t h i s Chapter we consider arbitrary olynomial nonlinear partial diff erent i a l equations w i t h continuous coef {icient s
where Q c (Rn i s nonvoid open, ci,f U : 0 + (R i s the unknown function.
E
P ( Q ) are given,
pij
E
INn,
while
The aim of the Chapter i s t o establish a necessary and sufficient condition for the existence of generalized solutions for partial differential equations in (3.1.1). This characterization happens t o be of a simple and purely algebraic nature and i t i s given by a special case of the aeutris coadit ion. Obviously, the nonlinear partial differential equations i n (3.1.1 are partly more general than the arbitrary analytic nonlinear partial different i a l equations. To that extent, the existence of generalized solutions, more precisely, t h e i r characterization, proved i n t h i s Chapter i s an extension of the global version of the Caucl~y-Kovalevskaia theorem presented earlier i n Chapter 2 . The reason we consider the particular, polynomial form of nonlinear partial differential equations in (3.1 . l ) i s that they allow f o r a rather simple and direct algebraic treatment leading to the mentioned neutrix characterization for the existence of generalized solutions. A t the cost of certain technical complications , a similar result may be obtained for arbitrary continuous nonlinear partial differential equations
where F i s any real valued function, continuous i n a l l i t s arguments. Such a result will obviously give a f u l l scale extension of the mentioned global version of the Cauchy-Kovalevskaia theorem. The question remains open whether existence results such as those i n t h i s Chapter can be obtained within the particular framework in Colombeau [1,2].
.
E. E Rosinger
132
52.
THE NOTION OF GENERALIZED SOLVTION
For convenience, and in order t o make the presentation of t h i s Chapter more selfcontained, we recall the relevant eneral e n t i t i e s defined in Chapter 1, and do so by using again a simplifie notation.
f
Let us denote b T(D2 the pol nomial nonlinear partial differential operator i n the l e t han term of 6 , l . l ) and define i t s order by
The spaces of generalized functions used i n the sequel w i l l be quotient vector spaces
or quotient algebras
A = A/Z
(3.2.3) where we have
with V and S vector subspaces, and respectively (3.2.5) where A
1c A c (co(Q))'
i s a subalgebra, while 1 i s an ideal in A.
Further, we shall require that the inclusion diagrams hold
and
(3.2.7)
where 0 denotes the null vector subspace, while in (C~(Q))', that i s
P(R)
i s the diagonal
Algebraic characterization
and similarly for #'(a). with u($) = ((, ,. . . ,$, . . .) s (~(a))', that P(Q) c @(!I) and both are subalgebras.
Note
Finally, each of the inclusion diagrams (3.2.6) and (3.2.7) is supposed to satisfy the neutriz condition, that is
respectively
It is easy to see that (3.2.9) and (3.2.10) respectively are the necessary and sufficient conditions for the existence of the vector space and algebra embedd ings
and
We recall that from a geometric point of view, the neutrix condition (3.2.9) means that the vector subspace V is off diagonal in the cartesian Similarly, the neutrix conditon (3.2.10) means that the product (c"(o)'. ideal 2' is off diagonal in the cartesian product (p(Q)) IN . Assuming now that
we can obviously extend the classical partial derivative operators
by defining them according to
E.E. Rosinger
where
D ~ = S
(DPs0,. . . , D ~ S . . .) ~ ,for every s = ( s o , . . . , s v , . . .)
E
(C~(Q))~.
I t follows that, as an extension of the classical partial differential operator
T (D) :
(3.2.17)
(9) -, C ' (Q)
we can define the mapping
where T(D)s = (T(D)so,.. . ,T(D)sV,.. .)
for every s = ( s
0'
. . . , S v , - ..)
E ( C m ( ) ) . Indeed, it i s easy to see that the definition i n (3.2.18),
(3.2.19)
correct. For that, we note the then i n view of (3.1.1), we obtain s,v E ( ~ ( n ) ) ' , (3.2.20)
is
T(D)(s + v) = T(D)s + C s,.
following.
Let
P
D "v
a
where s, are products of ci, D Pij s and possibly gPijv, while p" are some of the pij. In this way, if s E S and v E V, then the inclusions (3.2.13) imply that
therefore (3.2.22)
T(D)(s + v) - T(D)s E 2
hence (3.2.19) i s a valid definition. Now, as an extension of the notion of classical solution, we can define the eneralized solutions for the nonlinear partial differential equation i n 3.1.1) as being given by a l l generalized functions
7
(3.2.23)
U=S+VEE=S/V
such that, i n the sense of (3.2.18), we have
Algebraic characterization
where we note that, in view of (3.2.12), we have f
E
A.
As shown in Rosinger [1,2,3] and later in Chapter 6, this notion of generalized solution is an extension of the notion of distributional or weak solution. In particular, one can construct the spaces of generalized functions in (3.2.2) and (3.2.3) in such a way that they contain the distributions, that is Moreover, in (3.2.18), and therefore in (3.2.23 , E itself can be an algebra. In fact, one can have E = A, in whic case they are differential algebras and contain the distributions.
1
However, for the purposes of this Chapter, it is convenient to allow for the generality of the framework in (3.2.18) and (3.2.23). We note that within this framework, the only connection needed between E and A is that in (3.2.13), which for convenience we shall denote by
In rest, E can be arbitrary within the conditions (3.2.2), (3.2.4), (3.2.6) and (3.2.9) . We shall denote by
the set of all such quotient vector spaces E of generalized functions. Similarly, A can be arbitrar , provided that it satisfies the conditions (3.2.3), (3.2.5), (3.2.7) and (3.2.10). And we denote by
the set of all these quotient algebras A of generalized functions.
$3. THE PROBLEM OF SOLVABILITY OF NONLINEAR PARTIAL DIFFERENTIAL EqUATIONS Given the polynomial nonlinear partial continuous coefficients in (3.1.1) (3.3.1)
T(D)U(X)
=
f(x),
x
E
differential equation with
n
the problem of its solvability will be formulated as follows. For sequences of smooth functions
E.E. Rosinger
136
we shall find the necessary and sufficient condition for the existence of spaces of generalized functions E = S/V E vsm(fl) and A = A/Z E AL(O) with
and such that the generalized function
defined by the sequence differential equation
s
in (3.3.2), satisfies the nonlinear partial
in the sense of (3.2.18) . For convenience, we shall denote the nonlinear partial differential equation in (3.3.5) by E.
Let us now make more explicit the above solvability problem in (3.3.2)Condition (3.3.3) is nothing but (3.2.13), while condition 1: ' is equivalent to
SES
(3.4.1)
Finally, in view of (3.2.19), condition (3.3.5) is equivalent to ws = T(D)s - ~ ( f )E I
(3.4.2)
It follows that the solvability problem in (3.3.2)- (3.3.5) is equivalent to findin E = S / V E V S ~ ( ~ ~and ) A = A / I E A L ( ~ ) such that (3.2.13), (3.4.17 and (3.4.2) are satisfied. In view of the fact that condition (3.4.2) only involves the equation E, the sequence s and the ideal 3, without relating to E = S/V or A, we shall deal with it first. For that purpose, given a sequence of conDI tinuous functions w E (cO(fl)) , let us define the quotient algebra
where
Aw
is the subalgebra in
(CO(0))'
generated by
{w) U lo(fl)
,
Algebraic characterization
while
$
(3.4.4)
is the ideal in
A, generated by w, thus
Zw = w.Aw
It follows easily that (3.4.5)
Aw
=
Aw/ZwE AL(Q)
H
Zwfl @ (Q)
The interest in the quotient algebra characterization.
Aw
=
0
comes from the following
If w E (p(Q)) DI then the three conditions below are equivalent (3.4.6)
3A
(3.4.7)
Aw
(3.4.8)
zwn @ (Q)
= =
AIZ E AL(O)
:
w E Z
Aw/ZwE AL(Q) =
0
Further, if (3.4.6) holds, then we have the inclusion diagram
Proof In view of (3.4.5) the conditions (3.4.7) and (3.4.8) are equivalent.
. Then (3.2.7) yields (w)u@(Q) c Z U A c A Let us assume Further, in view of (3.4.4) we obviously have therefore A, c 13-4.6) WEZ*W.ACZ+W.A~CZ*Z~CZ Thus in view of (2.10) we obtain Zw n @(R) c Z n @(Q) = 0 and then (3.4.5) ends the proof of (3.4.8). Meanwhile we note that (3.4.9) has been proved as well.
E.E. Rosinger
138
Conversely, i f (3.4.8) holds, then we can take A = A"
in (3.4.6).
D
Returning t o condition (3.4.2), it i s now obvious that it can be written in the equivalent neuirix or off diagonal form (3.4.10)
zw n UO (Q) = 0 S
and in view of (3.4.9),
ZwS
is the smallest ideal f o r which (3.4.10) has
t o hold. In order t o further explicitate the condition (3.4.10) it i s useful t o impose the following rather natural and mild r e s t r i c t i o n on the plynomial nonlinear p a r t i a l d i f f e r e n t i a l operators T(D) in E . We c a l l T(D) nontrivial on Q , if and only i f , when r e s t r i c t e d t o any Cm(Q'), with 8' c 8 nonvoid and open, the range of the mapping T(D) in (3.2.17) is an i n f i n i t e subset of CO (Q') . I t i s equally convenient t o i m ose a similarly natural and mild r e s t r i c t i o n on the sequences s in (3.3.2!, which through (3.3.4) are supposed t o give the generalized solutions for E . For t h a t , we shall replace (3.3.2) by the condition
Here ( ~ ( 9 ) ) :i s the subset of a l l the sequences not s a t i s f y the condition
s
E
( ~ ( f i ) ) ' which do
3 Q' c Q nonvoid, open: (3.4.12)
3 t subsequce in s , g V VEIN: T(D)tV = g , on 9'
P r o ~ o s i t i o n2 If
T(D) i s nontrivial on Q, then
E
P(fi'), g # f :
Algebraic characterization
Proof Assume that
(p(Q))=;4 then (3.4.12) implies that
3 Q' c Q nonvoid, open
:
3 t subsequence in s, g E C'(Q),
g # f
:
3 vEN: T(D)tu = g, on Q' But (3.4.14) obviously implies that the range of the mapping T(D) (3.2.17), when restricted to P(Q'), is a finite subset of CO(Q')
in
At this stage we are led to introduce, see Rosinger [2, p. 391
2(a>
(3.4.15)
as the set of all the sequences condition does no2 hold
w
E
(CO (Q)'
for which the following
3 Q' c Q nonvoid, open: (3.4.16)
3 z subsequence in w, h
E
CO(Q'),
h # 0
:
V YEIN: zu = h, on Q'
Indeed, we obviously have the property
However, the important pro erty of the set of sequences Z(Q) condition (3.4.10) is the ollowing.
!
Pro~osition 3 If w
E
Z(Q)
then
related to
E.E. Rosinger
Proof
It i s easy t o see that an element of the intersection i n (3.4.18) has the form
where I E IN and $,+b0,... ,$( E f ( 9 ) . Therefore in order t o prove (3.4.18), it suffices to show that $ = 0 on 9. Assume that t h i s i s not the case and 9' c 9 i s nonvoid, open, such that
Denoting by wv, with v E DI, the continuous functions on 9 that are the terms i n the sequence w, the relation (3.4.19) written term by term, yields (3.4.21)
( W ~ ( X ) ) ~ + ~ .t.. $ ~.t( X wv(x).$ ) (x) + (-$(x)) = 0, V v 0
E
N,
x E 9
Therefore (3.4.20) w i l l iniply that the infinite matrix
has rank a t most
l
t
1, for any given x
E
9'
Now, a well-known roperty of Vandermonde determinants implies that the infinite sequence o numbers
P
contains a t most l + 1 different terms, for any given x E 9'. Therefore Lemma 1 below will grant the existence of a closed, nowhere dense subset I" c 9', such that each x E Q'\I" possesses an open neighbourhood 9" c fl'\I", with the property that the infinite sequence of functions
Algebraic characterization
when restricted t o Q", contains only a finite number of different functions. In other words, there exists a subsequence w" in w and 6'' E C"(Q") such that
Now, u E 'R(Q) w i l l imply that $" = 0 on Q". gether with (3.4.21) w i l l contradict (3.4.20).
And then (3.4.22) to-
Lemma 1 Suppose the sequence w = (wO,.. . , w y , . . .) of continuous functions on i s such that f o r any given x E Q, the sequence of numbers
Q
contains only a finite number of different terms. Then there e x i s t s a closed, nowhere dense subset r c 9 , such that the sequence of functions
restricted t o a suitable neighbourhood of any given only a finite number of different terms.
x E 9\r,
contains
Proof Denote by functions
I'
the s e t of a l l points
x
when restricted t o any neighbourhood of different terms. It i s easy t o see that remains t o prove that I' has no interior.
E
r
Q
such that the sequence of
x, contains i n f i n i t e l y many i s closed. Therefore it only It suffices t o show that
Indeed, denote 9' = i n t r and assume Q' # 4. Then I" corresponding as above t o Q', will s a t i s f y l" = Q ' , hence contradicting (3.4.23). In order t o obtain (3.4.23), the Baire category argument will be used in tuo successive steps. F i r s t , f o r p E N,
define the closed set
E. E. Ros inger
V
VEIN,
v > p + l :
A ={XEQ 3 AEN, ~ < p : P .,(x) = wA(x)
then obviously
therefore the Baire category argument implies t h a t int A f f o r a certain p E M.
n
(3.4.24)
Denote Q' = i n t A P.
We shall prove that
(fi\r) # 4.
Denote f o r p E IN
then we have
Indeed, denote f o r x E 52'
Mx and take p E IN,
= {(A,") E IN
x
I
IN
A < v 5 p, wA(x) f w,(x))
such that
I l ( p + l ) i min{lwA(x) - wv(x) l then obviously x E A' P' Now we show that (3.4.26)
1
A' closed, V p E IN P
Indeed, denoting M = {(A,") E N
we have
x
IN
I
A < v
(A,")
E
Mxl
Algebraic characterization
A' = u ( u {X E Q' I w ~ ( x -) w ~ ( x ) I i / ( p + l ) } P KcM (A ,v)EK
n
But (3.4.25) and (3.4.26) together with the Baire category argument imply that i n t A; # ( f o r a certain p E N . Denote then Q" = i n t A' P' will obviously be complete i f we show that (3.4.27)
The proof of (3.4.24)
n J / c n\r
Assume therefore x E 52' ' and V c 52' ' an open, connected neighbourhood of x. We s h a l l prove that the sequence of functions
when restricted t o V , contains a t most p t 1 different terms. Indeed, i f v E IN, v > p t 1, then wy(x) = wA(x), f o r a certain A E N, A 5 p , since
B u t then
Assume indeed that (3.4.28) i s f a l s e . y E V. Denote
V' V" then x E V ' , y closed. But V"
= {x' E V = {x" E
E
Then wV(y) # wA(y), f o r a certain
I
wy(x') = wA(x1)) V ( w~(x")# w~(x"))
V.", V = V' U V", V' n V" i s also closed, since
=
4 and V'
i s obviously
the inclusion J being obvious, while the converse r e s u l t s a s follows. then there exists o E N, a 5 p, such that Take x" E V" , wv(x' I) = wg(xr'), since v ? p t 1 and
E.E. Rosinger
144
# wA(x"),
Hence w,,(x")
therefore
a, A
and
w i l l imply t h a t
and t h i s completes t h e proof of (3.4.29).
As t h e decomposition V = V' U V" t h a t has been obtained c o n t r a d i c t s t h e connectedness of V , i t follows t h a t (3.4.28) holds. Now, (3.4.28) implies (3.4.27), which completes t h e proof of (3.4.24). Thus f i n a l l y (3.4.23) has been proved. Remark 1 The extent t o which Lemma 1 above is sharp can be seen in the examples presented i n Appendix 1. For t h e sake of completeness, we note t h e following obvious property of t h e s e t 2(9) of sequences of continuous functions on 9 (3.4.30)
v
W E
(co(n))'
w E qn) H
where, f o r z = (z
0'
:
I
V 9' c 9 nonvoid open, z subsequence in w :
.. .z , . . .
E
(
( DI , we denote
with
being t h e usual r e s t r i c t i o n of t h e function subset Z c (CO(a))' l e t us denote
Based on (3.4.30)
,
we obtain
k
to
9'.
Further, f o r any
Algebraic characterization
Corollarv 1
If w
E
(p(Q))'
then
V Q' c Q nonvoid open, z subsequence in w (3.4.31)
E
1~1~'
2(~) #
n
rP(nf) = u
I
:
Proof The implication '=st follows easily from (3.4.18) and (3.4.30). Then in view of (3.4.16) we obtain a Conversely, assume that w f 2(Q). subsequence z in w, such that for a certain nonvoid and open Q' c Q and h E f?'(Q'), h # 0, we have
which contradicts the hypotesis. Connected with the neutrix condition (3.4.10) we can now obtain the following result. Corollarv 2 If s
E
iu then, for every Q' c Q nonvoid and open, we have (C~(Q))~
Proof It follows from (3.4.17), (3.4.18) and (3.4.31) We note that in view of (3.4.17), we have the following correspondent of property (3.4.30)
v
s E~)Q('"C(
:
(3.4.33) s
which leads to
E
(cm(n))F
I
V Q' c Q nonvoid open, t subsequence in s: w
E.E. Rosinger
Corollarv 3 If
s
E
( ~ ( n ) ) ' then
V 62' c 62 nonvoid open, t subsequence in s (3.4.34)
s E (cm(n))FH
n ~ ( n . )= a
I
:
Proof The implication
'
I
follows from (3.4.32) and (3.4.33).
Conversely, (3.4.31) implies that ws
E
V )
thus (3.4.17) completes the proof We can summarize the above as follows. P r o ~ o s iion t 4 If
T(D)
i s nontrivial on 62, then
(cm(n)); # ( and
Proof
It follows from Proposition 2 and Corollary 2
o
A t t h i s point we can turn t o the conditions (3.3.3) and (3.4.1). Obviously, it i s easy t o find satisfied. Therefore, (3.2.13).
we concentrate on
E = S/V E ~ ~ ~ ( 6 such 2 ) that (3.4.1)
is
(3.3.3) , that
is
satisfying
condition
For t h a t , it is useful t o note that the existence of a vector space E in 3.2.2) and of an algebra A in (3.2.3) satisfying the conditions 3.2.13), (3.4.1) and (3.4.2) i s equivalent t o the existence of an algebra A in (3.2.3), satisfying the following two conditions
Algebraic characterization
In this case we can define E = S/V by
and
Indeed, it is easy to see that
Therefore, we prove now that the neutrix condition in the right hand term of (3.4.40) does indeed hold. For that, assume the relation v = u($) , for a certain v E Y and $ E Q). Then, for p = (0,.. . ,0) E Bin, the relation (3.4.39) yields v E 1. &us
since $ E cm(Q) c Co (Q) . Now, in view of (3.4.36) and hence (3.2.lo), we obtain v E 0. Finally, we are near to the solution of the roblem (3.3.2)- (3.3.5). Indeed, in view of Proposition 1, the conditions 6.4.36) and (3.4.37) are equivalent to the existence of the inclusion diagram
b satisfying the additional two conditions (3.4.42) and
{D'S~P
E ' ",
Ipl
< m)
c A
E .E. Ros i n g e r
We show now t h a t i n c a s e (3.4.41)- (3.4.43) h o l d , one can f i n d t h e s m a l l e s t s u b a l e b r a A and i d e a l Z i n A which s a t i s f y t h e s e c o n d i t i o n s . ~ n d e e % ,l e t u s denote by
(P (Q))N g e n e r a t e d by
t h e subalgebra i n (3.4.45)
{ ~ ~ s El N", p
Ipl
5 m)
U
P(Q)
Then (3.4.42) and ( 3 . 2 . 7 ) imply t h a t (3.4.46)
As c A
Let u s now denote by (3.4.47)
Is
the ideal in
As
g e n e r a t e d by
Zw
.
Then o b v i s o u s l y
Is i s t h e v e c t o r
S
subspace g e n e r a t e d by .As
(3.4.48) S
t h e r e f o r e (3.4.41) r e s u l t s i n t h e i n c l u s i o n (3.4.49)
Zs
C
Z
I n t h i s way, whenever (3.4.41) h o l d s , it can always be augmented t o become t h e f o l l o w i n g i n c l u s i o n diagram
Algebraic characterization
which will automatically satisfy (3.4.42), as a consequence of (3.4.45) . In conclusion, the existence of inclusion diagrams (3.4.41) which satisfy the conditions (3.4.42) and (3.4.43) is equivalent to the neutriz condition
zs n uO (n) = o
(3.4.51)
Now we can summarize the results in this section and obtain the following neutrix characterization for the existence of generalized solutions. Theorem 1 If T(D)
is nontrivial on 0 , then
Given any sequence of functions
there exist E
=
S/V E VS~(Q) and
A
= A/Z E
and (3.4.56)
T(D)U = f (see (3.2.18))
AL(Q)
such that
150
E.E. Rosinger
if and only if the neutrix condition is satisfied (see (3.4.47))
Remark 2 Let us recapitulate the main steps leadin to the above algebraic characterization in (3.4.57) for the existence o generalized solutions.
I
Given on fi a nontrivial polynomial nonlinear PDE & of order m with continuous coefficients
and
and a sequence of functions
one constructs As
in (3.4.44) and Is in (3.4.47).
condition
is necessary and sufficient for having (3.4.61)
As
=
As/IsE AL(fi)
in which case one can take (3.4.62)
A
=
As
Further, one can take (3.4.63)
E
=
Es
where (3.4.64) with
E,
=
sS/vsE vsm(n)
Then, the neutrix
Algebraic characterization
In this way one obtains
and for the generalized function
the equation E namely (3.4.69)
T(D)U = f
is satisfied. The above construction, which summarizes the proof of Theorem 1 ives in and fact a particular pair of s aces of generalized functions E I A = A . The existence o f other pairs which satisfy the conditions (3.4.54)- (3.4.56) can be obtained from a detailed study of the stability, general it y and exactness of generalized solutions, see Rosinger [2, . 13-16, 163-1721, Rosinger [3, pp. 224-2291, and also Oberguggenberger [6Pp
!/v
Remark 3 Concerning the problems of uniqueness, regularity - or more generally, coherence, see Colombeau [1,2], Rosinger [2,3], Oberguggenberger [6] and Chapter 4 next - of the generalized solutions whose existence is characterized in Theorem 1, the situation at present is rou hly as follows. Yithin the more particular framework of Colombeau [I ,2f, rather strong uniqueness, and coherence results have been obtained for large classes of linear an nonlinear PDEs. On the other hand, in Oberguggen, see the next Chapter, the framework used is still more general bergerin t is Chapter, in the sense that the mappings (3.2.18) are replaced than by mappings
[61
regularitdr
where Ez = S/V E VSO(fl) quotient algebra.
is a quotient vector space which need not be a
Within this more general setting (3.4.70), and for rather general semilinear hyperbolic systems in two independent variables, existence and uniqueness is proved for the Cauchy problem with rough initial data. Rather surprisingly, precisely because of the more general nature of the framework in (3.4.70), particularly strong coherence results are proved for the unique generalized solutions of the mentioned type of Cauchy problems, results which are shown to be impossible within the framework of Colombeau
E . E . Rosinger
152
[I,2] , this being one of the outstanding features of Oberguggenberger [6] as seen next in Chapter 4. However, the choice of the setting in (3.2.18), (3.4.70) or that in Colombeau [1,2] may be influenced by considerations other than the stron est possible coherence of generalized solutions. Indeed, for the globa version of the Cauchy-Kovalevskaia theorem in Chapter 2, the setting which proves to be useful is the following particular form of (3.2.18)
f
where both A l
and Az
are quotient algebras of generalized functions.
Remark 4 The results in Chapter 2 which, within the setting of (3.4.71), prove the global existence of generalized solutions on the whole of the domain of analyticity of arbitrary analytic nonlinear partial differential equations, can be seen as a particular case of the result in Theorem 1 above. Indeed, the essence of the proofs in Chapter 2 comes down to the fact that one can construct sequences of functions s in (3.3.2)- (3.3.5) such that
where we recall that ind(fl) is the set of all the sequences w E (~'(11))~ which satisfy the condtion, see (2.2.4)
(3.4.73)
3 I' c fl closed, nowhere dense : v x E n\r : 3 p E IN, V c fl\r neighbourhood of x V v ~ M , v > p , y ~ V : w,(Y) = 0
:
The important fact is that lnd(fl) is an ideal in (~'(fl)', called the nourhere dense ideal, which satisfies the neutrix conditon (3.2.10) , see . This makes it possible to easily secure the neutrix condition in Theorem 1, and then set up the framework in (3.4.71).
$5. THE NEIITBIX CONDITION AS A DENSELY VANISHING CONDITION ON IDEALS
The neutrix conditon (3.2.10) first comes into the picture as the rather trivial necessary and sufficient condition for the algebra embedding in (3.2.12), see also (1.6.14). Then, in (3.4.57 , it nevertheless proves to give the characterization for the existence o generalized solutions of a large class of nonlinear partial differential equations.
1
Algebraic characterization
153
This power of the neutrix conditon (3.2.10) when applied to ideals Z in subal$ebras A of (P(Q)' should not come as a surprise. Indeed, as seen in Rosinger [2,pp. 75-88] and Rosinger [3, pp. 306-3151, as well as in Chapter 6 In this volume, the neutrix condition happens also to characterize the existence of a very lar e class of chains of differential algebras of generalized functions, c ains which contain the L. Schwartz distributions, and incorporate as particular cases various distribution multiplications encountered in the literature, such as for instance that in Colombeau [1,2], for details on such multiplications see the references in Oberguggenberger [I] .
f
The form of the neutrix conditon (3.2.10) has the significant advantage of being a particularly simple algebraic condition on the ideal 2, with the clear geometric meaning that Z is off diagonal in (co(R))'. However, except for that, the neutrix conditon (3.2.10 does not give an explicit insight into the structure of the respective i eals Z. It is therefore of special interest to obtain alternative characterizations for the ideals Z which satisfy the neutrix condition, characterizations which can give deeper insight into their structure. Indeed, as seen in (3.4.2) and (3.4.57), the structure of the ideals Z which appear in quotient algebras A = ,411 E AL R) can give a direct and explicit understanding of the conditions o solvability of nonlinear partial differential equations.
d
I
In this Section, such an alternative characterization is presented, according to which, in a certain sense specified later, the sequences of functions w E Z have to vanish asymptotically on dense subsets of R. Let us now turn to the details. inclusion diagram
Our problem is the following. Given an
where A is a subalgebra and Z neutrix condition
is an ideal in A which satisfies the
we want to find equivalent conditions on Z, which give a better explicit understanding of the structure of Z. It should be noted from the beginnin that this problem of structural characterizations of Z seems to be ighly nontrivial. Indeed, it can often happen that when constructing generalized solutions for linear or nonlinear partial differential equations, we shall encounter in (3.5.1) the situation where
f
E.E . Rosinger
and (3.5.4)
I is not an ideal in (~(fl))"
Since obviously
the situation in (3.5.31, (3.5.4) means that we are dealing with the structure of ideals I in strict subalgebras A of a ring of continuous functions, namely C(N x Q) , a problem well known for its difficulty , see Gillrnan E Jerison, Walker. Finally, we should note that from the point of view of the structure of 2, the problem in (3.5.1), (3.5.2) has the following equivalent and simpler formulation, in which the subalgebra A does no longer appear : find the structure of subalgebras (3.5.6)
I c
(fl (a))'
which satisfy the conditions
and
Obviously, the neutrix condition (3.5.8) will imply that Z is a strict subalgebra in (C(Q) 1'. And now, we can turn to the 'dense vanishing' characterization of subalgebras 2 in (3.5.6)- (3.5.8) . The conjecture about it has emerged earlier, followin the study of the essence of various srf icient conditions for obtaining b.5.6)- (3.5.8), see for details Rosinger 1, pp. 39-59, 132-1381, Rosinger 12, pp. 173-1981. Indeed, for w E (~(n))', let us define its vanishing set by
f
(3.5.9)
" " = {x E
inf Iw,(x)I v€!N
=
0)
Then, the following sufficient condition for (3.5.8) results easily.
Algebraic characterization
P r o ~ o s iion t 5 Given any subset I c (P (n))', (3.5.10)
v
"
if
wEZ: i s dense in Q
then
Proof Assume that f o r certain w
E
w = ~($1
then
$(XI = WJX), hence (3.5.9) implies that (3.5.12)
Z and $ E b ( Q ) we have
v
V E
N,
X E
n
I$(x)I = inf Iw,(x)I = 0, V x E flu UEN
Now (3.5.12) and (3.5.10) as well as the continuity of yield $(x) = 0 ,
V X
E
$
on
il
will
n
It i s easy t o see that Ind(R) defined in (3.4.73), does s a t i s f y the 'densely vanishing' conditon (3.5.10). As the main result of t h i s Section, in Theorem 2 it w i l l be shown that the 'dense vanishing' condition (3.5.10) i s equivalent to the nestrix condition, within a rather large and natural class of inclusion diagrams (3.5.1). The basic property used f o r that purpose i s presented now.
If
w e ( b (Q))'
3
and Qw
a' c
i s not dense in 0 ,
Q nonvoid, open,
V VEN,
X E n ' :
then
c > 0 :
E.E. Rosinger
"
Proof As
is not dense in R, it follows that
Define now 6
:
(3.5.15)
51
by
+ [O,m)
6(~) = inf
UEDI
)Y(,~I
1, v Y
E 8
Obviously 6 is upper semicontinuous. Therefore, the set D c R of discontinuities of 6 is of first Baire category in R. In this way it follows that
thus we can find E B(x,c) such that 6 is continuous at y. But then, in view of (3.5.14y, we have S(Y) > 0 hence (3.5.13) follows by taking (3.5.15) into account. At this stage, we shall restrict the class of inclusion diagrams (3.5.1), (3.5.2) by noting that many of the spaces of generalized functions have a sheaf structure, see Rosinger [3, pp. 131-1331. For instance, in the case of the L. Schwartz distributions, we have a natural mapping (3.5.16)
R
Rt open
3
which turns pt(R) Seebach et. al.
Hp l ( Q J )
into a sheaf of sections over 52, see Appendix 2 and
In view of the above, a subalgebra 2 in (~(fl))', called l o c a l , if and only if
V R' c R nonvoid, open
see (3.5.6), will be
:
(3.5.17)
Here we recall that we have
where, for w
=
(w
0'
. . . ,w,,. . .) E (P(0))DI ,
we denoted
Algebraic characterization
with
p(n) 3 h - h being the usual restriction of It follows easily that Znd(9) is local. Finally, a subalgebra A in (~(9))'
V (3.5.18)
9' c 9 nonvoid, open, t E A : 3 c>o: 1 V , x : ] dVcAl9/
ltv(x) Obviously
A
(~(n))'
=
the full algebra
is called full, if and only if
IL
c
is full. Therefore lnd(Q)
is a local ideal in
(C"(9)'.
We also note that conditions (3.5.13 and (3.5.18), obviously recall (3.4.12), (3.4.16) and the definition o a nontrivial T(D) in Section 4, in all of which an arbitrary nonvoid, open 9' c Q is present. And now, the main result in this Section. Theorem 2 Given
A a j v l l subalgebra in
(~(9))'
and 1 a loeol ideal in A.
Then, the neut rix condition
and the 'densely vanishing' conditon
v
W E z :
(3.5.20) Ow is dense in 9 are equivalent. Proof The implication (3.5.20)
4
(3.5.19) follows from Proposition 5.
E .E . Rosinger
158
Conversely, assume that Qw i s not dense in Q, have (3.5.13). B u t obviously
f o r some w
E
I. Then we
therefore (3.5.18) implies that
I t follows that
is an ideal in A But (3.5.21) contradicts (3.5.17).
As mentioned in Appendix 4 t o Chapter 1, the 'neutrix calculus', in particular, the n e ~rti x conditon (3.2. l o ) , has f i r s t been introduced in Van der Corput, in connection with an abstract model f o r the study of l a r e classes of asymptotic expansions, and the s e t t i n g i s in essence t e following. Given an Abelian group G and an arbitrary i n f i n i t e s e t X , a subgroup
f
i s called a neutrix, i f and only if
in which case the functions f
E
N w i l l be called K- negligible.
However, the power of the neutrix condition (3.2.10) comes into play in a significant manner within the more particular framework of (3.2.7), when it is applied t o ideals 2, see Rosinger [2, pp. 75-88], Rosinger [3, pp. 306-3151, a s well as Chapter 6 below.
Algebraic characterization
$6. DENSE VANISHING IN THE CASE OF SMOOTH IDEALS In a rather sur rising manner, it happens that the 'densely vanishin ' property (3.5.201 can be significantly strenghtened in the case of su%algebras i n (L"(Q))'. Indeed, as in Theorem 2, suppose given a f u l l subalgebra A i n ( ~ ( n ) ) ' and a local ideal 1 in A. Let us define
which in view of the Leibnitz rule of product derivative, w i l l be an ideal in
It follows easily t h a t , as an example, we have
i
As seen i n Rosinger 1-31, as well as i n Chapters 2 and 7 i n t h i s volume, the ideal Znd(Q) p ays a crucial role i n the construction of generalized solutions for wide classes of nonlinear partial differential equations. Now, for w E Z00 l e t us denote (3.6.3)
Q(W)= {X E
n
V pEln: inf I ~ P w ~ (( x =) 0 ~€1
)= n p ~ d ' DPW Q
Theorem 3 The ideal i? (3.6.4)
s a t i s f i e s the following 'densely vanishing' condition
v
W E P :
Q(w) i s dense in Q
Proof Assume (3.6.4) i s f a l s e and take w E p , (3.6.5)
B(x,E) n Q(w)=
4
x E Q and
E
> 0 such that
E.E. Rosinger
160
For p E INn we define 6 P
6,(~)
=
fl
:
+
[O,m) by
inf ID~W~(Y) I, VEIN
v
Y E fl
Then $ is upper semicontinuous. Therefore the set Dp c fl discontinuities of 6 is of first Baire category in a. In this way P D = U D
of
is of first Baire category in Q
It follows that we can take Y
E
B(x,c)\D
in which case (3.6.5) yields
3 P E P : Y
~
Q
DPw
and in view of (3.6.6), we obtain dp(y) > 0 thus (3.5.13) will hold for t But
w E '?i
= D~W.
and (3.6.1) imply that ~=DPWEZCA
and we can apply (3.5.18), obtaining the relation 1
Aln/
Then, as in (3.5.21), it follows that u(1)
E
1
and (3.5.17) is contradicted. The 'densely vanishing' conditon (3.6.4) can further be strengthened. A subalgebra 1 in (C' (Q))' is called circled, if and only if
Algebraic characterization
Obviously Znd(Q)
is circled.
Given a vector subspace W c 200 let us denote
Theorem 4 Under the conditions in Theorem 3 , suppose that 1 is circled. Then the ideal ?l
satisfies the following 'densely vanishing' condition
V W c 200 countably infinite dimensional vector subspace
:
(3.6.9) Q(W)
is dense in Q
Proof Assume that (3.6.9) is false for a certain countably infinite dimensional vector subspace W c 200 generated by a Hamel basis w0 ,...,W m ,... E N
(3.6.10)
Let us take x E St and (3.6.11) For m
E
B(X,C) n H and p E INn
urn,p=
(3.6.12) and 6,
,P
:
Q
+
In this way S "',P
E
> 0 such that
a(#)
=
4
we dedfine
1
JD~WO
+. . .+JD~W"'J E 1
[O ,m) by
are upper semicontinuous. Therefore, denoting by
the set of discontinuities of 6,
,P'
it follows that
E .E . Rosinger
Dm,~
is of f i r s t Baire category in Q
In t h i s way
D = U U D is of f i r s t Baire category in Q meN m3P and we can take (3.6.14)
Y E B(x,f)\D
Then (3.6.11) implies that
B u t according t o (3.6.10), we obtain
f o r suitable A ...,Am 0' Lemma 2 below, we have
E
R.
Therefore, in view of (3.6.12), (3.6.16) and
hence (3.6.15) implies that
and then (3.6.13) will give
I t follows that (3.5.13) holds for t = w m 9 p . t = wm3P
E
1c A
Thus in view of (3.5.18) we obtain
But (3.6.12) implies that
Algebraic characterization
and similar to (3.5.21), the relation results
which contradicts (3.5.17) Lemma 2 ~f w E
(e(n)f" R
(3.6.17)
then IwI
=
Qw
More generally, if uO,.. . ,wm E
and A , .. . A
(C"(Q))'
E R then
Proof In view of (3.5.9), the relation (3.6.17) is obvious. Take now v
E
N and x E R , then lJo(~O)u(x)
r
(3.6.19)
I A ~ I . I ( ~ ~ ) ~ +...+ (x)I I
5 IJl~(I(~O),(x)I for every A
E
R,
Jm(wm)"(x)l
+..at
5
~ ~ I (J'"'$,x)I -I 5
l(."),(x)I)
such that
m={lAol,...,lAml} 5 IJI But ( 3 . 5 . 9 ) , (3.6.17) and (3.6.19) obviously imply (3.6.18)
E. E . Ros inger
164
57.
THE CASE OF NORMAL IDEALS
The increasingly stronger 'densely vanishing' conditions (3.5.20) , (3.6.4) and (3.6.9) seem t o point t o a deeper property involved, whose f u l l explicitation i s s t i l l an qpen problem. This i s i l l u s t r a t e d f o r instance by the f a c t that the above densely vanishing' conditions (3.5.20), (3.6.4) and (3.6.9) can be obtained under the following alternative assumptions, when I i s a subalgebra in ( ~ ( n ) ) ' which s a t i s f i e s the neutrix condition '
and i n addition, it is also normal, Kothe, that i s , it has the property
a
Indeed, the proofs of Theorems 2-4 w i l l o through with the following modification. When obtaining (3.5.13) i n t e respective proofs, we no longer use (3.5.18). Instead we note that we can use the property
which in view of (3.7.2) w i l l imply
Then owing t o dicted.
*) in (3.7.3), the neutrix condition (3.7.1) i s contra-
We note as an example that
Ind(Q) i s obviously normal.
Algebraic characterization
$8. CONCLUSIONS The results on 'dense vanishingy obtained in Sections 5 and 7 can further be strengthened and systematized in the following way, indicated by M. Oberguggenberger in a private communication. Given a subalgebra
let us consider the following three properties encountered in Sections 5 and 7: (DS) V w
I : nw dense in
E
fl
(LC) I local (see (3.5.7))
Then the following implications hold. Theorem 5 We always have (3.8.2)
(DS)
+
(LC)
3
(NX)
If I is normal, see (3.7.2), then (3.8.3)
(DS)
Further, if 1 (3.5.18), then (3.8.4)
#
(LC)
#
(NX)
is an ideal in a fvll subalgebra
(DS)
#
(LC)
=i
(DS)
=i(LC) #
see
(NX)
Finally, if I is an ideal in a subalgebra A c (CO(n)', then (3.8.5)
A c (CO,')n(
and A I 11°
(n) ,
(NX)
Proof Let us prove (3.8.2). This implication (LC) =, (NX) follows from the definition (3.5.17). For the implication (DS) =i (LC), let us take 0. c fl nonvoid, open. Given w s (~'(il)', it is obvious from (3.5.9) that
E. E. Rosinger
166
51;
(3.8.6)
=
nw n Q'
1 Q' Now, if
(DS)
holds f o r 2, V w
E
than (3.8.6) yields
Z : 52;
dense in Q'
1 Q' thus Propisition 5 implies that
therefore Z i s indeed local. We prove now (3.8.3). We recall (3.8.2) and i n addition, we prove the implication (NX) + (DS).
(DS). Then i t follows that For that, l e t us assume the failure of (3.5.13) holds. But i n view of (3.7.3) and (3.7.2), we obtain u(a) E Z n @ ( n ) . And then *) i n (3.7.3) will contradict (NX). For the proof of (3.8.4) it suffices t o show the implication (LC) =+ (DS), which follows obviously from Theorem 2 . Finally, in order t o prove (3.8.5), we only have t o show the implication (NX) ? (LC). For t h a t , l e t us assume the failure of (LC). Then we can take a' c 52 nonvoid, open, w E 1 and 10 E P ( Q ' ) , such that
and
Let us take we have
a
I t follows that
while also
E
P(Q)
such that
supp a
c
Q'
and for certain
x c Q, 0
Algebraic characterization
hence U(B).W
E
z n P(n)
thus in view of ( 3 . 8 . 7 ) , the assumption (NX)
is contradicted.
A convenient way to suntmarize the results in Theorem 5 above is as follows. Corollarv 4 If one of the following two conditions holds (3.8.8)
Z is a normal subalgebra in ( c o ( R ) ) ~
or (3.8.9)
Z is an ideal in a full subalgebra A c (co(Q))', with A c
UO (R) ,
then for 1 we have the equivalences (3.8.10)
(DS) w (LC) B (NS)
E .E . Rosinger
APPENDIX 1 ON THE SHARPNESS OF LEMU 1 IN SECTION 4 In view of the fact that, on the one hand, Lemma 1 in Section 4 plays a fundamental role in the algebraic characterization of the solvability of nonlinear partial differential equations given in Theorem 1 in Section 4, while on the other hand, it appears to be unknown in the earlier literature, it is useful to try to analyze its sharpness. For that purpose, we shall present two examples, with Q open subsets in W.
There exist sequences w = (w ,. . . w ,. . . E ( 0 ( ) of continuous functions on O, such that the following three conditions are satisfied:
while for a certain v E 0, we have V
V c O, V neighbourhood of v
(3.A1.2)
1
I
:
vE#}=a,
as well as
v (3.A1.3)
x ~ n , ~ # v : 3 V c fly V neighbourhood of x
Indeed, take Q = W , Define then w E (~o(S2))'
w,(x)
=
v
= 0 E
S2
and
:
cu E CO(W),
with
supp cu c [0,1].
by a((u + I)((u
+ 2)x - I)),
v E I , x E fl
It follows easily that (3.Al.l), (3.A1.2) and (3.A1.3) are satisfied. The above example shows the sharpness of Lemma 1, in the case of Q connected. Indeed, in view of (3.A1.2), the relation (3.A1.3) only holds for x # Y = 0 E Q . In other words, for w in Example 1, we must have (3.A1.4)
rf d
Algebraic characterization
169
w i t h the notation in Lemma 1, since the relation (3.A1.2) obviously implies V = O E ~ .
On the other hand, relation (3.Al.l) shows that w i n Example 1 s a t i s f i e s the assumption of Lemma 1 i n a way which i s most inconvenient for (3.A1.2) and (3.A1.4) to happen, yet these two l a t t e r relations s t i l l hold. For D not connected, we have:
There exist sequences w = (wo,. . . .wv,.. .) E (co(D))' of continuous functions on D such that the following three conditions are satisfied:
car{wv(x)l v
E
IN}
<2
and w i t h the notation i n Lemma 1
while in the same time
Indeed, l e t us take
and define w
for v
E
IN.
E
(co(Q))' by
Then (3.A1.5)-(3.A1.7) follow easily.
The interest i n Example 2 comes from the fact that, although (3.Al. 5) and 3.A1.6) are most inconvenient for ( 3 . At. 7) t o happen, that l a t t e r relation oes nevertheless hold.
d
E .E . Rosinger
APPENDIX 2
SHEAVES OF SECTIONS The localization property of the Schwartz distributions 3' (Rn) , see Appendix 1 t o Chapter 5 , gives them a structure of sheaf of sections over lRn, as follows from the definition below. It i s easy t o see that various classical spaces of functions, such as cP, with p E OJ, as well as the analytic functions have a similar structure. I t should be recalled t h a t , as mentioned i n Rosinger [3 , one encounters a localization principle on the very level of the usua reduction of the integro-differential balance equations of physics t o the corresponding partial differential equations. And the use of such a localization principle seems t o be unavoidable if the continuous formulation of physical laws i s used, see Abbott for the history of discrete and continuous formulations of Newtonian laws.
I
To the extent that local and global phenomena are interrelated in continuously formulated physical laws, the presence of a sheaf structure on various spaces of functions and generalized functions can be particularly useful. Indeed, as pointed out for instance i n Seebach e t . a l , sheaf theory i s an effective tool i n areas where problems have t o be approached based on local structure and informat ion. For convenience, here we recall the definition of a sheaf of sect ions. For details, as well as for the definition of the associated notion of sheaf of germs, one can consult Seebach e t . a l . , which presents a convenient introduct ion aimed a t a larger, mathematically trained readership. Suppose give a topological space given a mapping (3.A2.1)
X 3 U open
X
and a set
6u(U)
S of spaces S.
Suppose
= S E S
W e c a l l S = u(U) a section over U. Finally, suppose that f o r each pair of open sets we have rest rict ion mappings (3.A2.2)
P u , ~: 0 )
U,V c X,
For every open U c X we have P
U c V,
0 )
Then ( r , ~ ~ ,i s~ called ) a sheaf of sections over X , following four conditions are satisfied:
(3.A2.3)
with
~ = i, d u~( ~ ): u(U)
-+
u(U)
if and only if the
Algebraic c h a r a c t e r i z a t i o n
For every open U,V,W c X such t h a t (3.A2.4)
P
~ O ,P ~~ = ,P ~~
For every f a m i l y of open Ui E X,
U c V c W,
,
we have
~
with
i E I,
s,t
and
E
u ( U Ui), id
we
have
where U = U Ui. id And f i n a l l y , f o r every family of open i E I , we have t h e p r o p e r t y : i f
Ui c X ,
and si E u(Ui),
with
then
Now, i n o r d e r t o show t h a t t h e Schwartz d i s t r i b u t i o n s have a n a t u r a l sheaf of s e c t i o n s s t r u c t u r e , we s h a l l t a k e with t h e above n o t a t i o n s
(3.A2.9)
u(Q) = 3' (Q) , f o r open fi c R"
F i n a l l y , f o r open Q c A c iRn, (3.A2.10)
P Q , ~: V/ (A)
we d e f i n e
P/(a)
is t h e r e s t r i c t i o n of t h e d i s t r i b u t i o n where FI, s u b s e t Q c A.
F
E
P'(A)
t o t h e open
It is easy t o check t h a t (3.A2.8)- (3.A2.11) s a t i s f y (3.A2.3)- (3.A2.7).
This Page Intentionally Left Blank
CHAPTER 4 GENERALIZED SOLUI'IONS OF SEYILINEAR WAVE EOUATIONS WITH ROUGE INITIAL VALUES $1.
INTRODUCTION
As mentioned in Remark 3, Section 4, Chapter 3 - see 3.4.70) - it can be useful t o further extend the concept of generalized so ution introduced in Section 10, Chapter 1, concept which proved t o be so effective in the results presented in Chapters 2 and 3.
\
The aim of t h i s Chapter i s t o indicate one possible such extension, introduced recently in Oberguggenberger [ 6 ] . This extension, made i n the s p i r i t of (3.4.70), turns out t o be particularly effective in solving f o r rough i n i t i a l va ues semilinear hyperbolic systems of the form
Here
i s the unknown function, while the given a r e the diagonal, of functions
n
x
n
matrix
and the right hand term
with both A and F being C'-smooth. The semilinear hyperbolic system (4.1.1) i s supposed t o be solved with the i n i t i a l value problem (4.1.2)
U = u at
t = O
The interest in the problem (4.1. I ) , (4.1.2) comes from the f a c t t h a t the i n i t i a l value
in (4.1.2) can be chosen in a quite rough manner, namely u l , . . . ,un can be rather arbitrary generalized functions on the domain R of the space variable x. Then, owing t o the nonlinearity of the system (4.1.1), one i s
E.E. Rosinger
174
faced with the hi hly nontrivial problem of establishin the precise way of singalorities in the solution ~(t,xf, with t 2 0 and the propagation x E R, singularities caused at t = 0 by the rough initial value u.
07
Recently, a particular case of rough initial values leading to the so called delta uraves has been studied in Oberguggenberger [6] and Rauch k Reed. The extension of the concept of generalized solution introduced in Obergueenberger 161 and presented in this Chapter proves to have two rather striking qualities. First, it offers existence, uniqueness and regularity or coherence results which contain, and in fact o much beyond the similar earlier results. Secondly, the method of proo is unusually simple and transparent, giving thus a particularly clear understanding of the basic mathematical phenomena involved, which - as previously in Chapters 2 and 3 - prove to be of an algebraic nature, related to properties of rings of sequences of continuous or smooth functions on Euclidean spaces.
I
2 . THE GENERAL EXISTENCE AND UNIQUENESS RESULT First we start with the customary type of conditions on the semilinear hyperbolic system (4.1.1). Concerning the n (4.2.1)
x
n diagonal matrix of functions A, we assume that
A or DxA is bounded on R2
This condition is sufficient for the existence of the characteristic curves for all time t E R. The nonlinear term F is assumed to satisfy the bounded gradient condition V
K c R2 compact
:
3 C>O: (4.2.2)
V
(t,x) E K, u = (u1 ,...,un) IDU F(t,x,u) I < C i
E
R", 1 5 i
:
which guarantees that (4.1 .I), (4.1.2) has a unique global solution U E (C"(R~))~, for every intial value u E (COO@))". Now we can turn to the construction of the suitable spaces of generalized functions, and to the appropriate concept of generalized solution. For a convenient formulation of (4.1.1), let us define on following nonlinear partial differential operator
R2
the
Rough semi1inear waves
Then obviously
where we assume t h a t OJ + 1 = OJ. In order t o deal with t h e i n i t i a l value problem (4.1.2), l e t us define t h e following linear operator (4.2.5)
BU(t,x) = U(O,x),
(t,x)
E R2
I t follows that
Now (4.1.1), (4.1.2) can be written in the equivalent form
Concerning (4.2.7), we could t r y t o follow the method f o r systems of nonlinear p a r t i a l d i f f e r e n t i a l equations presented in Sect ion 14, Chapter 1, noting that with the notation there, we would have
Furthermore, a s shown in Oberguggenberger [6], it will be convenient t o take
However, a s it stands, the method in Section 14, Chapter 1 would need t o assume that F in (4.1.1) or (4.2.3) i s polynomial. Fortunately, t h i s and several other assumptions made i n Section 14, Chapter 1 can be done away with, owing t o the f a c t that the concept of generalized solution used in Oberguggenberger [6] i s more general than t h a t in Section 10, Chapter 1. In t h i s respect, instead of extensions of the type (1.14.14), we shall construct f o r T(D) in (4.2.3) above extensions, see (3.4.70)
where, see (1.6.7)
are suitable quotient vector spaces, while
E .E . Ros inger
that i s , 7 i s the vector space of a l l P-smooth functions on with values in Rn.
IR2
and
>
I t should be noted that for n 2, that i s , for nontrivial systems, the usual, point wise operations on functions f : IR2 --+ IRn w i l l only yield a vector space structure on 3 , and not one of algebra. Therefore, we could not use 7 , i n case one of the two spaces of generalized functions El or E2 i n (4.2.11) would have t o be an algebra. In t h i s respect, one of the advanta es of the extension (4.2.10) used in Oberguggenberger [6] i s precisely in t e fact that none of the two spaces El or E2 need t o be an algebra.
B
Now, in view of (1.6.5), (1.6.6), it follows that
w i t h the neutrix property
for
i~{1,2)
Turnin t o the i n i t i a l value problem (4.2.8), we shall construct for B (4.2.57 extensions of the type
in
where (4.2.16)
Eo = So/Vo
E
VS
a, (091)
i s a suitable quotient vector space, while (4.2.17)
a = P(IR,IR~)
in other words, X i s the vector space of a l l p-smooth functions on R I and with values i n IRn. Again, therefore, if n 2 2, then X: i s not an algebra with the usual pointwise operations on functions. Similar t o (4.2.13) and (4.2.14), we shall have
Rough semilinear waves
a s well a s t h e neutrix property (4.2.19)
"0
%,(o,I) '
We can proceed now with t h e d e t a i l s of t h e construction of extensions (4.2.10) and (4.2.15). For t h a t purpose, it is convenient t o s p l i t t h e nonlinear operator i n (4.2.3) i n t o its l i n e a r p a r t
T(D)
and i t s remaining nonlinear p a r t , which f o r simplicity w i l l again be denoted by F, t h a t i s (4.2.21)
FU(t ,x) = F ( t ,x,U(t ,x)) , ( t ,x)
E
R2
Then, similar t o (4.2.4) we obtain
(4.2.23)
F ( c ~ ( R ~c) )( c~ ( ( R ~ ) ) ~ ,e E R
In p a r t i c u l a r , i n view of (4.2.12) (4.2.24)
L(D)3 c 3, F3
On the other hand, (4.2.6), (4.2.25)
, we
have
c 3
(4.2.12) and (4.2.17) yield
B3 c X
Now, based on (4.2.24) and (4.2.25), we simply obtain t h e extensions (4.2.26)
L(D) : 3(Oj1)
+
(4.2.27)
B : 3(0,1)
K(o,l)
+
3(091),
F
:
3(O>l) + 3(Oy1)
by defining termwise the respective mappings, t h a t i s , given
E.E. Rosinger
we have
and
Finally, we can come to the choice of the vector spaces of generalized functions E l ,Ez and Eo in (4.2.10) and (4.2.15). For that, first, we shall take the vector subspaces
so that the following four conditions are satisfied
Si c Sz c Sz
L(D)Sl
FSl c 51 BS1
So
C
Next, we shall take the vector subspaces v1 C
S1, vz c Sz and Vo c So
in a way which satisfies the four conditions
(4.2.29)
L(D)Y1
c vz
F(s+v)
-
F(s)
E
V1, for s
E
51, v
E
VI
In the examples in Section 3 next, which include the known results in literature obtained until recently, it will be shown how the above conditions (4.2.28) and (4.2.29) can be satisfied. The point in these eight conditions 4.2.28) and (4.2.29), considered for the first time in Oberguggenberger \6], is that we can now define the
Rough semilinear waves
179
following four mappings between the respective spaces of generalized functions. First, the two linear mapping
and
then the nonlinear mapping
and finally, the linear mapping
Consequently, the mappings ( 4 . 2 . 3 0 ) , (4.2.31) and (4.2.32) allow us the definition of the nonlinear mapping
In this way, the problem of constructing the extensions in (4.2.10) and (4.2.15) got solved by (4.2.34) and (4.2.33) respectively. We note that the condition (4.2.36)
s1 n v2 = vl
is necessary and sufficient for the canonical embedding i in (4.2.30) to be injective, in which case, we shall consider that the inclusion holds (4.2.37)
El c
E2
We can now define the concept of generalized solution introduced in Oberguggenberger [6] , once the framework (4 -2.11)- (4.2.35) is given. Namely, a generalized function
E. E. Ros inger
180
is c a l l e d an (El -+ E2, Eo)-sequential s o l u t i o n of t h e s e m i l i n e a r hyperb o l i c system
with t h e a s s o c i a t e d i n i t i a l value problem
i f and only i f t h e mappings (4.2.34) and (4.2.33) s a t i s f y t h e conditions (4.2.41)
T(D)U = 0
and (4.2.42)
BU = u
where t h e i n i t i a l value
u
is given such t h a t
Remark 1
It is obvious from t h e above construction i n (4.2.11)- (4.2.35) t h a t t h e concept of (El -r E2, Eo)-sequential s o l u t i o n j u s t defined i s by no means limited t o t h e p a r t i c u l a r p a r t i a l d i f f e r e n t i a l equation i n (4.1.1) o r (4.2.39). Indeed, f o r a given arb i t rary nonlinear p a r t i a l d i f f e r e n t i a l operator T(D), one can e a s i l y a r r i v e a t a corresponding concept of (El + EP, Eo)-sequential s o l u t i o n , simply by adapting accordingly t h e conditions (4.2.28) and (4.2.29). S i m i l a r l y , one can use i n f i n i t e index s e t s A o t h e r than ( 0 , l ) c R, chosen i n (4.2.9). F i n a l l y , one can use vector subspaces 7 and X: o t h e r than t h o s e given i n (4.2.12) and (4.2.17) r e s p e c t i v e l y
.
A t t h i s s t a g e , we can now t u r n t o t h e question of existence and uniqueness of an (El + Ez , Eo)- s e q u e n t i a l s o l u i t i o n f o r our rough i n i t i a l value problem f o r t h e semilinear hyperbolic system (4.1. I ) , (4.1.2).
As is known, under t h e condition t h a t (4.2.44)
A,F,u
a r e P-smooth
t h e problem (4.1. I ) , (4.1.2) has a unique c l a s s i c a l s o l u t i o n
Rough semilinear waves
provided t h e (4.2.1) and (4.2.2) hold. I n o r d e r t o o b t a i n a general existence and uniqueness r e s u l t f o r rosgh initial values such a s i n (4.2.43), t h e followin tuo conditions a r e fundamental. F i r s t we assume t h a t , given any (,yJr E ( 0 , l ) ) E So, i f $€, with E ( 0 1 ) is t h e unique c l a s s i c a l s o l u t i o n of (4.1.1) f o r t h e i n i t i a l value problem $€(O,X) = ~ ~ ( x x) ,E R, then
Secondly, we assume t h a t , given any t ,z and Bt-Bz E V,, then (4.2.47)
t-z E
E
SI, such t h a t T(D) t ,T(D)z
E V2
V1
Under t h e above conditions ( 4 2 . 1 ) , (4.2.2) , (4.2.44) , (4.2.46) and (4.2.47), we o b t a i n t h e following general existence and uniqueness resslt. Theorem 1 Given an a r b i t r a r y i n i t i a l value system (4.2.48)
u E E,.
Then t h e s e m i l i n e a r hyperbolic
T(D)U = 0
with t h e rough i n i t i a l values (4.2.49)
BU = u
has a uniqve (El
+
E 2 , Eo)-sequential s o l u t i o n U
E
El.
Proof Assume t h a t we have t h e r e p r e s e n t a t i o n u = (,y,lc
E
( 0 , l ) ) + Vo E Eo = So/Vo
with (x,l,
E
(091)) E
$0
Then, with t h e r e s p e c t i v e construct ion preceeding (4.2.46), we o b t a i n S =
($,If
E ( 0 , l ) ) E Si
Now, from (4.2.34) and (4.2.33), it follows e a s i l y t h a t
E .E . Rosinger
182
is indeed an (El
E2 , Eo)- sequential solution of (4.2.48) , (4.2.49) .
The uniqueness of U in (4.2.50) follows at once from (4.2.47)
n
Remark 2 In view of the significant generality of the existence and uniqueness result in Theorem 1 above, it is p a r t i c u l a r l y important to establish the coherence p r o p e r t i e s , see Colombeau [1,2], of the unique generalized solutions given by this theorem. In other words, we have to establish the way in which these unique generalized solutions are related to the earlier known classical, distributional and generalized solutions. This coherence property will be illustrated next, in Sections 3 and 4, in the case of the and delta wave solutions. earlier known
eloc
Remark 3 It is important to note the fact that both the insight and the result in Theorem 1, gained by the general construction in this Section, are highly nontrivial. Indeed, on the one hand, they contain and unify in a clear and elegant manner the essential algebraic and analytic aspects of earlier known results. Here, to be more precise, we should mention that, at a closer study, the construction in this Section gives the obvious impression of requiring the minirnumminimorum of the algebraic and analytic conditions for bringing about the existence and uniqueness result in Theorem 1. Let us be more specific, by noting the following. One of the s t r o n g p o i n t s of Theorem 1 is that, as seen later in (4.4.I ) , the vector space EO of the i n i t i a l v a l u e s can be quite l a r g e , for instance, it can contain all the distributions in g' (W) . Now, as is obvious, the essence of the construction in this Section is to choose the s i x v e c t o r s u b s p a c e s SI, S2, So, VI , V2 and Vo in such a way that the conditions (4.2.28) and (4.2.29) are satisfied. However, since Eo = So/Vq, it follows that l a r g e Eo means l a r g e So and s m a l l VO. And condition (4.2.28) does not prevent So from being large. On the other hand, conditon (4.2.29) may easily prevent Vo from being small. Which means that the construction of a large EO = So/Vo is not a t r i v i a l matter. It is precise1 here that the mathematical difficulties involved in securing the resu t in Theorem 1 come to be manifested. And in view of (4.2.28), (4.2.29), these difficulties take the particularly simple, obvious and minimal form of e i h t i n c l u s i o n s involving vector spaces, in two of which linear partial dif erential operators are present.
P
9
In this way, the e n a b l i n g power of the respective framework for solvin systems of nonlinear partial differential equations is larger than that o customary functional analytic approaches which, owing to possible unnecessary topological ideosyncrasies, may require more stringent conditions.
'i
Rough semilinear waves
On the other hand, the mentioned general construction opens the door to a remarkably large variety of spaces of generalized functions in which one can search for the solutions of large classes of systems of nonlinear partial differential equations, see the comment in Remark 1. It should be noted that the use of the various Sobolev spaces had offered during the last decades a most impressive opening in the study of linear, and certain nonlinear partial differential equations. The difficulty however with this functional analytic method is in its near exclusive reliance on the topologies on the respective spaces of generalized functions. Indeed, as is known, Dacorogna, most of the even simplest nonlinear operations are not eont inuous in a large variety of such topologies. Therefore, a functional analytic approach to nonlinear partial differential equations often necessitates stringent part ieelarizations, in order to be able to overcome such difficulties. In more precise, technical terms, the enabling power of the framework in this Section comes from the large variety of the possiblities in the choice of the vector subspaces in (4.2.28) and (4.2.29). Indeed, as seen next in Sections 3 and 4, suitable choices of these vector subspaces make it easy to account for various analytical properties of generalized solutions of nonlinear partial differential equations. At this stage of the ongoing research, with the opening given by the construction in this Section, one can proceed further and elaborate appropriate methods - in the basic algebraic and analytic spirit of this construction - which can be applied to various classes of systems of nonlinear partial differential equations. As in Oberguggenberger (61, this Chapter presents one such method, specifically deviced for semilinear hyperbolic systems, see in particular Section 4 below.
$3.
COHEBENCE VITH Lioc SOLUTIONS
As is well known, for u E .CiOc(lR,F?), the semilinear hyperbolic system (4.1.1), with the initial value problem (4.1.2) has a unique solution u E c(I,L~~~(D,~')) We show now that these solutions are obtained by Theorem 1, Section 2 as well. For that purpose, let
E.E. Rosinger
Further, let us take (4.3.2)
V l c SI c ?(Oyl)
with S1 being the set of all convergent sequences in c ( R , L ~ ~ ~ ( ~ , ~ ~ ) ) , and V I being the subset of those sequences which converge to--;ero. In other words, if for instance v = t E (0,l)) E Vt , then by convergence to zero of the sequence v we mean that dt -+ 0 in c(~,c)~~(wf')) when t -+ 0. Similarly for sequences s E S1. Further, let us take
where S2 is the set of all convergent sequences in 2 (R2,Rn, while V 2 is its subset of sequences convergent to zero. Finally, we ta e
Here So is the set of all convergent sequences in C~~(IR,R~)and Yo is the subset of the sequences convergent to zero. It follows easily that
Now, owing to the bounded gradient condition (4.2.2), we shall have (4.2.28) and (4.2.29) satisfied. Further, as is known, the distributional solutions depend continuously on the initial values condition (4.2.46) follows easily.
u
E
U
E
~(~,~)~~(01,01"))
From this, L~~~(IR,P).
Finally, condition (4.2.47) follows from the fact that the Lioc solutions are unique. In this way, Theorem 1 does indeed contain the unique, Lioc solutions.
Rough semilinear waves
$4. THE DELTA VAVE SPACE We construct spaces of generalized functions E l , Ez, E, lowing properties :
with the fol-
(4.4.1)
El, E2, E, contain the 3'
distributions
(4.4.2)
for every initial value u E E,, ther exists a unique (El -+ E2, E,)- se uential solution U E El for the problem (4.1.1), 94.1.2)
(4.4.3)
Lie,
solutions are (El -+ E2 ,E,)- sequential solutions
certain dela wave solutions are (El -+ E2 ,EO)- sequentia1 solutions From (4.4.1) follows in particular that the initial value problem (4.1.2) for the semilinear hyperbolic system (4.1.1) admits solutions for as rough initial values as can be given by arbitrary distributions. However, as seen immediately, one can in fact use initial values which are quite a bit more rough. In view of the above, the space of generalized functions El Q e l ta wave space, see Oberguggenberger [6].
is called the
Now let us proceed as follows. The space
is equipped with the seminorms
Then, we define the space, see (4.2.20)
and define on this space the Frkchet topology given by the finest locally convex topology which makes continuous the mapping (4.4.8)
I@)
:
c
-
CL(,)
Obviously, this F'rhchet space on
c~(D)
is given by the seminorms
E.E. Rosinger
186
I n order t o follow t h e construction i n Section 2, we take again
while on t h e other hand, t h i s time we take (4.4.11)
sl s2= 7(Oy1), so
.
.
VZ
C $0
x(Osl)
Finally (4.4.12)
v1
C 31,
C s2,
Yo
w i l l be taken a s t h e respective s e t s of sequences which converge t o zero in
C'
c ~ ( ~and ) ljoc(~,O .
I n t h i s way, we obtain t h e vector spaces of generalized functions
Next, we have t o verify t h a t t h e conditions (4.2.28), and (4.2.47) a r e s a t i s f i e d .
(4.2.29),
F i r s t we note t h a t i n view of (4.4.11) and (4.4.12), (4.2.28) and (4.2.46) a r e t r i v i a l l y s a t i s f i e d .
(4.2.46)
t h e conditions
Now we t u r n t o t h e v e r i f i c a t i o n of t h e remaining two conditions (4.2.29) and (4.2.47). For t h a t , f o r
13 i
n,
function on t h e diagonal of l e t us denote by
l e t us denote by
A
in ( 4 . 1 1 )
ai E p(lR2 ,R)
Then, f o r given
the i-t h ( t ,x) E R 2 ,
t h e parametric representation of t h e i - t h c h a r a c t e r i s t i c curve of t h e operator L(D). In other words, q i ( t , x , r ) , with r E R , i s precisely t h a t i n t e g r a l curve of t h e vector f i e l d Dt + ai(t,x)Dx which passes through t h e point x a t t h e time r = t. Further, we note t h a t t h e matrix d i f f e r e n t i a l operator has a r i g h t inverse J given by
L(D)
i n (4.4.8)
Rough semilinear waves
for
16 E C and ( t ,x)
E
IR2
.
Moreover, it is easy t o v e r i f y t h a t
J :C+C
(4.4.16)
is continuous. We a l s o have Lemma 1 J maps C"'(IR2 ,Rn) continuously i n topology induced by C L(D) '
C,
when t h e former space has t h e
Proof Consider t h e mapping ( : C + C which t r a n s l a t e s i n i t i a l values along c h a r a c t e r i s t i c curves, according t o t h e r e l a t i o n
1
with $ E C and ( t , x E R2. Then i n view of (4.4.6), it i s obvious t h a t ( i s continuous. It f o lows t h a t
We a l s o note t h e inclusion
Therefore, given such t h a t
$ E f'(lR2,Rn),
L(D)x = But J
c
> 0 and k
d, P ~ ( x )< qe(d)
is t h e r i g h t inverse of
+
E
I, one can f i n d y,
'
L(D) , thus
L(D)(J$ - ,y) = 0 i n P ' ( R ~ , I R ~ ) This means t h a t
J$
-
x is
constant along t h e c h a r a c t e r i s t i c curves.
E
C,
E.E. Rosinger
188
However, (4.4.15) implies that
Therefore
which means that
Since c > 0 is arbitrary, one obtains
We turn now to the verification of the remainin two conditions (4.2.29) and (4.2.47), which are essential for the app ication of Theorem 1 in Section 2.
f
The inclusion Vl c V2 in (4.2.29) follows from the continuity of J (4.4.16) , the continuity of J : C?'(IR2 ,IRn) + C proved in Lemma 1, and decomposition of the identity mapping on C?'(IR2 ,iRn) into L(D)J. other inclusions in (4.2.29) follow easily from (4.4.11) , (4.4.12) (4.2.2).
in the The and
Finally, the verification of condition (4.2.47) is a bit more involved. Suppose given s,z E Sl = ( P(iR ,iRn2 ) o y l , such that T(D)S,T(D)ZEV~ and also Bs-Bz E Yo. Then, by efinition, there exist v E V , and w E V2, such that
Now, in order to verify condition (4.2.47), it only remains to show that
Assume that s
=
($,If E (0,1)),
z =
(xfIf E
(0,l))
Rough semilinear waves
Let us f i x k E DI and estimate t h e seminorm pk(#, -
x,)
For a s u i t a b l e T > 0, l e t
KT c R2 be t h e domain of determinacy of
L(D)
which contains [- k,k] x [- k,k] c R2 and i s bounded by t h e l i n e s t = + T , a s well a s t h e slowest and f a s t e s t c h a r a c t e r i s t i c s respective1 , passing through t h e endpoints of a s u f f i c i e n t l y l a r g e x- interval [a, c I, a t t = 0. Let [aT,pT] be t h e xinterval obtained by intersecting KT with . . t h e l i n e t = 7 , with -T r T. Then (4.4.17) gives, f o r t E [-T,T] and each coordinate 1 i n, t h e i n e q u a l i t i e s
,poj
< < < <
where C follows from (4.2.2) f o r KT. Now, i n view of Lemma 1, we note t h a t t h e i n t e g r a l involving J i s bounded uniformly i n t E [-T,T] , by ql((o,) .) , f o r a s u i t a b l e ! k. For the -I
other two terms i n the l a s t inequality we note t h a t
E.E. Rosinger
Theref ore, we obtain
Pt
j I ($€I.1 ( t , x )
-
(xc) ( t , x ) 1
I& s
"t
Using Gronwall's inequality, it follows t h a t
f o r a l l t E [-T,T] , provided t h a t C1 > 0 i s suitably chosen. I n t h i s way, we obtain indeed (4.4.18), which completes t h e proof of (4.2.47). We conclude t h a t t h e framework constructed i n (4.4.10)- (4.4.13) does indeed s a t i s f y t h e conditions i n Theorem 1, Section 2. I n p a r t i c u l a r , we have, therefore, obtained t h e r e s u l t claimed i n (4.4.2)
.
We can now proceed with t h e proof of t h e remaining r e s u l t s claimed i n (4.4.1), (4.4.3) and (4.4.4). F i r s t we prove (4.4.1). For t h a t purpose, we take any fixed sequence ( ( o E 1 c E ( 0 , l ) ) E (P(IR,IR)) ( 0 ~ 1 ) which converges i n P1(IR,R) t o t h e Dirac d e l t a d i s t r i b u t i o n 6, when c --+ 0. As i s well known, we could take f o r instance
Rough semilinear waves
where y
E
is given in such a way that
V(R,lR)
Now, we define the mapping (4.4.19)
V' (IR ,lRn) A Eo = So/Vo
with the help of the convolution
*,
as follows
Similarly, we define the mapping
where
Then, in view of 4.4.12), it follows immediately that a and ,8 above are linear embed i n g s . It follows that El and Eo do contain the 9' distributions. The fact that E2 also contains the 9' distributions follows from (4.2.37) and Lemma 1. Therefore (4.4.1) holds indeed.
d
The proof of (4.4.3) follows from
Given u t C)~~(R,R") and U E c(R,c)~~(R,R~)) the unique solution of the semilinear hyperbolic system (4.1.1) with the initial value problem (4.1.2). Then, for the initial value v = a(u) (El 4 E2 ,Eo)- sequential solution V E system (4.1.1). Moreover, we have the coherence property (4.4.23)
v = P(u)
E El
Eo , there corresponds a unique for the semilinear hyperbolic
E.E. Rosinger
Proof I n view o f (4.4.20), we o b t a i n v = (u*v,Ic E ( 0 , i ) ) + Vo
E
Eo = So/Vo
Then, according t o t h e proof of Theorem 1 i n S e c t i o n 2 , t h e unique solut i o n V E El can be obtained as
V = (V,)€ E ( 0 , l ) ) +
E, = Sl/Vl
v1 E
where V, is t h e unique c l a s s i c a l , i n f a c t , p-smooth s o l u t i o n of (4.1 . l ) with t h e i n i t i a l value u * ~ , . But VE
+
U i n C,
when
0
c
owing t o t h e well known continuous dependence property of c l a s s i c a l solut i o n s of ( 4 . 1 . 1 ) , ( 4 . 1 . 2 ) . Also we have obviously U*$, + U i n C,
when
6 -r
0
I n t h i s way, we o b t a i n (V, - U*$,lr
E (0,l)) E
V1
which, i n view of (4.4.22), y i e l d s (4.4.23) F i n a l l y , we t u r n t o t h e proof of property (4.4.4) concerning d e l t a uaves For t h a t purpose we s h a l l have t o make t h e following two a d d i t i o n a l assumpt i o n s on t h e semilinear hyperbolic system ( 4 . 1 . 1 ) , namely (4.4.24)
A
is c o n s t a n t
and F is bounded, more p r e c i s e l y V K c !R2
compact :
Concerning t h e rough i n i t i a l values
u,
we assume t h a t
Rough semilinear waves
(4.4.26)
u
E
P1(lR,e) and supp u is finite.
We recall now the following result, Oberguggenberger [2,9], Rauch & Reed. Let U6, with E (0,l) be the classical (?-smooth unique solution of (4.1.1) corresponding to the initial value u*(oC. Then, there exist
V E P' (lR2 ,lRn) and W E Cm(lR2 ,lRn) , such that
with V being the distributional solution of the linear hyperbolic initial value problem
while W is the classical solution of the semilinear hyperbolic initial value problem
Owing to (4.4.27), one calls V + W the delta uiave solu2ion of the semilinear hyperbolic initial value problem, see (4.1.I), (4.1.2)
Its decomposition property (4.4.28) and (4.4.29) is quite surprising and remarkable, since the rough initial value u, see (4.4.26), only influences the 1 inear part (4.4.28) of the semilinear hyperbolic system (4.4.30). Furthermore, this delta wave solution V property as well.
t
W has the following coherence
The delta wave solution V t W is precisely the unique (El --+ E2 ,Eo)sequential solution of the semilinear hyperbolic initial value problem (4.4.30).
E.E. Rosinger
Proof With the notations in (4.4.27)- (4.4.29)
Since A is constant and obtain for c E (0,l)
V
, let us
consider
is a solution of (4.4.28), we obviously
L(D)Vf = 0
Further, it follows, Oberguggenberger [2,9] , that U, - V, - W But W is
--r
0 in C, when
c
0
P-smooth, therefore W - W*$,+O
in C, when 6 - 0
In this way
(u, -
(V+W)*@,lc
E
(091))
E
VI
and then, according to (4.4.22), we obtain for relation
U
=
(U,(c
E
(0,l))
the
The question of coherence with more general types of delta urave solu2ions developed in Oberguggenberger [2,9] and Rauch 8 Reed remains open. As shown in Ober uggenberger [6], the choice of the very general framework in (4.2.10) and $4.2.15) where E l , E2 and E, are not necessarily algebras, has at least one critically important advantage, namely, it can offer coherence properties of generalized solutions, such as for instance in Propositions 1 and 2 in Section 4. Indeed, in case these three spaces are taken for instance as the differential a1 ebras constructed in Colombeau [1,2], then the coherence property (4.4.317 will in general fail, even in the case two dimensional, linear, variable coefficient hyperbolic initial value problems.
Rough semi1inear waves
1
The conditions (4.2.28) and (4.2.29 which define the essence of the framework in Section 2, are obviously re ated to the notions of stability, generality and exactness of generalized solutions, notions defined in Section 11, Chapter I . Indeed, large V 1 means high stability with respect to the possible perturbations of a given representative s defining the solution U, see (4.2.50). The generality property of solutions U increases with the size of El = S l / V l . Therefore, it means large S1 and small V l . According to (4.4.1), El is large enough to contain the 3' distributions. At that point, one can already note that stability and generality are conflicting. In case we omit condition (1.11.14) from the definition of exactness and also omit the requirement 'large A' in condition (1.11.15), then this concept of exactness can be applied to the quotient vector space EZ = S Z / Y Z as well. In this case, better ezactness will mean smaller V 2 . Since however V 1 c V z , see (4.2.29), we can note that stability will also conf1 ic t with exactness.
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CHAPTER 5 DISCONTINUOUS, SHOCK, WEAK AND GENERALIZED SOLUTIONS OF BASIC NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS 1 . THE NEED FOR NONCLASSICAL SOLUTIONS: THE EXAMPLE OF THE NONLINEAR
SHOCK WAVE EQUATIONS
It i s interesting t o note that many of the nonlinear partial differential equations of physics
are defined by highly regular, i n particular analytic functions F. I n + 191 of the partial derivatives i n fact, in many cases, the order (5.1.1) does not exceed 2 , while t e nonlinearities are polynomial, or even quadratic, as for instance i n the equations of fluid dynamics, general relativity, etc. I t can however happen that the i n i t i a l and/or boundary value problems associated w i t h (5.1.1 will no longer be given by analytic functions. Yet, under suitable we 1- posedness conditions satisfied by (5.1.1), such i n i t i a l and/or boundary values may be replaced by analytic approximations. I n t h i s way, it may appear that we may r e s t r i c t our attent ion to analytic partial differential equations and i n i t i a l or boundary values. This of course would be a major advantage, as we could for instance use the classical Cauchy-Kovalevskaia theorm, which guarantees the existence of an analytic - therefore, classica2 - solution for every noncharacteristic analytic i n i t i a l value problem.
Ipl
i
Unfortunately, t h i s and similar, Oleinik, Colton and the references mentioned there, existence results are of a local nature, i . e . , they guarantee the existence of classical, in particular analytic, sollitions only i n a neighbourhood of the noncharacteristic hypersurface on which the i n i t i a l values are given. And t h i s situation i s tuice unsatisfactory: f i r s t , in many physical problems we are interested i n solutions which exist on much larger domains than those granted by the above mentioned local existence results, then secondly, classical solutions w i l l i n general fail t o exist A particularly relevant, on the larger domains of physical interest. simple, yet important example i n this respect i s given by the conservation law
w i t h the i n i t i a l value problem
Obviously (5.1.2) i s an analytic nonlinear partial differential equation which i s of f i r s t order and has a polynomial, actually quadratic nonlinearity. Let us assume that the function u defining the i n i t i a l value
E.E. Rosinger
198
problem (5.1.3) i s analytic on R. It is easy t o see t h a t the c l a s s i c a l , in f a c t analytic solution U of (5.1.2), (5.1.3) w i l l be given by the implicit equation (5.1.4)
U(t,x) = ~ ( -x tU(t,x)),
t
> 0,
x
E
R
Hence, according t o t h e implicit function theorem, if (5.1.5)
tu' (x - tU(t ,x)) + 1 # 0
we can obtain U(s,y) from (5.1.4), f o r s and y in suitable neighbourhoods of t and x respectively. Obviously (5.1.5 i s s a t i s f i e d f o r t = 0, hence, there exists a neighbaurhood Q c [O,rn] x R of the x-axis R, so t h a t U(t ,x) e x i s t s f o r ( t , x ) E Q. However, if f o r a no matter how small interval I c R we have (5.1.6)
u'(x) < 0,
x
E
I
then the condition (5.1.5) may be violated f o r certain t > 0. This can happen irrespective of the extent of the domain of analyticity of u. = sin x f o r instance i s analytic not only f o r r e a l but also fIndeed, o r a l l comp u(xlex x, yet, it s a t i s f i e s (5.1.6) on every interval I = ((2k+l)a,(2k+2)r) c R, with k E R. Now, it is well known, Lax, that the violation of (5.1.5) can mean that the classical solution U no longer e x i s t s f o r the respective t and x. In other words, we can have Q $ [O,W] x R , i . e . , f o r certain x E R, the classical solution U(t,xj w i l l cease t o exist f o r sufficiently large t > 0. In particular, it ollows that
in other words, the equation (5.1.2) fails t o have classical solutions on the uhole of i t s domain of definition. However, from physical point of view, it is precisely the points ( t , x ) E [O,m) x R\Q which present interest in connection with the possible appearance and propagation of what are called shock waves. Fortunately, under rather eneral conditions, Lax, Schaef f e r , one can define certain generalized so ulions f o r a l l t 2 0 and x E IR
k
which are physically meaningful, and which are in f a c t classical solutions, except f o r points ( t ,x) E ,'l where 'I c [ O p ) x R consists of certain families of curves called shock fronts. For c l a r i t y , l e t us consider the following example, when the i n i t i a l value u in (5.1.3) i s given by
Shocks and distributions
in which case we have the shock front
and for 0
while for t
>
< 1, we have the classical solution
1 , we have a generalized solution
with U(t,x) defined a t the above example solution for a l l in (5.1.9) i s not fact that u i n
for
(t,x) E I'. It should be noted that in the failure of U t o be a classical does not come from the fact that u for instance analytic, but from the on I = ( 0 , l ) .
Before we go further and see the ways eneralized solutions could be defined, it should be noted that w i t h i n tk,e linear theory of distributions, the above generalized solution (5.1.12) cannot be dealt w i t h in a satisfactory way. Indeed, across the shock front I' i n (5.1. l o ) , the generalized solution U in (5.1.12) has a jump discontinuity of the type the Heaviside function has a t x = 0, i . e . ,
hence, i t s partial derivative Ux i n (5.1.2) will have across I' a singul a r i t y of the type the Dirac 6 distribution has a t x = 0. And then, the product UxU i n (5.1.2) when simplified t o one dimension, i s of the type H.6, which as i s known, cannot be dealt with within the Schwartz distribution theory, since both the factors are singular a t the same point x = 0. Let u s deal in some more detail w i t h t h i s difficulty. For instance, one can naturally ask whether, nevertheless the nonclassical solvtions U in (5.1.7) of (5.1.2) could perhaps be dealt with within the framework of the distributions. For instance, we could perhaps assume that, nevertheless, we may have
Indeed, as i s well known, Lax, Schaeffer, i n many important cases, the nonclassical solutions U of (5.1.2), (5.1.3) are in fact smooth functions on
E. E. Rosinger
200
(0,m) x R , with the exception of certain smooth curves 'I c (0,m) other words, we often have
x
R.
In
Furthermore, the nonclassical solutions U have f i n i t e jump discontinuit i e s across the curves I". In other words, i f we assume that (5.1.16)
'I = { ( t , x )
E
(O,CO)x R ( t = ~ ( x ) ) , with
7
E
p(R)
then
where
and
i s the Heaviside function. In such a case, using the distributional derivatives, one obtains from (5.1.16)- (5.1.19) the following relations in 2' ( ( 0 , ~ ) R)
where (5.1.21)
6
E
Tf (R)
i s the Dirac distribution, which i s the distributional derivative of the Heaviside function, that is (5.1.22)
6 = H'
in Vf (R)
The inappropriateness of dealing with nonclassical solutions of non 2 inear partial differential equations within the distributional framework becomes now obvious. Indeed, i f we t r y t o replace (5.1.17), (5.1.20) into (5.1.2) in order t o check whether or not U in 5.1.17) i s indeed a solution, t h i s simply cannot be done within Vf((O,m x IR), since the nonlinear term U.Ux in (5.1.2) would lead t o the s ingu Ear product
\
Shocks and d i s t r i b u t i o n s
which, as we mentioned, is not d e f i n e d w i t h i n t h e d i s t r i b u t i o n s . Needless t o s a y , i f i n s t e a d of t h e f i r s t o r d e r , polynomial n o n l i n e a r part i a l d i f f e r e n t i a l e q u a t i o n i n (5.1.2) we have a second o r d e r , polynomial n o n l i n e a r p a r t i a l d i f f e r e n t i a l equation - a s i t u a t i o n f r e q u e n t l y o c c u r i n i n p h y s i c s - and we t r y t o check whether o r n o t a n o n c l a s s i c a l s o l u t i o n is a d i s t r i b u t i o n a l s o l u t i o n , we can i n a d d i t i o n t o (5.1.23) end up w i t h y e t more s i n g u l a r p r o d u c t s , such as f o r i n s t a n c e
1
which a r e even l e s s d e f i n a b l e w i t h i n t h e d i s t r i b u t i o n s T'. I n some c a s e s of n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s , such as f o r ins t a n c e conservative ones, a s is t h e c a s e of (5.1.2) as w e l l , t h e problem of having t o d e a l with n o n c l a s s i c a l s o l u t i o n s c a n be approached by r e p l a c i n g t h e r e s p e c t i v e n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s w i t h more g e n e r a l , s o c a l l e d weak e q u a t i o n s . For i n s t a n c e , i n t h e c a s e o f ( 5 . 1 . 2 ) , it is obvious t h a t any f u n c t i o n U E Ci((O,m) x IR) which s a t i s f i e s i t , w i l l a l s o s a t i s f y t h e weak e q u a t i o n
which is obviously more g e n e r a l t h a n ( 5 . 1 . 2 ) , s i n c e it can admit s o l u t i o n s
The obvious 5.1.2) - it n t h i s way, t i o n s U as
t
advantage o f t h e weak e q u a t i o n (5.1.25) is t h a t - u n l i k e only c o n t a i n s U but none of t h e p a r t i a l d e r i v a t i v e s of U. t h e weak equation (5.1.25) can accommodate nonclassical soluwell.
However, many important n o n l i n e a r e q u a t i o n s of p h y s i c s a r e not i n a conserv a t i v e form and t h e r e f o r e , t h e y do not admit convenient weak g e n e r a l i z a t i o n s . One of t h e s i m p l e s t such examples is t h e f o l l o w i n g system which models t h e coupling between t h e v e l o c i t y U and s t r e s s C i n a one dimens i o n a l homogeneous medium o f c o n s t a n t d e n s i t y
with k > 0 d e ending on t h e medium, and where t h e second e q u a t i o n - owing - is not i n c o n s e r v a t i v e form. t o t h e term
E. E. Ros i n g e r
It f o l l o w s t h a t t h e weak e u a t i o n s have two d e f i c i e n c i e s : f i r s t , t h e y cannot always be used t o r e p a c e n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s , and secondly, even when t h e y can be used, t h e y a r e more g e n e r a l t h a n , and not n e c e s s a r i l y e q u i v a l e n t with t h e n o n l i n e a r p a r t i a l d i f f e r e n t i a l equat ions they replace.
P
A d e t a i l e d approach t o g e n e r a l i z e d s o l u t i o n s corresponding t o d i s e o n t i n u o u s f u n c t i o n s , such as f o r i n s t a n c e i n (5.1.17), is p r e s e n t e d i n Chapter 7. T h i s approach can d e a l with an a r b i t r a r y number of independent v a r i a b l e s and with r a t h e r l a r g e c l a s s e s of n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s , which i n c l u d e many of t h e e q u a t i o n s of physics.
$2.
INTEGRAL VERSUS PARTIAL DIFFERENTIAL EQUATIONS
There a p p e a r s as w e l l t o e x i s t deeper reasons f o r c o n s i d e r i n g n o n c l a s s i c a l solui! i o n s f o r l i n e a r and n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s . We r e c a l l t h a t most o f t h e b a s i c e q u a t i o n s of p h y s i c s which a r e d i r e c t e x p r e s s i o n s of p h y s i c a l laws, a r e balance e q u a t i o n s , v a l i d on s u f f i c i e n t l y r e g u l a r domains of space- t i m e , and as such, t h e y a r e w r i t t e n as integrod i f f e r e n t i a 1 e q u a t i o n s on t h e r e s p e c t i v e domains, Fung, E r i g e n , Peyret & Taylor. S i n c e a l o c a l , space-time, point-wise d e s c r i p t i o n of t h e s t a t e of a p h y s i c a l system is o f t e n considered t o be p r e f e r a b l e from t h e p o i n t of view of sat i s f a c t o r y o r h o p e f u l l y s u f f i c i e n t informat i o n , t h e r e s p e c t i v e i n t e r o - d i f f e r e n t i a l e q u a t i o n s a r e reduced - under s u i t a b l e a d d i t i o n a l regu a r i t y assumptions - t o p a r t i a l d i f f e r e n t i a l e q u a t i o n s whose c l a s s i c a l , function solutions
7
a r e supposed t o d e s c r i b e t h e s t a t e of t h e r e s p e c t i v e p h y s i c a l system. It f o l l o w s t h a t many of t h e b a s i c p a r t i a l d i f f e r e n t i a l e q u a t i o n s of p h y s i c s a r e consequences of p h y s i c a l laws and a d d i t i o n a l mathematical t y p e r e u l a r i t y c o n d i t i o n s needed i n t h e r e d u c t i o n of t h e primal integrod i k f e r e n t ial e q u a t i o n s t o t h e mentioned p a r t i a l d i f f e r e n t i a l e q u a t i o n s . These a d d i t i o n a l assumptions o r c o n d i t i o n s can be seen as c o n s t i t u t i n g a l o c a l i z a t i o n p r i n c i p l e , Eringen, which under s u i t a b l e forms, p l a y s a c r u c i a l r o l e i n v a r i o u s not i o n s o f weak, d i s t r i b u t i o n a l and g e n e r a l i z e d s o l u t i o n s . I n f a c t , t h i s l o c a l i z a t i o n p r i n c i p l e d e t e r m i n e s a n important sheaf s t r u c t u r e , Seebach, e t . a l . , on t h e r e s p e c t i v e s p a c e s of d i s t r i b u t i o n s and g e n e r a l i z e d f u n c t i o n s , s e e Appendex 2 , Chapter 3 .
A good example i n connection with t h e above is g i v e n by c o n s e r v a t i o n laws. Suppose a s c a l a r p h y s i c a l system occupying a f i x e d space domain A c R~ is such t h a t t h e change i n t i m e i n t h e t o t a l amount of t h a t p h y s i c a l e n t i t y i n any given s u f f i c i e n t l y r e g u l a r subdomain G c A is due t o t h e f l u x of t h a t p h y s i c a l e n t i t y a c r o s s t h e boundary aG of G , and t a k e s p l a c e a c c o r d i n g t o the relation
Shocks and d i s t r i b u t i o n s
where U(t , x ) E R is t h e d e n s i t y of t h e p h y s i c a l e n t i t y at t i m e t and at t h e s p a c e p o i n t x E G , w h i l e F ( t , x ) E Rm is t h e f l u x of t h a t p h y s i c a l e n t i t y a t t i m e t and at t h e space p o i n t x E dG.
As is known, i n c a s e U and F a r e assumed t o be s u f f i c i e n t l y r e g u l a r , f o r i n s t a n c e C1- smooth, t h e integro- d i f f e r e n t i a l e q u a t i o n (5.2.2) c a n be reduced t o t h e p a r t i a l d i f f e r e n t i a l equation
G , (5.2.2) y i e l d s
Indeed, i n view of Gauss' formula and t h e r e g u l a r i t y of
and t h e n , t h e a r b i t r a r i n e s s of
G c A will imply (5.2.3).
However, it is important t o n o t e t h a t t h e integro- d i f f e r e n t i a l e q u a t i o n (5.2.2) which is t h e direct e x p r e s s i o n of t h e c o n s e r v a t i o n law c o n s i d e r e d , is more general t h a n t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n ( 5 . 2 . 3 ) which was o b t a i n e d from (5.2.2) under t h e mentioned a d d i t i o n a l r e u l a r i t y assumptions on U and F, assumptions which a r e not r e q u i r e d on t e l e v e l of (5.2.2), a r e l a t i o n v a l i d f o r nonsmooth b u t i n t e g r a b l e U and F.
i?
Nevertheless, i f we make use of test functions $ E C1 (IR x A) w i t h compact s u p p o r t , equation (5.2.3) y i e l d s a f t e r an i n t e g r a t i o n by p a r t s
T h i s e u a t i o n , when assumed t o hold f o r every $ E V(R x A), is t h e weak form (5.2.3) and obviously, it i s more general t h a n ( 5 . 2 . 3 ) , although not n e c e s s a r i l y more g e n e r a l t h a n (5.2.2). However, f o r many of t h e non1i n e a r integro- d i f f e r e n t ial e q u a t i o n s , one cannot o b t a i n a corresponding convenient weak form, but only a n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n which, as mentioned, is o b t a i n e d by assumin among o t h e r s , s u i t a b l e regul a r i t y c o n d i t i o n s on t h e s t a t e f u n c t i o n of t h e r e s p e c t i v e p h y s i c a l system.
07
f
I n t h i s way, we can d i s t i n g u i s h two levels of localization : f i r s t , l o c a l i z a t i o n on compact s u b s e t s of t h e space- t i m e domain, and second, l o c a l i z a t i o n at points of t h e space- time domain. The f i r s t l o c a l i z a t i o n l e a d s t o weak forms of t h e p h y s i c a l b a l a n c e equat i o n s , as f o r i n s t a n c e i n (5.2.5) and does not depend on a d d i t i o n a l
204
E. E. Ros inger
regularity assumptions, but rather on specific features of the physical system i t s e l f , such as conservation properties, for instance. The second localization leads to linear or nonlinear part ial differential equations and does require additional regularity conditions, as for instance in the above example, where the partial differential equation was obtained from a physical law and certain additional matheregularity conditions. In general, the situation can be more complex, as it happens for instance in fluid dynamics, where i n addition to the physical laws and certain mathematical regularity conditions, one also needs specific assumptions on the mechanical properties of the fluid, usually called const itut ive equat ions, such as those concerning the stress- strain relationship, Fung. A typical and important example i s that of the so called Newtonian viscous fluids described by the Navier-Stokes equations, where the balance or conservation of mass, momentum and energy give the respective int egrodifferential equations, Peyret & Taylor
where A c R3 i s a domain occupied by the f l u i d , U : A -+ R3 i s the velocity of the fluid, while p , u , F, E and q are the density, stress tensor, internal volume force, t o t a l ener y and heat flux, respectively. If we assume now the suitable conditions of regularity on the above physical e n t i t i e s , as well as the usual constitutive equations, Fung, Peyret & Taylor, then, for an incompressible fluid, we obtain the nonlinear partial differential sgstem
w i t h t L 0 and x E A , where U = (9,Uz ,U3)? F = (FI ,F2 ,F3), while P :A IR i s the pressure and u i s the kinematic viscosity.
Shocks and distributions
205
Although, as seen above, many of the partial differential equations of physics are more particular than the i n i t i a l , direct expressions of the respective physical laws, such as for instance the balance, integro- differential equations, we are often obliged to deal only with these linear or nonlinear partial differential equations, since the less restrictive weak forms may f a i l t o exist under convenient form. Indeed, when looking for local, point- wise informat ion in space- time on the s t a t e functions ( 5 . 2 . I ) , partial differential equations seem t o be irreplacable. For the sake of completeness however, we should remember the following. The continuous - in particular, integro-differential-modelling of physical laws originated with Euler, while Newton's i n i t i a l formulation of the second law of dynamics has been discreie, see Abbott, pp. 220-227, and the literature cited there. The difference between these two kind of formulations i s that in certain cases - such as for instance irreversible physical processes - the discrete models are more general than the continuous ones. As seen above, nonclassical, i n particular generalized solutions appear i n a necessary way i n the study of some of the simplest nonlinear partial differential equations. I n addition, we have to remember the following. When replacing with partial differential equations the integro-differential balance equations which are the primary expressions of physical laws, we assumed certain additional regularity conditions on the s t a t e functions U. Usually these regularity conditions require a smoothness of U of an order which turns U into a classical solution of the resultin partial differential equations. Therefore, i n order t o avoid the possi\ility of eliminating physically meaningful nonclassical solutions when we reduce the original integro- differential equations t o partial differential equations, we have to sufficiently general means allowing for the incorporation o nonclassical, in particular generalized solutions of these partial differential equations.
provide
As we are mainly interested in generalized solutions for nonlinear partial differential equations, we shall in the next Section direct our attention towards two of the methods which have so f a r proved t o be most suited, see for instance Sobolev [1,2], Lions 121. However, in order to make clear the most ba.sic ideas involved, it is useful to consider a few more, simple, linear examples. The equation
has the classical solution
where (5.2.13)
u
E
C1 (IR)
E.E. Rosinger
206
and it describes the propagation of the space wave defined b u, alon physicak the characteristic lines x + t = constant. It follows that !rom point of view, there is no any justification for the regularity condition (5.2.13), as the mentioned kind of wave propagation can as well make sense for functions u $ C1 (R). In this way, we should be able to obtain for In such a (5.2.11) solutions 5.2.12 , when we no lon er have (5.2.13). in (5.2.12) wi 1 be a g e n e r a l i z e d s o l r t i o n . case, the correspon ing This indeed can be obtained if, similar to (5.2.5) , we replace (5.2.11) by its more general weak form
!i 1
Indeed, for every satisfy (5.2.14) .
u
E
f
Cis, (R) ,
the corresponding
U
in (5.2.11) will
In addition to its generality, the weak form (5.2.14) draws attention upon another important property of eneralized solutions, first pointed out and used in a systematic way in So olev [1,2]. Namely, if U,, with v E #, is a sequence of generalized solutions of (5.2.11) obtained from (5.2.14), and this sequence c o n v e r g e s uniform1 on compacts in R2 to a function U, then U will again satisf (5.2.143, and hence, it will be a generalized solution of (5.2.11). In !act, it is obvious that much more eneral types of convergence will still exhibit the above c l o s u r e p r o p e r t y o generalized solutions. On the other hand, the c l a s s i c a l solutions of (5.2.11) obviously f a i l to have this closure property. Indeed, if U,, with v E #, are
B
f
C1-smooth classical solutions of (5.2.11) , it may happen that U even if the convergence Uv
-+
E C?\C1
,
U is uniform on compacts in R2.
The above situation generalizes entirely to the customary linear wave equation
which has the classical solution
where (5.2.17)
u,v E C2 (R)
As the solution (5.2.16) describes the superposition of the propagation of the space waves u and v, along the characteristic lines x - t = constant and x + t = constant, respectively, it is obvious that from
Shocks and distributions
physical point of view, the regularity condition (5.2.17) i s not necessary. Indeed, the ~ e a kform of (5.2.16) i s
thus, f o r every u,v E L1 (R), the corresponding U i n (5.2.16) w i l l [31 satisfy (5.2.18), hence it w i l l be a generalized solution of (5.2.15). In view of (5.2.18), it i s obvious that the respective generalized solutions will again have the above closure property. And as before, the classical solutions of (5.2.15) will f a i l t o have the mentioned closure property. In view of the above, it would appear that a proper way t o proceed i s t o define the generalized solutions as solutions of the associated weak form of linear or nonlinear partial differential equations. However, such an approach proves t o have several deficiencies. Indeed, in the case of non1 inear partial differential equations, the explicit expression of the associated weak form cannot be obtained, except for particular cases, such as conservation laws for instance. Furthermore, even when the weak form i s available, it i s not so easy t o solve it in the unknown function U. But above a l l , even i n the case of linear partial differential equations, the weak form proves t o have an insufficient generality in order t o ha,ndle the whole range of useful generalized solutions, such as for instance the Creen junction or elementary solution asasociated with a linear, constant coeff icient partial differential equation. Indeed, l e t us consider the linear wave equation (5.2.15) i n the following more general form
where F E L1 (IR2). Then, the associated weak form i s c3j
It i s well known, Hormander, that i n case F has compact support, a generalized - i n f a c t , distribution - solution of (5.2.19) can be obtained by (5.2.21)
U = E*F
where * i s the convolution operator and E i s the of (5.2.15), i . e . , it i s the distribution solution of
elementary solution
where 6 i s the Dirac delta distribution. Now, although (5.2.22) resembles (5.2.19), it jails t o have an associated weak form (5.2.20), w i t h F a function, owing t o a basic property of the Dirac delta distribution,
E.E. Rosinger Schwartz [I . It follows that the elementary solution E of (5.2.22) cannot be o tained from the weak form (5.2.20) of (5.2.19), for any choice of the function F.
d
These examples can offer no more than a first illustration of the way generalized solutions will necessarily arise in the study of partial differential equations. For a rather im ressive account of their utility both in linear and nonlinear partial dif erential equations, a large literature is available, among others, Hormander and Lions [2] , cited above.
P
$3. CONCEPTS OF GENERALIZED SOLWIONS
Let us resume the main facts mentioned in the previous Section, with a view towards a suitable definition of generalized solutions for nonlinear partial differential equations. Within the sphere of analytic - thus classical - solutions of analytic partial differential equations, we can perform partial derivatives of arbitrary order as well as sufficiently general nonlinear operations, in particular multiplicalions. A basic deficiency encountered is that we are quite often interested in solutions on larger domains than those on which the analytic solutions prove to exist, and the solutions on such larger domains may fail to be classical solutions. It follows that a desirable concept of generalized solution should allow the followin three things : partial derivability of sufficiently high order, sufnciently general nonlinear operations, in particular unrestricted multiplication, and finally, existence of such generalized solutions on sufficiently large domains. It is particularly important to note that indefinite partial derivability of generalized functions, although highly desirable, is not absolutely necessary, as long as we deal with the usual linear or nonlinear partial differential equations which are of course of finite order. Indeed, all what is in fact required in these usual cases is that a generalized solution has partial derivatives up to, and including, the order of the respective partial differential equation. This relaxation in the requirements on generalized solutions may be particularly welcome in the light of the conflict between insufficient smoothness, multiplication and differentiation, see Section 4, Chapter 1. Historically, tuo basic ideas concernin the definition of generalized solutions have emerged. The first, in ~ 8 o l e v[I ,2], which may be called the sequential approach, has not known a sufficiently general and systematic theoretical development, yet, it led to a wide range of efficient, even if somewhat ad-hoc, solution methods especially for nonlinear partial differential equations, see for instance Lions [2]. The second idea, in Schwartz [ I ] , which can be called the linear junctional analytic approach,
Shocks and distributions
209
has been extensively developed from theoretical point of view, although i t s major power i s restricted to linear partial differential equations. Here we shall recall the main idea of the sequential approach which i s a t the basis of the nonlinear method presented i n t h i s volume. Suppose given a nonlinear partial differential equation
Then we construct an infinite seqauence of 'approximating' equations
in such a way that V, are classical solutions of ( 5 . 3 . 2 ) and they converge i n a certain weak sense t o U. For instance, we have
where 3 c C ? ( Q ~ i s a set of sufficiently smooth t e s t functions. i s defined as t e '?-weak limit' of V,, with Y E N.
Then, U
Here, the sequential and 1inear functional analytic approaches have a common point, as f o r many choices of 3, the relation ( 5 . 3 . 3 ) means that V, converges to U, when v 4 m, within a certain space of distributions. However, a s seen i n t h i s volume, the seqauential approach does possess a significant potential for extentions leading to systematic nonlinear theories, while the linear functional analytic approach does not seem t o do so. Moreover, as follows from the stability paradoxes in Section 11, Chapter 1, a proper way for a systematic nonlinear extension of the sequential approach needs a careful reassessment of i t s above mentioned common point w i t h the 1inear functional analytic approach.
$4. WHY USE DISTRIBUTIONS?
Although Schwartz's 1 inear theor of distributions has severe limitation in even solving linear partial di ferential equations, see Section 5 next, there are significant advantages in using that theory. Here we shortly mention some of the more important ones.
Y
It should also be mentioned that the linear functional analytic approach of Schwartz [I] had as main priority the indefinite partial derivability of generalized solutions, Hormander, and then, in view of the so called impossibility result of Schwartz [2], had to suffer from certain limitations on i t s capability to accommodate unrestricted nonlinear operations,
E.E. Rosinger
210
in particular unrestricted multiplication. It should however be pointed out that, as seen in this volume, there are various ways one can circumvent the restrictions which appear when we would like to have both indefinite partial derivabilit and unrestricted multiplication. In this respect, the !unctional analytic approach in Schwartz [I] is but one classical, linear ! of the many possible ones, and it seems to be less suited for a systematic study of nonlinear partial differential equations. Nevertheless, from the point of view of partial derivability, the space P' of distributions possesses a canonical structure. Indeed, let us consider the chain of inclusions (5.4.1)
P C... c C e c ... c C O c P 1 , P E N
where only the elements of P are indefinitely partial derivable in the classical sense. As is known, Schwartz [I], V' is the set of all linear functionals T : V + C which are continuous in the usual topology on P. In particular, the embedding CO 3 f H Tf E 2' is defined by
which in fact holds also for f E Lioc. In this way, the elements of P' will again be indefinite1 partial derivable, although no longer in a classical sense, but in the ollowing more general, ueak sense : suppose given T E 2)'(lRn) and p E N", then DPT f D' ' (lRn) is defined by
p
Obviously, if f
E
Ce , with t
E
IN, and p E INn, Ip[ 5 t, then
i.e., the weak and classical partial derivatives coincide for sufficiently smooth functions. The canonical property of 9' is the following
where D~ is the weak partial derivative in (5.4.3).
In other words D'
Shocks and distributions
211
is a minimal extension of CO in the sense that locally, every distribution i s a weak partial derivative of a continuous function.
The above roperty makes 3' sufficiently large in order t o contain the Dirac 6 Bistribution defined by
which, linear stance sional
among others, i s essential i n the study of constant coefficient partial differential ( 5 . 2 . 2 2 ) . Indeed, for simplicity, l e t us case, when n = 1, and l e t us define x+ E
elementary solutions of equations, see for inconsider the one dimenCo(IRn) by
then in view of (5.4.2) and (5.4.3), the weak derivative of
x+ is given
by
where H i s the Heaviside function
Similarly, a further weak derivation yields
hence, 6 i s the second weak derivative of a contineous function, and ( 5 . 4 . 1 0 ) holds globally on IR and not only locally. The use of the space 3' of distributions has a particularly important justification in the study of linear, constant coefficient partial differential equations. The basic result i n t h i s respect, f i r s t obtained in Ehrenpreis and Malgrange, concerns the proof of the existence of an elementary solution for every such equation, a result which has a wide range of useful consequences and applications, Hormander, Treves [2] , and which alone would fully justify the use of distributions. It should however be pointed out that the existence of generalized solutions for inhomogeneous linear constant coefficient partial differential equations can easily be obtained i n other spaces of generalized functions as well. For instance, usin elementary ring theoretical methods, Gutterman proved such existence resu t s within the space of Mikusinski operators. In f a c t , simil a r simple algebraic arguments together with some from classical Fourier analysis can deliver the mentioned Ehrenpreis-Malgrange results, see
k
212
E.E. Rosinger
Struble. A whole range of other linear applications of the V' distributions can be found in the literature, among others, in the last three monographs cited above, as well as in Treves [3]. A recent useful, yet easy to read account of many of the more important linear applications can be found in Friedlander. Although the space 2' of distributions has the above mentioned canonical property, various linear extensions of it, iven for instance by spaces of hyperfunctions, Sato et. al., have been stufied in the literature.
$5. TEE LEWY INEXISTENCE RESULT Soon after the foundation of the modern linear theory of distributions. Schwartz [I], and the proof of the existence of an elementary solution for every linear constant coefficient partial differential equation, Malgrange, Ehrenpreis, a very simple example of a linear variable coefficient partial differential equation given by Lew , showed that the linear theory of distributions is not sufficient even ?or the study of linear partial differential equations. Lewy's example is the following surprisingly simple equation
which for a large class of f E p(lR3), fails to have distribution solutions U E 3' in any neighbourhood of any point x E R3. In this way, it follows that the solution of (5.5.1) requires spaces of of the Schwartz distributions.
generalized f u ~ c tions which are larger than the space 3'
The interesting thing about Lewy's equation (5.5.1) is that it is not a kind of art ificial , counter-example type of equation, but it appears naturally in connection with certain studies in complex functions of several variables, Krantz. The phenomenon of insr ff iciency of the 2' distributional framework, pointed out by Lewy's example, became the object of several subsequent studies. We shall shortly relate the result of one of them. For that purpose we need several notations. Suppose given a domain 0 c Rn and an m-th order linear variable coefficient partial differential operator
with the coefficients c E p(8), for p P part of P(x,D) is by definition
E
!Jln, lpl 5 m.
The principal
Shocks and distributions
(5.5.3)
c c (x)& pE~n I P I=m
P~(X,~)=
E 0, ( E
and its complex conjugate is
finally, the commutator of P(x,D)
is defined by
which obviously is a polynomial of degree 2m C"- smooth coefficients.
-
1 in ( and it has real,
A basic necessary condition for solvability is the following. Theorem 1 (Hormander) Suppose the linear partial differential equation (5.5.6)
P(x,D)U
=
f, x E Q
has a solution U E T' (Q) , for every f E P(Q), then
It is easy to see that in Lewy's exa,mple (5.5.1) we have n
=
3, D = R3,
m = 1 and
= - 2x2 ,
hence, for x E R3 and
(1
(5.5.9)
0, Cl(~,t)
Pl(X,C)
=
which contradicts (5.5.7)
(2
*
=
0
2x1 and
(3
=
1, we have
E .E . Rosinger
with p E INn, lpl = m, In general, if the coefficients of P,, i.e., c P' is identically zero, hence (5.5.7) is are real, then obviously, CZmsatisfied. Unfortunately however, (5.5.7) is only a necessary and not also a sufficient condition for solvability, Treves [I]. However, in the case of first order linear partial differential equations with COD- smooth coefficients, Nirenber E ~rbvescould obtain a necessary and sufficient condition for solvabi ity. In the general case of (5.5.2), a certain strengthened form of (5.5.7) proves to be both necessary and sufficient for local solvability, provided that multiple characteristics are not present, Hormander .
7
It is interesting to note that in case we deal with the easier problem of solvability in the neighbourhood of a iven, fixed point x E Wn, we can find still simpler linear partial dif erential equations toi lhost distribut ion solutions. Indeed, Grushin showed that the equation
B
fails to have distribution solutions with suitably chosen f E p(IR2), U E V' (Q) in any neighbourhood Q c IR2 of x0 = (0,O) E R2 .
Finally, it should be mentioned that linear variable coefficient partial differential equations fail to have solutions in various 1inear extensions of the Schwartz P' distributions, such as for instance, spaces of hyperfunctions, Sato, et. al, Hormander. A result in this respect can be found in Shapira. In view of the above, it is important to note that in Colombeau [3] was for the first time proved the existence of generalized solutions for systems of arbitrary linear partial differential equations with L?- smooth coefficients, within the differential algebra of generalized functions, mentioned later in Chapter 8, see for details Rosinger [3, pp. 169-1771 .
Shocks and distributions
APPENDIX 1 YULTIPLICATION, LOCALIZATION AND REGULARIZATION OF DISTRIBUTIONS We short1 recall a few important properties of the V' distributions, which w i l be useful in the sequel, Schwartz [I], Friedlander.
1
As i s well known, one can multiply every smooth function x E P(IRn) with every distribution T E V'(IRn) and obtain as product the distribution (5.Al.l)
S = x.T
E
V'(IRn)
defined by
However, i f x E 2' (R") \f?'(IRn) , then we s h a l l have X. 9 6 V(IRn) , for certain $ E V(IRn) . Hence, the right hand term in (5.A1.2) w i l l no longer be defined, and then, we cannot use t h i s relation in order t o define the product in (5. A1 .1) . As an application t o the one dimensional case of the distributions in 9' (IR) , we have
Indeed, (5.Al.l) and (5.A1.2) w i l l give f o r y5 E V(R), the relations
Suppose now qiven Q c lRn open and l e t us define $(I)) as the s e t of a l l linear functionals T : P(S2) -+ C which are continuous in the usual topology on V(0). The relation between V'(IR) and P'(R) is an example of a localization principle corresponding t o a sheaf structure specified next. Obviously we have an embedding of vector spaces
defined as follows : if $ E V(Q), then we can consider $ E V(IRn), with # vanishing on IRn\Q. Now (5.A1.4) yields the embedding of vector spaces
defined by the r e s t r i c t i o n mapping
E .E . Ros i n g e r
V' (R")
3
T
H
T
I
E
V' (Q)
V(Q)
However, we a l s o have t h e f o l l o w i n g , less t r i v i a l , converse a s p e c t of t h e localization principle. Suppose g i v e n Ql c Q open, with such t h a t
whenever Qi
n
Q. # J
4, w i t h i ,j
Then, t h e r e e x i s t s a unique T
E
i
E
E
I,
and
Ti
E
V1(Qi),
with
i
E
I,
I . F u r t h e r , l e t u s suppose t h a t
P'(Q),
such t h a t
It f o l l o w s i n p a r t i c u l a r t h a t a d i s t r i b u t i o n T E V'rn) is uniquely determined i f i t s r e s t r i c t i o n TI t o a neighbourhood of an a r b i t r a r y
p o i n t x E R" is known. F u r t h e r we n o t e t h a t (5.A1.7)- (5.A1.9) allow u s t o d e f i n e t h e support supp T of a d i s t r i b u t i o n T E T ( Q ) as t h e c l o s e d s u b s e t i n Q which is t h e complementary of t h e l a r g e s t open s u b s e t i n Q on which T vanishes.
As an important a p p l i c a t i o n of t h e above, we p r e s e n t t h e regularization of c e r t a i n f u n c t i o n s o r d i s t r i b u t i o n s a c r o s s s i n g u l a r i t i e s . T h i s can h e l p i n b e t t e r understanding t h e d i f f i c u l t i e s involved i n t h e problems solved i n Chapter 7. Suppose g i v e n I' c Utn nonvoid, c l o s e d , t h e n Q = IRn\r is open. It is easy t o s e e t h a t i n g e n e r a l , t h e i n c l u s i o n (5.A1.5) is s t r i c t . T h e r e f o r e , l e t us define
and c a l l S t h e r - regularizat ion o f T . Obviously, even if Q is dense S1 and Sz o f i n lRn, i . e . , I' h a s no i n t e r i o r , two I'-regularizations T can be d i f f e r e n t and t h e i r d i f f e r e n c e Sz-S1 E V'(IRn) w i l l be a d i s t r i b u t i o n with s u p p o r t contained i n r. It f o l l o w s t h a t whenever it
Shocks and distributions
T
exists, the r-regularization of supported by r.
217
i s unique modzllo a distribution
Let us clarify the above by an example. then Q = R\I' = (-m,O)U(O,m) Suppose I' = (0) c R, Further, suppose given f E C"(Q) defined by f(x) = l n l x J , x
(5.A1.11) Obviously f
E
R.
Q
L:ioc(R), hence (5.4.2) yields a distribution
Tf
(5.A1.12)
E
i s dense in
E
PI (R)
Now, w i t h the usual derivative of smooth functions, we have
g(x) = Df(x) = l/x,
(5.A1.13) Hence
g
$!
Lio,(R),
x E Q
therefore (5.4.2) cannot be applied t o
Nevertheless, distribution
we have
(5. A1.14)
Tg E PI (Q)
g
E
C"(Q) c Lioc(Q),
hence
(5.4.2)
g
on
R.
yields a
We show now the stronger property (5.11.15) i.e.,
Tg
E
T(Q)
T admits a I'-regularization S g
E
P'(Rj.
Indeed, i n view of (5.A1.12), l e t us take (5.A1.16)
S = DTf E P/(R)
where D i s the distributional derivative in P' (R) . Then, (5.4.4) applied t o f E p ( Q ) , together with (5.A1.13) w i l l obviously give (5.A1.15). With usual notations, the above can be sumarized tion l / x which i s singular a t x = 0 and it i s W, can nevertheless be regularized a t x = 0 by S = D(enlx1) E P'(IR). I n view of t h i s , we shall and thus obtain
by saying that the funcnot local1 integrable on the d i s t r i ution identi j y l / x w i t h S,
r~
E.E. Rosinger
218
As mentioned above, the regularization (5.A1.17) of l/x is unique, modulo c W. These distributions are known to distributions uith support 'I = (0) ~. b e o f the form B C ~ D ~with ~ , ( E M , c E E . Finally, in view of P os~se (5.A1.l) and (5.A1.17) we can define the product
and we shall show that (5.A1.19)
(x). (11x1 = 1
Indeed, according to (5.4.2) and (5.4.3), we have for relations
$ E
P(R)
the
which completes the proof of (5.A1.19) It should be noted that the multiplication of two distributions which have common singularities does pose a considerable problem in the sense that, on the one hand, simple and natural definitions, such as for instance in (5.A1.1) and (5.A1.2) are no longer available, while on the other hand, there appears to be a large variety of other possible definitions with no natural or canonical candidate. Details in this connection can be found in Section 4, Chapter 1, as well as in Rosinger 1,2,3] and the literature cited there. Here, we should only like to recal the simple example of the product
f
which according to various interpretations, can be shown to be no longer a distribution, see details in Rosinger [I, pp. 11,29-311, Rosinger [2, pp. 66, 115-1181, and Mikusinski [2]. Finally, for Q c Rn open and C E I, let us denote by
the set of all sequences s E (cC(Q))' which converge in pf(Q), C respectively the set of all sequences v E S (0) which converge in T(Q) to zero. As is well known, Schwartz [I], we have the qvotient vector space representation given by the linear isomorphism
Shocks and d i s t r i b u t i o n s
where f o r s = ($, 1 v
E IN) E
C S ( Q ) , we have
It follows t h a t we can obtain the following q v o t i e n l represent a t ion of the L . Schwartz d i s t r i b u t i o n s
v e c t o r space
This Page Intentionally Left Blank
CHAPTER 6
CHAINS OF ALGEBBAS OF GENEUIZED FUNCTIONS $1. BESTRICTIONS ON EMBEDDINGS OF THE DISTRIBUTIONS INTO
QUOTIENT ALGEBRAS
The aim of this Chapter is to present the basic results concerning the necessary structure of the nonlinear theories of generalized functions originated in Rosinger [7,8] and developed in Rosinger [I ,2,3] , as well as in their more particular form, in Colombeau [1,2]. The starting point of these theories are the quotient algebras of generalized functions as defined in (1.5.26) and (1.6.9), and subsequently used directly in Chapters 2 and 3. The fundamental problem in connection with these quotient algebras A = A/Z is of course the clarification of their deeper structure, beyond that which is so simply given by their definition through the tu~o conditions (1.6.10) and (1.6.11). Surprisingly, to a good extent such a clarification can be reduced to the understanding of the structure of the ideals I alone.
A first set of results in this respect was already presented in Sections 5-8 in Chapter 3, in the nature of various densely vanishing conditions which characterize many of such ideals I.
In this Chapter we clarify three further issues. The first one, of a most general nature, is about the detailed necessary structure of the inclusion dia rams (1.6.10), and it is presented in Section 1, culminating in (6.1.271 and (6.1.33). The second issue dealt with in this Chapter is an alternative, most simple characterization of a large class of the mentioned ideals I. This characterization is obtained in Theorem 4, Section 3. The third and last issue dealt with in this Chapter centers around the fact presented in Section 4, according to which a natural way to deal with the conf 1 ict between insufficient smoothness, multiplication and differentiation is to allow the nonlinear partial differential operators to act between possibly infinite chains of quotient algebras of generalized functions. As seen in Chapter 1, in particular in Sections 1-4 and 8, a nonlinear theory of generalized functions has to face two major problems, namely: -
the conflict between insufficient smoothness, multiplication and differentiation, whose special case is the so called Schwartz impossibility result, and
E.E. Rosinger
-
the nonlinear s t a b i l i t y paradoxes.
In giving a solution t o the l a t t e r problem, we have been led - see Sections 5, 6, 9- 12 in Chapter 1 - t o quotient algebras of generalized functions, introduced i n (1.6.9)- (1.6.14), and t o the respective interplay between the s t a b i l i t y , generality and exactness properties of generalized solutions of nonlinear partial differential equations. In t h i s Chapter, a general and natural way i s given in order t o deal w i t h the constraints imposed by the Schwartz impossibility result and in the same time t o avoid the s t a b i l i t y paradoxes. As seen, t h i s i s obtained from a detailed analysis of the way partial derivatives and certain products involving the Dirac 6 distribution can be defined i n algebras of generalized functions. One should not be surprised in t h i s connection, since the derivative and a product involving the Dirac 6 distribution are the central elements involved in the Schwartz impossibility results i t s e l f . After a l l , 6 i s a highly nonsmooth element, and therefore, i t s multiplication and derivatives are likely t o provide aggravated instances of the conf 1 ict studied in Chapter 1. In view of the already classical role played by the Schwartz distributisons i n solving various 1inear or nonlinear partial differential equations, we shall require a level of generality not below the distributional one. I n other words, we shall require that the spaces - i n fact algebras - of geneof the Schwartz ralized functions we deal w i t h , contain the space ' distributions, see for instance (1.11.6). For the sake of simplicity, we shall only deal with the case when D = IRn and therefore, no explicit mention of the domain w i l l be made in the notation. The extension t o general domains D c IRn i s immediate. In order t o reformulate for the mentioned level of generality the basic algebraic clarification and solution of the s t a b i l i t y paradoxes - see i n particular (1.8.54 - (1.8.58) - we shall use a slight extension of the representat ion of istributions
d
which was given in (5 .A1.22). Suppose iven an arbitrary infinite index set A Further, suppose given I E 01. ft i s easy t o see that there exist vector subspaces
with linear surjections
such that, see (1.6.4)
Chains of differential algebras
and the mapping ( 6 . 1 . 3 ) i s an extension of, see ( 5 . 4 . 2 )
Denoting by
the kernel of the mapping ( 6 . 1 . 3 ) , we obtain the following vector space isomorph ism
hence the representation of distributions
which satisfy ( 6 . 1 . 2 ) - ( 6 . 1 . 8 ) The existence of V: and S; trated i n Example 1 a t the end of Sect ion 4.
i s illus-
Now, it i s obvious that a simple way for the embedding of 2' into quotient algebras of generalized functions would be by constructing inc lusion diagrams
w i t h A subalgebra i n
( c ' ) ~and
I
ideal in A,
such that
Indeed, owing t o ( 6 . 1 . 1 0 ) , an inclusion diagram ( 6 . 1 . 9 ) would give a mapping
which would be a linear embedding of
3' into the algebra A .
E. E. Rosinger
224
Unfortunately, inclusion diagrams such as in (6.1.9) , (6.1.10) cannot be constructed even when A = N and C = a, since in view of (1.8.57) we have
which contradicts (6.1.10). The alternative simple way l e f t is t o turn constructing inclusion diagrams
with A subalgebra in
P'
into a quotient algebra by
and I ideal in A,
such that
in which case we would obtain the linear injection of the algebra A = A / I on20 P', given by
Unfortunately again, inclusion dia rams such as in (6.1.13)- (6.1.15) cannot be constructed even in the case o f A = N and e = 0. The d i f f i c u l t y i s with condition (6.1.15) needed for the surjectivity of the linear injection (6.1.16). Indeed, t h i s follows from the next classical r e s u l t , see Rosinger [2, pp. 66, 115-1181. Lemma 1 Suppose given a sequence s of continuous functions on R, supp s, (6.1.17) Then
s E SO
shrinks t o 0 E R, and < s , - >= b
when u -, m
such that
Chains of d i f f e r e n t i a l algebras
225
In view of the above, we are obliged t o go t o a next level of more involved inclusion diagrams of the form
with A subalgebra in ( c ' ) ~ , spaces in (CC ) A , such that (6.1.20)
I
ideal in
A and V,
S
vector sub-
I ~ S = Y
The interest i n inclusion diagrams (6.1.19)- (6.1.22) i s in the following. First we note that (6.1.4)- (6.1.6) and (6.1.20) yield
hence, in view of the neutrix condition (1.6.11), we have, see (1.6.9)
--
Further, it follows easily that the mappings
c e
(6.1.25)
9' = SA/VA
e
s+VA -s+VIi som define a 1 inear embedding of functions A = ,411.
A = A/I
S/V
9'
I in, . i.n j.
t
s + I
into the quotient algebra of generalized
The role of the inclusion
in (6.1.19) i s obvious.
Indeed, in view of (6.1.25), it follows t h a t the
E.E. Rosinger multiplication in A = A / Z , when restricted to Ce , will coincide with the usual multiplication of Ce-smooth functions. A similar property concerning partial derivatives also follows, see Theorem 7 in Section 4. However, in view of results such as those in Propositions 2, 3 and 4 in Section 4, Chapter 1, it will be appropriate to consider the following type of inclusion diagrams, which are more general than those in (6.1.19), since they do not require condition (6.1.26) :
where
It is obvious that the 1 inear embedding (6.1.25) of V' into the quotient algebra of generalized functions A = ,411 will still hold for the above more general inclusion diagrams in (6.1.27)- (6.1.31). Remark 1 1.
The intermediate quotient space S/V in (6.1.25) obviously plays the role of a regvlarization of the representation of the distributions V' given in (6.1.8).
2.
It is important to note that the form of the inclusion diagram (6.1.27) is necessary in the following sense. Suppose that for a the inclusion quotient algebra A = A/Z E AL c ,A
(6.1.32)
V' c A
holds in the sense of the existence of a commutative diagram
Chains of d i f f e r e n t i a l algebras
s 3 s I
b
(6.1.33)
<s,.> E a' ]inj
s + l ~ A with a suitably chosen vector subspace S,
such t h a t
Then, i f we take (6.1.35)
V = I ~ S
we obtain an inclusion diagram (6.1.27)- (6.1.31)
The inclusion diagram (6.1.27)- (6.1.31) contains f o u r undetermined spaces, t h a t i s , V , S, I and A. I n view of (6.1.29), V can be obtained from I and S, thus only the three spaces S, I and A a r e a r b i t r a r y .
It i s p a r t i c u l a r l y important t o note however t h a t a s shown i n t h e sequel, the construction of such inclusion diagrams (6.1.27)- (6.1.31) can o f t e n be reduced t o t h e choice of the i d e a l s I alone, see a l s o Rosinger [1,2,3]. For convenience, we s h a l l only deal with t h e case when L = m, however, it i s obvious t h a t the theory i n t h i s Section remains v a l i d f o r a l l L E IA. In t h a t case (6.1.8) becomes
The general case of
L
E
We s t a r t with c e r t a i n
I is
presented i n d e t a i l i n Rosinger [2].
5 c 5c
(elA
a s given i n (6.1.2)- (6.1.6).
Further, we assume given a subalgebra A c (Cm)
A
,
such t h a t
conditions which a r e obviously s a t i s f i e d i n t h e p a r t i c u l a r case when A = (P)'.
E.E . Rosinger
228
The basic notion in the general theory of embeddings (6.1.25) is presented in : Definition 1 Given vector subspaces V and S in d call
fl
q
and an ideal Z in A, we
4
a regularization of the representation (6.2.1 of the P' distributions, if and only if (6.1.27)- (6.1.31) are satis ied. For convenience, the representation (6.2.1) being assumed given, it will no longer be mentioned when dealing with regularizations (6.2.4) . In case we also have satisfied the condition
then (V,S,Z,d) is called a
COO- smooth
regularization.
It is possible to introduced the following simp 1 if ical ion: Definition 2 An ideal Z in A is called regular, if and only if there exist vector such that (V,S,Z,A) is a regularisubspaces V and S in zation.
An3
The ideal Z is called Coo-smooth regular, if and only if (V,S,Z,A) is a COO- smooth regularization. The first basic result which is an extension of Theorem 4 in Rosinger [2, p. 751 and gives a useful structural characterization of regularizations and (?-smoothregularizations is the following. Theorem 1 Suppose Z is an ideal in in A fl such that
q
A and there exist vector subspaces S' and 7
Chains of differential algebras
Then, for every vector subspace V in 2 n (V,V
(6.2.9)
@ Sf@
$
we have that
l,Z,A) is a regularization
thus 2 is regular. Conversely, every regularization (V,S,I,A) has the form (6.2.6)- (6.2.9) . If in addition to (6.2.6)- (6.2.8), we also have
then (V,V @ S' @ 7,Z,d) is a P-smooth regularization
(6.2.11) thus Z is
Psmooth regular.
smooth re ularization Converse1 again, every Pin (6.2.6y- (6.2.8), (6.2.10), (6.2.117.
(V,S,Z,A) has the form
Proof Let us denote (6.2.12)
S=V@S'~T
and prove that (V,S,Z,A) is indeed a regularization First we note that (6.1.27) is obviously satisfied. Further, (6.1.30) and (6.1.31) follow easily from (6.2.8). Now we prove (6.1.29). For that we note that the above choice of S in (6.2.12) yields the inclusions
the last inclusion being implied by (6.2.7).
Now (6.2.8) yields
ZnScVel thus (6.1.29) follows from (6.2.6) and the inclusion V c 2.
E. E . Rosinger
230
Conversely, l e t us assume t h a t , (V,S,l,Ai i s a given regularization. us then take Sf = 0 and 7 a vector su space in S, such that (6.2.13)
Let
S = V @ l
Then obviously
the l a s t inclusion being implied by (6.1.30).
which gives the second equality in (6.2.6).
In t h i s way we obtain
Further we note that
Zn7cZnScV the l a s t inclusion resulting from (6.1.29).
Thus
Z n l c V n 7 c O which proves (6.2.6). We also have (6.2.8), since obviously
the l a s t inclusion being obtained from (6.1.31). Further, (6.1.31) yields
hence (6.2.13) gives
which proves (6.2.7)
.
Let us now assume that (6.2.10) also holds, then (6.2.11) w i l l obviously satisfy (6.1.19). Conversely, if we assume that (V,S,Z,A) i s a C"- smooth regularization, then we can again take S' = 0, while 7 w i l l be taken a s a vector subspace in S, such t h a t , see (6.2.13)
Chains of differential algebras
which is possible, since (6.1.19) gives
while in view of (6.1.21) and (6.1.23), we have
\
Now, we have 6.2.10) since by the above choice of (6.2.14), it fo lows that
Sf
and
1
in
Based on the above structural characterization of regularizations, we can identify a class of regular ideals Z given in the following: Definition 3 An ideal Z in A is called small, if and only if (6.2.15)
codim Z n
I$ < codim Z n I$
'5
In$
A useful pro erty of small ideals, which is an extension of Proposition 1 in Rosinger f2, p. 771 , is presented now. Pro~osition 1 Suppose the ideal Z in A n such that
7c A
5
(6.2.17)
is small. Then there exist vector subspaces
$ + (Ins)=
qeT
Proof Let us take
E=q,A = c , B = Z n $ in Lemma 2 below. Then taking
E. E. Rosinger
the proof i s completed Lemma 2 Suppose A and B are vector subspaces i n E and (6.2.18)
codim A n B 5 codim A B A
nB
Then there exist vector subspaces C i n E ,
n
n
(6.2.19)
A
(6.2.20)
A + B = A @ C
C = B
such that
C = 0 (the null subspace)
Proof Assume that (aili E I), (bjlj
E J)
are algebraic vector space bases in A and B respectively, such that
K, with
K = I n J , c k = a k = b k , for ~ i s an algebraic vector space base in A
E
K
n B.
In view of the hypothesis, there exists an injective mapping 0:
(J\K) +(I\K)
and then it follows easily that (ao(j)
+ b. 1 j J
E
J\K)
i s linear independent
We show that we can take C
as the vector subspace generared by the above family of linear independent vectors. Indeed if x E A n C, then
Chains of differential algebras
hence
which implies
thus x = 0. Now if x
E
B n C, then x = X A.b. = jEJ
+ b.)
" a j j€J\K
j
hence
implying that p.=O, J
j E J \ K
since a is injective. Thus again x that
Assume x
E
=
0. Finally, it suffices to show
B. Obviously
hence
therefore, indeed x = A
t
C
o
It can be seen that if E is finite-dimensional, the condition in the above inequality of codimensions is essential. Indeed, if E = A t B and dim A < dim B, then A @ C = E implies B fl C # 0, as otherwise we have the contradiction dim E 2 dim B
t
dim C = dim B
t
dim E
-
dim A > dim E
The basic property concerning the existence of regular ideals is given in:
E .E. Ros inger
Theorem 2 Every small ideal I in A i s regular. Proof Let us take ? given by Proposition 1. Then (6.2.6) and (6.2.7) are satisfied in view of (6.2.16) and (6.2.17). Furthermore, owing t o (6.2.3), we can choose vector subspaces S' in A n $ such that (6.2.8) will also hold. In t h i s way, Theorem 1 implies that
Z i s regular
o
Remark 2 1.
The existence of large classes of Z which are nontrivial, that i s pp. 81-88], in the case of A = I notation in (2.2.3), lnd(lRn) n regular ideal.
small and therefore reguZar ideals Z # 0, i s proved i n Rosinger [2, and .t E N. For instance, w i t h the n # i s a small and thus ((m(l ) ) ,
2.
The regular ideal mentioned i n pct. regular.
1. above i s also (m- smooth
$3. NEUTRIX CHARACTERIZATION OF REGULAR IDEALS As seen i n (1.6.11) and (6.1.23), the neutriz or off diagonality condition
i s a necessary condition for every ideal Z of an inclusion diagram 6.1.19) used in the construction of the quotient algebras of generalized unctions, see (6.1.24)
d
(6.3.2)
,
A = A/Z E AL C ,A
As a fundamental algebraic characteriza2ion of these algebras of generalized functions, we prove i n t h i s Section that, for a large class of such algebras, the neutrix condition does i n fact characterlze the regu lar ideals Z in the respective quotient algebras A = A/Z. Indeed, i n the case of the index set A = I and for
e
=
CO,
if we take
Chains of differential algebras
then the neutriz condition (6.3.1) will characterize the regular and C"'-smooth regular ideals 2, within a large class of so called co-final invariant ideals. It is easy to see that the neutrix characterization presented in Theorem 4 below, extends in the obvious way to all l E [A. This neutrix characterization of regular and COD-smooth regular ideals, first obtained in Rosinger [2], is presented here in its main features. For convenience, we shall use the framework in (6.1 .l) with l is, the distributions are given by the quotient representation
A useful class of ideals in specified in: Definition 4 An ideal 1 in
(~(fl))'
is called vanishing, if and only if
v (6.3.5)
wEI,pED(: 3 UEDI, U > ~ , ~ E R " : wu(x) = 0
With the definition in (2.2.4), it is obvious that (6.3.6)
Indn (~(l"))'
is a vanishing ideal
The basic property of vanishing ideals is presented now. Theorem 3 Every vanishing ideas1 2 is small and therefore regular. Proof In view of (6.2.15), we only have to show that (6.3.7)
codim Z n V" 5 codim Z n V"
2fw"
P'
For that, we note that obvious inequalities
= m,
that
E.E. Rosinger
codim I
n V" 5
dim I n
s*
< car I n s*
Ins* But
and car CO (iRn) = car IR Therefore codim I
n V" 5
(car
!RICar
'
= car !R
Ins* Now in order t o prove (6.3.7), it suffices t o show that (6.3.8)
codim I
n V"
2 car R
v" For t h a t , we define va E
p, with a
( v ) ( ) = a ,
u E
W,
E (0, I ) , X E
by
!Rn
Then it follows that (6.3.9)
(v,b E (091))
i s linear independent in V". Let us denote by V
the vector subspace in V" generated by (6.3.9). (6.3.10)
Z ~ V = O
Indeed, assume w
E
I
with suitable h E N ,
n V.
Then w E V
implies
A i E C and ai E ( 0 , l )
Then
Chains of differential algebras
But w
E
(6.3.13)
I, hence (6.3.5), (6.3.12) imply
C
i(ai)Y = 0
f o r infinitely many v E 91
l
II =
...
=
Ah
0
-
Indeed, assume h = 1. Then A1 = 0, h 2 2 . We can further aassume that O<@l
<
...
since
01 E
(0,l).
Assume now
and (6.3.15)
Dividing (6.3.13) by (6.3.16)
l _ ( i < h
Ai#O,
ah,
(@I/ah)'
f o r infinitely many v
E
we obtain +
IN.
-
7
.
+
Ah-l(ah-l/"h)"
+
Ih = O
Since
0 < ailah < 1 , 1
<
i 5 h-1
the relation (6.3.16) yields
which contradicts (6.3.15) and thus the proof of (6.3.14) i s completed. Obviously (6.3.11) and (6.3.14) yield (6.3.10). In view of (6.3.9), we have dim V = car R hence (6.3.10) implies (6.3.8) and the proof of (6.3.7) is completed. I t follows that I i s a small ideal in Theorem 2 i n Section 2, it i s regular.
(P(R"))',
hence in view of
A simple and useful characterization of vanishing ideals can be obtained within the following large class of ideals.
E.E. Rosinger
238
Definition 5 An ideal I in (p(Rn))'
is called eofinal invariant, if and only if
n I:, ' V w E (P(R 1) 3 wfEZ,pEO1: V VEIN, v > p : * W E 1 \ WV = w; I In view of (2.2.4), it is easy to see that (6.3.18)
Indr l (C?'(IRn)IN
is a cofinal invariant ideal
Pro~osition 2 A cofinal invariant ideal I is vanishing if and only if it is a proper ideal in (P(R n ) )01 . Proof Assume I is not vanishing. Then (6.3.5) yields w E I and p that
Let us define w'
(6.3.21) But (6.3.19), (6.3.22)
I , such
(P(R n) ) IN by
E
for v E I and x and (6.3.20) yield
E
E
Rn. Since Z was assumed cofinal invariant, (6.3.17)
w' E I (6.3.20) obviously imply llw' E (P(R
n 01
Now (6.3.21), (6.3.22) give
1)
Chains of differential algebras
which means that is obvious.
I cannot be a proper ideal in
(c(lRn))'.
The converse
The fundamental result given by the neulriz characterization of regular and Coo- smooth regular ideals which are cof inal invariant is a s follows. Theorem 4 For a eofinal invariant ideal ditions are equivalent :
I in
(6.3.23)
I is regular
(6.3.24)
I is f'-smooth regular
(P(lRn))'
the following three con-
in which case I w i l l also be a vanishing ideal. Proof We obviously have the implications (6.3.24) =, (6.3.23)
* (6.3.25)
with the l a s t implication resulting from (6.1.28) and (6.3.1). Therefore, we only have t o prove the implication (6.3.25) =, (6.3.24) Assume (6.3.25) holds, then obviously
1 i s a proper ideal in
n DI (P(R ) )
Hence, in view of Proposition 2, 1 is a vanishing ideal, therefore, according t o Theorem 3, I i s a small ideal. Now, Proposition 1 in Section 2 w i l l yield vector subspaces
Tcs" which s a t i s f y (6.2.16) and (6.2.17). Let us assume f o r the time being that satisfy
7
can be chosen so as also t o
E.E. Rosinger
If we take a vector subspace
such that
then we obtain (6.3.28)
( P e 9)
n
U' = 0
since (Per) nU'c ((Per) n U = (TnU
)nU'=O
) nUf =
e",N
c",N the l a t t e r two equalities being implied by (6.3.26) and (6.3.27). In view of (6.3.27), (6.3.28), we can choose vector subspaces S' c
s"
which w i l l s a t i s f y (6.2.8) and
Then, according t o Theorem 1 in Section 2, regular ideal.
1
i s indeed a Coo-smooth
Now a l l that is l e f t , is t o prove (6.3.26). For t h a t , we shall use Lemma 2 in Section 2 , as well as Lemma 3 below. us denote (6.3.30)
E=s",
A=P,B=II
and C = Z n s "
c",M
Then we have (6.3.31)
A n B = B n C = O
( t h e n u l l space)
Let
Chains of differential algebras
the l a t t e r equality being implied by (6.3.25). Lemma 3 below, we have
Our aim i s t o apply Lemma 2 t o A, B c E. the relation codim A
(6.3.33)
B
nB5
codim A n
K
24 1
Now, with the notation i n
To do so, we have f i r s t t o prove
B
Indeed, an a r ument similar t o that used in the proof of Theorem 3 in order t o establish $6.3.7), yields codim
E
K nE
< dim B <
car R
Now i n order t o obtain (6.3.33) it suffices t o show that codim A
(6.3.34)
A
nB
> car R
To do so, we shall again use the sequences of functions va E P, with a E (O,l), defined in the proof of Theorem 3, as well as the vector subspace V generated by them, see (6.3.9) , and prove the relation
Indeed, assume ii E B
-
n
V.
Then ii
E
V implies that
'
= lsish liVai
with h E IN,
Ai
E
R and ai E (0,1), hence
-
(6.3.36)
w u (n) =
l5lSh
But ii
E
B
~ ~ ( a u~ E )IN, , x E Rn
and (6.3.32) yield
E .E. Rosinger
with w E I n
f' and 3
E l?"(IRn).
The point is t h a t
Indeed, assume (6.3.38) i s f a l s e and t h a t non-void open, a r e such t h a t #(XI < -26,
x E
6
> 0 and
n'
E IRn,
with
n'
n1
Then i n view of (6.3.36) and (6.3.37) we have (6.3.39)
3 PEN; ,EN, v>p, x ~ n l :
v
Now define w' E ( P ( Rn ) ) IN by
Then obviously (6.3.40)
w' E I
since w E 1, and Z i s cofinal invariant. (6.3.41)
x
E P(f"),
and i n view of (6.3.39)
with
,
sup
define t E
Take then any
x c n1 (69(lR n ) )IN by
Then (6.3.40) gives
Therefore, i n view of (6.3.25), we have x E IRn
~ ( x )= 0, which i s absurd, since
x
can be chosen a r b i t r a r i l y within t h e condition
Chains of differential algebras
(6.3.41).
This completes the proof of (6.3.28).
Now (6.3.37) and (6.3.38) imply that
the l a t t e r e uality resulting from (6.3.10) i n the proof of Theorem 3, as well as the act that - as we have noticed - Z i s vanishing. This completes the proof of (6.3.35).
1
We can therefore conclude that (6.3.34) holds, since (val a E ( 0 , l ) ) i s an algebraic base i n V .
V c
PcA
and
Since (6.3.34) implies (6.3.33), we can finally apply Lemma 2 t o A and B given in (6.3.32) and obtain the existence of vector subspaces C i n E = 9,such that K n C = B n C = o and
Then (6.3.31) and Lemma 3 below impply the existence of vector subspaces D c E = P , such that
Taking, finally, (6.3.26)
7 = D,
the l a t t e r two relations in (6.3.42) w i l l yield
Lemma 3 If
A, B and C are vector subspaces in E and
A nB =B
n
C = 0 (the null space),
then the following two properties are equivalent
3 D c E vector subspace :
A n D = C n D = O A + C = A @ D
B n (A+C) = B n D
E.E. Rosinger
and
3 C c E vector subspace (6.3.44)
:
AnC=EnC=o A + E = K e C
where
Proof Assume (6.3.44) holds. Then (6.3.43) results by direct verification, if one takes D = (B n (A+C)) e C. Assuming (6.3.43) and taking any vector subspace C c E such that D = (BnD) e C, direct verification will yield (6.3.44) The following consequence will be particularly useful in Chapter 7 in the construction of chains of a1 ebras of generalized functions used in the resolution of singularities of solutions of nonlinear partial differential equations. Corollary 1
lndn (F(R"))' is a C"-smooth regular ideal in
n) ) # . (F(R
Proof It follows from (6.3.18), Theorem 4 and (2.2.4)
$4. THE UIILITY OF CUAINS OF ALGEBRAS OF GENERALIZED FUNCTIONS We come now to the high point of the theoretical ar ument in this Chapter, concerning the necessary structure for nonlinear t eories of generalized functions.
Rh
From the very beginning, in (1.1.5) it was already noted that in order to avoid certain aspects of the conflict between discontinuity, multiplication and differentiation, we may have to allow that the derivative operators act between two differenl algebras
Chains of differential algebras
That, of course, would imply the same for the nonlinear partial different i a l operators T(D) containing such derivative operators, see for instance (2.3.6). However, in Chapter 8, we shall see a more particular situation within the nonlinear theory given by the coupled calculus on Colombeau's differential algebra of generalized functions c(IRn). When dealing i n t h i s theory with linear or polynomial nonlinear partial differential equations
one i s i n fact dealing w i t h one single differential algebra G(Rn) which i s both the domain and the range of the respective partial differential operator, that i s
I n other words, a l l the algebraic and differential operat ions connected w i t h T(D) are performed i n the same, one single differential algebra G(@). That situation can obviously present advantages i n so f a r that it allows the maximum economy i n the number of spaces of generalized functions involved. However, it may as well present some disadvantages. Indeed, Colombeau's coupled calculus on the algebra G(Rn), although precludes the s t a b i l i t y paradoxes of type (1.8.1), nevertheless, it can exhibit them i n the milder forms, see Rosinger [3, pp. 197, 1991. And obviously, a reason for the presence of the mllder s t a b i l i t paradoxes i s the fact that in (6.4.3), one single space of generalized unctions i s involved. Indeed, as seen in (1.9. I ) , if (6.4.3) i s replaced w i t h the more general framework
2
involving two different spaces E and A of generalized functions, the need for a coupled calculus disappears, and so do the various forms of s t a b i l i t y paradoxes, see Section 12, Chapter 1.
I n addition, as seen i n Chapters 2 and 3, when solving linear or nonlinear partial differential equations, it i s often convenient t o use the general framework (6.4.4) w i t h two different spaces of generalized functions. A further disadvantage of a framework such as i n (6.4.3) which involves one single differential algebra may come from the limitations it can impose on the interplay between the s t a b i l i t y , generality and exactness properties of generalized solutions, see Sect ion 11, Chapter 1. Finally, we also have t o face the fact that a differential algebra, that i s an algebra w i t h arbitrary order differentiation, will by necessity have a peculiar multiplication. Indeed, as an example of that we have for instance the fact that none of the inclusions
E.E. Rosinger
is an inclusion of a1 ebras. More precisely, the multiplications in the two algebras in (6.4.57 are different, and we have an inclusion of algebras only in the case of
(6.4.6)
P(R~) c G(R")
The above difficulty in (6.4.5) is in fact the very reason for the need of the coupled calculus on i7(iRn). It is particularly important to note that the difficulty in (6.4.5) is to a large extent unavoidable, and it is not only a particular accident, specific to Colombeau's coupled calculus. Indeed, as seen in Sections 1-4 in Chapter 1, the difficulty in (6.4.5) is but one expression of the conflict between insufficient smoothness, multiplication and differentiation. The above then leads us to the idea of using more than one space of generalized functions, and possibly not both of them being differential algebras, that is, with arbitrary order differentiation. The hope is that in such a way we could avoid the above mentioned difficulties. Fortunately, as seen next, that hope can be achieved with the help of chains of algebras of generalized funelions
where Am is a differential algebra, while the algebras ,'A satisfy
with I E IN,
The arrows + in (6.4.7) are algebra with the notation in (1.9.11). homomorphisms, which have a number of convenient properties extending the classical situation of the chain of inclusions
The question as to what extent can inclusions of algebras such as in (6.4.5) be achieved within the chains of algebras (6.4.7) is not completely answered. A variety of partial positive ansers can be found in Rosinger [ 5 , pp. 88-104, 110-1121. See also Section 6 in the sequel. Given an m-th order linear or polynomial nonlinear partial differential operator T(D), it will be possible to consider it in the following frameworks
Chains of differential algebras
which is similar with (6.4.3), or more generally
which corresponds to (6.4.4). Concerning the multiplication of smooth functions, we have the inclusion of algebras
therefore, the algebras
C-smooth functions.
,'A
with
e E l, extend the multiplication of
Finally, concerning the multiplication of nonsmooth functions or distributions, we can have
for e E DI, [2, pp. 229, &by.
E
DIn,
(pl > l ,
if p
does not hold, see Rosinger
We proceed now to the construction of a large class of chains of algebras (6.4.7). Suppose given an arbitrary infinite index set A, together with vector spaces
satisfying (6.1.2)- (6.1.6), which means the commutativity of the diagram q3s
I
together with the relation (6.4.16)
5 = ker B
We shall also assume that
Finally, suppose given
0 lin, sur
TEP'(lRn)
E.E. Rosinger
(6.4.18)
n) ) A Z a p-smooth regular ideal in ( p(lR
In view of Example 1 at the end (6.4.14)- (6.4.18) can be satisfied. Our aim is to constract for each C (6.4.19)
E IN,
of a
this Section, conditions
COD- smooth regularization
(Vp Sp Zl, A[)
of
and then obtain the algebras in the chain (6.4.7) by the method given in (6.1.24), that is
We now proceed to construct the C"- smooth regularizations (6.4.19) from the given p-smooth regular ideal Z in (6.4.18). In view of (6.4.18) and Theorem 1 in Section 2, we can further assume given (6.4.22)
n DI (V,VG,Z,(C"(IR ) ) ) a
p-smooth regularization
of (6.4.20), such that
see (6.1.20), (6.2.10) and (6.2.11). Now, several auxiliary definitions and notat ions are first needed
A subset
X c (p(IRn))'
is called derivative invariant, if and only if
Obviously X = (?ln))' is derivative invariant. Also the intersection of any family o derivative invariant subsets is again derivative invariant. Given the vector subspaces V,S c
(P(R"))~and e
E IA, we denote
Chains of d i f f e r e n t i a l algebras
Further, we denote by (6.4.26)
A,(v,s>
the derivative invariant subalgebra generated by Ve + S in
n A ( P ( w )) .
Finally, we denote by (6.4.27)
ZL(v,s)
the ideal generated by Ve in Ae(V,S). Obviously, owing t o (6.4.23), Ae(V,S) has a unit element, therefore we have (6.4.28)
(Z,(V,S)
i s the vector space generated by Ve.Ae(V,S)
With t h e above notations, we s h a l l take (6.4.19) a s given by
f o r C E [A. Theorem 5 Given Z in (6.4.18) and V, S in (6.4.22), (6.4.23), then, with the construction in (6.4.29), it follows f o r 1 E IA, that (6.4.30)
(Ve,Se,Ze,Ae)
is a
P-smooth
regularization
E.E. Rosinger
Proof The inclusions in (6.1.19) are obvious. For the relation (6.1.20) we note that
Indeed, (6.4.25) and (6.4.22) yield
But Z is an ideal in p(R n) )h , hence (6.4.33) and (6.4.27) will indeed give (6.4.32), if we reca 1 the notation in (6.4.29).
I
Now obviously
Ien Se c I n (Ve @ S) c Ve the last inclusion being implied by (6.4.23). The relations (6.1.21), (6.1.22) are direct consequences of (6.4.22) In view of Theorem 5 , we indeed obtain quotient algebras of generalized funcl ions
In the rest of this Section, we shall present some of the more important properties of the algebras (6.4.34) needed in order to establish (6.4.7), Further details can be found in Rosinger 6.4.8) and (6.4.10)- (6.4.13).
Theorem 6 Given 1 in (6.4.18) and V, S in 6.4.22), (6.4.23), then, with the construction in (6.4.29), we have the fo lowing:
i
1 = Ae/Ze, with l (1) ' with unit element.
E
R, is an associative and commutative algebra
when restricted to p(Rn), (2) For I E 01, the multiplication in ,'A coincides with the usual multiplication of functions, that is, we have the inclusion of algebras
Chains of differential algebras
(6.4.35) (3)
F(R~)
c ~ l ,e
For h,k,l E IA,
h _< k
where ylk, ykh,
ye,
E
< t,
ri the following diagram is commutative
are algebra homomorphisms, defined as follows
and similarly for ykh, yehy injective, defined by (6.1.25).
while
te, ck,
ch
are linear
Proof It is useful to note that (6.4.25) yields
hence in view of (6.4.29) we obtasin
Now (6.4.26)- (6.4.29) yield
In this way (3) is immediate. (1)
follows easily
(2)
follows from (6.4.23)
In view of Theorem 6 above, the chain of algebras (6.4.7) obtains a detailed validation. It is easy to see that the algebra homomorphis
E .E. Rosinger
k (6.4.41) : [A -.A , I,k is anjective, if and only if
k
N, k
''
The basic result, presented next, concerns the way partial derivatives and therefore, linear or polynomial nonlinear partial differential operators can be defined on the quotient algebras of generalized functions ,'A with e E A. Theorem 7 With the assumptions in Theorem 6, the following hold: (I)
'A ;Ak,
I,k,m
E
N, k+n
-
<
l
(see (1.9.11))
(2) The partial derivatives are the linear mappings (6.4.43)
DP
: 'A
Ak, t,k L A, p
E
Mn, k + /pl 5 !
defined by
and when restricted to P(lRn), derivatives of functions.
they coincide with the usual partial
(3) The partial derivaties (6.4.43) satisfy the Leibnitz rule of product derivative, that is
where
and I,k E A, p
E
INn, k + Ipl
< 1.
(4) For l,l',k,kf E N , I 5 , k'< k, pEINn, k' + / p ( < l' , the following diagram is commutative
k + (pl < e,
Chains of d i f f e r e n t i a l algebras
Proof In view (6.4.25), it is easy t o see that we have (6.4.47)
gPvC c
vk,
l , k E R,
p E IN", k +
lpl
j
e
But (6.4.26) and (6.4.38) yield
hence, in view of (6.4.29), we obtain
therefore (6.4.47) and (6.4.48) yield
Now (1) follows from (6.4.48), (6.4.49) and (1.9.11).
I n view of (6.4.48), (6.4.49), it i s obvious that (6.4.43), (6.4.44) is a correct definition. The inclusion
in (6.4.23) and (6.4.29) complete the proof of (2) .
(3) and (4) follow from (6.4.44) by direct verification The result in pct. (2) in Theorem 7 above can be improved in suitable circumstances a s specified next. Suppose given (6.4.50)
A
a derivative invariant subalgebra in ( P ( R " ) ) ~ s a t i s f i e s (6.2.2), (6.2.3), and suppose t h a t
which
E.E . Rosinger
(6.4.51)
Z is a derivative invariant, C?- smooth regular ideal in A
Then in view of Theorem 1 in Section 2 , we can assume given (6.4.52)
(V,V
@
S , 1, A)
a C@'-smooth
regularization
of ( 6 . 4 . 2 0 ) , such that ( 6 . 4 . 2 3 ) holds. It is easy to see that the constructions in ( 6 . 4 . 2 5 ) - ( 6 . 4 . 2 9 ) remain valid and we shall have
Furthermore, Theorems 5 , 6 and 7 remain also valid. Let us now assume that (6.4.54)
V is derivative invariant
Then obviously
hence
which means that
It follows from ( 6 . 4 . 2 9 ) that
In this way thechain of algebras in ( 6 . 4 . 7 ) will collapse into the one sin le differential algebra in the chain, which is Am. Indeed, ( 6 . 4 . 3 4 ) yie ds
I
Then ( 6 . 4 . 3 6 ) becomes the 1 inear injective mapping
Chains of d i f f e r e n t i a l algebras
and (6.4.43), (6.4.44) and (6.4.46) become (6.4.61)
DP : Am + Am,
p E INn
with
Finally, in addition t o (6.4.14)- (6.4.17), l e t us assume t h a t the representation of distributions in (6.4.20), that i s
is such t h a t , given any distribution T
E
P'(IRn) and its representation
then the distributional p a r t i a l derivatives w i l l have the representation
D ~ TE a' (IRn) , with
p
E
INn,
We note that in view of (6.4.17), the relation (6.4.65) is well defined. Theorem 8 If in addition t o (6.4.50)-(6.4.52), (6.4.54) and (6.4.63)- (6.4.65) we also have (6.4.66)
V e S is derivative invariant
then t h e part iaZ derivatives
defined in (6.4.62), coincide with the distributional p a r t i a l derivatives, when restricted t o P' (Rn) , according t o the embedding (6.4.60)
E .E . Rosinger
Proof We r e c a l l t h a t i n view of (6.1.25),
em
i n (6.4.60)
is d e f i n e d by
t h a t is
Let u s t a k e now T
E
'D'(IRn),
Thus i n view of (6.4.65),
t h e n i n view of (6.4.68),
we o b t a i n
we o b t a i n
s i n c e according t o (6.4.58)
But (6.4.66) a p p l i e d t o (6.4.70) y i e l d s
F i n a l l y , (6.4.69) the relation
a p p l i e d t o (6.4.70)- (6.4.72)
g i v e s i n view of (6.4.42)
which completes t h e proof Example 1 If
c a r A = c a r IN,
t h e n with t h e n a t a t i o n i n (6.6.1),
and = with ( E M , obviously b e s a t i s f i e d .
we can t a k e V eA = V'
and t h e c o n d i t i o n s (6.1.2)-(6.1.6)
will
Chains of d i f f e r e n t i a l algebras
If
car A > car W ,
we take
o : W +A
(6.4.74)
injective
and define the i n j e c t i v e a l g e b r a homomorphism
sn i f
I
= o(n)
if
I
E A\o(N)
(us)A =
(6.4.76)
0
f o r some n E W
then obviously
Now we define
a s well a s the l i n e a r s v r j e c t i o n
Sl ~ S H T E P I ( R ~ )
(6.4.79) where, f o r (6.4.80)
s = o se + u ( $ ) ,
s e ES1,
$ E Ce( Rn )
we have
with
Tg given i n (5.4.2) and
v'(Rn).
T~
being the weak limit of
se
in
In view of (6.4.78), the definition (6.4.79) is coorrect.
By taking a s the kernel of the mapping (6.4.79), it i s obvious that I e VA and Sh s a t i s f y (6.1.2)-(6.1.6). In view of (6.4.78), it i s easy t o see that (6.4.82)
DPS; c
s;,
I ,
E
,
E
mn,
k
+
/PI 5
e
E.E. Rosinger
in particular (6.4.83)
gP$'c$',
p EM"
Similarly we obtain (6.4.84)
k l,k gPvAe c VA,
E
!I, p
€
Wn, k
+
IPI
I
e
When considering regularizations of (6.4.86)
P'
e e (P) SA/VA
with arbitrary I
E
!I, the following construction may be useful.
Exam~le2 Suppose car A > car W and we are given
satisfying (6.1.2)-(6.1.6), that is, we have the commutative diagram
and (6.4.89)
<
= ker q
In view of Example 1, we can also assume that relations (6.4.83), (6.4.85) Let us take Ao (6.4.90)
E
A and
r :A
+
A\{Ao}
and define the injective a l g e b r a homomorphism
,
satisfy the
Chains of d i f f e r e n t i a l algebras
s ( r ~ =) ~ 0
(6.4.92)
if
A = r(p) f o r some p
if
A =
A0
Then obviously
We can now take
and define the l i n e a r s u r j e c t i o n d 33ss T E E'(lRn)
(6.4.95)
in the following way: (6.4.96)
if
s = rsm+u($),
s m € q ,$E~"(R")
then
Finally, we take
V l = ker
(6.4.98) Obviously,
Vi
8
and l$will s a t i s f y (6.1.2)- (6.1.6)
E A
E. E . Rosinger
260
$5.
NONLINEAR PARTIAL DIFFERENTIAL OPERATORS IN CHAINS OF ALGEBRAS
Suppose given the polynomial nonlinear p a r t i a l d i f f e r n t i a l operator
where h,ki E H, p. . E iNn l? T(D) is by definition
and
ci E P ( R n ) .
provided t h a t none of the
ci,
with
[R"
1
We recall that the order of
h,
vanishes identically on
.
If we assume the conditions in Theorem 6, Section 4, then it follows easily that T(D) w i l l operate according t o
see pct . (1) and (2) in Theorem 6 , Section 4 , as well as Section 9, Chapter 1. In the next Chapter, we shall be interested in the following type of situation: We are given the polynomial nonlinear p a r t i a l d i f f e r e n t i a l equation T(D)U(x) = f (x) , x E Rn
(6.5.4) where f
E
P(Rn).
We can construct a sequence of smooth functions
such t h a t , see (6.4.29)
hence
Moreover, in the sense of (6.5.3), we have
Chains of d i f f e r e n t i a l algebras
In such a case, the sequence of smooth functions called a chain generalized solution of (6.5.4).
s
i n (6.5.5) w i l l be
In the particular case when A = IN and (6.5.9)
s =
($,I,
E
IN)
E
SdD
s w i l l be called a chain weak solution of (6.5.4). This l a t t e r case w i l l be the one encountered in the next Chapter. In view of Section 13, Chapter 1 , it is easy t o see t h a t the chains of algebras (6.4.7) can be adapted in such a way that (6.5.3) s t i l l holds f o r general nonlinear p a r t i a l d i f f e r e n t i a l operators. Similar adaptations can be made in order t o accommodate systems of nonlinear p a r t i a l d i f f e r e n t i a l equations. Details can be found i n Rosinger [2] .
96.
LIMITATIONS ON TIIE EMBEDDING OF SMOOTU FUNCTIONS INTO CDAINS OF ALGEBRAS
We have seen in (6.4.6) and pct . (2) in Theorems 6 and 7 i n Section 4, that within the chains of algebras containing the distributions, see (6.4.7)
we can have inclusions
which preserve both the algebra and differential structure of C"(iRn) . In other words, the multiplication and p a r t i a l derivatives in AP, when res t r i c t e d t o P(IRn), will coincide with the usual multiplication and part i a l derivatives of P- smooth functions. For short, we s h a l l say t h a t the inclusions (6.6.2) preserve the algebra and d i f f e r e n t i a l structures, o r that the spaces involved in the inclusions have compatible algebra and d i f f e r e n t i a l structures. There e x i s t s a natural interest in extending the l e f t hand term in the inclusions (6.6.2) in such a way that t h e algebra and/or d i f f e r e n t i a l structures involved w i l l s t i l l be compatible. Therefore, we a r e led t o the problem of finding the mazimal extensions of such kind. F i r s t we note that we may have t o consider separtely t h e algebra and diff erent i a l structures involved.
262
E.E. Rosinger
Indeed, concerning differentiation, the natural candidate i s the space of the Schwartz distributions which i s closed under the differentiation of distributions and i s contained i n t h e algebras of the chains (6.6.1), t h a t i s , we have
However, as 2" ( p ) does not have a natural algebra structure, the inclusion (6.6.3) can only be considered from the point of view of the preservation of the d i f f e r e n t i a l structure. A positive answer i s obtained in (6.4.67), and i n particular, i n the case of the inclusion, see Chapter 8
However t h i s happens when the chain of algebras (6.6.1) i s collapsed into one single d i f f e r e n t i a l algebra, see (6.4.59) and Chapter 8. Therefore, the question a r i s e s whether inclusions (6.6.3 with the preservation of the d i f f e r e n t i a l structures can be obtained w i t out collapsing the chain of algebras (6.6.1). This question is so f a r open.
h
Concerning the algebra structure, there are two natural candidates
since we obviously have the inclusions
and lioC(Rn) i s not an algebra.
However, neither C0(P) nor qo,(lRn)
have a natural differentiation. Nevertheless, C O ( p ) i s connected t o the usual differentiation of smooth functions through
that i s , CO(tRn) i s t h e range space of the usual p a r t i a l derivatives of smooth functions. In view of the above, we shall concentrate on inclusions
and t r y t o find out t o what extent they can preserve both the algebra and differential structures involved.
Chains of differential algebras
The main answer w i l l be that it i s not possible t o obtain inclusions
which would preserve b o t h the algebra and differential structure. As seen below, t h i s limitation is of such a simple and general nature that i t s validity oes beyond the framework of the chains of algebras (6.6.1). In other wor s , t h i s limitation i s not specific t o these chains, as it does not a r i s e from the way such chains are constructed in Section 4.
f
The result in (6.6.10) follows from an extension of Proposition 2 i n Section 4, Chapter 1 , an extension which goes along similar lines w i t h the extension of the Schwartz impossiblity result presented in Rosinger [3, p. 311 . Indeed, suppose given the algebras and algebra homomorphisms (6.6.11)
A4 -+ A3 + A2
4
A1
-4
A0
together w i t h the derivatives, see (1.1.15)
such that A3, A 2 , A ' , A0
are commutative
(6.6.13) and A'
, A0 are associative
and the following diagrams are commutative
Suppose that
and 1 i s the u n i t element i n l i , w i t h 1 < i < 3, while 1 , x, x,, are invariants of the respective algebra homomorphisms in (6.6.11) .
x-
E. E. Rosinger
Further, suppose that (6.6.17)
x + + x - = x in A , with 2 s i s 3
(6.6.19)
D x = 1 in A ~ , with
(6.6.20)
D(x+)~= 2.x+, D ( x - ) ~= 2.x-
1( i s 3 in A3
We note that
and the algebraic operations in (6.6.17), (6.6.18) coincide with those in C"(R), while the derivatives in (6.6.19), (6.6.20) coincide with those in c1(R)
.
We can also recall that
in the sense of the distributional derivaties in P'(R). Prooosition 3 Within the conditions (6.6.11)- (6.6.20), it follows that
Proof For convenience, l e t us again denote (6.6.24)
a = x,, b = x-
thus (6.6.18) yields (6.6.25)
x . a = a 2 , x . b = b 2 in A4
which by derivation gives
Chains of d i f f e r e n t i a l algebras
if we take into account (6.6.16) , (6.6.19) and (6.6.20) x.Da = a , x.Db = b in A3 (6.6.26)
I n view of (6.6.19), (6.6.16),
, hence
a derivation of (6.6.26) gives
Da + x.D2a = Day Db + x.D2b = Db
in A2
hence (6.6.27)
x-D2a = x.D2b = 0
in A2
Now the derivation of (6.6.25) also yields
owing t o (6.6.13), obtain (6.6.28)
(6.6.16) and (6.6.19).
a.Da = a ,
Hence, in view of (6.6.26), we
b-Db = b in A3
In view of (6.6.17), (6.6.26) and (6.6.28), we have
b.Da = (x-a).Da = x.Da - a-Da = 0 i n A3 that is (6.6.29)
a.Db = b.Da = 0 in A3
which by derivation and in view of (6.6.13), (6.6.16) yields (6.6.30)
DasDb + a.D2b = DaeDb + b.D2a = 0 in A2
In view of (6.6.17) and (6.6.27) we have (6.6.31)
a.D2b = -b.D2b, b.D2a = -a.D2a in A2
Then (6.6.17) and (6.6.19) give Da + Db = 1 in A2 hence (6.6.16) yields
E.E. Rosinger
266
which together with (6.6.30), (6.6.31) give
We show that
with the a1 ebra homomorphism A2 -+ A1 in (6.6.11). derivation o? (6.6.29) gives through (6.6.13) the relations
Indeed, the
and then, by one more derivation, we obtain
Now we note that applying t o in (6.6.11), and in view of
Multiplying (6.6.34) by we obtain
homomorphism we obtain
A2
-+
A'
x and taking into account (6.6.13) and (6.6.35),
On the other hand, the derivation of (6.6.27) gives through (6.6.19) the relations
which multiplied by b and (6.6.13) the relations
a
respectively, give in view of (6.6.36) and
Now again, i f we apply t o (6.6.32) the algebra homomor hism A2 -+ A' (6.6.11) and take into account (6.6.14)- (6.6.16), then f6.6.37) implies
in
Chains of differential algebras
hence i n particular (6.6.33)
.
From (6.6.38) it follows that (Da)2 = Da,
(Db)2 = Db i n A'
hence (6.6.13) yields for p E DI, (6.6.39)
( ~ a =) Da, ~
>1
p
( ~ b =) Db ~ i n A'
One more derivation applied to relations
(6.6.39) gives through
(6.6.13)
the
p ( ~ a ) p - 1 . ~=2 aD2a, p ( ~ b ) p - 1 . ~ =2 bD2b in A0 for p
E
N,
p
>
2,
thus i n view of (6.6.39) we obtain
hence
I t follows that for p,q E N ,
p,q
> 2,
which completes the proof of (6.6.23)
we have
o
Now the result mentioned i n (6.6.10) can be specified w i t h i n the following framework, which i s more general than that of the chains of algebras constructed i n Section 4. For simplicity, we shall only formulate it i n the one dimensional case. Suppose given the commutative and associative algebras and algebra homomorph isms (6.6.40)
A ~ + ' -+
A$
e
,M
and derivatives, see (1.1.15)
with the following commutative diagrams
E. E . Ros inger
The algebras (6.6.40)- (6.6.42) a r e called a d i f f e r e n t i a l c h a i n o f a l g e b r a s . Obviously, the chains of algebras constructed in Section 4 a r e n-dimensional versions of d i f f e r e n t i a l chains of algebras, when considered f o r l E N only. Given !E #,
k E
a,
l,k
>
1, we say t h a t t h e inclusion
p r e s e r v e s t h e a l g e b r a and d i f f e r e n t i a l s t r u c t u r e s , i f and only if f o r h,p E IN, p 5 h < !, we have the commutative diagrams
'A
A'i s defined in t h e obvious way with r = max{k,h), where A are algebra by (6.6.40)- (6.6.42), while c'(R) c A' and c'-~(N) c embeddings with t h e constant function 1 being t h e unit element in t h e
-
DP respective algebras, f i n a l l y , er(lR) order derivative of cr- smooth functions.
c
~
(
) i s t h e usual p-th
Theorem 9 Within d i f f e r e n t i a l chains of algebras (6.6.40)- (6.6.42), one cannot have f o r any C E IN, e 4, an inclusion
>
which preserves the algebra and d i f f e r e n t i a l structures, u n l e s s
Further, one cannot have f o r any
e
E IN,
!. >
2,
an inclusion
Chains of d i f f e r e n t i a l algebras
which preserves the algebra and d i f f e r e n t i a l structures, unless f o r a E A0 we have
Proof
It follows easily from Proposition 3 above and Theorem 2 in Rosinger [3, p. 311 Remark 3 In view of (6.6.22), which by i t s simplicity and important role would be desirable in a nonlinear theory of eneral ized functions, the relation (6.6.46) does not seem t o be desirab e. Indeed, i f we have any vector space embedding
f
(6.6.49)
P' (R)
c A0
then (6.6.50)
6€Pf(R), ~ # O E T ' ( W )
w i l l necessarily yield
Therefore, (6.6.22) and (6.6.46) can only mean that there i s a rather sharp difference between the distributional derivatives and the derivatives (6.6.41), (6.6.42) in the d i f f e r e n t i a l chain of algebras t o which A0 belongs. This i s precisely the meaning of the impossiblity in (6.6.10) or (6.6.45) The implication in (6.6.48), which i s an extension of t h e Schwartz imposs i b i l i t y result (1.2.11), seems t o be of a lesser concern. Indeed, just as in (6.6.49)- (6.6.51), it cannot mean that
and it can only mean that in A O , the singularity of higher than that of l / x , see (6.4.13).
6
at
x = 0
is
For further d e t a i l s see Sections 2 and 4 in Chapter 1 , a s well a s Rosinger [2, pp. 88-104, 100-1121.
This Page Intentionally Left Blank
RESOLWION OF SINGULARITIES OF WE& SOLUTIONS FOR POLYNOYIIAL NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS $1.
INTRODUCTION
In t h i s Chapter the general method developed in Chapter 2 will be applied t o the resolution of closed, nowhere dense singularit ies of sequential solutions for polynomial nonlinear partial differential equations, w i t h i n the framework of the chains of algebras of generalized functions constructed in Chapter 6, Section 4. We shall deal w i t h the important particular case of sequential solutions given by usual weak solut ions. The interest in these kind of solutions comes from a wide range of applications, see for instance Sections 3-8 i n the sequel. I t should be mentioned that the results i n t h i s Chapter can easily be extended t o larger classes of sequential solutions, as well as to general nonlinear partial differential equations and systems. An illustration of that was given in Chapter 2, w i t h the global version of the CauchyKovalevskaia theorem. Another illustration which presents a strengthening of that global result can be seen i n Section 9 i n t h i s Chapter. Futther details concerning possible extensions of the classes of weak solutions and nonlinear partial differential equations dealt w i t h i n t h i s Chapter can be found i n Rosinger [2, pp. 121- 1621. Stated simply, the resolution of singularities means that the weak solutions considered w i l l satisfy the respective polynomial nonlinear partial differential equations i n the usual algebraic sense, that i s , w i t h the ~ ~ u iplication lt and partial derivatives as they are defined within the chains of algebras of generalized functions i n Chapter 6, Section 4. In particular, the stability paradoxes will be avoided in t h i s way. One of the effects of the above i s that for a l l practical purposes, the weak solutions considered w i l l behave as global and classical solutions of the I n this respective polynomial nonlinear partial differential equations. way, the results i n t h i s Chapter can be seen as originating a Polynomial Nonlinear Operational Calculus for the respective types of nonlinear partial differential equations Concerning ueak solutions of nonconservaiive nonlinear partial differential equations, see Sections 5-8, it was for the first time i n the literaature that i n Rosinger [2] a rigorous treatment was given which, among others, eliminated the possibility of stabil ity paradoxes. This shows the interest i n the mentioned Polynomial Nonlinear Operational Calculus. Since we only consider weak solutions, t h e i r singularities will have t o be concentrated on closed nowhere dense subsets w i t h zero Lebesque measure. Yet, as seen with the mentioned global version of the Cauchy-Kovalevskaia theorem, the second condition above, that i s , of zero Lebesque measure, i s only of convenience and not of necessity. The fact however i s that the weak
E.E. Rosinger
272
s o l u t i o n s considered i n t h i s Chapter are s u f f i c i e n t l y g e n e r a l i n o r d e r t o i n c l u d e as r a t h e r simple p a r t i c u l a r c a s e s many of t h e known t y p e s of s i n u l a r i t i e s of s o l u t i o n s of f i r s t and second o r d e r n o n l i n e a r p a r t i a l d i f e r e n t ial e q u a t i o n s i n a p p l i c a t i o n s .
f
There is however no l i m i t a t i o n on t h e o r d e r of polynomial n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s which can be d e a l t with. Indeed, i n S e c t i o n s 5-8 i n t h e s e q u e l , junctions condit ions a r e found f o r r a t h e r g e n e r a l c l a s s e s of polynomial n o n l i n e a r p a r t i a l d i f f e r e n t i a l equat i o n s and t h e i r weak s o l u t i o n s . These j u n c t i o n c o n d i t i o n s a r e r a t h e r wide r a n i n g nonconservative e x t e n s i o n s o f t h e c l a s s i c a l Rankine-Hu o n i o t shock c o n f i t l o n s and t h e y determine t h e c l a s s of resoluble systems o polynomial n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s , which c o n t a i n s as p a r t i c u l a r c a s a e s many of t h e e q u a t i o n s of p h y s i c s , such as t h o s e of f l u i d dynamics, g e n e r a l r e l a t i v i t y and magnetohydrodynamics.
f
For s i m p l i c i t y , we s h a l l d e a l with t h e c a s e when t h e domain of t h e independent v a r i a b l e s is Rn. The g e n e r a l c a s e of a domain g i v e n by an a r b i t r a r y open s e t 0 c Rn does not involve a d d i t i o n a l complications. I n view of t h a t , we s h a l l f r e e l y switch t o t h e c a s e of a r b i t r a r y domains, whenever needed. F i n a l l y , s i n c e we o n l y d e a l with weak s o l u t i o n s , we s h a l l u s e t h e index s e t A = # when c o n s t r u c t i n g v a r i o u s c h a i n s of a l g e b r a s a c c o r d i n g t o t h e g e n e r a l method i n Chapter 6 , S e c t i o n 4. The Chapter ends with a strengthened form of t h e global Cauchy- Kovalevskaia theorem i n Chapter 2 , which shows t h e e x i s t e n c e o f global chain generalized solutions. That global existence r e s u l t is a l s o a first i n t h e l i t e r a t u r e , s e e Rosinger [3] . I n t h i s Chapter we s h a l l p r e s e n t t h e main concepts and r e s u l t s only. For t h e p r o o f s and f u r t h e r d e t a i l s one can c o n s u l t Rosinger [2, pp. 121-1621 and Rosinger [3, pp. 349- 3901 .
$2.
SIMPLE POLYNOHIAL NONLINEAR PDEs AND RESOLUTION OF SINGULARITIES
An m-th o r d e r polynomial n o n l i n e a r p a r t i a l d i f f e r e n t i a l o p e r a t o r is c a l l e d simple, i f and only i f it can be w r i t t e n i n t h e form
where
Li(D)
a r e m-th o r d e r l i n e a r p a r t i a l d i f f e r e n t i a l o p e r a t o r s with
Cm- smooth c o e f f i c i e n t s , while Ti
a r e polynomials of t h e form
Resolution of singularities
with cij
E
cm(!Rn)
In the present Section and the next two, we shall deal with nonlinear partial differential equations corresponding to (7.2. I), (7.2.2), that is, having the form (7.2.3) where f
T(D) U(x) E
cm(IRn)
=
f (x) , x c IRn
is given.
The nonlinear hyperbolic conservation laws, as well as the nonlinear second order wave equations studied in Sections 3 and 4, are obviously of the above form (7.2.3) . In general, the following large class of quasilinear partial differential operators
where cD E Cm(Rn) and T'(D) is an (m- 1)-th order simple polynomial nonlinear partial differential operator, are obviously of the above form (7-2.11, (7.2.2).
A function
is called a piece loise Cm-smooth ureak. solution of the simple polynomial nonlinear partial differential eqauation in (7.2.3), if and only if the following five conditions are satisfied: There exists a family G of Coo-smooth mappings 7 the set (7.2.5)
r
= {X E
~l
3 7
E
G : Y(X) =
is closed, has zero Lebesque measure
-
oE
:
IRn
g
IR
-4
y,
such that
~$7)
therefore it is nowhere dense
-
and
E. E. Ros inger
Further
where b = max{bi 11 5 i 5 a ) , with n o t a t i o n i n (7.2.2), following weak s o l u t i o n property holds:
*
where Li(D) Finally,
for
is t h e formal a d j o i n t of each
7 c 6,
B
(7.2.9)
{f '(B?) 17
7
of
E
Li(D).
there
0 E IRg7,
neighbourhood
and moreover, t h e
exists
a bounded
and
balanced
such t h a t
C) is l o c a l l y f i n i t e i n IRn
Solutions of important c l a s s e s of nonlinear hyperbolic conservation laws, a s well a s nonlinear second-order wave equations a r e known t o be p i e c e wise smooth weak s o l u t i o n s i n t h e above sense, s e e Sections 3 and 4.
A f i r s t r e s u l t is presented next on t h e r e s o l u t i o n o f s i n g u l a r i t i e s o f weak s o 1 u t i o n s f o r nonlinear p a r t i a l d i f f e r e n t i a l equations. The proof can be found i n Rosinger [3, pp . 353- 3601 . Theorem 1 Suppose U : IRn + IR is a p i e c e w i s e COD-smooth weak s o l u t i o n of t h e m-th order simple polynomial nonlinear p a r t i a l d i f f e r e n t i a l equation (7.2.3) . Then it is p o s s i b l e t o construct r e g u l a r i z a t i o n s (6.4.22) and a l g e b r a s (6.4.34) such t h a t ( 1 U = s + II(V,S) E A e , I (2)
E
IN,
where
s
E
S does not depend on I
U s a t i s f i e s (7.2.3) i n t h e usual a l g e b r a i c sense, with m u l t i p l i c a t i o n i n Ak and t h e p a r t i a l d e d r i v a t i v e o p e r a t o r s DP : + ilk, p s ln, ipl 5 m with ~ , rk R, k+m 5 e
In view of p c t . ( I ) , s i s a c h a i n weak s o l u t i o n of (7.2.3), s e e Section 5 , Chapter 6.
Resolution of singularit ies
Remark 1 Theorem 1 above remains valid f o r arbitrary open Q c IRn.
53.
RESOLUTION OF SINGULARITIES OF NONLINEAR SHOCK WAVES
Suppose given the nonlinear hyperbolic conservation law (7.3.1)
Ut(t,x) + c(U(t,x)).Ux(t,x) = 0,
t > 0, x
E
IR
with the i n i t i a l condition
We s h a l l suppose that the function
in (7.3.1) i s an arbitrary polynomial. Then it i s obvious that (7.3.1) i s a f irsl- order simple polynomial nonlinear p a r t i a l d i f f e r e n t i a l equation on fl = (0,m) x R c R 2 . Indeed, the l e f t hand term in (7.3.1) can be written in the form in (7.2. I ) , provided that we take a = 2, Ll(D) = D t , L2(D) = Dx,
T1U
= U
and
T2U =
b(U)
where
i s a primitive of the function in (7.3.3), and thus again a polynomial.
It is known that under rather general conditions, Schaeffer , Golubitsky & Schaeffer, f o r Coo-smooth or piece wise smooth i n i t i a l data u, the equation (7.3.1) has shock wave solutions U : Q + IR, with the following properties . There exists a finite set G of Coo-smooth functions '7 : Q COO- smooth curves (7.3.5)
r '7 =
{X
E Q(?(x) = 0)
which describe the propagation of the shocks, and such that (7.3.6)
U
E
P(Q\I'), where
l' = U l' ~ E G Y'
-+
IR,
defining
E.E. Rosinger
276
U is locally bounded on S2
(7.3.7)
Obviously, such a solution U will be a p i e c e w i s e (?'-smooth weak s o l u 2 i o n of the partial differential equation in (7.3.1 , in the sense of the definition in Section 2. Therefore Theorem 1 in ection 2, will yield the following result.
4
Theorem 2 Suppose U : (0,m) x IR -+ W is a shock wave solution of the nonlinear hyperbolic conservation law in (7.3.1) and that it satisfies the conditions 7.3.8). Then it is possible to construct Cm-smooth regulari6.4.22) and algebras (6.4.34) , such that (1)
u = s + Ie(v,s)
E ,'A
e ,I
where s E S does n o t depend on
e
(2) U satisfies (7.3.1) in the u s u a l a l g e b r a i c s e n s e , with multiplications in Ak and the partial derivative operators k Dt,Dx : -+ A , with l,k E R, kt1 5 e
'A
In view of pct. (I), 5, Chapter 6.
s is a c h a i n weak s o l u t i o n of (7.3. I ) , see Section
$4. RESOLUTION OF SINGULARITIES OF KLEIN-GORDON TYPE NONLINEAR WAVES Suppose given the Klein-Gordon type nonlinear wave equation
with the initial conditions
where T(D) is a first-order Cm-smooth simple polynomial nonlinear partial differential operator. Then it is obvious that (7.4.1) is a second o r d e r (?'-smooth s i m p l e p o l p o m i a l n o n l i n e a r partial differential equation on n = (0,m) x W c W2, since it has the form in (7.2.4).
Resolution of s i n g u l a r i t i e s
277
the 6 hJ, ReedIR,& Berning, with P I c Q
It is known t h a t under general conditions, Reed 2 , 3 : equation (7.4.1) has l o c a l o r global solutions open, which have t h e following properties. There e x i s t a f i n i t e number of points light cones with t h e boundaries given by
r; r with
1
a,
= { ( t , ~ )E
{(t,) E
R I ~ X-
x
xa + t =
,. . . ,xu
E
R,
which o r i g i n a t e
01
- x - t -0)
such t h a t
U E Cm(Pi\r) , where (7.4.5)
XI
-+
r
u
=
( r i u):?J
1<1~$7
U is l o c a l l y bounded on QJ
where it has been assumed t h a t T(D) i n (7.4.1) has t h e form (7.2. I ) , and L;(D) is t h e formal adjoint of Li(D). Obviously, such a solution U w i l l be a piece lvise Cm-smooth weak solution of (7.4.1), in t h e sense of t h e d e f i n i t i o n in Section 2. Therefore Theorem 1 in Section 2 w i l l yield t h e following r e s u l t , s i m i l a r t o t h a t of Theorem 2 in Section 3: Theorem 3 Suppose U : P -+ IR, with R c (0,m) x IR, i s a solution of t h e KleinGordon nonlinear wave equation (7.4.1) and t h a t it s a t i s f i e s t h e conditions (7.4.4)- (7.4.6). Then i t is possible t o construct C ~ s m o o t h regularizat ions (6.4.22) and algebras (6.4.34) , such t h a t
e
(1)
U = s + Ie(V,S) E A , e E W where s E S does not depend on l!
(2)
U s a t i s f i e s (7.4.1) in t h e usual algebraic sense, with multipliand t h e p a r t i a l derivative operators cation in DP : A' -+ Ak, p E N 2 , Ipl 5 2 , with l , k E W, kt2 5 t .
E.E. Rosinger
278
In view of pct (I), Chapter 6.
55.
s is a chain weak solution of (7.4.1), see Section 5 ,
JUNCTION CONDITIONS AM) BESOLUTION OF SINCULARITIES OF WEAK SOLUTIONS FOR TBE EQUATIONS OF ~GNETOBYDRODYNMCSAND GENERAL RELATIVITY
The problem of finding junction conditions across hypersurfaces of discontinuities for solutions of the equations of magnetohydrodynamics or general relativity is usually approached either by applying integral conditions or by introducing certain simplifying assumptions. Such methods present obvious deficiences when compared with direct methods which would be based on the idea of obtaining the junction conditions from weak solut ion conditions formulated across the hypersurfaces of discontinuities. As the nonlinearity of the equations involved will imply the presence of products of the Heaviside function with the Dirac 6 distribution and its partial derivatives, such direct methods cannot be implemented within the distributional framework. In the present Section, the polynomial nonlinear operations on the singular distributions mentioned will be performed within the quotient algebras containing the distributions. The resultin method has the major advantage that among others, a clear and algebraica ly simple insight is obtained into the structure o the nonlinearities of a particularly large class of systems of polynomia nonlinear partial differential equations encountered in the study of physics. This result is presented under its general form in Section 6. The method of establishing the junction conditions presented in the sequel will at the same time yield the resolution of singularities across the hypersurfaces involved in these junction conditions.
?
I n order to avoid rather trivial technical complications and also to make it possible at the same time to point out the essential underlying algebra ie phenomena, on1 the case of polynomial nonl inearities will be dealt
g
with, a case which o viously covers the situation in magnetohydrodynamics, as well as general relativity. The fact that in the sequel we shall deal with systems and not single nonlinear partial differential equations need not cause concern. Indeed, as seen in Section 14, Chapter 1, see also Rosinger [2, pp. 32-34], the study of sequential solutions of such systems is obtained as an easy and direct generalization of the situation for single nonlinear partial differential equations, see Section 10, Chapter 1. For convenience, we shall only consider the case of Coo-smooth coefficients. The general case of continuous coefficients is in principle similar and is dealt with in detail in Rosinger [2, pp. 139-1621. Suppose we are given a system of polynomial nonlinear partial differential equations
Resolution of singularities
'
(7.5.1)
I T
'Pi(')
where
U = (UI ,
. . . ,Ua)
(x) = ( x ) x E
D lS.i5kPi
15i(hP :
Q
Q
j
are unknown functions, while
-+
cPi,
P E P(n)are given.
f
The system in (7.5.1) is called of type (MH), associated partial differential operators
if and only if each of the
can be written in the form
where L (D) are linear partial differential operators with P-smooth PP coefficients and order while P (D) are linear partial differen-
"PP,
PP
tial operators with C'-smooth coefficients and order at most one. Sometimes it will be convenient to write the conditions (7.5.3) under the form
XEQ, is05b where P (D) are of the same type with P (D) . Ppa@' PP As can easily be seen, the equations of magnetohydrodynamics as well as those of general relativity are of type (MH). The Navier-Stokes equations are also of type (MH). If the order of the system (7.5.1) is m
=
~ ~ { I P5 P~ 5 ~b, ~1 5I i I5 hP, ~ I5j
then, if (7.5.1) is of type (MH), m
= 1
+max{m
it may be assumed that
PP11 5 p 5 b y
1 5 p 5 rP}
Suppose now given a hypersurface l' c il defined by (7.5.5)
r
= {X E qy(x) = 0)
< kPI. I
E .E . Rosinger
where 7 : R + R, y E COO. s h a l l a l s o suppose t h a t
r
(7.5.6)
r
Obviously t h e s u b s e t
in R
is c l o s e d .
We
h a s z e r o Lebesque measure
We s h a l l b e i n t e r e s t e d i n f i n d i n g oeak solutions U = (UI, . . . ,Ua) : R f o r an (MH) t y p e system (7.5.1) such t h a t d i s c o n t i n u o u s a c r o s s t h e hypersurface r.
U is Cm- smooth on R \ r
+
Ra
and
It f o l l o w s immediately t h a t U w i l l have t h e form
e,
where U- ,U+ : R + Ra,u- ,U+ E a r e c l a s s i c a l s o l u t i o n s f o r t h e system ( 7 5 . 1 ) w h i l e H : R -+ R d e f i n e d by
is t h e Heaviside f u n c t i o n a s s o c i a t e d with t h e r e p r e s e n t a t i o n hypersurface I' given i n (7.5.5).
of
the
The problem can now be formulated as f o l l o w s : f i n d t h e n e c e s s a r y and/or s u f f i c i e n t jvnct ion conditions on t h e system (7.5.I), t h e h y p e r s u r f a c e r and i t s r e p r e s e n t a t i o n through y i n (7.5.5), as well as on t h e c l a s s i c a l s o l u t i o n s U- and U+ of (7.5.1), such t h a t U g i v e n i n (7.5.7) is a weak s o l u t i o n f o r (7.5.1). I f t h e above problem i s considered w i t h i n t h e framework of t h e c h a i n s a l g e b r a s c o n t a i n i n g t h e d i s t r i b u t i o n s c o n s t r u c t e d i n S e c t i o n 4 , Chapter 6, it is easy t o o b t a i n necessary and sufficient j u n c t i o n c o n d i t i o n s . To d o s o , we s h a l l f i r s t c o n s t r u c t Cm-smooth r e g u l a r i z a t i o n s of t h e Heaviside f u n c t i o n (7.5.8), a c c o r d i n g t o t h e method p r e s e n t e d i n t h e f o l l o w i n g lemma. Lemma 1 q :R
Suppose g i v e n any f u n c t i o n
We d e f i n e s E (cm(i2))' 4
(7.5.10)
s
rlv
-+
[O,11 , q
E
Cm, such t h a t
by
(x) = q((v+l)y(x)), u E
IN,
xE R
Resolution of singularities
Then the following two relations hold
Proof The case 1 = 1 is obvious. If 1 > 1, it is easy to see that q ) : IR + 0 1 ? (1)' E Cm and (1))' also satisfies (7.5.9), ore the problem is reduced to the case C = 1.
there-
The Cm-smooth regularizations of H obtained above will generate Cm-smooth regularizations
of the intended weak solution U in (7.5.7) , given by the relation (7.5.14)
s a x = U a x
-
(U-)a(x))~qy(~),
l
At the stage the problem is to identify the conditions which will make s given in (7.5.13) a weak solution for the system (7.5.1). In this connection, the following result will be useful. Suppose given on R a first order, linear and homogeneous partial differential operator P(D) with Cm-smooth coefficients. For given functions $ ,$+ ,$; ,9; , E Cm(fl) define the regularizations t ,t' E tux =
with v E N, x
E
-
(1
(cm(fl))' +
+
as follows -
C (X))S~~(X)
R, assuming that q is as in (7.5.9).
Then obviously
E. E. Rosinger
P r o ~ o s iion t 1 The following r e l a t i o n s hold tP(D)t' E Sm
(7.5.16)
Proof -
It i s easy t o see t h a t f o r each v t,P(D)t;
= (d-
+
(d+- C )s,)P(D)
(P
+
($+- :-C)sqv)P(D)d:
=
= CP(D)$'
+
E
DI, t h e following r e l a t i o n holds
(dL+(di- d:)sqv) = +
(P(D) ($1- S3sq,
(LP(D)(l?'+($i-di)
+
(($+-$-)~(D)($;-$:))(S~~)~
+
(d+- d- (di- d~)sqvP(D)sgv
But since P(D)
+
+
+
($;-
d:)P(D)sqv)
(6+-L)P(D)P')sqv)
d- ($;-~:)p(D)sqv
=
+
+
is of f i r s t order, l i n e a r and homogeneous, we have
1 sqvP(D)s 17" = 7 p(D)(sqv)2 Now t h e r e l a t i o n (7.5.11) i n Lemma 1 w i l l imply
= ) a l s o s a t i s f i e s (7.5.11). where (7.5.12) i n emma 1 is taken i n t o account.
This completes t h e proof, i f D
Corollarv 1 I f P(D) is a l i n e a r p a r t i a l d i f f e r e n t i a l operator on D with Coo-smooth c o e f f i c i e n t s and order at most one, then t h e following r e l a t i o n s hold
Resolution of singularit i e s
where Q(D)
is the f i r s t order homogeneous part of
P(D).
Proof Assume P(D) has the form P(D)*(x) = Q(D)*(x)
+
d(x)*(x)
+
e(x), x
E Q
where Q(D) is the f i r s t - o r d e r homogeneous part of P(D), while d,e E e(0).Then relation (7.5.18) follows easily from (7.5.16). Further, f o r given v E M,
tst;
=
(C
+
we have
(*+-*-)sqv)d(*:
+
(*;-C)svu) =
therefore, in view of Lemma 1 < t d t t ,.> = $-d$! + ($+dlbJ - $-d$L)H Finally, it i s easy t o see that
= Tb_e + ($+e - $-e)H
The l a s t two relations (7.5.19).
together with
(7.5.17)
will
obviously yield
The result on junction conditions f o r discontinuous solutions of systems of p a r t i a l differential equations of type (MH) will be presented in Theorem 4. F i r s t we need the following r e s u l t .
Suppose
U- ,U+
: Q -+
IZa
are two Cm- smooth solutions of the m- th order
polynomial nonlinear system of type (MH) in 7.5.1 . Then f o r any Cmsmooth regularization s given in (7.5.13), t e f o lowing relations hold for every 1 < p < b
k 1
E. E. Ros inger
284
+
-
where Q
c
2
1
( Y ),.I
Lpp(D)((
c
l
((U+),+(U-),)((U+I,/-
)QBp,,/ (D)H)
is t h e f i r s t order homogeneous p a r t of
PP@@/(D)
Pop,,.
(D) .
Proof I n view of (7.5.4) and (7.5.13), it follows t h a t
b (D)s
= 1
(D)s,')
But (7.5.18) implies t h e r e l a t i o n s sopppoo' (D)s,, therefore t h e l i n e a r i t y of
E
L
sm
PP (D)
w i l l give
a s well a s =
b'
P
l
LDp(D) (
l
<s,PbPP,/ (D)s,/, .>)
Now (7.5.19) w i l l give <s,pgp,/
(D)s,'
'>
= (u- ),PBPO,f
(u- )a'
- (Y),Pgp,,'(D)(U-),')H 1 + +
+
q
((U+)QpBP17rZ/ +
+-
(U-),/)QDpoo/(D)H
-
Resolution of s i n g u l a r i t i e s
where QBpaaf(D) i s t h e f i r s t o r d e r homogeneous p a r t o f follows t h a t f o r 1 5 p 5 b, t h e following r e l a t i o n holds
-
Since U. we have
(Y),f))9ppaaf
285
Pppoo' (D)
It
(D)H)
was supposed t o be a c l a s s i c a l s o l u t i o n of t h e system (7.5. I), T (D)U- = 0 on Q,
'b
which completes t h e proof Before p r e s e n t i n g t h e r e s u l t i n Theorem 4, we need t h e f o l l o w i n g d e f i n i t ion. Given two f u n c t i o n s
U- ,U+ : R -+ lRa, U- ,U+ E Cm, we d e f i n e
U
:
R --+ Ra
by U(x) = U- (x)
+
(U+(x) - U- (x))H(x) , x E
and u s e t h e n o t a t i o n
Then t h e f u n c t i o n s U- , U+ w i l l be c a l l e d independent on r, i f and only i f f o r any .A E R, with a E I , t h e f o l l o w i n g i m p l i c a t i o n is v a l i d :
I f a = 1, t h e n U-, U+ a r e t r i v i a l l y independent on r, which i s why t h e above c o n d i t i o n was not demanded i n Theorem 1 i n S e c t i o n 2. I n terms of t h e system i n (7.5.11, t h e c a s e a = 1 corresponds t o t h e s i t u a t i o n when one unknown f u n c t i o n U : Q + lR h a s t o s a t i s f y b p a r t i a l d i f f e r e n t ial e q u a t i o n s .
E. E. Ros inger
U E Cm then U-, U,
Obviously, if Moreover,
U
E
are independent on
r.
Cm i f and only i f
Theorem 4 Suppose U- ,U+ : 0 -+ tRa are two Cm-smooth solutions of t h e m-th order polynomial nonlinear system of (MH) in (7.5.1) and suppose given a COO- smooth hypersurface (7.5.5). Then the function
where H i s the Heaviside function (7.5.8) associated with the hypersurface (7.5.5), i s a weak solution of the system (7.5.1), i f and only i f the following junction conditions a r e s a t i s f i e d f o r each 1 < b
<
+
where Q
PP@Q/(D)
&
- (Y)&'))QgPoo'(D)H =
fg
i s the f i r s t order homogeneous part of
I f (7.5.23) is s a t i s f i e d and the functions
P,jpoo' (Dl
U- , U+ a r e independent on I',
it i s possible t o construct Cm-smooth regularizations (6.4.22) and algebras (6.4.34) , such that
(7.5.24)
(7.5.25)
U = s, + Ie(V,S) where s, E S ,
E
with
AC , 1 1
a,
a,
C
E
do not depend on C
U s a t i s f i e s each of the equations of t h e system (7.5.1) in the vsual algebraic sense, with multiplication in Ak and the p a r t i a l derivative operators D~ : A' + A ~ p, E INn, Jpl 5 m y with
C,k E I, ktm !. C
Resolution of singularities
Proof
1
If the junction conditions (7.5.23 hold, then in view of Proposition 2, the function (7.5.22) will be a wea solution of the system (7.5.1) Conversely, assume U in t.5.22) is a weak solution of the system (7.5 1 ) The operations in t e definition of U can be performed with D1(n), however the same does not hold for the operations on U performed with 1 < j b, as these operations will involve products by Tp(D), H.P PPQQ'(D)H. Nevertheless, according to (7.5.20) the cm-smooth regularization s of U constructed in (7.5.14) has the property that TB(D)s is weakly convergent for every 1 j P < b. Therefore the assumption that U is a weak solution of the system (7.5.1) implies that
This, in view of the relations (7.5.21), converse.
completes the proof of the
For the rest of the proof, see Rosinger [3, pp. 373-3771.
$6. BESOLUBLE SYSTEMS OF POLYNOHIAL NONLINEAR PARTIAL DIFFERENTIAL EqUATIONS The necessary and sufficient junction conditions across hypersurfaces of discontinuities of weak solutions for systems of type (ME) and the resolution of the corresponding singularities presented in the previous Section are extended here to a large class of systems of nonlinear partial differential equations which contains many of the equations modelling various physical phenomena. For convenience, we shall only deal with the case of Cm-smooth coefficients. The general case of continuous coefficients can be treated in a similar way, see Rosinger [2,pp. 152-1621 . Definition 1 The s stem of polynomial nonlinear partial differential equations in 67.5.1j is called resoluble, if and only if each of the associated partial ifferential operators in (7.5.2) can be written in the form
whenever
E.E. Rosinger
288
where $ , x : D-IRa, u : R-. R, I , , y Y w ~ C mpPp~INn, , l E N and TPp PP are mip-th order polynomial nonlinear partial differential operators in $ and X. The pair (m' ,mu),
where
is called the split o r d e r of the resoluble system (7.5.1). Pro~osition3
A system of type (MH) is resoluble.
Proof Assume the partial differential operators in (7.5.2) corresponding to the system (7.5.1) are of form (7.5.3). Let us take U : R-*lRa given in (7.6.2). Then (7.5.3) or equivalently (7.5.4) yields
But
(I,
+
X,.W)P~~,,'(D)($,/
where Pgpuo/ (Dl therefore
+
x p )
=
is the first order homogeneous part of
Ppp,,,
(D),
Now the relation (7.6.1) follows easily The characteristic behaviour of the r e s o l v b 1 e systems of polynomial nonlinear partial differential equations with respect to weak solutions with discontinuities across hypersurfaces is presented now.
Resolution of singularities
Theorem 5 Suppose the m-th order polynomial nonlinear system of partial differential equations in (7.5.1) is resoluble. Given
U- ,U+ : D
-r
lRa,
two Cm- smooth solutions of (7.5.1), and a
Cm- smooth hypersurface (7.5.5), define the function
(7.6.4)
u- (XI= u- (XI
+
(U+(X) -
u- (x))H(x)
Y
U
: fl -+
x
E fl
lRa
by
where H is the Heaviside function (7.5.8) associated with the hypersurface (7.5.5)
.
Then U is a weak solution of (7.5.1 , if and only if the following jrnction conditions are satisfied for each I 5 g < b
In that case, if the functions U- ,U+ are independent on l', it is possible to construct Cm- smooth regularizations (6.4.22) and algebras (6.4.34) such that Ua (7.6.6)
=
sn + IL(V,S) E A
e,
where sa E S, with 1
(7.6.7)
< a < a, e E IA < a < a, do not depend on 1
L
U satisfies each of the equations of the system (7.5.1) in the usual algebraic sense, with the multiplication in and the partial derivative D~ :A' Ak , p E INn, Ipl < my with l,k E N, k+m 5 l
-
Proof For U given in (7.6.4) let us consider s given in (7.5.14) . Then, for each 1 < 3/ < b, the following relations hold
E .E. Rosinger
290
Indeed, in view of (7.6.1) we have for 1 tions
and v
E
M, the rela-
But, in view of (7.5.11), we have
Moreover
therefore the products Tpp(D) (U-,U+- U) -DPpplI in (7.6.20) are well defined in DJ(n). In this way (7.6.18) and (7.6.19) will follow easily from (7.5.12).
6
Assume now that the junction conditons 7.6.15) hold. Then in view of (7.6.18) and (7.6.19), u will obviously e a weak solution of (7.5.1). Conversely, if U is a weak solution of (7.5.1) then (7.6.18) and (7.6.19) will obviously imply the junction conditions (7.6.15). The second part of the proof is similar to that given for Theorem 4 in Section 5.
$7. COPPUTATION OF THE JUNCTION CONDITIONS The junction conditions (7.6.15) contain as a particular case the junction conditions (7.5.23), and at the same time have a more compact form, therefore we shall present here only the way they can be computed explicitely. We shall assume the notations and conditions in Theorem 5, Section 6. The basic relations used in the sequel are (7.7.1)
DPII= ( ~ ~ 7 ) - 6 P, E P , Ipl = 1
and (7.7.2) as well as
D~(D'~)
=
(DP~).(D'~'s),
p
E
P, (pl = 1,!. E M
Resolution of singularities
where 6 is the Dirac distribution concentrated on the hypersurface r in ( 7 . 5 . 5 ) , which is represented by the mapping 7 : S2 -+ R , 7 E COO. It is well known that the relations ( 7 . 7 . 1 ) and ( 7 . 7 . 2 ) are valid i f for instance grad 7(x) # 0, x E r It is easy to see that the relations ( 7 . 7 . 1 ) and ( 7 . 7 . 2 ) will yield
where K (D) are polynomial nonlinear pasrtial differential operators of !P order l? + 1 , which can be obtained from the recurrent relations
Substituing now the relations ( 7 . 7 . 6 ) , ( 7 . 7 . 7 ) into the junction conditions 67.6.15) and rearranging the terms according to the increasing order of the erivatives of the Dirac 6 distribution, we have
where Gpc(D) are polynomial nonlinear partial differential operators in U- , U+ - U- and 7. With the help of the relations ( 7 . 7 . 3 ) , ( 7 . 7 . 4 ) , it is possible to eliminate in ( 7 . 7 . 8 ) the Dirac 6 distribution and its lower derivatives and finally obtain relations of the form
with T'(D), TU(D) polynomial nonlinear partial differential operators in P P U-, U+ - U- and 7, relations which can be seen as the ezplicii form of the junction conditions in ( 7 . 6 . 1 5 ) . Indeed, as U- and U, were assumed to be known solutions of the resoluble system ( 7 . 5 . 1 ) , the relations ( 7 . 7 . 9 ) will give condtions on 7 in the form of polynomial nonlinear
E.E. Rosinger
292
p a r t i a l d i f f e r e n t i a l equations, i .e., conditions on t h e possible hypersurfaces of discontinuities of weak solutions of t h e resoluble system (7.5.1.)
58.
EXAMPLES OF BESOLUBLE SYSTElIS OF POLYNOYIbT, NONLINEAR PARTIAL DIFFEBENTIAI, EqUATIONS
The aim of t h i s section i s t o e x p l i c i t a t e i n a p a r t i c u l a r case of low order the conditions (7.6.1)- (7.6.3) defining t h e resoluble systems of p a r t i a l d i f f e r e n t i a l equations. Given t h e system of polynomial nonlinear p a r t i a l d i f f e r e n t i a l equations i n (7.5.1), i t s order of nonlinearity i s by d e f i n i t i o n (7.8.1)
r=max{k
.I1 ( _ D l b,
Dl
1 (_ i <_ h )
P
Obviously, t h e system of p a r t i a l d i f f e r e n t i a l equations i n (7.5.1) is nonl i n e a r , i f and only i f s ) 2 . The order of t h e system of p a r t i a l d i f f e r e n t i a l equations i n (7.5.1) i s called t h e integer
We consider the simplest n o n t r i v i a l systems of polynomial nonlinear p a r t i a l d i f f e r e n t i a l equations i n (7.5.1) which correspond t o
and which contain the nonlinearity exhibited by t h e Navier-Stokes equations. Obviously, t h e associated p a r t i a l d i f f e r e n t i a l operators i n (7.5.2) have t h e form
will
Resolution of s i n g u l a r i t i e s
where
Di = DPi , with pi =
(7.8.5)
i (O,...,m, 0 ,... ) ,
1 _< i
<
n
Now, a d i r e c t and easy computation w i l l show t h a t the p a r t i a l d i f f e r e n t i a l operators in (7.8.4) s a t i s f y t h e condition (7.6.1), i f and only i f the associated p a r t i a l d i f f e r e n t i a l operators
a l s o s a t i s f y t h a t condition. Substituting (7.6.2) into (7.8.6), it follows t h a t a l l t h e r e s u l t i n g terms in (7.8.6) can be written i n the form required in (7.6.1), with the exception of the terms in
Therefore, the system (7.5.1) w i l l be resoluble, i f and only i f
Obviously, equations.
the
condition
(7.8.8)
is
satisfied
by
the
Navier- Stokes
More complicated examples of resoluble systems of polynomial nonlinear p a r t i a l d i f f e r e n t i a l equations can be found i n Rosinger [2, pp. 157- 1621 .
E. E. Ros inger
GLOBAL VERSION OF THE CAUCHY-KOVALEVSKAIA THEOREM I N CHAINS OF ALGEBRAS OF GENERALIZED FUNCTIONS
59.
We recall that the classical Cauch -Kovalevskaia theorem has a wealth of outstanding features which during t e last four or five decades, with the emer ence of theories of generalized solutions, has somewhat been over ooked.
i
P
Let us review some of these features. First perhaps i s the fact that the mentioned theorem i s by f a r the most general nonlinear method in obtaining existence of solutions. The depth of insight t h i s theorem offers i s illustrated by the 'non hard' character of i t s proof, which only uses classical - i n fact rather elementary and straightforward - analysis, in which the only price paid i s in some computation involvin majorants of power series i n several complex variables, a computation w ich i s known to boil down to summin a eometric series. Indeed, that proof ives a clear understanding of i%e fact that the result in the Cauchy-Kova evskaia theorem i s locally the best possible in the given conditions. This fact i s further confirmed by the Holmgren uniqueness result.
f
P
I n t h i s way we obtain one of the best possible local existence, uniqueness and regularity results concerning classical solutions of very general nonlinear partial differential equations. The limitations of the result in the Cauchy-Kovalevskaia theorem are as follows. It only deals with noncharacterist ic i n i t i a l value problems. The fact however that the i n i t i a l values have to be given on a noncharacteristic hypersurface i s not so much a limitation but rather a necessity, as simple counter examples can i l l u s t r a t e i t . The limitation i s rather i n the fact that it does not apply to mixed, that i s , i n i t i a l and boundary value problems.
A second limitation i s that both the equation and i n i t i a l values have t o be analytic, as well as the noncharacterist i c hypersurf ace involved. This may present inconvenience i n the case of 1inear or nonlinear partial dif f erent i a l equations
where T(D) i s analytic, but f or the associated i n i t i a l values are not. However, in many - if not most - of such situations which present interest, one can assume that a l l the functions involved are piece wise analytic. And then, by breaking up the i n i t i a l domain 52 in smaller subdomains, one can get back to the analyticity conditions required i n the Cauchy-Kovalevskaia theorem. The rest i s a job i n patching up the analytic pieces of solutions obtained, which can be done for instance, i n the way mentioned i n Chapter 2 , or the sequel.
Resolution of singular it ies
295
Finally, the third limitation i s that we only obtain local solutions, that i s , in a neighbourhood of the noncharacteristic hypersurface on which the i n i t i a l values are specified. Here the feeling that the local character of the resulting solutions i s a limitation rather seems t o come from the conservative estimates used in the proof of the Cauchy-Kovalevskaia theorem i n order t o obtain the conver ence of the power series giving the solution. Indeed, these estimates w i l yield domains of convergence - and thus existence of solutions - which can in fact be much smaller than the respective maximal domains may actually be. But the other fact i s equally obvious: in many - i f not most - of the cases of analytic equations and noncharacterist i c analytic i n i t i a l values, we cannot expect the existence of global classical solutions. A good example for that i s offered by the shock wave equations. Therefore the legitimate object ion should rather be that the proof of the Cauchy- Kovalevskaia theorem i s not geared t o give information on the maximal domain of existence of solutions.
I
There i s also sometime the objection that the proof of the Cauchy-Kovalevskaia theorem does not give indications about the dependence of solutions on the i n i t i a l values. This however may be included into the third limitation above. Indeed, we already have a difficulty in establishing the dependence of the maximal size of the domain of existence of solutions on the i n i t i a l values. So that, perhaps owing t o the general and elementary nature of the proof, we cannot expect t o obtain more. The above suggest the possibilit that a wide range of global solutions of interest for nonlinear partial h f f e r e n t i a l equations with i n i t i a l and/or boundary value problems may belong to the following class: they are of a analytic on each of the pair wise disjoint subdomains Qi c Q suitable countable family (Qili E I ) , Qi n Q . = J
0
for
i,j
E
I,
i # j
such that the closed set
I' = Q\ u Qi i€I
i s nowhere dense i n Q
On each Qi, w i t h i E I , the local solution may be given by the CauchyKovaleskaia theorem, subjected t o the Holmgren uniqueness. The existence of such kind of solutions i s quaranteed by Chapter 2 , where it i s shown that one can further assume satisfied the condition (7.9.4)
mes
r
= 0
However, wishing t o allow greater enerality, we shall not assume t h i s l a t t e r condition, since the method o resolution of singularities we use i s powerful enough t o work without i t .
d
Such solutions, while unique and analytic locally, that i s , on w i t h i E I , could have globally a great f l e x i b i l i t y , owing t o b i l i t i e s allowed by the two conditions in (7.9.2) and (7.9.3), obviously sufficiently large in order to include for instance
each Q;, the possiwhich are Cantor- set I
E.E. Rosinger
296
type of 'I or fractals, which have lately been associated with the study of turbulence and chaos.
It i s i n t h i s way, among others, that the patching up of local solutions into a global one can be of interest. This problem was approached and given one of the possible solutions i n Chapter 2, where further comments on the relevance of t h i s patching up method are presented. . Here we give a stronger version of that globalized Cauchy-Kovalevskaia A" if sequential theorem, which i n Chapter 2 only constructed an solution. Indeed, we shall t h i s time construct a sequence s of C"o-smooth functions which i s a chain generalized solution in suitable chains of algebras of generalized function, see Section 5, Chapter 6 . We can s t a r t with a general analytic nonlinear partial differential equation as i n (2.5.1)
where R c [ R " i s nonvoid, open, t E I R , y q E iNn-l, p + Iql 5 m and G i s analytic.
6IRn-l,
m > 1, P E N , p < m ,
As a consequence of the mentioned result i n Theorem 2, Section 6, Chapter 2, there exist I' c R w i t h (7.9.6)
I' closed, nowhere dense i n R
(7.9.7)
mes I' = 0
and U : R\I'
-+
Q: an analytic solution of (7.9.5) on Q\I'
Now using the construction in (2.5.5) - (2.5.9) we obtain the sequence of Cw-smooth functions
which s a t i s f i e s (2.5.14) w i t h
Our construction of chains of algebras of generalized functions i n which a lobal version of the Cauchy-Kovalevskaia theorem holds, w i l l be based on 1 in Section 3, Chapter 6, according t o which
corollary
Resolution of s i n g u l a r i t i e s
i s a P - smooth regular ideal in (7.9.11)
( ~ ( n ) ) ' . Let us take
V
a vector subspace in I r l V'". Then, according t o Theorem 1 in Section 2, Chapter 6, there exist P-smooth regu larizat ions
f o r a suitable choice of the vector subspaces S c S'". IN Let us take any derivative invariant subalgebra A c (C'"(S2)) , such t h a t (7.9.14)
S E A
(7.9.15)
I+(V@S)cA
In view of (7.9.12) and (7.9.15) it follows easily that (7.9.16)
(V,V
@
S, 1,A)
i s a COO-smooth regularizatioli
hence (2.2.6) and (7.9.10) yield (7.9.17)
I is a derivative invariant, COO- smooth regular ideal in A
Now we can modify (6.4.25)- (6.4.29) as follows. F i r s t , l e t us denote by (7.9.18)
Iw S
the ideal in ( ~ ( n ) ) ' generated by (7.9.10) yield (7.9.19)
Iw c I S
{ D ~1 WE ~M") .
Then (2.2.6) and
E.E. Rosinger
Now, instead of (6.4.26) (7.9.20)
, we
define f o r C
E Il
Ae(V,S)
a s being t h e derivative invariant subalgebra generated by Iw + Ve + S + s S in A .
Further, instead of (6.4.27), we define f o r C
(7.9.21)
IA
E
Ie(v,s>
a s being t h e ideal generated by I
t
s
Ye
in Ae(V,S).
It is easy t o see t h a t similar t o Theorem 6 i n Section 4, Chapter 6, we obtain f o r C E IA, t h a t (7.9.22)
(Ve,Se,Ia,Ae)
is a
C"-smooth
regularization
while in view (7.9.18)- (7.9.21), it follows that f o r C (7.9.23)
wS€It,
(7.9.24)
Iw
C
E
IA,
we have
SEAC
Ie C I
S
Furthermore, Theorems 6 and 7 in Section 4, Chapter 6, w i l l hold f o r t h e Goo- smooth regularizations (7.9.22) and the corresponding quotient algebras of generalized functions
which form the chain of algebras of generalized functions
Now we come t o the following global version of the Cauchy-Kovalevskaia theorem, which f o r convenience is only formulated in t h e polynomial nonlinear case. Theorem 6 Suppose given the analytic and polynomial nonlinear p a r t i a l d i f f e r e n t i a l equation
Resolution of singular it ies
where RclRn i s open, O E Q , t E R , y ER"-', P E N , p < my q ~ i N ~ - ' , p + J q J < m. Further, suppose given the analytic i n i t i a l values
where V c R"-'
is open.
Then, one can construct sequences of functions
and @'-smooth regularizations (7.9.16) (7.9.25) , such that
with the corresponding algebras
-
satisfy the equation (7.9.27) in the usual algebraic sense, with multipliA k , p E iNn, (pi i a , and partial derivatives oP : cation in C,k E ! I k+m , C
A'
Morevover, Ul E A e , with e E IA, are usual analytic functions on an open set containing (0) x V , and they satisfy (7.9.28).
It follows that Chapter 6.
s
is a chain weak solution of (7.9.27), see Section 5,
Proof -
It follows from (7.9.23) and the extended versions of Theorem 6 and 7 in Section 4 , Chapter 6, corresponding t o the @'-smooth regularizations in (7.9.22).
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CHAPTER 8 THE PARTICULAR CASE OF COLOMBEAU'S ALGEBRAS 1
SMOOTH APPROXIMATIONS AND REPRESENTATIONS
Recently, a particularly efficient, yet simple and elementary nonlinear theory of generalized functions has been introduced by J.F. Colombeau . This nonlinear theory is based on the construction of certain differential of generalized functions, algebras which happen to be algebras ~(IR") particular cases of the construction in Chapter 6, see for details Section 3 in the sequel. The special interest in Colombeau's algebras g(IRn) comes from their rather natural and central role within the general nonlinear theory constructed in Chapter 6, see for details Appendices 1 and 3, at the end of this Chapter. Although results such as in Chapters 2, 3 and 4 could so far not been possible to reproduce in Colombeau's a1 ebras, nevertheless, the natural and central role of these algebras a1 ow the proof of the existence in Colombeau's algebras of generalized solutions for lar e classes of earlier unsolved or unsolvable linear or nonlinear partial di ferential equations, some of the latter having for instance a basic role in quantum field interaction theory. A first systematic account of this theory was presented in , where the method employed was based on the De Silva Colombeau 41 , a differentia in locally convex spaces. Soon after, in Colombeau completely elementary presentation of the mentioned nonlinear t eory of generalized functions has been achieved. In view of its rather surprising ease, which makes it readily accessible to wide groups of mathematicians, plysicists, engineers, etc., we shall only deal with that latter version. It is particularly interesting to note that the version in Colombeau [2, 41, although needs some rudiments of the linear theory of distributions, is in fact much simpler than any nontrivial presentation of that latter linear theory known up until now, presentation which would go at least as far as the solution of variable coefficient linear partial differential equations.
P
9
I'\
r,
In fact, Colombeau's theory appears to lead to one of the ultimate possible simplification in the study of linear and nonlinear partial differential equations. Indeed, it only uses two basic tools: elementary calculus and topology in Euclidean spaces, as well as quotient structures in commutative rin s of smooth functions. The effect is the reduction of most of the matfematics involved to usual partial derivatives and multiple integrals, respectively to chasing arrows in diagrams involving rings and ideals of smooth functions. What turns all that into a surprisingly efficient method for solving linear and nonlinear partial differential equations is an asymptotic interpretation, based on the specific structure of the index set - see (8.1.13) - which is used in the definition of Colombeau's generalized functions in (8.1.18)- (8.1.21). See also Appendices 2 and 4 at the end of this Chapter. That asymptotic interpretation proves to be much more
E.E. Rosinger
302
efficient than topological properties on various rather sophisticated spaces of functions. In this wa , we are presented with one of the deeper insight into the workings invo ved in connection with the solution of linear and nonlinear partial differential equations.
I
In this Chapter, we shall only present a short account of Colombeau's nonlinear theory. For further details one can consult Colombeau [I, 21, Biagioni [2] , and Rosinger [3, pp. 4% 1921 . Before we present Colombeau's method, let us in essence recapitulate the classical and linear distributional framework used in the study of linear or nonlinear partial differential equations, and for simplicity, let us assume that the domain of the independent variables is Q = R". Then, the spaces of possible solutions are (8.1.1) where by
An c
Cm
c
... c C O c Lie, c P1
An we denoted the analytic functions.
From the point of view of nonlinear operations, in particular unrestricted multiplication, the above spaces, except for Lf--p,, and 9' , are particularly suitable, since they are associative and commutative a1 ebras with unit element, moreover, they are closed under a wide range o nonlinear operations of appropriate smoothness.
P
From the point of view of partial derivability, only allow indefinite iterations of such operations.
An,
C' and
V'
It follows that in (8.1.1), An and Cm alone are the suitable kind of differential algebras for a convenient study of nonlinear partial differential equations. Unfortunately however, as seen in Chapter 5, An and d are not sufficiently large for the above purpose, and in fact, we would need some differential algebras A, sufficiently large in order to contain the V 1 distributions, i.e.,
It is important to note that in view of Section 5, Chapter 5, extensions of the space 3' of distributions are needed even for the solution of linear variable coefficient partial differential equations. And then, let us present the way Colombeau is constructing an extension such as in (8.1.2). The basic idea is very simple: as is well known, Schwartz [ I ] , every distribution can be approximated by C'-smooth functions, in particular
Colombeau ' s particular algebra
according to the following ' convolution with a 6-sequence' procedure, see also Mikusinski [I] . Suppose given an arbitrary, fixed distribution T For any $
and
E E
E
E
ID'.
P with
(0,m) , let us define ylE E ID by
Then, using the convolution of distributions, we define
and obtain that (8.1.6)
fE E COO,
E E
(0,m)
But $ + 6 in ID', when c + 0 hence, in view of a basic property convolution we obtain the following smooth approximaiion property (8.1.7)
fc+T
in P',
when c - 0
In this way, to each distribution T E ID' , one can associate sequences of functions (fclc E (0,m)) E (C)(',~) which converge to T in 1. In other words, we obtain
where the so called inclusion ' c ' - which is not yet a proper inclusion means the above mu 1 2 iva 1 ent association
-
is obviously a differential The importance of (8.1.8) is that (f?)(Oym) algebra with the term-wise operations on the sequences of functions. Therefore (8.1.81 is nearly one of the desired extension (8.1.2) of the distributions. ndeed, all we need is to mana e to take away the quotation marks in ( 8 . 8 ) that is, to replace (8.1.8 by a convenient rnivalent mapping .
304
E. E. Rosinger
I n Colombeauts method, t h i s i s done by a p a r t i c u l a r c h o i c e o f an index set A , of a s u b a l g e b r a A i n ( P ) ~ and f i n a l l y of a n i d e a l Z i n A, which t o g e t h e r g i v e u s t h e f o l l o w i n g e x t e n s i o n , i .e . , embedding
of t h e d i s t r i b u t i o n s 9' i n t o t h e differential algebra A = A/Z. I n t h i s way, we o b t a i n , among o t h e r s , a representation of d i s t r i b u t i o n s as c l a s s e s of sequences of smooth f u n c t i o n s . One of t h e s p e c i a l f e a t u r e s of Colomb e a u t s method is t h e c h o i c e of t h e above index set A, t o which we t u r n now. I n o r d e r t o d e f i n e t h e index s e t we d e n o t e
A,
we proceed as f o l l o w s .
For
m E IN+ = IN\{O),
The avove c o n d i t i o n *) is of course t h e same w i t h (8.1.3) and is needed i n o r d e r t o o b t a i n ' & s e q u e n c e s ' by t h e method i n ( 8 . 1 . 4 ) . The c o n d i t i o n **) w w i l l be needed i n connection with t h e embedding i n t h e diagram (8.2.27). From (8.1.11) it f o l l o w s obviously t h a t we have t h e i n c l u s i o n s
We s h a l l t a k e a s o u r b a s i c index s e t
The n o n t r i v i a l i t y of
B r e s u l t s from
Lemma 1
It should be noted t h a t , while p r o p e r t y (8.1.14 is of c o u r s e e s s e n t i a l f o r Colombeau's method, p r o p e r t y (8.1.15) will not e used i n t h e s e q u e l and it was only given as a g e n e r a l information connected with (8.1.12).
b
As a last remark on t h e index s e t Q, we n o t e t h a t , f o r m E R+ and w i t h t h e d e f i n i t i o n i n ( 8 . 1 . 4 ) , we have
Colombeau' s particular algebra
italics elitesequences which is the framework of Colombeau's method is given by
which is obviously an associative, commutative differential algebra with the term-wise operations on the respective sequences of functions. Let us now define the following subalgebra A in &[LRn], such that of all f E &[LRn]
v K c L R ~ compact, p
which consists
E Mn :
3 mEDI+: V (Elm:
3 q,c
I~~f((~,x)l
C s7 €
where DP is the usual partial derivative of order p E INn, of the Cmsmooth function
Further, let us define the following ideal 1 in A, such that
v K c L R ~ compact,
p E DIn
:
3 eEM+,pEB:
(8.1.19)
m ~ M + , m > e (€4,: , 3 v,c
V
lDPf OE,x) I S crB(m)-
e
where we denoted (8.1.20)
*) **)
increasing lim p(m) = m
given by all f
E
A
E.E. Rosinger
306
Finally, the algebra of generalized junctions of Colombeau i s given by
An easy direct check shows that A i s indeed a subalgebra in £[lRn] and Z is an ideal in A, therefore, E(lRn) i s an associative, commutative a lgebru. Further, we obviously have
therefore, we can define the partial derivative operators
Obviously, the above partial derivative operators on E(lRn) are linear and satisfy the Leibnitz rule of product derivative. There are various rather natural motivations for the particular, somewhat complicated and unexpected way the algebra A and ideal Z were defined above. These motivations become apparent as Colombeau ' s theory and i t s applications are presented. A rather direct, functional analytic motivation for the necessary strgcture of A and Z i s given in Appendix 1, a t the end of t h i s Chapter. Here we should only like to oint out a certain analogy with the quotient space respresentation o f distributions i n (5.A1.24). Indeed, similar t o P, the ideal 1 has to model a certain 'convergence t o zero' property, which i n (8.1.19) i s one of the weakest polynomial in 6. This possible for an ideal, thus subalgebra, bein ra, in particular ideal, makes Z in a way the largest possible su stability of generalized which i s convenient from the point of view solutions. But as Z has t o be an ideal in A, this sets a constraint on the possible growth of the elements in A. From here, we obtain the growth condition (8.1.18) which i s polynomial i n 116. The role layed by the other parameters, such as K , p, m, n, e t c . , in (8.1.18) and (8.1.19), w i l l become clear in the sequel, see also the mentioned Appendix 1. Before we go further, it is useful t o note the nontriviality of which follows among others from the fact that (8.1.25)
A
f
&[lRn] and I i s not an ideal i n &[lRn]
Indeed, l e t us define f : B
x
lRn
-+
C by
A and Z,
Colombeau ' s particular algebra
where d(#) = sup{llx-ylIlx,~E suPP # ) Y Further, l e t us define g :
x
f
E
E
[Itn + C by
g(#,x) = l / f ( # , x ) , Since d ( # f ) = ~ d ( # ) , with #
#
E
# I,
E E
a,
x
E
> 0, it follows easily that
£[!Rn]\~, g E 1 , f - g = 1 $ Z
hence (8.1.25) . Finally, we should note that the above definitions of A and Z may suggest the following question: I s n ' t it that A i s precisely the s e t of Cauchy sequences in a certain topology on p(lRn), and furthermore, Z is precisely the set of sequences convergent t o zero in the same topology? In which case g(!Rn) given in (8.1.21) would obviously be the completion of e"(IRn) in the mentioned topology?
As seen in Appendix 2 a t the end of t h i s Chapter, the answer i s negative, insofar as A and Z are not the Cauchy and convergent t o zero sequences in any topology on P ( R " ) . In particular, the quotient structure which defines ~(IR") in (8.1.21) is not a usual topological construction of a completion of P(!R") .
Of course, our f i r s t interest is t o prove the embeddings
and establish the way the algebraic and/or differential operations on CO and P' extend t o g(!Rn).
e",
In t h i s respect, it w i l l be more convenient t o move from particular t o general, as in t h i s way we s h a l l have the opportunity t o become more familiar with the differential algebra g(IRn) . Therefore, l e t us s t a r t with the easiest embedding
E.E. Rosinger
308
which i s defined by the mapping (8.2.3)
C?
3
f
?+z
E
P(R")
where
In order t o show that the mapping (8.2.3) i s well defined, we have t o prove that (8.2.5)
c " 3 f ~ ? € A
which follows easily from (8.1.181, by noticing that
Further, we note that the mapping (8.2.3) i s injective, as in view of (8.1.19) and (8.2.6), we obtain
I n view of (8.2.4) and (8.2.5), it i s obvious that the mapping (8.2.3) i s an algebra homomorphism. Finally, in view of (8.1.23) , (8.1.24) , (8.2.3) and (8.2.4), it follows that the partial derivative operators D ~ ,with p E INn, on b'(iRn), w i l l coincide with the usual partial derivatives of functions, when restricted t o P ( R " ) . We can resume the above in the following: Theorem 1 The embedding P ( R " ) c b'(lRn) defined i n (8.2.3) i s an embedding of di fferential algebras, and the function 1 E P(iRn) i s the unit in the o algebra P(iRn) . Let us now proceed further and define the embedding
by mapping
Colombeau ' s particular algebra
As above, i n order t o prove that (8.2.9) i s well defined, we have t o show that
First we note that i n view of (8.2. l o ) , we have T E £[!Rn], since 4 E 9(iRn). Further obviously
hence (8.2.11f) follows easily from (8.1.18). (8.2.12), it ollows easily that (8.2.13)
Now, from (8.1.19) and
( f ~ p , T ~ I ) + f = o
hence the mapping (8.2.9) i s injective.
I n t h i s way we obtain:
The embedding C' ( R ~ c) G(iRn) defined i n (8.2.9) i s an embedding of v e c t o r spaces. Before we study further properties of the above embedding, it i s useful t o define the embedding (8.2.14)
V f c G(Pn)
by the mapping
where we define
d
Obviously (8.2.15) i s an extension of 8.2.9), as it reduces t o the l a t t e r when the distribution T i s generate by a continuous function f . The fact that fT E &[iRn], with T E P , follows easily from basic results
E. E. Rosinger
310
concerning the convolution of distributions, Schwartz [I]. one obtains
In the same way
and then (8.1.18) yields (8.2.18)
' D 1 3 T ~ f T E A
thus the mapping (8.2.15) i s well defined. From (8.2.12) and (8.1.19) also follows that (8.2.19)
(T E P', f T E I)
*T = 0
which means that the mapping (8.2.15) is injective Finally, in view of (8.1.23), (8.2.15), and (5.4.3), it follows that the p a r t i a l derivative operators DP, with p E DI", on 4(lRn), w i l l coincide with the distributional p a r t i a l derivatives, when r e s t r i c t e d t o 'D'(lRn). In particular, DP on 4(lRn), coincides with the usual p a r t i a l derivative DP of smooth functions, when restricted t o c'(lRn), with l E 01, 2 lpl.
'
We can conclude as follows: Theorem 3 The embedding P(R") c 4(lRn) defined in (8.2.15) is an embedding of vector spaces which extends the distributional p a r t i a l derivatives . Remark 1 As seen in Example 1 below, the vector space embedding C"(lRn) c 4(lRn) in (8.2.9) is not an embedding of algebras, i .e., the multiplication in P((R"), when restricted t o ~ ( I R " ) , does not alwags coincide with the usual multiplication of continuous functions.
It is particularly important t o note that many of the e f f e c t s of the above deficiency concerning the multiplication in ii(lRn) of the usual continuous functions w i l l be eliminated with the introduction of the coupled calculus presented in Section 5. This coupled calculus is a specific and essential feature of Colombeau's method and it is one way t o overcome the conflict between insufficient smoothness, multiplication and differentiation presented in Chapter 1. However, as seen in (1.1.15), the use of one single
Colombeau's particular algebra
differential algebra in (8.1.23) may lead to undesirable basic 1 imitations, which so far, can be overcome only by usin chains of algebras, such as those in Chapter 6, or going even beyond t at framework, as is done in Chapter 4.
1
Now, let us consider several examples which help to clarify the above embedding properties. For simplicity, we consider the one dimensional case, when n = 1.
In connection with Remark 1, let us take fl ,fa E
Then, with the product in (8.2.21)
fl -f2 = 0
However, in G(R) (8.2.22)
(lRn) , we have
we shall have
(TI + 1).(f2 + 1) = g+Z E G(R)
where, in view of (8.1. lo), we have
and then, in view of (8.1.19), it follows that (8.2.24)
g e l
since we have
Obviously (8.2.22) and (8.2.24) yield
CO (Rn) defined by
E .E . Rosinger
312
In t h i s way, the product of the continuous functions x- and x+ in
CO (R) ,
i s zero
but it i s no longer zero in G(IR) .
Returning now t o the above general embedding results, one can note t h a t , while the embedding CO c G i s a particular case of the embedding 3' c E, the embedding COD c G does not a t f i r s t seem t o be a particular case of these l a t t e r two ones. Indeed, both (8.2.10) and (8.2.16) are the same kind of convolution formulas, while (8.2.4) i s obviously not. However, t h i s difference i s only apparent, since we have the following comm~rtative diagram
Indeed, one can prove, see Rosinger [3] , that
We show that Colombeau ' s algebra of generalized functions, see (8.1.21)
i s of the type (6.1.24), and in particular, Z i s a C"-smooth regular ideal i n the sense of Definition 2 in Section 2 , Chapter 6. In view of (8.1.18) and (8.1.19) we shall take, see (8.1.13) (8.3.2)
A = 4
and e = CO. That places us within the framework i n Section 2, Chapter 6, since A i s a subalgebra i n and Z i s an ideal in A. Furthermore, i n view of (8.2.2)- (8.2.7), condition (6.1.27) i s obviously satisfied. Within t h i s Section, it will be convenient t o consider the elements of ( P ) ~ as given by functions
Colombeau ' s particular algebra
such that
f(#,.)
P(R"),
f o r 4 E @.
f E (P)'
such that
E
Now, l e t us define
as the set of a l l
3 T E v'(IRn) : V D c IRn open, bounded :
3 mE!N+:
where the limit is taken in the sense of the weak topolo y of 'D'(D). Obviously, the conditions (6.1.2)- (6.1.5) w i l l be s a t i s f i e f . Therefore, with $' in (8.3.3), the kernel
of the linear surjection (6.1.3) w i l l s a t i s f y (6.1.6) and (6.1.7).
q,3,
With A and Z given as above, it only remains t o define S and V in order t o be able t o enter within the framework of Section 2, Chapter 6. And then, we take
In view of (8.2.14)- (8.2.19), it follows that
and the mapping (8.3.8)
S
3
f
w
T E 3' (lRn)
i s a l i n e a r s u r j e c t i o n which s a t i s f i e s (6.1.5). (8.3.9)
V
be the kernel of (8.3.8) .
Finally, l e t
E.E. Rosinger
Pro~osition1 The following relations hold (8.3.10)
Z ~ S = V
(8.3.12)
S c q
Proof s
n c result as follows. Let f E 1
i
n S.
Then f E l yields
hence (8.3.8), (8.3.9) yield f E V. Conversely, if f E V then (8.3.8), (8.3.9) yield T = 0 E 9'(lRn) , hence T = 0 E . But V c S c A, thus f E A. Now f + Z = T = 0 E G(R") yields f E 1.
c(lRn)
(8.3.11). Take f E Z, then (8.1.19), (8.3.4) and (8.3.5) yield f E I$.
)
,
then (8.3.6) implies
Assume given R c lRn 3021
open and $
E
V(Q)
.
Then, as seen in Rosinger [3, p.
Colombeau ' s particular algebra
And (8.3.16), (8.3.17) obviously imply that 'that for f E S, T E PI (IRn) , such that
5:.",.c!I?
$.
the relations (8.3.15),
V Q c lRn open : 3 mED(+: l i m f(/,,.) €10 In =
(8.3.18)
f E
(8.3.16) provide for
Tla
where the l i m i t i s taken in the sense of the weak topology on P'(Q). If T = O E P ' ( R " ) . Hence (8.3.18) and f E V then (8.3.9) implies that (8.3.5) yield f E Thus, we obtain the inclusion
5.
f:.:;i2;s
5
f s
Conversely, if yields f E V .
then (8.3.5) gives
T = 0 E I (R") , hence (8.3.9)
from (8.3.11)
Before going further, we should note that
hence (8.3.20)
${
A
Indeed, let us take o,-y : (0,m) (8.3.21) and define f
-*
( 0 , ~ ) such that
l i m o(c)/el/' = l i m -yn(e)/o(e) = €10 €10 097
: Q
x
IRn
-*
4: by
w
E.E. Rosinger
where d(4) is given in Section 1. Then, in view of (8.3.21), it follows easily that
However (8.3.24)
fa,y !f A
since, for p E INn, E > 0, x E Rn, we have
hence (8.1.18) and (8.3.24) will obviously imply (8.3.24).
A useful consequence of Proposition 1 is the following: Corollarv 1 We have (8.3.25)
codim Z fl
In3
5=0
therefore (8.3.26)
Z is a small ideal in
A
Proof The relation (8.3.25) follows directly from (8.3.14). obvious, in view of (6.2.15).
Now (8.3.26) is 51
Now we can show that Colombeau's differential algebra of generalized functions g(lRn) = All, see (8.3.1), is a particular case of the quotient algebras of generalized functions constructed in (6.1.24), Section 1, Chapter. Indeed, we obtain the following result. Theorem 4 Given Colombeau ' s differential algebra of generalized functions
Colombeau ' s particular algebra
then, with the notations in (8.3.3), (8.3.5), that (8.3.28)
(8.3.6) and (8.3.9), we have
(V,S,Z,A) is a C"-smooth regularization
of the representation of distributions
therefore (8.3.30)
1 is a C"- smooth regular ideal in A
Furthermore, we have the inclusion diagram
which satisfies the relations (8.3.32)
rns=v
Proof First we prove the inclusions in (8.3.31).
The inclusions
r -+ A -+ (plA follow from (8.1.21).
The inclusions
follow from (8.3.6)' (8.3.7) and (8.3.9). The inclusion
s-+q
E. E. Ros inger
318
follows from (8.3.3) and ( 8 . 3 . 5 ) . The inclusions
v+$,
S+$
follow from (8.3.13) and (8.3.12) respectively.
Finally, the inclusions
follow from (8.3.10) and (8.3.11) Now (8.3.32) respectively.
and
(8.3.33) are the same w i t h
(8.3.10)
and
(8.3.13)
Concerning ( 8 . 3 . 3 4 ) , the inclusion c follows from ( 8 . 3 . 3 1 ) . The converse inclusions i s obtained easily. Indeed, defined in ( 8 . 3 . 3 ) satisfies ( 6 . 1 . 3 ) , that i s , with the notation i n ( 8 . 3 . 4 ) , the mapping
i s a linear surjection. Now, i t suffices to take into account (8.3.8) and ( 8 . 3 . 5 ) , and the proof of (8.3.34) is completed. The relations (8.3.31)- (8.3.34) will obviously yield (8.3.28) and (8.3.30)
Remark 2 The weaker version of the above property (8.3.30) according to which
Z i s a regular ideal i n A can obtained i n a direct way from (8.3.26) and Theorem 2 i n Section 2 , Chapter 6. Remark 3 Concerning the fact that Colombeau's algebra P(IR') is a collapsed case of the chains of algebras (6.4.7), we note the following. In view of the results i n this Section, it i s easy to see that i n the case of Colombeau's algebra of generalized functions
the conditions required i n Theorem 8 i n Section 4 , Chapter 6 , are satisfied. In this way, the results i n 8.2.14) and 8 . 2 . 2 ) are particular cases of Theorem 8 and pct. (2) in heorem 6 i n (S ection 4 , Chapter 6.
$
Colombeau ' s particular algebra
319
Indeed, we can take (6.4.52) as given by (8.3.28). Further we can replace Je and Ae in (6.4.58) with 1 and A respectively and still obtain
C?- smooth regularizations
In this way, we shall obtain
and the results in Theorems 5, 6, 7 and 8 in Section 4, Chapter 6, will still hold.
94. INTEGRALS OF GENERALIZED FUNCTIONS As mentioned in Section 2, the inconvenience of the fact that the embedding co(lRn) c P(lRn) is only an embedding of vector spaces and not of algebras, will to a good extent be overcome by a c o u p l e d c a l u c l u s introduced in Section 5. One of the prerequisites of this coupled c a l c u l u s is the notion of the v a l u e of a generalized function F E G(R") at a point x E lRn. In order to define the v a l u e notion, we need an e x t e n s t i o n of the complex numbers. It is interesting to note that the way this extension is made, recalls certain basic constructions in Nonstandard Analysis, Schmieden & Laughwitz, Stroyan & Luxemburg, etc. Indeed, suppose given a generalized function
then
Therefore, for given, fixed x E lRn, it is natural to define the value F(x) of F at x, as follows: let us fix x in (8.4.2) and thus obtain the mapping
and define F(x) as generated in a suitable way by h in (8.4.3). In F in (8.4.1) is of course this respect, we note that the mapping f t-
E. E. Rosinger
320
- Fixl.
not injective, hence, we can expect the same w i t h the mappin It follows that just as in 8.4.1), a l l we have t o do i s t o actor out t e noninjectivity of h H F x), by using a suitable quotient structure, similar t o (8.1.21).
!
Then, l e t us denote
which i s obviously and associative and commutative algebra. us denote by the set of a l l h E 80, such that
Ih(4J
I 5
I t i s easy t o see that by lo the set of a l l h
E
C
-iii E
i s a subalgebra i n Eo. Finally, l e t us denote A , such that
It follows that 10 i s an ideal in A , (8.4.7)
&I
5 &O
Further, l e t
and similar t o (8.1.25), we have
and Zo i s not an ideal in Eo
Now, the associative and cummutat ive algebra numbers i s defined by
C of generalized complex
Obviously F may depend on the dimension n of the underlying Euclidean space lRn. I n other words, we may have variotts s e t s of generalized complex numbers, corresponding t o different values of n . However, that will not be of interest i n the sequel. Let us define the embedding
Colombeau' s particular algebra
by the mapping
where (8.4.11)
z(() = z ,
( E
hence (8.4.10) i s obviously injective.
In t h i s way, we obtain:
P r o ~ oist ion 2 The embedding C c C defined i n (8.4.10) i s an embedding of algebras. An essential operation, in fact binary relation, needed i n the sequel, i s the association of a usual complex number z E C w i t h some of the genewhich i s defined as follows. The usual ralized complex numbers z E z E Q: i s said t o be associated with the generalized complex number complex number z E C, i n which case we shall denote Z + z , if there i s a representation z = h + Z E F = &/Io,such that
r,
l i m h(dC) = z 6
lo
It i s important t o note that not every 5
E
F
has an associated
z
E
C.
In view of the above, l e t us denote
Given a generalized function
and xEIRn, the value lized complex number
F(x) of
F at
x,
i s by definition thegenera-
E. E. Ros inger
322
Now, we present the property of the embedding cO(lRn) c G(lRn), which is one of the essential features of the coupled calculus in Colombeau's method : Theorem 5 Suppose given a generalized function which corresponds t o a continuous function, i . e . , F = f E co(!In) c G(!Rn). If x E lRn, then (8.4.17)
F (x)
E
to and F(x)
+ f (x)
that i s , the generalized complex number F(x) which is the value of the eneralized function F a t x, has a s associated usual complex number , which is the usual value of the continuous function f a t x. In other words, we have the following commutative diagram
where (1) i s the usual computation of the value of f a t x, (2) i s defined by (8.2.9), (3 i s defined by (8.4.15), and (4) is the relation of association + , de ined in (8.4.12).
1
A further prere u i s i t e of the coupled calculus i s the notion of integral of a generalized unction, which is defined as follows. Suppose given a generalized function
P
and K c lRn compact. Then we define the integral of generalized complex number
where
F on
K,
as the
Colombeau ' s particular algebra
It is easy t o see t h a t t h i s i s a correct definition. Indeed, f ( d , . ) E p(lRn), with 4 E O, since f E A. Further, (8.4.21) yields
Finally, hence i n view of (8.1.18) and (8.4.5), we obtasin h E Ao. (8.4.20) does not depend on f in (8.4.19), since (8.1.19) and (8.4.22) w i l l obviously yield f E Z h E lo.
*
In the particular case of a generalized function which corresponds t o a p - smooth functions, i . e . , F = f E P ( R " c G(lRn) , t h e above notion of integral coincides with the usual one. In eed, in view of (8.2.3), we have
1
with (8.4.24)
T(4,x) = f ( x ) ,
4
E O,
x
E
lRn
and then, (8.4.20) and (8.4.21) yield
with, see (8.4. l o ) , the relation
thus, in view of the embedding (8.4.10), we can write
With t h e help of the relation of association + , t h e above generalizes t o con2 inuous functions. Indeed, we have: Theorem 6 Suppose given a generalized function which corresponds t o a continuous function, i .e. F = f E co(lRn) c G(lRn) . Then f o r every K c lRn compact, we have
E. E. Ros inger
(8.4.28)
I
F (x) dx
E
and
K
F(x) dx c
K
I
f (x) dx
K
i . e . , the integral over K of the generalized function F is a generalized complex number which has a s associated usual complex number the usual integral of f over K.
It follows that we have the commutative diagram Co(lRn)3f
t
(1)
(8.4.29)
F (x) d x ~ &
(7
where (1) is the usual integral of f over K , is t h e embedding (8.2.9) , (3) i s the integral (8.4.20) of the generalize function F over K , and (4) is the association F(x)dx c f (x)dx defined in (8.4.12). K K
I
Remark 4 The following two properties w i l l be useful in the sequel. Suppose given F E G(R"), $ E p(iRn) amd K c IRn compact, with supp c K. Then, w i t h the product $.F defined in !2(lRn), the integral of $.F over K , a s defined in (8.4.20), does not depend on K . Therefore, we s h a l l denote
Suppose given T,S supp $ c 61. Then
E
p'(lRn),
$ E 2(IRn)
and
fl c lRn
where the products $-T and $.S a r e computed in (8.4.31) follows easily from (8.2.16) and (8.4.20).
open,
6(!Rn).
such that
Indeed,
An extension of (8.4.27) which is essential in t h e coupled calculus defined in the next Section, is presented i n :
Colombeau' s p a r t i c u l a r a l g e b r a
Theorem 7 Suppose g i v e n $ E V(lRn) c(lRn), we have
and
T E V'(lRn),
t h e n , w i t h t h e product i n
As f o l l o w s from Chapter 1, no s i n g l e d i f f e r e n t i a l a l g e b r a is f u l l y s u i t e d t o handle i n a s u f f i c i e n t l y g e n e r a l wa t h e conf 1i c t ing i n t e r p l a y between arbitrary multiplication and indefinite derivability o r partial d e r i v a b i l i t y of g e n e r a l i z e d f u n c t i o n s . I n view of t h a t , it f o l l o w s t h a t addi t iona 1 s t r u e t u r e s may be r e q u i r e d on such d i f f e r e n t i a l a l g e b r a s . T h i s of c o u r s e a p p l i e s t o 6(IRn) a s well. And t h e n , Colombeauls method d e f i n e s an a d d i t i o n a l s t r u c t u r e by turo s p e c i a l equivalence r e l a l i o n s on 6(lRn), which t o g e t h e r with t h e u s u a l e q u a l i t y , a r b i t r a r y m u l t i p l i c a t i o n and i n d e f i n i t e p a r t i a l d e r i v a t i o n on 6(lRn), can be seen as a coupled calculus
A motivation f o r t h e way t h i s coupled c a l c u l u s is d e f i n e d , is presented first.
One of t h e b a s i c f e a t u r e s of t h e 1 i n e a r t h e o r y of d i s t r i b u t i o n s , Schwartz [I] , is t h e f o l l o w i n g . Given a f i x e d d i s t r i b u t i o n T E V'(lRn) , t h e n , f o r every t e s t f u n c t i o n $ E V(lRn) , one can d e f i n e t h e i n t egra 1 of t h e produc t $.T E 'D' (lRn) , by
Indeed, i n t h e p a r t i c u l a r c a s e when have 9.T = $.f E C ) ~ , ( R ~ c) V' (lRn) , sense of ( 5 . 4 . 2 ) .
It f o l l o w s t h a t t h e d i s t r i b u t i o n s T
T = f E ,CiOc(lRn) c l ( l R n ) hence (8.5.1)
we s h a l l
h o l d s i n t h e usual
E V'(lRn) can be c h a r a c t e r i z e d by t h e i n t e g r a l s (8.5.1) of t h e i r d i s t r i b u t i o n a l p r o d u c t s $.T with a r b i t r a r y
E. E. Rosinger
test functions $ E 3(IRn). Indeed, when 3(lRn), the corresponding numbers
~
in (8.5.1) ranges over a l l of
offer a local characterization of the fixed distribution T. And then, through the converse of the distribution localization principle mentioned i n (5.A1.7)- (5.A1.9), we obtain an overall, global characterization of the fixed distribuiton T.
It should be noted that within the linear theory of distributions, no point value T(x), a t x E lRn, i s associated with arbitrary distributions T E 3'(lRn), therefore the above local characterization in (8.5.1) does indeed play a special role. In particular, we obtain
In various applications of the generalized functions in G(lRn) -. for instance, the solution of linear and nonlinear partial different la1 equations, as presented i n the sequel - it proves t o be particularly useful to e2t end the above properties (8.5.1) and (8.5.2) from 3' (lRn) t o G(lRn). In f a c t , t h i s i s the essence of the coupled calculus i n the method of Colombeau. Let us now present the three concepts involved i n the mentioned extension. Then, we shall present their basic properties, which w i l l elucidate t h e i r role, as well as the way Colombeau's co%pled calvclus operates. It is particularly important t o point out that, although the next three definitions and related basic properties, as well as those in the previous Sections 3- 5, may a t f i r s t seem somewhat unusual and involved, Colombeau's coupled calculus i s i n fact by far the simplest way known so f a r i n the literature in order t o overcome the constraints inherent in any nonlinear theory of generalized functions, mentioned i n Chapter 1. However, i n view of Chapter 6 , that simplicity of approach may as well happen the impinge on i t s effectiveness. The extent t o which that may indeed be the case i s illustrated by the fact that the results i n Chapters 2-4 and 7 , could not so f a r been possible t o reproduce within Colombeau's algebra G(lRn). I t may appear that a convenient extension of (8.5.2) would be given by the following definition. A generalized function F E G(lRn) i s called test nu1 1, denoted F 0, if for every $ E 3(lRn) we have N
Colombeau ' s particular algebra
where t h e product is computed i n of (8.4.30).
4(IRn) and the integral i s in the sense
Two generalized functions FI ,F2 E 4(IRn) FI FZ, i f Fz - F1 i s t e s t n u l l , i . e . ,
a r e called test equal, denoted Fz - FI 0.
Remark 5 Obviously,
N
i s an equivalence relation on G(IRn).
However, as seen below in Theorem 7, the equivalence r e l a t i o n is too r e s t r i c t i v e , therefore, we need a more general equivalence relation defined as follows. N
A distribution T E p'(IRn) i s said t o be associated with a generalized function F E 4(IRn) , in which case we denote F 11- T, i f , f o r every $ E p(IRn), we have
where both products a r e computed in in the sense of (8.4.30) .
4(IRn), while the integrals a r e taken
Finally, two generalized functions Fl ,F2 E G'(IRn) a r e said t o be associated, denoted F1 m F2, i f F2 - F1 has 0 E v(IRn) a s associated distribution, i . e . , F2 - FI 11- 0. Remark 6 Obviously,
G
is an eqsivalence relation on G'(lRn).
As suggested by (8.5.4), the binary relation neither reflexive, nor symmetric.
11-
c G(IRn)
x
3' (01")
is
We s h a l l now present the basic properties connected with the above definitions. The properties will, among others, s e t t l e the relation between the usual function, respectively distribution multiplications
E.E. Rosinger
and their corresponding versions in
4(lRn)
First, we note that in view of (8.5.3) and (8.5.4), we have for F1 ,F2 E G(R") the relation
Similarly, if F (8.5.8)
E
g(lRn) and T E 2)'(lRn),
then
FII-TBFMT
Further, in view of (8.4.32), we have for every distribution T (8.5.9)
TNOJT=O
(8.5.10)
TwO+T=O
E
V' (R")
in other words, the equivalence relations and :: defined on 4(lRn), coincide with the usual equality = when restricted to V' (iRn) . Now, we can present the relation between the classical product of continuous functions in (8.5.5) and their product in 4(lRn). Theorem 8 Suppose given two continuous functions fl ,f2 E Co(iRn). product fl .f:! E CO (lRn) being a continuous function, distribution, i.e., fl,f2 E v'(lRn). On the other hand, we the generalized functions F1 ,F2 E 4(lRn) which correspond to respectively, according to the embedding (8.2.8) . Then, the product F1 'F2 in 4(iRn) is such that
E
Their usual is also a can consider fl and f2
4(iRn) of these generalized functions computed
i.e. the distribution fl sf2 Fl 'F2.
is associated with the generalized function
In other words, we have the commutative diagram
Colombeau ' s particular algebra
where (1) i s the usual product of continuous functions, ( 2 ) i s defined by (8.2.9), (3) i s the product of generalized functions i n 6(lRn), and (4) i s the relation of association 11defined i n (8.5.4), or the equivalence relation a I t i s particularly important t o note that the result in Theorem 7 concerning the relation between products of continuous functions effectuated i n C.o(lRn) and G(lRn) does not hold if i n (8.5.11 we replace x by . In fact, t h i s i s one of the main reasons why Colom eau's coupled calculus uses the ueaker equivalence relation a .
1
Concerning the relation between the distributional product i n (8.5.6) and i s version i n 6(Rn), we have: Theorem 9 Suppose given y, E P ( R " ) and T E P'(Rn) and l e t us denote by S = X - T E P'(lRn) t h e i r distributional product. On the other hand, considering their product as generalized functions i n G(lRn), we have F = xoT E G(Rn), where for the sake of clarity within the present theorem, we denoted by . the multiplication i n 2" (lRn), see (8.5.6), and by 0 the multiplication i n G(lRn). Then
i . e . , the distributional product generalized function product F = ,y,T.
S = ,y-T
i s associated with
the
I n other words, we have the commutative diagram
where (1) i s the distributional product (8.5.6), (2) i s defined by (8.2.2)
E .E. Rosinger
and (8.2.14), (3) is the product of generalized functions in ~ ( p ) ,and o , defined in (8.5.3). (4) is the equivalence relation N
An important property of the equivalence relations and z is their compatibility with the partial derivatives in P(P).Indeed we have: N
Theorem 10 If F l ,Fz E
~(p) and
p
E
p , then
We can recapitulate by noting the following components of Colombeau's coupled c a l c u l u s :
(1) g(IRn) is an associative and commutative algebra, with arbitrary partial derivative operators
which are linear mapping and satisfy the Leibnitz rule of product derivatives. (2) The vector space embedding
is such that the partial derivative operators (8.5.17) coincide with the usual distributional partial derivatives , when restricted to p' (IR"). In particular, DP in (8.5.17) coincides with the usual partial derivative D~ of smooth functions, when restricted to , ) n o ' . ? ( with !€ , 1 2 IpI. (3) The particular case of (8.5.18) given by
is an embedding of differential algebras.
(4) The particular vector space embedding
Colombeau's particular algebra
defined b (8.5.18) is not an embedding of algebras. This fact is unavoidab e, owin! to the conflict between insufficient smoothness, multiplication an differentiation, in particular, owing to the so called Schwartz impossibility and other related results, see Chapter
!
1.
Colombeau's coup1 ed c a l c u l u s aims, amongh others, to overcome the difficulty in (4) above. This is done in the following way. An equivalence relation N is defined on F,G E g(lRn), by the relation
4(lRn),
i.e., for arbitrary
is compatible with the vector space structure This equivalence relation of 4(lRn) and the partial derivative operators (8.5.17) on G(R"). Moreover, if T,S E P' (lRn) then
The interest in the equivalence relation then property: if x E P(lRn), T E V'(lRn)
where 0 and respectively.
.
N
comes from the following
denote the multiplications in
4(lRn) and
9'(!Rn)
However, in order to handle the difficulty in (4) above, the equivalence relation is too strong. Therefore, a weaker equivalence relation F: is defined on 4(!Rn), i.e. for arbitrary F,C E g(lRn), in the following way
This equivalence relation a is again compatible with the vector space structure of 4(Ifln), as well as the partial derivative operators (8.5.17) on 4(lRn). Further, if T,S E V' (lRn) then again
332
E.E. Rosinger
The e s s e n t i a l property of the equivalence relation x which settles the issue connected with (4) above, is the following. If f ,g E co(Rn) and F,G E i(Rnl are the generalized functions which corresponds to f and g respective y, then
where the multiplications in the left and right hand terms are in 9(Rn) and Co (Rn) respectively . In this way, Colombeau's coupled c a l c u l u s on the differential algebra 9(Rn) means in fact the a d d i t i o n a l consideration of the tloo equivalence r e l a t i o n s on 9(Rn) given by N and x . One can obviously ask whether it would be convenient to factor 9(!Rn) by and/or x , and in view of (8.5.22), (8.5.25), obtain the following embeddings of vector spaces
and hence do away with the additional complication brought about by Colombeau ' s coupled calculus. However, it is easy to see that 9(Rn)/, Rosinger [3] .
and 9(!Rn)/_
are not algebras,
Nevertheless, the equivalnce relation , and especially a , prove to be particularly useful in the stud of e n e r a l i z e d s o l u t i o n s of nonlinear partial differential equations. fnde!, let us take for instance, the shock wave equation
Even in case U E C1 ((o,~)xR)~P((o,~)xR) is a classical solution of , it is likely that will not be a generalized solution of when considered with the multiplication in G((O,m)xF), owing to in (4) above. But, in view of (8.5.26), U will obviously satisfy
with the multiplication in ((0,)~). It follows that in order to find the classical, weak, generalized, etc., solutions of the usual nonlinear partial differential equation (8.5.28), we have t o s o l v e the equiva 1 ence r e l a t i o n in (8.5.30), within G((0,m)xlR). Details are presented in Rosinger [3] .
Colombeau' s particular algebra
$6. GENERALIZED SOLUTIONS OF NONLINEAR WAVE EQUATIONS IN QUANTUM FIELD
INTERACTION
For simplicity we shall only consider the scalar valued case of nonlinear wave equations. The class of equations are of the form
where F : IR
--+
R are C"- smooth functions, such that F(0) = 0
(8.6.2) and
I t follows that F need not be bounded. F(u) = au + b sin u ,
(8.6.4) w i t h given a,b equation.
E
R,
u
For instance, we can have E
R
i n which case (8.6.1) i s a version of the Sine-Gordon
Before oing further, we have to note that since F i s only defined on R, it can %e applied but t o r e a l v a l u e d generalized functions U which are defined as follows. The generalized function G E G(Rn) i s called real valued, if and only if there exists a representation
such that x E Rn.
g((,x)
i s real vlued for a l l real valued
In view of the above we have G E G(iRn).
F(G)
E
G(R"),
( E 9,
and a l l
for every real valued
The nonlinear wave equation (8.6.1) w i l l be considered w i t h the Cauchy i n i t i a l value problem
E.E. Rosinger
where uo ,ul tions.
E
G(R3)
are arbitrary, given real valued generalized func-
Within the above rather general framework, we have the following existence result. Theorem 11 (Colombeau [2,4] ) The initial value problem
U(0 ,x) = uo (x) , x
(8.6.8)
a
E
R3
U(0,x) = u, (x) , x E IR3
has real valued generalized function solutions U E G(R4), for every pair of real valued generalized functions uo ,ui E G(R3) . Theorem 12 (Colombeau [2,4] ) The solution U E G(R4) of the problem (8.6.7)- ( 8 . 6 . 9 ) in Theorem 10 is o unique. The above uniqueness result shows that inspite of the generality of the framework in which the nonlinear wave equation and initial value problem 8.6.7)- (8.6.9) are considred the solution method is rather foeassed, as it elivers a unique solution within that general framework.
d
In order to see the appropriateness of that focussing it is useful to compare the unique generalized solutions with the unique classical solutions whenever the latter exist. In this respect, we shall consider the following tuo cases. Case 1 The classical P-smooth situation when uo, ul E P ( R 3 ) and we have a unique classical solution V E P ( R 4 ) . Then, in view of the embedding (8.2.2), it is obvious that
where U
E
P(IR4) is the unique generalized solution.
Colombeau's particular algebra
Case 2 When uo E C3(IR3), u1 E C2(IR2) and we have a unique classical solution V E Ca(IR4), see Colombeau [2, p. 2201. In that case we have the following coherence result. Theorem 13 (Colombeau [2, 41) Suppose V E C2(R;) and U E $(iR4) are the unique classical and generalized solutions o the nonlinear wave equation and initial value problem (8.6.7)- (8.6.9), corresponding to
Then U
E
P' (IR4)
and
and for every to E IR, we have
8 7.
vI
E
p'(IR3)
as well as
t=to
GENERALIZED SOLUTIONS FOR LINEAR PARTIAL DIFFEBENTIAI, EqUATIONS.
As mentioned in Chapter 5, the early history of the linear theory of the Schwartz distributions had known quite a number of momentous events, both for the better and for the worse. One of the first major successes was the proof of the existence of an elementary solution for every linear constant coefficient partial differential equation, which was obtained in the early fifties by Ehrenpreis, and independently, Palgrange. Soon after, in 1954, came the famous and improperly understood, so called impossibility result in Schwartz [2] . Another, rather anecdotic event, is mentioned in Treves [4], who in 1955 was given the theses problem to prove that every linear partial differential equation with (?-smooth coefficients not vanishing identically at a point, has a distribution solution in a neighbourhood of that point. The particularly instructive aspect involved is that the thesis director who suggested the above thesis problem was at the time, and for quite a while after, one of the leading analysts. That can only show the fact that around i955, there was hardly any understanding of the problems involved in the local distributional solvability of linear partial differential equations with (?- smooth coefficients, see Treves [4].
E.E. Rosinger
336
As mentioned in Chapter 5, a very simple and clear negative answer to the above thesis problem was soon given b Lewy in 1957, who showed that the following quite simple linear partial iifferential equation
cannot have distribution solutions in any neighbourhood of any point
x
E
W3, if f
E
p(W3)
belongs to a rather large class.
The solvability of linear partial differential equations with Coo-smooth coefficients failed to be achieved even later when, the Schwartz 3' distributions were extended by other linear spaces of peneral ized functions, such as the hyperfunctions, Sato et. al. That ailure was proved for instance in 1967, in Shapira. As mentioned in Chapter 5, a sufficiently eneral characterization of solvability, and thus of unsolvability, for inear partial differential equations with C?-smooth coefficients has not yet been obtained within the framework of Schwartzts linear theory of distributions. And that inspite of several quite far reaching partial results which make use of rather hard tools from linear functional analysis as well as complex functions of several variables.
I
d
In view of the above it is the more remarkable that or the first time ever in the study of various generalized functions, Colom eauts nonlinear theory does yield local generalized solutions for practically arbitrary systems of 1inear partial differential equations. Furthermore, under certain natural growth conditions on coefficients, one can also obtain global generalized solutions for large classes of sys l ems of 1inear partial differential equations, systems which contain as particular cases most of the so far unsolvable linear, P- smooth coefficient partial differential equations, see Colombeau [3,4] . Without going into the full details - which can be found in Rosin er [3] and Colombeau [3,4] - we shall present the main results and a few il ustrations.
k
The systems of linear partial differential equations whose generalized solutions will be obtained within Colombeau's nonlinear theory, contain as particular cases systems of the form
with the initial value problem
Colombeau ' s particular algebra
where P i j c LNn are f i n i t e , while aijpy bi and ui are P-smooth. I t i s well known, Treves [2], that under very general conditions, arbitrary systems of linear partial differential equations with P - smooth coeff icients and i n i t i a l value problems can be written i n the equivalent form (8.7.2), (8.7.3). Our aim i s t o find generalized functions
which in a suitable sense, are solutions of (8.7.2) and (8.7.3), or of even more general systems, see (8.7.19) .
A basic remark which conditions much of the way such generalized solutions are and can be found i s the following. The system (8.7.2), (8.7.3) c a n n o t i n general have solutions (8.7.41, if D t y D: and the respective equality relation = are considered i n G(fUnili, i n the usual way defined i n t h i s Chapter. The argument for that i s rat e r classical - see for details Colombeau [3] - and it i s based on a contradiction between Holmgren type uniqueness and general nonuniqueness results i n the P-smooth coefficient case, see also Treves [Z] and Colombeau [4] .
-
Furthermore, if we replace the equality relation = by the equivalence relation or i n E ( R " + ~ ) , the system (8.7.3), (8.7.4) w i l l s t i l l f a i l t o have solutions (8.7.4), see again Colombeau [3] .
-
A way our from t h i s impasse i s t o replace the partial derivatives
in D! defined next. The 6(IRnt1) by the more smooth partial derivatives effect of such a replacement i s very simple, yet crucial. Indeed, as seen in 6 coincide w i t h earlier i n t h i s Chapter the partial derivatives D!
h~E
the classical partial derivative when restricted t o P-smooth functions. Therefore, they are not sufficiently smooth in the following well known sense that an f'-type bound on a p-smooth function does not imply any I?- type bound on i t s classical derivatives. Contrary t o that situation, the more smooth partial derivatives h ~ : on G(IRntl), defined next, w i l l have the property (8.7.15) . Suppose given a function
E. E. Ros inger
338
(8.7.5)
h : ( 0 , ~ )-, (0,m) with
l i m h(c) = 0 €10
such a s f o r instance l/en(l/f) (8.7.6)
h ( ~ )=
if
c E
(0,l)
arbitrary otherwise
Such a function h satisfying (8.7.5) will be called a derivation rate. Before defining the smooth derivatives, we note the following property
3 $
E
(8.7.7) *)
a($),
6
>0
:
diam supp 9 = 1
and f o r given , one obtains $ and in a unique way. In view of t h a t , we s h a l l in the sequel often exchange with $€, according t o condition **) in (8.7.7). Given 1 < i
< n,
as follows:
if
we define the h-partial derivative
then
where
and * is the usual convolution of functions. It is easy t o see t h a t t h e above definition is correct, since indeed g E A, while hDxiF does not depend on f E A in (8.7.9).
Colombeau' s particular algebra
Obviously (8.7.11) yields
Given now p = (pl ,. . . ,pn) h-partial derivative
E
DI",
with
lpl ? 1 ,
we can define the
as the iteration
of the h- partial derivatives i n (8.7.8). I t i s easy t o see that the above definition i s correct, since the h- partial derivatives (8.7.8) are corn mutative. Obviously, the h- partial derivatives (8.7.8), and thus (8.7.13), are 1inear mappings. The essential smoothing property of the h- partial derivatives defined above i s apparent in (8.7.12). Indeed, given a compact K E iRn, we obviously have for 4 E @(IRn) the estimate
where K' = K + h(6)supp $ c iRn
i s compact and
In t h i s way, an I?-type bound for g((,.) on K can be obtained in terms on a bounded neighbourhood of K. of an I?-type bound for f ( $ € , .) Therefore, a representative g of the h- partial derivative hD F i s x I: locally bounded by a representative f of F. I t should be noted that in view of (8.7.7), we obviously have fi, c , r and hence h(6 in (8.7.151 depending only on ( and not on K. Finally, the justi ication of t definition (8.7.11 i s i n (8.7.12), which leads t o (8.7.15). And a l l that i s based on the c assical property of convolution t o commute with partial derivatives .
?
j
E. E. Rosinger
340
Now, we present several basic properties which show the coherence between the h-partial derivatives and the usual partial derivatives, when both are applied to important classes of generalized functions i n G(R"). Theorem 14 If
T E 9' (!Rn)
and p E INn, then
Theorem 15 For x E P(!Rn) and T E V'(!Rn) let us denote by XT E V'(!Rn) and x.T E g(!Rn) the classical product in V', respectively the product i n E. Then XT N x.T, see (8.5.13). Furthermore, for p E INn we have
Corollarv 2 For
,y
E
P(!Rn) and
T
E
I(!Rn),
the h-partial derivatives
hDxi,
with
1 < i < n , satisfy the following version of the Leibnitz rule of product derivatives
The basic result which uses h-partial derivatives and leads to the existence of generalized solutions for systems containing those i n (8.7.2 , (8.7.3) i s presented now. I t suffices to formulate i t for one sing e linear partial differential operator of the type
i
where P c Eln i s f i n i t e , while a E P(!Rn), with p E P. P
Colombeau ' s particular algebra
Theorem 16 If
U E 71t(lRn) then the following three conditions are equivalent:
With the classical multiplication i n P'(ln)
For any family
(hplp E P)
we have
of derivation rates, we have w i t h the
multiplication in E(!Rn)
I E P) P g(IRn) we have
There exists a family multiplication i n
(h
of derivation rates, such that with the
In view of Theorem 15 above, we are naturally led to the following: Definition 1 Suppose given the linear partial differential equation
where P c 01"
i s f i n i t e , ap s F(IR"), w i t h p
E
01,
and b
E
Vt(lRn).
A generalized function U E E(lRn) i s called a Colombeau weak s o l u t i o n of (8.7.23), if and only if there exists a family (h ( p E P) of derivation P rates such that
Indeed with t h i s definition we obtain:
E.E. Rosinger
Corollarv 3
A distribution U E P' (Rn) is a Colombeau weak solution of (8.7.23) , if and only if it satisfies that equation in the sense of the classical operations in P' (Rn) . The above definition and corollary extend in an obvious way to linear systems such as (8.7.2). As is known for instance from the mentioned example of Lewy, linear partial differential equations (8.7.23) do not in general have distribution solutions. However, as seen next, systems (8.7.2), (8.7.3) have Colombeau weak solutions under rather general conditions. These solutions are global on IRn+', that is in t and x. Theorem 17 Suppose that each of the coefficients a and bi in (8.7.2) satisfies ijp the condition written generically for c
for every bounded I c R and q E Hn+l Further, let us suppose that each of the initial values satisfies the condition written generically for u
for every p
E
ui
in (8.7.3)
Hn.
Then there exists a Colombeau weak solution
for the system (8.7.2), which also satisfies the initial value conditions (8.7.3). o As an easy consequence we obtain the following l o c a l existence result which does not require any boundedness conditions on coefficients or initial values.
Colombeau' s particular algebra
Corollarv 4 In every s t r i p
with L > 0 ,
there exists a Colombeau weak solution
which s a t i s f i e s (8.7.2) in A, and also s a t i s f i e s the i n i t i a l value conditions (8.7.3) f o r x E IRn, 1x1 < L The power of the local existence result in Corollary 4 above can easily be seen, as it yields Colombeau weak solutions in every strip of type (8.7.28), f o r various distributionally unsolvable linear p a r t i a l different i a l equations with COO-smooth coefficients. For instance, Lewy's equation (8.7.1) can be written in the equivalent form
which w i l l have Colombeau weak solutions in every s t r i p A=IRx{xER~~< [ xL )} c R 3 , L > O . Similarly, Grushin ' s equation can be equivalently written as (8.7.31)
D t U = -itDxU+ f , t E IR,
x E R
hence it w i l l have Colombeau weak solutions in every s t r i p A = I R x [-L,L] c R 2 , with L > 0. The same applies t o the i n i t i a l value problem f o r the Cauchy-Riemann equation (8.7.32)
Dt = -iDxU, t E R, x
E
R
which, as is well known, cannot have distribution solutions even locally if uo E P(W) i s not anlytic, since the only distribution solutions of (8.7.32) a r e a n a l y t i c in z = t + ix.
It should be noted that Lewy's equation (8.7.30) does not s a t i s f y the boundedness conditions (8.7.25 , owing t o the coefficient 2 i ( t + ixl ) . Hence, with the methods in t h i s ection, we cannot obtain f o r i t global Colombeau weak solutions.
d
344
E.E. Rosinger
1
On the other hand, Grushin's equation (8.7.31 satisfies (8.7.25), if and only if f satisfies that condition, in whic case we have g l o b a l Colombeau weak solutions for it. Thus, The Cauchy-Riemann equation (8.7.32) obviously satisfies (8.7.25). if uo satisfies (8.7.26), then (8.7.32), (8.7.33) will have g l o b a l Colombeau weak solutions. Concernin the coherence between the Colombeau weak solutions obtained by the metho! in the proof of Theorem 16 and known classical or distributional solutions, a series of examples of familiar linear partial differential equations are studied in Colombeau [3,4]. Here we mention the following coherence results. If (8.7.2) is constant coefficient hyperbolic, the Colombeau weak solutions coincide with the classical ones. If (8.7.2), (8.7.3) is analytic, the Colombeau weak solutions coincide with the classical analytic ones. Similar results hold for classes of parabolic or elliptic equations As mentioned, see also Treves [2], one cannot expect uniqueness results in Theorem 16 or Corollary 4, since systems (8.7.2), (8.7.3) can even have nonunique P-smooth solutions. The method of proof for Theorem 16 can be extended to systems (8.7.2), (8.7.3) with more general coefficients and intial values, which are no longer P-smooth, but can be distributions or even generalized functions. In fact, the method of proof for Theorem 16 is nonlinear, thus it can yield Colombeau weak solutions for n o n l i n e a r s y s t e m s of partial differential equations, see Colombeau [3,6] .
Colombeau ' s p a r t i c u l a r a l g e b r a
APPENDIX 1 THE NATURAL CHARACTER OF COLOMBEAU'S DIFFERENTIAL ALGEBRA It is i n t e r e s t i n g t o n o t e t h a t t h e d e f i n i t i o n s of t h e a l g e b r a A and i d e a l 1 g i v e n i n (8.1.18) and (8.1.19) r e s p e c t i v e l y , have a r a t h e r n a t u r a l c h a r a c t e r , i n s p i t e o f what at f i r s t s i g h t may appear t o be a n ad- hoc one. Indeed, l e t u s r e c a l l a few well known p r o p e r t i e s from t h e l i n e a r t h e o r y of d i s t r i b u t i o n s , Rudin, w i t h i n t h e framework of a g i v e n Euclidean s p a c e Rn. Let u s d e f i n e t h e mappping
Obviously, LT is a well d e f i n e d l i n e a r mapping o f has t h e following t h r e e properties
V
into
p, which
(8.A1.3)
LT is continuous with t h e u s u a l t o p o l o g i e s on 2 and Cm
(8.A1.4)
commutes with every t r a n s l a t i o n LT x E Rn
(8.A1.5)
LT commutes with every p a r t i a l d e r i v a t i v e
T~ :
Rn
-+
Rn , with
D ~ with , p
E
These p r o p e r t i e s a r e i n f a c t well known p r o p e r t i e s of t h e convolution of d i s t r i b u t i o n s .
~1"
*
The n o n t r i v i a l f a c t is given by t h e f o l l o w i n g two converse p r o p e r t i e s . Suppose g i v e n a l i n e a r mapping
which is continuous i n t h e u s u a l t o p o l o g i e s on s a t i s f i e s (8.A1.4) t h e n t h e r e e x i s t s T E V' such t h a t S i m i l a r l y , suppose given a l i n e a r mapping
and CO L = LT.
If
L
E.E. Rosinger
which is continuous in the usual topologies on V and f'. If L satisfies (8.A1.5) then there exists T E 'D' such that L = LT. In short, the convolution * of distributions is the unique bilinear form which commutes with translations or partial derivatives. Let us now recall the arguments in Section 1 on smooth approximations and representations which were expressed in (8.1.8) by the so called inclusion
It was further argued that all what was to be done was to replace it by a proper inclusion or embedding, as for instance in (8.1.10). In view of that, we are obviously interested in suitable mappings
For convenience, let us simplify the issue, by only considering the following restriction of (8.A1.9)
which is equivalent with
Now, if we request that for each c E (0,a) , the mappings in (8.A1 .11) are cant inuous with the usual topologies on V and C?, and that they corn muute urith a1 1 partial derivatives, then according to the above, there exist Tc E V', with 6 E (O,m), such that
But in view of the argument in Section 1, it is natural to require that t6 -+ T in 3' , when +0 This means in view of (8.A1.12), that we can further assume the property (8.A1.13)
TE + 6 in a', when c -+ 0
Therefore we can in fact assume that
in which case (8.A1.12) can be extended to T initial question of the mappings (8.A1.9).
E
V',
and that answers the
Colombeau ' s particular algebra
Recapitulating the above, we are led t o a general form for (8.A1.9), given by mappings
with
since i n the previous argument we could take 9 = {TfI E
E
(0,m)).
The essential points so f a r are i n the condition (8.A1.16) on the i n d e x s e t 9, and i n the presence of the c o n v o l u t i o n * of distributions i n (8.A1.17). Here it should be noted that, as i t follows from (8.2.10), i n Colombeau's theory the convolution in (8.A1.17) i s replaced by the following one
where
J(x) = ( ( - x ) ,
for ( E P ,
X E W "
Now, we are in the position t o obtain the needed insight into the n e c e s s a r y s t r u c t u r e of the sets 9, A and 1, which are fundamental i n Colombeau's theory.
\
Let us proceed f i r s t w i t h 9. I n view of (8.A1.13) and 8.A1.14), it i s natural t o ask condition *) i n (8.1.11), as well as the f o l owing one
was used. Condition **) in (8.1.11) as mentioned i n Section 1.
where the notation i n required for the diagrm
is
Turning t o A, it i s now obvious that it has t o be a partial derivative which contains a t least a l l the mappings invariant subalgebra in (P)'
But
for
f
EC",
PEN",
/ €9,
E
E (0,m)
and
XEIR".
Hence the elements
E .E. Rosinger
of d can exhibit a polynomial growth in I/€, depending on p # E a.
E
Nn and
A usual way to measure such a growth is to restrict the above p-smooth functions I?(#, .) and DP~(#,.) to compact subsets K c Rn. That being done, the condition in the definition (8.1.18) of d will follow now in a natural way.
Concerning I, it obviously has to be a partial derivative invariant ideal in A, subject to the additional conditon in (8.2.28). That latter conditions means
where
for f E COD(@). In particular, in the one dimensional case, when n we have for given m E N+
(8.A1.24)
D~%(#,,X)
=
"DPtqf(x) c q
I
=
1,
1 (T)~#(Y)$ + R
€
€
E N and suitably chosen 0 E (0,l). Hence, with 4 E a, x E R, then in (8.1.11) implies that all the integrals if 4 E under the above sum will vanish, and we remain with
In this way the elements of Z can behave as polynomials in E, depending E and p EN". on .EN+, Then, an argument similar with the one used above for A, will lead us to the definition of I given in (8.1.19). We can shortly recapitulate as follows. If we want to have (8.Al. 11) under any form, such as for instance
Colombeau s particular algebra
with have
I
(8.A1.27)
an infinite index s e t , and if we want that f o r every : P
( E i
we
P linear, continuous 0
P3Tk
*ti€C"
1
(8.A1.28)
commutes for p
n 3 DPT c--- D P ~ (E @i
then there exist T E 'D',
d
with
4
E
i,
E
INn
P such that
and the rest of the above argument will follow. In particular, differential algebras containing the 2' distribution and which are based on sequential smooth approximations as i n (8.A1.26), yet are different from Colombeau's algebras will f a i l t o satisfy a t least one of the conditions (8.A1.27) or (8.~1.28). Such algebras are studied under their general form i n Rosinger [1,2,3] and Chapters 2-7 in t h i s volume. Their u t i l i t y becomes apparent, among other, in connection with the possibility of increased stab i 1 i t y of generalized solutions of nonlinear partial differential equations.
It should be pointed out that a different but not less convincing argument about the natural character of Colombeauts differential algebras i s presented in Colombeau [I, pp 50- 661 . For convenience we present it here i n a summary version. Let us remember that the linear space 3' (IR") of the Schwartz distributions i s the set of a l l 1 inear and continuous mappings
Therefore, if we look a t distributions as complex valued functions defined on P(lRn) , it i s natural t o try to define the product of two distributions
as the usual product of complex valued functions, that is
E.E. Rosinger
where
t h a t is t h e s e t of I n t h i s way it is n a t u r a l t o t r y t o embed 'D'(Rn), 1i n e a r and continuous mappings (8. A1.30) , i n t o a d i f f e r e n t i a l a l g e b r a f ( ' D f R n ) ) of f - s m o o t h complex valued f u n c t i o n s on ' D R . However, a n embe d i n g
does r a i s e two immediate d i f f i c u l t i e s . F i r s t , one h a s t o f i n d a s u i t a b l e concept of p a r t i a l d e r i v a t i o n f o r complex valued f u n c t i o n s on p(Rn) , such t h a t (8.A1.34) w i l l h o l d and t h e p a r t i a l d e r i v a t i o n of f u n c t i o n s i n f('D(iRn)) w i l l extend t h a t of d i s t r i b u t i o n i n 'D'(iRn). T h i s problem h a s been d e a l t with i n Colombeau [5]. The second d i f f i c u l t y is more elementary and it is a l s o more b a s i c . Indeed, t h e m u l t i p l i c a t i o n i n (8. A1.32) does not even g e n e r a l i z e d t h e u s u a l m u l t i p l i c a t i o n of f u n c t i o n s i n f ( l l I n ) . I n o r d e r t o s e e t h a t , l e t us assume t h a t it does, and let u s t a k e f l , f 2 E P ( [ f n ) , and d e n o t e by
t h e usual product of f u n c t i o n s . Let u s denote by T I , T2, T E 9' (iRn) t h e d i s t r i b u t i o n s g e n e r a t e d by f l , f 2 and f r e s p e c t i v e l y , a c c o r d i n g t o (5.4.2). Then i n view o f (8.A1.35), we should have (8.A1.36)
T = Ti .T2
i n t h e s e n s e of (8.A1.32). Hence according t o (8.A1.33) we would o b t a i n
which is obviously f a l s e f o r a r b i t r a r y f l ,f2 E f ( i R n ) . However, t h i s second d i f f i c u l t y need not be f a t a l : indeed, one can n a t u r a l l y t h i n k about u s i n a s u i t a b l e quotient structure on f('D(iRn)) which f a c t o r s o u t t h e d i f e r e n c e between t h e r i g h t and l e f t hand terms i n
f
Colombeau' s particular algebra
351
(8.A1.37). Fortunate1 , such a quotient structure - which because of the multiplication involvedl i n (8.Al.37) should rather be a ring or an algebra - can easily be constructed. Indeed, let us f i r s t notice that V(lln) is dense i n E'(IRn), Schwartz [I], ,and E'(lRn) i s a De Silva space, Colombeau [5,1], therefore the restriction mapping
is injective.
Hence we can consider the e m b e d d i n g
The idea i s f i r s t t o try to correct on the smaller space C?'(f'(IRn)) the lack of identity i n (8.A1.37). For that we recall the relation E' ' (lRn) = e(lRn) , Schwartz [I] . Hence, each linear and continuous functional F E E" (lRn) can be identified with the function f E Cf'(lRn) defined by
where Sx i s the Dirac form defined by duality.
S distribution at x and < , > i s the bilinear
The fact of interest which follows now i s that in view of (8.A1.40), the elements of E" (iRn) can easily be multiplied according to (8.A1.41)
F = F1.F2,
Fl,F2
E &"(iRn)
where we define F by
which i s nothing b u t the usual multiplication of the corresponding funct ions i n P ( R " ) . Now we note that
according, for instance, to the partial derivation on E'(IRn) used i n Colombeau [5, I] . Hence (8.A1.40) suggests the definition of an equivalence relation : on Cf'(&'(!Rn)) as follows:
E. E. Rosinger
given FI ,F2
E
then we d e f i n e
P(f'(lRn)),
is obvious.
The u t i l i t y of t h i s equivaslence r e l a t i o n d e f i n e t h e l i n e a r mapping
Indeed, let us
i i n (8.A1.40). which is t h u s an extension of t h e canonical mappin is easy t o s e e t h a t a is well defined. Indeed, t e mapping Rn 3 x e dX E &' (R") is C"-smooth, Colombeau [5,1] , hence
R.
a(F) E P(lRn) , f o r F
E
It
P(&'(Rn) ) .
Obviously, we o b t a i n t h e following commu la2 ive diagram
and furthermore (8.A1.48)
k e r a = {F
E
P ( E ' (lRn)) IF
5
0)
It follows t h a t (8.A1.49)
P(&'(lRn))/ker a ,
and P(lRn) a r e isomorphic a l g e b r a s
I n t h i s way we have found a s u i t a b l e f a c t o r i z a t i o n on P ( E ' (lRn)) by t h e ideal ker a , which does indeed c o r r e c t t h e d i f f i c u l t y mentioned i n (8.A1.37). to a Now it only remains t o extend t h e i d e a l k e r a of P(&'(lRn)) o r i n an a p p r o p r i a t e subs i m i l a r l y s u i t a b l e i d e a l 1 i n C"(9(lRn)),
Colombeau' s particular algebra
P(P(iRn)) . T h i s means that the following condition has t o
algebra A of be satisfied (8.A1.50)
1n P ( & ' (IRn)) = ker a
which i s necessary and sufficient for the existence of a canonical embedding
The way t o obtain that i s indicated by the following basic result, Colombeau [ I , pp 57- 601 . Suppose given F
E
P(&'(IRn)), then F
V K c
where (
6 ,x
mn
compact, q
E
E
ker a if and only if
U+,
(y) = ( ( ( y - x ) / ~ ) / ~ nwith , y
E
E
lq:
IRn
We are now nearing the end of our argument. Indeed, we only have t o recall that for certain F E P(O(R")) and suitable ( E i , F(df) can exhibit a very f a s t rowth in 1 see the example followin (8.1.25). Then (8.61.52) o%viously implles that the extended ideal 1 &,as t o be an ideal i n a s t r i c t l y smaller subalgebra A of P(P(IRn ) , such that for F E A. F((€) does not grow faster in I / E than a po ynomial. In t h i s way one can easily arrive a t the definitions (8.1.18) and (8.1.19), if one also remembers (8.1.22) .
?
E. E. Rosinger
APPENDIX 2 ASYHPTWICS WITBOUT A TOPOLOGY The space 9(lRn) of generalized functions was constructed i n (8.1.21) in a way which involved b o t h some algebra and topology. Indeed, on the one hand, (8.1.21) i s a purely a1 ebraic quotient construction, where A is an algebra and 1 is an i eal in A. On the other hand, the definitions of A and 1 in (8.1.18) and (8.1.19) respectively, do obviously involve some kind of topology, owing t o the respective asymp t o t i c conditions when m -+ ao and c -, 0.
d
In view of t h a t , one may ask whether o r not the construction of 6(lRn) could be seen f o r instance as a usual completion of C"(lRn) in a certain vector space topology 7 on C"(lRn), in which case A would be t h e s e t of Cauchy nets in C"(lRn), while 1 would be the s e t of t h e nets convergent t o zero in C"(R") within the uniform topology 7. We shall show that there i s no such a uniform topology 7 on C"(lRn]. The argument is quite simple and straightforward and is based on a we1 known result in general topology, Kelley, on the necessary and sufficient condition on a convergence class in order t o be identical with the convergence generated by a topology. For convenience, we repeat that r e s u l t here. Suppose give a nonvoid s e t X and a class C of pairs (S,x), where S is a net i a X and x E X. Then there exists a topology 7 on X such that (8.A2.1) if and only i f
(S,x)
E
C
(3
S converges t o x in 7
C s a t i s f i e s the following four conditions:
(8.A2.2)
If
S is the constant net x then
(8.A2.3)
If
(S,x)
(8.A2.4)
I f (S,x) f? C then there x i s t s a subnet S' of S such that f o r every subnet S" of S' , we have (S" ,x) f? C
E
C then
(S',x)
E
(S,x)
E
C
C f o r every subnet S'
Finally, given a directed s e t D and a family of directed s e t s d E D , l e t us consider the directed s e t
Ed
of
S
with
Colombeau ' s p a r t i c u l a r a l g e b r a
and l e t u s denote G = U ({d)
ED
x
Ed)
and t h e n d e f i n e
by R d,q) = ( d , q ( d ) ) . F u r t h e r , f o r S : G + X and d Sd : + X by Sd(e) = S ( d , e ) . I n t h a t c a s e we have
dd
(8.A2.5)
E
D l e t us define
I f (Sd,xd) E C, with d E D, and (T,x) E C, where T(d) = x -d , with d E D , t h e n (S,R,x) E C
Obviously, t h e above c h a r a c t e r i z a t i o n of convergence c l a s s e s does r e f e r t o a e n e r a l , p o s s i b l y nonuniform topology 7 on X. However, w i t h a s l i g h t mo i f i c a t i o n , t h e mentioned c h a r a c t e r i z a t i o n can be a p p l i e d t o o u r c a s e , when
f
Indeed, as is well known i n t h e c a s e of a v e c t o r space topology 7 on X, t h e c l a s s 2 of n e t s convergent t o z e r o determines i n a unique way t h e c l a s s C of a l l convergent sequences, according t o t h e r e l a t i o n
It f o l l o w s t h a t i n o u r c a s e when X is g i v e n by (8.A2.6) and assumed t o be t h e c l a s s of n e t s convergent t o z e r o , t h e c l a s s C convergent n e t s would c o n s i s t of a l l
where S : @ + P(lRn), p5 E P(tRn) , and i f we d e f i n e f : @ f ( / , x ) = (S(d))(x) - $ ( x ) , t h e n
x
IRn
Z is of a l l
4
Q: by
I n p a r t i c u l a r , it f o l l o w s t h a t all n e t s i n C a r e d e f i n e d on t h e index s e t 4. Then, as a f i r s t remark, it f o l l o w s t h a t @ should be a d i r e c t e d s e t , a l though no such e x p l i c i t p r o v i s i o n is made o r is even needed i n Colombeau's theory. However, as mentioned above, t h e c o n d i t i o n (8.1.19) d e f i n i n g Z does involve an asymptotic p r o p e r t y f o r m -+ oo and -P 0, which could e v e n t u a l l y suggest a d i r e c t e d o r d e r on 4.
E.E. Rosinger
356
Nevertheless, even i f iP could be made into a directed s e t , the class C defined in (8.A2.8) would s t i l l obviously f a i l t o s a t i s f y contion (8.A2.5) Therefore, there is no topology on P(!Rn) in which Z would be the class of nets conver ent t o zero. Consequently, t h e quotient structure in (8.1.21), aceorbing t o which
cannot be seen a s a completion of
P(R")
in any vectore topology.
Be should however note t h a t there may exist a vector space topology on C"(!Rn) with Cauchy nets B, and nets convergent t o zero 3, such that
and possibly (8.A2.12)
Z c 3 , AcB,
9 n A = Z
The interesting thing however i s t h a t , even without such o r any other topology on f'(!Rn), t h e direct and explicit, as lvell as natrral asymptotics in the definitions of A and I , can o f f e r Colombeau's method a surprising efficiency in solving large classes of l i n e a r and nonlinear p a r t i a l d i f f e r e n t i a l equations. See further Appendix 4.
Colombeau's particular algebra
APPENDIX 3
CONNECTIONS WITH PREVIOUS ATTEMPTS IN DISTRIBITION MULTIPLICATION There exists a considerable literature on a large varieity of attempts to define suitable distribution multiplications. This literature, published both before and after the so called Schwartz impossibility result, Schwartz 121, has mainly developed along rather independent, pure mathematical ines, and the results obtained could hardly be used in order to set up sufficiently general nonlinear theories dealing with nonlinear partial differential equations. An account of most of that literature can be found for instance in Rosinger . Two recent papers with some of the most relevant results in that iel are Ambrose and Oberguggenberger [I]. The latter paper is the best account so far of the essence of the mentioned literature and we shall present here shortly its main results which establish the relationship between four of the most important earlier distribution multiplications and the multiplication in Colombeau's algebra Full details concerning proofs can be found in Oberguggenberger P(R~). [I], as well as the references cited there.
P3d
Suppose given two arbitrary distributions S,T
E
DJ(P).
The Ambrose product - which extends the product in Hikmander [2] denoted by S-T, and exists by definition, if and only if
3 V open neighbourhood of x
(Al) 3(o!3)F1(fl)
-
E
-
is
:
f 1(lRn)
(A3) the linear mapping 3(V)
3
a
j
3(a~)~~(/3T)dx r C is continuous,
lRn
where 3 and 3-I denote the direct and inverse Fourier transformas respectively. It is easy to see that, if (8.A3.1)
@ = 1 on supp a
then the linear mapping in (A3) does not depend on 0, hence it defines a In this way, S-T is defined as the distribution distribution in D'(V).
E .E . Rosinger
in P' (lRn) generated by (A3) and (8. A3.1). The Mikusinski [3] , Hirata- Ogata product (MHfl)
[S] [TI
e x i s t s , i f and only i f
l i m (a,*S) (pv*T) e x i s t s in V1 (iRn) U*
f o r every &sequence (avlu E PI)
E
(V(lRn))'
(8.A3.2)
cr,LO,
(8. A3.3)
supp a,
(aVlv E IN)
(Bvlv
and
E
IN).
We recall that
is called a &sequence, i f and only if VEIN +
(0) c IRn,
when v -.
b
I t follows easily t h a t , i f it exists, the limit in (MHO does not depend on the 6- sequences (@,I v E PI) and (BVlv E IN) . Hence [ ] [TI is defined a s the limit in (MHO), whenever it exists.
A f i r s t result i s the following implication
and i f (Al) holds, thus S-T exists, then S-T = [S] [TI
(8. A3.6)
It should however be noted that the existence of [S] [TI does not imply i s continuous a t the existence of S-T. For instance, i f f E /?(lRn) x = 0 E lRn, without being continuous in a whole neighbourhood, then [f] [6] e x i s t s , but f - 6 does not e x i s t . The Vladimirov product
v
SOT exists, if and only i f
X E R "
3 V open neighbourhood of (VL)
*)
x,
8 = 1 on V
**) F(pS)*F(pT)
E
S' (R")
It i s easy t o see that the linear mapping
E
V(lRn) :
Colombeau ' s particular algebra
is continuous.
In t h i s way,
SOT E 9'(lRn)
i s defined by (8.A3.7).
It can be shown t h a t (All w (VL)
(8.A3.8)
and whenever S-T exists, we have (8.A3.9)
S-T = SOT
We come now t o the Kaminski A-product. We c a l l (o,lv E IN) E (9(lRn))' A- sequence, i f and only if it s a t i s f i e s (8. A3.3), (8. A3.4) , a s well a s
where
supp o v c B(o,c,) clRn
and
0
when
v+m,
with
a
B(x,r)
denoting the ball of radius r > 0 around x E lRn. The Kaminski A-product (KA)
S A T exists, i f and only if
1 ( * S ) ( * T ) e x i s t s in v*m
ZJ'(P)
f o r every A- sequences (a,l v E IN) and (@,I v E IN). Again, it can be seen that when it e x i s t s , the limit in (KA) does not depend on the A-sequences involved, hence it is denoted by SAT. It can be shown t h a t if (8.A3.11)
[S] [TI
exists, then so does SAT and we have
SAT = [S] [TI
The converse however i s not true.
where r > 2 ,
then SAT = 0,
bui
Indeed, if we take
[S] [TI
does not e x i s t .
E.E. Rosinger
360
Finally, the relation'with Colombeau's multiplication i s obtained by taking particular A- sequences in (KA). Then one can show that if SAT E 2)' (lRn) exists, we have (8.A3.13)
ST w SAT
where ST E E(IRn) denotes the product of S and T in Colombeau's algebra G(IR~), and w i s the relation of association defined i n Sect ion 4. We can recapitulate as follows: (8. A3.14)
S.T exists
SOT exists
in which case
Further (8.A3.16)
S.T exists
+
[S][T]
exists and S.T = [S][T]
also (8.A3.17)
[S] [TI
exists =, SAT exists and
[S] [TI = SAT
finally (8. A3.18)
SAT exists
+ ST w
SAT
In t h i s way, the products S.T and SOT are identical and also the least general. The product [S] [TI i s a further generalization, while the product SAT i s the most general among the four mentioned products, which a l l lead again t o distributions i n P'(IR"). The product i n Colombeau's algebra E(IRn) i s related t o a l l of the above four products by being related t o the most general of them, that i s the Kaminski A- product, as seen i n (8. A3.18) .
Colombeau's particular algebra
AN INTUITIVE ILLUSTRATION OF THE STRUCTURE OF COLOMBEAU'S ALGEBRAS It suffices to consider the one dimensional case of the algebra G(R). As seen in Theorem 1, Section 2, the inclusion
is an inclusion of differential a1 ebras. In other words, the algebra structure and the derivatives on GfR), when restricted to functions in P(R), do perfectly coincide with the respective classical operations on C?- smooth functions. Concerning the differential structure of G(R) , that situation goes much further. Indeed, in view of Theorem 3, Section 2, the inclusion
still preserves the differential structure. That is, the derivatives on G(R), when restricted to P'(R), are precisely the usual distributional derivatives. So that the peculiarity about the structure of G(R) only appears in connection with the way it extends the usual multiplication of non Coo-smooth functions or distributions, whenever the latter are defined. Indeed, since a useful multiplication has to be compatible with differentiation - through the Leibnitz rule of product derivative, for instance it all comes down to the possible relationships between multiplication and differentiation in the case of non C"-smooth functions or distributions. Here, the conflict between insufficient smoothness, multiplication and differentiation, in particular, the so called Schwartz impossibility result, come to set up some of the basic limitations on such possible relationships. Fortunately, very simple examples can clearly illustrate the difficulties involved in extending the usual multiplication of non Coo-smooth functions or distributions, see Appendix 1, Chapter 1. And then, a rather direct analysis of these difficulties can easily lead to an intuitive understanding of the st ruct ure of G(Rn) . In view of the fact that - as mentioned above - the problem centers around the multiplication and differentiation of non Coo- smooth functions and distributions, it means that a good illustration can be obtained if we consider products and derivatives of discontinuous functions. Indeed, applying differentiation to a non Coo-smooth function for a suitable finite number of times, we must end up with a continuous function whose derivative
E.E. Rosinger
362
does no l o n e e r e x i s t as a continous function and may f o r i n s t a n c e e x i s t a s a d i s t r i b u t i o n given b a discontinuous f u n c t i o n s . And a s G(R) has a sheaf s t r u c t u r e , it s u f i c e s t o consider a l l t h e above l o c a l l y only, t h a t is, i n t h e neighbourhood of a d i s c o n t i n u i t y point of such a f u n c t i o n .
l
Now obviously, t h e simplest n o n t r i v i a l example is given by t h e continuous and non C1- smooth f u n c t i o n
whose usual d e r i v a t i v e no longer e x i s t s , and it only has t h e d i s t r i b u t i o n a l d e r i v a t i v e given by t h e Heaviside f u n c t i o n
We n o t e t h a t
and P ( R ) is an a l g e b r a with t h e usual m u l t i p l i c a t i o n of f u n c t i o n s . I n t h i s way, it s u f f i c e s t o study t h e r e l a t i o n s h i p between t h e m u l t i p l i c a t i o n i n f'(IR) and G(R) , and do s o with r e s p e c t t o t h e d e r i v a t i v e on P(R), and i n p a r t i c u l a r , on P' (R) . F i r s t , it is obvious t h a t , f o r m E N,,
we have
However, a s a power, H ~ , with m E #, m > 2, is not defined i n P' (R) , s i n c e it involves t h e m 2 f a c t o r product H.. .H. Yet, i n view of (8.A4.6)) (8.A4.5), f o r m E M, m > 2, we have t h e r e l a t i o n
>
(8.A4.7)
H~ = H i n 2' (R)
only v i a t h e r e l a t i o n (8.A4.6), t h a t is, i f
H~
is computed i n t h e a l g e b r a
P(R) .
As we a l s o have H E G(IR), it is obvious t h a t Hm, with m E Y,, defined i n G(R) , but nevertheless, f o r m E N , m > 2, we have
is
Colombeau ' s particular algebra
Indeed, if we had equality in (8.A4.8), then we would obtain by differentiation that
Let us take m,p
E
NI,
then we can compute
H~+PDH in two ways: f i r s t , we have
or, we can also have
which means that 1 1 m + p + ZD H = m + l j F i
But DH # 0
E
4(R).
DH,
m,p
E
IN+
Thus we obtain the absurd result that
which ends the proof of (8. A4.8). The difference between (8.A4.7) and (8.A4.8) gives us a very simple example, and i t s following analysis can offer an intuitive insight into the structure of G(R). In view of (8.A4.4), we have - among many other possible ones lowing representat ion
-
the fol-
E. E. Rosinger
Then, in view of the way multiplication is defined in 4(W),
we obtain
Now (8.A4.10) yields f o r ( E i the relation 0 if l i m h(d€,x) = 1 if €10
(8.A4.12) while f o r
4
Hence, f o r m (8.A4.14)
E
h,
E
NI ,
6
> 0 and x E I R ,
and ( E 4 ,
xO weobtain
we have
0 if l i m hm(dC,x) = 1 if €10
xO
and
with
E
> 0 and x E W.
In view of (8.A4.12) and the continuity of m ~ # ,m 2 2 , ( E Q and c > O
h ( ( € , - ) , we obviously have f o r
which i s expected t o happen, owing t o (8.A4.8). The crucial oint of t h e analysis is the comparison of the relations (8.114.7) and f 8 . ~ 4 . 8 ) , via the relations (8.14.6) . Suppose given $ E P(R) , then (8. A4.12)- (8. A4.15) and Lebesque ' s bounded convergence theorem give
Colombeau ' s particular algebra
for
d E @,
(8.A4.18) for m
E
m
E #+.
H~
8
This is in fact identical with H in P(R)
#+
But when seen in P'(R), the relation (8.A4.17) has the followin different meaning: in view of (8.114.12)- (8.14.15) and Lebesquets bounkd convergence theorem, it follows that (8.A4.19)
1
h , )
E 10
=
1 h m ( ) = H in Vt(R) €10
and hence, as also follows from (8.A4.17), we have (8.A4.20)
lim (hm(#,,-) E1 0
-
h((€,.))
= 0 in
?'(I)
where all the three limits above are in the sense of the weak topology on P'(R), and hold for m E #+ and 4 E @. We can now conclude that, although H ~ , with m E #, m L 2, is not definable as a power in V' (R) and is only defined via (8.A4.6), nevertheless, IIm and H are indistinguishable in V'(R), just as they are in I!?'@). In other words, Pt(IR) cannot retain an information on h or hm in (8.A4.19), except to register their common limit H . It follows that or in general the way discontinuities of functions such as in I!?'@), ( R ) appear in V'(R), its too simple in order to allow for a suitable relation between multiplication and differentiation, such as given by the Leibnitz rule of product derivative. This excessive simplicity in dealing with discontinuities is apparent in the following general situation: given any distribution T E V'(R), there exists families of functions fE E P(R), with E > 0, see (8.1.7), such that (8.A4.21)
lim fE = T in Vt(R) 6 10
in the sense of the weak topology on Vt(R),
which means that
NOW in view of (8.A4.22), it is obvious that, just as with (8.A4.18), the only thing retained in V' (R) from (8.A4.21) is the 1 imit value given by the distribution T, all other information about fE, with 6 > 0, being
E. E. Rosinger
366
lost. I n particular, the averaging process (8.A4.22) involving arbitrary test functions # E V(W), i s too coarse i n order to be able t o accommodate the discrimination i n (8.A4.16).
On the other hand, the picture i n G(IR), as given by (8.A4.9), (8.A4.11), is more so histicated. Indeed, H and iim are defined by h and hm respective y, through the quotient representations in the mentioned two relations. Morevoer, the relation between h and H , as well as hm and H~ i s not through a limit or convergence process - see Appendix 2 - but through an asymptotic interpretation. And as seen i n (8.A4.16), (8.A4.18), and of course (8.A4.8) , that asymptotic interpretation can distinguish between H and Hm, precisely because it does retain sufficient informat ion on h and hm.
f
The above may serve as an instructive example i n illustrating the fact that asymptotic interpretations can be more sophisticated - and thus useful than limit, convergence or topological processes.
FINAL REMARKS With this volume, the presentation of basic features of the 'algebra f i r s t ' approach t o a systematic and comprehensive nonlinear theory of generalized functions, needed for the solution of large classes of nonlinear partial differential equations, comes t o a certain completion.
A f i r s t s t a e of this 'algebra f i r s t ' a proach was started in Rosinger [7,8], and %eveloped later i n Rosinger {,2,31. One of i t s particular, nevertheless uite natural and rather centra cases, were presented i n Colombeau [ I , 27 . This f i r s t stage focused on the 'near embedding'
mentioned i n (8.1.8) for instance, which leads i n a natural way to the idea of constructing embeddings, see (1.5.25)
where (3)
A i s a subalgebra i n
(P(Wn ) )A
while (4)
I i s an ideal i n A
and A i s a suitable, infinite index set. The important point to note w i t h (2)- (4) inclusions
i s that we have the obvious
if we consider A w i t h the discrete topology. In that case however A x lRn w i l l become a completely regular topological space, and i n view of ( 5 ) , we have
i n other words, A i s a subalgebra i n the algebra C(A nuous functions on A x lRn, while I i s an ideal i n A.
x
IR") of conti-
Now we can recall the well known r i g i d i t y property between the t o p o l o g i c a l properties of a completely regular space X and the a l g e b r a i c properties of i t s ring of continuous functions C(X) , Gillman & Jerison. For instance, two real compact spaces X and Y are topologically homeomorphic,
E.E. Rosinger
368
i f and only if C(X) and C(Y) are isomorphic algebras. It follows that a good deal of the algebraic properties of the quotient algebras A = A/Z in (2) may well depend on the simple topolo ical properties of A x R". The extent t o which that proved indeed t o e the case i s presented in Rosinger [1,2,3], Colombeau [1,2] and Chapters 2-8 of t h i s volume.
%
In short, one may sa that the essence of t h i s 'algebra f i r s t ' approach centers around the f o lowing: Fortunate Inversion: We have schematically the situation:
i
nonl inear PDEs
linear algebra (semigroups , vector s p a c e s y u p s
functional analysis
linear1 PDEs
nonli ;ear a1 ebra (rings, etc.7 where an arrow
indicates the direction of increasing generality.
Nevertheless, in view of the mentioned rigidity property of rings of continuous functions, we can establish a rather powerful inverse connect ioa : nonlinear PDEs
nonlinear algebra
/
Concerning the second stage of that 'algebra f i r s t ' approach, l e t us note the following. The embeddin (2) has of course differential aspects as well, which may o be ond the afgebraic ones involved in (6) for instance. Indeed, one uou d li e t o have on A = A/Z partial derivatives which extend the distributional ones on 2)'(lRn). Historically, that issue has led t o a long lasting misundertandin started with a misinterpretation of L. Schwartz's so called impossi%ility result of 1954.
1:
k
However, as seen in Sections 1-4 and Appendix 1 in Chapter 1 , the difficult i e s with the differential structure on A = A/Z happen t o have a most simple algebraic nature and center around a conflict between discontinuity, multiplication and abstract differentiation.
Final remarks
369
I t i s precisely the clarification in Chapter 1 of that second algebraic phenomenon which, together with the earlier dealt w i t h algebraic aspects involved i n (1)- (6), bring to a certain completion the mentioned 'algebra f i r s t ' approach.
And now, where do we go from here? Well, perhaps not so surprisingly, one most promising direction seems to be that along the lines of further desescalation i n the sense mentioned i n the Foreword. Indeed, the 'algebra f i r s t ' a proach has already brought with it a signif icant desescalatlon from the unctional analytic methods, so much customary in the study of partial differential equations during the last four or five decades, to the 'nonlinear algebra' of rings of continuous functions. And one of the aspects of that desescalation which i s particularly important, yet it i s seldom noticed, i s that - unlike in functional analysis - the 'nonlinear a1 ebra' of rings i s i n fact a particular and enriched case of the 'linear a gebra' of vector spaces, grou s or semigroups. In this way, the 'algebra f i r s t ' approach in the study o nonlinear partial differential equations leads to more particular and not t o more general algebraic structures !
P
k
P
But now, i n view of the hierarchy
-
set theory
-
binary relations, order
-
algebra
-
topology
-
functional analysis
-
etc.
i t i s time for one furiher desescalation, namely, from 'algebra f i r s t ' to 'order f i r s t ' ! And the fact i s that there exists a precedent for i t , for nearly a decade by now, Browsowski. Indeed, in that paper, the linear Dirichlet problem
i s given a method of solution based on the Dedekind order completion of the Riesz space C ( a ) of continuous functions on the compact space iM1 c R".
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Unfortunately, as it stands, that method is not applicable to nonlinear partial differential equations. However, one can proceed along different lines in that 'order first' approach.
A promising direction is offered by an extension of the Cauchy-Kovalevskaia theorem to continuous nonlinear partial differential equations, using a Dedekind order completion of spaces of smooth functions, a joint result obtained in collaboration with I. Oberguggenberger, which is to be published elsewhere.
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