Nonlinear Partial Differential Equations: Sequential and Weak Solutions

NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

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NORTH-HOLLAND MATHEMATICS STUDIES

44

Notas de Matematica (73) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Nonlinear Partial Differential Equations Sequential and Weak Solutions

ELEMER E. ROSINGER National Research Institute for Mathematical Sciences Pretoria, South Africa

1980

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

NEW YORK

OXFORD

North-Holland Publishing Company, I980

All rights rrscwerl. No purt of rhispublicurioti mu,v he, rrprvducecl. siori>clin N retrievtrlsy.stett~. or trunsmirtrd, in uny form or by uny nwuns, c+c~trotiic, mc~chanicd. photocopying. rcw)rtling or othrrwisc.. without the prior pertnissioti of the, copvrighr owtwr.

ISBN: 0 4 4 4 8 6 0 5 5 ~

Pu hlishm: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK O X F O R D Sokc disirihurors for ihr U.S.A. und Cmnudri: ELSEVIER NORTH-HOLLAND. INC. 5 2 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

PRINTED IN T H E NETHERLANDS

Dedicated t o my parents i n law, Renee a n d Max Neufeld

FOREWORD The p r e s e n t volume i s a sequel t o t h a t p u b l i s h e d i n t h e s e r i e s S p r i n g e r L e c t u r e Notes i n Mathematics, as v o l . 684, i n 1978. D u r i n g t h e l a s t two years, a number o f new r e s u l t s o b t a i n e d by t h e a u t h o r have c o n t r i b u t e d t o t h e c l a r i f i c a t i o n o f t h e general framework on which t h e p r e v i o u s volume was based, and have come t o e n r i c h and l i f t t o a new l e v e l o f g e n e r a l i t y t h e r e s u l t s p u b l i s h e d e a r l i e r . An a t t e m p t i s made here t o p r e s e n t these developments i n a s e l f - c o n t a i n e d manner. I n developing t h e p r e s e n t method, t h e a u t h o r ' s main concern has been t o t r y t o f i n d a u n i f i e d way o f d e a l i n g w i t h weak s o l u t i o n s o f general non=

l i n e a r PDEs, a-ich would e x h i b i t a c e r t a i n n a t u r a l , o b j e c t i v e t r a i t , thus going beyond t h e somewhat adhoc appearance t h e customary f u n c t i o n a l a n a l y t i c methods, w i t h t h e i r w e a l t h o f l i n e a r t o p o l o g i c a l s t r u c t u r e s and f u n c t i o n spaces used, m i g h t now and t h e n suggest. I n t h i s respect, a way presented i t s e l f n a t u r a l l y by n o t i c i n g t h a t t h e s e q u e n t i a l , i n p a r t i c u l a r t h e weak s o l u t i o n s o f a n o n l i n e a r PDE T(D)u(x) = f ( x ) ,

x

E

R, Q domain i n Rn,

can be seen as elements i n t h e a l g e b r a o f continuous f u n c t i o n s C(N x Q ) . Consequently, t h e known r i i d i t y between t h e t o p o l o g i c a l p r o p e r t i e s o f a completely r e g u l a r space+-ti-an t e a l g e b r a i c p r o p e r t i e s o f i t s continuous f u n c t i o n s , a p p l i e d t o N x R, r e s p e c t i v e l y C(N x Q), might make i t p o s s i b l e t o t r a n s l a t e t h e t o p o l o g i c a l p r o p e r t i e s i n t o t h e a l g e b r a i c ones and v i c e versa, a procedure which c o u l d prove t o be an advantage i n t h e s t u d y o f n o n l i n e a r PDEs. F o r i n s t a n c e , one c o u l d expect t h a t a good deal o f t h e d i s c u s s i o n o f s e q u e n t i a l s o l u t i o n s m i g h t be k e p t on an a l g e b r a i c l e v e l , s i n c e these s o l u t i o n s a r e elements i n C ( N x Q), which p a r t i c i p a t e s m a i n l y through i t s a l g e b r a i c s t r u c t u r e . Moreover, t h e n a t u r a l t o p o l o g i c a l pro= p e r t i e s o f N x R would p r o v i d e t h e o b j e c t i v e c h a r a c t e r i s t i c o f t h e method aimed a t . A d d i t i o n a l , i m p o r t a n t aspects might obvious1 be a i n e d from t h e presenceof t h e chains o f d i f f e r e n t i a l subalgebras ( C X (a)) 8 i n C(N x Q), w i t h II E N, which appear as t h e n a t u r a l domains o f d e f i n i t i o n f o r t h e n o n l i n e a r PDOs i n v o l v e d , when t h e i r a c t i o n i s extended t o sequences o f f u n c t i o n s . F o r t u n a t e l y , these e x p e c t a t i o n s c o u l d be f u l f i l l e d t o a c e r t a i n e x t e n t . However, t h e method presented here o f s t u d y i n g s e q u e n t i a l and weak s o l u = t i o n s o f n o n l i n e a r PDEs can o n l y be considered as a f i r s t s t e p towards f u l l use o f t h e power o f f e r e d b y t h e t h e o r y o f a l g e b r a s o f continuous,functions a p p l i e d t o t h e p a r t i c u l a r case o f C(N x a ) , w i t h R a domain i n R

.

I am happy t o acknowledge here my s p e c i a l g r a t i t u d e t o P r o f . L. Nachbin, E d i t o r o f Notas de Mathematica i n N o r t h - H o l l a n d Mathematical S t u d i e s , f o r suggesting t h a t t h i s volume should be w r i t t e n . F o r d i s c u s s i o n s on v a r i o u s aspects o f t h i s work, g r a t e f u l thanks a r e a l s o due t o M.C. Reed and E. Schechter of Duke U n i v e r s i t y , C. M. Dafermos and W.A. Strauss o f Brown U n i v e r s i t y , H.C. Kranzer o f Adelphi U n i v e r s i t y and J. Horvath o f C o l l e g e Park. vi

vi i

FOREWORD

Work on the book was begun in Israel and continued during short-term v i s i t s to several universities in Switzerland and the U S A . The second part was completed in Pretoria, South Africa. The author i s most grateful t o a l l those who helped him with t h e i r coments a t that stage, special thanks being due t o Professor D.H. Jacobson, Director of the National Research I n s t i t u t e for Mathematical Sciences of the CSIR, Pretoria, who was kind enough t o extend a longer invitation t o his Institute and place a t the author’s disposal a l l the necessary f a c i l i t i e s f o r research. A l s o sincere t h a n k s are due t o Dr. E.

Fredriksson, Publisher a t NorthHolland, f o r his courteous and e f f i c i e n t collaboration.

The task of editing and typing a manuscript i s one f o r which an author cannot be thankful enough: I can only express my special admiration t o Mr. F.R. Baudert and Mrs. M. Russouw and acknowledge my heavy indebtedness t o them, f o r t h e i r patience and efficiency.

E.E.R.

Pretoria

PRELIMINARIES The method o f s o l v i n g n o n l i n e a r PDEs b y c o n s t r u c t i n g 'weak s o l u t i o n s ' i s w i d e l y used. One reason f o r t h i s i s t h a t r a t h e r b a s i c and s i m p l e n o n l i n e a r PDEs w i t h r e g u l a r i n i t i a l o r boundary c o n d i t i o n s may l a c k r e g u l a r s o l u t i o n s . A well-known example i s t h e case o f shock wave s o l u t i o n s o f n o n l i n e a r hy= p e r b o l i c c o n s e r v a t i o n laws. I n such cases, owing t o t h e presence o f singu= l a r i t i e s such as d i s c o n t i n u i t i e s o r l a c k o f d e r i v a t i v e s o f s u f f i c i e n t l y h i g h o r d e r , t h e s o l u t i o n s o b t a i n e d w i l l s a t i s f y t h e n o n l i n e a r PDEs i n a weak sense o n l y . U s u a l l y , these weak s o l u t i o n s can be i n t e r p r e t e d as d i s = t r i b u t i o n s . However, i n t h e case o f n o n l i n e a r PDEs, d i s t r i b u t i o n s o l u t i o n s may f a i l t o s a t i s f y t h e equations i n a d i s t r i b u t i o n a l sense, s i n c e even elementary n o n l i n e a r o p e r a t i o n s on d i s t r i b u t i o n s , f o r i n s t a n c e products, may l e a d o u t s i d e t h e d i s t r i b u t i o n s . I n t h a t way, t h e d i s t r i b u t i o n a l frame= work proves t o be r a t h e r narrow when s o l v i n g n o n l i n e a r PDEs. Moreover, as t h e well-known example o f H. Lewy shows, even l i n e a r PDEs may f a i l t o have distribution solutions.

A s u f f i c i e n t l y wide framework f o r s o l v i n g n o n l i n e a r PDEs w i l l be presented i n t h i s work b y c o n s i d e r i n g t h e weak s o l u t i o n s , i n p a r t i c u l a r t h e d i s t r i = b u t i o n s o l u t i o n s as ' s e q u e n t i a l s o l u t i o n s ' , i.e. b y c o n s i d e r i n g them as g i v e n b y sequences o f continuous, o r more r e g u l a r f u n c t i o n s . A b a s i c ad= vantage o f t h i s approach i s t h a t t h e ' s e q u e n t i a l s o l u t i o n s ' a r e i n a natu= r a l way elements o f c e r t a i n a s s o c i a t i v e and commutative algebras o f 'gene= ralized functions' containing the distributions D'cAmC

... c A P c ... c A o , p € N n

previously introduced by the author. These a l g e b r a s possess 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 D ~ : A P - + A P ' ~ ,P E N " ,

EN^,

q < p ( i i = ~ u

{m))

which s a t i s f y t h e L e i b n i t z r u l e o f p r o d u c t d e r i v a t i v e s . I n t h a t way, as shown i n t h e p r e s e n t work, t h e p o l y n o m i a l n o n l i n e a r PDEs Pij Z ci(x) D U(X) = f ( x ) , x R c Rn , l G i Q h l<JGki w i t h g i v e n c., f E C"(f2) and unknown u , can under q u i t e general c o n d i t i o n s f i n d d i s t r i b h i o n s o l u t i o n s , weak s o l u t i o n s o r ' s e q u e n t i a l s o l u t i o n s ' w i t h = i n t h e algebras Ap, w i t h p E , p " p i j , f o r 1 G i Q h , 1 Q j Q ki.

mn

The need f o r d e a l i n g with t h e whole c h a i n o f a l g e b r a s Ap, w i t h p E In , a r i s e s because c e r t a i n p r o d u c t s c o n t X i T 5 i g s i n g u l a r f a c t o r s , such as t h e D i r a c 6 d i s t r i b u t i o n and i t s d e r i v a t i v e s , may l e a d t o d i f f e r e n t r e s u l t s depending on t h e a l g e b r a s t h e y a r e considered i n . T h e r e f o r e i n o r d e r t o o b t a i n s u i t a b l e p r o p e r t i e s o f t h e m u l t i p l i c a t i o n one has t o choose t h e p r o p e r a l g e b r a s . F o r i n s t a n c e , as can be shown, based on a well-known counterexample g i v e n b y L. Schwartz i n t h e one-dimensional case n = 1, under n a t u r a l c o n d i t i o n s concerning pKoducts o f c" f u n c t i o n s and d e r i v a = t i v e s o f Lo1 f u n c t i o n s , t h e a l g e b r a A cannot accommodate t h e r e a l t i o n x

6(x) = 0 viii

PRELIMINARIES

ix

which gives an gpper bound on the order of singularity exhibited in x = 0 E R' by the Dirac 6 distribution, and which will hold in the algebra A" The counterexample mentioned was occasionally interpreted as i l l u s = trating t h a t a suitable distribution multiplication i s impossible, i .e. that i t i s impossible t o embed the distributions into differential alge= bras. However, as mentioned above, such embeddings are possible and use= ful i f chains of algebras are considered.

.

I t i s particularly important t o notice t h a t from the p o i n t of view of solving polynomial nonlinear PDEs, the essential features of the above algebras of 'generalized functions' are the following. F i r s t , the algebras are l a r e enough t o contain the usual weak or distribution solutions. Actua Tp y , as seen in Chapter 2 they will accommodate the existence of 'sequential solutions' for wide classes of nonlinear PDEs.

Secondly, olynomial nonlinear operations can freely be performed under the best a ge raic circumstances granted by associativity and comutativity.

eb--

Thirdly, the partial derivative operators s a t i s f y the Leibnitz rule of product derivative. Finally, the algebras are obtained in a simple, constructive way, a s quotient algebras of sequences of continuous or more regular functions. Therefore t h e i r elements, the 'generalized functions', in particular the 'sequential solutions' weak solihions or distributions, are classes of sequences of continuous or more regular functions, much in the s p i r i t of the usual approximation methods, a circumstance that will be specially helpful when constructing solutions for nonlinear PDEs and establishing t h e i r properties. The fact t h a t with the exception of the algebra Am, the elements of the other algebras Ap are not indefinitely derivable, i s no particular dis= advantage as long as the polynomial nonlinear PDEs considered are of f i n i t e order and can therefore be dealt w i t h i n the algebras Ap, with p f i n i t e and greater than the order of the equations involved. I n t h i s connection i t i s worth noticing t h a t the indefinite derivability of the distributions in D' does n o t compensate f o r the lack of suitable nonlinear operations within the distributions. I n conclusion i t may be remarked t h a t as long as one deals with PDEs of f i n i t e order, the essential fea= tures of the spaces of 'generalized functions' are t h e i r large size as well as algebra structure, and n o t necessarily the indefinite derivability of t h e i r elements. The study of general linear PDEs i n i t i a t e d two major departures from the classical framework. The f i r s t , due t o L. Schwartz, brought an increase in the 'reservoir' of b o t h data and possible solutions f o r linear PDEs, by introducing the distributions. T h a t departure proved t o be particu= l a r l y f r u i t f u l in the study of linear PDEs with constant coefficients. The second departure, due t o J . J . Kohn and L. Nirenberg, one of the aims of which was t o deal with linear PDEs with variable coefficients, extended the class of equations considered , by introducing pseudodifferential equations, an extension s t i l l within the linear framework. The approach t o nonlinear PDEs suggested in t h i s work represents a further departure along the direction initiated by L . Schwartz in that the equations considered are classical , while the possible solutions can be=

PRELIMINARIES

X

l o n g t o algebras which include the distributions. The method presented for polynomial nonlinear PDEs can easily be extended t o much wider classes of classical nonlinear PDEs.

The present work deals with the following problems related t o 'sequential solutions' of polynomial nonlinear PDEs : existence, s t a b i l i t y , resolu= tion of singularities and regularity. The characteristic idea of the approach i s f i r s t t o develop the consequences resulting from the basic algebraic properties of particular differential algebras of classes of sequences of regular functions on domains in Euclidean spaces. Here an important role will be played by the basic algebraic, topological and cardinality properties of algebras of sequences of continuous functions on domains in Euclidean spaces. Analytic methods are employed during l a t e r stages only. Methods based on the properties of linear topological structures on various function spaces will not be used in the present work. More precisely, the 'sequential solutions' being elements of dif= ferential algebras (Cm ( Q ) ) N contained in the algebra of continuous functions C ( N x R), the idea i s t o pursue the study of polynomial non= linear PDEs as f a r as possible, using algebraic properties of C(N x Q) and t o ological properties of N x R, which - owing t o the known ri i d i t be= tween t e topological properties of a completely regular space an the algebraic properties of C ( X ) - determine each other uniquely. The consequ= ent development will therefore have a certain objective character result= ing from the natural topological properties of N x n, in contrast t o the somewhat arbitrary or ad hoc nature of the developments based on the use of various 1 inear topological structures on various function and 'genera= lized function' spaces.

+

-%-$

Lately there has been an awareness that important results on PDEs can be deduced from basic, rather simple algebraic and possibly analytic or topo= logical properties of several classical function spaces , w i t h o u t recourse to the customary wealth of linear topological structures. In the case of linear PDEs with constant coefficients for instance, R. Struble could prove the existence of fundamental solutions, using basic algebraic properties of a special module structure on the space of smooth functions. I n view of the algebraic c l a r i t y of t h i s result and several other similar ones, i t may perhaps be considered t h a t a multivariable operational cal= culus has been developed. The role of purely algebraic structures proves t o be substantial also when important properties o f nonlinear PDEs are being investigated. A recent survey of Yu. Manin demonstrated in detail the interplay between differential a1 gebra and the classical or geometri= cal methods in the study of a wide class of nonlinear PDEs related t o the Korteweg-de Vries equation, with the resulting important role played by the basic algebraic structures involved. When studying PDEs, there are essentially three reasons f o r considering topologies on the associated function or 'generalized function' spaces: f i r s t , in order t o define partial derivatives; secondly, in order t o approximate existing exact solutions; and f i n a l l y , in order t o define 'weak solutions' as elements in completions of certain function spaces. In the study of exact solutions of polynomial nonlinear PDEs w i t h i n the framework of quotient algebras of sequences of functions, the considera= tion of topologies on function spaces can t o a certain extent be avoided. Indeed, the partial derivatives being definednterm by term on sequences of functions, only the standard topology on R will be involved. Further,

PRELIMINARIES

xi

t h e a l g e b r a i c s t u d y o f e x a c t s o l u t i o n s does n o t n e c e s s a r i l y r e q u i r e appro= x i m a t i o n methods. F i n a l l y , t h e ' r e s e r v o i r ' o f p o s s i b l e ' g e n e r a l i z e d s o l u = t i o n s ' can always be k e p t l a r g e enough, w i t h o u t recourse t o t o p o l o g y , s i m p l y b y c o n s i d e r i n g s u i t a b l e q u o t i e n t a l g e b r a s o f sequences o f f u n c t i o n s . I n t h i s connection, t h e method i n t h i s work aims t o reassess t h e balance between a l g e b r a and t o p o l o g y i n t h e s t u d y o f n o n l i n e a r PDEs: f i r s t , b y a ' b l o w u p ' o f t h e e s s e n t i a l a l g e b r a i c phenomena i n v o l v e d , f o l l o w e d b y t h e use o f t h e necessary l e v e l o f t o p o l o g y , which up t o a s i g n i f i c a n t ex= t e n t w i l l n o t exceed t h a t customary i n t h e s t u d y o f t h e a l g e b r a o f con= t i n u o u s f u n c t i o n s C(N x a). Other t o p o l o g i c a l s t r u c t u r e s m i g h t become useful i n c o n n e c t i o n w i t h t h e r e g u l a r i t y p r o p e r t i e s o f ' s e q u e n t i a l s o l u = tions'. As i s w e l l known, t h e s t u d y o f n o n l i n e a r PDEs has r e v e a l e d t h r e e fundamen= t a l phenomena, v i z t u r b u l e n c e , s o l i t a r y waves and shock waves. The p r e s e n t work d e a l s w i t h aspects o f t h e l a t t e r phenomenon, considered under i t s general form o f d i s t r i b u t i o n , weak o r ' s e q u e n t i a l s o l u t i o n s ' o f n o n l i n e a r PDEs. The f a c t t h a t these s o l u t i o n s a r e e l e m e n t s ' o f spaces o f ' g e n e r a l i z e d f u n c t i o n s ' possessing a s t r u c t u r e o f a s s o c i a t i v e and commutative algebra, makes i t p a r t i c u l a r l y easy t o deal w i t h polynomial n o n l i n e a r PDEs. How= ever, i t i s i m p o r t a n t t o n o t i c e t h a t t h e presence o f a s t r u c t u r e o f a l g e = b r a on t h e ' g e n e r a l i z e d f u n c t i o n s ' - u n l i k e t h e v e c t o r space s t r u c t u r e o f d i s t r i b u t i o n s , w i t h m u l t i p l i c a t i o n defined p a r t i a l l y , f o r special p a i r s o f f a c t o r s o n l y - o f f e r s a n o t h e r advantage, i n t h a t , i t y i e l d s a b a s i c and simple algebraic t o o l f o r modelling s i n g u l a r i t i e s i n the s o l u t i o n o f polynomial n o n l i n e a r PDEs. Indeed, i4TI-e i n groups, o r v e c t o r spaces as t h e case may be, a l l t h e o p e r a t i o n s - e x c e p t m u l t i p l i c a t i o n w i t h t h e scalar zero - are i n v e r t i b l e , t h e rings, o r e l s e algebras, a r e the next s i m p l e s t a l g e b r a i c s t r u c t u r e s which owing t o t h e presence o f z e r o d i v i s o r s , possess a n o n t r i v i a l n o n - i n v e r t i b l e , t h u s ' s i n g u l a r ' o p e r a t i o n , v i z m u l t i = p l ic a t i on. I n s h o r t , one may say t h a t t h e method presented i n t h i s work e s t a b l i s h e s a n a t u r a l and s t r o n g c o n n e c t i o n between t h e s t u d y o f general n o n l i n e a r PDEs and t h e s t u d y o f a l g e b r a s o f continuous f u n c t i o n s C ( N x a ) , where R C Rn i s a domain. Chapter 1 s t a r t s w i t h an i m p o r t a n t m o t i v a t i o n f o r t h e i n t r o d u c t i o n and s t r u c t u r e o f t h e a l g e b r a s o f ' g e n e r a l i z e d f u n c t i o n s ' , prompted b y s t a b i l i = t y c o n s i d e r a t i o n s c o n c e r n i n g weak o r ' s e q u e n t i a l s o l u t i o n s ' . As p o i n t e d o u t i n t h e case o f n o n l i n e a r PDEs, t h e s t a b i l i t y o f weak o r ' s e q u e n t i a l s o l u t i o n s ' - u n l i k e t h e s t a b i l i t y o f s o l u t i o n s o f l i n e a r PDEs, which i s t r i v i a l and assured a u t o m a t i c a l l y - becomes an e s s e n t i a l and n o n t r i v i a l problem, even i f q u i t e o f t e n overlooked. A f t e r d e f i n i n g t h e n o t i o n o f ' s e q u e n t i a l s o l u t i o n ' , a method f o r 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 ' s e q u e n t i a l s o l u t i o n s ' i s presented, and i t i s shown t h a t i n s p i t e o f t h e g e n e r a l i t y o f t h e framework, t h e r e g u l a r i t y o f s o l u t i o n s o f e l l i p t i c o r h y p o e l l i p t i c l i n e a r PDEs i s preserved.

I n Chapter 2 necessary and/or s u f f i c i e n t c o n d i t i o n s f o r t h e e x i s t e n c e o f ' s e q u e n t i a l s o l u t i o n s ' a r e g i v e n f o r l a r g e c l a s s e s o f n o n l i n e a r PDEs. By i n t r o d u c i n g c e r t a i n s i m p l e c o n d i t i o n s o f t o p o l o g i c a l t y p e , we a r r i v e a t r e s u l t s which o f f e r an i n s i g h t i n t o t h e s t r u c t u r e o f t h e a l g e b r a s o f 'generalized functions'. Chapter 3 p r e s e n t s t h e c o n s t r u c t i o n o f t h e a l g e b r a s o f ' g e n e r a l i z e d func=

PREL IMI N ARI ES

xi i

tions' containing the distributions used in the sequel. In Chapter 4 , a general result i s given concerning the resolution of singularities of distribution solutions for polynomial nonlinear PDEs. This result i s then applied t o the case of known weak or distribution solutions of f i r s t and second-order nonlinear wave equations. Further, a general necessary and sufficient junction condition across hypersurfaces of discontinuities i s established for weak solutions of a large class o f systems of PDEs, containing among others the equations of magnetohydro= dynamics and general relativity. An extension of that result gives an insight into the structure of nonlinearities of a particularly large class o f systems of PDEs, containing many o f the equations encountered in phys i cs

.

Chapter 5 deals with the s t a b i l i t y of weak or distribution solutions given in sequential form. Results are presented on maximal s t a b i l i t y , as well as the interplay between s t a b i l i t y and exactness. The study o f maximal s t a b i l i t y leads in a natural way t o a characteriza= tion of the necessary structure of the algebras of 'generalized functions' containing the distributions. Based on the properties of 'zero f i l t e r s ' , results are presented in Chapter 6 that specialize and strengthen those arrived a t in the theory of algebras of continuous functions. I n Chapter 7 , within the a1 gebras of 'generalized functions ' , a scattering problem i s solved with potentials arbitrary positive powers of the Dirac 6 distributions, i .e. exhibiting the strongest local singularities studied in the literature. The wave function solutions obtained in the one and three-dimensional cases are regular, except on the support of the poten= t i a l s , where the discontinuities accompanying the transmission and reflec= tion of waves are determined. I n view of the special role played in the construction of the algebras of 'generalized functions' by certain products involving the Dirac 6 d i s t r i = bution and i t s derivatives, Chapters 8 and 9 are devoted t o the study of such products, also o f certain unusual properties of the Dirac 6 d i s t r i = bution within the mentioned algebras.

An essential and characteristic property of multiplication within the a l = gebras of 'generalized function' which i s extensively used throughout the present work, i s i t s local character, i.e. the property that the product of two 'generalized functions' on an open subset depends only on the restriction of the two factors t o t h a t subset. In order t o clarify t h a t property as well as several related ones, including certain aspects o f division within the algebras of 'generalized functions' , Chapter 10 i s devoted t o the study of support and local properties of 'generalized functions'. The work ends w i t h an appendix which presents in some detail various con= nections between the algebras of 'generalized functions' and related notions and constructions in the Calculus.

TABLE OF CONTENTS

CHAPTER 1 : S e q u e n t i a l S o l u t i o n s o f N o n l i n e a r PDEs

1

0

Introduction

1

1

The Problem o f S t a b i l i t y o f S e q u e n t i a l S o l u t i o n s

2

2

D e f i n i n g t h e Q u o t i e n t Spaces and Algebras

5

3

The N o t i o n o f S e q u e n t i a l S o l u t i o n

10

4

Exactness o f S e q u e n t i a l S o l u t i o n s

12

5

I n t e r p l a y between S t a b i l i t y , G e n e r a l i t y and Exactness

13

The D e f i c i e n c y o f t h e D i s t r i b u t i o n a l Approach Concerning Exactness

17

7

R e s o l u t i o n o f Nowhere Dense S i n g u l a r i t i e s

19

8

P r e s e r v a t i o n o f E l 1 i p t i c i ty an3 Hypoel 1 i p t i c i t y

24

9

General N o n l i n e a r PDEs

25

10

V a r i a b l e Transforms

27

11

Stronger Conditions f o r P a r t i a l Derivative Operators

30

12

Systems o f N o n l i n e a r PDEs

32

13

A Cesaro C h a r a c t e r i z a t i o n o f t h e Nowhere Dense

6

Ideal CHAPTER 2 : Necessary and/or S u f f i c i e n t C o n d i t i o n s f o r t h e Existence o f Sequential Solutions

34 37

Introduction

37

Subsequence Q u a s i I n v a r i a n t S e q u e n t i a l S o l u t i o n s

38

A p p l i c a t i o n s t o L i n e a r and N o n l i n e a r PDEs

44

Subsequence I n v a r i a n t S e q u e n t i a l S o l u t i o n s

46

Resolvent Sets

51

Domains o f Sol v a b i 1 it y

55

Remarks on Lemma 2

62 xiii

xiv

TABLE

OF CONTENTS

CHAPTER 3 : Algebras 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

65

0

Introduction

65

1

Embedding t h e D i s t r i b u t i o n s i n t o Q u o t i e n t A1 gebras

65

2

S i m p l e r I n c l u s i o n Diagrams

68

3

R e g u l a r i z a t i o n s and Chains o f Q u o t i e n t Algebras Containing the D i s t r i b u t i o n s

69

4

Existence, C h a r a c t e r i z a t i o n and C o n s t r u c t i o n o f Regularizations

75

A d d i t i o n a l Chains o f Q u o t i e n t Algebras C o n t a i n i n g Distributions

88

Q u o t i e n t Algebras C o n t a i n i n g Large Subspaces o f Distributions

97

5 6

7

Q u o t i e n t Algebras Possessing S p e c i a l P r o p e r t i e s

104

8

Definition o f Positive Generalized Functions

107

9

Powers f o r C e r t a i n

L i m i t a t i o n s on Embedding Smooth Functions i n t o Chains o f Q u o t i e n t Algebras

110

10

Special Classes o f Regular I d e a l s

113

11

The P r o o f o f Lemma 1

115

12

Example o f S h r i n k i n g Non-6 Sequence

117

13

I n e x i s t e n c e o f t h e L a r g e s t Regular I d e a l s

118

CHAPTER 4 : R e s o l u t i o n o f S i n a u l a r i t i e s o f Weak S o l u t i o n s f o r Polynomi a1 N o i l inear PDEs

121

Introduction

121

The Case o f Simple Polynomial N o n l i n e a r PDEs

122

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 N o n l i n e a r Shock Waves

131

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 Klein-Gordon Type Nonl in e a r Waves

133

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 l u t i o n s f o r General Polynomial Nonl inear PDEs

134

J u n c t i o n C o n d i t i o n s and 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 l u t i o n s f o r t h e Equations o f Magneto-hydrodynamics and General R e l a t i v i t y

139

TABLE OF CONTENTS

xv

6

Resoluble Systems o f Polynomial N o n l i n e a r PDEs

152

7

Computation o f t h e J u n c t i o n C o n d i t i o n s

155

8

Examples o f Resoluble Systems o f PDEs

156

CHAPTER 5: S t a b i l i t y and Exactness o f S e q u e n t i a l and Weak S o l u t i o n s f o r Polynomial N o n l i n e a r PDEs

0

Introduction

163 163

1 B a s i c Facts and Remarks Concerning S t a b i l i t y

163

2

Maximal S t a b i l i t y

165

3

I n t e r p l a y between S t a b i l i t y and Exactness

167

4

Remarks i n t h e F i n i t e Smoothness Case

16 7

5

S t a b i l i t y o f Regular Weak S o l u t i o n s

168

6

S t a b i l i t y o f S o l u t i o n s w i t h Jump D i s c o n t i n u i t i e s f o r Resoluble Systems o f PDEs

170

CHAPTER 6: C h a r a c t e r i z a t i o n o f t h e Necessary S t r u c t u r e o f t h e Algebras C o n t a i n i n g t h e D i s c r i b u t i o n s Introduction

173

Zero F i l t e r s

174

D e n s i t y C h a r a c t e r i z a t i o n o f Z-Fi 1t e r s

176

Large C"-Regular I d e a l s

178

F u r t h e r P r o p e r t i e s o f Z-Fi 1t e r s

182

C o n s t r u c t i o n and C h a r a c t e r i z a t i o n o f a Class o f Vanishing I d e a l s

185

Countable Reduced Products o f V e c t o r Spaces

194

CHAPTER 7: Quantum S c a t t e r i n g i n P o t e n t i a l s P o s i t i v e Powers o f the Dirac 6 D i s t r i b u t i o n

0

173

Introduction

199 199

1 Wave F u n c t i o n S o l u t i o n s and J u n c t i o n R e l a t i o n s

199

2

Weak Wave F u n c t i o n S o l u t i o n s

20 1

3

Smooth Representations f o r t h e D i r a c 6 D i s t r i b u = t i o n and Smooth Weak Wave F u n c t i o n S o l u t i o n s

2 10

Wave F u n c t i o n S o l u t i o n s i n t h e Algebras Contain= ing the Distributions

214

4

TABLE OF CONTENTS

xvi

5

S t a b i l i t y and Exactness o f t h e Wave F u n c t i o n Sol u t i ons

CHAPTER 8 : Products w i t h D i r a c 6 D i s t r i b u t i o n s

220 22 3

0

Introduction

223

1

A Class o f R e g u l a r i z a t i o n s

223

2

Products w i t h D i r a c 6 D i s t r i b u t i o n s

229

3

Application t o a R i c c a t i D i f f e r e n t i a l Equation

234

4

Val id i t y o f Formulas i n Quantum Mechanics

2 35

5

The E x i s t e n c e o f t h e Sequences o f F u n c t i o n s i n Z6

242

Remark on N i l p o t e n t Elements i n t h e Q u o t i e n t A1 gebras

249

L i n e a r Independent F a m i l i e s o f D i r a c 6 D i s t r i = butions a t a Point

251

0

Introduction

251

1

Compatible Q u o t i e n t Algebras and Independent V a r i a b l e Transforms

251

L i n e a r Independent F a m i l i e s o f D i r a c 6 D i s t r i b u = tions a t a Point

255

Generalized D i r a c 6 D i s t r i b u t i o n s

257

6 CHAPTER 9

2 3

CHAPTER 10: Support and Local P r o p e r t i e s

265

0

Introduction

26 5

1

Support o f Elements i n Q u o t i e n t Algebras

265

2

L o c a l i z a t i o n w i t h i n t h e Q u o t i e n t Algebras

270

3

Equivalence between S=O and supp

4

Domains o f Sol vabi 1it y f o r Polynomi a1 Nonl i n e a r PDEs

S=0

278 2 79

FINAL REMARKS

281

APPENDIX 1: N e u t r i x C a l c u l u s and N e g l i g i b l e Sequences o f Functions

285

APPENDIX 2: The Embedding I m p o s s i b i l i t y R e s u l t o f L.Schwartz

2 89

APPENDIX 3: A N o n l i n e a r E x t e n s i o n of t h e Lax-Richtmyer Equivalence between S t a b i l i t y and Convergence o f

xvi i

TABLE OF CONTENTS

APPENDIX 4: REFERENCES

D i f f e r e n c e Schemes

29 3

The Cauchy-Bolzano Q u o t i e n t Algebra Construe= t i o n o f t h e Real Numbers

30 1 30 5

NOTE TO THE READER A t a f i r s t l e c t u r e , t h e Reader may c o n c e n t r a t e on t h e f o l l o w i n g s e c t i o n s : Foreword Prel i m i naries Chapter 1, Sections 0-5,

7 , 9 , 10, 12

Chapter 3, Sections 0-4, 7, 8, 10 Chapter 4, Sections 0-6 Chapter 5, Sections 0-6 Chapter 7, S e c t i o n s 0-5 However, f o r a deeper understanding o f t h e method presented i n t h i s work, t h e f o l l o w i n g s e c t i o n s a r e important: Chapter 1, Sections 6, 11 Chapter 2, Sections 0, 1, 3-5 Chapter 3, Sections 5, 6, 9, 13 Chapter 6, Sections 0-5 F i n a l Remarks Appendices 1, 2, 4

xviii

NOTAT I ONS

N = I0,1,2,

R

=

[PI

N

u

= p1 +

...... } {m}

...

P G q * Pi 4 q i

+ pnY where p = (pl, Y 1

iG n

...,p,)

where P

xix

E

N- n

= (pl

,... y p n ) y q

=

( 4 ly...q,)E -n N

This Page Intentionally Left Blank

CHAPTER 1 SEQUENTIAL SOLUTIONS OF NONLINEAR PDEs

0.

Introduction

The a i m of t h e p r e s e n t c h a p t e r i s twofold, v i z t o s e t u and m o t i v a t e t h e general framework used i n t h e sequel i n t h e s t u d y & u e n t i m a r t i = c u l a r weak o r d i s t r i b u t i o n s o l u t i o n s o f polynomial n o n l i n e a r PDEs. I n t h e f o l l o w i n g chapters, t h a t framework w i l l u s u a l l y be r e s t r i c t e d t o chains o f algebras containing the d i s t r i b u t i o n s

D' c Amc

... c

Ap c

... c A" ,

p E

8;

however, t o understand f u l l y t h e phenomena i n v o l v e d i t i s a d v i s a b l e t o c o n s i d e r t h e framework i n i t s f u l l g e n e r a l i t y . The s e q u e n t i a l s o l u t i o n s w i l l be elements o f v e c t o r spaces o r algebras o f ' g e n e r a l i z e d f u n c t i o n s ' which a r e c l a s s e s o f sequences o f f u n c t i o n s on domains i n Euclidean spaces. There a r e a t l e a s t two e q u a l l y i m p o r t a n t reasons f o r c o n s t r u c t i n g v a r i o u s spaces o f ' g e n e r a l i z e d f u n c t i o n s ' . The f i r s t , as mentioned i n t h e P r e l i = m i n a r i e s , i s connected w i t h t h e way 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 d e f i n e d f o r ' g e n e r a l i z e d f u n c t i o n s ' . The second reason, which w i l l be t h e f i r s t t o become atmarent as t h i s c h a p t e r i s read, i s t h e i n t e r p l a y between the s t a b i l i t e n e r a l i t and exactness p r o p e r t i e s o f s e q u e n t i a l i o l u t i o n s of p ~ ' n E s . Given t h e n e c e s s i t y o f d e a l i n g w i t h v a r i o u s spaces o f ' g e n e r a l i z e d func= t i o n s ' , t h i s c h a p t e r p r e s e n t s t h e b a s i c a l g e b r a i c ideas i n v o l v e d i n t h e c o n s t r u c t i o n s encountered w i t h i n t h e n e x t chapters. General a l g e b r a i c r e s u l t s on t h e s t a b i l i t y , 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 and r e g u l a r i t y o f s e q u e n t i a l s o l u t i o n s f o r polynomial n o n l i n e a r PDEs w i l l be presented. I n l a t e r c h a p t e r s some o f t h e s e r e s u l t s w i l l be strengthened w i t h t h e h e l p o f a d d i t i o n a l a n a l y t i c o r t o p o l o g i c a l methods a p p l i e d t o v a r i o u s p a r t i c u = l a r cases o f PDEs. T h i s c h a p t e r a c t u a l l y p r e s e n t s a 'blow u p ' o f t h e e s s e n t i a l a l g e b r a i c phenomena i n v o l v e d i n t h e s t u d y o f s e q u e n t i a l s o l u t i o n s o f polynomial non= l i n e a r PDEs, p o i n t i n g o u t

-

t h e wa p a r t i a l d e r i v a t i v e s can be d e f i n e d on a l g e b r a s o f ' g e n e r a l i z e d i ions' func?

as w e l l as

1

2

E.E.

-

Rosinger

+

t h e i n t e r l a between t h e s t a b i l i t y , g e n e r a l i t y and exactness proper= t i e s o sequential solutions.

The d i s c u s s i o n amounts t o a reassessment o f t h e balance between a l g e b r a and t o p o l o g y i n t h e s t u d y o f PDEs, s t r e s s b e i n g placed on t h e r i g h t stage a t which t h e t o p o l o g y s h o u l d be i n t r o d u c e d , and i t s c o r r e c t l e v e l . Indeed, i t becomes obvious t h a t t o o e a r l y a c o n s i d e r a t i o n o f t o p o l o g y can p r e c l u d e e s s e n t i a l a l g e b r a i c o p t i o n s r e l a t e d t o t h e above-mentioned i n t e r p l a y . Moreover, t h e c h o i c e o f t h e r i g h t t o p o l o g i c a l s t r u c t u r e s m i g h t prove t o be p a r t i c u l a r l y i m p o r t a n t .

A somewhat l e s s r e l a t e d example i s o f f e r e d i n t h e Appendix 3, where t h e Lax-Richtmyer equivalence between t h e s t a b i l i t y and convergence of d i f f e r = ence schemes f o r l i n e a r e v o l u t i o n equations i s extended t o t h e correspon= d i n g n o n l i n e a r case b y n o t i c i n g t h a t s t a b i l i t y as w e l l as convergence a r e p r o p e r t i e s which do n o t n e c e s s a r i l y r e q u i r e t h e completeness o f t h e u n d e r l y i n g t o p o l o g i c a l s t r u c t u r e , and t h a t t h e r e f o r e t h e i n i t i a l p r o o f i n t h e l i n e a r case, based on t h e u n i f o r m boundedness p r i n c i p l e o f l i n e a r o p e r a t o r s i n Banach spaces, can be r e p l a c e d b y a s i m p l e r , r a t h e r t r i v i a l one, o n l y i n v o l v i n g c o n t i n u i t y , compactness and boundedness i n normed v e c t o r spaces , a p r o o f t h a t w i l l be v a l i d i n t h e n o n l i n e a r case as w e l l . 1.

The Problem o f S t a b i l i t y o f Sequential S o l u t i o n s

The p r e s e n t work w i l l deal m a i n l y w i t h t h e c l a s s o f p o l y n o m i a l n o n l i n e a r PDEs :

(1)

z:

T-r-

Ci(X)

1 Q i G h

where u : R + R ' i s t h e unknown f E C o ( Q ) a r e given; and ci, partial derivatives.

(2)

Dpij

U(X) = f ( x )

,

x E

R

l G j G k i u n c t i o n , w h i l e R c Rn i s non-void, open , w i t h pij E Nn , denote t h e usdal

,b,j

-

The o r d e r o f t h e PDE i n (1) i s

m=max{lpij]

1 1 G i G h

,

l Q j < k i l

p r o v i d e d t h a t none o f t h e c o e f f i c i e n t s ci vanishes everywhere on R. I t w i l l be convenient t o a s s o c i a t e w i t h t h e PDE i n ( l ) , t h e corresponding PDO

(3)

T(D) =

Z

ci(x)

I G i G h

Dpij

1 G J Gki

which o b v i o u s l y generates a mapping

(4)

T ( D ) : Cm(R) * c" (a).

A c l a s s i c a l s o l u t i o n o f (1) i s any f u n c t i o n

(5)

JI

E

cm(Q)

f o r which t h e mapping (4) s a t i s f i e s

,

X E R

SEQUENTIAL SOLUTIONS

(6 1

3

T(D)$ = f on R.

I n case one i s i n t e r e s t e d i n s o l u t i o n s o f (1) which a r e n o t s u f f i c i e n t l y smooth a l l o v e r R i n o r d e r t o be c l a s s i c a l s o l u t i o n s , a usual procedure i s t o c o n s t r u c t weak o r i n general, s e q u e n t i a l s o l u t i o n s g i v e n b y s u i t a b l e sequences o f f u n Z Z n s , f o r i n s t a n c e (7)

s =

... , $,

($0,+1,

...), w i t h $,

E

Cm(Q), f o r v

E

N.

The meaning a s s o c i a t e d w i t h a s e q u e n t i a l s o l u t i o n ( 7 ) i s t h a t i t 'conver= ges' t o an ' i d e a l ' s o l u t i o n S o f ( l ) , supposed t o b e l o n g t o a c e r t a i n ' c o m p l e t i o n ' o f a s u i t a b l e f u n c t i o n space on n. F o r i n s t a n c e , i n case s i s a usual weak s o l u t i o n , t h e n S i s a d i s t r i b u t i o n . When c o n s t r u c t i n g s e q u e n t i a l s o l u t i o n s ( 7 ) o f ( l ) , t h e c o r r e c t and f u l l i n f o r m a t i o n on t h e ' i d e a l ' s o l u t i o n S o b v i o u s l y r e q u i r e s a knowledge o f a l l t h e sequences o f f u n c t i o n s t = s + v =

(8)

+

XI,

( X O Y

= ($0

+

($0,

$1,

..., $,,

... x,, . . . I

xo,

Y

$1

-t x 1 ,

...) t

=

. . . I

$,

+

x,,

..

with (8.1)

v = (XO,

x1,

... , X Z , .. . ) , where

xvE

which sequences t a l s o 'converge' t o t h e same ' i d e a l ' s o l u t i o n S . I n o t h e r words, t h e s i g n i f i c a n c e o f a weak o r s e q u e n t i a l s o l u t i o n o f (1) g i v e n b y a s i n l e sequence of f u n c t i o n s ( 7 ) can p r o p e r l y be a p p r e c i a t e d o n l y i f i t s s t a i l i t y p r o p e r t y (8) i s known as w e l l .

-9b

I n t h e case o f a l i n e a r PDE w i t h Cm-smooth c o e f f i c i e n t s and t h e sequen= t i a l s o l u t i o n (7)?EFGiny a d i s t r i b u t i o n S E U'(R), t h e s t a b i l i t y p r o p e r = t y (8) can e a s i l y be e s t a b l i s h e d . Indeed, owing t o t h e l i n e a r i t y and c o n t i n u i t y o f t h e corresponding PDO, one can t a k e i n (8.1) any sequence o f f u n c t i o n s weakly convergent t o z e r o i n U'(R). However, i n t h e case o f a n o n l i n e a r PDE, t h e problem o f t h e s t a b i l i t y o f a s e q u e n t i a l s o l u t i o n i s no l o n g e r t r i v i a l and i t s s t u d y i s e s s e n t i a l , even i f i t i s q u i t e o f t e n overlooked. Indeed, t h e p r o p e r t i e s o f a sequen= t i a l s o l u t i o n o b t a i n e d b y a p a r t i c u l a r c o n s t r u c t i o n o f a s i n l e sequence o f f u n c t i o n s ( 7 ) , c o u l d happen t o be a r a t h e r i r r e l e v a n t a c c i e n t , a r e s u l t o f t h e d i s c o n t i n u i t o f t h e c o r r e s p o n d i n g n o n l i n e a r PDO on a c e r = t a i n space o f i s r i u i o n s .

+

d

The f o l l o w i n g v e r y s i m p l e example i l l u s t r a t e s t h e above-mentioned p o i n t . Consider i n t h e one-dimensional case, ( n = l ) , w i t h R = R', t h e p o l y n o m i a l n o n l i n e a r POE o f o r d e r m=O:

(9)

( ~ ( x ) ) '= 1

,

x E R'.

A d i r e c t computation w i l l show t h a t t h e sequence of f u n c t i o n s (10)

s =

($0,

$1,

...,

$"Y

...I

4

E.E.

Rosinger

with (10.1)

$,(x)

=

&?-' cos vx,

for

v

E N,

x

E

R',

g i v e s a usual weak s o l u t i o n o f ( 9 ) - i . e . t h e f u n c t i o n s ( $ ) 2 converge i n However, t h e sequence o f f u n c t i a n s (10) i s a l s o P ' ( R ' ) t o 1, when v m. a usual weak s o l u t i o n o f t h e e q u a t i o n -f

(11)

U ( X ) = 0,

x

E

R'.

As seen i n t h e l i t e r a t u r e , [ 1 3 5 , 160, 96, 126, 1251 weak s o l u t i o n s f o r

n o n l i n e a r PDEs a r e o f t e n o b t a i n e d as l i m i t s - i n s u i t a b l e d i s t r i b u t i o n spaces - o f sequences o f f u n c t i o n s ( 7 ) , whose terms qv s a t i s f y l i n e a r i z e d o r r e g u l a r i z e d v e r s i o n s o f t h e i n i t i a l n o n l i n e a r PDEs. U s u a l l y , o n l y p a r t i c u l a r sequences o f f u n c t i o n s ( 7 ) a r e e x h i b i t e d , and t h e y r e s u l t from m o n o t o n i c i t y , f i x e d p o i n t o r compactness arguments. The l a s t - m e n t i o n e d case appears t o be most i n need o f subsequent s t u d y i n o r d e r t o determine t h e s t a b i l i t y p r o p e r t i e s o f t h e s e q u e n t i a l s o l u t i o n s presented. The non-uniqueness o f weak s o l u t i o n s s a t i s f y i n g n o n l i n e a r PDEs t o g e t h e r w i t h t h e usual i n i t i a l o r boundary c o n d i t i o n s and o b t a i n e d as d i s t r i b u t i o n a l l i m i t s o f p a r t i c u l a r l y c o n s t r u c t e d sequences o f f u n c t i o n s ( 7 ) may a l s o r e q u i r e a s t a b i l i t y a n a l y s i s i n t h e above-mentioned sense. I t follows t h a t specifying a sequential s o l u t i o n (7) together w i t h i t s s t a b i l i t y p r o p e r t y ( 8 ) , means t o g i v e a c l a s s o f sequences o f f u n c t i o n s i . e . an element

S = s t U = { t ( t = s t v , v E U} i n a q u o t i e n t space (12)

E

(then s

S/U

E

S, S E E )

where U c S a r e s u i t a b l e v e c t o r spaces o f sequences o f f u n c t i o n s on Q . v a r i o u s i n s t a n c e s , i t w i l l be c o n v e n i c n t t o deal w i t h p a r t i c u l a r cases o f t h e q u o t i e n t spaces (12), g i v e n by q u o t i e n t a l g e b r a s (13)

A

In

= A/l

where A i s an a l g e b r a o f sequences o f f u n c t i o n s on R, and 1 i s an i d e a l i n A. The most general space o f f u n c t i o n s considered i n t h e sequel i s t h e alge= b r a M(R) o f a l l r e a l - v a l u e d , mefisurable and a.e. f i n i t e f u n c t i o n s on R. T f o l l o w s t h a t t h e space (M(R)) o f a l l sequences o f f u n c t i o n s i n M(R), i s i n a n a t u r a l way an a s s o c i a t i v e and commutative algebra, when consider= e d w i t h t h e term-by-term o p e r a t i o n s on sequences of f u n c t i o n s . The most general cases o f q u o t i e n t spaces ( 1 2 ) and q u o t i e n t a l g e b r a s (13) However, i t w i l l be u s e f u l t o deal a l s o w i t h t h e subalgebras

w i l l be o b t a i n e d when V , S , I , A c(M(R))~.

R

E

w

of (M(n))N, and r e s t r i c t U ,

S,

I,

(C"Q))N

,

A i n (12) and (13) a c c o r d i n g l y .

5

SEQUENTIAL SOLUTIONS

One o f t h e b a s i c advantages o f d e f i n i n g t h e ' g e n e r a l i z e d f u n c t i o n s ' as e l e = ments o f t h e q u o t i e n t spaces (12) o r q u o t i e n t algebras ( 1 3 ) i s t h a t i t o f f e r s t h e p o s s i b i l i t y o f a d i r e c t and e x p l i c i t s t a b i l i t y a n a l y s i s o f t h e s e q u e n t i a l s o l u t i o n s o f polyE%%T n o n l i n e a r PDEs. 2.

D e f i n i t i o n o f t h e Q u o t i e n t Spaces and Algebras

As a l r e a d y mentioned, and w i l l now become e v i d e n t , t h e r e a r e several rea= sons f o r c o n s t r u c t i n g v a r i o u s q u o t i e n t spaces ( 1 2 ) and q u o t i e n t a l g e b r a s (13) when d e a l i n g w i t h s e q u e n t i a l s o l u t i o n s o f PDEs.

+

One o f t h e main reasons i s t h e s p e c i f i c i n t e r l a , i n t h e case o f each p a r t i c u l a r problem, between t h e c o n f l i c t i n g s t a i l i t y , g e n e r a l i t y and exactness p r o p e r t i e s o f t h e s e q u e n t i a l s o l u t i o n s i n v o l v e d . Indeed, t h e s t a b i l i t y p r o p e r t y o f s e q u e n t i a l s o l u t i o n s r e s u l t s i n an i n = t e r e s t i n q u o t i e n t spaces ( 1 2 ) and q u o t i e n t a l g e b r a s (13), h a v i n g v e c t o r spaces V , o r e l s e i d e a l s 7 , t h a t a r e l a r g e , p o s s i b l y maximal w i t h i n t h e a d d i t i o n a l c o n d i t i o n s t h e y w i l l have t o s a t i s f y . On t h e o t h e r hand, i n o r d e r t o i n c l u d e among t h e s e q u e n t i a l s o l u t i o n s a l l t h e c l a s s i c a l , weak o r d i s t r i b u t i o n s o l u t i o n s , t h u s s e c u r i n g t h e g e n e r a l i t y o f t h e n o t i o n o f s e q u e n t i a l s o l u t i o n , one i s i n t e r e s t e d i n l a r e q u o t i e n t spaces (12) and q u o t i e n t algebras (13). C l a s s i c a l s o l u t i o n s w i 1 ob= v i o u s l y be i n c l u d e d among t h e s e q u e n t i a l s o l u t i o n s , i n case embedding c o n d i t i o n s o f t h e f o l l o w i n g form a r e r e q u i r e d :

+

F c E

(14)

= S/V

or else

(15)

G

C

A = A/Z

where F and G a r e s u i t a b l e v e c t o r subspaces, o r e l s e subalgebras i n M ( R ) . I n o r d e r t o i n c l u d e t h e weak o r d i s t r i b u t i o n s o l u t i o n s among t h e sequen= t i a l s o l u t i o n s , t h e f o l l o w i n g embedding c o n d i t i o n s w i l l be r e q u i r e d :

(16)

D'(Q) c

(17)

U'(n) c A

E = S/V = A/I.

Obviously, t h e above requirements t h a t b o t h V and E i n (12), o r e l s e b o t h Z and A i n ( 1 3 ) s h o u l d be l a r g e , a r e t o a c e r t a i n e x t e n t c o n f l i c t i n g . The impact o f t h e exactness p r o p e r t y o f s e q u e n t i a l s o l u t i o n s w i l l be presented i n S e c t i o n 4. Meanwhile, i t i s u s e f u l t o r e w r i t e t h e above embeddings (14) and (15) under t h e form o f t h e i n j e c t i v e mappings F 3 +

(18)

-> u($)

+

V

E

E = S/V

o r else

(19)

G

3

JI >-

u($) t

rE

A =

AIL

6

Rosinger

E.E.

where for each $ u($)

E

M(R) we denote

...

= ($,$,

y

$,

.

.

.

by .

I

.

.

1

the sequence of functions with a l l the terms equal t o the function $. Obviously

u

=

I

tu(Ji)

E

M(R)I

i s a subalgebra in ( M ( Q ) ) N , which through the mapping $ u($) i s isomor= phic with M ( i 2 ) . Moreover, Q = { u ( o ) } i s the null ideal, while u(1) i s the unit element in (M(R))N. +

Now, i t i s easy t o see t h a t (18) i s equivalent t o the existence of the inclusion diagram:

satisfying (20.1)

V n UF =

Q

and (19) to:

2 -' ' G satisfying (21.1)

I n

uG = Q

where we used the notation:

'F = {u($)

I

$

F 1

and similarly for UG.

I t i s worth mentioning t h a t , even i f n o t in an explicit way, conditions of type (20.1), (21.1) on inclusion diagrams (20), (21) were used in [2161 connected with the 'neutrix calculus' developed i n i t i a l l y in order to simp1 ify methods i n asymptotical expansions. With the terminology in 12161, the sequences of functions i n V o r I are 'negligible'. In t h i s sense the meaning of (20.1) for instance, i s that the difference JI -x, w i t h $,x E F, i s not 'negligible' unless $ = x. Therefore the quotient space E = S/V 'distinguishes' between different functions in F. Similar= ly for the condition ( 2 1 . 1 ) . More details concerning the 'neutrix calcu= lus' can be found in Appendix 1. I t i s specially important t o point out that in the case of inclusion dia=

SEQUENTIAL SOLUTIONS

7

grams ( 2 1 ) , t h e r a t h e r simple condition ( 2 1 . 1 ) possesses a p a r t i c u l a r l y g r e a t power. Indeed, as seen l a t e r in Chapter 6 , f o r l a r g e and important c l a s s e s o f inclusion diagrams ( 2 1 ) , the condition (21.1) means t h a t the ' n e g l i g i b l e ' sequences of functions w E 7 have t o vanish densely on R , i.e. the sets

R

W

=

{x

E

R

I

3v

N : w v ( x ) = 03

E

a r e dense in R , where wv, with w E N , denotes t h e functions t h a t a r e t h e terms o f the sequence o f functions w. In the sequel, i t will be convenient s l i g h t l y t o r e s t r i c t t h e inclusion diagrams ( 2 0 ) and ( 2 1 ) , i . e . the q u o t i e n t spaces ( 1 2 ) and q u o t i e n t alge= bras ( 1 3 ) , which give t h e spaces of 'generalized f u n c t i o n s ' . Suppose given a vector subspace F C M ( R ) and a subalgebra G C M ( R ) . We s h a l l denote by

rs

the e t o f a l l q u o t i e n t spaces E in F and

v->

= S/V,

where V and S a r e vector subspaces

s

i

with

V n

UF

= O_.

W e s h a l l denote by ALG

the s e t o f a l l q u o t i e n t algebras A I i s an ideal i n A and I->

=

A / 7 , where A i s a subalgebra in G

N

,

A

i

with

7 n UG =

!!

Obviously F = UF/Q

E

VSF

G

= UG/Q E

ALG

The space of d i s t r i b u t i o n s D'(n) w i l l i n t h e sequel be t h e u o t i e n t space o f b a s i c i n t e r e s t . I t w i l l be convenient t o use several equiva ent con= s t r u c t i o n s giving D'(n). To i l l u s t r a t e , given a vector subspace F c M ( Q ) , denote by SF the s e t of a l l sequences s E FN of measurable functions weak= l y convergent in D'(n), and by V F t h e s e t of a l l sequences V E S F weakly convergent t o zero i n D ' ( n ) , i . e . c o n s t i t u t i n g the kernel of the l i n e a r mapping

s7____

(24.1)

SF 3 5

> G , .>E

D'(Q)

E.E. Rosinger

8

where (24.2)

,..., +", ..... ) .

assuming t h a t s = ($o, J1l

-

Obviously, i n case F i s l a r e enough i.e. s e q u e n t i a l l y dense i n V(Q)f o r instance, c o n t a i n s a e polynomials, o r c o n t a i n s qQ),e t c . , t h e mapping (24) i s a l i n e a r s u r j e c t i o n , t h e r e f o r e t h e mapping

rrft

(25)

SF/WF 3

+

(5

< S,'

W F ) >-

w i l l be a v e c t o r space isomorphism. spaces (26)

D'(Q2) =

>E

P'(Q)

I n t h a t way, we o b t a i n t h e q u o t i e n t

S F / V F E VSF

p r o v i d e d t h a t F c M(R) i s s u f f i c i e n t l y l a r g e , as mentioned above. The cases o f s p e c i a l i n t e r e s t w i l l be those where F = CR (n), w i t h R E fl given. I n t h e n e x t .chapters, we f u n c t i o n s ' from Al. , i . e . c o n s t r u c t them i n t u c h a r e s u l t i n g from (23) t h e we s h a l l have G = $(a),

s h a l l deal o n l y with spaces o f ' g e n e r a l i z e d w i t h q u o t i e n t algebras A = MI, and we s h a l l way t h a t a p a r t from t h e embedding c o n d i t i o n (15) embedding c o n d i t i o n ( 1 7 ) w i l l a l s o h o l d . U s u a l l y , witha. E s u i t a b l y chosen.

w

A second main, reason f o r c o n s t r u c t i n g v a r i o u s q u o t i e n t spaces and algebras when d e a l i n g w i t h s e q u e n t i a l s o l u t i o n s o f PDEs i s t h e way 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 d e f i n e d on algebras c o n t a i n i n g t h e E,i t i s i m p o r t a n t t h a t i t s h o u l d n o t be assumed t h a t each o f t h e alge= 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 i s i n v a r i a n t under a r b i t r a r y p a r t i a l d e r i v a t i o n , even i f some o f them w i l l be. I n o t h e r words, 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 s h o u l d be a1 lowed t o a c t between d i f f e r e n t algebras. Moreover, t h e e x i s t e n c e o f p a r t i a l d e r i v a t i v e s s h o u l d b e assumed o n l y 9 t o a c e r t a i n order.

I n t h i s connection, we s h a l l d e f i n e 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 w i t h = i n t h e s l i g h t l y more general framework. (27)

Dp :

E

-+

A

,

p E N',

IpI Q R

where E = S/W E VSF, A = A / l E AL a r e s u i t a b l y chosen and R E fl may depend on E and A. As seen i n S e c t i g n 11, t h e assumption t h a t E = A and II 2 1 i n ( 2 7 ) , which a u t o m a t i c a l l y i m p l i e s t h a t R = m, may under r a t h e r general c o n d i t i o n s l e a d t o a somewhat p a r t i c u l a r m u l t i p l i c a t i o n i n t h e a l g e b r a A, g i v i n g f o r i n s t a n c e

62 = 6.D6 = 0 where 6 i s t h e D i r a c d i s t r i b u t i o n . I f i t i s r e q u i r e d t h a t 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 i n ( 2 7 ) should t a k e values i n q u o t i e n t algebras, t h i s requirement w i l l p r o v i d e t h e oppor= t u n i t y o f e x t e n d i n g t h e mappings ( 4 ) generated b y t h e {olynomial n o n l i n e a r PDO i n ( 3 ) t o mappings T(D) : E + A between spaces o f g e n e r a l i z e d func= t i o n s ' , as w i l l be seen i n S e c t i o n 3. I n t h a t way a p r o p e r framework i s

SEQUENTIAL SOLUTIONS

9

o b t a i n e d f o r f i n d i n g s e q u e n t i a l s o l u t i o n s f o r polynomial n o n l i n e a r PDEs. The q u o t i e n t s t r u c t u r e o f E and A i n ( 2 7 ) suggests d e f i n i n g 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 as t h e term-by-term p a r t i a l d e r i v a t i v e s of sequences o f f u n c t i o n s . F o r instance, i n t h e case

sc

(C"(n))N

one can d e f i n e

Dp : E

-f

A,

p

Nn,

E

Ip

(s+vR)E ~ ' ( n ) v, s

E

sR,

i s surjective Corollary 1 Suppose $ E Cm(C2\ r ) i s a s o l u t i o n onmR \ r o f t h e m-th o r d e r polynomial i s C -smooth and nowhere dense. Then n o n l i n e a r PDE i n ( l ) , where r C $, b y means o f ( 5 9 ) , d e f i n e s a s e q u e n t i a l s o l u t i o n i n A l d And o f t h e PDE -f

) and r has z e r o Lebeseque measure,$ , b y means o f ( 5 9 ) , d e f i n e s a s e q u e n t i a l s o l u t i o n i n U,id + And o f t h e PDE i n ( l ) , i n (1).

I f $ E Lioc(Q \

r

w i t h U;ld g i v e n b y ( 7 3 ) , f o r R = m. Proof T h i s f o l l o w s f r o m Theorem 1 and ( 6 3 ) , ( 6 1 ) , (62) and ( 7 4 ) .

0

24

E.E. Rosinger

Remark 3 ( a ) F o r t h e method f o r t h e r e g u l a r i z a t i o n o f s i n g u l a r i t i e s presented i n t h i s s e c t i o n , t h e assumption t h a t t h e s i n g u l a r i t i e s a r e concentrated on nowhere-dense subsets i s e s s e n t i a l a t two stages. F i r s t , i n o r d e r t o e s t a b l i s h t h a t lnd i s an i d e a l i n (Co(n))N(see ( 6 6 ) ) , use i s made o f t h e f a c t t h a t t h e union o f two nowhere-dense subsets i s again nowhere-dense. Secondly, i n o r d e r t o e s t a b l i s h (67), use i s made o f t h e f a c t t h a t a con= t i n u o u s f u n c t i o n v a n i s h i n g o u t s i d e a nowhere-dense subset i s i d e n t i c a l l y zero. Obviously, b o t h r e l a t i o n s (66) and ( 6 7 ) a r e e s s e n t i a l i n o r d e r t o obtain (68).

(b) The power o f Theorem 1 l i e s i n t h e f a c t t h a t o n l y nowhere-dense s i n g u l a r i t i e s a r e r e q u i r e d , o f which i t i s known t h a t they may have arbi= t r a r y p o s i t i v e Lebesque measures. ( c ) The method f o r t h e r e g u l a r i z a t i o n o f s i n g u l a r i t i e s presented i n t h i s s e c t i o n i s a c t u a l l y t h e framework f o r more s p e c i a l i s e d cases o f a p p l i c a = t i v e i n t e r e s t , presented i n Chapter 4. 8. P r e s e r v a t i o n o f E l l i p t i c i t y and H y p o e l l i p t i c i t y I t i s shown i n t h e p r e s e n t s e c t i o n t h a t i n s p i t e o f notion o f sequential s o l u t i o n introduced i n Section s o l u t i o n s f o r e l l i p t i c and h y p o e l l i p t i c l i n e a r PDEs f o r i n s t a n c e i n t h e case o f s e q u e n t i a l s o l u t i o n s i n

the generality o f the 3, t h e r e g u l a r i t y o f w i l l be preserved, DAd + And.

Suppose g i v e n t h e 1i n e a r PDO (76)

L(D) =

C

1 u:

L ( D ) s v = f on V . I t 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 = <s, . > E D ' ( R ) s a t i s f i e s t h e PDE i n (77) on t h e open, dense subset R' c 0. Thus i n view o f t h e e l l i p t i c i t y o f t h e PDO i n ( 7 6 ) , S w i l l be a n a l y t i c on R' .

n

Theorem 3 Suppose t h e m-th o r d e r l i n e a r PDO i n ( 7 6 ) i s h y p o e l l i p t i c . Thgn any And o f t h e PDE i n ( 7 7 ) i s a C -smooth s e q u e n t i a l s o l u t i o n i n UAd(fi) +

f u n c t i o n on a c e r t a i n open, dense subset o f R, p r o v i d e d t h a t f and UAd(fi) i s g i v e n by ( 7 3 ) , w i t h R m.

E

c"(Q)

Proof T h i s i s s i m i l a r t o t h a t f o r Theorem 2.

9.

General N o n l i n e a r PDEs

The method f o r s o l v i n g PDEs i n q u o t i e n t a l g e b r a s o f c l a s s e s o f sequences o f f u n c t i o n s can be used n o t o n l y i n t h e case o f polynomial n o n l i n e a r PDEs, b u t a l s o f o r a much w i d e r c l a s s o f n o n l i n e a r PDEs, as shown i n t h i s section. An m-th o r d e r continuous n o n l i n e a r PDE i s b y d e f i n i t i o n o f t h e form

(78)

F(x,u(x),

...,DPU ( X ) ,...)

where p E N n y I p I Gm, f E

(78.1)

R= carCp E Nnl

=

f(x), x

C"(n) and F

E

E

R,

C"(nxR'),

with

I p I Gm}.

Obviously, an m-th o r d e r p o l y n o m i a l n o n l i n e a r PDE o f t y p e (1) i s a p a r t i = c u l a r case o f t h e PDE i n ( 7 8 ) . I n o r d e r t o s o l v e PDEs o f t y p e (78) w i t h i n q u o t i e n t a l g e b r a s , we s h a l l

26

Rosinger

E.E.

d e f i n e mappings o f t y p e ( 3 3 ) f o r t h e a s s o c i a t e d PDOs (79)

. ,Dpu(x) ,... ),

F ( D ) u ( x ) = F(x,u(x),..

x E

a.

I n t h i s connection, suppose g i v e n a q u o t i e n t space E = S / V E VS; quotient algebra A = A / l

E

and a

ALG, where F and G a r e r e s p e c t i v e l y a v e c t o r

subspace and a subalgebra i n M(a). We then use t h e n o t a t i o n : F(D) (80 1 E 6 A only i f (80.1)

F(D) ( F n cm(n)) c G

(80.2)

F(D)(S n (cm(n)lN) c A

(80.3)

F(D)(t+v)

-

F(D)t

E

I, V t

E

S n (Cm(R))N, v E V n (Cm(fi))N

where F(D) in, t h e above r e l a t i o n s means t h e term-by-term a p p l i c a t i o n o f t h e mapping F(D) : Cm(n)

-f

(n)

C"

generated b y (79) , t o sequences o f f u n c t i o n s . Now i f t h e r e l a t i o n s (80) h o l d , one can d e f i n e t h e mapping (81)

F(D) : E

-+

A

by (81.1)

F(D)(s+V) = F ( D ) t + l ,

Y s E S

where (81.2)

t

E

sn(cm(n))N, s-t

E

v.

With t h e h e l p o f t h e mapping ( 8 1 ) , we can d e f i n e t h e n o t i o n o f s e q u e n t i a l s o l u t i o n o f a continuous n o n l i n e a r PDE. Suppose g i v e n t h e m-th o r d e r continuous n o n l i n e a r PDE i n (78) and a se= quence s E ( M ( R ) ) N o f measurable f u n c t i o n s on R. Then s i s c a l l e d a s e q u e n t i a l s o l u t i o n o f (78), o n l y i f f o r a c e r t a i n v e c t o r subspace F and subalgebra G i n M(R), t h e r e e x i s t a q u o t i e n t space E = S / V E VSF and a q u o t i e n t a l g e b r a A = A / l EALGy w i t h E F&D) A , such t h a t (82.1)

s E

S

and t h e mapping (81) maps S (82.2)

+

(then S = s

F(D)(s+v) = u ( f )

E

V E E = S/v)

E into f

E

A;

.e.

+ r.

I n view o f ( 8 1 ) , (80) and ( 3 0 ) , t h e c o n d i t i o n ( 8 2 ) i s o b v i o u s l y e q u i v a l e n t to

27

SEQUENTIAL SOLUTIONS

t E

sn

(cm(n)lN :

(83) s-t E V , F(D)t

-

u(f) E 7

which i s t h e same as t h e f o l l o w i n g two: (84.1)

s E S

and

vt

E

s

n (cm(n)lN :

(84.2) s - t E V * F(D)t

-

u ( f ) E 7.

The c r u c i a l p o i n t i n t h e above d e f i n i t i o n s i s t h e r e l a t i o n ( 8 0 ) , more e x a c t = l y the c o n d i t i o n (80.3). I n t h i s connection, t h e n o n - t r i v i a l examples given i n the next proposition are important. Proposition 5 Suppose g i v e n t h e m-th o r d e r continuous n o n l i n e a r PDO i n ( 7 9 ) . following r e l a t i o n s hold: F(D) F(D) And And, D',d(R) And.

Then t h e

Proof I n view o f ( 6 4 ) , t h e r e l a t i o n s (80.1) and (80.2) o b v i o u s l # h o l d i n b o t h cases. Regarding t h e r e l a t i o n (80.3), assume t E (Cm(n)) and v

E

Ind n (Cm(R))N.

where-dense subset V X E R \

1 p

E

Then, a c c o r d i n g t o ( 6 9 ) , t h e r e e x i s t s a closed, no=

r

c R, such t h a t

r :

N, V neighb. o f x :

V v ~ N , v ) > p : v

V

= 0 on V .

Therefore F(D)(tv+vV)

-

F ( D ) t v = 0 on V, V v

E

N, v > p

which means t h a t F(D)(t+v)

-

F(D)t

E

Ind;

and t h i s completes t h e p r o o f o f (80.3).

10.

0

V a r i a b l e Transforms

As seen i n t h e p r e v i o u s s e c t i o n , t h e q u o t i e n t spaces and a l g e b r a s o f c l a s s e s

E .E.

28

Rosinger

of sequences o f f u n c t i o n s can i n a n a t u r a l way accommodate n o t o n l y alge= b r a i c o p e r a t i o n s w i t h p a r t i a l d e r i v a t i v e s b u t a l s o a wide range o f such continuous o p e r a t i o n s . I n t h i s s e c t i o n , a n a t u r a l procedure i s presented o f e f f e c t i n g dependent and independent v a r i a b l e t r a n s f o r m s w i t h i n t h e framework o f t h e q u o t i e n t spaces and algebras mentioned. These v a r i a b l e t r a n s f o r m s w i l l be used i n Chapters 3,7 and 8. Suppose g i v e n a q u o t i e n t space E = S / V c V S F and a q u o t i e n t a l g e b r a A =A/IEALG, where F and G a r e r e s p e c t i v e l y a v e c t o r subspace and a subal= gebra i n M( n).

Dependent v a r i a b l e t r a n s f o r m s from E t o A can be d e f i n e d as f o l l o w s . pose g i v e n a mapping (85)

Sup=

T : R’ + R’

such t h a t

F

(86.1)

T o $ E G,

V $

(86.2)

T o s E A,

V s E S

where (-ros),(x) (86.3)

=

E

T ( S ~ ( X ) ) V, v E N , x E R, and f i n a l l y ,

T o ( s + v ) - T O s E 7 , V s E S, v E V .

Then o b v i o u s l y i t i s p o s s i b l e t o d e f i n e a mapping (87)

E

T :

+ A

by (87.1)

T(s+V) = T o s

+

7, V s

E S.

A mapping (85) s a t i s f y i n g t h e c o n d i t i o n s ( 8 6 ) , w i l l be c a l l e d an pendent v a r i a b l e t r a n s f o r m .

E

-t

A-de=

A simple example i s g i v e n by t h e mapping T

: R’

-+

R’, w i t h ~ ( y =) ay

+ by

f o r y E R’,

where a,b E R’ a r e given, which i s o b v i o u s l y an E + A-dependent v a r i a b l e transform, provided t h a t G c o n t a i n s F , as w e l l as t h e c o n s t a n t f u n c t i o n s on Q, and E ---I A (see ( 4 4 ) ) .

I f t h e mapping (85) i s n o t defined everywhere on R ’ , f o r i n s t a n c e , t h e p o s i ti ve power mapping (88)

T~

:

[O,m)

+

R’,

w i t h ~ ~ ( =yyay ) for y E [Op),

where c1 E ( O , m ) b e i n g g i v e n (see i t s a p p l i c a t i o n s i n Chapters 3 and 7 ) , i t i s p o s s i b l e t o proceed i n a more general way as f o l l o w s . Suppose g i v e n a mapping

SEQUENTIAL SOLUTIONS (89)

A+

T :

R',

29

A C R 1

and t h e r e e x i s t s ST c S , s a t i s f y i n g t h e c o n d i t i o n s (90.1)

u

(90.2)

-r 0 s E A , V s

-

v

s , S ' E ST : s-s' E

s = s'

€ST.

Using t h e n o t a t i o n (91)

E

T

I

= {(stv) E E

s

E

ST}

i t i s o b v i o u s l y p o s s i b l e t o d e f i n e a mapping

(92)

T : E

+ A

T

by (92.1)

o s + I, V s E S ~ .

T ( s + V ) = -r

A mapping ( 8 9 ) s a t i s f y i n g t h e c o n d i t i o n s ( 9 0 ) w i l l be c a l l e d an E dent v a r i a b l e p a r t i a l t r a n s f o r m .

-t

A-depen=

To i l l u s t r a t e , we show how a r b i t r a r y p o s i t i v e powers can be d e f i n e d w i t h i n t h e above framework. We use t h e n o t a t i o n (93)

F

+

I

F

= {$ E

It f o l l o w s t h a t i f G 3 {$ E F n

-rCL o $ E G , V

C'(n),

e (a)

I

CL

E

(0,m)I (see ( 8 8 ) )

then

2 0, w x E n ~ F+. c

IJe a l s o use t h e n o t a t i o n

(94)

s,

=

Is

E

s

n (F+)

Nl

T~ o s E A , V

CL

E

(0,m)I.

Suppose g i v e n a s p l i t t i n g

(95)

s

=V@S'

s a t i s f y i n g the condition (95.1)

UF

c S'

which i s p o s s i b l e i n view o f (22). O b v i o u s l y S+ n S ' s a t i s f i e s ( g o ) , t h e r e f o r e u s i n g t h e n o t a t i o n (96)

E,

=

{(StV)E

E

one can, f o r each C L E(O,m), (97) by

yCL

: E,

+

A

sES+ n S ' } d e f i n e t h e mapping

30

E.E.

(97.1)

= Ta0 S

Ta(S+V)

+ I, v

S

Rosinger ES+n S ' .

In view of ( 9 5 . 1 ) , we have

and the p o s i t i v e power mappings ( 9 7 ) coincide with t h e usual p o s i t i v e powers of f u n c t i o n s , when applied t o E F+.

+

We now turn t o independent v a r i a b l e transforms. Suppose given a quotient space E = s/V E VSF and a quotient algebra A = A / I E A L ~ , where F i s a vector subspace i n M(R), while G i s a subalgebra i n M ( R ' ) , w i t h Q cR", R' c R n ' , non-void, open. Suppose given a mapping (99)

w : R +R'

such t h a t (100.1)

+o

(100.2)

s o w

w E G, V E

where ( s o w),(x) (100.3)

v o w

+

A, V s

E

F

E

S

= sv(w(x)), E

r, v

v

E

V v EN, x E

n, a n d f i n a l l y

V.

Then obviously i t i s possible t o define a mapping (101)

w :

E

+

A

by

(101.1)

W(S+V)

=

s o

w

+

r, v

s E

s.

A mapping ( 9 9 ) s a t i s f y i n g t h e conditions ( l o o ) , will be c a l l e d an E dependent variable transform. 11.

+

A-in=

Stronger Conditions f o r P a r t i a l Derivative Operators

I t i s shown in t h i s s e c t i o n t h a t t h e r e a r e very simple a l g e b r a i c considera= t i o n s which govern the way p a r t i a l operators can be defined on the quotient spaces or algebras. Namely i t i s necessary t o assume p a r t i a l d i f f e r e n t i a l operators t h a t a c t between d i f f e r e n t quotient spaces and a1 gebras, other= wise one automatically ends up with a p a r t i c u l a r m u l t i p l i c a t i o n f o r the Dirac d i s t r i b u t i o n and i t s d e r i v a t i v e s . This i s t h e main reason f o r deal= ing w i t h chains of quotient algebras when studying polynomial nonlinear

PDEs. In order t o simplify the presentation we s h a l l consider t h e one-dimensional case ( n = 1 ) only.

Suppose A is an a s s o c i a t i v e and commutative a l g e b r a , containing

-

t h e real-valued polynomials on R'

-

t h e d i s t r i b u t i o n s i n U'(R')

with support a f i n i t e number of

SEQUENTIAL SOLUTIONS

31

p o i n t s , i .e. l i n e a r combinations o f D i r a c d i s t r i b u t i o n s and t h e i r d e r i v a t i ves

.

Suppose a l s o t h a t

( 102 )

t h e m u l t i p l i c a t i o n on A induces on t h e p o l y n o m i a l s t h e usual m u l t i p l i c a t i o n and t h a t polynomial $ ( x ) = 1, V x E R ’ , i s t h e u n i t element i n t h e a l g e b r a A;

(103)

t h e r e e x i s t s a l i n e a r mapping D : A ( 2 7 ) ) such t h a t

(103.1)

D i s i d e n t i c a l w i t h t h e usual d e r i v a t i v e , when a p p l i e d t o p o l y = nomials o r d i s t r i b u t i o n s w i t h s u p p o r t a f i n i t e number o f p o i n t s

-f

A, c a l l e d d e r i v a t i v e (see

i n R’; D s a t i s f i e s the L e i b n i t z r u l e f o r product d e r i v a t i v e s

(103.2)

D(a.b) = (Da).b t a.(Db), W a, b

A

E

and f i n a l l y (see P r e l i m i n a r i e s ) (x-x,).

(104)

6x

=

0 E A , Y xOE R ’ ,

0

denotes t h e D i r a c d i s t r i b u t i o n w i t h s u p p o r t i n xo E R ’ .

where 6x 0

Proposition 6 W i t h i n t h e a l g e b r a A, t h e f o l l o w i n g r e l a t i o n s h o l d : (105)

( x - x ~ ) ~ . D= ~0~E A,

I x0

E

R’,

p, q E N, p > q

xO

(106)

+

(p+l).DP6

( X - X ~ ) . D ~= ~0~E&A,~ tf xo

(107)

6

R’,

p E N

0

xO

( X - X ~ ) ~ . ( )‘D ~= ~0 E A, tf x 0 E R’,

p,q E

N, q 2 2

xO

(108)

= 0 E A, Y xo E R ’ .

( 6 x ) z = dX .Ddx 0

0

0

Proof A p p l y i n g D t o (104) and t a k i n g i n t o account ( 1 0 3 ) , we have (109)

+

6

(X

-xo).D6

xO

= 0 E A , Y x0 E xO

R’

-

which, m u l t i p l i e d b y ( x - x ) , g i v e s i n view o f (104) - t h e r e l a t i o n (x-x,)’ D6, = 0 E A, V xo E R ’ . ‘Applying D t o t h e l a t t e r r e l a t i o n and t h e n mul= t i p ? y i n g b y ( x - x o ) , we have i n t h e same way t h e r e l a t i o n ( X - X ~ ) ~ . D =’ ~ ~ 0

0

E

A , V xo

E

R’.

Repeating t h e procedure, we have (105).

The r e l a t i o n (106) i s t h e r e s u l t o f repeated a p p l i c a t i o n o f D t o (109). Now, m u l t i p l y i n g (106) b y ( X - X ~ ) we ~ , have (p+l)(X-Xo) P .D P

.

+ (x-x,) p+l DP+lBI(.Q

= 0 E A, V xo E R’,

p

E

N.

32

E.E.

Rosinger

M u l t i p l y i n g t h i s r e l a t i o n b y (DP6x )q-l

and t a k i n g i n t o account (105), we

0

have (107). F i n a l l y , t a k i n g p=O and q=2 i n (107), we have ( 6

X

)' = 0

E

A , V xo E R ' .

A p p l i c a t i o n o f D t o t h i s l a s t r e l a t i o n completes ?he p r o o f o f (108). 12.

0

Systems o f Polynomial N o n l i n e a r PDEs

Suppose, i n s t e a d o f t h e polynomial n o n l i n e a r PDE i n ( I ) , we a r e given a svstem o f such PDEs (110)

1 G i p t 1 and

Hence w,(x")

,

# w,(x")

x" E

v"

C

V

C

t h e r e f o r e n, A

R"

C


u:

v

R':

wv(x) = 0. Proposition 5 Indc

2

c R

(see ( 6 5 ) , Chapter 1 )

and a l l t h r e e s e t s of sequences o f f u n c t i o n s a r e subsequence i n v a r i a n t . Proof The i n c l u s i o n 'Indc

2

i s obvious.

Assume now w E R \ R. Then, t h e r e e x i s t S2' c S2 non-void, subsequence i n w and $ ' E c O ( R ' ) , such t h a t

(34.1)

w ' = u ( $ ' ) on

S2'

open, w ' a

E.E. Rosinger

50 (34.2)

+ ' ( x ) # 0, V x E

n'.

But i n view o f (33) i t f o l l o w s t h a t N , v 2 ~ :'3 xu E R ' : W ' ( X ) = 0. v v which c o n t r a d i c t s Now (34.1) w i l l i m p l y t h a t + ' ( x ) = 0, V v E N , v

3

N : V v

PIE

E

(34.2).

V

I t i m n e d i a t e l y f o l l o w s t h a t lnd, 2 and

+

R s a t i s f y (28).

0

We now discuss t h e b a s i c necessar c o n d i t i o n on subsequence i n v a r i a n t s e q u e n t i a l s o l u t i o n s , w h i c is an analog o f t h e corresponding p a r t o f Theorem 1, S e c t i o n 1. Theorem 3 i s a subsequence i n v a r i a n t Suppose t h e sequence o f f u n c t i o n s s E s e q u e n t i a l s o l u t i o n o f t h e PDE i n (l), Chapter 1, such t h a t

(35) f o r a c e r t a i n subsequence i n v a r i a n t , h e r e d i t a r y and f u l l q u o t i e n t a l g e b r a A = A/1 E ALSB H F.

c" I

d

Then ws s a t i s f i e s t h e v a n i s h i n g c o n d i t i o n

(36)

ws

E

2.

Proof Assume t h a t (36) i s f a l s e . Then i n view o f (33) t h e r e e x i s t v o i d , open and a subsequence s ' i n s , such t h a t

(37)

# 0, V v

w;(x)

E

N, x

E

R' c

n

non-

a'

where, f o r t h e sake o f s i m p l i c i t y , we have used t h e n o t a t i o n

w'

wsl.

As w ' i s o b v i o u s l y a subsequence i n ws, t h e c o n d i t i o n (35) w i l l i m p l y ( 38)

W'E

7

s i n c e A=M i s subsequence i n v a r i a n t . (38), (37) and (31) w i l l y i e l d u(1)

I?'

).(1/(w' = (w'iRl

But A = A / 1 i s a l s o f u l l , t h e r e f o r e ,

))

E

1

In. Aln' I;.

which o b v i o u s l y i m p l i e s t h a t

thus c o n t r a d i c t i n g t h e f a c t t h a t A = A/I i s h e r e d i t a r y .

c 7

CONDITIONS FOR SOLUTIONS

51

Next we have a simple sufficient condition on subsequence invariant sequen= t i a l solutions, which i s also an analog of the respective implication in Theorem 1 , Section 1. Theorem 4 Suppose the sequence o f functions s

E

(6"(Q ) ) ~s a t i s f i e s

the condition

Then s i s a subsequence invariant sequential solution of the PDE in ( l ) , Chapter 1. Moreover, there exists u sequence invariant, hereditary and , such t h a t full quotient algebra A = A/l E AL sBy'yp

r(n)

ws

E

1

Proof

Taking A 4.

=

A n d , t h i s follows from (64), Chapter 1.

0

Resolvent Sets

The results i n the previous section establish an interest in subse uence invariant sequential solutions, and in view of Proposition 4 +e s ow t a t subsequence invariant quotient a1 gebras offer the natural framework for f i n d i n g such solutions. Let us use the notation R~~ = u

r

where the union i s taken over a l l the subsequence invariant quotient a l = SB gebras A = A j l E ALC,(n). Obviously (39)

T(D)-'(u(f)

+

RsB)

c

(C?(n))N

will be the s e t o f all subsequence invariant sequential solutions of the PDE in ( l ) , C h a p t e r T I t follows that the solution in the above sense of the PDE mentioned in= volves two steps. The i r s t , independent of the PDE, consists in suitable c h a r a c t z z a t i o n s of R SB The second, dependent on the PDE considered, consists in suitable characterizations of the inverse image in (39), primarily answering the question as t o whether t h a t inverse image i s nonvoid.

.

I t i s easy to see t h a t Theorem 1 yields the following:

Corollary 1 RsB

C

R.

I n view of the above, we shall next introduce a general definition.

52

E.E. Rosinger

Given a p r o p e r t y P, v a l i d f o r c e r t a i n q u o t i e n t algebras A = A / I E A L C O ( n ) , den0 t e by

t h e s e t o f a l l those q u o t i e n t a l g e b r a s h a v i n g t h e p r o p e r t y P. A sequence o f f u n c t i o n s S E w i l l be c a l l e d a P-sequential s o l u t i o n o f t h e PDE i n ( l ) , Chapter 1, o n l y i f

f o r a c e r t a i n A = A / l EAL

P C o p ) '

The s e t o f sequences o f continuous f u n c t i o n s on R

P

= U

a, given

by

I

where t h e u n i o n i s taken o v e r t h e q u o t i e n t a l g e b r a s A = A / r E A L P

,

is

c a l l e d the P-resolvent set. NOW, t h e necessary, o r e l s e s u f f i c i e n t c o n d i t i o n s on subsequence i n v a r i a n t s e q u e n t i a l s o l u t i o n s o b t a i n e d i n Theorems 3 and 4, w i l l r e s u l t i n t h e f o l 1owing c o r o l 1a r y .

Corol l a r y 2

I t f o l l o w s t h a t a b e t t e r c h a r a c t e r i z a t i o n o f t h e r e s o l v e n t s e t R SB ,H ,F

pre-supposes a n a r r o w i n g o f t h e gap between Ind and R . w i l l be presented i n Chapter 6.

Related r e s u l t s

However, i t should be n o t e d t h a t t h e simultaneous demand f o r t h e t h r e e p r o p e r t i e s 'subsequence i n i a r i a t ' , ' h e r e d i t a r y ' and ' f u l l ' c o n c e E i K j s e q u e n t i a l s o l u t i o n s s E(C ( Q ) ) [ m i g h t l e a d t o r a t h e r p a r t i c u l a r - s o l u = t i o n s . Indeed, i n view o f t h e v a n i s h i n g c o n d i t i o n ( 3 3 ) d e f i n i n g R , t h e inclusion

i n C o r o l l a r y 2 means t h a t t h e e r r o r sequence ws E R S B y H * F corresponding t o s , w i l l vanish r a t h e r o f t e n on R , v i z V fi' c R non-void, open :

3

(40)

P E N :

tt v

E

N, v 2 ~ :

R ' n Z(T(D)sV-f)

# 9.

I n o t h e r words, t h e zero-sets Z(T(D)sv-f) w i t h v

E

N, a r e ' a s y m p t o t i c a l l y

CONDITIONS

FOR SOLUTIONS

53

dense' i n R, a s i t u a t i o n which i n a way i s more p a r t i c u l a r t h a n t h e u n i = form convergence o f T(D)sv t o f on R, when v -+ m, as seen n e x t i n t h e r e l a = t i o n (48). Now among t h e above-mentioned t h r e e p r o p e r t i e s , t h e f i r s t two a r e o f topo= l o g i c a l n a t u r e , w h i l e t h e l a s t i s a l g e b r a i c . The demand f o r t h e f i r s t 3 2 t h e b s e q u e n c e i n v a r i a n t ' p r o p e r t y , seems t o be j u s t i f i e d , n o t l e a s t because t h e n o t i o n o f s o l u t i o n used i n S e c t i o n s 1 and 2 proves t o be r a t h e r general. T h e r e f o r e i f c o n d i t i o n (40) appears t o be t o o s t r o n g , t h e 'here= d i t a r y ' o r ' f u l l ' p r o p e r t i e s c o u l d be r e l i n q u i s h e d . I n t h i s case, i n view o f (31) and (32), t h e q u o t i e n t a l g e b r a s A = A / l i n v o l v e d , m i g h t have A n o t l a r g e enough, a s i t u a t i o n which i n view o f (43.2), Chapter 1, w i l l n o t be most f a v o u r a b l e t o t h e exactness o f t h e corresponding s e q u e n t i a l so= 1u t i o n s . Remark 1 The d i f f i c u l t y o f t h e problem concerning t h e r e l a t i o n between t h e r e s o l = vent s e t s encountered above i s i l l u s t r a t e d i n t h e r e l a t i o n s

F

(41)

RsB

I t A

proved b y t h e f o l l o w i n g examples. I t i s easy t o see t h a t i t i s p o s s i b l e t o con= Assume n = l and R = ( 0 , l ) . s t r u c t a sequence o f continuous f u n c t i o n s w €(C'(n))N, such t h a t

o

(43)

z(wV) = { ( 2 i t 1 ) / 2 ~ l /

(44)

W ~ ~ ( tXW) ~ * ~ ( X=) 1, tl v E N, x

v

< ip:

T(D)tV(Y) = T ( W p ( Y ) The s e t RE

~

A w i l l be c a l l e d t h e domain o f l o c a l s o l v a b i l i t y i n E -+ A o f

t h e PDE i n ( l ) , Chapter 1, w h i l e t h e s e t rE

-+

A

=n\

QE+A

w i l l be c a l l e d t h e l o c a l s i n g u l a r i t y i n E

(55)

t e

''+A=

Wt

n

s n(cm(n))N

cl

-+

A o f t h e mentioned PDE.

u u cNve N

A (T(D)tv

-

T(D)tp)

v>u

E I

where we denoted A(g) = R \ Z ( g ) ,

f o r g E C" ( Q )

Proof It s u f f i c e s t o show (54), t h e r e s t o f P r o p o s i t i o n 6 r e s u l t i n g e a s i l y .

Assume t h a t x belongs t o t h e l e f t hand t e r m i n (54). t E S , w i t h wt E 1 , such t h a t x

E

int u p E

n Z(T(D)t" N vEN v> p

-

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

T(D)tp)

Then

v > p

w i l l be a neighbourhood o f x.

Moreover, i f y

3 p E N ; V V E

N ,v>u :

T ( D ) t V ( y ) = T(D)t,,(Y)

E

V then o b v i o u s l y

CONDITIONS FOR SOLUTIONS

therefore x

E

RE

-+

57

A

Conversely, assume t h a t x obtain that

E

RE

+

A.

Then, w i t h t h e n o t a t i o n s i n (53), we

v2l-l

therefore

v2l.l The r e l e v a n c e o f t h e domain o f l o c a l s o l v a b i l i t y and o f t h e l o c a l s i n g u l a r < = ty w i l l be p r e s e n t e d i n Theorems 6 and 7.

F i r s t , we denote by

'E

-+

A

the set o f a l l the points x E

V neighbourhood o f x, p

(56)

3 t

E

(56.1)

wt

1

(56.2)

V v E N, v 2 p:

E

S,

T(D)tv = T(D)tp

-+

N :

E

on V

Qi

A w i l l be c a l l e d t h e domain o f s t r o n g l o c a l s o l v a b i l i t y i n A o f t h e PDE i n ( l ) , Chapter 1, w h i l e t h e s e t

The s e t E

satisfying the condition

~

w i l l be c a l l e d t h e s t r o n g l o c a l s i n g u l a r i t y i n

E

+

A o f t h e mentioned PDE.

Proposition 7

Ri

-+

A i s open,

i-;

+

A i s c l o s e d and

Proof Again i t s u f f i c e s t o show t h a t (57) i s v a l i d .

Assume t h a t x belongs t o t h e r i g h t - h a n d t e r m i n (57). Then t h e r e e x i s t s t E S n(Cm(Q))N, w i t h wt E 1 , as w e l l as p E N, such t h a t

58

E.E. x Eint

- T(D)tp)

Z(T(D)t\,

n

Rosinger

v E N v > p

Then

n

V = int

V E

- T(D)tp)

Z(T(D)tv N

v > p

w i l l be a neighbourhood o f x, w i t h t h e p r o p e r t y t h a t T(D)tv = T ( D ) t u therefore x

S

RE

E

-+

on V, V v

N, v > p

A

Conversely, assume t h a t x E RE obtain that

n

V C

E

Z(T(D)tv

A.

+

Then, w i t h t h e n o t a t i o n s i n (56), we

- T(D)tu)

v E N

v>u theref o r e x

int

E

n v E N v

Z(T(D)tv

-

T(D)tu)

>!J

0

Theorem 6

rE+A

$-+AcRE+A’ (59)

E‘

-+

c

r:

+

A and

A “ ~ - + A = ‘ ~ + A ‘

r E + A i s nowhere dense i n R ,

i n o t h e r words

(60)

$i

~

A i s dense i n R E

-+

A

Proof The i n c l u s i o n s as w e l l as t h e e q u a l i t y i n (59) a r e obvious. Therefore, i t o n l y remains t o show t h a t (60) i s v a l i d . B u t (60) f o l l o w s e a s i l y f r o m Lemma 2, S e c t i o n 1. 0 An example sented now.

o f r e g u l a r i t y p r o p e r t y o f s e q u e n t i a l s o l u t i o n s w i l l be p r e =

Call a subset H C (M(n))N c o f i n a l i n v a r i a n t , only i f V w E(M(Q))N :

(61)

3 w ’ E H , p E N (V;r:i~ap:

*

W E

H

CONDITIONS FOR SOLUTIONS

59

Theorem 7 Suppose t h e q u o t i e n t a l g e b r a A = A / 7 has I c o f i n a l i n v a r i a n t . I f x E Q;

A, t

~

E

c o n d i t i o n s (56.1-2)

S, an open neighbourhood

LI E

N s a t i s f y the

, then

T ( D ) t v = f on V, U v

(62)

V o f x and

E

N, v 2~

i n o t h e r words, t, E Cm(Q), w i t h v E N, v 2 p, a r e c l a s s i c a l s o l u t i o n s o f t h e PDE i n ( l ) , Chapter 1. Proof We d e f i n e $

E

C" (Q) b y $ = T(D)t

(63)

LI

Then, i n view of (56.2),

wtv = T(D)tv - f = $ - f on V, V v

(64) Assume now

i t follows t h a t

x

(65 1

E

E

N, v 2 L.

c" (a), such t h a t SUPP

x

v

c

Then (66)

wt.u(x)

E

7 . uco

C I.A C

7

B u t (64) and (65) w i l l i m p l y t h a t

(67)

W ~ . X= ( $ - f ) x

,V v

E

N, v 2

LI

Now, i n view o f t h e f a c t t h a t 7 i s c o f i n a l i n v a r i a n t , (66) and (67) w i l l yield u((dJ-f)x)

E

7

Then, a c c o r d i n g t o (23), Chapter 1, i t f o l l o w s t h a t

(VJ-f)

x

= 0

t h e r e f o r e , i n view o f t h e f a c t t h a t

x i s a r b i t r a r y , we can conclude t h a t

$ = f which t o g e t h e r w i t h (63) w i l l y i e l d ( 6 2 ) .

0

In view o f (60) i n Theorem 6, i t s u f f i c e s t o know t h e s i z e o f t h e domain o f l o c a l s o l v a b i l i t y nE -f A. I t i s obvious t h a t t h e s i z e o f RE

I f we denote

f

A i n c r e a s e s t o g e t h e r w i t h S and I.

E .E. Rosi nger

60

then ( 5 4 ) can o b v i o u s l y be w r i t t e n i n t h e form

therefore

- wu )

n

u

P E N

vEN v >u

11 w

lTYf, R' c R non-void, open:

A (wv

i s dense i n R

or, e q u i v a l e n t l y E

3 X E

R' :

Moreover, i n view o f t h e i n c l u s i o n

U

'T,f

l i m Z(wv) c RE int v -+m

-+

A

Concerning t h e p o s s i b l e n a t u r e o f t h e f a m i l y ( Z ( w ) I v E N ) o f subsets i n , where w E I i s given, t h e f o l l o w i n g two exampyes p r e s e n t i n t e r e s t i n g cases.

Q

(e( Q ) ) Nwhich s a t i s f i e s m Z(WV) = b

Example 1 : w E

(73)

v-+m

the conditions

CONDITIONS FOR SOLUTIONS

U

(74)

61

R' c R non-void, open:

3 P E N :

UvGN,v>u: Z(WV) n R' # 0 We s h a l l c o n s i d e r t h e c a w R = R', s i n c e t h e c o n s t r u c t i o n can e a s i l y be ex= tended t o a r b i t r a r y R C Rn n o n - v o i d and open. Suppose ( x Iv E N) i s dense i n R and c r e a s i n g t8 zero, so t h a t

(75)

{xV +

Nln{xv V +

~ E~

E

E

E~

u IV E

> 0, w i t h v E N, a r e s t r i c t l y de=

N} = flyV A,p

f o r i n s t a n c e , ( x v l v E N) a r e t h e r a t i o n a l numbers and v E N. We d e f i n e w

= (x-x

w,(x)

o

-E

v

)...(X-X~-E~),

U x E R

,v

E

Then o b v i o u s l y

..

{ X ~ + E ~ , . , x ~ + E ~V ~v , E

Z(wW) =

N,

t h e r e f o r e , ( 7 3 ) r e s u l t s e a s i l y from ( 7 5 ) . Suppose now g i v e n R'

x

u'

f o r a certain X

u'

f o r a suitable

u' E +

C

R non-void and open.

Then

R'

E

N.

Ev

u"

~.r = max

E

E

N.

Therefore

R', U v

E N,

w >u''

Taking now

{u', ~ " 1

i t obviously follows t h a t X

1J-I

=

E~

(c" (R))N by

E

+

E Z(wv)

n a', Y

v E

N,

v 2

u

and t h e p r o o f o f (74) i s completed. Example 2 : w €(C?

(n))N which s a t i s f i e s t h e c o n d i t i o n s

l i m mes A (w,) = 0 v-+int l i m Z(wv) = D (78) v +m where ( x v l v E N) i s dense i n n.

N, X # 1-1

E

N

J2/(v+l), with

62

Rosinger

E.E.

We s h a l l c o n s i d e r t h e case R = ( 0 , l ) c R', C Rn non-void and open b e i n g obvious.

the extension t o a r b i t r a r y

R

Denote f o r v

E

N

t (XI, - xol

6\,= min and t a k e w

E

0

G v } /(v+l)

( C ' ( S ~ ) )such ~ that

u

(O,l)\

Z(wV)

(79)

0 Gp
o :

carEwv

R'

3

r

:

C" (R) such t h a t supp w

=

I€ u

V E

and d e f i n e w E (C"

,

Define

E

lw

NI

E

< 2,

NI

U x

R1, which s a t i s f i e s the c o n d i t i o n s E

R

= car N

Indeed, take

R

[O,ll

E N

R n (0,~).

=

=

{ O l and (81) as w e l l as (82) are v a l i d .

car{wv(x) Iw

(84)

where RE

E

w E (C' (n))N, w i t h R C

Example 4:

(83)

r

R

w((~tl)((~+2)~-l)),V x E R, w

=

Then obviously

E

I v E N 1 = car N

Indeed, take R = R ' and w w E (C' (Q))N by w,(x)

x

(1/(2w+2),1/ 2vtl) ) N by

1if x

E

(1/ 2vt2),1/ (2w+l))

wv(x) = 0 if x E n \ then i t i s easy t o see t h a t (83-85

This Page Intentionally Left Blank

CHAPTER 3 ALGEBRAS CONTAINING THE DISTRIBUTIONS

0.

Introduction

As pointed out in Section 5, Chapter 1, a proper study of the sequential, in particular the weak or distribution solutions of PDEs, requires a s u i t = able determination of the interplay between the s t a b i l i t y , generality and exactness properties of the solutions involved. In view of the already traditional role played by distributions in solving linear or nonlinear PDEs, we shall from now on aim a t a level of generality n o t below the distributional one. Having defined t h i s aim, certain n a t u r a l methods will follow of dealing with the interplay between s t a b i l i t and exactness, leading t o chains of quotient algebras containing d s t r i bu = tions, o f the form

(1)

D'(n) c Am c

..... c A'c ..... c A O ,

II

E

4,

with the PDO in ( 3 ) , Chapter 1, acting between quotient algebras

(2)

T ( D ) : A'

with R , k E 1.

d,

R

+

Ak

- k > m y where m i s the order of T ( D ) .

Embedding the Distributions into Quotient Algebras

We shall start from the quotient space representation of the distributions (3)

o'(n)

=

s2/ v' ,

R E

fi

introduced in ( 3 1 ) , ( 2 4 ) , (25) and (26), Chapter 1.

A simple way of embedding the distributions into quotient algebras would be to construct inclusion diagrams of the form

r

9

- A

(C"(n))

65

N

E.E. Rosinger

66

R

w i t h A a subalgebra i n ( C ( Q ) )

r

(4.1)

nsR=

N

and I an i d e a l i n A, such t h a t

fi

g e n e r a t i n g the l i n e a r embedding of U'(R) i n t o t h e q u o t i e n t algebra A = A/7 AL R d e f i n e d by

c

(81,

P'(Q) =

sR/fi3( s

$1

t

(s

-f

t

r)

E A =

MI,

However, i n c l u s i o n diagrams o f t y p e ( 4 ) cannot be constructed, s i n c e

(fi.VR)

(5)

ve,

nsR F

w R€W

as can be seen u s i n g t h e argument i n t h e p r o o f o f (56), Chapter 1. Another simple way o f embedding t h e d i s t r i b u t i o n s i n t o q u o t i e n t algebras would be t o c o n s t r u c t i n c l u s i o n diagrams o f t h e form > A

7

w i t h A a subalgebra i n

>

(CR(WN

and I an i d e a l i n A, such t h a t

f i n A = 7 and R 9. V t A = S

(6.1) (6.2)

g e n e r a t i n g t h e l i n e a r i n j e c t i o n o f t h e algebra A = 4 / I o n t o U ' ( n ) , defined by A = wi 3

(S t

r)

+

(s

t

v')

E

D'(Q)

=

sR/vR.

Here, t h e problem a r i s e s connected w i t h t h e c o n d i t i o n (6.2) , which prevents A from c o n t a i n i n g some o f t h e f r e q u e n t l y used ' 6 sequences', as i s e v i d e n t from t h e lemma given below (see p r o o f i n S e c t i o n 11). Lemma 1 Suppose g i v e n s

E

(C"(S2))

N

such t h a t supp sv s h r i n k s t o xo

E

R, when v

-+

m.

I f s i s a sequence o f non-negative f u n c t i o n s , t h e f o l l o w i n g two p r o p e r t i e s are equivalent:

(7) lim

(8)

I

s,(x)dx

= 1;

v + m Q

If s

E

and

, then

<s,*> = 6 xO

s2

6 9.

ALGEBRAS CONTAINING THE DISTRIBUTIONS

67

However, the f a i l u r e of inclusion diagrams of types ( 4 ) and ( 6 ) can be overcome. Indeed, i n [169 - 1761 i t has been proved ( s e e a l s o Theorem 1 , Section 3, as well as t h e r e s u l t s i n Section 4 ) t h a t the following, more complicated inclusion diagrams can be constructed: 7

' (c-7 n) ) N

> A

I

w i t h A a sukalgekra i n ( C " ( R ) j N ; 1 an ideal in A ; spaces in V , S r e s p e c t i v e l y , such t h a t

(9.1)

I ~ S = V ;

(9.2)

V

(9.3)

sm =

m

n S = V;

and v , s vector sub=

and

Vrn t S;

and t h e r e f o r e generating t h e l i n e a r embedding of D'(n) i n t o t h e q u o t i e n t algebra A = A/I E AL c=(n), defined by U'(0) =

sm/vw

(10)

S/

W

s

+

v

A

111

vw-s

t

i som

=

S/7 W

v

s tl.

1 in , i n j

The intermediate q u o t i e n t space S / V plays t h e r o l e of a r e g u l a r i z a t i o n of the q u o t i e n t space representation of d i s t r i b u t i o n s in (3). The construe= t i o n of q u o t i e n t algebras containing t h e d i s t r i b u t i o n s e s s e n t i a l l y depends on the choice o f the r e g u l a r i z a t i o n s S/V , which a l s o determine t h e s t a b i = lity and exactness o f t h e sequential s o l u t i o n s in t h e q u o t i e n t algebras mentioned.

The next f i v e s e c t i o n s w i l l t h e r e f o r e deal with t h e way r e g u l a r i z a t i o n s and the corresponding q u o t i e n t algebras containing t h e d i s t r i b u t i o n s can be constructed. Remark 1

The form of t h e inclusion diagrams ( 9 ) i s necessary i n the following sense. the I f we assume t h a t f o r a c e r t a i n quotient a m = A/IsALy (n) ' following r e l a t i o n holds:

E. E

68 D'(Q

C A

. Rosi nger

;

1

i . e . i f we assume t h e e x i s t e n c e o f a commutative diagram sur < s,*> E D'(Q)

S3s\

s

inj

7 A~

t

f o r a s u i t a b l e v e c t o r subspace S c A n Sm, then t a k i n g

u = I n s we o b t a i n an i n c l u s i o n diagram ( 9 ) s a t i s f y i n g a l l t h e p r o p e r t i e s , except perhaps f o r the i n c l u s i o n U

c"(n)

2.

S*

Simpler I n c l u s i o n Diagrams

The choice o f t h e r e g u l a r i z a t i o n S/U i n t h e i n c l u s i o n diagram ( 9 ) c o u l d be reduced t o the choice o f S only, as U would r e s u l t from ( 9 . 2 ) . However, i t w i l l be more convenient f i r s t t o split S according t o (11)

S

=U@S'

where S ' i s a v e c t o r subspace i n S, and then t o r e p l a c e t h e problem o f t h e choice o f S b y t h e problem o f t h e choice o f t h e p a i r (U,S'),which under c e r t a i n c o n d i t i o n s (see ( 2 1 ) ) w i l l from now on be termed r e g u l a r i z a = tions. Indeed, assuming t h a t t h e s p l i t t i n g (11) holds, t h e r e l a t i o n s (9.2) and (9.3) become

Now the problem o f f u l f i l l i n g (9.1) remains. I t i s obvious t h a t i f t h e r e e x i s t s an i d e a l 7 i n A which s a t i s f i e s (9.1) then the s m a l l e s t i d e a l i n A c o n t a i n i n g U , i . e .

(13)

l ( U , A ) = t h e i d e a l i n A generated b y V

w i l l a l s o s a t i s f y (9.1).

Since u ( 1 ) has a simple s t r u c t u r e , v i z

E

U

C"(Q)

c A i n (9), t h e ideal l ( U , A )

.

7 ( U , A ) = t h e v e c t o r subspace i n A generated by U . A N I n t h e p a r t i c u l a r case when A = (C"(n)) , we s h a l l f o r t h e sake o f s i m p l i = c i t y use t h e n o t a t i o n :

(14)

(14.1)

l(U) = 7(V,

(C"(n))N.

ALGEBRCSCONTAINING THE DISTRIBUTIONS

69

Owing t o t h e s i m p l e s t r u c t u r e o f t h e i d e a l 7 ( V , A ) , we g a i n a good i n s i g h t i n t o the s t r u c t u r e o f the q u o t i e n t algebra A = A / 7 ( V , A ) c o n t a i n i n g the d i s t r i b u t i o n s . T h e r e f o r e t h e i n c l u s i o n diagrams ( 9 ) w i l l o n l y be c o n s i = dered under t h e p a r t i c u l a r f o r m

with (15.1)

7(V,A) n (V

(15.2)

S" =

0s')

= V

V" 0 s ' .

I t i s u s e f u l t o n o t i c e t h a t (15.1) can be w r i t t e n i n t h e s i m p l e r , equiva= l e n t form

(15.3)

7 ( V , A ) n S' =

2.

-

I n t h e case o f t h e i n c l u s i o n diagrams ( 1 5 ) , t h e l i n e a r embedding o f D'(n) i n t o q u o t i e n t a l g e b r a s i n ( l o ) , w i l l assume t h e f o l l o w i n g p a r t i c u l a r form:

D'(n)

= S"/V"

s

+

A = A/7(V,A)

+(V@S')/V

Urn - s isom

t V

-

s t I(V,A).

lin,inj

I n c o n s t r u c t i n g t h e i n c l u s i o n diagrams ( 1 5 ) , we have n o t o n l y t h e problem o f choosing t h e p a i r s ( V , S I ) , b u t a l s o t h a t o f choosing t h e subalgebras A A s i m p l e s o l u t i o n of t h i s l a t t e r problem w i l l however, be g i v e n i n t h e n e x t s e c t i o n . Therefore t h e problem o f c o n s t r u c t i n g c h a i n s o f quo= t i e n t a l g e b r a s ( I ) , 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 , w i l l be reduced t o t h e problem o f c o n s t r u c t i n g s u i t a b l e r e g u l a r i z a t i o n s ( V , S ' ) .

.

3, R e g u l a r i z a t i o n s and Chains o f Q u o t i e n t Algebras C o n t a i n i n g t h e Distributions I n t h i s section, D e f i n i t i o n 1 presents t h e b a s i c n o t i o n o f r e g u l a r i z a t i o n , which l e a d s t o t h e b a s i c r e s u l t o f Theorem 1 on t h e c o n s t r u c t i o n o f c h a i n s

70

E.E.

Rosinger

of q u o t i e n t algebras 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 . T h i s r e s u l t w i l l be used i n t h e n e x t two chapters i n t h e s t u d y o f polynomial n o n l i n e a r PDEs. A second way o f c o n s t r u c t i n g chains o f q u o t i e n t algebras c o n t a i n i n g l a r g e v e m u b s p a c e s o f d i s t r i b u t i o n s i s presented i n Sections 5 and 6. We f i r s t i n t r g d u c e s e v e r a l u s e f u l a u x i l i a r y n o t a t i o n s and d e f i n i t i o n s . subset HC (C (n))N i s c a l l e d d e r i v a t i v e i n v a r i a n t , o n l y i f

(17)

i?H

Obviously, Given V

cH,W

PE

A

Nn.

H = (C"(n))N i s d e r i v a t i v e i n v a r i a n t .

,S

C

( 18)

(Cm('d))

{v

VL=

N and L E V

I

E

8 , we use t h e n o t a t i o n s :

P C v E V , V

PE

Nn,

(PI ~ , X E Q ' : WV(X)

- $(x)>

E

.

Now d e f i n e w ' E (Cm(n))N b y

Then o b v i o u s l y

r

(65)

W' E

since w E 1

, and I

i s cofinal invariant.

Take t h e n any

E. E . Rosi nger

84

x

(66)

E

?(a), w i t h

supp

x

c R'

and i n view o f (64), d e f i n e t E (C"(R))N by

I x

tV

i f v < p

=I

Then (65) gives

Therefore

i n view o f (50) x ( x ) = 0,

u

, we

have

x E R

which i s absurd, s i n c e x can be chosen a r b i t r a r i l y w i t h i n t h e c o n d i t i o n (66). T h i s completes t h e p r o o f of ( 6 3 ) . Now (62) and (63) i m p l y t h a t

i

=

w E

B

n

v

n i n s"

c

r

n

v

=

Q

t h e l a t t e r e q u a l i t y r e s u l t i n g from (44) i n t h e p r o o f o f P r o p o s i t i o n 1, as w e l l as t h e f a c t t h a t - as we have n o t i c e d - I i s vanishing. T h i s com= p l e t e s t h e p r o o f of (60). We can t h e r e f o r e conclude t h a t (59) h o l d s , s i n c e V c V" c i s an a l g e b r a i c base i n V .

fi and (vale.

(0,l))

Since (59) i m p l i e s ( 5 8 ) , we can f i n a l l y a p p l y Lemma 2 t o fi agd B given i n (57) and o b t a i n t h e e x i s t e n c e o f v e c t o r subspaces i n E = S , such t h a t

c

a n c = B n c = 0 and a + B = A @ E. Then (56)mand Lemma 3 below i m p l y t h e e x i s t e n c e of v e c t o r subspaces D c E = S , such t h a t

A nD (67)

C nD = 0;

A t C = A @ D ;

and

Bn(AtC) =BnD. Taking, f i n a l l y , T = D, t h e l a t t e r two r e l a t i o n s i n (67) w i l l y i e l d (51). The converse f o l l o w s from C o r o l l a r y 1. Lemna 3 I f A, B and C a r e v e c t o r subspaces i n E and

0

ALGEBRAS CONTAINING THE DISTRIBUTIONS

A nB

85

B n C = 0 ( t h e null space),

=

then trle following two properties are equivalent: 3 D c E vector subspace: A n D = C n D = O

(68) A + C = A @ D

+

B n (A

C) = B n D

and

E

3 (69)

c E vector subspace:

Ant=@ n t = o A + B = A @ C

where

A

=

B

=

@ c @ A

(B n (A t (B n (A

c))

+ c)).

Proof Assume (69) holds. Then (68) results by direct verification, i f one takes 0 = ( B n ( A + C ) ) @ C. Assuming (68) and taking any vector subspace C C E such t h a t D = ( B n D) @ C, direct verification will yield ( 6 9 ) . 0 The cofinal invariant regular ideals actually s a t i s f y stronger conditions than those in (33-36), as can be seen in the following corollary. Corollary 2 If 7 i s p cofinal invariant regular ideal, there exist vector subspaces c S such t h a t

S, J

(70)

S"=

(71)

7 n T = V" n J = p ;

(72)

V"

V"@S@T;

+ (7

n.!?) = V"

@

J;

and

(73) in which case, for any vector subspace V c 7 n Vm (74)

(V, S

@ J)

will be a regularization.

86

E.E.

Rosinger

Proof See (41) i n P r o p o s i t i o n 1 and (54) i n t h e p r o o f o f Theorem 5. Several f u r t h e r general r e s u l t s on r e g u l a r i d e a l s a r e now presented. Theorem 6 N An i d e a l I i n (C"(n)) i s r e g u l a r , o n l y i f t h e r e e x i s t v e c t o r spaces T c Sm s a t i s f y i n g t h e c o n d i t i o n s ( 3 3 ) and (M),as w e l l as

wm n

(75)

t T) =

(Ucm(,)

Q

.

Proof Assume t h a t I i s r e g u l a r ; that

then i n view o f ( 3 3 ) , (35) and ( 3 6 ) , i t f o l l o w s

Now an element o f t h e l e f t - h a n d term i n (75) has t h e form v = u(IJJ) t t w i t h v E Urn, IJJ E

C"(n) and

Thus i n view o f (76) we have

t E T.

u(q)

= v

-

t E

u(IJJ)

= v

-

t = t'

~

~

m

n(

(~w)" @T )

CT,

i.e. (77) with t '

E

T.

Now Wm n T =

Q

i n ( 3 4 ) a p p l i e d t o (77) w i l l y i e l d

v = u ( o ) , t ' = -t and t h i s completes t h e p r o o f o f ( 7 5 ) . Conversely, assume t h a t (75) h o l d s .

We then show t h a t (76) i s v a l i d .

Indeed, an element o f t h e l e f t - h a n d t e r m i n (76) has t h e form (78) w i t h IJJ E

U($)

C"(n) , v

= v t t E

v = U(IJJ)

Vm and t

-

t E

E T.

vm n

Therefore t

( U ~ W ( ~ )

T).

Hence, i n view o f ( 7 5 ) , we have v = u ( o ) . Now (78) w i l l y i e l d ( 7 6 ) . I t o n b remains t o show t h a t (36) h o l d s f o r s u i t a b l e v e c t o r subspaces To do so, we t a k e a v e c t o r subspace

S c S t h a t a l s o s a t i s f y (35). U' c Ucm(n) such t h a t

ALGEBRAS CONTAINING THE DISTRIBUTIONS

(79)

UC"(

n)

= ( %"(

n)

nT)

@

87

U I .

the l a t t e r inclusion resulting from ( 7 6 ) . T h u s , ( 7 9 ) will yield ( 8 0 ) . NOW, in view of (80), there e x i s t vector subspaces S c Sm satisfying (35) and such t h a t U ' c S. I n t h a t case, ( 7 9 ) will obviously imply (36). 0 Theorem 7

N If a vanishing ideal 7 in (Cm(n)) s a t i s f i e s the condition (81)

(v"

+ 1) n

~

~

m

=(

Q~ , )

then 'I i s a regular ideal and there e x i s t regularizations (I n

v",~

@ T)

satisfying the conditions (70-72) as well as (82)

(see (73) and ( 3 6 ) ) .

UC"(S-2)

Proof Since 7 i s vanighing, Proposition 1 will grant the existence of vector subspaces T c S t h a t s a t i s f y (71) and ( 7 2 ) . B u t condition (81) i s ob= viously equivalent t o

%"( n) n

(v" +

( 7 n s"))

=

Q

;

or, in view of ( 7 2 ) , equivalent t o the condition uCm(n)n (v"

@ T) =

Q

.

Now the existence o f vector subspaces S c Tosatisfying (70) and (82) easily follows. Proposition 3 The ideal (see (65) , Chapter 1) (83)

I;d

= 7,d

(Cm(n))N = Ind

0

88

E.E. Rosinger

i s v a n i s h i n g , c o f i n a l i n v a r i a n t and r e g u l a r , and s a t i s f i e s t h e c o n d i t i o n (81). Proof It s u f f i c e s t o show t h a t

w = v

(84) with w

E

zd;,

v

E

s a t i s f i e s (81).

Assume

t U($)

V m and $ E C”(n).

Then o b v i o u s l y w

E

Sm and < w,.>

=$

.

Now ( 6 9 ) i n Chapter 1 w i l l i m p l y t h a t $(x) = 0, V x E Q. Then i n view o f 0 ( 8 4 ) , we have w = v E Vm, which completes t h e p r o o f o f (81). Remark 4 a) The c h a r a c t e r i z a t i o n o b t a i n e d i n Theorem 4 e L t a b l i s h e s an e q u i v a l e n c e between r e g u l a r i z a t i o n s and r e g u l a r i d e a l s i n ( C ( Q ) ) N , The r e g u l a r i d e a l s 1 - i n t h e c o f i n a l i n v a r i a n t case - have t h e s i m p l e c h a r a c t e r i z a t i o n pre= sented i n Theorem 5, v i z

I t i s i m p o r t a n t t o n o t i c e t h a t i n view o f c o n d i t i o n (23) i n Chapter 1 de= f i n i n g q u o t i e n t a l g e b r a s A = A / l E ALCm(n), t h e above c o n d i t i o n o b t a i n e d

i n Theorem 5 i s t h e b e s t o s s i b l e . T h i s i n d i c a t e s t h a t as we advanced from i n c l u s i o n diagrams (g+lusion diagrams (23) and q u o t i e n t a l g e = b r a s ( 2 4 ) 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 , no undue r e s t r i c t i o n was imposed. I t a l s o i n d i c a t e s t h e abundance o f r e g u l a r i d e a l s and c o r r e s p o n d i n g l y o f r e g u l a r i z a t i o n s , which can t h e r e f o r e be r e s t r i c t e d i n a meaningful way i n accordance w i t h t h e requirements o f t h e p a r t i c u l a r problems b e i n g c o n s i = dered (see Chapters 4,5 and 7 ) . b ) Various r e g u l a r i z a t i o n s can be a s s o c i a t e d w i t h a r e g u l a r i d e a l i n a c o n s t r u c t i v e way. The c o n s t r u c t i o n presented i n t h e p r o o f s o f P r o k o s i t i o n 1 and Theorem 5, o n l y i n v o l v e s t h e c h o i c e o f v e c t o r subspaces i n S , supposed t o s a t i s f y s i m p l e decomposition p r o p e r t i e s . S p e c i f i c i n s t a n c e s o f t h e c o n s t r u c t i o n o f r e g u l a r i z a t i o n s w i l l be presented i n t h e n e x t chap= t e r s , i n connection w i t h t h e s t u d y o f p a r t i c u l a r t y p e s o f p o l y n o m i a l non= l i n e a r PDEs, as w e l l as a s c a t t e r i n g problem i n p o t e n t i a l s p o s i t i v e powers o f the Dirac d i s t r i b u t i o n . 5.

A d d i t i o n a l Chains o f Q u o t i e n t Algebras c o n t a i n i n g D i s t r i b u t i o n s

As seen i n 3), Theorem 2, and 4 ) , Theorem 3, t h e q u o t i e n t a l g e b r a s con= t a i n i n g t h e d i s t r i b u t i o n s d e f i n e d i n ( 2 4 ) i n d u c e on C ( Q ) t h e usual m u l t i = p l i c a t i o n and p a r t i a l d e r i v a t i v e s f o r f u n c t i o n s . As seen i n t h e p r e s e n t s e c t i o n , i t i s p o s s i b l e t o c o n s t r u c t c h a i n s - i n t h i s case, f i n i t e ones t o o - o f q u o t i e n t a l g e b r a s i n such a way t h a t t h e y w i l l i n d u c e - € E - G s u a l - m u l t i p l i c a t i o n and p a r t i a l d e r i v a t i v e s on t h e l a r e r spaces o f f u n c t i o n s CK(n), w i t h 9. E These q u o t i e n t algebras, under t e c o n d i t i o n s presented i n Theorem 16, S e c t i o n 6, w i l l a l s o c o n t a i n l a r g e v e c t o r subspaces o f t h e d i s t r i b u t i o n s i n U’(n).

n.

-+

ALGEBRAS CONTAINING THE DISTRIBUTIONS Suppose g i v e n II

E

R.

A subset H c ( C R ( n ) ) (85)

89

N

i s c a l l e d C'-derivative

invariant, only i f

c H , V p E Nn.

Dp(H n ( C . e + l P I : f i ) ) N )

Obviously, H = (CR( Q ) ) ~i s C' - d e r i v a t i v e i n v a r i a n t . Given V ,S c ( C o ( n ) ) N , we use t h e n o t a t i o n s (86.1)

~1%

=

s

n(~'(n))~

and (86.2)

VII

= {v E V I I I

1

Dpv

E

V

,V

p €Nn,

[PI

p:

r

supp wv c R \

v.

neighbourhood o f x :

Therefore i n view o f (130.1) i t f o l l o w s t h a t V X E R \

3 p' V

E

r:

N , V'c R v>u':

\r

neighbourhood o f x:

vEN,

supp w;

c R \ V'

which o b v i o u s l y i m p l i e s (132) if ( 6 5 ) , Chapter 1 i s taken i n t o account. F u r t h e r , we show t h a t (133)

W"

- w'

E

v.

Indeed, assume $ E D(R). inequalities

Then i n view o f (130.2) we have f o r v E N, t h e

f lwp)-";(") I * I + ( x ) I d x G ( f I J l ( x ) I d x ) / ( v + l ) . R R Now t h e r e l a t i o n s (129), (133) and (132) w i l l y i e l d w E

v"

t

( I n d n sm)

which completes t h e p r o o f o f (126) f o r a r b i t r a r y II

E

17.

ALGEBRAS CONTAINING THE DISTRIBUTIONS

99

Lemma 4

.

Given JI E C" (R w i t h supp $ c R compact, and c o n s t r u c t E d"(n) such t h a t (134.1)

supp

(134.2)

1

x

+ supp

c B(O,E)

-

x(x)

$(XI1

Gn

Y

E.

q > 0, i t i s p o s s i b l e t o

$ and

tl x

R.

E

Proof Obvious y , i t can be assumed w i t h o u t l o s s o f genera i t y t h a t 0 supp $ c R. We assume f u r t h e r p > 0, such t h a t B(O,E)

I

s u p { ~ $ ( x ) - ~I ) x and t a k e w

E

E

R, y

E

Q n B(X,P))

E

R and

rl ;

satisfying the conditions

C"(R)

2 0 , tf x E R;

.(X)

supp w c B(0,p);

and

w(x)dx = 1.

R

x

We n e x t d e f i n e

x(x) =

E

Cm(R) b y

I NY)dX-Y)dY.

R

Now (134.1) w i l l o b v i o u s l y f o l l o w . Ix(x)-+(x) =

R

I

I

R

Moreover l4X-Y)dY =

I$(X)-$(Y)

I$(x)-$(y) I w ( x - y ) d y G r i

, tf

x E

Q

n B(x,P)

and t h i s completes t h e p r o o f o f (134.2).

0

We now prove a r e s u l t i m p l y i n g t h a t t h e c h a i n s o f q u o t i e n t a l g e b r a s (93) can a l s o c o n t a i n t h e d i s t r i b u t i o n s i n U ' ( n ) , w i t h t h e p o s s i b l e e x c e p t i o n o f f u n c t i o n s n o t s u f f i c i e n t l y smooth. Theorem 14

N R Suppose t h g i d e a l I i n ( C o ( R ) ) i s v a n i s h i n g , i s C -smooth f o r a c e r t a i n g i v e n R E N, and s a t i s f i e s t h e c o n d i t i o n (135)

( v" + I ) n Uco(n) =

Q.

Then t h e r e e x i s t v e c t o r subspaces R,T c SR

(136)

1n T = V nT=Q

(137)

V t t n T ) = v @ ~

, satisfying

the conditions

100

E.E.

Rosinger

Therefore using the n o t a t i o n

we o b t a i n a r e g u l a r i z a t i o n

Moreover

f o r e v e r y v e c t o r subspace V c I n V " .

Proof Taking E =

s',

and B = 7 n S R

A = y'

and u s i n g t h e same argument as i n t h e p o o f o f P r o p o s i t i o n 1, we o b t a i n t h e e x i s t e n c e o f v e c t o r subspaces T c S s a t i s f y i n g

E

II

2 and ( I n sR)= v'@ T .

(142)

I n T = V nT

(143)

V'

+

=

Now (142) w i l l o b v i o u s l y i m p l y (136), s i n c e T c SR and Vo n S' Also, t h e r e l a t i o n (143) o b v i o u s l y i m p l i e s t h a t (144)

vo

t

( r n sR)=

i f we t a k e i n t o account v" n T =

(145)

v

+ ( I ns') c

vo

@

2in vo

=

T

(136).

But

+ ( I n S O ) c V" + ( I n's')

t h e l a t t e r i n c l u s i o n r e s u l t i n g from (126). (145) w i l l i m p l y (137).

Now t h e r e l a t i o n s (144) and

I n view o f (135) and (137), we have

I t f o l l o w s t h a t t h e r e e x i s t v e c t o r subspaces

R'c So such t h a t

Then, t a k i n g

i n Lemna 5 below, i t f o l l o w s t h a t t h e r e e x i s t v e c t o r subspaces

C = R c F = SR

vR .

ALGEBRAS CONTAINING THE DISTRIBUTIONS

101

T h i s completes t h e p r o o f o f (138). Now i n view o f Theorem 10, t h e r e l a t i o n s (136-139) w i l l o b v i o u s l y i m p l y (140).

R

Finally, noticing that i n view o f (86.1).

@T

, the

c 'S

r e l a t i o n (141) f o l l o w s e a s i l y , 0

Lemma 5 I f F, A and B a r e v e c t o r subspaces i n E and

(146)

E = A t F

(147)

A n B = 0 ( t h e n u l l space)

then t h e r e e x i s t v e c t o r subspaces C c F, such t h a t (148)

A n C = O

(149)

A

@B

= A

@

C.

Proof Assume, i n view o f (146), t h a t (150)

E=A@F'

f o r a s u i t a b l e v e c t o r subspace F ' C F. F u r t h e r , assume t h a t ( a . l i E I ) , ( b j [ j E J ) and ( f k l k E K) a r e a l g e b r a i c bases i n A, B and F ' r k p e c t i v e l y . Then i n view o f (146)

c

b. = J

Xji

c

ai +

p j k fk,

Y

j

E

J,

k E K

~

E

I

hji

# 0 and p j k # 0

where

f o r o n l y a f i n i t e number o f i E I , k v e c t o r subspace i n F ' generated by c

=

j

c

pjk

fk,

E

K respectively.

with j

E

Now t a k e C as t h e

J.

k E K

Then, i n view o f (150), t h e r e l a t i o n (148) f o l l o w s e a s i l y . i s obvious t h a t

Moreover, i t

102

E.E. A t B C A

+

Rosinger

C.

F i n a l l y , we demonstrate t h e i n c l u s i o n C c A t

Indeed, i f x

E

B.

C, then

where p. # 0 f o r o n l y a f i n i t e number o f j E J .

I t now f o l l o w s e a s i l y t h a t

J

x =

1 Pjbj1 oj j € J j E J

z

i € I

hji ai E A

+ B;

and i n view o f (147) and ( 1 4 8 ) , t h i s completes t h e p r o o f o f (149).

0

The analog o f Theorems 1 and 8 i s presented i n t h e f o l l o w i n g theorem. Theorem 15

Q

If ( V , S + T ) i s a r e g u l a r i z a t i o n g i v e n as i n (139-140), t h e n f o r each k E fly k . .R , i t i s p o s s i b l e t o c o n s t r u c t t h e i n c l u s i o n diagrams

which w i l l s a t i s f y t h e c o n d i t i o n

R

@ T)

= Vk

o r equivalently, the simpler condition

(151.2)

Ik(V,S

@

T)

n (U k

c

(0)

@ R @ T)

=

2

.

Proof T h i s i s s i m i l a r t o t h e p r o o f o f Theorem 1.

0

We now g i v e t h e main r e s u l t i n t h i s s e c t i o n , i n d i c a t i n g t h e v e c t o r subspaces

ALGEBRAS CONTAINING THE DISTRIBUTIONS

103

o f d i s t r i b u t i o n s which can be embedded i n t o t h e q u o t i e n t a l g e b r a s ( 9 3 ) . Given a v e c t o r subspace H c S o y we use t h e n o t a t i o n

DpL)

(152)

=

{S = < s , * >

s E HI.

Theorem 16

Lt'

I f (V,S T ) i s a r e g u l a r i z a t i o n g i v e n as i n (139-140), t h e n t h e f o l = l o w i n g embeddings h o l d :

Ck(n)

(153)

9

c Ak(V,

UQT(R)

S

@ T),

U k

E

w,

k c o n t a i n e d i n t h e r e s p e c t i v e q u o t i e n t a l g e b r a s ( 9 3 ) , can e a s i l y be apprecia= ted, i f i t i s n o t e d t h a t i n view o f (138), we have (155)

D'(Q) =

CO(i-2)

@ Dk

0

$2).

T h e r e f o r e t h e o n l y d i s t r i b u t i o n s t h a t m i g h t n o t be c o n t a i n e d i n t h e quo= t i e n t algebras k

A (V,S

@ T),

with k E

8, k

O , V x E Rn

c;(n)

(167)

= )I{

1.

E CR(Q)

**)

($1"

E

cR(n),

R C e r t a i n s i m p l e f a c t s about C+(R) s h o u l d be noted. $ E C R ( R ) and $ ( x ) > 0, V x E R

exist

)I E

C (R), w i t h

R, then $

$(x) 2 0 , V x

example o f such a f u n c t i o n i s

E

E

2 $ ( x ) = x1

v

a

E

For instance, i f

Cf(n). But i f

R, such t h a t

.....xn,2

(0,m)

$ $

R 2 1, t h e r e

Ct(R).

V x = (xl,

An

...,xn)

E

R,

i f 0 E R. However, t h e r e a r e non-negativemCm-smooth f u n c t i o n s on n t h a t v a n i s h on subsets o f 8 and y e t belong t o C + ( R ) . F o r example, i f 0 E R and

I

=

1

exp(-l/(xl +

0

... t

x,)) f o r xi > 0 , w i t h 1 < i < n

otherwise

108

E.E.

then $

E C,"(

Rosinger

Q).

F o r a v e c t o r subspace P c So we use t h e n o t a t i o n :

D ; , ~ , ~ ( Q )=

(168)

c

< s , - x P(Q)

and c a l l such d i s t r i b u t i o n s P-

I

s E P n (C;(Q))~I R C -non-negative.

Now, u s i n g t h e i d e a of t h e general method f o r dependent v a r i a b l e t r a n s f o r m , q i v e n i n ( 8 9 - 9 8 ) , ChaDter 1, i t i s D o s s i b l e t o o b t a i n t h e f o l l o w i n a r e s u l t on p o s i t i v e powers w i t h i n t h e q u o t i e n t algebras c o n t a i n i n g t h e d i s t r i b u = tions. F i r s t , we c o n s i d e r t h e case of chains o f q u o t i e n t algebras c o n t a i n i n g t h e distributions, introduced i n (24). Theorem 1 7 Suppose g i v e n a C m - r e g u l a r i z a t i o n ( V J ) satisfying the condition

(169)

U($)

P,

E

v

s

and a v e c t o r subspace p c

$ E CY(n).

Then

Cy(n) c

1)

,(Q) c A ' ( V , s ) ,

V R

E

w.

Moreover, i n t h e case o f p o s i t i v e power a l g e b r a s i t f o l l o w s t h a t :

2)

For R

(170)

E

1 and

c1 E ( O , m ) ,

we can d e f i n e t h e mapping c a l l e d t h e a - t h power

D;),m,

,(a)

-+

3 S

Scl E AR(V,.S)

by

(170.1)

Sa = s a t

Z R ( V , S ) , w i t h S = < s,.> , s

E

P n (C,"(n))N

.

The mapping ( 1 7 0 ) w i l l have t h e p r o p e r t i e s

1 = S;

(171.1)

S

(171.2)

SatB

(171.3)

(Sa)m = SaWm

If p E Nn and I p I

= Sa.SBy V a,@ E ( 0 , m ) ;

,V

a

(Op), m

E

N \ {O}.

= 1, 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 (see

Dp : AR(V,S)

+.

Ak(V,S), k E

R,

(27))

k ;

(174)

=

6.

Now we can d e f i n e a r b i t r a r y p o s A t i v e powers o f 6 w i t h t h e h e l p o f t h e map= p i n g (170), p r o v i d e d t h a t t h e C - r e g u l a r i z a t i o n ( V , S ) s a t i s f i e s t h e condi= tion

s

(175)

E

s.

T h i s procedure, i n r a t h e r more general form, w i l l be used i n Chapter 7. We now p r e s e n t t h e analog o f Theorem 17, i n t h e case o f c h a i n s of q u o t i e n t a1 gebras (93). Theorem 18 Suppose g i v e n a C " - r e g u l a r i z a t i o n ( V , S ) and a v e c t o r subspace P c S , satisfying

( 176 1

u($) E

P,

$

E

c p .

Then

Moreover, i n t h e case o f p o s i t i v e power a l g e b r a s , i t f o l l o w s t h a t :

2) F o r R

E

R and

C ~ (EO , m ) ,

we can d e f i n e t h e mapping c a l l e d t h e

power

(177)

Sa

E

AR( V ,s)

&

E . E . Rosinger

110

by (177.1)

SO1 = s'

t

IR(V,.S), w i t h S = < s , * >

, sE

p n ( c (' sl))N.

The mapping (177) w i l l a l s o have t h e p r o p e r t i e s g i v e n i n (171.1-4) 3)

The mapping (177) a p p l i e d t o f u n c t i o n s i n Cf(sl) i s i d e n t i c a l t o t h e usual a - t h power o f f u n c t i o n s .

Proof 1)

f o l l o w s from (176) and ( 9 2 ) .

2)

f o l l o w s from (176),

3)

f o l l o w s from 1 ) and 2).

(92) and ( 9 5 ) . 0

Remark 8 a)

The c o n d i t i o n (176) can e a s i l y be f u l f i l l e d , s i n c e t h e C"-regulariza= t i o n ( v , ~ ) s a t i s f i e s (90.3).

b)

I n view o f Theorem 16, we can d e f i n e a r b i t r a r y p o s i t i v e powers o f t h e D i r a c 6 d i s t r i b u t i o n a l s o w i t h i n t h e chains o f q u o t i e n t a l g e b r a s i n t r o = duced i n ( 9 3 ) , by u s i n g t h e procedure g i v e n i! (172-175). More pre= cisely, i f E $(a) f o r a c e r t a i n g i v e n RE N , t h e a r b i t r a r y p o s i t i v e powers o f 6 w i l l be d e f i n e d i n each o f t h e q u o t i e n t a l g e b r a s

+

k

A

(us),

with k €

8,

kGR.

D e t a i l s o f t h i s procedure a r e presented i n Chapter 7. 9.

L i m i t a t i o n s on t h e Embedding o f Smooth Functions i n t o Chains o f Q u o t i e n t A1 gebras

I t i s shown i n t h i s s e c t i o n t h a t , i n c o n t r a s t t o t h e case o f t h e chains o f q u o t i e n t algebras i n (24) which c o n t a i n a l l t h e d i s t r i b u t i o n s i n D ' ( n ) , smooth f u n c t i o n embeddings o f t h e form

( 178)

L!-~(R)

c

A (us) , R

E

N, R

2,

w i t h i n t h e chains o f q u o t i e n t algebras ( 9 3 ) a r e not possible. T h i s r e s u l t f o l l o w s from an a d a p t a t i o n o f L . Schwartz's well-known c o u n t e r example - see Appendix 2 - and f o r t h e sake o f s i m p l i c i t y i s presented o n l y f o r t h e one-dimensional case n = l , w i t h R = R

'.

Suppose g i v e n t h e a1 gebras (179)

A

2

, A1

and A'

t o g e t h e r w i t h t h e 1 i n e a r mappings c a l l e d d e r i v a t i v e s D D A2 + A1 -+ Ao (180) which s a t i s f y t h e L e i b n i t z r u l e f o r p r o d u c t d e r i v a t i v e s .

ALGEBRAS CONTAINING THE DISTRIBUTIONS

111

Suppose f u r t h e r t h a t t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d : (181)

A

2

contains the functions x, x ( L n l x 1 - 1 )

and x 2 ( a n 1x1 -1)

.

and t h e m u l t i p l i c a t i o n

in A

2

9

i s such t h a t

2 ( x ( a n l XI -1)l.x = x (an1 XI - 1 ) ;

A

(182)

1 contains the functions 1, x and x(Enlx1-1)

1 and t h e c o n s t a n t f u n c t i o n 1 i s t h e u n i t element i n A ; A'

(183)

i s a s s o c i a t i v e and c o n t a i n s t h e f u n c t i o n s 1 and x

and t h e c o n s t a n t f u n c t i o n 1 i s t h e u n i t element i n A';

( 184 1

t h e mapping D : A' A 2 x and x (an 1x1-1)

1

+

applied t o the functions

i s t h e usual d e r i v a t i v e on C'(R ') ; t h e mapping D : A1

(185)

+

A'

applied t o the function

X

i s t h e usual d e r i v a t i v e on C ' ( R ' ) . Theorem 19 F o r any (186) where (187 1

6 E A',

the relation

x.6 = 0,

. signifies

m u l t i p l i c a t i o n i n A',

implies

6 = 0.

Proof The i d e a i s t o a p p l y t h e method used i n t h e p r o o f o f Theorem 1, Appendix 2, w i t h i n t h e above, more general framework. To do so, we f i r s t p r o v e t h a t t h e f o l l o w i n g r e l a t i o n h o l d s i n A': (188)

(D2(x(hlxl-l))).x

= 1.

Indeed, i n view o f t h e L e i b n i t z r u l e s a t i s f i e d b y t h e mapping D : A* we have

*) f o r x = 0, b o t h o f t h e f u n c t i o n s have b y d e f i n i t i o n t h e v a l u e z e r o .

+

A

1

,

E.E.

112

Rosinger

D((x(9.,n(x(-l)).x) = ( D ( x ( a n \ x \ - l ) ) ) . x t x (E.nlx(-l) i f (184) i s a l s o t a k n i n t o o a c c o u n t . Then (185) and t h e f a c t t h a t t h e + A also satisfies the Leibnitz rule, w i l l y i e l d l i n e a r mapping D : A

D2( (x(9.n

x l - l ) ) . x ) = (D2 (x(Rnlx

-l))).x

+

2 D(x(anlx1-1)).

It follows that

(189)

2 2 ( D2( x ( Rn x l - l ) ) ) . x = D ( x ( a n l x - 1 ) )

- 2

D(x(Rnlx1-1))

B u t (184) w i l l y i e l d

if (181) i s taken i n t o account.

2 D(x ( a n l x l - 1 ) ) = 2 x(Rnlx1-1)

+

x;

hence, (182), (185) and t h e l i n e a r i t y o f t h e mapping D : A’ (190)

D 2 ( x2 ( a n l x l - 1 ) ) = 2 D ( x ( a n l x 1 - 1 ) )

E

A’

and assuming (186) f o r a c e r t a i n 6

E

0 = x

-1

.(x.6)

= (x

-1

yield

Thus u s i n g t h e n o t a t i o n

(183) and (188) w i l l i m p l y

A’,

.x).6

A’

+ 1.

Now (189) and (190) w i l l o b v i o u s l y y i e l d (188). x - l = D2(x(Rnlxl-1))

-f

=

6.

0

I t f o l l o w s from t h e r e s u l t i n Theorem 19 t h a t t h e a l g e b r a A’ cannot c o n t a i n t h e D i r a c 6 d i s t r i b u t i o n , known t o s a t i s f y ( 1 8 6 ) , a r e l a t i o n g i v i n g an upper bound on t h e s i n g u l a r i t y e x h i b i t e d a t x = 0 b y t h a t d i s t r i b u t i o n . The e s s e n t i a l rea on why t h e c h a i n o f algebras (180) have t h a t i n c o n v e n i e n t f e a t u r e i s t h a t A i s supposed t o c o n t a i n t h e f u n c t i o n

3

(191)

x(Rnlx1-1)

E

C(R1) \

C1(R’).

I t i s i n t h i s sense t h a t Theorem 19 i m p l i e s t h e i m p o s s i b i l i t y o f t h e embed= dings (178). T h i s i m p o s s i b i l i t y r e s u l t i s i m p o r t a n t because none o f t h e algebras i n (180) was supposed commutative and o n l y t h e a l g e b r a Ao was r e = q u i r e d t o be a s s o c i a t i v e . Moreover, t h e f u n c t i o n i n (191) has a s i n g u l a r i = t y o n l y a t x = 0, i . e .

x(Rnlx1-1)

E

C ~ ( R \COI); ’

i n o t h e r words (192)

s i n g supp x(Rnlx1-1) =

{ol.

The i n t e r e s t i n g b u t undecided q u e s t i o n remains as t o whether smooth func= t i o n embeddings o f t h e form (193)

CR-’(Q)

are possible o r not.

C AR(V,S),

R E N, R 2 1

ALGEBRAS CONTAINING THE DISTRIBUTIONS 10.

113

Special Classes of Regular Ideals

In order t o obtain l a t e r in Section 4, Chapter 4, the general r e s u l t s on the resolution of s i n g u l a r i t i e s of weak s o l u t i o n s f o r polynomial nonlinear PDEs, i t i s useful t o consider special c l a s s e s of regular i d e a l s and e s = t a b l i s h those of their p r o p e r t i e s t h a t will be needed. These special c l a s s e s of regular i d e a l s w i l l be obtained by p a r t i c u l a r i z i n g t h e notions introduced i n Definition 3, Section 4 , o r e l s e Definition 6 , Section 5 . Definition 9 Suppose given a vector subspace R c Sm such t h a t (194)

R ) n V"

(Ucm(Q) t

=

2.

An ideal I in (C"(Q)) N ismcalled C", R-regular, only i f f o r s u i t a b l e vector subspaces S , T c S , the following r e l a t i o n s hold:

v"

(195)

I nT

(196)

rnsrnc V"@T

(197)

S m C V"

( 198)

UC"(Q)

=

nT =

0T

@ S t

2

R C S B T .

Obviously, the above notion i s a p a r t i c u l a r case of t$at introduged i n Definition 3, Section 3. Indeed, i f an ideal 1 i n ( C (g))Ni s C , R-regu= l a r , thgn i t i s a l s o C -regular. Moreover, I will be C -regular only i f i t i s C , R-regular, f o r R = 2. I t i s a l s o easy t o see t h a t (194) follows from (197) and (198). In a s i m i l a r way, by p a r t i c u l a r i z i n g Definition 6 , Section 5 , we can define

C , R-regular i d e a l s .

In order t o shorten the presentation, we s h a l l deal only w i t h C", R-regu= l a r i d e a l s and notice t h a t a l l t h e r e s u l t s i n this s e c t i o n obtained f o r them are valid a l s o f o r C? , R-regular i d e a l s . A necessary condition on C", R-regular i d e a l s i s obtained in the following extension of Corollary 1, Section 4.

Proposition 7 I f the ideal 1 in (C"(Q))N ( 199 1

I n

i s C", t

R-regular, then

R ) = 9.

Proof The r e l a t i o n s (196) and (198) imply t h a t I

(UC"(S-2)

n (S

t

@ T).

R) c (I n

s")

n ( u ~ " ( ~t) R )

c (v"

0T ) n

114

E.E. Rosinger

Therefore (197) w i l l y i e l d

which, i n view o f (195), w i l l i m p l y (199). We now p r e s e n t a s u f f i c i e n t c o n d i t i o n on C", f o l l o w i n g e x t e n s i o n o f Theorem /, S e c t i o n 4.

R-regular i d e a l s i n the

Theorem 20

I f a v a n i s h i n g i d e a l T i n (Cm(n)) ( 200 1

N

1 ) n (Ucm(n)

(Vm t

s a t i s f i e s the condition

+ R)

=

2,

then I i s Cm, R-regular, and t h e r e e x i s t C m - r e g u l a r i z a t i o n s

( r n v", s @

T)

satisfying the conditions (201)

I ~ T = V " ~ T =

(202 1

v"

(203)

Sm=Vm@S@

(204)

Ucm(n)

t

(r

n

s")

i- R C

=

2

vm @

T

T S.

Proof Since 'I i s vanishing, E r o p o s i t i o n 1, S e c t i o n 4 w i l l g r a n t t h e e x i s t e n c e o f v e c t o r subspaces T C S t h a t s a t i s f y (201) and (202). B u t c o n d i t i o n (200) i s equivalent t o ( ~ ~ m t ( ~ R) )

n (vm t

( r n S" 1)

=

2;

t h e r e f o r e , i n view o f (202) i t i s e q u i v e l e n t t o t h e c o n d i t i o n

From these c o n s i d e r a t i o n s t h e e x i s t e n c e o f v e c t o r subspaces S c S m s a t i s f y = i n g (2g3) and (204) f o l l o w s e a s i l y . Then i n view o f Theorem 4, S e c t i o n 4, 0 (I n V , S T ) w i l l indeed be a C m - r e g u l a r i z a t i o n .

0

Remark 9

a

The between t h e necessarx c o n d i t i o n i n P r o p o s i t i o n 7 and t h e s u f f i c i e n t c o n d i t i o n i n Theorem 20 on C , R - r e g u l a r i d e a l s , can e a s i l y be seen by comparing t h e r e l a t i o n s (199) and (200). As seen l a t e r i n Sections 1-5, Chapter 4, i n c o n n e c t i o n w i t h 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 i m p o r t a n t c l a s s e s o f weak s o l u t i o n s f o r p o l y n o m i a l n o n l i n e a r PDEs, t h i s gap t u r n s o u t t o be i r r e l e v a n t , s i n c e c o n d i t i o n (200) w i l l be a u t o m a t i c a l l y s a t i s f i e d .

ALGEBRAS CONTAINING THE DISTRIBUTIONS

11.

115

The Proof o f Lemma 1

The p r o o f 0s Lemma 1 i n S e c t i o n 1 i s g i v e n h e r e . The e q u i v a l e n c e between ( 7 ) and (8) f o l l o w s e a s i l y . The second p a r t i s proved as f o l l o w s . = Rn, xo = 0 E Rn.

For s i m p l i c i t y we s h a l l assume t h a t

R

For a E R ' and v

E

N denote

=

I x E Rnl s v ( x ) > a 1

E(a,v)

F i r s t , we prove t h e r e l a t i o n

K

(205)

v

-, ~0

i

s v ( x ) d x 2 1, U a

E

R'

E(a,v)

Assume i t i s f a l s e

.

Then

3 a € R'

,E >O,

U V E N

, v >

I

N :

p'€

p':

Sv(x)dx

= 6 and supp s ( v ) s h r i n k s t o 0 E Rn, when w -t Therefore, assuming $ E lI(Rn) and $ = 1 on a neighbourhood o f 0 E Rn, one obtains ~0

1

=

$(O)

= lim

In sv(x)$(x)dx = l i m I

v+-tRR"

v+mR

s (x)dx

It follows t h a t

1

-

E/Z

r

sv(x)dx

Rn

Now, f o r v E N, t h e r e l a t i o n s h o l d

< 1

sv(x)dx

+

E(a,v)

a

I

dx

SUPP SV

Therefore, one o b t a i n s f o r v E N, v 2 max {IJ',~

1 - ~ / 2 < 1 - ~ + a J SUPP sv

dx

" 3the

inequality

E.E. Rosinger

116

which i s absurd since supp sv s h r i n k s t o 0 E R’, completed.

and t h e p r o o f o f (205) i s

We prove now t h a t t h e r e e x i s t av E [ O p ) , w i t h v E N, such t h a t (206)

l i m av v+cQ

and Tiiii

m

v +

sv x)dx 2 1

E(av,v)

m

Indeed, a c c o r d i n g t o (205), t h e r e e x i s t v E N, w i t h (207)

vo < v1
O ,

F i r s t we d e f i n e

x

: R’

-+

V v E N, x

< l/v

l / ( v + l ) < 1x1

if

and v

E

Mi01

otherwise

0 E

R’

R’ by

(-l)vlxl

and we d e f i n e t

E

by

(R’-+R’)N

otherwise

0

Then

while

1/ (2v+2)

1/ (2v+2) (216)

I

tv(x)dx =

R’

v + l Qu<m

6u2 + 6 A =

We d e f i n e now t ’ tb(X)

E

I

- 1/2v+2)

X(x)dx = 2

0

+1 c (R’

v -+

> R’)N

= tv(x)/cv,

O by Y v

I

E

N, x E R’

X(x)dx =

E.E. Rosinger

118 Then (216) w i l l i m p l y j l t$

(217)

( x ) d x = 1, V v

E

N

R

w h i l e (215) w i l l i m p l y t $ ( x ) x ( x ) d x = l / l 2 ( ~ + l ) ~,cV~ v

I

(218)

E

N

R'

B u t an easy computation w i l l y i e l d t h e i n e q u a l i t y

the r e f o r e

Now, i n view o f (218), i t f o l l o w s t h a t

1 t $ ( x ) x ( x ) d x > ~ ~ / 4 ( v + 1,) V~ v

E

N\{O)

R'

hence

(219)

fi

J l t \ ; ( x ) x ( x ) d x 2 1/4 > 0

= x(0)

v + m R A p p l y i n g a simple smoothing t o t ' and x and t a k i n g i n t o account (217 and (219), i t i s o b v i o u s l y p o s s i b l e t o o b t a i n s E (C "(R'))N and IJJ E D(R ) so t h a t (211-213) w i l l be v a l i d .

1

13.

I n e x i s t e n c e o f L a r g e s t Regular I d e a l s

Connected w i t h t h e c o n d i t i o n

(220)

I

n Uc"(R)

Q

N on i d e a l s 7 i n (C"(Q)) encountered i n C o r o l l a r y 1, Theorem 5 and Remark 4, S e c t i o n 4, i t can be n o t i c e d t h a t t h e r e a r e no l a r g e s t b u t o n l y maximal such i d e a l s . Indeed, i n case R = R'

, define

w ' , w"

wV' ( x ) = 1 + s i n v x , w;(x)

=

E

(C"(R)) N by

1

t cosvx,

V v E N, x E R

Then 7 ' = w'.(C"(R))N,

I"

= w".(C"(R))N

a r e i d e a l s i n (C"(R))N which s a t i s f y (220).

I

=

7'

+ I"

However

ALGEBRAS CONTAINING THE DISTRIBUTIONS

119

i s an i d e a l i n (Cm(Q))N which does n o t s a t i s f y (220). Indeed w = w ' t w"

1

E

and wV ( x ) = 2 t s i n v x

t

cosvx > 0, V v

E

N, x E

n

therefore

u ( i ) = w.(i/w)

E

r.(cm(n))N

c

r

I t f o l l o w s t h a t no i d e a l i n (Cm(n))N which c o n t a i n s fy (220).

1' and I", w i l l s a t i s =

Obviously, a s i m p l e m o d i f i c a t i o n o f t h e above example w i l l i m p l y t h e -= istence o f ideals 7 i n (C(n))N satisfying the condition

(221)

I n Uc o ( n )

=

Q

encountered i n Theorem 11, S e c t i o n 5 . The above p r o p e r t y can be seen as a reason f o r t h e i n e x i s t e n c e o f a unique, ' c a n o n i c a l ' c h a i n o f q u o t i e n t a l g e b r a s (24), r e s p e c t i v e l y (93), and there= f o r e can be seen as an a d i i t i o n a l reason - besides t h e one o f f e r e d by t h e i n t e r p l a y between c o n f l i c t i n g s t a b i l i t y , g e n e r a l i t y and exactness i n t e r e s t s concerning s e q u e n t i a l s o l u t i o n s - f o r d e a l i n g w i t h v a r i o u s chains o f quo= t i e n t a l g e b r a s (24) o r (93).

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CHAPTER 4 RESOLUTION OF SINGULARITIES OF WEAK SOLUTIONS FOR POLYNOMIAL NONLINEAR PDEs

0.

Introduction

The aim o f t h e p r e s e n t c h a p t e r i s t o a p p l y t h e general method f o r t h e reso= l u t i o n o f nowhere-dense s i n g u l a r i t i e s o f s e q u e n t i a l s o l u t i o n s f o r p o l y n o m i a l n o n l i n e a r PDEs p r e s e n t e d i n S e c t i o n 7, Chapter 1, t o t h e i m p o r t a n t p a r t i c u = l a r case o f weak s o l u t i o n s o f t h e PDEs mentioned. I n o r d e r t o deal w i t h weak s o l u t i o n s , t h e s i n g u l a r i t i e s w i l l have t o be r e s t r i c t e d t o nowhere-dense subsets w i t h z e r o Lebesque measure. However, t h e r e s u l t s presented i n t h i s c h a p t e r remain s u f f i c i e n t l y general t o i n c l u d e as r a t h e r s i m p l e p a r t i c u l a r cases most o f t h e known types o f s i n g u l a r i t i e s e x h i b i t e d by weak s o l u t i o n s o f f i r s t - and second-order n o n l i n e a r PDEs, [ 3, 14-17, 67, 87-90, 115,117, 149, 161-163, 1781, f o r i n s t a n c e those o f shock wave s o l u t i o n s f o r n o n l i n e a r h y p e r b o l i c c o n s e r v a t i o n laws, K l e i n Gordon wave equations, as w e l l as t h e e q u a t i o n s o f magnetohydrodynamics o r general r e l a t i v i t y . S t a t e d s i m p l y , 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 l u t i o n s o f t h e PDE i n (1) means t h a t t h e weak s o l u t i o n s considered w i l l s a t i s f y t h e PDE i n ( l ) , i n t h e usual a l g e b r a i c sense, w i t h m u l t i p l i c a t i o n and 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 d e f i n e d w i t h i n t h e c h a i n s o f q u o t i e n t algebras (24) o r Tv3) , Chapter 3. Thus t h e r e s u l t s p r e s e n t e d i n t h i s c h a p t e r can be seen as b e l o n g i n g t o a ' p o l y n o m i a l n o n l i n e a r o p e r a t i o n a l c a l c u l u s ' f o r PDEs. T h i s ' o p e r a t i o n a l c a l c u l u s ' o f f e r s t h e p o s s i b i l i t y , among o t h e r s , o f i d e n t i = f y i n g t h e c l a s s o f r e s o l u b l e systems o f p o l y n o m i a l n o n l i n e a r PDEs, charac= t e e d b y an upper bound on t h e c o m p l e x i t y o f t h e i r n o n l i n e a r i t i e s , a c l a s s which i n c l u d e s many - i f n o t most - o f t h e e q u a t i o n s of p h y s i c s . I n S e c t i o n 1, a p a r t i c u l a r case w i l l be c o n s i d e r e d o f t h e general polyno= m i a l n o n l i n e a r PDE (see ( l ) , Chapter 1)

and a c o r r e s p o n d i n g r e s u l t 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 w i l l be e s t a b l i s h e d i n Theorem 1. T h i s r e s u l t w i l l be f u r t h e r p a r t i c u l a r i z e d , i n S e c t i o n s 2 and 3, t o t h e case of shock wave s o l u t i o n s o f n o n l i n e a r h y p e r b o l i c c o n s e r v a t i o n laws, o r K l e i n-Gordon t y p e n o n l i n e a r waves.

121

E.E. Rosinger

122

A much more general r e s u l t 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 l u = t i o n s i s presented i n S e c t i o n 4. I n S e c t i o n 5 a c h a r a c t e r i z a t i o n i s presented o f weak s o l u t i o n s by j u n c t i o n c o n d i t i o n s across the hypersurfaces o f t h e i r d i s c o n t i n u i t i e s , f o r a l a r g e c l a s s o f systems o f polynomial n o n l i n e a r PDEs i n c l u d i n g among o t h e r s t h e equations o f magnetohydrodynamics and general r e l a t i v i t y . T h i s r e s u l t i s then extended i n S e c t i o n 6 t o t h e c l a s s o f r e s o l u b l e systems o f polynomial n o n l i n e a r PDEs, a c l a s s which i n c l u d e s many o f t h e equations o f physics. The PDE i n (1) can be considered w i t h i n t h e framework o f t h e chains o f q u o t i e n t algebras i n (24) o r (93), Chapter 3, corresponding t o t h e smooth= ness o f t h e c o e f f i c i e n c t s c. and r i g h t - h a n d t e r m f. I n t h e case ci, f€C"(n), f o r i n s t a n c e both types Ehains o f q u o t i e n t a l g e b r a s may be used. Other= wise, i n case c . , f E C ( Q ) , w i t h R E N, t h e most obvious procedure would be t o use t h e c i a i n s o f q u o t i e n t algebras (93) , Chapter 3.

01

The choice o f t h e framework i s a l s o i n f l u e n c e d by t h e smoothness e x h i b i t e d a p a r t from t h e i r s i n g u l a r i t i e s by t h e weak s o l u t i o n s b e i n g considered.

To a v o i d a l e n g t h y p r e s e n t a t i o n , t h e p r o o f s w i l l be g i v e n w i t h i n one o f t h e two frameworks o n l y , t h e necessary m o d i f i c a t i o n s b e i n g mentioned when= e v e r t h e o t h e r framework c o u l d a l s o be used. As mentioned i n d), Remark 6, S e c t i o n 6, Chapter 3, i t i s u s e f u l t o make a p a r a l l e l s t u d y o f t h e problems considered, w i t h i n b o t h t y p e s o f chains o f q u o t i e n t algebras (24) and (93), Chapter 3, b e c a u s e c h t y p e has c e r t a i n advantages compared with the other. We now g i v e a simple and u s e f u l c l a s s i f i c a t i o n o f t h e polynomial n o n l i n e a r PDEs i n (1). Given k E

1.

fly t h e PDE i n ( 1 ) i s c a l l e d C k -smooth, o n l y i f ci,

f E C

k

(n).

The Case o f Simple Polynomial N o n l i n e a r PDEs

An m-th o r d e r polynomial n o n l i n e a r PDO i n ( 3 ) , Chapter 1, i s c a l l e d simple, o n l y i f i t can be w r i t t e n i n t h e form

where L. (D) a r e m-th o r d e r l i n e a r PDOs w i t h continuous c o e f f i c i e n t s , w h i l e Ti a r e b o l y n o m i a l s o f t h e form Tiu(x)

c. . ( x ) ( u ( x ) ) j ,

C

=

1 G j But u

E

C"(Q \

r)

(37)

r



nowhere-dense

i n view o f (6), t h e r e f o r e

SUPP

since Q \

supp

r

c

i s open, as according t o ( 5 ) ,

r

i s closed.

It follows t h a t

u = $ o n ~ r. \

But i n view o f ( 5 ) ,

r

has zero Lebesque measure, w h i l e i n view o f ( 7 )

u E qoc(Q) since i t may be assumed t h a t b >l i n ( 7 ) otherwise the PDE i n ( 4 ) becomes trivial.

D'(Q) the r e l a t i o n

Thus (37) w i l l y i e l d i n u =

IJJ E

C"(Q)

c o n t r a d i c t i n g (10) and thus completing t h e p r o o f o f (36). (36) there e x i s t vector subspaces S c S , such t h a t

Now i n view o f

s;

(38)

SE

(39)

UCm(Q)

(40)

Sm =

c

and

S;

Vm@

S

@ J.

Now (33), (34), (40) and (39) w i l l , according t o Theorem 4, Section 4, Chapter 3, i m p l y t h a t

SO

(v, J) i s a Cm-reguIarization (41) f o r any vector subspace V n 1 C V" Since by d e f i n i t i o n {Dpwlp E N n I C lw C I, i t i s obviously possible, i n view o f (26), t o choose V so t h a t t h e f o l l o w = i n g condition i s satisfied:

(42)

(Dpwlp

E

NnI

t a k i n g f o r instance V = 7

C V;

r~ V".

F i n a l l y , i n view o f Theorems 2 and 3, Section 3, Chapter 3, the r e l a t i o n s 0 (38), (17) and (42) complete the p r o o f o f Theorem 1. The above r e s u l t o f Theorem 1 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 z a t i o n of weak s o l u t i o n s f o r n o n l i n e a r PDEs, i n t h e case o f i n f i n i t e smoothness, has i t s counterpart f o r f i n i t e smoothness, presented i n the f o l l o w i n g theorem.

130

E.E.

Rosinger

Theorem 2 k, R 1 i s a piecewise C-imooth weak s o l u t i o n o f t h e m-th o r d e r If smooth s i m p l e polynomial n o n l i n e a r PDE i n ( 3 ) , and kl 2..

;Sqpose u

: R

-f

(43) then

C"

i t i s possible t o construct (?-regularizations

1) where s does

2)

(n)

u = s t IR(V,S)

not depend

AR(V,S),

E

'b R

E

N, il

( V , S ) such t h a t Q

kl

on 2;

u s a t i s f i e s t h e PDE i n ( 3 ) , i n t h e usual a l g e b r a i c sense, w i t h m u l t i p l i c a t i o n i n Ak(V,S) and t h e p a r t i a l d e r i v a t i v e opera t o r s DP : A ~ ( v , s ) with R, k

-f

A~(w,s), p

E

N",

\PI d m

N, R-k 2 m y R G kl and k d minCkl-m,k2).

E

Proof We use the n o t a t i o n h = min{k -m k I . can c o n s t r u c t a r e g u l a r i z i n g l e q i e i c e (44)

s

u =

E $1,

As i n t h e p r o o f o f Theorem 1, we

<S,.>

and t h e corresponding e r r o r sequence (45 1

b o t h s a t i s f y i n g (19). (46)

-

w = T(D)s

{DPwIp

u ( f ) E Vh

Therefore E

Nn,

h l c Ind n V'.

IpI

Assume now g i v e n any i d e a l I i n ( C

(47)

IDpwlp

E

such t h a t h) c I c I n d

Nn Y

and I i s Ck'-smooth (see (126), Chapter 3). Obviously, such a c h o i c e o f I i s p o s s i b l e , i n view o f (46) and P r o p o s i t i o n 6, S e c t i o n 6, Chapter 3. Using now t h e same argument as t h a t i n t h e p r o o f o f Theorem 14, S e c t i o n 6, Cheyter 3, f o r R=kl,it f o l l o w s t h a t t h e r e e x i s t v e c t o r subspaces T C S satisfying

v"

(48)

I nT

(49 1

V'

(50)

(v" @ T )

t

=

nT =

2;

( 7 nso) = V @ 7; n UC" ( a ) =

and

Q.

B u t i n view o f ( 4 3 ) , and u s i n g an argument s i m i l a r t o t h a t which e s t a b l i s h e d (36), we have

(51)

sB

v"

@uco

(a)

0T .

RESOLUTION OF SINGULARITIES NOW, i n view o f (51), t h e r e e x i s t v e c t o r subspaces R '

u"

S" =

@

@

UC" (n)

0R '

R's

131 C

So , such t h a t

@ 7.

Thus t a k i n g i n Lemma 5, S e c t i o n 6, Chapter 3

v" @ Up(n) 0 + R's @ J, B

E = S o , F =Ski A =

= R'

i t f o l l o w s t h a t t h e r e e x i s t v e c t o r subspaces

C=R

F = Sk l

C

satisfying

@

(V'

Up(no@ R's @ 7) n R

=

9

and

s"

(52)

=

u"

0 Uco

@

R's

0R

@ 7.

Introducing the notation (53)

S

=

Up (n) @ R's @ R

t h e r e l a t i o n s (48), (49) and (52) w i l l i m p l y t h a t (54)

(V, S

@

T ) i s a c" - r e g u l a r i z a t i o n

f o r any v e c t o r subspace V S e c t i o n 5, Chapter 3.

C

1 n 'V

, ifwe

t a k e i n t o account Theorem 10,

B u t i n view o f (46), i t i s p o s s i b l e t s choose V so as t o s a t i s f y

w E Vh.

(55)

Thus i n v i e w o f t h e v e r s i o n s o f Theorems 2,3 and 9, Chapter 3, correspon= d i n g t o C " - r e g u l a r i z a t i o n s , t h e r e l a t i o n s (53), (45) and (55) complete t h e p r o o f o f Theorem 2. 0 2.

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 N o n l i n e a r Shock Waves

Suppose g i v e n t h e n o n l i n e a r h y p e r b o l i c c o n s e r v a t i o n law (56)

ut(x,t)

t

a(u(x,t)).ux(x,t)

t 0,

x

E

R',

t > 0,

w i t h the i n i t i a l condition (56.1)

u(x,O) = u o ( x ) , x

E

R'.

We s h a l l suppose t h a t t h e f u n c t i o n (57)

a : R' -+ R'

i n (56) i s an a r b i t r a r y p o l y n o m i a l . Then i t i s obvious t h a t (56) i s a f i r s t - o r d e r C--smooth s i m p l e p o l y n o m i a l n o n l i n e a r PDE on n = R'x(0,m) c R2. Indeed, t h e l e f t - h a n d t e r m i n ( 5 6 ) can be w r i t t e n i n t h e fonn i n (21, p r o =

E.E. Rosinger

132

v i d e d t h a t we t a k e a=2, L1(D) = Dty L2(D) = D x y T1u = u and T2u = b ( u ) where (58)

b : R'

R'

-+

i s a p r i m i t i v e o f t h e f u n c t i o n i n ( 5 7 ) , and thus a g a i n a polynomial. I t i s known t h a t under r a t h e r general c o n d i t i o n s [ 67,1781 f o r Cm-smooth o r piecewise smooth i n i t i a l d a t a u o y t h e e q u a t i o n (56) has shock wave so u=

tions u :

w i t h the following properties.

R',

+

Thgre e x i s t s a f i n i t e s e t G o f -smooth c u r v e s

Cm-smooth f u n c t i o n s

y:

R

+

R',

defin

C

r

Y

=

IX E n

I

Y ( X ) = 01

which d e s c r i b e t h e p r o p a g a t i o n o f t h e shocks, and such t h a t

r),

u E C

(60)

u i s l o c a l l y bounded on R ;

I

(61)

n

\

where

r

(59)

(u(x,t)$(x,t) t

t

=

y g G r y ;

b(u(x,t))$(x,t))dx

d t = 0, tf $ E D ( f i ) .

X

Obviously, such a s o l u t i o n u w i l l be a i e c e w i s e Cm-smooth weak s o l u t i o n o f t h e PDE i n (56), i n t h e sense o f t h e S e f i n i t i o n i n S e c t i o n 1. There= f o r e Theorems 1 and 2 w i l l y i e l d t h e f o l l o w i n g r e s u l t . Theorem 3 Suppose u : R' x(0,m) -+ R1 i s a shock wave s o l u t i o n o f t h e n o n l i n e a r h y p e r b o l i c c o n s e r v a t i o n law i n (56)mti s a t i s f i e s the conditions (59-61). Then i t i s p o s s i b l e t o c o n s t r u c t C - r e g u l a r i z a t i o n s (V,s), such that

1) 2)

tt R

u = s t lR(V,S) E AR(V,S),

E

8,

n o t depend o n R ; where s E S does u s a t i s f i e s t h e PDE i n (56), i n t h e usual a l g e b r a i c sense, w i t h m u l t i p l i c a i i o n i n Ak(V,S) and 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 Dt, Dx : A (V,S) -+ Ak(V,S) w i t h R , k E \, R-k > 1.

If

(62)

u .$

c" (n)

then one can c o n s t r u c t

3)

u = s t lR(V,.S) depend on R ;

Y

C " - r e g u l a r i z a t i o n s ( V , S ) , such t h a t 6

AR(V,s),

tt

RE

w,

where s

E

s

does not

RESOLUTION OF SINGULARITIES u s a t i s f i e s t h e PDE i n ( 5 6 ) , i n t h e usual a l g e b r a i c sense, w i t h m u l t i p l i c t t i o n i n Ak(V,S) and 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 Dt,Dx : A (V,S) + Ak(V,S) with1,kE N, e-k > 1.

4)

3.

133

R e s o l u t i o n o f S i n q u l a r i t i e s o f Klein-Gordon Type N o n l i n e a r Waves

Suppose g i v e n t h e Klein-Gordon type n o n l i n e a r wave e q u a t i o n (63)

u t t ( x y t ) - u x x ( ~ , t ) = T(D)u(x,t),

x

E

R',

t > 0,

w i t h t h e in i ti a1 condi ti ons (63.1)

u(x,O)

(63.2)

ut(x,O)

= f(x),

x

= q(x),

E

x

E

R'

and

R',

where T(D) i s a f i r s t - o r d e r Cm-smooth s i m p l g p o l y n o m i a l n o n l i n e a r PDO. Then i t i s obvious t h a t (63) i s a second-order C -smooth s i m p l e p o l y n o m i a l non= l i n e a r PDE on R = R' x(0,m) C R', s i n c e i t has t h e form i n (4). I t i s known t h a t under general c o n d i t i o n s [ 161-1631 t h e e q u a t i o n (63) has l o c a l o r g l o b a l s o l u t i o n s u : R' + R', w i t h R' c R open, which have t h e following properties.

There e x i s t a f i n i t e number o f p o i n t s xl, cones w i t h the-ari es g i v e n b y

r-c1

= {(x,t)

rta =

{(x,t)

E E

R'I R'I

...,xo E

R',

originating light-

X-xa t t = 01 , x-xa

-

t = 01

, with

1
m

now consider the finite-smooth case. (see (70)).

0

Suppose given k l , k 2 E N , w i t h

A distribution

E U'(Q)\ c" (Q) i s called a Ckl-regular weak solution of an m - t h order C"-smooth PDE i n (68), only i f there e x i s t s a sequence o f functions s E Skl satisfying the following three conditions:

(75)

s

=

< s,

.>

(76)

w

=

T(D)s

-

(77)

TW i s a C?

u(f)

E

Vhy w i t h h = min

Ikl-m,k21;

R-regular ideal

where

(77.1)

R

(77.2)

W I

=

R!s;

= the ideal i n

(e

generated by IDPwIp

E

Nn,lpl

hl.

E . E . Rosinger

136

The f o l l o w i n c ] theorem shows t h a t t h e above n o t i o n extends t h a t o f t h e f i n i t e - s m o o t h case o f piecewise smooth weak s o l u t i o n s , d e f i n e d i n S e c t i o n 1. Theorem 6

Ifa {iecewise Ck'-smooth weak s o l u t i o n u : R + R' o f an m-th o r d e r C k z smoot s i m p l e polynomial n o n l i n e a r P D t s a t i s f i e s t h e c o n d i t i o n

u 4 C"

(78)

(a),

then u i s a Ckl-regular

weak s o l u t i o n o f t h a t PDE.

Proof T h i s f o l l o w s e a s i l y f r o m (44), ( 4 5 ) , (46), (51) and (49), as used i n the p r o o f o f Theorem 2, S e c t i o n 1.

0

We now g i v e t h e general r e s u l t 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 l u t i o n s f o r PO-1 n o n l i n e a r PDEs. We d e a l f i r s t w i t h t h e i n f i n i t e smooth case. Theorem 7 Suppose S E D'(Q) i s a Cm-regular weak s o l u t i o n o f t h e m-th o r d e r Cm-2mooth polynomial n o n l i n e a r PDE i n ( 6 8 ) . Then i t i s p o s s i b l e t o c o n s t r u c t C -regu= l a r i z a t i o n s ( V , S ) , such t h a t

I)

s

= s t

rR(v,s) E

A ~ ( v , s ) , 4' R E

R,

where s E S does n o t depend on R;

2)

S s a t i s f i e s t h e PDE i n (68), i n t h e usual a l g e b r a i c sense, w i t h

m u l t i p l i c a t i o n i n Ak(V,S)

Dp : AR(V,S)

+

Ak(V,S),

and 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 p E Nn,

I p I Gm, w i t h R, k

E

m,

R-k > m .

Proof T h i s f o l l o w s e a s i l y f r o m (71-73) above, and (195-198), S e c t i o n 10, a l s o Theorem 4, S e c t i o n 4, Chapter 3, i f account i s a l s o taken o f Theorems 2 and 3, S e c t i o n 3, Chapter 3.

0

I n a s i m i l a r way, t h e f o l l o w i n g general r e s u l t can be o b t a i n e d 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 l u t i o n s f o r polynomial n o n l i n e a r PDEs, i n t h e case o f f i n i t e smoothness. Theorem 8 Suppose S E D ' ( R ) i s a Ckl-regular weak s o l u t i o n o f t h e m-th o r d e r ck2smooth polynomial n o n l i n e a r PDE i n (68). Then i t i s p o s s i b l e t o c o n s t r u c t c" - r e g u l a r i z a t i o n s ( V , S ) such t h a t

1)

R R S = s + 1 ( V , S ) E A ( V , S ) , 4' R E N, R d kl where s

e s does not

depend on R ; 2)

S s a t i s f i e s t h e PDE i n (68), i n t h e usual a l g e b r a i c sense, w i t h

RESOLUTION OF SINGULARITIES

137

m u l t i p l i c a t i o n i n A k ( V , S ) and 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 Dp : AR(V,S)

II G kl,

* Ak(V,S),

p E Nn,

I p I G m , w i t h a, k

E

N, a-k > m y

k G min Ikl-m,k22.

The s u f f i c i e n t c o n d i t i o n on r e g u l a r i d e a l s g i v e n i n Theorem 2 0 , S e c t i o n 10, Chapter 3, can be t r a n s l a t e d i n t o a s u f f i c i e n t c o n d i t i o n on r e g u l a r weak s o l u t i o n s , a s p r e s e n t e d now, i n t h e case o f i n f i n i t e smoothness. Theorem 9 Suppose themPDE i n (68) i s Cm-smooth, and t h a t we a r e g i v e n a d i s t r i b u t i o n S E t ? ' ( Q ) \ C (a). I f t h e r e e x i s t s a sequence o f f u n c t i o n s s E S , such t h a t (79)

s

(80)

w = T(D)s

(81)

lw i s a vanishing ideal

= < S,.

then S i s a (?-regular

>

;

- u(f)

E V";

(see (39), Chapter 3);

and

weak s o l u t i o n f o r t h e Cm-smooth PDE i n (68).

Proof I n view o f (81) and (82) above, as w e l l as t h e p r e v i o u s l y mentioned Theo= 0 rem 20, S e c t i o n 10, Chapter 3, i t f o l l o w s t h a t (73) h o l d s . Remark 1 The r e s u l t o f Theorem 9 y i e l d s i m p o r t a n t i n s i g h t i n t o t h e e s s e n t i a l p r o = p e r t i e s o f r e g u l a r weak s o l u t i o n s , and t h e r e f o r e i n p a r t i c u l a r , p i e c e w i s e smooth weak s o l u t i o n s . Indeed, t h e c o n d i t i o n s (79) and (80) a r e t h e n a t u r a l requirements t h a t t h e r e q u l a r weak s o l u t i o n S s h o u l d be generated b y a r e u l a r i z i n g sequence s t h a t y i e l d s an e r r o r sequence w weakly convergent +TT- t o zero. However, these two c o n d i t i o n s a r e t h e consequence o f t h e p a r t i c u l a r way t h e e n e r a l i t y o f s e q u e n t i a l s o l u t i o n s (see S e c t i o n 5, Chapter 1) was s o l v %-Tie w en c o n s t r u c t i n g c h a i n s o f q u o t i e n t a l g e b r a s ( 2 4 ) and (93), Chap= t e r 3, (see S e c t i o n 0 , Chapter 3 ) . T h i s means t h a t i t i s p o s s i b l e t o dispense w i t h these c o n d i t i o n s when general s e q u e n t i a l s o l u t i o n s a r e b e i n g studied. On t h e o t h e r hand, t h e c o n d i t i o n s (81) and ( 8 2 ) a w e s s e n t i a l w i t h i n t h e framework of any k i n d o f q u o t i e n t a l g e b r a , as d e f i n e d i n Chapter 1. I n = deed, i n view o f P r o p o s i t i o n 1 below, t h e c o n d i t i o n (81) means t h a t t h e e r r o r sequence w, t o g e t h e r w i t h i t s p a r t i a l d e r i v a t i v e s o f a r b i t r a r y f i x e d order

Dpw, p

E Nn,

IpI G

Q.

have t o v a n i s h s i m u l t a n e o u s l y . F i n a l l y , c o n d i t i o n (82) demands a separa= , generatemr= t h e r e g u l a r i z i n g sequence s and t h e i d e a l 1

t i o n between

b i t r a r y p a r t i a l d e r i v a t i v e s o f t h e e r r o r sequence w .

138

E.E.

Rosinger

I n s h o r t , the two c o n d i t i o n s (81) and (82) mean r e s p e c t i v e l y t h e v a n i s h i n g o f t h e e r r o r sequence and t h e s e p a r a t i o n between t h e ' n e g l i g i b l e ' e r r o r sequence (see S e c t i o n 2, Chapter 1) and t h e r e g u l a r i z i n g sequence.

Proposition 1 The i d e a l I,

(see (73.2)) i s vanishing, o n l y i f

Vu,

RE

N :

3 vEN,v>p,xEQ: V p

(83)

E

Nn,

IpI GR:

DPwv(x) = 0. Proof Assume Iw i s v a n i s h i n g .

Obviously, f o r any g i v e n R E N

c

i=

( D P w ) ~E

rw.

p E Nn IPI G!z Therefore a p p l y i n g (39), Chapter 3, t o

i E Iw,

we e a s i l y o b t a i n ( 8 3 ) .

The converse i s obvious.

11

I n t h e case o f f i n i t e smoothness, i t i s easy t o o b t a i n t h e c o r r e s p o n d i n g v e r s i o n s o f Theorem 9 and P r o p o s i t i o n 1, p r e s e n t e d next, as w e l l as t h e corresponding remarks on t h e s u f f i c i e n t c o n d i t i o n s on r e g u l a r weak solu= ti ons. Theorem 10 Suppose t h e m-th o r d e r PDE i n (68) i s Ck2-smooth, and a d i s t r i b u t i o n S E P'(R)\ C(Q) i s given. I f t h e r e e x i s t s a sequence o f f u n c t i o n s s w i t h kl 2 m, such t h a t (84)

s

(85)

w = T(D)s

(86)

Iw i s a v a n i s h i n g i d e a l ;

(87)

(V

E

ski

<s, *>;

=

t

-

u(f)

rw)n(uc"

E Vh,

t

R!S)

w i t h h = min {kl-m,k2}; and =

p;

then S i s a C k ' - r e g u l a r weak s o l u t i o n of t h e m-th o r d e r Ck2-smooth PDE i n (68)

-

Proposition 2 The i d e a l Iw (see (77.2)) i s vanishing, o n l y i f

RESOLUTION OF SINGULARITIES

5.

13Y

J u n c t i o n C o n d i t i o n s and R e s o l u t i o n o f S i n q u l a r i t i e s o f Weak S o l u t i o n s

f l The problem o f f i n d i n g j u n c t i o n c o n d i t i o n s across h y p e r s u r f a c e s f o r s o l u = t i o n s o f t h e e q u a t i o n s o f magnetohydrodynamics o r qeneral r e l a t i v i t y i s u s u a l l y approached e i t h e r by a p p l y i n g i n t e g r a l c o n d i t i o n s o r b y i n t r o d u c i n g ce r t a in s impl ify ing ass ump t ions

.

Both methods p r e s e n t obvious d e f i c i e n c e s when compared w i t h t h e d i r e c t method suggested i n [161 and based on t h e i d e a o f o b t a i n i n g t h e j u n c t i o n c o n d i t i o n s from weak s o l u t i o n c o n d i t i o n s f o r m u l a t e d across t h e h y p e r s u r f a = As t h e n o n l i n e a r i t y o f t h e e q u a t i o n s i n v o l v e d ces o f d i s c o n t i n u i t i e s . w i l l i m p l y t h e presence of p r o d u c t s o f t h e H e a v i s i d e f u n c t i o n w i t h t h e D i r a c 6 d i s t r i b u t i o n and i t s p a r t i a l d e r i v a t i v e s , t h e l a t t e r method cannot be implemented w i t h i n t h e d i s t r i b u t i o n a l framework. For t h i s reason, s p e c i a l r e g u l a r i z a t i o n procedures were suggested i n [ 10,14-171 , amounting t o the c o n s t r u c t i o n o f subalgebras c o n t a i n e d i n t h e d i s t r i b u t i o n s and c o n t a i n i n g s i n g u l a r d i s t r i b u t i o n s , such as t h e H e a v i s i d e f u n c t i o n , t h e D i r a c 6 d i s t r i b u t i o n , as w e l l as i t s d e r i v a t i v e s . I n t h e p r e s e n t s e c t i o n , t h e polynomial n o n l i n e a r o p e r a t i o n s on t h e singu= l a r d i s t r i b u t i o n s mentioned w i l l be performed w i t h i n t h e q u o t i e n t 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 . The r e s u l t i n g a l t e r n a t i v e o f t h e method suggested i n [161 has t h e major advantage, among o t h e r s , t h a t a c l e a r and a l g e b r a i c a l l y simple i n s i g h t i s obtained i n t o the s t r u c t u r e o f the n o n l i = n e a r i t i e s o f a p a r t i c u l a r l y l a r g e c l a s s o f systems o f polynomial n o n l i n e a r PDts encountered i n t h e s t u d y o f p h y s i c s , a r e s u l t p r e s e n t e d under i t s general f o r m i n S e c t i o n 6. The method o f e s t a b l i s h i n g t h e j u n c t i o n c o n d i = t i o n s p r e s e n t e d w i l l a t t h e same t i m e y i e l d 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 across t h e h y p e r s u r f a c e s i n v o l v e d i n these j u n c t i o n c o n d i t i o n s . I n o r d e r t o a v o i d r a t h e r t r i v i a l t e c h n i c a l c o m p l i c a t i o n s (see S e c t i o n 9, Chapter 1 ) and a l s o t o make i t p o s s i b l e a t t h e same t i m e t o p o i n t o u t t h e e s s e n t i a l u n d e r l y i n g a l g e b r a i c phenomena, o n l y t h e case o f p o l y n o m i a l non= l i n e a r i t i e s w i l l be d e a l t w i t h , a case which o b v i o u s l y covers t h e s i t u a t i o n i n magnetohydrodynamics, as w e l l as general r e l a t i v i t y . Suppose we a r e g i v e n a system o f polynomial n o n l i n e a r PDEs (see (110), Chapter 1):

l = H, V 2,! E N \ { O } .

Proof The case ~ = il s okvious. I f R > 1, i t i s easy t o see t h a t (I-,)' : R1-+[O,l], (q)'E Cm and ( q ) a l s o s a t i s f i e s (95), t h e r e f o r e t h e problem i s reduced t o t h e case R=l. 0 The Cm-smooth r e g u l a r i z a t i o n s of regularizations

(99)

H o b t a i n e d above w i l l generate p-smooth

s = (sl,...YSa~((Cm(~))N)a

142

E.E. Rosinger

o f t h e p o t e n t i a l weak s o l u t i o n u i n (93) given by t h e r e l a t i o n (see (119), Chapter 1): (99.1)

-

sav(x) = (u-),(x)

+ ( ( U ~ ) ~ ( X )(u-)a(xj)snv(x)y

V 1 Q a = $-

E

+

C'(Q) i n ( 9 5 ) . (+t -+-)Hy

Then o b v i o u s l y

< t', *> = $ ' t

($i-$L)H.

Proposition 3 The f o l l o w i n q r e l a t i o n s h o l d :

(101)

t P(D)t' E

(m21

< tP(D)t',

So ->

P ( ~ ) + -i

= $- P(D)+L t

+-

+

bt

+

mJt +J+;- +L)P(D)H.

P(D)+L)H +

+

Proof -

It i s easy t o see t h a t f o r each v

E

N, t h e f o l l o w i n 9 r e l a t i o n h o l d s :

RESOLUTION OF SINGULARITIES

143

= stv

(SJZ

where 5 = (n)' also s a t i s f i e s ( 9 5 ) . Lemma 1 i s taken into account.

This completes the proof i f (98) in 0

Corollary 1 If P ( D ) i s a linear PDO on R w i t h continuous coefficients and order a t most one, then the following relations hold: (103)

t P(D)t'E

(104)

< t P(D)t', - > = 9- P(D)$'-

+

-

-t

So

f

($+ p(D)$;

+

3

$-

P(D)$'-) H

- $y?(D)H;

($+ + +-)($;

where Q ( D ) i s the first-order homogeneous p a r t of P ( D ) . Proof Assume P ( D ) has the form P ( D ) $ ( X I = Q(D)$(x)+ d(x)$(x) + e ( x ) , x E Q ,

where Q ( D ) i s the first-order homogeneous part of P ( D ) , while d,e Then relation (103) follows easily from (101). Further, for given v

t, d t; = $- d :$ +

($+

-

+

Co(n).

N, we have

E

= ($-

E

+

(qt - $_)Snw)d($l (+; f

-

($- d (+;

$y($; -

($+

+

-

$-Id

- $'-bnw) = $1)Srl,

+

$LXSqwY;

therefore, in view of Lemma 1,

< t d t ' , .>

d :$

= $-

+

($+ d $;

-

$- d $'-)H.

Finally, i t i s easy t o see that

< te , * >

= $-

e

+

(qt e

- $-

e)H*

The l a s t two relations together with (102) will obviously yield (104). The result on junction conditions for discontinuous solutions of systems of PDEs of type (MH) will be presented next in Theorems 11 and 1 2 . F i r s t we consider the i n f i n i t e l y smooth case, i . e . the case when the co= efficients c . and right-hand terms f inoothesystem (89) as well as y defining in 742) the hypersurface r age c -smooth. Moreover, the func= (95) used in the regularizations (96) and ( 9 9 ) will also be tions assumed C -smooth.

0

144

E. E. Rosi nger

Proposition 4

n

Ra a r e two Cm-smooth s o l u t i o n s o f t h e m-the o r d e r (MH) i n (89). Then f o r any c -smooth r e g u l a r i z a t i o n s given i n (99), t h e f o l l o w i n g r e l a t i o n s h o l d f o r e v e r y 1 < B < b: Sjppose u-,

:

u+

+

C -smoothmpolynomial n o n l i n e a r system o f PDEs o f t y p e

( 105 1

TB(D)s

( 106 1

< TB(D)s,

where

Q

BWcl

, (D)

sm

E

*

1)

(uJa1

>

=

(D)H)

QBpaal

i s t h e f i r s t - o r d e r homogeneous p a r t of P

B paa

I

(0).

Proof I n view o f (91.1) and (99), i t f o l l o w s t h a t

B u t (103) i n C o r o l l a r y 1 i m p l i e s t h e r e l a t i o n s m

SapBpaal

(D)SaI

t h e r e f o r e the l i n e a r i t y o f L TB(D)s

E

E

BP

s ; (D) w i l l g i v e

sm

as w e l l as

Now (104) i n C o r o l l a r y 1 w i l l g i v e

where Q

8P a

I

(D) i s t h e f i r s t - o r d e r homogeneous p a r t o f P

BPW

I t f o l l o w s t h a t f o r 1 Q g Q b, t h e f o l l o w i n g r e l a t i o n h o l d s :

, (D).

RESOLUTION OF SINGULARITIES

145

Since u was supposed t o be a c l a s s i c a l s o l u t i o n s o f t h e system (89), we have T (D)u

e

-

= 0 on

R,

which completes t h e p r o o f .

U

B e f o r e p r e s e n t i n g t h e r e s u l t i n Theorem 11, we need t h e f o l l o w i n g d e f i n i = tion. Given two f u n c t i o n s u-, u+ : R

+

Ra,

u-,

U+ E

U ( X ) = u-( x ) + ( ~ + ( x )- u-(x))H(x),

c

m

we d e f i n e u : R X E

+

Ra by

R,

and use t h e n o t a t i o n :

< a,

u @ c"(n)>. a 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 Cm-independent on 11, o n l y i f f o r any ha E R1, w i t h C I , t h e f o l l o w i n g i m p l i c a t i o n i s v a l i d :

I =

{all < a

I f a = l , then u-, u+ a r e t r i v i a l l y C"-independent on r, which i s why t h e above c o n d i t i o n was n o t demanded i n Theorem 1 i n S e c t i o n 1. I n terms o f t h e system o f PDEs i n (89), t h e case a = l 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 : R R ' has t o s a t i s f y b PDEs. -f

Obviously, ifu Moreover, u

E

C" t h e n u-, u+ a r e

c" o n l y i f D~(u-),(x) = D~(u,),(x),

C"-independent

on

r.

E

tl 1 G a

< a,

x

E

r,

p

E

Nn

.

Theorem 11 Suppose u , ut : R -f Ra a r e two C"-smooth s o l u t i o n s o f t h e m-th o r d e r C"'-smoothpolynom~al n o n l i n e a r system o f PDEs o f t y p e (MH) i n (89) and suppose g i v e n a C -smooth h y p e r s u r f a c e (92). Then t h e f u n c t i o n

(107)

~ ( x )= u-(x)

+ (u+(x)

-

~ - ( x ) ) H ( x ) , x E R,

146

E . E. Rosi nger

where H i s t h e Heaviside f u n c t i o n (94) a s s o c i a t e d w i t h t h e hypersurface (92), i s a weak s o l u t i o n o f t h e system (89), o n l y i f t h e f o l l o w i n g j u n c t i o n c o n d i t i o n s a r e s a t i s f i e d f o r each 1 Q B b:

where Q

BPclcl

I

( D ) i s t h e f i r s t - o r d e r homogeneous p a r t o f P

B Pas ' (D).

I n t h a t case, i f t h e f u i c t i o n s u , u+ a r e Cm-independent on p o s s i b l e t o c o n s t r u c t C - r e g u l a r T z a t i o n s ( V , S ) , such t h a t

( 109 1

ua

= s

a

where scl (110)

t

rk(v,s)E A'(v,s), E S,

with 1 Q a

Q

v

1

m .

E

N",

/PI

my

Proof I f t h e j u n c t i o n c o n d i t i o n s (108) hold, then i n view o f P r o p o s i t i o n 4, t h e f u n c t i o n (107) w i l l be a weak s o l u t i o n o f t h e system ( 8 9 ) .

Conversely, assume u i n (107) i s a weak s o l u t i o n o f t h e system ( 8 9 ) . Since we a r e i n t h e Po-smooth case, t h e o p e r a t i o n s i n t h e d e f i n i t i o n o f u can be performed w i t h i n D'(n), however t h e same does n o t h o l d f o r t h e o p e r a t i o n s on u performed by t h e PDOs T (D) , w i t h 1 Q B Q b y as these o p e r a t i o n s w i l l I (D)ff. Nevertheless, a c c o r d i n g t o (105) i n Propoi n v o l v e p r o d u c t s H.P s i t i o n 4, t h e (?-smo&%egularization s o f u, c o n s t r u c t e d i n (99) has t h e p r o p e r t y t h a t T (D)s i s weakly convergent f o r e v e r y 1 Q f 3 Q b . Therefore t h e assumption !hat u i s a weak s o l u t i o n o f t h e system (89) i m p l i e s t h a t < TB(D)s,* >

= fB , Y 1

Q B Qb.

This, i n view o f t h e r e l a t i o n s (106), completes t h e p r o o f o f t h e converse. Assuming now t h a t t h e j u n c t i o n c o n d i t i o n s (108) a r e s a t i s f i e d and t h e s o l u t i o n s u , u a r e (?-independent on r, we proceed w i t h t h e c o n s t r u c t i o n o f f ' - r e g u T a r i t a t i o n s ( V , S ) such t h a t (109) and (110) w i l l h o l d . Obviously, we can assume t h a t

(111)

u q

c?.

We use the n o t a t i o n :

RESOLUTION OF SINGULARITIES

(112 1

w

B = TB(D)s - u ( f B ) , f o r 1 G 5

Q

147

b

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

Indeed, assume V C 0 \ r such t h a t V i s an open b a l l and i t s c l o s u r e does n o t i n t e r s e c t r. Then, i n view o f t h e c o n t i n u i t y o f Y, t h e r e e x i s t s E > 0 such t h a t one and o n l y one o f t h e f o l l o w i n g two c o n d i t i o n s i s s a t i s f i e d :

(114)

~ ( xQ )

(115)

Y(X)

E,

-

> E, v

tt

X E

x

E

V ; or

v.

Assume t h a t (114) h o l d s and t a k e u E N such t h a t (Pt1)E G -1. view o f ( 9 5 ) , ( 9 6 ) and (99.1) i t f o l l o w s t h a t

(116)

U 1

sav(x) = (u&(x),

Q

a, v

E

N, v 2 1-1

,x

E

Then i n

V.

I n t h e same way i t can be shown t h a t (115) w i l l i m p l y

(117)

sav(x) = ( U + ) ~ ( X ) ,V 1 Q a

Q

a, v

E

N, v 2

u,

x

E

V.

T h e r e f o r e t h e f a c t t h a t u and ut a r e c l a s s i c a l s o l u t i o n s o f t h e system (89) w i l l r e s u l t i n t h e r e l a t i o n s

(118)

wBV(x) = 0, U 1 G 6 G b y v

E

N,

V

2 U , x E V,

c o m p l e t i n g t h e p r o o f o f (112), as r i s nowhege de se i n a, a c c o r d i n g t o (92.1). L e t us denote by 1 , t h e i d e a l i n ( C (a)) generated by n {DPwB 11 G B b, p E N 1 . Then t h e r e l a t i o n (118) w i l l o b v i o u s l y i m p l y that

a

(119)

'w

'nd.

Assume now g i v e n any i d e a l 1 i n (Cm(Q))N

( 120 1

such t h a t

1 c 1 c lnd. W

Then, as seen i n t h e p r o o f 02 Theorem 7 i n S e c t i o n 4, Chapter 3, t h e r e e x i s t v e c t o r subspaces T c S s a t i s f y i n g t h e c o n d i t i o n s

(121)

i n T = v"n T = Q;

( I22 1

V"

(123)

(Ip"

(1 n S")

t

a

= V"

@

T)n U

P ( Q =)

T;

and

Q-

B u t assumption (111) means t h a t

31 G

a : u

a

7 c"(n).

Thus r e a s o n i n g s i m i l a r t o t h a t used i n t h e p r o o f o f (36) w i l l y i e l d

b' l G a Q a :

148

E . E . Rosinger

However, we s h a l l need and prome t h e f o l l o w i n g s t r o n g e r r e s u l t . by S t h e v e c t o r subspace i n S generated by Esa 1 < a d a, ua then'

I

su)n(f @

(124)

( ~ ~ m t( ~ )

7

Denote C"(R)};

= O_

Indeed assuming t h a t o u r s u p p o s i t i o n i s f a l s e and t h a t

where J, E Cm(n), I = {a 11 < IY. < a, uCI $! Cm(n } , X, E R' , v E Vm and t E T , t h e n i n view o f ( l o o ) , (120) and (120 , as w e l l as (69), Chapter 1, t h e f o l l o w i n g r e l a t i o n i n D'(n) i s obtained:

c

J, t

X, ua

= < t, - >

, supp



nowhere dense.

a €1

But u a €

Cm(n \ r ) , V 1 < supp

since R

\r

< t, - > c

i s open, as

r

a, as a consequence o f (107).

Therefore

r i s closed.

Then

X,ua = O o n R \ r €1 NOW, s i n c e r has zero Lebesque measure (see ( 9 2 . 1 ) ) , t h e r e l a t i o n (126) w i l l give i n U ' ( Q ) the e q u a l i t y (126)

C

J,t

CI

t h e r e f o r e (125) w i l l i m p l y v + t

E

v",

which i n view o f (121) w i l l r e s u l t i n (128)

t = u(0) E Q,

But t h e r e l a t i o n (127) which i s v a l i d i n U'(Q) o b v i o u s l y i m p l i e s XCIuCI=-J,onR

Z CIE

s i n c e J, EC"(R),

I

Xa

C CIE

uCIE Cm(R \

I

r)

and, i n view o f (92.1),

r is

nowhere dense i n R . Thus i t f o lows t h a t

s i n c e u-, g i ve

J, = 0 on R and

Xa

u+ were assumed

C

v = u(0) E Q,

= 0, V C I E m

-

I

ndependent on

r.

Now

125) and (128) w i l l

RESOLUTION OF SINGULARITIES

149

which completes the proof of ( 1 2 4 ) . In view of (124) there e x i s t vector subspaces s c s" such t h a t (129)

UC"(S2)

(130)

S"

=

+

c s;

v" @ S @

T.

Then (121), ( 1 2 2 ) , (130) and (129) will , according t o Theorem 4, Section 4, Chapter 3, imply t h a t (131)

0 T)

s

(v,

i s a c"-regularization

for any vector subspace v c I nV'. B u t by definition {DPwB [ I d6 d b y P E Nn 1 c I, c I ; and moreover, the junction condition (108) means t h a t

w

E P ,V l d B 6 b ;

B therefore i t i s obviously possible to choose

(132)

I

{DPwB


. a Then the functions u-, u+ will be called (?-independent on r , only i f f o r any Xa E R *, w i t h a E J , the following implication i s valid: (

Xaua€ P Q ) ) -(Acl=

C

0, +'a E J ) .

clEJ

Obviously, u-, u+ are (?-independent on Tin case a = l or u over u E C "only i f u-(x)

=

u+(x),

v

x

Er.

E P.

More=

150

E . E . Rosinger

Theorem 12 ose u , u : R -+ Ra a r e two ckl-smooth s o l u t i o n s o f t h e m-th o r d e r $ @ % n o o t l i pofynomial n o n l i n e a r k s y s t e m o f PDEs o f t y p e (MH) i n (89) and kl m. Suppose a l s o q i v e n a c '-smooth hypersurface (92). Then t h e function (133)

+ (u,(x)-u-(x))H(x),

U ( X ) = u-(x)

x

E Q,

where H i s t h e H e a v i s i d e f u n c t i o n (94) a s s o c i a t e d w i t h t h e hypersurface (92), i s a weak s o l u t i o n o f t h e system (89), o n l y i f t h e j u n c t i o n condi= t i o n s (108) a r e s a t i s f i e d f o r each 1 Q B G b . I n t h a t case, i f t h e f u n c t i o n s u , u+ a r e C"-independent on p o s s i b l e t o c o n s t r u c t C " - r e q u l a r i z a t i o n s ( V , S ) such t h a t :

r,

it is

(134)

IR(V,S) EAR(U,S), tl 1 G a G a y R E N, & Q k l where s a E S, w i t h 1 Q a G a y do n o t depend on R;

(135)

u s a t i s f i e s each o f t h e PDEs o f t h e system (8i3) i n t h e usual a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n i n A ( V , S ) a n d W p a r t i a l d e r i v a t i v e operators

uQ

=

sat

DP:AR(UyS) -Ak(V,S), p E Nny Ip I Q m , w i t h R L-k > m y R Qkl and k G m i n I kl-rn,k21.

,k

E

N,

Proof The p r o o f t h a t u i s a weak s o l u t i o n o n l y i f (108) holds, i s s i m i l a r t o t h e p r o o f o f Theorem 11, To prove t h e second p a r t o f t h e theorem we use t h e n o t a t i o n h = m i n ( kl-my

k2].

Obviously, we can assume t h a t (136)

u q

c

I t can e a s i l y be seen t h a t t h e sequence s c o n s t r u c t e d i n (99) s a t i s f i e s

(137)

s

€skl,<s Q

w h i l e t h e sequences w (138)

W

B

B

= TB(D)S

ay

. > = u

a

,

V

I

Q

~

Q

~

c o n s t r u c t e d i n (112) s a t i s f y

-

U(fS)

E

Ind nvh

v

1 "6

Qb.

Therefore (139)

{DPwB 11 Q B Qb, p

ENn,

Assume t h e n g i v e n any i d e a l I i n (2'

( 140)

{DPwB 11 Q B Qb, p

ENny

I p I Q h ) C Ind n Lf'. N such t h a t

(n))

IpI Qh}c I

c Ind

and I i s 81-smooth (see (126), Chapter 3) which i s p o s s i b l e i n view o f

RESOLUTION OF SINGULARITIES

151

(139) and P r o p o s i t i o n 6, S e c t i o n 6, Chapter 3 . I n t h i s case, t h e argument i n t h e p r o o f o f Theorem 14, S e c t i o n 6, ChapteL 3, a p p l i e d f o r L = kl w i l l y i e l d t h e e x i s t e n c e o f v e c t o r subspaces T c s which s a t i s f y (141)

l n T = V " n T = q ;

(142)

v"

(143)

(f @

( r n s o ) = v"

t

T) n

@ T; and

(n)

UCo

=

0.

I f we denote by Su t h e v e c t o r subspace i n s" generated by Isc, 1 1 c, G a, % 9 C" (Q)}, t h e n s i n c e u , us were assumed C" -independent on r, t h e argument used t o o b t a i n (124) in-the p r o o f o f Theorem 11, w i l l now y i e l d

(144)

Wc"(q

t SUP

(f'

0 T ) = 0..

Now, i n view o f (144), t h e r e e x i s t v e c t o r subspaces R' c So such t h a t

f'

So =

0 (UC" (Q) + Su) 0 R '

@

T.

Therefore, t a k i n g i n Lemma 5, S e c t i o n 6, Chapter 3

F=Skl

E=S',

9

+ Su) @T

A = f' @ ( U c o ( n )

,B

=

R',

i t f o l l o w s t h a t t h e r e e x i s t v e c t o r subspaces

C=

CF=Skl

R

which s a t i s f y

W c . (a)+ S u )

(Lp

@ T) n R

=

0,

and (145)

So =

v"

a(Ue(Q) + Su)

0R0

T.

Introducing the notation

s = (UC (Q1 + SU) @ R (146) t h e r e l a t i o n s (141), (142) and (145) w i l l i m p l y t h a t (147)

(v,

s

@T)

i s a e-regularization

f o r any v e c t o r subspace V Chapter 3.

C

1 n V'

, according

t o Theorem 10, S e c t i o n 5,

B u t i n view o f (139), i t i s p o s s i b l e t o choose V such t h a t (148)

we

E

Vh,

V 1 G 6 G b.

Now t h e r e l a t i o n s (146), (138) and (148) w i l l complete t h e p r o o f , i f we t a k e i n t o account t h e v e r s i o n s o f Theorems 2,3 and 9, Chapter 3, correspon=

E . E . Rosinger

152

d i n g t o C" - r e g u l a r i z a t i o n s . 6.

Resoluble Systems o f Polynomial N o n l i n e a r PDEs

The necessary and s u f f i c i e n t j u n c t i o n c o n d i t i o n s across hypersurfaces o f d i s c o n t i n u i t i e s o f weak s o l u t i o n s f o r systems o f PDEs o f type (MH) and the r e s o l u t i o n o f t h e corresponding s i n g u l a r i t i e s presented i n t h e pre= vious s e c t i o n a r e extended here t o a l a r g e c l a s s o f systems o f PDEs which c o n t a i n s many - i f n o t most - o f t h e equations m o d e l l i n g p h y s i c a l pheno= mena. The r e s u l t o b t a i n e d can be i n t e r p r e t e d as an upper bound on t h e degree o f n o n l i n e a r i t y t h a t may be expected i n most o f t h e equations o f physics. I t i s , p a r t i c u l a r l y i n t e r e s t i n g t h a t t h e r e s u l t i n g c l a s s o f systems o f PDEs - which we s h a l l h e r e a f t e r r e f e r t o as r e s o l u b l e - i s c h a r a c t e r i z e d by i t s s p e c i a l behaviour i n r e g a r d t o weak s o l u t i o n s w i t h d i s c o n t i n u i t i e s across hypersurfaces. More p r e c i s e l y , t h e upper bound on t h e degree o r c o m p l e x i t y o f n o n l i n e a r i t y , p r e v i o u s l y mentioned, i s i n f a c t a s u f f i c i e n t c o n d i t i o n f o r o b t a i n i n g simple, a l g e b r a i c j u n c t i o n c o n d i t i o n s c h a r a c t e r i z i n g weak s o l u t i o n s w i t h d i s c o n t i n u i t i e s across hypersurfaces. The s i g n i f i c a n t advantage o f d e a l i n g w i t h t h i s problem w i t h i n t h e frame= work o f t h e q u o t i e n t algebras 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 soon becomes apparent. I n f a c t , one o f t h e m a j o r advantages o f o u r method o f d e a l i n g w i t h n o n l i n e a r PDEs w i t h i n t h e framework o f t h e q u o t i e n t algebras i s t h e i d e n t i f i c a t i o n o f t h e c l a s s o f r e s o l u b l e systems o f PDEs. Definition 1 The system o f polynomial n o n l i n e a r PDEs i n (89) i s c a l l e d r e s o l u b l e , o n l y i f each o f t h e a s s o c i a t e d PDOs i n 90) can be w r i t t e n i n t h e form TB(D)u(x) =

(149)

C l < p < r

B

whenever (149.1)

u(x) = $(x) t x(x).o(x),

where I ), x : Q and T a r e m' BP

x E a,

R a y w : n + R', $, X, w E Cm E Nn, R , P P - t h o r d e r polynomial n o n l i n e a r PD8s i n $ an!'X.

+

BP

The p a i r ( m u , ml'), where

m' = m a x {m'

BP

(1989b, lGp

Proof I t f o l l o w s e a s i l y from (71-73), Chapter 4.

NOW, i n o r d e r t o p r e s e n t t h e r e s u l t on t h e maximal s t a b i l i t y o f S-sequential s o l u t i o n s o f t h e PDE i n ( 6 ) , i t i s convenient t o i n t r o d u c e t h e f o l l o w i n g d e f in i t i on. Definition 2

A sequence o f f u n c t i o n s s E ( C ~ ( R ) )i ~s c a l l e d a maximally s t a b l e S-sequen= t i a l s o l u t i o n o f t h e PDE i n ( 6 ) , o n l y i f t h e r e e x i s t s a c m - r e g u l a r i z a t i o n (V,s),w i t h V maximal, and s a t i s f y i n g ( 9 ) and ( 1 0 ) . Theorem 1 Any s - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 ) i s maximally s t a b l e . Proof t h e s e t o f a l l t h e v e c t o r subspaces V C Vm f o r which (V,s) Denotemby RV i s a C - r e g u f a r i z a t i o n . We s h a l l show t h a t R Vs i s c b a i n complete. Indeed, i n view o f (21.1), Chapter 3, a v e c t o r subspace U CmV belongs t o RVS , o n l y i f I( V ) n S = Q , where I( V ) i s t h e i d e a l i n ( C (f2))N g e n y a t e d b y V. But, i t i s easy t o see t h a t I( V ) i s t h e v e c t o r subspace i n ( C (f2))N generated b y V. (Cm(n))N. Therefore, RVS w i l l indeed be c h a i n complete. NOW, i n view o f ( 9 ) and ( l o ) , as w e l l as Z o r n ' s Lemma, t h e p r o o f i s com= pleted.

0

Remark 1 R e s u l t s on t h e s t r u c t u r e o f t h e maximal v e c t o r subspaces V w h i c h d e f i n e t h e m a x i m a l l y s t a b l e s e q u e n t i a l s o l u T E 6 - 3 n d whose e x i s t e n c e i s g r a n t e d by Theo= rem 1, w i l l be presented i n t h e n e x t chapter.

STABILITY AND EXACTNESS

3.

167

I n t e r p l a y between S t a b i l i t y and Exactness

As seen i n Sections 4 and 5, Chapter 1, t h e exactness p r o p e r t y o f sequen= t i a l , weak o r d i s t r i b u t i o n s o l u t i o n s f o r polynomial n o n l i n e a r PDEs c o n s i = dered w i t h i n t h e framework o f t h e c h a i n s o f q u o t i e n t a l g e b r a s ( 2 4 ) , Chapter 3, o r ( 1 ) above, i s d i r e c t l y expressed by t h e size o f the ideals

(13)

ZR(V,S) , R

E

R

E

h'

and algebras (14)

A'(Y,S)

,R

t h a t i s , b e t t e r exactness means s m a l l e r i d e a l s ( 1 3 ) and l a r g e r a l g e b r a s ( 1 4 ) . I n e x t r e m i s , maximal exactness means minimal i d e a l s ( 1 3 ) and maximal a l g e = bras (14). NOW, i n view o f (17-20), Chapter 3, where t h e i d e a l s ( 1 3 ) and a l g e b r a s ( 1 4 ) were d e f i n e d , i t i s easy t o see t h a t t h e i d e a l s (13) decrease i n t h e i r s i z e t o g e t h e r w i t h V , w h i l e i n a c o n f l i c t i n manner, t h e a l g e b r a s ( 1 4 ) i n c r e a s e t h e i r size together w i t h V . a t concerns S , i t was seen i n Lemma 1, Sec= t i o n 1, t h a t n e i t h e r i n c r e a s e n o r decrease i s p o s s i b l e .

+

More p r e c i s e l y , suppose t h e PDO i n ( 7 ) i s considered t o a c t as f o l l o w s (see p c t . 2 and 4, Remark 3, S e c t i o n 3, Chapter 3): T(D) : AR(V,S)

+

Ak(V,S)

w i t h 2 , k E 4 , k < R , R-k > m. Then, maximal s t a b i l i t y f o r t h e s e q u e n t i a l , w&ak o r d i s t r i b u t i o n s o l u t i o n s o f t h e m)E i n (6)maximal ideals 7 ( V , S ) , t h a t i s , maximal V . On t h e p t h e r s i d e , maxima7 exactness f t h e same s o l u t i o n s m e a m m a 1 i d e a l s Z ( V , S ) , and maximal a l g e b r a s A ( V , S ) , t h a t i s , i n t h e same t i m e minimal and maximal V .

R

The way t h e above c o n f l i c t i s s e t t l e d r e q u i r e s i n each p a r t i c u l a r case o f s e q u e n t i a l , weak o r d i s t r i b u t i o n s o l u t i o n considered f o r t h e PDE i n ( 6 ) , a s p e c i f i c a n a l y s i s o f t h e s t a b i l i t y and exactness i n t e r e s t s i n v o l v e d . 4.

Remarks i n t h e F i n i t e l y Smooth Case k2 Suppose, i h e PDE i n ( 6 ) i n C -smooth, f o r a c e r t a i n g i v e n k2 E N, t h a t i s , c., f E C 2(Q). We s h a l l be i n t e r e s t e d i n t h e s t a b i l i t y o f s e q u e n t i a l s a l u t i o n o f t h e in-order PDE i n ( 6 ) , g i v e n b y sequences o f f u n c t i o n s s €(Ck1(Q))N, w i t h k l E N, kl > m. The framework w i l l be g i v e n t h i s t i m e by t h e chains o f q u o t i e n t algebras ( 9 3 ) , Chapter 3. More p r e c i s e l y , we s h a l l c o n s i d e r t h a t t h e PDO i n ( 7 ) a c t s as f o l l o w s (see Remark 5, S e c t i o n 5, Chapter 3): T(D) : A'( V,S) * Ak( V , S )

where ( V , S ) i s a C " - r e g u l a r i z a t i o n , k Q min {kl-m,k23.

w h i l e R, k

E

N, R-k > m y "kl,

I t i s easy t o see t h a t t h e remarks i n S e c t i o n 1, remain v a l i d , under t h e i r corresponding form, f o r C" - r e g u l a r i z a t i o n s and t h e c h a i n s o f q u o t i e n t a l = gebras ( 9 3 ) , Chapter 3, w h i c h t h e y generate.

168

E.E.

Rosinger

Therefore, given a v e c t o r subspace S c So which s a t i s f i e s t h e c o n d i t i o n s

so

(15)

v"

=

@S

Uco(n)

y

cs

kl N we c a l l a sequence o f f u n c t i o n s s E ( C (a)) an S - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 ) , o n l y i f t h e r e e x i s t s a C " - r e g u l a r i z a t i o n ( V , S ) such that

, tf

(16)

s

(17)

w = T(D)s-u(f)

E

A'(v,s)

R E N, R G kl E

k I (V,S), V k

E

a-k

N,

> m,k < min{kl-m,k2}

I n t h a t case, one o b v i o u s l y o b t a i n s t h a t S = s

+ I

R

( V , S ) E A ( V , S ) , U R E N, R

< kl

and S s a t i s f i e s t h e PDE i n ( 6 ) , i n t h e usual a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n i n Ak( V , S ) and 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

DP : A ~ ( v , s ) f o r R, k

E

N, R-k

+

> my a

A ~ ( v , s ) , ~E Nn , I P I

Q

kl,

Q

my

k G min{kl-m,k21

k Moreover, ifa d i s t r i b u t i o n S E P ' ( R ) \ C " ( R ) i s a C ' - r e g u l a r weak s a l u t i o n o f t h e PDE i n ( 6 ) , then, t h e r e e x i s t s a sequence o f f u n c t i o n s s E S and and a v e c t o r subspace S c So s a t i s f y i n g (15), such t h a t S = < s, s i s an S - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 ) . kl N Now, c a l l i n g a sequence o f f u n c t i o n s s E ( C (n)) a maximally s t a b l e S - s e q u e n t i a l s o l u t i o n o f t h e PDE i n ( 6 ) , o n l y i f t h e r e l a t i o n s (16) and ( 1 7 ) h o l d f o r a c e r t a i n C " - r e g u l a r i z a t i o n ( V , S ) , w i t h V maximal, one can e a s i l y o b t a i n t h e corresponding v e r s i o n o f Theorem 1. F i n a l l y , i t i s easy t o see t h a t t h e remarks i n S e c t i o n 3 can a l s o be adap= t e d t o t h e case o f f i n i t e smoothness. 5.

S t a b i l i t y o f Regular Weak S o l u t i o n s

Two general classes o f weak s o l u t i o n s f o r polynomial n o n l i n e a r PDEs were t i r s t , i n S e c t i o n 4, Chapter 4, t h e r e g u l a r weak s t u d i e d i n Chapter 4 s o l u t i o n s f o r a r b i t r a r y polynomial n o n l i n e a r PDEs were i n t r o d u c e d as an e x t e n s i o n o f t h e p i e c e w i s e smooth weak s o l u t i o n s o f s i m p l e polynomial n o n l i n e a r PDEs s t u d i e d i n S e c t i o n s 1-3, Chapter 4. Then, i n S e c t i o n 6, Chapter 4, t h e weak s o l u t i o n s w i t h d i s c o n t i n u i t i e s across hypersurfaces were s t u d i e d i n t h e case of r e s o l u b l e systems o f polynomial n o n l i n e a r PDEs, systems e x t e n d i n g t h e t y p e o f e q u a t i o n s o f magnetohydrodynamics and general r e l a t i v i t y , s t u d i e d i n S e c t i o n 5, Chapter 4.

.

The r e s u l t s on t h e mentioned weak s o l u t i o n s presented i n Chapter 4, con= cerned t h e r e s o l u t i o n o f t h e i r s i n g u l a r i t i e s and were o b t a i n e d b y construe= t i n g a r t i c u l a r i n s t a n c e s o f t h e chains o f q u o t i e n t algebras (24) o r (93), C h a p t b 3 T 5 T u c h a way t h a t t h e weak s o l u t i o n s s a t i s f i e d t h e polynomial n o n l i n e a r PDEs i n t h e usual a l g e b r a i c sense, w i t h t h e m u l t i p l i c a t i o n and 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 i n t h e c h a i n s o f q u o t i e n t algebras.

STABILITY AND EXACTNESS

169

The aim o f t h i s s e c t i o n i s t o p r e s e n t a b a s i c s t a b i l i t y r e s u l t c o n c e r n i n g t h e r e g u l a r weak s o l u t i o n s f o r a r b i t r a r y p o l y n o m i a l n o n l i n e a r PDEs by con= s t r u c t i n g l a r g e classes o f c h a i n s o f q u o t i e n t a l g e b r a s ( 2 4 ) o r ( 9 3 ) , Chap= t e r 3, e x t e n d i n g those p a r t i c u l a r ones used i n Chapter 4. I n t h i s way, t h e r e s u l t s c o n c e r n i n g 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 p r e s e n t e d i n Chapter 4 w i l l a l s o be improved. We s h a l l o n l y deal w i t h t h e case o f i n f i n i t e smoothness. The r e s u l t s o b t a i n e d can e a s i l y be r e f o r m u l a t e d i n t h e f i n i t e l y smooth case.

A s i m p l e and u s e f u l t o o l f o r e s t a b l i s h i n g t h e mentioned s t a b i l i t y p r o p e r t y o f t h e r e g u l a r weak s o l u t i o n s i s t h e s u f f i c i e n t c o n d i t i o n on t h e s e s o l u = t i o n s p r e s e n t e d i n Tbeorem 9, S e c t i o n 4, Chapter 4, as w e l l as t h e s u f f i = c i e n t c o n d i t i o n on C , R - r e g u l a r i d e a l s presented i n Theorem 20, S e c t i o n 10, Chapter 3. Supposg t h e PDE i n ( 6 ) i s Cm-smooth and t h e d i s t r i b u t i o n S E U'(Q)\C"(Q) i s a C - r e g u l a r weak s o l u t i o n f o r t h e PDE i n ( 6 ) which admits a sequence S" s a t i s f y i n g t h e c o n d i t i o n s o f functions s

(18)

s

=

(19)

w

=

(20)

lW i s a v a n i s h i n g i d e a l i n (C"(Q))

(21)

(v"

< 5, ' >

T(D)s

-

u(f)

E

t zw)n(ucm(n)

V"

+ R.S) 1

=

N

o_

Chapter 4 )

where (see ( 7 3 . 2 ) ,

lw = t h e i d e a l i n (C"(Q))N generated by IDP,

(22)

Ip E Nnl

Then, we denote by iDs t h e s e t o f a l l v a n i s h i n g i d e a l s 7 i n (Cm(Q))N, such t h a t

(23)

lw c 7

(24)

(vm +

i)n(ucm(n) + R.S) 1 =

2

Theorem 2 Under t h e above c o n d i t i o n s , t h e f o l l o w i n g p r o p e r t i e s h o l d :

1)

The s e t of i d e a l s i D s i s c h a i n complete.

2)

Each i d e a l 1 E i D s i s Cm, R - r e g u l a r , where R =

3)

F o r each i d e a l I E i D s t h e r e e x i s t C m - r e g u l a r i z a t i o n s

(25)

(Invm>@)

satisfying the conditions

1

R s.

170

E . E . Rosinger

4)

Each Cm-regula r i z a t i o n (V,S')win (25) y i e l d s t h e following resolu= t i o n of s i n g u l a r i t i e s f o r t h e C -regular weak solution S of t h e PDE in ( 6 ) :

4.1)

S = s + l R ( V , S ' ) E AR(V,S'), \d R E 1, where s i s given i n (18), s E S ' and does

4.2)

S satisfies t h

not depend

on R

PDE in ( 6 ) in t h e usual a l g e b r a i c sense, with multi= p l i c a t i o n in A ( V , S ' ) and the p a r t i a l d e r i v a t i v e operators

Dp : A"(V,S')

R

+

Ak(V,S'), p

E

N n y IpI Q m , w i t h II, k E

R,

R-k am

Proof 1)

I t follows e a s i l y from (23) and ( 2 4 )

2 ) and 3 ) follow from the s u f f i c i e n t condition on Coo, R-regular i d e a l s presented in Theorem 20, Section 10, Chapter 3.

4)

I t follows from 2 ) and 3 ) i f account i s taken of Theorem 7 , Section 4, Chapter 4.

Remark 2

+

From the point of view of t h e s t a b i l i t of the (?-regular weak solution S , Theorem 2 above presents a specia i n t e r e s t . Indeed, the s e t of ideals iD, being chain complete, t h e application of Zorn's lemma w i l l y i e l d t h e fol 1 owing property \d I E

iDs :

-

1 E iD, maximal:

ic 1 Then, t h e cha ns of quotient algebras

see ( 2 4 ) , Chapter 3 )

AR

associated t o l a r e gr maximal i d e a l s E iDS will o f f e r b e t t e r s t a b i l i t y s o l u t i o n S. properties f o r t e C -rG@TFweak

-8

6.

S t a b i l i t y of Solutions with Jump Discontinuities f o r Resoluble Systems of PDE S

Suppose t h e m-th order Coo-smooth polynomial nonlinear system of PDEs in (89) Chapter 4 , i s resoluble ( s e e (149),Chapter 4 ) .

STABILITY AND EXACTNESS Suppose u ,ut : R system of-PDts.

Ra a r e two

+

cm-smooth s o l u t i o n s o f t h e mentioned

F i n a l l y , suppose t h a t t h e f u n c t i o n u : R

+

U ( X ) = u-(x)

(31)

171

-

(u+(x)

+

Ra d e f i n e d b y

u-( x ) ) H ( x ) , x E 0,

where H i s t h e H e a v i s i d e f u n c t i o n ( 9 4 ) , Chapter 4, a s s o c i a t e d w i t h t h e C -smooth h y p e r s u r f a c e r i n ( 9 2 ) , Chapter 4, i s a weak s o l u t i o n o f t h e mentioned system o f PDEs, t h e r e f o r e s a t i s f y i n g on r t h e necessary and s u f = f i c i e n t j u n c t i o n c o n d i t i o n s (108) o r (151), Chapter 4. We s h a l l a l s o assume t h a t u- and u+ a r e em-independent on 5, Chapter 4 ) .

r

(see S e c t i o n

Then s = ( S ~ , . . . , S , ) E ( ~ " ) ~ c o n s t r u c t e d i n ( 9 9 ) , Chapter 4, has t h e proper= tY

<sa

(32)

ua =

(33)

wB = TB(D)s

,a>

-

, V 1< a < u(fB)

a

v",

E

V 1< B


u:

supp sv c 0' Theorem 4 Suppose t h e ( ? - r e g u l a r i z a t i o n i n v a r i a n t . Then

( V , S ) i s o f l o c a l t y p e and V @S

......3 X

Z ( v ) n (Cp,p+l, f o r any v

E

U , 1-1

i s cofinal

n') i s i n f i n i t e

N and R' c R non-void, open.

E

Proof Assume v

V, p

E

E

N and R' c R n o n - v o i d open, such t h a t

# 0, Y v

v,(x)

E

N,

v 2 p, x

E

R'

Obviously, we can a l s o assume t h a t (26.2) h o l d s f o r t h e above p and R ' . D e f i n e t h e n w E (C'(R))N by if x

0

9 R'

wV ( x ) = sv(x)/vv(x) whenever v

E

vV.wv =

(27)

>u . sV , V v

N, v

if x E

R'

It follows t h a t E

N, v 2 p

therefore (28)

v.w

s i n c e s E V @S But v (29)

E

E V

@

and V

@

s S i s cofinal invariant.

V w i l l imply v.w

E

l(V)

and (28), (29) above t o g e t h e r w i t h (21.1) i n Chapter 3, w i l l y i e l d v.w which i n view o f (27) w i l l c o n t r a d i c t (26.1).

E

V, 0

184

E.E.

Rosinger

Remark 1 The c o n d i t i o n s i n Theorem 4 a r e s a t i s f i e d i n t h e case o f i m p o r t a n t classes o f C"-regularizations. Indeed: a ) I t i s easy t o see t h a t U @ S w i l l be c o f i n a l i n v a r i a n t whenever U has t h a t p r o p e r t y . What concerns s e c u r i n q t h a t l a t t e r s i t u a t i o n , i t i s o b v i o u s l y s u f f i c i e n t t o c o n s i d e r c o f i n a l i n v a r i a n t C" - r e q u l a r i d e a l s I and use t h e procedure i n Theorem 10, S e c t i o n 5, Chapter 3, f o r construe= t i n q V which i n t h i s case can be assumed c o f i n a l i n v a r i a n t . b ) I n view of the c o n d i t i o n (90.2) i n t h e d e f i n i t i o n o f C " - r e q u l a r i z a t i o n s , i t fol lows t h a t

v x En: 3 s

ua s:

E

= 6,

<s,*>

(the Dirac 6 d i s t r i b u t i o n concentrated a t x)

t h e r e f o r e , t h e c o n d i t i o n ( 2 6 ) w i l l be s a t i s f i e d i n case

tfx

En:

I s E V @ S : * ) < s,*>

-

- &x

supp sv s h r i n k s t o {XI, when v

**)

-+ m

However, t h e c o n d i t i o n s i n Theorem 4 can be relaxed, as seen i n t h e n e x t two theorems. Theorem 5 Suppose g i v e n a C " - r e g u l a r i z a t i o n (U,S) and U Then Z(V) n f o r any v

E

u

u p . Now a r e a s o n i n g s i m i l a r t o t h e one i n t h e second 0 p a r t o f t h e p r o o f o f Theorem 4, w i l l l e a d t o a c o n t r a d i c t i o n . I n t h e same way one can a l s o prove: Theorem 6 Then

Suppose ( V , S ) i s a C " - r e q u l a r i z a t i o n .

u

Z(v) n

({VI

vEN f o r any v

5.

E

V and s E V @

x

suop s v ) P 0

D'(n).

S,< s,*> # 0 E

C o n s t r u c t i o n and C h a r a c t e r i z a t i o n o f a Class o f V a n i s h i n q I d e a l s

The s t a n d a r d r e s u l t on z - f i l t e r s a s s o c i a t e d t o C" - r e g u l a r i z a t i o n s p r e s e n t e d i n Theorem 1, S e c t i o n 1, has been improved i n two d i r e c t i o n s : f i r s t , i n S e c t i o n 2 , by t h e d e n s i t y c h a r a c t e r i z a t i o n o f t h e p r o j e c t i o n on R o f t h e z - f i l t e r s , and then i n S e c t i o n 4, where l o w e r bounds on t h e s i z e o f t h e elements i n t h e z - f i l t e r s have been g i v e n i n Theorems 4,5 and 6. The aim o f t h i s s e c t i o n i s t o p r e s e n t a c o n s t r u c t i o n and c h a r a c t e r i z a t i o n o f t h e c l a s s o f E-vanishing i d e a l s i n ( C " ( I 2 ) ) N d e f i n e d below. The t o o l used i n t h i s c o n n e c t i o n i s a p r o p e r t y o f reduced p r o d u c t s o f c o u n t a b l e f a = m i l i e s o f v e c t o r spaces, p r e s e n t e d i n t h e n e x t s e c t i o n . The i n t e r e s t o f t h e men i o n e d c o n s t r u c t i o n i s t h a t i t s t a r t s f r o m v e c t o r subs aces i n s a t i s f y i n g the E-vanishing condition ( s e e T 3 e w i t h (C" (39) ,Chapter 3 ) . Given a sequence o f f u n c t i o n s w E ( C " ( R ) ) ZX(w) = Cv E

N Iw,(x)

N and x

E

R, denote

= 01

and c a l l i t t h e zero s e t a t x o f w. Obviously Zx(w) = p r (Z(w) "(N N

x(x1))

and p r Z(w) =

n

Cx

E

R IZx(w) # 81

Given a s e t o f sequences o f f u n c t i o n s H Zx(H) =

CZ,(w)I

and c a l l i t t h e zero f a m i l y a t x

w

E

HI

o f H.

I f E i s a s e t o f subsets i n R, denote

C

( e ( a ) )N

denote

E.E.

186

Rosinger

V E E

WE =

{w E (CO (R))

E:

3 x E E : I

Examples o f h e r e d i t a r y s e t s o f subsets i n R are t h e s e t Ef o f non-void and f i n i t e subsets i n R , as w e l l as t h e s e t E, o f non-void and c o u n t a b l e subsets i n fi

.

The main n o t i o n i n t h i s s e c t i o n i s presented i n t h e f o l l o w i n g d e f i n i t i o n . A v e c t o r subspace V i n ( C D ( R ) ) N i s c a l l e d E-vanishing, o n l y i f

The s e t E o f subsets i n R i s c a l l e d dense i n R

only i f

It i s easy t o see t h a t Ef and Ec a r e dense i n R, and i n general, E w i l l be dense i n R, o n l y i f

V R ' c R non-void, open : (32)

3 E E

E:

E c Rl The i m p o r t a n t p r o p e r t y connected w i t h E-vanishing, w i t h E dense i n R, i s presented now. Proposition 4

I f 7 i s a c o f i n a l i n v a r i a n t E-vanishing i d e a l and E i s dense i n n, then 7 i s a (?-regular i d e a l . Proof I t i s a d i r e c t consequence o f (30), Chapter 3.

(31) above and Theorem 11, S e c t i o n 5,

The c o n s t r u c t i o n o f E-vanishing i d e a l s i s based on t h e p r o p e r t y o f i s h i n g v e c t o r subspaces presented n e x t .

0

E-van=

Proposition 5 N Suppose V i s an E - v a n i s h i n g v e c t o r subspace i n ( e ( Q ) )f o r a c e r t a i n E C Ec. Then, f o r any vl,.. . ,vh E V , t h e r e l a t i o n h o l d s

STRUCTURE OF THE ALGEBRAS

V E

(33)

187

E E :

3 x E E :

...

zx(vl)

zx(vh) # 0

I f E i s h e r e d i t a r y then t h e s t r o n g e r r e l a t i o n h o l d s

(3

1

V E

EE:

3 F

CE:

*) car F **) V

X E

=

car E

F

:

Proof The r e l a t i o n (33) f o l l o w s e a s i l y from C o r o l l a r y 1 i n S e c t i o n 6 . Assume now t h a t E i s h e r e d i a t r y and t a k e E

Then ( 3 3 ) i m p l i e s t h a t

E

E.

F

= E = ixl)

3 x 1 E E :

.

Denote then El = E \ I x l )

I f El = 0 then

completed. Otherwise El E E , s i n c e E i s h e r e d i t a r y . be a p p l i e d t o El and i t f o l l o w s t h a t 3 x 2 E El

Denoting E2 = E l \Cx21 t a k e F = E = Cx1,x21 repeated.

.

T h e r e f o r e (33) can

:

,in

case E2 = 0 t h e p r o o f i s completed as one can Otherwise a g a i n E 2 E E and t h e procedure can be

I n case E i s f i n i t e , one w i l l end up w i t h F = E .

F

and t h e p r o o f i s

Otherwise one o b t a i n s

= { x ~ ~ x ~ 1~ C, . E .

which w i l l a l s o s a t i s f y * ) i n (34), s i n c e E i s countable.

0

The r e s u l t i n P r o p o s i t i o n 5 l e a d s t o t h e f o l l o w i n g method f o r c o n s t r u c t i n g E - v a n i s h i n g , c" - r e g u l a r i d e a l s . Theorem 7

I f U i s an E - v a n i s h i n g vectorsrbspace i n (c" (Q))N, w i t h E (see (89.1),

C

Chapter 3) i s an E - v a n i s h i n g i d e a l i n ( c " ( Q ) ) N .

Ec, then I ( V )

188

E.E. Rosinger

I f i n ' a d d i t i o n V i s c o f i n a l i n v a r i a n t and E i s dense i n C" - r e g u l a r i d e a l .

n, t h e n

7(V) i s a

Proof

N N Obviously l ( V ) i s t h e v e c t o r subspace i n ( C " ( R ) ) generated b y V.(C"(n)) , t h e r e f o r e any w E I ( V ) can be w r i t t e n under t h e f o r m (35)

c

w =

1 G i p,

Then o b v i o u s l y

N.

Zx(wl) n { p , p + l

(52)

F i r s t we show t h a t

,....I

= ZX(w) n ~p,p+l,,...I,

V x E E

J implies that Zx(w) E Z x ( I ) ,

V x

E

E,

t h e r e f o r e , i n view o f (52), we have Zx(w') i n other

words, w '

E

E

fix, 4 x

E

E

J.

Since t h e i n c l u s i o n 7 C J

f o l l o w s e a s i l y from (50) i n P r o p o s i t i o n 6, t h e p r o o f i s completed.

0

F i n a l l y , a c h a r a c t e r i z a t i o n o f E-vanishing subsequence i n v a r i a n t i d e a l s can be o b t a i n e d as f o l l o w s ,

STRUCTURE OF THE ALGEBRAS

19 3

Theorem 13 Suppose I i s an E - v a n i s h i n g i d e a l . I f I i s a subsequence i n v a r i a n t i d e a l , then

(53)

Zx(7)

Hf,

C

V x E E

Conversely, i f ( 5 3 ) holds t h e n

(54)

I c i(nf)

and i(h4) i s a subsequence i n v a r i a n t E - v a n i s h i n g i d e a l , where hi = ( h i x = h i f ] x E E)

Proof I f Z i s subsequence i n v a r i a n t then, i n view o f Theorem 10,ZI w i l l have t h e required property.

N F o r p r o v i n g t h e converse, assume w E i(hl) and w ' E (C" (Q)) i s a subsequence i n w, such t h a t W;=W

, V v E N liV

Then, i n view o f t h e h y p o t h e s i s V

X E

E :

3 v E N : { p , ptl,..

.. .

1 c

ix(w)

But, a c c o r d i n g t o t h e d e f j n i t i o n o f a subsequence, i t f o l l o w s t h a t Y u C N :

llv 2

The r e f o r e V

X E

E :

3 X E N : {A, X t l ,

....1

c Z,(Wl)

which means t h a t w ' E i ( M ) . F i n a l l y , t h e i n c l u s i o n (54) w i l l f o l l o w e a s i l y f r o m ( 5 3 ) .

194

E.E.

6.

Rosinger

Countable Reduced Products o f Vector Spaces

Suppose g i v e n a non-void f a m i l y (Xi

I

i E I)

o f n o n - t r i v i a l v e c t o r spaces on R’ t o g e t h e r w i t h a s e t 8 o f subsets i n t h e i n d e x s e t I . We denote then by YB t h e s e t o f a l l elements

which s a t i s f y t h e c o n d i t i o n 3 8 EB: (55)

B c B(x)

where

(55.1)

B(x) = Ii E I

xi = 0

E

Xi}

I t i s easy t o see t h a t YB

i s a p r o p e r v e c t o r subspace i n X, i . e . Y C X, B f o n l y i f B i s a f i l t e r base on I . r o d u c t of t h e f a m i l y ( X . 1 i E I ) o f v e c t o r spaces B i s t h e q u o t l e n t v e c t o r space.

As known, t h e reduced on R’ accordin-ase

’ ’ 8

xi

= X/Y,

We s h a l l c a l l a p r o p e r v e c t o r subspace Y i n X reduced subspace, o n l y i f Y C Y f o r a c e r t a i n f i l t e r base B on I . Then, t h e f o l l o w i n g r e s u l t can be prove1 e a s i l y . Lemma 1

A p r o p e r v e c t o r subspace Y i n X i s a reduced subspace, o n l y i f By = I B ( x )

IX E

Yl

i s a f i l t e r generator on I .

+

As known, t h e p r o e r i d e a l s i n a r b i t r a r y powers o f R ’ o r C’ a r e reduced subspaces. That a c t among others, e x p l a i n s t h e necessary r o l e p l a y e d i n Non-standard A n a l y s i s by reduced powers, i n p a r t i c u l a r u l t r a powers. I n t h i s connection, t h e q u e s t i o n a r i s e s whether t h e p r o p e r v e c t o r subspaces i n C a r t e s i a n products o f v e c t o r spaces on R’ a r e also reduced subspaces. An a f f i r m a t i v e answer w i l l be o b t a i n e d i n t h i s s e c t i o n under r a t h e r general c o n d i t i o n s i n t h e case o f countable C a r t e s i a n p r o d u c t s o f v e c t o r spaces on The c o u n t a b i l i t y c o n d i t i o n proves t o be e s s e n t i a l , as can be seen R’ i n a s i m p l e c o u n t e r example presented i n Remark 2.

.

F i r s t , we need t h e f o l l o w i n g d e f i n i t i o n .

A v e c t o r subspace Y i n X i s c a l l e d

STRUCTURE OF THE ALGEBRAS

195

vanishing, o n l y i f B(x) # 0

(57)

,U

X E

Y

t h e r e f o r e , a v a n i s h i n g v e c t o r subspace i s a p r o p e r v e c t o r subspace. Obviously, a reduced v e c t o r subspace i s a v a n i s h i n g v e c t o r subspace. I n t h e case o f c o u n t a b l e C a r t e s i a n p r o d u c t s , t h e converse o f t h e above p r o p e r = ty i s e s t a b l i s h e d i n : Theorem 14 Any v a n i s h i n g v e c t o r subspace i n a c o u n t a b l e C a r t e s i a n p r o d u c t o f v e c t o r spaces on R’ i s a reduced v e c t o r subspace. Proof Assume Y i s suffices t o e a s i l y from X + Xi

7:

a v a n i s h i n g v e c t o r subspace i n X. According t o Lemma 1, i t show t h a t B i s a f i l t e r g e n e r a t o r on I. T h a t p r o p e r t y f o l l o w s Lemma 3 belxw, by t a k i n g t h e r e T~ as t h e p r o j e c t i o n mapping and I = I11

Remark 2 I n case o f an uncountable C a r t e s i a n p r o d u c t o f v e c t o r spaces on R’ , t h e r e s u l t i n Theorem above does n o t n e c e s s a r i l y h o l d . Indeed, suppose I = R1 and X . = R ’, w i t h i E I. D e f i n e x = (xi[ i E I ) , y = (yiI i E I ) E X = 1

xi

= i, yi

= 1t i

( ~ / 2- a r c t g i),U i E I

Then o b v i o u s l y (58)

B(x)nB(y) = 0

Denote now by Y t h e v e c t o r subspace i n X generated by {x,y). Then i t can e a s i l y be seen t h a t Y i s a v a n i s h i n g v e c t o r subspace i n X. However, i n view o f (58), Y cannot be a reduced v e c t o r subspace. An a d d i t i o n a l c h a r a c t e r i z a t i o n connected w i t h reduced v e c t o r subspaces i s given in: Lemma 2 Suppose Y i s a p r o p e r v e c t o r subspace i n X. t e r base B on I,o n l y i f

Then Y = YB f o r a c e r t a i n f i l =

E . E . Rosinger

196 Proof

Assume Y s a t i s f i e s (59), then o b v i o u s l y Y = Y 1. Therefore By i s a f i l t e r base on I , s i n c e space i n X .

w i t h t h e n o t a t i o n s i n Lemma

BY

Y i s a p r o p e r v e c t o r sub=

The converse i s obvious.

0

The p r o o f o f Theorem 14 was based on t h e f o l l o w i n g lemma o f a r a t h e r gene= r a l interest. Suppose g i v e n a f a m i l y o f l i n e a r mappings : X

Ti

+

Xi,

for i E I

.

between a r b i t r a r y v e c t o r spaces on R '

Define then t h e mapping

I T ~ X=

B ( x ) = {iE I

X 3 x >-

0

E

Xi}

C

I

Suppose f u r t h e r t h a t I i s a s e t o f n o n - v o i d and countable subsets i n I and denote n B(x) #

= { x E X IJ

XI

0,

'J J E 11

3 -Lemma Suppose Y i s a v e c t o r subspace i n X and Y set

IX E

B Y y J = CJ n B ( x )

XI.

C

Then, f o r each J E I,t h e

Yl

i s a f i l t e r generator on J. Proof Assume t h e statement i n Lemma 3 i s f a l s e . xl,...,xh E Y such t h a t

(60)

J n B(xl)

n

... n B(xh)

=

Then t h e r e e x i s t J E I and

0

D e f i n e t h e mapping

R

h

3

X = (X l,..., Ah) + y y x =

XIXl

+ ...

t

Xh xh

E

Y

and t h e n t h e mapping

J Obviously hi,

(61)

3

i

+

hi = { X E R h

lTi

yX =

o E xi)

w i t h i E J , a r e v e c t o r subspaces i n R

c R~

, 'J

#

Indeed, (60) i m p l i e s t h a t

i

E

J

c R~

h . Moreover

STRUCTURE OF THE ALGEBRAS

197

b r i € J :

3 ki

E {ly...,h}

:

#

‘i ‘ki t h e r e f o r e by t a k i n g

xi

= (0,

...,0,1,0 ,... ,O) E

where t h e o n l y c o o r d i n a t e 1 i n

xi @

Ai,

xi

R

h

, for

i E 3,

i s i n t h e ki-th

position, i t follows t h a t

br i E J

Now t h e r e l a t i o n (61) and t h e B a i r e c a t e g o r y argument a p p l i e d t o R imp 1y t h a t

h

will

s i n c e J i s countable.

Assume then

It follows obviously t h a t 7i

yx

# 0, Y i E J

therefore

B u t yA

E

Y

c X I and thus ( 6 2 ) c o n t r a d i c t s the property defining Xl

n

As seen i n t h e previous s e c t i o n , a p a r t i c u l a r i n t e r e s t presents t h e follow= ing consequence of Lemma 3 above ( s e e [ 1761 , Lemma 1, Section 3 , Chapter 8) *

Given a non-void s e t X , denote by W the s e t of a l l the functions + R'. For w E W and x E X denote

w : NxX

ZX(w) = {v

N J W(U,X) = 01

E

F i n a l l y , f o r a s e t Y of non-void an countable subsets in X denote b r Y E Y t

wy

=

{ W E

w

3yEY: ZY(W) f

1 0

E . E . Rosinger

198 the r e l a t i o n holds V Y E Y :

(63)

3 yEY: '~('1)

' * *

zy(vh) #

0

Proof Take i n Lemma 3 above

x=w I = N x X

(64) Xi=

I =

R' { N x Y I Y E Y ]

where t h e n o t a t i o n s i n t h e r i g h t - h a n d t e r n o f t h e above e q u a l i t i e s a r e those i n C o r o l l a r y 1. F u r t h e r , d e f i n e t h e mappings T

i

= X

+

Xi,

for iE I,

i n Lemma 3, by TiX

= w(v,x)

where a c c o r d i n g t o t h e n o t a t i o n s i n (64) x E X corresponds t o w i E I corresponds t o (v,x) E N x X . Then t h e r e l a t i o n (63) w i l l f o l l o w e a s i l y .

E

W and 0

CHAPTER 7 QUANTUM SCATTERING IN POTENTIALS POSITIVE POWERS OF THE DIRAC 6 DISTRIBUTION

0.

Introduction

Recently, t h e r e has been an i n t e r e s t in quantum s c a t t e r i n g i n p o t e n t i a l s with strong local s i n g u l a r i t i e s [4,29,36,37,151,152,180,188, see a l s o 174, 1761 . The s t r o n g e s t local s i n g u l a r i t i e s of t h e p o t e n t i a l s considered a r e those o f measures which may f a i l t o be absolutely continuous with respect t o t h e Lebesque measure [ 361.

The p o t e n t i a l s t r e a t e d in t h i s chapter, given by a r b i t r a r y p o s i t i v e powers with CL E R' a n d 0 < m < m, of t h e Dirac 6 d i s t r i b u t i o n , present the s t r o n g e s t local s i n g u l a r i t i e s d e a l t with in l i t e r a t u r e . The wave function s o l u t i o n s W obtainsd have t h e s c a t t e r i n g property of con= s i s t i n g from p a i r s Y , Y+ of usual c -smooth s o l u t i o n s of the p o t e n t i a l f r e e motions, each v a l i d on the respective s i d e of the p o t e n t i a l s and con= nected by special junction r e l a t i o n s , on t h e support of t h e p o t e n t i a l s . In case of t h e p a r t i c u l a r p o t e n t i a l rr. 6 , corresponding t o m = l , which i s t h e only one of t h a t type t r e a t e d in l i t e r a t u r e [ 571, t h e junction r e l a t i o n obtained here i s i d e n t i c a l w i t h t h e known one.

1. Wave Function Solutions and Junction Relations The one dimensional wave function Y i s t h e s o l u t i o n of t h e equation (1)

Y" (x) t

(k - U ( X ) ) Y ( X )

=

0, x

E

R',

k E R' ,

where t h e p o t e n t i a l i s defined by (2)

U(X) =

a(s(x))'", x

E

R',

E

R', m

E

(O,m),

In view of t h e f a c t t h a t t h e p o t e n t i a l s ( 2 ) a r e concentrated in x = 0 E R ' (see Chapters 8 and l o ) , t h e problem i s t o find wave function solutionsYof (1,2) having t h e form

(3)

Y(X)

where Y-, (4)

'+"I

Y+E

(x)

Cm(R') t

Y-(x)

if x < 0

Y,(x)

if x > 0

=

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

k "(x) = 0,

x ER' , 199

200

E.E. Rosinger

R', which w i l l a c t u a l l y and s a t i s f y c e r t a i n j u n c t i o n r e l a t i o n s i n x = 0 express t h e s c a t t e r i n g p r o p e r t i e s o f t h e p o t e n t i a l s ( 2 ) . As known [ 5 7 ] , t h a t i s t h e s i t u a t i o n i n t h e p a r t i c u l a r case o f m = l , when t h e j u n c t i o n r e l a t i o n i n x = 0 E R1 between Y- and y, i s g i v e n by

I n t h e general case o f an a r b i t r a r y p o s i t i v e power m t h r e e problems a r i s e :

E

R', o f t h e O i r a c

1)

t o d e f i n e t h e power ( & ( x ) ) ~ ,x

2)

t o prove t h a t t h e h y p o t h e s i s ( 3 ) i s c o r r e c t , and

3)

t o o b a t i n a j u n c t i o n r e l a t i o n extending (5)

E

(0,m),

the following

6 distribution,

The t h i r d problem w i l l be s o l v e d f i r s t , by a usual 'weak s o l u t i o n ' approach presented i n S e c t i o n 2 . T h a t approach w i l l a l s o suggest t h e way t h e f i r s t two problems can be s o l v e d w i t h i n t h e q u o t i e n t algebras 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 . A f t e r p r e l i m i n a r y c o n s t r u c t i o n s presented i n S e c t i o n 3, t h e s o l u t i o n o f t h e f i r s t two problems w i l l be g i v e n i n S e c t i o n 4. The j u n c t i o n r e l a t i o n s i n x = 0 E R 1 between o f t h e form

y- and Y+ i n ( 3 ) , w i l l be

1

(

J(IY = ~ ) O) a 1

for a E R' (see [57] and ( 5 ) )

20 1

QUANTUM SCATTERING

with u

=

f

1 and

-

m

G K G t

m

arbitrary.

The interpretation o f the wave function solutions (3,6) corresponding t o the one dimensional q u a n t u m scattering (1) in potentials ( 2 ) i s as follows the potential ( 2 ) has no effect on the motion.

1)

For m

2)

For m = l , the known [ 5 7 ] motion i s obtained.

3)

If m=2, there are two cases: f i r s t , for the discrete levels of the potential we1 1

E

(O,l),

,x

U ( x ) = -(vlT)2 ( 6 ( x ) ) 2

(7.1)

E

.....

R’, v = 1,3,5,7,

the wave function solutions suffer a sign change when scattering through the potential , while the discrete levels o f the potential we1 1 (7.2)

U(X) =

-(VV)~(~(X))’

,x

E

R’,

2,4,6,8,

v =

......

have no effect on the motion. For m

4) (8)

E(2,m)

U(X) =

the scattering t h r o u g h the potential well

c 1 ( 6 ( ~ ) ) x~ , E

R’,

ci € ( - m y

O),

exhibits an indeterminary involving the two arbitrary parameters and K in (6.4).

u

As known [ 5 7 ] , the problem of the three dimensional spherically symmetric quantum scattering with no angular momentum, has the radial wave function solution R given by

( r z R ’ ( r ) ) ’t r2(k-U(r))R(r) = 0 , r

(9)

k

E(O,m),

E

R’,

I n case the potential i s concentrated on the sphere of radius a i s given by a positive power of the Dirac 6 distribution

(10)

U(r)

=

a(s(r-a))m, r

E (~,m),

a

E

R’, m E

E

(0,m)

and

( ~ , m ) ,

the scattering problem (9,lO) can be reduced t o the i n i t i a l one (1,2), with the consequent interpretation for the radial wave function solution R. 2.

Weak Wave Function Solutions

There are two important advanta es in solving the scattering problem (1,2) within the quotient alge ras containing the distributions. F i r s t , the arbitrary positive powers of‘ the Dirac 6 distribution in the potentials ( 2 ) can be defined in a convenient way, as seen in Section 8, Chapter 3. Secondly, owing t o the automatically granted s t a b i l i t y property o f the sequential solutions obtained within the quotient algebras, i t i s possible t o construct the wave function solutions (3,6) using the most convenient particular weak solutions which correspond to the most convenient particular weak representations of the Dirac 6 distribution. Indeed, the s t a b i l i t y property will imply that the wave function solutions obtained do not depend on the particular way they were constructed.

b-9-

202

E.E.

Rosinger

The weak r e p r e s e n t a t i o n o f t h e D i r a c 6 d i s t r i b u t i o n , employed f o r t h e s i m p l e r computation o f b o t h t h e weak s o l u t i o n s ( 3 ) and j u n c t i o n r e l a t i o n s ( 6 ) , i s given i n 6(x) = l i m v+m

(11)

V(wV, l / w v ,x),

x E R’

,

where l i m wV

(12)

=

0 and wv

E

(0,m) f o r v

E

N,

v + m

while

V(o,K,x)

(13)

where w

E

(0,~) and K

E

K if x

E

(0,~)

0 if x

E

R’\(O,o)

=

R’.

The advantage o f t h e weak r e p r e s e n t a t i o n (11) i s t h a t i t i s g i v e n by p i e c e wise c o n s t a n t f u n c t i o n s which make i t easy t o compute t h e corresponding weak s o l u t i o n s . Moreover, t h e weak r e p r e s e n t a t i o n (11) i n v o l v e s o n l y one m o b i l e p o i n t on t h e x - a x i s , namely x = ,w, E (0,~). I t i s i m p o r t a n t t o mention t h a t t h e nons mmetric c h a r a c t e r o f t h e weak r e = p r e s e n t a t i o n (11) besides t h e a d v a n t T j d h G T v i n g o n l y one m o b i l e p o i n t x = w , a l s o corresponds t o a c e r t a i n n e c e s s i t y i m p l i e d by t h e way t h e Dirac’6 d i s t r i b u t i o n can be represented i n s e v e r a l classes o f q u o t i e n t a l = gebras 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 ( f o r d e t a i l s , see Chapters 8 and 9 ) .

We s h a l l now proceed t o t h e c o n s t r u c t i o n o f t h e weak wave f u n c t i o n s o l u t i o n s o f (1,2) corresponding t o t h e weak r e p r e s e n t a t i o n (11) o f t h e D i r a c 6 d i s = tri b u t i on. Suppose g i v e n m E ( O p ) , a (1) w i t h t h e p o t e n t i a l (14)

E

R’ and v E

m UV(x) = V(wvs a / ( u v ) sx), x

N. Then t h e usual s o l u t i o n yV o f

E

R’,

w i l l obviously s a t i s f y the condition

(15)

yV

E

c”(R’\

to,Wv)

j n c’(R’)

I t f o l l o w s t h a t Yv r e s t r i c t e d t o ( - m y 0 ) i s t h e s o l u t i o n o f t h e p o t e n t i a l f r e e e q u a t i o n (4), t h e r e f o r e i t w i l l be convenient t o s t a t e t h e i n i t i a l c o n d i t i o n s on Yv a t x = 0 E R’, i . e . ,

Now t h e problem i s t o f i n d t h e c o n d i t i o n s on m E ( 0 , ~ ) and a ER’ which w i l l g r a n t t h e weak convergence on R ’ o f t h e sequence o f f u n c t i o n s

QUANTUM SCATTERING

(17)

Y o y Y p...y Y v

203

,.....

p r o v i d e d t h a t wv, w i t h v

E

N, s a t i s f y i n g ( 1 2 ) have been s u i t a b l y chosen,

Denote t h e r e f o r e b y M(k) t h e s e t o f a l l (m,a)E(O,m) x t i s f y i n g ( 1 2 ) and such t h a t

(“wv))

1i m v + m

( 18)

R ’ f o r which t h e r e e x i s t wv , w i t h V E N , sa=

(::)

=

e x i s t s and i t i s f i n i t e , f o r

y;(””)

any yo,yl

E

C’

i n (16)

The i n t e r e s t i n t h e s e t M(k) i s due t o t h e f o l l o w i n g reason. The f u n c t i o n Y r e s t r i c t e d t o ( w ,a) is a l s o t h e s o l u t i o n o f t h e p o t e n t i a l f r e e e q u a t i o n T h e r e f o r e , as’seen i n Theorem 2 below, t h e sequence o f f u n c t i o n s ( 1 7 ) w i l l converge weakly on R ’ , o n l y if t h e r e l a t i o n (18) h o l d s , i n which case t h e wave f u n c t i o n s o l u t i o n ( 3 ) w i l l be determined by t h e c o n d i t i o n s

(t).

w h i l e t h e j u n c t i o n m a t r i x J(m,a) i n ( 6 ) w i l l have t o s a t i s f y

where yp,yA r e su l t r o

yl;

a r e a r b i t r a r y b o t h i n (19) and (20), w h i l e z o y z l

E

cl

We s h a l l now e s t a b l i s h t h e s t r u c t u r e o f t h e s e t M(k). Theorem 1 The s e t M(k) does n o t depend on k E R 1 and M = ((O,l] x R’)u(121 x {-n2, - 4 ~ ’-%’, ~

.... ~ ) U ( ( ~ y ~ ) x ( - ~ , ~ ) ) U ( ( ~ y ~ ) X I o ) )

Proof Ifx

E

Cm(R’)

x”

i s t h e unique s o l u t i o n o f

( x ) t h X ( X ) = 0, x

with the i n i t i a l conditions

E

R’, h

E

R’,

E.E.

204

( I :) [ =

,a

Rosinger

E

R1, b,c

E

C1

x'(a)

then

where = e x p ( x Ah)

W(h,x) and

Therefore (my&

M(k), o n l y i f

lim

(22)

A p p l y i n g ( 2 1 ) t o t h e f u n c t i o n s yV we o b t a i n

x R1.

Assume now (m,a)E(O,m)

W(k-a/(wV)

m

,wV) = J(k,m,a)

e x i s t s and i t i s f i n i t e ,

\ ) + a

f o r s u i t a b l e uv

, with

v E N, s a t i s f y i n g (12).

I t f o l l o w s t h a t t h e o n l y problem l e f t i s t o make t h e c o n d i t i o n i n ( 2 2 ) ex= p l i c i t i n terms o f k , m and CI .

F i r s t assume

CI

> 0.

k

-

a/(wv)

Then i t can a l s o be assumed t h a t

m

< 0

s i n c e wV > 0 and uV + 0, a c c o r d i n g t o ( 1 2 ) . exPLv

(23)

W(k

-

c1/(wV)

m

Therefore

exp(-Lv)

-

exp(-Lv))

V

#uv 1 = $

Hv( expLv -exp( -Lv)) where

1

+XPLV

expLv

t exp( -Lv)

205

QUANTUM SCATTERING

=

(-k

t

a/(wv) m ) t

lim v+m

H

=

+

H

V

,

Lv

= wv HV

But obviously (24)

V

m

and

I L

lim v+m

(25)

0

if m

E

(0,2)

=

V

Therefore M(k) n([2,m)

(25)

Indeed, i f m

t

(27)

E [

x (0,m)) =

0

2,m) t h e n (24) and (25) i m p l y t h a t t h e t e r m

-

Hv(expLv

i n (23) tends t o t

m

exp(-Lv))

t o g e t h e r w i t h v.

L e t us now assume t h a t m E (0,2). Then t h e t h r e e terms i n (23) e x c e p t t h e one i n (27) have a f i n i t e l i m i t when v + m. Concerning t h e t e r m i n (27), i t i s easy t o see t h a t 0

$ Hv(expLv

lim v + m

-

if m ~ ( 0 ~ 1 )

i f m = l

exp(-Lv)) =

t m i f m E

Therefore M(k)n((l,2)

(28)

x (0,m)) =

0

and (29)

(0,ll x ( 0 , m ) c M(k)

Assume now

~1

< 0.

k

-

Then i t can a l s o be assumed t h a t

d(~" > 0) ~

s i n c e wv > 0 and wv

(30)

W(k-a/(w")

where t h is t i me

m

-f

0, a c c o r d i n g t o ( 1 2 ) .

,av) =

Therefore

(1,2)

.

E E . Rosi nger

206

lim

(33)

L~

if m~(0,2)

(-a)$

if m = 2

+

if m

=

x + m

lim

0

(-HV s i n L v )

m

-

(2,m)

i f m = l

cx

v * m

E

a,

if

m

E

(1,2)

Therefore M(k)n((1,2) x

(35)

=

(-m,O))

B

and (36)

( 0 , l I x (-m,O)

C

M(k)

Then again t h e t h r e e terms i n ( 3 0 ) e x c e p t t h e one Assume now t h a t m = 2. i n ( 3 4 ) have a f i n i t e l i m i t when v + m , w h i l e t h e t e r m i n ( 3 4 ) has t h e f o l l o w i n g behaviour

10

lim

i f a =

IT)^, w i t h p

1,2,3,4,

.....

(-Hv sinLv) =

v-tm +m

otherwise

Therefore (37)

M(k)n({2} x

(-m,O))

I 2 1 x {-(pn)’Ip

1,2,3,4,

..... 1

F i n a l l y , assume t h a t m E ( 2 , ~ ) . Then ( 3 2 ) and ( 3 3 ) i m p l y t h a t HV = lim L = + m V v+m t h e r e f o r e , i n view o f t h e t e r m ( 3 4 ) i n (30), a necessary c o n d i t i o n f o r ob= t a i n i n g J(k,m,a) f i n i t e i n ( 2 2 ) , i s t h a t (38)

lim

v+m

20 7

QUANTUM SCATTERING

lim s i n Lv = 0 v+m T h i s time, owing t o t h e r e l a t i o n s ( 3 1 ) , ( 3 8 ) and ( 3 9 ) , i t f o l l o w s t h a t be= s i d e s t h e c o n d i t i o n (12), wv,.with v E N, w i l l have t o s a t i s f y some a d d i = t i o n a l c o n d i t i o n which i s e a s i l y suggested b y ( 3 9 ) . Indeed, i n view o f ( 3 8 ) , t h e necessary and s u f f i c i e n t c o n d i t i o n f o r (39) i s t h a t (39)

lim (Lv v'm where nv E N, w i t h v

(40)

lim

n

-

nvn) = 0

E

N, and

+

=

v-tm

m

Denote t h e n ev = L v

-

nVn , f o r v

E

N.

I t f o l 1ows t h a t

(41)

lim

ev = 0

V'W

Now t h e problem i s t o o b t a i n wvy w i t h v (42)

+ eV , w i t h v

L v = nv IT

where L, i s given i n (31). such t h a t

k

-

E

E

N, as s o l u t i o n s o f t h e e q u a t i o n s

N.

That can be done as f o l l o w s .

a/wm > 0, V w E (0,A)

and assume B > 0 such t h a t t h e f u n c t i o n 0: (0,A)

-+

(B,m)

d e f i n e d b y (see ( 3 1 ) ) O ( w ) = w(k-a/wm)'

1

,

w

E

(0,A)

has t h e p r o p e r t i e s

0 i s s t r i c t l y d e c r e a s i n g on (0,A) lim

O(w) =

m y

w-+o

lim

0(w) = B

o + A

Then, t h e i n v e r s e f u n c t i o n

e x i s t s , i s s t r i c t l y d e c r e a s i n g on (B,m) and

Assume A > 0

208

E.E. Rosinger

NOW, in view of ( 4 3 ) , t h e unique solution of t h e equation (42) w i l l be given by -1 w = o (nvn t e v ) , with v E N, (45) V and ( 4 0 ) , ( 4 1 ) together with (43) w i l l imply t h a t wV, with v E N , s a t i s f y ( 1 2 ) . In t h a t way, the necessary condition (39) f o r the f i n i t e n e s s of J ( k,m,cl) i s s a t i s f i e d .

I n order t o secure the f i n i t e n e s s of J(k,m,a) i t s u f f i c e s t o choose ov, i n other words nv and ev i n ( 4 5 ) , i n such a way t h a t (46)

lim

( - H v sinLV) e x i s t s and i t i s f i n i t e

v-tm

and (47)

lim #+a2

cosL

v

exists

The condition (47) is easy t o f u l f i l l , s i n c e i n view of (45) and ( 3 1 ) , i t fol 1 ows t h a t Lv = O ( W V ) = nvr (48) therefore, i t suffices that

n V' with v (49) B u t (48) a l s o y i e l d s

E N,

sinLv = s i n e

V

t

ev

have constant p a r i t y

, for v

E

N,

hence, i n view of (38) and ( 4 1 ) , t h e condition ( 4 6 ) i s f u l f i l l e d , only i f (50)

lim ev HV

exists and i t i s f i n i t e

v + m

Since i t i s e a s i e r , we s h a l l compute t h e l i m i t o f the square i n (50), which in view of ( 4 0 ) , (41) and (44) can be obtained as follows

as sv E ( O , l ) , w i t h v E N , and a l l the above operations a r e v a l i d , provided that ev # 0 , f o r v E N (5') B u t , i n view o f (43),(40) and ( 4 1 ) , t h e limit

209

QUANTUM SCATTERING

can assume any v a l u e i n [ O + a ] , depending on a s u i t a b l e c h o i c e o f nv and T h e r e f o r e , i n view o f ( 4 6 ) , ( 4 7 ) , (49) and (22), t h e r e l a t i o n (6.4) e., w i l l f o l l o w e a s i l y , p r o v i d e d t h a t ( 5 1 ) h o l d s . I n o t h e r words (52)

(2,m)

x

M(k)

(-a ,O)C

Now, t h e r e l a t i o n s ( 2 5 ) , (28), ( 2 9 ) , ( 3 5 ) , ( 3 6 ) , (37) and (52) complete the proof.

0

The necessary and s u f f i c i e n t c o n d i t i o n f o r t h e e x i s t e n c e o f weak s o l u t i o n s o b t a i n e d w i t h t h e h e l p o f t h e s e t M c h a r a c t e r i z e d i n Theorem 1, can be pre= sented now. I t i s w g r t h n o t i c i n g t h a t t h e weak s o l u t i o n s o b t a i n e d a r e u n i f o r m l i m i t s o f C -smooth f u n c t i o n s on compacts i n R ' \ I O ) . Theorem 2 Suppose g i v e n m (53)

( 0 , ~ ) and a

E

E

Then t h e sequence o f f u n c t i o n s

R'.

.....

Yo,Y p..., Yv,..

c o n s t r u c t e d i n ( 1 5 ) converges weakly on R' f o r s u i t a b l y chosen s a t i s f y i n g ( 1 2 ) , o n l y i f m,a)E M.

w

V"""

woywl,...,

* *

I n t h a t case, t h e sequence o f f u n c t i o n s ( 5 3 ) has t h e f o l l o w i n g two proper= ties (54) (55)

vv

=

lim v+m

v- on I,

(-m

,O], V v

E

N

= I, u n i f o r m l y on i n t e r v a l s [a,-),

w i t h a z 0.

where Y- and Y+ a r e g i v e n i n ( 3 ) , ( 4 ) and ( 6 ) . Proof I n view o f (12) we can assume t h a t wV < 1 , Y v

EN,

i n which case ( 1 4 ) and (24) w i l l y i e l d t h e r e l a t i o n

(yl: 1;)

= W( k

(

) W ( k ,-wV)

):L

Y V E N

Then i t i s obvious t h a t t h e r e l a t i o n (18) w i l l h o l d , o n l y i f (56)

v

~

e x i s~t s and ~i t i s ~ f i n i t e , ~ for any ) yo,yl

E

(16)

C' i n

210

E.E.

Rosinger

But a p p l y i n g again ( 1 4 ) and ( 2 4 ) , i t f o l l o w s t h a t

t h e r e f o r e , t h e sequence o f f u n c t i o n s Y0,Y1'...'Y"

Y

. .......

w i l l converge weakly on R 1 , o n l y i f ( 5 6 ) h o l d s , which as seen above, i s e q u i v a l e n t t o t h e c o n d i t i o n (m,a)€ M. The r e l a t i o n ( 5 4 ) i s an obvious consequence o f t h e form o f t h e p o t e n t i a l i n (14). F i n a l l y , (55) f o l l o w s e a s i l y f r o m ( 5 7 ) . 3.

U

Smooth Representations f o r t h e D i r a c 6 D i s t r i b u t i o n and Smooth Weak Wave F u n c t i o n Sol u t i o n s

As mentioned a t t h e b e g i n n i n g o f S e c t i o n 2 , t h e r e a r e i m p o r t a n t advantages i n s o l v i n g t h e s c a t t e r i n g problem (1,2) w i t h i n t h e q u o t i e n t a l g e b r a s con= t a i n i n g t h e d i s t r i b u t i o n s . I n view o f t h e way t h e c h a i n s o f q u o t i e n t alge= bras ( 2 4 ) o r ( 9 3 ) i n Chapter 3 were c o n s t r u c t e d and 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 a r e a c t i n g between t h e q u o t i e n t algebras o f these chains, i n o r = der t o s o l v e t h e s c a t t e r i n g problem (1,2) w i t h i n t h e mentioned chains o f q u o t i e n t algebras i t s u f f i c e s t o smooth t h e weak s o l u t i o n s i n Theorem 2, S e c t i o n 2 , by ' r o u n d i n g o f f t h e c o r n e r s ' o f Yv a t t h e p o i n t s x = 0 and x = wY , w i t h w E N. I n &he case o f t h e chains o f q u o t i e n t a l g e b r a s ( 2 4 ) i n Chapter 3, we need a C -smoothing w h i l e i n t h e case o f t h e chains o f q u o t i e n t algebras (93) i n Chapter 3, a C2-smoothing w i l l be s u f f i c i e n t . I n view o f ( 1 4 ) and ( 1 5 ) , i t i s obvious t h a t a Cm-smoothing o f t h e represen= t a t i o n o f t h e D i r a c 6 d i s t r i b u t i o n i n ( 1 1 ) w i l l i m p l y a Cm-smoothing o f t h e weak wave f u n c t i o n s o l u t i o n s Y,, , w i t h v E N . T h i s smoothing o f t h e re= p r e s e n t a t i o n ( 1 1 ) i s now c o n s t r u c t e d by ' r o u n d i n g o f f t h e c o r n e r s ' o f V(wv,l/wvy*) a t x = 0 and x = wV, w i t h v E N , w i t h t h e h e l p o f two a r b i = t r a r y f u n c t i o n s 6, y E Cr(R')(see (167), Chapter 3) which s a t i s f y t h e following conditions

* ) B = 0 on

(-my

-11

* * ) 0 G B G M on ( - 1 , l ) 1581 , *

* * * ) B = 1 on [ 1,m) * * * * ) Dp ~ ( 0 #) 0, Y p E N

respectively

* ) y = 1 on (59)

(-m,-ll

* * ) 0 G y G 1 on ( - 1 , l )

* * * ) y = 0 on [ 1,m)

211

QUANTUM SCATTERING

The existence of such functions B and y results from Lemma 1, a t the end of this section. We shall suppose now given

any sequence

(m,a) E M and

which s a t i s f i e s the condition ( 1 2 ) and (18) and generates the weak wave function solution (Yv

I

v

N)

E

through (1) and (14), for a certain given i n i t i a l condition

(16).

We shall take any two sequences (w;

I

I

v E N ) and (w;

v

E

N)

which s a t i s f y the conditions

I

I

WoYW1,..

(62)

lim

6

(w;

YW'

VY"""

+

are pair wise different

=

W ;

0 , where

m=

Then we can define the sequence of functions s6 (63)

max{ l , m }

v+m

s b V ( x ) = B(~/o~)Y((x-w,,)/w;

E (Cm(R'))N

)/uVy4 v

E

given by

N , x E R'

I t i s easy t o see that the following properties result

(65)

11 - L1Sbv(x)dx

2((M+1)5 +

W ;

)/wV ,4 u

E

N

therefore, in view of ( 6 2 ) , i t follows t h a t (66)

s6 E

sm n(C;(R')) N ,

<s6,.>

= 6

Now, the Cm-smooth re presentation o f the Dirac 6 distribution obtained i n (63) and1 replace the representation in (11) and correspgndingly, the weak wave function solutions in (15) will be replaced by the C -smooth weak wave function solutions xv of ( 1 ) generated by the potentials ~ ( s , ~ ( x ) )x~E, R', v E N , conditions given by (see ( 1 6 ) )

E.E. Rosinger

212

where (68.1)

xo < i n f

{-LO;

1

v

E

N 1

Indeed, t h e f o l l o w i n g r e s u l t holds. Theorem 3 I t i s p o s s i b l e t o choose t h e sequences

(u;

I

v E

N) and (u; 1 v

E

N)

s a t i s f y i n g t h e c o n d i t i o n s (60-62) and such t h a t t h e sequence o f Cm-smooth functions

xo

(69)

Y

XI,.

. . ,xv , . . . . . . .

converges i n D ' ( R ' ) t o Y g i v e n i n ( 3 ) , ( 4 ) and ( 6 ) . Proof We s h a l l use a Gronwa 1 i n e q u a l t y argument. and x E R', we denote

BVb) =

/

I n t h i s connection, f o r v

0

- k t 4s,v(x))m

and f i n a l l y = F"(X)

N

1

0

\

HJx)

E

-

Gvb)

Then, o b v i o u s l y t h e f o l l o w i n g r e l a t i o n s w i l l h o l d

QUANTUM SCATTERING

213

and = B,(x)G,(x),

G:(x)

x E

R'

The above r e l a t i o n s w i l l y i e l d X

HV(x) =

X

f

(A,(E)-B,(E))F,(

=

I

and (75) with

X; ( x )

-t

(K

-

~(s,,(x))~)x,(x)

= 0, Y v

E

N, x E

R',

E.E. Rosinger

216

f o r any g i v e n (76.1)

xo< i n f { - w ' l v € N ) V

where y, y- and Y+ a r e g i v e n i n ( 3 ) , ( 4 ) and ( 6 ) . Now, t h e i d e a o f t h e p r o o f i s t o c o n s t r u c t C z r e g u l a r i z a t i o n s ( v , S ) such that (77)

S6 E

s

(see ( 6 3 ) )

and (78)

s ES

I n t h i s case (66) w i l l a c c o r d i n g t o Theorem 17, S e c t i o n 8, Chapter 3, y i e l d w i t h i n p o s i t i v e power a l g e b r a s t h e r e l a t i o n (79)

6 = s6 + + ( ! I S )E AR(V,S), Y

E

fl

as w e l l as (80)

( s 6 ) " t I'(v,s)

6"'

E

A ~ ( v , s ) ,v

mE

(o,m),

R

E

R

Therefore, (74), (75), ( 7 8 ) and ( 8 0 ) w i l l y i e l d w i t h i n t h e same p o s i t i v e power a1 gebras t h e r e 1a t i o n (81)

Y"

+

(k

-

~ ( 6 ) ~ ) =" 0

thus c o m p l e t i n g t h e p r o o f o f 1) and 2 ) . The c o n s t r u c t i o n o f t h e C m - r e g u l a r i z a t i o n s (V,S) f o r which (77) and (78) hold, w i l l proceed a c c o r d i n g t o t h e method presented i n Theorem 4, S e c t i o n 4, Chapter 3. We denote b y

t h e s e t o f a l l t h e sequences o f f u n c t i o n s w conditions

Y x ER' : (82)

3 P E N :

Y VEN,

v > p :

w (x) = 0 V

and

E

(Cw(R'))N

which s a t i s f y the

QUANTUM SCATTERING 3F

C

217

F finite :

R',

G open, G 3 F :

Y G c R',

3 p 1 E N :

(83)

Y v E N, v

>)I':

supp wy c G i n o t h e r words, wv vanishes a t each x E R ' , f o r v s u f f i c i e n t l y l a r g e and furthermore, supp E~ s h r i n k s t o a f i n i t e subset o f R', when v + m. I t i s N o n t r i v i a l examples o f easy t o see t h a t l6 i s an icJeal i n ( C (R'))N. sequences o f f u n c t i o n s w E l 6 w i l l be g i v e n i n ( 6 ) and ( 9 ) , Chapter 8. F u r t h e r , we denote by T t h e g e c t o r subspace i n sequences o f f u n c t i o n s t E S o f t h e form tv(x)

(84)

D P ~ 6 v ( ~ - ~ o V) , v E N, x

=

where xo E R ' and p

E

E

F g e n e r a t e d by a l l t h e R',

N are a r b i t r a r y .

We s h a l l prove t h a t 1; and T s a t i s f y t h e c o n d i t i o n s ( 3 3 ) and (34) i n Theo= rem 4, S e c t i o n 4, Chapter 3 . I n view o f (84) and ( 6 6 ) , t h e f o l l o w i n g r e l a = t i o n i s obvious

vmn T =

(85)

F u r t h e r , t h e re1 a t i on

f o l l o w s from (83),(84) and ( 6 6 ) as w e l l as t h e w e l l known p r o p e r t y t h a t any d i s t r i b u t i o n w i t h s u p p o r t a f i n i t e subset i s a l i n e a r combination o f t h e D i r a c 6 d i s t r i b u t i o n and i t s d e r i v a t i v e s . We prove now t h e r e l a t i o n

, then

Assume indeed t h a t w E 1: n T

where F

C

E

N and X E R'. Assume now g i v e n xO x q Then, i n view o f ( 6 3 ) , (58) a8d ( 5 9 ) , t h e r e e x i s t s

R ' i s a f i n i t e subset, p E

xo E F and f i x e d . )I

obviously

N, such t h a t t h e r e l a t i o n (88) s i m p l i f i e s a t x = xo, as f o l l o w s

xO

(89)

wv(xo) =

c q E N q QPxo

X

Dq S ~ ~ ( OV ) v, 'oq

E

N, v >pxo

2 18

E.E. Rosinger

u,

But, owing t o (82), we can assume t h a t

was chosen i n such a way t h a t 0

wv(x0) = 0, V u

(90)

N, v

E

>

.

11, 0

F u r t h e r , (63) and (58-60) w i l l o b v i o u s l y y i e l d

,

D q ~ 6 u ( 0 ) = DqS(0)/uu(u:)q

(91)

V u

E

N

Therefore, (89-91) w i l l i m p l y t h e r e l a t i o n s

Considering t h e f i r s t p, ding t o v with q E

N,

%px

0

, as

q G p,

1 o f t h e above r e l a t i o n s , i . e . those correspon=

t

N, uxo< v

E

t p,

unknowns ,

, and

considering

5

= h Dqf3(0), q xoq t h e d e t e r m i n a n t o f t h e r e s u l t i n g homo=

0

0

geneous system o f l i n e a r equations w i l l be t h e f o l l o w i n g Vandermonde d e t e r = m i nant det((l/u(Jq

I

G u Gp,

p ,O

which i n view o f (61), w i l l h

DqS(0) = 0, Y q

t

0

vanish. E

,O

N, q Gp, 0

****) i n (58) w i l l i m p l y t h e r e l a t i o n s

which i n view o f

=O, YqEN,

h

)

Gq G p

0

T h e r e f o r e , (92) w i l l y i e l d

,Oq

(93)

,0

p,

qGp,

xOq

0

Since xo E F was chosen a r b i t r a r i l y , t h e r e l a t i o n s (93) and (88) w i l l y i e l d W E

2

thus c o m p l e t i n g t h e p r o o f o f (87). NOW, t h e r e l a t i o n s (85-87) o b v i o u s l y i m p l y t h a t 1; and T s a t i s f y t h e con= d i t i o n s (33) and (34) i n Theorem 4, S e c t i o n 4, Chapter 3. T h e r e f o r e , i t o n l y remains t o c o n s t r u c t v e c t o r subspaces S' c S" which s a t i s f y t h e condi= t i o n s (35) and (36) i n t h e mentioned theorem, and such t h a t

w i l l a l s o h o l d (see (77) and ( 7 8 ) ) . I n t h i s c o n n e c t i o n we n o t i c e t h a t i n view o f (84), t h e r e l a t i o n f o l l o w s easily

QUANTUM SCATTERING

219

(95) Moreover (96)

S

6

vm(+-\ucm(Rl)

J

s i n c e (74) h o l d s and Y was supposed non smooth. Now, t h e r e l a t i o n s (95)mand ( 9 6 ) w i l l o b v i o u s l y g r a n t t h e e x i s t e n c e o f v e c t o r subspaces S' c S s a t i s f y i n g t h e c o n d i t i o n s

v" i;)

(97)

Sm C

(98)

c s 3 u UC"(R')

S'

'
T

c S'

I n t h a t case ( V , S ) , w i t h S g i v e n i n (94) and V a v e c t o r subspace i n 1; n w i l l be t h e needed C - r e g u l a r i z a t i o n .

v", 0

I n t h e case o f f i n i t e smoothness, i . e . of t h e chains o f q u o t i e n t algebras (93) i n Chapter 3, t h e r e s u l t i n Theorem 4 takes t h e f o l l o w i n g form. Theorem 5 Suppose g i v e n t h e s c a t t e r i n g problem

+

(k

-

a(6(~))~)'? = (0, ~)

where k E R ' and ( m y % )

E

M.

'4' '

(99)

(x)

x

E

R',

I n case t h e wave f u n c t i o n s o l u t i o n Y i s n o t continuous a t x = 0

E

R',

it

i s p o s s i b l e t o c o n s t r u c t C " - r e g u l a r i z a t i o n s ( V , S ) , such t h a t t h e a s s o c i a t e d chain o f

o s i t i v e ower q u o t i e n t algebras 5i-n-Fo owing p r o p e r t i e s :

(see S e c t i o n s 7 and 8, Chapter 3)

w i l l have t e 1)

Y

= s

+ lR(V,S) E

where s E S does

2)

AR(V,S),

not

V R E

depend on R

4,

.

I s a t i s f i e s t h e e q u a t i o n (99) i n t h e usual a l g e b r a i c sense, w i t h mul= t i p 1 i c a t i o n and p o s i t i v e power i n Ak( V , S ) and t h e d e r i v a t r v e o p e r a t o r s k 1-k 2 2 . + A ( V , S ) , p E N, p < 2 , R, k E

Dp: A'(V,S)

m,

Proof -

I t i s s i m i l a r t o t h e p r o o f o f Theorem 4, w i t h t h e f o l l o w i n g m o d i f i c a t i o n s . F i r s t , i n s t e a d o f 7; we c o n s i d e r t h e s e t 7: o f a l l t h e sequences o f func= t i o n s w €(C'(R'))N which s a t i s f y t h e c o n d i t i o n s (82) a n d (83). Then, t h e relations

220

E.E. Rosinger

w i l l r e s u l t i n a way s i m i l a r t o (85-87) and ( 9 5 ) . F u r t h e r , s i n c e Y i s n o t continuous, i t w i l l f o l l o w t h a t

Therefore, t h e r e e x i s t v e c t o r subspaces S' cS" which s a t i s f y t h e c o n d i t i o n s

Then, t a k i n g

and V a v e c t o r subspace i n I" n V " , one obtaines i n view o f Theorem 10, S e c t i o n 5, Chapter 3, t h e neided C " - r e g u l a r i z a t i o n ( V , S ) .

0

Remark 1 The smooth r e p r e s e n t a t i o n o f t h e D i r a c 6 d i s t r i b u t on g i v e n b y s 6 ( 6 3 ) has t h e f o l l o w i n g p r o p e r t y

( 100)

Dp s s v ( 0 ) # 0,

E

S" i n

Y VEN, p E N

c a l l e d t h e s t r o n g presence on t h e s u p p o r t I01 t i a l r o l e i n e s t a b l i s h i n g Theorems 4 and 5 .

C

R'

which p l a y e d an essen=

Obviously, t h i s p r o p e r t y imp1 i e s among o t h e r s t h a t symmetric represents= t i o n s f o r t h e D i r a c 6 d i s t r i b u t i o n cannot be used, s i n c e D ssV(O) # 0, YvEN.

The general n-dimensional v e r s i o n o f t h e p r o p e r t y o f s t r o n g presence on t h e support, as w e l l as s e v e r a l o f i t s consequences w i l l be presented i n Chap= t e r s 8 and 9. 5.

S t a b i l i t y and Exactness o f t h e Wave F u n c t i o n S o l u t i o n s

The aim o f t h i s s e c t i o n i s t o e s t a b l i s h t h e s t a b i l i t y and exactness proper= t i e s o f t h e wave f u n c t i o n s o l u t i o n s o b t a i n e d w i t h i n t h e p a r t i c u l a r i n s t a n = ces o f chains o f q u o t i e n t algebras c o n s t r u c t e d i n Theorems 4 and 5, S e c t i o n 4. T h i s time, i n t h e case o f t h e wave f u n c t i o n s o l u t i o n s I o f t h e s c a t t e r i n g problem (1,2), t h e c o n d i t i o n s f o r s t a b i l i t y and exactness a r e b e t t e r and s i m p l e r t h a n those f o r t h e weak s o l u t i o n s o f polynomial n o n l i n e a r PDEs pre= sented i n Sections 5 and 6, Chapter 5. Indeed, t h e r e t h e problem was t o c o n s t r u c t r e g u l a r i z a t i o n s ( V , S ) which would se a r a t e (see Remark 1, S e c t i o n 4, Chapter 4) t h e ' r e g u l a r i z i n g sequence' s rom t e ' e r r o r sequence' w, a c c o r d i n g t o t h e r e l a t i o n s

-I;---h (101)

w

E V , S

E S

Here, t h e problem i s s i m p l e r s i n c e t h e ' e r r o r sequence' w i s i d e n t i c a l l y zero. Indeed, w i t h t h e ' r e g u l a r i z i n g sequence'

221

QUANTUM SCATTERING

s = (&yX 1 ' " . ' X "

(102)

Y

.....)

E S",

<S,*>

=

Y

f o r t h e wave f u n c t i o n s o l u t i o n Y g i v e n i n Theorem 3, S e c t i o n 3, and t h e smooth r e p r e s e n t a t i o n o f t h e O i r a c 6 d i s t r i b u t i o n . (103)

S6

E

S",

<S6,*,

=

6

g i v e n i n (63), t h e r e g u l a r i z e d v e r s i o n o f t h e e q u a t i o n (1,Z) f i e d e x a c t l y (see ( 7 5 ) )

(104)

D2s + ( k

-

w i l l be s a t i s =

a ( ~ , ) ~ ) s= ~ ( 0 ) E Q

T h i s somewhat p a r a d o x i c a l advantage o v e r t h e case o f t h e weak s o l u t i o n s o f polynomial n o n l i n e a r PDEs s t u d i e d i n Chapters 4 and 5, i s due t o t h e f a c t t h a t i n t h e case o f t h e s c a t t e r i n g problem (1,2) b o t h t h e e q u a t i o n and t h e wave f u n c t i o n s o l u t i o n s have t o be r e g u l a r i z e d a n m a t can be done, as seen i n (104), i n an e r r o r l e s s manner. However, t h e c o n d i t i o n (101) w i l l n o t s i m p l i f y i n a t r i v i a l way, s i n c e t h i s t i m e t h e ' r e g u l a r i z i n g sequence' s 6 o f t h e D i r a c 6 d i s t r i b u t i o n w i l l a l s o have t o be taken i n t o account. Nevertheless, as seen i n t h e p r o o f o f Theorems 4 and 5, S e c t i o n 4, t h e new c o n d i t i o n on t h e r e g u l a r i z a t i o n s ( V , S ) w i l l be (see ( 7 7 ) and ( 7 8 ) ) (105)

s,

S6

E S

which i s not o f s e p a r a t i o n type, l e a v i n g t h e r e f o r e ample space f o r t h e c h o i c e o f ( V,S) as seen n e x t . The method used w i l l be s i m i l a r t o t h e one i n S e c t i o n s 2 and 3, Chapter 5. Suppose t h e r e f o r e t h a t , under t h e c o n d i t i o n s i n Theorem 4, S e c t i o n 4, t h e sequences (102) and (103) have been c o n s t r u c t e d . Then, t h e wave f u n c t i o n s o l u t i o n Y i s c a l l e d an S-sequential s o l u t i o n o f t h e s c a t t e r i n g problem (1,2) o n l y i f t h e r e e x i s t s a (?-regularization ( V , S ) , such t h a t t h e corresponding c h a i n o f p o s i t i v e power q u o t i e n t algebras A'( V,S) = A'( V,S)/IR(V,S)

, with

R

E

R,

s a t i s f y the condition (106 1

s, s 6 E AR(V,S),

4 9.

E

a

Obviously, t h e c o n d i t i o n (106) w i l l h o l d , whenever t h e C m - r e g u l a r i z a t i o n ( V , S ) s a t i s f i e s (105). Now, t h e r e s u l t i n Theorem 4, S e c t i o n 4, can be extended as f o l l o w s . Theorem 6 Under t h e above c o n d i t i o n s , i f t h e wave f u n c t i o n s o l u t i o n Y i s an S;sequen= t i a l s o l u t i o n o f t h e s c a t t e r i n g problem (1,2), then, f o r a c e r t a i n C -regu= l a r i z a t i o n ( V , S ) , t h e f o l l o w i n g p r o p e r t i e s h o l d w i t h i n t h e corresponding c h a i n o f p o s i t i v e power q u o t i e n t algebras: 1)

I = s

+ IR(V,S) E AR(V,S), 4 R

E

a,

where s E AR(V,S) does n o t depend on R.

E.E.

222 2)

Rosinger

Y s a t i s f i e s t h e e q u a t i o n (1,2) i n t h e usual a l g e b r a i c sense, w i t h m u l t i p l i c a t i o n and p o s i t i v e power i n Ak'(V,S) and t h e d e r i v a t i v e opera= tors k Dp : AR(V,S) -+ A ( V , S ) , p E N, p G 2 , i l , k E R , k-k 2 2

Proof I t f o l l o w s from t h e p r o o f o f Theorem 4, S e c t i o n 4.

U

I f t h e wave f u n c t i o n s o l u t i o n Y i s an S - s e q u e n t i a l s o l u t i o n o f t h e s c a t t e r = i n g problem (1,21, then we say t h a t Y i s maximally s t a b l e , o n l y i n case t h e r e e x i s t s a C - r e g u l a r i z a t i o n ( V , S ) , w i t h V maximal, and such t h a t (106) holds

.

Theorem 7 Any wave f u n c t i o n s o l u t i o n Y which i s an S-sequential s o l u t i o n o f t h e s c a t = t e r i n g problem (1,2) i s maximally s t a b l e . Proof S i m i l a r t o t h e p r o o f o f Theorem 1, S e c t i o n 2, Chapter 5.

0

Remark 2

1)

Concerning t h e exactness p r o p e r t i e s o f t h e wave f u n c t i o n s o l u t i o n s o f t h e s c a t t e r i n g p r o b 1 , 2 ) , t h e s i t u a t i o n i s s i m i l a r t o t h e one d e s c r i b e d i n S e c t i o n 3, Chapter 5.

2)

The v e r s i o n o f t h e above r e s u l t s corresponding t o t h e case o f f i n i t e smoothness, i . e . , t o chains o f q u o t i e n t a l g e b r a s ( 9 3 ) i n Chapter 3, can e a s i l y be e s t a b l i s h e d (see a l s o S e c t i o n 4, Chapter 5 ) .

CHAPTER 8 PRODUCTS WITH DIRAC 6 DISTRIBUTIONS

0.

Introduction

The following r e l a t i o n s a r e well known in the theory of d i s t r i b u t i o n s ( x - x ~ ) ~ . D6x~ ( x ) = 0, V x0

E

Rn,

p, q

E

Nn, q $ p

0

where 6 , i s t h e Dirac 6 d i s t r i b u t i o n concentrated in xo. Their importance, among o t l e r s , i s i n the upper bounds they give on the order of s i n g u l a r i t i e s exhibited by t h e Dirac 6 d i s t r i b u t i o n and i t s p a r t i a l d e r i v a t i v e s . As seen in Sections 2 and 11, Chapter 1, Section 9 , Chapter 3 and Appendix 2 , the r e l a t i o n s (1) play an e s s e n t i a l r o l e i n determining the way p a r t i a l d e r i v a t i v e operators can be defined on q u o t i e n t algebras containing t h e distributions.

The aim of t h i s chapter i s t o construct chains of q u o t i e n t algebras ( 2 4 ) and (93) i n Chapter 3, i n which r e l a t i o n s of type (1) a r e v a l i d . Several f u r t h e r p r o p e r t i e s of these chains of q u o t i e n t algebras, f o r i n s t a n c e , the Val i d i t y of known formulas in Quantum Mechanics i nvol vi n g s i ngul a r products with Dirac and Heisenberg d i s t r i b u t i o n s , w i l l be presented. The construction of t h e s e chains of q u o t i e n t algebras i s based on a non= t r i vial a1 gebrai c argument i nvol vi ng t h e computation o f a general i zed Van= dermonde determinant. T h a t computation was offered by R . C . K i n g , who was s o kind t o promptly answer an inquiry of the author and c o r r e c t and prove t h e a u t h o r ' s i n i t i a l conjecture. The r e s u l t s w i l l be presented only i n t h e case o f t h e chains of q u o t i e n t algebras (24) i n Chapter 3, s i n c e they can e a s i l y be reformulated f o r t h e chains of q u o t i e n t algebras ( 9 3 ) i n Chapter 3. 1.

A Class of Regularizations

The construction of t h e chains of q u o t i e n t algebras mentioned i n Section 0 i s a c t u a l l y a generalization t o t h e n-dimensional case of t h e construction o f the chains o f q u o t i e n t algebras encountered i n Theorems 4 and 5 , Section 4 , Chapter 7. For given R

R , we denote by

E

6 223

E.E. Rosinger

224

t h e s e t o f a l l t h e sequences o f f u n c t i o n s w f o l l o w i n g two c o n d i t i o n s

E

N satisfy the (C R( R ) ) ~ which

Y X E R ~ :

(2)

3

W E N :

Y v

>u:

v

EN,

wv(x) = 0 and

3 F c R ~ ,F f i n i t e : Y G c R",

(3)

G open, G

3

F :

N :

3

Y v E N, v >p':

supp wv c G a n N Obviously 1: i s an i d e a l i n (C (R ) ) and

v

s

E

I: n sR:

(4) supp <s,

> i s a f i n i t e subset i n Rn

Examples o f n o n t r i v i a l sequences o f f u n c t i o n s w E I: can be o b t a i n e d as f o l l o w s . Suppose Y E D(Rn) and

(5)

0

+

SUPP y

t h e n we d e f i n e w w,(x)

E

(C"(Rn))N by

= Y((vtl)x),

V v E N, x

E

Rn

It follows easily that

(6)

w E l 6R

y

v

LEN

I f we suppose i n a d d i t i o n t h a t

In Y(x)dx = 1 R and d e f i n e now w E (C"(Rn))N b y (7)

(8)

wv(x) = ( v + l ) n Y ( ( v t l ) x ) ,

then obviously (9) and

w

E

I;n&

v

RE

N

Y w

E

N,

x E Rny

225

PRODUCTS

(10)




I t can be noticed t h a t t h e condition ( 5 ) i s the one which implies thafi w i n ( 6 ) o r ( 9 ) s a t i s f y the vanishing condition ( 2 ) in the d e f i n i t i o n of 16. In order t o c o c s t r u c t t h e corresponding n-dimensional version of the vector subspace T c S used in the proof of Theorems 4 a n d 5 , Section 4 , Chapter 7 , several preliminary constructions a r e needed. For given k

E

N , we denote

1

P(n,a)

=

n(a)

car P ( n y a )

{p

E

\ P I
and =

I t i s easy t o s e e t h a t t h e r e e x i s t s a l i n e a r order 4 on N n such t h a t

Nn

{p(l),p(2),

=

p(1) 4

.....1

p(2) 4

.......

and P ( n , k ) = I p ( 1 ) ,..., p ( n ( k ) ) l , V

N

Given now a sequence of functions s E ( C m ( R n ) ) N , we s h a l l a s s o c i a t e t o i t the following Wronskian type i n f i n i t e matrix of functions, using t h e above l i n e a r order 4 on Nn :

Dp(’)so(x).

;)..

/. x)

..

,

xERn

Further, l e t us denote by

L t h e s e t of a l l t h e i n f i n i t e vectors N A = (Ap

I

pEN)ER

which have only a f i n i t e number of non zero components A

An i n f i n i t e matrix

!J

.

E . E . Rosinger

226

A = (a

W

1

v,p E N )

of real numbers w i l l be c a l l e d column wise non s i n g u l a r , only i f

V A

L:

E

A A

E L * A = O

Finally, we denote by L

6

the s e t of a l l the sequences of functions s i n g three conditions

E

Sm which s a t i s f y t h e follow=

(10)

< s,

(11)

supp s v shrinks t o to1 c R n , when v

*>

=

6 + m

and (12)

W,(O)

i s column wise non s i n g u l a r

The existence of sequences of functions s

E

Z6 w i l l be proved i n Section

5.

In view of the conditions (10) and ( l l ) , the sequences s quences.

E

Z6 a r e &se=

The condition (12) s a t i s f i e d by t h e &sequences S E Z ~i s c a l l e d strong pre= sence on t h e support I01 c Rn in view of i t s meaning i n the following p a r t i = c u l a r case ( s e e a l s o ( l o o ) , Chapter 7 ) . Suppose, we a r e i n t h e one dimen= sional case n = l and given Y E D ( R ' ) such t h a t

Then, we define s

E Sm by

I t i s easy t o see t h a t in t h i s case s will s a t i s f y the condition ( 1 2 ) , only if (13)

Dp Y(0) # 0,

V p E N,

Connected with the meaning of the property o f strong presence on support i t i s relevant t o compare the above condition (13) w i t h i t s opposite i n ( 5 ) . As seen l a t e r i n (101), the n-dimensional version o f t h e condition (13) w i l l be a d i r e c t g e n e r a l i z a t i o n , replacing the d e r i v a t i v e s by p a r t i a l deri vati ves

.

NOW, with t h e help o f t h e &sequences s E Z g we s h a l l define vector sub= spaces T E S as follows. Suppose given s E Z6. Then we denote by TS

227

PRODUCTS

t h e v e c t o r subspace i n ? generated b y a l l t h e sequences o f f u n c t i o n s t E o f t h e form t u ( x ) = DPsV(x-xo), V u

(14) where x

0

E

Rn and p

E

N,

x

E

sm

R",

Nn a r e a r b i t r a r y .

E

Prooosition 1 l;, and Ts s a t i s f y t h e f o l l o w i n g c o n d i t i o n s : (15)

I

m

6

n T = T nv"=p s

(16)

vm t

(17)

( V"@T;)

s

(1; n

sm)= v"@

n ucm(Rn) =

T~

p

Proof The r e l a t i o n n

v"

=

p

f o l l o w s e a s i l y from ( 1 4 ) . The r e l a t i o n ( 1 7 ) f o l l o w s from (16) and ( 1 4 ) . The i n c l u s i o n _C i n (16) f o l l o w s e a s i l y from ( 4 ) and ( 1 4 ) . 3 i n (16) i t s u f f i c e s t o show t h a t I n o r d e r t o prove t h e i n c l u s i o n -

(18)

D p s E f +

1;,Vp€Nn

I t i s easy t o see t h a t thc sequence o f f u n c t i o n s i n (8-10) has t h e p r o p e r t y

(19)

DPw E 1; n Sm, = 0'6,

V p E Nn

Therefore, (18) w i l l r e s u l t from ( 1 9 ) . Now, i t remains t o prove t h e r e l a t i o n (20)

f ' n ~= q 6

Assume t h a t w

(21)

s

E

1: n T S . Then

wv(x) =

c xo E F

C A Dpsv(x-xo), p E Nn 'oP I P I Gmx

V w

E

N, x

E

R',

0

where F i s a f i n i t e s u b s e t i n Rn, m

E xO

Assume g i v e n xo

E

F fixed.

N and

h

E

R'.

xOP

Then, i n view o f ( 1 1 ) , t h e r e e x i s t s px E

N,

0

such t h a t t h e r e l a t i o n (21) w r i t t e n f o r x = x0 w i l l t a k e t h e f o l l o w i n g sim=

228

E.E.

Rosinger

p l e r form

was chosen i n such a way

But, i n view o f ( 2 ) , i t can be assumed t h a t p x 0

that

Then ( 2 2 ) and ( 2 3 ) w i l l y i e l d

Now, we d e f i n e t h e i n f i n i t e v e c t o r A = ( A '

u

(25)

A;

I

E

N)

E

R

N

by

=

otherwise Then, o b v i o u s l y A

E

L

and ( 2 4 ) i s e q u i v a l e n t t o t h e r e l a t i o n

ws(o)

A

E

L

Therefore, i n view o f ( 1 2 ) , i t f o l l o w s t h a t A = 0 which a c c o r d i n g t o (25) w i l l y i e l d (26)

x xOP

= 0, Y p E N ~ , 1p1 <mx 0

As x E F was chosen a r b i t r a r i l y , t h e r e l a t i o n s ( 2 6 ) and ( 2 1 ) w i l l complete 0 t h e p r o o f of ( 2 0 ) . The p r o p e r t i e s o f t h e i d e a l 1 : and o f t h e v e c t o r subspaces T e s t a b l i s h e d i n P r o p o s i t i o n 1, o f f e r t h e p o s s i b i l i t y f o r c o n s t r u c t i n g the'needed r e g u l a r i = zations, Indeed, i n view of (15-17) above, as w e l l as Themorem 4, S e c t i o n 4, Chapter 3, o b v i o u s l y t h e r e e x i s t v e c t o r subspaces s c S , such t h a t

229

PRODUCTS

and

(VSC t)Ts )

(28)

i s a C"-regularization

f o r any v e c t o r subspace V C l i n V"

+

However, from t h e p o i n t of view of t h e s t a b i l i t o f t h e r e l a t i o n s (1) w i t h i n t h e chains of q u o t i e n t algebras c o n t a i n i n g t e i s t r i b u t i o n s , i t i s u s e f u l t o c o n s i d e r a c l a s s o f C " - r e g u l a r i z a t i o n s which i s l a r g e r t h a n t h e one i n (28) and which can be o b t a i n e d as f o l l o w s . For given s

E

Z

6

we denote by

i vs t h e s e t o f a l l t h e p a i r s ( 7 , T ) where I i s an i d e a l i n (C"(Rn))N and T i s a v e c t o r subspace i n s", such t h a t t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d :

(29)

1 3 1;

(30)

I

(31)

znsrncvm@~

(32)

V"

I

TITs

n T = T n V"

=

2

n ( U c m ( ~ n )+ T ) =

2

I n view o f P r o p o s i t i o n 1, o b v i o u s l y

Given now ( 1 , ~ )E !V , i t f o l l o w s from Theorem 6, S e c t i o n 4, Chapter 3, t h a t t h e r e e x i s t v e c t o r sabspaces s c Smsuch t h a t

(v, s

(33)

@ T)

i s a C"-regularization

f o r any v e c t o r subspace V c 1 n

v" .

We s h a l l denote b y RGS

t h e s e t o f a l l t h e C " - r e g u l a r i z a t i o n s o f t y p e ( 3 3 ) , which o b v i o u s l y c o n t a i n as p a r t i c u l a r cases those g i v e n i n ( 2 8 ) . 2.

Products w i t h D i r a c 6 D i s t r i b u t i o n s

P r o p e r t i e s o f t y p e (1) concerning p r o d u c t s w t h D i r a c 6 d i s t r i b u t i o n s and t h e i r p a r t i a1 d e r i v a t i ves a r e p r e s e n t e d now. Theorem 1 Suppose g i v e n a C " - r e g u l a r i z a t i o n

( V,

s @T

E

RGS and

E.E.

2 30 I; n v" c

(34)

Rosinger

v

Then t h e p a r t i a l d e r i v a t i v e s o f t h e D i r a c 6 d i s t r i b u t i o n have t h e f o l l o w i n g properties

# 0

DPBx

1)

E

-

AR(V, Sc)T ) , Y xo E R n y p E Nn,

R E

R

0

Y ( x - x ~ ) . D (~x~) ~= 0

2)

f o r every Y

0 E

E

AR(V,S

Y x,xOE

Rn,

p E Nn,R

E

1,

Cm(Rn) which s a t i s f i e s t h e c o n d i t i o n

v

q

E

( X - X ~ ) ~ . D( 'x ~) ~= 0

E

AR(V,S

D ~ Y ( O =) 0,

(35)

0T ) ,

N ~ q, ~p

or lql G R

I n particular

3)

(;> T ) , V

x,x0

E

Rn,p,q

E

N

n

REN,q $ p

0

and I q l > R Proof 1)

I n view o f (10) and (29) above, as w e l l as Theorem 3, S e c t i o n 3, Chap= t e r 3, i t f o l l o w s t h a t Dp6 = DPs t IR(V,S (.) T ) , V p

t h e r e f o r e , f o r g i v e n p E Nn and R DP6 = 0

E

AR(V,S

E

E

R,

Nn,

R E

1

the r e l a t i o n holds

@ T)

only i f Dps

(36)

E

IR(V,S

@ T)

But, o b v i o u s l y

Dps

E

Ts

C

T

hence, (36) w i l l y i e l d (37)

DPS E

I('V,S

@ T)

nT c I nT =

Q

i f (30) i s taken i n t o account.

Now, t h e r e l a t i o n s (37) and (10) o b v i o u s l y c o n t r a d i c t each o t h e r , thus the, p r o o f o f 1) i s completed f o r xo = 0 E Rn. I n t h e case o f a r b i t r a r y xo E R the proof i s s i m i l a r .

2) (38)

As above i n l ) , t h e r e l a t i o n f o l l o w s e a s i l y

YY.DP6= u(Y).DPs t IR(V,S @ T ) ,

Suppose g i v e n p (39)

E

Nn and R

w = u(Y).DPs

E

1.

V p

E

NnyR E

4

We d e f i n e w E (Cm(Rn))N by

231

PRODUCTS

Then

w

(40)

E

zJ

rR(v,s

T)

Indeed, f i r s t we prove t h a t D ~ W E

(41)

ri, Y

q E N ~ , lg

I a.

I n t h i s c o n n e c t i o n we n o t i c e t h a t (39) and t h e L e i b n i t z r u l e f o r p r o d u c t derivative yield

r I

and k E N, k 21, t h e r e l a t i o n holds

@ T)

only i f (45)

(DPs)k E I'l(V,S

But, i n view o f (18-20),

@

I R (v,s

T)

@ T)

Chapter 3, t h e h y p o t h e s i s (44) y i e l d s

c 1;

t h e r e f o r e (45) w i l l i m p l y t h a t ( D P S ) ~E

11

Now, t h e c o n d i t i o n ( 2 ) a p p l i e d t o (DPs)k w i l l y i e l d (46)

DpsV(O) = 0,

f o r a c e r t a i n 1-1

E

>u

Y v E N, v

N.

But (46) and (12) w i l l o b v i o u s l y c o n t r a d i c t each o t h e r .

0

An expected p r o p e r t y o f t h e p r o d u c t o f two p a r t i a l d e r i v a t i v e s o f t h e D i r a c d i s t r i b u t i o n concentrated i n d i f f e r e n t p o i n t s i s presented now. Theorem 3 Suppose t h e C m - r e g u l a r i z a t i o n (V,S ( 3 4 ) . Then

Dp 6, .Dq6 0

f o r any xo,yo

E

Rn,

@ T)

= 0 E AR(V,S yo xo # yo, and p,q

E

RG,

s a t i s f i e s the condition

@ T) E

Nn,R

E

fl.

Proof As i n t h e p r o o f o f l ) , Theorem 1, we o b t a i n t h a t

DP6

= DPs

xO

Dq6 YO

t lR(V,S

@ T),

Y x0

E

Rn,

t IR(V,S

@ T),

V yo

E

Rn, q

p E Nn,&

E

R

xO

= Dqs YO

E

Nn,& E fl

233

PRODUCTS

,s

where s x

E

S m a r e defined by

yo

0

(47) Therefore

-

DPBX .Dp6 = DPs .Dqs + lR(V,S(5) T ) , V xo,y0 0 yo yo yo R EN

(48)

E

Rn,p,q

€Nn,

L e t us denote w = DPs

.Dqs xo

yo

Since xo # yo, t h e r e l a t i o n s ( 4 7 ) and (11) w i l l y i e l d = 0, Y v

w,(x)

(49)

f o r a c e r t a i n li

w

E

EN.

EN,

v 2 1-1

,x

E

Rn,

B u t ( 4 7 ) , (11) and (49) o b v i o u s l y i m p l y t h a t

11n vm

t h e r e f o r e , i n view of t h e above h y p o t h e s i s ( 3 4 ) , as w e l l as t h e n o t a t i o n i n (18-20), Chapter 3, i t f o l l o w s t h a t

w

E

v

c lR(V,S

@ 7)

Now, i n view o f ( 4 8 ) , t h e p r o o f i s completed.

0

An e x t e n s i o n o f t h e p r o p e r t y i n (1) , r e s p e c t i v e l y i n 3) , Theorem 1, concern= i n g products i n v o l v i n g p o l y n o m i a l s and t h e D i r a c 6 d i s t r i b u t i o n o r i t s par= t i a l d e r i v a t i v e s i s presented n e x t . T h a t p r o p e r t y i s ty i c a l f o r t h e m u l t i = p l i c a t i o n i n t h e c h a i n s o f q u o t i e n t a l g e b r a s c o n t a i n i n-%a g t e istributions and cannot be o b t a i n e d w i t h i n t h e framework o f d i s t r i b u t i o n s . F o r t h e sake o f s i m p l i c i t y we s h a l l deal o n l y w i t h t h e one-dimensional case n = l . Theorem 4 Suppose t h e C m - r e g u l a r i z a t i o n ( V , S Then

@ T)

( X - X ~ ) ~ (( xD) ~) ~=~ 0~ E AR(V,S

E RG,

@ T),

s a t i s f i e s t h e c o n d i t i o n (34).

Y x0 E R',

p, k

E

N , i E N,

0

p > R +

1, k > 2

Proof I n view o f t h e L e i b n i t z r u l e f o r p r o d u c t d e r i v a t i v e s (see 4) i n Theorem 3, S e c t i o n 3, Chapter 3) s a t i s f i e d b y t h e d e r i v a t i v e o p e r a t o r s a c t i n g between t h e q u o t i e n t algebras

E . E . Rosinger

234

i t follows e a s i l y t h a t the r e l a t i o n

D ( ( X - X ~ ) ~ ~ (~x( ) D) ~~ )=~ (,p + l ) ( x - x o ) p ( D q 6 x ( x ) ) ~+ 0

0

(50) t

k(x-xo)Ptl(Dqsx

(x))

k- l D q t l

6,

0

k>

R

holds i n AR(V,S T ) , f o r any p,q,k, i n Theorem 1, i t f o l l o w s t h a t

0

(x)

'Dq6,

(XI

0

E

E

AR(V,S

N, k 2 2 .

But, i n view o f 3)

(.I> T )

0

whenever p r e 1a t i on

t

1 > q and p

t

1

L.

Therefore, ( 5 0 ) and ( 5 1 ) i m p l y t h a t t h e

il

i s v a l i d i n A ( V , S It) -. T ) , f o r any p,q,k,

il E N,

k > 2 and p >max

{q,il}.

But t h e p r o d u c t (x-xo)pt1(Dq6x

( x) ) ~ 0

i n t h e l e f t hand term o f ( 5 2 ) i s computed i n A'+'(V,S i n view o f 3 ) i n Theorem 1, i t f o l l o w s t h a t (x-xo)pt1(Dq6x

(53)

whenever p t 1

T ) , therefore,

G;

( x ) ) ~= 0 E ALtl(V,S

0 > q and p t 1 > il t

t

T)

1.

NOW, t a k i n g q = p i n ( 5 2 ) and ( 5 3 ) , t h e p r o o f w i l l be completed.

3.

0

Application t o a R i c c a t i D i f f e r e n t i a l Equation

An a p p l i c a t i o n o f t h e r e l a t i o n s ( l ) , r e s p e c t i v e l y 3) i n Theorem 1, S e c t i o n 2, t o t h e s o l u t i o n o f a R i c c a t i d i f f e r e n t i a l e q u a t i o n i s presented now. Theorem 5 The R i c c a t i d i f f e r e n t i a l e q u a t i o n y ' = xptqy(ytl)

t Dqt16(x)

,x

E

R',

w i t h p,q E N, p 2 1, has t h e general s o l u t i o n

1 xP+q+l

Y(X) = ce-

t D%(x),

x

E

R', c

E

( - ~ , o I,

p+q+l -1

i n each o f t h e q u o t i e n t algebras AR(V,S

@T),

R

E

N, R G q ,

provided t h a t t h e Cm-regularization (V,S (34).

&) T )

E

RG,

s a t i s f i e s the condition

PRODUCTS

235

Proof Assume c E R ’ and d e f i n e Y

E

Cm(R’) by

xP+q+l ~ ( x =) ce-

p+q+l -1 , x

6

R’

Then

l/Y

E

Cm(R’),

Y

c E (-m,O]

therefore, denoting T = 1 / Y t Dq6 we o b t a i n t h a t

G>

But A (V,S T i i s an a s s o c i a t i v e and commutative a l g e b r a w i t h t h e u n i t element 1 E C ( R ), t h e r e f o r e t h e f o l l o w i n g r e l a t i o n s a r e v a l i d w i t h i n i t

~P+~T(T+I)

= xP+q(o%s)*

+

2 x p + q o q q 1 / ~ )+

(54)

+

X ~ + ~ ( ~ +/ Y xp+qDq6 ) ~

+

X ~ + ~ ( ~ / Y )

p r o v i d e d t h a t c E (-m,O1, NOW, i n view o f 3) i n Theorem 1, S e c t i o n 2, i t f o l l o w s t h a t (55)

I? P xp+qDq6 = 0 E A ( V , S V)

whenever p 2 1 and R

T)

< q.

The r e l a t i o n s (54) and (55) w i l l y i e l d w i t h i n t h e q u o t i e n t a l g e b r a AR(V,S @ T) t h e r e l a t i o n (56)

x P + ~ T ( T + ~=) x P + q ( i / y ) ( i

+ i/q, Y

c E ( - ~ , 0 1,

p r o v i d e d t h a t p 2 1 and R G q . B u t (56) o b v i o u s l y i m p l i e s t h a t T i s t h e s o l u t i o n o f t h e R i c c a t i d i f f e r e n = 11 t i a l e q u a t i o n considered. 4.

V a l i d i t y o f Formulas i n Quantum Mechanics

I n t h e one dimensional case n = l , t h e D i r a c 6 d i s t r i b u t i o n and t h e Heisen= berg d i s t r i b u t i o n s

6+

=

6- =

f (6

1 1 + -(-)) 711 x

11 4 (6 - -(-)) 711 x

a r e u s u a l l y assumed i n Quantum Mechanics t o s a t i s f y t h e f o l l o w i n g r e l a t i o n s :

E.E.

2 36

-

1

(57)

A2

(58)

: 6 = -

( 5 9)

6- -

(60)

(-)A

1

=

X

1 1 1 (x)2 = - ; ; I (F) 1

1

2 -

Rosinger

D6

-&

1

D6 1

1

(yT)

1

- v;;T (3)

D 6

These r e l a t i o n s were proved i n [ 68,140,71 ,using s p e c i a l , p a r t i c u l a r regu= l a r i z a t i o n s f o r those o f t h e o p e r a t i o n s i n v o l v e d which cannot be d e f i n e d i n t h e framework o f t h e d i s t r i b u t i o n s . F o r i n s t a n c e , t h e expression ti2

- y71

1 2

($

i n t h e l e f t hand t e r m o f (57), was r e g u l a r i z e d ' e n b l o c k ' i n [140,1 o u t r e g u l a r i z i n g any o f i t s two terms

, with=

1

62 o r (y)z

The aim o f t h i s s e c t i o n i s t o prove t h a t t h e above r e l a t i o n s (57-60) a r e v a l i d w i t h i n t h e framework o f a l a r g e c l a s s o f chains o f q u o t i e n t algebras 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 . The i n t e r e s t i n t h i s r e s u l t i s i n t h e f a c t t h a t - as usual - t h e l a r g e s i z e o f t h e mentioned c l a s s w i l l i m p l y a con= v e n i e n t s t a b i l i t y p r o p e r t y o f t h e r e l a t i o n s (57-60), thus making them i n = de endent o f t h e p a r t i c u l a r r e g u l a r i z a t i o n s i n v o l v e d . I n a d d i t i o n , w i z i n t e ramework o f t h e mentioned chains o f q u o t i e n t algebras, t h e r e l a t i o n s (57-60) w i l l be v a l i d i n t h e usual a l g e b r a i c sense, w i t h a l l t h e o p e r a t i o n s involved - addition, substraction, m u l t i p l i c a t i o n , derivative - effectuated w i t h i n these a1 gebras

T-5-

.

The r e s u l t i n t h i s c o n n e c t i o n i s presented i n Theorem 6, a f t e r s e v e r a l pre= 1im i n a r y d e f i n i t i o n s .

z6.

Then we d e f i n e t h e sequence Suppose g i v e n a sequgnce o f f u n c t i o n s s E o f f u n c t i o n s t, E ( C (R1))N by t h e c o n v o l u t i o n s i n D ' ( R ' ) tsv= s V

1 Y * (1)

v

E

N,

which a r e o b v i o u s l y w e l l d e f i n e d , s i n c e , i n view o f (11)

V v

s y E P(R'),

E

N,

while

(5)

E D'(R1)

Moreover (61)

ts E Srn,

We denote b y

.

= ($1

PRODUCTS

237

t h e v e c t o r subspace i n Sm generated by Dpts, with p E N .

I n view o f Lemmas 1 and 2 below, i t f o l l o w s t h a t t h e r e l a t i o n h o l d s (see (16) and ( 1 7 ) )

(v"

(62)

'+

~

~

~T ~~ n)) pS =

m ( + j(

Q

Now, i n view o f ( 6 2 ) i t f o l l o w s t h a t t h e r e e x i s t C m - r e g u l a r i z a t i o n ( V , S , + T ) E RG, such t h a t (63)

Ps'

sic.,\

T

We s h a l l denote by RG;

the s e t o f a l l the Cm-regularizations ( V , S

c+

T ) E RG,

which s a t i s f y ( 6 3 ) .

Theorem 6

Z6 such t h a t t h e r e l a t i o n s (57-60)

There e x i s t sequences o f f u n c t i o n s s E a r e v a l i d w i t h i n t h e q u o t i e n t algebras

(?

AR(V,S

T ) ,R

_I

Cm-regu a r i z a t i ons

f o r any

u,

E

V,S

i+>

T)

E

RG;.

Proof Assume

\y

E P(R'

such t h a t dx = 1

and (13) h o l d s .

We d e f i n e t h e n s

E

s"

Sv(X) = ( v + l ) Y ( ( v + l ) x ) , U v

by E

N, x E R '

As seen i n S e c t i o n 1, i t f o l l o w s t h a t

s E Z6 F o r p E N, l e t us denote

I

Mp = sup(lDPY(x)l

x E R'I

and l e t us assume t h a t

supp Y c for a suitable L

E

[ -L,L

I

( 0 , ~ ) . Then, i t f o l l o w s e a s i l y t h a t

I x ~ + ~ D ~ s ~ (< x )M I .Lp,Y v t h e r e f o r e s i s a '&sequence'

E N , x E R', p E N P i n t h e sense of [ 138,140,7].

2 38

E.E.

Rosinger

(VS @ T )

Assume now g i v e n a C " - r e g u l a r i z a t i o n (29) and ( 6 3 ) , we o b t a i n t h a t

Then, i n view o f

E RG;.

E)

s, t s E S

T

t h e r e f o r e (10) and (61) w i l l y i e l d t h e r e l a t i o n s (64)

6 , = $(s

1 + ;;r t s ) + IR(V,S

GT),V R

(65)

6- = $(s

- ;;1r t s ) +

G-)

(66)

a),( 1

= s ts +

IR(V,S

E

8,

T), Y R E

8,

T), V R E

IR(V,S

We d e f i n e now t h e sequences o f f u n c t i o n s t ' , t " (67)

tl = ( s

+

1

t" = (s

tJ,

711

-

1

711

8

,t", E(Cm(R'))N by

t S ) 2 , t"' = s ts

and r e c a l l t h a t t h e f o l l o w i n g r e l a t i o n s were proved i n [ 140,71 (68)

t ' ,t" , t " I



E



= y t a

and k(m)

>;

Now we choose vo = k(m)

t

l , , . . , vu = k(mt1)

Then t h e c o n d i t i o n s *) and **) i n (104) a r e o b v i o u s l y s a t i s f i e d . Moreover, i n view o f (98.2) as w e l l as Theorem 8 below, i t f o l l o w s t h a t t h e c o n d i t i o n **? i n (104) i s a l s o s a t i s f i e d . T h e r e f o r e A i s indeed column w i s e nonsingu= lar. I n t h i s case, t h e r e l a t i o n s (102) and (103) t o g e t h e r w i t h Lemma 4 below im= p l y t h a t W,(O) i s column wise n o n s i n g u l a r o n l y i f t h e c o n d i t i o n (101) i s satisfied. 0 The main r e s u l t i n t h i s s e c t i o n i s presented now. Theorem 7 The s e t Z6 o f sequences o f f u n c t i o n s i s n o t v o i d .

245

PRODUCTS

Proof I n view of P r o p o s i t i o n 2 , i t s u f f i c e s t o show t h e e x i s t e n c e o f Y which s a t i s f y t h e c o n d i t i o n s ( 9 2 ) and (101). I n t h i s connection, we d e f i n e a E Cm(Rn) by xl+. . .+xn , V x = (x1,...,xn) a(x) = e and assume

qRn)

E

E

E

aRn)

Rn

such t h a t

f3 2 0 on Rn and f o r a c e r t a i n neighbourhood V o f 0

E

Rn

B = l o n V Then o b v i o u s l y

I

K =

I f we d e f i n e now \y

a(x)B(x)dx > 0

Rn

=

\y

clf3

E

D(Rn) b y

/K

t h e n t h e c o n d i t i o n ( 9 2 ) w i l l o b v i o u s l y be s a t i s f i e d , w h i l e t h e r e l a t i o n s Dp\y(0) = 1/K > 0, Y p EN’,

w i l l imply (101).

0

I n case a r b i t r a r y p o s i t i v e powers o f t h e D i r a c 6 d i s t r i b u t i o n a r e t o be d e f i n e d w i t h i n t h e chains o f q u o t i e n t 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 , the following r e s u l t i s useful. Corollary 1

Proof I f we choose i n t h e p r o o f o f Theorem 7,

B

E

CY(Rn)

then i t follows t h a t \y E

CY(Rn)

therefore

s

E

zg “(CY(R

n N

1)

Lemma 4 The i n f i n i t e m a t r i x o f complex numbers A = (aVu

I

v,u

E

N ) i s column wise

E.E.

246

Rosinger

nonsingular, o n l y i f Y;,a€N:

..............a vO‘

I :

Ii

# O

Proof I t f o l l o w s e a s i l y from t h e d e f i n i t i o n i n S e c t i o n 1.

0

And now, t h e theorem on general ized Vandermonde determinants ( f o r n o t a t i o n s see S e c t i o n 1) whose p r e s e n t form, as w e l l as p r o o f was o f f e r e d by R.C.King. Theorem 8 Suppose g i v e n n E N

a

n 2 1.

Then f o r each a E Nn, a 2 e = (1, holds

I

:

...*1) E Nn

and II E N,R 21, t h e r e l a t i o n

I

where p ( j ) = (pl(j) ¶...,pn( j ) ) , f o r 1 G j Q R. Remark 1 The value o f t h e d e t e r m i n a n t depends o n l y on n,a,p( 1) depend on a. Proof

L e t s consider the determinant

.... ,p(R)

and does not

PRODUCTS

Al = d e t ( ( a + p ( o ) ) P ( T ) ) ,

247

where 1 Q o r T Q II

F o r 1 < T < R y t h e -c-th column i n Al i s

... x

(altpl(l))P1(T)x

C,(d

=

x i f a = ( a ly...,

( a n t p n ( l ) ) Pn

an)

...

'

x (antpn(a)) Pn

.

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

p(

R)P(T) where 0' = 1 whenever i t occurs. We o b t a i n t h e n (105)

c

C2(T) = C i ( T )

A

where t h e sum C i s taken f o r a l l 1 Q A Q R s u c h t h a t I p ( X ) I A

Introducing t h e determinant A2 = d e t ( p ( ~ ) ' ( ~ , where 1 < u y T < 1,

i t f o l l o w s from (105) t h a t A2 = Al, s i n c e C ~ ( T )i s t h e r - t h column i n A 2 . We s h a l l now s i m p l i f y A2 w i t h t h e h e l p o f t h e f u n c t i o n F : N x N -+ N d e f i n e d by

1

i f k = O

F(h,k) = h(h-1) ...(h - k t l )

if k 2 1

which o b v i o u s l y s a t i s f i e s t h e c o n d i t i o n s

( 106 1

F(h,k)

= 0 * h

- k + 1 Then, t h e e x p r e s s i o n o f V g i v e n i n (112) y i e l d s DPvi

E

7,;

Y 0 < iG h , p

E

N , p GII

250

Rosinger

E.E.

hence, i n view o f c o n d i t i o n ( 2 ) , we o b t a i n (117)

Dpvi ( 0 ) = 0, U 0 G i l , v E

N, x E R'

hence (119)

DPtV(0) = p Dp-lsv(0),

Therefore,

i n case R 21, t h e r e l a t i o n s (118) and (119) w i l l i m p l y

V p

Dqs (0) = 0, Y g E N , q V

E

N, p 2 1, v E N

4 R -1, v

EN,

v 2p

which o b v i o u s l y c o n t r a d i c t s (12) and thus ends t h e p r o o f o f (115).

Now, we s h a l l show t h a t (120)

sm = o

EA~(v,s@

TI, Y m,RE

N, m > R

Indeed, i n view o f (114) we o b t a i n

sm=

tmt

I~(v,s@ T)

v

E A ~ ( V , S @ T), m

EN,

m

RE II

hence, according t o (113) and (10) i t f o l l o w s t h a t

sm=

xm(61m E A ~ ( V , S @ T),V

But t h e m u l t i p l i c a t i o n i n A'(

sm =

V,S@

m E N , m 2 1 , R~ ~

T) i s a s s o c i a t i v e , thus

xm 6 ( 6 p - 1

and i n view o f 3) i n Theorem 1, S e c t i o n 2 xm6 =

o

E A ~ v , s @

T),U m

EN,

m

>I,

m >R

which means t h a t t h e p r o o f o f (120) i s completed. F i n a l l y , t h e r e l a t i o n s (115) and (120) w i l l o b v i o u s l y y i e l d (111).

CHAPTER 9 LINEAR INDEPENDENT FAMILIES OF D I R A C 6 DISTRIBUTIONS AT A POINT

0.

Introduction

The r e p r e s e n t a t i o n s o f t h e D i r a c 6 d i s t r i b u t i o n c o n s t r u c t e d andmused i n Chapter 7 and 8, were g i v e n b y weakly convergent sequences o f C -smooth f u n c t i o n s which s a t i s f y t h e c o n d i t i o n o f s t r o n g presence on t h e s u p p o r t (see (100) i n Chapter 7 and ( 1 2 ) i n Chapter 8 ) . An immediate consequence o f t h i s c o n d i t i o n was t h e nonsymmetr o f t h e mentioned r e p r e s e n t a t i o n s , which means t h a t t h e D i r a c 6 i s t r i u t i o n and i t s p a r t i a l d e r i v a t i v e s a r e n o t i n v a r i a n t under t h e independent v a r i a b l e t r a n s f o r m (see S e c t i o n 10, Chapter 1).

+

(1)

Rn3 x

+

a x E Rn, w i t h a = -1,

w i t h i n the chains o f q u o t i e n t algebras containing t h e d i s t r i b u t i o n s

.

The aim o f t h i s c h a p t e r i s t o p r e s e n t t h e f o l l o w i n g s t r o n g e r r e s u l t : Apply= i n g t o any g i v e n p a r t i a l d e r i v a t i v e DP6, w i t h p E N", o f t h e D i r a c 6 d i s = t r i b u t i o n t h e f o l l o w i n g independent v a r i a b l e t r a n s f o r m s w

(2)

Rn3 x

-+a a x E Rn

, with

a E R1 \ { O )

we o b t a i n l i n e a r independent elements (3)

Dp6( aox) ,. .. ,Dp6( a,x)

w i t h i n t h e chains o f q u o t i e n t 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 , pro= v i ded t h a t (4)

ao,.

. . ,a m E R'

\

{Ol a r e p a i r w i s e d i f f e r e n t

I n S e c t i o n 3, t h i s r e s u l t i s extended i n o r d e r t o i n c l u d e a l s o g e n e r a l i z e d D i r a c 6 d i s t r i b u t i o n s which have t h e f o r m (5)

lim a+m

an

s(ax)

1. Compatible Q u o t i e n t Algebras and Independent V a r i a b l e Transforms W i t h i n t h i s Chapter, i t w i l l be c o n v e n i e n t t o c o n s i d e r t h e independent v a r i a b l e t r a n s f o r m d e f i n e d i n S e c t i o n 10, Chapter 1, under an a l t e r n a t i v e form p r e s e n t e d now. Suppose g i v e n an independent v a r i a b l e t r a n s f o r m

251

E.E. Rosinger

252

(6)

0:

* Rn,

Rn

w

E

cm

then we can o b v i o u s l y d e f i n e t h e a l g e b r a homomorphism w : ( CTRn)IN *

(7)

( C"pn))N

by ( W ( S ) ) ~ ( X=) s V ( w ( x ) ) , V

(7.1)

S

E ( C TRn))N, v EN,

XE

Rn

We s h a l l say t h a t t h e independent v a r i a b l e t r a n s f o r m w i s i n v e r t i b l e , o n l y if (8)

: Rn

w

-+

Rn e x i s t s and w - l E Cm

Suppose now g i v e n a q u o t i e n t a l g e b r a (see S e c t i o n 2 , Chapter 1)

A = A / I E AL cTRn)

(9)

We s h a l l say t h a t t h e q u o t i e n t a l g e b r a A and t h e independent v a r i a b l e t r a n s = form w a r e compatible, o n l y i f (see (100.2) and (100.3), Chapter 1) w(A) c A and w ( I )

(10) where

c 7

i s t h e mapping d e f i n e d i n ( 7 ) .

Proposition 1 I f t h e q u o t i e n t algebra A i n ( 9 ) and t h e independent t r a n s f o r m w i n ( 6 ) a r e compatible, t h e n t h e mapping

d e f i n e d by

i n an a l g e b r a homomorphism. Proof U

I t i s obvious.

W i t h i n t h i s chapter, we s h a l l o n l y deal w i t h chains o f q u o t i e n t algebras o f t y p e (24), Chapter 1, g i v e n i n

(11)

where (see (33)

(11.1)

(3r),a E N ,

A'(v,s

(v,S

, (34)

0

T ) E RGs and

while

(11.2)

s

E

and (44) , Chapter 8)

zg

v

=

11n

v"

D I RAC

253

D ISTRI BUT I ONS

A u s e f u l c h a r a c t e r i z a t i o n o f c o m p a t i b i l i t y between t h e q u o t i e n t algebras (11) and independent v a r i a b l e t r a n s f o r m s i s presented now. Proposition 2

A q u o t i e n t a l g e b r a i n (11) A'( V,S

@ T)

= A'( V,S

e)

G)

T ) / I R ( V,S

T)

and an i n v e r t i b l e independent v a r i a b l e t r a n s f o r m w i n ( 6 ) a r e compatible, only i f (12)

(9T ) i s an i n v a r i a n t o f

A'(V,S

w i n (7)

Proof The c o n d i t i o n (12) i s by d e f i n i t i o n necessary. We s h a l l show now t h a t i t i s a l s o s u f f i c i e n t . I n t h i s r e s p e c t we o n l y need t o prove t h a t

rR(v,s(5T ) \

(13)

i s an i n v a r i a n t o f w i n ( 7 )

B u t , i n view o f (20), i n Chapter 3, IR(V,S generated by "R

I

= {V E V

DPv

c)

T ) i s t h e i d e a l i n AR(V@T)

V , Y p E Nn,

E

IpI < a } 0

Therefore, i n view o f Lemma 1 below, (13) i s v a l i d . Lemma 1

I f w i s an i n v e r t i b l e independent v a r i a b l e t r a n s f o r m and V = I: then, f o r each R E N , VQ =

V

(V E

I

D'v

E

V

,Y

p

E

Nn,

IpI

n V"

El

i s an i n v a r i a n t o f t h e mapping w i n ( 7 ) . Proof Since w i s i n v e r t i b l e , t h e r e l a t i o n f o l l o w s e a s i l y (14)

w(vm )c

V"

Moreover, t h e r e l a t i o n i s a l s o v a l i d

(15)

w( q c

1;

Indeed, assume t h a t w E I: and denote w ' = w(w). F i r s t we show t h a t w ' s a t i s f i e s t h e c o n d i t i o n ( 2 ) i n Chapter 8. Assume t h e r e f o r e t h a t x E Rn and denote x ' = w(x ) . Then w and x; s a t i s f y t h e Hence mentioned conditpon, s i n c e w E 1; aRd x; €ORn.

(16)

w,(x;)

= 0,

f o r a certain u E N.

4 v

E

N, v 2 p

E.E.

254

Rosinger

so s a t i s f i e s t h e c o n d i t i o n ( 3 ) i n Chapter 8. L e t us denote But w F ' = w . ( F ) , then o b v i o u s l y F ' c Rn i s a f i n i t e subset. Assume now g i v e n an open subset G ' c Rn, such t h a t F ' c G ' , then o b v i o u s l y G = w ( G ' ) c Rn i s open and F c G. T h e r e f o r e (17)

supp wv c G, V v

for a certain plies (18)

E N.

N,v

E

> P I ,

B u t , i t i s easy t o see t h a t t h e r e l a t i o n (17) i m =

supp w; c G I , V v

N, v

E

Now, t h e r e l a t i o n s (16) and (18) w i l l i m p l y ( 1 5 ) . F i n a l l y , t h e r e l a t i o n s (14) and (15) y i e l d (19)

w ( v ) = ~ ( 1 :n

v") c

Assume now g i v e n R E \ and v

~ ( 1 : )n w ( v " ) c

E

11 n v"

=

v

VR and denote v ' = w ( v ) .

I n view o f (19) i t f o l l o w s t h a t E

v

c

DpvI

E

V",

v'

v"

therefore (20)

V p E Nn

B u t (19) w i l l a l s o i m p l y t h a t v'

E

v c

1;

t h e r e f o r e , i t i s easy t o n o t i c e t h a t (21)

Dpv' s a t i s f i e s t h e c o n d i t on ( 3 ) i n Chapter 8, Y p E Nn

F u r t h e r , by d e f i n i t i o n , t h e r e l a t i o n v Dpv E V c li, V p

E

Nn,

E

VR y i e l d s

PI G R

therefore (22)

Dpv s a t i s f i e s t h e c o n d i t i o n ( 2 ) i n Chapter 8, V p

E

N, I p I

p :

DqvV(O)

0

=

B u t m d II by the hypothesis, hence, the relations (30) and (33) will y i e l d

v >p

Further, in view o f (11.2) we obtain t h a t

(35)

4 q

E Nn,

3 v

E

U E

N :

N, v > a :

D4Sv(O)

# 0

since the matrix Ws(0) i s column wise nonsingular (see (12) in Chapter 8). Then the relations (34) and (35) will y i e l d

c

Xi ( a i ) k = 0 , V k

E

N, k

p '

f o r a c e r t a i n u' E N, s i n c e V T+i m" II bu x o 1 I = 03, where I1 1I i s any g i v e n norm on Rn. I n case x = 0 E Rn, t h e r e l a t i o n (56) f o l l o w s d i r e c t l y from t h e f a c t t h a t w s a t i s ? i e s t h e c o n d i t i o n ( 2 ) i n Chapter 8.

w,

We prove now t h a t w ' s a t i s f i e s t h e c o n d i t i o n ( 3 ) i n Chapter 8. Since s a t i s f i e s t h e mentioned c o n d i t i o n , t h e r e e x i s t s a bounded s u b s e t B c R a neighbourhood V c B o f x = 0 E Rn, and p " E N, such t h a t (57)

SUPP

(58)

v

wU c 6, Y u

E

n SUPP w V c (01

N, v > p "

c R ~ Y,

EN,

Now, t h e r e l a t i o n s ( 5 3 ) , ( 5 5 ) , (57) and (58) w i l l i m p l y t h a t (59)

SUPP W;

c {O}, Y u

f o r a s u i t a b l e p " ' E N. i s completed.

E N,u > p " '

And i n view o f ( 5 6 ) and ( 5 9 ) , t h e p r o o f o f (54)

F u r t h e r , we s h a l l prove t h e r e l a t i o n (60)

(v)

ab

Assume t h a t v E V . f i c e s t o show t h a t

Then v

(61)

v"

qJV) E

B u t i n view o f (50.1), (62)

E

:I n V",

t h e r e f o r e i n view o f ( 5 4 ) , i t s u f =

t h e r e l a t i o n (61) i s e q u i v a l e n t t o

1 l i m In v v ( x ) Y (6 x ) d x = 0, V Y v+mR V

E

U(Rn)

Now, i n o r d e r t o p r o v e ( 6 2 ) , f i r s t we n o t i c e t h a t v E 1; and t h e condi t i on ( 3 ) i n Chapter 8, i m p l y t h e e x i s t e n c e o f a bounded subset B c Rn and U ' E N, such t h a t (63)

supp

vV c

6, Y v E

N,

v >p'

L e t us t a k e x E D(Rn) such t h a t X = 1 on B . r e l a t i o n (62) i s e q u i v a l e n t t o

(64)

lim

1

V + m $

For given Y E

D

1 v ( X ) X ( X ) Y ( ~x ) d x = 0, Y Y

Then i n view o f (63), t h e E

D(Rn)

V

(Rn).we d e f i n e t h e sequence o f f u n c t i o n s s

E

n N (D(R ) ) by

1 s V ( x ) = X ( X ) \ ~ ( x ) , Y v E N, x E Rn LV Then i t i s easy t o see t h a t t h e f o l l o w i n g r e l a t i o n h o l d s i n U(Rn)

E.E.

262

(65)

l i m sv

Rosinger

Y(0)x

=

v'm

NOW, i n view o f a w e l l known p r o p e r t y o f b i l i n e a r forms on V ( R n ) x D(Rn)), t h e r e l a t i o n VE v" and (65) w i l l i m p l y ( 6 4 ) . I n t h i s way, t h e p r o o f o f (60) i s completed. F i n a l l y , we can prove t h e r e l a t i o n (66 1

%(VJ

c VR

y

Y R E

a

Indeed, assume g i v e n R E and v E V and denote v ' = % ( v ) . and t h e obvious i n c l u s i o n VQ c V w i l e y i e l d

Then (60)

v"

v' E V c therefore (67)

Dpv' E

V",

Y p E Nn

But s i m i l a r l y t o (59) we o b t a i n t h a t (68)

c{O}, Y v E N, v 2 p '

SUPP V;

therefore (69)

DpvI s a t i s f i e s t h e c o n d i t i o n ( 3 ) i n Chapter 8, Y p E Nn

Further, by the d e f i n i t i o n o f (70)

DPV E

v

Suppose now g i v e n x

c 0

r: , v E

Rn.

$ we

obtain

p E N ~ ,

GR

I f xo # 0 E Rn t h e n (68) w i l l y i e l d

DpvI s a t i s f i e s t h e c o n d i t i o n ( 2 ) i n Chapter 8, a t xo (71)

E

Rn\\O},

Yp€Nn

If xo = 0 E Rn then (50.1) and t h e f a c t t h a t i n view o f ( 7 0 ) , Dpv w i t h I p I G R , s a t i s f i e s t h e c o n d i t i o n ( 2 ) i n Chapter 8, a t xo = 0 E Rn, w i l l imply t h a t P + E NnY

Dpv' s a t i s f i e s t h e c o n d i t i o n ( 2 ) i n Chapter 8 a t xo = 0 E Rn, (72)

V p

E

Nn,

IpI < R

But t h e r e l a t i o n s ( 6 9 ) , (71) and (72) o b v i o u s l y i m p l y (66). I n casevlimm bv

=

-

m,

t h e p r o o f o f (66) i s s i m i l a r .

I n view o f P r o p o s i t i o n 3, i t f o l l o w s t h a t t h e d e f i n i t i o n o f M t r a n s f o r m algebras i n S e c t i o n 1, can n a t u r a l l y be extended t o t h e case when M con= t a i n s n o t o n l y independent v a r i a b l e transforms b u t a l s o a l g e b r a hornomor= phism o f t y p e ( 5 0 ) . I n o r d e r t o d e f i n e t h e g e n e r a l i z e d D i r a c 6 d i s t r i b u t i o n s i n (49), l e t us denote

0

D I R A C DISTRIBUTIONS

Mb = Mo

263

U {Wb)

I t f o l l o w s t h a t i n case A'( V 9 - S (J + I T ) , with LEB, a r e Mb t r a n s f o r m a l g e b r a s , t h e n cob DP6 E AR(V,S

( + T) , Y R E

N,

C+, T),V

L

p E Nn,

since DP6

and

E

D'(Rn) C A'(V,S

E

W,

p E Nn,

W b : AL(V,St.+: J) -+ AE(V,S (tj T),V II E 1 The e x t e n s i o n o f Theorem 1 i n S e c t i o n 2 i s p r e s e n t e d now.

Theorem 2 The t r a n s f o r m s o f t h e p a r t i a l d e r i v a t i v e s o f any g i v e n o r d e r p E Nn o f t h e Dirac 6 d i s t r i b u t i o n DP6(aox) ,.. . ,DP6(amx) and t h e g e n e r a l i z e d D i r a c 6 d i s t r i b u t i o n p a r t i a l d e r i v a t i v e

a r e l i n e a r independent w i t h i n t h e Mb t r a n s f o r m a l g e b r a s A L E N , p r o v i d e d t h a t m G J ? and ao,...,am E R1\{O) are Proof Assume t h a t i t i s f a l s e and X o , . . . , ~ , X (73) (74)

A,

D P6(aox) t

X= 0

=r

.. . +

( 3 io E {O,

A,

E R ' a r e such t h a t

DP6(amx) + hob DP6 = 0 E A'( V,S

...,m l

: Xi

@ T)

# 0) 0

B u t i n view o f (11) above, as w e l l as Theorem 3, S e c t i o n 3, Chapter 3, i t follows that

t h e r e f o r e , a c c o r d i n g t o ( 7 ) and (50) we o b t a i n t h e r e l a t i o n s (75)

Xi

DP6(aix) = A1. w ai DPs

Y i

E

-+ IR(V,S @

T ) E AR(V,S

...,m l

{O,

and (76)

Awb DP6 = A w ~DpS

+

IR(VyS

@ T) E

We d e f i n e now v E (Cm(Rn))N by (77)

c

v = O < i

A1 . w ai DPs Gm

t h e n (73), (75) and (76) w i l l y i e l d

+

Xub D's

A'(VyS

@ T)

@

T),

264

E.E.

Rosinger

t h e r e f o r e , as i n t h e p o o f o f Theorem 1, S e c t i o n 2 , i t f o l l o w s t h a t v sa= t i s f i e s t h e c o n d i t i o n 33), which t o g e t h e r w i t h (77) w i l l y i e l d (79) (

c

Qm

OGi

Ai(ai)

ql+

hlbvln+lql)Dp+qs V (O)=O,V

q

E

Nn, 191 G R,v

E

N,

v>p

But (35) and (79) w i l l y i e l d

where f o r any g i v e n u E N i t i s p o s s i b l e t o choose vo,...,vk (81)

vO,...,vR

E N, such t h a t

>a

...,

As ho, h , A , a o , ..., am a r e g i v e n f i x e d constants, t h e r e l a t i o n s (80) and (81) t o g e t h b w i t h ( 5 1 ) w i l l i m p l y t h a t

x =o

(82) since u

E

N i n (81) i s a r b i t r a r y .

Now (80) and (82) w i l l y i e l d C OGi

+(ai)

Qm

k

= 0, V k E N, k Q t

which as i n t h e p r o o f o f Theorem 1, S e c t i o n 2, w i l l f i n a l l y i m p l y (83)

x0

=

... =

Am

0

Since (82) and (83) c o n t r a d i c t ( 7 4 ) , t h e p r o o f i s completed.

0

Corollary 3 The f a m i l y (DP6(ax)

I

a E R'\(01)

o f t r a n s f o r m s o f t h e p a r t i a l d e r i v a t i v e s o f any g i v e n o r d e r p E Nn o f t h e Dirac 6 d i s t r i b u t i o n , together w i t h the generalized Dirac 6 d i s t r i b u t i o n p a r t i a1 d e r i v a t i v e w

DP6

a r e l i n e a r independent w i t h i n t h e M,, t r a n s f o r m a l g e b r a s Am(V,S

@ T).

CHAPTER 10 SUPPORT AND LOCAL PROPERTIES

0.

Introduction

An e s s e n t i a l p r o p e r t y o f t h e p r o d u c t w i t h i n t h e q u o t i e n t a l g e b r a s encoun= t e r e d so f a r , i s i t s l o c a l c h a r a c t e r . T h i s p r o p e r t y extends t h e known p r o p e r t 4 o f those p r o m which can be 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 i n U'(R ) , f o r i n s t a n c e

Y S

(1)

E

U'(Rn)), Y E Cm(Rn)

:

supp (yc.S) c supp Y n supp S

o r more general (see [ 17,76,82-85,101])

Y S,S',T,T'

E

U'(Rn)), G c Rn non-void, open : =*

S.T = SIT on G

I n order t o present the l o c a l properties o f the product w i t h i n the quotient algebras i t i s useful t o e x t e n d t h e n o t i o n o f s u p p o r t o f a d i s t r i b u t i o n t o t h e case o f elements o f these q u o t i e n t algebras. W i t h t h e h e l p o f t h e mentioned extended n o t i o n o f s u p p o r t s e v e r a l o t h e r l o c a l p r o p e r t i e s o f elements i n q u o t i e n t a l g e b r a s w i l l be presented.

1.

Support o f Elements i n Q u o t i e n t Algebras

Suppose f o r a c e r t a i n f i x e d 11 S e c t i o n 2, Chapter 1)

A = A/Z

(3) If S

E

E

AL R

c

E

8,

we a r e g i v e n a q u o t i e n t a l g e b r a (see

(0)

A and E c R , we s h a l l say t h a t S vanishes on E, o n l y i f

3 s €A:

(4)

* ) S = s + Z

**) s v = 0 on E, Y v

E

N, v 2 p

f o r a c e r t a i n ~.rE N. The s u p p o r t o f S w i l l be c a l l e d t h e c l o s e d subset

265

266

Rosinger

E.E.

supp S = R

(5)

\ {x E

n

I

S vanishes on a neighbourhood o f x }

Proposition 1

R (n) t h e above n o t i o n o f s u p p o r t i s i d e n t i c a l w i t h t h e

For functions i n C usual one. Proof

I t f o l l o w s from (23) i n Chapter 1.

0

The l o c a l c h a r a c t e r o f t h e m u l t i p l i c a t i o n and a d d i t i o n i n t h e q u o t i e n t a l = g e b r n presented i n t h e n e x t two theorems , w h i c h e extensions o f t h e p r o p e r t i e s i n (1) and ( 2 ) Theorem 1 I f S,T E A t h e n T ) c supp S u supp T

1)

supp ( S

2)

supp (S.T) c supp S n supp T

t

Proof I t f o l l o w s from a d i r e c t v e r i f i c a t i o n .

0

Before p r e s e n t i n g Theorem 2, we s h a l l i n t r o d u c e t h e f o l l o w i n g d e f i n i t i o n . Given S,S' E A and E c R, we say t h a t S = S ' on E

(6) only i f S

-

S ' vanishes on E.

Theorem 2 Suppose S,S',T,T'

E

A and E c

1)

S t T = S ' t T ' o n E

2)

S.T = S ' . T ' on E

n. I f

S = S ' and T = T ' on

E, then

Proof I t f o l l o w s from a d i r e c t v e r i f i c a t i o n .

0

Corollary 1 The e q u a l i t y on E o f elements i n A i s an equivalence r e l a t i o n on A which i s compatible w i t h i t s a l g e b r a s t r u c t u r e .

A u s e f u l e x p l i c i t e x p r e s s i o n o f t h e s u p p o r t i s presented now. Theorem 3 If S E A =

(7)

A / l , then

-

supp S = w €n7 c l v l ~ m msupp (sv t wv)

26 7

SUPPORT

f o r any

SE

A , such t h a t S = s

t

I

E

A.

Proof F i r s t , we prove t h e i n c l u s i o n t o t h e r i g h t hand t e r m o f ( 7 ) .

v

(8)

n supp ( s , + w),

C

=

.

Assume g i v e n X E Q which does n o t b e l o n g Then, t h e r e e x i s t s w E 1 , such t h a t

8, YV

E

N, v > i ~ ,

where V i s a c e r t a i n neighbourhood o f x and

i~

E N i s s u i t a b l y chosen.

B u t t h e r e l a t i o n ( 8 ) o b v i o u s l y i m p l i e s t h a t S vanishes on V , t h e r e f o r e x g supp s. supp S . Then The converse i n c l u s i o n 3 r e s u l t s as f o l l o w s . Assume t h a t x by d e f i n i t i o n t h e r e e x i s t s a neighbourhood V o f x and a sequence o f func= t i o n s s E A, such t h a t (see ( 4 ) ) S = s + I and s v = O o n V , Y v ~ N ,v>p,

(9)

for a certain p

E

N.

But the r e l a t i o n ( 9 ) obviously implies t h a t x

+

-

cl lim v+m

supp s v

t h e r e f o r e x cannot b e l o n g t o t h e r i g h t hand t e r m o f ( 7 ) .

0

I n t h e case o f t h e chains o f q u o t i e n t a l g e b r a s used i n Chapter 7, 8 and 9, some a d d i t i o n a l p r o p e r t i e s o f t h e s u p p o r t a r e presented now. F i r s t , an i m p o r t a n t f e a t u r e o f t h e s u p p o r t o f t h e D i r a c 6 d i s t r i b u t i o n and i t s p a r t i a i d e r i v a t i v e s w i l l r e s u l t i n case t h i s d i s t r i b u t i o n i s represen= t e d b y a c -smooth 6 -sequence which s a t i s f i e s t h e c o n d i t i o n o f s t r o n g sence on t h e s u p p o r t (see ( 1 2 ) i n Chapter 8 ) . T h i s f e a t u r e - p=*r e s e n t e n e x t i n Theorem 4 - g i v e s an a l t e r n a t i v e i n s i g h t i n t o t h e meaning o f t h e mentioned c o n d i t i o n of s t r o n g presence on t h e s u p p o r t . Theorem 4 Within t h e chains o f q u o t i e n t algebras

A'(

v,s

@ T),

witha

E

R,

@

c o r r e s p o n d i n g t o C m - r e g u l a r i z a t i o n ( v,s T ) d e f i n e d i n ( 3 3 ) , Chapter 8, t h e p a r t i a l d e r i v a t i v e s Dp6, o f any g i v e n o r d e r p E Nn, of t h e D i r a c 6 d i s = t r i b u t i o n have t h e f o l l o w i n g p r o p e r t i e s :

1)

supp DP6 = ( O I E Rn

2) 3)

DP6 vanishes on any Ec Rny such t h a t 0

(10)

v c 1; n vm DP6 does n o t v a n i s h on f O I C Rn, i n case t h a t t h e f o l l o w i n g c o n d i t i o n i s s a t i s f i e d (see (44) i n Chapter 8):

4)

4

cl E

Dp6 does n o t v a n i s h on Rn\ EO3, i n case t h a t t h e f o l l o w i n g c o n d i t i o n i s s a t i s f i e d (see (34) i n Chapter 8):

268

Rosinger

E.E.

vcI;n!r

(11) Proof

Assume t h a t (see S e c t i o n 1, Chapter 8) (V,S

6;J ) E

RG,

. Then, i n view o f (10) and (29) i n Chapter 8, f o r a c e r t a i n given s E as w e l l as Theorem 3, S e c t i o n 3, Chapter 3, we o b t a i n t h e r e l a t i o n

@ T) E

DPg = Dps t IR(V,s

(12)

AR(V,s

(5J ) , V R

E

Now, 1) and 2) w i l l r e s u l t e a s i l y from (12) above, as w e l l as (11) i n Chapter 8. 3)

Assume t h a t i t i s f a l s e and DPg vanishes on Rn tion, there e x i s t s a representation DPg = t t

(13)

IR(v,s @ T ) ,

with t

E

\@I. Then, by d e f i n i =

AR(V,s

&?J )

such t h a t t

(14)

=

o

on R~ \ l o } ,

v

R E N, v 2

u

f o r a c e r t a i n P E N. B u t t i s a sequence o f continuous f u n c t i o n s on Rn, t h e r e f o r e (14) y i e l d s = 0 on R

t,

n

Then o b v i o u s l y t E 11 n

(15)

t

vc

E

, V R E N, v > u v",

I%,S

hence i n view o f ( l o ) , we o b t a i n t h a t

@T)

But t h e r e l a t i o n s (13) and ( 1 5 ) w i l l c o n t r a d i c t 1 ) i n Theorem 1, S e c t i o n 2, Chapter 8. 4) (16)

Assume t h a t i t i s f a l s e and t h e r e e x i s t s a r e p r e s e n t a t i o n (13) such that t,(O)

f o r a suitable

(17)

u

w = t

= 0,

V R E N, v > 1.I

E N.

L e t us denote

-

,

DPs

then (12) and (13) w i l l y i e l d

w

E

IR(V.S

@ J)

t h e r e f o r e , i n view o f (20) i n Chapter 3, we o b t a i n v . .w i i R f o r c e r t a i n vi E VL and wi E A (V,S

(18)

W =

C

O < i < h

B u t i n view o f ( l l ) , i t f o l l o w s t h a t

@ J).

SUPPORT

269

m

vi

E VR

c v c 76

therefore, the condition ( 2 ) in Chapter 8, applied t o vO,...,vh viV(O) = 0, Y

(19)

V E

N,

V

E

N.

f o r a certain suitably chosen U'

will yield

>, 1-I'

NOW, the relations (17-19) will imply t h a t

DpsV(O)= 0 , V v

N, v

E

> u' 0

which will obviously contradict (12) in Chapter 8.

We present now in the framework of the chains of quotient algebras in Theorem 4 , an additional information on the general e x p l i c i t expression of the support obtained in Theorem 3. Theorem 5 Suppose the chains of quotient algebras in Theorem 4 s a t i s f y the condition (11). Suppose given for a certain I? E R , an element s = s + 7 R (V,S (u,

N.

L e t us denote w = s 1 - s2 then (23) y i e l d s

(26)

W E 7

But, i n view of (24) and ( 2 5 ) we o b t a i n f o r v E N, v 2 p

slv(x) - sZv(x)

if x

E

SlV(X)

if x

E

Now, i f we t a k e E = G1 \ G 2 and F = tain d(E,

a \F)

= d(G1\G2,

a \(G2\G1)

\(G1

, the U

relation

G2)

G2\ G1

then i n view o f ( 2 2 ) we ob=

G2\G1) > 0

t h e r e f o r e a c c o r d i n g t o Lemma 1 below, t h e r e e x i s t s Y E Cm(Q) such t h a t

E.E.

272

-4

if x

E

Rosinger

G2\ GI

Y(x) =

(28)

i f x E G1 \ G2

.$ Then we denote

w ' = U(Y).W

(29)

and i n view o f (26), as w e l l as (23) i n Chapter 1, we o b t a i n 1

W'E

Therefore s = (sl t s 2 ) / 2 t

(30)

w'E

A

and i n view o f (24) and (25) s = s t 7 E A = A / 1

(31)

B u t t h e r e l a t i o n s (27-30) y i e l d

(32)

s,,

= 0 on G1 U

G2, V v

E

N,

>

v

and (31) t o g e t h e r w i t h (32) w i l l i m p l y t h a t

G1

U

G2

Gs

E

Lemma 1 Suppose g i v e n t h e subsets E C F

C

R such t h a t

d(E, Q \ F) > 0 then there e x i s t s Y

E

C"(R) w i t h t h e f o l l o w i n g p r o p e r t i e s

1)

O Q Y G l o n a

2)

Y = l o n E

3) 4)

Y = O o n R \ F Y E D(Q) i f E i s bounded

Proof L e t us d e f i n e

x

: Rn

x

(0,m)

-+

R ' by

KE exp(E2/(llxl12

X(X,E)

E'))

i f IIxII
0

then

G i u G2

E

GS

Proof We s h a l l use t h e n o t a t i o n s i n t h e p r o o f o f Theorem 7 I n view o f Lemma 1, t h e r e e x i s t s

-j

if x

E

\y E

G2 \ G1

Y(x) =

4

C"(Q)

if x E Gi

such t h a t

GI

2 74

E.E.

Rosinger

I n t h i s case, t h e r e l a t i o n (32) i n t h e p r o o f o f Theorem 7 w i l l y i e l d sV

= 0 on G i

U

G2, Y v E N , v 2~ a

which completes t h e proof. Theorem 9 Suppose g i v e n S E A = f ies t h e c o n d i t i o n s

and an open subset G c R \ supp S which s a t i s =

A/7

(36)

cl G n supp S = fl

(37)

c l G i s compact

then

G

E

GS

Proof I n view o f (36) i t f o l l o w s t h a t Y x E c l G : (38)

]EX

:

E(0,m)

B ( x , E ~ )n R

E

Gs

where we denote B ( x , p ) = {y E Rn

for x

E

Rn,

I

Ily-xll < p }

p E ( O , m ) and II II t h e E u c l i d e a n norm on Rn.

But i n view o f (37) we o b t a i n

f o r c e r t a i n xO,

...,xh

E c l G, which can be assumed p a i r wise d i f f e r e n t .

Now, i f h=O t h e n t h e p r o o f i s completed, s i n c e (38) and (39) w i l l y i e l d G

C

cl G

C

B ( x ~ , E /2) ~ nR 0

Assume then h = l .

E

Gs

We s h a l l denote

G I = B ( x ~ , E ~n~R, ) Gi = B ( x ~ , E ~ ~n/ R~ )

) nR

G2 = B(xo,cx 0

I n view o f (38) t h e s e t s G1,Gi and G2 s a t i s f y t h e c o n d i t i o n s i n Theorem 8, t h e r e fo r e

275

SUPPORT

i f (39) i s taken i n t o account, thus t h e p r o o f i s a g a i n completed.

1 denote

Assume f u r t h e r t h a t h=2.

= B ( x ~ , E /~2 ) 2

G2 = ( B ( x

fli?

/2)) n n 1 follows that

,E

EX

xo Then, i n view o f (38) and O

Thus, t h e s e t s G , G j and G2 s a t i s f y t h e c o n d i t i o n s i n Theorem 8. T h e r e f o r e t h e r e l a t i o n ( 4 0 j w i l l f o l l o w again, i f (39) i s t a k e n i n t o account. I n t h i s way, t h e p r o o f i s a g a i n completed. I t i s easy t o see t h a t t h e above procedure can b e used f o r any h E N, h >D3 i n (39).

Corollarv 3 Suppose g i v e n S E A = A / l perties are equivalent:

(41)

and a s u b s e t H c R, t h e n t h e f o l l o w i n g two p r o =

supp S c H

and

Y

(42)

K

R \H, K compact:

c

3GEGS: K C G

Proof Assume t h a t ( 4 2 ) i s v a l i d and x E R \ H. Then o b v i o u s l y x E G, f o r a c e r = supp S, s i n c e G i s open b y d e f i n i t i o n . I t f a l = tain G E G Therefore x lows i n th?s way t h a t ( 4 1 ) h o l d s .

.

The converse r e s u l t s from Theorem 9, n o t i c i n g t h a t R

\ supp S i s open.

Corollary 4 Suppose g i v e n S E A = lent:

(43)

supp

s

=

A/l

, then

t h e f o l l o w i n g two c o n d i t i o n s a r e equiva=

g

and

Y G c R , G open, bounded :

(44)

G

E

GS

E.E. Rosinger

2 76 Proof I t f o l l o w s from C o r o l l a r y 3.

0

Theorem 10 Suppose g i v e n S E A =

v

Y E

4 / 1 and a c l o s e d subset F c R.

Then, t h e c o n d i t i o n

D(R) :

(45) C R \F * Y . S

supp Y

= 0 € A

implies the condition (46)

supp S C F

Moreover, i n case I i s c o f i n a l i n v a r i a n t (see (61) i n Chapter 2 ) , t h e con= d i t i o n s (45) and (46) a r e e q u i v a l e n t . Proof Assume t h a t (45) i s v a l i d and x E R \F. Since F i s closed, i t f o l l o w s t h a t t h e r e e x i s t s Y E D(Q), w i t h supp Y c R \ F and a neighbourhood V o f x, such t h a t (47)

Y = l o n V

But, i n view o f t h e h y p o t h e s i s \y.S = 0

E

A

t h e r e f o r e , g i v e n any r e p r e s e n t a t i o n (48)

S = s t l

E

A =

A/I,withsEA

we o b t a i n (49)

u(Y).s

E

A

Now, t h e r e l a t i o n s (48) and (49) y i e l d

(50)

s

= u(i-i)s t

r

E

A

L e t us denote t = u(1-Y)s

t h e n (47) y i e l d s tv=OonV,VvEN which t o g e t h e r w i t h (50) i m p l i e s t h a t x p a r t i s completed.

$ supp

S and t h e p r o o f o f t h e f i r s t

Assume now t h a t (46) i s v a l i d and 7 i s c o f i n a l i n v a r i a n t . Then l e t us t a k e Y E D(Q) such t h a t supp Y C il \F. B u t supp Y i s compact, t h e r e f o r e C o r o l =

SUPPORT

277

l a r y 3 i m p l i e s t h e e x i s t e n c e o f G E Gs such t h a t supp Y c G

(51)

S i n c e b y d e f i n i t i o n S vanishes on G t i s f i e s the condition

sv

(52)

= 0 on G, V

for a certain p sume t h a t

(53)

E

N.

v

% we

E

can assume t h a t s i n (48) sa=

N, v > p

E

Moreover, s i n c e 7 i s c o f i n a l i n v a r i a n t , we can as=

p = o

Then, t h e r e l a t i o n s (51-53) w i l l y i e l d = u(0) E

U(Y).S

Q

hence (48) w i l l imply Y . S = 0

E

A.

Corollary 5 Suppose g i v e n S

v

Y

E

E

A =

u(n)

supp Y n

I f 7 i s c o f i n a l i n v a r i a n t then

A/7. : SUPP

S =

0

=*

\y.S =

0 E A

Proof 0

I t f o l l o w s from t h e second p a r t o f Theorem 10.

An i m p o r t a n t decomposition p r o p e r t y f o r t h e elements o f q u o t i e n t a l g e b r a s i s presented now. Theorem 11 Suppose g i v e n S

(54)

E

A = A/7

and I i s c o f i n a l i n v a r i a n t .

If

supp S = F U K w i t h F closed, K compact and F n K =

0

then t h e f o l l o w i n g decomposition h o l d s

s

=

SF

SK

where SF, SK

E

A =

A/I

(55)

and

(55.1)

supp SF n supp SK =

(55.2)

K nsupp

sF

0, w i t h

= F n supp

supp SK compact

sK= 0

Proof Assume t h e open subsets G1,G2,G3,G4

(56)

K c Gl,cl pact

G1

C

C

R such t h a t

G2, c l G2 C G3, c l G3

C

G4, F n c l G4 = 0, c l G4 conp

278

E.E.

Rosinger

L e t us denote K1 = ( c l G4)\G1 then obviously K1 n supp S =

0 and K1 i s compact

t h e r e f o r e , i n view o f C o r o l l a r y 3, t h e r e e x i s t s G Then b y d e f i n i t i o n , t h e r e e x i s t s a r e p r e s e n t a t i o n

+ I€ A

S = s

E

Gs such t h a t

K1

c G.

W l , with s E A

=

such t h a t = 0 on G,

sv

for a certain p t h a t p = 0.

N.

E

V

V E

N, v

>p,

But 7 i s c o f i n a l i n v a r i a n t , t h e r e f o r e we can assume

Now i n view o f ( 5 6 ) , Lemma 1 g r a n t s t h e e x i s t e n c e o f YF YK E D(n) such t h a t

YF = 1 on n \G4,

YF = 0 on c l G3

YK = 1 on c l G1,

YK = 0 on

E

C"(n)

and

R \G2

Then o b v i o u s l y

s = U(YF).S +

U(YK).S

t h e r e f o r e , i f we d e f i n e SF = u(YF)s

+ 1

SK

+

= u(YK)s

E A

7 E A

t h e r e l a t i o n s ( 5 5 ) , (55.1) and (55.2) w i l l obvious h o l d . 3.

Equivalence between S=O and supp S=0

Given S E A = A l l , t h e i m p l i c a t i o n S = 0 E

A =. supp

S = 0

i s obvious. The converse i m p l i c a t i o n i s proved t o h o l d under t h e f o l l o w i n g c o n d i t i o n s . Theorem 12 Suppose g i v e n S E (57)

A

= A l l and 7 i s c o f i n a l i n v a r i a n t . Then t h e c o n d i t i o n

S = O E A =

A/I

i s e q u i v a l e n t t o t h e f o l l o w i n g two ones (58)

supp

s

=

0

SUPPORT

(59)

2 79

S vanishes o u t s i d e o f a compact subset o f

R

Proof Assume t h a t (58) and (59) a r e v a l i d .

Then t h e r e e x i s t s a r e p r e s e n t a t i o n

I E A = A / l , w i t h s ~A ,

S = s +

such t h a t

su

R \K, Y u

= 0 on

E

N, v > p ,

f o r c e r t a i n p E N and compact subset K c R. since I i s cofinal invariant.

B u t one can assume t h a t 1-1 = 0,

L e t us now t a k e Y E o(R), such t h a t Y = l o n K then obviously

s =

U(Y).S

therefore '4.S = S E

(60)

A

0

B u t supp Y n supp S = 2, w i l l i m p l y t h a t

lJ.S = 0

E

= A/I

i n view o f ( 5 8 ) .

Therefore Corollary 5 i n Section

A

which t o g e t h e r w i t h (60) completes t h e p r o o f o f ( 5 7 ) . The converse i s obvious.

4.

0

Domains o f S o l v a b i l i t y f o r Polynomial N o n l i n e a r PDEs

The n o t i o n o f s u p p o r t d e f i n e d f o r t h e elements o f q u o t i e n t a l g e b r a s ( 3 ) o f = f e r s a n a t u r a l way f o r d e f i n i v g domains of s o l v a b i l i t y f o r p o l y n o m i a l non= l i n e a r PDEs. Suppose g i v e n t h e m-th o r d e r polynomial n o n l i n e a r PDE i n (1) , Chapter 1, which we s h a l l c o n s i d e r w i t h i n t h e f o l l o w i n g framework (see S e c t i o n 3, Chap= t e r 1 and S e c t i o n s 0 and 5, Chapter 2 ) : (61)

T(D) :

E * A

where

,A

E

= S/V E VSF

(61.2)

E

m QA

(61.3)

F

v e c t o r subspace i n M ( R ) , G subalgebra i n M(R) and G 3

(61.1)

Then, t h e c l o s e d s u b s e t i n

= A / l E ALG

R given by

C"(n)

zao

E.E. Rosinger

rE+ A

(62)

supp (T(D)S-f)

n

=

S E E

will be c a l l e d the s i n g u l a r i t y i n E

+

A of t h e mentioned PDE.

Obviously, in case t h e mentioned PDE has a sequential s o l u t i o n i n E + A , then rE A = 0. +

The open subset in (63)

given by

n E + A = n \

rE

+ A

w i l l be c a l l e d the domain of s o l v a b i l i t y i n E

-+

A of t h e mentioned PDE.

The r e s u l t s on support obtained i n Sections 1-3 lead t o the following ex= l i c i t expressions f o r t h e subsets o f s i n g u l a r i t y and domain of s o l v a b n i t y h o l y n o m i a l nonlinear PDE, expressions which a r e p a r t i c u l a r l y conve= n i e n t in order t o study t h e v a r i a t i o n of t h e mentioned subsets when t h e i r dependence on E and A i s considered. Theorem 13 The following r e l a t i o n s hold (64)

rE

+

A

E + A

(65)

n

= s Q s w Q 7 cl v - +SUPP ~ (T(D)sV

=

-

f

-

wV)

u u int lim (n \ s u p p ( T ( D ) s V - f SESWE'I v - f m

- wv)

Proof The r e l a t i o n (64) follows e a s i l y from ( 6 2 ) , as well as Theorem 3 in Section 1. The r e l a t i o n (65) follows e a s i l y from (63) and ( 6 4 ) .

0

In view of Theorem 13, i t i s obvious t h a t the domain of s o v a b i l i t y w i l l increase and correspondingly the s i n g u l a r i t y w i l l decrease , whenever S and l i n ( 6 1 . 1 ) increase.

F I N A L REMARKS

The quotient algebras of the chains ( 2 4 ) or (93) in Chapter 3, used within the Chapters 3-9, are particular cases o f the quotient algebras (1)

A = A/I

defined in Section 2 , Chapter 1, by the inclusion diagrams

(2)

i-iP-

+

> G

N

.rnuG=q UG

Indeed, besides the f a c t t h a t in the case of the quotient algebras of the mentioned chains we have (3)

G =

c"(n) or G

c'(n)

the essential particularity i s t h a t the ideals I in A s a t i s f y also the con= di t i on (4)

I i s an ideal in G

N

As seen in Sections 4 and 5 , Chapter 3, as well as in Chapter 6 , t h i s par= t i c u l a r feature of the ideals 1 in (1) and ( 2 ) proves t o be specially con= venient in establishing basic properties of the quotient algebras of the mentioned chains, properties which depend c r i t i c a l ly on the structure of the ideals I . However, as seen in Proposition 2 , Appendix 4 , the rather simple case of the Cauchy-Bolzano quotient algebra (1,2) defining the real numbers, leads t o an ideal 1 in A which does not s a t i s f y the above condition ( 4 ) , f o r G = Q. In other words, the sequential completion of Q i n the usual metric topology requires the f u l l generality of the quotient algebras ( 1 . 2 ) . I t i s therefore natural t o assume that a deeper study of the sequentia1,in particular weak solutions of polynomial nonlinear PDEs will also require quotient algebras (1-3) o f a general form, not necessarily satisfying the condition ( 4 ) . The d i f f i c u l t problem arising here i s the lack of s u f f i c i e n t knowledge concerning the structure of ideals in subalgebras of the algebra of continuous functions on a completely regular topological space.

Even in the case of general quotient algebras (1-3) a further objection 283

E.E. Rosinger

282

m i g h t be r a i s e d . o f functions

(5)

Indeed, as seen i n S e c t i o n 4, Chapter 1, t h e sequences

W E 1

a r e ' e r r o r sequences o r ' n e g l i g i b l e ' sequences o f f u n c t i o n s see S e c t i o n 2, Chapter 1 , as we 1 as Appendix l ) , w h i l e t h e sequences o f f u n c t i o n s (6)

Z G A

a r e t h e ' a d m i s s i b l e ' sequences o f f u n c t i o n s i n t h e q u o t i e n t a gebra ( 1 , 2 ) . Therefore, t h e c o n d i t i o n r e q u i r e d i n (1,2) t h a t (7)

7

i s an i d e a l i n A

m i g h t i n c e r t a i n cases prove t o be t o o s t r o n g , s i n c e i t means t h a t t h e pro= d u c t between a ' n e g i g i b l e ' sequence o f f u n c t i o n s ( 5 ) and an ' a d m i s s i b l e ' sequence o f f u n c t i o n s ( 6 ) i s always a ' n e g l i g i b l e ' sequence o f f u n c t i o n s . However, t h e l i k e l i h o o d o f t h e above o b j e c t i o n seems t o be r a t h e r s m a l l , s i n c e n a t u r a l and minimal assumptions on Iw i l l i m p l y t h a t I s a t i s f i e s t h e c o n d i t i o n ( 7 ) . Indeed, l e t us suppose t h a t (8)

G i s a subalgebra i n

M(n)

and (9)

N 7 i s a subalgebra i n G

which s a t i s f i e s t h e c o n d i t i o n s

(10)

inuG=2

(11)

7 . UG c 7

Then, w i t h t h e n o t a t i o n s i n S e c t i o n 6, Chapter 1, i t f o l l o w s t h a t

(12)

1 i s an i d e a l i n AG(T)

and t h e i n c l u s i o n diagram i s v a l i d

t h e r e f o r e , we o b t a i n t h e q u o t i e n t a l g e b r a

Moreover, i n t h e sense s p e c i f i e d i n (50-52), Chapter 1, each q u o t i e n t a l = gebra (1,2) i s o f t h e f o r m (14). I t i s w o r t h n o t i c i n g t h a t t h e v a r i a n t i n (12) o f t h e c o n d i t i o n ( 7 ) was i m = p l i e d b y t h e assumptions on 7 g i v e n i n (9-11), assumptions w h i c h a r e natu=

FINAL REMARKS

283

r a l and m i n i m a l . Indeed, (10) i s t h e ' n e u t r i x ' c o n d i t i o n (see Appendix 4) which i s anyhow assumed i n ( 2 ) . F u r t h e r , t h e c o n d i t i o n (11) means t h a t t h e p r o d u c t between a ' n e g l i g i b l e ' sequence o f f u n c t i o n s and an ' a d m i s s i b l e ' f u n c t i o n i s always a ' n e g l i g i b l e ' sequence o f f u n c t i o n s . F i n a l l y , t h e con= d i t i o n ( 9 ) which m i g h t seem t o be t h e s t r o n g e s t and thus t h e most q u e s t i o n = a b l e , means t h a t t h e p r o d u c t o f two ' n e g l i g i b l e ' sequences o f f u n c t i o n s i s always a ' n e g l i g i b l e ' sequence o f f u n c t i o n s . I n case t h e c o n d i t i o n s (9-11) a r e a c c e p t a b l e , an o b j e c t i o n m i g h t y e t a r i s e of t h e subalgebra AG ( I ) . Indeed, i n s p i t e o f t h e connected w i t h t h e

size

f a c t t h a t & ( ? ) i s t h e l a r g e s t subalgebra i n GN f o r which t h e i n c l u s i o n diagram (13) i s v a l i d , i t s s i z e m i g h t prove t o be t o o small from some p o i n t s of view, such as f o r i n s t a n c e t h e g e n e r a l i t y p r o p e m s e c t i o n 5, Chap= t e r 1) o f t h e s e q u e n t i a l , i n p a r t i c u l a r weak s o l u t i o n s f o r polynomial non= l i n e a r PDEs

.

This Page Intentionally Left Blank

APPENDIX 1 NEUTRIX CALCULUS AND NEGLIGIBLE SEQUENCES OF FUNCTIONS

Connected w i t h t h e s t u d y o f v a r i o u s a s y m p t o t i c expansions, J.G. Van d e r Corput, 12161, developed a ' n e u t r i x c a l c u l u s ' meant t o deal i n a general and u n i f i e d way w i t h ' n e g l i g i b l e ' q u a n t i t i e s . The b a s i c i d e a o f h i s method presented i n t h e sequel proved t o be o f a w i d e r i n t e r e s t , b e i n g f o r i n = stance u s e f u l i n t h e t h e o r y o f d i s t r i b u t i o n s [53,541.

-

I n t h e c o n d i t i o n s (20.1) and (21.1) i n S e c t i o n 2, Chapter 1, d e f i n i n g t h e q u o t i e n t spaces r e s p e c t i v e l y algebras o f c l a s s e s o f sequences o f f u n c t i o n s on domains i n t u c l i d i a n spaces, we have a l s o made use o f t h e n o t i o n o f ' n e g l i g i b l e ' sequences o f f u n c t i o n s . An o t h e r example can be seen i n Ap= pendix 4, where R ' i s d e f i n e d as a q u o t i e n t a l g e b r a o f c l a s s e s o f sequences o f r a t i o n a l numbers, a c c o r d i n g t o t h e Cauchy-Bolzano method. I n t h a t case t h e ' n e g l i g i b l e ' sequences o f r a t i o n a l numbers w i l l c o i n c i d e w i t h t h e se= quences o f r a t i o n a l numbers convergent t o zero. And now, t h e d e f i n i t i o n o f n e u t r i x . Suppose g i v e n an a r b i t r a r y non-void s e t X and an A b e l i a n group G. j e c t of o u r s t u d y w i l l be t h e f u n c t i o n s

f : X

+

The ob=

G

i n o t h e r words, t h e elements o f t h e C a r t e s i a n p r o d u c t

which i s i n a n a t u r a l way a l s o an A b e l i a n group. The problem i s t o d e f i n e i n a s u i t a b l e and general way t h e n o t i o n o f ' n e g l i = g i b l e ' f u n c t i o n f E Gx.

A g i v e n subgroup (2)

N C Gx

w i l l be c a l l e d a n e u t r i x , only i f YfEN,yEG:

(3)

i'

;(I)x=:J

=+

Y

=

0

i n which case t h e f u n c t i o n s f E N w i l l be c a l l e d N - n e g l i g i b l e .

285

-

2 86

E.E.

Rosinger

The i n t e r e s t i n t h e above n o t i o n comes from t h e f a c t t h a t i n case X has a d i r e c t e d p a r t i a 1 o r d e r < , one can d e f i n e a n e u t r i x l i m i t f o r f u n c t i o n s i n GX as f o l l o w s . Suppose g i v e n a n e u t r i x N c GX, a functi'on f E Gx and y E G. Then we d e f i n e N-

l i m f(x) = y v + w X o n l y i f t h e f u n c t i o n g E G d e f i n e d by

(4)

-

g(x) = f ( x )

yt x E

x,

i s knegligible. I n view o f ( 3 ) i t i s easy t o see t h a t t h e l i m i t ( 4 ) i s unique, whenever i t exists. The c o n d i t i o n ( 3 ) d e f i n i n g a n e u t r i x can be g i v e n t h e f o l l o w i n g a l g e b r a i c c h a r a c t e r i z a t i on. L e t us d e f i n e t h e group monomorphi sm: u : G

+

Gx

and denote UG = u(G) Then UG i s t h e subgroup o f -_constant functions i n G

X

.

L e t us denote by

2 t h e n u l l subgroup i n G

X

Proposition 1

A subgroup N c GX i s a n e u t r i x o n l y i f t h e i n c l u s i o n diagram NL

G

T

x

s a t i s f i e s the condition

o r e q u i v a l e n t l y t h e mapping

d e f i n e d by

NEUTRIX CALCULUS

287

i s a group monomorphism, where 8 i s t h e canon c a l q u o t i e n t epimorph sm. Proof It follows easily.

0

I t i s w o r t h n o t i c i n g t h a t t h e c o n d i t i o n ( 7 ) i s t h e o p p o s i t e o f t h e condi= t i o n t h a t t h e c h a i n o f group homomorphisms U

---+G

X

e

x /N-0

(8)

0 +G

-+G

i s exact.

Indeed, ( 7 ) i s e q u i v a l e n t t o ( 6 ) , w h i l e ( 8 ) i s e q u i v a l e n t t o

(9)

UG = N

X I n view o f ( 7 ) and ( 4 ) , we can i n t e r p r e t G / N as t h e s e q u e n t i a l c o m p l e t i o n o f G o b t a i n e d by u s i n g sequences i n G w i t h i n d i c e s i n t h e d i r e c t e d s e t X . I f we t a k e now

X = N and G = M(R)

then t h e i n c l u s i o n diagrams (20) and (21) i n Chapter 1 a r e p a r t i c u l a r cases o f ( 5 ) above, w h i l e t h e c o n d i t i o n s (20.1) and (21.1) i n Chapter 1 a r e iden= t i c a l w i t h ( 6 ) above. Moreover t h e s e q u e n t i a l s o l u t i o n s of polynomial n o n l i n e a r PDEs d e f i n e d i n S e c t i o n 3, Chapter 1, can be seen as n e u t r i x l i m i t s i n t h e sense o f ( 4 ) above. Indeed, suppose g i v e n t h e m-th o r d e r polynomial n o n l i n e a r PDE (see (1) i n Chapter 1) (10)

T(D)u(x) = f ( x ) , x E R,

w i t h continuous c o e f f i c i e n t s and r i g h t hand t e r m and l e t us c o n s i d e r T(D) : E

-+

A

where

m E

S/V

E

VSmm

c (Q)

,A

= A / 7 E AL e(n) and

E


0 , such t h a t Vu,v

E

B, A t

E

Further, f o r each bounded

(0,TI:

An important c l a s s of e x p l i c i t d i f f e r e n c e schemes s a t i s f y i n g t h e conditions ( 6 . 1 ) and ( 6 . 2 ) a r e those of the form (7)

CAtu = u

+ Atf(At,u), f o r

U E

x,

where f : [ O , T l x X + X i s continuous and s a t i s f i e s the local Lipschitz con= dition

VB c X bounded :

IIf( & , u )

-

f(At,v)ll

< M(B)llu-vll.

Usually, C u represents a function of d i s c r e t i z e d space v a r i a b l e , associa= ted t o t h e A h n c t i o n u E Xo of continuous space v a r i a b l e . In t h a t case,Cat

295

NONLINEAR STABILITY AND CONVERGENCE

w i l l a l s o depend on t h e f i n i t e space increments

and t h e aim o f t h e s t a b i l i t y a n a l y s i s i s t o e s t a b l i s h a r e l a t i o n between Ax and A t , which w i l l g r a n t t h e convergence o f t h e d i f f e r e n c e scheme ( 6 ) t o t h e s o l u t i o n o f ( l ) , p r o v i d e d t h a t A t -+ 0. The d i f f e r e n c e scheme ( 6 ) i s c a l l e d c o n s i s t e n t w i t h t h e problem ( l ) , o n l y i f f o r any compact K c Xo and E > 0, t h e r e e x i s t s ~ ( K , E ) > 0, such t h a t Yu E K, t E [O,T],

A t E (O,T]

:

(8) At

0, such t h a t Yu E K, t (10)

n where Cat

At,

[O,T],

N,

A t E (O,T],

n

E

1 t - n . A t l < q ( k , ~ ) * IIC:t~

-

E ( t ) u l l <E

E

m o a t

denotes t h e n - t h successive i t e r a t i o n o f CAt:

GT :

Xo

-+

Xo.

The above d e f i n i t i o n o f convergence c o i n c i d e s i n t h e l i n e a r case w i t h t h e usual one [ 116,1651. Indeed, i n t h a t case, a convergent d i f f e r e n c e scheme w i l l under t h e c o n d i t i o n s o f Lax-Richtmyer's theorem b e s t a b l e and, as seen i n t h e p r o o f of t h a t theorem, s t a b i l i t y w i l l g r a n t u n i f o r m convergence on [ 0,Tl and bounded B C Xo. The p r a c t i c a l importance o f t h e e q u i v a l e n c e between t h e convergence and s t a b i l i t y of a d i f f e r e n c e scheme i s i n t h e f a c t t h a t t h e s t a b i l i t y o f a d i f f e r e n c e scheme o n l y depends on t h e d i f f e r e n c e scheme i t s e l f and, u n l i k e convergence. i t i s n o t r e l a t e d to t h e p n s s i h l y unknown s o l u t i o n ( 2 ) o r ( 3 ) o f problem (1). I n t h e l i n e a r case 116,165 , t h e s t a b i l i t y o f t h e d i f f e r = ence scheme (6) means t h e uniform boundedness o f t h e family o f l i n e a r opera= tors C,t: w i t h A t E (O,T], n E N, n.At G T . (11) The way t h e e x t e n s i o n i s done i n t h e n o n l i n e a r case i s g i v e n i n t h e f o l l o w = i n g d e f i n i t i o n . The d i f f e r e n c e scheme ( 6 ) i s c a l l e d s t a b l e , o n l y i f f o r

296

E.E.

any compact K c Xo,

Rosinger

0, such t h a t

there e x i s t s L(K)

The e x t e n s i o n o f t h e Lax-Richtmyer equivalence [ 116,1651 between convergence and s t a b i l i t y i s o b t a i n e d as f o l l o w s : Theorem 1 Suppose t h e problem (1) i s p r o p e r l y posed and t h e d i f f e r e n c e scheme ( 6 ) i s c o n s i s t e n t w i t h i t . Then, t h e d i f f e r e n c e scheme ( 6 ) i s convergent t o t h e s o l u t i o n ( 2 ) o r ( 3 ) o f problem ( l ) , o n l y i f i t i s s t a b l e . Proof The i m p l i c a t i o n " c o n v e r g e n t * s t a b l e " . Assume K C Xo compact and denote

B = titu

I

UE

K,At

E ( O , T l , n E N, n A t < T I .

We s h a l l prove t h a t

(13)

BC

Xo bounded.

Indeed, o t h e r w i s e one o b t a i n s u . E K, J such t h a t n A . t g T, j J

(14)

with

j E

A. t E ( O , T l ,

J

N,

n . E N, f o r j, J

N

Since K i s compact and i n view o f ( 1 4 ) , one can e v e n t u a l l y by passing t o subsequences, assume t h a t t h e r e l a t i o n s h o l d

lim

(16)

j+

u ;u E K j

l i m n.A.t = t J J

(17)

j + m

Assume now, t h a t a l s o

then ( 1 5 ) g i v e s n. IICA;,U~ Since u j

E

K, w i t h j

E

-

E ( t ) u . l l t IIE(t)u.lI, w i t h j J J

E

N.

N, t h e r e l a t i o n s ( 1 7 ) and ( 1 8 ) , t o g e t h e r w i t h ( l o ) ,

NONLINEAR STABILITY AND CONVERGENCE

29 7

w i l l give n, j

Jim

+ rn

-

IIC J u . A j t J

E(t)u.ll = J

o

F u r t h e r , (16) and ( 4 ) r e s u l t i n (21)

l i m eE(t)ujll

=

IIE(t)ull

j,,

B u t , ( 2 0 ) and ( 2 1 ) c o n t r a d i c t ( 1 9 ) , thus, t h e assumption ( 1 8 ) i s n o t c o r = r e c t . Then, e v e n t u a l l y b y p a s s i n g t o a subsequence, one can assume (22)

ajt

' 6 > 0,

w i t h j E N.

NOW, ( 1 4 ) and (22) i m p l y nj

' T/6,

N

with j E

hence, b y p a s s i n g e v e n t u a l l y t o a subsequence, one can assume t h a t = n = c o n s t a n t , w i t h j E N. j I n t h a t case, ( 1 5 ) i m p l i e s

(23)

n

(24)

j < IIC,.~U~II n G J n

llC:.tuj - CA.t~II n , w i t h j E N. J J hence (24) i s absurd, s i n c e t h e compact s e t IIC,.tn

UII

I f n=O, t h e n Ca.t = i d X , J 0 w i l l a l s o be bounded. Otherwise, n

w i l l give

t l i m A . t = T~ @,TI,

+

J

> 1 together

K

w i t h ( 1 7 ) , ( 2 2 ) , and (23)

hence, i n view o f (16) and ( 6 . 1 ) one o b t a i n s

j + m J

Iim

IIC,.~UII n

=

IICtn uII -

J

j + c u

n

l i m I I Cn A . ~ U -~ C:.~UII J J

= 0.

j + m

NOW, (25) and (26) w i l l c o n t r a d i c t (24) and t h e p r o o f o f (13) i s completed. I n o r d e r t o p r o v e t h a t t h e a r b i t r a r y compact K c Xo s a t i s f i e s ( 1 2 ) , we s h a l l a p p l y (6.2) t o B g i v e n i n ( 1 3 ) . Assume t h e r e f o r e , u,v E K, A t E (O,T] , n E N, n.At < T. Then, (6.2) a p p l i e d s u c c e s s i v e l y , w i l l g i v e

-

IICitu


11

therefore

T

x = O , V A E N , A > ~

x

I t f o l l o w s i n t h i s way t h a t f o r any g i v e n x

IIT xII

A

,with A

E

X, t h e f a m i l y o f numbers

E N,

i s bounded b y max {IIT xi11 A where

,,

E

x

E

N, 1

,,I

N depends o n l y on x E X.

As t h e normed v e c t o r space (X,II II) i s o b v i o u s l y n o t complete, we can see t h a t t h e r e s u l t i n g framework i s not s u f f i c i e n t f o r g r Z X i n g t h e v a l i d i t y o f t h e

300

E.E.

Rosinger

principle o f uniform boundedness o f linear operators.

APPENDIX 4 THE CAUCHY-BOLZANO QUOTIENT ALGEBRA CONSTRUCTION OF THE REAL NUMBERS

A c l a s s i c a l example o f q u o t i e n t a l g e b r a c o n s t r u c t i o n i s g i v e n b y t h e w e l l known Cauchy-Bolzano method, [218] , f o r c o n s t r u c t i n g t h e s e t o f r e a l num= b e r s R 1 f r o m t h e s e t o f r a t i o n a l numbers Q. L e t us denote A N t h e subalgebra i n Q o f a l l t h e Cauch sequences r = (ro,rl,. r a t i o n a l numbers and l e t us denote b y

. . ,rv,.. . ..)

r t h e i d e a l i n A o f a l l t h e sequences z = numbers which converge t o zero.

(Z~,Z~,...,Z~,

....)

o f rational

We s h a l l denote b y UQ

t h e subalgebra i n A o

Q t h e null i d e a l i n Q

if .a l l

t h e c o n s t a n t sequences.

F i n a l l y , we denote b y

Then t h e f o l l o w i n g i n c l u s i o n diagram i s v a l i d

(1)

+-I Q

i

> QN

’ 0‘

and i t s a t i s f i e s t h e c o n d i t i o n

(2)

rnuQ=Q

Moreover, a c c o r d i n g t o Cauchy-Bolzano

(3)

R’ and A = A / l a r e i s o m o r p h i c f i e l d s .

As s en i n Appendix 1, t h e c o n d i t i o n ( 2 ) above means t h a t 1 i s a n e u t r i x i n Qg and t h e c o r r e s p o n d i n g n e u t r i x l i m i t i s i d e n t i c a l w i t h t h e u s u a l l i m i t f o r r a t i o n a l numbers, i . e . , t h e r e l a t i o n h o l d s

301

of

302

Rosinger

E.E.

I

(4)

-

lim

r

W

V

= lim

r

W

.

whenever r = ( royrl ,.. ,rv , . . . . )

E

v N Q and one o f t h e l i m i t s i n ( 4 ) e x i s t s .

The i d e a l I has s e v e r a l i m p o r t a n t 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 n o t i c e t h a t

(5)

7 i s a maximal i d e a l i n A

since A = A / 7 i s a f i e l d . L e t us now denote by

B t h e subalgebra i n Q obviously

N

o f a l l t h e bounded sequences o f r a t i o n a l numbers. Then

A i s a subalgebra i n B

(6)

The s p e c i a l r e l a t i o n between 7 and B i s presented i n t h e n e x t two p r o p o s i = tions. Proposition 1 7 i s a maximal subsequent i n v a r i a n t i d e a l i n B (see (28) i n Chapter 2 )

Proof I _

Assume t h a t i t i s f a l s e and J i s a subsequence i n v a r i a n t i d e a l i n 8, such that

(71

# L e t us t a k e

(8)

B

7C.3 C

#

then

..., V Y "

z =(zoyzl,

...)E J \ I

Since J c 8, t h e r e l a t i o n (8) g i v e s a p o i n t 5 ) i n t = (zoyzl, zv VY""'

(9)

lim w

z'

v

E

R'\COI

..., ,...... ),

2' = ( t ~ , 2 ~ , . . . , Z '

=

and a subsequence such t h a t

5

But

s i n c e 3 i s subsequence assume t h a t

nvariant.

Moreover, i n view o f ( 9 ) we can o b v i o u s l y

I z ' l 2 I 5 1'2 > 0, V V E N . ....) Therefore d e f i n i n g z " = ( z ~ , ~ ~ ,..J;,.

E

QN b y

CAUCHY-BALZANO QUOTIENT ALGEBRA

30 3

Now, t h e r e l a t i o n s (10-12) w i l l y i e l d

1

z'.z" E J . B

=

C

J

hence (13)

J = B

s i n c e J i s an i d e a l B . Since ( 7 ) and (13) c o n t r a d i c t each o t h e r , t h e p r o o f i s completed. Proposition 2

B i s a maximal subalgebra i n Q

N

i n which 7 i s an i d e a l .

Proof Assume t h a t i t i s f a l s e and C i s a subalgebra i n Q

(14)

B

C

N

such t h a t

C

# (15)

I i s an i d e a l i n C

L e t us t a k e t h e n

z

,..., zV,......)

= (z0,z1

E

C \ B

I t f o l l o w s t h a t t h e r e e x i s t s a subsequence z ' (zo,zl ,..., zv ,.... ) , such t h a t

z

(16)

lim lJ-

I

Izv

=

= (zv 0

,..., zv

,z vl

,.....) i n lJ

m

lJ

Obviously, we can assume t h a t (17)

v 0 < v1