Distributions and Nonlinear Partial Differential Equations
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684 Elemer E. Rosinger
Distributions and Nonlinear Partial Differential Equations
Springer-Verlag Berlin Heidelberg New York 1978
Author
Elemer E. Rosinger Department of Computer Science Technion City Haifa/Israel
AMS Subject Classifications (1970): 35Axx, 35 Dxx, 46 Fxx
ISBN 3-540-08951-9 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08951-9 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 64 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-643210
to my wife
HERMONA
PRE F ACE
The nonlinear method in the theory of distributions presented in this work is based on embeddings of the distributions in
V'
into associative and commutative algebras whose elements are classes of sequences of smooth functions on Rn. The embeddings define various distribution multiplications. Positive powers can also be defined for cer tain distributions, as for instance the Dirac 0 function. A framework is in that way obtained for the study of nonlinear partial differential equations with weak or distribution solutions as well as for a whole range of irregular operations on distributions, encountered for instance in quantum mechanics. In chapter 1, the general method of constructing the algebras containing the distributions and basic properties of these algebras are presented. The way the algebras are constructed can be interpreted as a sequential completion of the space of smooth funcn. tions on R In chapter 2, based on an analysis of classes of singularities of piece wise smooth functions on Rn, situated on arbitrary closed subsets of Rn with smooth boundaries, for instance, locally finite families of smooth surfaces, the so called Dirac algebras, which prove to be useful in later applications are introduced. Chapter 3 presents a first application. A general class of nonlinear partial differential equations, with polynomial nonlinearities is considered. These equations include among others, the nonlinear hyperbolic equations modelling the shock waves as well as well known second order nonlinear wave equations. It is shown that the piece wise smooth weak solutions of the general nonlinear equations considered, satisfy the equations in the usual algebraic sense, with the multiplication and derivatives in the algebras containing the distributions. It follows in particular that the same holds for the piece wise smooth shock wave solutions of nonlinear hyperbolic equations. A second application is given in chapter 4, Where one and three dimensional quantum particle motions in p
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