Oberguggenberger M. Multiplication of Distributions and Applications to Partial Differential Equations

Harlow: Longman Higher Education, 1992. - 332p.This book addresses the theory and applications of mathematical models that involve
(a) nonlinear operations;
(b) differentiation; and
(c) the presence of singular objects (like measures or non-differentiable functions).
Classical nonlinear analysis can certainly deal with (a) and (b) , while the theory of distributions has proved to be successful in handling (b) and (c) simultaneously. First attempts in defining nonlinear operations within distribution theory go back to the early fifties, pressed by the renormalization problem in Quantum Field Theory as it was seen then. It was soon observed that the combination of (a) and (b) with (c) entails that either the differentiation operators or the nonlinear operations cannot be consistent with their classical counterparts. As a consequence, research on "multiplication of distributions" for a long period concentrated on devising particular methods combining (a) and (c), but neglecting (b) to a large extent, while the theory of nonlinear partial differential equations took off in a different direction (predominantly
avoiding (c)).
The subject entered a new era in the late seventies and early eighties when theories of differential algebras larger than the space of distributions were developed, based on the "sequential approach", which subsequently found rapid applications to partial differential equations. These theories can accommodate most of the particular definitions of nonlinear operations with distributions, but more importantly provide
a framework where (a), (b), and (c) can be treated in full generality and where the consistency problem can be clearly understood and dealt with. A new way of looking at generalized functions and partial differential equations emerged: Sequences of regularizations are interpreted as new objects and are studied as members of algebraic entities. More information on the regularization process is kept than in distribution theory, and this information is structured by the particular model under consideration. It should be remarked that Nonstandard Analysis to some extent follows a similar road, and in this respect can be seen as a particular instance of the general theory, also capable of multiplying distributions.
This book deals mainly with these new theories of algebras of generalized functions, focussing above all on those of J.F. Colombeau and also of E.E.Rosinger, and the applications to partial differential equations obtained from 1985 onwards. One chapter is devoted to the "classical" approaches. This appeared appropriate, as no systematic presentation is available elsewhere, and their interrelations and relations with algebras
containing the space of distributions have been clarified only recently.
Though the emphasis of this book is on algebras and their applications to partial differential equations, I have attempted to put many of the related approaches into place and to connect and compare them with each other, thus indicating one of the possible roads through the immense variety of particular methods that have accumulated by now.
The book is intended as an introduction and panorama on the level of current research. It is aimed at people interested in the fields of partial differential equations, nonlinear analysis, generalized functions, theoretical physics, and their applications. The reader is assumed to be familiar with classical distribution theory.