Review of Some Essentials in Weak Solutions and Distributions. Classical Versus Distribution Solutions. Impossibility and Degeneracy Results in Distributions. Limitations of the Linear Distribution Theory.
Colombeau's Nonlinear Theory of Generalized Functions. The Differential Algebra G as an Extension of the D' Distributions. Generalized Solutions of Nonlinear Partial Differential Equations. Generalized Solutions for Linear Partial Differential Equations.
A Necessary Structure for Nonlinear Theories of Generalized Functions. Stability, Generality and Exactness of Generalized Solutions. Chains of Algebras of Generalized Functions. Resolution of Singularities of Weak Solutions for Polynomial Nonlinear Partial Differential Equations.
During the last few years, several fairly systematic nonlinear theories of generalized solutions of rather arbitrary nonlinear partial differential equations have emerged. The aim of this volume is to offer the reader a sufficiently detailed introduction to two of these recent nonlinear theories which have so far contributed most to the study of generalized solutions of nonlinear partial differential equations, bringing the reader to the level of ongoing research.
The essence of the two nonlinear theories presented in this volume is the observation that much of the mathematics concerning existence, uniqueness regularity, etc., of generalized solutions for nonlinear partial differential equations can be reduced to elementary calculus in Euclidean spaces, combined with elementary algebra in quotient rings of families of smooth functions on Euclidean spaces, all of that joined by certain asymptotic interpretations. In this way, one avoids the complexities and difficulties of the customary functional analytic methods which would involve sophisticated topologies on various function spaces. The result is a rather elementary yet powerful and far-reaching method which can, among others, give generalized solutions to linear and nonlinear partial differential equations previously unsolved or even unsolvable within distributions or hyperfunctions.
Part 1 of the volume discusses the basic limitations of the linear theory of distributions when dealing with linear or nonlinear partial differential equations, particularly the impossibility and degeneracy results. Part 2 examines the way Colombeau constructs a nonlinear theory of generalized functions and then succeeds in proving quite impressive existence, uniqueness, regularity, etc., results concerning generalized solutions of large classes of linear and nonlinear partial differential equations. Finally, Part 3 is a short presentation of the nonlinear theory of Rosinger, showing its
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- © North Holland 1987
- 1st November 1987
- North Holland
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Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa