Generalized solutions of nonlinear partial differential equations

✍ Scribed by Rosinger, Elemer E

• Publisher: North-Holland, Sole distributors for the U.S.A. and Canada, Elsevier Science Pub
• Year: 1987
• Tongue: English
• Leaves: 422
• Series: North-Holland mathematics studies 146.; Notas de matemática (Rio de Janeiro Brazil) no. 119
• Edition: 1st
• Category: Library

✦ Synopsis

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

✦ Table of Contents

Content: 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.