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Nonlinear Theory of Generalized Functions



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ISBN 9780849306495
Published March 16, 1999 by Chapman and Hall/CRC
400 Pages

 
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Book Description

Questions regarding the interplay of nonlinearity and the creation and propagation of singularities arise in a variety of fields-including nonlinear partial differential equations, noise-driven stochastic partial differential equations, general relativity, and geometry with singularities. A workshop held at the Erwin-Schrödinger International Institute for Mathematical Physics in Vienna investigated these questions and culminated in this volume of invited papers from experts in the fields of nonlinear partial differential equations, structure theory of generalized functions, geometry and general relativity, stochastic partial differential equations, and nonstandard analysis.

The authors provide the latest research relevant to work in partial differential equations, mathematical physics, and nonlinear analysis. With a focus on applications, this books provides a compilation of recent approaches to the problem of singularities in nonlinear models. The theory of differential algebras of generalized functions serves as the central theme of the project, along with its interrelations with classical methods.

Table of Contents

Preface
Partial Differential Equations
On the Structure of Singularities in Solutions of the Nonlinear Schrödinger Equation for the Critical Case, p = 4/n , J. Angulo, J. L. Bona, F. Linares, M. Scialom
The Schrödinger Equation with Point Interaction in an Algebra of New Generalized Functions, A. Antonevich
Polynomial a Priori Estimates for Some Evolution PDE and Generalized Solutions, H. A. Biagioni, T. Gramchev
Shift Differentials of Maps in BV Spaces, A. Bressan, M. Lewicka
Calculation of the Singularity Dynamics for Quadratic Nonlinear Hyperbolic Equations. Example: The Hopf Equation, V. G. Danilov, G. A. Omel'yanov
Vanishing Viscosity Boundary Layers for Nonlinear Hyperbolic Systems, O. Guès
Ordinary Differential Equations and Generalized Functions, R. Hermann, M. Oberguggenberger
Conservation Laws, Delta Shocks and Singular Shocks, B. L. Keyfitz
Nonlinear Singular Schrödinger Type Equations, H. Lange, m. Poppenberg, H. Teismann
Non-Analytic Solutions of Nonlinear Wave Models, Y. A. Li, P. J. Olver, P. Rosenau
The Dirichlet Problem and Compact Operators in Colombeau Theory, D. Scarpalezos
Highly Oscillatory Shock Waves, Y. Wang
Structure Theory
Sharp Topologies on (C, E, P)-Algebras, A. Delcroix, D. Scarpalezos
(C, E, P)-Sheaf Structures and Applications, J.-A. Marti
Local and Microlocal Analysis in the Space of Colombeau Generalized Functions, S. Pilipovic
Basics of a General Spectral Theory of Banach Modules, N. Y. Radyno
Extensions of Algebras, Memofunctions and Their Applications, Y. V. Radyno
On the Multiplication of Periodic Hyperfunctions of One Variable, V. Valmorin
Geometry, General Relativity
Distributional Aspects of General Relativity: The Example of the Energy-Momentum Tensor of the Extended Kerr-Geometry, H. Balasin
Lie Symmetries of Differential Equations in a Generalized Functions Setting, M. Kunzinger
Arbitrary Global Lie Group Actions on Generalized Solutions of Nonlinear PDEs and an Answer to Hilbert's Fifth Problem, E. E. Rosinger
Distributional Description of Impulsive Gravitational Waves, R. Steinbauer
Non-Linear Generalized Functions in General Relativity, J. A. Vickers
Stochastic Analysis
A White Noise Approach to Stochastic Differential Equations Driven by Wiener and Poisson Processes, H. Holden, B. Øksendal
White Noise Driven Stochastic Partial Differential Equations: Triviality and Non-Triviality, F. Russo, M. Oberguggenberger
Measurement Methods Related to Differential Equations, J. Ubøe
On the Small Time Asymptotics of Solutions of Linear and Non-Linear Stochastic Differential Equations, T. Zhang
Nonstandard Methods
The Global Control of Shock Waves, J.E. Rubio
Pointwise Values and Fundamental Theorem in the Algebra of Asymptotic Functions, T. D. Todorov

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