1st Edition

Nonlinear Theory of Generalized Functions

    400 Pages
    by Chapman & Hall

    392 Pages
    by Chapman & Hall

    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.

    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


    Michael Oberguggenberger (Inst Mathematic/Geometrie, Innsbruck, Austria) (Edited by) Michael Grosser (University of Vienna, Vienna, Austria) (Edited by) Michael Kunzinger (Edited by) Gunther Hormann (Institut fur Mathematic, Vienna, Austria) (Edited by)