Nonlinear Theory of Generalized Functions

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.

Preface Partial Differential EquationsOn 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. ScialomThe Schrödinger Equation with Point Interaction in an Algebra of New Generalized Functions, A. AntonevichPolynomial a Priori Estimates for Some Evolution PDE and Generalized Solutions, H. A. Biagioni, T. GramchevShift Differentials of Maps in BV Spaces, A. Bressan, M. LewickaCalculation of the Singularity Dynamics for Quadratic Nonlinear Hyperbolic Equations. Example: The Hopf Equation, V. G. Danilov, G. A. Omel'yanovVanishing Viscosity Boundary Layers for Nonlinear Hyperbolic Systems, O. GuèsOrdinary Differential Equations and Generalized Functions, R. Hermann, M. OberguggenbergerConservation Laws, Delta Shocks and Singular Shocks, B. L. KeyfitzNonlinear Singular Schrödinger Type Equations, H. Lange, m. Poppenberg, H. TeismannNon-Analytic Solutions of Nonlinear Wave Models, Y. A. Li, P. J. Olver, P. RosenauThe Dirichlet Problem and Compact Operators in Colombeau Theory, D. ScarpalezosHighly Oscillatory Shock Waves, Y. WangStructure TheorySharp Topologies on (C, E, P)-Algebras, A. Delcroix, D. Scarpalezos(C, E, P)-Sheaf Structures and Applications, J.-A. MartiLocal and Microlocal Analysis in the Space of Colombeau Generalized Functions, S. PilipovicBasics of a General Spectral Theory of Banach Modules, N. Y. RadynoExtensions of Algebras, Memofunctions and Their Applications, Y. V. RadynoOn the Multiplication of Periodic Hyperfunctions of One Variable, V. ValmorinGeometry, General Relativity Distributional Aspects of General Relativity: The Example of the Energy-Momentum Tensor of the Extended Kerr-Geometry, H. BalasinLie Symmetries of Differential Equations in a Generalized Functions Setting, M. KunzingerArbitrary Global Lie Group Actions on Generalized Solutions of Nonlinear PDEs and an Answer to Hilbert's Fifth Problem, E. E. RosingerDistributional Description of Impulsive Gravitational Waves, R. SteinbauerNon-Linear Generalized Functions in General Relativity, J. A. VickersStochastic Analysis A White Noise Approach to Stochastic Differential Equations Driven by Wiener and Poisson Processes, H. Holden, B. ØksendalWhite Noise Driven Stochastic Partial Differential Equations: Triviality and Non-Triviality, F. Russo, M. OberguggenbergerMeasurement Methods Related to Differential Equations, J. UbøeOn the Small Time Asymptotics of Solutions of Linear and Non-Linear Stochastic Differential Equations, T. ZhangNonstandard MethodsThe Global Control of Shock Waves, J.E. RubioPointwise Values and Fundamental Theorem in the Algebra of Asymptotic Functions, T. D. Todorov