Non-Standard Analysis, Multiplication of Schwartz Distributions and Delta-Like Solution of Hopf's Equation

2008/09/25 by Guy Berger, Berger, Guy
Mathematics · #26E35 #30G06 #35D05 #35L65. #35L67 #46F10 #46F30 #46S10 #46S20 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical and Theoretical Analysis #advanced mathematical theories #math.FA #msc:26E35 #msc:30G06 #msc:35D05 #msc:35L65. #msc:35L67 #msc:46F10 #msc:46F30 #msc:46S10 #msc:46S20

paper · pdf · doi:10.48550/arxiv.0809.4322

This is a Master thesis of Guy Burger, under the supervision of Todor D. Todorov, defended in Mathematics Department of California Polytechnic State Universiy, San Luis Obispo, CA-93407, in Cal Poly Library, LD729.6.S 52 M3 B47, 2005

openalex publication_date 2008/09/25 · arxiv created 2008/10/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct an algebra of generalized functions ^*E(ℝ^d). We also construct an embedding of the space of Schwartz distributions D^′(ℝ^d) into ^*E(ℝ^d) and thus present a solution of the problem of multiplication of Schwartz distributions which improves J.F. Colombeau's solution. As an application we prove the existence of a weak delta-like solution in ^*E(ℝ^d) of the Hopf equation. This solution does not have a counterpart in the classical theory of partial differential equations. Our result improves a similar result by M. Radyna obtained in the framework of perturbation theory.

Related

• Distributions with dynamic test functions and multiplication by discontinuous functions
• Generalized functions with infinitesimals
• An extension of the Eilenberg - Mac Lane concept of category in the case of nonrigid structures
• A non-archimedean Algebra and the Schwartz impossibility theorem
• Lecture Notes: Non-Standard Approach to J.F. Colombeau's Theory of Generalized Function: University of Vienna, Austria, May 2006