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

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

2008-09-25

arxiv.org/abs/0809.4322v2

Mathematics

Functional Analysis

This is a Master thesis of Guy Burger, under the supervision of Todor D. Todorov, defended in Mathematics Department of Califo

Scientific paper

We construct an algebra of generalized functions $^*\mathcal{E}(\mathbb{R}^d)$. We also construct an embedding of the space of Schwartz distributions $\mathcal{D}^\prime(\mathbb{R}^d)$ into $^*\mathcal{E}(\mathbb{R}^d)$ 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 ${^*\mathcal{E}(\mathbb{R}^d)}$ 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.

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