Solutions of nonlinear PDEs in the completion of uniform convergence spaces

TL;DR

This paper develops a framework for solving nonlinear PDEs using generalized function spaces formed as completions of uniform convergence spaces, providing existence results and analyzing solution properties.

Contribution

It introduces a novel approach to solving nonlinear PDEs via completions of uniform convergence spaces, extending classical methods with new existence and regularity results.

Findings

Existence of solutions in generalized function spaces.

Analysis of solution regularity and structure.

Detailed discussion on the completion of uniform convergence spaces.

Abstract

This paper deals with the solution of large classes of systems of nonlinear partial differential equations (PDEs) in spaces of generalized functions that are constructed as the completion of uniform convergence spaces. The existence result for the mentioned systems of equations are obtained as an application of a basic approximation result, which is formulated entirely in terms of usual real valued functions on open subsets of Euclidean n-space. The structure and regularity properties of the solutions are explained at the hand of suitable results relating to the structure of the completion of uniform convergence spaces that are defined as initial structures. In this regard, we include also a detailed discussion of the completion of initial uniform convergence spaces in general.