The order completion method for systems of nonlinear PDEs revisited

TL;DR

This paper advances the theory of order completion by constructing generalized function spaces that contain solutions to all continuous nonlinear PDE systems, offering new insights into existence, uniqueness, and regularity of solutions.

Contribution

It introduces enriched spaces of generalized functions that encompass solutions to all continuous nonlinear PDEs, extending the basic order completion theory.

Findings

Constructed spaces contain generalized solutions to all continuous nonlinear PDEs

Established existence and uniqueness results for these solutions

Interpreted solutions as regularity results for PDE systems

Abstract

In this paper we presents further developments regarding the enrichment of the basic Theory of Order Completion. In particular, spaces of generalized functions are constructed that contain generalized solutions to all systems of continuous, nonlinear PDEs. In terms of the existence and uniqueness results previously obtained for such systems of equations, one may interpret the existence of generalized solutions presented here as a regularity result.