Generalized solutions to nonlinear first order Cauchy problems

TL;DR

This paper extends the Order Completion Method to establish the existence and regularity of generalized solutions for a broad class of nonlinear first order Cauchy problems, using completion of uniform convergence spaces.

Contribution

It introduces a new framework for analyzing nonlinear first order Cauchy problems by constructing solution spaces as completions of uniform convergence spaces.

Findings

Proves the existence of generalized solutions for a wide class of nonlinear PDEs.

Establishes regularity properties of these solutions.

Provides a unified approach to solving nonlinear first order Cauchy problems.

Abstract

The recent significant enrichment of the Order Completion Method for nonlinear Systems of PDEs resulted in the global existence of generalized solutions to a large class of such equations. In this paper we investigate the existence and regularity of the generalized solutions to a family of nonlinear first order Cauchy problems. The spaces of generalized solutions are obtained as the completions of suitably constructed uniform convergence spaces.