Wave equations and symmetric first-order systems in case of low regularity

TL;DR

This paper develops a method using Colombeau generalized functions to analyze the transition between wave equations and symmetric hyperbolic systems with low regularity coefficients, enabling new insights into solvability and uniqueness.

Contribution

It introduces a Colombeau-based framework for transferring solvability and uniqueness results between wave equations and hyperbolic systems under low regularity conditions.

Findings

Established transfer of solvability between wave equations and hyperbolic systems.

Proved new uniqueness results for wave equations with non-smooth coefficients.

Provided a framework for analyzing equations with irregular coefficients.

Abstract

We analyse an algorithm of transition between Cauchy problems for second-order wave equations and first-order symmetric hyperbolic systems in case the coefficients as well as the data are non-smooth, even allowing for regularity below the standard conditions guaranteeing well-posedness. The typical operations involved in rewriting equations into systems are then neither defined classically nor consistently extendible to the distribution theoretic setting. However, employing the nonlinear theory of generalized functions in the sense of Colombeau we arrive at clear statements about the transfer of questions concerning solvability and uniqueness from wave equations to symmetric hyperbolic systems and vice versa. Finally, we illustrate how this transfer method allows to draw new conclusions on unique solvability of the Cauchy problem for wave equations with non-smooth coefficients.