Symmetrisers and generalised solutions for strictly hyperbolic systems with singular coefficients

TL;DR

This paper develops a framework for solving strictly hyperbolic systems with non-smooth coefficients using Colombeau generalized functions, extending classical results to cases where traditional solutions may not exist.

Contribution

It introduces generalized strict hyperbolicity, constructs symmetrisers, and proves existence, uniqueness, and regularity of solutions in the Colombeau algebra for systems with singular coefficients.

Findings

Established existence and uniqueness of generalized solutions.

Linked generalized solutions to classical solutions under regularity conditions.

Extended symmetric hyperbolic system results to non-smooth coefficient cases.

Abstract

This paper is devoted to strictly hyperbolic systems and equations with non-smooth coefficients. Below a certain level of smoothness, distributional solutions may fail to exist. We construct generalised solutions in the Colombeau algebra of generalised functions. Extending earlier results on symmetric hyperbolic systems, we introduce generalised strict hyperbolicity, construct symmetrisers, prove an appropriate G\r{a}rding inequality and establish existence, uniqueness and regularity of generalised solutions. Under additional regularity assumptions on the coefficients, when a classical solution of the Cauchy problem (or of a transmission problem in the piecewise regular case) exists, the generalised solution is shown to be associated with the classical solution (or the piecewise classical solution satisfying the appropriate transmission conditions).