Relative entropy in hyperbolic relaxation for balance laws

TL;DR

This paper develops a new framework for approximating hyperbolic balance laws using relaxation systems, with a focus on relative entropy estimates and convergence proofs, especially for systems with source terms in materials science.

Contribution

It introduces a novel relaxation approach with a detailed relative entropy analysis for hyperbolic balance laws, extending previous methods to systems with source terms.

Findings

Established convergence in the smooth regime for a broad class of systems.

Provided a framework applicable to materials science models with source terms.

Extended the relative entropy method to hyperbolic balance laws with additional challenges.

Abstract

We present a general framework for the approximation of systems of hyperbolic balance laws. The novelty of the analysis lies in the construction of suitable relaxation systems and the derivation of a delicate estimate on the relative entropy. We provide a direct proof of convergence in the smooth regime for a wide class of physical systems. We present results for systems arising in materials science, where the presence of source terms presents a number of additional challenges and requires delicate treatment. Our analysis is in the spirit of the framework introduced by Tzavaras [A. Tzavaras, Commun. Math. Sci., 3-2, 2005] for systems of hyperbolic conservation laws.