The Relative Entropy Method for Inhomogeneous Systems of Balance Laws

TL;DR

This paper extends the relative entropy method to inhomogeneous hyperbolic systems of balance laws, providing a unified framework for stability, uniqueness, and convergence analysis despite spatial and temporal inhomogeneities.

Contribution

It develops new hypotheses enabling the application of the relative entropy framework to inhomogeneous systems with diverse characteristics.

Findings

Establishes measure-valued weak vs strong uniqueness theorem

Proves stability of viscous solutions

Demonstrates convergence as viscosity tends to zero

Abstract

General hyperbolic systems of balance laws with inhomogeneity in space and time in all constitutive functions are studied in the context of relative entropy. A framework is developed in this setting that contributes to a measure-valued weak vs strong uniqueness theorem, a stability theorem of viscous solutions and a convergence theorem as the viscosity parameter tends to zero. The main goal of this paper is to develop hypotheses under which the relative entropy framework can still be applied. Examples of systems with inhomogeneity that have different charateristics are presented and the hypotheses are discussed in the setting of each example.