An extension of Bakhvalov's theorem for systems of conservation laws with damping

TL;DR

This paper extends Bakhvalov's theorem to include damping terms in 2x2 conservation law systems, ensuring global solutions and decay properties for periodic initial data, with applications to gas dynamics and relativistic systems.

Contribution

It introduces a class of damping terms that preserve global solutions in systems satisfying Bakhvalov conditions, expanding the theorem's applicability to physical and non-physical gamma ranges.

Findings

Existence of global solutions with damping for large BV initial data

Decay of periodic solutions over time

Application to isentropic gas dynamics and relativistic systems

Abstract

For $2\X2$ systems of conservation laws satisfying Bakhvalov conditions, we present a class of damping terms that still yield the existence of global solutions with periodic initial data of possibly large bounded total variation per period. We also address the question of the decay of the periodic solution. As applications we consider the systems of isentropic gas dynamics, with pressure obeying a $\gamma$-law, for the physical range $\gamma\ge1$, and also for the "non-physical" range $0<\gamma<1$, both in the classical Lagrangian and Eulerian formulation, and in the relativistic setting. We give complete details for the case $\gamma=1$, and also analyze the general case when $|\gamma-1|$ is small. Further, our main result also establishes the decay of the periodic solution.