Liouville type theorem and kinetic formulation for 2x2 systems of conservation laws

TL;DR

This paper establishes a kinetic formulation and a Liouville-type theorem for bounded entropy solutions of 2x2 conservation law systems, revealing their regularity properties outside a small singular set.

Contribution

It introduces a kinetic representation for solutions with convex entropy and proves a Liouville theorem for nonlinear systems, characterizing solution regularity.

Findings

Solutions satisfy nonlocal kinetic equations

Finite entropy solutions are regular outside a small set

Liouville theorem characterizes solution behavior

Abstract

We study $\mathbf L^\infty$ entropy solutions to $2\times 2$ systems of conservation laws. We show that, if a uniformly convex entropy exists, these solutions satisfy a pair of kinetic equations (nonlocal in velocity), which are then shown to characterize all solutions with finite entropy production. Next, we prove a Liouville-type theorem for genuinely nonlinear systems, which is the main result of the paper. This implies in particular that for every finite entropy solution, every point $(t,x) \in \mathbb R^+\times \mathbb R\setminus \br J$ is of vanishing mean oscillation, where $\br J \subset \mathbb R^+\times \mathbb R$ is a set of Hausdorff dimension at most 1.