Strong time regularity and decay of $L^\infty$ solutions to $2\times 2$ systems of conservation laws

TL;DR

This paper proves regularity and decay properties of bounded solutions to certain 2x2 conservation law systems, using kinetic formulation to unify the analysis of nonlinear and viscous solutions.

Contribution

It establishes continuity in time for entropy solutions and derives decay estimates for viscosity solutions, advancing understanding of solution behavior in nonlinear conservation laws.

Findings

Finite entropy solutions are continuous in time with respect to local L^1 norm.

Vanishing viscosity solutions exhibit dispersive decay over time.

Kinetic formulation effectively unifies regularity and decay analysis.

Abstract

We consider $\mathbf L^\infty$ solutions to $2\times 2$ systems of conservation laws. For genuinely nonlinear systems we prove that finite entropy solutions (in particular entropy solutions, if a uniformly convex entropy exists) belong to $C^0(\mathbb R^+; \mathbf L^1_{loc}(\mathbb R))$. Our second result establishes a dispersive-type decay estimate for vanishing viscosity solutions. Both results are unified by the use of a kinetic formulation.