On optimal regularity estimates for finite-entropy solutions of scalar conservation laws

TL;DR

This paper establishes optimal regularity estimates for finite-entropy solutions of scalar conservation laws with convex flux functions, using a novel characterization involving a specific cost function.

Contribution

It introduces a new regularity estimate for finite-entropy solutions under degenerate convexity and doubling measure conditions on the flux's second derivative.

Findings

Characterization of finite-entropy solutions via an optimal regularity estimate.

Extension of regularity results to flux functions with degenerate convexity.

Use of a cost function framework originally introduced by Golse and Perthame.

Abstract

We consider finite-entropy solutions of scalar conservation laws $u_t +a(u)_x =0$, that is, bounded weak solutions whose entropy productions are locally finite Radon measures. Under the assumptions that the flux function $a$ is strictly convex (with possibly degenerate convexity) and $a''$ forms a doubling measure, we obtain a characterization of finite-entropy solutions in terms of an optimal regularity estimate involving a cost function first used by Golse and Perthame.