On the structure of entropy dissipation and regularity for quasi-entropy solutions to 1d scalar conservation laws and to isentropic Euler system with $\gamma=3$

TL;DR

This paper studies the structure of entropy dissipation and regularity properties of quasi-entropy solutions to 1D scalar conservation laws and the isentropic Euler system with γ=3, revealing concentration on rectifiable sets and improved regularity.

Contribution

It introduces a Lagrangian framework for quasi-entropy solutions and proves concentration of entropy dissipation on rectifiable sets, also enhancing regularity results for the Euler system.

Findings

Entropy dissipation measures are concentrated on 1-rectifiable sets in 1D.

Lagrangian representation for quasi-entropy solutions is established.

Fractional regularity of solutions is slightly improved using the sign of kinetic measures.

Abstract

In this paper, we first investigate quasi-entropy solutions to scalar conservation laws in several space dimensions. In this setting, we introduce a suitable Lagrangian representation for such solutions. Next, we prove that, in one space dimension and for fluxes $f$ satisfying a general non-degeneracy condition, the entropy dissipation measures of quasi-entropy solutions are concentrated on a 1-rectifiable set. The same result is obtained for the isentropic Euler system with $\gamma = 3$, for which we also slightly improve the available fractional regularity by exploiting the sign of the kinetic measures.