Regularity of solutions to scalar conservation laws with a force

TL;DR

This paper establishes improved regularity estimates for entropy solutions to scalar conservation laws with a force, utilizing a novel velocity-variable decomposition based on the flux's non-degeneracy.

Contribution

It introduces a new decomposition of entropy solutions in the velocity variable, enabling finer control of flux degeneracy and improved regularity estimates.

Findings

Entropy solutions exhibit higher regularity under the new decomposition.

The entropy dissipation measure has locally finite singular moments.

The method applies to scalar conservation laws with forcing terms.

Abstract

We prove regularity estimates for entropy solutions to scalar conservation laws with a force. Based on the kinetic form of a scalar conservation law, a new decomposition of entropy solutions is introduced, by means of a decomposition in the velocity variable, adapted to the non-degeneracy properties of the flux function. This allows a finer control of the degeneracy behavior of the flux. In addition, this decomposition allows to make use of the fact that the entropy dissipation measure has locally finite singular moments. Based on these observations, improved regularity estimates for entropy solutions to (forced) scalar conservation laws are obtained.