On the structure of $L^\infty$-entropy solutions to scalar conservation laws in one-space dimension

TL;DR

This paper analyzes the structure of entropy solutions to scalar conservation laws in one dimension, showing that entropy dissipation concentrates on countably many Lipschitz curves and detailing the characteristics' structure.

Contribution

It provides a detailed structural analysis of entropy solutions, revealing the measure concentration of entropy dissipation and the behavior of characteristic curves, with sharpness demonstrated through counterexamples.

Findings

Entropy dissipation is concentrated on countably many Lipschitz curves.

Characteristic curves are segments outside a countably 1-rectifiable set.

Initial data is taken in a strong sense, with counterexamples confirming sharpness.

Abstract

We prove that if $u$ is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics. \\ In particular the characteristic curves are segments outside a countably 1-rectifiable set and the left and right traces of the solution exist in a $C^0$-sense up to the degeneracy due to the segments where $f"=0$. We prove also that the initial data is taken in a suitably strong sense and we give some counterexamples which show that these results are sharp.