Structure of entropy solutions to general scalar conservation laws in one space dimension

TL;DR

This paper investigates the detailed regularity and structure of entropy solutions to scalar conservation laws in one dimension, revealing their continuity properties and representing their solutions through a wave-based Lagrangian framework.

Contribution

It extends wave representation techniques to entropy solutions, providing near-optimal regularity estimates and a detailed structural understanding of solution level sets.

Findings

Entropy solutions are continuous outside a 1-rectifiable set.

Near the rectifiable set, solutions are left/right continuous in regions.

Wave representation can be extended to entropy solutions, revealing fine structure.

Abstract

In this paper, we show that the entropy solution of a scalar conservation law is - continuous outside a $1$-rectifiable set $\Xi$, - up to a $\mathcal H^1$ negligible set, for each point $(\bar t,\bar x) \in \Xi$ there exists two regions where $u$ is left/right continuous in $(\bar t,\bar x)$. We provide examples showing that these estimates are nearly optimal. In order to achieve these regularity results, we extend the wave representation of the wavefront approximate solutions to entropy solution. This representation can the interpreted as some sort of Lagrangian representation of the solution to the nonlinear scalar PDE, and implies a fine structure on the level sets of the entropy solution.