# On the regularity of weak solutions to Burgers equation with finite entropy production

## TL;DR

This paper investigates the regularity of weak solutions to Burgers' equation with finite entropy production, showing that the set of non-Lebesgue points has Hausdorff dimension at most one and establishing conditions for Hölder regularity.

## Contribution

It demonstrates that weak solutions with finite entropy production have a controlled set of irregular points and provides conditions for Hölder regularity based on entropy measure densities.

## Key findings

- Non-Lebesgue points form a set of Hausdorff dimension at most one.
- Solutions are Hölder continuous where the entropy measure has finite density.
- Small entropy measure implies the solution is close to an entropy solution.

## Abstract

Bounded weak solutions of Burgers' equation $\partial_tu+\partial_x(u^2/2)=0$ that are not entropy solutions need in general not be $BV$. Nevertheless it is known that solutions with finite entropy productions have a $BV$-like structure: a rectifiable jump set of dimension one can be identified, outside which $u$ has vanishing mean oscillation at all points. But it is not known whether all points outside this jump set are Lebesgue points, as they would be for $BV$ solutions. In the present article we show that the set of non-Lebesgue points of $u$ has Hausdorff dimension at most one. In contrast with the aforementioned structure result, we need only one particular entropy production to be a finite Radon measure, namely $\mu=\partial_t (u^2/2)+\partial_x(u^3/3)$. We prove H\"older regularity at points where $\mu$ has finite $(1+\alpha)$-dimensional upper density for some $\alpha>0$. The proof is inspired by a result of De Lellis, Westdickenberg and the second author : if $\mu_+$ has vanishing 1-dimensional upper density, then $u$ is an entropy solution. We obtain a quantitative version of this statement: if $\mu_+$ is small then $u$ is close in $L^1$ to an entropy solution.