Optimal regularizing effect for scalar conservation laws

Abstract

We investigate the regularity of bounded weak solutions of scalar conservation laws with uniformly convex flux in space dimension one, satisfying an entropy condition with entropy production term that is a signed Radon measure. We prove that all such solutions belong to the Besov space $B_{\infty ,\mathrm{loc}}^{1/3,3}$. Since C. de Lellis and M. Westdickenberg [11] have proved the existence of such solutions that do not belong to $B_{q,\mathrm{loc}}^{s,p}$ if either $s>1/\max (p,3)$ or $s=1/3$ and $1\le q<p<3$ or $s=1/p$ with $p\ge 3$ and $q<\infty$, this regularizing effect is optimal. The proof is based on the kinetic formulation of scalar conservation laws and on an interaction estimate in physical space.