Preprint

Inserted: 20 sep 2026
Last Updated: 20 sep 2026

Year: 2026

Abstract:

We consider bounded entropy solutions of $\operatorname{div}_xA(u)=0$ on an open subset of $\mathbb R^N$, $N\geq2$. The flux $A\in C^m(\mathbb R;\mathbb R^N)$, $m\geq\max\{N,3\}$, is only assumed to satisfy the qualitative non-degeneracy condition that $\xi\cdot A$ is non-constant on every non-empty open interval of states, for every $\xi\neq0$. We prove that entropy solutions are continuous outside the jump set of De Lellis, Otto, and Westdickenberg (Arch. Ration. Mech. Anal., 2003). We also prove an $L^1$-to-$L^\infty$ stability estimate: on a ball, the uniform distance between an entropy solution and a continuous function is controlled by their $L^1$ distance on a larger ball, through a modulus depending only on the flux and on the modulus of continuity of the reference function. The nonlinearity assumption is sharp and strictly weaker than the power-law non-degeneracy in the continuity result of Silvestre (Comm. Pure Appl. Math., 2019) and in the stability result of Golding (C. R. Math. Acad. Sci. Paris, 2024), at the price of more regularity of the flux. The key observation for the proof is that the qualitative condition yields quantitative non-degeneracy on a subinterval of every interval of values of the solution, where velocity-averaging estimates can be localized.