On the time continuity of entropy solutions

TL;DR

This paper proves that entropy solutions to a convection diffusion equation are continuous in time with respect to local L^1 convergence, without relying on solution uniqueness.

Contribution

It establishes time continuity of entropy solutions for convection diffusion equations without assuming uniqueness, broadening understanding of solution regularity.

Findings

Entropy solutions are continuous in time in the local L^1 sense.

Continuity proof does not depend on solution uniqueness.

Results apply to convection diffusion equations with general flux and diffusion terms.

Abstract

We show that any entropy solution $u$ of a convection diffusion equation $\partial_t u + \div F(u)-\Delta\phi(u) =b$ in $\OT$ belongs to $C([0,T),L^1_{Loc}(\o\O))$. The proof does not use the uniqueness of the solution.