$L^1$-Contraction Property of Entropy Solutions for Scalar Conservation Laws with Minimal Regularity Assumptions on the Flux

Abstract

This paper is concerned with entropy solutions of scalar conservation laws of the form $\partial_{t}u+\diver f=0$ in ${\mathbb{R}}^d\times (0,\infty )$. The flux $f=f(x,u)$ depends explicitly on the spatial variable $x$. Using an extension of Kruzkov's method, we establish the $L^1$-contraction property of entropy solutions under minimal regularity assumptions on the flux.