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

This paper is concerned with entropy solutions of scalar conservation laws of the form ${\mathrm{\partial }}_{t}u+\text{diver}f=0$ in ${\mathbb{R}}^d\times (0,\mathrm{\infty })$. 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.