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

TL;DR

This paper proves the $L^1$-contraction property for entropy solutions of scalar conservation laws with fluxes that have minimal regularity, extending Kruzhkov's method to more general settings.

Contribution

It extends the $L^1$-contraction property to scalar conservation laws with less regular flux functions, broadening the applicability of existing theory.

Findings

Established $L^1$-contraction for entropy solutions with minimal flux regularity

Extended Kruzhkov's method to more general flux functions

Provided theoretical foundation for less regular conservation laws

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.