$L^1$-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 + \operatorname {div} f = 0 \quad \text {in } \mathbb {R}^d \times (0, \infty ), $

where the flux $ f = f(x, u) $ depends explicitly on the spatial variable $ x $. Using an extension of Kruzkov’s doubling variable method, we establish contraction properties of entropy solutions under minimal regularity assumptions on the flux, as well as the uniqueness of entropy solutions. The flux is assumed to be locally Lipschitz, along with some additional conditions.

Keywords

Partial differential equations / Scalar conservation laws / Entropy solutions for scalar conservation laws / $L^1$-contraction / Uniqueness of entropy solutions / 35Axx / 35Bxx / 35Qxx

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Funding

Israel Science Foundation(No. 569/21)

Ben-Gurion University

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