$L^2$-stability $\&$ Minimal Entropy Conditions for Scalar Conservation Laws with Concave-Convex Fluxes

TL;DR

This paper establishes $L^2$-stability and uniqueness of solutions for scalar conservation laws with non-convex, concave-convex fluxes under minimal entropy conditions, using $a$-contraction with shifts and a modified front tracking method.

Contribution

It introduces $a$-contraction with shifts for stability and proves a minimal entropy condition-based uniqueness theorem for weak solutions.

Findings

Proves $L^2$-stability of shocks with large and small amplitudes.

Provides estimates on the weight coefficient $a$ in different regimes.

Establishes a uniqueness theorem under minimal entropy conditions.

Abstract

In this paper, we study stability properties of solutions to scalar conservation laws with a class of non-convex fluxes. Using the theory of $a$-contraction with shifts, we show $L^2$-stability for shocks among a class of large perturbations, and give estimates on the weight coefficient $a$ in regimes where the shock amplitude is both large and small. Then, we use these estimates as a building block to show a uniqueness theorem under minimal entropy conditions for weak solutions to the conservation law via a modified front tracking algorithm. The proof is inspired by an analogous program carried out in the $2 \times 2$ system setting by Chen, Golding, Krupa, $\&$ Vasseur.