Sharp a-contraction estimates for small extremal shocks

TL;DR

This paper establishes sharp a-contraction estimates for small extremal shocks in 1D hyperbolic conservation laws with convex entropy, crucial for proving stability and uniqueness results.

Contribution

It demonstrates that the weight coefficient a can be proportionally chosen to the shock size, advancing the understanding of shock stability under large perturbations.

Findings

a can be chosen proportional to shock size

Supports stability and uniqueness in 2x2 systems

Provides a key estimate for BV-weak stability

Abstract

In this paper, we study the $a$-contraction property of small extremal shocks for 1-d systems of hyperbolic conservation laws endowed with a single convex entropy, when subjected to large perturbations. We show that the weight coefficient $a$ can be chosen with amplitude proportional to the size of the shock. The main result of this paper is a key building block in the companion paper, [{arXiv:2010.04761}, 2020], in which uniqueness and BV-weak stability results for $2\times 2$ systems of hyperbolic conservation laws are proved.