Sharp a-contraction estimates for small extremal shocks

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.

arXiv comment41 pages, 2 figures, minor correction to lemma 4.2 and corresponding changes to section 4

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• $L^2$-stability $\&$ Minimal Entropy Conditions for Scalar Conservation Laws with Concave-Convex Fluxes
• Theory of shifts, shocks, and the intimate connections to $L^2$-type a posteriori error analysis of numerical schemes for hyperbolic problems
• Uniqueness & Weak-BV Stability in the Large for Isothermal Gas Dynamics