Orbital stability of undercompressive viscous shock waves under $L^1\cap H^4$ perturbation

TL;DR

The paper addresses the orbital stability of viscous shock waves of Lax and undercompressive types under $L^1\cap H^4$ perturbations for partially symmetric hyperbolic–parabolic systems, assuming Evans-function stability ($\mathcal{D}$). It introduces a novel vertical estimate and minimal Strichartz-type bounds to replace stronger localization and to handle both inward and outward characteristic modes, yielding global nonlinear stability with explicit decay rates and a phase shift $\delta(t)$. The main contributions include extending Mascia–Zumbrun’s results to undercompressive shocks, simplifying the linear/nonlinear analysis, and providing a framework that works for relaxation systems with nonscalar equilibria. This advances the understanding of shock stability under low-regularity initial data and offers a robust method to analyze discontinuous or more complex relaxation shocks in multi-component systems.

Abstract

By the use of a new vertical estimate introduced by the authors in the context of relaxation shocks for shallow water flow, we both simplify and extend the basic $L^1\cap H^3$ stability results of Mascia and Zumbrun for viscous shock waves, in particular extending their results for Lax waves to the undercompressive case.

Theorems & Definitions (10)

• Theorem 1.1
• Remark 1.2
• Proposition 2.1: MZ1RZ
• Remark 2.2
• Proposition 2.3: Auxiliary integral bounds
• proof
• Proposition 2.4: Nonlinear damping estimate Z1
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['main1']}