Non-linear stability of shock profiles in dissipative hyperbolic-hyperbolic systems
Matthias Sroczinski, Kevin Zumbrun
TL;DR
The paper proves nonlinear time-asymptotic stability of smooth planar shock profiles for dissipative hyperbolic-hyperbolic systems in dimensions $d\ge 2$ under a spectral stability (Evans-function) condition. The authors introduce a paradifferential damping framework that works without Kawashima-type symmetrizability, bridging hyperbolic-hyperbolic systems with relaxation/Jin-Xin-type structures and enabling nonlinear stability analysis in multi-D relativistic viscous gas models. They develop a full linear–nonlinear program: low/medium-frequency Kreiss-symmetrizer methods and high-frequency paradifferential energy estimates yield resolvent and semigroup bounds, while nonlinear damping closes the iteration and yields decay rates $\|u(t)-\bar{u}\|_{H^s}\lesssim (1+t)^{-(d-1)/4}$ and $\|u(t)-\bar{u}\|_{L^p}\lesssim (1+t)^{-(d-1)/2(1-1/p)}$. The results reduce nonlinear stability to spectral verification and offer a versatile toolset applicable to relativistic gas models (FT14/FT17/BDN18/BDN22) and general second-order relaxation-type systems, with clear directions for removing small-amplitude restrictions and extending to large-amplitude, multi-D settings.
Abstract
We give the first proof of nonlinear stability for smooth shock profiles of second-order dissipative hyperbolic-hyperbolic systems under the assumption of spectral stability, showing stability of smooth small-amplitude profiles in dimensions greater than or equal to two. This class of systems notably includes the two types of causal viscous relativistic gas models introduced respectively by Freistühler-Temple and Bemfica-Disconzi-Noronha, and (the equivalent second-order form of) a class of first-order numerical relaxation systems generalizing the well-known Jin-Xin relaxation schemes. A significant technical innovation is a new para-differential type of nonlinear damping estimate similar to that used by the first author to study stability of constant states, allowing the treatment of systems far from the symmetric structure required for the standard ``Kawashima-type'' energy estimates that are typically used for that purpose.