2025/10/10 by Matthias Sroczinski, Sroczinski, Matthias, Kevin Zumbrun +1
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2510.09287
openalex publication_date 2025/10/10 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28
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.