Asymptotic Stability of Rarefaction Waves for the Hyperbolized Navier-Stokes-Fourier System

TL;DR

This paper proves the global stability and convergence of rarefaction waves in a generalized hyperbolic Navier-Stokes-Fourier system with finite signal speeds, using energy methods and relative entropy.

Contribution

It extends classical stability results to a hyperbolic system with Maxwell and Cattaneo laws, showing global existence and asymptotic convergence.

Findings

Global existence of solutions for small initial perturbations

Uniform convergence to rarefaction wave over time

System exhibits finite signal propagation speeds

Abstract

This paper investigates the asymptotic stability of rarefaction waves for a one-dimensional compressible fluid system, where the Newton's law of viscosity and Fourier's law of heat conduction are replaced by Maxwell's law and Cattaneo's law, respectively. The system, which generalizes the classical Navier-Stokes-Fourier equations, features finite signal propagation speeds. We consider the Cauchy problem in Lagrangian coordinates with initial data connecting two different constant states via a rarefaction wave of the corresponding Euler system. Our main result proves that, provided the initial perturbation and wave strength are sufficiently small, the relaxation system admits a unique global solution. Furthermore, this solution converges uniformly to the background rarefaction wave as time approaches infinity. The proof is established through a combination of the relative entropy method and usual energy estimates.