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

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.