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

Yuxi Hu, Mengran Yuan, Jie Zhang

TL;DR

The paper proves the asymptotic stability of a centered 1-rarefaction wave for a one-dimensional compressible flow described by a hyperbolic relaxation system with Maxwell's law for heat flux and Cattaneo's law for viscous stress. The authors construct a smooth approximate rarefaction, reformulate the Cauchy problem around this profile, and apply a combination of relative entropy and energy methods to obtain uniform a priori estimates. They establish global existence and show that the perturbation decays in time, yielding uniform convergence to the background rarefaction. This work extends known results for isentropic or classical NS-Fourier systems to the non-isentropic, finite-propagation-speed setting, highlighting the role of thermo-mechanical couplings in stability analysis.

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.

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

TL;DR

The paper proves the asymptotic stability of a centered 1-rarefaction wave for a one-dimensional compressible flow described by a hyperbolic relaxation system with Maxwell's law for heat flux and Cattaneo's law for viscous stress. The authors construct a smooth approximate rarefaction, reformulate the Cauchy problem around this profile, and apply a combination of relative entropy and energy methods to obtain uniform a priori estimates. They establish global existence and show that the perturbation decays in time, yielding uniform convergence to the background rarefaction. This work extends known results for isentropic or classical NS-Fourier systems to the non-isentropic, finite-propagation-speed setting, highlighting the role of thermo-mechanical couplings in stability analysis.

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.
Paper Structure (6 sections, 9 theorems, 197 equations)

This paper contains 6 sections, 9 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Smooth approximation of the rarefaction wave
  3. Reformulation of the Problem
  4. Uniform a Priori Estimates
  5. $L^2$ estimates
  6. High-order estimate

Key Result

Theorem 1.1

Let $(v_-, u_-, \theta_-) \in R_1 (v_+, u_+, \theta_+)$ and $\delta=|v_+-v_-|+|u_+-u_-|+|\theta_+-\theta_-|$. Assume the initial data $(v_0, u_0, \theta_0, q_0, S_0)$ satisfy and there exists a positive constant $\epsilon_0$ such that if $I_0+\delta < \epsilon_0$, then the initial value problem 1.3-1.4 has a unique global solution in time satisfying Moreover, this solution approaches the rarefac

Theorems & Definitions (16)

  • Theorem 1.1
  • Lemma 2.1
  • Theorem 3.1
  • Proposition 3.2
  • proof : Proof of Theorem \ref{['thm3.1']}
  • proof : Proof of Theorem \ref{['th1.1']}
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 6 more