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Stability of viscous shock for the Navier-Stokes-Fourier system: outflow and impermeable wall problems

Xushan Huang, Hobin Lee, HyeonSeop Oh

Abstract

We investigate the time-asymptotic stability of solutions to the one-dimensional Navier-Stokes-Fourier system in the half-space, focusing on the outflow and impermeable wall problems. When the prescribed boundary and far-field conditions form an outgoing viscous shock, we prove that the solution converges to the viscous shock profile, up to a dynamical shift, provided that the initial perturbation and the shock amplitude are sufficiently small. In order to obtain our results, we employ the method of $a$-contraction with shifts. Although the impermeable wall problem is technically simpler to analyze in Lagrangian mass coordinates, the outflow problem leads to a free boundary in that framework. Therefore, we use Eulerian coordinates to provide a unified approach to both problems. This is the first result on the time-asymptotic stability of viscous shocks for initial-boundary value problems of the Navier-Stokes-Fourier system for the outflow and impermeable wall cases.

Stability of viscous shock for the Navier-Stokes-Fourier system: outflow and impermeable wall problems

Abstract

We investigate the time-asymptotic stability of solutions to the one-dimensional Navier-Stokes-Fourier system in the half-space, focusing on the outflow and impermeable wall problems. When the prescribed boundary and far-field conditions form an outgoing viscous shock, we prove that the solution converges to the viscous shock profile, up to a dynamical shift, provided that the initial perturbation and the shock amplitude are sufficiently small. In order to obtain our results, we employ the method of -contraction with shifts. Although the impermeable wall problem is technically simpler to analyze in Lagrangian mass coordinates, the outflow problem leads to a free boundary in that framework. Therefore, we use Eulerian coordinates to provide a unified approach to both problems. This is the first result on the time-asymptotic stability of viscous shocks for initial-boundary value problems of the Navier-Stokes-Fourier system for the outflow and impermeable wall cases.

Paper Structure

This paper contains 27 sections, 15 theorems, 243 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Literature review
  3. Main results
  4. Main ideas for the proof
  5. Organization of paper
  6. Preliminaries
  7. Viscous shock waves
  8. Poincaré-type inequality
  9. A priori estimates and proof of the main theorem
  10. Local existence of solutions
  11. Construction of weight function
  12. Construction of shift
  13. A priori estimates
  14. Proof of Theorem \ref{['thm:outflow']}
  15. Zeroth order estimates
  16. ...and 12 more sections

Key Result

Theorem 1.1

For a given constant state $(\rho_+,u_+, \theta_+) \in\Omega^{-}_{sub}\cup \Gamma_{trans}^{-}$, there exist positive constants $\delta_0,\varepsilon_0 >0$ small enough such that the following holds. For any $(u_-,\theta_-)$ satisfying $0>u_->u_+$ and $(u_-,\theta_-) \in S_3^{P}(\rho_+,u_+,\theta_+)$ let $\rho_->0$ be the (unique) constant state such that $(\rho_-,u_-,\theta_-) \in S_3(\rho_+,u_+,\

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (29)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.1: Outflow problem
  • Theorem 1.2: Impermeable wall problem
  • Remark 1.3
  • Lemma 2.1
  • Remark 2.1
  • Lemma 2.2
  • Proposition 3.1
  • proof
  • ...and 19 more