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Stability of strong viscous shock wave under periodic perturbation for 1-D isentropic Navier-Stokes system in the half space

Lin Chang, Lin He, Jin Ma

TL;DR

The paper addresses the stability of large-amplitude viscous shocks for the 1-D isentropic Navier–Stokes system in a half-space under space-periodic perturbations. It constructs a composite ansatz blending two viscous shocks with two periodic profiles and deploys the anti-derivative method with evolving shifts to capture the phase interaction and far-field oscillations. The main result proves global existence and asymptotic convergence of the actual solution to a shifted viscous shock $(V_2^S,U_2^S)(x-st-eta)$, with the shift $eta$ determined by the periodic data, and crucially allows arbitrarily large shock strength, extending prior small-shock stability results. This work advances the theory of shock stability under boundary-driven periodic forcing and provides a robust framework for analyzing isentropic gas dynamics in a half-space with impermeable boundary conditions.

Abstract

In this paper, a viscous shock wave under space-periodic perturbation of 1-D isentropic Navier-Stokes system in the half space is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover, the strength of {the} shock wave could be arbitrarily large. This result essentially improves the previous work " A. Matsumura, M. Mei, Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary. Arch. Ration. Mech. Anal. 146 (1999), no. 1, 1-22." where the strength of shock wave is sufficiently small and the initial periodic oscillations vanish.

Stability of strong viscous shock wave under periodic perturbation for 1-D isentropic Navier-Stokes system in the half space

TL;DR

The paper addresses the stability of large-amplitude viscous shocks for the 1-D isentropic Navier–Stokes system in a half-space under space-periodic perturbations. It constructs a composite ansatz blending two viscous shocks with two periodic profiles and deploys the anti-derivative method with evolving shifts to capture the phase interaction and far-field oscillations. The main result proves global existence and asymptotic convergence of the actual solution to a shifted viscous shock , with the shift determined by the periodic data, and crucially allows arbitrarily large shock strength, extending prior small-shock stability results. This work advances the theory of shock stability under boundary-driven periodic forcing and provides a robust framework for analyzing isentropic gas dynamics in a half-space with impermeable boundary conditions.

Abstract

In this paper, a viscous shock wave under space-periodic perturbation of 1-D isentropic Navier-Stokes system in the half space is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover, the strength of {the} shock wave could be arbitrarily large. This result essentially improves the previous work " A. Matsumura, M. Mei, Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary. Arch. Ration. Mech. Anal. 146 (1999), no. 1, 1-22." where the strength of shock wave is sufficiently small and the initial periodic oscillations vanish.
Paper Structure (15 sections, 20 theorems, 172 equations, 1 figure)

This paper contains 15 sections, 20 theorems, 172 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and the Main Theorem
  3. Preliminaries
  4. Ansatz
  5. Location of The Shift ${\mathcal{X}}(t)$ and ${\mathcal{Y}}(t)$.
  6. the Main Result
  7. Reformulation of the Original Problem
  8. A Priori Estimate
  9. Low Order Estimates.
  10. High Order Estimates.
  11. Proof of Theorem \ref{['theorem']}
  12. Proof of Theorem \ref{['theorem']}
  13. Proof of Lemma \ref{['Lem-shift']} and Lemma \ref{['Lem-F']}
  14. Proof of Lemma \ref{['Lem-shift']}
  15. Proof of Lemma \ref{['Lem-F']}

Key Result

Lemma 2.1

There exists a unique viscous shock $(V^{S}_{2}, U^{S}_{2})(\xi_2)$ up to a shift satisfying as $\xi_{2} \rightarrow \pm \infty,$ where $\theta=\left|v_{+}-v_{-}\right|$, $c_\pm=\frac{v_{\pm}^{\alpha+1}}{s} |p'(v_\pm)+s^2|$, $s=\frac{-u_+}{v_+-v_-}.$

Figures (1)

  • Figure :

Theorems & Definitions (32)

  • Lemma 2.1: mm1999
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.1
  • Lemma 3.1
  • Remark 3.1
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 4.1
  • ...and 22 more