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Stability of composite Wave of Planar Viscous Shock and Rarefaction for 3D Barotropic Navier-Stokes Equations

Jiajin Shi, Yi Wang

TL;DR

This work establishes nonlinear time-asymptotic stability of a composite wave formed by a planar rarefaction and a planar viscous shock for the three-dimensional compressible barotropic Navier–Stokes equations under generic $H^2$ perturbations without zero-mass restrictions. The authors develop an $a$-contraction framework with time-dependent shifts and a weighted energy functional, augmented by an effective velocity to exploit viscous dissipation in Eulerian coordinates. Under small composite-wave strength, they prove global existence of a strong solution and convergence to the composite wave up to a sublinear shift, with precise control of wave interactions and decay in the transverse directions. This advances multidimensional stability results for composite wave patterns in viscous compressible flows and extends the $a$-contraction approach to the 3D barotropic NS system with physical viscosities in a periodic transverse setting.

Abstract

We prove the nonlinear time-asymptotic stability of the composite wave consisting of a planar rarefaction wave and a planar viscous shock for the three-dimensional (3D) compressible barotropic Navier-Stokes equations under generic perturbations, in particular, without zero-mass conditions. It is shown that if the composite wave strength and the initial perturbations are suitably small, then 3D Navier-Stokes system admits a unique global-in-time strong solution which time-asymptotically converges to the corresponding composite wave up to a time-dependent shift for planar viscous shock. Our proof is based on the $a$-contraction method with time-dependent shift and suitable weight function.

Stability of composite Wave of Planar Viscous Shock and Rarefaction for 3D Barotropic Navier-Stokes Equations

TL;DR

This work establishes nonlinear time-asymptotic stability of a composite wave formed by a planar rarefaction and a planar viscous shock for the three-dimensional compressible barotropic Navier–Stokes equations under generic perturbations without zero-mass restrictions. The authors develop an -contraction framework with time-dependent shifts and a weighted energy functional, augmented by an effective velocity to exploit viscous dissipation in Eulerian coordinates. Under small composite-wave strength, they prove global existence of a strong solution and convergence to the composite wave up to a sublinear shift, with precise control of wave interactions and decay in the transverse directions. This advances multidimensional stability results for composite wave patterns in viscous compressible flows and extends the -contraction approach to the 3D barotropic NS system with physical viscosities in a periodic transverse setting.

Abstract

We prove the nonlinear time-asymptotic stability of the composite wave consisting of a planar rarefaction wave and a planar viscous shock for the three-dimensional (3D) compressible barotropic Navier-Stokes equations under generic perturbations, in particular, without zero-mass conditions. It is shown that if the composite wave strength and the initial perturbations are suitably small, then 3D Navier-Stokes system admits a unique global-in-time strong solution which time-asymptotically converges to the corresponding composite wave up to a time-dependent shift for planar viscous shock. Our proof is based on the -contraction method with time-dependent shift and suitable weight function.

Paper Structure

This paper contains 21 sections, 19 theorems, 379 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main result
  3. Planar rarefaction wave
  4. Approximate rarefaction wave
  5. Planar viscous shock
  6. Notations
  7. Main result
  8. Proof of the main result
  9. Some inequalities
  10. Relative quantities
  11. Construction of the weight function and the time-dependent shift
  12. Proof of Theorem \ref{['thm:main-ch2']}
  13. Proof of Proposition \ref{['prop:apriori-estimate']}
  14. Reformulation of the problem
  15. Estimates on $\left\| {(v-\bar{v},h-\bar{h})} \right\|$
  16. ...and 6 more sections

Key Result

Lemma 2.1

Let $w_m>w_-$ and $\tilde{w}:= w_m-w_-$, then the Cauchy problem eq:smooth Burgers admits a unique global smooth solution $w^R(t, x_1)$ satisfying (1) $w_- < w^R(t, x_1) < w_m$ and $w^R_{x_1} >0$, for $x_1 \in \mathbb{R}$ and $t\geq 0$. (2) For any $t > 0$ and $p \in [1,+\infty]$, there exists a p (3) For $x_1 \geq w_m t,\; t\geq 0$, it holds that (4) For $x_1 \leq w_-t,\; t\geq 0$, it holds

Theorems & Definitions (25)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Proposition 3.1: Local existence
  • ...and 15 more