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Time-asymptotic stability of composite weak planar waves for a general $n\times n$ multi-D viscous system

Jiayun Meng

TL;DR

This work extends the time-asymptotic stability of composite waves from 1-D to multi-D for a general $n\times n$ viscous system by proving stability of a planar viscous $1$-shock in combination with either a planar $n$-rarefaction or a planar viscous $n$-shock under small perturbations. The authors deploy the $a$-contraction method with shifts, augmented by weight functions and a relative entropy framework, to obtain $L^2$ and $H^m$ estimates that control the interaction between waves and higher-order terms. A key contribution is the construction of wave-adapted bases and a Poincaré-type inequality that reconcile dissipation with hyperbolic dynamics in multiple transverse directions, enabling global-in-time stability and convergence to a shifted composite wave. The results apply to a broad class of multi-D viscous systems, including the 3-D Brenner-Navier-Stokes equations, and provide a rigorous pathway to understanding the long-time behavior of complex wave superpositions in high dimensions.

Abstract

We prove the time-asymptotic stability of the superposition of a weak planar viscous 1-shock and either a weak planar n-rarefaction or a weak planar viscous n-shock for a general $n\times n$ multi-D viscous system. In 2023, Kang-Vasseur-Wang [11] showed the stability of the superposition of a viscous shock and a rarefaction for 1-D compressible barotropic Navier-Stokes equations and solved a long-standing open problem officially introduced by Matsumura-Nishihara [23] in 1992. Our work is an extension of [11], where a general $n\times n$ multi-D viscous system is studied. Same as in [11], we apply the $a$-contraction method with shifts, an energy based method invented by Kang and Vasseur in [9], for both viscous shock and rarefaction at the level of the solution. In such a way, we can work with general perturbations and compositions of waves. Finally, a technique to classify and control higher-order terms is developed to work in multi-D.

Time-asymptotic stability of composite weak planar waves for a general $n\times n$ multi-D viscous system

TL;DR

This work extends the time-asymptotic stability of composite waves from 1-D to multi-D for a general viscous system by proving stability of a planar viscous -shock in combination with either a planar -rarefaction or a planar viscous -shock under small perturbations. The authors deploy the -contraction method with shifts, augmented by weight functions and a relative entropy framework, to obtain and estimates that control the interaction between waves and higher-order terms. A key contribution is the construction of wave-adapted bases and a Poincaré-type inequality that reconcile dissipation with hyperbolic dynamics in multiple transverse directions, enabling global-in-time stability and convergence to a shifted composite wave. The results apply to a broad class of multi-D viscous systems, including the 3-D Brenner-Navier-Stokes equations, and provide a rigorous pathway to understanding the long-time behavior of complex wave superpositions in high dimensions.

Abstract

We prove the time-asymptotic stability of the superposition of a weak planar viscous 1-shock and either a weak planar n-rarefaction or a weak planar viscous n-shock for a general multi-D viscous system. In 2023, Kang-Vasseur-Wang [11] showed the stability of the superposition of a viscous shock and a rarefaction for 1-D compressible barotropic Navier-Stokes equations and solved a long-standing open problem officially introduced by Matsumura-Nishihara [23] in 1992. Our work is an extension of [11], where a general multi-D viscous system is studied. Same as in [11], we apply the -contraction method with shifts, an energy based method invented by Kang and Vasseur in [9], for both viscous shock and rarefaction at the level of the solution. In such a way, we can work with general perturbations and compositions of waves. Finally, a technique to classify and control higher-order terms is developed to work in multi-D.
Paper Structure (20 sections, 14 theorems, 219 equations)

This paper contains 20 sections, 14 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Viscous shock wave
  4. Construction of approximate rarefaction wave
  5. Construction of weight functions, shift functions and the superposition wave
  6. Proof of Theorem \ref{['thm']}
  7. Local-in-time estimates and a priori estimates
  8. Global existence and estimates
  9. Time-asymptotic behavior
  10. Energy estimate
  11. Wave interaction estimates
  12. Higher derivatives estimates
  13. Relative entropy method
  14. Poincaré type inequality
  15. $L^2$ estimates
  16. ...and 5 more sections

Key Result

Theorem 1.1

For any $\ul \in \mathbb{R}^n$, there exist constants $\delta_0, \epsilon_0>0$ such that the following is true. Let $U_m \in \mathcal{S}_1(\ul)$ and $U_+ \in \mathcal{S}_n(U_m)$ or $\mathcal{R}_n(U_m)$ be such that Let $\mathbf{S}_1$ be the viscous 1-shock wave solution to (eq for sol shock) with end states $\ul$ and $U_m$, and $\mathbf{W}_n$ be the viscous n-shock wave solution $\mathbf{S}_n$ to

Theorems & Definitions (21)

  • Theorem 1.1
  • Proposition 1.2
  • Lemma 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 3.1
  • Proposition 3.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 11 more