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Asymptotic stability of the symmetric flow via inviscid damping and enhanced dissipation

Qi Chen, Hao Li, Shunlin Shen, Zhifei Zhang

Abstract

In this paper, we establish the inviscid damping and enhanced dissipation estimates for the linearized Navier-Stokes system around the symmetric flow in a finite channel with the non-slip boundary condition. As an immediate consequence, we prove the asymptotic stability of the symmetric flow in the high Reynolds number regime. Namely, if the initial velocity perturbation $u^{\mathrm{in}}$ satisfies $\Vert u^{\mathrm{in}}-(U(y),0) \Vert_{H^5}\leq c ν^{\frac{2}{3}}$, then inviscid damping and enhanced dissipation estimates also hold for the solution to the Navier-Stokes system.

Asymptotic stability of the symmetric flow via inviscid damping and enhanced dissipation

Abstract

In this paper, we establish the inviscid damping and enhanced dissipation estimates for the linearized Navier-Stokes system around the symmetric flow in a finite channel with the non-slip boundary condition. As an immediate consequence, we prove the asymptotic stability of the symmetric flow in the high Reynolds number regime. Namely, if the initial velocity perturbation satisfies , then inviscid damping and enhanced dissipation estimates also hold for the solution to the Navier-Stokes system.
Paper Structure (25 sections, 42 theorems, 475 equations)

This paper contains 25 sections, 42 theorems, 475 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Key ideas and outline of the proof
  4. Coercive estimates for Rayleigh equation
  5. Resolvent estimates with Navier-slip boundary condition
  6. Resolvent estimates for $\lambda\in \mathbb{R}\setminus (U(0),U(1))$
  7. Resolvent estimates for $\lambda\in [U(0), U(1)]$
  8. Exterior energy estimates
  9. Interior energy estimates
  10. Resolvent estimates for $F\in L^{2}$ and $F\in H^{-1}$
  11. Weak-type resolvent estimates
  12. Resolvent estimates for $F\in H_{0}^{1}$
  13. Summary of main results
  14. Boundary layer corrector
  15. Estimates on $w_{cor,1}$ and $w_{cor,2}$
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $U(y)\in \mathrm{S}$ and $(\omega,u)$ be the solution to the linearized NS system equ:omega,linear,NS with $\int_{\mathbb{T}}\omega^{\mathrm{in}}(x,y)dx=0$. There exist positive constants $\nu_0$, $\varepsilon_0$, such that for $\nu\in(0,\nu_0]$, $\varepsilon\in[0,\varepsilon_{0}]$, the followin

Theorems & Definitions (79)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: Coercive estimates
  • proof
  • Lemma 2.2: Hardy-type inequality
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • ...and 69 more