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Asymptotic stability threshold of the 2-D monotone shear flow with no-slip boundary condition

Zhen Li, Shunlin Shen, Zhifei Zhang

Abstract

In this paper, we investigate the asymptotic stability threshold problem for the 2-D Navier-Stokes equations in a finite channel with no-slip boundary conditions, around monotone shear flow $(U(t,y),0)$. We establish that the flow is asymptotically stable under perturbations satisfying $\|u^{\mathrm{in}}\|_{H^2}\leq cν^{\frac12}$. To achieve the stability threshold $ν^{\frac{1}{2}}$, the key ingredients of the proof include: sharp resolvent estimates for the vorticity based on weak-type resolvent bounds; weighted space-time estimates for the vorticity; pointwise estimates for the velocity. Furthermore, we handle the nonlinear term through a divergence formulation, which facilitates the sharp application of the aforementioned space-time estimates.

Asymptotic stability threshold of the 2-D monotone shear flow with no-slip boundary condition

Abstract

In this paper, we investigate the asymptotic stability threshold problem for the 2-D Navier-Stokes equations in a finite channel with no-slip boundary conditions, around monotone shear flow . We establish that the flow is asymptotically stable under perturbations satisfying . To achieve the stability threshold , the key ingredients of the proof include: sharp resolvent estimates for the vorticity based on weak-type resolvent bounds; weighted space-time estimates for the vorticity; pointwise estimates for the velocity. Furthermore, we handle the nonlinear term through a divergence formulation, which facilitates the sharp application of the aforementioned space-time estimates.
Paper Structure (11 sections, 15 theorems, 207 equations)

This paper contains 11 sections, 15 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Resolvent estimates of the linearized operator
  3. Resolvent estimates with Naiver-slip boundary condition
  4. Resolvent estimates with no-slip boundary condition
  5. Space-Time estimates of the linearized NS
  6. Space-time estimates for high frequencies
  7. Space-time estimates for low frequencies: frozen-time case
  8. Space-time estimates for low frequencies: heat case
  9. Space-time estimates for the full problem
  10. Nonlinear stability
  11. Auxiliary estimates

Key Result

Theorem 1.1

Let $(\omega,\psi)$ be the solution to equ: omli11. There exist positive constants $\nu_0$, $\epsilon_{0}$ and $c$, such that if the initial perturbation satisfies $\|u^{\mathrm{in}}\|_{H^2}\leq c\nu^{\frac{1}{2}}$, $0<\nu\leq \nu_{0}$, then the solution $(\omega,u)$ satisfies the global stability e where the stability norm is given by with $f_k(t,y)=:\int_{\mathbb{T}} f(t,x,y) e^{-ikx}dx$. Here,

Theorems & Definitions (27)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3: Weak-type resolvent estimates
  • proof
  • Lemma 2.4
  • proof
  • proof : Proof of Proposition \ref{['lemma:non-slip boundary,resolvent']}
  • Proposition 3.1
  • Proposition 3.2
  • ...and 17 more