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Nonlinear stability for 3-D plane Poiseuille flow in a finite channel

Qi Chen, Shijin Ding, Zhilin Lin, Zhifei Zhang

Abstract

In this paper, we study the nonlinear stability for the 3-D plane Poiseuille flow $(1-y^2,0,0)$ at high Reynolds number $Re$ in a finite channel $\mathbb{T}\times [-1,1 ]\times \mathbb{T}$ with non-slip boundary condition. We prove that if the initial velocity $v_0$ satisfies $\|v_0-(1-y^2,0,0)\|_{H^{4}}\leq c_0 Re^{-\frac{7}{4}}$ for some $c_0>0$ independent of $Re$, then the solution of 3-D Naiver-Stokes equations is global in time and does not transit away from the plane Poiseuille flow. To our knowledge, this is the first nonlinear stability result for the 3-D plane Poiseuille flow and the transition threshold is accordant with the numerical result by Lundbladh et al. \cite{LHR}.

Nonlinear stability for 3-D plane Poiseuille flow in a finite channel

Abstract

In this paper, we study the nonlinear stability for the 3-D plane Poiseuille flow at high Reynolds number in a finite channel with non-slip boundary condition. We prove that if the initial velocity satisfies for some independent of , then the solution of 3-D Naiver-Stokes equations is global in time and does not transit away from the plane Poiseuille flow. To our knowledge, this is the first nonlinear stability result for the 3-D plane Poiseuille flow and the transition threshold is accordant with the numerical result by Lundbladh et al. \cite{LHR}.
Paper Structure (40 sections, 73 theorems, 792 equations, 1 figure, 1 table)

This paper contains 40 sections, 73 theorems, 792 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Transition threshold problem
  3. Main result
  4. Ideas and sketch of the proof
  5. Reformulation of nonlinear system
  6. Resolvent estimates
  7. Space-time estimates
  8. Energy functional and nonlinear stability
  9. Resolvent estimates with Navier-slip boundary condition
  10. Resolvent estimate for $F\in L^2$
  11. Weak-type resolvent estimate for $F\in L^2$
  12. Resolvent estimate for $F\in {H_k^{-1}}$
  13. Weak-type resolvent estimate for $F\in H^{-1}_k$
  14. Resolvent estimate for $F\in H^1_0$
  15. Orr-Sommerfeld equation without nonlocal term
  16. ...and 25 more sections

Key Result

Theorem 1.1

Assume that $u_0\in H^1_0\cap H^{4}(\Omega)$ with $\mathrm{div}u_0=0$. There exist constants $\nu_0,c_0,c,C>0$ independent of $\nu$ such that if $\|u_0\|_{H^{4}}\leq c_0 \nu^{\frac{7}{4}}, 0<\nu \leq \nu_0$, then the solution $u$ of the problem pertur-ins-no-slip is global in time with the following

Figures (1)

  • Figure 1: Plane Poiseuille flow $(1-y^2,0,0)$

Theorems & Definitions (133)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Corollary 2.1
  • proof
  • Corollary 2.2
  • Remark 2.1
  • proof
  • Lemma 2.1
  • proof
  • ...and 123 more