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Stability for the 2-D plane Poiseuille flow in finite channel

Shijin Ding, Zhilin Lin

Abstract

In this paper, we study the stability for 2-D plane Poiseuille flow $(1-y^2,0)$ in a channel $\mathbb{T}\times (-1,1)$ with Navier-slip boundary condition. We prove that if the initial perturbation for velocity field $u_0$ satisfies that $\|u_0\|_{H^{\frac{7}{2}+}} \leq ε_1 ν^{2/3}$ for some suitable small $0<ε_1 \ll 1$ independent of viscosity coefficient $ν$, then the solution to the Navier-Stokes equations is global in time and does not transit from the plane Poiseuille flow. This result improves the result of \cite{DL1} from $3/4$ to $2/3$.

Stability for the 2-D plane Poiseuille flow in finite channel

Abstract

In this paper, we study the stability for 2-D plane Poiseuille flow in a channel with Navier-slip boundary condition. We prove that if the initial perturbation for velocity field satisfies that for some suitable small independent of viscosity coefficient , then the solution to the Navier-Stokes equations is global in time and does not transit from the plane Poiseuille flow. This result improves the result of \cite{DL1} from to .
Paper Structure (5 sections, 5 theorems, 75 equations)

This paper contains 5 sections, 5 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Sketch and key ideas of the proof
  3. Resolvent estimates for Orr-Sommerfeld equation
  4. Space-time estimates for linearized Navier-Stokes equations around the Poiseuille flow
  5. Nonlinear stability

Key Result

Theorem 1.1

Suppose that $\mathrm{div} u_0=0, u_0\in H^{\frac{7}{2}+}$ with $\|u_0\|_{H^{\frac{7}{2}+}} \leq \epsilon_1 \nu^{2/3}$ for some suitable small $0<\epsilon_1 \ll 1$ independent of $\nu$, then vor-ns with Naver-slip-bc admits a global in time solution $\omega$ with where the energy functional $\mathcal{E}_k$ is defined as

Theorems & Definitions (11)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 1 more