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arXiv · 2310.11694

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

Qi Chen·Shijin Ding·Zhilin Lin·Zhifei Zhang
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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}.

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Qi Chen, Shijin Ding, Zhilin Lin, Zhifei Zhang. 2023-10-18. Nonlinear stability for 3-D plane Poiseuille flow in a finite channel. https://arxiv.org/abs/2310.11694

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