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

2023/10/18 by Chen, Qi, Ding, Shijin, Lin, Zhilin +1 · 3 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2310.11694

Abstract

In this paper, we study the nonlinear stability for the 3-D plane Poiseuille flow (1-y²,0,0) at high Reynolds number Re in a finite channel \mathbbT× [-1,1 ]× \mathbbT with non-slip boundary condition. We prove that if the initial velocity v₀ satisfies ‖v₀-(1-y²,0,0)‖_H⁴≤ c₀ Re^-(7)/(4) for some c₀>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. \citeLHR.

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