Year: 2026
Type: article
Abstract: In this paper, we study the nonlinear stability for the 3-D plane Poiseuille flow ( 1 − y 2 , 0 , 0 ) (1-y^2,0,0) at high Reynolds number R e Re in a finite channel T × [ − 1 , 1 ] × T \mathbb {T}\times [-1,1 ]\times \mathbb {T} with non-slip boundary condition. We prove that if the initial velocity v 0 v_0 satisfies ‖ v 0 − ( 1 − y 2 , 0 , 0 ) ‖ H 4 ≤ c 0 R e − 7 4 \begin{equation*} \|v_0-(1-y^2,0,0)\|_{H^{4}}\leq c_0 Re^{-\frac {7}{4}} \end{equation*} for some c 0 > 0 c_0>0 independent of R e Re , then the solution of 3-D Naiver-Stokes equations is global in time and does not transition 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 consistent with the numerical result by Lundbladh et al. [ Threshold amplitudes for transition in channel flows , Transition, Turbulence and Combustion, 309–318, Springer, Dordrecht, 1994].
DOI: 10.1090/memo/1640
Authors: Qi Chen, Shijin Ding, Zhilin Lin, Zhifei Zhang
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4387800753