Transition threshold for the 2-D Couette flow in a finite channel

2018/08/27 by Qi Chen, Te Li, Chen, Qi +5
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1808.08736

openalex publication_date 2018/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the transition threshold problem for the 2-D Navier-Stokes equations around the Couette flow (y,0) at large Reynolds number Re in a finite channel. We develop a systematic method to establish the resolvent estimates of the linearized operator and space-time estimates of the linearized Navier-Stokes equations. In particular, three kinds of important effects: enhanced dissipation, inviscid damping and boundary layer, are integrated into the space-time estimates in a sharp form. As an application, we prove that if the initial velocity v₀ satisfies ‖v₀-(y, 0)‖_H²≤ cRe^{-\frac 12} for some small c independent of Re, then the solution of the 2-D Navier-Stokes equations remains within O(Re^{-\frac 12}) of the Couette flow for any time.

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