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

TL;DR

This paper investigates the transition threshold for 2-D Couette flow in a finite channel at high Reynolds numbers, establishing resolvent estimates and demonstrating stability under small perturbations.

Contribution

It introduces a systematic method to derive resolvent and space-time estimates incorporating key effects like dissipation, damping, and boundary layers, advancing understanding of flow stability.

Findings

Proves stability of Couette flow for perturbations of order Re^{-1/2}.

Develops sharp resolvent estimates for the linearized operator.

Integrates dissipation, damping, and boundary layer effects into stability analysis.

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 $v_0$ satisfies $\|v_0-(y, 0)\|_{H^2}\le cRe^{-\frac 12}$ 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.