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

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 $\Vert v_0-(y,0){\Vert }_{H^2}\le cR{e}^{-\frac{1}{2}}$ for some small $c$ independent of $Re$, then the solution of the 2-D Navier-Stokes equations remains within $O(R{e}^{-\frac{1}{2}})$ of the Couette flow for any time.

46 pages