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

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