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In this paper, we study the transition threshold problem for the 2-DNavier-Stokes equations around the Couette flow $(y,0)$ at large Reynoldsnumber $Re$ in a finite channel. We develop a systematic method to establishthe resolvent estimates of the linearized operator and space-time estimates ofthe linearized Navier-Stokes equations. In particular, three kinds of importanteffects: enhanced dissipation, inviscid damping and boundary layer, areintegrated into the space-time estimates in a sharp form. As an application, weprove that if the initial velocity $v_0$ satisfies $\|v_0-(y, 0)\|_{H^2}\lecRe^{-\frac 12}$ for some small $c$ independent of $Re$, then the solution ofthe 2-D Navier-Stokes equations remains within $O(Re^{-\frac 12})$ of theCouette flow for any time.

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