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 high 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 a 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 ≦ c R e - 1 2 for some small c independent of Re , then the solution of the 2-D Navier–Stokes equations remains within O ( R e - 1 2 ) of the Couette flow for any time.