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 v0\ of the Couette flow for any time.