Enhanced dissipation for the 2D couette flow in critical space

We consider the 2D incompressible Navier-Stokes equations on T x R, with initial vorticity that is delta close in H-x(log) L-y(2) to -1(the vorticity of the Couette flow (y, 0)). We prove that if delta << nu(1/2), where nu denotes the viscosity, then the solution of the Navier-Stokes equation approaches some shear flow which is also close to Couette flow for time t >> nu(-1/3) by a mixing-enhanced dissipation effect and then converges back to Couette flow when t -> +infinity. In particular, we show the nonlinear enhanced dissipation and the inviscid damping results in the almost critical space H-x(log) L-y(2) subset of L-x,y(2).