Enhanced dissipation for the 2D Couette flow in critical space

We consider the 2D incompressible Navier-Stokes equations on $\mathbb{T}\times \mathbf{R}$, 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 $\delta \ll {\nu }^{1\mathrm{/}2}$, 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 $t\gg {\nu }^{-1\mathrm{/}3}$ by a mixing-enhanced dissipation effect and then converges back to Couette flow when $t\to +\mathrm{\infty }$. In particular, we show the nonlinear enhanced dissipation and the inviscid damping results in the almost critical space $H^{log}xL^2{y}\subset L^2_{x,y}$.