We consider the 2D incompressible Navier-Stokes equations on $\mathbb{ T}\times \mathbf{ R}$, with initial vorticity that is $\delta$ close in $H^{ log}_xL^2_{ y}$ to $-1$(the vorticity of the Couette flow $(y,0)$). We prove that if $\delta\ll \nu^{ 1/2}$, 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 $t\gg \nu^{ -1/3}$ by a mixing-enhanced dissipation effect and then converges back to Couette flow when $t\to +\infty$. In particular, we show the nonlinear enhanced dissipation and the inviscid damping results in the almost critical space $H^{ log}_xL^2_{ y}\subset L^2_{ x,y}$.
Categories: math.AP