Enhanced dissipation for the 2D couette flow in critical space

We consider the 2 D incompressible Navier-Stokes equations on (Formula presented.) with initial vorticity that is δ close in (Formula presented.) to −1(the vorticity of the Couette flow (Formula presented.)). We prove that if (Formula presented.) where ν denotes the viscosity, then the solution of the Navier-Stokes equation approaches some shear flow which is also close to Couette flow for time (Formula presented.) by a mixing-enhanced dissipation effect and then converges back to Couette flow when (Formula presented.) In particular, we show the nonlinear enhanced dissipation and the inviscid damping results in the almost critical space (Formula presented.).