Abstract
We consider the 2 D incompressible Navier-Stokes equations on with initial vorticity that is δ close in
to −1(the vorticity of the Couette flow
). We prove that if
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
by a mixing-enhanced dissipation effect and then converges back to Couette flow when
In particular, we show the nonlinear enhanced dissipation and the inviscid damping results in the almost critical space