Enhanced dissipation for the 2D couette flow in critical space. Issue 12 (20th July 2020) - Masmoudi, Nader; Zhao, Weiren
Abstract: We consider the 2 D incompressible Navier-Stokes equations on T × R, with initial vorticity that is δ close in H x log L y 2 to −1(the vorticity of the Couette flow ( y, 0 ) ). We prove that if δ ≪ ν 1 / 2, 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 t ≫ ν − 1 / 3 by a mixing-enhanced dissipation effect and then converges back to Couette flow when t → + ∞ . In particular, we show the nonlinear enhanced dissipation and the inviscid damping results in the almost critical space H x log L y 2 ⊂ L x, y 2 .