Enhanced dissipation for the 2D couette flow in critical space

Abstract

We consider the 2 D incompressible Navier-Stokes equations on $\mathbb{T}\times \mathbf{R},$ with initial vorticity that is δ close in $H_x^{\mathit{log}}{L}_y^2$ to −1(the vorticity of the Couette flow $(y,0)$). We prove that if $\delta \ll {\nu }^{1/2},$ 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 $t\gg {\nu }^{-1/3}$ 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 $H_x^{\mathit{log}}{L}_y^2\subset L_{x,y}^2.$

Keywords:

• Couette flow; critical space; enhanced dissipation; inviscid damping, Navier Stokes

Additional information

Funding

The work of N. M is supported by NSF grant DMS-1716466 and by Tamkeen under the NYU Abu Dhabi Research Institute grant of the center SITE.