Stability threshold of the 2D Couette flow in Sobolev spaces

Abstract

We study the stability threshold of the 2D Couette flow in Sobolev spaces at high Reynolds number $Re$. We prove that if the initial vorticity ${\Omega }_{in}$ satisfies $\Vert {\Omega }_{in}-(-1){\Vert }_{H^{\sigma }}\le \epsilon R{e}^{-1/3}$, then the solution of the 2D Navier-Stokes equation approaches to some shear flow which is also close to Couette flow for time $t\gg R{e}^{1/3}$ by a mixing-enhanced dissipation effect and then converges back to Couette flow when $t\to +\infty$.