Stability threshold of two-dimensional Couette flow in Sobolev spaces

Abstract

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