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 \operatorname{Re} . We prove that if the initial vorticity \Omega_{\operatorname{in}} satisfies \|\Omega_{\operatorname{in}}-(-1)\|_{H^{\sigma}}\leq \varepsilon \operatorname{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 \operatorname{Re}^{1/3} by a mixing-enhanced dissipation effect, and then converges back to Couette flow when t\to +\infty .

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