Abstract

We study the stability of the Couette flow under the 2D Navier–Stokes equations with full dissipation or only vertical dissipation. It was proved that if the initial velocity $v_0$ satisfies $\parallel v_0-(y,0){\parallel }_{H^3}\le c{\nu }^{1∕3}$ for some small $c$ independent of the viscosity coefficient $\nu$, then the solution of the 2D Navier–Stokes equations rapidly converges to some shear flow which is close to Couette flow for $t\gg {\nu }^{-1∕3}$. Moreover, we prove the optimal enhanced dissipation estimate of the vorticity and inviscid damping estimate of the velocity. To this end, we introduce two new ideas: (1) we make the energy estimates in short time scale $t\le {\nu }^{-1∕6}$ and long time scale $t\ge {\nu }^{-1∕6}$ separately; (2) we construct a new multiplier to control the growth caused by echo cascades.

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