Asymptotic stability threshold of the 2D Couette flow in a finite channel

Abstract

In this paper we study the asymptotic stability threshold of the Couette flow for the 2D Navier–Stokes equations in a finite channel $\Omega =\mathbb{T}\times [-1,1]$ with Navier-slip boundary condition. It was proved that if the initial velocity $v_0$ satisfies $\Vert v_0-(y,0){\Vert }_{H^4}\le c{\nu }^{1/3}$ 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 close to Couette flow for $t\gg {\nu }^{-1/3}$. Moreover, we prove the optimal enhanced dissipation and inviscid damping estimates. To this end, we develop a new approach that does not rely on the construction of the Fourier multiplier. Therefore, our approach opens a way toward the asymptotic stability threshold problem for other laminar flows in a domain with a physical boundary.