Stability threshold for 2D shear flows near Couette of the Navier-Stokes equation

Abstract

In this paper, we consider the stability threshold of the 2D shear flow $(U(y),0{)}^{\top }$ of the Navier-Stokes equation at high Reynolds number $Re$. When the shear flow is near in Sobolev norm to the Couette flow $(y,0{)}^{\top }$ in some sense, we prove that if the initial data $u_0$ satisfies $\Vert u_0-(U(y),0{)}^{\top }\Vert \le \epsilon R{e}^{-1/3}$, then the solution of the 2D Navier-Stokes equation approaches to some shear flow which is also close to the Couette flow for $t\gg R{e}^{1/3}$, as $t\to \infty$.