Asymptotic stability of the symmetric flow via inviscid damping and enhanced dissipation

Abstract

In this paper, we establish the inviscid damping and enhanced dissipation estimates for the linearized Navier-Stokes system around the symmetric flow in a finite channel with the non-slip boundary condition. As an immediate consequence, we prove the asymptotic stability of the symmetric flow in the high Reynolds number regime. Namely, if the initial velocity perturbation $u^{\mathrm{in}}$ satisfies $\Vert u^{\mathrm{in}}-(U(y),0){\Vert }_{H^5}\le c{\nu }^{\frac{2}{3}}$, then inviscid damping and enhanced dissipation estimates also hold for the solution to the Navier-Stokes system.