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

TL;DR

This paper proves the asymptotic stability of symmetric flows in a finite channel by establishing inviscid damping and enhanced dissipation estimates for the linearized Navier-Stokes system, especially at high Reynolds numbers.

Contribution

It introduces new stability results for symmetric flows using inviscid damping and enhanced dissipation estimates in the context of the Navier-Stokes equations with non-slip boundary conditions.

Findings

Proves inviscid damping for the linearized Navier-Stokes system.

Establishes enhanced dissipation estimates in a finite channel.

Demonstrates asymptotic stability of symmetric flows at high Reynolds numbers.

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 $\Vert u^{\mathrm{in}}-(U(y),0) \Vert_{H^5}\leq c \nu^{\frac{2}{3}}$, then inviscid damping and enhanced dissipation estimates also hold for the solution to the Navier-Stokes system.