Inviscid damping near the Couette flow in a channel

TL;DR

This paper proves the asymptotic stability of the Couette flow in a 2D channel for the Euler equations, showing that small smooth perturbations decay and the flow converges to a nearby shear flow over time.

Contribution

It establishes the nonlinear stability of Couette flow in a channel setting for the 2D Euler equations, extending understanding of inviscid damping in bounded domains.

Findings

Velocity converges strongly to a shear flow.

Vorticity remains supported in the interior and converges weakly to zero.

Perturbations decay over time, confirming stability.

Abstract

We prove asymptotic stability of shear flows in a neighborhood of the Couette flow for the 2D Euler equations in the domain $\T\times[0,1]$. More precisely we prove that if we start with a small and smooth perturbation (in a suitable Gevrey space) of the Couette flow, then the velocity field converges strongly to a nearby shear flow. The vorticity, which is initially assumed to be supported in the interior of the channel, will remain supported in the interior of the channel, will be driven to higher frequencies by the linear flow, and will converge weakly to $0$ as $t\to\infty$, modulo the shear flows (zero mode in $x$).