Nonlinear inviscid damping near monotonic shear flows

TL;DR

This paper proves the nonlinear stability of certain monotonic shear flows in 2D Euler equations, showing solutions converge to nearby shear flows over time, under specific spectral and smoothness conditions.

Contribution

It establishes the first nonlinear asymptotic stability result for Euler flows around general steady solutions with non-explicit linearized flow solutions.

Findings

Solutions converge strongly to nearby shear flows as time approaches infinity.

Stability holds for shear flows with Gevrey smoothness and specific spectral properties.

The result applies to flows linear outside a compact subset of the channel.

Abstract

We prove nonlinear asymptotic stability of a large class of monotonic shear flows among solutions of the 2D Euler equations in the channel $\mathbb{T}\times[0,1]$. More precisely, we consider shear flows $(b(y),0)$ given by a function $b$ which is Gevrey smooth, strictly increasing, and linear outside a compact subset of the interval $(0,1)$ (to avoid boundary contributions which are incompatible with inviscid damping). We also assume that the associated linearized operator satisfies a suitable spectral condition, which is needed to prove linear inviscid damping. Under these assumptions, we show that if $u$ is a solution which is a small and Gevrey smooth perturbation of such a shear flow $(b(y),0)$ at time $t=0$, then the velocity field $u$ converges strongly to a nearby shear flow as the time goes to infinity. This is the first nonlinear asymptotic stability result for Euler equations around general steady solutions for which the linearized flow cannot be explicitly solved.