Nonlinear inviscid damping near monotonic shear flows

Volume 230, Issue 2 (2023), pp. 321–399

Alexandru D. Ionescu

^{ }Hao Jia^{ }
Pub. online: 18 July 2023
Type: Article

Received

28 January 2020

28 January 2020

Revised

9 June 2021

9 June 2021

Accepted

20 July 2021

20 July 2021

Published

18 July 2023

18 July 2023

#### Abstract

We prove non-linear 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 non-linear asymptotic stability result for Euler equations around general steady solutions for which the linearized flow cannot be explicitly solved.