Nonlinear inviscid damping near monotonic shear flows
Volume 230, Issue 2 (2023), pp. 321–399
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.