Nonlinear inviscid damping near monotonic shear flows

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.

Full Text (PDF format)

The first author was supported in part by NSF grant DMS-1600028. The second author was supported in part by NSF grants DMS-1600779 and DMS-1945179.

Received 28 January 2020

Received revised 9 June 2021

Accepted 20 July 2021

Published 18 July 2023