Linear Inviscid Damping in Sobolev and Gevrey Spaces

TL;DR

This paper offers a concise proof of linear inviscid damping stability in Gevrey spaces for shear flows, extending results to both finite and infinite channels with various perturbation conditions.

Contribution

It provides an alternative, shorter proof of Gevrey regularity stability based on high Sobolev regularity, covering finite/infinite channels and finite order vanishing perturbations.

Findings

Stability in Gevrey regularity follows from high Sobolev regularity.

Applicable to both finite and infinite channel settings.

Handles perturbations vanishing of finite order.

Abstract

In a recent article Jia established linear inviscid damping in Gevrey regularity for compactly supported Gevrey regular shear flows in a finite channel, which is of great interest in view of existing nonlinear results. In this article we provide an alternative very short proof of stability in Gevrey regularity as a consequence of stability in high Sobolev regularity. Here, we consider both the setting of a finite channel with compactly supported perturbations and of an infinite channel without this restriction. Furthermore, we consider the setting where perturbations vanish only of finite order.