Linear inviscid damping in Sobolev and Gevrey spaces

Abstract

In a recent article (Jia, 2020) 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 (Deng and Masmoudi, 2018; Masmoudi and Masmoudi, 2015; Ionescu and Jia, 2019). In this article we provide an alternative short proof of stability in Gevrey regularity for those flows which admit an approach by a Fourier-based Lyapunov functional. For these flows we show that stability in L2 by Fourier methods as in Zillinger (2017a) and Zillinger (2016) immediately upgrades to stability in Gevrey regularity. Furthermore, in the setting of a finite channel we do not need to assume compact support but only vanishing of infinite order and also establish Sobolev stability results for perturbations vanishing to finite order.