Linear inviscid damping in Sobolev and Gevrey spaces. (December 2021) - Zillinger, Christian
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 L 2 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.