Linear inviscid damping in Sobolev and Gevrey spaces. (December 2021)

• Record Type: Journal Article
• Authors: Zillinger, Christian
• Is Part Of: Nonlinear analysis. Volume 213(2021)
• Publication Date: 2021-12
• Subjects: 76E05 -- 35Q31 -- 76B03 -- 35B40; Inviscid damping -- Gevrey regularity -- Shear flows -- Euler equations -- Asymptotic stability; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112492
• Languages: English
• ISSNs: 0362-546X

Abstract

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.