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Linear inviscid damping for monotone shear flows


Author: Christian Zillinger
Journal: Trans. Amer. Math. Soc. 369 (2017), 8799-8855
MSC (2010): Primary 76E05; Secondary 35Q31, 35Q35, 76B03
DOI: https://doi.org/10.1090/tran/6942
Published electronically: June 27, 2017
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Abstract: In this article, we prove linear stability, scattering and inviscid damping with optimal decay rates for the linearized 2D Euler equations around a large class of strictly monotone shear flows, $ (U(y),0)$, in a periodic channel under Sobolev perturbations. Here, we consider the settings of both an infinite periodic channel of period $ L$, $ \mathbb{T}_{L}\times \mathbb{R}$, as well as a finite periodic channel, $ \mathbb{T}_{L} \times [0,1]$, with impermeable walls. The latter setting is shown to not only be technically more challenging, but to exhibit qualitatively different behavior due to boundary effects.


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Christian Zillinger
Affiliation: Mathematisches Institut, Universitat Bonn, 53115 Bonn, Germany
Email: zill@math.uni-bonn.de, zillinge@usc.edu

DOI: https://doi.org/10.1090/tran/6942
Keywords: 2D Euler, incompressible, linear inviscid damping, shear flows, asymptotic stability, Lyapunov functional, boundary effects, blow-up
Received by editor(s): September 8, 2015
Received by editor(s) in revised form: March 21, 2016
Published electronically: June 27, 2017
Article copyright: © Copyright 2017 American Mathematical Society