Long-time asymptotics for two-dimensional exterior flows with small circulation at infinity

Gallay, Thierry; Maekawa, Yasunori

doi:10.2140/apde.2013.6.973

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Long-time asymptotics for two-dimensional exterior flows with small circulation at infinity

Thierry Gallay and Yasunori Maekawa

Vol. 6 (2013), No. 4, 973–991

DOI: 10.2140/apde.2013.6.973

Abstract

We consider the incompressible Navier–Stokes equations in a two-dimensional exterior domain $Ω$ , with no-slip boundary conditions. Our initial data are of the form $u_{0} = α Θ_{0} + v_{0}$ , where $Θ_{0}$ is the Oseen vortex with unit circulation at infinity and $v_{0}$ is a solenoidal perturbation belonging to $L^{2} {(Ω)}^{2} \cap L^{q} {(Ω)}^{2}$ for some $q \in (1, 2)$ . If $α \in ℝ$ is sufficiently small, we show that the solution behaves asymptotically in time like the self-similar Oseen vortex with circulation $α$ . This is a global stability result, in the sense that the perturbation $v_{0}$ can be arbitrarily large, and our smallness assumption on the circulation $α$ is independent of the domain $Ω$ .

Keywords

Navier–Stokes equations, exterior domains, long-time asymptotics, Oseen vortices

Mathematical Subject Classification 2010

Primary: 35B35, 35Q30, 76D05, 76D17

Milestones

Received: 28 February 2012

Accepted: 6 August 2012

Published: 21 August 2013

Authors

Thierry Gallay
Institut Fourier Université de Grenoble I 100 rue des Maths B.P. 74 38402 Saint-Martin-d’Hères France
http://www-fourier.ujf-grenoble.fr/~gallay
Yasunori Maekawa
Department of Mathematics Graduate School of Science Kobe University 1-1 Rokkodai, Nada-ku Kobe 657-8501 Japan
http://www.math.kobe-u.ac.jp/home-j/yasunori.html

Vol. 6, No. 4, 2013