Vol. 6, No. 4, 2013

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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
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 u0 = αΘ0 + v0, where Θ0 is the Oseen vortex with unit circulation at infinity and v0 is a solenoidal perturbation belonging to L2(Ω)2 Lq(Ω)2 for some q (1,2). If α 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 v0 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