#### Vol. 6, No. 4, 2013

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
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 $\Omega$, with no-slip boundary conditions. Our initial data are of the form ${u}_{0}=\alpha {\Theta }_{0}+{v}_{0}$, where ${\Theta }_{0}$ is the Oseen vortex with unit circulation at infinity and ${v}_{0}$ is a solenoidal perturbation belonging to ${L}^{2}{\left(\Omega \right)}^{2}\cap {L}^{q}{\left(\Omega \right)}^{2}$ for some $q\in \left(1,2\right)$. If $\alpha \in ℝ$ is sufficiently small, we show that the solution behaves asymptotically in time like the self-similar Oseen vortex with circulation $\alpha$. 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 $\alpha$ is independent of the domain $\Omega$.

##### 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