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

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 $\Omega$.