LONG-TIME ASYMPTOTICS FOR TWO-DIMENSIONAL EXTERIOR FLOWS WITH SMALL CIRCULATION AT INFINITY

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(Omega)(2) boolean AND L-q(Omega)(2) for some q is an element of (1, 2). If alpha is an element of 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.