  Both experimental and numerical studies of fluid motion indicate that
initially localized regions of vorticity tend to evolve into isolated vortices
and that these vortices then serve as organizing centers for the flow. In this
paper we prove that in two dimensions localized regions of vorticity do evolve
toward a vortex. More precisely we prove that any solution of the
two-dimensional Navier-Stokes equation whose initial vorticity distribution is
integrable converges to an explicit self-similar solution called ``Oseen's
vortex''. This implies that the Oseen vortices are dynamically stable for all
values of the circulation Reynolds number, and our approach also shows that
these vortices are the only solutions of the two-dimensional Navier-Stokes
equation with a Dirac mass as initial vorticity. Finally, under slightly
stronger assumptions on the vorticity distribution, we also give precise
estimates on the rate of convergence toward the vortex.
