  Motivated by applications to vortex rings, we study the Cauchy problem for
the three-dimensional axisymmetric Navier-Stokes equations without swirl, using
scale invariant function spaces. If the axisymmetric vorticity is integrable
with respect to the two-dimensional measure dr dz, where (r,\theta,z) denote
the cylindrical coordinates in R^3, we show the existence of a unique global
solution, which converges to zero in L^1 norm as time goes to infinity. The
proof of local well-posedness follows exactly the same lines as in the
two-dimensional case, and our approach emphasizes the similarity between both
situations. The solutions we construct have infinite energy in general, so that
energy dissipation cannot be invoked to control the long-time behavior. We also
treat the more general case where the initial vorticity is a finite measure
whose atomic part is small enough compared to viscosity. Such data include
point masses, which correspond to vortex filaments in the three-dimensional
picture.
