On three-dimensional Navier-Stokes equations with axi-symmetric vortex rings as initial vorticity

Abstract

We consider the classical Cauchy problem for the 3d Navier-Stokes equation with the initial vorticity \omega_0 concentrated on a circle, or more generally, a linear combination of such data for circles with common axis of symmetry. We show that natural approximations of the problem obtained by smoothing the initial data satisfy good a-priori estimates which enable us to conclude that the original problem with the singular initial distribution of vorticity has a solution.We impose no restriction on the size of the initial data.