On the Cauchy Problem for Axi-Symmetric Vortex Rings

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.