On the Cauchy problem for axi-symmetric vortex rings (Q2257782)

This paper deals with the existence of a solution of a Cauchy problem for Navier Stokes equations with rotational symmetry and initial vorticity supported on a circle. It is shown that the initial value problem has a global solution, which is smooth for \(t>0\); the initial condition for the vorticity is satisfied in the weak sense. The main ideas involved in this study depend on the conservation of vorticity and momentum, axi-symmetric vector fields with no swirl, and the Nash-type estimates that yield decay laws even when the initial condition is a Dirac distribution. The uniqueness of the solution still remains an open question; however, it is conjectured that it may be established in some classes of axi-symmetric solutions without swirl.