On the Cauchy problem for axi-symmetric vortex rings

TL;DR

This paper proves the existence of solutions to the 3D Navier-Stokes equations with initial vorticity concentrated on a circle, using approximation methods that handle singular initial data without size restrictions.

Contribution

It establishes the existence of solutions for axi-symmetric vortex rings with singular initial vorticity, extending previous results to more general initial conditions.

Findings

Solutions exist for initial vorticity on a circle

Good a-priori estimates are obtained for approximations

No restrictions on initial data size

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.