Time evolution of vortex rings with large radius and very concentrated vorticity

Abstract

We study the time evolution of an incompressible fluid with axial symmetry without swirl when the vorticity is sharply concentrated on $N$ annuli of radii $\approx$ $r_0$ and thickness $\epsilon$. We prove that when $r_0=|\log \epsilon {|}^{\alpha },\alpha >2$, the vorticity field of the fluid converges as $\epsilon \to 0$ to the point vortex model, at least for a small but positive time. This result generalizes a previous paper that assumed a power law for the relation between $r_0$ and $\epsilon$.