Time evolution of concentrated vortex rings

Abstract

We study the time evolution of an incompressible fluid with axisymmetry without swirl when the vorticity is sharply concentrated. In particular, we consider $N$ disjoint vortex rings of size $\epsilon$ and intensity of the order of $|\log \epsilon {|}^{-1}$. We show that in the limit $\epsilon \to 0$, when the density of vorticity becomes very large, the movement of each vortex ring converges to a simple translation, at least for a small but positive time.

24 pages. This updated version provides a new Appendix B, containing the corrected proof of Lemma 3.1. For the sake of clarity, this proof has already been included in arXiv:2102.07807 (where the results of this article have been extended)

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