# Time evolution of concentrated vortex rings

## TL;DR

This paper investigates the evolution of concentrated vortex rings in an incompressible fluid, showing that as their size diminishes, their motion converges to simple translations over short times.

## Contribution

It provides a rigorous analysis of the limiting behavior of multiple vortex rings with concentrated vorticity as their size approaches zero.

## Key findings

- Vortex rings move as translations in the limit of small size.
- The vorticity density becomes very large as size shrinks.
- The results hold for multiple disjoint 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 $\varepsilon$ and intensity of the order of $|\log\varepsilon|^{-1}$. We show that in the limit $\varepsilon\to 0$, 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.