Global time evolution of concentrated vortex rings

TL;DR

This paper analyzes the long-term evolution of concentrated vortex rings in an incompressible fluid, showing that as their thickness shrinks, each ring moves steadily along the axis, with precise localization of vorticity support.

Contribution

It extends previous work by proving convergence of vortex ring motion to translation for arbitrary times, with sharp radial localization and axial concentration properties.

Findings

Vortex rings translate steadily along the axis as thickness approaches zero.

Radial localization of vorticity support is sharp at any time.

Axial concentration is maintained over arbitrary time intervals.

Abstract

We study the time evolution of an incompressible fluid with axial symmetry without swirl, assuming initial data such that the initial vorticity is very concentrated inside $N$ small disjoint rings of thickness $\varepsilon$ and vorticity mass of the order of $|\log\varepsilon|^{ -1}$. When $\varepsilon \to 0$ we show that the motion of each vortex ring converges to a simple translation with constant speed (depending on the single ring) along the symmetry axis. We obtain a sharp localization of the vorticity support at time $t$ in the radial direction, whereas we state only a concentration property in the axial direction. This is obtained for arbitrary (but fixed) intervals of time. This study is the completion of a previous paper arXiv:1904.04785 [math-ph], where a sharp localization of the vorticity support was obtained both along the radial and axial directions, but the convergence for $\varepsilon\to 0$ worked only for short times.