Global dynamics of a single vortex ring

TL;DR

This paper rigorously analyzes the long-term behavior of vortex rings in 3D Euler flows, confirming the vortex filament conjecture and revealing a universal filamentation mechanism causing instability.

Contribution

It provides the first global-in-time validation of the vortex filament conjecture for generic initial data and uncovers a universal filamentation process in vortex ring dynamics.

Findings

Vorticity remains sharply concentrated and propagates along the axis.

Leading-order speed matches the Kelvin--Hicks formula.

Thin vortex rings are dynamically unstable under general conditions.

Abstract

We study the global-in-time dynamics of vortex rings for the three-dimensional incompressible Euler equations, under the assumption of axisymmetric flows without swirl. For a broad class of initial data sharing only the macroscopic invariants with a thin vortex ring, we prove that the vorticity remains sharply concentrated and propagates along the symmetry axis with leading-order speed given by the Kelvin--Hicks formula, providing the first global-in-time validation of the vortex filament conjecture for a single vortex ring arising from generic initial data. We further identify a universal filamentation mechanism driven by the competition between rapid core translation and slower local induction. This mechanism gives linear-in-time stretching of the vortex support under very general assumptions on the data, yielding dynamical instability of any thin vortex ring configurations in the $W^{2,\infty}$ norm.