Stability of oppositely-propagating pair of Hill's spherical vortices

TL;DR

This paper proves the stability of a pair of Hill's spherical vortices moving apart in 3D Euler flows, showing each remains close to Hill's vortex profile under small initial interaction energy.

Contribution

It introduces a novel stability analysis for oppositely-propagating Hill's vortices using energy interaction and variational methods.

Findings

Each vortex remains close to Hill's vortex profile over time.

The speed of vortex propagation is estimated and shown to be optimal.

Stability holds when initial interaction energy is sufficiently small.

Abstract

We establish the stability of a pair of Hill's spherical vortices moving away from each other in 3D incompressible axisymmetric Euler equations without swirl. Each vortex in the pair propagates away from its odd-symmetric counterpart, while keeping its vortex profile close to Hill's vortex. This is achieved by analyzing the evolution of the interaction energy of the pair and combining it with the compactness of energy-maximizing sequences in the variational problem concerning Hill's vortex. The key strategy is to confirm that, if the interaction energy is initially small enough, the kinetic energy of each vortex in the pair remains so close to that of a single Hill's vortex for all time that each vortex profile stays close to the energy maximizer: Hill's vortex. An estimate of the propagating speed of each vortex in the pair is also obtained by tracking the center of mass of each vortex. The estimate can be understood as optimal in the sense that the power exponent of the $\varepsilon$--the small perturbation measured in the ($L^1\cap L^2$+impulse) norm--appearing in the error bound cannot be improved.