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

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.