Desingularization of vortices for the incompressible Euler equation on a sphere

TL;DR

This paper constructs stable, rotating solutions to the incompressible Euler equations on a sphere that approximate pairs of point vortices, using variational methods and symmetry considerations.

Contribution

It introduces a novel variational approach to construct and analyze stable vortex solutions on a sphere that approximate point vortices.

Findings

Constructed global solutions with vortex pair asymptotics.

Proved stability of solutions under symmetric perturbations.

Analyzed asymptotic behavior of energy-maximizing configurations.

Abstract

In this paper, we construct a family of global solutions to the incompressible Euler equation on a standard 2-sphere. These solutions are odd-symmetric with respect to the equatorial plane and rotate with a constant angular speed around the polar axis. More importantly, these solutions ``converges" to a pair of point vortices with equal strength and opposite signs. The construction is achieved by maximizing the energy-impulse functional relative to a family of suitable rearrangement classes and analyzing the asymptotic behavior of the maximizers. Based on their variational characterization, we also prove the stability of these rotating solutions with respect to odd-symmetric perturbations.