Regularization for point vortices on $\mathbb S^2$

TL;DR

This paper constructs smooth vortex patch solutions on the sphere that approximate point vortex systems, using Lyapunov--Schmidt reduction, and demonstrates their properties and convergence to classical vortex configurations.

Contribution

It introduces the first regularization method for point vortex equilibria on the sphere via patch solutions and proves their existence and boundary regularity.

Findings

Existence of symmetric patch solutions approximating point vortices.

Construction of solutions near critical points of the Kirchhoff--Routh function.

Vortex patch boundaries are $C^1$ close curves, perturbations of small ellipses.

Abstract

We construct a series of patch type solutions for incompressible Euler equation on $\mathbb S^2$, which constitutes the regularization for steady or traveling point vortex systems. We first prove the existence of $k$-fold symmetric patch solutions, whose limit is the well-known von K\'arm\'an point vortex street on $\mathbb S^2$; then we consider the general steady case, where besides a non-localized part induced by the sphere rotation, $j$ positive and $k$ negative patches are located near a nondegenerate critical point of the Kirchhoff--Routh function on $\mathbb S^2$. Our construction is accomplished by Lyapunov--Schmidt reduction argument, where the traveling speed or vortex patch location are used to eliminate the degenerate direction of a linearized operator. We also show that the boundary of each vortex patch is a $C^1$ close curve, which is a perturbation of a small ellipse in the spherical coordinates. As far as we know, this is the first attempt for a regularization of the point-vortex equilibria on $\mathbb S^2$.