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

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 $\mathbb S^2$.