$C^1$ type regularization for point vortices on $\mathbb S^2$

TL;DR

This paper develops a $C^1$ regularization method for point vortex solutions on the sphere, using tangent mappings and Lyapunov--Schmidt reduction, providing new insights into vortex dynamics and stability.

Contribution

It introduces a novel $C^1$ regularization framework for vortex solutions on $ ext{S}^2$, combining geometric and analytical techniques.

Findings

Vortices are located near nondegenerate critical points of Kirchhoff--Routh function.

The scaled stream function is a perturbation of the ground state for a plasma problem.

Qualitative and quantitative estimates for the regularization series are established.

Abstract

We construct a series of classic vorticity solutions for incompressible Euler equation on $\mathbb S^2$, which constitute the $C^1$ type regularization for a general traveling point vortex system. The construction is accomplished by applying tangent mapping on $\mathbb S^2$ and Lyapunov--Schmidt reduction argument. Using the fixed-point theorem and a finite dimensional equation on vortex dynamics, we prove that the vortices are located near a nondegenerate critical point of Kirchhoff--Routh function. Moreover, in the tangent space at each vortex center, the scaled stream function is verified as a perturbation of the ground state for generalized plasma problem. Some other qualitative and quantitative estimates for the regularization series are also obtained in this paper.