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

Takashi Sakajo; Changjun Zou

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arXiv · 2411.15176

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

Takashi Sakajo·Changjun Zou

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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.

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Takashi Sakajo, Changjun Zou. 2024-11-18. $C^1$ type regularization for point vortices on $\mathbb S^2$. https://arxiv.org/abs/2411.15176

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