\(\boldsymbol{C^1}\) Type Regularization for Point Vortices on \(\mathbb S^2\)

Abstract. We construct a series of classic vorticity solutions for the incompressible Euler equation on [Formula: see text], which constitute the [Formula: see text] type regularization for a general traveling point vortex system. The construction is accomplished by applying a tangent mapping on [Formula: see text] and the Lyapunov–Schmidt reduction argument. By introducing a normalized condition to deal with the rotation invariance on [Formula: see text], we prove that the vortices will be located near a 1-dimensional degenerate normalized critical point of the 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 a generalized plasma problem. Some other qualitative and quantitative estimates for the regularization series are also obtained in this paper.