Localization of point vortices under curvature perturbations

Abstract

We discuss the effect of curvature on the dynamics of a two-dimensional inviscid incompressible fluid with initial vorticity concentrated in N small disjoint regions, that is, the classical point vortex system. We recall some results about point vortex dynamics on simply connected surfaces with constant curvature K, that is, plane, spherical, and hyperbolic surfaces. We show that the effect of curvature can be treated as a smooth perturbation to the Green's function of the equation related to the stream function in the planar case. Then we obtain as a main result that the localization property of point vortices, already proved for the plane, is preserved also under the effect of curvature perturbation.