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.

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Mathematical Subject Classification 2010

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