On the collapse of three point vortices on surfaces

Abstract

Point vortices represent an important reduced model describing two-dimensional ideal fluid dynamics. It is well known that there exist three-vortex configurations on the Euclidean plane $\mathbb{R}^2$ that exhibit finite-time singularities, i.e., collapse to a single point. Moreover, in $\mathbb{R}^2$, such collapses occur only self-similarly. Here, we investigate the extent to which this phenomenon persists on curved surfaces. We show that self-similar collapse is a universal feature of surfaces of nonnegative constant curvature, namely the plane and the sphere. In contrast, on the hyperbolic plane, it is shown that self-similar collapsing solutions do not exist with respect to any distance variable defined by an analytic function of the geodesic distance. Finally, we establish the existence of nearly self-similar collapse of three vortices on arbitrary smooth surfaces embedded in $\mathbb{R}^3$.