2026/07/17 by Theodore D. Drivas, Boris A. Khanikati, Valeriya A. Khanikati
#physics.flu-dyn #math-ph #math.MP
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 ℝ2 that exhibit finite-time singularities, i.e., collapse to a single point. Moreover, in ℝ2, 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 ℝ3.