On the Finite-Time Singularities of Point-Vortex Systems I: The Half-Plane

Abstract

In this paper, the first in a series devoted to finite-time singularities of point-vortex systems in domains with boundaries, we establish the existence of finite-time interior collapse for three vortices in the upper half-plane $\mathbb{H}^+$. Our approach treats the boundary-induced interactions as a perturbation of the classical self-similar Gröbli collapse in the full plane. Under suitable conditions on the vortex circulations and collapse profile, we prove the existence of a one-dimensional local strong stable manifold of the augmented rescaled system, which generates exact solutions in which three distinct vortices collapse simultaneously at any prescribed interior point $z^*\in\mathbb{H}^+$. Moreover, these solutions are asymptotically self-similar, with the characteristic square-root collapse rate $O(\sqrt{T-t})$.