Collisions of Four Point Vortices in the Plane

Abstract

This paper addresses the question of existence of (not necessarily self-similar) solutions to the 4-vortex problem that lead to total or partial collision. We begin by showing that energy considerations alone imply that, for the general $N$-vortex problem, the virial being zero is a necessary condition for a solution to both evolve towards total collision and satisfy certain regularity condition. For evolutions assumed to be bounded, a classification for asymptotic partial collision configurations is offered. This classification depends on inertia and vorticity considerations. For non-necessarily bounded evolutions, we discuss the relationship between partial and (non-regular) total collisions when the virial is not zero and a generic condition on the vorticities holds. Finally, we give a canonical transformation that, for a class of 4-vortex systems satisfying a restriction on the vorticities, allows to formally apply the averaging principle in order reduce the dynamics when two of the vorticities are near a binary collision. The reduced system is a one-degree of freedom hamiltonian system.