On Finiteness of Stationary Configurations of the Planar Five-vortex Problem

TL;DR

This paper investigates the finiteness of stationary configurations in the planar five-vortex problem, establishing upper bounds on equilibria and rigid configurations, and showing finiteness of relative equilibria and collapses under certain conditions.

Contribution

It provides new bounds on the number of stationary configurations and proves finiteness results for the five-vortex problem, including special cases with same-sign vorticities.

Findings

Maximum of 6 equilibria

Maximum of 24 rigidly translating configurations

Finiteness of relative equilibria and collapse configurations under generic conditions

Abstract

The finiteness problem of stationary configurations for the planar five-vortex problem is considered in this paper. The numbers of equilibria and rigidly translating configurations are shown to be at most 6 and 24 respectively. The numbers of relative equilibria and collapse configurations are shown to be finite, except perhaps if the 5-tuple of vorticities belongs to a given codimension 2 subvariety of the vorticity space. In particular, if the vorticities are of the same sign, the number of stationary configurations is finite.