Finiteness of Stationary Configurations of the Planar Four-vortex Problem. II

TL;DR

This paper proves that the planar four-vortex problem has finitely many collapse configurations, extending previous results and providing improved bounds, which is surprising given the infinite collapse configurations in other vortex problems.

Contribution

It establishes the finiteness of collapse configurations in the four-vortex problem without prescribing a collapse constant, using advanced singularity analysis methods.

Findings

Finitely many collapse configurations in the four-vortex problem.

Improved upper bounds for collapse configurations.

Contrasts with infinite collapse configurations in N=3 and N=5 cases.

Abstract

In an earlier paper \cite{yu2021Finiteness}, we showed that there are finitely many stationary configurations (consisting of equilibria, rigidly translating configurations, relative equilibria and collapse configurations) in the planar four-vortex problem. However, we only established finiteness of collapse configurations in the sense of prescribing a collapse constant. In this paper, by developing ideas of Albouy-Kaloshin and Hampton-Moeckel to do an analysis of the singularities, we further show that there really are finitely many collapse configurations in the four-vortex problem. This is an unexpectedly result, because the $N$-vortex problem has infinitely many collapse configurations for $N= 3$ and for $N= 5$. %to complete the work on finiteness of stationary configurations of the planar four-vortex problem. We also provide better upper bounds for collapse configurations than that in \cite{yu2021Finiteness}.