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arXiv · 2111.07292

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

Xiang Yu
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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}.

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Xiang Yu. 2021-11-14. Finiteness of Stationary Configurations of the Planar Four-vortex Problem. II. https://arxiv.org/abs/2111.07292

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