Finiteness of Stationary Configurations of the Planar Four-vortex Problem

TL;DR

This paper proves that the planar four-vortex problem has only finitely many stationary configurations, including equilibria, translations, rotations, and collapses, and provides upper bounds for each class.

Contribution

It establishes the finiteness and bounds of stationary configurations in the four-vortex problem, a novel result in vortex dynamics.

Findings

Finitely many stationary configurations exist in the four-vortex problem.

Upper bounds are provided for each class of configurations.

Includes equilibria, translations, rotations, and collapse configurations.

Abstract

For the planar four-vortex problem, we show that there are finitely many stationary configurations consisting of equilibria, rigidly translating configurations, relative equilibria (uniformly rotating configurations) and collapse configurations. We also provide upper bounds for these classes of stationary configurations.