Collisions and regularization for the 3-vortex problem

TL;DR

This paper analyzes the dynamics of three vortices in a fluid, proving that only self-similar motions lead to total collisions and providing a regularization method for certain configurations.

Contribution

It establishes the conditions for total collisions and introduces a regularization technique for the reduced vortex system near collinear configurations.

Findings

Total collisions are only self-similar

No binary collisions occur in the system

Regularization smooths dynamics near collinear configurations

Abstract

We study the dynamics of 3 point-vortices on the plane for a fluid governed by Euler's equations, concentrating on the case when the moment of inertia is zero. We prove that the only motions that lead to total collisions are self-similar and that there are no binary collisions. Also, we give a regularization of the reduced system around collinear configurations (excluding binary collisions) which smoothes out the dynamics.