Global existence and collisions for symmetric configurations of nearly parallel vortex filaments

TL;DR

This paper proves large-time existence and collision phenomena for symmetric configurations of nearly parallel vortex filaments modeled by a Schrödinger system with Newton-type interactions, extending previous two- and three-filament results.

Contribution

It establishes large-time existence results for specific configurations of four or more nearly parallel vortex filaments and describes conditions leading to collisions.

Findings

Large-time existence for four and N filaments

Existence of traveling wave dynamics

Configurations leading to filament collisions

Abstract

We consider the Schr\"odinger system with Newton-type interactions that was derived by R. Klein, A. Majda and K. Damodaran [18] to modelize the dynamics of N nearly parallel vortex filaments in a 3-dimensional homogeneous incompressible uid. The known large time existence results are due to C. Kenig, G. Ponce and L. Vega [17] and concern the interaction of two filaments and particular configurations of three filaments. In this article we prove large time existence results for particular configurations of four nearly parallel filaments and for a class of configurations of N nearly parallel filaments for any N\geq 2. We also show the existence of travelling wave type dynamics. Finally we describe configurations leading to collision.