Nearly parallel helical vortex filaments in the three dimensional Euler equations

TL;DR

This paper rigorously justifies a formal asymptotic model describing the evolution of nearly parallel helical vortex filaments in 3D Euler equations, focusing on specific symmetric configurations.

Contribution

It provides a rigorous mathematical validation of the asymptotic motion law for vortex filaments in two symmetric configurations.

Findings

Validation of the asymptotic model for regular polygon configurations

Extension to configurations with a central straight filament

Mathematical proof of the model's accuracy in specified setups

Abstract

Klein, Majda, and Damodaran have previously developed a formalized asymptotic motion law describing the evolution of nearly parallel vortex filaments within the framework of the three-dimensional Euler equations for incompressible fluids. In this study, we rigorously justify this model for two configurations: the central configuration consisting of regular polygons of $N$ helical-filaments rotating with constant speed, and the central configurations of $N+1$ vortex filaments, where an $N$-polygonal central configuration surrounds a central straight filament.