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Nearly parallel helical vortex filaments in the three dimensional Euler equations

Ignacio Guerra, Monica Musso

TL;DR

The paper provides a rigorous derivation of the Klein–Majda–Damodaran nearly-parallel vortex filament model from the 3D Euler equations for two central configurations: (i) N helices forming a regular polygon and (ii) N helices surrounding a central straight filament. It constructs ε-concentrated vorticity along the helices via a screw-symmetric, inner-outer gluing framework built on Liouville-type inner profiles and a carefully chosen scale μ, achieving a globally defined approximate stream function that concentrates near the vortex curves. The main results show the existence of smooth Euler solutions whose vorticity converges to a sum of delta-regularized filaments with center dynamics dictated by the KMD law, with the leading-order frequency α_ε matching the KMD prediction $\alpha_ε = 2(1/h^{2} - (N-1)/r^{2}) + o(1)$ (and an analogous expression for the N+1 configuration). This work provides a rigorous foundation for the KMD model in Euler flow and demonstrates a robust inner-outer gluing construction for forming and controlling multiple nearly parallel helices in 3D fluids.

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.

Nearly parallel helical vortex filaments in the three dimensional Euler equations

TL;DR

The paper provides a rigorous derivation of the Klein–Majda–Damodaran nearly-parallel vortex filament model from the 3D Euler equations for two central configurations: (i) N helices forming a regular polygon and (ii) N helices surrounding a central straight filament. It constructs ε-concentrated vorticity along the helices via a screw-symmetric, inner-outer gluing framework built on Liouville-type inner profiles and a carefully chosen scale μ, achieving a globally defined approximate stream function that concentrates near the vortex curves. The main results show the existence of smooth Euler solutions whose vorticity converges to a sum of delta-regularized filaments with center dynamics dictated by the KMD law, with the leading-order frequency α_ε matching the KMD prediction (and an analogous expression for the N+1 configuration). This work provides a rigorous foundation for the KMD model in Euler flow and demonstrates a robust inner-outer gluing construction for forming and controlling multiple nearly parallel helices in 3D fluids.

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 helical-filaments rotating with constant speed, and the central configurations of vortex filaments, where an -polygonal central configuration surrounds a central straight filament.

Paper Structure

This paper contains 11 sections, 16 theorems, 300 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on the operator $\nabla \cdot ( K \, \nabla \, )$
  3. The approximate stream function
  4. The approximate stream function near the vortices.
  5. Global definition for the approximate stream function.
  6. Choice of $\mu$
  7. Choice of the non-linearity and construction of the first approximate solution
  8. Choice of the non-linearity $F$
  9. Estimate of the error function
  10. The inner-outer gluing system
  11. Solving the inner-outer gluing system

Key Result

Theorem 1

Let be given. Let $\gamma_j$ be the helices parametrized by helices with $\nu= {1\over h}$, for $j=1,\ldots,N$. Then there exist $\alpha_\varepsilon >0$ and a smooth solution $\vec{\omega}_\varepsilon (x,t)$ to $(euler)$ such that in sense of distributions. Here and ${\textbf{t}}_{\gamma_j^\varepsilon }$ is the tangent vector to $\gamma_j^\varepsilon.$

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.1
  • Proposition 3.2
  • ...and 25 more