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Dynamics of helical vortex filaments in non viscous incompressible flows

Martin Donati, Christophe Lacave, Evelyne Miot

TL;DR

The paper proves that for 3D Euler flows with vorticity initially concentrated around helices in a swirl-free, helical-symmetric setting, the concentration persists near filaments and each filament translates and rotates as a helix, with radii remaining distinct. The authors reduce the 3D problem to a 2D vorticity equation with a matrix-valued operator $K$, derive a sharp Biot–Savart decomposition through a Green’s function $ abla_x^ot \mathcal{G}_{K,\mathcal{U}}$, and analyze a reduced single-filament problem in an exterior field. They establish weak and strong localization results: the vorticity remains tightly confined to annuli around filament centers, and the centers follow a Da Rios-type motion law with a velocity coefficient $\nu_i$ determined by filament geometry. The combination of a precise Green’s-function decomposition and a Mar̆y PLC-type localization argument yields a rigorous description of vortex-filament dynamics for generic concentrated data and clarifies the role of the exterior-field interactions. These results provide a rigorous connection between 3D vortex dynamics under helically symmetric reduction and the predicted filament motion, with potential implications for vortex wakes and rotor/aircraft wake modeling.

Abstract

In this paper we study concentrated solutions of the three-dimensional Euler equations in helical symmetry without swirl. We prove that any helical vorticity solution initially concentrated around helices of pairwise distinct radii remains concentrated close to filaments. As suggested by the vortex filament conjecture, we prove that those filaments are translating and rotating helices. Similarly to what is obtained in other frameworks, the localization is weak in the direction of the movement but strong in its normal direction, and holds on an arbitrary long time interval in the naturally rescaled time scale. In order to prove this result, we derive a new explicit formula for the singular part of the Biot-Savart kernel in a two-dimensional reformulation of the problem. This allows us to obtain an appropriate decomposition of the velocity field to reproduce recent methods used to describe the dynamics of vortex rings or point-vortices for the lake equation.

Dynamics of helical vortex filaments in non viscous incompressible flows

TL;DR

The paper proves that for 3D Euler flows with vorticity initially concentrated around helices in a swirl-free, helical-symmetric setting, the concentration persists near filaments and each filament translates and rotates as a helix, with radii remaining distinct. The authors reduce the 3D problem to a 2D vorticity equation with a matrix-valued operator , derive a sharp Biot–Savart decomposition through a Green’s function , and analyze a reduced single-filament problem in an exterior field. They establish weak and strong localization results: the vorticity remains tightly confined to annuli around filament centers, and the centers follow a Da Rios-type motion law with a velocity coefficient determined by filament geometry. The combination of a precise Green’s-function decomposition and a Mar̆y PLC-type localization argument yields a rigorous description of vortex-filament dynamics for generic concentrated data and clarifies the role of the exterior-field interactions. These results provide a rigorous connection between 3D vortex dynamics under helically symmetric reduction and the predicted filament motion, with potential implications for vortex wakes and rotor/aircraft wake modeling.

Abstract

In this paper we study concentrated solutions of the three-dimensional Euler equations in helical symmetry without swirl. We prove that any helical vorticity solution initially concentrated around helices of pairwise distinct radii remains concentrated close to filaments. As suggested by the vortex filament conjecture, we prove that those filaments are translating and rotating helices. Similarly to what is obtained in other frameworks, the localization is weak in the direction of the movement but strong in its normal direction, and holds on an arbitrary long time interval in the naturally rescaled time scale. In order to prove this result, we derive a new explicit formula for the singular part of the Biot-Savart kernel in a two-dimensional reformulation of the problem. This allows us to obtain an appropriate decomposition of the velocity field to reproduce recent methods used to describe the dynamics of vortex rings or point-vortices for the lake equation.
Paper Structure (17 sections, 26 theorems, 240 equations)

This paper contains 17 sections, 26 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. The helical symmetry framework
  3. Some known facts on the 2D reduction and the well-posedness of the 3D Euler in helical symmetry
  4. Some useful properties of the matrix K
  5. Expansion of the Biot-Savart law
  6. Decomposition of the velocity
  7. Reduction of the problem to a single vortex
  8. Single vortex in an exterior field
  9. Preliminary computations
  10. Estimates on the local energy
  11. Estimate on radial vorticity moments
  12. Estimates on the energy
  13. Estimates on first and second vorticity moments
  14. Weak localization
  15. Strong localization
  16. ...and 2 more sections

Key Result

Theorem 1.1

Assume that there exists $R_{\mathcal{U}}>0$ such that $B(0,R_{\mathcal{U}})\subset \mathcal{U}$. Let $(z_{i,0})_{1 \leqslant i \leqslant N}$ be $N$ points in $B(0,R_{\mathcal{U}})$ such that $|z_{i,0}| \neq |z_{j,0}|$ for every $i \neq j$. Let $\gamma_i \in \mathbb{R}^*$. For every $\varepsilon >0$ For $T>0$, let $(v^\varepsilon,\omega^\varepsilon)$ be the unique weak solution of eq:Helico2D_resc

Theorems & Definitions (49)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Lemma 2.7
  • ...and 39 more