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Helical vortex filaments with compactly supported cross-sectional vorticity for the incompressible Euler equations in $\mathbb{R}^3$

Averkios Averkiou, Monica Musso

TL;DR

This work constructs smooth helical vortex filaments for the 3D incompressible Euler equations with helical symmetry and no swirl, achieving cross-sectional vorticity that remains compactly supported in $\mathbb{R}^2$ for all times. The authors implement an inner–outer gluing scheme guided by a carefully regularized Green’s function for the helical operator, driven by a semilinear elliptic core $\Gamma$ with a compactly supported nonlinearity to concentrate vorticity near a point on the helix. A precise linear theory for inner and outer problems, together with a projected inner problem and a solvability-tuned reduced constraint on the rotation speed $\alpha$, yields a global-in-time, smooth filament that concentrates to a Dirac measure on the helix as $\varepsilon\to 0$, and extends to polygonal multi-filament configurations. The results sharpen prior variational and patch-based approaches by delivering uniform-in-time confinement of cross-sectional vorticity and providing a detailed asymptotic description of vorticity cores in the helical setting, with potential implications for vortex dynamics in unbounded and pipe-like domains.

Abstract

We revisit the vortex filament conjecture for three-dimensional inviscid and incompressible Euler flows with helical symmetry and no swirl. Using gluing arguments, we provide the first construction of a smooth helical vortex filament in the whole space $\mathbb{R}^3$ whose cross-sectional vorticity is compactly supported in $\mathbb{R}^2$ for all times. The construction extends to a multi-vortex solution comprising several helical filaments arranged along a regular polygon. Our approach yields fine asymptotics for the vorticity cores, thus improving related variational results for smooth solutions in bounded helical domains and infinite pipes, as well as non-smooth vortex patches in the whole space.

Helical vortex filaments with compactly supported cross-sectional vorticity for the incompressible Euler equations in $\mathbb{R}^3$

TL;DR

This work constructs smooth helical vortex filaments for the 3D incompressible Euler equations with helical symmetry and no swirl, achieving cross-sectional vorticity that remains compactly supported in for all times. The authors implement an inner–outer gluing scheme guided by a carefully regularized Green’s function for the helical operator, driven by a semilinear elliptic core with a compactly supported nonlinearity to concentrate vorticity near a point on the helix. A precise linear theory for inner and outer problems, together with a projected inner problem and a solvability-tuned reduced constraint on the rotation speed , yields a global-in-time, smooth filament that concentrates to a Dirac measure on the helix as , and extends to polygonal multi-filament configurations. The results sharpen prior variational and patch-based approaches by delivering uniform-in-time confinement of cross-sectional vorticity and providing a detailed asymptotic description of vorticity cores in the helical setting, with potential implications for vortex dynamics in unbounded and pipe-like domains.

Abstract

We revisit the vortex filament conjecture for three-dimensional inviscid and incompressible Euler flows with helical symmetry and no swirl. Using gluing arguments, we provide the first construction of a smooth helical vortex filament in the whole space whose cross-sectional vorticity is compactly supported in for all times. The construction extends to a multi-vortex solution comprising several helical filaments arranged along a regular polygon. Our approach yields fine asymptotics for the vorticity cores, thus improving related variational results for smooth solutions in bounded helical domains and infinite pipes, as well as non-smooth vortex patches in the whole space.

Paper Structure

This paper contains 18 sections, 10 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Statement of the main result
  4. Reduction of the problem
  5. Notation
  6. Structure of the paper
  7. Construction of the approximate stream function
  8. Globally defined approximate regularisation
  9. Choice of Nonlinearity and Error of Approximation
  10. Inner-Outer Gluing Scheme
  11. Solving the Outer problem
  12. Inner and Outer Linear Theories
  13. Linear Outer Theory
  14. Linear Inner Theory
  15. Projected Linearised Inner Problem
  16. ...and 3 more sections

Key Result

Theorem 1

Let $r_0>0,\,h>0,$ and consider the helix $\mathcal{G}(\tau)$ parametrised by parhelix, as well as the rotation matrices $R_{\theta}$ and $Q_{\theta}$ in rotmatr. Then, there exist a constant $c>0$ and a smooth, global-in-time solution $\vec{\omega}_{\varepsilon}(x,\tau)$ to euler with compactly sup and in the sense of measures, where $\delta_{\mathcal{G}(\tau)}$ denotes a uniform Dirac delta sup

Theorems & Definitions (20)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 10 more