The Euler vortex-filament limit for a general moving curve
Let $\gamma : [0,T] \times (\mathbb{R}/L\mathbb{Z}) \to \mathbb{R}^3$ be a smooth family of embedded closed curves, parametrized by arclength, solving the binormal curvature flow $$\partial_t \gamma = \partial_s \gamma \times \partial_{ss} \gamma.$$ Da Rios derived this law formally as the motion of a thin vortex filament. With time rescaled by $4\pi/|\log(\epsilon/L)|$, Jerrard and Seis proved that if the vorticity of a conservative weak solution of the Euler equations is initially concentrated, in a flat-norm sense on the scale $\epsilon$, around $\gamma(0)$, with initial energy close to the least compatible with that concentration, and at every later time stays so concentrated around some Lipschitz curve of length $L$ with uniformly bounded generalized curvature, then those curves stay within a distance of order $|\log\epsilon|^{-1/2}$ of the binormal curvature flow $\gamma$. Dávila, del Pino, Musso and Wei constructed solutions, defined for all times, whose vorticity concentrates near a translating and rotating helix. For axisymmetric flows without swirl, Guo, Jeong and Zhao proved that vorticity sharing the invariants of a thin vortex ring stays concentrated for all times and travels along the axis with the Kelvin–Hicks speed to leading order.
Problem. Can one construct, for every sufficiently small $\epsilon > 0$, a smooth finite-energy Euler velocity $u^\epsilon$, solving $\partial_\tau u^\epsilon + (u^\epsilon \cdot \nabla) u^\epsilon + \nabla p^\epsilon = 0$, $\operatorname{div} u^\epsilon = 0$ on physical times $0 \le \tau \le 4\pi T/|\log\epsilon|$, whose initial vorticity $\omega^\epsilon = \nabla \times u^\epsilon$ is supported within distance $C\epsilon$ of $\gamma(0)$, with $C$ independent of $\epsilon$, with $\int |\omega^\epsilon(0,x)| \, dx \to L$, and such that for every compactly supported smooth vector field $\phi$ $$\sup_{0 \le t \le T} \left| \int \omega^\epsilon\!\left(\frac{4\pi t}{|\log\epsilon|}, x\right) \cdot \phi(x) \, dx - \int_0^L \partial_s \gamma(t,s) \cdot \phi(\gamma(t,s)) \, ds \right| \to 0$$ as $\epsilon \to 0$?
Source: The vortex filament conjecture, after Da Rios (1906); named and formulated in this direction by Jerrard and Seis (2017). Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, binormal curvature flow, Euler equations, vortex filaments
References
@article{darios1906,
author = {Da Rios, Luigi Sante},
title = {Sul moto d'un liquido indefinito con un filetto vorticoso di forma qualunque},
journal = {Rend. Circ. Mat. Palermo},
volume = {22},
number = {1},
year = {1906},
pages = {117--135},
doi = {10.1007/BF03018608},
}
@article{jerrard2017,
author = {Jerrard, Robert L. and Seis, Christian},
title = {On the vortex filament conjecture for {E}uler flows},
journal = {Arch. Ration. Mech. Anal.},
volume = {224},
number = {1},
year = {2017},
pages = {135--172},
doi = {10.1007/s00205-016-1070-3},
eprint = {1603.00227},
archivePrefix = {arXiv},
}
@article{davila2022,
author = {D{\'a}vila, Juan and del Pino, Manuel and Musso, Monica and Wei, Juncheng},
title = {Travelling helices and the vortex filament conjecture in the incompressible {E}uler equations},
journal = {Calc. Var. Partial Differential Equations},
volume = {61},
number = {4},
year = {2022},
pages = {119},
doi = {10.1007/s00526-022-02217-4},
}
@misc{guo2026,
author = {Guo, Dengjun and Jeong, In-Jee and Zhao, Lifeng},
title = {Global dynamics of a single vortex ring},
year = {2026},
eprint = {2602.20131},
archivePrefix = {arXiv},
}
@misc{aim-vortex-filament-binormal,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The Euler vortex-filament limit for a general moving curve}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/167-vortex-filament-binormal.md},
note = {Entry 167 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00915, "The Euler vortex-filament limit for a general moving curve", clopen.solutions.