Smooth autonomous fast dynamo action on the three-torus
Abstract
We construct a nonempty Ck-open family of smooth, autonomous, divergence-free velocity fields on 𝕋3 that generate fast dynamos. The proof proceeds by first establishing fast dynamo action for a large class of smooth, time-periodic velocity fields exhibiting the classical stretch--fold--shear mechanism. We then realize the associated period map within a smooth autonomous flow using carefully tuned return dynamics to a global Poincaré section. In both settings, the proof first isolates unstable distributional eigenmodes for the ideal dynamo operator on an anisotropic Banach space of distributions, and then shows that this spectral instability persists under the singular perturbation ε Δ. The resulting estimates are uniform as ε→0 and stable under Ck perturbations of the velocity field, yielding an open set of smooth fast dynamo vector fields. This result resolves the Fast Dynamo Conjecture of Zeldovich and Sakharov, as recorded in Arnold's book of problems.
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BibTeX
@misc{zelati2026smooth,
title = {{Smooth autonomous fast dynamo action on the three-torus}},
author = {Michele Coti Zelati and Massimo Sorella and David Villringer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.04153},
eprint = {2609.04153},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04153},
}
RIS
TY - UNPB
TI - Smooth autonomous fast dynamo action on the three-torus
AU - Michele Coti Zelati
AU - Massimo Sorella
AU - David Villringer
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.04153
UR - https://arxiv.org/abs/2609.04153
AB - We construct a nonempty C^k-open family of smooth, autonomous, divergence-free velocity fields on 𝕋^3 that generate fast dynamos. The proof proceeds by first establishing fast dynamo action for a large class of smooth, time-periodic velocity fields exhibiting the classical stretch--fold--shear mechanism. We then realize the associated period map within a smooth autonomous flow using carefully tuned return dynamics to a global Poincaré section. In both settings, the proof first isolates unstable distributional eigenmodes for the ideal dynamo operator on an anisotropic Banach space of distributions, and then shows that this spectral instability persists under the singular perturbation ε Δ. The resulting estimates are uniform as ε→0 and stable under C^k perturbations of the velocity field, yielding an open set of smooth fast dynamo vector fields. This result resolves the Fast Dynamo Conjecture of Zeldovich and Sakharov, as recorded in Arnold's book of problems.
ER -