Autonomous Lipschitz Fast Dynamo on the Three-Torus

Does there exist a single real-valued, divergence-free, time-independent Lipschitz velocity field $u\in W^{1,\infty}(\mathbb T^3;\mathbb R^3)$ , chosen independently of magnetic diffusivity, that is a fast dynamo for the kinematic induction equation on the flat three-torus? The author constructs such a field and constants $\varepsilon_0,\gamma_0>0$ such that, for every $0<\varepsilon\le\varepsilon_0$ , the induction operator has an eigenvalue $\lambda_\varepsilon$ with $\operatorname{Re}\lambda_\varepsilon\ge\gamma_0$ . Thus every sufficiently small diffusivity admits a nonzero real divergence-free magnetic field with exact exponential $L^2$ growth. The velocity is Lipschitz but not $C^1$ , so this settles only the Lipschitz regularity variant; Arnold's smooth autonomous fast-dynamo problem on $\mathbb T^3$ remains open.

Result: Proved(see note)
Status: Variant only
AI contribution: AI-assisted
Method: Argument
Field: Dynamo theory; spectral PDE
Posed by: V. I. Arnold
Year posed: 1994
Years open: 32y
Solved: 2026-08-03
Model: GPT-5.5 Pro, GPT-5.6 Sol
Vendor: OpenAI
Collaborators: Lukas Niebel
Verification: Unreviewed
Publication: Preprint
Significance: 30 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

One variant case of Arnold's 1994 fast-dynamo problem, not the problem itself. Arnold asks for a single velocity field on T^3 that is smooth, divergence-free, autonomous and deterministic, fixed independently of the magnetic diffusivity, and that grows the magnetic field exponentially at every small enough diffusivity. The field constructed here is all of that except smooth: it is Lipschitz, not C^1. The sibling entry on this site relaxes the opposite hypothesis, keeping a smooth field but making it random and time-dependent. Neither settles Arnold's problem as posed, which remains open.

What the AI did

During exploration, GPT-5.5 Pro and GPT-5.6 Sol helped identify a candidate fast-dynamo construction. They were later used to check calculations; identify errors, inconsistencies, and gaps in preliminary arguments; support development of some arguments; and assist with drafting and revision. The author states that he critically reviewed and verified every mathematical claim, calculation, and AI-generated suggestion and takes full responsibility for the manuscript.

Verification

Single-author arXiv preprint; not yet peer-reviewed, Lean-verified, or independently expert-checked.

Source

PaperarXiv:2608.02586 - An autonomous Lipschitz fast dynamo on the three-torus

Related entries

RelatedSmooth random fast dynamo

Submitted by October on 05 Aug 2026