Autonomous Lipschitz Fast Dynamo on the Three-Torus
openconfidence 70%
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Precise statement
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.
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fidelity, correctness, priority, or novelty.
What AI did
GPT-5.5 Pro, GPT-5.6 Sol
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.
arXiv:2608.02586 - An autonomous Lipschitz fast dynamo on the three-torus
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.
Known method families
argument (source-reported)
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
What remains uncertain
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VibeMath has not independently audited the mathematical statement, proof, or novelty claim.