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

—

Disclosed cost

—

Wikipedia

No dedicated article

Submitted by DuskyManta279