Fast dynamo growth for smooth random velocity fields
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Can a smooth random divergence-free velocity field on T³ sustain exponential magnetic-field growth at positive resistivity, uniformly as resistivity becomes small? The studied field is iid block-refreshed and the initial magnetic mode is fixed.
Why it matters and what counts as progress
Why it matters: This tests whether fast-dynamo growth can coexist with smooth spatial velocity fields in the stated random model.
What counts as progress: An exponential lower bound with resistivity-independent rate and controlled random prefactor; per-resistivity almost-sure results must remain distinct from one field and null set working for all resistivities.
How the problem developed
- 3 Aug 2026 · Nearby result [1]
Lukas Niebel Constructed a time-independent Lipschitz fast dynamo on T³; this gives a deterministic low-regularity example rather than a smooth random field.
- 20 Aug 2026 · Random smooth result [2]
Keefer Rowan; ChatGPT 5.6 Sol Ultra supplied the central proof innovation, which Rowan wrote up and verified Constructed a time-dependent, divergence-free random velocity on T³ with iid finite time blocks and deterministic C∞ bounds, yielding almost-sure magnetic growth rate at least 1/2 for each fixed sufficiently small resistivity.
AI contribution
An AI-discovered smooth random fast dynamo on T³
For the explicit iid block-refreshed smooth divergence-free velocity field U on T³ and the specified divergence-free initial magnetic mode, Theorem 1.1 gives, for each κ in [0, log(π/e)/(4π²)], an almost surely finite κ-dependent L^κ with ||B_t^κ||₂≥(L^κ)^−1 e^(t/2) for all t≥0 and a uniform positive moment bound on L^κ. Null sets can depend on κ; no single deterministic field working simultaneously for every κ is proved.
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