Does there exist an integer k≥ 1 and a nonempty C^k-open family of smooth, autonomous, divergence-free velocity fields mathbf u colon T^3→ R^3 such that every mathbf u in the family generates a fast dynamo? More precisely, for the magnetic-induction equation partial_t mathbf B=nabla×(mathbf u×mathbf B)+varepsilon Deltamathbf B, with nabla·mathbf B=0, does there exist a constant c>0, independent of sufficiently small magnetic diffusivity varepsilon>0, and nonzero solutions mathbf B_varepsilon whose exponential growth rates satisfy liminf_t→∞t^-1log lVertmathbf B_varepsilon(t)rVert≥ c as varepsilon→0?

Fast Dynamo Conjecture of Zeldovich and Sakharov

Does there exist an integer $k\ge 1$ and a nonempty $C^k$ -open family of smooth, autonomous, divergence-free velocity fields $\mathbf u\colon\mathbb T^3\to\mathbb R^3$ such that every $\mathbf u$ in the family generates a fast dynamo? More precisely, for the magnetic-induction equation $\partial_t\mathbf B=\nabla\times(\mathbf u\times\mathbf B)+\varepsilon\Delta\mathbf B$ , with $\nabla\cdot\mathbf B=0$ , does there exist a constant $c>0$ , independent of sufficiently small magnetic diffusivity $\varepsilon>0$ , and nonzero solutions $\mathbf B_\varepsilon$ whose exponential growth rates satisfy $\liminf_{t\to\infty}t^{-1}\log\lVert\mathbf B_\varepsilon(t)\rVert\ge c$ as $\varepsilon\to0$ ?

References

Primary source

arXiv

Additional references

Smooth autonomous fast dynamo action on the three-torus — arXiv

dynamical-systems magnetohydrodynamics partial-differential-equations

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A September preprint claims the conjecture is solved by constructing a robust smooth steady flow, but no independent verification has been recorded.

The conjecture asks whether fast magnetic-field growth can occur for smooth, steady, divergence-free flows on the three-torus, robustly under perturbations. The scan gives no definite proposer or original date.

Known results

Zel'dovich and Ruzmalkin (1980): fast dynamo action is possible in three-dimensional motion, but not in planar or spherical two-dimensional motion.
April 2025: an autonomous Lipschitz fast dynamo was claimed on $\mathbb{R}^3$ , but it is neither smooth nor periodic.
August 2026: an autonomous Lipschitz fast dynamo was constructed on $\mathbb{T}^3$ ; the source explicitly leaves the smooth case open.
2025–2026: time-dependent or subsequential constructions establish weaker forms, not the stated conjecture.

September 2026 claimed resolution

On September 3, 2026, Smooth autonomous fast dynamo action on the three-torus claimed a smooth, divergence-free, time-independent flow and a nonempty open family with uniform fast growth, by realizing a stretch–fold–shear construction through a global Poincare section. This would settle the problem, but the preprint is unrefereed and independently unconfirmed.

Current status (as of September 2026): A preprint claims the full smooth autonomous open-family result, but absent independent verification the conjecture remains unsettled.

Sources

No solutions have been posted yet.