Smooth Random Fast Dynamo on the Three-Torus

Arnold's fast-dynamo problem asks for a smooth divergence-free velocity field on $\mathbb{T}^3$, chosen independently of the magnetic diffusivity, that drives exponential growth of the magnetic field at every sufficiently small diffusivity. This constructs a genuinely $C^\infty$ field with that behaviour: random and time-dependent, refreshing iid on finite time blocks, for which the almost sure exponential growth rate is at least $1/2$ at each fixed small enough resistivity, with a time-uniform lower bound whose random prefactor has a resistivity-uniform inverse-moment bound. The field is neither autonomous nor deterministic, so Arnold's smooth autonomous problem on $\mathbb{T}^3$ remains open.

Constructed an explicit class of smooth random, time-dependent incompressible velocity fields on T^3, obtained by alternating smooth shear flows with iid random phases on finite time blocks. For every fixed sufficiently small resistivity, the magnetic field has an almost-sure exponential growth rate at least 1/2, together with a time-uniform lower bound whose random prefactor has a resistivity-uniform inverse-moment estimate. This is one variant case of Arnold's 1994 fast-dynamo problem, not the problem itself: Arnold asks for a field that is smooth, autonomous and deterministic all at once, and this one keeps the smoothness while giving up the other two. The sibling entry on this site relaxes the opposite hypothesis, keeping an autonomous deterministic field at Lipschitz regularity. Neither settles Arnold's problem as posed, which remains open.