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.

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.