An autonomous Lipschitz fast dynamo on the three-torus

Abstract

We construct a single real-valued, divergence-free, time-independent velocity field $u\in\mathrm{W}^{1,\infty}(\mathbb{T}^3;\mathbb{R}^3) $ that is a fast dynamo for the kinematic induction equation on the flat three-torus. For every sufficiently small positive magnetic diffusivity, the corresponding induction operator has an eigenvalue whose real part is bounded below by a positive constant independent of the diffusivity. For each such diffusivity, there is a non-zero real-valued, divergence-free solution of the induction equation whose $\mathrm{L}^2 $-norm obeys an exact exponential growth law with a uniformly positive rate. For the same velocity field, the particle flow and its inverse have Lipschitz constants growing at most linearly in time, every time map has zero topological entropy, and the ideal induction group has zero exponential growth rate in operator norm. The velocity is differentiable everywhere and smooth away from a single circle, but is not $\mathrm{C}^1 $.