Smooth steady fast dynamos on the three-torus

Abstract

The fast dynamo problem asks whether the motion of an electrically conducting fluid can amplify a magnetic field exponentially, at a rate that remains uniformly positive as resistivity vanishes. In this talk, I will present a construction of smooth, time-independent, incompressible flows on the three-dimensional torus with this property, resolving the fast dynamo conjecture of Zeldovich and Sakharov. 
The construction proceeds by building anisotropic Banach spaces of distributions adapted to the dynamics, yielding favourable spectral properties for the non-resistive problem. From there, we treat the Laplacian perturbatively using techniques from singular perturbation theory. 
Our construction also yields robustness under sufficiently small smooth perturbations of the velocity field.
This is joint work with Michele Coti Zelati and Massimo Sorella.