Title: Instability in the smooth Ponomarenko dynamo and the MHD system
Abstract: In this work we study the kinematic dynamo equations for a passive vector describing the evolution of a magnetic field that is transported by a prescribed velocity field. We establish the existence of solutions that exhibit exponential growth in time for a large class of helical velocity fields. We construct an unstable eigenmode via resolvent estimates of the linear operator, which we carry out by introducing suitable Green’s functions that capture the local behaviour of the system. This provides a mathematical justification for the physically conjectured mechanism by which helical flows can sustain magnetic field generation in the Ponomarenko dynamo. As a by-product, we prove that the nonlinear 3D MHD equations are unstable in Lp for any p > 1 around the Taylor-Couette flow and the zero magnetic field. This is joint work with D. Villringer (Imperial).