Nonlinear instability for the 3D MHD equations around the Taylor–Couette flow

Abstract We study the three-dimensional (3D) magnetohydrodynamics (MHD) equations in an annular cylinder, perturbed around the explicit steady state given by the 3D Taylor–Couette velocity field and zero magnetic field. Combining a recent linear instability result for the magnetic field with the framework of Friedlander et al (2006 Commun. Math. Phys. 264 335–47), we prove nonlinear instability of the solution around this steady state in L p , for any p > 1. In particular, our results are, to the best of our knowledge, the first rigorous instability results for 3D MHD without forcing, in which the instability is produced as a result of the (exponential) growth of the magnetic field. Furthermore, we offer a mathematical proof of the physically conjectured transfer of energy from the velocity field to the magnetic field in the MHD system.