MHD equilibria in toroidal geometries

The computation of 3D magnetohydrodynamics (MHD) equilibria is of major importance for magnetic confinement devices such as tokamaks or stellarators. In this talk I will present recent results on the existence of stepped pressure MHD equilibria in 3D toroidal domains, where the plasma current exhibits an arbitrary number of current sheets. The toroidal domains where these equilibria are shown to exist do not need to be small perturbations of an axisymmetric domain, and in fact they can have any knotted topology. The proof involves three main ingredients: a Cauchy-Kovalevskaya theorem for Beltrami fields, a Hamilton-Jacobi equation on the two-dimensional torus, and a KAM theorem for divergence-free fields in three dimensions. This is based on joint work with A. Enciso and A. Luque.