Alberto Enciso

ICMAT, Spain

MHD equilibria with nonconstant pressure in toroidal domains

In the talk we will discuss the existence of piecewise smooth MHD equilibria in three-dimensional toroidal domains of \(\mathbb R^3\) where the pressure is constant on the boundary but not in the interior. The pressure is piecewise constant and the plasma current exhibits 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 building blocks we use in our construction are analytic toroidal domains satisfying a certain nondegeneracy condition, which roughly states that there exists a force-free field that is ergodic on the surface of the domain. The proof involves three main ingredients: a gluing construction of piecewise smooth MHD equilibria, a Hamilton–Jacobi equation on the two-dimensional torus that can be understood as a nonlinear deformation of a cohomological equation (so the nondegeneracy assumption plays a major role in the corresponding analysis), and a new KAM theorem tailored for the study of divergence-free fields in three dimensions. The talk is based on joint work with A. Luque and D. Peralta-Salas.