Imagen: Álvaro Minguito/ICMAT

To describe the macroscopic behaviour of electrically conducting fluids—such as plasmas, liquid metals, and astrophysical fluids—it is necessary to combine fluid dynamics and electromagnetism. This framework is provided by magnetohydrodynamics (MHD), whose fundamental equations model the interaction between the fluid velocity, the magnetic field, and the pressure, and play a central role in plasma physics, astrophysics, and controlled nuclear fusion.

From a mathematical standpoint, these equations form a highly constrained and geometrically intricate nonlinear system of partial differential equations, especially in three-dimensional domains. The existence and regularity of solutions depend very sensitively on the topology of the domain and on the prescribed boundary conditions. In toroidal geometries—which are particularly relevant for magnetic confinement devices such as tokamaks and stellarators—these difficulties are further compounded by the lack of symmetry and by the complex structure of magnetic field lines.

When one restricts attention to fluids at rest, known as stationary or equilibrium states, the system no longer evolves in time and an exact balance between magnetic and pressure forces is imposed, leading to a simplified but still highly nontrivial set of equations. In this context, the present article proves the existence of piecewise smooth MHD equilibria in three-dimensional toroidal domains of Euclidean space, where the pressure is constant on the boundary but varies in the interior. It also establishes the existence of free-boundary steady states surrounded by vacuum and supported by an external surface current. The toroidal domains in which these equilibria are shown to exist need not be small perturbations of axisymmetric domains and may, in fact, exhibit arbitrary knotted topology.

The building blocks of the construction are analytic toroidal domains satisfying a suitable nondegeneracy condition, which roughly asserts the existence of a force-free field that is ergodic on the surface of the domain. The proof relies on three main ingredients: a gluing construction of piecewise smooth MHD equilibria; a Hamilton–Jacobi equation on the two-dimensional torus, which can be interpreted as a nonlinear deformation of the cohomological equation and for which the nondegeneracy assumption plays a crucial role; and a new KAM theorem tailored to the study of divergence-free vector fields in three dimensions whose Poincaré map cannot be computed explicitly. Together, these results yield counterexamples to the celebrated Grad conjecture in the setting of stepped-pressure equilibria far from rotational symmetry.

Overall, this work provides a rigorous mathematical foundation for a class of equilibrium configurations that are physically relevant yet mathematically challenging, and it significantly advances our understanding of MHD equilibria in realistic three-dimensional settings.

Reference

Enciso, A., Luque, A., & Peralta-Salas, D. (2025). MHD equilibria with nonconstant pressure in nondegenerate toroidal domains. Journal of the European Mathematical Society, 27(6), 2251–2291.