Global regularity for the deterministic horizontally dissipative MHD system

Establish global regularity of classical solutions for the two-dimensional deterministic magnetohydrodynamics system with horizontal velocity dissipation and horizontal magnetic diffusion.

Background

The paper studies a stochastic two-dimensional anisotropic magnetohydrodynamics system on the torus, with dissipation acting only in the horizontal direction. Its introduction places this problem in the context of deterministic MHD systems with partial or anisotropic dissipation.

The unresolved issue concerns the corresponding deterministic system: although conditional estimates and related global results are known, global regularity of classical solutions has not been established. The stochastic results proved in the paper concern global martingale and probabilistically weak solutions, together with pathwise uniqueness, and therefore do not resolve this deterministic classical-regularity problem.