Global unique solvability of 3D MHD equations in a thin periodic domain

We study magnetohydrodynamic equations for a viscous incompressible resistive fluid in a thin 3D domain. We prove the global existence and uniqueness of solutions corresponding to a large set of initial data from Sobolev type space of the order 1/2 and forcing terms from L2 type space. We also show that the solutions constructed become smoother for positive time and prove the global existence of (unique) strong solutions.