Regularity criteria for the magnetohydrodynamic equations with partial viscous terms and the Leray-$\alpha$-MHD model

We prove some regularity conditions for the MHD equations withpartial viscous terms and the Leray-$\alpha$-MHD model. Since thesolutions to the Leray-$\alpha$-MHD model are smoother than that ofthe original MHD equations, we are able to obtain better regularityconditions in terms of the magnetic field $B$ only.