BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity

In this paper we derive a criterion for the breakdown of classicalsolutions to the incompressible magnetohydrodynamic equations withzero viscosity and positive resistivity in $\mathbb{R}^3$. Thisresult is analogous to the celebrated Beale-Kato-Majda's breakdowncriterion for the inviscid Eluer equations of incompressiblefluids. In $\mathbb{R}^2$ we establish global weak solutions tothe magnetohydrodynamic equations with zero viscosity and positiveresistivity for initial data in Sobolev space $H^1(\mathbb{R}^2)$.