Regularity Criteria for the Three-dimensional Magnetohydrodynamic Equations

This paper studies the three-dimensional density-dependent incompressible magnetohydrodynamic equations. First, a regularity criterion is proved which allows the initial density to contain vacuum. Then we establish another blow-up criterion in the Besov space Ḃ0 ∞,2 when the positive initial density is bounded away from zero. Third, we prove a global nonexistence result for initial density with highly decreasing at infinity. Fourth, we obtain a regularity criterion to the density-dependent incompressible magnetohydrodynamic equations in a bounded domain. Finally, we also give some remarks on the regularity criteria for the three-dimensional full compressible magnetohydrodynamic equations in a bounded domain and for the incompressible homogeneous magnetohydrodynamic equation in the whole space R3.