Regularity Criteria for Solutions to the Three-dimensional Axial Symmetric Incompressible MHD Equations

To study the regularity criteria of solutions, the three-dimensional axially symmetric incompressible magnetohydrodynamics (MHD ) equations with swirl are presented. Two types of particular weak solutions to the three-dimensional axially symmetric incompressible MHD equations u^θ =0, B^r=B^z=0, and B^r=B^z=0 are considered. By using the energy method and the embedding method, the regularity criteria for the two types of weak solutions to the three-dimensional axially symmetric incompressible MHD equations are obtained if the radial direction component u^r satisfies some conditions.