Regularity results for weak solutions of the 3D MHD equations

We study regularity of general and axisymmetric weak solutions of the 3D MHD equations with dissipation and resistance. A general weak solution is shown to be smooth if it satisfies a Serrin condition. The regularity of axisymmetric weak solutions is analyzed through the MHD equations in cylindrical coordinates, whose concrete form is derived here using Gibbs' notion of dyadic product. We establish that it is sufficient to impose conditions on certain components (in cylindrical coordinates) of an axisymmetric weak solution in order for the solution to be regular.