Regularity criteria of MHD system involving one velocity and one current density component

We study a regularity criteria of the three-dimensional magnetohydrodynamics system. In particular, we obtain a regularity criteria that involves one velocity component and one current density component. Its immediate consequence is a regularity criteria in terms of one velocity component and two entries from the Jacobian matrix of the magnetic vector field, in contrast to many previous results that involve three entries all from the Jacobian matrix of the velocity vector field. Moreover, the conditions on the current density component is in the norm with scaling dimension zero; i.e. Serrin-class.