Regularity criteria for 3D MHD equations via mixed velocity-magnetic gradient tensors

Abstract

Abstract A well-known regularity criterion in He and Xin (2005 J. Differ. Equ. 213 235–54) shows that if the velocity gradient tensor belongs to L q ( 0 , T ; L p ( R 3 ) ) with 2 / q + 3 / p = 2 , 3 / 2 < p ⩽ ∞ and 1 ⩽ q < ∞ , then the corresponding weak solution to the 3D magnetohydrodynamic (MHD) equations is smooth on ( 0 , T ] . In this paper, we prove that the role of the velocity gradient tensor can be replaced by the combination of the diagonal part of the velocity gradient tensor and the non-diagonal part of the magnetic gradient tensor or by the combination of the diagonal part of the magnetic gradient tensor and the non-diagonal part of the velocity gradient tensor. The main interest among others is that the diagonal part of a gradient tensor is related to the deformation while the non-diagonal part is related to the rotation. Thus, our theorems may provide us with a new viewpoint to understand the potential singularity of the 3D MHD flow.