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.
Funder
National Natural Science Foundation of China
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics