In this paper, we consider the regularity criteria for the 3D MHD equations. It is proved that if
then the solution actually is smooth. This extends the previous results given by Guo and Gala (Anal. Appl. 10:373-380, 2013), Gala (Math. Methods Appl. Sci. 33:1496-1503, 2010).
MSC: 35B65, 35Q35, 76D03.
Keywords:MHD equations; regularity criteria; regularity of solutions
In this paper, we consider the following three-dimensional (3D) magnetohydrodynamic (MHD) equations:
where is the fluid velocity field, is the magnetic field, π is a scalar pressure, and are the prescribed initial data satisfying in the distributional sense. Physically, (1) governs the dynamics of the velocity and magnetic fields in electrically conducting fluids such as plasmas, liquid metals, and salt water. Moreover, (1)1 reflects the conservation of momentum, (1)2 is the induction equation, and (1)3 specifies the conservation of mass.
Besides its physical applications, MHD system (1) is also mathematically significant. Duvaut and Lions  constructed a global weak solution to (1) for initial data with finite energy. However, the issue of regularity and uniqueness of such a given weak solution remains a challenging open problem. Many sufficient conditions (see, e.g., [2-16] and the references therein) were derived to guarantee the regularity of the weak solution. Among these results, we are interested in regularity criteria involving only partial components of the velocity u, the magnetic field b, the pressure gradient ∇π, etc.
In , Jia and Zhou used an intricate decomposition technique and delicate inequalities to obtain the following regularity criterion:
that is, if (2) holds, then the solution of (1) is smooth. Applying a more subtle decomposition technique (see [, Remark 3]), Zhang, Li, and Yu  could be able to prove smoothness condition (2) in the case . Noticeably, Zhang  treated the range in a unified approach.
As far as one directional derivative of the velocity field is concerned, Cao and Wu  proved the following regularity criterion:
Jia and Zhou  showed that if
then the solution is regular. These results were improved by Zhang  to be
We would like to give another contribution in this direction and prove that if
Notice that our result extends
which is in  for the Navier-Stokes equations. At this moment, for MHD equations (1), we could not be able to add the regularity condition on one directional derivative of the velocity field only, since we need to convert (1) into a symmetric form. We also improve the result
of  in the sense that we need only one directional derivative of or to ensure the smoothness of the solution.
Before stating the precise result, let us recall the weak formulation of MHD equations (1).
(2) (1)1,2,3,4 are satisfied in the distributional sense;
(3) the energy inequality
Now, our main results read as follows.
We have the following scaling properties:
where BMO is the homogeneous space of bounded mean oscillations associated with the semi-norm
Furthermore, we have the following strict imbeddings:
which could be justified simply as
3 Proof of Theorem 2
In this section, we shall prove Theorem 2. As we will see in the proof below, we need only to consider the case that
First, let us convert (1) into a symmetric form. Writing
we find by adding and subtracting (1)1 with (1)2,
We now estimate I as
Plugging (6) into (5), invoking the Gronwall inequality, and noticing (3), we deduce
Utilizing interpolation inequalities and the following multiplicative Gagliardo-Nireberg inequalities
it follows that
Putting (10) into (8), utilizing the Gronwall inequality, and noticing (7), we deduce
The classical Serrin-type regularity criterion  then concludes the proof of Theorem 2.
The authors declare that they have no competing interests.
We declare that all authors collaborated and dedicated the same amount of time in order to perform this article.
This work was partially supported by the Youth Natural Science Foundation of Jiangxi Province (20132BAB211007), the Science Foundation of Jiangxi Provincial Department of Education (GJJ13658, GJJ13659) and the National Natural Science Foundation of China (11326138, 11361004). The authors thank the referees for their careful reading and suggestions which led to an improvement of the work.
Cao, CS, Wu, JH: Two regularity criteria for the 3D MHD equations. J. Differ. Equ.. 248, 2263–2274 (2010). Publisher Full Text
Chen, QL, Miao, CX, Zhang, ZF: On the regularity criterion of weak solutions for the 3D viscous magneto-hydrodynamics equations. Commun. Math. Phys.. 284, 919–930 (2008). Publisher Full Text
Duan, HL: On regularity criteria in terms of pressure for the 3D viscous MHD equations. Appl. Anal.. 91, 947–952 (2012). Publisher Full Text
Fan, JS, Jiang, S, Nakamura, G, Zhou, Y: Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations. J. Math. Fluid Mech.. 13, 557–571 (2011). Publisher Full Text
He, C, Xin, ZP: On the regularity of weak solutions to the magnetohydrodynamic equations. J. Differ. Equ.. 213, 235–254 (2005). Publisher Full Text
Ji, E, Lee, J: Some regularity criteria for the 3D incompressible magnetohydrodynamics. J. Math. Anal. Appl.. 369, 317–322 (2010). Publisher Full Text
Jia, XJ, Zhou, Y: A new regularity criterion for the 3D incompressible MHD equations in terms of one component of the gradient of pressure. J. Math. Anal. Appl.. 396, 345–350 (2012). Publisher Full Text
Jia, XJ, Zhou, Y: Regularity criteria for the 3D MHD equations involving partial components. Nonlinear Anal., Real World Appl.. 13, 410–418 (2012). Publisher Full Text
Ni, LD, Guo, ZG, Zhou, Y: Some new regularity criteria for the 3D MHD equations. J. Math. Anal. Appl.. 396, 108–118 (2012). Publisher Full Text
Zhang, ZJ: Remarks on the regularity criteria for generalized MHD equations. J. Math. Anal. Appl.. 375, 799–802 (2011). Publisher Full Text
Zhang, ZJ, Li, P, Yu, GH: Regularity criteria for the 3D MHD equations via one directional derivative of the pressure. J. Math. Anal. Appl.. 401, 66–71 (2013). Publisher Full Text
Gala, S, Lemarié-Rieusset, PG: Multipliers between Sobolev spaces and fractional differentiation. J. Math. Anal. Appl.. 322, 1030–1054 (2006). Publisher Full Text