Remarks on the regularity criteria for the 3D MHD equations in the multiplier spaces
© Zhang et al.; licensee Springer. 2013
Received: 27 September 2013
Accepted: 22 November 2013
Published: 12 December 2013
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.
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.
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.
then the solution actually is smooth. Here, is the multiplier spaces, which is strictly larger than (see Section 2 for details).
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).
(1)1,2,3,4 are satisfied in the distributional sense;
- (3)the energy inequality
Now, our main results read as follows.
then the solution is smooth on .
The proof of Theorem 2 will be given in Section 3. In Section 2, we introduce the multiplier spaces and recall their fine properties.
3 Proof of Theorem 2
The rest of the proof is organized as follows. In the first step, we dominate and the time integral of , while the second step is devoted to controlling .
Step 1. estimates.
Step 2. estimates.
The classical Serrin-type regularity criterion  then concludes the proof of Theorem 2.
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.
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