- Open Access
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.
- MHD equations
- regularity criteria
- regularity of solutions
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.
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.
- Duvaut G, Lions JL: Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Ration. Mech. Anal. 1972, 46: 241-279.MathSciNetView ArticleMATHGoogle Scholar
- Cao CS, Wu JH: Two regularity criteria for the 3D MHD equations. J. Differ. Equ. 2010, 248: 2263-2274. 10.1016/j.jde.2009.09.020MathSciNetView ArticleMATHGoogle Scholar
- Chen QL, Miao CX, Zhang ZF: On the regularity criterion of weak solutions for the 3D viscous magneto-hydrodynamics equations. Commun. Math. Phys. 2008, 284: 919-930. 10.1007/s00220-008-0545-yMathSciNetView ArticleMATHGoogle Scholar
- Duan HL: On regularity criteria in terms of pressure for the 3D viscous MHD equations. Appl. Anal. 2012, 91: 947-952. 10.1080/00036811.2011.556626MathSciNetView ArticleMATHGoogle Scholar
- Fan JS, Jiang S, Nakamura G, Zhou Y: Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations. J. Math. Fluid Mech. 2011, 13: 557-571. 10.1007/s00021-010-0039-5MathSciNetView ArticleMATHGoogle Scholar
- He C, Xin ZP: On the regularity of weak solutions to the magnetohydrodynamic equations. J. Differ. Equ. 2005, 213: 235-254. 10.1016/j.jde.2004.07.002MathSciNetView ArticleMATHGoogle Scholar
- Ji E, Lee J: Some regularity criteria for the 3D incompressible magnetohydrodynamics. J. Math. Anal. Appl. 2010, 369: 317-322. 10.1016/j.jmaa.2010.03.015MathSciNetView ArticleMATHGoogle Scholar
- 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. 2012, 396: 345-350. 10.1016/j.jmaa.2012.06.016MathSciNetView ArticleMATHGoogle Scholar
- Jia XJ, Zhou Y: Regularity criteria for the 3D MHD equations involving partial components. Nonlinear Anal., Real World Appl. 2012, 13: 410-418. 10.1016/j.nonrwa.2011.07.055MathSciNetView ArticleMATHGoogle Scholar
- Jia XJ, Zhou Y: Regularity criteria for the 3D MHD equations via partial derivatives. Kinet. Relat. Models 2012, 5: 505-516.MathSciNetView ArticleMATHGoogle Scholar
- Ni LD, Guo ZG, Zhou Y: Some new regularity criteria for the 3D MHD equations. J. Math. Anal. Appl. 2012, 396: 108-118. 10.1016/j.jmaa.2012.05.076MathSciNetView ArticleMATHGoogle Scholar
- Zhang ZJ: Remarks on the regularity criteria for generalized MHD equations. J. Math. Anal. Appl. 2011, 375: 799-802. 10.1016/j.jmaa.2010.10.017MathSciNetView ArticleMATHGoogle Scholar
- Zhang ZJ, Li P, Yu GH: Regularity criteria for the 3D MHD equations via one directional derivative of the pressure. J. Math. Anal. Appl. 2013, 401: 66-71. 10.1016/j.jmaa.2012.11.022MathSciNetView ArticleMATHGoogle Scholar
- Zhou Y: Remarks on regularities for the 3D MHD equations. Discrete Contin. Dyn. Syst. 2005, 12: 881-886.MathSciNetView ArticleMATHGoogle Scholar
- Zhou Y, Fan JS: Logarithmically improved regularity criteria for the 3D viscous MHD equations. Forum Math. 2012, 24: 691-708.MathSciNetView ArticleMATHGoogle Scholar
- Zhou Y, Gala S: Regularity criterion for the solutions to the 3D MHD equations in the multiplier space. Z. Angew. Math. Phys. 2011, 61: 193-199.MathSciNetView ArticleMATHGoogle Scholar
- Zhang, ZJ: Remarks on the regularity criteria for the 3D MHD equations in terms of the gradient of the pressure (submitted)Google Scholar
- Zhang, ZJ: Mageto-micropolar fluid equations with regularity in one direction (submitted)Google Scholar
- Guo ZG, Gala S: A regularity criterion for the Navier-Stokes equations in terms of one directional derivative of the velocity field. Anal. Appl. 2013, 10: 373-380.MathSciNetView ArticleMATHGoogle Scholar
- Gala S: Extension criterion on regularity for weak solutions to the 3D MHD equations. Math. Methods Appl. Sci. 2010, 33: 1496-1503.MathSciNetMATHGoogle Scholar
- Gala S, Lemarié-Rieusset PG: Multipliers between Sobolev spaces and fractional differentiation. J. Math. Anal. Appl. 2006, 322: 1030-1054. 10.1016/j.jmaa.2005.07.043MathSciNetView ArticleMATHGoogle Scholar
- Lemarié-Rieusset PG: Recent Developments in the Navier-Stokes Problem. Chapman & Hall, London; 2002.View ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.