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A New Regularity Criterion for the 3D MHD Equations Involving Partial Components

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Abstract

In this paper, we establish a new regularity criterion for the 3D incompressible MHD equations involving partial components of the velocity gradient and magnetic fields.

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References

  1. Cao, C., Wu, J.: Two regularity criteria for the 3D MHD equations. J. Differ. Equ. 248(9), 2263–2274 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cao, C., Titi, E.S.: Regularity criteria for the three-dimensional Navier-Stokes equations. Indiana Univ. Math. J. 57(6), 2643–2661 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, X., Gala, S., Guo, Z.: A new regularity criterion in terms of the direction of the velocity for the MHD equations. Acta Appl. Math. 113(2), 207–213 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. He, C., Xin, Z.: On the regularity of solutions to the magnetohydrodynamic equations. J. Differ. Equ. 213(2), 235–254 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ji, E., Lee, J.: Some regularity criteria for the 3D incompressible magnetohydrodynamics. J. Math. Anal. Appl. 369(1), 317–322 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Jia, X., Zhou, Y.: Regularity criteria for the 3D MHD equations involving partial components. Nonlinear Anal., Real World Appl. 13(1), 410–418 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Jia, X., Zhou, Y.: Regularity criteria for the 3D MHD equations via partial derivatives. Kinet. Relat. Models 5, 505–516 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kukavica, I., Ziane, M.: Navier-Stokes equations with regularity in one direction. J. Math. Phys. 48(6), 065203 (2007). 10 pp.

    Article  MathSciNet  Google Scholar 

  9. Ni, L., Guo, Z., Zhou, Y.: New regularity criteria for the 3D MHD equations. J. Math. Anal. Appl. 396, 108–118 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36(5), 635–664 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  11. Zhou, Y.: Remarks on regularities for the 3D MHD equations. Discrete Contin. Dyn. Syst. 12, 881–886 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Zhou, Y.: Regularity criteria for the 3D MHD equations in terms of the pressure. Int. J. Non-Linear Mech. 41, 1174–1180 (2006)

    Article  MATH  Google Scholar 

  13. Zhou, Y., Gala, S.: Regularity criteria for the solutions to the 3D MHD equations in the multiplier space. Z. Angew. Math. Phys. 61, 193–199 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Zhou, Y., Pokorný, M.: On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component. J. Math. Phys. 50(12), 123514 (2009). 11 pp.

    Article  MathSciNet  Google Scholar 

  15. Zhou, Y., Pokorný, M.: On the regularity of the solutions of the Navier-Stokes equations via one velocity component. Nonlinearity 23, 1097–1107 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. Zhou, Y.: A new regularity criterion for weak solutions to the Navier-Stokes equations. J. Math. Pures Appl. 84(11), 1496–1514 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Zhou, Y.: A new regularity criterion for the Navier-Stokes equations in terms of the gradient of one velocity component. Methods Appl. Anal. 9(4), 563–578 (2002)

    MATH  MathSciNet  Google Scholar 

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Acknowledgements

This work was partially supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ12A01009, and NSFC under Grant No. 11301394.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, Chongqing, P.R. China

    Xiaochun Chen

  2. College of Mathematics and Information Science, Wenzhou University, Wenzhou, 325035, P.R. China

    Zhengguang Guo

  3. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, P.R. China

    Mingxuan Zhu

Authors
  1. Xiaochun Chen
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  2. Zhengguang Guo
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  3. Mingxuan Zhu
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Correspondence to Zhengguang Guo.

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Chen, X., Guo, Z. & Zhu, M. A New Regularity Criterion for the 3D MHD Equations Involving Partial Components. Acta Appl Math 134, 161–171 (2014). https://doi.org/10.1007/s10440-014-9876-1

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  • DOI: https://doi.org/10.1007/s10440-014-9876-1

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