Table of Contents
International Journal of Partial Differential Equations
Volume 2014 (2014), Article ID 213083, 5 pages
http://dx.doi.org/10.1155/2014/213083
Research Article

MHD Equations with Regularity in One Direction

School of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, China

Received 3 July 2014; Accepted 6 October 2014; Published 27 October 2014

Academic Editor: Antonin Novotny

Copyright © 2014 Zujin Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. G. Duvaut and J.-L. Lions, “Inéquations en thermoélasticité et magnétohydrodynamique,” Archive for Rational Mechanics and Analysis, vol. 46, pp. 241–279, 1972. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. E. Hopf, “Üer die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen,” Mathematische Nachrichten, vol. 4, pp. 213–231, 1951. View at Google Scholar · View at MathSciNet
  3. J. Leray, “Sur le mouvement d'un liquide visqueux emplissant l'espace,” Acta Mathematica, vol. 63, no. 1, pp. 193–248, 1934. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. C. He and Z. Xin, “On the regularity of weak solutions to the magnetohydrodynamic equations,” Journal of Differential Equations, vol. 213, no. 2, pp. 235–254, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. Y. Zhou, “Remarks on regularities for the 3D MHD equations,” Discrete and Continuous Dynamical Systems Series A, vol. 12, no. 5, pp. 881–886, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. J. T. Beale, T. Kato, and A. Majda, “Remarks on the breakdown of smooth solutions for the 3-D Euler equations,” Communications in Mathematical Physics, vol. 94, no. 1, pp. 61–66, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. H. Beirão da Veiga, “A new regularity class for the Navier-Stokes equations in Rn,” Chinese Annals of Mathematics B, vol. 16, no. 4, pp. 407–412, 1995. View at Google Scholar · View at MathSciNet
  8. D. Chae and J. Lee, “Regularity criterion in terms of pressure for the Navier-Stokes equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 46, no. 5, pp. 727–735, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. Q. Chen, C. Miao, and Z. Zhang, “On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations,” Communications in Mathematical Physics, vol. 284, no. 3, pp. 919–930, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. L. Escauriaza, G. Seregin, and V. Šverák, “Backward uniqueness for parabolic equations,” Archive for Rational Mechanics and Analysis, vol. 169, no. 2, pp. 147–157, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. C. He and Y. Wang, “On the regularity criteria for weak solutions to the magnetohydrodynamic equations,” Journal of Differential Equations, vol. 238, no. 1, pp. 1–17, 2007. View at Publisher · View at Google Scholar · View at Scopus
  12. H. Kozono, T. Ogawa, and Y. Taniuchi, “The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations,” Mathematische Zeitschrift, vol. 242, no. 2, pp. 251–278, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. J. Serrin, “On the interior regularity of weak solutions of the Navier-Stokes equations,” Archive for Rational Mechanics and Analysis, vol. 9, pp. 187–195, 1962. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. J. Wu, “Regularity criteria for the generalized MHD equations,” Communications in Partial Differential Equations, vol. 33, no. 1–3, pp. 285–306, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. J. Wu, “Regularity results for weak solutions of the 3D MHD equations,” Discrete and Continuous Dynamical Systems, vol. 10, no. 1-2, pp. 543–556, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. Z. Zhang, Z.-a. Yao, M. Lu, and L. Ni, “Some Serrin-type regularity criteria for weak solutions to the Navier-Stokes equations,” Journal of Mathematical Physics, vol. 52, no. 5, Article ID 053103, 7 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. Y. Zhou, “Regularity criteria for the generalized viscous {MHD} equations,” Annales de l'Institut Henri Poincaré. Analyse Non Lineairé, vol. 24, no. 3, pp. 491–505, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. Y. Zhou, “Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain,” Mathematische Annalen, vol. 328, no. 1-2, pp. 173–192, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. C. S. Cao, “Sufficient conditions for the regularity to the 3D Navier-Stokes equations,” Discrete and Continuous Dynamical Systems. Series A, vol. 26, no. 4, pp. 1141–1151, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. C. Cao and E. S. Titi, “Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor,” Archive for Rational Mechanics and Analysis, vol. 202, no. 3, pp. 919–932, 2011. View at Publisher · View at Google Scholar · View at Scopus
  21. C. S. Cao and E. S. Titi, “Regularity criteria for the three-dimensional Navier-Stokes equations,” Indiana University Mathematics Journal, vol. 57, no. 6, pp. 2643–2661, 2008. View at Publisher · View at Google Scholar · View at Scopus
  22. Z. Zhang, D. Zhong, and L. Hu, “A new regularity criterion for the 3D Navier-Stokes equations via two entries of the velocity gradient tensor,” Acta Applicandae Mathematicae, vol. 129, pp. 175–181, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. Z. Zhang, F. Alzahrani, T. Hayat, and Y. Zhou, “Two new regularity criteria for the Navier-Stokes equations via two entries of the velocity Hessian tensor,” Applied Mathematics Letters, vol. 37, pp. 124–130, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  24. Z. Zhang, “A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component,” Communications on Pure and Applied Analysis, vol. 12, no. 1, pp. 117–124, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. Z. L. Zhang, Z. A. Yao, P. Li, C. C. Guo, and M. Lu, “Two new regularity criteria for the 3D Navier-Stokes equations via two entries of the velocity gradient tensor,” Acta Applicandae Mathematicae, vol. 123, pp. 43–52, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. Y. Zhou, “A new regularity criterion for the Navier-Stokes equations in terms of the gradient of one velocity component,” Methods and Applications of Analysis, vol. 9, no. 4, pp. 563–578, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. Y. Zhou, “A new regularity criterion for weak solutions to the Navier-Stokes equations,” Journal de Mathématiques Pures et Appliquées, vol. 84, no. 11, pp. 1496–1514, 2005. View at Publisher · View at Google Scholar
  28. Y. Zhou and M. Pokorný, “On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component,” Journal of Mathematical Physics, vol. 50, no. 12, Article ID 123514, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. Y. Zhou and M. Pokorný, “On the regularity of the solutions of the Navier-Stokes equations via one velocity component,” Nonlinearity, vol. 23, no. 5, pp. 1097–1107, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. C. Cao and J. Wu, “Two regularity criteria for the 3D MHD equations,” Journal of Differential Equations, vol. 248, no. 9, pp. 2263–2274, 2010. View at Publisher · View at Google Scholar · View at Scopus
  31. Z. J. Zhang, P. Li, and G. H. Yu, “Regularity criteria for the 3D MHD equations via one directional derivative of the pressure,” Journal of Mathematical Analysis and Applications, vol. 401, no. 1, pp. 66–71, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. Z. Zhang, Z.-a. Yao, and X. Wang, “A regularity criterion for the 3D magneto-micropolar fluid equations in Triebel-Lizorkin spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 6, pp. 2220–2225, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. Z. J. Zhang, “Magneto-micropolar fluid equations with regularity in one direction,” submitted.