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Abstract and Applied Analysis
Volume 2014, Article ID 234809, 4 pages
http://dx.doi.org/10.1155/2014/234809
Research Article

Regularity Criterion for the Nematic Liquid Crystal Flows in Terms of Velocity

1Department of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China
2School of Mathematics and Information Science, Shaoguan University, Shaoguan 512005, China

Received 7 February 2014; Accepted 15 May 2014; Published 30 June 2014

Academic Editor: Gaohang Yu

Copyright © 2014 Ruiying Wei et al. 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. J. L. Ericksen, “Conservation laws for liquid crystals,” Transactions of the Society of Rheology: Journal of Rheology, vol. 5, no. 1, pp. 23–34, 1961. View at Publisher · View at Google Scholar · View at MathSciNet
  2. J. L. Ericksen, “Continuum theory of nematic liquid crystals,” Res Mechanica, vol. 21, no. 4, pp. 381–392, 1987. View at Google Scholar · View at Scopus
  3. J. L. Ericksen, “Liquid crystals with variable degree of orientation,” Archive for Rational Mechanics and Analysis, vol. 113, no. 2, pp. 97–120, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  4. F. Leslie, Theory of Flow Phenomenum in Liquid Crystals, vol. 4, Springer, New York, NY, USA, 1979.
  5. F.-H. Lin, “Nonlinear theory of defects in nematic liquid crystals: phase transition and flow phenomena,” Communications on Pure and Applied Mathematics, vol. 42, no. 6, pp. 789–814, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. F.-H. Lin and C. Liu, “Nonparabolic dissipative systems modeling the flow of liquid crystals,” Communications on Pure and Applied Mathematics, vol. 48, no. 5, pp. 501–537, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. Q. Liu, J. Zhao, and S. Cui, “A regularity criterion for the three-dimensional nematic liquid crystal flow in terms of one directional derivative of the velocity,” Journal of Mathematical Physics, vol. 52, no. 3, Article ID 033102, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. Fan and B. Guo, “Regularity criterion to some liquid crystal models and the Landau-Lifshitz equations in R3,” Science in China A: Mathematics, vol. 51, no. 10, pp. 1787–1797, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Q. Liu and J. Zhao, “Logarithmically improved blow-up criteria for the nematic liquid crystal flows,” Nonlinear Analysis: Real World Applications, vol. 16, pp. 178–190, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  10. T. Huang and C. Wang, “Blow up criterion for nematic liquid crystal flows,” Communications in Partial Differential Equations, vol. 37, no. 5, pp. 875–884, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. F. Lin, J. Lin, and C. Wang, “Liquid crystal flows in two dimensions,” Archive for Rational Mechanics and Analysis, vol. 197, no. 1, pp. 297–336, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Z. Zhang, X. Wang, and Z.-A. Yao, “Remarks on regularity criteria for the weak solutions of liquid crystals,” Journal of Evolution Equations, vol. 12, no. 4, pp. 801–812, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Z. Zhang, T. Tang, and L. Liu, “An Osgood type regularity criterion for the liquid crystal flows,” Nonlinear Differential Equations and Applications, vol. 21, no. 2, pp. 253–262, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  14. G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, vol. 1, Springer, New York, NY, USA, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  15. G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, vol. 2, Springer, New York, NY, USA, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  16. P. Penel and M. Pokorný, “Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity,” Applications of Mathematics, vol. 49, no. 5, pp. 483–493, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. C. 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 Zentralblatt MATH · View at MathSciNet
  18. S. Gala, “Remarks on regularity criterion for weak solutions to the Navier-Stokes equations in terms of the gradient of the pressure,” Applicable Analysis, vol. 92, no. 1, pp. 96–103, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. X. Jia and Y. Zhou, “Remarks on regularity criteria for the Navier-Stokes equations via one velocity component,” Nonlinear Analysis: Real World Applications, vol. 15, pp. 239–245, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  20. 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 Zentralblatt MATH · View at MathSciNet
  21. 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 Zentralblatt MATH · View at MathSciNet
  22. 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, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. R. A. Adams, Sobolev Spaces, Academic Press, New York, NY, USA, 1975.
  24. X. Zheng, “A regularity criterion for the tridimensional Navier-Stokes equations in term of one velocity component,” Journal of Differential Equations, vol. 256, no. 1, pp. 283–309, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  25. O. A. Ladyzhenskaya, Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, NY, USA, 2nd edition, 1969, English translation. View at MathSciNet