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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 184674, 9 pages
http://dx.doi.org/10.1155/2012/184674
Research Article

A Note on the Regularity Criterion of Weak Solutions of Navier-Stokes Equations in Lorentz Space

School of Science, Tianjin Polytechnic University, Tianjin 300387, China

Received 3 July 2012; Accepted 7 August 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Xunwu Yin. 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. 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
  2. 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 Zentralblatt MATH
  3. Z. M. Chen and Z. Xin, “Homogeneity criterion for the Navier-Stokes equations in the whole spaces,” Journal of Mathematical Fluid Mechanics, vol. 3, no. 2, pp. 152–182, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. Q. Chen and Z. Zhang, “Space-time estimates in the Besov spaces and the Navier-Stokes equations,” Methods and Applications of Analysis, vol. 13, no. 1, pp. 107–122, 2006. View at Zentralblatt MATH
  5. Z.-M. Chen and W. G. Price, “Blow-up rate estimates for weak solutions of the Navier-Stokes equations,” Proceedings of the Royal Society A, vol. 457, no. 2015, pp. 2625–2642, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. Z.-M. Chen and W. G. Price, “Morrey space techniques applied to the interior regularity problem of the Navier-Stokes equations,” Nonlinearity, vol. 14, no. 6, pp. 1453–1472, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. B. Dong, G. Sadek, and Z. Chen, “On the regularity criteria of the 3D Navier-Stokes equations in critical spaces,” Acta Mathematica Scientia Series B, vol. 31, no. 2, pp. 591–600, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. B.-Q. Dong and Z.-M. Chen, “Regularity criterion for weak solutions to the 3D Navier-Stokes equations via two velocity components,” Journal of Mathematical Analysis and Applications, vol. 338, no. 1, pp. 1–10, 2008. View at Publisher · View at Google Scholar
  9. 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 · View at Zentralblatt MATH
  10. 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
  11. I. Kukavica and M. Ziane, “One component regularity for the Navier-Stokes equations,” Nonlinearity, vol. 19, no. 2, pp. 453–469, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. 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
  13. B.-Q. Dong and Z.-M. Chen, “Regularity criteria of weak solutions to the three-dimensional micropolar flows,” Journal of Mathematical Physics, vol. 50, no. 10, Article ID 103525, 2009. View at Publisher · View at Google Scholar
  14. B.-Q. Dong, Y. Jia, and Z.-M. Chen, “Pressure regularity criteria of the three-dimensional micropolar fluid flows,” Mathematical Methods in the Applied Sciences, vol. 34, no. 5, pp. 595–606, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. B.-Q. Dong and W. Zhang, “On the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 7, pp. 2334–2341, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, The Netherlands, 1977.
  17. H. Kozono and N. Yatsu, “Extension criterion via two-components of vorticity on strong solutions to the 3D Navier-Stokes equations,” Mathematische Zeitschrift, vol. 246, no. 1-2, pp. 55–68, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. Z. Zhifei and C. Qionglei, “Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in R3,” Journal of Differential Equations, vol. 216, no. 2, pp. 470–481, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. J. Bergh and J. Löfström, Interpolation Spaces, Springer, New York, NY, USA, 1976.
  20. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, vol. 18, North-Holland, Amsterdam, The Netherlands, 1978.
  21. R. O'Neil, “Convolution operators and L(p,q) spaces,” Duke Mathematical Journal, vol. 30, pp. 129–142, 1963. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. H. Fujita and T. Kato, “On the Navier-Stokes initial value problem. I,” Archive for Rational Mechanics and Analysis, vol. 16, pp. 269–315, 1964.