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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 395420, 13 pages
http://dx.doi.org/10.1155/2012/395420
Research Article

Logarithmical Regularity Criteria of the Three-Dimensional Micropolar Fluid Equations in terms of the Pressure

School of Mathematical Sciences, Anhui University, Hefei 230601, China

Received 21 May 2012; Accepted 19 June 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Yan Jia 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. A. C. Eringen, “Theory of micropolar fluids,” Journal of Mathematics and Mechanics, vol. 16, pp. 1–18, 1966.
  2. S. Popel, A. Regirer, and P. Usick, “A continuum model of blood flow,” Biorheology, vol. 11, pp. 427–437, 1974.
  3. R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, The Netherlands, 1977.
  4. G. P. Galdi and S. Rionero, “A note on the existence and uniqueness of solutions of the micropolar fluid equations,” International Journal of Engineering Science, vol. 15, no. 2, pp. 105–108, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, Mass, USA, 1999.
  6. Q. Chen and C. Miao, “Global well-posedness for the micropolar fluid system in critical Besov spaces,” Journal of Differential Equations, vol. 252, no. 3, pp. 2698–2724, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. Z.-M. Chen and W. G. Price, “Decay estimates of linearized micropolar fluid flows in 3 space with applications to L3-strong solutions,” International Journal of Engineering Science, vol. 44, no. 13-14, pp. 859–873, 2006. View at Publisher · View at Google Scholar
  8. B.-Q. Dong and Z. Zhang, “Global regularity of the 2D micropolar fluid flows with zero angular viscosity,” Journal of Differential Equations, vol. 249, no. 1, pp. 200–213, 2010. View at Publisher · View at Google Scholar
  9. M. A. Rojas-Medar, “Magneto-micropolar fluid motion: existence and uniqueness of strong solution,” Mathematische Nachrichten, vol. 188, pp. 301–319, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. J. Chen, Z.-M. Chen, and B.-Q. Dong, “Uniform attractors of non-homogeneous micropolar fluid flows in non-smooth domains,” Nonlinearity, vol. 20, no. 7, pp. 1619–1635, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. B.-Q. Dong and Z.-M. Chen, “Global attractors of two-dimensional micropolar fluid flows in some unbounded domains,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 610–620, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. B.-Q. Dong and Z.-M. Chen, “On upper and lower bounds of higher order derivatives for solutions to the 2D micropolar fluid equations,” Journal of Mathematical Analysis and Applications, vol. 334, no. 2, pp. 1386–1399, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. B.-Q. Dong and Z.-M. Chen, “Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows,” Discrete and Continuous Dynamical Systems Series A, vol. 23, no. 3, pp. 765–784, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. 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, 13 pages, 2009. View at Publisher · View at Google Scholar
  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. J. Yuan, “Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations,” Mathematical Methods in the Applied Sciences, vol. 31, no. 9, pp. 1113–1130, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. 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
  18. B. Yuan, “On regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space,” Proceedings of the American Mathematical Society, vol. 138, no. 6, pp. 2025–2036, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. Y. Jia, W. Zhang, and B.-Q. Dong, “Remarks on the regularity criterion of the 3D micropolar fluid flows in terms of the pressure,” Applied Mathematics Letters, vol. 24, no. 2, pp. 199–203, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. L. C. Berselli and G. P. Galdi, “Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations,” Proceedings of the American Mathematical Society, vol. 130, no. 12, pp. 3585–3595, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. Q. Chen and Z. Zhang, “Regularity criterion via the pressure on weak solutions to the 3D Navier-Stokes equations,” Proceedings of the American Mathematical Society, vol. 135, no. 6, pp. 1829–1837, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. J. Fan, S. Jiang, and G. Ni, “On regularity criteria for the n-dimensional Navier-Stokes equations in terms of the pressure,” Journal of Differential Equations, vol. 244, no. 11, pp. 2963–2979, 2008. View at Publisher · View at Google Scholar
  23. Y. Zhou, “On regularity criteria in terms of pressure for the Navier-Stokes equations in 3,” Proceedings of the American Mathematical Society, vol. 134, no. 1, pp. 149–156, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. J. Fan, S. Jiang, G. Nakamura, and Y. Zhou, “Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations,” Journal of Mathematical Fluid Mechanics, vol. 13, no. 4, pp. 557–571, 2011. View at Publisher · View at Google Scholar
  25. Y. Zhou, “Regularity criteria for the 3D MHD equations in terms of the pressure,” International Journal of Non-Linear Mechanics, vol. 41, no. 10, pp. 1174–1180, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. 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 Zentralblatt MATH
  27. Y. Meyer, “Oscillating patterns in some nonlinear evolution equations,” in Mathematical Foundation of Turbulent Viscous Flows, M. Cannone and T. Miyakawa, Eds., vol. 1871 of Lecture Notes in Mathematics, pp. 101–187, Springer, Berlin, Germany, 2006. View at Publisher · View at Google Scholar