Table of Contents
ISRN Applied Mathematics
Volume 2014, Article ID 564758, 4 pages
http://dx.doi.org/10.1155/2014/564758
Research Article

Some Regularity Criteria for the 3D Boussinesq Equations in the Class

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

Received 2 December 2013; Accepted 15 January 2014; Published 23 February 2014

Academic Editors: Y. Dimakopoulos and X.-S. Yang

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. E. Hopf, “Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen,” Mathematische Nachrichten, vol. 4, pp. 213–231, 1951. View at Google Scholar · View at MathSciNet
  2. P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002. View at Publisher · 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
  4. J. Serrin, “On the interior regularity of weak solutions of the Navier-Stokes equations,” Archive for Rational Mechanics and Analysis, vol. 9, pp. 187–191, 1962. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. Serrin, “The initial value problem for the Navier-Stokes equations,” in Nonlinear Problems, pp. 69–98, University of Wisconsin Press, Madison, Wis, USA, 1963. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. H. B. da Veiga, “A new regularity class for the Navier-Stokes equations in n,” Chinese Annals of Mathematics B, vol. 16, no. 4, pp. 407–412, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. S. Gala, “A remark on the blow-up criterion of strong solutions to the Navier-Stokes equations,” Applied Mathematics and Computation, vol. 217, no. 22, pp. 9488–9491, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. X. W. He and S. Gala, “Regularity criterion for weak solutions to the Navier-Stokes equations in terms of the pressure in the class L20,T;B˙,-13,” Nonlinear Analysis: Real World Applications, vol. 12, no. 6, pp. 3602–3607, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. Neustupa, A. Novotný, and P. Penel, “An interior regularity of a weak solution to the Navier-Stokes equations in dependence on one component of velocity,” in Topics in Mathematical Fluid Mechanics, vol. 10 of Quaderni di Matematica, pp. 163–183, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Z. J. Zhang, “A logarithmically improved regularity criterion for the 3D Boussinesq equations via the pressure,” Acta Applicandae Mathematicae, 2013. View at Publisher · View at Google Scholar
  11. Z. J. Zhang, “A remark on the regularity criterion for the 3D Navier-Stokes equations involving the gradient of one velocity component,” Journal of Mathematical Analysis and Applications, 2014. View at Publisher · View at Google Scholar
  12. Z. J. 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
  13. Z. J. Zhang, P. Li, and D. X. Zhong, “Navier-Stokes equations with regularity in two entries of the velocity gradient tensor,” Applied Mathematics and Computation, vol. 228, pp. 546–551, 2014. View at Publisher · View at Google Scholar
  14. Z. J. 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 Zentralblatt MATH · View at MathSciNet
  15. Z. J. Zhang, D. X. 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, no. 1, pp. 175–181, 2014. View at Publisher · View at Google Scholar
  16. 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 · View at MathSciNet
  17. 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, 11 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. 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
  19. J. S. Fan and Y. Zhou, “A note on regularity criterion for the 3D Boussinesq system with partial viscosity,” Applied Mathematics Letters, vol. 22, no. 5, pp. 802–805, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. N. Ishimura and H. Morimoto, “Remarks on the blow-up criterion for the 3-D Boussinesq equations,” Mathematical Models & Methods in Applied Sciences, vol. 9, no. 9, pp. 1323–1332, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. H. Qiu, Y. Du, and Z. Yao, “Blow-up criteria for 3D Boussinesq equations in the multiplier space,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 1820–1824, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. H. Qiu, Y. Du, and Z. Yao, “Serrin-type blow-up criteria for 3D Boussinesq equations,” Applicable Analysis, vol. 89, no. 10, pp. 1603–1613, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Z. Zhang, “A remark on the regularity criterion for the 3D Boussinesq equations involving the pressure gradient,” Abstract and Applied Analysis, vol. 2014, Article ID 510924, 4 pages, 2014. View at Publisher · View at Google Scholar
  24. Y. Meyer, P. Gerard, and F. Oru, “Inégalités de Sobolev précisées,” in Séminaire sur les Équations aux Dérivées Partielles, 1996-1997, Exp. No. 4, p. 8, École Polytechnique, Palaiseau, France, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet