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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 184674, 9 pages
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.
- J. Leray, “Sur le mouvement d'un liquide visqueux emplissant l'espace,” Acta Mathematica, vol. 63, no. 1, pp. 193–248, 1934.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- I. Kukavica and M. Ziane, “One component regularity for the Navier-Stokes equations,” Nonlinearity, vol. 19, no. 2, pp. 453–469, 2006.
- 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.
- 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.
- 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.
- 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.
- R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, The Netherlands, 1977.
- 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.
- Z. Zhifei and C. Qionglei, “Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in ,” Journal of Differential Equations, vol. 216, no. 2, pp. 470–481, 2005.
- J. Bergh and J. Löfström, Interpolation Spaces, Springer, New York, NY, USA, 1976.
- H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, vol. 18, North-Holland, Amsterdam, The Netherlands, 1978.
- R. O'Neil, “Convolution operators and spaces,” Duke Mathematical Journal, vol. 30, pp. 129–142, 1963.
- 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.