×

Logarithmical regularity criteria of the three-dimensional micropolar fluid equations in terms of the pressure. (English) Zbl 1246.76140

Summary: This paper is devoted to the regularity criterion of the three-dimensional micropolar fluid equations. Some new regularity criteria in terms of the partial derivative of the pressure in the Lebesgue spaces and the Besov spaces are obtained which improve the previous results on the micropolar fluid equations.

MSC:

76S05 Flows in porous media; filtration; seepage
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. C. Eringen, “Theory of micropolar fluids,” Journal of Mathematics and Mechanics, vol. 16, pp. 1-18, 1966. · Zbl 0145.21302
[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. · Zbl 0383.35057
[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. · Zbl 0351.76006 · doi:10.1016/0020-7225(77)90025-8
[5] G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, Mass, USA, 1999. · Zbl 0923.76003
[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. · Zbl 1234.35193 · doi:10.1016/j.jde.2011.09.035
[7] Z.-M. Chen and W. G. Price, “Decay estimates of linearized micropolar fluid flows in \Bbb R3 space with applications to L3-strong solutions,” International Journal of Engineering Science, vol. 44, no. 13-14, pp. 859-873, 2006. · Zbl 1213.76012 · doi:10.1016/j.ijengsci.2006.06.003
[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. · Zbl 1402.35220 · doi:10.1016/j.jde.2010.03.016
[9] M. A. Rojas-Medar, “Magneto-micropolar fluid motion: existence and uniqueness of strong solution,” Mathematische Nachrichten, vol. 188, pp. 301-319, 1997. · Zbl 0893.76006 · doi:10.1002/mana.19971880116
[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. · Zbl 1155.37043 · doi:10.1088/0951-7715/20/7/005
[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. · Zbl 1103.76008 · doi:10.1016/j.amc.2006.04.024
[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. · Zbl 1158.35074 · doi:10.1016/j.jmaa.2007.01.047
[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. · Zbl 1170.35336 · doi:10.3934/dcds.2009.23.765
[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. · Zbl 1283.76016 · doi:10.1063/1.3245862
[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. · Zbl 1194.35322 · doi:10.1016/j.na.2010.06.029
[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. · Zbl 1137.76071 · doi:10.1002/mma.967
[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. · Zbl 1219.35189 · doi:10.1002/mma.1383
[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. · Zbl 1191.35217 · doi:10.1090/S0002-9939-10-10232-9
[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. · Zbl 1210.35189 · doi:10.1016/j.aml.2010.09.003
[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. · Zbl 1075.35031 · doi:10.1090/S0002-9939-02-06697-2
[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. · Zbl 1126.35047 · doi:10.1090/S0002-9939-06-08663-1
[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. · Zbl 1143.35081 · doi:10.1016/j.jde.2008.02.030
[23] Y. Zhou, “On regularity criteria in terms of pressure for the Navier-Stokes equations in \Bbb R3,” Proceedings of the American Mathematical Society, vol. 134, no. 1, pp. 149-156, 2006. · Zbl 1075.35044 · doi:10.1090/S0002-9939-05-08312-7
[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. · Zbl 1270.35339 · doi:10.1007/s00021-010-0039-5
[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. · Zbl 1160.35506 · doi:10.1016/j.ijnonlinmec.2006.12.001
[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. · Zbl 1190.35046 · doi:10.1016/j.jde.2009.09.020
[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. · Zbl 1358.35096 · doi:10.1007/11545989_4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.