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On regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space. (English) Zbl 1191.35217
Summary: The regularity of weak solutions and blow-up criteria for smooth solutions to the micropolar fluid equations in three dimensional space are studied in the Lorentz space \(L^{p,\infty}(\mathbb R^3)\). We obtain that if \(u\in L^q(0,T;L^{p,\infty}(\mathbb R^3))\) for \(\frac 2q+\frac 3p\leq 1\) with \(3<p\leq\infty\), or if \(\nabla u\in L^q(0,T;L^{p,\infty}(\mathbb R^3))\) for \(\frac 2q+ \frac 3p\leq 2\) with \(\frac 32<p\leq\infty\), or if the pressure \(P\in L^q(0,T;L^{p,\infty}(\mathbb R^3))\) for \(\frac 2q+ \frac 3p\leq 2\) with \(\frac 32<p\leq\infty\), or if \(\nabla P\in L^q(0,T;L^{p,\infty}(\mathbb R^3))\) for \(\frac 2q+\frac 3p\leq 3\) with \(1<p\leq\infty\), then the weak solution \((u,\omega)\) satisfying the energy inequality is a smooth solution on \([0,T)\).

MSC:
35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35B44 Blow-up in context of PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
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[1] Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. · Zbl 0344.46071
[2] Luigi C. Berselli and Giovanni P. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations, Proc. Amer. Math. Soc. 130 (2002), no. 12, 3585 – 3595. · Zbl 1075.35031
[3] Dongho Chae and Jihoon Lee, Regularity criterion in terms of pressure for the Navier-Stokes equations, Nonlinear Anal. 46 (2001), no. 5, Ser. A: Theory Methods, 727 – 735. · Zbl 1007.35064 · doi:10.1016/S0362-546X(00)00163-2 · doi.org
[4] Jianwen Chen, Bo-Qing Dong, and Zhi-Min Chen, Pullback attractors of non-autonomous micropolar fluid flows, J. Math. Anal. Appl. 336 (2007), no. 2, 1384 – 1394. · Zbl 1152.37030 · doi:10.1016/j.jmaa.2007.03.044 · doi.org
[5] Jianwen Chen, Zhi-Min Chen, and Bo-Qing Dong, Uniform attractors of non-homogeneous micropolar fluid flows in non-smooth domains, Nonlinearity 20 (2007), no. 7, 1619 – 1635. · Zbl 1155.37043 · doi:10.1088/0951-7715/20/7/005 · doi.org
[6] Qionglei Chen, Changxing Miao, and Zhifei Zhang, The Beale-Kato-Majda criterion for the 3D magneto-hydrodynamics equations, Comm. Math. Phys. 275 (2007), no. 3, 861 – 872. · Zbl 1138.76066 · doi:10.1007/s00220-007-0319-y · doi.org
[7] Bo-Qing Dong and Zhi-Min Chen, On upper and lower bounds of higher order derivatives for solutions to the 2D micropolar fluid equations, J. Math. Anal. Appl. 334 (2007), no. 2, 1386 – 1399. · Zbl 1158.35074 · doi:10.1016/j.jmaa.2007.01.047 · doi.org
[8] A. Cemal Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966), 1 – 18.
[9] L. C. F. Ferreira and E. J. Villamizar-Roa, On the existence and stability of solutions for the micropolar fluids in exterior domains, Math. Methods Appl. Sci. 30 (2007), no. 10, 1185 – 1208. · Zbl 1117.35065 · doi:10.1002/mma.838 · doi.org
[10] Giovanni P. Galdi and Salvatore Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Internat. J. Engrg. Sci. 15 (1977), no. 2, 105 – 108. · Zbl 0351.76006 · doi:10.1016/0020-7225(77)90025-8 · doi.org
[11] Cheng He and Yun Wang, On the regularity criteria for weak solutions to the magnetohydrodynamic equations, J. Differential Equations 238 (2007), no. 1, 1 – 17. · Zbl 1220.35117 · doi:10.1016/j.jde.2007.03.023 · doi.org
[12] Cheng He and Zhouping Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations 213 (2005), no. 2, 235 – 254. · Zbl 1072.35154 · doi:10.1016/j.jde.2004.07.002 · doi.org
[13] Hideo Kozono, Takayoshi Ogawa, and Yasushi Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z. 242 (2002), no. 2, 251 – 278. · Zbl 1055.35087 · doi:10.1007/s002090100332 · doi.org
[14] Hideo Kozono and Yukihiro Shimada, Bilinear estimates in homogeneous Triebel-Lizorkin spaces and the Navier-Stokes equations, Math. Nachr. 276 (2004), 63 – 74. · Zbl 1078.35087 · doi:10.1002/mana.200310213 · doi.org
[15] Hideo Kozono and Yasushi Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z. 235 (2000), no. 1, 173 – 194. · Zbl 0970.35099 · doi:10.1007/s002090000130 · doi.org
[16] O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. · Zbl 0184.52603
[17] H. Lange, The existence of instationary flows in incompressible micropolar fluids, Arch. Mech. (Arch. Mech. Stos.) 29 (1977), no. 5, 741 – 744. · Zbl 0378.76009
[18] P. G. Lemarié-Rieusset, Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, Chapman & Hall/CRC, Boca Raton, FL, 2002. · Zbl 1034.35093
[19] Pierre-Louis Lions, Mathematical topics in fluid mechanics. Vol. 1, Oxford Lecture Series in Mathematics and its Applications, vol. 3, The Clarendon Press, Oxford University Press, New York, 1996. Incompressible models; Oxford Science Publications. · Zbl 0866.76002
[20] G. G. Lorentz, Some new functional spaces, Ann. of Math. (2) 51 (1950), 37 – 55. · Zbl 0035.35602 · doi:10.2307/1969496 · doi.org
[21] Miao, C. X., Harmonic analysis and application to partial differential equations. 2nd ed., Beijing: Science Press, 2004.
[22] Richard O’Neil, Convolution operators and \?(\?,\?) spaces, Duke Math. J. 30 (1963), 129 – 142. · Zbl 0178.47701
[23] Giovanni Prodi, Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl. (4) 48 (1959), 173 – 182 (Italian). · Zbl 0148.08202 · doi:10.1007/BF02410664 · doi.org
[24] Marko A. Rojas-Medar and José Luiz Boldrini, Magneto-micropolar fluid motion: existence of weak solutions, Rev. Mat. Complut. 11 (1998), no. 2, 443 – 460. · Zbl 0918.35114 · doi:10.5209/rev_REMA.1998.v11.n2.17276 · doi.org
[25] James Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems (Proc. Sympos., Madison, Wis., 1962) Univ. of Wisconsin Press, Madison, Wis., 1963, pp. 69 – 98.
[26] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. · Zbl 0232.42007
[27] Michael Struwe, On a Serrin-type regularity criterion for the Navier-Stokes equations in terms of the pressure, J. Math. Fluid Mech. 9 (2007), no. 2, 235 – 242. · Zbl 1131.35060 · doi:10.1007/s00021-005-0198-y · doi.org
[28] H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in \?\(^{n}\), Chinese Ann. Math. Ser. B 16 (1995), no. 4, 407 – 412. A Chinese summary appears in Chinese Ann. Math. Ser. A 16 (1995), no. 6, 797. · Zbl 0837.35111
[29] E. J. Villamizar-Roa and M. A. Rodríguez-Bellido, Global existence and exponential stability for the micropolar fluid system, Z. Angew. Math. Phys. 59 (2008), no. 5, 790 – 809. · Zbl 1155.76009 · doi:10.1007/s00033-007-6090-2 · doi.org
[30] Norikazu Yamaguchi, Existence of global strong solution to the micropolar fluid system in a bounded domain, Math. Methods Appl. Sci. 28 (2005), no. 13, 1507 – 1526. · Zbl 1078.35096 · doi:10.1002/mma.617 · doi.org
[31] Yuan, B. Q., Regularity of weak solutions to magneto-micropolar fluid equations. Acta Math. Sci. Ser. B Engl. Ed., 2010, 30B. · Zbl 1240.35421
[32] Baoquan Yuan and Bo Zhang, Blow-up criterion of strong solutions to the Navier-Stokes equations in Besov spaces with negative indices, J. Differential Equations 242 (2007), no. 1, 1 – 10. · Zbl 1134.35089 · doi:10.1016/j.jde.2007.07.009 · doi.org
[33] Yong Zhou, Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain, Math. Ann. 328 (2004), no. 1-2, 173 – 192. · Zbl 1054.35062 · doi:10.1007/s00208-003-0478-x · doi.org
[34] Yong Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in \Bbb R&sup3;, Proc. Amer. Math. Soc. 134 (2006), no. 1, 149 – 156. · Zbl 1075.35044
[35] Yong Zhou, On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in \Bbb R^\Bbb N, Z. Angew. Math. Phys. 57 (2006), no. 3, 384 – 392. · Zbl 1099.35091 · doi:10.1007/s00033-005-0021-x · doi.org
[36] Yong Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst. 12 (2005), no. 5, 881 – 886. · Zbl 1068.35117 · doi:10.3934/dcds.2005.12.881 · doi.org
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