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Pressure regularity criteria of the three-dimensional micropolar fluid flows. (English) Zbl 1219.35189

The authors study the regularity of weak solutions to the three-dimensional micropolar fluid flows. By the energy argument, the authors show that the weak solution is regular under the assumptions that the pressure field belongs to some Lebesgue space, or Morrey space, or multiplier space, or BMO space, or Besov space respectively.
Reviewer: Cheng He (Beijing)

MSC:

35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35B65 Smoothness and regularity of solutions to PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
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[1] Eringen, Theory of micropolar fluids, Journal of Mathematics and Mechanics 16 pp 1– (1966) · Zbl 0145.21302
[2] Doi, The Theory of Polymer Dynamics (1986)
[3] Popel, A continuum model of blood flow, Biorheology 11 pp 427– (1974)
[4] Galdi, A note on the existence and uniqueness of solutions of micropolar fluid equations, International Journal of Engineering Science 14 pp 105– (1977) · Zbl 0351.76006
[5] Łukaszewicz, Modeling and Simulation in Science, Engineering and Technology, in: Micropolar Fluids. Theory and Applications (1999)
[6] Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids (1969) · Zbl 0184.52603
[7] Chen Q Miao C Global well-posedness for the micropolar fluid system in the critical Besov spaces 2010 · Zbl 1234.35193
[8] Chen, Decay estimates of linearized micropolar fluid flows in R3 space with applications to L3-strong solutions, International Journal of Engineering Science 44 pp 859– (2006) · Zbl 1213.76012
[9] Rojas-Medar, Magneto-micropolar fluid motion: existence and uniqueness of strong solution, Mathematische Nachrichten 188 pp 301– (1997) · Zbl 0893.76006
[10] Dong, Global regularity for the 2D micropolar fluid flows with zero angular viscosity, Journal of Differential Equations 249 pp 200– (2010) · Zbl 1402.35220
[11] Chen, Uniform attractors of non-homogeneous micropolar fluid flows in non-smooth domains, Nonlinearity 20 pp 1619– (2007) · Zbl 1155.37043
[12] Dong, Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows, Discrete and Continuous Dynamics Systems 23 pp 765– (2009) · Zbl 1170.35336
[13] Serrin, On the interior regularity of weak solutions of the Navier Stokes equations, Archive for Rational Mechanics and Analysis 9 pp 187– (1962) · Zbl 0106.18302
[14] Beirão da Veiga, A new regularity class for the Navier Stokes equations in Rn, Chinese Annals of Mathematics 16 pp 407– (1995) · Zbl 0837.35111
[15] Kozono, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Mathematische Zeitschrift 242 pp 251– (2002) · Zbl 1055.35087
[16] Kozono, Bilinear estimates in BMO and the Navier-Stokes equations, Mathematische Zeitschrift 235 pp 173– (2000) · Zbl 0970.35099
[17] Chae, Regularity criterion in terms of pressure for the Navier-Stokes equations, Nonlinear Analysis TMA 46 pp 727– (2001) · Zbl 1007.35064
[18] Berselli, Regularity criteria involving the pressure for the weak solutions of the Navier-Stokes equations, Proceedings of the American Mathematical Society 130 pp 3585– (2002) · Zbl 1075.35031
[19] Zhou, Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain, Mathematische Annalen 328 pp 173– (2004) · Zbl 1054.35062
[20] Zhou, On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in Rn, Zeitschrift für Angewandte Mathematik und Physik 57 pp 384– (2006) · Zbl 1099.35091
[21] Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in R3, Proceedings of the American Mathematical Society 134 pp 149– (2006) · Zbl 1075.35044
[22] Seregin, Navier-Stokes equations with lower bounds on the pressure, Archive for Rational Mechanics and Analysis 163 pp 65– (2002) · Zbl 1002.35094
[23] Fan, On regularity criteria for the n-dimensional Navier-Stokes equations in terms of the pressure, Journal of Differential Equations 244 pp 2963– (2006) · Zbl 1143.35081
[24] Chen, Regularity criterion via the pressure on weak solutions to the 3D Navier-Stokes equations, Proceedings of the American Mathematical Society 135 pp 1829– (2007) · Zbl 1126.35047
[25] Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, International Journal of Non-linear Mechanics 41 pp 1174– (2006) · Zbl 1160.35506
[26] Dong, Regularity criteria of weak solutions to the three-dimensional micropolar flows, Journal of Mathematical Physics 50 pp 103525– (2009) · Zbl 1283.76016
[27] Dong, On the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces, Nonlinear Analysis, TMA 73 pp 2334– (2010) · Zbl 1194.35322
[28] Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem (2002) · Zbl 1034.35093
[29] Triebel, Theory of Function Spaces (1983) · Zbl 1235.46002
[30] Beale, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Communications in Mathematical Physics 94 pp 61– (1984) · Zbl 0573.76029
[31] Kato, Strong Lp solutions of the Navier-Stokes equations in Rm with applications to weak solutions, Mathematische Zeitschrift 187 pp 471– (1984) · Zbl 0545.35073
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