×

Regularity criterion for the 3D micropolar fluid equations in terms of pressure. (English) Zbl 1416.35059

Summary: In this paper, we consider the regularity of weak solutions to the 3D incompressible micropolar fluid equations. It is proved that if the one directional derivative of the pressure, say \(\partial _{3}P\), satisfies \[ \partial_3 P\in L^\beta(0,T;L^\alpha(\mathbb{R}^3)) \text{ with }\frac{2}{\beta}+ \frac{3}{\alpha} \leq 2,\frac{3}{2}\leq\alpha<\infty, \] then the corresponding weak solution \((u,\omega )\) is regular on \([0, T]\).

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
35Q30 Navier-Stokes equations
35D30 Weak solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams, R., Fournier, J.: Soblev Spaces, 2nd edn. Academic, New York (2003)
[2] Berselli, L., Galdi, G.: Regularity criteria involving the pressure for the weak solutions of the Navier-Stokes equations. Proc. Am. Math. Soc. 130, 3585-3595 (2002) · Zbl 1075.35031 · doi:10.1090/S0002-9939-02-06697-2
[3] Cao, C., Titi, E.: Regularity criteria for the three-dimensional Navier-Stokes equaitons. India Univ. Math. J. 57, 2643-2661 (2008) · Zbl 1159.35053 · doi:10.1512/iumj.2008.57.3719
[4] Cao, C., Wu, J.: Two new regularity criteria for the 3D MHD equations. J. Differ. Equ. 248, 2263-2274 (2010) · Zbl 1190.35046 · doi:10.1016/j.jde.2009.09.020
[5] Chemin, J.: Perfect Incompressible Fluids. Oxford University Press, New York (1998) · Zbl 0927.76002
[6] da Veiga, H.B.: A new regularity class for the Navier-Stokes equations in \[{\mathbb{R}}^nRn\]. Chin. Ann. Math. Ser. B 16, 407-412 (1995) · Zbl 0837.35111
[7] da Veiga, H.B.: A sufficient condition on the pressure for the regularity of weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 2, 99-106 (2000) · Zbl 0970.35105 · doi:10.1007/PL00000949
[8] Dong, B., Chen, Z.: Regularity criteria of weak solutions to the three-dimensional micropolar flows. J. Math. Phys. 50, 103525 (2009) · Zbl 1283.76016 · doi:10.1063/1.3245862
[9] Dong, B., Jia, Y., Chen, Z.: Pressure regularity criteria of the three-dimensional micropolar fluid flows. Math. Methods Appl. Sci. 34, 595-606 (2011) · Zbl 1219.35189 · doi:10.1002/mma.1383
[10] Dong, B., Li, J., Wu, J.: Global well-posedness and large-time decay for the 2D micropolar equations. J. Differ. Equ. 262, 3488-3523 (2017) · Zbl 1361.35143 · doi:10.1016/j.jde.2016.11.029
[11] Dong, B., Zhang, W.: On the regularity criterion for the three-dimensional micropolar fluid flows in Besov spaces. Nonlinear Anal. 73, 2334-2341 (2010) · Zbl 1194.35322 · doi:10.1016/j.na.2010.06.029
[12] Dong, B., Zhang, Z.: Global regularity of the 2D micropolar fluid flows with zero angular viscosity. J. Differ. Equ. 249, 200-213 (2010) · Zbl 1402.35220 · doi:10.1016/j.jde.2010.03.016
[13] Dong, B., Zhang, Z.: On the weak-strong uniqueness of Koch-Tataru’s solution for the Navier-Stokes equations. J. Differ. Equ. 256, 2406-2422 (2014) · Zbl 1285.35064 · doi:10.1016/j.jde.2014.01.007
[14] Eringen, A.: Theory of micropolar fluids. J. Math. Mech. 16, 1-18 (1966)
[15] Escauriaza, L., Seregin, G., S̆verák, \[V.: L^{3,\infty }-\] L3,∞-solutions of Navier-Stokes equations and backward uniqueness. Russ. Math. Surv. 58, 211-250 (2003) · Zbl 1064.35134 · doi:10.1070/RM2003v058n02ABEH000609
[16] Fan, J., Jiang, S., Nakamura, G., Zhou, Y.: Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations. J. Math. Fluid Mech. 13, 557-571 (2011) · Zbl 1270.35339 · doi:10.1007/s00021-010-0039-5
[17] Gala, S.: On the regularity criteria for the three-dimensional micropolar fluid equations in the critical Morrey-Campanato space. Nonlinear Anal. Real World Appl. 12, 2142-2150 (2011) · Zbl 1225.35041 · doi:10.1016/j.nonrwa.2010.12.028
[18] Gala, S.: A remark on the logarithmically improved regularity criterion for the micropolar fluid equations in terms of pressure. Math. Methods Appl. Sci. 34, 1945-1953 (2011) · Zbl 1230.35100 · doi:10.1002/mma.1488
[19] Galdi, G., Rionero, S.: A note on the existence and uniqueness of solutions of micropolar fluid equations. Int. J. Eng. Sci. 14, 105-108 (1977) · Zbl 0351.76006 · doi:10.1016/0020-7225(77)90025-8
[20] Jia, X., Zhou, Y.: A new regularity criterion for the 3D incompressible MHD equations in terms of one compenent of the gredient of pressure. J. Math. Anal. Appl. 396, 345-350 (2012) · Zbl 1256.35060 · doi:10.1016/j.jmaa.2012.06.016
[21] Jia, Y., Zhang, W., Dong, B.: Remarks on the regularity of the 3D micropolar fluid flows in terms of the pressure. Appl. Math. Lett. 24, 199-203 (2011) · Zbl 1210.35189 · doi:10.1016/j.aml.2010.09.003
[22] Lemarié-Rieusset, P.G.: Recent Developments in the Navier-Stokes Problem. Chapman and Hall/CRC, London (2002) · Zbl 1034.35093 · doi:10.1201/9781420035674
[23] Lukaszewicz, G.: Micropolar Fluids: Theory and Applications, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston, Boston (1999) · Zbl 0923.76003 · doi:10.1007/978-1-4612-0641-5
[24] Prodi, G.: Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. 48, 173-182 (1959) · Zbl 0148.08202 · doi:10.1007/BF02410664
[25] Rojas-Medar, M.: Magnato-microplar fluid motion: existence and uniqueness of strong solution. Math. Nachr. 188, 301-319 (1997) · Zbl 0893.76006 · doi:10.1002/mana.19971880116
[26] Serrin, J.: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 9, 187-195 (1962) · Zbl 0106.18302 · doi:10.1007/BF00253344
[27] Sohr, H.: A generalization of Serrin’s regularity criterion for the Navier-Stokes equations. Quaderni Di Math. 10, 321-347 (2002) · Zbl 1062.35075
[28] Struwe, M.: On a Serrin-type regularity criterion for the Navier-Stokes equaitons in terms of the pressure. J. Math. Fluid Mech. 9, 235-242 (2007) · Zbl 1131.35060 · doi:10.1007/s00021-005-0198-y
[29] Xiang, Z., Yang, H.: On the regularity criteria for the 3D magneto-micropolar fluids in terms of one directional derivative. Bound. Value Probl. 2012, 139 (2012) · Zbl 1280.35118 · doi:10.1186/1687-2770-2012-139
[30] Yamaguchi, N.: Existence of global strong solution to the micropolar fluid system in a bounded domain. Math. Methods in Appl. Sci. 28, 1507-1526 (2005) · Zbl 1078.35096 · doi:10.1002/mma.617
[31] Zhang, Z., Li, P., Yu, G.: Regularity criteria for the 3D MHD equations via one directional derivative of the pressure. J. Math. Anal. Appl. 401, 66-71 (2013) · Zbl 1364.35290 · doi:10.1016/j.jmaa.2012.11.022
[32] Zhang, Z.: A regularity criterion for the three-dimensional micropolar fluid system in homogeneous Besov spaces. Electron. J. Differ. Equ. 104, 1-6 (2016) · Zbl 1399.35307
[33] Zhou, Y.: Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain. Math. Ann. 328, 173-192 (2004) · Zbl 1054.35062 · doi:10.1007/s00208-003-0478-x
[34] Zhou, Y.: On the regularity criteria in terms of pressure for the Navier-Stokes equations in \[\mathbb{R}^3\] R3. Proc. Am. Math. Soc. 134, 149-156 (2006) · Zbl 1075.35044 · doi:10.1090/S0002-9939-05-08312-7
[35] Zhou, Y.: On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in \[\mathbb{R}^3\] R3. Z. Angew. Math. Phys. 57, 384-392 (2006) · Zbl 1099.35091 · doi:10.1007/s00033-005-0021-x
[36] Zhou, Y.: Regularity criteria for the 3D MHD equations in terms of the pressure. Int. J. Non Linear Mech. 41, 1174-1180 (2006) · Zbl 1160.35506 · doi:10.1016/j.ijnonlinmec.2006.12.001
[37] Zhou, Y., Gala, S.: Regularity criteria for the solutions to the 3D MHD equations in the multiplier space. Z. Angew. Math. Phys. 61, 193-199 (2010) · Zbl 1273.76447 · doi:10.1007/s00033-009-0023-1
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.