# American Institute of Mathematical Sciences

March  2015, 14(2): 637-655. doi: 10.3934/cpaa.2015.14.637

## Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain

 1 Dipartimento di Matematica, Via F. Buonarroti 2, 56126 Pisa, Italy 2 Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 210037

Received  January 2014 Revised  July 2014 Published  December 2014

In this paper, we prove some logarithmically improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain.
Citation: Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure and Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637
##### References:
 [1] R. A. Adams and J. F. Fournier, Sobolev Spaces, 2nd ed., Elsevier/Academic Press, Amsterdam, 2003. [2] H. Beir ao da Veiga, Existence and asymptotic behavior for strong solutions of the Navier-Stokes equations in the whole space, Indiana Univ. Math. J., 36 (1987), 149-166. doi: 10.1512/iumj.1987.36.36008. [3] H. Beir ao da Veiga and F. Crispo, Sharp inviscid limit results under Navier type boundary condition. An $L^p$ theory, J. Math. Fluid Mech., 12 (2010), 397-411. doi: 10.1007/s00021-009-0295-4. [4] L. C. Berselli, On a regularity criterion for the solutions to the 3D Navier-Stokes equations, Differential Integral Equations, 15 (2002), 1129-1137. [5] L. C. Berselli, Some criteria concerning the vorticity and the problem of global regularity for the 3D Navier-Stokes equations, Ann. Univ. Ferrara Sez. VII Sci. Mat., 55 (2009), 209-224. doi: 10.1007/s11565-009-0076-2. [6] L. C. Berselli and R. Manfrin, On a theorem by Sohr for the Navier-Stokes equations, J. Evol. Equ., 4 (2004), 193-211. doi: 10.1007/s00028-003-1135-2. [7] L. C. Berselli and S. Spirito, On the Boussinesq system: Regularity criteria and singular limits, Methods Appl. Anal., 18 (2011), 391-416. doi: 10.4310/MAA.2011.v18.n4.a3. [8] S. Chandrasekhar, Liquid Crystals, Cambridge University Press, 2nd edition, 1992. [9] R. Danchin, Density-dependent incompressible fluids in bounded domains, J. Math. Fluid Mech., 8 (2006), 333-381. doi: 10.1007/s00021-004-0147-1. [10] J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Ration. Mech. Anal., 9 (1962), 371-378. [11] J. Fan, Y. Fukumoto and Y. Zhou, Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations, Kinet. Relat. Models, 6 (2013), 545-556. doi: 10.3934/krm.2013.6.545. [12] J. Fan, H. Gao and B. Guo, Regularity criteria for the Navier-Stokes-Landau-Lifshitz system, J. Math. Anal. Appl., 363 (2010), 29-37. doi: 10.1016/j.jmaa.2009.07.047. [13] J. Fan, S. Jiang, G. Nakamura and Y. Zhou, Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations, J. Math. Fluid Mech., 13 (2011), 557-571. doi: 10.1007/s00021-010-0039-5. [14] J. Fan and T. Ozawa, Regularity criteria for the 3D density-dependent Boussinesq equations, Nonlinearity, 22 (2009), 553-568. doi: 10.1088/0951-7715/22/3/003. [15] F. Guillén-González, M. A. Rojas-Medar and M. A. Rodríguez-Bellido, Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model, Math. Nachr., 282 (2009), 846-867. doi: 10.1002/mana.200610776. [16] H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal., 37 (2006), 1417-1434. doi: 10.1137/S0036141004442197. [17] F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810. [18] X. Li and D. Wang, Global strong solution to the density-dependent incompressible flow of liquid crystals,, \emph{Trans Amer. Math. Soc.}, (). [19] F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math, 48 (1995), 501-537. doi: 10.1002/cpa.3160480503. [20] A. Lunardi, Interpolation Theory, Lecture Notes. Scuola Normale Superiore di Pisa (New Series), 2nd ed., Edizioni della Normale, Pisa, 2009. [21] T. Ogawa, Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow, SIAM J. Math. Anal., 34 (2003), 1318-1330. doi: 10.1137/S0036141001395868. [22] T. Ogawa and Y. Taniuchi, A note on blow-up criterion to the 3D Euler equations in a bounded domain, J. Diff. Equations, 190 (2003), 39-63. doi: 10.1016/S0022-0396(03)00013-5. [23] H. Sohr, A regularity class for the Navier-Stokes equations in Lorentz spaces, J. Evol. Equ., 1 (2001), 441-467. doi: 10.1007/PL00001382. [24] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983. doi: 10.1007/978-3-0346-0416-1. [25] Y. Zhou and S. Gala, Logarithmically improved regularity criteria for the Navier-Stokes equations in multiplier spaces, J. Math. Anal. Appl., 356 (2009), 498-501. doi: 10.1016/j.jmaa.2009.03.038. [26] Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708 doi: 10.1515/form.2011.079.

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##### References:
 [1] R. A. Adams and J. F. Fournier, Sobolev Spaces, 2nd ed., Elsevier/Academic Press, Amsterdam, 2003. [2] H. Beir ao da Veiga, Existence and asymptotic behavior for strong solutions of the Navier-Stokes equations in the whole space, Indiana Univ. Math. J., 36 (1987), 149-166. doi: 10.1512/iumj.1987.36.36008. [3] H. Beir ao da Veiga and F. Crispo, Sharp inviscid limit results under Navier type boundary condition. An $L^p$ theory, J. Math. Fluid Mech., 12 (2010), 397-411. doi: 10.1007/s00021-009-0295-4. [4] L. C. Berselli, On a regularity criterion for the solutions to the 3D Navier-Stokes equations, Differential Integral Equations, 15 (2002), 1129-1137. [5] L. C. Berselli, Some criteria concerning the vorticity and the problem of global regularity for the 3D Navier-Stokes equations, Ann. Univ. Ferrara Sez. VII Sci. Mat., 55 (2009), 209-224. doi: 10.1007/s11565-009-0076-2. [6] L. C. Berselli and R. Manfrin, On a theorem by Sohr for the Navier-Stokes equations, J. Evol. Equ., 4 (2004), 193-211. doi: 10.1007/s00028-003-1135-2. [7] L. C. Berselli and S. Spirito, On the Boussinesq system: Regularity criteria and singular limits, Methods Appl. Anal., 18 (2011), 391-416. doi: 10.4310/MAA.2011.v18.n4.a3. [8] S. Chandrasekhar, Liquid Crystals, Cambridge University Press, 2nd edition, 1992. [9] R. Danchin, Density-dependent incompressible fluids in bounded domains, J. Math. Fluid Mech., 8 (2006), 333-381. doi: 10.1007/s00021-004-0147-1. [10] J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Ration. Mech. Anal., 9 (1962), 371-378. [11] J. Fan, Y. Fukumoto and Y. Zhou, Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations, Kinet. Relat. Models, 6 (2013), 545-556. doi: 10.3934/krm.2013.6.545. [12] J. Fan, H. Gao and B. Guo, Regularity criteria for the Navier-Stokes-Landau-Lifshitz system, J. Math. Anal. Appl., 363 (2010), 29-37. doi: 10.1016/j.jmaa.2009.07.047. [13] J. Fan, S. Jiang, G. Nakamura and Y. Zhou, Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations, J. Math. Fluid Mech., 13 (2011), 557-571. doi: 10.1007/s00021-010-0039-5. [14] J. Fan and T. Ozawa, Regularity criteria for the 3D density-dependent Boussinesq equations, Nonlinearity, 22 (2009), 553-568. doi: 10.1088/0951-7715/22/3/003. [15] F. Guillén-González, M. A. Rojas-Medar and M. A. Rodríguez-Bellido, Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model, Math. Nachr., 282 (2009), 846-867. doi: 10.1002/mana.200610776. [16] H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal., 37 (2006), 1417-1434. doi: 10.1137/S0036141004442197. [17] F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810. [18] X. Li and D. Wang, Global strong solution to the density-dependent incompressible flow of liquid crystals,, \emph{Trans Amer. Math. Soc.}, (). [19] F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math, 48 (1995), 501-537. doi: 10.1002/cpa.3160480503. [20] A. Lunardi, Interpolation Theory, Lecture Notes. Scuola Normale Superiore di Pisa (New Series), 2nd ed., Edizioni della Normale, Pisa, 2009. [21] T. Ogawa, Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow, SIAM J. Math. Anal., 34 (2003), 1318-1330. doi: 10.1137/S0036141001395868. [22] T. Ogawa and Y. Taniuchi, A note on blow-up criterion to the 3D Euler equations in a bounded domain, J. Diff. Equations, 190 (2003), 39-63. doi: 10.1016/S0022-0396(03)00013-5. [23] H. Sohr, A regularity class for the Navier-Stokes equations in Lorentz spaces, J. Evol. Equ., 1 (2001), 441-467. doi: 10.1007/PL00001382. [24] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983. doi: 10.1007/978-3-0346-0416-1. [25] Y. Zhou and S. Gala, Logarithmically improved regularity criteria for the Navier-Stokes equations in multiplier spaces, J. Math. Anal. Appl., 356 (2009), 498-501. doi: 10.1016/j.jmaa.2009.03.038. [26] Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708 doi: 10.1515/form.2011.079.
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