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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

Luigi C. Berselli 1, and  Jishan Fan 2,

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.
Keywords: Regularity criteria, MHD, Boussinesq equations, liquid crystals.
Mathematics Subject Classification: 35Q30, 76D03, 76D09.
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 & 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,, 2$^{nd}$ ed., (2003).   Google Scholar

[2]

H. Beir ao da Veiga, Existence and asymptotic behavior for strong solutions of the Navier-Stokes equations in the whole space,, \emph{Indiana Univ. Math. J.}, 36 (1987), 149.  doi: 10.1512/iumj.1987.36.36008.  Google Scholar

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H. Beir ao da Veiga and F. Crispo, Sharp inviscid limit results under Navier type boundary condition. An $L^p$ theory,, \emph{J. Math. Fluid Mech.}, 12 (2010), 397.  doi: 10.1007/s00021-009-0295-4.  Google Scholar

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L. C. Berselli, On a regularity criterion for the solutions to the 3D Navier-Stokes equations,, \emph{Differential Integral Equations}, 15 (2002), 1129.   Google Scholar

[5]

L. C. Berselli, Some criteria concerning the vorticity and the problem of global regularity for the 3D Navier-Stokes equations,, \emph{Ann. Univ. Ferrara Sez. VII Sci. Mat.}, 55 (2009), 209.  doi: 10.1007/s11565-009-0076-2.  Google Scholar

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L. C. Berselli and R. Manfrin, On a theorem by Sohr for the Navier-Stokes equations,, \emph{J. Evol. Equ.}, 4 (2004), 193.  doi: 10.1007/s00028-003-1135-2.  Google Scholar

[7]

L. C. Berselli and S. Spirito, On the Boussinesq system: Regularity criteria and singular limits,, \emph{Methods Appl. Anal.}, 18 (2011), 391.  doi: 10.4310/MAA.2011.v18.n4.a3.  Google Scholar

[8]

S. Chandrasekhar, Liquid Crystals,, Cambridge University Press, (1992).   Google Scholar

[9]

R. Danchin, Density-dependent incompressible fluids in bounded domains,, \emph{J. Math. Fluid Mech.}, 8 (2006), 333.  doi: 10.1007/s00021-004-0147-1.  Google Scholar

[10]

J. L. Ericksen, Hydrostatic theory of liquid crystals,, \emph{Arch. Ration. Mech. Anal.}, 9 (1962), 371.   Google Scholar

[11]

J. Fan, Y. Fukumoto and Y. Zhou, Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations,, \emph{Kinet. Relat. Models}, 6 (2013), 545.  doi: 10.3934/krm.2013.6.545.  Google Scholar

[12]

J. Fan, H. Gao and B. Guo, Regularity criteria for the Navier-Stokes-Landau-Lifshitz system,, \emph{J. Math. Anal. Appl.}, 363 (2010), 29.  doi: 10.1016/j.jmaa.2009.07.047.  Google Scholar

[13]

J. Fan, S. Jiang, G. Nakamura and Y. Zhou, Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations,, \emph{J. Math. Fluid Mech.}, 13 (2011), 557.  doi: 10.1007/s00021-010-0039-5.  Google Scholar

[14]

J. Fan and T. Ozawa, Regularity criteria for the 3D density-dependent Boussinesq equations,, \emph{Nonlinearity}, 22 (2009), 553.  doi: 10.1088/0951-7715/22/3/003.  Google Scholar

[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,, \emph{Math. Nachr.}, 282 (2009), 846.  doi: 10.1002/mana.200610776.  Google Scholar

[16]

H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations,, \emph{SIAM J. Math. Anal.}, 37 (2006), 1417.  doi: 10.1137/S0036141004442197.  Google Scholar

[17]

F. M. Leslie, Some constitutive equations for liquid crystals,, \emph{Arch. Ration. Mech. Anal.}, 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar

[18]

X. Li and D. Wang, Global strong solution to the density-dependent incompressible flow of liquid crystals,, \emph{Trans Amer. Math. Soc.}, ().   Google Scholar

[19]

F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, \emph{Comm. Pure Appl. Math}, 48 (1995), 501.  doi: 10.1002/cpa.3160480503.  Google Scholar

[20]

A. Lunardi, Interpolation Theory,, Lecture Notes. Scuola Normale Superiore di Pisa (New Series), (2009).   Google Scholar

[21]

T. Ogawa, Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow,, \emph{SIAM J. Math. Anal.}, 34 (2003), 1318.  doi: 10.1137/S0036141001395868.  Google Scholar

[22]

T. Ogawa and Y. Taniuchi, A note on blow-up criterion to the 3D Euler equations in a bounded domain,, \emph{J. Diff. Equations}, 190 (2003), 39.  doi: 10.1016/S0022-0396(03)00013-5.  Google Scholar

[23]

H. Sohr, A regularity class for the Navier-Stokes equations in Lorentz spaces,, \emph{J. Evol. Equ.}, 1 (2001), 441.  doi: 10.1007/PL00001382.  Google Scholar

[24]

H. Triebel, Theory of Function Spaces,, Birkh\, (1983).  doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[25]

Y. Zhou and S. Gala, Logarithmically improved regularity criteria for the Navier-Stokes equations in multiplier spaces,, \emph{J. Math. Anal. Appl.}, 356 (2009), 498.  doi: 10.1016/j.jmaa.2009.03.038.  Google Scholar

[26]

Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations,, \emph{Forum Math.}, 24 (2012), 691.  doi: 10.1515/form.2011.079.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. F. Fournier, Sobolev Spaces,, 2$^{nd}$ ed., (2003).   Google Scholar

[2]

H. Beir ao da Veiga, Existence and asymptotic behavior for strong solutions of the Navier-Stokes equations in the whole space,, \emph{Indiana Univ. Math. J.}, 36 (1987), 149.  doi: 10.1512/iumj.1987.36.36008.  Google Scholar

[3]

H. Beir ao da Veiga and F. Crispo, Sharp inviscid limit results under Navier type boundary condition. An $L^p$ theory,, \emph{J. Math. Fluid Mech.}, 12 (2010), 397.  doi: 10.1007/s00021-009-0295-4.  Google Scholar

[4]

L. C. Berselli, On a regularity criterion for the solutions to the 3D Navier-Stokes equations,, \emph{Differential Integral Equations}, 15 (2002), 1129.   Google Scholar

[5]

L. C. Berselli, Some criteria concerning the vorticity and the problem of global regularity for the 3D Navier-Stokes equations,, \emph{Ann. Univ. Ferrara Sez. VII Sci. Mat.}, 55 (2009), 209.  doi: 10.1007/s11565-009-0076-2.  Google Scholar

[6]

L. C. Berselli and R. Manfrin, On a theorem by Sohr for the Navier-Stokes equations,, \emph{J. Evol. Equ.}, 4 (2004), 193.  doi: 10.1007/s00028-003-1135-2.  Google Scholar

[7]

L. C. Berselli and S. Spirito, On the Boussinesq system: Regularity criteria and singular limits,, \emph{Methods Appl. Anal.}, 18 (2011), 391.  doi: 10.4310/MAA.2011.v18.n4.a3.  Google Scholar

[8]

S. Chandrasekhar, Liquid Crystals,, Cambridge University Press, (1992).   Google Scholar

[9]

R. Danchin, Density-dependent incompressible fluids in bounded domains,, \emph{J. Math. Fluid Mech.}, 8 (2006), 333.  doi: 10.1007/s00021-004-0147-1.  Google Scholar

[10]

J. L. Ericksen, Hydrostatic theory of liquid crystals,, \emph{Arch. Ration. Mech. Anal.}, 9 (1962), 371.   Google Scholar

[11]

J. Fan, Y. Fukumoto and Y. Zhou, Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations,, \emph{Kinet. Relat. Models}, 6 (2013), 545.  doi: 10.3934/krm.2013.6.545.  Google Scholar

[12]

J. Fan, H. Gao and B. Guo, Regularity criteria for the Navier-Stokes-Landau-Lifshitz system,, \emph{J. Math. Anal. Appl.}, 363 (2010), 29.  doi: 10.1016/j.jmaa.2009.07.047.  Google Scholar

[13]

J. Fan, S. Jiang, G. Nakamura and Y. Zhou, Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations,, \emph{J. Math. Fluid Mech.}, 13 (2011), 557.  doi: 10.1007/s00021-010-0039-5.  Google Scholar

[14]

J. Fan and T. Ozawa, Regularity criteria for the 3D density-dependent Boussinesq equations,, \emph{Nonlinearity}, 22 (2009), 553.  doi: 10.1088/0951-7715/22/3/003.  Google Scholar

[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,, \emph{Math. Nachr.}, 282 (2009), 846.  doi: 10.1002/mana.200610776.  Google Scholar

[16]

H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations,, \emph{SIAM J. Math. Anal.}, 37 (2006), 1417.  doi: 10.1137/S0036141004442197.  Google Scholar

[17]

F. M. Leslie, Some constitutive equations for liquid crystals,, \emph{Arch. Ration. Mech. Anal.}, 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar

[18]

X. Li and D. Wang, Global strong solution to the density-dependent incompressible flow of liquid crystals,, \emph{Trans Amer. Math. Soc.}, ().   Google Scholar

[19]

F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, \emph{Comm. Pure Appl. Math}, 48 (1995), 501.  doi: 10.1002/cpa.3160480503.  Google Scholar

[20]

A. Lunardi, Interpolation Theory,, Lecture Notes. Scuola Normale Superiore di Pisa (New Series), (2009).   Google Scholar

[21]

T. Ogawa, Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow,, \emph{SIAM J. Math. Anal.}, 34 (2003), 1318.  doi: 10.1137/S0036141001395868.  Google Scholar

[22]

T. Ogawa and Y. Taniuchi, A note on blow-up criterion to the 3D Euler equations in a bounded domain,, \emph{J. Diff. Equations}, 190 (2003), 39.  doi: 10.1016/S0022-0396(03)00013-5.  Google Scholar

[23]

H. Sohr, A regularity class for the Navier-Stokes equations in Lorentz spaces,, \emph{J. Evol. Equ.}, 1 (2001), 441.  doi: 10.1007/PL00001382.  Google Scholar

[24]

H. Triebel, Theory of Function Spaces,, Birkh\, (1983).  doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[25]

Y. Zhou and S. Gala, Logarithmically improved regularity criteria for the Navier-Stokes equations in multiplier spaces,, \emph{J. Math. Anal. Appl.}, 356 (2009), 498.  doi: 10.1016/j.jmaa.2009.03.038.  Google Scholar

[26]

Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations,, \emph{Forum Math.}, 24 (2012), 691.  doi: 10.1515/form.2011.079.  Google Scholar

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