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

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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

Abstract
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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
[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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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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