A remark on blow up criterion of three-dimensional nematic liquid crystal flows

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June 2016, 5(2): 337-348. doi: 10.3934/eect.2016007

A remark on blow up criterion of three-dimensional nematic liquid crystal flows

1.	School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

Received January 2016 Revised March 2016 Published June 2016

Abstract
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In this paper, we study the initial value problem for the three-dimensional nematic liquid crystal flows. Blow up criterion of smooth solutions is established by the energy method, which refines the previous result.

Keywords: blow up criterion., smooth solutions, Nematic liquid crystal flows.

Mathematics Subject Classification: Primary: 76A15; Secondary: 76B0.

Citation: Yinxia Wang. A remark on blow up criterion of three-dimensional nematic liquid crystal flows. Evolution Equations & Control Theory, 2016, 5 (2) : 337-348. doi: 10.3934/eect.2016007

References:

[1]	G. Brown and W. Shaw, The Mesomorphic State, Liquid Crystals,, Chem. Rev., 57 (1957), 1049. doi: 10.1021/cr50018a002. Google Scholar
[2]	Q. Chen, Z. Tan and G. Wu, LPS's criterion for incompressible nematics liquid crystal flows,, Acta Mathematica Scientia, 34 (2014), 1072. doi: 10.1016/S0252-9602(14)60070-9. Google Scholar
[3]	J. Chemin, perfect Incompressible Fluids,, Oxford Lecture Ser. Math. Appl. 14, 14 (1998). Google Scholar
[4]	J. Ericksen, Hydrostatic theory of liquid crystal,, Arch. Rational Mech. Anal., 9 (1962), 371. doi: 10.1007/BF00253358. Google Scholar
[5]	J. Ericksen and D. Kinderlehrer (eds.), Theory and Applications of Liquid Crystals,, The IMA Volumes in Mathematics and its Applications, 5 (1987), 21. doi: 10.1007/978-1-4613-8743-5. Google Scholar
[6]	J. Ericksen, Equilibrium theory of liquid crystals,, Advances in Liquid Crystals, 2 (1976), 233. doi: 10.1016/B978-0-12-025002-8.50012-9. Google Scholar
[7]	F. Frank, On the theory of liquid crystals,, Discussions Faraday Soc., 25 (1958), 19. Google Scholar
[8]	P. G. de Gennes, The Physics of Liquid Crystals,, Oxford, (1974). Google Scholar
[9]	Y. Hao and X. Liu, The existence and blow-up criterion of liquid crystals system in critical Besov space,, Commun. Pure Appl. Anal., 13 (2014), 225. Google Scholar
[10]	R. Hardt, D. Kinderlehrer and F. Lin, Existence and partial regularity of static liquid crystal configurations,, Comm. Math. Phys., 105 (1986), 547. doi: 10.1007/BF01238933. Google Scholar
[11]	M. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in $\mathbbR^2$,, Cal. Var. Part. Differ. Equ., 40 (2011), 15. doi: 10.1007/s00526-010-0331-5. Google Scholar
[12]	T. Huang and C. Wang, Blow up criterion for nematic liquid crystal flows,, Comm. Part. Differ.Equ., 37 (2012), 875. doi: 10.1080/03605302.2012.659366. Google Scholar
[13]	T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 891. doi: 10.1002/cpa.3160410704. Google Scholar
[14]	H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations,, Math. Z., 242 (2002), 251. doi: 10.1007/s002090100332. Google Scholar
[15]	F. Leslie, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265. doi: 10.1007/BF00251810. Google Scholar
[16]	F. Leslie, Theory of flow phenomemum in liquid crystal,, In: Brown (ed.) Advances in Liquid Crystals, 4 (1979), 1. Google Scholar
[17]	F. Lin, Nonlinear theory of defects in nematic liquid crystal: Phase transition and flow phenomena,, Comm. Pure Appl. Math., 42 (1989), 789. doi: 10.1002/cpa.3160420605. Google Scholar
[18]	F. Lin and C. Liu, Nonparabolic Dissipative Systems Modeling the Flow of Liquid Crystals,, Comm. Pure Appl. Math., 48 (1995), 501. doi: 10.1002/cpa.3160480503. Google Scholar
[19]	F. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions,, Arch. Rational Mech. Anal., 197 (2010), 297. doi: 10.1007/s00205-009-0278-x. Google Scholar
[20]	F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals,, Chin. Ann. Math. Ser. B, 31 (2010), 921. doi: 10.1007/s11401-010-0612-5. Google Scholar
[21]	A. Majda and A. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press: Cambridge, (2002). Google Scholar
[22]	C. Oseen, Die anisotropen Fliissigkeiten, Tatsachen und Theorien,, Forts. Chemie, 21 (1929), 25. Google Scholar
[23]	H. Triebel, Theory of Function Spaces,, Monograph in Mathematics, 78 (1983). doi: 10.1007/978-3-0346-0416-1. Google Scholar
[24]	Y. Wang and Y. Wang, Blow up criterion for three-dimensional nematic liquid crystal flows with partial viscosity,, Math. Meth. Appl. Sci. 36 (2013), 36 (2013), 60. doi: 10.1002/mma.2569. Google Scholar
[25]	Y. Wang, A logarithmically improved blow up criterion for three-dimensional nematic liquid crystal flows with partial viscosity,, Scienceasia, 39 (2013), 73. Google Scholar
[26]	Y. X. Wang, Blow-up criteria of smooth solutions to the three-dimensional magneto-micropolar fluid equations,, Boundary Value Problems, 2015 (2015). doi: 10.1186/s13661-015-0382-9. Google Scholar
[27]	H. Wen and S. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals,, Nonlinear Analysis: Real World Appl., 12 (2011), 1510. doi: 10.1016/j.nonrwa.2010.10.010. Google Scholar
[28]	Y. Zhang, Z. Tan and G. Wu, Blow up criterion for incompressible nematics liquid crystal flows,, preprint, (). Google Scholar
[29]	Z. Zhang, S. Liu, J. Pan and L. Ma, A refined blow up criterion for the nematics liquid crystals,, Int. J. Contemp. Math. Sciences, 9 (2014), 441. doi: 10.12988/ijcms.2014.4438. Google Scholar

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References:

[1]	G. Brown and W. Shaw, The Mesomorphic State, Liquid Crystals,, Chem. Rev., 57 (1957), 1049. doi: 10.1021/cr50018a002. Google Scholar
[2]	Q. Chen, Z. Tan and G. Wu, LPS's criterion for incompressible nematics liquid crystal flows,, Acta Mathematica Scientia, 34 (2014), 1072. doi: 10.1016/S0252-9602(14)60070-9. Google Scholar
[3]	J. Chemin, perfect Incompressible Fluids,, Oxford Lecture Ser. Math. Appl. 14, 14 (1998). Google Scholar
[4]	J. Ericksen, Hydrostatic theory of liquid crystal,, Arch. Rational Mech. Anal., 9 (1962), 371. doi: 10.1007/BF00253358. Google Scholar
[5]	J. Ericksen and D. Kinderlehrer (eds.), Theory and Applications of Liquid Crystals,, The IMA Volumes in Mathematics and its Applications, 5 (1987), 21. doi: 10.1007/978-1-4613-8743-5. Google Scholar
[6]	J. Ericksen, Equilibrium theory of liquid crystals,, Advances in Liquid Crystals, 2 (1976), 233. doi: 10.1016/B978-0-12-025002-8.50012-9. Google Scholar
[7]	F. Frank, On the theory of liquid crystals,, Discussions Faraday Soc., 25 (1958), 19. Google Scholar
[8]	P. G. de Gennes, The Physics of Liquid Crystals,, Oxford, (1974). Google Scholar
[9]	Y. Hao and X. Liu, The existence and blow-up criterion of liquid crystals system in critical Besov space,, Commun. Pure Appl. Anal., 13 (2014), 225. Google Scholar
[10]	R. Hardt, D. Kinderlehrer and F. Lin, Existence and partial regularity of static liquid crystal configurations,, Comm. Math. Phys., 105 (1986), 547. doi: 10.1007/BF01238933. Google Scholar
[11]	M. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in $\mathbbR^2$,, Cal. Var. Part. Differ. Equ., 40 (2011), 15. doi: 10.1007/s00526-010-0331-5. Google Scholar
[12]	T. Huang and C. Wang, Blow up criterion for nematic liquid crystal flows,, Comm. Part. Differ.Equ., 37 (2012), 875. doi: 10.1080/03605302.2012.659366. Google Scholar
[13]	T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 891. doi: 10.1002/cpa.3160410704. Google Scholar
[14]	H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations,, Math. Z., 242 (2002), 251. doi: 10.1007/s002090100332. Google Scholar
[15]	F. Leslie, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265. doi: 10.1007/BF00251810. Google Scholar
[16]	F. Leslie, Theory of flow phenomemum in liquid crystal,, In: Brown (ed.) Advances in Liquid Crystals, 4 (1979), 1. Google Scholar
[17]	F. Lin, Nonlinear theory of defects in nematic liquid crystal: Phase transition and flow phenomena,, Comm. Pure Appl. Math., 42 (1989), 789. doi: 10.1002/cpa.3160420605. Google Scholar
[18]	F. Lin and C. Liu, Nonparabolic Dissipative Systems Modeling the Flow of Liquid Crystals,, Comm. Pure Appl. Math., 48 (1995), 501. doi: 10.1002/cpa.3160480503. Google Scholar
[19]	F. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions,, Arch. Rational Mech. Anal., 197 (2010), 297. doi: 10.1007/s00205-009-0278-x. Google Scholar
[20]	F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals,, Chin. Ann. Math. Ser. B, 31 (2010), 921. doi: 10.1007/s11401-010-0612-5. Google Scholar
[21]	A. Majda and A. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press: Cambridge, (2002). Google Scholar
[22]	C. Oseen, Die anisotropen Fliissigkeiten, Tatsachen und Theorien,, Forts. Chemie, 21 (1929), 25. Google Scholar
[23]	H. Triebel, Theory of Function Spaces,, Monograph in Mathematics, 78 (1983). doi: 10.1007/978-3-0346-0416-1. Google Scholar
[24]	Y. Wang and Y. Wang, Blow up criterion for three-dimensional nematic liquid crystal flows with partial viscosity,, Math. Meth. Appl. Sci. 36 (2013), 36 (2013), 60. doi: 10.1002/mma.2569. Google Scholar
[25]	Y. Wang, A logarithmically improved blow up criterion for three-dimensional nematic liquid crystal flows with partial viscosity,, Scienceasia, 39 (2013), 73. Google Scholar
[26]	Y. X. Wang, Blow-up criteria of smooth solutions to the three-dimensional magneto-micropolar fluid equations,, Boundary Value Problems, 2015 (2015). doi: 10.1186/s13661-015-0382-9. Google Scholar
[27]	H. Wen and S. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals,, Nonlinear Analysis: Real World Appl., 12 (2011), 1510. doi: 10.1016/j.nonrwa.2010.10.010. Google Scholar
[28]	Y. Zhang, Z. Tan and G. Wu, Blow up criterion for incompressible nematics liquid crystal flows,, preprint, (). Google Scholar
[29]	Z. Zhang, S. Liu, J. Pan and L. Ma, A refined blow up criterion for the nematics liquid crystals,, Int. J. Contemp. Math. Sciences, 9 (2014), 441. doi: 10.12988/ijcms.2014.4438. Google Scholar

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