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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

Yinxia Wang 1,

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

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 and 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), 1049-1157. doi: 10.1021/cr50018a002.

[2]

Q. Chen, Z. Tan and G. Wu, LPS's criterion for incompressible nematics liquid crystal flows, Acta Mathematica Scientia, 34 (2014), 1072-1080. doi: 10.1016/S0252-9602(14)60070-9.

[3]

J. Chemin, perfect Incompressible Fluids, Oxford Lecture Ser. Math. Appl. 14, The Clarendon Press / Oxford Univ. Press, New York, 1998.

[4]

J. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378. doi: 10.1007/BF00253358.

[5]

J. Ericksen and D. Kinderlehrer (eds.), Theory and Applications of Liquid Crystals, The IMA Volumes in Mathematics and its Applications, 5, Springer-Verlag, New York, 1987, Papers from the IMA workshop held in Minneapolis, Minn., January 21-25, 1985. doi: 10.1007/978-1-4613-8743-5.

[6]

J. Ericksen, Equilibrium theory of liquid crystals, Advances in Liquid Crystals, 2 (1976), 233-298. doi: 10.1016/B978-0-12-025002-8.50012-9.

[7]

F. Frank, On the theory of liquid crystals, Discussions Faraday Soc., 25 (1958), 19-28.

[8]

P. G. de Gennes, The Physics of Liquid Crystals, Oxford, 1974.

[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), 225-236, arXiv:1305.1395v2.

[10]

R. Hardt, D. Kinderlehrer and F. Lin, Existence and partial regularity of static liquid crystal configurations, Comm. Math. Phys., 105 (1986), 547-570. doi: 10.1007/BF01238933.

[11]

M. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in $\mathbb{R}^2$, Cal. Var. Part. Differ. Equ., 40 (2011), 15-36. doi: 10.1007/s00526-010-0331-5.

[12]

T. Huang and C. Wang, Blow up criterion for nematic liquid crystal flows, Comm. Part. Differ.Equ., 37 (2012), 875-884. doi: 10.1080/03605302.2012.659366.

[13]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.

[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), 251-278. doi: 10.1007/s002090100332.

[15]

F. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810.

[16]

F. Leslie, Theory of flow phenomemum in liquid crystal, In: Brown (ed.) Advances in Liquid Crystals, Academic Press, New York, 4 (1979), 1-81.

[17]

F. Lin, Nonlinear theory of defects in nematic liquid crystal: Phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605.

[18]

F. 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.

[19]

F. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x.

[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), 921-938. doi: 10.1007/s11401-010-0612-5.

[21]

A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press: Cambridge, 2002.

[22]

C. Oseen, Die anisotropen Fliissigkeiten, Tatsachen und Theorien, Forts. Chemie, Phys. und Phys. Chemic, 21 (1929), 25-113.

[23]

H. Triebel, Theory of Function Spaces, Monograph in Mathematics, vol. 78. Birkhauser: Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[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), 60-68. doi: 10.1002/mma.2569.

[25]

Y. Wang, A logarithmically improved blow up criterion for three-dimensional nematic liquid crystal flows with partial viscosity, Scienceasia, 39 (2013), 73-78.

[26]

Y. X. Wang, Blow-up criteria of smooth solutions to the three-dimensional magneto-micropolar fluid equations, Boundary Value Problems, 2015 (2015), 10pp. doi: 10.1186/s13661-015-0382-9.

[27]

H. Wen and S. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Analysis: Real World Appl., 12 (2011), 1510-1531. doi: 10.1016/j.nonrwa.2010.10.010.

[28]

Y. Zhang, Z. Tan and G. Wu, Blow up criterion for incompressible nematics liquid crystal flows, preprint, arXiv:1210.4140v2.

[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), 441-446. doi: 10.12988/ijcms.2014.4438.

show all references

References:
[1]

G. Brown and W. Shaw, The Mesomorphic State, Liquid Crystals, Chem. Rev., 57 (1957), 1049-1157. doi: 10.1021/cr50018a002.

[2]

Q. Chen, Z. Tan and G. Wu, LPS's criterion for incompressible nematics liquid crystal flows, Acta Mathematica Scientia, 34 (2014), 1072-1080. doi: 10.1016/S0252-9602(14)60070-9.

[3]

J. Chemin, perfect Incompressible Fluids, Oxford Lecture Ser. Math. Appl. 14, The Clarendon Press / Oxford Univ. Press, New York, 1998.

[4]

J. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378. doi: 10.1007/BF00253358.

[5]

J. Ericksen and D. Kinderlehrer (eds.), Theory and Applications of Liquid Crystals, The IMA Volumes in Mathematics and its Applications, 5, Springer-Verlag, New York, 1987, Papers from the IMA workshop held in Minneapolis, Minn., January 21-25, 1985. doi: 10.1007/978-1-4613-8743-5.

[6]

J. Ericksen, Equilibrium theory of liquid crystals, Advances in Liquid Crystals, 2 (1976), 233-298. doi: 10.1016/B978-0-12-025002-8.50012-9.

[7]

F. Frank, On the theory of liquid crystals, Discussions Faraday Soc., 25 (1958), 19-28.

[8]

P. G. de Gennes, The Physics of Liquid Crystals, Oxford, 1974.

[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), 225-236, arXiv:1305.1395v2.

[10]

R. Hardt, D. Kinderlehrer and F. Lin, Existence and partial regularity of static liquid crystal configurations, Comm. Math. Phys., 105 (1986), 547-570. doi: 10.1007/BF01238933.

[11]

M. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in $\mathbb{R}^2$, Cal. Var. Part. Differ. Equ., 40 (2011), 15-36. doi: 10.1007/s00526-010-0331-5.

[12]

T. Huang and C. Wang, Blow up criterion for nematic liquid crystal flows, Comm. Part. Differ.Equ., 37 (2012), 875-884. doi: 10.1080/03605302.2012.659366.

[13]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.

[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), 251-278. doi: 10.1007/s002090100332.

[15]

F. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810.

[16]

F. Leslie, Theory of flow phenomemum in liquid crystal, In: Brown (ed.) Advances in Liquid Crystals, Academic Press, New York, 4 (1979), 1-81.

[17]

F. Lin, Nonlinear theory of defects in nematic liquid crystal: Phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605.

[18]

F. 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.

[19]

F. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x.

[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), 921-938. doi: 10.1007/s11401-010-0612-5.

[21]

A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press: Cambridge, 2002.

[22]

C. Oseen, Die anisotropen Fliissigkeiten, Tatsachen und Theorien, Forts. Chemie, Phys. und Phys. Chemic, 21 (1929), 25-113.

[23]

H. Triebel, Theory of Function Spaces, Monograph in Mathematics, vol. 78. Birkhauser: Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[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), 60-68. doi: 10.1002/mma.2569.

[25]

Y. Wang, A logarithmically improved blow up criterion for three-dimensional nematic liquid crystal flows with partial viscosity, Scienceasia, 39 (2013), 73-78.

[26]

Y. X. Wang, Blow-up criteria of smooth solutions to the three-dimensional magneto-micropolar fluid equations, Boundary Value Problems, 2015 (2015), 10pp. doi: 10.1186/s13661-015-0382-9.

[27]

H. Wen and S. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Analysis: Real World Appl., 12 (2011), 1510-1531. doi: 10.1016/j.nonrwa.2010.10.010.

[28]

Y. Zhang, Z. Tan and G. Wu, Blow up criterion for incompressible nematics liquid crystal flows, preprint, arXiv:1210.4140v2.

[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), 441-446. doi: 10.12988/ijcms.2014.4438.

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