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Continuous maximal regularity on singular manifolds and its applications
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 |
References:
[1] |
G. Brown and W. Shaw, The Mesomorphic State, Liquid Crystals,, Chem. Rev., 57 (1957), 1049.
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.
doi: 10.1016/S0252-9602(14)60070-9. |
[3] |
J. Chemin, perfect Incompressible Fluids,, Oxford Lecture Ser. Math. Appl. 14, 14 (1998).
|
[4] |
J. Ericksen, Hydrostatic theory of liquid crystal,, Arch. Rational Mech. Anal., 9 (1962), 371.
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 (1987), 21.
doi: 10.1007/978-1-4613-8743-5. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
F. Leslie, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265.
doi: 10.1007/BF00251810. |
[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. |
[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. |
[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. |
[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. |
[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, 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. |
[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. |
[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. |
[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. |
[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. |
show all references
References:
[1] |
G. Brown and W. Shaw, The Mesomorphic State, Liquid Crystals,, Chem. Rev., 57 (1957), 1049.
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.
doi: 10.1016/S0252-9602(14)60070-9. |
[3] |
J. Chemin, perfect Incompressible Fluids,, Oxford Lecture Ser. Math. Appl. 14, 14 (1998).
|
[4] |
J. Ericksen, Hydrostatic theory of liquid crystal,, Arch. Rational Mech. Anal., 9 (1962), 371.
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 (1987), 21.
doi: 10.1007/978-1-4613-8743-5. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
F. Leslie, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265.
doi: 10.1007/BF00251810. |
[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. |
[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. |
[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. |
[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. |
[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, 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. |
[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. |
[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. |
[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. |
[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. |
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