- Previous Article
- EECT Home
- This Issue
-
Next Article
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), 1049-1157.
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 $\mathbbR^2$, 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, ().
|
[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 $\mathbbR^2$, 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, ().
|
[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. |
[1] |
Sili Liu, Xinhua Zhao, Yingshan Chen. A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4515-4533. doi: 10.3934/dcdsb.2020110 |
[2] |
Yang Liu, Sining Zheng, Huapeng Li, Shengquan Liu. Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3921-3938. doi: 10.3934/dcds.2017165 |
[3] |
Jihong Zhao, Qiao Liu, Shangbin Cui. Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows. Communications on Pure and Applied Analysis, 2013, 12 (1) : 341-357. doi: 10.3934/cpaa.2013.12.341 |
[4] |
Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455 |
[5] |
Yu-Zhu Wang, Weibing Zuo. On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1327-1336. doi: 10.3934/cpaa.2014.13.1327 |
[6] |
Yan Jia, Xingwei Zhang, Bo-Qing Dong. Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion. Communications on Pure and Applied Analysis, 2013, 12 (2) : 923-937. doi: 10.3934/cpaa.2013.12.923 |
[7] |
Yang Liu, Xin Zhong. On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5219-5238. doi: 10.3934/cpaa.2020234 |
[8] |
Francisco Guillén-González, Mouhamadou Samsidy Goudiaby. Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4229-4246. doi: 10.3934/dcds.2012.32.4229 |
[9] |
Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161 |
[10] |
Hao Wu. Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 379-396. doi: 10.3934/dcds.2010.26.379 |
[11] |
Bagisa Mukherjee, Chun Liu. On the stability of two nematic liquid crystal configurations. Discrete and Continuous Dynamical Systems - B, 2002, 2 (4) : 561-574. doi: 10.3934/dcdsb.2002.2.561 |
[12] |
M. Gregory Forest, Hongyun Wang, Hong Zhou. Sheared nematic liquid crystal polymer monolayers. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 497-517. doi: 10.3934/dcdsb.2009.11.497 |
[13] |
Shanshan Guo, Zhong Tan. Energy dissipation for weak solutions of incompressible liquid crystal flows. Kinetic and Related Models, 2015, 8 (4) : 691-706. doi: 10.3934/krm.2015.8.691 |
[14] |
Stefano Bosia. Well-posedness and long term behavior of a simplified Ericksen-Leslie non-autonomous system for nematic liquid crystal flows. Communications on Pure and Applied Analysis, 2012, 11 (2) : 407-441. doi: 10.3934/cpaa.2012.11.407 |
[15] |
Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure and Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225 |
[16] |
Qiang Tao, Ying Yang. Exponential stability for the compressible nematic liquid crystal flow with large initial data. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1661-1669. doi: 10.3934/cpaa.2016007 |
[17] |
Junyu Lin. Uniqueness of harmonic map heat flows and liquid crystal flows. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 739-755. doi: 10.3934/dcds.2013.33.739 |
[18] |
Boling Guo, Yongqian Han, Guoli Zhou. Random attractor for the 2D stochastic nematic liquid crystals flows. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2349-2376. doi: 10.3934/cpaa.2019106 |
[19] |
Xin Zhong. A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4603-4615. doi: 10.3934/dcdsb.2020115 |
[20] |
Zhenlu Cui, M. Carme Calderer, Qi Wang. Mesoscale structures in flows of weakly sheared cholesteric liquid crystal polymers. Discrete and Continuous Dynamical Systems - B, 2006, 6 (2) : 291-310. doi: 10.3934/dcdsb.2006.6.291 |
2020 Impact Factor: 1.081
Tools
Metrics
Other articles
by authors
[Back to Top]