American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in $R^N$
  • CPAA Home
  • This Issue
  • Next Article
    On the trace regularity results of Musielak-Orlicz-Sobolev spaces in a bounded domain
September  2016, 15(5): 1661-1669. doi: 10.3934/cpaa.2016007

Exponential stability for the compressible nematic liquid crystal flow with large initial data

Qiang Tao 1, and  Ying Yang 1,

1. 

School of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China, China

Received  September 2015 Revised  April 2016 Published  July 2016

In this paper, we consider the asymptotic behavior of the global spherically or cylindrically symmetric solution for the compressible nematic liquid crystal flow in multi-dimension with large initial data. Using the uniform point-wise positive lower and upper bounds of the density, we obtain the exponential stability of the global strong solution.
Keywords: global strong solution, exponential stability., Nematic liquid crystal flow.
Mathematics Subject Classification: Primary: 35B40, 35Q3.
Citation: Qiang Tao, Ying Yang. Exponential stability for the compressible nematic liquid crystal flow with large initial data. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1661-1669. doi: 10.3934/cpaa.2016007
References:
[1]

H. J. Choe and H. Kim, Global existence of the radially symmetric solutions of the Navier-Stokes equations for the isentropic compressible fuids,, \emph{Math. Meth. Appl. Sci.}, 28 (2005), 1.  doi: 10.1002/mma.545.  Google Scholar

[2]

P. G. de Gennes, The Physics of Liquid Crystals,, Oxford, (1974).   Google Scholar

[3]

S. J. Ding, J. Y. Lin, C. Y. Wang and H. Y. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D,, \emph{Discrete Contin. Dyn. Syst.}, 32 (2012), 539.  doi: 10.3934/dcds.2012.32.539.  Google Scholar

[4]

S. J. Ding, C. Y. Wang and H. Y. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 15 (2011), 357.  doi: 10.3934/dcdsb.2011.15.357.  Google Scholar

[5]

J. L. Ericksen, Hydrostatic theory of liquid crystal,, \emph{Arch. Rational Mech. Anal.}, 9 (1962), 371.   Google Scholar

[6]

J. C. Gao, Q. Tao and Z. A. Yao, A blowup criterion for the compressible nematic liquid crystal flows in dimension two,, \emph{J. Math. Anal. Appl.}, 415 (2014), 33.  doi: 10.1016/j.jmaa.2014.01.039.  Google Scholar

[7]

B. L. Guo and Y. Q. Han, Global regular solutions for Landau-Lifshitz equation,, \emph{Front. Math. China}, 4 (2006), 538.  doi: 10.1007/s11464-006-0027-5.  Google Scholar

[8]

R. Hardt, D. Kinderlehrer and F. Lin, Existence and partial regularity of static liquid crystal configurations,, \emph{Commun. Math. Phys.}, 105 (1986), 547.   Google Scholar

[9]

X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals,, \emph{SIAM J. Math. Anal.}, 45 (2013), 2678.  doi: 10.1137/120898814.  Google Scholar

[10]

J. R. Huang and S. J. Ding, Spherically symmetric solutions to compressible hydrodynamic flow of liquid crystals in N dimensions,, \emph{Chin. Ann. Math.}, 33B (2012), 453.   Google Scholar

[11]

T. Huang, C. Y. Wang and H. Y. Wen, Strong solutions of the compressible nematic liquid crystal,, \emph{J. Differential Equations}, 252 (2012), 2222.  doi: 10.1016/j.jde.2011.07.036.  Google Scholar

[12]

T. Huang, C. Y. Wang and H. Y. Wen, Blow up criterion for compressible nematic liquid crystal flows in dimension three,, \emph{Arch. Rational Mech. Anal.}, 204 (2012), 285.  doi: 10.1007/s00205-011-0476-1.  Google Scholar

[13]

X. D. Huang and Y. Wang, A Serrin criterion for compressible nematic liquid crystal flows,, \emph{Math. Meth. Appl. Sci.}, 36 (2013), 1363.  doi: 10.1002/mma.2689.  Google Scholar

[14]

F. Jiang, S. Jiang and D. H. Wang, On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain,, \emph{J. Funct. Anal.}, 265 (2013), 3369.  doi: 10.1016/j.jfa.2013.07.026.  Google Scholar

[15]

F. M. Leslie, Some constitutive equations for liquid crystals,, \emph{Arch. Rational Mech. Anal.}, 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar

[16]

F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena,, \emph{Commun. Pure Appl. Math.}, 42 (1989), 789.  doi: 10.1002/cpa.3160420605.  Google Scholar

[17]

F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, \emph{Commun. Pure Appl. Math.}, 48 (1995), 501.  doi: 10.1002/cpa.3160480503.  Google Scholar

[18]

F. H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid cyrstals,, \emph{Discrete Contin. Dyn. Syst.}, 2 (1996), 1.   Google Scholar

[19]

F. H. Lin and C. Y. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals,, arXiv:1408.4138., ().   Google Scholar

[20]

J. Y. Lin, B. S. Lai and C. Y. Wang, Global finite energy weak solutions to the compressible nematic liquid crystal flow in dimension three,, \emph{SIAM J. Math. Anal.}, 47 (2015), 2952.  doi: 10.1137/15M1007665.  Google Scholar

[21]

Y. M. Qin and L. Huang, Global existence and regularity of a 1D liquid crystal system,, \emph{Nonlinear Anal. Real World Appl.}, 15 (2014), 172.  doi: 10.1016/j.nonrwa.2013.07.003.  Google Scholar

[22]

Q. Tao, J. C. Gao and Z. A. Yao, Global strong solutions of the compressible nematic liquid crystal flow with the cylinder symmetry,, \emph{Commun. Math. Sci.}, 13 (2015), 2065.  doi: 10.4310/CMS.2015.v13.n8.a5.  Google Scholar

show all references

References:
[1]

H. J. Choe and H. Kim, Global existence of the radially symmetric solutions of the Navier-Stokes equations for the isentropic compressible fuids,, \emph{Math. Meth. Appl. Sci.}, 28 (2005), 1.  doi: 10.1002/mma.545.  Google Scholar

[2]

P. G. de Gennes, The Physics of Liquid Crystals,, Oxford, (1974).   Google Scholar

[3]

S. J. Ding, J. Y. Lin, C. Y. Wang and H. Y. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D,, \emph{Discrete Contin. Dyn. Syst.}, 32 (2012), 539.  doi: 10.3934/dcds.2012.32.539.  Google Scholar

[4]

S. J. Ding, C. Y. Wang and H. Y. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 15 (2011), 357.  doi: 10.3934/dcdsb.2011.15.357.  Google Scholar

[5]

J. L. Ericksen, Hydrostatic theory of liquid crystal,, \emph{Arch. Rational Mech. Anal.}, 9 (1962), 371.   Google Scholar

[6]

J. C. Gao, Q. Tao and Z. A. Yao, A blowup criterion for the compressible nematic liquid crystal flows in dimension two,, \emph{J. Math. Anal. Appl.}, 415 (2014), 33.  doi: 10.1016/j.jmaa.2014.01.039.  Google Scholar

[7]

B. L. Guo and Y. Q. Han, Global regular solutions for Landau-Lifshitz equation,, \emph{Front. Math. China}, 4 (2006), 538.  doi: 10.1007/s11464-006-0027-5.  Google Scholar

[8]

R. Hardt, D. Kinderlehrer and F. Lin, Existence and partial regularity of static liquid crystal configurations,, \emph{Commun. Math. Phys.}, 105 (1986), 547.   Google Scholar

[9]

X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals,, \emph{SIAM J. Math. Anal.}, 45 (2013), 2678.  doi: 10.1137/120898814.  Google Scholar

[10]

J. R. Huang and S. J. Ding, Spherically symmetric solutions to compressible hydrodynamic flow of liquid crystals in N dimensions,, \emph{Chin. Ann. Math.}, 33B (2012), 453.   Google Scholar

[11]

T. Huang, C. Y. Wang and H. Y. Wen, Strong solutions of the compressible nematic liquid crystal,, \emph{J. Differential Equations}, 252 (2012), 2222.  doi: 10.1016/j.jde.2011.07.036.  Google Scholar

[12]

T. Huang, C. Y. Wang and H. Y. Wen, Blow up criterion for compressible nematic liquid crystal flows in dimension three,, \emph{Arch. Rational Mech. Anal.}, 204 (2012), 285.  doi: 10.1007/s00205-011-0476-1.  Google Scholar

[13]

X. D. Huang and Y. Wang, A Serrin criterion for compressible nematic liquid crystal flows,, \emph{Math. Meth. Appl. Sci.}, 36 (2013), 1363.  doi: 10.1002/mma.2689.  Google Scholar

[14]

F. Jiang, S. Jiang and D. H. Wang, On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain,, \emph{J. Funct. Anal.}, 265 (2013), 3369.  doi: 10.1016/j.jfa.2013.07.026.  Google Scholar

[15]

F. M. Leslie, Some constitutive equations for liquid crystals,, \emph{Arch. Rational Mech. Anal.}, 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar

[16]

F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena,, \emph{Commun. Pure Appl. Math.}, 42 (1989), 789.  doi: 10.1002/cpa.3160420605.  Google Scholar

[17]

F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, \emph{Commun. Pure Appl. Math.}, 48 (1995), 501.  doi: 10.1002/cpa.3160480503.  Google Scholar

[18]

F. H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid cyrstals,, \emph{Discrete Contin. Dyn. Syst.}, 2 (1996), 1.   Google Scholar

[19]

F. H. Lin and C. Y. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals,, arXiv:1408.4138., ().   Google Scholar

[20]

J. Y. Lin, B. S. Lai and C. Y. Wang, Global finite energy weak solutions to the compressible nematic liquid crystal flow in dimension three,, \emph{SIAM J. Math. Anal.}, 47 (2015), 2952.  doi: 10.1137/15M1007665.  Google Scholar

[21]

Y. M. Qin and L. Huang, Global existence and regularity of a 1D liquid crystal system,, \emph{Nonlinear Anal. Real World Appl.}, 15 (2014), 172.  doi: 10.1016/j.nonrwa.2013.07.003.  Google Scholar

[22]

Q. Tao, J. C. Gao and Z. A. Yao, Global strong solutions of the compressible nematic liquid crystal flow with the cylinder symmetry,, \emph{Commun. Math. Sci.}, 13 (2015), 2065.  doi: 10.4310/CMS.2015.v13.n8.a5.  Google Scholar

[1]

Jihong Zhao, Qiao Liu, Shangbin Cui. Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows. Communications on Pure & Applied Analysis, 2013, 12 (1) : 341-357. doi: 10.3934/cpaa.2013.12.341
[2]

Bagisa Mukherjee, Chun Liu. On the stability of two nematic liquid crystal configurations. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 561-574. doi: 10.3934/dcdsb.2002.2.561
[3]

Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
[4]

Yang Liu, Sining Zheng, Huapeng Li, Shengquan Liu. Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3921-3938. doi: 10.3934/dcds.2017165
[5]

M. Gregory Forest, Hongyun Wang, Hong Zhou. Sheared nematic liquid crystal polymer monolayers. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 497-517. doi: 10.3934/dcdsb.2009.11.497
[6]

Zhiyuan Geng, Wei Wang, Pingwen Zhang, Zhifei Zhang. Stability of half-degree point defect profiles for 2-D nematic liquid crystal. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6227-6242. doi: 10.3934/dcds.2017269
[7]

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 & Continuous Dynamical Systems - A, 2012, 32 (12) : 4229-4246. doi: 10.3934/dcds.2012.32.4229
[8]

Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455
[9]

Blanca Climent-Ezquerra, Francisco Guillén-González. Global in time solution and time-periodicity for a smectic-A liquid crystal model. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1473-1493. doi: 10.3934/cpaa.2010.9.1473
[10]

Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157
[11]

T. Tachim Medjo. On the existence and uniqueness of solution to a stochastic simplified liquid crystal model. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2243-2264. doi: 10.3934/cpaa.2019101
[12]

Qiao Liu, Ting Zhang, Jihong Zhao. Well-posedness for the 3D incompressible nematic liquid crystal system in the critical $L^p$ framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 371-402. doi: 10.3934/dcds.2016.36.371
[13]

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
[14]

Hao Wu. Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 379-396. doi: 10.3934/dcds.2010.26.379
[15]

Patricia Bauman, Daniel Phillips. Analysis and stability of bent-core liquid crystal fibers. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1707-1728. doi: 10.3934/dcdsb.2012.17.1707
[16]

Stefano Bosia. Well-posedness and long term behavior of a simplified Ericksen-Leslie non-autonomous system for nematic liquid crystal flows. Communications on Pure & Applied Analysis, 2012, 11 (2) : 407-441. doi: 10.3934/cpaa.2012.11.407
[17]

Qiumei Huang, Xiaofeng Yang, Xiaoming He. Numerical approximations for a smectic-A liquid crystal flow model: First-order, linear, decoupled and energy stable schemes. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2177-2192. doi: 10.3934/dcdsb.2018230
[18]

Shijin Ding, Changyou Wang, Huanyao Wen. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 357-371. doi: 10.3934/dcdsb.2011.15.357
[19]

Shengquan Liu, Jianwen Zhang. Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2631-2648. doi: 10.3934/dcdsb.2016065
[20]

Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Isotropic-nematic phase transitions in liquid crystals. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 565-579. doi: 10.3934/dcdss.2011.4.565

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences