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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 and 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, Math. Meth. Appl. Sci., 28 (2005), 1-28. doi: 10.1002/mma.545.

[2]

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

[3]

S. J. Ding, J. Y. Lin, C. Y. Wang and H. Y. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D, Discrete Contin. Dyn. Syst., 32 (2012), 539-563. doi: 10.3934/dcds.2012.32.539.

[4]

S. J. Ding, C. Y. Wang and H. Y. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 357-371. doi: 10.3934/dcdsb.2011.15.357.

[5]

J. L. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378.

[6]

J. C. Gao, Q. Tao and Z. A. Yao, A blowup criterion for the compressible nematic liquid crystal flows in dimension two, J. Math. Anal. Appl., 415 (2014), 33-52. doi: 10.1016/j.jmaa.2014.01.039.

[7]

B. L. Guo and Y. Q. Han, Global regular solutions for Landau-Lifshitz equation, Front. Math. China, 4 (2006), 538-568. doi: 10.1007/s11464-006-0027-5.

[8]

R. Hardt, D. Kinderlehrer and F. Lin, Existence and partial regularity of static liquid crystal configurations, Commun. Math. Phys., 105 (1986), 547-570.

[9]

X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals, SIAM J. Math. Anal., 45 (2013), 2678-2699. doi: 10.1137/120898814.

[10]

J. R. Huang and S. J. Ding, Spherically symmetric solutions to compressible hydrodynamic flow of liquid crystals in N dimensions, Chin. Ann. Math., 33B (2012), 453-0478.

[11]

T. Huang, C. Y. Wang and H. Y. Wen, Strong solutions of the compressible nematic liquid crystal, J. Differential Equations, 252 (2012), 2222-2265. doi: 10.1016/j.jde.2011.07.036.

[12]

T. Huang, C. Y. Wang and H. Y. Wen, Blow up criterion for compressible nematic liquid crystal flows in dimension three, Arch. Rational Mech. Anal., 204 (2012), 285-311. doi: 10.1007/s00205-011-0476-1.

[13]

X. D. Huang and Y. Wang, A Serrin criterion for compressible nematic liquid crystal flows, Math. Meth. Appl. Sci., 36 (2013), 1363-1375. doi: 10.1002/mma.2689.

[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, J. Funct. Anal., 265 (2013), 3369-3397. doi: 10.1016/j.jfa.2013.07.026.

[15]

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

[16]

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

[17]

F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Commun. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503.

[18]

F. H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid cyrstals, Discrete Contin. Dyn. Syst., 2 (1996) 1-22.

[19]

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

[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, SIAM J. Math. Anal., 47 (2015), 2952-2983. doi: 10.1137/15M1007665.

[21]

Y. M. Qin and L. Huang, Global existence and regularity of a 1D liquid crystal system, Nonlinear Anal. Real World Appl., 15 (2014), 172-186. doi: 10.1016/j.nonrwa.2013.07.003.

[22]

Q. Tao, J. C. Gao and Z. A. Yao, Global strong solutions of the compressible nematic liquid crystal flow with the cylinder symmetry, Commun. Math. Sci., 13 (2015), 2065-2096. doi: 10.4310/CMS.2015.v13.n8.a5.

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, Math. Meth. Appl. Sci., 28 (2005), 1-28. doi: 10.1002/mma.545.

[2]

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

[3]

S. J. Ding, J. Y. Lin, C. Y. Wang and H. Y. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D, Discrete Contin. Dyn. Syst., 32 (2012), 539-563. doi: 10.3934/dcds.2012.32.539.

[4]

S. J. Ding, C. Y. Wang and H. Y. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 357-371. doi: 10.3934/dcdsb.2011.15.357.

[5]

J. L. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378.

[6]

J. C. Gao, Q. Tao and Z. A. Yao, A blowup criterion for the compressible nematic liquid crystal flows in dimension two, J. Math. Anal. Appl., 415 (2014), 33-52. doi: 10.1016/j.jmaa.2014.01.039.

[7]

B. L. Guo and Y. Q. Han, Global regular solutions for Landau-Lifshitz equation, Front. Math. China, 4 (2006), 538-568. doi: 10.1007/s11464-006-0027-5.

[8]

R. Hardt, D. Kinderlehrer and F. Lin, Existence and partial regularity of static liquid crystal configurations, Commun. Math. Phys., 105 (1986), 547-570.

[9]

X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals, SIAM J. Math. Anal., 45 (2013), 2678-2699. doi: 10.1137/120898814.

[10]

J. R. Huang and S. J. Ding, Spherically symmetric solutions to compressible hydrodynamic flow of liquid crystals in N dimensions, Chin. Ann. Math., 33B (2012), 453-0478.

[11]

T. Huang, C. Y. Wang and H. Y. Wen, Strong solutions of the compressible nematic liquid crystal, J. Differential Equations, 252 (2012), 2222-2265. doi: 10.1016/j.jde.2011.07.036.

[12]

T. Huang, C. Y. Wang and H. Y. Wen, Blow up criterion for compressible nematic liquid crystal flows in dimension three, Arch. Rational Mech. Anal., 204 (2012), 285-311. doi: 10.1007/s00205-011-0476-1.

[13]

X. D. Huang and Y. Wang, A Serrin criterion for compressible nematic liquid crystal flows, Math. Meth. Appl. Sci., 36 (2013), 1363-1375. doi: 10.1002/mma.2689.

[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, J. Funct. Anal., 265 (2013), 3369-3397. doi: 10.1016/j.jfa.2013.07.026.

[15]

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

[16]

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

[17]

F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Commun. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503.

[18]

F. H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid cyrstals, Discrete Contin. Dyn. Syst., 2 (1996) 1-22.

[19]

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

[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, SIAM J. Math. Anal., 47 (2015), 2952-2983. doi: 10.1137/15M1007665.

[21]

Y. M. Qin and L. Huang, Global existence and regularity of a 1D liquid crystal system, Nonlinear Anal. Real World Appl., 15 (2014), 172-186. doi: 10.1016/j.nonrwa.2013.07.003.

[22]

Q. Tao, J. C. Gao and Z. A. Yao, Global strong solutions of the compressible nematic liquid crystal flow with the cylinder symmetry, Commun. Math. Sci., 13 (2015), 2065-2096. doi: 10.4310/CMS.2015.v13.n8.a5.

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