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

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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

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

Received September 2015 Revised April 2016 Published July 2016

Abstract
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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

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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

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