January  2016, 15(1): 73-90. doi: 10.3934/cpaa.2016.15.73

Large-time behavior of liquid crystal flows with a trigonometric condition in two dimensions

1. 

Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 210037

2. 

College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 361000

Received  January 2015 Revised  August 2015 Published  December 2015

In this paper, we study the large-time behavior of weak solutions to the initial-boundary problem arising in a simplified Ericksen-Leslie system for nonhomogeneous incompressible flows of nematic liquid crystals with a transformation condition of trigonometric functions (called by trigonometric condition for simplicity) posed on the initial direction field in a bounded domain $\Omega\subset \mathbb{R}^2$. We show that the kinetic energy and direction field converge to zero and an equilibrium state, respectively, as time goes to infinity. Further, if the initial density is away from vacuum and bounded, then the density, and velocity and direction fields exponential decay to an equilibrium state. In addition, we also show that the weak solutions of the corresponding compressible flows converge {an equilibrium} state.
Citation: Jishan Fan, Fei Jiang. Large-time behavior of liquid crystal flows with a trigonometric condition in two dimensions. Communications on Pure and Applied Analysis, 2016, 15 (1) : 73-90. doi: 10.3934/cpaa.2016.15.73
References:
[1]

M. A. Abdallah, F. Jiang and Z. Tan, Decay estimates for isentropic compressible magnetohydrodynamic equations in bounded domain, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 2211-2220. doi: 10.1016/S0252-9602(12)60171-4.

[2]

S. J. Ding, J. R. Huang, F. G. Xia, H. Y. Wen and R. Z. Zi, Incompressible limit of the compressible nematic liquid crystal flow, J. Funct. Anal., 264 (2013), 1711-1756. doi: 10.1016/j.jfa.2013.01.011.

[3]

E. Feireisl and H. Petzeltová, Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow, Arch. Ration. Mech. Anal., 150 (1999), 77-96. doi: 10.1007/s002050050181.

[4]

M. Grasselli and H. Wu, Long-time behavior for a hydrodynamic model on nematic liquid crystal flows with asymptotic stabilizing boundary condition and external force, SIAM J. Math. Anal., 45 (2013), 965-1002. doi: 10.1137/120866476.

[5]

J. L. Hineman and C. Y. Wang, Well-posedness of Nematic liquid crystal flow in $L_{u l o c}^3(R^3)$, Arch. Ration. Mech. Anal., 210 (2013), 177-218. doi: 10.1007/s00205-013-0643-7.

[6]

M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36. doi: 10.1007/s00526-010-0331-5.

[7]

X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals, SIAM J. Math. Analysis, 252 (2013), 2678-2699. doi: 10.1137/120898814.

[8]

X. P. Hu and H. Wu, Long-time dynamics of the nonhomogeneous incompressible flow of nematic liquid crystals, Commun. Math. Sci., 11 (2013), 779-806. doi: 10.4310/CMS.2013.v11.n3.a6.

[9]

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

[10]

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

[11]

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.

[12]

F. Jiang, S. Jiang, and D. H. Wang, Global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions, Arch. Ration. Mech. Anal., 214 (2014), 403-451. doi: 10.1007/s00205-014-0768-3.

[13]

F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system, Math. Methods Appl. Sci., 32 (2009), 2243-2266. doi: 10.1002/mma.1132.

[14]

F. Jiang and Z. Tan, On the domain dependence of solutions to the Navier-Stokes equations of a two-dimensional compressible flow, Math. Methods Appl. Sci., 32 (2009), 2350-2367. doi: 10.1002/mma.1138.

[15]

J. Li, Z. H. Xu and J. W. Zhang, Global well-posedness with large oscillations and vacuum to the three-dimensional equations of compressible nematic liquid crystal flows, preprint, arXiv:1204.4966.

[16]

J. K. Li, Global strong and weak solutions to inhomogeneous nematic liquid crystal flow in two dimensions, Nonlinear Anal., 99 (2014), 80-94. doi: 10.1016/j.na.2013.12.023.

[17]

X. L. Li and D. H. Wang, Global solution to the incompressible flow of liquid crystals, J. Differential Equations, 252 (2012), 745-767. doi: 10.1016/j.jde.2011.08.045.

[18]

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

[19]

F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x.

[20]

F. H. 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.

[21]

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

[22]

F. H. Lin and C. Y. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimensions three, preprint, arXiv:1408.4146.

[23]

F. H. Lin and C. Y. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130361, 18 pp.

[24]

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.

[25]

P. Lions, Mathematical Topics in Fluid Mechanics: Incompressible models, Oxford University Press, New York, 1996.

[26]

Q. Liu, On the temporal decay of solutions to the two-dimensional nematic liquid crystal flows, preprint, arXiv:1409.8475.

[27]

X. G. Liu and Z. Y. Zhang, Existence of the flow of liquid crystals system, Chinese Ann. Math. Ser. A, 30 (2009), 1-20.

[28]

D. G. Matteis and G. E. Virga, Director libration in nematoacoustics, Physical Review E, 83 (2011), 011703. doi: 10.1103/PhysRevE.83.011703.

[29]

A. Novotnỳ and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, Oxford, 2004.

[30]

M. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88 (1985), 209-222. doi: 10.1007/BF00752111.

[31]

L. Simon, Asymptotics for a class of nonlinear evolution equation, with applications to geometri problems, Ann. of Math.(2), 118 (1983), 525-571. doi: 10.2307/2006981.

[32]

C. Y. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Ration. Mech. Anal., 200 (2011), 1-19. doi: 10.1007/s00205-010-0343-5.

[33]

C. Y. Wang and X. Xu, On the rigidity of nematic liquid crystal flow on $\mathbbS^2$, J. Funct. Anal., 266 (2014), 5360-5376. doi: 10.1016/j.jfa.2014.02.023.

[34]

D. H. Wang and C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals, Arch. Ration. Mech. Anal., 204 (2012), 881-915. doi: 10.1007/s00205-011-0488-x.

[35]

H. Y. Wen and S. J. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. Real World Appl., 12 (2011), 1510-1531. doi: 10.1016/j.nonrwa.2010.10.010.

[36]

H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 26 (2010), 379-396. doi: 10.3934/dcds.2010.26.379.

[37]

H. Wu, X. Xu and C. Liu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties, Calc. Var. Partial Differential Equations, 45 (2012), 319-345. doi: 10.1007/s00526-011-0460-5.

[38]

Y. Zhou, J. S. Fan and G. Nakamura, Global strong solution to the density-dependent 2-D liquid crystal flows, Abstr. Appl. Anal., Art. ID 947291 (2013), 5pp.

show all references

References:
[1]

M. A. Abdallah, F. Jiang and Z. Tan, Decay estimates for isentropic compressible magnetohydrodynamic equations in bounded domain, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 2211-2220. doi: 10.1016/S0252-9602(12)60171-4.

[2]

S. J. Ding, J. R. Huang, F. G. Xia, H. Y. Wen and R. Z. Zi, Incompressible limit of the compressible nematic liquid crystal flow, J. Funct. Anal., 264 (2013), 1711-1756. doi: 10.1016/j.jfa.2013.01.011.

[3]

E. Feireisl and H. Petzeltová, Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow, Arch. Ration. Mech. Anal., 150 (1999), 77-96. doi: 10.1007/s002050050181.

[4]

M. Grasselli and H. Wu, Long-time behavior for a hydrodynamic model on nematic liquid crystal flows with asymptotic stabilizing boundary condition and external force, SIAM J. Math. Anal., 45 (2013), 965-1002. doi: 10.1137/120866476.

[5]

J. L. Hineman and C. Y. Wang, Well-posedness of Nematic liquid crystal flow in $L_{u l o c}^3(R^3)$, Arch. Ration. Mech. Anal., 210 (2013), 177-218. doi: 10.1007/s00205-013-0643-7.

[6]

M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36. doi: 10.1007/s00526-010-0331-5.

[7]

X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crystals, SIAM J. Math. Analysis, 252 (2013), 2678-2699. doi: 10.1137/120898814.

[8]

X. P. Hu and H. Wu, Long-time dynamics of the nonhomogeneous incompressible flow of nematic liquid crystals, Commun. Math. Sci., 11 (2013), 779-806. doi: 10.4310/CMS.2013.v11.n3.a6.

[9]

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

[10]

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

[11]

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.

[12]

F. Jiang, S. Jiang, and D. H. Wang, Global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions, Arch. Ration. Mech. Anal., 214 (2014), 403-451. doi: 10.1007/s00205-014-0768-3.

[13]

F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system, Math. Methods Appl. Sci., 32 (2009), 2243-2266. doi: 10.1002/mma.1132.

[14]

F. Jiang and Z. Tan, On the domain dependence of solutions to the Navier-Stokes equations of a two-dimensional compressible flow, Math. Methods Appl. Sci., 32 (2009), 2350-2367. doi: 10.1002/mma.1138.

[15]

J. Li, Z. H. Xu and J. W. Zhang, Global well-posedness with large oscillations and vacuum to the three-dimensional equations of compressible nematic liquid crystal flows, preprint, arXiv:1204.4966.

[16]

J. K. Li, Global strong and weak solutions to inhomogeneous nematic liquid crystal flow in two dimensions, Nonlinear Anal., 99 (2014), 80-94. doi: 10.1016/j.na.2013.12.023.

[17]

X. L. Li and D. H. Wang, Global solution to the incompressible flow of liquid crystals, J. Differential Equations, 252 (2012), 745-767. doi: 10.1016/j.jde.2011.08.045.

[18]

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

[19]

F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x.

[20]

F. H. 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.

[21]

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

[22]

F. H. Lin and C. Y. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimensions three, preprint, arXiv:1408.4146.

[23]

F. H. Lin and C. Y. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130361, 18 pp.

[24]

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.

[25]

P. Lions, Mathematical Topics in Fluid Mechanics: Incompressible models, Oxford University Press, New York, 1996.

[26]

Q. Liu, On the temporal decay of solutions to the two-dimensional nematic liquid crystal flows, preprint, arXiv:1409.8475.

[27]

X. G. Liu and Z. Y. Zhang, Existence of the flow of liquid crystals system, Chinese Ann. Math. Ser. A, 30 (2009), 1-20.

[28]

D. G. Matteis and G. E. Virga, Director libration in nematoacoustics, Physical Review E, 83 (2011), 011703. doi: 10.1103/PhysRevE.83.011703.

[29]

A. Novotnỳ and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, Oxford, 2004.

[30]

M. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 88 (1985), 209-222. doi: 10.1007/BF00752111.

[31]

L. Simon, Asymptotics for a class of nonlinear evolution equation, with applications to geometri problems, Ann. of Math.(2), 118 (1983), 525-571. doi: 10.2307/2006981.

[32]

C. Y. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Ration. Mech. Anal., 200 (2011), 1-19. doi: 10.1007/s00205-010-0343-5.

[33]

C. Y. Wang and X. Xu, On the rigidity of nematic liquid crystal flow on $\mathbbS^2$, J. Funct. Anal., 266 (2014), 5360-5376. doi: 10.1016/j.jfa.2014.02.023.

[34]

D. H. Wang and C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals, Arch. Ration. Mech. Anal., 204 (2012), 881-915. doi: 10.1007/s00205-011-0488-x.

[35]

H. Y. Wen and S. J. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. Real World Appl., 12 (2011), 1510-1531. doi: 10.1016/j.nonrwa.2010.10.010.

[36]

H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 26 (2010), 379-396. doi: 10.3934/dcds.2010.26.379.

[37]

H. Wu, X. Xu and C. Liu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties, Calc. Var. Partial Differential Equations, 45 (2012), 319-345. doi: 10.1007/s00526-011-0460-5.

[38]

Y. Zhou, J. S. Fan and G. Nakamura, Global strong solution to the density-dependent 2-D liquid crystal flows, Abstr. Appl. Anal., Art. ID 947291 (2013), 5pp.

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