# American Institute of Mathematical Sciences

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 & Applied Analysis, 2016, 15 (1) : 73-90. doi: 10.3934/cpaa.2016.15.73
##### References:
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Wang and H. Y. Wen, Strong solutions of the compressible nematic liquid crystal flow,, J. Differential Equations, 252 (2012), 2222. doi: 10.1016/j.jde.2011.07.036. Google Scholar [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. doi: 10.1016/j.jfa.2013.07.026. Google Scholar [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. doi: 10.1007/s00205-014-0768-3. Google Scholar [13] F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system,, Math. Methods Appl. Sci., 32 (2009), 2243. doi: 10.1002/mma.1132. Google Scholar [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. doi: 10.1002/mma.1138. Google Scholar [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, (). Google Scholar [16] J. K. Li, Global strong and weak solutions to inhomogeneous nematic liquid crystal flow in two dimensions,, Nonlinear Anal., 99 (2014), 80. doi: 10.1016/j.na.2013.12.023. Google Scholar [17] X. L. Li and D. H. Wang, Global solution to the incompressible flow of liquid crystals,, J. Differential Equations, 252 (2012), 745. doi: 10.1016/j.jde.2011.08.045. Google Scholar [18] F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: phase transition and flow phenomena,, Comm. Pure Appl. Math., 42 (1989), 789. doi: 10.1002/cpa.3160420605. Google Scholar [19] F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in two dimensions,, Arch. Rational Mech. Anal., 197 (2010), 297. doi: 10.1007/s00205-009-0278-x. Google Scholar [20] F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, Comm. Pure Appl. Math., 48 (1995), 501. doi: 10.1002/cpa.3160480503. Google Scholar [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. Google Scholar [22] F. H. Lin and C. Y. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimensions three,, preprint, (). Google Scholar [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). Google Scholar [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. doi: 10.1137/15M1007665. Google Scholar [25] P. Lions, Mathematical Topics in Fluid Mechanics: Incompressible models,, Oxford University Press, (1996). Google Scholar [26] Q. Liu, On the temporal decay of solutions to the two-dimensional nematic liquid crystal flows,, preprint, (). Google Scholar [27] X. G. Liu and Z. Y. Zhang, Existence of the flow of liquid crystals system,, Chinese Ann. Math. Ser. A, 30 (2009), 1. Google Scholar [28] D. G. Matteis and G. E. Virga, Director libration in nematoacoustics,, Physical Review E, 83 (2011). doi: 10.1103/PhysRevE.83.011703. Google Scholar [29] A. Novotnỳ and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow,, Oxford University Press, (2004). Google Scholar [30] M. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations,, Arch. Ration. Mech. Anal., 88 (1985), 209. doi: 10.1007/BF00752111. Google Scholar [31] L. Simon, Asymptotics for a class of nonlinear evolution equation, with applications to geometri problems,, Ann. of Math.(2), 118 (1983), 525. doi: 10.2307/2006981. Google Scholar [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. doi: 10.1007/s00205-010-0343-5. Google Scholar [33] C. Y. Wang and X. Xu, On the rigidity of nematic liquid crystal flow on $\mathbbS^2$,, J. Funct. Anal., 266 (2014), 5360. doi: 10.1016/j.jfa.2014.02.023. Google Scholar [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. doi: 10.1007/s00205-011-0488-x. Google Scholar [35] H. Y. Wen and S. J. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals,, Nonlinear Anal. Real World Appl., 12 (2011), 1510. doi: 10.1016/j.nonrwa.2010.10.010. Google Scholar [36] H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows,, Discrete Contin. Dyn. Syst., 26 (2010), 379. doi: 10.3934/dcds.2010.26.379. Google Scholar [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. doi: 10.1007/s00526-011-0460-5. Google Scholar [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). Google Scholar

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##### 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. doi: 10.1016/S0252-9602(12)60171-4. Google Scholar [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. doi: 10.1016/j.jfa.2013.01.011. Google Scholar [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. doi: 10.1007/s002050050181. Google Scholar [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. doi: 10.1137/120866476. Google Scholar [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. doi: 10.1007/s00205-013-0643-7. Google Scholar [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. doi: 10.1007/s00526-010-0331-5. Google Scholar [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. doi: 10.1137/120898814. Google Scholar [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. doi: 10.4310/CMS.2013.v11.n3.a6. Google Scholar [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. doi: 10.1007/s00205-011-0476-1. Google Scholar [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. doi: 10.1016/j.jde.2011.07.036. Google Scholar [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. doi: 10.1016/j.jfa.2013.07.026. Google Scholar [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. doi: 10.1007/s00205-014-0768-3. Google Scholar [13] F. Jiang and Z. Tan, Global weak solution to the flow of liquid crystals system,, Math. Methods Appl. Sci., 32 (2009), 2243. doi: 10.1002/mma.1132. Google Scholar [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. doi: 10.1002/mma.1138. Google Scholar [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, (). Google Scholar [16] J. K. Li, Global strong and weak solutions to inhomogeneous nematic liquid crystal flow in two dimensions,, Nonlinear Anal., 99 (2014), 80. doi: 10.1016/j.na.2013.12.023. Google Scholar [17] X. L. Li and D. H. Wang, Global solution to the incompressible flow of liquid crystals,, J. Differential Equations, 252 (2012), 745. doi: 10.1016/j.jde.2011.08.045. Google Scholar [18] F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: phase transition and flow phenomena,, Comm. Pure Appl. Math., 42 (1989), 789. doi: 10.1002/cpa.3160420605. Google Scholar [19] F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in two dimensions,, Arch. Rational Mech. Anal., 197 (2010), 297. doi: 10.1007/s00205-009-0278-x. Google Scholar [20] F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, Comm. Pure Appl. Math., 48 (1995), 501. doi: 10.1002/cpa.3160480503. Google Scholar [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. Google Scholar [22] F. H. Lin and C. Y. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimensions three,, preprint, (). Google Scholar [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). Google Scholar [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. doi: 10.1137/15M1007665. Google Scholar [25] P. Lions, Mathematical Topics in Fluid Mechanics: Incompressible models,, Oxford University Press, (1996). Google Scholar [26] Q. Liu, On the temporal decay of solutions to the two-dimensional nematic liquid crystal flows,, preprint, (). Google Scholar [27] X. G. Liu and Z. Y. Zhang, Existence of the flow of liquid crystals system,, Chinese Ann. Math. Ser. A, 30 (2009), 1. Google Scholar [28] D. G. Matteis and G. E. Virga, Director libration in nematoacoustics,, Physical Review E, 83 (2011). doi: 10.1103/PhysRevE.83.011703. Google Scholar [29] A. Novotnỳ and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow,, Oxford University Press, (2004). Google Scholar [30] M. Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations,, Arch. Ration. Mech. Anal., 88 (1985), 209. doi: 10.1007/BF00752111. Google Scholar [31] L. Simon, Asymptotics for a class of nonlinear evolution equation, with applications to geometri problems,, Ann. of Math.(2), 118 (1983), 525. doi: 10.2307/2006981. Google Scholar [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. doi: 10.1007/s00205-010-0343-5. Google Scholar [33] C. Y. Wang and X. Xu, On the rigidity of nematic liquid crystal flow on $\mathbbS^2$,, J. Funct. Anal., 266 (2014), 5360. doi: 10.1016/j.jfa.2014.02.023. Google Scholar [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. doi: 10.1007/s00205-011-0488-x. Google Scholar [35] H. Y. Wen and S. J. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals,, Nonlinear Anal. Real World Appl., 12 (2011), 1510. doi: 10.1016/j.nonrwa.2010.10.010. Google Scholar [36] H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows,, Discrete Contin. Dyn. Syst., 26 (2010), 379. doi: 10.3934/dcds.2010.26.379. Google Scholar [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. doi: 10.1007/s00526-011-0460-5. Google Scholar [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). Google Scholar
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