zbMATH — the first resource for mathematics

Finite-dimensional global attractor for a system modeling the \(2D\) nematic liquid crystal flow. (English) Zbl 1242.35057
The authors consider a 2D system that models the nematic liquid crystal flow through the Navier-Stokes equations suitably coupled with a transport-reaction-diffusion equation for the averaged molecular orientations. This system has been proposed as a reasonable approximation of the well-known Ericksen-Leslie system. Taking advantage of previous well-posedness results and proving suitable dissipative estimates, here the authors show that the system endowed with periodic boundary conditions is a dissipative dynamical system with a smooth global attractor of finite fractal dimension.

35B41 Attractors
35Q35 PDEs in connection with fluid mechanics
76A15 Liquid crystals
76D05 Navier-Stokes equations for incompressible viscous fluids
35K57 Reaction-diffusion equations
Full Text: DOI arXiv
[1] Bosia, S.: Well-posedness and long term behavior of a simplified Ericksen–Leslie non-autonomous system for nematic liquid crystal flow. Commun. Pure Appl. Anal. (to appear) · Zbl 1270.35119
[2] Cavaterra, C., Rocca, E.: On a 3D isothermal model for nematic liquid crystals accounting for stretching terms, preprint arXiv:1107.3947v1 (2011), pp. 1–14
[3] Climent-Ezquerra B., Guillen-González F., Rojas-Medar M.: Reproductivity for a nematic liquid crystal model. Z. Angew. Math. Phys. 57, 984–998 (2006) · Zbl 1106.35058 · doi:10.1007/s00033-005-0038-1
[4] Climent-Ezquerra B., Guillen-González F., Moreno-Iraberte M.J.: Regularity and time-periodicity for a nematic liquid crystal model. Nonlinear Anal. 71, 530–549 (2009) · Zbl 1194.35088 · doi:10.1016/j.na.2008.10.092
[5] Coutand D., Shkoller S.: Well-posedness of the full Ericksen–Leslie model of nematic liquid crystals. C. R. Acad. Sci. Paris Ser. I Math. 333, 919–924 (2001) · Zbl 0999.35078 · doi:10.1016/S0764-4442(01)02161-9
[6] de Gennes P.G., Prost J.: The Physics of Liquid Crystals, 2nd edn. Oxford Science Publications, Oxford (1993)
[7] Ericksen J.L.: Conservation laws for liquid crystals. Trans. Soc. Rheol. 5, 22–34 (1961) · doi:10.1122/1.548883
[8] Ericksen J.L.: Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113, 97–120 (1991) · Zbl 0729.76008 · doi:10.1007/BF00380413
[9] Fan J., Ozawa T.: Regularity criteria for a simplified Ericksen–Leslie system modeling the flow of liquid crystals. Discrete Contin. Dyn. Syst. 25, 859–867 (2009) · Zbl 1179.35085 · doi:10.3934/dcds.2009.25.859
[10] Guillen-González F., Rodríguez-Bellido M.A., Rojas-Medar M.A.: Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model. Math. Nachr. 282, 846–867 (2009) · Zbl 1173.35033 · doi:10.1002/mana.200610776
[11] Hu X., Wang D.: Global solution to the three-dimensional incompressible flow of liquid crystals. Commun. Math. Phys. 296, 861–880 (2010) · Zbl 1192.82080 · doi:10.1007/s00220-010-1017-8
[12] Kato T., Ponce G.: Commutator estimates and the Euler and Navier–Stokes equations. Commun. Pure Appl. Math. 41, 891–907 (1988) · Zbl 0671.35066 · doi:10.1002/cpa.3160410704
[13] Leslie, F.M.: Theory of flow phenomena in liquid crystals. In: Advances in Liquid Crystals, vol. 4, pp. 1–81. Academic Press, New York (1979)
[14] Lin F.-H., Liu C.: Nonparabolic dissipative system modeling the flow of liquid crystals. Commun. Pure Appl. Math. 48, 501–537 (1995) · Zbl 0842.35084 · doi:10.1002/cpa.3160480503
[15] Lin F.-H., Liu C.: Existence of solutions for the Ericksen–Leslie system. Arch. Ration. Mech. Anal. 154, 135–156 (2000) · Zbl 0963.35158 · doi:10.1007/s002050000102
[16] Lin P., Liu C., Zhang H.: An energy law preserving C 0 finite element scheme for simulating the kinematic effects in liquid crystal dynamics. J. Comput. Phys. 227, 1411–1427 (2007) · Zbl 1133.65077 · doi:10.1016/j.jcp.2007.09.005
[17] Liu C., Shen J.: On liquid crystal flows with free-slip boundary conditions. Discrete Contin. Dyn. Syst. 7, 307–318 (2001) · Zbl 1014.76005 · doi:10.3934/dcds.2001.7.307
[18] Liu C., Walkington N.J.: Approximation of liquid crystal flows. SIAM J. Numer. Anal. 37, 725–741 (2000) · Zbl 1040.76036 · doi:10.1137/S0036142997327282
[19] Liu C., Walkington N.J.: Mixed methods for the approximation of liquid crystal flows. M2AN Math. Model. Numer. Anal. 36, 205–222 (2002) · Zbl 1032.76035 · doi:10.1051/m2an:2002010
[20] Pata V., Zelik S.: A result on the existence of global attractors for semigroups of closed operators. Commun. Pure Appl. Anal. 6, 481–486 (2007) · Zbl 1152.47046 · doi:10.3934/cpaa.2007.6.481
[21] Shkoller S.: Well-posedness and global attractors for liquid crystals on Riemannian manifolds. Comm. Partial Differ. Equ. 27, 1103–1137 (2001) · Zbl 1011.35029 · doi:10.1081/PDE-120004895
[22] Sun H., Liu C.: On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete Contin. Dyn. Syst. 23, 455–475 (2009) · Zbl 1156.76007 · doi:10.3934/dcds.2009.23.455
[23] Temam R.: Navier–Stokes equations and nonlinear functional analysis, 2nd edn. CBMS-NSF Reg. Conf. Ser. Appl. Math., 66. SIAM, Philadelphia (1995) · Zbl 0833.35110
[24] Wu H.: Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows. Discrete Contin. Dyn. Syst. 26, 379–396 (2010) · Zbl 1185.35178 · doi:10.3934/dcds.2010.26.379
[25] Wu, H., Xu, X., Liu, C.: Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties, preprint arXiv:0901.1751v2, pp. 1–26 (2010)
[26] Wu, H., Xu, X., Liu, C.: On the general Ericksen–Leslie system: Parodi’s relation, well-posedness and stability, preprint arXiv:1105.2180v3, pp. 1–34 (2011)
[27] Zelik S.: The attractor for a nonlinear reaction–diffusion system with a supercritical nonlinearity and its dimension. Rend. Acad. Naz. Sci. XL Mem. Mat. Appl. 24, 1–25 (2000)
[28] Zheng, S.: Nonlinear Evolution Equations. Pitman series Monographs and Survey in Pure and Applied Mathematics, vol. 133, Chapman&Hall/CRC, Boca Raton (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.