# American Institute of Mathematical Sciences

March  2012, 11(2): 407-441. doi: 10.3934/cpaa.2012.11.407

## Well-posedness and long term behavior of a simplified Ericksen-Leslie non-autonomous system for nematic liquid crystal flows

 1 Dipartimento di Matematica F. Brioschi'', Politecnico di Milano, Milano, Italy

Received  October 2010 Revised  December 2010 Published  October 2011

We analyze a simplified Ericksen-Leslie model for nematic liquid crystal flows firstly introduced in [18] with non-autonomous forcing bulk term and boundary conditions on the order parameter field. We obtain existence of weak solutions in the two- and three-dimensional cases. We prove uniqueness, continuous dependence on initial conditions, forcing and boundary terms and also existence of strong solutions in the 2D case. Focusing on the 2D case, we then study the long term behavior of solutions by obtaining existence of global attractors for normal forcing terms (according to [21]). Finally, we prove the existence of exponential attractors for quasi-periodic forcing terms in the 2D model.
Citation: Stefano Bosia. Well-posedness and long term behavior of a simplified Ericksen-Leslie non-autonomous system for nematic liquid crystal flows. Communications on Pure and Applied Analysis, 2012, 11 (2) : 407-441. doi: 10.3934/cpaa.2012.11.407
##### References:
 [1] H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681. doi: 10.1016/0362-546X(80)90068-1. [2] V. Chepyzhov and M. Vishik, "Attractors for Equations of Mathematical Physics," Amer. Math. Soc. Colloq. Publ. 49, Providence RI, 2002. [3] B. Climent-Ezquerra, F. Guillen-González and M. Rojas-Medar, Reproductivity for a nematic liquid crystal model, Z. Angew. Math. Phys. ZAMP, 71 (2006), 984-998. doi: 10.1007/s00033-005-0038-1. [4] B. Climent-Ezquerra, F. Guillen-González and M. Moreno-Iraberte, Regularity and time-periodicity for a nematic liquid crystal model, Nonlinear Anal., 71 (2009), 530-549. doi: 10.1016/j.na.2008.10.092. [5] P. De Gennes and J. Prost, "The Physics of Liquid Crystals," 2nd edition, Clarendon Press, Oxford, 1993. [6] K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin Heidelberg, 1985. [7] A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations," Research in Applied Mathematics, Masson/John Wiley co-publication, Paris, 1994. [8] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $R^3$, C. R. Acad. Sci. Paris, Sér. I, 330 (2000), 713-718. [9] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh, Sect. A, 135 (2005), 703-730. [10] L. Evans, "Partial Differential Equations," Grad. Stud. Math. 19, Amer. Math. Soc., Providence RI, 1998. [11] J. Fan and T. Ozawa, Regularity criteria for a simplified Ericksen-Leslie system modelling the flow of liquid crystals, Discrete Contin. Dyn. Syst., 25 (2009), 859-867. doi: 10.3934/dcds.2009.25.859. [12] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer Verlag, Berlin Heidelberg New York, 1983. [13] F. Guillén-González and M. Rojas-Medar, Global solution of nematic liquid crystals models, C. R. Acad. Sci. Paris, Sér. I, 335 (2002), 1085-1090. [14] F. Guillén-González, M. Rodríguez-Bellido and M. Rojas-Medar, Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model, Math. Nachr., 282 (2009), 846-867. doi: 10.1002/mana.200610776. [15] X. Hu and D. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals, Commun. Math. Phys., 296 (2010), 861-880. doi: 10.1007/s00220-010-1017-8. [16] F. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transitions and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605. [17] F. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x. [18] F. Lin and C. Liu, Nonparabolic dissipative systems modelling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503. [19] J.-L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications," Volume 1, Dunod, Paris, 1968. [20] C. Liu and J. Shen, On liquid crystal flows with free-slip boundary conditions, Discrete Contin. Dyn. Syst., 7 (2001), 307-318. doi: 10.3934/dcds.2001.7.307. [21] S. Lu, H. Wu and C. Zhong, Attractors for nonautomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719. doi: 10.3934/dcds.2005.13.701. [22] S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces, J. Differential Equations, 230 (2006), 196-212. doi: 10.1016/j.jde.2006.07.009. [23] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations, Evolutionary Partial Differential Equations" Vol. 4 (eds. C. Dafermos and M. Pokorny), Elsevier, Amsterdam (2008), 103-200. doi: 10.1016/S1874-5717(08)00003-0. [24] R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Anal., 32 (1998), 71-85. doi: 10.1016/S0362-546X(97)00453-7. [25] S. Shkoller, Well-posedness and global attractors for liquid crystals on Riemannian manifolds, Comm. Partial Differential Equations, 27 (2002), 1103-1137. doi: 10.1081/PDE-120004895. [26] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. [27] I. Stewart, "The Static and Dynamic Continuum Theory of Liquid Crystals, A Mathematical Introduction," Taylor & Francis, London and New York, 2004. [28] H. Sun and C. Liu, On energetic variational approaches in modelling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 23 (2009), 455-475. doi: 10.3934/dcds.2009.23.455. [29] L. Tartar, "An Introduction to Sobolev Spaces and Interpolation Spaces," Springer Verlag, Berlin Heidelberg, 2007. [30] R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Appl. Math. Sci., 68, Springer Verlag, New York Berlin Heidelberg, 1997. [31] R. Temam, "Navier-Stokes Equations: Theory and Numerical Analysis," Reprint of the 1984 edition, AMS, Chelsea Publishing, Providence RI, 2001. [32] E. Virga, "Variational Theories for Liquid Crystals," Applied Mathematics and Mathematical Computations, 8, Chapman & Hall, London, 1994. [33] H. Wu, Long-time behaviour for a nonlinear hydrodynamic system modelling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 20 (2010), 379-396. [34] H. Wu, X. Xu and C. Liu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties,, preprint, ().

show all references

##### References:
 [1] H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681. doi: 10.1016/0362-546X(80)90068-1. [2] V. Chepyzhov and M. Vishik, "Attractors for Equations of Mathematical Physics," Amer. Math. Soc. Colloq. Publ. 49, Providence RI, 2002. [3] B. Climent-Ezquerra, F. Guillen-González and M. Rojas-Medar, Reproductivity for a nematic liquid crystal model, Z. Angew. Math. Phys. ZAMP, 71 (2006), 984-998. doi: 10.1007/s00033-005-0038-1. [4] B. Climent-Ezquerra, F. Guillen-González and M. Moreno-Iraberte, Regularity and time-periodicity for a nematic liquid crystal model, Nonlinear Anal., 71 (2009), 530-549. doi: 10.1016/j.na.2008.10.092. [5] P. De Gennes and J. Prost, "The Physics of Liquid Crystals," 2nd edition, Clarendon Press, Oxford, 1993. [6] K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin Heidelberg, 1985. [7] A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations," Research in Applied Mathematics, Masson/John Wiley co-publication, Paris, 1994. [8] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $R^3$, C. R. Acad. Sci. Paris, Sér. I, 330 (2000), 713-718. [9] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh, Sect. A, 135 (2005), 703-730. [10] L. Evans, "Partial Differential Equations," Grad. Stud. Math. 19, Amer. Math. Soc., Providence RI, 1998. [11] J. Fan and T. Ozawa, Regularity criteria for a simplified Ericksen-Leslie system modelling the flow of liquid crystals, Discrete Contin. Dyn. Syst., 25 (2009), 859-867. doi: 10.3934/dcds.2009.25.859. [12] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer Verlag, Berlin Heidelberg New York, 1983. [13] F. Guillén-González and M. Rojas-Medar, Global solution of nematic liquid crystals models, C. R. Acad. Sci. Paris, Sér. I, 335 (2002), 1085-1090. [14] F. Guillén-González, M. Rodríguez-Bellido and M. Rojas-Medar, Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model, Math. Nachr., 282 (2009), 846-867. doi: 10.1002/mana.200610776. [15] X. Hu and D. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals, Commun. Math. Phys., 296 (2010), 861-880. doi: 10.1007/s00220-010-1017-8. [16] F. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transitions and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605. [17] F. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x. [18] F. Lin and C. Liu, Nonparabolic dissipative systems modelling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503. [19] J.-L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications," Volume 1, Dunod, Paris, 1968. [20] C. Liu and J. Shen, On liquid crystal flows with free-slip boundary conditions, Discrete Contin. Dyn. Syst., 7 (2001), 307-318. doi: 10.3934/dcds.2001.7.307. [21] S. Lu, H. Wu and C. Zhong, Attractors for nonautomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719. doi: 10.3934/dcds.2005.13.701. [22] S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces, J. Differential Equations, 230 (2006), 196-212. doi: 10.1016/j.jde.2006.07.009. [23] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations, Evolutionary Partial Differential Equations" Vol. 4 (eds. C. Dafermos and M. Pokorny), Elsevier, Amsterdam (2008), 103-200. doi: 10.1016/S1874-5717(08)00003-0. [24] R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Anal., 32 (1998), 71-85. doi: 10.1016/S0362-546X(97)00453-7. [25] S. Shkoller, Well-posedness and global attractors for liquid crystals on Riemannian manifolds, Comm. Partial Differential Equations, 27 (2002), 1103-1137. doi: 10.1081/PDE-120004895. [26] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. [27] I. Stewart, "The Static and Dynamic Continuum Theory of Liquid Crystals, A Mathematical Introduction," Taylor & Francis, London and New York, 2004. [28] H. Sun and C. Liu, On energetic variational approaches in modelling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 23 (2009), 455-475. doi: 10.3934/dcds.2009.23.455. [29] L. Tartar, "An Introduction to Sobolev Spaces and Interpolation Spaces," Springer Verlag, Berlin Heidelberg, 2007. [30] R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Appl. Math. Sci., 68, Springer Verlag, New York Berlin Heidelberg, 1997. [31] R. Temam, "Navier-Stokes Equations: Theory and Numerical Analysis," Reprint of the 1984 edition, AMS, Chelsea Publishing, Providence RI, 2001. [32] E. Virga, "Variational Theories for Liquid Crystals," Applied Mathematics and Mathematical Computations, 8, Chapman & Hall, London, 1994. [33] H. Wu, Long-time behaviour for a nonlinear hydrodynamic system modelling the nematic liquid crystal flows, Discrete Contin. Dyn. Syst., 20 (2010), 379-396. [34] H. Wu, X. Xu and C. Liu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties,, preprint, ().
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