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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

Stefano Bosia 1,

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.
Keywords: non-autonomous dynamical systems, Liquid crystal flow, Navier-Stokes equations, global attractor, exponential attractor..
Mathematics Subject Classification: Primary: 35B41, 35Q35, 76A15; Secondary: 76D0.
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 & 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.  Google Scholar

[2]

V. Chepyzhov and M. Vishik, "Attractors for Equations of Mathematical Physics," Amer. Math. Soc. Colloq. Publ. 49, Providence RI, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

P. De Gennes and J. Prost, "The Physics of Liquid Crystals," 2nd edition, Clarendon Press, Oxford, 1993. Google Scholar

[6]

K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin Heidelberg, 1985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

L. Evans, "Partial Differential Equations," Grad. Stud. Math. 19, Amer. Math. Soc., Providence RI, 1998.  Google Scholar

[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.  Google Scholar

[12]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer Verlag, Berlin Heidelberg New York, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J.-L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications," Volume 1, Dunod, Paris, 1968.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

I. Stewart, "The Static and Dynamic Continuum Theory of Liquid Crystals, A Mathematical Introduction," Taylor & Francis, London and New York, 2004. Google Scholar

[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.  Google Scholar

[29]

L. Tartar, "An Introduction to Sobolev Spaces and Interpolation Spaces," Springer Verlag, Berlin Heidelberg, 2007.  Google Scholar

[30]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Appl. Math. Sci., 68, Springer Verlag, New York Berlin Heidelberg, 1997.  Google Scholar

[31]

R. Temam, "Navier-Stokes Equations: Theory and Numerical Analysis," Reprint of the 1984 edition, AMS, Chelsea Publishing, Providence RI, 2001.  Google Scholar

[32]

E. Virga, "Variational Theories for Liquid Crystals," Applied Mathematics and Mathematical Computations, 8, Chapman & Hall, London, 1994.  Google Scholar

[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.  Google Scholar

[34]

H. Wu, X. Xu and C. Liu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties,, preprint, ().   Google Scholar

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.  Google Scholar

[2]

V. Chepyzhov and M. Vishik, "Attractors for Equations of Mathematical Physics," Amer. Math. Soc. Colloq. Publ. 49, Providence RI, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

P. De Gennes and J. Prost, "The Physics of Liquid Crystals," 2nd edition, Clarendon Press, Oxford, 1993. Google Scholar

[6]

K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin Heidelberg, 1985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

L. Evans, "Partial Differential Equations," Grad. Stud. Math. 19, Amer. Math. Soc., Providence RI, 1998.  Google Scholar

[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.  Google Scholar

[12]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer Verlag, Berlin Heidelberg New York, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J.-L. Lions and E. Magenes, "Problèmes aux limites non homogènes et applications," Volume 1, Dunod, Paris, 1968.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

I. Stewart, "The Static and Dynamic Continuum Theory of Liquid Crystals, A Mathematical Introduction," Taylor & Francis, London and New York, 2004. Google Scholar

[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.  Google Scholar

[29]

L. Tartar, "An Introduction to Sobolev Spaces and Interpolation Spaces," Springer Verlag, Berlin Heidelberg, 2007.  Google Scholar

[30]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Appl. Math. Sci., 68, Springer Verlag, New York Berlin Heidelberg, 1997.  Google Scholar

[31]

R. Temam, "Navier-Stokes Equations: Theory and Numerical Analysis," Reprint of the 1984 edition, AMS, Chelsea Publishing, Providence RI, 2001.  Google Scholar

[32]

E. Virga, "Variational Theories for Liquid Crystals," Applied Mathematics and Mathematical Computations, 8, Chapman & Hall, London, 1994.  Google Scholar

[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.  Google Scholar

[34]

H. Wu, X. Xu and C. Liu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties,, preprint, ().   Google Scholar

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