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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 |
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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