On a double penalized Smectic-A model

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December 2012, 32(12): 4171-4182. doi: 10.3934/dcds.2012.32.4171

On a double penalized Smectic-A model

Blanca Climent-Ezquerra ^1, and Francisco Guillén-González ^1,

1.	Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla

Received September 2011 Published August 2012

Abstract
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In smectic-A liquid crystals, a unity director vector $\boldsymbol{n}$ appear modeling an average preferential direction of the molecules and also the normal vector of the layer configuration. In the E's model [5], the Ginzburg-Landau penalization related to the constraint $|\boldsymbol{n}|=1$ is considered and, assuming the constraint $\nabla\times \boldsymbol{n}=0$, $\boldsymbol{n}$ is replaced by the so-called layer variable $\varphi$ such that $\boldsymbol{n}=\nabla\varphi$.
In this paper, a double penalized problem is introduced related to a smectic-A liquid crystal flows, considering a Cahn-Hilliard system to model the behavior of $\boldsymbol{n}$. Then, the issue of the global in time behavior of solutions is attacked, including the proof of the convergence of the whole trajectory towards a unique equilibrium state.

Keywords: Smectic-A liquid crystals, Cahn-Hilliard sys-tem, Navier-Stokes equations, coupled non-linear parabolic system, convergence to equilibrium..

Mathematics Subject Classification: Primary: 76A15; Secondary: 35A35, 35Q35, 35K30, 76D05, 76A10, 76D0.

Citation: Blanca Climent-Ezquerra, Francisco Guillén-González. On a double penalized Smectic-A model. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4171-4182. doi: 10.3934/dcds.2012.32.4171

References:

[1]	F. Bethuel, H. Brezis and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. Partial Differential Equations, 1 (1993), 123-148. doi: 10.1007/BF01191614.
[2]	B. Climent-Ezquerra, F. Guillén-González and M. J. Moreno-Iraberte, Regularity and time-periodicity for a nematic liquid crystal model, Nonlinear Analysis, 71 (2009), 539-549
[3]	B. Climent-Ezquerra, F. Guillén-González and M. A. Rodrĺguez Bellido, Stability for nematic liquid crystals with stretching terms, International Journal of Bifurcations and Chaos, 20 (2010), 2937-2942. doi: 10.1142/S0218127410027477.
[4]	B. Climent-Ezquerra and F. Guillén-González, Global in time solutions and time-periodicity for a Smectic-A liquid crystal model, Communications on Pure and Applied Analysis, 9 (2010), 1473-1493. doi: 10.3934/cpaa.2010.9.1473.
[5]	W. E, Nonlinear continuum theory of smectic-A liquid crystals, Arch. Rat. Mech. Anal., 137 (1997), 159-175. doi: 10.1007/s002050050026.
[6]	M. Grasselli and H. Wu, Long-time behavior for a nematic liquid crystal model with asymptotic stabilizing boundary condition and external force, preprint.
[7]	F. H. Lin and C. Liu, Non-parabolic dissipative systems modelling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503.
[8]	C. Liu, Dynamic Theory for Incompressible Smectic Liquid Crystals: Existence and Regularity, Discrete and Continuous Dynamical Systems, 6 (2000), 591-608. doi: 10.3934/dcds.2000.6.591.
[9]	A. Segatti and H. Wu, Finite dimensional reduction and convergence to equilibrium for incompressible Smectic-A liquid crystal flows, preprint, arXiv:1011.0358.
[10]	H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows, Discrete and Continuous Dynamical System, 26 (2010), 379-396. doi: 10.3934/dcds.2010.26.379.
[11]	S. Zheng, "Nonlinear Evolution Equations," Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 133, Chapman & Hall/CRC, Boca Raton, FL, 2004.

show all references

References:

[1]	F. Bethuel, H. Brezis and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. Partial Differential Equations, 1 (1993), 123-148. doi: 10.1007/BF01191614.
[2]	B. Climent-Ezquerra, F. Guillén-González and M. J. Moreno-Iraberte, Regularity and time-periodicity for a nematic liquid crystal model, Nonlinear Analysis, 71 (2009), 539-549
[3]	B. Climent-Ezquerra, F. Guillén-González and M. A. Rodrĺguez Bellido, Stability for nematic liquid crystals with stretching terms, International Journal of Bifurcations and Chaos, 20 (2010), 2937-2942. doi: 10.1142/S0218127410027477.
[4]	B. Climent-Ezquerra and F. Guillén-González, Global in time solutions and time-periodicity for a Smectic-A liquid crystal model, Communications on Pure and Applied Analysis, 9 (2010), 1473-1493. doi: 10.3934/cpaa.2010.9.1473.
[5]	W. E, Nonlinear continuum theory of smectic-A liquid crystals, Arch. Rat. Mech. Anal., 137 (1997), 159-175. doi: 10.1007/s002050050026.
[6]	M. Grasselli and H. Wu, Long-time behavior for a nematic liquid crystal model with asymptotic stabilizing boundary condition and external force, preprint.
[7]	F. H. Lin and C. Liu, Non-parabolic dissipative systems modelling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503.
[8]	C. Liu, Dynamic Theory for Incompressible Smectic Liquid Crystals: Existence and Regularity, Discrete and Continuous Dynamical Systems, 6 (2000), 591-608. doi: 10.3934/dcds.2000.6.591.
[9]	A. Segatti and H. Wu, Finite dimensional reduction and convergence to equilibrium for incompressible Smectic-A liquid crystal flows, preprint, arXiv:1011.0358.
[10]	H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows, Discrete and Continuous Dynamical System, 26 (2010), 379-396. doi: 10.3934/dcds.2010.26.379.
[11]	S. Zheng, "Nonlinear Evolution Equations," Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 133, Chapman & Hall/CRC, Boca Raton, FL, 2004.

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