# American Institute of Mathematical Sciences

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

## On a double penalized Smectic-A model

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

Received  September 2011 Published  August 2012

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.
Citation: Blanca Climent-Ezquerra, Francisco Guillén-González. On a double penalized Smectic-A model. Discrete & Continuous Dynamical Systems - A, 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. 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. [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. 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. 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. 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. 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. 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, (). [10] H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows,, Discrete and Continuous Dynamical System, 26 (2010), 379. 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 (2004).

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##### 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. 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. [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. 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. 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. 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. 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. 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, (). [10] H. Wu, Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows,, Discrete and Continuous Dynamical System, 26 (2010), 379. 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 (2004).
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