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On essential coexistence of zero and nonzero Lyapunov exponents
On a double penalized Smectic-A model
1. | Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla |
  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.
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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