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December  2012, 32(12): 4229-4246. doi: 10.3934/dcds.2012.32.4229

Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows

Francisco Guillén-González 1, and  Mouhamadou Samsidy Goudiaby 2,

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla
2. 

LANI, UFR SAT, Université Gaston Berger, Saint-Louis, BP 234, Senegal

Received  July 2010 Revised  May 2012 Published  August 2012

In this paper, we study a conditional long-time stable fully discrete finite element scheme for a Ginzburg-Landau model for nematic liquid crystal flow. We also obtain its time asymptotic convergence (when number of time steps go to infinity, fixed time step and mesh size) towards a unique critical point of the elastic energy subject to the finite element subspace. Finally, we estimate some convergence rates towards this limit critical point. To prove convergence of the whole sequence, a Lojasiewicz type inequality is used.
    Moreover, we extend these results to other schemes given in [3] and [10].
Keywords: finite element method, Lojasiewicz inequality, Navier-Stokes Equations, convergence to equilibrium., Nematic liquid crystal flow.
Mathematics Subject Classification: Primary: 35Q35, 35K55, 35B40; Secondary: 65M6.
Citation: Francisco Guillén-González, Mouhamadou Samsidy Goudiaby. Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4229-4246. doi: 10.3934/dcds.2012.32.4229
References:
[1]

R. A. Adams, "Sobolev Spaces,'' Academic Press, New York, 1975.

[2]

S. Bartels and A. Prohl, Constraint preserving implicit finite element discretization of harmonic map heat flow into spheres, Math. Comp., 76 (2007), 1847-1859. doi: 10.1090/S0025-5718-07-02026-1.

[3]

R. Becker, X. Feng and A. Prohl, Finite element approximations of the Ericken-Leslie model for nematic liquid crystal flow, SIAM J. Numer. Anal., 46 (2008), 1704-1731. doi: 10.1137/07068254X.

[4]

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. doi: 10.1016/j.na.2008.10.092.

[5]

B. Climent-Ezquerra, F. Guillén-González and M. A. Rodríguez-Bellido, Stability for nematic liquid crystal with stretching terms, International Journal of Bifurcation and Chaos, 20 (2010), 2937-2942. doi: 10.1142/S0218127410027477.

[6]

J. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica., 21 (1967), 381-392.

[7]

V. Girault and R. A. Raviart, "Finite Element Approximation for Navier-Stokes Equations: Theory and Algorithms,'' Springer, Berlin, Heidelberg, New York, 1981.

[8]

M. Grasseli, H. Wu and S. Zheng, Convergence to equilibrium for parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations, SIAM J. Math. Anal., 40 (2009), 2007-2033. doi: 10.1137/080717833.

[9]

M. Grasseli, H. Petzeltová and G. Schimperna, Asymptotic behaviour of a nonisothermal viscous Cahn-Hilliard equation with inertial term, J. Diff. Equ., 239 (2007), 38-60. doi: 10.1016/j.jde.2007.05.003.

[10]

F. Guillén-González and J. V. Gutiérrez-Santacreu, A linearmixed finite element scheme for a nematic Eriksen-Leslie liquidcrystal model, Submitted.

[11]

F. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810.

[12]

F. H. Lin and C. Liu, Nonparabolic dissipative systemsmodeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503.

[13]

C. Liu and N. J. Walkington, Approximation of liquid crystal flows, SIAM J. Numer. Anal., 37 (2000), 725-741. doi: 10.1137/S0036142998344512.

[14]

C. Liu and N. J. Walkington, Mixed methods for the approximation of liquid crystal flows, M2AN Math. Model. Numer., 36 (2002), 205-222. doi: 10.1051/m2an:2002010.

[15]

C. Liu, H. Wu and X. Xu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties, preprint, arXiv:0901.1751.

[16]

S. Lojasiewicz, Une propriotopologique des sous-ensembles analytiques rls, In "Les Equations aux Divs Partielles (Paris 1962)'' Editions du Centre National de la Recherche Scientifique, Paris, (1963), 87-89.

[17]

S. Lojasiewicz, Ensemble semi-analytiques, I. H. E. S. Notes, 1965.

[18]

B. Merlet and M. Pierre, Convergence to equilibrium for the Backward Euler Scheme and Applications, Comm. Pure Appl. Anal., 9 (2010), 685-702.

[19]

L. Simon, Asymptotics for a class of non-linear evolution equations with applications to geometric problems, Ann. Of Math., 118 (1983), 525-571. doi: 10.2307/2006981.

[20]

H. Wu, Long time Behaviour for Nonlinear Hydrodynamic System modeling the Nematic Liquid Cristal Flows, Discrete and Contin. Dyn. Syst., 26 (2010), 379-396. doi: 10.3934/dcds.2010.26.379.

[21]

H. Wu, M. Grasseli and S. Zheng, Convergence to equilibrium for a parabolic-hyperbolic phase field system with dynamical boundary condition, J. Maht. Anal. Appl., 329 (2007), 948-976. doi: 10.1016/j.jmaa.2006.07.011.

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces,'' Academic Press, New York, 1975.

[2]

S. Bartels and A. Prohl, Constraint preserving implicit finite element discretization of harmonic map heat flow into spheres, Math. Comp., 76 (2007), 1847-1859. doi: 10.1090/S0025-5718-07-02026-1.

[3]

R. Becker, X. Feng and A. Prohl, Finite element approximations of the Ericken-Leslie model for nematic liquid crystal flow, SIAM J. Numer. Anal., 46 (2008), 1704-1731. doi: 10.1137/07068254X.

[4]

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. doi: 10.1016/j.na.2008.10.092.

[5]

B. Climent-Ezquerra, F. Guillén-González and M. A. Rodríguez-Bellido, Stability for nematic liquid crystal with stretching terms, International Journal of Bifurcation and Chaos, 20 (2010), 2937-2942. doi: 10.1142/S0218127410027477.

[6]

J. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica., 21 (1967), 381-392.

[7]

V. Girault and R. A. Raviart, "Finite Element Approximation for Navier-Stokes Equations: Theory and Algorithms,'' Springer, Berlin, Heidelberg, New York, 1981.

[8]

M. Grasseli, H. Wu and S. Zheng, Convergence to equilibrium for parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations, SIAM J. Math. Anal., 40 (2009), 2007-2033. doi: 10.1137/080717833.

[9]

M. Grasseli, H. Petzeltová and G. Schimperna, Asymptotic behaviour of a nonisothermal viscous Cahn-Hilliard equation with inertial term, J. Diff. Equ., 239 (2007), 38-60. doi: 10.1016/j.jde.2007.05.003.

[10]

F. Guillén-González and J. V. Gutiérrez-Santacreu, A linearmixed finite element scheme for a nematic Eriksen-Leslie liquidcrystal model, Submitted.

[11]

F. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810.

[12]

F. H. Lin and C. Liu, Nonparabolic dissipative systemsmodeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503.

[13]

C. Liu and N. J. Walkington, Approximation of liquid crystal flows, SIAM J. Numer. Anal., 37 (2000), 725-741. doi: 10.1137/S0036142998344512.

[14]

C. Liu and N. J. Walkington, Mixed methods for the approximation of liquid crystal flows, M2AN Math. Model. Numer., 36 (2002), 205-222. doi: 10.1051/m2an:2002010.

[15]

C. Liu, H. Wu and X. Xu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties, preprint, arXiv:0901.1751.

[16]

S. Lojasiewicz, Une propriotopologique des sous-ensembles analytiques rls, In "Les Equations aux Divs Partielles (Paris 1962)'' Editions du Centre National de la Recherche Scientifique, Paris, (1963), 87-89.

[17]

S. Lojasiewicz, Ensemble semi-analytiques, I. H. E. S. Notes, 1965.

[18]

B. Merlet and M. Pierre, Convergence to equilibrium for the Backward Euler Scheme and Applications, Comm. Pure Appl. Anal., 9 (2010), 685-702.

[19]

L. Simon, Asymptotics for a class of non-linear evolution equations with applications to geometric problems, Ann. Of Math., 118 (1983), 525-571. doi: 10.2307/2006981.

[20]

H. Wu, Long time Behaviour for Nonlinear Hydrodynamic System modeling the Nematic Liquid Cristal Flows, Discrete and Contin. Dyn. Syst., 26 (2010), 379-396. doi: 10.3934/dcds.2010.26.379.

[21]

H. Wu, M. Grasseli and S. Zheng, Convergence to equilibrium for a parabolic-hyperbolic phase field system with dynamical boundary condition, J. Maht. Anal. Appl., 329 (2007), 948-976. doi: 10.1016/j.jmaa.2006.07.011.

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