American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Semilinear elliptic systems involving multiple critical exponents and singularities in $\mathbb{R}^N$
  • DCDS Home
  • This Issue
  • Next Article
    Global conservative solutions to the Camassa--Holm equation for initial data with nonvanishing asymptotics
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 & Continuous Dynamical Systems - A, 2012, 32 (12) : 4229-4246. doi: 10.3934/dcds.2012.32.4229
References:
[1]

R. A. Adams, "Sobolev Spaces,'', Academic Press, (1975).   Google Scholar

[2]

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

[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.  doi: 10.1137/07068254X.  Google Scholar

[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.  doi: 10.1016/j.na.2008.10.092.  Google Scholar

[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.  doi: 10.1142/S0218127410027477.  Google Scholar

[6]

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

[7]

V. Girault and R. A. Raviart, "Finite Element Approximation for Navier-Stokes Equations: Theory and Algorithms,'', Springer, (1981).   Google Scholar

[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.  doi: 10.1137/080717833.  Google Scholar

[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.  doi: 10.1016/j.jde.2007.05.003.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

C. Liu, H. Wu and X. Xu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties,, preprint, ().   Google Scholar

[16]

S. Lojasiewicz, Une propriotopologique des sous-ensembles analytiques rls,, In, (1963), 87.   Google Scholar

[17]

S. Lojasiewicz, Ensemble semi-analytiques,, I. H. E. S. Notes, (1965).   Google Scholar

[18]

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

[19]

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

[20]

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

[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.  doi: 10.1016/j.jmaa.2006.07.011.  Google Scholar

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces,'', Academic Press, (1975).   Google Scholar

[2]

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

[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.  doi: 10.1137/07068254X.  Google Scholar

[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.  doi: 10.1016/j.na.2008.10.092.  Google Scholar

[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.  doi: 10.1142/S0218127410027477.  Google Scholar

[6]

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

[7]

V. Girault and R. A. Raviart, "Finite Element Approximation for Navier-Stokes Equations: Theory and Algorithms,'', Springer, (1981).   Google Scholar

[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.  doi: 10.1137/080717833.  Google Scholar

[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.  doi: 10.1016/j.jde.2007.05.003.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

C. Liu, H. Wu and X. Xu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties,, preprint, ().   Google Scholar

[16]

S. Lojasiewicz, Une propriotopologique des sous-ensembles analytiques rls,, In, (1963), 87.   Google Scholar

[17]

S. Lojasiewicz, Ensemble semi-analytiques,, I. H. E. S. Notes, (1965).   Google Scholar

[18]

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

[19]

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

[20]

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

[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.  doi: 10.1016/j.jmaa.2006.07.011.  Google Scholar

[1]

Yinnian He, Yanping Lin, Weiwei Sun. Stabilized finite element method for the non-stationary Navier-Stokes problem. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 41-68. doi: 10.3934/dcdsb.2006.6.41
[2]

Qiang Tao, Ying Yang. Exponential stability for the compressible nematic liquid crystal flow with large initial data. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1661-1669. doi: 10.3934/cpaa.2016007
[3]

Carlo Morosi, Livio Pizzocchero. On the constants in a Kato inequality for the Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 557-586. doi: 10.3934/cpaa.2012.11.557
[4]

Roberta Bianchini, Roberto Natalini. Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations. Kinetic & Related Models, 2019, 12 (1) : 133-158. doi: 10.3934/krm.2019006
[5]

Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907
[6]

Hi Jun Choe, Hyea Hyun Kim, Do Wan Kim, Yongsik Kim. Meshless method for the stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 495-526. doi: 10.3934/dcdsb.2001.1.495
[7]

Yinnian He, R. M.M. Mattheij. Reformed post-processing Galerkin method for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 369-387. doi: 10.3934/dcdsb.2007.8.369
[8]

Kaitai Li, Yanren Hou. Fourier nonlinear Galerkin method for Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 497-524. doi: 10.3934/dcds.1996.2.497
[9]

Hi Jun Choe, Do Wan Kim, Yongsik Kim. Meshfree method for the non-stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 17-39. doi: 10.3934/dcdsb.2006.6.17
[10]

Takayuki Kubo, Ranmaru Matsui. On pressure stabilization method for nonstationary Navier-Stokes equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2283-2307. doi: 10.3934/cpaa.2018109
[11]

Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207
[12]

M. Gregory Forest, Hongyun Wang, Hong Zhou. Sheared nematic liquid crystal polymer monolayers. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 497-517. doi: 10.3934/dcdsb.2009.11.497
[13]

Bagisa Mukherjee, Chun Liu. On the stability of two nematic liquid crystal configurations. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 561-574. doi: 10.3934/dcdsb.2002.2.561
[14]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[15]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
[16]

Evrad M. D. Ngom, Abdou Sène, Daniel Y. Le Roux. Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller. Evolution Equations & Control Theory, 2015, 4 (1) : 89-106. doi: 10.3934/eect.2015.4.89
[17]

G. M. de Araújo, S. B. de Menezes. On a variational inequality for the Navier-Stokes operator with variable viscosity. Communications on Pure & Applied Analysis, 2006, 5 (3) : 583-596. doi: 10.3934/cpaa.2006.5.583
[18]

Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3107-3123. doi: 10.3934/dcds.2016.36.3107
[19]

Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3421-3438. doi: 10.3934/dcdsb.2017173
[20]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences