-
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
Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows
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 |
  Moreover, we extend these results to other schemes given in [3] and [10].
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, ().
|
[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, ().
|
[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. |
[1] |
Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222 |
[2] |
Yinnian He, Yanping Lin, Weiwei Sun. Stabilized finite element method for the non-stationary Navier-Stokes problem. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 41-68. doi: 10.3934/dcdsb.2006.6.41 |
[3] |
Eid Wassim, Yueqiang Shang. Local and parallel finite element algorithms for the incompressible Navier-Stokes equations with damping. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022022 |
[4] |
Qiang Tao, Ying Yang. Exponential stability for the compressible nematic liquid crystal flow with large initial data. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1661-1669. doi: 10.3934/cpaa.2016007 |
[5] |
Sili Liu, Xinhua Zhao, Yingshan Chen. A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4515-4533. doi: 10.3934/dcdsb.2020110 |
[6] |
Carlo Morosi, Livio Pizzocchero. On the constants in a Kato inequality for the Euler and Navier-Stokes equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 557-586. doi: 10.3934/cpaa.2012.11.557 |
[7] |
Roberta Bianchini, Roberto Natalini. Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations. Kinetic and Related Models, 2019, 12 (1) : 133-158. doi: 10.3934/krm.2019006 |
[8] |
Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907 |
[9] |
Hi Jun Choe, Hyea Hyun Kim, Do Wan Kim, Yongsik Kim. Meshless method for the stationary incompressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2001, 1 (4) : 495-526. doi: 10.3934/dcdsb.2001.1.495 |
[10] |
Yinnian He, R. M.M. Mattheij. Reformed post-processing Galerkin method for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 369-387. doi: 10.3934/dcdsb.2007.8.369 |
[11] |
Kaitai Li, Yanren Hou. Fourier nonlinear Galerkin method for Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 1996, 2 (4) : 497-524. doi: 10.3934/dcds.1996.2.497 |
[12] |
Hi Jun Choe, Do Wan Kim, Yongsik Kim. Meshfree method for the non-stationary incompressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 17-39. doi: 10.3934/dcdsb.2006.6.17 |
[13] |
Takayuki Kubo, Ranmaru Matsui. On pressure stabilization method for nonstationary Navier-Stokes equations. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2283-2307. doi: 10.3934/cpaa.2018109 |
[14] |
Bagisa Mukherjee, Chun Liu. On the stability of two nematic liquid crystal configurations. Discrete and Continuous Dynamical Systems - B, 2002, 2 (4) : 561-574. doi: 10.3934/dcdsb.2002.2.561 |
[15] |
M. Gregory Forest, Hongyun Wang, Hong Zhou. Sheared nematic liquid crystal polymer monolayers. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 497-517. doi: 10.3934/dcdsb.2009.11.497 |
[16] |
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 and Continuous Dynamical Systems, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207 |
[17] |
Mouhamadou Samsidy Goudiaby, Ababacar Diagne, Leon Matar Tine. Longtime behavior of a second order finite element scheme simulating the kinematic effects in liquid crystal dynamics. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3499-3514. doi: 10.3934/cpaa.2021116 |
[18] |
Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 |
[19] |
Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149 |
[20] |
Ling-Bing He, Li Xu. On the compressible Navier-Stokes equations in the whole space: From non-isentropic flow to isentropic flow. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3489-3530. doi: 10.3934/dcds.2021005 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]