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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, (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.
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.
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.
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.
doi: 10.1142/S0218127410027477. |
[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).
|
[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. |
[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. |
[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. |
[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. |
[13] |
C. Liu and N. J. Walkington, Approximation of liquid crystal flows,, SIAM J. Numer. Anal., 37 (2000), 725.
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.
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, (). Google Scholar |
[16] |
S. Lojasiewicz, Une propriotopologique des sous-ensembles analytiques rls,, In, (1963), 87.
|
[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.
|
[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. |
[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. |
[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. |
show all references
References:
[1] |
R. A. Adams, "Sobolev Spaces,'', Academic Press, (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.
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.
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.
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.
doi: 10.1142/S0218127410027477. |
[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).
|
[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. |
[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. |
[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. |
[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. |
[13] |
C. Liu and N. J. Walkington, Approximation of liquid crystal flows,, SIAM J. Numer. Anal., 37 (2000), 725.
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.
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, (). Google Scholar |
[16] |
S. Lojasiewicz, Une propriotopologique des sous-ensembles analytiques rls,, In, (1963), 87.
|
[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.
|
[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. |
[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. |
[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. |
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