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Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions
1. | Politecnico di Milano - Dipartimento di Matematica "F. Brioschi", Via Bonardi 9, 20133 Milano |
2. | Dipartimento di Matematica, Università di Modena e Reggio Emilia, Via Campi 213/B, I-41125 Modena, Italy |
3. | Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 6086 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex |
References:
[1] |
G. Caginalp, An analysis of a phase field model of a free boundary,, Arch. Rational Mech. Anal., 92 (1986), 205.
doi: 10.1007/BF00254827. |
[2] |
P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation. Partial differential equations and applications,, Discrete Contin. Dynam. Systems, 10 (2004), 211.
doi: 10.3934/dcds.2004.10.211. |
[3] |
H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition,, Phys. Rev. Letters, 79 (1997), 893.
doi: 10.1103/PhysRevLett.79.893. |
[4] |
H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows,, Europhys. Lett., 42 (1998), 49.
doi: 10.1209/epl/i1998-00550-y. |
[5] |
H. P. Fischer, J. Reinhard, W. Dieterich, J.-F. Gouyet, P. Maass, A. Majhofer and D. Reinel, Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall,, J. Chem. Phys., 108 (1998), 3028.
doi: 10.1063/1.475690. |
[6] |
C. G. Gal, M. Grasselli and A. Miranville, Robust exponential attractors for singularly perturbed phase-field equations with dynamic boundary conditions,, NoDEA Nonlinear Differential Equations Appl., 15 (2008).
doi: 10.1007/s00030-008-7029-9. |
[7] |
C. G. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions,, in, 29 (2008).
|
[8] |
S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions,, in, 251 (2006), 149.
|
[9] |
M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials,, Discrete Contin. Dynam. Systems, 28 (2010), 67.
doi: 10.3934/dcds.2010.28.67. |
[10] |
O. A. Ladyžhenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1967).
|
[11] |
A. Miranville and S. Zelik, Robust exponential attractors for singularly perturbed phase-field type equations,, Electron. J. Differential Equations, 2002 ().
|
[12] |
A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions,, Math. Meth. Appl. Sci., 28 (2005), 709.
doi: 10.1002/mma.590. |
[13] |
A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions,, Discrete Contin. Dynam. Systems, 28 (2010), 275.
doi: 10.3934/dcds.2010.28.275. |
[14] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", 2nd edition, 68 (1997).
|
[15] |
E. Zeidler, "Nonlinear Functional Analysis and its Applications. Part I. Fixed-Point Theorems,", Springer-Verlag, (1986).
|
show all references
References:
[1] |
G. Caginalp, An analysis of a phase field model of a free boundary,, Arch. Rational Mech. Anal., 92 (1986), 205.
doi: 10.1007/BF00254827. |
[2] |
P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation. Partial differential equations and applications,, Discrete Contin. Dynam. Systems, 10 (2004), 211.
doi: 10.3934/dcds.2004.10.211. |
[3] |
H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition,, Phys. Rev. Letters, 79 (1997), 893.
doi: 10.1103/PhysRevLett.79.893. |
[4] |
H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows,, Europhys. Lett., 42 (1998), 49.
doi: 10.1209/epl/i1998-00550-y. |
[5] |
H. P. Fischer, J. Reinhard, W. Dieterich, J.-F. Gouyet, P. Maass, A. Majhofer and D. Reinel, Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall,, J. Chem. Phys., 108 (1998), 3028.
doi: 10.1063/1.475690. |
[6] |
C. G. Gal, M. Grasselli and A. Miranville, Robust exponential attractors for singularly perturbed phase-field equations with dynamic boundary conditions,, NoDEA Nonlinear Differential Equations Appl., 15 (2008).
doi: 10.1007/s00030-008-7029-9. |
[7] |
C. G. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions,, in, 29 (2008).
|
[8] |
S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions,, in, 251 (2006), 149.
|
[9] |
M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials,, Discrete Contin. Dynam. Systems, 28 (2010), 67.
doi: 10.3934/dcds.2010.28.67. |
[10] |
O. A. Ladyžhenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1967).
|
[11] |
A. Miranville and S. Zelik, Robust exponential attractors for singularly perturbed phase-field type equations,, Electron. J. Differential Equations, 2002 ().
|
[12] |
A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions,, Math. Meth. Appl. Sci., 28 (2005), 709.
doi: 10.1002/mma.590. |
[13] |
A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions,, Discrete Contin. Dynam. Systems, 28 (2010), 275.
doi: 10.3934/dcds.2010.28.275. |
[14] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", 2nd edition, 68 (1997).
|
[15] |
E. Zeidler, "Nonlinear Functional Analysis and its Applications. Part I. Fixed-Point Theorems,", Springer-Verlag, (1986).
|
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