-
Previous Article
The annulus as a K-spectral set
- CPAA Home
- This Issue
-
Next Article
Abstract reaction-diffusion systems with $m$-completely accretive diffusion operators and measurable reaction rates
Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions
1. | Université de La Rochelle, Laboratoire de Mathématiques Images et Applications EA 3165, Avenue Michel Crépeau, 17042 La Rochelle Cedex 1 |
2. | Dipartimento di Matematica, Università di Modena e Reggio Emilia, Via Campi 213/B, I-41125 Modena |
3. | Université de Poitiers, Mathématiques SP2MI, 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), 205-245.
doi: 10.1007/BF00254827. |
[2] |
L. Cherfils, S. Gatti and A. Miranville, Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials (Corrigendum, J. Math. Anal. Appl. 348 (2008), 1029-1030), J. Math. Anal. Appl., 343 (2008), 557-566.
doi: 10.1016/j.jmaa.2008.07.058. |
[3] |
L. Cherfils, S. Gatti and A. Miranville, Finite dimensional attractors for the Caginalp system with singular potentials and dynamic boundary conditions, Bull. Transilvania University Bra\csov-Series III: Mathematics, Informatics, Physics, 2 (2009), 25-34. |
[4] |
L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115.
doi: 10.1007/s10492-009-0008-6. |
[5] |
L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.
doi: 10.1007/s00032-011-0165-4. |
[6] |
P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dynam. Systems, 10 (2004), 211-238.
doi: 10.3934/dcds.2004.10.211. |
[7] |
H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Lett., 79 (1997), 893-896.
doi: 10.1103/PhysRevLett.79.893. |
[8] |
H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Lett., 42 (1998), 49-54.
doi: 10.1209/epl/i1998-00550-y. |
[9] |
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), 3028-3037.
doi: 10.1063/1.475690. |
[10] |
S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in "Differential Equations: Inverse and Direct Problems, Lecture Notes in Pure and Applied Mathematics," Taylor and Francis, (2006), 149-170.
doi: 10.1201/9781420011135.ch9. |
[11] |
G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912.
doi: 10.3934/cpaa.2009.8.881. |
[12] |
G. Gilardi, A. Miranville and G. Schimperna, Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chinese Ann. Math., Ser. B, 31 (2010), 679-712.
doi: 10.1007/s11401-010-0602-7. |
[13] |
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), 67-98.
doi: 10.3934/dcds.2010.28.67. |
[14] |
J. Málek and D. Prážak, Large time behavior via the method of $l$-trajectories, J. Diff. Eqns., 181 (2002), 243-279.
doi: 10.1006/jdeq.2001.4087. |
[15] |
A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582.
doi: 10.1002/mma.464. |
[16] |
A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735.
doi: 10.1002/mma.590. |
[17] |
A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations, Vol. IV" (eds. C.M. Dafermos and M. Pokorny), Elsevier/North-Holland, (2008), 103-200.
doi: 10.1016/S1874-5717(08)00003-0. |
[18] |
A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dynam. Systems, 28 (2010), 275-310.
doi: 10.3934/dcds.2010.28.275. |
[19] |
G. Ruiz Goldstein, A. Miranville and G. Schimperna, A Cahn-Hilliard equation in a domain with non-permeable walls, Phys. D, 240 (2011), 754-766.
doi: 10.1016/j.physd.2010.12.007. |
show all references
References:
[1] |
G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.
doi: 10.1007/BF00254827. |
[2] |
L. Cherfils, S. Gatti and A. Miranville, Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials (Corrigendum, J. Math. Anal. Appl. 348 (2008), 1029-1030), J. Math. Anal. Appl., 343 (2008), 557-566.
doi: 10.1016/j.jmaa.2008.07.058. |
[3] |
L. Cherfils, S. Gatti and A. Miranville, Finite dimensional attractors for the Caginalp system with singular potentials and dynamic boundary conditions, Bull. Transilvania University Bra\csov-Series III: Mathematics, Informatics, Physics, 2 (2009), 25-34. |
[4] |
L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115.
doi: 10.1007/s10492-009-0008-6. |
[5] |
L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.
doi: 10.1007/s00032-011-0165-4. |
[6] |
P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dynam. Systems, 10 (2004), 211-238.
doi: 10.3934/dcds.2004.10.211. |
[7] |
H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Lett., 79 (1997), 893-896.
doi: 10.1103/PhysRevLett.79.893. |
[8] |
H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Lett., 42 (1998), 49-54.
doi: 10.1209/epl/i1998-00550-y. |
[9] |
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), 3028-3037.
doi: 10.1063/1.475690. |
[10] |
S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in "Differential Equations: Inverse and Direct Problems, Lecture Notes in Pure and Applied Mathematics," Taylor and Francis, (2006), 149-170.
doi: 10.1201/9781420011135.ch9. |
[11] |
G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912.
doi: 10.3934/cpaa.2009.8.881. |
[12] |
G. Gilardi, A. Miranville and G. Schimperna, Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chinese Ann. Math., Ser. B, 31 (2010), 679-712.
doi: 10.1007/s11401-010-0602-7. |
[13] |
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), 67-98.
doi: 10.3934/dcds.2010.28.67. |
[14] |
J. Málek and D. Prážak, Large time behavior via the method of $l$-trajectories, J. Diff. Eqns., 181 (2002), 243-279.
doi: 10.1006/jdeq.2001.4087. |
[15] |
A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582.
doi: 10.1002/mma.464. |
[16] |
A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735.
doi: 10.1002/mma.590. |
[17] |
A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations, Vol. IV" (eds. C.M. Dafermos and M. Pokorny), Elsevier/North-Holland, (2008), 103-200.
doi: 10.1016/S1874-5717(08)00003-0. |
[18] |
A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dynam. Systems, 28 (2010), 275-310.
doi: 10.3934/dcds.2010.28.275. |
[19] |
G. Ruiz Goldstein, A. Miranville and G. Schimperna, A Cahn-Hilliard equation in a domain with non-permeable walls, Phys. D, 240 (2011), 754-766.
doi: 10.1016/j.physd.2010.12.007. |
[1] |
Franck Davhys Reval Langa, Morgan Pierre. A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 653-676. doi: 10.3934/dcdss.2020353 |
[2] |
Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6387-6403. doi: 10.3934/dcdsb.2021024 |
[3] |
Ciprian G. Gal, M. Grasselli. On the asymptotic behavior of the Caginalp system with dynamic boundary conditions. Communications on Pure and Applied Analysis, 2009, 8 (2) : 689-710. doi: 10.3934/cpaa.2009.8.689 |
[4] |
Maurizio Grasselli, Alain Miranville, Giulio Schimperna. The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 67-98. doi: 10.3934/dcds.2010.28.67 |
[5] |
Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5321-5335. doi: 10.3934/dcdsb.2020345 |
[6] |
I. D. Chueshov, Iryna Ryzhkova. A global attractor for a fluid--plate interaction model. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1635-1656. doi: 10.3934/cpaa.2013.12.1635 |
[7] |
Alexey Cheskidov, Susan Friedlander, Nataša Pavlović. An inviscid dyadic model of turbulence: The global attractor. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 781-794. doi: 10.3934/dcds.2010.26.781 |
[8] |
Rana D. Parshad, Juan B. Gutierrez. On the global attractor of the Trojan Y Chromosome model. Communications on Pure and Applied Analysis, 2011, 10 (1) : 339-359. doi: 10.3934/cpaa.2011.10.339 |
[9] |
Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195 |
[10] |
Nobuyuki Kenmochi, Noriaki Yamazaki. Global attractor of the multivalued semigroup associated with a phase-field model of grain boundary motion with constraint. Conference Publications, 2011, 2011 (Special) : 824-833. doi: 10.3934/proc.2011.2011.824 |
[11] |
Monica Conti, Stefania Gatti, Alain Miranville. Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 485-505. doi: 10.3934/dcdss.2012.5.485 |
[12] |
Ning Ju. The global attractor for the solutions to the 3D viscous primitive equations. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 159-179. doi: 10.3934/dcds.2007.17.159 |
[13] |
Brahim Alouini. Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4545-4573. doi: 10.3934/cpaa.2020206 |
[14] |
Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275 |
[15] |
Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717 |
[16] |
Messoud Efendiev, Anna Zhigun. On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 651-673. doi: 10.3934/dcds.2018028 |
[17] |
Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215 |
[18] |
Yirong Jiang, Nanjing Huang, Zhouchao Wei. Existence of a global attractor for fractional differential hemivariational inequalities. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1193-1212. doi: 10.3934/dcdsb.2019216 |
[19] |
Hiroshi Matano, Ken-Ichi Nakamura. The global attractor of semilinear parabolic equations on $S^1$. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1 |
[20] |
Yuncheng You. Global attractor of the Gray-Scott equations. Communications on Pure and Applied Analysis, 2008, 7 (4) : 947-970. doi: 10.3934/cpaa.2008.7.947 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]