-
Previous Article
Semigroup-theoretic approach to identification of linear diffusion coefficients
- DCDS-S Home
- This Issue
-
Next Article
Observability of $N$-dimensional integro-differential systems
Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions
1. | Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France |
2. | Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, 86962 Chasseneuil Futuroscope Cedex |
References:
[1] |
S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1085-1095.
doi: 10.1016/0001-6160(79)90196-2. |
[2] |
J. Berry, K. R. Elder and M. Grant, Simulation of an atomistic dynamic field theory for monatomic liquids: Freezing and glass formation, Phys. Rev. E, 77 (2008), 061506.
doi: 10.1103/PhysRevE.77.061506. |
[3] |
J. Berry, M. Grant and K. R. Elder, Diffusive atomistic dynamics of edge dislocations in two dimensions, Phys. Rev. E, 73 (2006), 031609.
doi: 10.1103/PhysRevE.73.031609. |
[4] |
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. |
[5] |
G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Ration. Mech. Anal., 92 (1986), 205-245.
doi: 10.1007/BF00254827. |
[6] |
F. Chen and J. Shen, Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn-Hilliard systems, Commun. Comput. Phys., 13 (2013), 1189-1208. |
[7] |
P. G. de Gennes, Dynamics of fluctuations and spinodal decomposition in polymer blends, J. Chem. Phys., 72 (1980), 4756-4763.
doi: 10.1063/1.439809. |
[8] |
P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics, Phys. Rev. E, 79 (2009), 051110, 11pp.
doi: 10.1103/PhysRevE.79.051110. |
[9] |
G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations, Phys. Rev. E, 47 (1993), 4289-4300.
doi: 10.1103/PhysRevE.47.4289. |
[10] |
G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations, Phys. Rev. E, 47 (1993), 4301-4312.
doi: 10.1103/PhysRevE.47.4301. |
[11] |
Z. Hu, S. M. Wise, C. Wang and J. S. Lowengrub, Stable finite difference, nonlinear multigrid simulation of the phase field crystal equation, J. Comput. Phys., 228 (2009), 5323-5339.
doi: 10.1016/j.jcp.2009.04.020. |
[12] |
R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth, Phys. D, 63 (1993), 410-423.
doi: 10.1016/0167-2789(93)90120-P. |
[13] |
M. Korzec, P. Nayar and P. Rybka, Global weak solutions to a sixth order Cahn-Hilliard type equation, SIAM J. Math. Anal., 44 (2012), 3369-3387.
doi: 10.1137/100817590. |
[14] |
M. Korzec and P. Rybka, On a higher order convective Cahn-Hilliard type equation, SIAM J. Appl. Math., 72 (2012), 1343-1360.
doi: 10.1137/110834123. |
[15] |
A. Miranville, Existence of solutions for a one-dimensional Allen-Cahn equation, J. Appl. Anal. Comput., 3 (2013), 265-277. |
[16] |
A. Makki and A. Miranville, Well-posedness for one-dimensional anisotropic Cahn-Hilliard and Allen-Cahn systems, Electron. J. Differential Equaytions, (2015), 1-15. |
[17] |
T. V. Savina, A. A. Golovin, S. H. Davis, A. A. Nepomnyashchy and P. W. Voorhees, Faceting of a growing crystal surface by surface diffusion, Phys. Rev. E, 67 (2003), 021606.
doi: 10.1103/PhysRevE.67.021606. |
[18] |
S. Torabi, J. Lowengrub, A. Voigt and S. Wise, A new phase-field model for strongly anisotropic systems, Proc. R. Soc. A, 465 (2009), 1337-1359.
doi: 10.1098/rspa.2008.0385. |
[19] |
J. E. Taylor and J. W. Cahn, Diffuse interfaces with sharp corners and facets: Phase-field models with strongly anisotropic surfaces, Phys. D, 112 (1998), 381-411.
doi: 10.1016/S0167-2789(97)00177-2. |
show all references
References:
[1] |
S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1085-1095.
doi: 10.1016/0001-6160(79)90196-2. |
[2] |
J. Berry, K. R. Elder and M. Grant, Simulation of an atomistic dynamic field theory for monatomic liquids: Freezing and glass formation, Phys. Rev. E, 77 (2008), 061506.
doi: 10.1103/PhysRevE.77.061506. |
[3] |
J. Berry, M. Grant and K. R. Elder, Diffusive atomistic dynamics of edge dislocations in two dimensions, Phys. Rev. E, 73 (2006), 031609.
doi: 10.1103/PhysRevE.73.031609. |
[4] |
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. |
[5] |
G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Ration. Mech. Anal., 92 (1986), 205-245.
doi: 10.1007/BF00254827. |
[6] |
F. Chen and J. Shen, Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn-Hilliard systems, Commun. Comput. Phys., 13 (2013), 1189-1208. |
[7] |
P. G. de Gennes, Dynamics of fluctuations and spinodal decomposition in polymer blends, J. Chem. Phys., 72 (1980), 4756-4763.
doi: 10.1063/1.439809. |
[8] |
P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics, Phys. Rev. E, 79 (2009), 051110, 11pp.
doi: 10.1103/PhysRevE.79.051110. |
[9] |
G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations, Phys. Rev. E, 47 (1993), 4289-4300.
doi: 10.1103/PhysRevE.47.4289. |
[10] |
G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations, Phys. Rev. E, 47 (1993), 4301-4312.
doi: 10.1103/PhysRevE.47.4301. |
[11] |
Z. Hu, S. M. Wise, C. Wang and J. S. Lowengrub, Stable finite difference, nonlinear multigrid simulation of the phase field crystal equation, J. Comput. Phys., 228 (2009), 5323-5339.
doi: 10.1016/j.jcp.2009.04.020. |
[12] |
R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth, Phys. D, 63 (1993), 410-423.
doi: 10.1016/0167-2789(93)90120-P. |
[13] |
M. Korzec, P. Nayar and P. Rybka, Global weak solutions to a sixth order Cahn-Hilliard type equation, SIAM J. Math. Anal., 44 (2012), 3369-3387.
doi: 10.1137/100817590. |
[14] |
M. Korzec and P. Rybka, On a higher order convective Cahn-Hilliard type equation, SIAM J. Appl. Math., 72 (2012), 1343-1360.
doi: 10.1137/110834123. |
[15] |
A. Miranville, Existence of solutions for a one-dimensional Allen-Cahn equation, J. Appl. Anal. Comput., 3 (2013), 265-277. |
[16] |
A. Makki and A. Miranville, Well-posedness for one-dimensional anisotropic Cahn-Hilliard and Allen-Cahn systems, Electron. J. Differential Equaytions, (2015), 1-15. |
[17] |
T. V. Savina, A. A. Golovin, S. H. Davis, A. A. Nepomnyashchy and P. W. Voorhees, Faceting of a growing crystal surface by surface diffusion, Phys. Rev. E, 67 (2003), 021606.
doi: 10.1103/PhysRevE.67.021606. |
[18] |
S. Torabi, J. Lowengrub, A. Voigt and S. Wise, A new phase-field model for strongly anisotropic systems, Proc. R. Soc. A, 465 (2009), 1337-1359.
doi: 10.1098/rspa.2008.0385. |
[19] |
J. E. Taylor and J. W. Cahn, Diffuse interfaces with sharp corners and facets: Phase-field models with strongly anisotropic surfaces, Phys. D, 112 (1998), 381-411.
doi: 10.1016/S0167-2789(97)00177-2. |
[1] |
Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127 |
[2] |
Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations and Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012 |
[3] |
Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations and Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517 |
[4] |
Quan Wang, Dongming Yan. On the stability and transition of the Cahn-Hilliard/Allen-Cahn system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2607-2620. doi: 10.3934/dcdsb.2020024 |
[5] |
Christopher P. Grant. Grain sizes in the discrete Allen-Cahn and Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 127-146. doi: 10.3934/dcds.2001.7.127 |
[6] |
Alain Miranville, Ramon Quintanilla, Wafa Saoud. Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2257-2288. doi: 10.3934/cpaa.2020099 |
[7] |
Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301 |
[8] |
Jie Shen, Xiaofeng Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1669-1691. doi: 10.3934/dcds.2010.28.1669 |
[9] |
Alain Miranville, Wafa Saoud, Raafat Talhouk. On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3633-3651. doi: 10.3934/dcdsb.2018308 |
[10] |
Haydi Israel. Well-posedness and long time behavior of an Allen-Cahn type equation. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2811-2827. doi: 10.3934/cpaa.2013.12.2811 |
[11] |
Dong Li. A regularization-free approach to the Cahn-Hilliard equation with logarithmic potentials. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2453-2460. doi: 10.3934/dcds.2021198 |
[12] |
Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure and Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461 |
[13] |
Jan Prüss, Vicente Vergara, Rico Zacher. Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 625-647. doi: 10.3934/dcds.2010.26.625 |
[14] |
Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703 |
[15] |
Irena Pawłow. Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1169-1191. doi: 10.3934/dcds.2006.15.1169 |
[16] |
Juan Wen, Yaling He, Yinnian He, Kun Wang. Stabilized finite element methods based on multiscale enrichment for Allen-Cahn and Cahn-Hilliard equations. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1873-1894. doi: 10.3934/cpaa.2021074 |
[17] |
Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207 |
[18] |
Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013 |
[19] |
Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033 |
[20] |
Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]