Figure 1.
The topological structure of attractor for $ n = 1 $
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July
2020, 25(7): 2607-2620.
doi: 10.3934/dcdsb.2020024
On the stability and transition of the Cahn-Hilliard/Allen-Cahn system
| a. | Department of Mathematics, Sichuan University, Chengdu 610065, China |
| b. | School of Data Sciences, Zhejiang University of Finance & Economics, Hangzhou 310018, China |
In this paper, the main objective is to study the stability and transition of the Cahn-Hilliard/Allen-Cahn system. By using the dynamic transition theory, combining with the spectral theorem for general linear completely continuous fields, we prove that the system undergoes a continuous transition and bifurcates from a trivial solution to an attractor as the control parameter crosses a certain critical value. In addition, for some special cases, i.e., the domain is $ n $-dimensional box $ (n = 1,2,3) $, we not only obtain the stability of the singular points of the attractors, the topological structure of the attractors is also illustrated.
Keywords:
Cahn-Hilliard/Allen-Cahn system,
dynamic transition theory,
attractor bifurcation,
topological structure.
Mathematics Subject Classification:
Primary: 35G60; Secondary: 35Q56.
Citation:
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
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References:
| [1] |
D. Brochet, D. Hilhorst and A. Novick-Cohen,
Finite-dimensional exponential attractor for a model for order-disorder and phase separation, Appl. Math. Lett., 7 (1994), 83-87.
doi: 10.1016/0893-9659(94)90118-X. |
| [2] |
J. Cahn and A. Novick-Cohen,
Evolution equations for phase separation and ordering in binary alloys, J. Statist. Phys., 76 (1994), 877-909.
doi: 10.1007/BF02188691. |
| [3] |
H. Chan and J. Wei,
Traveling wave solutions for bistable fractional Allen-Cahn equations with a pyramidal front, J. Differential Equations, 262 (2017), 4567-4609.
doi: 10.1016/j.jde.2016.12.010. |
| [4] |
P. Frank and J. Wei,
Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones, Journal of Functional Analysis, 264 (2013), 1131-1167.
doi: 10.1016/j.jfa.2012.03.010. |
| [5] |
M. Gokieli and A. Ito,
Global attractor for the Cahn-Hilliard/Allen-Cahn system, Nonlinear Analysis, 52 (2003), 1821-1841.
doi: 10.1016/S0362-546X(02)00303-6. |
| [6] |
M. Gokieli and L. Marcinkowski, Modelling phase transitions in alloys, Nonlinear Analysis, 63 (2005), e1143–e1153.
doi: 10.1016/j.na.2005.03.090. |
| [7] |
M. Kubo,
The Cahn-Hilliard equation with time-dependent constraint, Nonlinear Analysis, 75 (2012), 5672-5685.
doi: 10.1016/j.na.2012.05.015. |
| [8] |
C. Laurence and M. Alain,
Finite-dimensional attractors for a model of Allen-Cahn equation based on a microforce balance, Comptes Rendus de l' Academie des Sciences-Series I-Mathematics, 329 (1999), 1109-1114.
doi: 10.1016/S0764-4442(00)88483-9. |
| [9] |
S. Li and D. Yan,
On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3077-3088.
|
| [10] |
D. Li and C. Zhong,
Global attractor for the Cahn-Hilliard system with fast growing nonlinearity, J. Differential Equations, 149 (1998), 191-210.
doi: 10.1006/jdeq.1998.3429. |
| [11] |
H. Liu, T. Sengul and S. Wang, Dynamic transitions for quasilinear systems and Cahn-Hilliard equation with onsager mobility, J. Math. Phys., 53 (2012), 023518, 31 pp.
doi: 10.1063/1.3687414. |
| [12] |
H. Liu, T. Sengul, S. Wang and P. Zhang,
Dynamic transitions and pattern formations for a Cahn-Hilliard model with long-range repulsive interactions, Commun. Math. Sci., 13 (2015), 1289-1315.
doi: 10.4310/CMS.2015.v13.n5.a10. |
| [13] |
T. Ma and S. Wang, Phase Transition Dynamics,, Springer-Verlag, New York, 2014.
doi: 10.1007/978-1-4614-8963-4. |
| [14] |
T. Ma and S. Wang, Bifurcation Theory and Applications,, World Scientific, Singapore, 2005.
doi: 10.1142/5798. |
| [15] |
T. Ma and S. Wang,
Cahn-Hilliard equations and phase transition dynamics for binary systems, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 741-784.
doi: 10.3934/dcdsb.2009.11.741. |
| [16] |
A. Novick-Cohen,
Triple-junction motion for an Allen-Cahn/Cahn-Hilliard system, Physica D, 137 (2000), 1-24.
doi: 10.1016/S0167-2789(99)00162-1. |
| [17] |
L. Song, Y. Zhang and T. Ma,
Global attractor of the Cahn-Hilliard equation in $\text{H}^k$ spaces, J. Math. Anal. Appl., 355 (2009), 53-62.
doi: 10.1016/j.jmaa.2009.01.035. |
show all references
References:
| [1] |
D. Brochet, D. Hilhorst and A. Novick-Cohen,
Finite-dimensional exponential attractor for a model for order-disorder and phase separation, Appl. Math. Lett., 7 (1994), 83-87.
doi: 10.1016/0893-9659(94)90118-X. |
| [2] |
J. Cahn and A. Novick-Cohen,
Evolution equations for phase separation and ordering in binary alloys, J. Statist. Phys., 76 (1994), 877-909.
doi: 10.1007/BF02188691. |
| [3] |
H. Chan and J. Wei,
Traveling wave solutions for bistable fractional Allen-Cahn equations with a pyramidal front, J. Differential Equations, 262 (2017), 4567-4609.
doi: 10.1016/j.jde.2016.12.010. |
| [4] |
P. Frank and J. Wei,
Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones, Journal of Functional Analysis, 264 (2013), 1131-1167.
doi: 10.1016/j.jfa.2012.03.010. |
| [5] |
M. Gokieli and A. Ito,
Global attractor for the Cahn-Hilliard/Allen-Cahn system, Nonlinear Analysis, 52 (2003), 1821-1841.
doi: 10.1016/S0362-546X(02)00303-6. |
| [6] |
M. Gokieli and L. Marcinkowski, Modelling phase transitions in alloys, Nonlinear Analysis, 63 (2005), e1143–e1153.
doi: 10.1016/j.na.2005.03.090. |
| [7] |
M. Kubo,
The Cahn-Hilliard equation with time-dependent constraint, Nonlinear Analysis, 75 (2012), 5672-5685.
doi: 10.1016/j.na.2012.05.015. |
| [8] |
C. Laurence and M. Alain,
Finite-dimensional attractors for a model of Allen-Cahn equation based on a microforce balance, Comptes Rendus de l' Academie des Sciences-Series I-Mathematics, 329 (1999), 1109-1114.
doi: 10.1016/S0764-4442(00)88483-9. |
| [9] |
S. Li and D. Yan,
On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3077-3088.
|
| [10] |
D. Li and C. Zhong,
Global attractor for the Cahn-Hilliard system with fast growing nonlinearity, J. Differential Equations, 149 (1998), 191-210.
doi: 10.1006/jdeq.1998.3429. |
| [11] |
H. Liu, T. Sengul and S. Wang, Dynamic transitions for quasilinear systems and Cahn-Hilliard equation with onsager mobility, J. Math. Phys., 53 (2012), 023518, 31 pp.
doi: 10.1063/1.3687414. |
| [12] |
H. Liu, T. Sengul, S. Wang and P. Zhang,
Dynamic transitions and pattern formations for a Cahn-Hilliard model with long-range repulsive interactions, Commun. Math. Sci., 13 (2015), 1289-1315.
doi: 10.4310/CMS.2015.v13.n5.a10. |
| [13] |
T. Ma and S. Wang, Phase Transition Dynamics,, Springer-Verlag, New York, 2014.
doi: 10.1007/978-1-4614-8963-4. |
| [14] |
T. Ma and S. Wang, Bifurcation Theory and Applications,, World Scientific, Singapore, 2005.
doi: 10.1142/5798. |
| [15] |
T. Ma and S. Wang,
Cahn-Hilliard equations and phase transition dynamics for binary systems, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 741-784.
doi: 10.3934/dcdsb.2009.11.741. |
| [16] |
A. Novick-Cohen,
Triple-junction motion for an Allen-Cahn/Cahn-Hilliard system, Physica D, 137 (2000), 1-24.
doi: 10.1016/S0167-2789(99)00162-1. |
| [17] |
L. Song, Y. Zhang and T. Ma,
Global attractor of the Cahn-Hilliard equation in $\text{H}^k$ spaces, J. Math. Anal. Appl., 355 (2009), 53-62.
doi: 10.1016/j.jmaa.2009.01.035. |
Figure 2.
The topological structure of attractor for $ n = 2 $
Figure 3.
The topological structure of attractor for $ n = 3 $
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