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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

Quan Wang a, and  Dongming Yan b,,

a. 

Department of Mathematics, Sichuan University, Chengdu 610065, China
b. 

School of Data Sciences, Zhejiang University of Finance & Economics, Hangzhou 310018, China

* Corresponding author: Dongming Yan

Received  January 2019 Published  July 2020 Early access  April 2020

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 & Continuous Dynamical Systems - B, 2020, 25 (7) : 2607-2620. doi: 10.3934/dcdsb.2020024
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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M. Gokieli and L. Marcinkowski, Modelling phase transitions in alloys, Nonlinear Analysis, 63 (2005), e1143–e1153. doi: 10.1016/j.na.2005.03.090.  Google Scholar

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M. Kubo, The Cahn-Hilliard equation with time-dependent constraint, Nonlinear Analysis, 75 (2012), 5672-5685.  doi: 10.1016/j.na.2012.05.015.  Google Scholar

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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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

T. Ma and S. Wang, Phase Transition Dynamics,, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[14]

T. Ma and S. Wang, Bifurcation Theory and Applications,, World Scientific, Singapore, 2005. doi: 10.1142/5798.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

T. Ma and S. Wang, Phase Transition Dynamics,, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[14]

T. Ma and S. Wang, Bifurcation Theory and Applications,, World Scientific, Singapore, 2005. doi: 10.1142/5798.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The topological structure of attractor for $ n = 1 $
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Figure 2.  The topological structure of attractor for $ n = 2 $
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Figure 3.  The topological structure of attractor for $ n = 3 $
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