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April  2020, 19(4): 2257-2288. doi: 10.3934/cpaa.2020099

Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature

Alain Miranville 1, , Ramon Quintanilla 2, and  Wafa Saoud 3,

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. 

E.S.E.I.A.A.T. -U.P.C, Departament de Matemátiques, Colom 11, 08222 Terrassa, Barcelona, Spain
3. 

The International University of Beirut, Department of Mathematics and Physics, Tal Abbas Road, Halba-Akkar, Lebanon

Dedicated to Professor Tomás Caraballo on occasion of his sixtieth birthday

Received  November 2018 Revised  September 2019 Published  January 2020

The main goal of this paper is to study the asymptotic behavior of a coupled Cahn-Hilliard/Allen-Cahn system with temperature. The work is divided into two parts: In the first part, the heat equation is based on the usual Fourier law. In the second one, it is based on the type Ⅲ heat conduction law. In both parts, we prove the existence of exponential attractors and, therefore, of finite-dimensional global attractors.

Keywords: Allen-Cahn/Cahn-Hilliard equations, global attractor, exponential attractor, Fourier law, type Ⅲ heat conduction law.
Mathematics Subject Classification: 35B40, 35B41, 35B45, 35K05, 35K51.
Citation: Alain Miranville, Ramon Quintanilla, Wafa Saoud. Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2257-2288. doi: 10.3934/cpaa.2020099
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, 1st edition, Elsevier, Amsterdam, 1992.  Google Scholar

[2]

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

[3]

J.W. Cahn and A. Novick-Cohen, Evolution equations for phase separation and ordering in binary alloys, Statistical Phys., 76 (1994), 877-909.   Google Scholar

[4]

C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett., 94 (2005), 154301. Google Scholar

[5]

R. Dal Passo, L. Giacomelli and A. Novick-Cohen, Existence for an Allen-Cahn/Cahn-Hilliard system with degenerate mobility, Interfaces Free Boundaries, 1 (1999), 199-226.  doi: 10.4171/IFB/9.  Google Scholar

[6]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations, Research in Applied Mathematics, 37, Wiley, New York, 1994.  Google Scholar

[7]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $ {\rm I\!R}^3 $, C. R. Acad. Sci., Paris Sér.I, 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar

[8]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.  doi: 10.1002/mana.200310186.  Google Scholar

[9]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730.  doi: 10.1017/S030821050000408X.  Google Scholar

[10]

P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dyn. Syst., 10 (2004), 211-238.  doi: 10.3934/dcds.2004.10.211.  Google Scholar

[11]

A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Roy. Soc. Lond. A, 432 (1991), 171-194.  doi: 10.1098/rspa.1991.0012.  Google Scholar

[12]

A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Therm. Stresses, 15 (1992), 253-264.  doi: 10.1080/01495739208946136.  Google Scholar

[13]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208.  doi: 10.1007/BF00044969.  Google Scholar

[14]

A. E. Green and P. M. Naghdi, A new thermoviscous theory for fluids, J. Non-Newtonian Fluid Mech., 56 (1995), 289-306.   Google Scholar

[15]

J. Jiang, Convergence to equilibrium for a parabolic-hyperbolic phase-field model with Cattaneo heat flux law, J. Math. Anal. Appl., 341 (2008), 149-169.  doi: 10.1016/j.jmaa.2007.09.041.  Google Scholar

[16]

J. Jiang, Convergence to equilibrium for a fully hyperbolic phase field model with Cattaneo heat flux law, Math. Methods Appl. Sci., 32 (2009), 1156-1182.  doi: 10.1002/mma.1092.  Google Scholar

[17]

I. B. Krasnyuk, R. M. Taranets and M. Chugunova, Long-time oscillating properties of confined disordered binary alloys, Journal of Advanced Research in Applied Mathematics, 7 (2015), 1-16.  doi: 10.5373/jaram.2067.061814.  Google Scholar

[18]

P. C. Millett, S. Rokkam, A. El-Azab, M. Tonks and D. Wolf, Void nucleation and growth in irradiated polycrystalline metals: A phase-field model, Modelling Simul. Mater. Sci. Eng., 17 (2009), 0064003. Google Scholar

[19]

A. Miranville, Some mathematical models in phase transition, Discrete and Continuous Dynamical Systems S, 7 (2014), 271-306.  doi: 10.3934/dcdss.2014.7.271.  Google Scholar

[20]

A. Miranville, Exponential attractors for a class of evolutionary equation by a decomposition method, C. R. Acad. Sci. Paris Sér. I Math., 328, (1999) 145–150. doi: 10.1016/S0764-4442(99)80153-0.  Google Scholar

[21]

A. Miranville, Exponential attractors for a class of evolutionary equation by a decomposition method, C. R. Acad. Sci. Paris Sér. II, 1999. doi: 10.1016/S0764-4442(99)80295-X.  Google Scholar

[22]

A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo heat flux law, Nonlinear Anal. TMA, 71 (2009), 2278-2290.  doi: 10.1016/j.na.2009.01.061.  Google Scholar

[23]

A. Miranville, W. Saoud and R. Talhouk, Asymptotic behavior of a model for order-disorder and phase separation, Asympt. Anal., 103 (2017), 57-76.  doi: 10.3233/ASY-171419.  Google Scholar

[24]

A. Miranville, W. Saoud and R. Talhouk, On the Cahn-Hilliard/Allen-Cahn equations with singular potentials, Discrete Cont. Dynam. Systems Ser. B, 24 (2019), 3633-3651.   Google Scholar

[25]

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.  Google Scholar

[26]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam (2008), 103–200. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[27]

A. Novick-Cohen, Triple-junction motion for an Allen-Cahn/Cahn-Hilliard system, Phys. D, 137 (2000), 1-24.  doi: 10.1016/S0167-2789(99)00162-1.  Google Scholar

[28]

A. Novick-Cohen and L. Peres Hari, Geometric motion for a degenerate Allen-Cahn/Cahn-Hilliard system: The partial wetting case, Phys. D, 209 (2005), 205-235.  doi: 10.1016/j.physd.2005.06.028.  Google Scholar

[29]

R. Quintanilla, Damping of end effects in a thermoelastic theory, Appl. Math. Lett., 14 (2001), 137-141.  doi: 10.1016/S0893-9659(00)00125-7.  Google Scholar

[30]

R. Quintanilla, On existence in thermoelasticity without energy dissipation, J. Therm. Stresses, 25 (2002), 195-202.  doi: 10.1080/014957302753384423.  Google Scholar

[31]

R. Quintanilla, Impossibility of localization in linear thermoelasticity, Proc. Roy. Soci. Lond. A, 463 (2007), 3311-3322.  doi: 10.1098/rspa.2007.0076.  Google Scholar

[32]

R. Quintanilla and R. Racke, Stability in thermoelasticity of type Ⅲ, Discr. Cont. Dyn. Sys. Ser. B, 3 (2003), 383-400.  doi: 10.3934/dcdsb.2003.3.383.  Google Scholar

[33]

R. Quintanilla and B. Straughan, Growth and uniqueness in thermoelasticity, Proc. Roy. Soci. Lond. A, 456 (2000), 1419-1429.  doi: 10.1098/rspa.2000.0569.  Google Scholar

[34]

R. Quintanilla and B. Straughan, Energy bounds for some non-standard problems in thermoelasticity, Proc. Roy. Soci. Lond. A, 461 (2005), 1147-1162.  doi: 10.1098/rspa.2004.1381.  Google Scholar

[35]

R. Quintanilla and B. Straughan, A note on discontinuity waves in type Ⅲ thermoelasticity, Proc. Roy. Soci. Lond. A, 60 (2004), 1169-1175.  doi: 10.1098/rspa.2003.1131.  Google Scholar

[36]

R. Quintanilla and B. Straughan, Nonlinear waves in a Green-Naghdi dissipationless fluid, J. Non-Newtonian Fluid Mech., 154 (2008), 207-210.   Google Scholar

[37] J. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic Pdes and the Theory of Global Attractors, Cambridge University Press, USA, 2001.  doi: 10.1007/978-94-010-0732-0.  Google Scholar
[38]

S. Rokkam, A. El-Azab, P. Millett and D. Wolf, Phase field modeling of void nucleation and growth in irradiated metals, Modelling Simul. Mater. Sci. Eng., 17 (2009) 0064, 002. Google Scholar

[39]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 2nd edition, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[40]

M. R. Tonks, D. Gaston, P. C. Millett, D. Andrs and P. Talbot, An object-oriented finite element framework for multiphysics phase field simulations, Comput. Mater. Sci., 51 (2012), 20-29.   Google Scholar

[41]

L. Wang, J. Lee, M. Anitescu, A. E. Azab, L. C. Mcinnes, T. Munson and B. Smith, A differential variational inequality approach for the simulation of heterogeneous materials, in Proc. SciDAC, 2011. Google Scholar

[42]

Y. Xia, Y. Xu and C. W. Shu, Application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system, Commun. Comput. Phys., 5 (2009), 821-835.   Google Scholar

[43]

C. Yang, X. C. Cai, D. E. Keyes and M. Pernice, NKS method for the implicit solution of a coupled Allen-Cahn/Cahn-Hilliard system, Proceedings of the 21th International Conference on Domain Decomposition Methods, 2012.  Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, 1st edition, Elsevier, Amsterdam, 1992.  Google Scholar

[2]

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

[3]

J.W. Cahn and A. Novick-Cohen, Evolution equations for phase separation and ordering in binary alloys, Statistical Phys., 76 (1994), 877-909.   Google Scholar

[4]

C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett., 94 (2005), 154301. Google Scholar

[5]

R. Dal Passo, L. Giacomelli and A. Novick-Cohen, Existence for an Allen-Cahn/Cahn-Hilliard system with degenerate mobility, Interfaces Free Boundaries, 1 (1999), 199-226.  doi: 10.4171/IFB/9.  Google Scholar

[6]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations, Research in Applied Mathematics, 37, Wiley, New York, 1994.  Google Scholar

[7]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $ {\rm I\!R}^3 $, C. R. Acad. Sci., Paris Sér.I, 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar

[8]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.  doi: 10.1002/mana.200310186.  Google Scholar

[9]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730.  doi: 10.1017/S030821050000408X.  Google Scholar

[10]

P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dyn. Syst., 10 (2004), 211-238.  doi: 10.3934/dcds.2004.10.211.  Google Scholar

[11]

A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Roy. Soc. Lond. A, 432 (1991), 171-194.  doi: 10.1098/rspa.1991.0012.  Google Scholar

[12]

A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Therm. Stresses, 15 (1992), 253-264.  doi: 10.1080/01495739208946136.  Google Scholar

[13]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208.  doi: 10.1007/BF00044969.  Google Scholar

[14]

A. E. Green and P. M. Naghdi, A new thermoviscous theory for fluids, J. Non-Newtonian Fluid Mech., 56 (1995), 289-306.   Google Scholar

[15]

J. Jiang, Convergence to equilibrium for a parabolic-hyperbolic phase-field model with Cattaneo heat flux law, J. Math. Anal. Appl., 341 (2008), 149-169.  doi: 10.1016/j.jmaa.2007.09.041.  Google Scholar

[16]

J. Jiang, Convergence to equilibrium for a fully hyperbolic phase field model with Cattaneo heat flux law, Math. Methods Appl. Sci., 32 (2009), 1156-1182.  doi: 10.1002/mma.1092.  Google Scholar

[17]

I. B. Krasnyuk, R. M. Taranets and M. Chugunova, Long-time oscillating properties of confined disordered binary alloys, Journal of Advanced Research in Applied Mathematics, 7 (2015), 1-16.  doi: 10.5373/jaram.2067.061814.  Google Scholar

[18]

P. C. Millett, S. Rokkam, A. El-Azab, M. Tonks and D. Wolf, Void nucleation and growth in irradiated polycrystalline metals: A phase-field model, Modelling Simul. Mater. Sci. Eng., 17 (2009), 0064003. Google Scholar

[19]

A. Miranville, Some mathematical models in phase transition, Discrete and Continuous Dynamical Systems S, 7 (2014), 271-306.  doi: 10.3934/dcdss.2014.7.271.  Google Scholar

[20]

A. Miranville, Exponential attractors for a class of evolutionary equation by a decomposition method, C. R. Acad. Sci. Paris Sér. I Math., 328, (1999) 145–150. doi: 10.1016/S0764-4442(99)80153-0.  Google Scholar

[21]

A. Miranville, Exponential attractors for a class of evolutionary equation by a decomposition method, C. R. Acad. Sci. Paris Sér. II, 1999. doi: 10.1016/S0764-4442(99)80295-X.  Google Scholar

[22]

A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo heat flux law, Nonlinear Anal. TMA, 71 (2009), 2278-2290.  doi: 10.1016/j.na.2009.01.061.  Google Scholar

[23]

A. Miranville, W. Saoud and R. Talhouk, Asymptotic behavior of a model for order-disorder and phase separation, Asympt. Anal., 103 (2017), 57-76.  doi: 10.3233/ASY-171419.  Google Scholar

[24]

A. Miranville, W. Saoud and R. Talhouk, On the Cahn-Hilliard/Allen-Cahn equations with singular potentials, Discrete Cont. Dynam. Systems Ser. B, 24 (2019), 3633-3651.   Google Scholar

[25]

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.  Google Scholar

[26]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam (2008), 103–200. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[27]

A. Novick-Cohen, Triple-junction motion for an Allen-Cahn/Cahn-Hilliard system, Phys. D, 137 (2000), 1-24.  doi: 10.1016/S0167-2789(99)00162-1.  Google Scholar

[28]

A. Novick-Cohen and L. Peres Hari, Geometric motion for a degenerate Allen-Cahn/Cahn-Hilliard system: The partial wetting case, Phys. D, 209 (2005), 205-235.  doi: 10.1016/j.physd.2005.06.028.  Google Scholar

[29]

R. Quintanilla, Damping of end effects in a thermoelastic theory, Appl. Math. Lett., 14 (2001), 137-141.  doi: 10.1016/S0893-9659(00)00125-7.  Google Scholar

[30]

R. Quintanilla, On existence in thermoelasticity without energy dissipation, J. Therm. Stresses, 25 (2002), 195-202.  doi: 10.1080/014957302753384423.  Google Scholar

[31]

R. Quintanilla, Impossibility of localization in linear thermoelasticity, Proc. Roy. Soci. Lond. A, 463 (2007), 3311-3322.  doi: 10.1098/rspa.2007.0076.  Google Scholar

[32]

R. Quintanilla and R. Racke, Stability in thermoelasticity of type Ⅲ, Discr. Cont. Dyn. Sys. Ser. B, 3 (2003), 383-400.  doi: 10.3934/dcdsb.2003.3.383.  Google Scholar

[33]

R. Quintanilla and B. Straughan, Growth and uniqueness in thermoelasticity, Proc. Roy. Soci. Lond. A, 456 (2000), 1419-1429.  doi: 10.1098/rspa.2000.0569.  Google Scholar

[34]

R. Quintanilla and B. Straughan, Energy bounds for some non-standard problems in thermoelasticity, Proc. Roy. Soci. Lond. A, 461 (2005), 1147-1162.  doi: 10.1098/rspa.2004.1381.  Google Scholar

[35]

R. Quintanilla and B. Straughan, A note on discontinuity waves in type Ⅲ thermoelasticity, Proc. Roy. Soci. Lond. A, 60 (2004), 1169-1175.  doi: 10.1098/rspa.2003.1131.  Google Scholar

[36]

R. Quintanilla and B. Straughan, Nonlinear waves in a Green-Naghdi dissipationless fluid, J. Non-Newtonian Fluid Mech., 154 (2008), 207-210.   Google Scholar

[37] J. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic Pdes and the Theory of Global Attractors, Cambridge University Press, USA, 2001.  doi: 10.1007/978-94-010-0732-0.  Google Scholar
[38]

S. Rokkam, A. El-Azab, P. Millett and D. Wolf, Phase field modeling of void nucleation and growth in irradiated metals, Modelling Simul. Mater. Sci. Eng., 17 (2009) 0064, 002. Google Scholar

[39]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 2nd edition, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[40]

M. R. Tonks, D. Gaston, P. C. Millett, D. Andrs and P. Talbot, An object-oriented finite element framework for multiphysics phase field simulations, Comput. Mater. Sci., 51 (2012), 20-29.   Google Scholar

[41]

L. Wang, J. Lee, M. Anitescu, A. E. Azab, L. C. Mcinnes, T. Munson and B. Smith, A differential variational inequality approach for the simulation of heterogeneous materials, in Proc. SciDAC, 2011. Google Scholar

[42]

Y. Xia, Y. Xu and C. W. Shu, Application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system, Commun. Comput. Phys., 5 (2009), 821-835.   Google Scholar

[43]

C. Yang, X. C. Cai, D. E. Keyes and M. Pernice, NKS method for the implicit solution of a coupled Allen-Cahn/Cahn-Hilliard system, Proceedings of the 21th International Conference on Domain Decomposition Methods, 2012.  Google Scholar

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