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Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations
Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature
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 |
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.
References:
[1] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, 1st edition, Elsevier, Amsterdam, 1992. |
[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. |
[3] |
J.W. Cahn and A. Novick-Cohen,
Evolution equations for phase separation and ordering in binary alloys, Statistical Phys., 76 (1994), 877-909.
|
[4] |
C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett., 94 (2005), 154301. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
A. E. Green and P. M. Naghdi,
Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208.
doi: 10.1007/BF00044969. |
[14] |
A. E. Green and P. M. Naghdi,
A new thermoviscous theory for fluids, J. Non-Newtonian Fluid Mech., 56 (1995), 289-306.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[30] |
R. Quintanilla,
On existence in thermoelasticity without energy dissipation, J. Therm. Stresses, 25 (2002), 195-202.
doi: 10.1080/014957302753384423. |
[31] |
R. Quintanilla,
Impossibility of localization in linear thermoelasticity, Proc. Roy. Soci. Lond. A, 463 (2007), 3311-3322.
doi: 10.1098/rspa.2007.0076. |
[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. |
[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. |
[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. |
[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. |
[36] |
R. Quintanilla and B. Straughan,
Nonlinear waves in a Green-Naghdi dissipationless fluid, J. Non-Newtonian Fluid Mech., 154 (2008), 207-210.
|
[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.![]() ![]() ![]() |
[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. |
[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. |
[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.
|
[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. |
[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.
|
[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. |
show all references
References:
[1] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, 1st edition, Elsevier, Amsterdam, 1992. |
[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. |
[3] |
J.W. Cahn and A. Novick-Cohen,
Evolution equations for phase separation and ordering in binary alloys, Statistical Phys., 76 (1994), 877-909.
|
[4] |
C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett., 94 (2005), 154301. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
A. E. Green and P. M. Naghdi,
Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208.
doi: 10.1007/BF00044969. |
[14] |
A. E. Green and P. M. Naghdi,
A new thermoviscous theory for fluids, J. Non-Newtonian Fluid Mech., 56 (1995), 289-306.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[30] |
R. Quintanilla,
On existence in thermoelasticity without energy dissipation, J. Therm. Stresses, 25 (2002), 195-202.
doi: 10.1080/014957302753384423. |
[31] |
R. Quintanilla,
Impossibility of localization in linear thermoelasticity, Proc. Roy. Soci. Lond. A, 463 (2007), 3311-3322.
doi: 10.1098/rspa.2007.0076. |
[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. |
[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. |
[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. |
[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. |
[36] |
R. Quintanilla and B. Straughan,
Nonlinear waves in a Green-Naghdi dissipationless fluid, J. Non-Newtonian Fluid Mech., 154 (2008), 207-210.
|
[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.![]() ![]() ![]() |
[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. |
[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. |
[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.
|
[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. |
[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.
|
[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. |
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