\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

Dedicated to Professor Tomás Caraballo on occasion of his sixtieth birthday

Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: 35B40, 35B41, 35B45, 35K05, 35K51.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, 1st edition, Elsevier, Amsterdam, 1992.
    [2] D. BrochetD. 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 PassoL. 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. EfendievA. 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. EfendievA. 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. EfendievA. 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. FabrieC. GalusinskiA. 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. KrasnyukR. 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. MiranvilleW. 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. MiranvilleW. 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. RobinsonInfinite-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. TonksD. GastonP. C. MillettD. 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. XiaY. 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.
  • 加载中
SHARE

Article Metrics

HTML views(182) PDF downloads(396) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return