# American Institute of Mathematical Sciences

September  2014, 13(5): 1971-1987. doi: 10.3934/cpaa.2014.13.1971

## Asymptotic behavior of the conserved Caginalp phase-field system based on the Maxwell-Cattaneo law

 1 Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 6086 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex

Received  September 2013 Revised  September 2014 Published  June 2014

Our aim in this paper is to study the well-posedness and the asymptotic behavior, in terms of finite-dimensional attractors, for the conserved Caginalp phase-field system based on the Maxwell-Cattaneo law, instead of the usual Fourier law, for heat conduction. The system consists of the equation for the order parameter and that for the enthalpy, instead of the relative temperature or the thermal displacement variable.
Citation: Alain Miranville. Asymptotic behavior of the conserved Caginalp phase-field system based on the Maxwell-Cattaneo law. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1971-1987. doi: 10.3934/cpaa.2014.13.1971
##### References:
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Caginalp, The dynamics of a conserved phase-field system: Stefan-like, Hele-Shaw and Cahn-Hilliard models as asymptotic limits,, \emph{IMA J. Appl. Math.}, 44 (1990), 77. doi: 10.1093/imamat/44.1.77. Google Scholar [8] J. W. Cahn, On spinodal decomposition,, \emph{Acta Metall.}, 9 (1961), 795. Google Scholar [9] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy,, \emph{J. Chem. Phys.}, 28 (1958), 258. Google Scholar [10] C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media,, \emph{ Phys. Rev. Letters}, 94 (2005). Google Scholar [11] P. Colli, G. Gilardi, M. Grasselli and G. Schimperna, The conserved phase-field system with memory,, \emph{Adv. Math. Sci. Appl.}, 11 (2001), 265. Google Scholar [12] P. Colli, G. Gilardi, Ph. Laurençot and A. Novick-Cohen, Uniqueness and long-time behavior for the conserved phase-field system with memory,, \emph{Discrete Cont. Dynam. Systems}, 5 (1999), 375. doi: 10.3934/dcds.1999.5.375. Google Scholar [13] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations,, Research in Applied Mathematics, (1994). Google Scholar [14] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $R^3$,, \emph{C.R. Acad. Sci. Paris S\'erie I Math.}, 330 (2000), 713. doi: 10.1016/S0764-4442(00)00259-7. Google Scholar [15] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system,, \emph{Math. Nach.}, 272 (2004), 11. doi: 10.1002/mana.200310186. Google Scholar [16] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 13 (2005), 703. doi: 10.1017/S030821050000408X. Google Scholar [17] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation,, \emph{Discrete Contin. Dyn. Systems}, 10 (2004), 211. doi: 10.3934/dcds.2004.10.211. Google Scholar [18] S. Gatti, M. Grasselli and V. Pata, Exponential attractors for a conserved phase-field system with memory,, \emph{Phys. D}, 189 (2004), 31. doi: 10.1016/j.physd.2003.10.005. Google Scholar [19] G. Gilardi, On a conserved phase field model with irregular potentials and dynamic boundary conditions,, \emph{Istit. Lombardo Accad. Sci. Lett. Rend. A}, 141 (2007), 129. Google Scholar [20] M. Grasselli, G. Schimperna, A. Segatti and S. Zelik, On the 3D Cahn-Hilliard equation with inertial term,, \emph{J. Evol. Eqns.}, 9 (2009), 371. doi: 10.1007/s00028-009-0017-7. Google Scholar [21] M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term,, \emph{Commun. Partial Diff. Eqns.}, 34 (2009), 137. doi: 10.1080/03605300802608247. Google Scholar [22] M. Grasselli, G. Schimperna and S. Zelik, Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term,, \emph{Nonlinearity}, 23 (2010), 707. doi: 10.1088/0951-7715/23/3/016. Google Scholar [23] A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics,, \emph{Proc. Royal Society London A}, 432 (1991), 171. doi: 10.1098/rspa.1991.0012. Google Scholar [24] A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid,, \emph{J. Thermal Stresses}, 15 (1992), 253. doi: 10.1080/01495739208946136. Google Scholar [25] J. Jiang, Convergence to equilibrium for a parabolic-hyperbolic phase-field model with Cattaneo heat flux law,, \emph{J. Math. Anal. Appl.}, 341 (2008), 149. doi: 10.1016/j.jmaa.2007.09.041. Google Scholar [26] J. Jiang, Convergence to equilibrium for a fully hyperbolic phase field model with Cattaneo heat flux law,, \emph{Math. Methods Appl. Sci.}, 32 (2009), 1156. doi: 10.1002/mma.1092. Google Scholar [27] A. Miranville, On the conserved phase-field system,, \emph{J. Math. Anal. Appl.}, 400 (2013), 143. doi: 10.1016/j.jmaa.2012.11.038. Google Scholar [28] A. Miranville, A generalized conserved phase-field system based on type III heat conduction,, \emph{Quart. Appl. Math.}, (). doi: 10.1090/S0033-569X-2013-01331-1. Google Scholar [29] A. Miranville, A reformulation of the Caginalp phase-field system based on the Maxwell-Cattaneo law,, submitted., (). Google Scholar [30] A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo law,, \emph{Nonlinear Anal. TMA}, 71 (2009), 2278. doi: 10.1016/j.na.2009.01.061. Google Scholar [31] A. Miranville and R. Quintanilla, Some generalizations of the Caginalp phase-field system,, \emph{Appl. Anal.}, 88 (2009), 877. doi: 10.1080/00036810903042182. Google Scholar [32] A. Miranville and R. Quintanilla, A phase-field model based on a three-phase-lag heat conduction,, \emph{Appl. Math. Optim.}, 63 (2011), 133. doi: 10.1007/s00245-010-9114-9. Google Scholar [33] A. Miranville and R. Quintanilla, A type III phase-field system with a logarithmic potential,, \emph{Appl. Math. Letters}, 24 (2011), 1003. doi: 10.1016/j.aml.2011.01.016. Google Scholar [34] A. Miranville and R. Quintanilla, On a phase-field system based on the Cattaneo law,, \emph{Nonlinear Anal. TMA}, 75 (2012), 2552. doi: 10.1016/j.na.2011.11.001. Google Scholar [35] A. Miranville and R. Quintanilla, A conserved phase-field system based on the Maxwell-Cattaneo law,, \emph{Nonlinear Anal. Real World Appl.}, 14 (2013), 1680. doi: 10.1016/j.nonrwa.2012.11.004. Google Scholar [36] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in \emph{Handbook of Differential Equations, 4 (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar [37] G. Mola, Stability of global and exponential attractors for a three-dimensional conserved phase-field system with memory,, \emph{Math. Methods Appl. Sci.}, 32 (2009), 2368. doi: 10.1002/mma.1139. Google Scholar [38] A. Novick-Cohen, A conserved phase-field model with memory,, GAKUTO International Series. Mathematical Sciences and Applications, 5 (1995). Google Scholar [39] W. Shen and S. Zheng, On the coupled Cahn-Hilliard equation,, \emph{Comm. Partial Diff. Eqns.}, 18 (1993), 701. doi: 10.1080/03605309308820946. Google Scholar [40] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Second edition, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar

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##### References:
 [1] S. Aizicovici and H. Petzeltová, Convergence to equilibria of solutions to a conserved phase-field system with memory,, \emph{Discrete Cont. Dynam. Systems Ser. S}, 2 (2009), 1. doi: 10.3934/dcdss.2009.2.1. Google Scholar [2] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992). Google Scholar [3] D. Brochet, Maximal attractor and inertial sets for some second and fourth order phase field models,, \emph{Pitman Res. Notes Math. Ser.}, 296 (1993), 77. Google Scholar [4] D. Brochet, D. Hilhorst and A. Novick-Cohen, Maximal attractor and inertial sets for a conserved phase field model,, \emph{Adv. Diff. Eqns.}, 1 (1996), 547. Google Scholar [5] G. Caginalp, An analysis of a phase field model of a free boundary,, \emph{Arch. Rational Mech. Anal.}, 92 (1986), 205. doi: 10.1007/BF00254827. Google Scholar [6] G. Caginalp, Conserved-phase field system: Implications for kinetic undercooling,, \emph{Phys. Rev. B}, 38 (1988), 789. Google Scholar [7] G. Caginalp, The dynamics of a conserved phase-field system: Stefan-like, Hele-Shaw and Cahn-Hilliard models as asymptotic limits,, \emph{IMA J. Appl. Math.}, 44 (1990), 77. doi: 10.1093/imamat/44.1.77. Google Scholar [8] J. W. Cahn, On spinodal decomposition,, \emph{Acta Metall.}, 9 (1961), 795. Google Scholar [9] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy,, \emph{J. Chem. Phys.}, 28 (1958), 258. Google Scholar [10] C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media,, \emph{ Phys. Rev. Letters}, 94 (2005). Google Scholar [11] P. Colli, G. Gilardi, M. Grasselli and G. Schimperna, The conserved phase-field system with memory,, \emph{Adv. Math. Sci. Appl.}, 11 (2001), 265. Google Scholar [12] P. Colli, G. Gilardi, Ph. Laurençot and A. Novick-Cohen, Uniqueness and long-time behavior for the conserved phase-field system with memory,, \emph{Discrete Cont. Dynam. Systems}, 5 (1999), 375. doi: 10.3934/dcds.1999.5.375. Google Scholar [13] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations,, Research in Applied Mathematics, (1994). Google Scholar [14] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $R^3$,, \emph{C.R. Acad. Sci. Paris S\'erie I Math.}, 330 (2000), 713. doi: 10.1016/S0764-4442(00)00259-7. Google Scholar [15] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system,, \emph{Math. Nach.}, 272 (2004), 11. doi: 10.1002/mana.200310186. Google Scholar [16] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 13 (2005), 703. doi: 10.1017/S030821050000408X. Google Scholar [17] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation,, \emph{Discrete Contin. Dyn. Systems}, 10 (2004), 211. doi: 10.3934/dcds.2004.10.211. Google Scholar [18] S. Gatti, M. Grasselli and V. Pata, Exponential attractors for a conserved phase-field system with memory,, \emph{Phys. D}, 189 (2004), 31. doi: 10.1016/j.physd.2003.10.005. Google Scholar [19] G. Gilardi, On a conserved phase field model with irregular potentials and dynamic boundary conditions,, \emph{Istit. Lombardo Accad. Sci. Lett. Rend. A}, 141 (2007), 129. Google Scholar [20] M. Grasselli, G. Schimperna, A. Segatti and S. Zelik, On the 3D Cahn-Hilliard equation with inertial term,, \emph{J. Evol. Eqns.}, 9 (2009), 371. doi: 10.1007/s00028-009-0017-7. Google Scholar [21] M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term,, \emph{Commun. Partial Diff. Eqns.}, 34 (2009), 137. doi: 10.1080/03605300802608247. Google Scholar [22] M. Grasselli, G. Schimperna and S. Zelik, Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term,, \emph{Nonlinearity}, 23 (2010), 707. doi: 10.1088/0951-7715/23/3/016. Google Scholar [23] A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics,, \emph{Proc. Royal Society London A}, 432 (1991), 171. doi: 10.1098/rspa.1991.0012. Google Scholar [24] A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid,, \emph{J. Thermal Stresses}, 15 (1992), 253. doi: 10.1080/01495739208946136. Google Scholar [25] J. Jiang, Convergence to equilibrium for a parabolic-hyperbolic phase-field model with Cattaneo heat flux law,, \emph{J. Math. Anal. Appl.}, 341 (2008), 149. doi: 10.1016/j.jmaa.2007.09.041. Google Scholar [26] J. Jiang, Convergence to equilibrium for a fully hyperbolic phase field model with Cattaneo heat flux law,, \emph{Math. Methods Appl. Sci.}, 32 (2009), 1156. doi: 10.1002/mma.1092. Google Scholar [27] A. Miranville, On the conserved phase-field system,, \emph{J. Math. Anal. Appl.}, 400 (2013), 143. doi: 10.1016/j.jmaa.2012.11.038. Google Scholar [28] A. Miranville, A generalized conserved phase-field system based on type III heat conduction,, \emph{Quart. Appl. Math.}, (). doi: 10.1090/S0033-569X-2013-01331-1. Google Scholar [29] A. Miranville, A reformulation of the Caginalp phase-field system based on the Maxwell-Cattaneo law,, submitted., (). Google Scholar [30] A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo law,, \emph{Nonlinear Anal. TMA}, 71 (2009), 2278. doi: 10.1016/j.na.2009.01.061. Google Scholar [31] A. Miranville and R. Quintanilla, Some generalizations of the Caginalp phase-field system,, \emph{Appl. Anal.}, 88 (2009), 877. doi: 10.1080/00036810903042182. Google Scholar [32] A. Miranville and R. Quintanilla, A phase-field model based on a three-phase-lag heat conduction,, \emph{Appl. Math. Optim.}, 63 (2011), 133. doi: 10.1007/s00245-010-9114-9. Google Scholar [33] A. Miranville and R. Quintanilla, A type III phase-field system with a logarithmic potential,, \emph{Appl. Math. Letters}, 24 (2011), 1003. doi: 10.1016/j.aml.2011.01.016. Google Scholar [34] A. Miranville and R. Quintanilla, On a phase-field system based on the Cattaneo law,, \emph{Nonlinear Anal. TMA}, 75 (2012), 2552. doi: 10.1016/j.na.2011.11.001. Google Scholar [35] A. Miranville and R. Quintanilla, A conserved phase-field system based on the Maxwell-Cattaneo law,, \emph{Nonlinear Anal. Real World Appl.}, 14 (2013), 1680. doi: 10.1016/j.nonrwa.2012.11.004. Google Scholar [36] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in \emph{Handbook of Differential Equations, 4 (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar [37] G. Mola, Stability of global and exponential attractors for a three-dimensional conserved phase-field system with memory,, \emph{Math. Methods Appl. Sci.}, 32 (2009), 2368. doi: 10.1002/mma.1139. Google Scholar [38] A. Novick-Cohen, A conserved phase-field model with memory,, GAKUTO International Series. Mathematical Sciences and Applications, 5 (1995). Google Scholar [39] W. Shen and S. Zheng, On the coupled Cahn-Hilliard equation,, \emph{Comm. Partial Diff. Eqns.}, 18 (1993), 701. doi: 10.1080/03605309308820946. Google Scholar [40] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Second edition, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar
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