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 and Applied Analysis, 2014, 13 (5) : 1971-1987. doi: 10.3934/cpaa.2014.13.1971
References:
[1]

S. Aizicovici and H. Petzeltová, Convergence to equilibria of solutions to a conserved phase-field system with memory, Discrete Cont. Dynam. Systems Ser. S, 2 (2009), 1-16. doi: 10.3934/dcdss.2009.2.1.

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

[3]

D. Brochet, Maximal attractor and inertial sets for some second and fourth order phase field models, Pitman Res. Notes Math. Ser., 296 (1993), 77-85.

[4]

D. Brochet, D. Hilhorst and A. Novick-Cohen, Maximal attractor and inertial sets for a conserved phase field model, Adv. Diff. Eqns., 1 (1996), 547-578.

[5]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.

[6]

G. Caginalp, Conserved-phase field system: Implications for kinetic undercooling, Phys. Rev. B, 38 (1988), 789-791.

[7]

G. Caginalp, The dynamics of a conserved phase-field system: Stefan-like, Hele-Shaw and Cahn-Hilliard models as asymptotic limits, IMA J. Appl. Math., 44 (1990), 77-94. doi: 10.1093/imamat/44.1.77.

[8]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801.

[9]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.

[10]

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

[11]

P. Colli, G. Gilardi, M. Grasselli and G. Schimperna, The conserved phase-field system with memory, Adv. Math. Sci. Appl., 11 (2001), 265-291.

[12]

P. Colli, G. Gilardi, Ph. Laurençot and A. Novick-Cohen, Uniqueness and long-time behavior for the conserved phase-field system with memory, Discrete Cont. Dynam. Systems, 5 (1999), 375-390. doi: 10.3934/dcds.1999.5.375.

[13]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Applied Mathematics, Vol. 37, John-Wiley, New York, 1994.

[14]

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

[15]

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

[16]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 13 (2005), 703-730. doi: 10.1017/S030821050000408X.

[17]

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

[18]

S. Gatti, M. Grasselli and V. Pata, Exponential attractors for a conserved phase-field system with memory, Phys. D, 189 (2004), 31-48. doi: 10.1016/j.physd.2003.10.005.

[19]

G. Gilardi, On a conserved phase field model with irregular potentials and dynamic boundary conditions, Istit. Lombardo Accad. Sci. Lett. Rend. A, 141 (2007), 129-161.

[20]

M. Grasselli, G. Schimperna, A. Segatti and S. Zelik, On the 3D Cahn-Hilliard equation with inertial term, J. Evol. Eqns., 9 (2009), 371-404. doi: 10.1007/s00028-009-0017-7.

[21]

M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term, Commun. Partial Diff. Eqns., 34 (2009), 137-170. doi: 10.1080/03605300802608247.

[22]

M. Grasselli, G. Schimperna and S. Zelik, Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term, Nonlinearity, 23 (2010), 707-737. doi: 10.1088/0951-7715/23/3/016.

[23]

A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Royal Society London A, 432 (1991), 171-194. doi: 10.1098/rspa.1991.0012.

[24]

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

[25]

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.

[26]

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.

[27]

A. Miranville, On the conserved phase-field system, J. Math. Anal. Appl., 400 (2013), 143-152. doi: 10.1016/j.jmaa.2012.11.038.

[28]

A. Miranville, A generalized conserved phase-field system based on type III heat conduction, Quart. Appl. Math., to appear. doi: 10.1090/S0033-569X-2013-01331-1.

[29]

A. Miranville, A reformulation of the Caginalp phase-field system based on the Maxwell-Cattaneo law, submitted.

[30]

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

[31]

A. Miranville and R. Quintanilla, Some generalizations of the Caginalp phase-field system, Appl. Anal., 88 (2009), 877-894. doi: 10.1080/00036810903042182.

[32]

A. Miranville and R. Quintanilla, A phase-field model based on a three-phase-lag heat conduction, Appl. Math. Optim., 63 (2011), 133-150. doi: 10.1007/s00245-010-9114-9.

[33]

A. Miranville and R. Quintanilla, A type III phase-field system with a logarithmic potential, Appl. Math. Letters, 24 (2011), 1003-1008. doi: 10.1016/j.aml.2011.01.016.

[34]

A. Miranville and R. Quintanilla, On a phase-field system based on the Cattaneo law, Nonlinear Anal. TMA, 75 (2012), 2552-2565. doi: 10.1016/j.na.2011.11.001.

[35]

A. Miranville and R. Quintanilla, A conserved phase-field system based on the Maxwell-Cattaneo law, Nonlinear Anal. Real World Appl., 14 (2013), 1680-1692. doi: 10.1016/j.nonrwa.2012.11.004.

[36]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations, Evolutionary Partial Differential Equations (eds. C.M. Dafermos and M. Pokorny), Vol. 4, Elsevier, Amsterdam, 103-200, 2008. doi: 10.1016/S1874-5717(08)00003-0.

[37]

G. Mola, Stability of global and exponential attractors for a three-dimensional conserved phase-field system with memory, Math. Methods Appl. Sci., 32 (2009), 2368-2404. doi: 10.1002/mma.1139.

[38]

A. Novick-Cohen, A conserved phase-field model with memory, GAKUTO International Series. Mathematical Sciences and Applications, Vol. 5 (1995), Gakkotosho Co., Ltd., Tokyo.

[39]

W. Shen and S. Zheng, On the coupled Cahn-Hilliard equation, Comm. Partial Diff. Eqns., 18 (1993), 701-727. doi: 10.1080/03605309308820946.

[40]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

show all references

References:
[1]

S. Aizicovici and H. Petzeltová, Convergence to equilibria of solutions to a conserved phase-field system with memory, Discrete Cont. Dynam. Systems Ser. S, 2 (2009), 1-16. doi: 10.3934/dcdss.2009.2.1.

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

[3]

D. Brochet, Maximal attractor and inertial sets for some second and fourth order phase field models, Pitman Res. Notes Math. Ser., 296 (1993), 77-85.

[4]

D. Brochet, D. Hilhorst and A. Novick-Cohen, Maximal attractor and inertial sets for a conserved phase field model, Adv. Diff. Eqns., 1 (1996), 547-578.

[5]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.

[6]

G. Caginalp, Conserved-phase field system: Implications for kinetic undercooling, Phys. Rev. B, 38 (1988), 789-791.

[7]

G. Caginalp, The dynamics of a conserved phase-field system: Stefan-like, Hele-Shaw and Cahn-Hilliard models as asymptotic limits, IMA J. Appl. Math., 44 (1990), 77-94. doi: 10.1093/imamat/44.1.77.

[8]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801.

[9]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.

[10]

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

[11]

P. Colli, G. Gilardi, M. Grasselli and G. Schimperna, The conserved phase-field system with memory, Adv. Math. Sci. Appl., 11 (2001), 265-291.

[12]

P. Colli, G. Gilardi, Ph. Laurençot and A. Novick-Cohen, Uniqueness and long-time behavior for the conserved phase-field system with memory, Discrete Cont. Dynam. Systems, 5 (1999), 375-390. doi: 10.3934/dcds.1999.5.375.

[13]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Applied Mathematics, Vol. 37, John-Wiley, New York, 1994.

[14]

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

[15]

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

[16]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 13 (2005), 703-730. doi: 10.1017/S030821050000408X.

[17]

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

[18]

S. Gatti, M. Grasselli and V. Pata, Exponential attractors for a conserved phase-field system with memory, Phys. D, 189 (2004), 31-48. doi: 10.1016/j.physd.2003.10.005.

[19]

G. Gilardi, On a conserved phase field model with irregular potentials and dynamic boundary conditions, Istit. Lombardo Accad. Sci. Lett. Rend. A, 141 (2007), 129-161.

[20]

M. Grasselli, G. Schimperna, A. Segatti and S. Zelik, On the 3D Cahn-Hilliard equation with inertial term, J. Evol. Eqns., 9 (2009), 371-404. doi: 10.1007/s00028-009-0017-7.

[21]

M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term, Commun. Partial Diff. Eqns., 34 (2009), 137-170. doi: 10.1080/03605300802608247.

[22]

M. Grasselli, G. Schimperna and S. Zelik, Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term, Nonlinearity, 23 (2010), 707-737. doi: 10.1088/0951-7715/23/3/016.

[23]

A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Royal Society London A, 432 (1991), 171-194. doi: 10.1098/rspa.1991.0012.

[24]

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

[25]

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.

[26]

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.

[27]

A. Miranville, On the conserved phase-field system, J. Math. Anal. Appl., 400 (2013), 143-152. doi: 10.1016/j.jmaa.2012.11.038.

[28]

A. Miranville, A generalized conserved phase-field system based on type III heat conduction, Quart. Appl. Math., to appear. doi: 10.1090/S0033-569X-2013-01331-1.

[29]

A. Miranville, A reformulation of the Caginalp phase-field system based on the Maxwell-Cattaneo law, submitted.

[30]

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

[31]

A. Miranville and R. Quintanilla, Some generalizations of the Caginalp phase-field system, Appl. Anal., 88 (2009), 877-894. doi: 10.1080/00036810903042182.

[32]

A. Miranville and R. Quintanilla, A phase-field model based on a three-phase-lag heat conduction, Appl. Math. Optim., 63 (2011), 133-150. doi: 10.1007/s00245-010-9114-9.

[33]

A. Miranville and R. Quintanilla, A type III phase-field system with a logarithmic potential, Appl. Math. Letters, 24 (2011), 1003-1008. doi: 10.1016/j.aml.2011.01.016.

[34]

A. Miranville and R. Quintanilla, On a phase-field system based on the Cattaneo law, Nonlinear Anal. TMA, 75 (2012), 2552-2565. doi: 10.1016/j.na.2011.11.001.

[35]

A. Miranville and R. Quintanilla, A conserved phase-field system based on the Maxwell-Cattaneo law, Nonlinear Anal. Real World Appl., 14 (2013), 1680-1692. doi: 10.1016/j.nonrwa.2012.11.004.

[36]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations, Evolutionary Partial Differential Equations (eds. C.M. Dafermos and M. Pokorny), Vol. 4, Elsevier, Amsterdam, 103-200, 2008. doi: 10.1016/S1874-5717(08)00003-0.

[37]

G. Mola, Stability of global and exponential attractors for a three-dimensional conserved phase-field system with memory, Math. Methods Appl. Sci., 32 (2009), 2368-2404. doi: 10.1002/mma.1139.

[38]

A. Novick-Cohen, A conserved phase-field model with memory, GAKUTO International Series. Mathematical Sciences and Applications, Vol. 5 (1995), Gakkotosho Co., Ltd., Tokyo.

[39]

W. Shen and S. Zheng, On the coupled Cahn-Hilliard equation, Comm. Partial Diff. Eqns., 18 (1993), 701-727. doi: 10.1080/03605309308820946.

[40]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

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