April  2014, 7(2): 271-306. doi: 10.3934/dcdss.2014.7.271

Some mathematical models in phase transition

1. 

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

Received  March 2013 Revised  July 2013 Published  September 2013

Our aim in these notes is to discuss the well-posedness and the asymptotic behavior, in terms of finite-dimensional attractors, of models in phase transition. In particular, we focus on the Caginalp phase field model.
Citation: Alain Miranville. Some mathematical models in phase transition. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 271-306. doi: 10.3934/dcdss.2014.7.271
References:
[1]

S. Aizicovici, E. Feireisl and F. Issard-Roch, Long-time convergence of solutions to a phase-field system, Math. Methods Appl. Sci., 24 (2001), 277-287. doi: 10.1002/mma.215.  Google Scholar

[2]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Commun. Partial Diff. Eqns., 4 (1979), 827-868. doi: 10.1080/03605307908820113.  Google Scholar

[3]

S. M. Allen and J. W. Cahn, A microscopic theory of antiphase boundary motion and its application to antiphase domain coarsening, J. Chem. Phys., 28 (1957), 258-267. doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[4]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[5]

P. W. Bates and S. Zheng, Inertial manifolds and inertial sets for the phase-field equations, J. Dyn. Diff. Eqns., 4 (1992), 375-398. doi: 10.1007/BF01049391.  Google Scholar

[6]

D. Brochet, X. Chen and D. Hilhorst, Finite dimensional exponential attractors for the phase-field model, Appl. Anal., 49 (1993), 197-212. doi: 10.1080/00036819108840173.  Google Scholar

[7]

D. Brochet and D. Hilhorst, Universal attractor and inertial sets for the phase field model, Appl. Math. Letters, 4 (1991), 59-62. doi: 10.1016/0893-9659(91)90076-8.  Google Scholar

[8]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Applied Mathematical Sciences, 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[9]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 2 (1958), 258-267. doi: 10.1063/1.1744102.  Google Scholar

[10]

G. Caginalp, The role of microscopic anisotropy in the macroscopic behavior of a phase boundary, Ann. Physics, 172 (1986), 136-155. doi: 10.1016/0003-4916(86)90022-9.  Google Scholar

[11]

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.  Google Scholar

[12]

G. Caginalp, Stefan and Hele-shaw type models as asymptotic limits of the phase-field equations, Phys. Review A, 39 (1989), 5887-5896. doi: 10.1103/PhysRevA.39.5887.  Google Scholar

[13]

G. Caginalp and X. Chen, Phase field equations in the singular limit of sharp interface problems, in "On the Evolution of Phase Boundaries" (ed. M. Gurtin), IMA Vol. Math. Appl., 43, Springer, New York, (1992), 1-27. doi: 10.1007/978-1-4613-9211-8_1.  Google Scholar

[14]

G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, European J. Appl. Math., 9 (1998), 417-445. doi: 10.1017/S0956792598003520.  Google Scholar

[15]

G. Caginalp, X. Chen and C. Eck, Numerical tests of a phase field model with second order accuracy, SIAM J. Appl. Math., 68 (2008), 1518-1534. doi: 10.1137/070680965.  Google Scholar

[16]

G. Caginalp and E. A. Socolovsky, Efficient computation of a sharp interface spreading via phase field methods, Appl. Math. Letters, 2 (1989), 117-120. doi: 10.1016/0893-9659(89)90002-5.  Google Scholar

[17]

B. Chalmers, "Principles of Solidification," R. E. Krieger Publishing, Huntington, New York, 1977. doi: 10.1007/978-1-4684-1854-5_5.  Google Scholar

[18]

X. Chen, G. Caginalp and C. Eck, A rapidly converging phase field model, Discrete Cont. Dyn. Systems, 15 (2006), 1107-1034. Google Scholar

[19]

L. Cherfils and A. Miranville, Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. Math. Sci. Appl., 17 (2007), 107-129.  Google Scholar

[20]

L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115. doi: 10.1007/s10492-009-0008-6.  Google Scholar

[21]

R. Chill, E. Fašangovà and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462. doi: 10.1002/mana.200410431.  Google Scholar

[22]

C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Review Letters, 94 (2005), 154301. doi: 10.1103/PhysRevLett.94.154301.  Google Scholar

[23]

C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations, in "Free Boundary Value Problems" (Oberwolfach, 1989), Internat. Ser. Numer. Math., Vol. 95, Birkhäuser, Basel, (1990), 46-58.  Google Scholar

[24]

H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Review Letters, 79 (1997), 893-896. doi: 10.1103/PhysRevLett.79.893.  Google Scholar

[25]

H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Letters, 42 (1998), 49-54. Google Scholar

[26]

C. G. Gal, A Cahn-Hilliard model in bounded domains with permeable walls, Math. Methods Appl. Sci., 29 (2006), 2009-2036. doi: 10.1002/mma.757.  Google Scholar

[27]

C. G. Gal and M. Grasselli, On the asymptotic behavior of the Caginalp system with dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 689-710. doi: 10.3934/cpaa.2009.8.689.  Google Scholar

[28]

C. G. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions, in "Nonlinear Phenomena with Energy Dissipation: Mathematical Analysis, Modeling and Simulation," Gakuto Int. Ser. Math. Sci. Appl., Vol. {29}, Gakkōtosho, Tokyo, (2008), 117-139.  Google Scholar

[29]

S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in "Differential Equations: Inverse and Direct Problems" (eds. A. Favini and A. Lorenzi), Lecture Notes in Pure and Applied Mathematics, Vol. 251, Chapman & Hall, Boca Raton, FL, (2006), 149-170. doi: 10.1201/9781420011135.ch9.  Google Scholar

[30]

J. W. Gibbs, "Collected Works," Yale University Press, New Haven, 1948. Google Scholar

[31]

G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912. doi: 10.3934/cpaa.2009.8.881.  Google Scholar

[32]

M. Grasselli, A. Miranville, V. Pata and S. Zelik, Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, Math. Nachr., 280 (2007), 1475-1509. doi: 10.1002/mana.200510560.  Google Scholar

[33]

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.  Google Scholar

[34]

M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5.  Google Scholar

[35]

S. I. Hariharan and G. W. Young, Comparison of asymptotic solutions of a phase-field model to a sharp-interface model, SIAM J. Appl. Math., 62 (2001), 244-263. doi: 10.1137/S0036139900374908.  Google Scholar

[36]

B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity, 12 (1999), 1263-1275. doi: 10.1088/0951-7715/12/5/303.  Google Scholar

[37]

B. R. Hunt, T. Sauer and J. A. Yorke, Prevalence: A translation-invariant "almost every'' for infinite-dimensional spaces, Bull. Amer. Math. Soc., 27 (1992), 217-238. doi: 10.1090/S0273-0979-1992-00328-2.  Google Scholar

[38]

M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal., 153 (1998), 187-202. doi: 10.1006/jfan.1997.3174.  Google Scholar

[39]

G. Lamé and B. P. Clapeyron, Mémoire sur la solidification par refroidissement d'un globe solide, Ann. Chem. Phys., 47 (1831), 250-256. Google Scholar

[40]

L. D. Landau and E. M. Lifschitz, "Statistical Physics (Part 1)," Third edition, Pergamon, New York, 1980. Google Scholar

[41]

S. Łojasiewicz, "Ensembles Semi-Analytiques," IHES, Bures-sur-Yvette, 1965. Google Scholar

[42]

A. M. Meirmanov, The classical solution of a multidimensional Stefan problem for quasilinear parabolic equations (in Russian), Mat. Sb. (N.S.), 112 (1980), 170-192.  Google Scholar

[43]

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.  Google Scholar

[44]

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

[45]

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

[46]

A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Cont. Dyn. Systems, 28 (2010), 275-310. doi: 10.3934/dcds.2010.28.275.  Google Scholar

[47]

O. A. Oleinik, A method of solution of the general Stefan problem, Soviet Math. Dokl., 1 (1960), 1350-1353.  Google Scholar

[48]

O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43 (1990), 44-62. doi: 10.1016/0167-2789(90)90015-H.  Google Scholar

[49]

O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a "thermodynamically consistent'' phase-field model, Physica D, 69 (1993), 107-113. doi: 10.1016/0167-2789(93)90183-2.  Google Scholar

[50]

J. C. Robinson, Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces, Nonlinearity, 22 (2009), 711-728. doi: 10.1088/0951-7715/22/4/001.  Google Scholar

[51]

L. Rubinstein, On the solution of Stefan's problem, (in Russian) Izvestia Akad. Nauk SSSR, 11 (1947), 37-54.  Google Scholar

[52]

S. I. Serdyukov, N. M. Voskresenskii, V. K. Bel'nov and I. I. Karpov, Extended irreversible thermodynamics and generalization of the dual-phase-lag model in heat transfer, J. Non-Equilib. Thermodyn., 28 (2003), 1-13. doi: 10.1515/JNETDY.2003.013.  Google Scholar

[53]

L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. Math., 118 (1983), 525-571. doi: 10.2307/2006981.  Google Scholar

[54]

J. Sprekels and S. Zheng, Global smooth solutions to a thermodynamically consistent model of phase-field type in higher space dimensions, J. Math. Anal. Appl., 176 (1993), 200-223. doi: 10.1006/jmaa.1993.1209.  Google Scholar

[55]

J. Stefan, Uber einige Probleme der Theorie der Warmeleitung, S.-B. Wien Akad. Mat. Natur., 98 (1889), 173-484. Google Scholar

[56]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997.  Google Scholar

[57]

A. Visintin, Introduction to Stefan-type problems, in "Handbook of Differential Equations: Evolutionary Equations. Vol. IV" (eds. C. M. Dafermos and M. Pokorny), Elsevier/North Holland, Amsterdam, (2008), 377-484. doi: 10.1016/S1874-5717(08)00008-X.  Google Scholar

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S. Zelik, The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and its dimension, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 24 (2000), 1-25.  Google Scholar

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Z. Zhang, Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions, Commun. Pure Appl. Anal., 4 (2005), 683-693. doi: 10.3934/cpaa.2005.4.683.  Google Scholar

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S. Zheng, Global existence for a thermodynamically consistent model of phase field type, Diff. Integral Eqns., 5 (1992), 241-253.  Google Scholar

show all references

References:
[1]

S. Aizicovici, E. Feireisl and F. Issard-Roch, Long-time convergence of solutions to a phase-field system, Math. Methods Appl. Sci., 24 (2001), 277-287. doi: 10.1002/mma.215.  Google Scholar

[2]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Commun. Partial Diff. Eqns., 4 (1979), 827-868. doi: 10.1080/03605307908820113.  Google Scholar

[3]

S. M. Allen and J. W. Cahn, A microscopic theory of antiphase boundary motion and its application to antiphase domain coarsening, J. Chem. Phys., 28 (1957), 258-267. doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[4]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[5]

P. W. Bates and S. Zheng, Inertial manifolds and inertial sets for the phase-field equations, J. Dyn. Diff. Eqns., 4 (1992), 375-398. doi: 10.1007/BF01049391.  Google Scholar

[6]

D. Brochet, X. Chen and D. Hilhorst, Finite dimensional exponential attractors for the phase-field model, Appl. Anal., 49 (1993), 197-212. doi: 10.1080/00036819108840173.  Google Scholar

[7]

D. Brochet and D. Hilhorst, Universal attractor and inertial sets for the phase field model, Appl. Math. Letters, 4 (1991), 59-62. doi: 10.1016/0893-9659(91)90076-8.  Google Scholar

[8]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Applied Mathematical Sciences, 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[9]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 2 (1958), 258-267. doi: 10.1063/1.1744102.  Google Scholar

[10]

G. Caginalp, The role of microscopic anisotropy in the macroscopic behavior of a phase boundary, Ann. Physics, 172 (1986), 136-155. doi: 10.1016/0003-4916(86)90022-9.  Google Scholar

[11]

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.  Google Scholar

[12]

G. Caginalp, Stefan and Hele-shaw type models as asymptotic limits of the phase-field equations, Phys. Review A, 39 (1989), 5887-5896. doi: 10.1103/PhysRevA.39.5887.  Google Scholar

[13]

G. Caginalp and X. Chen, Phase field equations in the singular limit of sharp interface problems, in "On the Evolution of Phase Boundaries" (ed. M. Gurtin), IMA Vol. Math. Appl., 43, Springer, New York, (1992), 1-27. doi: 10.1007/978-1-4613-9211-8_1.  Google Scholar

[14]

G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, European J. Appl. Math., 9 (1998), 417-445. doi: 10.1017/S0956792598003520.  Google Scholar

[15]

G. Caginalp, X. Chen and C. Eck, Numerical tests of a phase field model with second order accuracy, SIAM J. Appl. Math., 68 (2008), 1518-1534. doi: 10.1137/070680965.  Google Scholar

[16]

G. Caginalp and E. A. Socolovsky, Efficient computation of a sharp interface spreading via phase field methods, Appl. Math. Letters, 2 (1989), 117-120. doi: 10.1016/0893-9659(89)90002-5.  Google Scholar

[17]

B. Chalmers, "Principles of Solidification," R. E. Krieger Publishing, Huntington, New York, 1977. doi: 10.1007/978-1-4684-1854-5_5.  Google Scholar

[18]

X. Chen, G. Caginalp and C. Eck, A rapidly converging phase field model, Discrete Cont. Dyn. Systems, 15 (2006), 1107-1034. Google Scholar

[19]

L. Cherfils and A. Miranville, Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. Math. Sci. Appl., 17 (2007), 107-129.  Google Scholar

[20]

L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115. doi: 10.1007/s10492-009-0008-6.  Google Scholar

[21]

R. Chill, E. Fašangovà and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462. doi: 10.1002/mana.200410431.  Google Scholar

[22]

C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Review Letters, 94 (2005), 154301. doi: 10.1103/PhysRevLett.94.154301.  Google Scholar

[23]

C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations, in "Free Boundary Value Problems" (Oberwolfach, 1989), Internat. Ser. Numer. Math., Vol. 95, Birkhäuser, Basel, (1990), 46-58.  Google Scholar

[24]

H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Review Letters, 79 (1997), 893-896. doi: 10.1103/PhysRevLett.79.893.  Google Scholar

[25]

H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Letters, 42 (1998), 49-54. Google Scholar

[26]

C. G. Gal, A Cahn-Hilliard model in bounded domains with permeable walls, Math. Methods Appl. Sci., 29 (2006), 2009-2036. doi: 10.1002/mma.757.  Google Scholar

[27]

C. G. Gal and M. Grasselli, On the asymptotic behavior of the Caginalp system with dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 689-710. doi: 10.3934/cpaa.2009.8.689.  Google Scholar

[28]

C. G. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions, in "Nonlinear Phenomena with Energy Dissipation: Mathematical Analysis, Modeling and Simulation," Gakuto Int. Ser. Math. Sci. Appl., Vol. {29}, Gakkōtosho, Tokyo, (2008), 117-139.  Google Scholar

[29]

S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in "Differential Equations: Inverse and Direct Problems" (eds. A. Favini and A. Lorenzi), Lecture Notes in Pure and Applied Mathematics, Vol. 251, Chapman & Hall, Boca Raton, FL, (2006), 149-170. doi: 10.1201/9781420011135.ch9.  Google Scholar

[30]

J. W. Gibbs, "Collected Works," Yale University Press, New Haven, 1948. Google Scholar

[31]

G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912. doi: 10.3934/cpaa.2009.8.881.  Google Scholar

[32]

M. Grasselli, A. Miranville, V. Pata and S. Zelik, Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, Math. Nachr., 280 (2007), 1475-1509. doi: 10.1002/mana.200510560.  Google Scholar

[33]

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.  Google Scholar

[34]

M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5.  Google Scholar

[35]

S. I. Hariharan and G. W. Young, Comparison of asymptotic solutions of a phase-field model to a sharp-interface model, SIAM J. Appl. Math., 62 (2001), 244-263. doi: 10.1137/S0036139900374908.  Google Scholar

[36]

B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity, 12 (1999), 1263-1275. doi: 10.1088/0951-7715/12/5/303.  Google Scholar

[37]

B. R. Hunt, T. Sauer and J. A. Yorke, Prevalence: A translation-invariant "almost every'' for infinite-dimensional spaces, Bull. Amer. Math. Soc., 27 (1992), 217-238. doi: 10.1090/S0273-0979-1992-00328-2.  Google Scholar

[38]

M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon, J. Funct. Anal., 153 (1998), 187-202. doi: 10.1006/jfan.1997.3174.  Google Scholar

[39]

G. Lamé and B. P. Clapeyron, Mémoire sur la solidification par refroidissement d'un globe solide, Ann. Chem. Phys., 47 (1831), 250-256. Google Scholar

[40]

L. D. Landau and E. M. Lifschitz, "Statistical Physics (Part 1)," Third edition, Pergamon, New York, 1980. Google Scholar

[41]

S. Łojasiewicz, "Ensembles Semi-Analytiques," IHES, Bures-sur-Yvette, 1965. Google Scholar

[42]

A. M. Meirmanov, The classical solution of a multidimensional Stefan problem for quasilinear parabolic equations (in Russian), Mat. Sb. (N.S.), 112 (1980), 170-192.  Google Scholar

[43]

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.  Google Scholar

[44]

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

[45]

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

[46]

A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Cont. Dyn. Systems, 28 (2010), 275-310. doi: 10.3934/dcds.2010.28.275.  Google Scholar

[47]

O. A. Oleinik, A method of solution of the general Stefan problem, Soviet Math. Dokl., 1 (1960), 1350-1353.  Google Scholar

[48]

O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43 (1990), 44-62. doi: 10.1016/0167-2789(90)90015-H.  Google Scholar

[49]

O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a "thermodynamically consistent'' phase-field model, Physica D, 69 (1993), 107-113. doi: 10.1016/0167-2789(93)90183-2.  Google Scholar

[50]

J. C. Robinson, Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces, Nonlinearity, 22 (2009), 711-728. doi: 10.1088/0951-7715/22/4/001.  Google Scholar

[51]

L. Rubinstein, On the solution of Stefan's problem, (in Russian) Izvestia Akad. Nauk SSSR, 11 (1947), 37-54.  Google Scholar

[52]

S. I. Serdyukov, N. M. Voskresenskii, V. K. Bel'nov and I. I. Karpov, Extended irreversible thermodynamics and generalization of the dual-phase-lag model in heat transfer, J. Non-Equilib. Thermodyn., 28 (2003), 1-13. doi: 10.1515/JNETDY.2003.013.  Google Scholar

[53]

L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. Math., 118 (1983), 525-571. doi: 10.2307/2006981.  Google Scholar

[54]

J. Sprekels and S. Zheng, Global smooth solutions to a thermodynamically consistent model of phase-field type in higher space dimensions, J. Math. Anal. Appl., 176 (1993), 200-223. doi: 10.1006/jmaa.1993.1209.  Google Scholar

[55]

J. Stefan, Uber einige Probleme der Theorie der Warmeleitung, S.-B. Wien Akad. Mat. Natur., 98 (1889), 173-484. Google Scholar

[56]

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