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June  2012, 32(6): 1997-2025. doi: 10.3934/dcds.2012.32.1997

On a phase field model for solid-liquid phase transitions

Sylvie Benzoni-Gavage 1, , Laurent Chupin 2, , Didier Jamet 3, and  Julien Vovelle 1,

1. 

Université de Lyon, CNRS UMR 5208 & Université Lyon 1, Institut Camille Jordan, 43 bd du 11 novembre 1918, F-69622 Villeurbanne cedex, France, France
2. 

Université Blaise Pascal & CNRS UMR 6620, Laboratoire de Mathématiques, Campus des Cézeaux, B.P. 80026, F-63177 Aubière cedex, France
3. 

CEA-Grenoble (DEN/DTP/SMTH), 17, rue des martyrs, F-38054 Grenoble cedex 9, France

Received  January 2011 Revised  July 2011 Published  February 2012

A new phase field model is introduced, which can be viewed as a nontrivial generalisation of what is known as the Caginalp model. It involves in particular nonlinear diffusion terms. By formal asymptotic analysis, it is shown that in the sharp interface limit it still yields a Stefan-like model with: 1) a generalized Gibbs-Thomson relation telling how much the interface temperature differs from the equilibrium temperature when the interface is moving or/and is curved with surface tension; 2) a jump condition for the heat flux, which turns out to depend on the latent heat and on the velocity of the interface with a new, nonlinear term compared to standard models. From the PDE analysis point of view, the initial-boundary value problem is proved to be locally well-posed in time (for smooth data).
Keywords: nonlinear diffusion., reaction-diffusion system, Diffuse interface, Caginalp model, sharp interface limit.
Mathematics Subject Classification: Primary: 80A22, 35K51; Secondary: 35R35, 35A0.
Citation: Sylvie Benzoni-Gavage, Laurent Chupin, Didier Jamet, Julien Vovelle. On a phase field model for solid-liquid phase transitions. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 1997-2025. doi: 10.3934/dcds.2012.32.1997
References:
[1]

D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, in "Annual Review of Fluid Mechanics," Vol. 30, Annual Reviews, Palo Alto, CA, (1998), 139-165.

[2]

E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Existence and boundedness of solutions for a singular phase field system, J. Differential Equations, 246 (2009), 3260-3295.

[3]

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.

[4]

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.

[5]

C. Cavaterra, C. G. Gal, M. Grasselli and A. Miranville, Phase-field systems with nonlinear coupling and dynamic boundary conditions, Nonlinear Anal., 72 (2010), 2375-2399. doi: 10.1016/j.na.2009.11.002.

[6]

P. Colli, D. Hilhorst, F. Issard-Roch and G. Schimperna, Long time convergence for a class of variational phase-field models, Discrete Contin. Dyn. Syst., 25 (2009), 63-81.

[7]

E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369.

[8]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[9]

M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98.

[10]

M. Grasselli, H. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72. doi: 10.4171/ZAA/1277.

[11]

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

[12]

A. Miranville and R. Quintanilla, A Caginalp phase-field system with a nonlinear coupling, Nonlinear Anal. Real World Appl., 11 (2010), 2849-2861. doi: 10.1016/j.nonrwa.2009.10.008.

[13]

Pierre Ruyer, "Modèle de Champ de Phase pour l'Étude de l'Ébullition," Ph.D thesis, École Polytechnique, 2006.

show all references

References:
[1]

D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, in "Annual Review of Fluid Mechanics," Vol. 30, Annual Reviews, Palo Alto, CA, (1998), 139-165.

[2]

E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Existence and boundedness of solutions for a singular phase field system, J. Differential Equations, 246 (2009), 3260-3295.

[3]

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.

[4]

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.

[5]

C. Cavaterra, C. G. Gal, M. Grasselli and A. Miranville, Phase-field systems with nonlinear coupling and dynamic boundary conditions, Nonlinear Anal., 72 (2010), 2375-2399. doi: 10.1016/j.na.2009.11.002.

[6]

P. Colli, D. Hilhorst, F. Issard-Roch and G. Schimperna, Long time convergence for a class of variational phase-field models, Discrete Contin. Dyn. Syst., 25 (2009), 63-81.

[7]

E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369.

[8]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[9]

M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98.

[10]

M. Grasselli, H. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72. doi: 10.4171/ZAA/1277.

[11]

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

[12]

A. Miranville and R. Quintanilla, A Caginalp phase-field system with a nonlinear coupling, Nonlinear Anal. Real World Appl., 11 (2010), 2849-2861. doi: 10.1016/j.nonrwa.2009.10.008.

[13]

Pierre Ruyer, "Modèle de Champ de Phase pour l'Étude de l'Ébullition," Ph.D thesis, École Polytechnique, 2006.

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