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Global existence for a phase separation system deduced from the entropy balance. (English) Zbl 1428.35163
Summary: This paper is concerned with a thermomechanical model describing phase separation phenomena in terms of the entropy balance and equilibrium equations for the microforces. The related system is highly nonlinear and admits singular potentials in the phase equation. Both the viscous and the non-viscous cases are considered in the Cahn-Hilliard relations characterizing the phase dynamics. The entropy balance is written in terms of the absolute temperature and of its logarithm, appearing under time derivative. The initial and boundary value problem is considered for the system of partial differential equations. The existence of a global solution is proved via some approximations involving Yosida regularizations and a suitable time discretization.

35K55 Nonlinear parabolic equations
35G31 Initial-boundary value problems for nonlinear higher-order PDEs
80A22 Stefan problems, phase changes, etc.
Full Text: DOI arXiv
[1] Barbu, V., Nonlinear Semigroups and Differential Equations in Banach Spaces (1976)
[2] Barbu, V., (Nonlinear Differential Equations of Monotone Types in Banach Spaces. Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics (2010), Springer: Springer New York) · Zbl 1197.35002
[3] Bonetti, E., A new approach to phase transitions with thermal memory via the entropy balance, (Mathematical Methods and Models in Phase Transitions (2005), Nova Sci. Publ.: Nova Sci. Publ. New York), 125-155
[4] Bonetti, E.; Bonfanti, G.; Rossi, R., Analysis of a temperature-dependent model for adhesive contact with friction, Physica D, 285, 42-62 (2014) · Zbl 1457.74034
[5] Bonetti, E.; Colli, P.; Fabrizio, M.; Gilardi, G., Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity, Discrete Contin. Dyn. Syst. Ser. B, 6, 1001-1026 (2006) · Zbl 1123.35309
[6] Bonetti, E.; Colli, P.; Fabrizio, M.; Gilardi, G., Global solution to a singular integro-differential system related to the entropy balance, Nonlinear Anal., 66, 1949-1979 (2007) · Zbl 1128.35056
[7] Bonetti, E.; Colli, P.; Fabrizio, M.; Gilardi, G., Existence and boundedness of solutions for a singular phase field system, J. Differential Equations, 246, 3260-3295 (2009) · Zbl 1170.35443
[8] Bonetti, E.; Colli, P.; Gilardi, G., Singular limit of an integrodifferential system related to the entropy balance, Discrete Contin. Dyn. Syst. Ser. B, 19, 1935-1953 (2014) · Zbl 1304.35083
[9] Bonetti, E.; Colli, P.; M, G., Frémond: A phase field model with thermal memory governed by the entropy balance, Math. Models Methods Appl. Sci., 13, 1565-1588 (2003) · Zbl 1053.80009
[10] Bonetti, E.; Frémond, M., A phase transition model with the entropy balance, Math. Methods Appl. Sci., 26, 539-556 (2003) · Zbl 1017.35039
[11] Bonetti, E.; Frémond, M.; Rocca, E., A new dual approach for a class of phase transitions with memory: existence and long-time behaviour of solutions, J. Math. Pures Appl., 88, 9, 455-481 (2007) · Zbl 1141.74011
[12] Bonetti, E.; Rocca, E., Global existence and long-time behaviour for a singular integro-differential phase-field system, Commun. Pure Appl. Anal., 6, 367-387 (2007) · Zbl 1140.45010
[13] Brezis, H., (Opérateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert. Opérateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Math. Stud., vol. 5 (1973), North-Holland: North-Holland Amsterdam) · Zbl 0252.47055
[14] Canevari, G.; Colli, P., Solvability and asymptotic analysis of a generalization of the Caginalp phase field system, Commun. Pure Appl. Anal., 11, 1959-1982 (2012) · Zbl 1275.80003
[15] Canevari, G.; Colli, P., Convergence properties for a generalization of the Caginalp phase field system, Asymptot. Anal., 82, 139-162 (2013) · Zbl 1278.35163
[16] Colli, P.; Colturato, M., Global existence for a singular phase field system related to a sliding mode control problem, Nonlinear Anal. RWA, 41, 128-151 (2018) · Zbl 06839539
[17] Colli, P.; Gilardi, G.; Grasselli, M.; Schimperna, G., The conserved phase-field system with memory, Adv. Math. Sci. Appl., 11, 265-291 (2001) · Zbl 0982.80006
[18] Colli, P.; Gilardi, G.; Grasselli, M.; Schimperna, G., Global existence for the conserved phase field model with memory and quadratic nonlinearity, Port. Math., 58, 159-170 (2001) · Zbl 0985.35094
[19] Colli, P.; Gilardi, G.; Laurençot, P.; Novick-Cohen, A., Uniqueness and long-time behavior for the conserved phase-field system with memory, Discrete Contin. Dyn. Syst., 5, 375-390 (1999) · Zbl 0981.74012
[20] Colli, P.; Kurima, S., Time discretization of a nonlinear phase field system in general domains, Comm. Pure Appl. Anal., 18, 3161-3179 (2019)
[21] Colturato, M., Well-posedness and longtime behavior for a singular phase field system with perturbed phase dynamics, Evol. Equ. Control Theory, 7, 217-245 (2018) · Zbl 1404.35268
[22] Conti, M.; Gatti, S.; Miranville, A.; Quintanilla, R., On a Caginalp phase-field system with two temperatures and memory, Milan J. Math., 85, 1-27 (2017) · Zbl 1372.35047
[23] Fabrizio, M.; Giorgi, C.; Morro, A., Internal dissipation relaxation properties and free energy in materials with fading memory, J. Elasticity, 40, 107-122 (1995) · Zbl 0853.73024
[24] Frémond, M., Non-smooth Thermomechanics (2002), Springer-Verlag: Springer-Verlag Berlin · Zbl 0990.80001
[25] Gilardi, G.; Miranville, A.; Schimperna, G., On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8, 881-912 (2009) · Zbl 1172.35417
[26] Gilardi, G.; Rocca, E., Convergence of phase field to phase relaxation models governed by an entropy equation with memory, Math. Methods Appl. Sci., 29, 2149-2179 (2006) · Zbl 1120.80007
[27] Gilardi, G.; Rocca, E., Well-posedness and long-time behaviour for a singular phase field system of conserved type, IMA J. Appl. Math., 72, 498-530 (2007) · Zbl 1141.80007
[28] Green, A. E.; Naghdi, P. M., A re-examination of the basic postulates of thermo-mechanics, Proc. R. Soc. A, 432, 171-194 (1991) · Zbl 0726.73004
[29] Gurtin, M. E., Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92, 178-192 (1996) · Zbl 0885.35121
[30] Gurtin, M. E.; Pipkin, A. C., A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal., 31, 113-126 (1968) · Zbl 0164.12901
[31] Ito, A.; Suzuki, T., Asymptotic behavior of the solution to the non-isothermal phase separation, Nonlinear Anal., 68, 1825-1843 (2008) · Zbl 1137.35395
[32] Jerome, J. W., (Approximations of Nonlinear Evolution Systems. Approximations of Nonlinear Evolution Systems, Mathematics in Science and Engineering, vol. 164 (1983), Academic Press Inc.: Academic Press Inc. Orlando)
[33] Kenmochi, N.; Niezgódka, M., Viscosity approach to modelling non-isothermal diffusive phase separation, Japan J. Indust. Appl. Math., 13, 135-169 (1996) · Zbl 0865.35062
[34] Kou, J.; Sun, S., Thermodynamically consistent modeling and simulation of multi-component two-phase flow with partial miscibility, Comput. Methods Appl. Mech. Engrg., 331, 623-649 (2018)
[35] Miranville, A.; Schimperna, G., Nonisothermal phase separation based on a microforce balance, Discrete Contin. Dyn. Syst. Ser. B, 5, 753-768 (2005) · Zbl 1140.80388
[36] Montanaro, A., On thermo-electro-mechanical simple materials with fading memory: restrictions of the constitutive equations in a Green-Naghdi type theory, Meccanica, 52, 3023-3031 (2017) · Zbl 1455.74021
[37] Müller, I., Thermodynamics of mixtures and phase field theory, Int. J. Solids Struct., 38, 1105-1113 (2001) · Zbl 1007.74010
[38] Nečas, J., Les Méthodes Directes en Théorie des Équations Elliptiques (1967), (French) · Zbl 1225.35003
[39] Podio-Guidugli, P., A virtual power format for thermomechanics, Contin. Mech. Thermodyn., 20, 479-487 (2009) · Zbl 1234.74022
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