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September  2012, 11(5): 1959-1982. doi: 10.3934/cpaa.2012.11.1959

Solvability and asymptotic analysis of a generalization of the Caginalp phase field system

Giacomo Canevari 1, and  Pierluigi Colli 2,

1. 

Dipartimento di Matematica ``F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy
2. 

Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, I-27100 Pavia

Received  July 2011 Revised  November 2011 Published  March 2012

We study a diffusion model of phase field type, which consists of a system of two partial differential equations involving as variables the thermal displacement, that is basically the time integration of temperature, and the order parameter. Our analysis covers the case of a non-smooth (maximal monotone) graph along with a smooth anti-monotone function in the phase equation. Thus, the system turns out a generalization of the well-known Caginalp phase field model for phase transitions when including a diffusive term for the thermal displacement in the balance equation. Systems of this kind have been extensively studied by Miranville and Quintanilla. We prove existence and uniqueness of a weak solution to the initial-boundary value problem, as well as various regularity results ensuring that the solution is strong and with bounded components. Then we investigate the asymptotic behaviour of the solutions as the coefficient of the diffusive term for the thermal displacement tends to zero and prove convergence to the Caginalp phase field system as well as error estimates for the difference of the solutions.
Keywords: asymptotic behaviour, regularity, error estimates., well-posedness, Phase field model.
Mathematics Subject Classification: Primary: 80A22; Secondary: 35K55, 35B30, 35B4.
Citation: Giacomo Canevari, Pierluigi Colli. Solvability and asymptotic analysis of a generalization of the Caginalp phase field system. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1959-1982. doi: 10.3934/cpaa.2012.11.1959
References:
[1]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,'', Noord\-hoff, (1976).   Google Scholar

[2]

H. Brezis, "Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,'', North-Holland Math. Stud. {\bf 5}, 5 (1973).   Google Scholar

[3]

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

[4]

P. Colli, G. Gilardi and M. Grasselli, Global smooth solution to the standard phase field model with memory,, Adv. Differential Equations, 2 (1997), 453.   Google Scholar

[5]

P. Colli, G. Gilardi and M. Grasselli, Well-posedness of the weak formulation for the phase-field model with memory,, Adv. Differential Equations, 2 (1997), 487.   Google Scholar

[6]

G. Duvaut, Résolution d'un problème de Stefan (fusion d'un bloc de glace à zéro degré),, C. R. Acad. Sci. Paris S閞. A-B, 276 (1973).   Google Scholar

[7]

M. Frémond, "Non-smooth Thermomechanics,'', Springer-Verlag, (2002).   Google Scholar

[8]

A.E. Green and P.M. Naghdi, A re-examination of the basic postulates of thermomechanics,, Proc. Roy. Soc. Lond. A, 432 (1991), 171.  doi: 10.1098/rspa.1991.0012.  Google Scholar

[9]

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

[10]

A.E. Green and P.M. Naghdi, Thermoelasticity without energy dissipation,, J. Elasticity, 31 (1993), 189.  doi: 10.1007/BF00044969.  Google Scholar

[11]

A.E. Green and P.M. Naghdi, A new thermoviscous theory for fluids,, J. Non-Newtonian Fluid Mech., 56 (1995), 289.  doi: 10.1016/0377-0257(94)01288-S.  Google Scholar

[12]

O.A. Ladyženskaja, V.A. Solonnikov, and N.N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'', Trans. Amer. Math. Soc. {\bf 23}, 23 (1968).   Google Scholar

[13]

J.L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,'', Dunod Gauthier-Villars, (1969).   Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

J. Simon, Compact sets in the space $L^p(0,T; B)$,, Ann. Mat. Pura Appl., 146 (1987), 65.  doi: 10.1007/BF01762360.  Google Scholar

show all references

References:
[1]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,'', Noord\-hoff, (1976).   Google Scholar

[2]

H. Brezis, "Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,'', North-Holland Math. Stud. {\bf 5}, 5 (1973).   Google Scholar

[3]

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

[4]

P. Colli, G. Gilardi and M. Grasselli, Global smooth solution to the standard phase field model with memory,, Adv. Differential Equations, 2 (1997), 453.   Google Scholar

[5]

P. Colli, G. Gilardi and M. Grasselli, Well-posedness of the weak formulation for the phase-field model with memory,, Adv. Differential Equations, 2 (1997), 487.   Google Scholar

[6]

G. Duvaut, Résolution d'un problème de Stefan (fusion d'un bloc de glace à zéro degré),, C. R. Acad. Sci. Paris S閞. A-B, 276 (1973).   Google Scholar

[7]

M. Frémond, "Non-smooth Thermomechanics,'', Springer-Verlag, (2002).   Google Scholar

[8]

A.E. Green and P.M. Naghdi, A re-examination of the basic postulates of thermomechanics,, Proc. Roy. Soc. Lond. A, 432 (1991), 171.  doi: 10.1098/rspa.1991.0012.  Google Scholar

[9]

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

[10]

A.E. Green and P.M. Naghdi, Thermoelasticity without energy dissipation,, J. Elasticity, 31 (1993), 189.  doi: 10.1007/BF00044969.  Google Scholar

[11]

A.E. Green and P.M. Naghdi, A new thermoviscous theory for fluids,, J. Non-Newtonian Fluid Mech., 56 (1995), 289.  doi: 10.1016/0377-0257(94)01288-S.  Google Scholar

[12]

O.A. Ladyženskaja, V.A. Solonnikov, and N.N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'', Trans. Amer. Math. Soc. {\bf 23}, 23 (1968).   Google Scholar

[13]

J.L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,'', Dunod Gauthier-Villars, (1969).   Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

J. Simon, Compact sets in the space $L^p(0,T; B)$,, Ann. Mat. Pura Appl., 146 (1987), 65.  doi: 10.1007/BF01762360.  Google Scholar

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