American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature
  • CPAA Home
  • This Issue
  • Next Article
    Symmetries and blow-up phenomena for a Dirichlet problem with a large parameter
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,'' Noordhoff, Leyden, 1976.  Google Scholar

[2]

H. Brezis, "Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,'' North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973.  Google Scholar

[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.  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), 453-486.  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), 487-508.  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), A1461-A1463.  Google Scholar

[7]

M. Frémond, "Non-smooth Thermomechanics,'' Springer-Verlag, Berlin, 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), 171-194. 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), 253-264. doi: 10.1080/01495739208946136.  Google Scholar

[10]

A.E. Green and P.M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208. 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), 289-306. 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. 23, Amer. Math. Soc., Providence, RI, 1968.  Google Scholar

[13]

J.L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,'' Dunod Gauthier-Villars, Paris, 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), 2278-2290. 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), 877-894. 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), 2849-2861. 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), 1003-1008. 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), 65-96. doi: 10.1007/BF01762360.  Google Scholar

show all references

References:
[1]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,'' Noordhoff, Leyden, 1976.  Google Scholar

[2]

H. Brezis, "Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,'' North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973.  Google Scholar

[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.  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), 453-486.  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), 487-508.  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), A1461-A1463.  Google Scholar

[7]

M. Frémond, "Non-smooth Thermomechanics,'' Springer-Verlag, Berlin, 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), 171-194. 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), 253-264. doi: 10.1080/01495739208946136.  Google Scholar

[10]

A.E. Green and P.M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208. 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), 289-306. 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. 23, Amer. Math. Soc., Providence, RI, 1968.  Google Scholar

[13]

J.L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,'' Dunod Gauthier-Villars, Paris, 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), 2278-2290. 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), 877-894. 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), 2849-2861. 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), 1003-1008. 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), 65-96. doi: 10.1007/BF01762360.  Google Scholar

[1]

Pavel Krejčí, Elisabetta Rocca. Well-posedness of an extended model for water-ice phase transitions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 439-460. doi: 10.3934/dcdss.2013.6.439
[2]

Michele Colturato. Well-posedness and longtime behavior for a singular phase field system with perturbed phase dynamics. Evolution Equations & Control Theory, 2018, 7 (2) : 217-245. doi: 10.3934/eect.2018011
[3]

Pierluigi Colli, Gianni Gilardi, Elisabetta Rocca, Jürgen Sprekels. Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 37-54. doi: 10.3934/dcdss.2017002
[4]

S. Gatti, Elena Sartori. Well-posedness results for phase field systems with memory effects in the order parameter dynamics. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 705-726. doi: 10.3934/dcds.2003.9.705
[5]

Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036
[6]

Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089
[7]

K. Domelevo. Well-posedness of a kinetic model of dispersed two-phase flow with point-particles and stability of travelling waves. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 591-607. doi: 10.3934/dcdsb.2002.2.591
[8]

Xiaoli Wang, Peter Kloeden, Meihua Yang. Asymptotic behaviour of a neural field lattice model with delays. Electronic Research Archive, 2020, 28 (2) : 1037-1048. doi: 10.3934/era.2020056
[9]

Kazuo Yamazaki, Xueying Wang. Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1297-1316. doi: 10.3934/dcdsb.2016.21.1297
[10]

Elissar Nasreddine. Well-posedness for a model of individual clustering. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2647-2668. doi: 10.3934/dcdsb.2013.18.2647
[11]

David M. Ambrose, Jerry L. Bona, David P. Nicholls. Well-posedness of a model for water waves with viscosity. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1113-1137. doi: 10.3934/dcdsb.2012.17.1113
[12]

Changyan Li, Hui Li. Well-posedness of the two-phase flow problem in incompressible MHD. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021090
[13]

Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321
[14]

Hyungjin Huh, Bora Moon. Low regularity well-posedness for Gross-Neveu equations. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1903-1913. doi: 10.3934/cpaa.2015.14.1903
[15]

Haibo Cui, Qunyi Bie, Zheng-An Yao. Well-posedness in critical spaces for a multi-dimensional compressible viscous liquid-gas two-phase flow model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1395-1410. doi: 10.3934/dcdsb.2018156
[16]

Hongwei Wang, Amin Esfahani. Well-posedness and asymptotic behavior of the dissipative Ostrovsky equation. Evolution Equations & Control Theory, 2019, 8 (4) : 709-735. doi: 10.3934/eect.2019035
[17]

Seung-Yeal Ha, Jinyeong Park, Xiongtao Zhang. A global well-posedness and asymptotic dynamics of the kinetic Winfree equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1317-1344. doi: 10.3934/dcdsb.2019229
[18]

Jan Prüss, Vicente Vergara, Rico Zacher. Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 625-647. doi: 10.3934/dcds.2010.26.625
[19]

Jan Prüss, Yoshihiro Shibata, Senjo Shimizu, Gieri Simonett. On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities. Evolution Equations & Control Theory, 2012, 1 (1) : 171-194. doi: 10.3934/eect.2012.1.171
[20]

Caochuan Ma, Zaihong Jiang, Renhui Wan. Local well-posedness for the tropical climate model with fractional velocity diffusion. Kinetic & Related Models, 2016, 9 (3) : 551-570. doi: 10.3934/krm.2016006

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences