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April  2011, 4(2): 409-421. doi: 10.3934/dcdss.2011.4.409

Variational inequalities for a non-isothermal phase field model

Kota Kumazaki 1, and  Masahiro Kubo 2,

1. 

Department of Mathematics, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, Aichi, 466-8555
2. 

Department of Mathematics, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555

Received  January 2009 Revised  June 2009 Published  November 2010

We study variational inequalities for a non-isothermal phase field model of the Penrose-Fife type. We consider time-dependent constraints for the order parameter and the Dirichlet boundary condition for the temperature.
Keywords: variational inequalities, Nonlinear parabolic PDE, phase transitions..
Mathematics Subject Classification: Primary: 35K55, 35K85; Secondary: 82C2.
Citation: Kota Kumazaki, Masahiro Kubo. Variational inequalities for a non-isothermal phase field model. Discrete and Continuous Dynamical Systems - S, 2011, 4 (2) : 409-421. doi: 10.3934/dcdss.2011.4.409
References:
[1]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland, Amsterdam, London, New York, 1973.

[2]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Appl. Math. Sci., 121, Springer-Verlag, New York, 1996.

[3]

X. Chen and C. M. Elliott, Asymptotics for a parabolic double obstacle problem, Proc. Roy. Soc. London Ser. A, 444 (1994), 429-445. doi: 10.1098/rspa.1994.0030.

[4]

A. Damlamian and N. Kenmochi, Le problème de Stefan avec conditions latérales variables, Hiroshima Math. J., 10 (1980), 271-293.

[5]

A. Ito and M. Kubo, Well-posedness for an extended Penrose-Fife phase field model with energy balance supplied by Dirichlet boundary conditions, Nonlinear Anal., 9 (2008), 370-383. doi: 10.1016/j.nonrwa.2006.11.005.

[6]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Educ. Chiba Univ., 30 (1987), 1-87.

[7]

N. Kenmochi and M. Kubo, Weak solutions of nonlinear systems for non-isothermal phase transitions, Adv. Math. Sci. Appl., 9 (1999), 499-521.

[8]

N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising from phase change problems, Nonlinear Anal., 22 (1994), 1163-1180. doi: 10.1016/0362-546X(94)90235-6.

[9]

N. Kenmochi and I. Pawlow, Parabolic-elliptic free boundary problems with time-dependent obstacles, Japan J. Appl. Math., 5 (1988), 87-121. doi: 10.1007/BF03167902.

[10]

M. Kubo, Well-posedness of initial boundary value problem of degenerate parabolic equations, Nonlinear Anal., 63 (2005), e2629-e2637. doi: 10.1016/j.na.2005.03.029.

[11]

M. Kubo and Q. Lu, Evolution equation for nonlinear degenerate parabolic PDE, Nonlinear Anal., 64 (2006), 1849-1859. doi: 10.1016/j.na.2005.07.027.

[12]

K. Kumazaki, A. Ito and M. Kubo, A non-isothermal phase separation with constraints and Dirichlet boundary condition for temperature, Nonlinear Anal., 71 (2009), 1950-1963. doi: 10.1016/j.na.2009.01.039.

[13]

K. Kumazaki, A. Ito and M. Kubo, Generalized solution of a non-isothermal phase separation model, Discrete Contin. Dyn. Syst. 2009, Dynamical Systems and Differential Equations, Proceedings of the 7th AIMS International Conference, suppl., 476-485.

[14]

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

[15]

Y. Yamada, On evolution equations generated by subdifferential operators, J. Fac. Sci., Univ. Tokyo, Sect. IA Math., 43 (1976), 491-515.

show all references

References:
[1]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland, Amsterdam, London, New York, 1973.

[2]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Appl. Math. Sci., 121, Springer-Verlag, New York, 1996.

[3]

X. Chen and C. M. Elliott, Asymptotics for a parabolic double obstacle problem, Proc. Roy. Soc. London Ser. A, 444 (1994), 429-445. doi: 10.1098/rspa.1994.0030.

[4]

A. Damlamian and N. Kenmochi, Le problème de Stefan avec conditions latérales variables, Hiroshima Math. J., 10 (1980), 271-293.

[5]

A. Ito and M. Kubo, Well-posedness for an extended Penrose-Fife phase field model with energy balance supplied by Dirichlet boundary conditions, Nonlinear Anal., 9 (2008), 370-383. doi: 10.1016/j.nonrwa.2006.11.005.

[6]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Educ. Chiba Univ., 30 (1987), 1-87.

[7]

N. Kenmochi and M. Kubo, Weak solutions of nonlinear systems for non-isothermal phase transitions, Adv. Math. Sci. Appl., 9 (1999), 499-521.

[8]

N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising from phase change problems, Nonlinear Anal., 22 (1994), 1163-1180. doi: 10.1016/0362-546X(94)90235-6.

[9]

N. Kenmochi and I. Pawlow, Parabolic-elliptic free boundary problems with time-dependent obstacles, Japan J. Appl. Math., 5 (1988), 87-121. doi: 10.1007/BF03167902.

[10]

M. Kubo, Well-posedness of initial boundary value problem of degenerate parabolic equations, Nonlinear Anal., 63 (2005), e2629-e2637. doi: 10.1016/j.na.2005.03.029.

[11]

M. Kubo and Q. Lu, Evolution equation for nonlinear degenerate parabolic PDE, Nonlinear Anal., 64 (2006), 1849-1859. doi: 10.1016/j.na.2005.07.027.

[12]

K. Kumazaki, A. Ito and M. Kubo, A non-isothermal phase separation with constraints and Dirichlet boundary condition for temperature, Nonlinear Anal., 71 (2009), 1950-1963. doi: 10.1016/j.na.2009.01.039.

[13]

K. Kumazaki, A. Ito and M. Kubo, Generalized solution of a non-isothermal phase separation model, Discrete Contin. Dyn. Syst. 2009, Dynamical Systems and Differential Equations, Proceedings of the 7th AIMS International Conference, suppl., 476-485.

[14]

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

[15]

Y. Yamada, On evolution equations generated by subdifferential operators, J. Fac. Sci., Univ. Tokyo, Sect. IA Math., 43 (1976), 491-515.

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