Existence and asymptotic analysis of a phase field model for supercooling
Authors:
Olaf Klein, Fabio Luterotti and Riccarda Rossi
Journal:
Quart. Appl. Math. 64 (2006), 291-319
MSC (2000):
Primary 80A22; Secondary 28A33, 35K55
DOI:
https://doi.org/10.1090/S0033-569X-06-01019-9
Published electronically:
May 2, 2006
MathSciNet review:
2243865
Full-text PDF Free Access
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Abstract: We prove an existence result for an initial-boundary value problem which models a perturbation of a phase transition phenomenon with supercooling effects. When the perturbation parameter goes to 0, an asymptotic analysis is performed. It leads to an existence result, in the framework of Young measures, for a slight modification of the original problem.
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- Masayasu Aso, Michel Frémond, and Nobuyuki Kenmochi, Phase change problems with temperature dependent constraints for the volume fraction velocities, Nonlinear Anal. 60 (2005), no. 6, 1003–1023. MR 2115030, DOI https://doi.org/10.1016/j.na.2004.08.041
Aso-Ken05 M. Aso and N. Kenmochi, A class of doubly nonlinear quasi-variational evolution problems. To appear in GAKUTO Internat. Ser. Math. Sci. Appl. 23, Gakkōtosho, Tokyo, 2005.
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- Erik J. Balder, Lectures on Young measure theory and its applications in economics, Rend. Istit. Mat. Univ. Trieste 31 (2000), no. suppl. 1, 1–69. Workshop on Measure Theory and Real Analysis (Italian) (Grado, 1997). MR 1798830
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Rossi-Savare05 R. Rossi and G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications. Preprint IMATI-CNR n. 7-PV (2004) 1-45, to appear on ESAIM Control Optim. Calc. Var..
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Aso-Fre-Ken04 M. Aso, M. Frémond, and N. Kenmochi, Quasi-variational evolution problems for irreversible phase change, Nonlinear partial differential equations and their applications, GAKUTO Internat. Ser. Math. Sci. Appl. 20, Gakkōtosho, Tokyo, 2004, pp. 517–535.
Aso-Fre-Ken05 ---, Phase change problems with temperature dependent constraints for the volume fraction velocities. Nonlinear Anal. 60 (2005), 1003–1023.
Aso-Ken05 M. Aso and N. Kenmochi, A class of doubly nonlinear quasi-variational evolution problems. To appear in GAKUTO Internat. Ser. Math. Sci. Appl. 23, Gakkōtosho, Tokyo, 2005.
Attouch84 H. Attouch, Variational convergence for functions and operators, Pitman Advance Publishing Program, Boston, MA, 1984.
Balder84 E. J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory. SIAM J. Control Optim. 22 (1984), 570–598.
Balder:lecture:notes98 ---, Lectures on Young measure theory and its applications in economics. Rend. Istit. Mat. Univ. Trieste 31 (2000), 1–69, Workshop on Measure Theory and Real Analysis (Italian) (Grado, 1997).
Barbu76 V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Noordhoff, Leiden, 1976.
Brezis73 H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973.
Colli92 P. Colli, On some doubly nonlinear evolution equations in Banach spaces. Japan J. Indust. Appl. Math. 9 (1992), 181–203.
colli-fremond-klein00 P. Colli, M. Frémond, and O. Klein, Global existence of a solution to a phase field model for supercooling. Nonlinear Anal. Real World Appl. 2 (2001), 523–539.
Colli-Visintin90 P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations. Comm. Partial Differential Equations 15 (1990), 737–756.
Dellacherie-Meyer79 C. Dellacherie and P.A. Meyer, Probabilities and Potential, North-Holland, Amsterdam, 1978.
fre M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002.
jerome83 J.W. Jerome, Approximation of nonlinear evolution systems, Number 164 in Math. Sci. Engrg. Academic Press, Orlando, 1983.
klein00 O. Klein, Two phase field systems modelling supercooling, Free boundary problems: Theory and applications, II (Chiba, 1999), GAKUTO Internat. Ser. Math. Sci. Appl. 14, Tokyo, 2000, 273–282.
Lions-Magenes72 J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. 1, Springer-Verlag, New York-Heidelberg, 1972.
Rossi-Savare05 R. Rossi and G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications. Preprint IMATI-CNR n. 7-PV (2004) 1-45, to appear on ESAIM Control Optim. Calc. Var..
Segatti04 A. Segatti, Global attractor for a class of doubly nonlinear abstract evolution equations, Discrete Cont. Dyn. Syst. 14 (2006), no. 4, 801–820.
Simon J. Simon, Compact Sets in the space $L^{p}(0,T;B)$. Ann. Mat. Pura Appl. (4) 146 (1987), 65–96.
Valadier90 M. Valadier, Young measures, Methods of nonconvex analysis (Varenna, 1989), Springer, Lect. Notes Math. 1446, Berlin, 1990, 152–188.
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Additional Information
Olaf Klein
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, D–10117 Berlin, Germany
Email:
klein@wias-berlin.de
Fabio Luterotti
Affiliation:
Dipartimento di Matematica, Università di Brescia, via Valotti 9, I–25133 Brescia, Italy
Email:
luterott@ing.unibs.it
Riccarda Rossi
Affiliation:
Dipartimento di Matematica, Università di Brescia, via Valotti 9, I–25133 Brescia, Italy
Email:
riccarda.rossi@ing.unibs.it
Keywords:
Phase field system,
supercooling,
doubly nonlinear equations,
Young measures.
Received by editor(s):
July 19, 2005
Published electronically:
May 2, 2006
Additional Notes:
The second and third author have been partially supported by the Italian COFIN project 2004 “Modellizzazione Matematica ed Analisi dei Problemi a Frontiera Libera”
Article copyright:
© Copyright 2006
Brown University
