Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



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
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Abstract | References | Similar Articles | Additional Information

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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  • 1. Masayasu Aso, Michel Frémond, and Nobuyuki Kenmochi, Quasi-variational evolution problems for irreversible phase change, Nonlinear partial differential equations and their applications, GAKUTO Internat. Ser. Math. Sci. Appl., vol. 20, Gakkōtosho, Tokyo, 2004, pp. 517–525. MR 2087495
  • 2. 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, https://doi.org/10.1016/j.na.2004.08.041
  • 3. M. Aso and N. Kenmochi, A class of doubly nonlinear quasi-variational evolution problems. To appear in GAKUTO Internat. Ser. Math. Sci. Appl. 23, Gakkotosho, Tokyo, 2005.
  • 4. H. Attouch, Variational convergence for functions and operators, Pitman Advance Publishing Program, Boston, MA, 1984. MR 0773850 (86f:49002)
  • 5. E. J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory. SIAM J. Control Optim. 22 (1984), 570-598. MR 0747970 (85k:49018)
  • 6. 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
  • 7. Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR 0390843
  • 8. H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). MR 0348562
  • 9. Pierluigi Colli, On some doubly nonlinear evolution equations in Banach spaces, Japan J. Indust. Appl. Math. 9 (1992), no. 2, 181–203. MR 1170721, https://doi.org/10.1007/BF03167565
  • 10. Pierluigi Colli, Michel Frémond, and Olaf Klein, Global existence of a solution to a phase field model for supercooling, Nonlinear Anal. Real World Appl. 2 (2001), no. 4, 523–539. MR 1858904, https://doi.org/10.1016/S1468-1218(01)00008-6
  • 11. P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations, Comm. Partial Differential Equations 15 (1990), no. 5, 737–756. MR 1070845, https://doi.org/10.1080/03605309908820706
  • 12. C. Dellacherie and P.A. Meyer, Probabilities and Potential, North-Holland, Amsterdam, 1978. MR 0521810 (80b:60004)
  • 13. Michel Frémond, Non-smooth thermomechanics, Springer-Verlag, Berlin, 2002. MR 1885252
  • 14. J.W. Jerome, Approximation of nonlinear evolution systems, Number 164 in Math. Sci. Engrg. Academic Press, Orlando, 1983. MR 0690582 (85g:35064)
  • 15. 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.
  • 16. J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR 0350177
  • 17. 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..
  • 18. Antonio Segatti, Global attractor for a class of doubly nonlinear abstract evolution equations, Discrete Contin. Dyn. Syst. 14 (2006), no. 4, 801–820. MR 2177098, https://doi.org/10.3934/dcds.2006.14.801
  • 19. J. Simon, Compact Sets in the space $ L^{p}(0,T;B) $. Ann. Mat. Pura Appl. (4) 146 (1987), 65-96. MR 0916688 (89c:46055)
  • 20. Michel Valadier, Young measures, Methods of nonconvex analysis (Varenna, 1989) Lecture Notes in Math., vol. 1446, Springer, Berlin, 1990, pp. 152–188. MR 1079763, https://doi.org/10.1007/BFb0084935

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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

DOI: https://doi.org/10.1090/S0033-569X-06-01019-9
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

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