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Boldrini, José Luiz; Planas, Gabriela

Weak solutions of a phase-field model for phase change of an alloy with thermal properties. (English) Zbl 1012.35049

Math. Methods Appl. Sci. 25, No. 14, 1177-1193 (2002).
The paper studies a model for phase change processes occurring in binary alloys with thermal effects which is expressed by a coupled system of nonlinear partial differential equations. A serious difficulty in treating the problem is the possibility of degeneracy of the parabolic character of the model. The existence of weak solutions is proved by using a regularization technique, compactness arguments and the Leray-Schauder theory.
Reviewer: Dumitru Motreanu (Perpignan)

Cited in 8 Documents

MSC:

35K65 Degenerate parabolic equations
35R35 Free boundary problems for PDEs
80A22 Stefan problems, phase changes, etc.
35K55 Nonlinear parabolic equations
82B26 Phase transitions (general) in equilibrium statistical mechanics
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35B25 Singular perturbations in context of PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)

Keywords:

compactness arguments; regularization; Leray-Schauder theory
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\textit{J. L. Boldrini} and \textit{G. Planas}, Math. Methods Appl. Sci. 25, No. 14, 1177--1193 (2002; Zbl 1012.35049)
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References:

[1] Caginalp, Physical Review E 48 pp 1897– (1993)
[2] Caginalp, Annals of Physics 237 pp 66– (1995)
[3] Lauren?ot, Quarterly Applied Mathematics 15 pp 739– (1997)
[4] Caginalp, Archive for Rational Mechanics Analysis 92 pp 205– (1986) · Zbl 0608.35080
[5] Hoffman, Numerical Functional Analysis and Optimization 13 pp 11– (1992)
[6] Moro?anu, Journal of the Mathematical Analysis Applications 237 pp 515– (1999)
[7] Penrose, Physica D 43 pp 44– (1990)
[8] Colli, Physica D 111 pp 311– (1998)
[9] Colli, Journal of Differential Equations 142 pp 54– (1998)
[10] Wheeler, Physical Review A 45 pp 7424– (1992)
[11] Warren, Acta Metallurgica et Materialia 43 pp 689– (1995)
[12] Rappaz, Mathematical Methods in the Applied Sciences 23 pp 491– (2000)
[13] Partial Differential Equation of Parabolic Type. Prentice-Hall: Englewood Cliffs, NJ, 1964.
[14] Partial Differential Equations. Mir: Moscow, 1978.
[15] Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society: Providence, 1968.
[16] Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol 840. Springer: Berlin, 1981.
[17] Simon, Annali di Matematica Pura ed Applicata 146 pp 65– (1987)
[18] Control of Distributed Singular Systems. Gauther-Villars: Paris, 1985.
[19] Introduction ? la Th?orie des Points Critiques. Math?matiques et Applications, vol. 13, Springer: Berlin, 1993.
[20] Marcus, Journal of Functional Analysis 33 pp 217– (1979)
[21] Marcus, Transactions of the American Mathematical Society 251 pp 187– (1979)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.