Weak solutions to a phase-field model with non-constant thermal conductivity
Author:
Ph. Laurençot
Journal:
Quart. Appl. Math. 55 (1997), 739-760
MSC:
Primary 35Q99; Secondary 35D05, 80A22
DOI:
https://doi.org/10.1090/qam/1486546
MathSciNet review:
MR1486546
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Abstract: We investigate the existence of weak solutions to a phase-field model when the thermal conductivity vanishes for some values of the order parameter. We obtain weak solutions for a general class of free energies, including non-differentiable ones. We also study the $\omega$-limit set of these weak solutions, and investigate their convergence to a solution of a degenerate Cahn-Hilliard equation.
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J. W. Cahn and J. E. Taylor, Surface motion by surface diffusion, Acta Metall. Mater. 42, 1045–1063 (1994)
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Ph. Laurençot, Long-time behaviour for a model of phase-field type, Proc. Roy. Soc. Edinburgh 126A, 167–185 (1996)
Ph. Laurençot, Degenerate Cahn-Hilliard equation as limit of the phase-field equations with non-constant thermal conductivity, preprint Institut Elie Cartan, Nancy
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969
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J. F. Blowey and C. M. Elliott, A phase-field, model with a double obstacle potential, in Motion by Mean Curvature and Related Topics, G. Buttazzo, A. Visintin, eds., de Gruyter, Berlin, 1994, pp. 1–22
H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North Holland, Amsterdam, 1973
D. Brochet, X. Chen, and D. Hilhorst, Finite dimensional exponential attractor for the phase field model, Applicable Anal. 49, 197–212 (1993)
G. Caginalp, An analysis of a phase field model for a free boundary; Arch. Rational Mech. Anal. 92, 205–245 (1986)
G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of the phase field equations, Phys. Rev. A 39, 5887–5896 (1989)
J. W. Cahn and J. E. Taylor, Surface motion by surface diffusion, Acta Metall. Mater. 42, 1045–1063 (1994)
C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal. 27, 404–423 (1996)
C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations, in Free Boundary Value Problems, K. H. Hoffmann, J. Sprekels, eds., Internat. Ser. Num. Math. 95, Birkhäuser, Basel, 1990, pp. 46–58
O. Kavian, Introduction à la théorie des points critiques, Mathématiques et Applications 13, SMAI, Springer-Verlag, Paris, 1993
N. Kenmochi and M. Niezgòdka, Evolution systems of nonlinear variational inequalities arising from phase change problems, Nonlinear Anal., TMA 22, 1163–1180 (1994)
M. Langlais and D. Phillips, Stabilization of solutions of nonlinear and degenerate evolution equations, Nonlinear Anal., TMA 9, 321–333 (1985)
Ph. Laurençot, Long-time behaviour for a model of phase-field type, Proc. Roy. Soc. Edinburgh 126A, 167–185 (1996)
Ph. Laurençot, Degenerate Cahn-Hilliard equation as limit of the phase-field equations with non-constant thermal conductivity, preprint Institut Elie Cartan, Nancy
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969
U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. Math. 3, 510–585 (1969)
J. Simon, Compact sets in the space $L^{p}\left ( 0, T; B \right )$, Ann. Mat. Pura Appl. 146, 65–96 (1987)
B. E. E. Stoth, The Cahn-Hilliard equation as degenerate limit of the phase-field equations, Quart. Appl. Math. 53, 695–700 (1995)
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Article copyright:
© Copyright 1997
American Mathematical Society