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The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy Part I: Mathematical analysis

Published online by Cambridge University Press:  16 July 2009

J. F. Blowey  and
C. M. Elliott
J. F. Blowey
Affiliation:
School of Mathematical and Physical Sciences, University of Sussex, Brighton BN1 9QH, UK
C. M. Elliott
Affiliation:
School of Mathematical and Physical Sciences, University of Sussex, Brighton BN1 9QH, UK
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Abstract

A mathematical analysis is carried out for the Cahn–Hilliard equation where the free energy takes the form of a double well potential function with infinite walls. Existence and uniqueness are proved for a weak formulation of the problem which possesses a Lyapunov functional. Regularity results are presented for the weak formulation, and consideration is given to the asymptotic behaviour as the time becomes infinite. An investigation of the associated stationary problem is undertaken proving the existence of a nontrivial stationary solution and further regularity results for any stationary solution. Stationary solutions are constructed in one and two dimensions; a formula for the number of stationary solutions in one dimension is derived. It is then natural to study the asymptotic behaviour as the phenomenological parameter λ→0, the main result being that the interface between the two phases has minimal area.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 2 , Issue 3 , September 1991 , pp. 233 - 280
Copyright
Copyright © Cambridge University Press 1991

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References

Birkhoff, G. & Rota, G. C. 1959 Ordinary Differential Equations. Wiley.Google Scholar
Cahn, J. W. 1961 On spinodal decomposition. Ada Metall. 9, 795–801.CrossRefGoogle Scholar
Cahn, J. W. & Hilliard, J. E. 1958 Free energy of a non-uniform system I. Interfacial free energy. J. Chem. Phys., 258–267.Google Scholar
Cahn, J. W. & Hilliard, J. E. 1971 Spinodal decomposition: a reprise. Acta Metall. 28, 151–161.CrossRefGoogle Scholar
Carr, J., Gurtin, M. & Slemrod, M. 1984 Structured phase transitions on a finite interval. Arch. Rat. Mech. Anal. 86, 317–351.CrossRefGoogle Scholar
Copetti, M. I. M. & Elliott, C. M. 1990 Kinetics of phase decomposition processes: numerical solutions to the Cahn—Hilliard equation. Materials Science and Technology 6, 273–283.CrossRefGoogle Scholar
Elliott, C. M. 1989 The Cahn–Hilliard model for the kinetics of phase separation. In Mathematical Models for Phase Change Problems, I.S.N.M. 88 (ed. Rodrigues, J. F.), Birkhäuser Verlag.Google Scholar
Elliott, C. M. & Luckhaus, S. 1990 Preprint.Google Scholar
Elliott, C. M. & Songmu, Z. 1986 On the Cahn–Hilliard equation. Arch. Rat. Mech. Anal. 96, 339–357.CrossRefGoogle Scholar
Federer, H. 1968 Geometric Measure Theory. Springer-Verlag.Google Scholar
Fleming, W. H. & Rishel, R. W. 1960 An integral formula for total gradient variation. Arch. Rat. 11, 218–222.Google Scholar
Giusti, E. 1984 Minimal Surfaces and Functions of Bounded Variation. Birkuäuser Verlag.CrossRefGoogle Scholar
Grisvard, P. 1985 Elliptic Problems in Non smooth Domains. Pitman.Google Scholar
Gunton, J. D., San-Miguel, M. & Sahni, P. S. 1983 The dynamics of first order phase transitions. In Phase Transitions and Critical Phenomena Edition (ed. Domb, C. and Lebowitz, J.), Academic Press.Google Scholar
Gurtin, M. E. 1985 On a theory of phase transitions with interfacial energy. Arch. Rat. Mech. Anal. 87 (3), 187–212.CrossRefGoogle Scholar
Lions, J. L. 1969 Quelques méthods de resolution des problèmes aux limit nonlinéaires. Dunod.Google Scholar
Luckhaus, S. & Modica, L. 1989 The Gibbs–Thompson relation within the gradient theory of phase transitions. Arch. Rat. Mech. Anal. 107 (1), 71–83.CrossRefGoogle Scholar
Marcus, M. & Mizel, V. J. 1973 Nemitsky operators on Sobolev spaces. Arch. Rat. Mech. Anal. 51, 347–370.CrossRefGoogle Scholar
Massari, U. & Miranda, M. 1984 Minimal Surfaces of Codimension One. North-Holland.Google Scholar
Modica, L. 1987 The gradient theory of phase transitions and the minimal interface criterion. Arch. Rat. Math. Anal. 98 (2), 123–142.CrossRefGoogle Scholar
Nicaelenko, B., Scheurer, B. & Témam, R. 1989 Some global dynamical properties of a class of pattern formation equations. Comms. P.D.E.s 14 (2), 245–297.CrossRefGoogle Scholar
Novick-Cohen, A. & Segel, L. A. 1984 Nonlinear aspects of the Cahn–Hilliard equation. Physica 10 (D), 277–298.Google Scholar
Oono, Y. & Puri, S. 1988 Study of phase separation dynamics by use of the cell dynamical systems. I. Modelling. Phys. Rev. A 38 (1), 434–453.CrossRefGoogle Scholar
Sicripov, V. P. & Skripov, A. P. 1979 Spinodal decomposition (phase transition via unstable states). Sov. Phys. Usp. 22, 389–410.CrossRefGoogle Scholar
Témam, R. 1977 Navier–Stokes Equations. North-Holland.Google Scholar
Témam, R. 1988 Infinite-DimensionalDynamical Systems in Mechanics and Physics. Springer-Verlag.CrossRefGoogle Scholar
Van, Der Waals J. D. 1893 The thermodynamic theory of capillarity flow under the hypothesis of a continuous variation of density. Verhandel/Konink. Akad. Weten. 1 (8) (in Dutch).Google Scholar
Von Wahl, W. 1985 On the Cahn–Hilliard equation u ′ + Δ2u — Δf(u) = 0. Delft Progress Report 10, 291–310.Google Scholar
Visintin, A. 1984 Stefan Problem with Surface Tension (Technical Report no. 424, Instituto di Analisi Numerica, Pavia, Italy).Google Scholar
Visintin, A. 1989 Stefan problem with surface tension. In Mathematical Models for Phase Change Problems, I.S.N.M. 88 (ed. Rodrigues, J. F.), Birkhäuser Verlag.Google Scholar
Zheng, Songmu 1986 Asymptotic behaviour of the solution to the Cahn–Hilliard eqvation. Applic. Anal. 23, 165–184.Google Scholar