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Some models of Cahn-Hilliard equationsin nonisotropic media

Published online by Cambridge University Press:  15 April 2002

Alain Miranville
Alain Miranville*
Affiliation:
Université de Poitiers, Mathématiques, SP2MI, Téléport 2, boulevard Marie et Pierre Curie, 86962 Chasseneuil Futuroscope Cedex, France. ([email protected])
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Abstract

We derive in this article some models ofCahn-Hilliard equations in nonisotropic media. These models, based onconstitutive equations introduced by Gurtin in [19], take the work ofinternal microforces and also the deformations of the material intoaccount. We then study the existence and uniqueness of solutions andobtain the existence of finite dimensional attractors.

Keywords

Cahn-Hilliard equationinternal microforcesdeformable continuumnonisotropic materialglobal attractorexponential attractor.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 3 , May 2000 , pp. 539 - 554
Copyright
© EDP Sciences, SMAI, 2000

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References

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of partial differential equations satisfying general boundary conditions I, II. Comm. Pure Appl. Math. 12 (1959) 623-727 ; 17 (1964) 35-92.
Babin, A. and Nicolaenko, B., Exponential attractors of reaction-diffusion systems in an unbounded domain. J. Dyn. Differential Equations 7 (1995) 567-590. CrossRef
A.V. Babin and M.I. Vishik, Attractors of evolution equations. North-Holland, Amsterdam (1991).
H. Brezis, Analyse fonctionnelle, théorie et applications. Masson (1983).
Cahn, J.W., On spinodal decomposition. Acta Metall. 9 (1961) 795-801. CrossRef
Cahn, J.W. and Hilliard, J.E., Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys. 2 (1958) 258-267. CrossRef
Carrive, M., Miranville, A., Piétrus, A. and Rakotoson, J.M., The Cahn-Hilliard equation for an isotropic deformable continuum. Appl. Math. Letters 12 (1999) 23-28. CrossRef
M. Carrive, A. Miranville and A. Piétrus, The Cahn-Hilliard equation for deformable elastic continua. Adv. Math. Sci. Appl. (to appear).
Chepyzhov, V.V. and Vishik, M. I., Attractors of nonautonomous dynamical systems and their dimension. J. Math. Pures Appl. 73 (1994) 279-333.
L. Cherfils and A. Miranville, Generalized Cahn-Hilliard equations with a logarithmic free energy (submitted).
Cholewe, J.W. and Dlotko, T., Global attractors of the Cahn-Hilliard system. Bull. Austral. Math. Soc. 49 (1994) 277-302. CrossRef
Debussche, A. and Dettori, L., On the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. TMA 24 (1995) 1491-1514. CrossRef
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations. Masson (1994).
Efendiev, M. and Miranville, A., Finite dimensional attractors for a class of reaction-diffusion equations in $\mathbb{R}^n$ with a strong nonlinearity. Disc. Cont. Dyn. Systems 5 (1999) 399-424.
C.M. Elliot and S. Luckhauss, A generalized equation for phase separation of a multi-component mixture with interfacial free energy. Preprint.
Fabrie, P. and Miranville, A., Exponential attractors for nonautonomous first-order evolution equations. Disc. Cont. Dyn. Systems 4 (1998) 225-240.
C. Galusinski, Perturbations singulières de problèmes dissipatifs : étude dynamique via l'existence et la continuité d'attracteurs exponentiels. Thèse, Université Bordeaux-I (1996).
Galusinski, C., Hnid, M. and Miranville, A., Exponential attractors for nonautonomous partially dissipative equations. Differential Integral Equations 12 (1999) 1-22.
Gurtin, M., Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D 92 (1996) 178-192. CrossRef
J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969).
D. Li and C. Zhong, Global attractor for the Cahn-Hilliard system with fast growing nonlinearity. J. Differential Equations (1998).
M. Marion and R. Temam, Navier-Stokes equations, theory and approximation, in Handbook of numerical analysis, P.G. Ciarlet and J.L. Lions eds. (to appear).
Miranville, A., Exponential attractors for nonautonomous evolution equations. Appl. Math. Letters 11 (1998) 19-22. CrossRef
Miranville, A., Exponential attractors for a class of evolution equations by a decomposition method. C. R. Acad. Sci. 328 (1999) 145-150. CrossRef
A. Miranville, Long time behavior of some models of Cahn-Hilliard equations in deformable continua. Nonlinear Anal. Series B (to appear).
Miranville, A., Exponential attractors for a class of evolution equations by a decomposition method. II. The nonautonomous case. C. R. Acad. Sci. 328 (1999) 907-912. CrossRef
Miranville, A., Equations de Cahn-Hilliard généralisées dans un milieu déformable. C. R. Acad. Sci. 328 (1999) 1095-1100. CrossRef
Miranville, A., A model of Cahn-Hilliard equation based on a microforce balance. C. R. Acad. Sci. 328 (1999) 1247-1252. CrossRef
Miranville, A., Piétrus, A. and Rakotoson, J.M., Dynamical aspect of a generalized Cahn-Hilliard equation based on a microforce balance. Asymptotic Anal. 16 (1998) 315-345.
Nicolaenko, B., Scheurer, B. and Temam, R., Some global dynamical properties of a class of pattern formation equations. Comm. Partial Differential Equations 14 (1989) 245-297. CrossRef
R. Temam, Infinite dimensional dynamical systems in mechanics and physics. 2nd. ed., Springer-Verlag, New-York (1997).