×

zbMATH — the first resource for mathematics

Long-time behavior of some models of Cahn-Hilliard equations in deformable continua. (English) Zbl 0989.35066
The author considers the generalized models of Cahn-Hilliard equations. Existence and uniqueness of weak solutions is shown in the case of the system being the coupling of the Cahn-Hilliard equation with the Navier equation. Also a weakly coupled system is defined for which the existence of a finite-dimensional attractor is proved.

MSC:
35K35 Initial-boundary value problems for higher-order parabolic equations
35B41 Attractors
35A15 Variational methods applied to PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. Agmon, A. Douglis, 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. · Zbl 0093.10401
[2] Babin, A.; Nicolaenko, B., Exponential attractors of reaction – diffusion systems in an unbounded domain, J. dynamics differential equations, 7, 4, 567-590, (1995) · Zbl 0846.35061
[3] Brezis, H., Analyse fonctionnelle, théorie et applications, (1983), Masson Paris
[4] Cahn, J.W., On spinodal decomposition, Acta metall., 9, 795-801, (1961)
[5] Cahn, J.W.; Hilliard, J.E., Free energy of a nonuniform system I, interfacial free energy, J. chem. phys., 2, 258-267, (1958)
[6] Carrive, M.; Miranville, A.; Piétrus, A.; Rakotoson, J.M., The cahn – hilliard equation for an isotropic deformable continuum, Appl. math. lett., 12, 2, 23-28, (1999) · Zbl 0939.35042
[7] M. Carrive, A. Miranville, A. Piétrus, The Cahn-Hilliard equation for deformable elastic continua, Adv. Math. Sci. Appl., to appear. · Zbl 0987.35156
[8] Chepyzhov, V.V.; Vishik, M.I., Attractors of nonautonomous dynamical systems and their dimension, J. math. pure appl., 73, 279-333, (1994) · Zbl 0838.58021
[9] L. Cherfils, A. Miranville, Generalized Cahn-Hilliard equations with a logarithmic free energy, Revista de la Real Academia de Ciencias, to appear. · Zbl 1002.35062
[10] Cholewe, J.W.; Dlotko, T., Global attractors of the cahn – hilliard system, Bull. austral. math. soc., 49, 277-302, (1994)
[11] Debussche, A.; Dettori, L., On the cahn – hilliard equation with a logarithmic free energy, Nonlinear anal. TMA, 24, 10, 1491-1514, (1995) · Zbl 0831.35088
[12] C. Dupaix, D. Hilhorst, Ph. Laurençot, Upper-semicontinuity of the attractor for a singularly perturbed phase-field model, Adv. Math. Sci. Appl., to appear. · Zbl 0905.35015
[13] Eden, A.; Foias, C.; Nicolaenko, B.; Temam, R., Exponential attractors for dissipative evolution equations, (1994), Masson Paris · Zbl 0842.58056
[14] Efendiev, M.; Miranville, A., Finite dimensional attractors for a class of reaction – diffusion equations in Rn with a strong nonlinearity, Discrete control dynamics systems, 5, 2, 399-424, (1999) · Zbl 0959.35025
[15] C.M. Elliot, S. Luckhauss, A generalized equation for phase separation of a multi-component mixture with interfacial free energy, preprint.
[16] Fabrie, P.; Miranville, A., Exponential attractors for nonautonomous first-order evolution equations, Discrete control dynamics systems, 4, 2, 225-240, (1998) · Zbl 0980.34051
[17] C. Galusinski, Thèse, Université Bordeaux - I 1996.
[18] Galusinski, C.; Hnid, M.; Miranville, A., Exponential attractors for nonautonomous partially dissipative equations, Differential integral equations, 12, 1, 1-22, (1999) · Zbl 1012.35010
[19] Gurtin, M., Generalized ginzburg – landau and cahn – hilliard equations based on a microforce balance, Physica D, 92, 178-192, (1996) · Zbl 0885.35121
[20] Haraux, A., Systèmes dynamiques dissipatifs et applications, (1991), Masson Paris
[21] Lions, J.L., Quelques Méthodes de Résolution des problèmes aux limites non linéaires, (1969), Dunod Paris
[22] Li, D.; Zhong, C., Global attractor for the cahn – hilliard system with fast growing nonlinearity, J. differential equations, 149, 2, 191-210, (1998) · Zbl 0912.35029
[23] Miranville, A., Exponential attractors for nonautonomous evolution equations, Appl. math. lett., 11, 2, 19-22, (1998) · Zbl 1337.34064
[24] Miranville, A., Exponential attractors for a class of evolution equations by a decomposition method, C.R. acad. sci. Série I, 328, 2, 145-150, (1999) · Zbl 1101.35334
[25] Miranville, A., A model of cahn – hilliard equation based on a microforce balance, C.R. acad. sci. Série I, 328, 12, 1247-1252, (1999) · Zbl 0932.35118
[26] Miranville, A.; Piétrus, A.; Rakotoson, J.M., Dynamical aspect of a generalized cahn – hilliard equation based on a microforce balance, Asymptotic anal., 16, 315-345, (1998) · Zbl 0936.35036
[27] Nicolaenko, B.; Scheurer, B.; Temam, R., Some global dynamical properties of a class of pattern formation equations, Comm. partial differential equations, 14, 2, 245-297, (1989) · Zbl 0691.35019
[28] Novick-Cohen, A.; Segel, L.A., Nonlinear aspects of the cahn – hilliard equation, Physica D, 10, 277-298, (1984)
[29] Temam, R., Infinite dimensional dynamical systems in mechanics and physics, (1997), Springer New York · Zbl 0871.35001
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.