zbMATH — the first resource for mathematics

Finite-dimensional attractor for the viscous Cahn-Hilliard equation in an unbounded domain. (English) Zbl 1115.35025
Summary: We consider the viscous Cahn-Hilliard equation in an infinite domain. Due to the noncompactness of operators, we use weighted Sobolev spaces to prove that the semigroup generated by this equation has the global attractor which has finite Hausdorff dimension.

35B41 Attractors
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
35K55 Nonlinear parabolic equations
35K35 Initial-boundary value problems for higher-order parabolic equations
Full Text: DOI
[1] Frédéric Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Differential Equations 83 (1990), no. 1, 85 – 108. · Zbl 0706.35058
[2] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623 – 727. · Zbl 0093.10401
[3] A. V. Babin, The attractor of a Navier-Stokes system in an unbounded channel-like domain, J. Dynam. Differential Equations 4 (1992), no. 4, 555 – 584. · Zbl 0762.35082
[4] Anatoli Babin and Basil Nicolaenko, Exponential attractors of reaction-diffusion systems in an unbounded domain, J. Dynam. Differential Equations 7 (1995), no. 4, 567 – 590. · Zbl 0846.35061
[5] A. V. Babin and M. I. Vishik, Attractors of evolution equations, Studies in Mathematics and its Applications, vol. 25, North-Holland Publishing Co., Amsterdam, 1992. Translated and revised from the 1989 Russian original by Babin. · Zbl 0778.58002
[6] A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A 116 (1990), no. 3-4, 221 – 243. · Zbl 0721.35029
[7] Ahmed Bonfoh and Alain Miranville, On Cahn-Hilliard-Gurtin equations, Proceedings of the Third World Congress of Nonlinear Analysts, Part 5 (Catania, 2000), 2001, pp. 3455 – 3466. · Zbl 1042.74553
[8] J.W. Cahn, On spinodal decomposition, Acta Metall. 9(1961), 795-801.
[9] J.W. Cahn and J.E. Hilliard, Free energy of a non-uniform system I. Interfacial free energy, J. Chem. Phys. 2(1958), 258-267.
[10] Messoud Efendiev and Alain Miranville, Finite-dimensional attractors for reaction-diffusion equations in \?\(^{n}\) with a strong nonlinearity, Discrete Contin. Dynam. Systems 5 (1999), no. 2, 399 – 424. · Zbl 0959.35025
[11] Messoud Efendiev, Alain Miranville, and Sergey Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr. 272 (2004), 11 – 31. · Zbl 1046.37047
[12] M. Hall Jr. and J. H. van Lint , Combinatorics, D. Reidel Publishing Co., Dordrecht-Boston, Mass.; Mathematical Centre, Amsterdam, 1974. NATO Advanced Study Institutes Series, Series C: Mathematical and Physical Sciences, Vol. 16.
[13] A. Miranville, A. Piétrus, and J. M. Rakotoson, Dynamical aspect of a generalized Cahn-Hilliard equation based on a microforce balance, Asymptot. Anal. 16 (1998), no. 3-4, 315 – 345. · Zbl 0936.35036
[14] Alain Miranville, Some generalizations of the Cahn-Hilliard equation, Asymptot. Anal. 22 (2000), no. 3-4, 235 – 259. · Zbl 0953.35055
[15] Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, 2nd ed., Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. · 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.