Math. Nachr. 272, 11 – 31 (2004) / DOI 10.1002/mana.200310186 Exponential attractors for a singularly perturbed Cahn-Hilliard system Messoud Efendiev ∗1 , Alain Miranville ∗∗2 , and Sergey Zelik ∗∗∗ 2 1 Universit¨ at Stuttgart, Mathematisches Institut A, Pfaffenwaldring 57, 70569 Stuttgart, Germany 2 Universit´ e de Poitiers, Laboratoire d’Applications des Math´ ematiques, SP2MI, T´ el´ eport 2, Avenue Marie et Pierre Curie, 86962 Chasseneuil Futuroscope Cedex, France Received 16 August 2002, accepted 16 August 2002 Published online 12 July 2004 Key words Exponential attractors, continuity, viscous Cahn-Hilliard system MSC (2000) 37L30, 35B40, 35B45 Our aim in this article is to give a construction of exponential attractors that are continuous under perturbations of the underlying semigroup. We note that the continuity is obtained without time shifts as it was the case in previous studies. Moreover, we obtain an explicit estimate for the symmetric distance between the perturbed and unperturbed exponential attractors in terms of the perturbation parameter. As an application, we prove the continuity of exponential attractors for a viscous Cahn-Hilliard system to an exponential attractor for the limit Cahn-Hilliard system. c  2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction The study of the long time behavior of systems arising from physics and mechanics is a capital issue, as it is important, for practical purposes, to understand and predict the asymptotic behavior of the system. For many parabolic and weakly damped wave equations, one can prove the existence of the finite dimensional (in the sense of the Hausdorff or the fractal dimension) global attractor, which is a compact invariant set which attracts uniformly the bounded sets of the phase space. Since it is the smallest set enjoying these properties, it is a suitable set for the study of the long time behavior of the system. We refer the reader to [2], [31], [32], [42] and [43] for extensive reviews on this subject. Now, the global attractor may present two major defaults for practical purposes. Indeed, the rate of attraction of the trajectories may be small and (consequently) it may be sensible to perturbations. In order to overcome these difficulties, Foias, Sell and Temam proposed in [25] the notion of inertial manifold, which is a smooth finite dimensional hyperbolic (and thus robust) positively invariant manifold which contains the global attractor and attracts exponentially the trajectories. Unfortunately, all the known constructions of inertial manifolds are based on a restrictive condition, the so-called spectral gap condition. Consequently, the existence of inertial manifolds is not known for many physically important equations (e.g. the Navier-Stokes equations, even in two space dimensions). A non-existence result has even been obtained by Mallet-Paret and Sell for a reaction-diffusion equation in higher space dimensions. Thus, as an intermediate object between the two ideal objects that the global attractor and an inertial manifold are, Eden, Foias, Nicolaenko and Temam proposed in [11] the notion of exponential attractor, which is a compact positively invariant set which contains the global attractor, has finite fractal dimension and attracts exponentially the trajectories. So, compared with the global attractor, an exponential attractor is expected to be more robust under perturbations and numerical approximations, due to the exponential attraction property (see [11], [24] and [28] for discussions on this subject). Another motivation for the study of exponential attractors comes from the ∗ e-mail: efendiev@mathematik.uni-stuttgart.de, Phone: +49 711 685 5541, Fax: +49 711 685 5599 ∗∗ Corresponding author: e-mail: miranv@mathlabo.univ-poitiers.fr, Phone: +33 5 49496891, Fax: +33 5 49496901 ∗∗∗ e-mail: zelik@mathlabo.univ-poitiers.fr, Phone: +33 5 49496891, Fax: +33 5 49496901 c  2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim