We consider the one-dimensional Cahn-Hilliard equation with an inertial term εu tt , for ε 0. This equation, endowed with proper boundary conditions, generates a strongly continuous semigroup S ε (t) which acts on a suitable phase-space and possesses a global attractor. Our main result is the construction of a robust family of exponential attractors {M ε }, whose common basins of attraction are the whole phase-space.  2005 Elsevier Inc. All rights reserved.

A first order nonlinear evolution equation is relaxed by means of a time convolution operator, with a kernel obtained by rescaling a given positive decreasing function. This relaxation produces an integrodifferential equation, the formal limit of which, as the scaling parameter (or relaxation time) ε tends to zero, is the original equation. The relaxed equation is equivalent to the widely studied hyperbolic relaxation when the memory kernel, in particular, is the decreasing exponential. In this work, we establish general conditions which ensure that the longterm dynamics of the two evolution equations are, in some appropriate sense, close, when ε is small. Namely, we prove the existence of a robust family of exponential attractors for the related dissipative dynamical systems, which is stable with respect to the singular limit ε → 0. The abstract result is then applied to Allen-Cahn and Cahn-Hilliard type equations.

We consider a modified version of the viscous Cahn–Hilliard equation governing the relative concentration u of one component of a binary system. This equation is characterized by the presence of the additional inertial term ωuttthat accounts for the relaxation of the diffusion flux. Here ω≥0 is an inertial parameter which is supposed to be dominated from above by the viscosity coefficient δ. Endowing the equation with suitable boundary conditions, we show that it generates a dissipative dynamical system acting on a certain phase-space depending on ω. This system is shown to possess a global attractor that is upper semicontinuous at ω = δ = 0. Then, we construct a family of exponential attractors εω,δ, which is a robust perturbation of an exponential attractor of the Cahn–Hilliard equation, namely the symmetric Hausdorff distance between εω,δand ε0, 0goes to 0 as (ω, δ) goes to (0, 0) in an explicitly controlled way. This is done by using a general theorem which requires the construction of another dynamical system, strictly related to the original one, but acting on a different phase-space depending on both ω and δ.

We analyze a Jeffreys type model ruling the motion of a viscoelastic polymeric solution with linear memory in a twodimensional domain with nonslip boundary conditions. For fixed values of the concentrations, we describe the asymptotic dynamics and we prove that, when the scaling parameter in the memory kernel (physically, the Weissenberg number of the flow) tends to zero, the model converges in an appropriate sense to the Navier-Stokes equations.

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