Global attractors for singular perturbations of the Cahn–Hilliard equations

Abstract

We consider the singular perturbations of two boundary value problems, concerning respectively the viscous and the nonviscous Cahn-Hilliard equations in one dimension of space. We show that the dynamical systems generated by these two problems admit global attractors in the phase space H 1 0 (0, ) × H −1 (0, ), and that these global attractors are at least upper-semicontinuous with respect to the vanishing of the perturbation parameter.

Singularly perturbed 1D Cahn–Hilliard equation revisited

“…We now report a well-posedness result for P which is proven in [7] (see also [22] and [46] Thanks to Theorem 2.2, we can define the semigroup…”

“…There the author proved the existence of the (weak) global attractor and its upper semicontinuity with respect to . These results were improved in [46], with different boundary conditions, by establishing the existence of the (strong) global attractor and its smoothness, provided that is small enough. The same authors proved in [45] the existence of an exponential attractor and an inertial manifold, always assuming small enough.…”

On the viscous Cahn-Hilliard equation with singular potential and inertial term

“…The more delicate point stands, of course, in dealing with the right hand sides. Indeed, we claim that there exists the limit 27) at least for almost every t 1 ∈ [0, t), surely including t 1 = 0. We just sketch the proof of this fact, which follows closely the lines of the argument given in [5, Section 3] to which we refer the reader for more details.…”

“…For instance, in space dimension N = 3 the existence of global in time strong solutions is, up to our knowledge, an open issue also in the case when f is a globally Lipschitz (nonlinear) function [17], whereas for N = 2 the occurrence of a critical exponent is observed in case f has a polynomial growth [16,18]. The situation is somehow more satisfactory in space dimension N = 1 (cf., e.g., [26,27]) due to better Sobolev embeddings (in particular all solutions taking values in the "energy space" are also uniformly bounded). It is however worth noting that, in the case when f is singular, even the existence of (global) weak solutions is a mathematically very challenging problem.…”

Global Attractor for Coupled Two-Compartment Gray-Scott Equations

“…There are many studies on the existence of global attractors for diffusion equations. For the classical results we refer the reader to [3,11,19,21,22,24]. Recently, based on the iteration technique for regularity estimates, combining with the classical existence theorem of global attractors, Song et al [17,18] considered the global attractor for some parabolic equations, such as Cahn-Hilliard equation, Swift-Hohenberg equation and so on, in H k (0 ≤ k ≤ ∞) space.…”