Long time behavior of a singular perturbation of the viscous Cahn–Hilliard–Gurtin equation

Abstract

SUMMARYWe consider a singular perturbation of the generalized viscous Cahn-Hilliard equation based on constitutive equations introduced by Gurtin. This equation rules the order parameter , which represents the density of atoms, and it is given on a n-rectangle (n 3) with periodic boundary conditions. We prove the existence of a family of exponential attractors that is robust with respect to the perturbation parameter >0, as goes to 0. In a similar spirit, we analyze the stability of the global attractor. If n = 1, 2, then we also construct a family of inertial manifolds that is continuous with respect to . These results improve and generalize the ones contained in some previous papers. Finally, we establish the convergence of any trajectory to a single equilibrium via a suitable version of the Łojasiewicz-Simon inequality, provided that the potential is real analytic.

Singularly perturbed 1D Cahn–Hilliard equation revisited

“…Also, we establish the upper semicontinuity of the global attractor at any , including the limit case = 0. Here we report an alternative proof along the lines of [17] (see also [2]) which differs from all the previous ones based on a contradiction argument. Section 6 is dedicated to analyze further properties of the global attractors under suitable assumptions on the nature of the stationary points.…”

“…The method used in [36,37] works essentially for equations involving linear self-adjoint operators only. In [2], we successfully applied the latter method to a singular perturbation of the standard viscous Cahn-Hilliard equation in one and two space dimensions. This method may also be applied to problem (2.2).…”

“…This method may also be applied to problem (2.2). However, our approach works for a larger class of nonlinear evolution dynamical systems, even for equations that contain a non-self-adjoint operator (see [3]). Moreover, our results do not need stronger assumptions on g as in [5,36,37].…”

“…The same authors proved in [45] the existence of an exponential attractor and an inertial manifold, always assuming small enough. We recall that inertial manifolds and their dependence on were also analyzed in [2,3] in the viscous case. The existence a family of robust exponential attractors with no restriction on was proven in [22] by using the new construction mentioned above which is no longer based on the squeezing property (see also [23] for a version with memory).…”

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

“…The new term induces additional regularity and some parabolic smoothing effects, and, for this reason, (1.4) is mathematically more tractable in comparison to (1.3). Indeed, existence, regularity and large time behavior of solutions have been analyzed in a number of papers (cf., e.g., [6,7,13,15,19] and references therein). In all these contributions, however, f is taken as a smooth function of at most polynomial growth at infinity.…”