Existence and Continuity of Inertial Manifolds for the Hyperbolic Relaxation of the Viscous Cahn-Hilliard Equation

摘要

We consider the hyperbolic relaxation of the viscous Cahn-Hilliard equation epsilon phi(tt) + phi(t) - Delta(delta phi(t) - Delta phi + g(phi)) = 0, (0.1) in a bounded domain of R-d with smooth boundary when subject to Neumann boundary conditions, and on rectangular domains when subject to periodic boundary conditions. The space dimension is d=1, 2 or 3, but it is required delta = epsilon = 0 when d = 2or 3; delta being the viscosity parameter. The constant e is an element of (0, 1] is a relaxation parameter, phi is the order parameter and g : R -> R is a nonlinear function. This equationmodels the early stages of spinodal decomposition in certain glasses. Assuming that e is dominated from above by delta when d = 2 or 3, we construct a family of exponential attractors for Eq. (0.1) which converges as (epsilon, delta) goes to (0, delta(0)), for any delta(0) is an element of [0, 1], with respect to a metric that depends only on epsilon, improving previous results where thismetric also depends on delta. Then we introduce two change of variables and corresponding problems, in the case of rectangular domains and d = 1 or 2 only. First, we set (phi) over tilde (t) = phi(root epsilon t) and we rewrite Eq. (0.1) in the variables ((phi) over tilde, (phi) over tilde (t)). We show that there exist an integer n, independent of both epsilon and delta, a value 0 (0) (n) 0 is arbitrary chosen. Moreover, we show the continuity of the inertial manifolds at delta = delta(0), for any delta(0) is an element of [0, (2 - eta)root epsilon] boolean OR [(2 + eta root epsilon, 1]. Second, we set phi(t) = -(2 epsilon)-1(I - delta Delta)phi + epsilon(-1/2)v and we rewrite Eq. (0.1) in the variables (phi, v). Then, we prove the existence of an inertial manifold of dimension that depends on delta, but is independent of epsilon, for any fixed delta is an element of (0, 1] and every epsilon is an element of (0, 3/16 delta(2)]. In addition, we prove the convergence of the inertial manifolds when epsilon -> 0(+).