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

We consider the hyperbolic relaxation of the viscous Cahn–Hilliard equation εϕtt+ϕt-Δ(δϕt-Δϕ+g(ϕ))=0,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 δ= ε= 0 when d= 2 or 3; δ being the viscosity parameter. The constant ε∈ (0 , 1] is a relaxation parameter, ϕ is the order parameter and g: R→ R is a nonlinear function. This equation models the early stages of spinodal decomposition in certain glasses. Assuming that ε is dominated from above by δ when d= 2 or 3, we construct a family of exponential attractors for Eq. (0.1) which converges as (ε, δ) goes to (0 , δ) , for any δ∈ [0 , 1] , with respect to a metric that depends only on ε, improving previous results where this metric also depends on δ. 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 ϕ~(t)=ϕ(εt) and we rewrite Eq. (0.1) in the variables (ϕ~ , ϕ~ _t). We show that there exist an integer n, independent of both ε and δ, a value 0 < ε~ (n) ≤ 1 and an inertial manifold of dimension n, for either ε∈ (0 , ε~] and δ=2ε or ε∈ (0 , ε~] and δ∈ [0 , 3 ε]. Then, we prove the existence of an inertial manifold of dimension that depends on ε, but is independent of δ and η, for any fixed ε∈ (0 , (η- 2) ²] and every δ∈[ε,(2-η)ε], for an arbitrary η∈ (1 , 2). Next, we show the existence of an inertial manifold of dimension that depends on ε and η, but is independent of δ, for any fixed ε∈(0,1(2+η)2] and every δ∈[(2+η)ε,1], where η> 0 is arbitrary chosen. Moreover, we show the continuity of the inertial manifolds at δ= δ, for any δ0∈[0,(2-η)ε]∪[(2+η)ε,1]. Second, we set ϕ_t= - (2 ε) ^{- 1}(I- δΔ) ϕ+ ε^{- 1 / 2}v and we rewrite Eq. (0.1) in the variables (ϕ, v). Then, we prove the existence of an inertial manifold of dimension that depends on δ, but is independent of ε, for any fixed δ∈ (0 , 1] and every ε∈(0,316δ2]. In addition, we prove the convergence of the inertial manifolds when ε→ 0 ⁺.