Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D

We consider a modified version of the viscous Cahn-Hilliard equation governing the relative concentration u of one component in a binary system. This equation is characterized by the presence of the additional inertial term ω∂_tt u that 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 E_(ω,δ), which is a robust perturbation of an exponential attractor of the Cahn-Hilliard equation, namely, the symmetric Hausdorff distance between E_(ω,δ) and E_0,0 goes 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.