We consider a modified version of the viscous Cahn–Hilliard equation governing the relative concentration u of one component of a binary system. This equation is characterized by the presence of the additional inertial term ωuttthat 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 εω,δ, which is a robust perturbation of an exponential attractor of the Cahn–Hilliard equation, namely the symmetric Hausdorff distance between εω,δand ε0, 0goes 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.
Global and exponential attractors for a Cahn–Hilliard equation with logarithmic potentials and mass source
