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

• Published in: Applied mathematics & optimization Vol. 84; no. 3; pp. 3339 - 3416
• Main Author: Bonfoh, Ahmed
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2021; Springer Nature B.V
• Subjects: Boundary conditions; Calculus of Variations and Optimal Control; Optimization; Control; Convergence; Domains; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Order parameters; Simulation; Smooth boundaries; Spinodal decomposition; Systems Theory; Theoretical; 82C26; 37L25; 35B25; 35B45; Exponential attractors; Inertial manifolds; Continuity; Hyperbolic relaxation; Viscous Cahn–Hilliard equation
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-021-09749-9

We consider the hyperbolic relaxation of the viscous Cahn–Hilliard equation 0.1 ε ϕ tt + ϕ t - Δ ( δ ϕ t - Δ ϕ + 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 0 ) , for any δ 0 ∈ [ 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 ( ε t ) 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 ~ 0 ( n ) ≤ 1 and an inertial manifold of dimension n , for either ε ∈ ( 0 , ε ~ 0 ] and δ = 2 ε or ε ∈ ( 0 , ε ~ 0 ] 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 0 , for any δ 0 ∈ [ 0 , ( 2 - η ) ε ] ∪ [ ( 2 + η ) ε , 1 ] . Second, we set ϕ t = - ( 2 ε ) - 1 ( I - δ Δ ) 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 , 3 16 δ 2 ] . In addition, we prove the convergence of the inertial manifolds when ε → 0 + .