Uniform global attractors for non-isothermal viscous and non-viscous Cahn–Hilliard equations with dynamic boundary conditions

Abstract

We consider a model of non-isothermal phase separation taking place in a confined container. The order parameter ϕ is governed by a viscous or non-viscous Cahn–Hilliard type equation which is coupled with a heat equation for the temperature θ . The former is subject to a non-linear dynamic boundary condition recently proposed by physicists to account for interactions with the walls, while the latter is endowed with a standard (Dirichlet, Neumann or Robin) boundary condition. We indicate by α the viscosity coefficient, by ε a (small) relaxation parameter multiplying ∂ t θ in the heat equation and by δ a small latent heat coefficient (satisfying δ ≤ λ α , λ > 0 ) multiplying Δ θ in the Cahn–Hilliard equation and ∂ t ϕ in the heat equation. We analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dynamical systems. We first prove that the model generates a strongly continuous semigroup on a suitable phase space Y K α (depending on the choice of the boundary conditions) which possesses the global attractor A ε , δ , α . Our main results allow us to show that a proper lifting A 0 , 0 , α , α > 0 , of the global attractor of the well-known viscous Cahn–Hilliard equation (that is, the system corresponding to ( ε , δ ) = ( 0 , 0 ) ) is upper semicontinuous at ( 0 , 0 ) with respect to the family { A ε , δ , α } ε , δ , α > 0 . We also establish that the global attractor A 0 , 0 , 0 of the non-viscous Cahn–Hilliard equation (corresponding to ( ε , α ) = ( 0 , 0 ) ) is upper semicontinuous at ( 0 , 0 ) with respect to the same family of global attractors. Finally, the existence of exponential attractors M ε , δ , α is also obtained in the cases ε ≠ 0 , δ ≠ 0 , α ≠ 0 , ( 0 , δ , α ) , δ ≠ 0 , α ≠ 0 and ( ε , δ , α ) = ( 0 , 0 , α ) , α ≥ 0 , respectively. This allows us to infer that, for each ( ε , δ , α ) ∈ [ 0 , ε 0 ] × [ 0 , δ 0 ] × [ 0 , α 0 ] , A ε , δ , α has finite fractal dimension and this dimension is bounded, uniformly with respect to ε , δ and α .