On the Cahn–Hilliard equation with dynamic boundary conditions and a dominating boundary potential

Abstract

The Cahn-Hilliard and viscous Cahn-Hilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some well-posedness and regularity results are proved.

A Boundary Control Problem for the Viscous Cahn–Hilliard Equation with Dynamic Boundary Conditions

“…In particular, the values of the state variable have to be bounded far away from the singularity of the bulk and boundary potentials in order that the solution to the linearized problem introduced below is smooth as well. Even though all this could be true (for smooth data) also in other situations, i.e., if the structure of the system is somehow different, we give a list of assumptions that implies the whole set of conditions listed in [10], since the latter surely guarantee all we need. We also assume the potentials to be slightly smoother than in [10], since this will be useful later on.…”

“…Even though all this could be true (for smooth data) also in other situations, i.e., if the structure of the system is somehow different, we give a list of assumptions that implies the whole set of conditions listed in [10], since the latter surely guarantee all we need. We also assume the potentials to be slightly smoother than in [10], since this will be useful later on. In order to avoid a heavy notation, we write f and f Γ in place of W and W Γ , respectively.…”

“…As far as the latter are concerned, the most common ones in the literature are the usual no-flux conditions for both y and w. However, different boundary conditions have been recently proposed: namely, still the usual no-flux condition for the chemical potential (∂ n w) Γ = 0 on Γ × (0, T ) (1.4) in order to preserve mass conservation, and the dynamic boundary condition (∂ n y) Γ + ∂ t y Γ − ∆ Γ y Γ + W ′ Γ (y Γ ) = u Γ on Γ × (0, T ) (1.5) where y Γ denotes the trace y Γ on the boundary Γ of Ω, ∆ Γ stands for the Laplace-Beltrami operator on Γ, W ′ Γ is a nonlinearity analoguous to W ′ but now acting on the boundary value of the order parameter, and finally u Γ is a boundary source term. We just quote, among other contributions, [5,18,21,23,24,28] and especially the papers [14] and [10]. In the former, the reader can find the physical meaning and free energy derivation of the boundary value problem given by (1.1) and (1.4)-(1.5), besides the mathematical treatment of the problem itself.…”

“…As mentioned above, the recent paper [10] contains a number of results that regard the problem obtained by complementing the equations (1.1) with the already underlined initial and boundary conditions, namely,…”

“…In this paper, we confine ourselves to the viscous case τ > 0 and avoid potentials like (1.6), in order to be able to apply all of the results proved in [10]. However, regular and singular potentials like (1.2) and (1.3) are allowed.…”