A Cahn–Hilliard model in a domain with non-permeable walls

We consider in this article a Cahn–Hilliard model in a bounded domain with non-permeable walls, characterized by dynamic-type boundary conditions. Dynamic boundary conditions for the Cahn–Hilliard system have recently been proposed by physicists in order to account for the interactions with the walls in confined systems and are obtained by writing that the total bulk mass is conserved and that there is a relaxation dynamics on the boundary. However, in the case of non-permeable walls, one should also expect some mass on the boundary. It thus seems more realistic to assume that the total mass, in the bulk and on the boundary, is conserved, which leads to boundary conditions of a different type. For the resulting mathematical model, we prove the existence and uniqueness of weak solutions and study their asymptotic behavior as time goes to infinity.

Research highlights► Cahn–Hilliard model in a domain with non-permeable walls. ► Boundary contribution to the total free energy. ► Existence and uniqueness of weak solutions. ► Long-time behavior (ωω-limits, attractors).