Cahn–Hilliard equations on an evolving surface

Abstract

We describe a functional framework suitable to the analysis of the Cahn–Hilliard equation on an evolving surface whose evolution is assumed to be given a priori. The model is derived from balance laws for an order parameter with an associated Cahn–Hilliard energy functional and we establish well-posedness for general regular potentials, satisfying some prescribed growth conditions, and for two singular non-linearities – the thermodynamically relevant logarithmic potential and a double-obstacle potential. We identify, for the singular potentials, necessary conditions on the initial data and the evolution of the surfaces for global-in-time existence of solutions, which arise from the fact that integrals of solutions are preserved over time, and prove well-posedness for initial data on a suitable set of admissible initial conditions. We then briefly describe an alternative derivation leading to a model that instead preserves a weighted integral of the solution and explain how our arguments can be adapted in order to obtain global-in-time existence without restrictions on the initial conditions. Some illustrative examples and further research directions are given in the final sections.

Regularisation and separation for evolving surface Cahn-Hilliard equations

“…This article on one hand complements [8] by proving instantaneous regularisation of the weak solutions and on the other hand extends the validity of the strict separation property for the local Cahn-Hilliard equation with constant mobility to the setting of evolving surfaces in R 3 in two cases. It is also worth observing that our approach to the estimates for solutions to the approximate problems allows us to forgo some of the assumptions made in [8] (see, e.g., [8, Assumption A P ]), generalising the results therein. For the sake of completeness, here we also include the proof of continuous dependence on the initial data, entailing uniqueness for the general system (1.2), which was omitted in [8].…”

“…The Cahn-Hilliard equation in an evolving surface setting has recently been studied in [8]. More precisely, having fixed T > 0, considering a family of closed, connected, oriented surfaces in R 3 such that its evolution is given a priori as a flow determined by the (sufficiently smooth) velocity field V, the evolving surface Cahn-Hilliard equation reads u…”

“…In the equations above, equipped with suitable initial conditions, ∇ Γ denotes the tangential gradient on the surface Γ(•), ∆ Γ is the Laplace-Beltrami operator and u denotes the material time derivative of u (see Section 2 for more details). The functional framework in this work is the same as in [8].…”

“…This is usually referred as a regular potential. This case was also taken into account in [8]. However, the polynomial approximation does not ensure the existence of physical solutions, that is, solutions whose values are in [−1, 1], due to the lack of comparison principles for the Cahn-Hilliard equation.…”

“…As far as the evolving surface version is concerned, both systems (1.1) and (1.2) are treated in [8], where the authors establish existence, uniqueness and stability of solutions for the different cases where F is a smooth, logarithmic or double obstacle potential, respectively. Some regularity results are also proved.…”