Phase field models on evolving surfaces

We study the asymptotic limit of some evolving surface partial differential equations. We first examine the setting of an evolving surface with prescribed velocity, extending the method of formally matched asymptotic expansions to account for the movement of the domain. We apply this method to the Cahn-Hilliard equation, considering various forms for the mobility and potential functions. In particular looking at different scalings of the mobility with respect to the interface thickness parameter. Mullins-Sekerka type problems are derived with additional terms which are due to the domain evolution. We then consider the evolving surface finite element method and applying it to the Cahn-Hilliard equation in an evolving surface setting. We do this so as to support the theoretical findings as well as to further explore some interesting behaviour of solutions. We finally examine the setting of an evolving surface with an unknown surface velocity, described by a geometric evolution equation coupled to intrinsic fields on the surface. The method of formally matched asymptotic expansions is further extended to account for the unknown surface. We apply the technique to a derived model for focal cell adhesion which aims to extend a known model from the literature. We finish with simulations of a reduced model of our derived version