Short time existence for coupling of scaled mean curvature flow and diffusion

Abstract

We prove a short time existence result for a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss a mean curvature flow scaled with a term that depends on a quantity defined on the surface coupled to a diffusion equation for that quantity. The proof is based on a splitting ansatz, solving both equations separately using linearization and a contraction argument. Our result is formulated for the case of immersed hypersurfaces and yields a uniform lower bound on the existence time that allows for small changes in the initial value of the height function.

Regularization and Separation for Evolving Surface Cahn–Hilliard Equations

“…In particular, as already noticed, taking inspiration from such models (see, e.g., [49,56]), we do not consider the elasticity of the surface, that would determine an evolution equation for the surface itself, which we assume to be given, but we only study the Cahn-Hilliard equation arising from such models. The natural direction for future work is then to consider the fully coupled system, where the evolution of the surface is itself part of problem, see for instance [1,2,49,56]. In equation (1.6), u is again to be thought of as the difference between the concentrations of the two components in the mixture.…”