Numerical analysis of Cahn-Hilliard type equations on evolving surfaces

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There has been much research into the Cahn-Hilliard equation since it was initially proposed by John W. Cahn and John E. Hilliard in 1958. Typically, this is studied on a stationary Euclidean domain. This research considers a twofold improvement, by looking at hypersurfaces (i.e. a non-Euclidean domain) which may be evolving in time. This adds non-trivial complications to the existing theory. Beyond mathematical curiosity, there is genuine reason to consider this set up, as there are physically relevant models coupling the Cahn-Hilliard equation to surface evolution equations (for example by a forced mean curvature flow). In this research I aim to: adapt results of the numerical analysis of the Cahn-Hilliard equation with a logarithmic potential energy to the case of (stationary) surface finite elements, and study similar equations on an evolving domain. This includes a fully discrete numerical scheme for the Cahn-Hilliard equation on an evolving surface with a sufficiently smooth potential - as current research only has considered semi-discrete schemes for this problem. Similarly, one can couple the Cahn-Hilliard equations to the Navier-Stokes equations to obtain a system which can be used to model richer phenomena. The evolving surface Navier-Stokes equations have been a recent topic of interest, but there is currently much less research regarding Navier-Stokes-Cahn-Hilliard systems on evolving surfaces. I aim to study a Navier-Stokes-Cahn-Hilliard system, similar to the well known "Model H" of Hohenberg and Halperin, on an evolving surface both analytically and numerically. These problems been studied for a stationary, flat domain but there is considerably less literature for evolving domains or hypersurfaces. There is currently much interest in partial differential equations on evolving hypersurfaces, this adds extra complexity and accuracy to the underlying physical model. As such, the numerical analysis of such models is necessary for practical schemes which will be implemented. Similarly, this serves nicely as an example of a nonlinear partial differential equation on an evolving surface for which we can obtain quantitative information. An example being recent work on the phenomenon of "separation from the pure phases". Funding is through the Engineering and Physical Sciences Research Council (EPSRC), which covers research in mathematical sciences. Moreover, the Cahn-Hilliard equation finds applications, outside of mathematics, to many real world problems such as in metallurgy, modelling chemotaxis, RNA/protein dynamics, and in image processing analysis. Likewise the Navier-Stokes-Cahn-Hilliard system has been used to model chemotaxis (with advection) and tumour growth, and the kinetics of lipid biomembranes.