Regularisation and separation for evolving surface Cahn-Hilliard equations, joint with Charlie Elliott, Maurizio Grasselli and Andrea Poiatti.

In this paper, we consider two models for the constant mobility Cahn-Hilliard equation with a logarithmic potential on an evolving surface and establish instantaneous regularisation results as well as a separation from pure phases property. This means that the (weak) solutions are bounded away from the singular values of the nonlinearity, which is essential to prove higher order regularity of the solutions. The techniques in this paper are an improvement of those in the article below and allow for the relaxation of some of the assumptions therein regarding Galerkin approximation methods for PDEs on evolving surfaces.

Cahn-Hilliard equations on an evolving surface, joint with Charlie Elliott 

In this paper, we study models for the constant mobility Cahn-Hilliard equation on a given evolving surface and establish existence, uniqueness, stability and (some) regularity of weak solutions, for the usual smooth and singular (logarithmic and double obstacle) potentials. The singular nonlinearities are the most interesting problem; the fact that the domains are allowed to evolve in time plays a role in the well-posedness results, which are now, as opposed to the classical fixed-domain case, dependent on an interplay between the evolution of the surfaces, the Cahn-Hilliard dynamics and the initial data.

This has now been accepted for publication in the European Journal for Applied Mathematics, click here.

Function spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs, joint with Amal Alphonse, Ana Djurdjevac and Charlie Elliott

In this paper, we develop an abstract framework to treat parabolic PDEs on time-dependent families of Banach spaces. We define an appropriate notion of weak time derivative which does not rely on an inner product structure and explore properties of evolving Bochner-Sobolev spaces. We also establish an Aubin-Lions compactness result for these spaces and explore some examples.

This has now been accepted for publication in the Journal of Differential Equations, click here.