On Cahn-Hilliard models with singular potentials and source terms

The Cahn-Hilliard (CH) equation is a fourth order nonlinear partial differential equation (PDE) originally used to model the separation of a mixture containing two phases of matter, such as water and oil, and is now an ubiquitous component of many mathematical models in applied sciences. It is formulated with a nonconvex potential function, usually taken as a smooth polynomial, that possesses two equal minima, say -1 and 1. Through the addition of appropriate source terms, the CH equation has seen recent applications in modelling the growth of tumours and in repairing damaged black- and-white images (also known as inpainting). However, an unfortunate feature with a polynomial potential is that the solution may not stay bounded in between -1 and 1, and undesirable effects, such as negative mass densities in tumour models or new shades of colour in black-and-white images, can occur.

A remedy is to employ singular potentials, such as a logarithmic-type function, to ensure that the solution to the model stays in between -1 and 1. In exchange, the mathematical analysis of these models with singular potentials become more involved compared to the polynomial potentials. For recent applications in tumour growth and inpainting, the new combination of singular potentials and source terms has not received much attention in the literature, and current analytical studies are mostly confined to establishing the existence of weak solutions.

In this project, we plan to expand the scope of the analysis for these new Cahn-Hilliard models with singular potentials and source terms by investigating the issues of uniqueness and regularity of solutions, as well as addressing the existence of stationary solutions analytically and also numerically. We believe that the proposed methodologies can yield both theoretical and practical contributions towards the active research areas of tumour growth and inpainting, in the form of a novel and alternative strategy to perform inpainting with the CH equation that is built upon a rigorous analytical basis, and a mathematical foundation for practitioners to make meaningful comparisons between tumour model simulations and experimental data