: 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. 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.
- H. Garcke, K.F. Lam and A. Signori. Sparse Optimal Control of a Phase Field Tumor Model with Mechanical Effects. SIAM J. Control Optim., 59:1555--1580 (2021)
- S. Frigeri, K.F. Lam and A. Signori. Strong well-posedness and inverse identification problem of a non-local phase field tumor model with degenerate mobilities. European Jnl. Appl. Math. (2021)
- M. Ebenbeck and K.F. Lam. Weak and stationary solutions to a Cahn-Hilliard-Brinkman model with singular potentials and source terms. Adv. Nonlinear Anal., 10:24--65 (2020)