Cell-cell adhesion is one the most fundamental mechanisms regulating collective cell migration during tissue development, homeostasis and repair, allowing cell populations to self-organise and eventually form and maintain complex tissue shapes. Adhesive forces are highly linked to the cell geometry and often, continuum models represent these by nonlocal attractive interactions.
In this talk, I will explain how such models can be approximated by Cahn-Hilliard type equations in the limit of short-range interactions. The resulting model is a local aggregation-diffusion equation, resembling a thin-film type equation, and numerical simulations in one and two dimensions reveal that it still shows the diversity of patterns observed both in experiments and in previously used nonlocal models. Moreover, this local equation can be written as gradient flow with respect to the 2-Wasserstein metric, which motivates the use of variational methods to prove existence. The existence results also rely on a bound from below of the associated free energy functional, which is given by a suitable functional inequality. Finally, I will discuss generalisations of the existence theory to the case of systems of interacting populations.