Nonlocal-to-local convergence of the Cahn–Hilliard equation with degenerate mobility and the Flory–Huggins potential

The Cahn–Hilliard equation is a fundamental model for phase separation phenomena. Its rigorous derivation from the nonlocal aggregation equation, motivated by the desire to link interacting particle systems and continuous descriptions, has received much attention in recent years. In the recent article, we showed how to treat the case of degenerate mobility for the first time. Here, we discuss how to adapt the exploited tools to the case of the mobility $m\left(u\right)=u(1-u)$ as in the original works of Giacomin–Lebowitz and Elliot–Garcke. The main additional information is the boundedness of $u$, implied by the form of mobility, which allows handling the nonlinear terms. We also discuss the case of (mildly) singular kernels and a model of cell–cell adhesion with the same mobility. Another supplementary finding of our work is the energy and entropy inequalities for the nonlocal equation where we give a precise meaning to the dissipation terms despite the singularity of the potential.