The relaxation of the Cahn-Hilliard equation for the modeling of tumors and its numerical simulation

The Cahn-Hilliard equation, arising from physics, describes the phase separation occurring in a material during a sudden cooling process and is the subject of many pieces of research [2]. An interesting application of this equation is its capacity to model cell populations undergoing attraction and repulsion effects. For this application, we consider a variant of the Cahn-Hilliard equation with a single-well potential and a degenerate mobility. This particular form introduces numerous di culties especially for numerical simulations. We propose a relaxation of the equation to tackle these issues and analyze the resulting system. Interestingly, this relaxed version of the degenerate Cahn-Hilliard equation bears some similarity with a nonlinear Keller-Segel model. We also describe a simple  nite element scheme that preserves the critical physical (or biological) properties using an upwind approach.