Jan-Frederik Pietschmann: On a cross-diffusion model for multiple species with nonlocal interaction and size exclusion

Abstract:
We study a PDE model for two diffusing species interacting by local size exclusion and global attraction. This leads to a nonlinear degenerate cross-diffusion system, for which we provide a global existence result as well as additional regularity in the case of equal
diffusivities. The analysis is motivated by the formulation of this system as a formal gradient flow for an appropriate energy functional consisting of entropic terms as well as quadratic nonlocal terms. Moreover, we investigate phase separation effects inherent in the cross-diffusion model by an analytical and numerical study of minimizers of the energy functional and their behaviour as the diffusivity tends to zero.