Cross-diffusion systems in physics and biology: population dynamics, biofilms and two-phase flow

In this talk we focus on the proof of the existence of global-in-time weak solutions to reaction-cross-diffusion systems for an arbitrary number of species.  The first model we discuss extends in the case of linear transition rates the two-species population model of Shigesada, Kawasaki, and Teramoto.  After presenting the existence analysis, we derive the model from a stochastic particle model and we link at the formal level the entropy structure of  the cross-diffusion system satisfying the detailed balance condition with the entropy structure of a reversible microscopic many-particle Markov process on a discretized space. Finally, we discuss two other cross diffusion models: one describing biofilm models for wastewater treatment,  and another one which describes a first result in the direction of two-phase flow modeling with cross diffusion effects. These results are based on joint works with L. Chen, X. Chen, L. Desvillettes, H. Dietert, A. Juengel, Pina Milisic and Nicola Zamponi.