On cross-diffusion systemsfor two populations subject to a common congestion effect

In this paper, we investigate the existence of solution for systems of Fokker-Planck equations coupled through a common nonlinear congestion. Two different kinds of congestion are considered: a porous media congestion or \textit{soft} congestion and the \textit{hard} congestion given by the constraint $\rho_1+\rho_2 \leqslant 1$. We show that these systems can be seen as gradient flows in a Wasserstein product space and then we obtain a constructive method to prove the existence of solutions. Therefore it is natural to apply it for numerical purposes and some numerical simulations are included.