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

TL;DR

This paper studies the existence of solutions for coupled Fokker-Planck systems modeling two populations with congestion effects, introducing a gradient flow approach in Wasserstein space and providing numerical methods.

Contribution

It introduces a novel gradient flow framework for coupled congestion-affected systems and offers a constructive existence proof along with numerical simulations.

Findings

Existence of solutions for systems with porous media and hard congestion.

A constructive method based on gradient flows in Wasserstein space.

Numerical simulations demonstrating the approach.

Abstract

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.