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On Cross-Diffusion Systems for Two Populations Subject to a Common Congestion Effect

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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 soft congestion and the 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.

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Acknowledgements

The author gratefully thanks G. Carlier for suggesting this problem and for fruitful discussions about this work.

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  1. Department of Mathematics and Statistics, McGill University, Montreal, Canada

    Maxime Laborde

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  1. Maxime Laborde
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Correspondence to Maxime Laborde.

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Laborde, M. On Cross-Diffusion Systems for Two Populations Subject to a Common Congestion Effect. Appl Math Optim 81, 989–1020 (2020). https://doi.org/10.1007/s00245-018-9527-4

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