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Persisting entropy structure for nonlocal cross-diffusion systems
Helge Dietert1; Ayman Moussa2
1 Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, F-75013 Paris, France. — Partly done while at: Institut für Mathematik, Universität Leipzig, D-04103 Leipzig, Germany
2 Sorbonne Université, CNRS, Université Paris Cité, Laboratoire Jacques-Louis Lions (LJLL), Département de Mathématiques et Applications (DMA), École Normale Supérieure (ENS-PSL), F-75005 Paris, France
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 1, pp. 69-104.

For cross-diffusion systems possessing an entropy (i.e. a Lyapunov functional) we study nonlocal versions and exhibit sufficient conditions to ensure that the nonlocal version inherits the entropy structure. These nonlocal systems can be understood as population models per se or as approximation of the classical ones. With the preserved entropy, we can rigorously link the approximating nonlocal version to the classical local system. From a modelling perspective, this gives a way to prove a derivation of the model and, from a PDE perspective, this provides a regularisation scheme to prove the existence of solutions. A guiding example is the SKT model [22]. In this context, we answer positively the question raised by [12] for the derivation and thus complete the derivation.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1762

Helge Dietert 1; Ayman Moussa 2

1 Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, F-75013 Paris, France. — Partly done while at: Institut für Mathematik, Universität Leipzig, D-04103 Leipzig, Germany
2 Sorbonne Université, CNRS, Université Paris Cité, Laboratoire Jacques-Louis Lions (LJLL), Département de Mathématiques et Applications (DMA), École Normale Supérieure (ENS-PSL), F-75005 Paris, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Helge Dietert; Ayman Moussa. Persisting entropy structure for nonlocal cross-diffusion systems. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 1, pp. 69-104. doi : 10.5802/afst.1762. https://afst.centre-mersenne.org/articles/10.5802/afst.1762/

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