Multispecies cross-diffusions: from a nonlocal mean-field to a porous medium system without self-diffusion

Sophie Hecht (1, 2) , Marie Doumic (3) , Marie Doumic (3) , Benoit Perthame (2, 4) , Diane Peurichard (2, 4)

1 CNRS - Centre National de la Recherche Scientifique
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
3 MERGE - Mathématiques pour l’évolution, la reproduction, la croissance et l’émergence
4 MAMBA - Modelling and Analysis for Medical and Biological Applications

Systems describing the long-range interaction between individuals have attracted a lot of attention in the last years, in particular in relation with living systems. These systems are quadratic, written under the form of transport equations with a nonlocal self-generated drift. We establish the localisation limit, that is the convergence of nonlocal to local systems, when the range of interaction tends to 0. These theoretical results are sustained by numerical simulations. The major new feature in our analysis is that we do not need diffusion to gain compactness, at odd with the existing literature. The central compactness result is provided by a full rank assumption on the interaction kernels. In turn, we prove existence of weak solutions for the resulting system, a cross-diffusion system of quadratic type.

Keywords: Cross-diffusion; Nonlocal interactions; Localisation limit; Mathematical biology; Multispecies models; Aggregation equation

Domains: General Mathematics [math.GM]

Sophie Hecht, Marie Doumic, Marie Doumic, Benoit Perthame, Diane Peurichard. Multispecies cross-diffusions: from a nonlocal mean-field to a porous medium system without self-diffusion. 2023. ⟨hal-04108050v1⟩