Cross-diffusion models in complex frameworks from microscopic to macroscopic

Abstract

This paper deals with the micro–macro derivation of models from the underlying description provided by methods of the kinetic theory for active particles. We consider the so-called exotic models according to the definition proposed in [ N. Bellomo, N. Outada, J. Soler, Y. Tao and M. Winkler, Chemotaxis and cross diffusion models in complex environments: Modeling towards a multiscale vision, Math. Models Methods Appl. Sci. 32 (2022) 713–792]. The first part of the presentation focuses on a survey and a critical analysis of some phenomenological models known in the literature. We refer to a selection of case studies, in detail, the transport of virus models, social dynamics, and Keller–Segel in a fluid. The second part shows how a Hilbert-type approach can be developed to derive models at the macroscale from the underlying description provided by the kinetic theory of active particles. The third part deals with the derivation of macroscopic models corresponding to the selected case studies. Finally, a forward look into the future research perspectives is proposed.