From a multiscale derivation of nonlinear cross-diffusion models to Keller–Segel models in a Navier–Stokes fluid

Abstract

This paper deals with a micro–macro derivation of a variety of cross-diffusion models for a large system of active particles. Some of the models at the macroscopic scale can be viewed as developments of the classical Keller–Segel model. The first part of the presentation focuses on a survey and a critical analysis of some phenomenological models known in the literature. The second part is devoted to the design of the micro–macro general framework, where methods of the kinetic theory are used to model the dynamics of the system including the case of coupling with a fluid. The third part deals with the derivation of macroscopic models from the underlying description, delivered within a general framework of the kinetic theory.

Kinetic‐fluid derivation and mathematical analysis of the cross‐diffusion–Brinkman system

“…This section aims to derive general macroscopic models using the micro-macro decomposition method following the line of the paper by Bellomo et al 26 Note that the authors in the aforementioned work 26 have derived macroscopic models of the Keller-Segel type, which describe the chemotaxis phenomenon. 37 First, we present the properties of the kinetic system, which leads to an equivalent micro-macro formulation.…”

“…Several contributions have investigated the asymptotic limit in the following cases: diffusion limit, [19][20][21][22] anomalous diffusion limit, 23,24 hyperbolic model, 25 and the Keller-Segel models of pattern formation in biological tissues. [26][27][28][29][30] Note that there are different approaches to construct such scheme for kinetic models in various contexts. For instance, Bellomo et al 31,32 developed the approach of continuum mechanics based on micro-macro derivation in biological tissues, and Bellouquid 33 developed a model for incompressible Navier-Stokes (see the interesting overview in the work of Banasiak and Lachowicz 34 for more details).…”

On Entropy Dynamics for Active “Living” Particles

“…The first one starts with a detailed modeling of the interactions involving not only social-biological states at the microscopic scale, but also localization of the interacting active particles; applications refer to crowd dynamics [9], vehicular traffic [5,6], and multicellular systems [27]. The second approach is based on the conjecture that space dynamics is induced by a velocity jump perturbation of the spatially homogeneous activity dynamics; see for example [25,30]. The latter has been introduced by the pioneer paper [47], where the aforementioned perturbation of the transport equation is used to derive diffusion models (parabolic) and models with finite speed of propagation (hyperbolic) [48].…”

“…General reasonings about entropy calculations have been developed in various papers, focusing, for instance, on biological systems [25][26][27] as well as social and financial markets [28,29]. Some papers show how models at the macroscopic scale can be derived from the underlying description at the microscopic scale delivered by kinetic theory models [25,30].…”

“…Some papers show how models at the macroscopic scale can be derived from the underlying description at the microscopic scale delivered by kinetic theory models [25,30]. Hence, it is quite natural to develop calculations analogous to those of the classical kinetic theory as well as to multiscale micro-macro problems.…”