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From a multiscale derivation of nonlinear cross-diffusion models to Keller-Segel models in a Navier-Stokes fluid. (English) Zbl 1353.35038

Summary: 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.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35K57 Reaction-diffusion equations
35Q30 Navier-Stokes equations
35K40 Second-order parabolic systems
92C17 Cell movement (chemotaxis, etc.)
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