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Kinetic and Moment Models for Cell Motion in Fiber Structures Read chapter Read first chapter

2019 | OriginalPaper | Chapter

Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits

Authors : José A. Carrillo, Katy Craig, Yao Yao

Published in: Active Particles, Volume 2

Publisher: Springer International Publishing

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Abstract

Given a large ensemble of interacting particles, driven by nonlocal interactions and localized repulsion, the mean-field limit leads to a class of nonlocal, nonlinear partial differential equations known as aggregation-diffusion equations. Over the past 15 years, aggregation-diffusion equations have become widespread in biological applications and have also attracted significant mathematical interest, due to their competing forces at different length scales. These competing forces lead to rich dynamics, including symmetrization, stabilization, and metastability, as well as sharp dichotomies separating well-posedness from finite time blow-up. In the present work, we review known analytical results for aggregation-diffusion equations and consider singular limits of these equations, including the slow diffusion limit, which leads to the constrained aggregation equation, and localized aggregation and vanishing diffusion limits, which lead to metastability behavior. We also review the range of numerical methods available for simulating solutions, with special attention devoted to recent advances in deterministic particle methods. We close by applying such a method—the blob method for diffusion—to showcase key properties of the dynamics of aggregation-diffusion equations and related singular limits.

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previous chapter Kinetic Models for Pattern Formation in Animal Aggregations: A Symmetry and Bifurcation Approach
next chapter High-Resolution Positivity and Asymptotic Preserving Numerical Methods for Chemotaxis and Related Models
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Metadata
Title
Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits
Authors
José A. Carrillo
Katy Craig
Yao Yao
Copyright Year
2019
Book
Active Particles, Volume 2

Print ISBN: 978-3-030-20296-5

Electronic ISBN: 978-3-030-20297-2

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-20297-2_3

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