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Active Particles, Volume 2Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits

Active Particles, Volume 2: Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular... [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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Active Particles, Volume 2Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits

Part of the Modeling and Simulation in Science, Engineering and Technology Book Series
Editors: Bellomo, Nicola; Degond, Pierre; Tadmor, Eitan
Carrillo, José A.; Craig, Katy; Yao, Yao
Active Particles, Volume 2 — Aug 23, 2019
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43 pages

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Publisher
Springer International Publishing
Copyright
© Springer Nature Switzerland AG 2019
ISBN
978-3-030-20296-5
Pages
65 –108
DOI
10.1007/978-3-030-20297-2_3
Publisher site
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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.]

Published: Aug 23, 2019

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