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

  • José A. Carrillo
  • Katy Craig
  • Yao YaoEmail author
Chapter
First Online:
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

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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Notes

Acknowledgements

We thank Matias Delgadino, Franca Hoffmann, Jingwei Hu, Francesco Patacchini, Ihsan Topaloglu, and Li Wang for useful discussions. JAC was partially supported by the EPSRC grant number EP/P031587/1. JAC is grateful to the Mittag-Leffler Institute for providing a fruitful working environment during the special semester Mathematical Biology. KC was supported by NSF DMS-1811012. YY was supported by NSF DMS-1715418. The authors acknowledge the American Institute of Mathematics (AIM) for supporting a visit during the early stages of this work. This work used the Extreme Science and Engineering Discovery Environment (XSEDE) Comet at the San Diego Supercomputer Center through allocation DMS180023.

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Authors and Affiliations

  1. 1.Imperial College LondonLondonUK
  2. 2.University of CaliforniaSanta BarbaraUSA
  3. 3.Georgia Institute of TechnologyAtlantaUSA

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