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Aggregation-diffusion equations: dynamics, asymptotics, and singular limits. (English) Zbl 1451.76117

Bellomo, Nicola (ed.) et al., Active particles, Volume 2. Advances in theory, models, and applications. Cham: Birkhäuser. Model. Simul. Sci. Eng. Technol., 65-108 (2019).
Summary: 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.
For the entire collection see [Zbl 1427.74004].

MSC:

76R50 Diffusion
76M99 Basic methods in fluid mechanics
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
35K57 Reaction-diffusion equations

Software:

NumPy; SciPy; Matplotlib
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Full Text: DOI arXiv

References:

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