A Nonlocal-to-Local Approach to Aggregation-Diffusion Equations

IF 10.8 1区数学 Q1 MATHEMATICS, APPLIED

SIAM Review Pub Date : 2025-05-08 DOI:10.1137/25m1726248

C. Falcó, R. E. Baker, J. A. Carrillo

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引用次数: 0

Abstract

SIAM Review, Volume 67, Issue 2, Page 353-372, May 2025.
Abstract.Over the past few decades, nonlocal models have been widely used to describe aggregation phenomena in biology, physics, engineering, and the social sciences. These are often derived as mean-field limits of attraction-repulsion agent-based models and consist of systems of nonlocal partial differential equations. Using differential adhesion between cells as a biological case study, we introduce a novel local model of aggregation-diffusion phenomena. This system of local aggregation-diffusion equations is fourth-order, resembling thin-film or Cahn–Hilliard type equations. In this framework, cell sorting phenomena are explained through relative surface tensions between distinct cell types. The local model emerges as a limiting case of short-range interactions, providing a significant simplification of earlier nonlocal models while preserving the same phenomenology. This simplification makes the model easier to implement numerically and more amenable to calibration to quantitative data. In addition, we discuss recent analytical results based on the gradient flow structure of the model, along with open problems and future research directions.

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聚集扩散方程的非局部到局部方法

SIAM评论，第67卷，第2期，第353-372页，2025年5月。摘要。在过去的几十年里，非局部模型被广泛用于描述生物、物理、工程和社会科学中的聚集现象。这些通常推导为基于吸引-排斥体模型的平均场极限，并由非局部偏微分方程系统组成。以细胞间的差异粘附为例，提出了一种新的局部聚集-扩散现象模型。这个局部聚集扩散方程组是四阶的，类似于薄膜或Cahn-Hilliard型方程组。在这个框架中，细胞分选现象是通过不同细胞类型之间的相对表面张力来解释的。局部模型作为短程相互作用的极限情况出现，在保留相同现象学的同时，为早期的非局部模型提供了重要的简化。这种简化使模型更容易在数值上实现，并且更易于校准定量数据。此外，我们还讨论了基于梯度流结构模型的最新分析结果，以及有待解决的问题和未来的研究方向。

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来源期刊

SIAM Review 数学-应用数学

CiteScore

16.90

自引率

0.00%

发文量

期刊介绍： Survey and Review feature papers that provide an integrative and current viewpoint on important topics in applied or computational mathematics and scientific computing. These papers aim to offer a comprehensive perspective on the subject matter. Research Spotlights publish concise research papers in applied and computational mathematics that are of interest to a wide range of readers in SIAM Review. The papers in this section present innovative ideas that are clearly explained and motivated. They stand out from regular publications in specific SIAM journals due to their accessibility and potential for widespread and long-lasting influence.