Collaborative Research: Kinetic Models of Aggregation and Dispersion

在物理现实的许多数学模型中，相干结构是由竞争影响的平衡形成和维持的。 一方面，聚焦、集中或聚集效应通常由非线性机制产生。 这些影响往往被其他过程所抵消，这些过程以某种方式分散，分散，碎片或分散事物。 该建议旨在开发几种新颖而有用的数学工具，用于分析这种竞争效应如何实现动态平衡。 要研究的特定聚集和分散模型特别出现在以下研究中：动物生态学，群体动力学，形状匹配，流体动力学和质量运输。 所学到的数学课程预计是基本的，并有助于一个身体的理解，承诺是有用的研究人员在一系列学科。 此外，还计划通过暑期学校、系列讲座、会议、研讨会等，对学生和年轻研究人员进行培训和交流，并传播成果。本研究计划以偏微分方程的应用和理论为主要研究对象，在4个领域进行动态行为的研究。第一个领域考虑了一个没有详细平衡的基本凝聚-破碎模型，该模型来自丹羽对海洋中部鱼群规模的大量经验数据的缩放分析。第二个领域将发展度量和测地线的人群配置路径和相关的流体动力学问题的形状距离提出的图像分析。第三个领域的重点是在一个新的模型的非局部色散和非线性浓度的长时间的动态和梯度结构，与固定点方程孤立波剖面。最后一个领域考虑随机粘性粒子动力学，寻求建立在PDE理论的最新进展，将粘性粒子动力学与守恒律的奇异解联系起来，以及随机冲击聚类的相关进展。实际应用包括动物生态学、图像分析、流体动力学和随机相互作用粒子系统等领域。

In many mathematical models of physical reality, coherent structures are formed and maintained by a balance of competing influences. On the one hand, focusing, concentration, or aggregation effects are typically produced by nonlinear mechanisms. These effects are often counterbalanced by other processes that disperse, defocus, fragment, or spread things out in some way. This proposal aims to develop several novel and useful mathematical tools for analyzing how such competing effects achieve dynamic balance. The particular models of aggregation and dispersion to be studied arise specifically in studies of: animal ecology, crowd dynamics, shape matching, hydrodynamics, and mass transportation. The mathematical lessons learned are expected to be fundamental, and contribute to a body of understanding that promises to be useful to researchers across a range of disciplines. Further, the investigators plan to be substantially engaged in training and interacting with students and young researchers, at summer schools, lecture series, and disseminating results at conferences, workshops, and seminars.The proposed research focuses on the study of dynamic behavior in four areas strongly motivated by applications and the theory of partial differential equations. The first area considers a fundamental coagulation-fragmentation model without detailed balance, coming from Niwa's scaling analysis of a large body of empirical data on fish school size in the mid-ocean. The second area will develop metrics and geodesics for crowd-configuration paths and related hydrodynamic problems for shape distances proposed by image analysts. The third area focuses on the long-time dynamics and gradient structure in a new model of nonlocal dispersion and nonlinear concentration, related to fixed-point equations for solitary wave profiles. The final area considers random sticky particle dynamics, seeking to build on recent advances in PDE theory that tie sticky particle dynamics to singular solutions of conservation laws, and on related progress for random shock clustering. Real-world applications include the fields of animal ecology, image analysis, fluid dynamics, and stochastic interacting particle systems.