Collaborative Research: Kinetic Models of Aggregation and Dispersion

在许多物理现实的数学模型中，连贯的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是由非线性机制产生的。这些效果通常会被其他分散、散焦、碎片或以某种方式分散事物的过程所抵消。本提案旨在开发一些新颖而有用的数学工具来分析这些竞争效应如何实现动态平衡。要研究的聚集和分散的特定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、流体动力学和大众运输。所学到的数学课程预计将是基础的，并有助于形成一个对各个学科的研究人员有用的理解体系。此外，研究人员计划在暑期学校、系列讲座、会议、讲习班和研讨会上大量参与培训和与学生和年轻研究人员的互动。提出的研究集中在四个领域的动力行为的研究强烈的应用和偏微分方程的理论。第一个领域考虑了一个基本的凝结-破碎模型，没有详细的平衡，来自Niwa对海洋中部鱼群规模的大量经验数据的缩放分析。第二个领域将发展人群配置路径的度量和测地线以及图像分析人员提出的形状距离的相关流体动力学问题。第三个领域的重点是非局部色散和非线性集中的新模型的长期动力学和梯度结构，涉及孤立波剖面的不动点方程。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定律的奇异解联系起来，以及随机激波聚类的相关进展。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.