Collaborative Research: Kinetic Models of Aggregation and Dispersion

合作研究：聚集和分散的动力学模型

• 批准号: 1514826
• 负责人: Jian-Guo Liu
• 金额: $ 22.35万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1514826&HistoricalAwards=false
• 关键词: Collaborative; Research; Kinetic; Models; Aggregation

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.

在物理现实的许多数学模型中，相干结构是通过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。 这些影响通常会被其他以某种方式分散、散焦、破碎或展开的过程所抵消。 该提案旨在开发几种新颖且有用的数学工具来分析这种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具体出现在以下研究中：动物生态学、人群动力学、形状匹配、流体动力学和公共交通。 所学到的数学课程预计将是基础性的，并有助于形成对跨学科研究人员有用的理解体系。 此外，研究人员计划在暑期学校、系列讲座中大量参与培训并与学生和年轻研究人员互动，并在会议、讲习班和研讨会上传播研究结果。拟议的研究重点是偏微分方程应用和理论强烈推动的四个领域的动态行为研究。第一个领域考虑了没有详细平衡的基本凝固-破碎模型，该模型来自丹羽对大洋中鱼群规模的大量经验数据的缩放分析。第二个领域将开发人群配置路径的度量和测地线以及图像分析师提出的形状距离的相关流体动力学问题。第三个领域侧重于非局域色散和非线性集中新模型中的长期动力学和梯度结构，与孤立波剖面的定点方程相关。最后一个领域考虑随机粘性粒子动力学，寻求建立在将粘性粒子动力学与守恒定律奇异解联系起来的偏微分方程理论的最新进展以及随机冲击聚类的相关进展的基础上。现实世界的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。