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: 1515400
• 负责人: Robert Pego
• 金额: $ 22.25万
• 依托单位: Carnegie-Mellon 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 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1515400&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.

在许多物理现实的数学模型中，相干结构是通过竞争影响的平衡来形成和维护的。 一方面，聚焦，浓度或聚集效应通常由非线性机制产生。 这些效果通常与以某种方式分散，散焦，碎片或分散的其他过程相抵消。 该建议旨在开发几种新颖且有用的数学工具，以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是在：动物生态学，人群动态，形状匹配，流体动力学和大规模运输的研究中出现的。 学到的数学课程将是基本的，并有助于一种理解体系，该理解有望对各种学科的研究人员有用。 此外，调查人员计划在暑期学校，讲座系列中与学生和年轻研究人员进行基本培训和互动，并在会议，研讨会和研讨会上传播结果。拟议的研究侧重于四个领域的动态行为研究由应用和偏微分方程的理论强烈动机。第一个领域考虑了一个基本的凝血碎裂模型，而没有详细的平衡，来自NIWA对中海洋中鱼类学校大小的大量经验数据的扩展分析。第二个区域将开发用于人群配置路径的指标和测量学，以及图像分析师提出的形状距离的相关流体动力问题。第三个领域的重点是在非局部分散和非线性浓度的新模型中的长期动力学和梯度结构，这与孤立波轮廓的固定点方程相关。最终区域考虑了随机粘性粒子动力学，试图基于PDE理论的最新进展，将粘性粒子动力学与保护定律的单数解决方案以及随机冲击聚类的相关进度。现实世界的应用包括动物生态学的领域，图像分析，流体动力学和随机相互作用的粒子系统。