Collaborative Research: Kinetic Models of Aggregation and Dispersion

在许多物理现实的数学模型中，连贯的结构是通过相互竞争的影响的平衡来形成和维持的。一方面，聚焦、集中或聚集效应通常是由非线性机制产生的。这些影响通常被以某种方式分散、散焦、碎片化或散开的其他过程所抵消。这项提议旨在开发几种新颖和有用的数学工具，用于分析这种竞争效应如何实现动态平衡。需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、流体动力学和质量传输。预计所学到的数学经验将是根本性的，并有助于建立一个有望对一系列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、系列讲座和研讨会上与学生和年轻研究人员进行大量培训和互动，并在会议、研讨会和研讨会上传播结果。拟议的研究重点是研究四个领域的动态行为，这些领域受到应用和偏微分方程理论的强烈推动。第一个领域考虑了一个基本的凝聚-碎裂模型，但没有详细的平衡，该模型来自Niwa对大洋中部鱼群大小的大量经验数据的比例分析。第二个领域将开发人群配置路径的度量和测地线，以及图像分析人员提出的形状距离的相关流体动力学问题。第三部分研究了一个新的非局域频散和非线性集中模型的长时间动力学和梯度结构，与孤波剖面的不动点方程有关。最后一个领域考虑随机粘性粒子动力学，寻求建立在PDE理论的最新进展的基础上，该理论将粘性粒子动力学与守恒定律的奇异解联系在一起，并建立在随机激波聚集的相关进展上。真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的粒子系统。

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.