Collaborative Research: Nonlocal Models of Aggregation and Dispersion

在物理现实的许多模型中，连贯的系统状态通过抵消影响的平衡出现并维持。集中、聚集或聚焦的效应是由非线性机制产生的。分散、分散或破碎的竞争影响通常是由于线性作用。该项目解决了几个涉及这种动态平衡的问题，其解决方案将需要开发新的有用的数学工具。这些问题与动物生态学、图像分析、最优运输、流体动力学和相互作用因子的随机模型的研究有关。即将发展的新的数学思想有望引起各学科研究人员的浓厚兴趣。该项目还将吸引学生和年轻研究人员，并在研讨会、会议、暑期学校和系列讲座上进行交流。特别是，本项目的研究将集中于理解与四个主题有关的动态现象，这些主题是由偏微分方程的应用和理论现状所强烈激发的：(1)在h -s中产生的群体大小分布的基本合并分裂模型。Niwa对海洋中部鱼群规模的实证数据分析(2)基于等密度人群配置建模动力学的保体积形状变形最小作用原理及其与（正则化）最优运输指标的联系；(3)基于正则化模型的一种新型弱奇异激波动力学的浅水方程描述了与水深相比大尺度的波浪；(4)在提出的非局部色散和非线性聚焦新模型中，相干波和模式如何通过类梯度动力学形成，这与各种非线性系统中计算波形的定点公式有关。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In many models of physical reality, coherent system states emerge and are maintained through a balance of countervailing influences. Effects that concentrate, aggregate, or focus are generated by nonlinear mechanisms. Competing influences that diffuse, disperse or fragment are often due to linear action. This project addresses several problems which involve a dynamic balance of this kind, whose solution will require the development of new and useful mathematical tools. These problems arise in connection with investigations in animal ecology, image analysis, optimal transportation, fluid dynamics, and stochastic models of interacting agents. The new mathematical ideas to be developed promise to be of strong interest to researchers in a variety of disciplines. The project will also engage students and young researchers, and lead to results to be communicated at seminars, conferences, summer schools, and lecture series.In particular, this project's research will focus on understanding dynamic phenomena that concern four topics strongly motivated by applications and the present state of theory in partial differential equations: (1) A fundamental merging-splitting model of group-size distributions, arising in H.-S. Niwa's analysis of empirical data on fish school size in the mid-ocean; (2) Least action principles for volume-preserving shape deformations and their connection to (regularized) optimal transport metrics, motivated by modeling dynamics of constant-density crowd configurations; (3) A novel type of weakly singular shock wave dynamics in a regularized model for shallow water equations describing waves on scales large compared to the water depth; (4) How coherent waves and patterns form through gradient-like dynamics in a proposed new model of nonlocal dispersion and nonlinear focusing, related to fixed-point formulations for computing wave shapes in a variety of nonlinear systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.