Collaborative Research: Nonlocal Models of Aggregation and Dispersion

在物理现实的许多模型中，相干系统状态出现并通过平衡的抵消影响来维持。 集中、聚集或聚焦的效果是由非线性机制产生的。 扩散、分散或分裂的竞争性影响力往往是由于线性作用。 这个项目解决了几个问题，涉及这种动态平衡，其解决方案将需要新的和有用的数学工具的发展。 这些问题的出现与动物生态学，图像分析，最佳运输，流体动力学和相互作用的随机模型的调查。 新的数学思想被开发的承诺是强烈的兴趣，研究人员在各种学科。 本项目还将吸引学生和年轻研究人员参与，并在研讨会、会议、暑期学校和系列讲座中交流成果。特别是，本项目的研究将侧重于理解与偏微分方程的应用和理论现状密切相关的四个主题有关的动态现象：（1）一个基本的合并分裂模型的群体规模分布，产生于H.- S. Niwa对海洋中部鱼群规模经验数据的分析;（2）保体积形状变形的最小作用量原理及其与（正则化）最佳运输指标，由恒定密度人群配置的建模动力学激发;（三）浅水方程正则化模型中的一种新型弱奇异冲击波动力学，水深;（4）相干波和模式如何通过拟提出的非局部色散和非线性聚焦的新模型中的类梯度动力学形成，与计算各种非线性系统中波形的定点公式有关。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.