Collaborative Research: Nonlocal Models of Aggregation and Dispersion

在物理现实的许多模型中，连贯的系统状态会出现，并通过相互抵消的影响的平衡来维持。集中、聚集或聚焦的效果是由非线性机制产生的。扩散、分散或碎片化的相互竞争的影响通常是由于线性作用。这个项目解决了涉及这种动态平衡的几个问题，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剂的随机模型的研究有关。即将开发的新数学思想有望引起不同学科的研究人员的强烈兴趣。该项目还将吸引学生和年轻的研究人员，并导致在研讨会、会议、暑期学校和系列讲座上交流结果。特别是，该项目的研究将侧重于理解涉及四个主题的动态现象，这些动态现象涉及偏微分方程组的应用和理论现状：(1)群体规模分布的基本合并-分裂模型，源于H.-S.Niwa对海洋中部鱼类种群大小的经验数据的分析；(2)保体积形状变形的最小作用量原理及其与(正则化)最优输运度量的联系，其动机是恒定密度人群构型的动力学建模：(3)描述比水深大的尺度上的波的浅水方程的正则化模型中的一种新型的弱奇异激波动力学；(4)在提出的非局部色散和非线性聚焦的新模型中，相干波和图案是如何通过类梯度动力学形成的，与计算各种非线性系统中的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优势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.