Collaborative Research: Nonlocal Models of Aggregation and Dispersion

合作研究：聚集和分散的非局部模型

基本信息

批准号：
1812609
负责人：
Robert Pego
金额：
$ 23万
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：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812609&HistoricalAwards=false
关键词：
Collaborative Research Nonlocal Models Aggregation

项目摘要

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.

在许多物理现实模型中，连贯的系统状态出现，并通过平衡反击影响来维持。非线性机制会产生浓缩，聚集或焦点的效果。竞争的影响，分散，分散或碎片通常是由于线性动作引起的。该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，最佳运输，流体动力学和相互作用剂的随机模型有关。要发展的新数学思想有望在各种学科中对研究人员产生强烈的兴趣。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，暑期学校和讲座系列中进行传达的结果。尤其是，该项目的研究将集中在理解动态现象上，这些现象涉及四个主题，这是由应用程序和应用程序强烈激励的四个主题当前的部分微分方程中的理论状态：（1）在H.-S中引起的基本合并分布模型。 Niwa对中海洋中鱼类学校大小的经验数据分析；（2）最小的动作原理，用于支撑形状变形及其连接（正则化）最佳传输指标，这是通过建模恒定密度人群配置的动力学来激励的；（3）与水深度相比，浅水方程的一种新型弱奇异的冲击波动力学，描述了大尺度上的波；（4）在提议的非局部分散和非线性聚焦的新模型中，如何通过类似梯度的动力学形成一致的波和模式，这与各种非线性系统中计算波形的固定点配方有关使用基金会的知识分子优点和更广泛的审查标准，通过评估被认为值得支持。