Nonlocal Reaction-Diffusion Equations and Wasserstein Gradient Flows

非局部反应扩散方程和 Wasserstein 梯度流

• 批准号: 2204722
• 负责人: Olga Turanova
• 金额: $ 23.08万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204722&HistoricalAwards=false
• 关键词: Nonlocal; Reaction; Diffusion; Equations; Wasserstein

Partial differential equations arise throughout science and engineering as models of biological and physical phenomena. This project concerns the theoretical and numerical analysis of an important class of equations, including ones that model the development of cancer tumors, describe the growth and spread of living organisms’ populations, and underlie certain algorithms for robotic swarms. The results are expected to shed light on the mechanisms driving these phenomena and have potential for application in medicine and engineering. The project has an interdisciplinary component, provides mentoring and training opportunities for undergraduate and graduate students, and includes outreach in the community to promote STEM disciplines among new generations. The research will focus on three main interconnected topics: (1) the study of qualitative properties of solutions to nonlocal reaction-diffusion equations arising in biology and ecology; (2) the development and convergence analysis of deterministic particle methods for partial differential equations, with an emphasis of their adaptation to robotic swarming, made in collaboration with engineers; and (3) the theory of gradient flows on the space of measures. A main aim of the project is to extend the gradient flow formulation to nonlocal partial differential equations and to partial differential equations, such as reaction-diffusion equations, that are not mass-preserving. The project will also lead to the development, implementation, and study of numerical methods for both local and nonlocal equations.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.

偏微分方程作为生物和物理现象的模型出现在整个科学和工程领域。该项目涉及一类重要方程的理论和数值分析，包括模拟癌症肿瘤发展的方程，描述生物体种群的生长和扩散，以及机器人群的某些算法的基础。这些结果有望揭示驱动这些现象的机制，并具有在医学和工程中应用的潜力。该项目有一个跨学科的组成部分，为本科生和研究生提供指导和培训机会，并包括在社区中推广STEM学科。研究将集中在三个主要的相互关联的主题：（1）在生物学和生态学中产生的非局部反应扩散方程的解的定性性质的研究;（2）偏微分方程的确定性粒子方法的发展和收敛性分析，重点是它们对机器人群集的适应，与工程师合作;（3）测度空间上的梯度流理论。该项目的一个主要目的是将梯度流公式扩展到非局部偏微分方程和偏微分方程，如反应扩散方程，这不是质量守恒的。该项目还将导致本地和非本地方程的数值方法的开发，实施和研究。该奖项反映了NSF的法定使命，并已被认为是值得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的支持。