Nonlocal Reaction-Diffusion Equations and Wasserstein Gradient Flows

偏微分方程作为生物和物理现象的模型出现在科学和工程中。这个项目涉及对一类重要方程的理论和数值分析，其中包括对癌症肿瘤的发展建模、描述生物种群的生长和扩散以及机器人群的某些算法的方程。这一结果有望阐明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，并包括在社区开展外联活动，在新一代人中推广STEM学科。这项研究将集中在三个相互关联的主题上：(1)生物学和生态学中出现的非局部反应扩散方程解的定性性质的研究；(2)偏微分方程解的确定性粒子方法的发展和收敛分析，重点是它们对机器人蜂群的适应；(3)测度空间上的梯度流理论。该项目的一个主要目标是将梯度流公式推广到非局部偏微分方程组，以及非质量守恒的偏微分方程组，如反应扩散方程。该项目还将导致开发、实施和研究本地和非本地方程的数值方法。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。

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.