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整个科学和工程中。该项目涉及对一类重要方程的理论和数值分析，包括对癌症肿瘤发展的模型，描述生物体种群的生长和传播，以及机器人群体的某些算法的基础。研究结果有望揭示驱动这些现象的机制，并在医学和工程方面具有应用潜力。该项目具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，并包括在社区中推广STEM学科的新一代。研究将集中在三个相互关联的主要主题上：(1)研究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；(2)与工程师合作，开发了求解偏微分方程的确定性粒子方法，并对其收敛性进行了分析，重点研究了其对机器人群的适应性；(3)测度空间上的梯度流动理论。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，如非质量保持的反应扩散方程。该项目还将导致局部和非局部方程的数值方法的发展，实施和研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.