Nonlocal Reaction-Diffusion Equations and Wasserstein Gradient Flows

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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.