Singular Limits and Gradient Flows: Analysis and Numerics

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Minimization of potential energy characterizes phenomena throughout the natural world. In certain systems, known as gradient flows, the principle of minimizing potential energy as quickly as possible completely determines the system's behavior. Examples include models of vortices in superconductors and of cells communicating via chemical signals. The present project will study gradient flows arising in physics, biology, and engineering through a unified approach combining mathematical analysis and numerical simulation. This research will be pursued in collaboration with undergraduate and graduate students.The gradient flow systems at the heart of the present research are nonlinear systems involving nonlocal interactions, which arise not only in the examples mentioned previously, but also in robotic control algorithms, geometric shape optimization, and quantum information theory. These gradient flows do not satisfy traditional convexity assumptions of the calculus of variations, and techniques from partial differential equations have been unable to compensate for these limitations in the general theory. This project aims to overcome these gaps through the development of more general notions of convexity, well-posedness results that link dynamics and asymptotic behavior, and novel spatial and temporal discretizations of the gradient flow problem. Two key motivating questions are singular limits of gradient flows: (1) the nonlocal approximation of diffusion and (2) the slow diffusion limit, a generalization of the mesa problem. In both cases, convexity and regularity deteriorate in the limit, bringing into focus the limitations of the existing theory.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.