Singular Limits and Gradient Flows: Analysis and Numerics

势能最小化是整个自然界现象的特征。在某些被称为梯度流的系统中，尽可能快地使势能最小化的原则完全决定了系统的行为。例子包括超导体中的涡旋模型和通过化学信号进行通信的细胞模型。本项目将通过数学分析和数值模拟相结合的统一方法来研究物理、生物和工程中产生的梯度流。这项研究将与本科生和研究生合作进行。本研究的核心是涉及非局部相互作用的非线性系统，不仅出现在前面提到的例子中，还出现在机器人控制算法、几何形状优化和量子信息理论中。这些梯度流不满足变分的传统凸性假设，而偏微分方程法无法弥补一般理论中的这些限制。这个项目旨在通过发展更一般的凸性概念、将动力学和渐近行为联系起来的适定性结果以及梯度流问题的新的空间和时间离散化来克服这些差距。两个关键的激励问题是梯度流的奇异极限：(1)扩散的非局部近似和(2)慢扩散极限，它是Mesa问题的推广。在这两种情况下，凸性和规律性在极限中恶化，突出了现有理论的局限性。这一裁决反映了NSF的法定使命，并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，被认为值得支持。

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.