Computer-assisted Proofs in Fluid Mechanics and Applications

这个项目的研究目标有两个：一方面追求最近在有限能量不可压缩流体中寻找奇点或整体存在的方法和思想，另一方面转移解决谱几何和数学物理中长期悬而未决的问题的技术。它成功的一个关键因素是拥有准确的数值，以及对正则性理论的深刻理解。因此，该项目的跨学科性质，包括数值计算、物理信息神经网络(PINN)、计算机辅助证明、偏微分方程组(PDE)的现代方法和调和分析，是其成功的关键因素。最近出现的将数学分析和深度学习结合在一起的技术为开发新的攻击角度和更好地理解数学中的经典问题提供了新的机会。本科生和研究生、博士后研究人员和研究来访者积极参与该项目的工作。特别是，来自数学科学中代表性不足群体的本科生受到指导，并参与到项目中不可或缺的研究经验中，关于丰富的、适合学习和部分进步的问题，无论是在数字上还是在分析上。该项目由三个部分组成，第一部分涉及不可压缩欧拉方程和其他水波模型；第二部分涉及表面准地转(SQG)、广义SQG方程和相关模型；第三部分涉及光谱几何和数学物理的应用。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。

The research goal of this project is twofold: on the one hand to pursue methods and ideas developed in recent work in the search for either singularities or global existence in incompressible fluids with finite energy and on the other to transfer the techniques to solve long-standing open problems in spectral geometry and mathematical physics. A key ingredient in its success is to have accurate numerics, together with a deep understanding of the regularity theory. Therefore, the interdisciplinary nature of the project, which involves numerical computations, physics informed neural networks (PINN), computer-assisted proofs, modern methods in partial differential equations (PDE) and harmonic analysis, is an essential ingredient for its successful outcome. The recent emergence of techniques blending mathematical analysis and deep learning offers a new opportunity for developing new angles of attack and better understanding classical problems in mathematics. Undergraduate and graduate students, postdoctoral researchers, and research visitors actively participate in the work of this project. In particular, undergraduate students from underrepresented groups in the mathematical sciences are mentored and involved in research experiences integral to the project, on problems that are rich and suitable for learning and partial progress, both numerically and analytically. The project is structured in three blocks, the first one involving the incompressible Euler equations, as well as other water wave models; the second on the surface quasi-geostrophic (SQG), the generalized-SQG equations, and related models; and a third one on applications to spectral geometry and mathematical physics.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.