Regularity, Stability, and Turbulence in Fluid Flows

这个项目将推进物理模型的知识，这些模型在工程、气象学或自然灾害（如洪水和飓风）中具有重要的基础意义。它将推动强大的数学工具的发展，基于最先进的分析，以及线性和非线性偏微分方程的理论。该项目的一个重要组成部分涉及新一代科学家和研究人员的培训和指导。该研究项目为博士生和博士后提供了充足的机会进行数学分析和物理建模应用于流体力学的培训。本项目使用强大的数学工具来解决流体力学中流动行为的几个基本问题。一个严谨的理论将被发展来研究不可压缩流如何在接近可能的爆炸时间时变得线性不稳定，这是湍流的典型特征。边界对不可压缩流动动力学的影响是至关重要的。它将在气象学和海洋学中使用的三维准地转方程模型上进行深入研究。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。

This project will advance the knowledge of physical models which are of fundamental importance in engineering, meteorology, or natural disaster, such as flooding and hurricanes. It will advance the development of powerful mathematical tools based on state of the art analysis, and the theory of linear and nonlinear partial differential equations. An important component of the project concerns the training and mentoring of a new generation of scientists and researchers. The research program provides ample opportunities for Ph.D. students and Postdocs to be trained in mathematical analysis and physical modeling applied to fluid mechanics. This project uses powerful mathematical tools to tackle several fundamental questions on the behavior of flows in fluid mechanic. A rigorous theory will be developed to study how incompressible flows become linearly unstable close to possible blow-up time, a typical signature of turbulence. The effect of boundaries on the dynamic of incompressible flows is of paramount importance. It will be thoroughly investigated on the 3D quasi-geostrophic equation, a model used in meteorology and oceanography.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.