Singularity Formation and Propagation in Incompressible Fluids

这个项目围绕着流体中某些类型的相干结构的数学分析。粗略地说，连贯结构是流体流动中定义良好的特征，可以很容易地区分并以简单的方式进行数学描述。一个例子是漩涡，所有的流体粒子围绕一个点旋转。另一个例子是尖锐锋，当流体由暖区和冷区组成时，可能会出现一个薄的移动过渡区。在自然界中存在着更多的连贯结构的例子，我们每天都会遇到它们。虽然给出一种有效的方法来描述一般流体流动似乎是一个极其艰巨的问题，但描述相干结构的演变似乎适合于数学分析。此外，由于在物理和数值实验中经常观察到相干结构占主导地位，描述它们的演变是非常重要的。在这方面，人们可以问一个简单的问题：在小扰动下，“强”涡旋会发生什么？漩涡是持续存在还是迅速解体？另一个有趣的问题是，良好的流体流动是否可以发展出具有奇点的连贯结构，即无限速度或速度梯度。这些问题是这个项目的核心。在数学上，主要研究不可压缩欧拉方程和相关模型的弱解和强解的动力学。该项目研究了一种新的方法来解决零粘度流体中有限时间奇点形成的经典问题，即在没有外部影响的情况下流体中出现某些奇异结构。先前的工作提供了在某些情况下不可压缩欧拉方程强解的有限时间奇点形成的一个例子。项目的一部分包括扩展和加强这些结果，以及研究方法对其他问题的适用性，包括二维欧拉方程中旋涡斑块的动力学。另一部分研究不可压缩欧拉方程的某些奇异弱解及其相关模型的稳定性，特别是奇异涡旋在光滑扰动下的稳定性。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 revolves around the mathematical analysis of certain types of coherent structures in fluids. Roughly speaking, a coherent structure is a well-defined feature in fluid flow that can be easily distinguished and mathematically described in a simple manner. One example is a vortex, where all fluid particles rotate about a single point. Another example is a sharp front, which may arise when a fluid consists of warm and cold regions separated by a thin moving transition region. Many more examples of coherent structures exist in the natural world and we encounter them daily. While it seems to be an extremely formidable problem to give an efficient way to describe general fluid flows, describing the evolution of coherent structures seems to be amenable to mathematical analysis. Moreover, since coherent structures are often observed to be dominant in physical and numerical experiments, describing their evolution is of great importance. A simple question that one could ask in this regard is: What happens to a "strong" vortex under small perturbations? Does the vortex persist or does it quickly disintegrate? Another question of interest is whether nice fluid flows can develop coherent structures that possess a singularity, that is, infinite velocity or velocity gradient. Such questions lie at the core of this project. Mathematically, the research focuses primarily on the dynamics of both weak and strong solutions to the incompressible Euler equations and related models. The project investigates a novel approach to the classical problem of finite-time singularity formation in fluids with zero viscosity -- that is, the emergence of certain singular structures in a fluid without external influence. Previous work provided an example of finite-time singularity formation for strong solutions to the incompressible Euler equations in certain settings. Part of the project involves extending and strengthening these results as well as investigating the applicability of the methods to other questions including the dynamics of vortex patches in the 2D Euler equation. Another part of the project studies the stability of certain singular weak solutions to the incompressible Euler equation and related models, specifically, the stability of singular vortices with respect to smoother perturbations.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.