Singularity and Small-Scale Formation for Model Equations of Fluid Dynamics

欧拉方程是一个微分方程组，用来描述像水和空气这样的流体的运动。它们与考虑了流体运动中摩擦力影响的纳维-斯托克斯方程一起，被广泛应用于各种自然和技术情况，例如，在模拟飞机机翼的升力或海洋中的水循环时。尽管这些方程早在两百多年前就被构想出来了，但它们的一些基本数学性质仍然没有得到很好的理解。困难在于，所有描述流体的方程都显示出强烈的“小规模地层”倾向。例如，在最终导致湍流的非常小的漩涡和不规则流动的形成中可以看到这一点。在这个研究项目中，研究者和合作者从数学的角度研究了流体流动中不规则性的形成。目的是详细分析导致小规模地层形成的机制。这些项目涉及对某些流体力学模型方程的几何奇点形成的详细研究。这些模型方程的灵感来自三维不可压缩流体流动的欧拉方程。总体目标是更好地理解在有限时间内导致奇点形成的复杂机制，以及涡度和涡度梯度等量的确切增长率。主要的困难来自于方程的非局部和非线性性质。在其中一个项目中，研究者考虑了二维修正表面准地转和Boussinesq方程的双曲流情景。目标是深入了解双曲流场景，这被认为是最终为三维欧拉方程创建有限时间爆破的一个很好的候选。在剩下的项目中，研究者考虑一维模型方程，其目标是尽可能详细地描述奇点的形成。一个重要的总体主题是使用屏障函数和考虑了解决方案结构的详细信息的先验估计将爆炸场景稳定到奇异时间。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。

The Euler equations are a system of differential equations that describe the motions of fluids like water and air. Together with the Navier-Stokes equations, which take into account the effect of friction in fluid motion, they are applied in a wide variety of natural and technical situations, for example when modeling the lift of an aircraft wing or the circulation of water in the oceans. Although these equations were first conceived more than two hundred years ago, some of their fundamental mathematical properties are still not well understood. The difficulty lies in the fact that all the equations describing fluids show a strong tendency for "small scale formation." This is seen, for example, in the formation of very small vortices and irregularities in the flow that ultimately cause turbulence. In this research project, the investigator and collaborators study the formation of irregularities in fluid flow from a mathematical point of view. The goal is to give a detailed analysis of the mechanisms that lead to small-scale formation.The projects concern detailed research on geometric singularity formation for certain model equations of fluid dynamics. These model equations are inspired by the Euler equations for three-dimensional, incompressible fluid flow. The overall goal is to gain a better understanding of the complex mechanisms leading to singularity formation in finite time, and also the exact growth rates of quantities like the vorticity and vorticity gradient. The main difficulty comes from the nonlocal and nonlinear nature of the equations. In one of the projects, the investigator considers the hyperbolic flow scenario for the modified surface quasi-geostrophic and Boussinesq equations in two dimensions. The goal is to obtain insight into the hyperbolic flow scenario, which is thought to be a good candidate to ultimately create finite-time blowup for the three-dimensional Euler equations. In the remaining projects, the investigator considers one-dimensional model equations, for which the goal is to describe the singularity formation in as much detail as possible. An important overall theme consists in stabilizing the blowup scenario up to the singular time using barrier functions and a priori estimates that take detailed information about the structure of the solution into account.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.