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

流体动力学模型方程的奇异性和小尺度形成

• 批准号: 1810687
• 负责人: Duy Nguyen Vu Hoang
• 金额: $ 10.15万
• 依托单位: University of Texas at San Antonio
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2021-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1810687&HistoricalAwards=false
• 关键词: Singularity; Small; Scale; Formation; Model

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.

欧拉方程是描述水和空气等流体运动的微分方程组。与考虑流体运动中摩擦效应的纳维-斯托克斯方程一起，它们被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水循环时海洋。尽管这些方程是在两百多年前首次提出的，但它们的一些基本数学特性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表现出强烈的“小规模形成”趋势。例如，这可以在流动中形成非常小的涡流和不规则性，最终导致湍流中看到。在这个研究项目中，研究人员和合作者从数学的角度研究了流体流动不规则性的形成。目标是对导致小规模形成的机制进行详细分析。该项目涉及对某些流体动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导致有限时间内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。主要困难来自方程的非局部和非线性性质。在其中一个项目中，研究人员考虑了二维修正表面准地转方程和 Boussinesq 方程的双曲流场景。目标是深入了解双曲流场景，该场景被认为是最终为三维欧拉方程创建有限时间爆炸的良好候选者。在其余项目中，研究人员考虑一维模型方程，其目标是尽可能详细地描述奇点形成。一个重要的总体主题包括使用障碍函数和考虑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 NSF 的法定使命，并通过评估被认为值得支持利用基金会的智力优势和更广泛的影响审查标准。