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Mathematical Analysis of Fluid Flow at High Reynolds Number from the Point of View of Turbulence

从湍流角度进行高雷诺数流体流动的数学分析

基本信息

批准号：
1514771
负责人：
Vlad Vicol
金额：
$ 17.71万
依托单位：
Princeton University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2015
资助国家：
美国
起止时间：
2015-08-01 至 2018-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1514771&HistoricalAwards=false
关键词：
Mathematical Analysis Fluid Flow Reynolds

项目摘要

While the equations describing the motion of incompressible fluids have been around for more than two centuries, the underlying mathematics is still not fully understood. Do the solutions of the Navier-Stokes equations correctly describe what we see in fluid experiments? Can we use these equations to make precise predictions about the flows, to predict tomorrow's weather, or to predict the macro-scale evolution of the Earth's climate? These questions are conjecturally related through the phenomenological theories of turbulence and the statistical properties of solutions to the Euler and Navier-Stokes equations. When attempting to give rigorous answers to these questions we are faced with new frontiers in mathematical analysis, and fundamental new ideas are needed to understand the underlying phenomena. In this project the investigator studies the relation between turbulence and the equations that are used to describe fluid flows. The project focuses on furthering our understanding of the hypothesized link between fluid turbulence and the Navier-Stokes equations. The complexity of turbulent flows observed in experiments translates into fundamental mathematical issues, chief among which are the problems of singularities, uniqueness, and stability in the fluid equations. The investigator and colleagues attack these problems from two intimately related angles: by analyzing the statistical properties of solutions to stochastic partial differential equations; and by studying the emergence of singularities in the Euler and related active scalar equations. In tackling these problems the investigator appeals to ideas from hypoellipticity (in the sense of Hormander), convex integration, Lagrangian particle adapted methods, and Landau damping. The goal is to develop nonlinear, solution-adapted methods.

虽然描述不可压缩流体运动的方程已经存在了两个多世纪，但其基础数学仍然没有完全理解。纳维尔-斯托克斯方程的解是否正确地描述了我们在流体实验中所看到的？我们能用这些方程来精确预测气流，预测明天的天气，或者预测地球气候的宏观演变吗？这些问题通过湍流的唯象理论和欧拉方程和纳维尔-斯托克斯方程解的统计性质在理论上相互关联。当试图给出这些问题的严格答案时，我们面临着数学分析的新领域，需要基本的新思想来理解潜在的现象。在这个项目中，研究人员研究湍流和用于描述流体流动的方程之间的关系。该项目的重点是进一步了解流体湍流和Navier-Stokes方程之间的假设联系。在实验中观察到的湍流的复杂性转化为基本的数学问题，其中主要是奇异性，唯一性和稳定性的问题，在流体方程。研究人员和同事从两个密切相关的角度攻击这些问题：通过分析随机偏微分方程解的统计特性;通过研究欧拉方程和相关的活动标量方程中奇点的出现。在解决这些问题的调查呼吁从hypoellipticity的想法（在意义上的霍曼德），凸积分，拉格朗日粒子适应的方法，和朗道阻尼。我们的目标是开发非线性，解决方案适应的方法。