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

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，但其背后的数学仍然没有完全被理解。纳维-斯托克斯方程的解正确地描述了我们在流体实验中看到的东西吗？我们能不能用这些方程来准确地预测水流，预测明天的天气，或者预测地球气候的宏观演变？这些问题通过湍流的唯象理论和欧拉方程和纳维-斯托克斯方程解的统计性质被猜想地联系起来。当我们试图对这些问题给出严格的答案时，我们面临着数学分析的新前沿，需要基本的新想法来理解潜在的现象。在这个项目中，研究人员研究了湍流与用于描述流体流动的方程之间的关系。该项目的重点是加深我们对流体湍流和纳维-斯托克斯方程之间假设联系的理解。在实验中观察到的湍流的复杂性转化为基本的数学问题，其中最主要的是流体方程中的奇异性、唯一性和稳定性问题。研究人员和同事从两个密切相关的角度来解决这些问题：通过分析随机偏微分方程解的统计性质；通过研究欧拉方程和相关活动标量方程中奇点的出现。在处理这些问题时，研究人员求助于亚椭圆性(在霍曼德意义下)、凸积分、拉格朗日粒子适应方法和朗道阻尼。其目标是开发非线性的、适用于解决方案的方法。

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.