Collaborative Research: Deterministic and Statistical Relations Between the Navier-Stokes Equations and Its Determining Forms

在这个项目中，主要研究人员研究与控制大气和海洋的基本方程有关的关键开放问题，以期在气象学和天气预报中应用。这些方程描述了气体和液体在相当一般的条件下的流体运动，从层流到湍流，尺度从不到一毫米到天文长度。因此，他们的研究在航空科学、气象学、石油工业、等离子体物理学以及最近的生物物理流体动力学中都有广泛的应用。这些方程固有的非线性和多尺度性质使得天气预报和理解气候演变的问题具有挑战性。研究人员开发分析和计算工具来研究大气和气候基本控制方程的长期行为。一个主要的重点是将最近发展起来的数据同化技术与这些方程的统计解理论联系起来。这反过来又有助于将长期收集的大量天气/气候数据纳入天气和气候的数学模型，从而有可能改进统计预测。研究生都在从事该项目的工作。研究人员在一类新的函数空间中研究了湍流的几个方面，包括光滑解的存在时间，统计性质，以及强或弱全局吸引子上的长期动力学，这类函数空间包含在所有的Sobolev类中。这类问题的一个重要特征是非线性发展方程满足一个几乎线性的微分不等式。这种泛函类具有吸收导数的性质，具有使非线性项变得更温和的效果，潜在地应用于一类广泛的非线性发展方程。对于某些类型的初始数据，这给出了比一些著名的Soblev泛函类更好的Navier-Stokes方程的存在时间。利用这类新的泛函空间，通过确定模式、结点和确定形式，研究了N-S方程统计解的有限维性质。此外，研究人员还开发了用于流体动力学方程统计解的数据同化技术。统计解与工程师所关心的平均物理量直接相关。它们提供了一座桥梁，从能量、能化和热传递等物理观测的时间平均到相空间上的系综测量。在这个项目中开发的技术应该立即在工程和科学上应用。

In this project, the principal investigators study key open problems concerning the fundamental equations governing the atmosphere and ocean with a view towards applications in meteorology and weather forecasting. These equations describe fluid motions for gases and liquids under quite general conditions, from laminar to turbulent flows, on scales ranging from below a millimeter to astronomical lengths. Consequently, their study has wide-ranging applications in aeronautical sciences, in meteorology, in the petroleum industry, in plasma physics, and more recently, in biophysical fluid dynamics. The inherent nonlinear and multi-scale nature of these equations make the problem of weather forecasting and understanding climate evolution challenging. The investigators develop analytical and computational tools to study the long-term behavior of the fundamental governing equations of the atmosphere and climate. A main focus is to connect recently developed techniques of data assimilation with the theory of statistical solutions of these equations. This in turn facilitates the incorporation of vast amounts of weather/climate data, collected over a long period of time, into mathematical models of weather and climate, potentially leading to improved statistical prediction. Graduate students are engaged in the work of the project. The investigators study several aspects of turbulent flows, including the time of existence of smooth solutions, statistical properties, and the long-term dynamics on the strong or weak global attractor, in a new class of functional spaces that is contained in all Sobolev classes. A key feature of this class is that the nonlinear evolution equation satisfies a differential inequality that is almost linear. This functional class has the property that it can absorb derivatives, which has the effect of making the nonlinear term milder, with potential applications to a wide class of nonlinear evolution equations. For certain types of initial data, this yields a better existence time for the Navier-Stokes equations than some well-known Sobolev functional classes. This new class of functional spaces is employed to study the finite-dimensional properties of statistical solutions of the Navier-Stokes equations, via the determining modes, nodes, and determining forms. Furthermore, the investigators develop data assimilation techniques for statistical solutions of hydrodynamic equations. Statistical solutions are directly related to the average physical quantities with which engineers are concerned. They provide a bridge from the time average of physical observables such as energy, enstrophy, and heat transfer to ensemble measures on the phase space. The techniques developed in this project should have immediate engineering and scientific applications.