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

在这个项目中，主要研究人员研究有关大气和海洋基本方程的关键开放问题，以期在气象学和天气预报中应用。 这些方程描述了从层流到湍流的一般条件下的气体和液体的流体运动，其尺度从低于一毫米到天文长度不等。 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具有广泛的应用。 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。 一个主要的重点是连接最近开发的数据同化技术与这些方程的统计解决方案的理论。 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。 研究生正在从事该项目的工作。研究人员研究湍流的几个方面，包括光滑解的存在时间，统计特性，以及强或弱全局吸引子的长期动力学，在一类新的函数空间中包含在所有Sobolev类中。 这类的一个关键特征是非线性发展方程满足几乎线性的微分不等式。 这个功能类的属性，它可以吸收的衍生物，这具有的效果，使非线性项温和，与潜在的应用广泛的一类非线性发展方程。 对于某些类型的初始数据，这产生了比一些著名的Sobolev函数类更好的Navier-Stokes方程的存在时间。这类新的函数空间被用来研究Navier-Stokes方程的统计解的有限维性质，通过确定模式，节点和确定形式。 此外，研究人员开发数据同化技术的统计解决方案的流体动力学方程。 统计解与工程师关心的平均物理量直接相关。 它们提供了一个桥梁，从时间平均的物理观测量，如能量，拟能，传热合奏措施上的相空间。 在这个项目中开发的技术应该有直接的工程和科学应用。

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.