Analysis of Nonlinear Stochastic Partial Differential Equations with Applications in Turbulence Theory and Climate Modeling

流体动力学面临的一个根本挑战是，基本控制方程没有考虑到由于经验、数值和物理来源的建模不确定性而产生的持续的、随机驱动的波动。鉴于在计算方面取得的广泛进展和概率方法在应用中的普遍存在，显然需要更好地理解随机偏微分方程（SPDEs）的一般数值和分析基础以及与流体动力学基本方程的关系。该项目旨在进一步发展spde的数值和分析工具，以期在气候和天气建模以及湍流研究方面取得新的应用。这项工作将特别侧重于三个新兴的非线性spde领域。首先，我们将讨论不变测度和统计稳态的理论。特别是，我们将开发工具来研究Navier-Stokes方程和地球物理和湍流应用中产生的相关模型的不变测度类中的非粘极限。我们还将识别和表征导致一些以前未处理的非线性spde类遍历和混合特性的噪声制度。该项目的第二部分将开发噪声非线性偏微分方程的参数估计和逆建模技术。该项目的第三部分致力于非线性spde的数值分析，并开发为此目的所需的相关分析工具。

A fundamental challenge faced in fluid dynamics is that the basic governing equations do not account for persistent, randomly driven fluctuations that arise due to uncertainties in modeling from empirical, numerical and physical sources. In view of the wide progress made in computation and given the ubiquity of probabilistic methods in applications there is a clear need to better understand the numerical and analytical underpinnings of stochastic partial differential equations (SPDEs) in general and in relation to the basic equations of fluid dynamics. This project seeks to further the development of numerical and analytical tools for SPDEs with a view towards novel applications in climate and weather modeling and for the study of turbulence.The work will focus in particular on three emerging areas of nonlinear SPDEs. Firstly we will address the theory of invariant measures and statistically steady states. In particular we will develop tools to study inviscid limits in the class of invariant measures for the Navier-Stokes equations and for related models arising in geophysical and turbulence applications. We will also identify and characterize noise regimes leading to ergodic and mixing properties for some previously unaddressed classes of nonlinear SPDEs. A second portion of the project will develop parameter estimation and inverse modeling techniques for noisy nonlinear PDEs. The third portion of the project is devoted to the numerical analysis of nonlinear SPDEs and to developing related analytical tools needed for this purpose.