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

非线性随机偏微分方程分析及其在湍流理论和气候建模中的应用

• 批准号: 1733909
• 负责人: Nathan Glatt-Holtz
• 金额: $ 1.8万
• 依托单位: Tulane University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-12-14 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1733909&HistoricalAwards=false
• 关键词: Analysis; Nonlinear; Stochastic; Partial; Differential

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.

流体动力学面临的一个根本挑战是，基本的控制方程不考虑持续的，随机驱动的波动，由于从经验，数值和物理来源建模的不确定性。鉴于在计算中取得的广泛进展，并给出了无处不在的概率方法在应用中有一个明确的需要，以更好地了解一般的随机偏微分方程（SPDE）的数值和分析基础，并与流体动力学的基本方程。该项目旨在进一步开发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实现新的应用，工作将特别侧重于非线性SPDE的三个新兴领域。 首先，我们将讨论不变测度和统计稳态理论。特别是，我们将开发工具来研究在Navier-Stokes方程和相关模型中产生的地球物理和湍流应用的不变措施类的无粘极限。我们还将确定和表征噪声制度导致遍历和混合性质的一些以前未处理的类的非线性SPDE。 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。 该项目的第三部分是致力于非线性SPDE的数值分析，并开发相关的分析工具，需要为此目的。