Analysis of stochastic partial differential equations with multiple scales

多尺度随机偏微分方程分析

基本信息

批准号：
1712934
负责人：
Sandra Cerrai
金额：
$ 36万
依托单位：
University of Maryland, College Park
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712934&HistoricalAwards=false
关键词：
Analysis stochastic partial differential equations

项目摘要

In the description of complex systems, both deterministic and stochastic, it is usually important to be able to have a simplified model of those systems, in order to make their analysis more approachable. Usually, such a simplification is realized by looking at a smaller number of factors that are considered more relevant for the evolution of the system and by neglecting other factors that are considered less relevant. However, such an approximation, that can be effective on some given time interval, is not effective on longer time scales and the neglected factors turn out to play a fundamental role in the description of the systems' behavior. This research will analyze a large class of equations used in these models. These are highly complex equations and an understanding of them is crucial for a deeper understanding of the main features of the model and for a better effectiveness in applications. This research will develop new methods and techniques ranging over many fields of mathematics. Education and training will also be a major part of the project. The main goal of this research project is the analysis of limit theorems for stochastic partial differential equations having multiple scales. In particular, the PI will study some generalizations of the Smoluchowskii-Kramers approximation for systems with an infinite number of degrees of freedom and its long-time effects, as well as the validity of the averaging and the large deviation principle for some classes of stochastic partial differential equations (SPDEs). Specifically, the PI will try to understand what happens in the regime where the noise is weak and almost white in space. Moreover, she will study the convergence of SPDEs defined on narrow channels or describing stochastic incompressible viscous fluids in the whole space to a new class of SPDEs defined on graphs and open books. These asymptotic results will be important not only to provide a simplified description of some relevant multi-scale SPDEs that arise e.g. in the study of molecular motors and fluid dynamics, but also because at the limit they provide new interesting mathematical objects that are worthy of investigation. What characterizes and unifies this approach to all of these asymptotic problems is the effort to understand how they all interplay and interact one with the other.

在确定性和随机性的复杂系统的描述中，能够具有这些系统的简化模型通常很重要，以使其分析更加平易近人。通常，通过查看较少数量的因素，这些因素被认为与系统的演变更相关，并忽略了被认为不太相关的其他因素，从而实现了这种简化。但是，这样的近似值在某些给定的时间间隔内可以有效，在更长的时间尺度上无效，被忽视的因素在描述系统行为中起着基本作用。这项研究将分析这些模型中使用的大型方程。这些是高度复杂的方程式，对它们的理解对于更深入地理解模型的主要特征以及在应用中的有效性更高至关重要。这项研究将开发出许多数学领域的新方法和技术。教育和培训也将是该项目的主要部分。该研究项目的主要目标是分析具有多个量表的随机部分微分方程的极限定理。特别是，PI将研究具有无限数量自由度及其长期效应的系统的Smoluchowskii-Kramer近似的一些概括，以及平均效果的有效性和对某些随机分类偏微分方程（SPDES）的平均偏差原理和大偏差原理。具体而言，PI将尝试了解噪声弱且几乎白色的制度中发生的情况。此外，她将研究在狭窄的通道上定义的SPDE的收敛性，或描述整个空间中不可压缩的粘性流体，以与图形和开放书籍定义的新类别SPDE。这些渐近结果不仅要简化一些相关的多尺度SPDE，例如在研究分子电动机和流体动力学的研究中，还因为它们在极限上提供了值得研究的新有趣的数学对象。对所有这些渐近问题的特征和统一的特征和统一的方法是努力了解它们如何相互作用和相互作用。