Propagation of Randomness in Nonlinear Evolution Equations

非线性演化方程中随机性的传播

基本信息

批准号：
2101381
负责人：
Andrea Nahmod
金额：
$ 23.63万
依托单位：
University of Massachusetts Amherst
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-15 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101381&HistoricalAwards=false
关键词：
Propagation Randomness Nonlinear Evolution Equations

项目摘要

We are all familiar with dispersive and wave phenomena since we observe it all the time in nature. It could be as simple as when we look at a rainbow: dispersion causes the spatial separation of white light into different colors. Or when we look at the ripples that form when we throw a pebble in the lake: the expanding ring is called a “wave-packet” and we note that waves travel at different speeds, the longest going fastest and the shortest ones slowest. But wave phenomena also arise in quantum mechanics, plasmas, fiber optics, ferromagnetism, atmospheric and water waves and many other settings. Because waves in nature interact in a nonlinear fashion as they propagate and have different properties such as amplitude, length, oscillation, speed, and position over time, it is important to understand how they may behave under certain noisy conditions or when taking measurements in certain media where small errors are unavoidable. Understanding the most efficient way to send a signal through a fiber optic cable or being able to anticipate the properties of a gas when the temperature approaches absolute zero (a Bose-Einstein condensate) are two very different phenomena in nature but are both aspects of solutions to the same nonlinear model. Being able to understand and describe the dynamical behavior of solutions to such models given an initial statistical ensemble and have a precise description of how the inherent randomness built in these models propagates is fundamental to accurately predict wave phenomena when studying the natural world. This project is aimed at answering several central questions about long-time dynamics and the propagation of randomness in the context of nonlinear dispersive and wave phenomena. At the same time, the work of the project is designed to foster the training of graduate students and junior researchers in the U.S.The synergy between deterministic and probabilistic approaches in the study of nonlinear evolution equations in the last few years has furthered our understanding of the dynamics of solutions in fundamental ways and opened the door to new paradigms that have moved research forward in various directions. To address important challenges at the cutting edge of current research, aimed at a quantitative understanding of the dynamical properties of generic wave phenomena, the principal investigator (PI) adopts an innovative approach based on the integration of methods and ideas from analysis, probability, statistical mechanics, dynamical systems, combinatorics and analytic number theory coupled with the impetus of recent new methods that were inspired by the spectacular advances in singular stochastic parabolic equations. As part of this project, the PI will explore several exciting directions in three areas of research at the forefront of nonlinear evolution equations, where the interplay of deterministic and probabilistic approaches is the key to make progress. The problems aim at studying the long-time dynamics of dispersive flows from a probabilistic viewpoint, the invariance of Gibbs measures for the nonlinear Hartree equation - arising from the mean field limit for the N-body Schrödinger equation - and for the nonlinear wave and Schrödinger equations on tori; and at the development of a new probabilistic quasilinear hyperbolic theory. The problems to be studied have the advantage that they are graded at different levels of difficulty, each leading to independent partial progress and deeper understanding. Some of the questions that will be pursued as part of this project lead to excellent research problems for graduate doctoral students and postdoctoral fellows. Furthermore, the PI’s work will lead to the development of new graduate topics courses, thus enriching the development of the new generation of researchers.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

我们所有人都熟悉分散现象，因为我们一直在自然界观察到它。它可能就像我们看彩虹时一样简单：分散会导致白光将白光分为不同的颜色。或者，当我们在湖上扔卵石时看着涟漪时：膨胀的环称为“波浪行动”，我们注意到波浪以不同的速度行驶，最长的速度最快，最短的速度最慢。波浪现象也会出现在量子机械，等离子体，光纤，铁磁，大气和水波以及许多其他环境中。因为自然界的波浪在传播时以非线性方式相互作用，并且具有不同的特性，例如放大器，长度，振荡，速度和位置，随着时间的流逝，重要的是要了解它们在某些噪声条件下如何在某些噪声条件下或在某些媒体中进行测量时如何行为，而在某些小错误不可避免的情况下。了解当温度接近绝对零（Bose-Einstein冷凝物）时，了解通过光纤电缆发送信号的最有效方法或能够预测气体的性质，本质上是两个截然不同的现象，但这都是相同非线性模型的解决方案的方面。能够理解和描述在初始统计合奏的情况下，解决此类模型的解决方案的动态行为，并确切地描述了这些模型中构建的继承随机性如何在研究自然世界时准确预测波浪现象至关重要。该项目的目的是在非线性分散和波浪现象的背景下回答有关长期动态以及随机性传播的几个核心问题。同时，该项目的工作旨在促进美国的研究生和初级研究人员的培训，而在过去几年的非线性演化方程研究中，确定性和概率方法之间的协同作用增强了我们对解决方案动态的理解，在基本方面，为新的范围开辟了各个方向研究的新范围。为了解决当前研究最前沿的重要挑战，旨在定量了解通用波浪现象的动态特性，主要研究者（PI）采用了一种创新的方法，基于分析，概率，概率，统计学，统计机制，动态系统，组合学和分析理论与新方法相关的方法和思想的整合，从抛物线方程。作为该项目的一部分，PI将在非线性演化方程的最前沿的三个研究领域探索几个令人兴奋的方向，在这种方程式的最前沿，确定性和概率方法的相互作用是取得进步的关键。该问题旨在从概率观点（gibbs的不变性）测量非线性hartree方程的gibbs度量的不变性 - 由nbodyschrödinger方程的平均场限制以及非线性波和schrödinger方程式在tori上的平均场限制而产生；并在发展新的概率的准双曲理论。要研究的问题的优势是它们在不同级别的难度上进行了评分，每种都会导致独立的部分进步和更深入的理解。作为该项目的一部分，将提出的一些问题为研究生博士生和博士后研究员带来了极好的研究问题。此外，PI的工作将导致新的研究生主题课程的发展，从而丰富新一代研究人员的发展。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响评估标准来评估值得支持。