New Challenges in the Study of Propagation of Randomness for Nonlinear Evolution Equations

非线性演化方程随机传播研究的新挑战

• 批准号: 2400036
• 负责人: Andrea Nahmod
• 金额: $ 38.85万
• 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: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2400036&HistoricalAwards=false
• 关键词: New; Challenges; Study; Propagation; Randomness

Waves are everywhere in nature. We observe them when we look at the ripples that form when we throw a pebble in a lake, the expanding ring called a wave-packet; or when we look at a rainbow that is formed when light wave passes through a prism or water droplet and note the spatial separation of white light into different colors. Partial differential equations (PDE) modeling wave propagation phenomena have played a fundamental role in understanding such physical and natural events as well as quantum mechanics, fiber optics, ferromagnetism, atmospheric and water waves, and many other physical models. In these cases, wave phenomena are never too smooth or too simple, and in fact the byproduct of nonlinear wave interactions as they propagate in time. Being able to understand and describe the dynamical behavior of such models under certain noisy conditions or 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 this context using analytical and probabilistic methodologies. The work of the project, and its connections to science, promotes interdisciplinary interactions and fosters the training of graduate students and junior researchers in the United States thus fundamentally contributing to its STEM workforce. The interplay of deterministic methods in nonlinear PDE and probabilistic ones naturally feed off each other and when combined contribute to a deep understanding of wave phenomena, which opens the door to new paradigms that move research forward in various directions. The Principal Investigator studies several projects at the forefront of current research. The problems, grouped in two interrelated directions, aim broadly at: (1) studying the out of equilibrium long-time dynamics of dispersive flows from a probabilistic viewpoint in energy subcritical regimes by means of suitable quantitative quasi-invariance, modified energies and stability theory of random structures; (2) establishing the invariance of Gibbs measures for the probabilistically critical three-dimensional nonlinear Schrödinger equation (also known as a model in constructive quantum field theory) in the context of equilibrium statistical mechanics; (3) establishing a suitable probabilistic local theory of the hyperbolic sine-Gordon equation on 2D tori and the invariance of its associated Gibbs measure; and (4) the development of the random tensor theory for the nonlinear wave equations and for non-Gaussian data. The research bridges between the dispersive and wave equations communities that specialize in stochastic equations and contributes to understanding in a fundamental way propagation of randomness in nonlinear wave phenomena.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.

波浪在自然界中无处不在。当我们看着鹅卵石在湖中扔卵石时，我们会观察它们，这是一个称为Wave acket的膨胀环。或者，当我们看着光波穿过棱镜或水滴时形成的彩虹并注意到白光分为不同颜色的空间分离。部分微分方程（PDE）建模波传播现象在理解此类物理和自然事件以及量子力学，光纤，铁磁性，大气和水波以及许多其他物理模型方面发挥了基本作用。在这些情况下，波浪现象永远不会太光滑或太简单，实际上，非线性波相互作用的副产品随着时间的流逝而传播。能够在某些噪声条件下理解和描述此类模型的动态行为，或给出初始统计合奏，并确切地描述这些模型中构建的继承随机性如何传播，这对于在研究自然世界时准确预测波浪现象至关重要。该项目旨在通过分析和概率方法回答有关长期动态和随机性传播的几个中心问题。该项目的工作及其与科学的联系促进了跨学科的互动，并促进了美国的研究生和初级研究人员的培训，从而从根本上为其STEM劳动力做出了贡献。在非线性PDE和有问题的方法中确定性方法的相互作用自然会相互融合，当组合时，对波浪现象有了深刻的了解，这为新范式打开了新范式，这些范式向前朝着各种方向发展研究。主要研究者研究了当前研究最前沿的几个项目。这些问题分组为两个相互关联的方向，广泛针对：（1）通过适当的定量准准危机，从概率的角度来研究分散流的长期动力学，从概率的角度来研究，通过适当的定量准定量抗病性，修改的能量和随机结构的稳定性理论； （2）在平衡统计机制的背景下，建立了gibbs测量值的gibbs测量值（在建设性量子场理论中也称为模型）； （3）建立在2D Tori上的双曲线正弦方程及其相关Gibbs测量的不变性的合适概率局部理论； （4）非线性波方程和非高斯数据的随机张量理论的发展。分散方程和波动方程之间的研究桥梁专门从事随机方程式，并以一种基本的方式对非线性波浪现象中随机性传播的理解有助于理解。本奖奖反映了NSF的法定任务，并通过使用基金会的智力和更广泛的影响来诚实地认为，通过评估诚实地将其视为诚实的支持。