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:
  • 金额:
    $ 23.63万
  • 依托单位:
  • 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

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.
我们都熟悉色散和波动现象,因为我们在自然界中一直观察到它。这可能就像我们看彩虹一样简单:色散导致白色光在空间上分离成不同的颜色。或者当我们观察向湖中扔石子时所形成的涟漪时:扩大的环被称为“波包”,我们注意到波以不同的速度传播,最长的最快,最短的最慢。但是波动现象也出现在量子力学、等离子体、光纤、铁磁性、大气和水波以及许多其他环境中。由于自然界中的波在传播时以非线性方式相互作用,并且随着时间的推移具有不同的特性,例如振幅,长度,振荡,速度和位置,因此了解它们在某些噪声条件下或在某些介质中进行测量时的行为是很重要的,其中小误差是不可避免的。了解通过光纤电缆发送信号的最有效方式,或者能够预测温度接近绝对零度时气体的性质(玻色-爱因斯坦凝聚)是自然界中两种非常不同的现象,但都是同一非线性模型解决方案的两个方面。能够理解和描述给定初始统计系综的此类模型的解的动力学行为,并精确描述这些模型中构建的固有随机性如何传播,这是在研究自然世界时准确预测波动现象的基础。该项目旨在回答关于长时间动态和非线性色散和波动现象背景下随机性传播的几个核心问题。与此同时,该项目的工作旨在促进美国研究生和初级研究人员的培训。在过去几年中,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进一步加深了我们对解决方案动力学的理解,并为新的范式打开了大门,这些范式推动了研究向各个方向发展。为了解决当前研究前沿的重要挑战,旨在定量了解一般波动现象的动力学特性,主要研究员(PI)采用了一种创新方法,该方法基于分析,概率,统计力学,动力系统,组合数学和解析数论加上最近的新方法的推动力,这些方法受到奇异随机抛物方程的惊人进展的启发。作为该项目的一部分,PI将在非线性演化方程前沿的三个研究领域探索几个令人兴奋的方向,其中确定性和概率方法的相互作用是取得进展的关键。这些问题的目的是从概率的角度研究色散流的长期动力学,非线性哈特里方程的吉布斯测度的不变性-来自N体薛定谔方程的平均场极限-以及环面上的非线性波和薛定谔方程;以及一个新的概率准线性双曲理论的发展。要研究的问题有一个优点,那就是它们被分为不同的难度级别,每个问题都导致独立的部分进展和更深入的理解。作为本项目的一部分,将探讨的一些问题将为研究生、博士生和博士后研究员带来出色的研究问题。此外,PI的工作将导致新的研究生课题课程的开发,从而丰富了新一代研究人员的发展。该奖项反映了NSF的法定使命,并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

项目成果

期刊论文数量(3)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Uniqueness of the 2D Euler equation on a corner domain with non-constant vorticity around the corner
拐角附近有非恒定涡度的拐角域上二维欧拉方程的唯一性
  • DOI:
    10.1088/1361-6544/ac586a
  • 发表时间:
    2022
  • 期刊:
  • 影响因子:
    1.7
  • 作者:
    Agrawal, Siddhant;Nahmod, Andrea R
  • 通讯作者:
    Nahmod, Andrea R
Random tensors, propagation of randomness, and nonlinear dispersive equations
  • DOI:
    10.1007/s00222-021-01084-8
  • 发表时间:
    2020-06
  • 期刊:
  • 影响因子:
    3.1
  • 作者:
    Yu Deng;A. Nahmod;H. Yue
  • 通讯作者:
    Yu Deng;A. Nahmod;H. Yue
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Andrea Nahmod其他文献

Andrea Nahmod的其他文献

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{{ truncateString('Andrea Nahmod', 18)}}的其他基金

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
  • 财政年份:
    2024
  • 资助金额:
    $ 23.63万
  • 项目类别:
    Standard Grant
FRG: Collaborative Research: New Challenges in the Derivation and Dynamics of Quantum Systems
FRG:协作研究:量子系统推导和动力学的新挑战
  • 批准号:
    2052740
  • 财政年份:
    2021
  • 资助金额:
    $ 23.63万
  • 项目类别:
    Standard Grant
Collaborative Research: Dynamics of Nonlinear Partial Differential Equations: Integrating Deterministic and Probabilistic Methods
合作研究:非线性偏微分方程的动力学:集成确定性和概率方法
  • 批准号:
    1800852
  • 财政年份:
    2018
  • 资助金额:
    $ 23.63万
  • 项目类别:
    Continuing Grant
FRG: Collaborative Research: Long-Term Dynamics of Nonlinear Dispersive and Hyperbolic Equations: Deterministic and Probabilistic Methods
FRG:协作研究:非线性色散和双曲方程的长期动力学:确定性和概率方法
  • 批准号:
    1463714
  • 财政年份:
    2015
  • 资助金额:
    $ 23.63万
  • 项目类别:
    Continuing Grant
New Challenges in Nonlinear PDEs.
非线性偏微分方程的新挑战。
  • 批准号:
    1201443
  • 财政年份:
    2012
  • 资助金额:
    $ 23.63万
  • 项目类别:
    Continuing Grant
Nonlinear Fourier Analysis and Partial Differential Equations
非线性傅里叶分析和偏微分方程
  • 批准号:
    0803160
  • 财政年份:
    2008
  • 资助金额:
    $ 23.63万
  • 项目类别:
    Standard Grant
Nonlinear Fourier Analysis And Geometric Dispersive Equations.
非线性傅里叶分析和几何色散方程。
  • 批准号:
    0503542
  • 财政年份:
    2005
  • 资助金额:
    $ 23.63万
  • 项目类别:
    Continuing Grant
Harmonic Analysis and Geometric Partial Differential Equations
调和分析与几何偏微分方程
  • 批准号:
    0202139
  • 财政年份:
    2002
  • 资助金额:
    $ 23.63万
  • 项目类别:
    Continuing Grant
Harmonic Analysis and Partial Differential Equations
调和分析和偏微分方程
  • 批准号:
    9971159
  • 财政年份:
    1999
  • 资助金额:
    $ 23.63万
  • 项目类别:
    Standard Grant

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Conference: 17th International Conference on Computability, Complexity and Randomness (CCR 2024)
会议:第十七届可计算性、复杂性和随机性国际会议(CCR 2024)
  • 批准号:
    2404023
  • 财政年份:
    2024
  • 资助金额:
    $ 23.63万
  • 项目类别:
    Standard Grant
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
  • 财政年份:
    2024
  • 资助金额:
    $ 23.63万
  • 项目类别:
    Standard Grant
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人口增长的健身景观中几何与随机性之间的相互作用
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    EP/X040089/1
  • 财政年份:
    2024
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    $ 23.63万
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    Research Grant
Development of self-organization model and verification of forecast accuracy of Baiu heavy rainfall systems based on the randomness of water content
基于含水量随机性的Baiu暴雨系统自组织模型建立及预报精度验证
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    22KJ1845
  • 财政年份:
    2023
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    Grant-in-Aid for JSPS Fellows
Robust Quantum Randomness for Industry
工业领域强大的量子随机性
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    Collaborative R&D
Randomness in High-Dimensional Combinatorics: Colorings, Robustness, and Statistics
高维组合中的随机性:着色、鲁棒性和统计
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    2247078
  • 财政年份:
    2023
  • 资助金额:
    $ 23.63万
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AF: Small: The Power of Randomness in Decision and Verification
AF:小:决策和验证中随机性的力量
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