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: 1764403
• 负责人: Gigliola Staffilani
• 金额: $ 24万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1764403&HistoricalAwards=false
• 关键词: Collaborative; Research; Dynamics; Nonlinear; Partial; Collaborative; Research; Dynamics; Nonlinear; Partial

We interact with waves all the time and everywhere. When we listen to music, when we use our cell phones, when we warm up a dinner in a microwave, when we look at the stars in the sky and when we relax on a sunny beach. But wave phenomena may also affect the lives of millions of people when earthquakes shake and propagate, tsunamis form or nuclear radiations get out of control. Indeed, waves naturally arise occur in a variety of physical systems such as nonlinear optics, atmosphere and ocean waves, quantum mechanics and plasmas. The study of waves is fundamental for the understanding of phenomena at both a very small scale, such as the Bose-Einstein Condensate, and at a very large one, such as collusion of galaxies. These expressions of nature are never too smooth and rarely too simple: interactions of small waves can produce very large outcomes, such as freak waves, while complicated objects such as solitons almost do not see each other when they cross. Phenomena such as these are the byproduct of nonlinear wave interactions, and understanding what are the possible outcomes, given the initial state of a system of waves, is fundamental to predict and to control it, hopefully to our advantage. In this NSF supported research the PIs present a series of projects at the cutting edge of research in nonlinear wave phenomena in which deterministic approaches, classically based on harmonic and Fourier analysis, are implemented alongside probabilistic ones to capture basic properties of wave phenomena. It has become clear in recent years that deterministic methods and probabilistic ones naturally feed off each other and when combined not only contribute to our understanding but also open the door to new paradigms to move research forward in various directions. More precisely, the PIs propose four projects at the forefront of nonlinear evolution equations, where the interplay of deterministic and probabilistic approaches is the key to make progress. The problems range from the study of weak turbulence for dispersive and fluid equations to the analysis of integrable structures, from the definition of Gibbs type measures to the probabilistic existence and stability of certain geometric flows enjoying null form nonlinearities. The probabilistic component of PIs' work in the last few years has contributed in bridging the dispersive and wave nonlinear equations community with that specialized in stochastic partial differential equations. This interaction has created ongoing collaborations between members of these two communities. The work that the PIs, their students and collaborators will generate in solving the problems described in this project will further solidify the interactions between these two vibrant communities. The broader impact component of the project aims at fostering the training of doctoral graduate students and junior researchers in the US, thus fundamentally contributing to the STEM workforce. It will also enhance dissemination and collaborative research.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.

我们一直与海浪互动。当我们听音乐时，当我们使用手机时，当我们在微波炉中加热晚餐时，当我们看着天空中的星星以及在阳光明媚的海滩上放松时。但是，当地震摇动和传播，海啸形成或核辐射失控时，波浪现象也可能影响数百万人的生活。 确实，自然出现的波浪发生在多种物理系统中，例如非线性光学，大气和海浪，量子力学和等离子体。对波的研究对于在很小的范围（例如玻色的凝结物）和非常大的情况（例如星系勾结）上对现象的理解至关重要。这些自然的表达永远不会太光滑，很少太简单了：小波的相互作用会产生很大的结果，例如怪胎波，而复杂的物体（例如唯一的物体）几乎在交叉时几乎看不到彼此。诸如此类的现象是非线性波相互作用的副产品，并且鉴于波浪系统的初始状态，理解可能的结果是什么，这是预测和控制它的基础，希望对我们有优势。在这项NSF支持的研究中，PI在非线性波浪现象中提出了一系列项目的一系列项目，在非线性波浪现象中，确定性方法是基于谐波和傅立叶分析的，与概率的方法实施，以捕获波浪现象的基本特性。近年来，确定性方法和概率的方法已经很明显，自然会互相融合，而当组合不仅有助于我们的理解，而且还为新范式打开了新的范式，以向各种方向发展研究。更确切地说，PIS提出了四个在非线性演化方程的最前沿的项目，确定性和概率方法的相互作用是取得进步的关键。问题范围从分散和流体方程的弱湍流的研究到对整合结构的分析，从吉布斯类型的定义到概率的存在以及某些几何流量的概率存在和稳定性，这些几何流量享有非线性。在过去几年中，PIS工作的概率组成部分在将分散性和波浪非线性方程社区与专门从事随机偏微分方程的专业架起。这种互动创造了这两个社区成员之间的持续合作。 PIS，他们的学生和合作者将在解决该项目中描述的问题方面产生的工作将进一步巩固这两个充满活力的社区之间的相互作用。该项目的更广泛的影响组成部分旨在促进美国博士研究生和美国初级研究人员的培训，从而从根本上为STEM劳动力做出贡献。这也将增强传播和协作研究。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛影响的评论标准来评估值得支持的。