Probabilistic and Deterministic Aspects of Nonlinear Dispersive and Wave Equations

非线性色散方程和波动方程的概率和确定性方面

• 批准号: 1800697
• 负责人: Dana Mendelson
• 金额: $ 16.18万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-08-01 至 2023-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800697&HistoricalAwards=false
• 关键词: Probabilistic; Deterministic; Aspects; Nonlinear; Dispersive

Nonlinear dispersive and wave equations model wave propagation phenomena for many physical systems, from water waves to the dynamics of quantum gases. Understanding the "typical" behavior of these systems has many applications, such as problems in material science involving signal degeneration in optic fibers. For the last few decades, research on these equations has centered around questions on the existence of solutions, their long time behavior, and the possibility of singularity formation. Fundamental progress has been made in many settings, yet in some regimes, the nonlinear interactions are so strong compared to the dispersion of the waves that typical methods break down. In those regimes, probabilistic tools have been instrumental in analyzing the behavior of these systems, enabling researchers to answer new and exciting questions in a variety of settings. This project aims to investigate the behavior of solutions to nonlinear wave and Schrodinger equations, and more generally Hamiltonian equations in infinite dimensions. More specifically, the PI will address several problems, including probabilistic existence and long-time behavior for power-type nonlinear wave and Schrodinger equations, the analysis of integrable structures, and the definition of invariant Gibbs-type measures. These problems have several common goals, many of them motivated by recent developments in the study of nonlinear dispersive and wave equations via probabilistic techniques. The PI will refine existing probabilistic tools and develop new ones in order to treat new settings, such as certain geometric flows. Additionally, to tackle these problems, the PI will employ a combination of tools from harmonic analysis, probability theory, spectral theory and the theory of Hamiltonian systems.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.

非线性色散方程和波动方程为许多物理系统的波传播现象建模，从水波到量子气体动力学。了解这些系统的“典型”行为有许多应用，例如材料科学中涉及光纤信号退化的问题。在过去的几十年里，对这些方程的研究主要集中在解的存在性、长时间行为和奇点形成的可能性等问题上。在许多情况下已经取得了根本性的进展，但在某些情况下，与波的色散相比，非线性相互作用如此之强，以至于典型的方法失败了。在这些制度中，概率工具在分析这些系统的行为方面发挥了重要作用，使研究人员能够在各种环境中回答新的和令人兴奋的问题。本计画旨在研究非线性波动与薛定谔方程，以及更一般的无限维哈密尔顿方程的解的行为。更具体地说，PI将解决几个问题，包括幂型非线性波和薛定谔方程的概率存在性和长时间行为，可积结构的分析，以及不变吉布斯型测度的定义。这些问题有几个共同的目标，其中许多是由最近的发展，通过概率技术的非线性色散和波动方程的研究动机。PI将改进现有的概率工具，并开发新的工具，以处理新的设置，如某些几何流。此外，为了解决这些问题，PI将采用谐波分析，概率论，频谱理论和哈密顿系统理论的组合工具。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。