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.

非线性色散和波动方程模拟了许多物理系统的波传播现象，从水波到量子气体的动力学。了解这些系统的“典型”行为有很多应用，例如材料科学中涉及光纤信号衰减的问题。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、它们的长期行为以及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进展，但在某些情况下，非线性相互作用与波的色散相比非常强烈，以至于典型的方法无法实现。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下回答新的、令人兴奋的问题。该项目旨在研究非线性波和薛定谔方程以及更一般的无限维哈密顿方程解的行为。更具体地说，PI 将解决几个问题，包括功率型非线性波和薛定谔方程的概率存在性和长期行为、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，其中许多目标是由通过概率技术研究非线性色散和波动方程的最新发展推动的。 PI 将完善现有的概率工具并开发新的工具，以处理新的设置，例如某些几何流。此外，为了解决这些问题，PI 将结合使用调和分析、概率论、谱理论和哈密顿系统理论的工具。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力评估进行评估，认为值得支持。优点和更广泛的影响审查标准。