Collaborative Research: On New Directions for the Derivation of Wave Kinetic Equations

合作研究：波动力学方程推导的新方向

• 批准号: 2306378
• 负责人: Gigliola Staffilani
• 金额: $ 32.49万
• 依托单位: Massachusetts Institute of Technology
• 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: 2024-09-01 至 2027-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2306378&HistoricalAwards=false
• 关键词: Collaborative; Research; New; Directions; Derivation

The beauty and power of mathematics is to recognize common features in a variety of phenomena that may look physically different. This is certainly the case when one studies wave turbulence theory. This theory is focused on the fundamental concept that when in a given physical system a large number of interacting waves are present, the description of an individual wave is neither possible nor relevant. What becomes important and practical is the description of the density and the statistics of the interacting waves. Arguably the most recognizable and fundamental objects within this theory are the wave kinetic equations. These equations, their solutions and their approximations have been used to study a variety of phenomena: water surface gravity and capillary waves, inertial waves due to rotation and internal waves on density stratifications, which are important in the study of planetary atmospheres and oceans; Alfvén wave turbulence in solar wind; planetary Rossby waves, which are important for the study of weather and climate evolutions; waves in Bose-Einstein condensates (BECs) and in nonlinear optics; waves in plasmas of fusion devices; and many others. This project will tackle foundational questions in wave turbulence theory through rigorous mathematical analysis. In addition, the project will promote collaborations, facilitate the dissemination of interdisciplinary research, and provide opportunities for undergraduate and graduate students to work on a multifaceted and forward-looking line of mathematical research.This project tackles challenging problems at the intersection of the physics and the mathematical analysis of nonlinear interactions of waves that are central in the study of wave turbulence theory. These problems include the rigorous derivation of wave kinetic equations, the analysis of the 4-wave kinetic equation for the Fermi- Pasta-Ulam-Tsingou (FPUT) chain and the well-posedness of a geometric wave equation via Feynman diagrams in the energy regime. The research proposed will not just address important open problems but will contribute to the interdisciplinary development of several new and complex tools both in mathematics and physics. The proposal aims at providing these tools by blending Feynman diagrams, harmonic analysis, probability, combinatorics, incidence geometry, kinetic theory, dispersive PDE, quantum field theory and the FPUT chain.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.

数学的美和力量是在各种现象中识别出可能在身体上不同的现象中的共同特征。当人们研究波湍流理论时，当然是这种情况。该理论的重点是基本概念，即在给定的物理系统中存在大量相互作用波，对单个波的描述既不可能也不相关。变得重要和实用的是对相互作用波的密度和统计数据的描述。可以说，该理论中最公认和最基本的对象是波动力学方程。这些方程，它们的溶液及其近似已被用来研究各种现象：水表面重力和毛细血管波，由于旋转而引起的惯性波和密度分层的内波，这对于行星气氛和海洋的研究很重要；太阳风中的Alfvén波湍流；行星罗斯比（Rossby Wave），这对于研究天气和气候演变很重要； Bose-Einstein冷凝物（BEC）和非线性光学元件中的波；融合装置平原上的波浪；还有许多其他。该项目将通过严格的数学分析来解决波动湍流理论中的基本问题。此外，该项目将促进合作，促进跨学科研究的传播，并为本科生和研究生提供机会，从而在数学研究的多方面和前瞻性的数学研究线上工作。该项目挑战物理学相交的问题，这些问题是物理学的交叉点，以及对波浪构成波动理论的无线互动的非线性相互作用的数学分析。这些问题包括波动动力学方程的严格推导，Fermi-Pasta-ulam-tsingou（FPUT）链的4波动力学方程的分析以及通过能量方面的Feynman图通过Feynman图的几何波浪等效性的良好性。提出的研究不仅会解决重要的开放问题，而且还将为数学和物理学中几种新的和复杂的工具的跨学科发展做出贡献。该提案旨在通过将Feynman图，谐波分析，概率，组合学，发病率几何形状，动力学理论，分散性PDE，量子现场理论和FPUT链融合来提供这些工具。该奖项反映了NSF的法规使命，并认为通过基金会的知识优点和广泛的actitia cribitia cribitia cripitia crocritia cribitia crocritia crocritia crocritia crocritia crocritia crocritia crocritia crocritia均值得通过评估。