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Questions in Wave Turbulence and Quantum Kinetic Theories

波湍流和量子动力学理论中的问题

基本信息

批准号：
2305523
负责人：
Minh-Binh Tran
金额：
$ 10.95万
依托单位：
Texas A&M University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-11-01 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2305523&HistoricalAwards=false
关键词：
Questions Wave Turbulence Quantum Kinetic

项目摘要

This research is devoted to developing mathematical methods for studying two different classes of physical phenomena: wave turbulence and quantum kinetics. Wave turbulence is manifested in many physical systems, including familiar water waves, water surface gravity and capillary waves, and a great variety of waves in plasmas (in particular, in fusion devices). A Bose-Einstein condensate is a state of matter that was first predicted theoretically in 1924 and produced experimentally in 1995. While quantum phenomena are exhibited on micro-scales, and nature is well described on macro-scales by classical mechanics, for a Bose-Einstein condensate macroscopic quantum phenomena become apparent. The quantum kinetic theory that describes some physical properties of this state of matter is the second subject of this research. Although wave turbulence and quantum kinetics are dramatically different as physical phenomena, their mathematical description uses similar equations. Despite the widespread applications of these kinetic equations, little is known rigorously in this field. The aim of the project is to develop mathematical techniques and apply them to concrete physical situations, leading to a more profound understanding of both wave turbulence and quantum kinetics.The project is aimed at developing the theory of the existence, uniqueness, finite-time condensation, and relaxation to equilibrium for strong solutions of the kinetic equations. The principal investigator will address the local-in-time existence via techniques of harmonic analysis and novel Strichartz-type estimates. For important questions related to 3-wave wave turbulence, regularity theory for the kinetic equation with a particular broadening (regularization) of the resonance set will be used. This research project is informed by several sub-fields of mathematics, physics, and chemistry, and the work has potential applications in optical turbulence, oceanography and atmospheric sciences, and quantum physics.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.

这项研究致力于发展数学方法来研究两种不同的物理现象：波动湍流和量子动力学。波湍流在许多物理系统中表现出来，包括熟悉的水波，水面重力和毛细波，以及等离子体中的各种波（特别是在聚变装置中）。玻色-爱因斯坦凝聚态是一种物质状态，在1924年首次被理论预测，并在1995年实验产生。虽然量子现象在微观尺度上表现出来，而经典力学在宏观尺度上很好地描述了自然，但对于玻色-爱因斯坦凝聚体，宏观量子现象变得明显。描述这种物质状态的一些物理性质的量子动力学理论是本研究的第二个主题。虽然波湍流和量子动力学作为物理现象是截然不同的，但它们的数学描述使用类似的方程。尽管这些动力学方程的广泛应用，很少有人知道严格在这一领域。该项目的目的是发展数学技术并将其应用于具体的物理情况，从而更深刻地理解波动和量子动力学。该项目的目的是发展动力学方程强解的存在性、唯一性、有限时间凝聚和松弛平衡的理论。主要研究者将通过谐波分析和新的Escherichartz型估计技术来解决当地的时间存在。对于与3波湍流相关的重要问题，将使用具有共振集的特定加宽（正则化）的动力学方程的正则性理论。该研究项目涉及数学、物理和化学的多个子领域，在光学湍流、海洋学和大气科学以及量子物理学方面具有潜在的应用价值。该奖项反映了NSF的法定使命，并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，被认为值得支持。