Questions in Wave Turbulence and Quantum Kinetic Theories

波湍流和量子动力学理论中的问题

• 批准号: 1814149
• 负责人: Minh-Binh Tran
• 金额: $ 10.95万
• 依托单位: University of Wisconsin-Madison
• 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: 2018-06-01 至 2018-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1814149&HistoricalAwards=false
• 关键词: Questions; Wave; Turbulence; Quantum; Kinetic

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.

这项研究致力于开发数学方法来研究两类不同的物理现象：波湍流和量子动力学。波湍流在许多物理系统中都有体现，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各样的波（特别是聚变装置中）。玻色-爱因斯坦凝聚是一种物质状态，于 1924 年首次在理论上被预测，并于 1995 年通过实验产生。虽然量子现象是在微观尺度上展示的，而经典力学在宏观尺度上很好地描述了自然，但对于玻色-爱因斯坦凝聚态来说，爱因斯坦凝聚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。尽管波湍流和量子动力学作为物理现象有很大不同，但它们的数学描述使用相似的方程。尽管这些动力学方程得到了广泛的应用，但人们对该领域的严格了解却很少。该项目的目的是开发数学技术并将其应用于具体的物理情况，从而使人们对波湍流和量子动力学有更深刻的理解。该项目旨在发展存在性、唯一性、有限时间凝聚的理论，以及动力学方程强解的松弛至平衡。首席研究员将通过调和分析技术和新颖的 Strichartz 型估计来解决局部时间存在问题。对于与三波湍流相关的重要问题，将使用具有特定共振集展宽（正则化）的动力学方程的正则理论。该研究项目涉及数学、物理和化学的多个子领域，该工作在光学湍流、海洋学和大气科学以及量子物理方面具有潜在的应用。该奖项反映了 NSF 的法定使命，并被认为值得支持通过使用基金会的智力优点和更广泛的影响审查标准进行评估。