Questions in Wave Turbulence and Quantum Kinetic Theories

本研究致力于发展数学方法来研究两类不同的物理现象：波动湍流和量子动力学。波浪湍流表现在许多物理系统中，包括我们熟悉的水波、水面重力和毛细波，以及等离子体（特别是聚变装置）中各种各样的波。玻色-爱因斯坦凝聚是物质的一种状态，最早在1924年被理论预测，1995年被实验证实。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。描述物质这种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。虽然波动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相似的方程。尽管这些动力学方程得到了广泛的应用，但在这一领域，人们对其知之甚少。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，从而对波湍流和量子动力学有更深刻的理解。本项目旨在发展动力学方程强解的存在性、唯一性、有限时间凝聚和平衡弛豫理论。首席研究员将通过谐波分析技术和新颖的斯特里查兹型估计来解决局部时间存在问题。对于与三波波动湍流有关的重要问题，将使用共振集具有特定展宽（正则化）的动力学方程的正则性理论。本研究项目以数学、物理和化学的几个子领域为基础，在光学湍流、海洋学和大气科学、量子物理等领域具有潜在的应用前景。该奖项反映了美国国家科学基金会的法定使命，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。

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.