Derivation of the Kinetic Wave Equation

运动波方程的推导

• 批准号: 1800840
• 负责人: Pierre Germain
• 金额: $ 27万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-06-01 至 2021-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800840&HistoricalAwards=false
• 关键词: Derivation; Kinetic; Wave; Equation

Turbulence is a universal phenomenon, occurring in a number of physical systems. The simplest one is the flow of a fluid, say water in a river. In some regimes, the flow is very smooth, but in other situations it appears very chaotic, with eddies at various scales, interacting in a very complicated manner: the flow is then said to be turbulent. While turbulent flows are very hard to understand, and constitute to this day a scientific riddle, an approach was suggested by Kolmogorov in 1941. Kolmogorov's approach is instead of trying to fully describe the flow focus on statistical quantities that can be measured in the flow. In other words, instead of understanding everyting about the flow, which might not be possible, certain averaged quantities should follow precise physical laws. While this approach was very successful in many respects, it remains mysterious in many others. In particular, at a very fundamental level, no rigorous justification of the laws of turbulence is known: these laws seem to be valid experimentally, but how they exactly arise from first principles is not known. The aim of the program is to investigate this very fundamental question, which is related to very practical concerns. While turbulence in fluid flows is the first example that comes to mind, another type of turbulence, known as weak turbulence, might be more tractable, and provide the right entry point. Weak turbulence describes turbulence as it arises in nonlinear wave equations (of which there are many instances, from waves propagating on the surface of the ocean to electromagnetic waves or quantum physics). It was conjectured by several scientists, in particular Zakharov in the 70's and 80's, that weak turbulence is described by a specific equation, known as the kinetic wave equation. The central aim of the PI is to investigate this conjecture with the help of mathematical tools: in particular, the theory of partial differential equations, in connection with probability theory. This effort will hopefully enable us to validate Zakharov's claim under appropriate conditions, which would then open the way to a theoretical and rigorous understanding of weak turbulence.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.

湍流是一种普遍现象，发生在许多物理系统中。最简单的是流体的流动，例如在河里的水。在某些机制中，流动非常平稳，但是在其他情况下，它看起来非常混乱，涡流的各种尺度都非常复杂：然后说流是湍流的。虽然湍流很难理解，并且构成了今天的科学谜语，但科尔莫格罗夫（Kolmogorov）在1941年提出了一种方法。科尔莫戈罗夫（Kolmogorov）的方法不是试图将流量集中在流量中可以测量的统计数量上。 换句话说，某些平均数量不应遵循精确的物理定律，而不是理解有关流动的所有信息，而不是不可能的。尽管这种方法在许多方面都非常成功，但在许多其他方面仍然是神秘的。特别是，在非常基本的层面上，尚无对湍流定律的严格理由：这些定律在实验上似乎是有效的，但是尚不清楚它们是如何准确的。该计划的目的是调查这个非常基本的问题，这与非常实际的问题有关。虽然流体流中的湍流是第一个想到的例子，但另一种类型的湍流（称为弱湍流）可能更容易处理，并提供正确的入口点。弱湍流描述了在非线性波方程中产生的湍流（其中有很多实例，从在海面传播的波到电磁波或量子物理学）。它是由几位科学家（尤其是70年代和80年代的Zakharov）猜想的，弱的湍流用特定方程式（称为动力波方程）描述。 PI的核心目的是借助数学工具来研究这种猜想：特别是与概率理论有关的部分微分方程理论。这项努力将有望使我们能够在适当条件下验证Zakharov的主张，然后这将为对弱势脆弱性的理论和严格理解开辟道路。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子的评估而被视为值得支持的。和更广泛的影响审查标准。