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，例如河流中的水。在某些情况下，流动非常平滑，但在其他情况下，它显得非常混乱，具有不同规模的涡流，以非常复杂的方式相互作用：因此流动被称为湍流。虽然湍流很难理解，并且至今仍然是一个科学谜题，但柯尔莫哥洛夫在 1941 年提出了一种方法。柯尔莫哥洛夫的方法不是试图完全描述流动，而是专注于可以在流动中测量的统计量。 换句话说，某些平均量应该遵循精确的物理定律，而不是理解有关流动的一切（这可能是不可能的）。虽然这种方法在许多方面都非常成功，但在其他许多方面仍然很神秘。特别是，在非常基本的层面上，我们不知道湍流定律的严格合理性：这些定律在实验上似乎是有效的，但它们到底是如何从第一原理中产生的尚不清楚。该计划的目的是调查这个非常基本的问题，它与非常实际的问题有关。虽然流体流动中的湍流是我想到的第一个例子，但另一种类型的湍流（称为弱湍流）可能更容易处理，并提供正确的进入点。弱湍流描述了非线性波动方程中出现的湍流（其中有很多例子，从海洋表面传播的波到电磁波或量子物理学）。一些科学家，特别是七十年代和八十年代的扎哈罗夫推测，弱湍流是由一个特定的方程来描述的，称为动力学波动方程。 PI 的中心目标是借助数学工具（特别是与概率论相关的偏微分方程理论）来研究这一猜想。这项工作有望使我们能够在适当的条件下验证扎哈罗夫的主张，从而为对弱湍流进行理论和严格的理解开辟道路。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。