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，比如说河流中的水。在某些状态下，流动是非常平滑的，但在其他情况下，它看起来非常混乱，具有各种尺度的漩涡，以非常复杂的方式相互作用：流动被称为湍流。虽然湍流是很难理解的，并构成了这一天的科学之谜，一种方法是由Kolmogorov在1941年提出的。Kolmogorov的方法不是试图完全描述流，而是关注可以在流中测量的统计量。 换句话说，某些平均量应该遵循精确的物理定律，而不是理解关于流动的一切，这可能是不可能的。虽然这种方法在许多方面非常成功，但在许多其他方面仍然神秘。特别是，在一个非常基本的水平上，没有严格的理由湍流定律是已知的：这些法律似乎是有效的实验，但他们究竟是如何产生的第一原理是未知的。该计划的目的是调查这个非常基本的问题，这是非常实际的问题。虽然流体流动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。弱湍流描述了非线性波动方程中出现的湍流（有许多例子，从海洋表面传播的波到电磁波或量子物理）。几位科学家，特别是扎哈罗夫在70年代和80年代提出，弱湍流由一个特定的方程描述，称为动力学波动方程。PI的中心目标是调查这个猜想的帮助下的数学工具：特别是，理论的偏微分方程，与概率论。这一努力有望使我们能够在适当的条件下验证Zakharov的主张，从而为弱湍流的理论和严格理解开辟道路。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。