Derivation of the Kinetic Wave Equation

湍流是一种普遍现象，发生在许多物理系统中。最简单的例子是流体的流动，比如河流中的水。在某些情况下，流动非常平滑，但在其他情况下，它看起来非常混乱，有各种规模的漩涡，以非常复杂的方式相互作用：然后说流动是湍流。虽然湍流很难理解，至今仍是一个科学难题，但柯尔莫哥洛夫在1941年提出了一种方法。Kolmogorov的方法不是试图完全描述流，而是专注于流中可以测量的统计量。换句话说，与其理解流体的一切（这可能是不可能的），某些平均量应该遵循精确的物理定律。虽然这种方法在许多方面非常成功，但在其他许多方面仍然很神秘。特别是，在一个非常基本的层面上，没有对湍流定律的严格证明是已知的：这些定律似乎在实验上是有效的，但它们究竟是如何从第一原理中产生的却不得而知。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，并且提供了正确的切入点。弱湍流描述了非线性波动方程中出现的湍流（其中有许多例子，从在海洋表面传播的波到电磁波或量子物理）。几位科学家，尤其是70年代和80年代的扎哈罗夫，推测弱湍流可以用一个特定的方程来描述，这个方程被称为动能波动方程。PI的中心目标是在数学工具的帮助下研究这一猜想：特别是与概率论有关的偏微分方程理论。这一努力有望使我们能够在适当的条件下验证扎哈罗夫的说法，从而为从理论上和严格地理解弱湍流开辟道路。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。

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.