Derivation of the Kinetic Wave Equation

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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.