Analytic Methods in Hydrodynamic and Wave Turbulence

湍流在自然界中无处不在，在从大气和海洋动力学到恒星诞生的众多物理理论中发挥着重要作用。这个项目致力于研究流体力学和波浪湍流，两者的核心都是预测能量级联。这项支持研究的主要目的是更好地理解这些能量级联。本项目分为两个项目：流体力学方程弱解的非唯一性研究，以及波浪湍流理论中动力波动方程的严格推导方法。第一个项目的重点将是利用凸积分的理论工具来建立理论能量级联，以解决长期存在的与Navier-Stokes方程弱解相关的开放问题。第二个项目的总体目标将是严格确定动力学波动方程的有效性，该方程被理论化以预测诸如水波、等离子体物理和气候科学等主机设置中的湍流现象。该项目将利用分析、数论和统计物理等多种数学工具。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 ubiquitous in nature, playing a fundamental role in a multitude of physical theories, from atmospheric and oceanic dynamics to the birth of stars. This project is dedicated to an investigation into hydrodynamic and wave turbulence, both of which, at the core predict cascades of energy. The principal aim of this supported research is to better understand these energy cascades. The project is divided into two projects: a study of non-uniqueness of weak solutions to equations arising in hydrodynamics, and a rigorous approach to the derivation of the kinetic wave equation in wave turbulence theory. The focus of the first project will be to utilize the theoretical tool of convex integration to build theoretical energy cascades in order to resolve long standing open problems related to weak solutions to the Navier-Stokes equations. The overarching goal of the second project will be to rigorously determine the validity of the kinetic wave equation which is theorized to predict turbulence phenomena in a host settings such as water waves, plasma physics and climate science. The project will draw on a diverse set of mathematical tools from analysis, number theory and statistical physics.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.