Analytic Methods in Hydrodynamic and Wave Turbulence

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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.