Singularity and Small-Scale Formation for Model Equations of Fluid Dynamics

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The Euler equations are a system of differential equations that describe the motions of fluids like water and air. Together with the Navier-Stokes equations, which take into account the effect of friction in fluid motion, they are applied in a wide variety of natural and technical situations, for example when modeling the lift of an aircraft wing or the circulation of water in the oceans. Although these equations were first conceived more than two hundred years ago, some of their fundamental mathematical properties are still not well understood. The difficulty lies in the fact that all the equations describing fluids show a strong tendency for "small scale formation." This is seen, for example, in the formation of very small vortices and irregularities in the flow that ultimately cause turbulence. In this research project, the investigator and collaborators study the formation of irregularities in fluid flow from a mathematical point of view. The goal is to give a detailed analysis of the mechanisms that lead to small-scale formation.The projects concern detailed research on geometric singularity formation for certain model equations of fluid dynamics. These model equations are inspired by the Euler equations for three-dimensional, incompressible fluid flow. The overall goal is to gain a better understanding of the complex mechanisms leading to singularity formation in finite time, and also the exact growth rates of quantities like the vorticity and vorticity gradient. The main difficulty comes from the nonlocal and nonlinear nature of the equations. In one of the projects, the investigator considers the hyperbolic flow scenario for the modified surface quasi-geostrophic and Boussinesq equations in two dimensions. The goal is to obtain insight into the hyperbolic flow scenario, which is thought to be a good candidate to ultimately create finite-time blowup for the three-dimensional Euler equations. In the remaining projects, the investigator considers one-dimensional model equations, for which the goal is to describe the singularity formation in as much detail as possible. An important overall theme consists in stabilizing the blowup scenario up to the singular time using barrier functions and a priori estimates that take detailed information about the structure of the solution into account.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.