Applications of Harmonic Analysis to the Study of Incompressible Flow

本研究项目的目的是研究初始数据非常弱的不可压缩流体运动方程解的行为。下面将研究数学流体力学中两个重要的开放问题。首先是临界Sobolev和Besov空间中具有初速度的不可压缩流体方程解的正则性。二是具有初始涡度的流体方程解在有界平均振荡函数空间中的行为。我们还将研究在这两种情况下粘度非常小的流体对无粘流体的近似。虽然这项研究的大部分将涉及二维流动，但也将考虑将二维结果扩展到三维轴对称设置。低粘度流体和粘度可以忽略的流体引起了科学家和工程师的极大兴趣。本研究项目的一个目标是更好地理解无粘性流体必须表现得多好，才能被小粘度流体合理地近似。此外，本项目旨在研究粘性和非粘性流动的必要假设，以将二维分析扩展到更复杂的三维环境，其中流动是围绕轴对称的。对这两个流体力学领域的理解的任何改进都将导致对不良流体更精确的数值模拟。

The aim of this research project is to study behavior of solutions to the equations of incompressible fluid motion with very weak initial data. The following two important open problems in mathematical fluid mechanics will be investigated. The first is the regularity of solutions to the incompressible fluid equations with initial velocity in critical Sobolev and Besov spaces. The second is the behavior of solutions to the fluid equations with initial vorticity in the space of functions of bounded mean oscillation. We will also study the approximation of inviscid fluids by fluids of very small viscosity in these two settings. While much of this research will address two-dimensional flows, extensions of two-dimensional results to the three-dimensional axisymmetric setting will also be considered. Low viscosity fluids and fluids in which viscosity is negligible are of great interest to scientists and engineers. A goal of this research project is to better understand how well an inviscid fluid must behave in order to be reasonably approximated by fluids of small viscosity. Moreover, this project aims to study the assumptions necessary on both viscous and inviscid flows to extend two-dimensional analysis to a more complicated three-dimensional setting where the flow is symmetric about an axis. Any improvement in the understanding of these two areas of fluid mechanics will lead to more accurate numerical simulations of badly behaved fluids.