Applications of Harmonic Analysis to the Study of Incompressible Flow

本研究的目的是研究不可压缩流体运动方程在非常弱的初始数据下的解的行为。 以下两个重要的开放问题在数学流体力学将被调查。 第一个是临界Sobolev和Besov空间中带初速度的不可压缩流体方程解的正则性。 第二类是有界平均振荡函数空间中具有初始涡量的流体方程解的性质。我们还将研究在这两种情况下，用粘度很小的流体来近似无粘流体。 虽然这项研究的大部分将解决二维流动，二维结果的三维轴对称设置的扩展也将被考虑。 低粘度流体和粘度可忽略的流体是科学家和工程师非常感兴趣的。 这个研究项目的目标是更好地了解如何以及一个无粘流体必须表现，以便合理地近似为小粘度的流体。 此外，该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设置，其中流动关于轴对称。 对流体力学这两个领域的理解的任何改进都将导致对行为不良的流体进行更精确的数值模拟。

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.