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.
