The principal investigator will study the mathematical problems of fluid motion. Methods of Littlewood-Paley theory and wavelets will be used to study weak singularities for the incompressible flows of an ideal fluid. The basic nonstationary problem for the Euler equations will be investigated in function classes with, generally, essentially unbounded vorticity. He will address the problem of linear vector transport by flows that are less than regular. These aspects of transport theory are directly applicable to questions of existence and uniqueness for the Euler equations.
The mathematical theory of fluid motion is fundamental both for applications in meteorology, geophysics, astrophysics and within mathematics itself. Most of the flows in nature such as ocean currents, or hurricanes, or the motion of matter in a galaxy, are highly irregular. The origin and evolution of singular structures such as rapid oscillations of a flow are still not well understood. In addition to this, numerical modelling of irregular flows is limited by instabilities due to the small scale structures constantly produced by the flow. The PI will investigate two related aspects of this subject. First, the basic mathematical nature of the flows with singularities. Second, transport of quantities such as the magnetic field by irregular flows in the solar photosphere.
