9622958 Majda The Vlasov-Poisson equations form the simplest system of equations which describe the microscopic properties of a plasma. Therefore, this system is the starting point for any prediction about the microscopic properties of a plasma. The Vlasov-Poisson equations also play another very important role. Recent research which involved the proposer demonstrated a close relationship between the Vlasov-Poisson equations and the two dimensional Euler equations in vorticity form. The Vlasov-Poisson equations have a simpler mathematical structure than the Euler equations which makes the Vlasov-Poisson equations easier to analyze and solve numerically, and leads to many explicit solutions. Consequently, the Vlasov-Poisson equations form an excellent system of model equations to study before considering any open problems for the two dimensional Euler equations. The proposer will consider some fundamental problems for the Vlasov-Poisson equations pertaining to the lack of uniqueness and singularity formation for weak solutions and to the theory of statistical equilibrium solutions. Many (but not all) of these problems are related to unsolved problems for the Euler equations. The proposer will also study some fundamental open problems for the Euler equations with vortex sheet initial data, using Vlasov-Poisson results to motivate and guide this work. Finally, the proposer will study some interesting issues regarding the numerical solution of both the Vlasov- Poisson and Euler equations. Fluid dynamics is the study of continuous media like air and water. Scientists formulate problems in this field by applying the fundamental laws of physics such as the conservation of mass, momentum and energy. For most problems this procedure leads to sets of mathematical equations which are too complicated to solve. Based on physical and/or mathematical intuition the scientist then tries to find a simplified formulation of the original problem which is simple enough to solve and sufficiently complicated to describe the essential features of the original problem. The mixing process between layers of fluid which have different densities, like air and water, or layers of fluid moving at different velocities is very complicated and occurs in a wide variety of important applications in fields including aerodynamics, meteorology and oceanography. For example, the flow of air over an airplane is very complicated. In order to predict the trailing wake behind an airplane, or the mixing process in a fluid, scientists often use a simplified model of a fluid called a vortex sheet. This is the simplest realistic description of this and many other problems. Unfortunately, despite the wide-spread use of vortex sheet models in many problems in fluid dynamics, many mathematical properties of vortex sheets remain poorly understood. The goal of this proposal is to provide answers to many of the fundamental questions about the mathematical properties of vortex sheets. Successful completion of the proposed work will lead to an increased understanding of the strengths and limitations when using vortex sheets to model complicated flows. The starting point for this work is the simplest set of equations which describe a plasma, a fluid in which the microscopic properties of charged particles are modeled. Therefore, this study should also lead to an increased understanding of the properties of a plasma containing a concentrated beam of electrons.
