On Singular Solutions of the Vlasov-Poisson Equations

Abstract

A vortex sheet is, loosely speaking, a surface in a fluid such that the normal velocity of the fluid is continuous along the surface but the tangential velocity of the fluid is discontinuous across the surface. Many theoretical, numerical and analytical investigations have been done to try to understand the properties of solutions to the incompressible Euler Equations with non-smooth (vortex sheet) initial data, see [1], [3], [4], [5], [6], [8], [9], [10], [13], [14], [15], [16], [17] and [18] for example. Despite this research, many fundamental open mathematical problems about the nature of these solutions still exist. In order to get a handle on some of these open problems, Andrew Majda proposed that one should study a simpler problem, namely, the one-component Vlasov Poisson Equations (VPE) from plasma physics. Hopefully, insight gained by studying this model problem will provide new insights into the original problems about incompressible fluids. In this paper I will present this system of equations and some connections between the VPE and the vorticity equation for a 2-D incompressible fluid with vortex sheet initial data.