Existence of global weak solutions to one‐component vlasov‐poisson and fokker‐planck‐poisson systems in one space dimension with measures as initial data

Abstract

AbstractWe consider Cauchy problems for the 1‐D one component Vlasov‐Poisson and Fokker‐Planck‐Poisson equations with the initial electron density being in the natural space of arbitrary non‐negative finite measures. In particular, the initial density can be a Dirac measure concentrated on a curve, which we refer to as “electron sheet” initial data. These problems resemble both structurally and functional analytically Cauchy problems for the 2‐D Euler and Navier‐Stokes equations (in vorticity formulation) with vortex sheet initial data. Here, we need to define weak solutions more specifically than usual since the product of a finite measure with a function of bounded variation is involved. We give a natural definition of the product, establish its weak stability, and existence of weak solutions follows. Our concept of weak solutions through the newly defined product is justified since solutions to the Fokker‐Planck‐Poisson equation, the analogue of Navier‐Stokes equation, are shown to converge to weak solutions of the Vlasov‐Poisson equation as the Fokker‐Planck term vanishes. The main difficulty is the aforementioned weak stability which we establish through a careful analysis of the explicit structure of these equations. This is needed because the problem studied here is beyond the range of applicability of the “velocity averaging” compactness methods of DiPerna‐Lions. © 1994 John Wiley & Sons, Inc.