CONCENTRATIONS IN THE ONE-DIMENSIONAL VLASOV-POISSON EQUATIONS .2. SCREENING AND THE NECESSITY FOR MEASURE-VALUED SOLUTIONS IN THE 2-COMPONENT CASE

Weak and measure-valued solutions for the two-component Vlasov-Poisson equations in a single space dimension are proposed and studied here;a s a simpler analogue problem for the limiting behavior of approximatio ns for the two-dimensional Euler equations with general vorticity of t wo signs. From numerical experiments and mathematical theory, it is kn own that much more complex behavior can occur in limiting processes fo r vortex sheets with general vorticity of two signs as compared with n on-negative vorticity. Here such behavior is confirmed rigorously for the simpler analogue problem through explicit examples involving singu lar charge concentration. For the two-component Vlasov-Poisson equatio ns, the concepts of measure-valued and weak solution are introduced. E xplicit examples with charge concentration establish that the limit of weak solutions in a dynamic process is necessarily a measure-valued s olution in some cases rather than the anticipated weak solution, i.e. no concentration-cancellation occurs. The limiting behavior of computa tional regularizations involving high resolution particle methods is p resented here both for the instances with measure-valued solutions and also for new examples with non-unique weak solutions. The authors dem onstrate that different computational regularizations can exhibit comp letely different limiting behavior in situations with measure-valued a nd/or non-unique weak solutions.