Concentrations in the one-dimensional Vlasov-Poisson equations. II. Screening and the necessity for measure-valued solutions in the two component case

Weak and measure-valued solutions for the two-component Vlasov-Poisson equations in a single space dimension are proposed and studied here as a simpler analogue problem for the limiting behavior of approximations for the two-dimensional Euler equations with general vorticity of two signs. From numerical experiments and mathematical theory, it is known that much more complex behavior can occur in limiting processes for vortex sheets with general vorticity of two signs as compared with non-negative vorticity. Here such behavior is confirmed rigorously for the simpler analogue problem through explicit examples involving singular charge concentration. For the two-component Vlasov-Poisson equations, the concepts of measure-valued and weak solution are introduced. Explicit examples with charge concentration establish that the limit of weak solutions in a dynamic process is necessarily a measure-valued solution in some cases rather than the anticipated weak solution, i.e. no concentration-cancellation occurs. The limiting behavior of computational regularizations involving high resolution particle methods is presented here both for the instances with measure-valued solutions and also for new examples with non-unique weak solutions. The authors demonstrate that different computational regularizations can exhibit completely different limiting behavior in situations with measure-valued and/or non-unique weak solutions.