Vlasov-poisson systems with measures as initial data

This short exposition is about joint work done with Andrew Majda (Commu. Pure Appl. Math., 41(1994), 1365-1401), and Andrew Majda and George Majda (Physica D, 74(1994), 268-300; 79(1994), 41-76). We are motivated by the study on the vortex sheet initial value problem for the 2-D incompressible. Euler equations. A. Majda proposed to study the Vlasov-Poisson systems with measures as initial data to help finding hints for the vortex sheets problem. We consider Cauchy problems for both the one- and two-component Vlasov-Poisson systems and the related Fokker-Planck-Poisson system with measures as initial data in one space dimension. The existence of global weak solutions for the one-component problems has been obtained by the author and A. Majda. For the two-component Vlasov-Poisson system, however, bona fide measure-valued solutions are found in the limit of weakly converging approximate solutions, which are obtained through a procedure typical in physics and numerical computations. Weak solutions of the one-component Vlasov-Poisson system can have very strong finite time singularities. We have an explicit example of an exact weak solution which develops a Dirac point charge concentration at finite time from an initial datum which concentrates on a straight segement (called "an electron sheet") in the position vs. velocity plane. Other explicit examples of exact solutions have been constructed to demonstrate that the Cauchy problems for the Vlasov-Poisson systems have in general multiple weak solutions. A highly efficient numerical algorithm, developed by G. Majda, enables us to show numerically and clearly that different regularizations of the problems select different weak solutions. In particular, solutions of the Fokker-Planck-Poisson system converge as the Fokker-Planck term vanishes to a weak solution which is different from the limit of solutions from smoothing the same initial datum. There seems therefore to be no selection principles.