Existence, uniqueness and asymptotic behavior of Wigner-Poisson and Vlasov-Poisson systems: A survey

Abstract

We present a review of the Wigner-Poisson system of equations, including its equivalence to a Schrödinger-Poisson system when such an equivalence exists, and including its relationship to the Vlasov equation in the classical limit h → 0. Existence and uniqueness for a particle cloud in all space, in a periodic setting, and with a nonlinear Poisson equation and periodic boundary conditions are all discussed. Finally, some results on asymptotic decay of solutions are mentioned for both the Wigner-Poisson system and the Vlasov-Poisson system in the repulsive case. These results are founded on an identity known in the quantum context as “pseudo-conformal conservation law”. The counterparts of this identity for the Vlasov-Poisson system and the classical N—body problem are presented.