Summary: The Wigner-Poisson-Fokker-Planck system: global-in-time
solution and dispersive effects
Anton Arnold, Elidon Dhamo, Chiara Manzini
Key words: Wigner equation, Fokker-Planck operator, Poisson equation, dispersive regularization.
AMS 2000 Subject Classification: 35A05, 35K55, 35Q40, 47B44, 81Q99, 81S30, 82D37
This paper is concerned with the Wigner-Poisson-Fokker-Planck system, a kinetic evolu-
tion equation for an open quantum system with a non-linear Hartree potential. Existence,
uniqueness and regularity of global solutions to the Cauchy problem in 3 dimensions are es-
tablished. The analysis is carried out in a weighted L2space, such that the linear quantum
Fokker-Planck operator generates a dissipative semigroup. The non-linear potential can be
controled by using the parabolic regularization of the system.
The main technical difficulty for establishing global-in-time solutions is to derive a-priori
estimates on the electric field: Inspired by a strategy for the classical Vlasov-Fokker-Planck
equation, we exploit dispersive effects of the free transport operator. As a "by-product" we
also derive a new a-priori estimate on the field in the Wigner-Poisson equation.
The goal of this paper is to prove the existence and uniqueness of global-in-time solutions
to the coupled Wigner-Poisson-Fokker-Planck (WPFP) system in three dimensions. This