 
Summary: The WignerPoissonFokkerPlanck system: globalintime
solution and dispersive effects
Anton Arnold, Elidon Dhamo, Chiara Manzini
Key words: Wigner equation, FokkerPlanck operator, Poisson equation, dispersive regularization.
AMS 2000 Subject Classification: 35A05, 35K55, 35Q40, 47B44, 81Q99, 81S30, 82D37
Abstract
This paper is concerned with the WignerPoissonFokkerPlanck system, a kinetic evolu
tion equation for an open quantum system with a nonlinear 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
FokkerPlanck operator generates a dissipative semigroup. The nonlinear potential can be
controled by using the parabolic regularization of the system.
The main technical difficulty for establishing globalintime solutions is to derive apriori
estimates on the electric field: Inspired by a strategy for the classical VlasovFokkerPlanck
equation, we exploit dispersive effects of the free transport operator. As a "byproduct" we
also derive a new apriori estimate on the field in the WignerPoisson equation.
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1 Introduction
The goal of this paper is to prove the existence and uniqueness of globalintime solutions
to the coupled WignerPoissonFokkerPlanck (WPFP) system in three dimensions. This
