Abstract
This paper is concerned with the Wigner–Poisson–Fokker–Planck system, a kinetic evolution 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 established. The analysis is carried out in a weighted L2-space, such that the linear quantum Fokker–Planck operator generates a dissipative semigroup. The non-linear potential can be controlled 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.
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