Research On Several Problems Of Wigner-Poisson-Fokker-Planck Equation

This paper study the Wigner-Poisson-Fokker-Planck(WPFP) equation which describes a quantum model of semiconductor device, in presence of a self-consistent potential, with periodic boundary conditions. Existence and uniqueness of local classical solutions to the periodic problem in three dimensions are established. The new ingredients are proving that the quantum Fokker Planck operator is a semigroup-generator in a weighted L2 space and the non-linear potential term is a Lipschitz perturbation of the quantum Fokker Planck term in three dimensional. In particular, the analysis of properties of WPFP equation in one dimension is carried out with the application of a Wigner function approach and the semiclassical limit. The conservation of mass and the decay rate of quantity of electric charge are established. The relationship of electron kinetic energy and Husimi kinetic energy is given.