Note on Dispersive Effects in Quantum Kinetic Equations

We present a global-in-time well-posedness result for the Wigner-Poisson-Fokker-Planck system, a kinetic evolution equation for an open quantum system with a non-linear Hartree potential. Our purely kinetic -analysis allows a unified treatment of the elliptic and hypo-elliptic cases. The crucial tool is to exploit in the quantum framework the dispersive effects of the free transport equation. This yields a local-in-time a-priori estimate for the electric field which allows a new nonlocal-in-time definition of the self-consistent potential. The -regularity of the Wigner function is established for positive times. [DOI: 10.1685 / CSC06011] About DO