Title: Dispersive effects in quantum kinetic equations

Authors: Anton Arnold, Elidon Dhamo and Chiara Manzini

Issue: Volume 56 (2007), Issue 3, 1299-1332

Abstract: This paper is concerned with a global-in-time well-posedness analysis for the Wigner-Poisson-Fokker-Planck system, a kinetic evolution equation for an open quantum system with a non-linear Hartree potential. The purely kinetic $L^2$-analysis here presented, allows a unified treatment of the elliptic and hypoelliptic cases. The crucial novel tool of the analysis is to exploit in the quantum framework the dispersive effects of the free transport equation. It yields an a-priori estimate on the electric field for all time which allows a new nonlocal-in-time definition of the self-consistent potential and field. Thus, one can circumvent the lacking $v$-integrability of the Wigner function, which is a central problem in quantum kinetic theory. Due to the (degenerate) parabolic character of this system, the $C^{\infty}$-regularity of the Wigner function, its macroscopic density, and the field are established for positive times.