On Semi-classical Limit of Spatially Homogeneous Quantum Boltzmann Equation: Weak Convergence

It is expected in physics that the homogeneous quantum Boltzmann equation with Fermi–Dirac or Bose–Einstein statistics and with Maxwell–Boltzmann operator (neglecting effect of the statistics) for the weak coupled gases will converge to the homogeneous Fokker–Planck–Landau equation as the Planck constant ħ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar $$\end{document} tends to zero. In this paper and the upcoming work (He et al. in On semi-classical limit of spatially homogeneous quantum Boltzmann equation: asymptotic expansion, preprint), we will provide a mathematical justification on this semi-classical limit. Key ingredients into the proofs are the new framework to catch the weak projection gradient, which is motivated by Villani (Arch Rational Mech Anal 143(3):273–307, 1998) to identify the H-solutions for Fokker–Planck–Landau equation, and the symmetric structure inside the cubic terms of the collision operators.