On Semi-classical Limit of Spatially Homogeneous Quantum Boltzmann Equation: Asymptotic Expansion

Abstract

We continue our previous work He et al. (Commun Math Phys 386: 143–223, 2021) on the limit of the spatially homogeneous quantum Boltzmann equation as the Planck constant \(\epsilon \) tends to zero, also known as the semi-classical limit. For general interaction potential, we prove the following: (i). The spatially homogeneous quantum Boltzmann equations are locally well-posed in some weighted Sobolev spaces with quantitative estimates uniformly in \(\epsilon \). (ii). The semi-classical limit can be further described by the following asymptotic expansion formula:

$$\begin{aligned} f^\epsilon (t,v)=f_L(t,v)+O(\epsilon ^{\vartheta }). \end{aligned}$$

This holds locally in time in Sobolev spaces. Here \(f^\epsilon \) and \(f_L\) are solutions to the quantum Boltzmann equation and the Fokker–Planck–Landau equation with the same initial data. The convergent rate \(0<\vartheta \le 1\) depends on the integrability of the Fourier transform of the particle interaction potential. Our new ingredients lie in a detailed analysis of the Uehling-Uhlenbeck operator from both angular cutoff and non-cutoff perspectives.