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

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 ϵ 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 ϵ . (ii). The semi-classical limit can be further described by the following asymptotic expansion formula: f ϵ ( t , v ) = f L ( t , v ) + O ( ϵ ϑ ) . This holds locally in time in Sobolev spaces. Here f ϵ 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 < ϑ ≤ 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.