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

We continue our previous work [Ling-Bing He, Xuguang Lu and Mario Pulvirenti, Comm. Math. Phys., 386(2021), no. 1, 143223.] 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: f(epsilon) (t, v) = fL(t, v) + O(epsilon(theta)). 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 this http URL convergent rate 0< theta <= 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.