On semi-classical limit of spatially homogeneous quantum Boltzmann equation: asymptotic expansion

TL;DR

This paper investigates the semi-classical limit of the spatially homogeneous quantum Boltzmann equation, establishing well-posedness and deriving an asymptotic expansion with explicit convergence rates as Planck's constant approaches zero.

Contribution

It proves local well-posedness of the quantum Boltzmann equation with uniform estimates and provides an asymptotic expansion describing the semi-classical limit with explicit convergence rates.

Findings

Established local well-posedness in weighted Sobolev spaces.

Derived an asymptotic expansion with rate depending on potential's Fourier transform.

Analyzed the Uehling-Uhlenbeck operator from angular cutoff and non-cutoff perspectives.

Abstract

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: $$ f^\epsilon(t,v)=f_L(t,v)+O(\epsilon^{\vartheta}).$$ 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 $0<\vartheta \leq 1$ 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.