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arXiv · 2309.00891

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

Ling-Bing He·Xuguang Lu·Mario Pulvirenti·Yu-Long Zhou
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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 $ε$ 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: $$ f^ε(t,v)=f_L(t,v)+O(ε^{\vartheta}).$$ 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 $0<\vartheta \leq 1$ 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.

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Ling-Bing He, Xuguang Lu, Mario Pulvirenti, Yu-Long Zhou. 2023-09-02. On semi-classical limit of spatially homogeneous quantum Boltzmann equation: asymptotic expansion. https://arxiv.org/abs/2309.00891

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