Communications in Mathematical Sciences

Volume 19 (2021)

Number 2

Asymptotic analysis of the Boltzmann equation with very soft potentials from angular cutoff to non-cutoff

Pages: 287 – 324



Ling-Bing He (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Zheng-An Yao (School of Mathematics, Sun Yat-Sen University, Guangzhou,

Yu-Long Zhou (School of Mathematics, Sun Yat-Sen University, Guangzhou,


Our focus is the Boltzmann equation in a torus under very soft potentials around equilibrium. We analyze the asymptotics of the equation from angular cutoff to non-cutoff. We first prove a refined decay result of the semi-group stemming from the linearized Boltzmann operator. Then we prove the global well-posedness of the equations near equilibrium, refined decay patterns of the solutions. Finally, we rigorously give the asymptotic formula between the solutions to cutoff and noncutoff equations with an explicit convergence rate.


Boltzmann equation, very soft potential, asymptotic analysis, angular cutoff, angular non-cutoff, short-range interaction, long-range interaction

2010 Mathematics Subject Classification

35B40, 35Q20, 82C40

Received 8 June 2020

Accepted 25 August 2020

Published 12 April 2021