# American Institute of Mathematical Sciences

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August  2021, 14(4): 705-724. doi: 10.3934/krm.2021020

## Lower bound for the Boltzmann equation whose regularity grows tempered with time

 1 Department of Mathematical Sciences, Tsinghua University, China 2 School of Mathematical Sciences, Center for Statistical Science, Peking University, China

* Corresponding author: Jie Ji

Received  January 2021 Revised  May 2021 Published  August 2021 Early access  June 2021

Fund Project: This work is supported by NSFC under Grant NO.11771236

As a first step towards the general global-in-time stability for the Boltzmann equation with soft potentials, in the present work, we prove the quantitative lower bounds for the equation under the following two assumptions, which stem from the available energy estimates, i.e. (ⅰ). the hydrodynamic quantities (local mass, local energy, and local entropy density) are bounded (from below or from above) uniformly in time, (ⅱ). the Sobolev regularity for the solution grows tempered with time.

Citation: Ling-Bing He, Jie Ji, Ling-Xuan Shao. Lower bound for the Boltzmann equation whose regularity grows tempered with time. Kinetic & Related Models, 2021, 14 (4) : 705-724. doi: 10.3934/krm.2021020
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