THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARD POTENTIAL

doi:10.1007/s10473-024-0205-0

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (2): 455-473.doi: 10.1007/s10473-024-0205-0

THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARD POTENTIAL

Lvqiao LIU¹, Juan ZENG^2,*

1. School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China;
2. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

收稿日期:2022-11-20 修回日期:2023-01-08 出版日期:2024-04-25 发布日期:2023-12-06
通讯作者: *Juan ZENG, E-mail: juan-zeng@whu.edu.cn
作者简介:Lvqiao LIU, E-mail: lvqiaoliu@whu.edu.cn
基金资助:
Liu's research was supported by the NSFC (12101012) and the PhD Scientific Research Start-up Foundation of Anhui Normal University. Zeng's research was supported by the NSFC (11961160716, 11871054, 12131017).

THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARD POTENTIAL

Lvqiao LIU¹, Juan ZENG^2,*

1. School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China;
2. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received:2022-11-20 Revised:2023-01-08 Online:2024-04-25 Published:2023-12-06
Contact: *Juan ZENG, E-mail: juan-zeng@whu.edu.cn
About author:Lvqiao LIU, E-mail: lvqiaoliu@whu.edu.cn
Supported by:
Liu's research was supported by the NSFC (12101012) and the PhD Scientific Research Start-up Foundation of Anhui Normal University. Zeng's research was supported by the NSFC (11961160716, 11871054, 12131017).

摘要/Abstract

摘要： In this article, we study the smoothing effect of the Cauchy problem for the spatially homogeneous non-cutoff Boltzmann equation for hard potentials. It has long been suspected that the non-cutoff Boltzmann equation enjoys similar regularity properties as to whose of the fractional heat equation. We prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with a sharp Gevrey index, depending on the angular singularity. Our proof relies on the elementary $L^2$ weighted estimates.

关键词: Boltzmann equation, Gevrey regularity, non-cutoff, hard potential

Abstract: In this article, we study the smoothing effect of the Cauchy problem for the spatially homogeneous non-cutoff Boltzmann equation for hard potentials. It has long been suspected that the non-cutoff Boltzmann equation enjoys similar regularity properties as to whose of the fractional heat equation. We prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with a sharp Gevrey index, depending on the angular singularity. Our proof relies on the elementary $L^2$ weighted estimates.

Key words: Boltzmann equation, Gevrey regularity, non-cutoff, hard potential

中图分类号:

35B65

Lvqiao LIU, Juan ZENG. THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARD POTENTIAL[J]. 数学物理学报(英文版), 2024, 44(2): 455-473.

Lvqiao LIU, Juan ZENG. THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARD POTENTIAL[J]. Acta mathematica scientia,Series B, 2024, 44(2): 455-473.

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THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARD POTENTIAL

THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARD POTENTIAL

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