Gelfand-Shilov smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off

Wei-Xi Li; Lvqiao Liu

doi:10.3934/krm.2020036

Article Contents

2020, Volume 13, Issue 5: 1029-1046. Doi: 10.3934/krm.2020036

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Gelfand-Shilov smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off

Wei-Xi Li^1, and
Lvqiao Liu^2, ,

1.
School of Mathematics and Statistics, Wuhan University & , Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China
2.
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

^* Corresponding author: Lvqiao Liu
^* Corresponding author: Lvqiao Liu

Received: December 31, 2019

Revised: April 30, 2020

Published: August 2020

The first author is supported by NSF grant Nos. 11871054, 11771342; Fok Ying Tung Education Foundation (151001) and the Natural Science Foundation of Hubei Province (2019CFA007)

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Abstract

In this work we consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation. For any given solution belonging to weighted Sobolev space, we will show it enjoys at positive time the Gelfand-Shilov smoothing effect for the velocity variable and Gevrey regularizing properties for the spatial variable. This improves the result of Lerner-Morimoto-Pravda-Starov-Xu [J. Funct. Anal. 269 (2015) 459-535] on one-dimensional Boltzmann equation to the physical three-dimensional case. Our proof relies on the elementary $ L^2 $ weighted estimate.

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Mathematics Subject Classification: Primary: 35B65; 35Q20; 35H10.

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