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.

中文翻译：

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空间非齐次玻尔兹曼方程的Gelfand-Shilov平滑效应

在这项工作中，我们考虑空间不均匀非截止Boltzmann方程的柯西问题。对于任何属于加权Sobolev空间的给定解，我们将证明它在正时具有速度变量的Gelfand-Shilov平滑效果和空间变量的Gevrey正则化属性。这改善了Lerner-Morimoto-Pravda-Starov-Xu的结果[J. 功能 肛门 269（2015）459-535]将一维Boltzmann方程应用于物理三维情况。我们的证明依赖于基本的$ L ^ 2 $加权估计。