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The Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off

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Abstract

In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cutoff. This equation is partially elliptic in the velocity direction and degenerates in the spatial variable. We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in the Gevrey class at positive time with the Gevrey index depending on the angular singularity. Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for the Cauchy problem of the linearized Boltzmann operator.

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Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant No. 11631011). The third author was supported by National Natural Science Foundation of China (Grant Nos. 11961160716, 11871054 and 11771342), the Natural Science Foundation of Hubei Province (Grant No. 2019CFA007) and the Fundamental Research Funds for the Central Universities (Grant No. 2042020kf0210).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Hua Chen, Xin Hu & Wei-Xi Li

  2. Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, China

    Hua Chen & Wei-Xi Li

  3. Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan, 430070, China

    Jinpeng Zhan

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  1. Hua Chen
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Correspondence to Hua Chen.

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Chen, H., Hu, X., Li, WX. et al. The Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off. Sci. China Math. 65, 443–470 (2022). https://doi.org/10.1007/s11425-021-1886-9

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