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The smoothing effect in sharp Gevrey space for the spatially homogeneous non-cutoff Boltzmann equations with a hard potential

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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 L2 weighted estimates.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Anhui Normal University, Wuhu, 241002, China

    Lvqiao Liu

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

    Juan Zeng

Authors
  1. Lvqiao Liu
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  2. Juan Zeng
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Correspondence to Juan Zeng.

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Conflict of Interest The authors declare no conflict of interest.

Additional information

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).

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Liu, L., Zeng, J. The smoothing effect in sharp Gevrey space for the spatially homogeneous non-cutoff Boltzmann equations with a hard potential. Acta Math Sci 44, 455–473 (2024). https://doi.org/10.1007/s10473-024-0205-0

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  • DOI: https://doi.org/10.1007/s10473-024-0205-0

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