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Regularizing effect and local existence for the non-cutoff Boltzmann equation

Radjesvarane Alexandre Yoshinori Morimoto Seiji Ukai Chao-Jiang Xu Tong Yang

Analysis of PDEs mathscidoc:1912.43931

Archive for rational mechanics and analysis, 198, (1), 39-123, 2010.10
The Boltzmann equation without Grads angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non-integrable angular singularity of the cross-section. However, even though this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo-differential operators, we prove the regularizing effect in all (time, space and velocity) variables on the solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial
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@inproceedings{radjesvarane2010regularizing,
  title={Regularizing effect and local existence for the non-cutoff Boltzmann equation},
  author={Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, and Tong Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210731293117495},
  booktitle={Archive for rational mechanics and analysis},
  volume={198},
  number={1},
  pages={39-123},
  year={2010},
}
Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, and Tong Yang. Regularizing effect and local existence for the non-cutoff Boltzmann equation. 2010. Vol. 198. In Archive for rational mechanics and analysis. pp.39-123. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210731293117495.
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