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Measure Valued Solutions to the Spatially Homogeneous Boltzmann Equation Without Angular Cutoff

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Abstract

A uniform approach is introduced to study the existence of measure valued solutions to the homogeneous Boltzmann equation for both hard potential with finite energy, and soft potential with finite or infinite energy, by using Toscani metric. Under the non-angular cutoff assumption on the cross-section, the solutions obtained are shown to be in the Schwartz space in the velocity variable as long as the initial data is not a single Dirac mass without any extra moment condition for hard potential, and with the boundedness on moments of any order for soft potential.

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Acknowledgements

Authors would like to express their hearty gratitude to anonymous referees for many suggestions and advises which improved the submitted manuscript. The research of the first author was supported in part by Grant-in-Aid for Scientific Research No.25400160, Japan Society for the Promotion of Science. The research of the third author was supported in part by the General Research Fund of Hong Kong, CityU No. 11303614.

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Authors and Affiliations

  1. Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, 606-8501, Japan

    Yoshinori Morimoto

  2. Department of Mathematics, City University of Hong Kong, Hong Kong, People’s Republic of China

    Shuaikun Wang & Tong Yang

  3. Department of Mathematics, Jinan University, Guangzhou, People’s Republic of China

    Tong Yang

Authors
  1. Yoshinori Morimoto
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  2. Shuaikun Wang
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  3. Tong Yang
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Correspondence to Tong Yang.

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Morimoto, Y., Wang, S. & Yang, T. Measure Valued Solutions to the Spatially Homogeneous Boltzmann Equation Without Angular Cutoff. J Stat Phys 165, 866–906 (2016). https://doi.org/10.1007/s10955-016-1655-0

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  • DOI: https://doi.org/10.1007/s10955-016-1655-0

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