The Fourier transforms for the spatially homogeneous Boltzmann equation and Landau equation

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Abstract

In this paper, we study the Fourier transforms for two equations arising in the kinetic theory. The first equation is the spatially homogeneous Boltzmann equation. The Fourier transform of the spatially homogeneous Boltzmann equation has been first addressed by Bobylev (Sov Sci Rev C Math Phys 7:111–233, 1988) in the Maxwellian case. Alexandre et al. (Arch Ration Mech Anal 152(4):327–355, 2000) investigated the Fourier transform of the gain operator for the Boltzmann operator in the cut-off case. Recently, the Fourier transform of the Boltzmann equation is extended to hard or soft potential with cut-off by Kirsch and Rjasanow (J Stat Phys 129:483–492, 2007). We shall first establish the relation between the results in Alexandre et al.  (2000) and Kirsch and Rjasanow (2007) for the Fourier transform of the Boltzmann operator in the cut-off case. Then we give the Fourier transform of the spatially homogeneous Boltzmann equation in the non cut-off case. It is shown that our results cover previous works (Bobylev 1988; Kirsch and Rjasanow 2007). The second equation is the spatially homogeneous Landau equation, which can be obtained as a limit of the Boltzmann equation when grazing collisions prevail. Following the method in Kirsch and Rjasanow (2007), we can also derive the Fourier transform for Landau equation.

Keywords

Fourier transform Boltzmann equation Landau equation 

Mathematics Subject Classification

76P05 82C40 

Notes

Acknowledgements

The authors would like to express their sincere thanks to the referees for valuable comments. This work is supported by the National Natural Science Foundation of China (Grant No. 11501292).

Compliance with ethical standards

Conflict of interest

No potential conflict of interest was reported by the authors.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of ScienceNanjing University of Posts and TelecommunicationsNanjingPeople’s Republic of China
  2. 2.Department of Mathematics, School of ScienceNanjing University of Science and TechnologyNanjingPeople’s Republic of China

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