Skip to main content
SpringerLink
Log in

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

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Alexandre, R., Desvillettes, L., Villani, C., Wennberg, B.: Entropy dissipation and long range interactions. Arch. Ration. Mech. Anal. 152(4), 327–355 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alexandre, R., Elsafadi, M.: Littlewood Paley decomposition and regularity issues in Boltzmann homogeneous equations. I. Non cutoff and Maxwell cases. Math. Models Methods Appl. Sci. 15, 907–920 (2005)

    Article  MathSciNet  Google Scholar 

  3. Alexandre, R., Villani, C.: On the Landau approximation in plasma physics. Ann. Inst. Henri Poincaré Anal. Non Linéaire 21(1), 61–95 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Arkeryd, L.: Intermolecular forces of infinite range and the Boltzmann equation. Arch. Ration. Mech. Anal. 77, 11–21 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arsen’Ev, A.A., Buryak, O.E.: On the connection between a solution of the Boltzmann equation and a solution of the Landau-Fokker-Planck equation. Math. Sb. 69(2), 465–478 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bobylev, A.V.: The theory of the nonlinear, spatially uniform Boltzmann equation for Maxwell molecules. Sov. Sci. Rev. C. Math. Phys. 7, 111–233 (1988)

    MathSciNet  MATH  Google Scholar 

  7. Bobylev, A.V., Rjasanow, S.: Difference scheme for the Boltzmann equation based on fast Fourier transform. Eur. J. Mech. B/Fluids 16(2), 293–306 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Cannone, M., Karch, G.: Infinite energy solutions to the homogeneous Boltzmann equation. Commun. Pure Appl. Math. 63, 747–778 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Carlen, E.A., Gabetta, E., Toscani, G.: Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Commun. Math. Phys. 199, 521–546 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Desvillettes, L.: About the use of the Fourier transform for the Boltzmann equation. In: Summer School on “Methods and Models of Kinetic Theory” (M&MKT 2002). Riv. Mat. Univ. Parma 2(7), 1–99 (2003)

  11. Desvillettes, L., Villani, C.: On the spatially homogeneous Landau equation for hard potentials. Part 1. Existence, uniqueness and smoothness. Commun. Partial Differ. Equ. 25(1–2), 179–259 (2000)

    Article  MATH  Google Scholar 

  12. Gabetta, E., Toscani, G., Wennberg, B.: Metrics for Probability distributions and the trend to equilibrium for solutions of the Boltzmann equation. J. Stat. Phys. 81, 901–934 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Goudon, T.: On Boltzmann equations and Fokker-Planck asymptotics: influence of grazing collisions. J. Stat. Phys. 89, 751–776 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Guo, Y.: The Landau equation in a periodic box. Commun. Math. Phys 231(3), 391–434 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ibragimov, I., Rjasanow, S.: Numerical solution of the Boltzmann equation on the uniform grid. Computing 69(2), 163–186 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kirsch, R., Rjasanow, S.: A weak formulation of the Boltzmann equation based on the Fourier Transform. J. Stat. Phys. 129, 483–492 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Morimoto, Y., Yang, T.: Smoothing effect of the homogeneous Boltzmann equation with measure valued initial datum. Ann. Inst. Henri Poincaré Anal. Non Linéaire. 32(2), 429–442 (2015)

  18. Pareschi, L., Perthame, B.: A Fourier spectral method for homogeneous Boltzmann equations. Transp. Theory Stat. Phys. 25, 369–382 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pareschi, L., Russo, G.: Numerical solution of the Boltzmann equation, I: spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal. 37(4), 1217–1245 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pareschi, L., Russo, G.: On the stability of spectral methods for the homogeneous Boltzmann equation. Transp. Theory Stat. Phys. 29(3–5), 431–447 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pareschi, L., Toscani, G., Villani, C.: Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit. Numer. Math. 93(3), 527–548 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pulvirenti, M., Toscani, G.: The theory of the nonlinear Boltzmann equation for Maxwell molecules in Fourier representation. Ann. Mat. Pura Appl. 4(171), 181–204 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tanaka, H.: Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 46(1), 67–105 (1978–1979)

  24. Toscani, G., Villani, C.: Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Stat. Phys. 94, 619–637 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Villani, C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal. 143, 273–307 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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).

Author information

Authors and Affiliations

  1. School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210003, People’s Republic of China

    Fei Meng

  2. Department of Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

    Fang Liu

Authors
  1. Fei Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fei Meng.

Ethics declarations

Conflict of interest

No potential conflict of interest was reported by the authors.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meng, F., Liu, F. The Fourier transforms for the spatially homogeneous Boltzmann equation and Landau equation. Anal.Math.Phys. 9, 655–674 (2019). https://doi.org/10.1007/s13324-018-0223-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13324-018-0223-y

Keywords

Mathematics Subject Classification

Search

Navigation