On a Boltzmann Equation for Compton Scattering from Non relativistic Electrons at Low Density

  • E. Cortés
  • M. EscobedoEmail author


A Boltzmann equation, used to describe the evolution of the density function of a gas of photons interacting by Compton scattering with electrons at low density and non relativistic equilibrium, is considered. A truncation of the very singular redistribution function is introduced and justified. The existence of weak solutions is proved for a large set of initial data. A simplified equation, where only the quadratic terms are kept and that appears at very low temperature of the electron gas, for small values of the photon’s energies, is also studied. The existence of weak solutions, and also of more regular solutions that are very flat near the origin, is proved. The long time asymptotic behavior of weak solutions of the simplified equation is described.


Kinetic equation Compton scattering Low density non relativistic electrons Kompaneets equation Weak solutions Long time behavior Dirac measures 



The research of the first author is supported by the Basque Government through the BERC 2014-2017 program, by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323, and by MTM2014-52347-C2-1-R of DGES. The research of the second author is supported by Grants MTM2014-52347-C2-1-R of DGES and IT641-13 of the Basque Government. The authors gratefully thank the referees for their careful reading of the manuscript and their valuable comments and recommendations.


  1. 1.
    Ballew, J., Iyer, G., Pego, R.L.: Bose–Einstein condensation in a hyperbolic model for the Kompaneets equation. SIAM J. Math. Anal. 48, 3840–3859 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barik, P.K., Giri, A.K., Laurençot, P.: Mass-conserving solutions to the Smoluchowski coagulation equation with singular kernel. (2018) ArXiv e-prints, arXiv:1804.00853
  3. 3.
    Birkinshaw, M.: The Sunyaev–Zeldovich effect. Phys. Rep. 310(2), 97–195 (1999)ADSCrossRefGoogle Scholar
  4. 4.
    Bogachev, V.I.: Measure Theory, vol. 2. Springer, Berlin (2007)CrossRefzbMATHGoogle Scholar
  5. 5.
    Brown, L.M., Feynman, R.P.: Radiative corrections to Compton scattering. Phys. Rev. 85, 231–244 (1952)ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Buet, C., Després, B., Leroy, T.: Anisotropic models and angular moments methods for the Compton scattering. E-prints, hal-01717173 (2018)Google Scholar
  7. 7.
    Caflisch, R.E., Levermore, C.D.: Equilibrium for radiation in a homogeneous plasma. Phys. Fluids 29(3), 748–752 (1986)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Camejo, C.C., Gröpler, R., Warnecke, G.: Regular solutions to the coagulation equations with singular kernels. Math. Methods Appl. Sci. 38(11), 2171–2184 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chane-Yook, M., Nouri, A.: On a quantum kinetic equation linked to the Compton effect. Transp. Theory Stat. Phys. 33(5–7), 403–427 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cortés, E., Escobedo, M.: On a system of equations for the normal fluid-condensate interaction in a Bose gas. ArXiv e-prints, (2018)Google Scholar
  11. 11.
    Dreicer, H.: Kinetic theory of an electron–photon gas. Phys. Fluids 7, 735–753 (1964)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Escobedo, M., Herrero, M.A., Velázquez, J.J.L.: A nonlinear Fokker–Planck equation modelling the approach to thermal equilibrium in a homogeneous plasma. Trans. Am. Math. Soc. 350(10), 3837–3901 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Escobedo, M., Mischler, S.: On a quantum Boltzmann equation for a gas of photons. J. Math. Pures Appl. 80(5), 471–515 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Escobedo, M., Mischler, S., Valle, M.A.: Homogeneous Boltzmann equation in quantum relativistic kinetic theory. Electron. J. Diff. Equ. Monogr. 4(2), 1–85 (2003)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Escobedo, M., Mischler, S., Velázquez, J.J.L.: Asymptotic description of Dirac mass formation in kinetic equations for quantum particles. J. Diff. Equ. 202(2), 208–230 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ferrari, E., Nouri, A.: On the Cauchy problem for a quantum kinetic equation linked to the Compton effect. Math. Comput. Modell. 43, 838–853 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Folland, G.B.: Real Analysis: Modern Techniques and Their Applications. Pure and Applied Mathematics. Wiley, New York (1999)zbMATHGoogle Scholar
  18. 18.
    Fournier, N., Laurençot, P.: Well-posedness of Smoluchowski’s coagulation equation for a class of homogeneous kernels. J. Funct. Anal. 233(2), 351–379 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Grachev, S.I.: Nonstationary radiative transfer: evolution of a spectrum by multiple compton scattering. Astrophysics 57(4), 550–558 (2014)ADSCrossRefGoogle Scholar
  20. 20.
    Kavian, O.: Remarks on the Kompaneets Equation, a Simplified Model of the Fokker–Planck Equation. Studies in Applied Mathematics. North-Holland (2002)Google Scholar
  21. 21.
    Klenke, A.: Probability Theory: A Comprehensive Course (Universitext). Springer, London (2013)Google Scholar
  22. 22.
    Kompaneets, A.S.: The establishment of thermal equilibrium between quanta and electrons. Soviet J. Exp. Theor. Phys. 4, 730–737 (1957)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Levermore, C.D., Liu, H., Pego, R.L.: Global dynamics of Bose–Einstein condensation for a model of the Kompaneets equation. SIAM J. Math. Anal. 48(4), 2454–2494 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mészáros, P., Bussard, R.W.: The angle-dependent Compton redistribution function in X-ray sources. Astrophys. J. 306, 238–247 (1986)ADSCrossRefGoogle Scholar
  25. 25.
    Norris, J.R.: Smoluchowski’s coagulation equation: uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent. Ann. Appl. Probab. 9(1), 78–109 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Weyman, R.: Diffusion approximation for a photon gas interacting with a plasma via the compton effect. Phys. Fluids 8, 2112–2114 (1965)ADSCrossRefGoogle Scholar
  27. 27.
    Zel’Dovich, Y.B.: Reviews of topical problems: interaction of free electrons with electromagnetic radiation. Sov. Phys. Uspekhi 18, 79–98 (1975)ADSCrossRefGoogle Scholar
  28. 28.
    Zel’Dovich, Y.B., Levich, E.V.: Bose condensation and shock waves in photon spectra. Sov. J. Exp. Theor. Phys. 28, 1287–1290 (1969)ADSGoogle Scholar
  29. 29.
    Zel’Dovich, Y.B., Levich, E.V., Syunyaev, R.A.: Stimulated compton interaction between Maxwellian electrons and spectrally narrow radiation. Sov. J. Exp. Theor. Phys. 35, 733–740 (1972)ADSGoogle Scholar
  30. 30.
    Zel’Dovich, Y.B., Syunyaev, R.A.: Shock wave structure in the radiation spectrum during bose condensation of photons. Sov. J. Exp. Theor. Phys. 35, 81–85 (1972)ADSGoogle Scholar

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Authors and Affiliations

  1. 1.BCAM - Basque Center for Applied MathematicsBilbaoSpain
  2. 2.Departamento de MatemáticasUniversidad del País VascoBilbaoSpain

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