Skip to main content
Log in

On the Fundamental Solution of a Linearized Uehling–Uhlenbeck Equation

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

In this paper we describe the fundamental solution of the equation that is obtained linearizing the Uehling–Uhlenbeck equation around the steady state of Kolmogorov type f(k) =  k −7/6. Detailed estimates on its asymptotics are obtained.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Abramowitz M., Stegun I. (1972). Handbook of Mathematical Functions. Dover, New York

    MATH  Google Scholar 

  2. Ajeizer A., Peletminsky S. (1980). Les méthodes de la physique statistique. MIR, Moscow

    Google Scholar 

  3. Balk A.M. (2000). On the Kolmogorov–Zakharov spectra of weak turbulence. Physica D 139, 137–157

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Balk, A.M., Zakharov, V.E.: Stability of weak-turbulence Kolmogorov spectra. Nonlinear Waves and Weak Turbulence (Ed. V. E. Zakharov). AMS Translations Series 2, vol. 182, pp. 1–81 (1998)

  5. Escobedo, M., Mischler, S., Velazquez, J.J.L.: Singular solutions for the Uehling–Uhlenbeck equation. Proc. R. Soc. Edinburgh (in press)

  6. Josserand C., Pomeau Y. (2001). Nonlinear aspects of the theory of Bose–Einstein condensates. Nonlinearity 14, R25–R62

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Kats A.V., Kantorovich V.M. (1973). Symmetry properties of the collision integral and nonisotropic stationary solutions in weak turbulence theory. JETP 37, 80–85

    ADS  Google Scholar 

  8. Kats A.V., Kantorovich V.M. (1974). Anysotropic turbulent distributions for waves with a nondecay dispersion law. JETP 38, 102–107

    ADS  Google Scholar 

  9. Lacaze R., Lallemand P., Pomeau Y., Rica S. (2001). Dynamical formation of a Bose–Einstein condensate. Physica D 152-153: 779–786

    Article  ADS  MathSciNet  Google Scholar 

  10. Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics, 3rd edn. Pergamon, 1977

  11. Levich E., Yakhot V. (1977). Time evolution of a Bose system passing through the critical point. Phys. Rev. B 15, 243–251

    Article  ADS  Google Scholar 

  12. Levich E., Yakhot V. (1978). Time development of coherent and superfluid properties in a course of a λ−transition. J. Phys. A Math. Gen. 11, 2237–2254

    Article  ADS  Google Scholar 

  13. Lu X.G. (2000). A modified Boltzmann equation for Bose–Einstein particles: isotropic solutions and long time behaviour. J. Stat. Phys. 98, 1335–1394

    Article  MATH  Google Scholar 

  14. Lu X.G. (2004). On isotropic distributional solutions to the Boltzmann equation for Bose- Einstein particles. J. Stat. Phys. 116, 1597–1649

    Article  MATH  Google Scholar 

  15. Muskhelishvili N.I. (1992). Singular Integral Equations. Dover, New York

    Google Scholar 

  16. Nordheim L.W. (1928). On the kinetic method in the new statistics and its applications in the electron theory of conductivity. Proc. R. Soc. London A 119: 689–698

    Article  ADS  Google Scholar 

  17. Reed M., Simon B. (1975). Methods of Mathematical Physics. Fourier Analysis and Self Adjointness, vol 2. Academic, New York

  18. Semikov D.V., Tkachev I.I. (1995). Kinetics of Bose condensation. Phys. Rev. Lett. 74, 3093–3097

    Article  ADS  Google Scholar 

  19. Semikov D.V., Tkachev I.I. (1997). Condensation of Bosons in the kinetic regime. Phys. Rev. D 55(2): 489–502

    Article  ADS  Google Scholar 

  20. Taylor M.E. (1981). Pseudodifferential Operators. Princeton University Press, Princeton

    MATH  Google Scholar 

  21. Uehling E.A., Uhlenbeck G.E. (1933). Transport phenomena in Einstein–Bose and Fermi–Dirac gases. Phys. Rev. 43, 552–561

    Article  MATH  ADS  Google Scholar 

  22. Zakharov V.E. (1972). Collapse of Langmuir waves. Sov. JETP 35, 908–914

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. Escobedo.

Additional information

Communicated by F. Otto

Rights and permissions

Reprints and permissions

About this article

Cite this article

Escobedo, M., Mischler, S. & Vélazquez, J.J.L. On the Fundamental Solution of a Linearized Uehling–Uhlenbeck Equation. Arch Rational Mech Anal 186, 309–349 (2007). https://doi.org/10.1007/s00205-007-0084-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-007-0084-2

Keywords

Navigation