Mathematical Notes

, Volume 64, Issue 1, pp 63–81 | Cite as

Logarithmic asymptotics of solutions of the large deviation problem for the boltzmann equation with small transfer of momentum

  • V. P. Maslov
  • A. M. Chebotarev


For the Boltzmann equation with small transfer of momentum, we derive a system of nonlinear integral-differential equations describing the logarithmic asymptotics of the solution to the Cauchy problem in the domain at the distanceO(1) from the support of the initial condition.

Key words

Boltzmann equation kinetic equation transfer equation momentum transfer WKB-expansion logarithmic asymptotics Laplace asymptotics large deviation problem 


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  1. 1.
    W. Braun and K. Hepp, “The Vlasov dynamics and its fluctuations in thel/N-limit of interacting classical particles,”Comm. Math. Phys.,56, 101–113 (1977).CrossRefMathSciNetGoogle Scholar
  2. 2.
    V. P. Maslov and O. Yu. Shvedov, “An asymptotic formula for theN-particle density function asN → ∞ and violation of the chaos hypothesis,”Russian J. Math. Phys.,2, No. 2, 217–234 (1994).MathSciNetGoogle Scholar
  3. 3.
    V. P. Maslov,Complex Markov Chains and the Feynman Path Integral [in Russian], Nauka, (1980).Google Scholar
  4. 4.
    H. Tanaka, “Probabilistic treatment of the Boltzmann equation of Maxwellian molecules,”Z. Wahrscheinlichkeitstheorie Verw. Gebiete,46, 67–105 (1978).CrossRefMATHGoogle Scholar
  5. 5.
    L. V. Lukshin, “Stochastic algorithms in the mathematical theory of space-inhomogeneous Boltzmann equation”,Matem. Modelirovanie,1, No. 2, 151–160 (1989).MATHMathSciNetGoogle Scholar
  6. 6.
    A. A. Arsen'ev,Lectures on Kinetic Theory [in Russian], Nauka, Moscow (1992).Google Scholar
  7. 7.
    V. P. Maslov,Asymptotic Methods and Perturbation Theory [in Russian], Nauka, Moscow (1988).Google Scholar
  8. 8.
    V. P. Maslov, “Asymptotics of solution ofN-particle Kolmogorov-Feller equations and the asymptotic behavior of the solution of the Boltzmann equation in the large deviations domain”.Mat. Zametki [Math. Notes],58, No. 5, 694–709 (1995).MATHMathSciNetGoogle Scholar
  9. 9.
    Ch. K. Birdsall and A. Bruce Langdon,Plasma Physics, via Computer Simulation, McGraw-Hill (1985).Google Scholar
  10. 10.
    I. I. Gikhman and A. V. Skorokhod,Introduction to the Theory of Stochastic Processes [in Russian], Nauka, Moscow (1977).Google Scholar
  11. 11.
    A. M. Chebotarev, “Sufficient conditions for the regularity of Markov jump processes,”Teor. Veroyatnost. i Primenen. [Theory Probab. Appl.],33, No. 1, 25–39 (1988).MATHMathSciNetGoogle Scholar
  12. 12.
    L. D. Landau,Kinetic Equations for the Coulomb Interaction [in Russian], Vol. 1, Nauka, Moscow (1969).Google Scholar
  13. 13.
    M. V. Fedoryuk,The Saddle-Point Method [in Russian], Nauka, Moscow (1977).Google Scholar
  14. 14.
    L. A. Skinner, “Note on the asymptotic behavior of multidimensional Laplace integrals,”SIAM J. Math. Anal.,11, No. 5, 911–917 (1980).MATHMathSciNetGoogle Scholar
  15. 15.
    V. P. Maslov, S. M. Frolovichev, and S. I. Chernykh, “Exact large deviation asymptotics of solutions to boundary value problems for differential equations,”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],296, No. 2, 275–279 (1987).MathSciNetGoogle Scholar
  16. 16.
    V. I. Piterbarg and V. R. Fataflov, “The Laplace method for probability measures in Banach spaces,”Uspekhi Mat. Nauk [Russian Math. Surveys],50, No. 6, 58–150 (1995).Google Scholar

Copyright information

© Plenum Publishing Corporation 1999

Authors and Affiliations

  • V. P. Maslov
    • 1
  • A. M. Chebotarev
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityUSSR

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