Abstract
The problem of constructing an asymptotic approximation to the solution of the kinetic Boltzmann equation is considered for the hydrodynamic region of low Knudsen numbers. The problem is linearized for one-dimensional perturbations in a gas at rest. The distribution function is sought in the form of a multiscale expansion of the Hilbert asymptotic series type. The construction of a solution uniformly suitable as t → ∞ is demonstrated with reference to a particular example of sonic wave propagation. It is shown that the multiscale technique makes it possible to extend the domain of applicability of the Hilbert expansion to the entire interval of dissipative relaxation.
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REFERENCES
D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Teubner, Leipzig (1912).
C. Cercignani, Theory and Application of the Boltzmann Equation, Scottish Acad. Press, Edinburgh (1975).
J. Ferziger and H. Kaper, Mathematical Theory of Transport Processes in Gases, North-Holland, Amsterdam (1972).
M.N. Kogan, Rarefied Gas Dynamics, Plenum Press, New York (1969).
A. Nayfeh, Perturbation Methods, Wiley, New York, etc. (1973).
D. Smith, Singular-Perturbation Theory, Cambridge Univ. Press, London (1985).
L. D. Landau and E. M. Lifshitz, Theoretical Physics. Vol. 6. Hydrodynamics [in Russian], Nauka, Moscow (1986).
V. S. Galkin and V. I. Nosik, “Modification of the Burnett equations with reference to the sound propagation problem, ” Izv. Ros. Akad. Nauk, Mekh. Zhidk. Gaza, No. 3, 126 (1999).
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Chekmarev, I.B., Chekmareva, O.M. Multiscale Expansion Method in the Hilbert Problem. Fluid Dynamics 38, 646–652 (2003). https://doi.org/10.1023/A:1026390231673
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DOI: https://doi.org/10.1023/A:1026390231673