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Asymptotic theory of the Boltzmann equation. (English) Zbl 0115.45006
For Part II, see “Rarefied gas dynamics, Proc. 3rd Int. Sympos., Palais de l’UNESCO, Paris, 1962, New York: Academic Press, Vol. I, 26–59 (1963).
Reviewer: R. S. Żelazny

MSC:
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
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[1] Grad, Communs. Pure and Appl. Math. 2 pp 331– (1949)
[2] H. Grad, ”Principles of the Kinetic Theory of Gases,” inHandbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1958), Vol. XII, Sec. 26.
[3] For a study of the shock slip problem see Y. S. Pan and R. F. Probstein, inProceedings of Third International Conference on Rarefied Gases, Paris, 1962(to be published).
[4] Hilbert, Math. Ann. 72 pp 562– (1912)
[5] alsoGrundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen(Chelsea Publishing Company, New York, 1953).
[6] Ikenberry, J. Ratl. Mech. Anal. 5 pp 1– (1956)
[7] The formal theory of these asymptotic expansions was presented at a Colloquium at Princeton University, March 1, 1961.
[8] J. Thurber and L. Sirovich, inProceedings of the Third International Conference on Rarefied Gases, Paris, 1962(to be published).
[9] Sirovich, Phys. Fluids 6 pp 218– (1963)
[10] C. S. Wang Chang and G. E. Uhlenbeck,On the Propagation of Sound in Monatomic Gases(University of Michigan, Ann Arbor, Michigan, 1952).
[11] J. E. McCune, T. F. Morse, and G. Sandri, inProceedings of the Third International Conference on Rarefied Gases, Paris, 1962(to be published).
[12] H. Grad, ”Asymptotic Theory of the Boltzmann Equation II,” inProceedings of the Third International Conference on Rarefied Gases, Paris, 1962(to be published); hereafter referred to as II. · Zbl 0105.42104
[13] There is no loss of generality in taking zero as the unperturbed velocity, uo = 0, so long as we disregard boundary value problems.
[14] Only in Sec. IX will the nonlinear equation be used.
[15] For the specific form that follows, see reference 2, Sec. 25.
[16] S. Chapman and T. G. Cowling,The Mathematical Theory of Non-Uniform Gases(Cambridge University Press, New York, 1939). · Zbl 0063.00782
[17] The statement in reference 15, Chap. 7 that \(\rho\)n = 0, n>0for all timeis meaningless. It is incorrect if it is intended to apply to a solution of the Boltzmann equation, and it is irrelevant with regard to the question of deriving the fluid partial differential equations.
[18] Friedrichs, Communs. Pure and Appl. Math. 7 pp 345– (1954)
[19] For simple power law molecules, the nondimensionalization is such that these coefficients are absolute constants. More generally, they will depend upon the temperature of the unperturbed gas (as well as molecular constants).
[20] The caret is used in this section to denote irreducibility, see Appendix.
[21] H. M. Mott-Smith, ”A New Approach in the Kinetic Theory of Gases,” Lincoln Laboratory MIT Rept. (1954).
[22] Pekeris, Phys. Fluids 5 pp 1608– (1962)
[23] Grad, Communs. Pure and Appl. Math. 2 pp 325– (1949)
[24] There should be no confusion in the use of the caret to indicate irreducibility of a tensor and also the normalization (3.19) of a distribution function.
[25] The complete theory is given by Weyl [H. Weyl,The Classical Groups(Princeton University Press, Princeton, New Jersey, 1946), Chap. V, Sec. 6]. But his presentation is unnecessarily complicated for our purposes because of the greater generality. Our presentation is an elaboration of that given in reference 2, Sec. 30.
[26] Ikenberry, Arch. Ratl. Mech. Anal. 9 pp 255– (1962)
[27] The comparison with the Sonine polynomials Sm(n) of Chapman and Cowling14is given by Srm = (-2)rr! Sm+12(12 \(\xi\)2).
[28] Hecke, Math. Ann. 12 pp 274– (1922)
[29] Pekeris, Proc. Natl. Acad. Sci. U.S. 41 pp 661– (1955)
[30] The subscripts in \(\lambda\)mr occur in the reversed order to that in the notation of Wang Chang and Uhlenbeck9.
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