Abstract
An abstract form of the spatially non-homogeneous Boltzmann equation is derived which includes the usual, more concrete form for any kind of potential, hard or soft, with finite cutoff. It is assumed that the corresponding “gas” is confined to a bounded domain by some sort of reflection law. The problem then considered is the corresponding initial-boundary value problem, locally in time.
Two proofs of existence are given. Both are constructive, and the first, at least, provides two sequences, one converging to the solution from above, the other from below, thus producing, at the same time as existence, approximations to the solution and error bounds for the approximation.
The solution is found within a space of functions bounded by a multiple of a Maxwellian, and, in this space, uniqueness is also proved.
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References
Grad, H.: Principles of the kinetic theory of gases. Handb. Phys., Vol. 12 (ed. S. Flügge). Berlin-Göttingen-Heidelberg: Springer 1958
Riesz, F., Sz-Nagy, B.: Functional analysis. New York: Frederick Ungar Publishing Company 1955
Schnute, J., Shinbrot, M.: Kinetic theorey and boundary conditions for fluids. Can. J. Math.25, 1183–1215 (1973)
Schnute, J.: Entropy and kinetic theory for a confined gas. Can. J. Math.27, 1271–1315 (1975)
Arkeryd, L.: On the Boltzmann equation. Part II. The full initial value problem. Arch. Rat. Mech. Anal.45, 17–34 (1972)
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Communicated by J. L. Lebowitz
Research supported, in part, by the National Research Council of Canada (NRC A8560)
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Kaniel, S., Shinbrot, M. The Boltzmann equation. Commun.Math. Phys. 58, 65–84 (1978). https://doi.org/10.1007/BF01624788
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DOI: https://doi.org/10.1007/BF01624788