Abstract
Existence problems for the Boltzmann equation constitute a main area of research within the kinetic theory of gases and transport theory. The present paper considers the spatially periodic case with L1 initial data. The main result is that the Loeb subsolutions obtained in a preceding paper are shown to be true solutions. The proof relies on the observation that monotone entropy and finite energy imply Loeb integrability of non-standard approximate solutions, and uses estimates from the proof of the H-theorem. Two aspects of the continuity of the solutions are also considered.
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References
R. M. Anderson, A non-standard representation for Brownian motion and Ito integration, Israel J. of Math. 25 (1976), 15–46.
L. Arkeryd, On the Boltzmann equation, Arch. Rational Mech. Anal. 45 (1972), 1–34.
L. Arkeryd, A non-standard approach to the Boltzmann equation, Arch. Rational Mech. Anal. 77 (1981), 1–10.
C. Cercignani, Theory and application of the Boltzmann equation, Academic Press (1975).
N. J. Cutland, Internal controls and relaxed controls, J. London Math. Soc. 27 (1983), 130–140.
N. J. Cutland, NonStandard measure theory and its applications, Bull. London Math. Soc. 15 (1983), 529–589.
P. A. Loeb, Conversion from non-standard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113–122.
C. Truesdell & R. G. Muncaster, Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas, Academic Press (1980).
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Arkeryd, L. Loeb solutions of the boltzmann equation. Arch. Rational Mech. Anal. 86, 85–97 (1984). https://doi.org/10.1007/BF00280649
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DOI: https://doi.org/10.1007/BF00280649