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Arkeryd, Leif

Loeb solutions of the Boltzmann equation. (English) Zbl 0598.76091

Arch. Ration. Mech. Anal. 86, 85-97 (1984).
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 \(L^ 1\) initial data. The main result is that the Loeb subsolutions [obtained by the author, ibid. 77, 1-10 (1981; Zbl 0547.76084)] 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.

Cited in 18 Documents

MSC:

76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82B40 Kinetic theory of gases in equilibrium statistical mechanics
45K05 Integro-partial differential equations

Keywords:

Existence problems; spatially periodic case; Loeb subsolutions; monotone entropy; finite energy; Loeb integrability; non-standard approximate solutions; H-theorem; continuity

Citations:

Zbl 0547.76084
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\textit{L. Arkeryd}, Arch. Ration. Mech. Anal. 86, 85--97 (1984; Zbl 0598.76091)
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References:

[1] R. M. Anderson, A non-standard representation for Brownian motion and Ito integration, Israel J. of Math. 25 (1976), 15-46. · Zbl 0353.60052
[2] L. Arkeryd, On the Boltzmann equation, Arch. Rational Mech. Anal. 45 (1972), 1-34. · Zbl 0245.76060
[3] L. Arkeryd, A non-standard approach to the Boltzmann equation, Arch. Rational Mech. Anal. 77 (1981), 1-10. · Zbl 0547.76084
[4] C. Cercignani, Theory and application of the Boltzmann equation, Academic Press (1975). · Zbl 0403.76065
[5] N. J. Cutland, Internal controls and relaxed controls, J. London Math. Soc. 27 (1983), 130-140. · Zbl 0495.49002
[6] N. J. Cutland, NonStandard measure theory and its applications, Bull. London Math. Soc. 15 (1983), 529-589. · Zbl 0529.28009
[7] P. A. Loeb, Conversion from non-standard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113-122. · Zbl 0312.28004
[8] C. Truesdell & R. G. Muncaster, Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas, Academic Press (1980).
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