Rigorous validity of the  Boltzmann equation for a thin layer of a rarefied gas

Raffaele Esposito; Mario Pulvirenti

doi:10.3934/krm.2010.3.281

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2010, Volume 3, Issue 2: 281-297. Doi: 10.3934/krm.2010.3.281

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Rigorous validity of the Boltzmann equation for a thin layer of a rarefied gas

Raffaele Esposito^1, and
Mario Pulvirenti^2,

1.
Dipartimento di Matematica pura ed Applicata, Università dell’Aquila, Via Vetoio - Coppito, L’Aquila, 67100
2.
Dipartimento di Matematica Guido Castelnuovo, Università La Sapienza, P.e Aldo Moro, 2, 67100 Roma

Received: December 31, 2009

Revised: January 31, 2010

Published: May 2010

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Abstract

We consider a thin layer of a rarified gas modeled by a large hard-sphere system and show that, as long as the thickness of the layer is much larger than the interaction length, the limiting behavior is described, at least for short times, by a Boltzmann equation with two-dimensional position variable and three-dimensional velocity. By the analysis of the Lorentz gas we argue that, if the thickness of the layer is of the same order of the interaction length, this is not the case.

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Mathematics Subject Classification: Primary: 76P05, 82C40; Secondary: 76A20.

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