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Comptes Rendus Mathématique
Volume 342, n° 11
pages 843-848 (juin 2006)
Doi : 10.1016/j.crma.2006.04.005
Received : 12 November 2005 ;  accepted : 4 April 2006
Strong solutions of the Boltzmann equation in one spatial dimension
Les solutions globales de lʼéquation de Boltzmann dans la géometrie uni-dimensionnelle
 

Andrei Biryuk 1 , Walter Craig 2 , Vladislav Panferov 1
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada 

Abstract

For the Boltzmann equation, the setting of a narrow shock tube implies that solutions   depend upon  , however they have one-dimensional spatial dependence. This Note discusses the case in which solutions are periodic in x , with controlled total energy and entropy, and such that the macroscopic density determined by the initial data is bounded. Our principal result is that the macroscopic density then remains bounded at all subsequent times, that is, this data gives rise to strong solutions which exist globally in time. Through a weak/strong uniqueness principle, these solutions are unique among the class of dissipative solutions. Additionally, we show that the flow of the Boltzmann equation propagates the moments in   and derivatives in both   and   of the solution  . Our main theorems are valid for Boltzmann collision kernels which are bounded, and which have a relative velocity cutoff. The proofs depend upon a new averaging property of the collision operator and integral inequalities based in turn on entropy and on the Bony functional. To cite this article: A. Biryuk et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

The full text of this article is available in PDF format.
Résumé

Dans un domaine qui représente un tube à choc, les solutions   de lʼéquation de Boltzmann dépendent de   mais elles ne dépendent que de  . Dans cette Note, on considère le cas de solutions périodiques en  , dont la densité macroscopique initiale est finie, et lʼénergie et lʼentropie totales sont bornées par une certaine constante C. Le résultat principal est que la densité macroscopique de la solution reste bornée pour tout temps  , cʼest-à-dire, les conditions initiales donnent lieu à des solutions fortes qui existent globalement en temps. Le résultat implique lʼunicité de nos solutions dans la classe de solutions dissipatives faibles. Ces solutions   conservent les propriétés de régularité en x et en v , et les moments finis en v . Les théorèmes principaux sont valables pour des noyaux de collision de Boltzmann bornés, et avec une troncature de vitesse relative. Les démonstrations dépendent dʼune propriété nouvelle de moyennisation de lʼopérateur de collision, et de deux inégalités intégrales basées sur lʼentropie et sur la fonctionnelle de Bony. Pour citer cet article : A. Biryuk et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

The full text of this article is available in PDF format.
1  Research supported in part by a CRC Postdoctoral Fellowship at McMaster University.
2  Research supported in part by the Canada Research Chairs Program and the NSERC through grant # 238452-01.


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