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Comptes Rendus Mathématique
Volume 334, n° 7
pages 557-562 (2002)
Doi : S1631-073X(02)02302-6
Received : 23 October 2001 ;  accepted : 4 February 2002
Velocity averaging in L1 for the transport equation
Moyennisation en vitesse dans L1 pour l'équation de transport
 

François Golse a, b , Laure Saint-Raymond b
a Institut Universitaire de France & École normale supérieure, DMA, 45, rue d'Ulm, 75005 Paris, France 
b Université Paris 6, Laboratoire d'analyse numérique, 175, rue du Chevaleret, 75013 Paris, France 

Note presented by Pierre-Louis Lions

Abstract

A new result of L1-compactness for velocity averages of solutions to the transport equation is stated and proved in this Note. This result, proved by a new interpolation argument, extends to the case of any space dimension Lemma 8 of Golse-Lions-Perthame-Sentis [J. Funct. Anal. 76 (1988) 110-125], proved there in space dimension 1 only. This is a key argument in the proof of the hydrodynamic limits of the Boltzmann or BGK equations to the incompressible Euler or Navier-Stokes equations. To cite this article: F. Golse, L. Saint-Raymond, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 557-562.

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Résumé

On énonce et démontre dans cette Note un nouveau résultat de compacité dans L1 pour les moyennes en vitesse des solutions de l'équation de transport. Ce résultat, établi par un nouvel argument d'interpolation, généralise à toute dimension d'espace le Lemme 8 de Golse-Lions-Perthame-Sentis [J. Funct. Anal. 76 (1988) 110-125], qui n'était jusqu'ici connu qu'en dimension 1 d'espace. C'est un point crucial dans les preuves des limites hydrodynamiques des équations de Boltzmann ou de BGK vers les équations de Navier-Stokes. Pour citer cet article : F. Golse, L. Saint-Raymond, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 557-562.

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


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