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Comptes Rendus Mathématique
Volume 348, n° 15-16
pages 867-871 (août 2010)
Doi : 10.1016/j.crma.2010.07.008
Received : 8 December 2009 ;  accepted : 8 July 2010
Global well-posedness theory for the spatially inhomogeneous Boltzmann equation without angular cutoff
Existence globale pour l'équation de Boltzmann sans troncature
 

Radjesvarane Alexandre a , Y. Morimoto b , S. Ukai c , Chao-Jiang Xu d , T. Yang e
a École navale, IRENAV, BRCM Brest, cc 600, 29240 Brest, France 
b Kyoto University, Japan 
c 17-26 Iwasaki-cho, Hodogaya-ku, Yokohama, Japan 
d Université de Rouen, France and Wuhan University, China 
e City University, Hong Kong 

Abstract

We present the first global well-posedness result for the Boltzmann equation without angular cutoff in the framework of weighted Sobolev spaces, in a close to equilibrium framework, and for Maxwellian molecules. These solutions become smooth for any positive time. An important ingredient of the proof rests on the introduction of a new norm, encoding both the singularity and the dissipation properties of the linearized collision operator.

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Résumé

Nous présentons le premier résultat d'existence globale pour l'équation de Boltzmann sans troncature angulaire, dans le cadre des espaces de Sobolev à poids, dans un cadre proche de l'équilibre, et pour des molécules maxwelliennes. Ces solutions devienent régulières pour tout temps positif. Un point important de la preuve consiste en l'introduction d'une nouvelle norme adaptée à la singularité et aux propriétés de dissipation de l'opérateur de collision linéarisé.

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© 2010  Published by Elsevier Masson SAS de la part de Académie des sciences.
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