Regularity of solutions for the Boltzmann equation without angular cutoff

Radjesvarane Alexandre; Yoshinore Morimoto; Seiji Ukai; Chao-Jiang Xu; Tong Yang

doi:10.1016/j.crma.2009.04.025

Partial Differential Equations

Regularity of solutions for the Boltzmann equation without angular cutoff
[Régularité de solutions pour l'équation de Boltzmann sans angulaire cutoff]

Présenté par : Pierre-Louis Lions

Radjesvarane Alexandre ¹ ; Yoshinore Morimoto ² ; Seiji Ukai ³ ; Chao-Jiang Xu ⁴ ; Tong Yang ⁵

¹ École Navale, BRCM Brest, 29240 Brest, France
² Kyoto University, Japan
³ Yokohama, Japan
⁴ Université de Rouen, France
⁵ City University, Hong Kong

Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 747-752.

Résumé
Abstract

Nous considérons l'équation de Boltzmann inhomogène sans hypothèse de troncature angulaire. Nous montrons que toute solution classique est $C^{\infty}$ par rapport à toutes les variables, localement en temps et en espace. La preuve s'appuie sur un principe d'incertitude généralisé, des bornes fonctionnelles précisées sur l'opérateur de collision, une estimation de coercivité, ainsi qu'une analyse de commutateurs avec cet opérateur, avec un choix approprié d'opérateurs pseudo-différentiels.

We prove that classical solution of the spatially inhomogeneous and angular non-cutoff Boltzmann equation is $C^{\infty}$ with respect to all variables, locally in the space and time variables. The proof relies on a generalized uncertainty principle, some improved upper bound and coercivity estimates on the nonlinear collision operator, and some subtle analysis on the commutators between the collision operators and some appropriately chosen pseudo-differential operators.

Reçu le : 2009-02-04
Accepté le : 2009-04-21
Publié le : 2009-05-23

DOI : 10.1016/j.crma.2009.04.025

Affiliations des auteurs :

Radjesvarane Alexandre ¹ ; Yoshinore Morimoto ² ; Seiji Ukai ³ ; Chao-Jiang Xu ⁴ ; Tong Yang ⁵

¹ École Navale, BRCM Brest, 29240 Brest, France
² Kyoto University, Japan
³ Yokohama, Japan
⁴ Université de Rouen, France
⁵ City University, Hong Kong

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     author = {Radjesvarane Alexandre and Yoshinore Morimoto and Seiji Ukai and Chao-Jiang Xu and Tong Yang},
     title = {Regularity of solutions for the {Boltzmann} equation without angular cutoff},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {747--752},
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     volume = {347},
     number = {13-14},
     year = {2009},
     doi = {10.1016/j.crma.2009.04.025},
     language = {en},
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Radjesvarane Alexandre; Yoshinore Morimoto; Seiji Ukai; Chao-Jiang Xu; Tong Yang. Regularity of solutions for the Boltzmann equation without angular cutoff. Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 747-752. doi : 10.1016/j.crma.2009.04.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.025/

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