Partial Differential Equations
Regularity of solutions for the Boltzmann equation without angular cutoff
Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 747-752.

We prove that classical solution of the spatially inhomogeneous and angular non-cutoff Boltzmann equation is C 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.

Nous considérons l'équation de Boltzmann inhomogène sans hypothèse de troncature angulaire. Nous montrons que toute solution classique est C 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.

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Published online:
DOI: 10.1016/j.crma.2009.04.025
Alexandre, Radjesvarane 1; Morimoto, Yoshinore 2; Ukai, Seiji 3; Xu, Chao-Jiang 4; Yang, Tong 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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Alexandre, Radjesvarane; Morimoto, Yoshinore; Ukai, Seiji; Xu, Chao-Jiang; Yang, Tong. 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. http://www.numdam.org/articles/10.1016/j.crma.2009.04.025/

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