We prove that classical solution of the spatially inhomogeneous and angular non-cutoff Boltzmann equation is 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 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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@article{CRMATH_2009__347_13-14_747_0, author = {Alexandre, Radjesvarane and Morimoto, Yoshinore and Ukai, Seiji and Xu, Chao-Jiang and Yang, Tong}, title = {Regularity of solutions for the {Boltzmann} equation without angular cutoff}, journal = {Comptes Rendus. Math\'ematique}, pages = {747--752}, publisher = {Elsevier}, volume = {347}, number = {13-14}, year = {2009}, doi = {10.1016/j.crma.2009.04.025}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2009.04.025/} }
TY - JOUR AU - Alexandre, Radjesvarane AU - Morimoto, Yoshinore AU - Ukai, Seiji AU - Xu, Chao-Jiang AU - Yang, Tong TI - Regularity of solutions for the Boltzmann equation without angular cutoff JO - Comptes Rendus. Mathématique PY - 2009 SP - 747 EP - 752 VL - 347 IS - 13-14 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.04.025/ DO - 10.1016/j.crma.2009.04.025 LA - en ID - CRMATH_2009__347_13-14_747_0 ER -
%0 Journal Article %A Alexandre, Radjesvarane %A Morimoto, Yoshinore %A Ukai, Seiji %A Xu, Chao-Jiang %A Yang, Tong %T Regularity of solutions for the Boltzmann equation without angular cutoff %J Comptes Rendus. Mathématique %D 2009 %P 747-752 %V 347 %N 13-14 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.04.025/ %R 10.1016/j.crma.2009.04.025 %G en %F CRMATH_2009__347_13-14_747_0
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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