[Régularité de solutions pour l'équation de Boltzmann sans angulaire cutoff]
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.
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.
Accepté le :
Publié le :
@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://archive.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://archive.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://archive.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, Tome 347 (2009) no. 13-14, pp. 747-752. doi : 10.1016/j.crma.2009.04.025. http://archive.numdam.org/articles/10.1016/j.crma.2009.04.025/
[1] Some solutions of the Boltzmann equation without angular cutoff, J. Statist. Phys., Volume 104 (2001), pp. 327-358
[2] Integral estimates for linear singular operator linked with Boltzmann operator. Part I: Small singularities , Indiana Univ. Math. J., Volume 55 (2006), pp. 1975-2021
[3] Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., Volume 152 (2000), pp. 327-355
[4] Uncertainty principle and kinetic equations, J. Funct. Anal., Volume 255 (2008), pp. 2013-2066
[5] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang, Regularizing effect and local existence for non-cutoff Boltzmann equation, preprint, 2008
[6] Littlewood Paley decomposition and regularity issues in Boltzmann equation homogeneous equations. I. Non-cutoff and Maxwell cases, Math. Models Methods Appl. Sci., Volume 15 (2005), pp. 907-920
[7] On the singularities of the global small solutions of the full Boltzmann equation, Monatsh. Math., Volume 131 (2000), pp. 91-108
[8] The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, vol. 67, Springer-Verlag, 1988
[9] Smoothness of classical solutions to the Vlasov–Poisson–Landau System, Kinetic Related Models, Volume 1 (2008) no. 3, pp. 369-386
[10] Y. Chen, L. Desvillettes, L. He, Smoothing effects for classical solutions of the full Landau equation, Arch. Ration. Mech. Anal., in press
[11] About the regularization properties of the non-cut-off Kac equation, Comm. Math. Phys., Volume 168 (1995), pp. 417-440
[12] Regularization properties of the 2-dimensional non-radially symmetric non-cutoff spatially homogeneous Boltzmann equation for Maxwellian molecules, Trans. Theory Stat. Phys., Volume 26 (1997), pp. 341-357
[13] Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff, Comm. Partial Differential Equations, Volume 29 (2004), pp. 133-155
[14] Regularity of entropy solutions for spatially homogeneous Boltzmann equation without angular cutoff, Kinetic Related Models, Volume 1 (2008), pp. 453-489
[15] Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff, Japan J. Appl. Math., Volume 1 (1984) no. 1, pp. 141-156
[16] A review of mathematical topics in collisional kinetic theory (Friedlander, S.; Serre, D., eds.), Handbook of Mathematical Fluid Dynamics, vol. I, North-Holland, Amsterdam, 2002, pp. 71-305
Cité par Sources :
