Uncertainty principle and regularity for Boltzmann type equations

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

doi:10.1016/j.crma.2007.10.032

Partial Differential Equations/Harmonic Analysis

Uncertainty principle and regularity for Boltzmann type equations
[Principe d'incertitude et régularité pour des équations de type Boltzmann]

Présenté par : Jean-Michel Bony

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

¹ IRENAv, French Naval Academy, 29240 Brest-Lanvéoc, France
² Kyoto University, Japan
³ City University, 83, Tat Chee Avenue, Kowloon, Hong Kong
⁴ Université de Rouen, avenue de l'université, technopôle du Madrillet, 76801 Saint Étienne du Rouvray cedex, France
⁵ Shanghai Jiao Tong University, PR China

Comptes Rendus. Mathématique, Volume 345 (2007) no. 12, pp. 673-677.

Résumé
Abstract

Nous montrons une version généralisée du principe d'incertitude, et l'appliquons à l'étude de propriétés de régularisation de solutions d'équations cinétiques. En particulier, nous considérons les versions linéarisée et non linéaire de l'équation de Boltzmann, sans faire l'hypothèse de troncature angulaire de Grad.

We give a generalized version of uncertainty principle, and apply it to the study of regularization properties of solutions to kinetic equations. In particular, both linearized and nonlinear space inhomogeneous Boltzmann equations without Grad's cutoff assumption are considered.

Reçu le : 2007-09-20
Accepté le : 2007-10-16
Publié le : 2007-11-26

DOI : 10.1016/j.crma.2007.10.032

Affiliations des auteurs :

Radjesvarane Alexandre ¹ ; Yoshinori Morimoto ² ; Seiji Ukai ³ ; Chao-Jiang Xu ⁴ ; Tong Yang ^{3, 5}

¹ IRENAv, French Naval Academy, 29240 Brest-Lanvéoc, France
² Kyoto University, Japan
³ City University, 83, Tat Chee Avenue, Kowloon, Hong Kong
⁴ Université de Rouen, avenue de l'université, technopôle du Madrillet, 76801 Saint Étienne du Rouvray cedex, France
⁵ Shanghai Jiao Tong University, PR China

@article{CRMATH_2007__345_12_673_0,
     author = {Radjesvarane Alexandre and Yoshinori Morimoto and Seiji Ukai and Chao-Jiang Xu and Tong Yang},
     title = {Uncertainty principle and regularity for {Boltzmann} type equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {673--677},
     publisher = {Elsevier},
     volume = {345},
     number = {12},
     year = {2007},
     doi = {10.1016/j.crma.2007.10.032},
     language = {en},
}

TY  - JOUR
AU  - Radjesvarane Alexandre
AU  - Yoshinori Morimoto
AU  - Seiji Ukai
AU  - Chao-Jiang Xu
AU  - Tong Yang
TI  - Uncertainty principle and regularity for Boltzmann type equations
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 673
EP  - 677
VL  - 345
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crma.2007.10.032
LA  - en
ID  - CRMATH_2007__345_12_673_0
ER  -

%0 Journal Article
%A Radjesvarane Alexandre
%A Yoshinori Morimoto
%A Seiji Ukai
%A Chao-Jiang Xu
%A Tong Yang
%T Uncertainty principle and regularity for Boltzmann type equations
%J Comptes Rendus. Mathématique
%D 2007
%P 673-677
%V 345
%N 12
%I Elsevier
%R 10.1016/j.crma.2007.10.032
%G en
%F CRMATH_2007__345_12_673_0

Radjesvarane Alexandre; Yoshinori Morimoto; Seiji Ukai; Chao-Jiang Xu; Tong Yang. Uncertainty principle and regularity for Boltzmann type equations. Comptes Rendus. Mathématique, Volume 345 (2007) no. 12, pp. 673-677. doi : 10.1016/j.crma.2007.10.032. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.032/

Bibliographie
Cité par

[1] R. Alexandre Integral estimates for linear singular operator linked with Boltzmann operator. Part I: small singularities $0 < ν < 1$ , Indiana Univ. J. Math., Volume 55–6 (2007)

[2] R. Alexandre, Y. Morimoto, S. Ukai, C.-Y. Xu, T. Yang, in preparation

[3] F. Bouchut Hypoelliptic regularity in kinetic equations, J. Math. Pure Appl., Volume 81 (2002), pp. 1135-1159

[4] C. Fefferman The uncertainty principle, Bull. Amer. Math. Soc., Volume 9 (1983), pp. 129-206

[5] C. Fefferman; D.H. Phong The uncertainty principle and sharp Gȧrding inequalities, Comm. Pure. Appl. Math., Volume 34 (1981), pp. 285-331

[6] Y. Morimoto The uncertainty principle and hypoelliptic operators, Publ. RIMS Kyoto Univ., Volume 23 (1987), pp. 955-964

[7] Y. Morimoto Estimates for degenerate Schrödinger operators and hypoellipticity for infinitely degenerate elliptic operators, J. Math. Kyoto Univ., Volume 32 (1992), pp. 333-372

[8] Y. Morimoto; T. Morioka The positivity of Schrödinger operators and the hypoellipticity of second order degenerate elliptic operators, Bull. Sci. Math., Volume 121 (1997), pp. 507-547

[9] Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff, preprint

[10] Y. Morimoto, C.-J. Xu, Hypoellipticity for a class of kinetic equations, J. Math. Kyoto. Univ. 47 (2007), in press

[11] C. Villani A review of mathematical topics in collisional kinetic theory (S. Friedlander; D. Serre, eds.), Handbook of Mathematical Fluid Dynamics, vol. I, North-Holland, Amsterdam, 2002, pp. 71-305

Cité par Sources :

Commentaires - Politique