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

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2007-09-20
Accepted: 2007-10-16
Published online: 2007-11-26

DOI: 10.1016/j.crma.2007.10.032

Author's affiliations:

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

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     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},
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     year = {2007},
     doi = {10.1016/j.crma.2007.10.032},
     language = {en},
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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/

References
Cited by

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

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[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

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