Existence of local solutions for the Boltzmann equation without angular cutoff

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

doi:10.1016/j.crma.2009.09.015

Ordinary Differential Equations

Existence of local solutions for the Boltzmann equation without angular cutoff
[Existence locale pour l'équation de Boltzmann sans troncature]

Présenté par : Pierre-Louis Lions

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

¹ École navale, IRENAV, BRCM Brest, cc 600, 29240 Brest, France
² Kyoto University, Japan
³ Yokohama, Japan
⁴ Université de Rouen, France
⁵ City University, Hong Kong

Comptes Rendus. Mathématique, Volume 347 (2009) no. 21-22, pp. 1237-1242.

Résumé
Abstract

Nous considérons l'équation de Boltzmann inhomogène sans hypothèse de troncature angulaire. Nous montrons l'existence de solutions locales classiques ainsi que leur unicité pour le problème de Cauchy, dans une classe de fonctions exponentiellement décroissantes du type Maxwellian, relativement à la variable de vitesse.

We consider the spatially inhomogeneous Boltzmann equation without angular cutoff. We prove the existence and uniqueness of local classical solutions to the Cauchy problem, in the function space with Maxwellian type exponential decay with respect to the velocity variable.

Reçu le : 2009-02-04
Accepté le : 2009-09-01
Publié le : 2009-10-01

DOI : 10.1016/j.crma.2009.09.015

Affiliations des auteurs :

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

¹ École navale, IRENAV, BRCM Brest, cc 600, 29240 Brest, France
² Kyoto University, Japan
³ Yokohama, Japan
⁴ Université de Rouen, France
⁵ City University, Hong Kong

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     author = {Radjesvarane Alexandre and Yoshinori Morimoto and Seiji Ukai and Chao-Jiang Xu and Tong Yang},
     title = {Existence of local solutions for the {Boltzmann} equation without angular cutoff},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1237--1242},
     publisher = {Elsevier},
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     number = {21-22},
     year = {2009},
     doi = {10.1016/j.crma.2009.09.015},
     language = {en},
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Radjesvarane Alexandre; Yoshinori Morimoto; Seiji Ukai; Chao-Jiang Xu; Tong Yang. Existence of local solutions for the Boltzmann equation without angular cutoff. Comptes Rendus. Mathématique, Volume 347 (2009) no. 21-22, pp. 1237-1242. doi : 10.1016/j.crma.2009.09.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.09.015/

Bibliographie
Cité par

[1] R. Alexandre Some solutions of the Boltzmann equation without angular cutoff, J. Statist. Phys., Volume 104 (2001) no. 1–2, pp. 327-358

[2] R. Alexandre; Y. Morimoto; S. Ukai; C.-J. Xu; T. Yang Regularity of solutions for the Boltzmann equation without angular cutoff, C. R. Acad. Sci. Paris, Ser. I, Volume 347 (2009) no. 13–14, pp. 747-752

[3] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang, Regularizing effect and local existence for non-cutoff Boltzmann equation, preprint, 2009, available via | HAL

[4] R. Alexandre; C. Villani On the Boltzmann equation for long-range interaction, Comm. Pure Appl. Math., Volume 55 (2002), pp. 30-70

[5] S. Ukai Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff, Japan J. Appl. Math., Volume 1 (1984) no. 1, pp. 141-156

[6] C. Villani On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal., Volume 143 (1998), pp. 273-307

[7] C. Villani A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, vol. I, North-Holland, Amsterdam, 2002, pp. 71-305

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