Regularity of solutions for the Boltzmann equation without angular cutoff

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

doi:10.1016/j.crma.2009.04.025

Partial Differential Equations

Regularity of solutions for the Boltzmann equation without angular cutoff
[Régularité de solutions pour l'équation de Boltzmann sans angulaire cutoff]

Présenté par : Pierre-Louis Lions

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

¹ École Navale, BRCM Brest, 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. 13-14, pp. 747-752.

Résumé
Abstract

Nous considérons l'équation de Boltzmann inhomogène sans hypothèse de troncature angulaire. Nous montrons que toute solution classique est $C^{\infty}$ 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 $C^{\infty}$ 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.

Reçu le : 2009-02-04
Accepté le : 2009-04-21
Publié le : 2009-05-23

DOI : 10.1016/j.crma.2009.04.025

Affiliations des auteurs :

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

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

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     title = {Regularity of solutions for the {Boltzmann} equation without angular cutoff},
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Radjesvarane Alexandre; Yoshinore Morimoto; Seiji Ukai; Chao-Jiang Xu; Tong Yang. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.025/

Bibliographie
Cité par

[1] R. Alexandre Some solutions of the Boltzmann equation without angular cutoff, J. Statist. Phys., Volume 104 (2001), pp. 327-358

[2] R. Alexandre Integral estimates for linear singular operator linked with Boltzmann operator. Part I: Small singularities $0 < ν < 1$ , Indiana Univ. Math. J., Volume 55 (2006), pp. 1975-2021

[3] R. Alexandre; L. Desvillettes; C. Villani; B. Wennberg Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., Volume 152 (2000), pp. 327-355

[4] R. Alexandre; Y. Morimoto; S. Ukai; C.-J. Xu; T. Yang 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] R. Alexandre; M. Safadi 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] L. Boudin; L. Desvillettes On the singularities of the global small solutions of the full Boltzmann equation, Monatsh. Math., Volume 131 (2000), pp. 91-108

[8] C. Cercignani The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, vol. 67, Springer-Verlag, 1988

[9] Y. Chen 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] L. Desvillettes About the regularization properties of the non-cut-off Kac equation, Comm. Math. Phys., Volume 168 (1995), pp. 417-440

[12] L. Desvillettes 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] L. Desvillettes; B. Wennberg Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff, Comm. Partial Differential Equations, Volume 29 (2004), pp. 133-155

[14] Z.H. Huo; Y. Morimoto; S. Ukai; T. Yang Regularity of entropy solutions for spatially homogeneous Boltzmann equation without angular cutoff, Kinetic Related Models, Volume 1 (2008), pp. 453-489

[15] 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

[16] 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

Cyril Imbert; Luis Silvestre Global regularity estimates for the Boltzmann equation without cut-off, Journal of the American Mathematical Society, Volume 35 (2021) no. 3, p. 625 | DOI:10.1090/jams/986
Christopher Henderson; Stanley Snelson; Andrei Tarfulea Local existence, lower mass bounds, and a new continuation criterion for the Landau equation, Journal of Differential Equations, Volume 266 (2019) no. 2-3, p. 1536 | DOI:10.1016/j.jde.2018.08.005
Luis Silvestre A New Regularization Mechanism for the Boltzmann Equation Without Cut-Off, Communications in Mathematical Physics, Volume 348 (2016) no. 1, p. 69 | DOI:10.1007/s00220-016-2757-x
R. Alexandre; Y. Morimoto; S. Ukai; C. -J. Xu; T. Yang Global Existence and Full Regularity of the Boltzmann Equation Without Angular Cutoff, Communications in Mathematical Physics, Volume 304 (2011) no. 2, p. 513 | DOI:10.1007/s00220-011-1242-9

Cité par 4 documents. Sources : Crossref

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