Hermite basis diagonalization for the non-cutoff radially symmetric linearized Boltzmann operator
Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 23, 10 p.

We provide some new explicit expressions for the linearized non-cutoff radially symmetric Boltzmann operator with Maxwellian molecules, proving that this operator is a simple function of the standard harmonic oscillator. A detailed article is available on arXiv [15].

DOI : 10.5802/slsedp.18
Lerner, N. 1 ; Morimoto, Y. 2 ; Pravda-Starov, K. 3 ; Xu, C.-J. 4

1 Institut de Mathématiques de Jussieu Université Pierre et Marie Curie (Paris VI) 4 Place Jussieu 75252 Paris cedex 05 France
2 Graduate School of Human and Environmental Studies Kyoto University Kyoto 606-8501 Japan
3 Université de Cergy-Pontoise CNRS UMR 8088 Département de Mathématiques 95000 Cergy-Pontoise France
4 School of Mathematics Wuhan university 430072 Wuhan P.R. China and Université de Rouen CNRS UMR 6085 Département de Mathématiques 76801 Saint-Etienne du Rouvray France
@article{SLSEDP_2011-2012____A23_0,
     author = {Lerner, N. and Morimoto, Y. and Pravda-Starov, K. and Xu, C.-J.},
     title = {Hermite basis diagonalization for the non-cutoff radially symmetric linearized {Boltzmann} operator},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:23},
     pages = {1--10},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2011-2012},
     doi = {10.5802/slsedp.18},
     language = {en},
     url = {http://www.numdam.org./articles/10.5802/slsedp.18/}
}
TY  - JOUR
AU  - Lerner, N.
AU  - Morimoto, Y.
AU  - Pravda-Starov, K.
AU  - Xu, C.-J.
TI  - Hermite basis diagonalization for the non-cutoff radially symmetric linearized Boltzmann operator
JO  - Séminaire Laurent Schwartz — EDP et applications
N1  - talk:23
PY  - 2011-2012
SP  - 1
EP  - 10
PB  - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://www.numdam.org./articles/10.5802/slsedp.18/
DO  - 10.5802/slsedp.18
LA  - en
ID  - SLSEDP_2011-2012____A23_0
ER  - 
%0 Journal Article
%A Lerner, N.
%A Morimoto, Y.
%A Pravda-Starov, K.
%A Xu, C.-J.
%T Hermite basis diagonalization for the non-cutoff radially symmetric linearized Boltzmann operator
%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:23
%D 2011-2012
%P 1-10
%I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
%U http://www.numdam.org./articles/10.5802/slsedp.18/
%R 10.5802/slsedp.18
%G en
%F SLSEDP_2011-2012____A23_0
Lerner, N.; Morimoto, Y.; Pravda-Starov, K.; Xu, C.-J. Hermite basis diagonalization for the non-cutoff radially symmetric linearized Boltzmann operator. Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 23, 10 p. doi : 10.5802/slsedp.18. http://www.numdam.org./articles/10.5802/slsedp.18/

[1] Alexandre, R.: Remarks on 3D Boltzmann linear operator without cutoff. Transport Theory Statist. Phys. 28-5, 433Ð473 (1999). | MR | Zbl

[2] R. Alexandre, L. Desvillettes, C. Villani, B. Wennberg, Entropy dissipation and long-range interactions, Arch. Rational Mech. Anal. 152 (2000) 327-355. | MR | Zbl

[3] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T.Yang, Uncertainty principle and kinetic equations, J. Funct. Anal. 255 (2008) 2013-2066. | MR | Zbl

[4] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff, Comm. Math. Phys. 304 (2011), 513-581. | MR | Zbl

[5] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T. Yang, Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential, J. Funct. Anal., Vol. 262 (2012), 915-1010. | MR | Zbl

[6] C. Cercignani, The Boltzmann Equation and its Applications, Applied Mathematical Sciences, vol. 67 (1988), Springer-Verlag, New York | MR | Zbl

[7] L. Desvillettes, About the regularization properties of the non cut-off Kac equation, Comm. Math. Phys. 168 (1995) 417-440. | MR | Zbl

[8] L. Desvillettes, G. Furioli, E. Terraneo, Propagation of Gevrey regularity for solutions of Boltzmann equation for Maxwellian molecules, Trans. Amer. Math. Soc. 361 (2009) 1731–1747. | MR | Zbl

[9] P.-T. Gressman, R.-M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc. 24 (2011), 771-847. | MR | Zbl

[10] L. Hörmander, The analysis of linear partial differential operators, vol. I–IV, (1985) Springer Verlag. | Zbl

[11] L. Hörmander, Symplectic classification of quadratic forms and general Mehler formulas, Math. Z. 219, 3 (1995) 413–449. | EuDML | MR | Zbl

[12] N. Lekrine, C.-J. Xu, Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac’s equation, Kinetic and Related Models, 2 (2009), 647-666. | MR | Zbl

[13] N. Lerner, Metrics on the phase space and non-selfadjoint pseudodifferential operators, (2010) Pseudo-Differential Operators. Theory and Applications, Birkhäuser Verlag, Basel. | MR | Zbl

[14] N. Lerner, Y. Morimoto, K. Pravda-Starov, Hypoelliptic estimates for a linear model of the Boltzmann equation without angular cutoff, (2011) to appear in Comm. Partial Differential Equations, arXiv:1012.4915. | MR | Zbl

[15] N. Lerner, Y. Morimoto, K. Pravda-Starov, C.-J. Xu, Diagonalization of the Linearized Non-Cutoff Radially Symmetric Boltzmann Operator, preprint, arXiv:1111.0423v1.

[16] Y. Morimoto, C.-J. Xu, Hypoellipticity for a class of kinetic equations, J. Math. Kyoto Univ. 47, no. 1 (2007), 129-152. | MR | Zbl

[17] Y. Morimoto, C.-J. Xu, Ultra-analytic effect of Cauchy problem for a class of kinetic equations, J. Differential Equations, 247 (2009) 596-617. | MR | Zbl

[18] C. Mouhot, Explicit coercivity estimates for the Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006) 1321-1348. | MR | Zbl

[19] C. Mouhot, R.M. Strain, Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff, J. Math. Pures Appl. (9) 87 (2007), no. 5, 515-535. | MR

[20] Y.P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. I, Comm. Pure Appl. Math. 27 (1974) 407-428. | MR | Zbl

[21] Y.P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. II, Comm. Pure Appl. Math. 27 (1974) 559-581. | MR | Zbl

[22] A. Unterberger, Oscillateur harmonique et opérateurs pseudodifférentiels, Ann. Institut Fourier, 29 (3) (1979), 201-221. | EuDML | Numdam | MR | Zbl

[23] C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, vol. I, North-Holland, Amsterdam (2002) 71-305. | MR | Zbl

[24] B. Wennberg, Regularity in the Boltzmann equation and the Radon transform, Comm. Partial Differential Equations, 19 (1994) 2057-2074. | MR | Zbl

Cité par Sources :