Hermite basis diagonalization for the non-cutoff radially symmetric linearized Boltzmann operator

N. Lerner[1]; Y. Morimoto[2]; K. Pravda-Starov[3]; C.-J. Xu[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

Séminaire Laurent Schwartz — EDP et applications (2011-2012)

  • Volume: 2011-2012, page 1-10
  • ISSN: 2266-0607

Abstract

top
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].

How to cite

top

Lerner, N., et al. "Hermite basis diagonalization for the non-cutoff radially symmetric linearized Boltzmann operator." Séminaire Laurent Schwartz — EDP et applications 2011-2012 (2011-2012): 1-10. <http://eudml.org/doc/251161>.

@article{Lerner2011-2012,
abstract = {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].},
affiliation = {Institut de Mathématiques de Jussieu Université Pierre et Marie Curie (Paris VI) 4 Place Jussieu 75252 Paris cedex 05 France; Graduate School of Human and Environmental Studies Kyoto University Kyoto 606-8501 Japan; Université de Cergy-Pontoise CNRS UMR 8088 Département de Mathématiques 95000 Cergy-Pontoise France; 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},
author = {Lerner, N., Morimoto, Y., Pravda-Starov, K., Xu, C.-J.},
journal = {Séminaire Laurent Schwartz — EDP et applications},
keywords = {non-cutoff Boltzmann equation; non-cutoff Kac equation; spectral analysis; microlocal analysis; harmonic oscillator.},
language = {eng},
pages = {1-10},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Hermite basis diagonalization for the non-cutoff radially symmetric linearized Boltzmann operator},
url = {http://eudml.org/doc/251161},
volume = {2011-2012},
year = {2011-2012},
}

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
PY - 2011-2012
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2011-2012
SP - 1
EP - 10
AB - 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].
LA - eng
KW - non-cutoff Boltzmann equation; non-cutoff Kac equation; spectral analysis; microlocal analysis; harmonic oscillator.
UR - http://eudml.org/doc/251161
ER -

References

top
  1. Alexandre, R.: Remarks on 3D Boltzmann linear operator without cutoff. Transport Theory Statist. Phys. 28-5, 433Ð473 (1999). Zbl0939.35147MR1705619
  2. R. Alexandre, L. Desvillettes, C. Villani, B. Wennberg, Entropy dissipation and long-range interactions, Arch. Rational Mech. Anal. 152 (2000) 327-355. Zbl0968.76076MR1765272
  3. R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, T.Yang, Uncertainty principle and kinetic equations, J. Funct. Anal. 255 (2008) 2013-2066. Zbl1166.35038MR2462585
  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. Zbl1230.35082MR2795331
  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. Zbl1232.35110MR2863853
  6. C. Cercignani, The Boltzmann Equation and its Applications, Applied Mathematical Sciences, vol. 67 (1988), Springer-Verlag, New York Zbl0646.76001MR1313028
  7. L. Desvillettes, About the regularization properties of the non cut-off Kac equation, Comm. Math. Phys. 168 (1995) 417-440. Zbl0827.76081MR1324404
  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. Zbl1159.76044MR2465814
  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. Zbl1248.35140MR2784329
  10. L. Hörmander, The analysis of linear partial differential operators, vol. I–IV, (1985) Springer Verlag. Zbl0601.35001
  11. L. Hörmander, Symplectic classification of quadratic forms and general Mehler formulas, Math. Z. 219, 3 (1995) 413–449. Zbl0829.35150MR1339714
  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. Zbl1194.35089MR2556716
  13. N. Lerner, Metrics on the phase space and non-selfadjoint pseudodifferential operators, (2010) Pseudo-Differential Operators. Theory and Applications, Birkhäuser Verlag, Basel. Zbl1186.47001MR2599384
  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. Zbl1251.35064MR2876831
  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. Zbl1319.35156
  16. Y. Morimoto, C.-J. Xu, Hypoellipticity for a class of kinetic equations, J. Math. Kyoto Univ. 47, no. 1 (2007), 129-152. Zbl1146.35027MR2359105
  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. Zbl1175.35024MR2523694
  18. C. Mouhot, Explicit coercivity estimates for the Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006) 1321-1348. Zbl1101.76053MR2254617
  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. Zbl05163855MR2322149
  20. Y.P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. I, Comm. Pure Appl. Math. 27 (1974) 407-428. Zbl0304.45010MR636407
  21. Y.P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. II, Comm. Pure Appl. Math. 27 (1974) 559-581. Zbl0304.45010MR636407
  22. A. Unterberger, Oscillateur harmonique et opérateurs pseudodifférentiels, Ann. Institut Fourier, 29 (3) (1979), 201-221. Zbl0396.47027MR552965
  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. Zbl1170.82369MR1942465
  24. B. Wennberg, Regularity in the Boltzmann equation and the Radon transform, Comm. Partial Differential Equations, 19 (1994) 2057-2074. Zbl0818.35128MR1301182

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.