Trend to equilibrium and spectral localization properties for the linear Boltzmann equation

Daniel Han-Kwan[1]; Matthieu Léautaud[2]

  • [1] CNRS and École Polytechnique Centre de Mathématiques Laurent Schwartz UMR7640 F91128 Palaiseau cedex
  • [2] Université Paris Diderot Institut de Mathématiques de Jussieu Paris Rive Gauche Bâtiment Sophie Germain 75205 Paris Cedex 13 France

Séminaire Laurent Schwartz — EDP et applications (2013-2014)

  • page 1-15
  • ISSN: 2266-0607

Abstract

top
The aim of this note is to present the results from [11, 12], which deal with the linear Boltzmann equation, set in a bounded domain and in the presence of an external force. A specificity of these works is that the collision operator is allowed to be degenerate in the following two senses: (1) the associated collision kernel may vanish in a large subset of the phase space; (2) we do not assume that it is bounded below by a Maxwellian at infinity in velocity.We study:the large time behavior of solutions of the linear Boltzmann equation, by giving criteria (inspired from control theory) which ensure converge towards an equilibrium and when possible, convergence at an exponential rate [11] ;some properties of localization for the spectrum of the associated operator [12].

How to cite

top

Han-Kwan, Daniel, and Léautaud, Matthieu. "Trend to equilibrium and spectral localization properties for the linear Boltzmann equation." Séminaire Laurent Schwartz — EDP et applications (2013-2014): 1-15. <http://eudml.org/doc/275675>.

@article{Han2013-2014,
abstract = {The aim of this note is to present the results from [11, 12], which deal with the linear Boltzmann equation, set in a bounded domain and in the presence of an external force. A specificity of these works is that the collision operator is allowed to be degenerate in the following two senses: (1) the associated collision kernel may vanish in a large subset of the phase space; (2) we do not assume that it is bounded below by a Maxwellian at infinity in velocity.We study:the large time behavior of solutions of the linear Boltzmann equation, by giving criteria (inspired from control theory) which ensure converge towards an equilibrium and when possible, convergence at an exponential rate [11] ;some properties of localization for the spectrum of the associated operator [12].},
affiliation = {CNRS and École Polytechnique Centre de Mathématiques Laurent Schwartz UMR7640 F91128 Palaiseau cedex; Université Paris Diderot Institut de Mathématiques de Jussieu Paris Rive Gauche Bâtiment Sophie Germain 75205 Paris Cedex 13 France},
author = {Han-Kwan, Daniel, Léautaud, Matthieu},
journal = {Séminaire Laurent Schwartz — EDP et applications},
language = {eng},
pages = {1-15},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Trend to equilibrium and spectral localization properties for the linear Boltzmann equation},
url = {http://eudml.org/doc/275675},
year = {2013-2014},
}

TY - JOUR
AU - Han-Kwan, Daniel
AU - Léautaud, Matthieu
TI - Trend to equilibrium and spectral localization properties for the linear Boltzmann equation
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2013-2014
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
SP - 1
EP - 15
AB - The aim of this note is to present the results from [11, 12], which deal with the linear Boltzmann equation, set in a bounded domain and in the presence of an external force. A specificity of these works is that the collision operator is allowed to be degenerate in the following two senses: (1) the associated collision kernel may vanish in a large subset of the phase space; (2) we do not assume that it is bounded below by a Maxwellian at infinity in velocity.We study:the large time behavior of solutions of the linear Boltzmann equation, by giving criteria (inspired from control theory) which ensure converge towards an equilibrium and when possible, convergence at an exponential rate [11] ;some properties of localization for the spectrum of the associated operator [12].
LA - eng
UR - http://eudml.org/doc/275675
ER -

References

top
  1. C. Bardos, G. Lebeau, and J. Rauch. Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim., 30:1024–1065, 1992. Zbl0786.93009MR1178650
  2. É. Bernard and F. Salvarani. On the convergence to equilibrium for degenerate transport problems. Arch. Ration. Mech. Anal., 208(3):977–984, 2013. Zbl1282.35055MR3048598
  3. É. Bernard and F. Salvarani. On the exponential decay to equilibrium of the degenerate linear Boltzmann equation. J. Funct. Anal., 265(9):1934–1954, 2013. Zbl06262364MR3084493
  4. L. Desvillettes and F. Salvarani. Asymptotic behavior of degenerate linear transport equations. Bull. Sci. Math., 133(8):848–858, 2009. Zbl1180.35102MR2569870
  5. L. Desvillettes and C. Villani. On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Comm. Pure Appl. Math., 54(1):1–42, 2001. Zbl1029.82032MR1787105
  6. L. Desvillettes and C. Villani. On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math., 159(2):245–316, 2005. Zbl1162.82316MR2116276
  7. R. J. DiPerna, P.-L. Lions, and Y. Meyer. L p regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire, 8(3-4):271–287, 1991. Zbl0763.35014MR1127927
  8. J. Dolbeault, C. Mouhot, and C. Schmeiser. Hypocoercivity for linear kinetic equations conserving mass. http://arxiv.org/abs/1005.1495, to appear in Trans. AMS, 2010. Zbl06429109
  9. F. Golse, P.-L. Lions, B. Perthame, and R. Sentis. Regularity of the moments of the solution of a transport equation. J. Funct. Anal., 76(1):110–125, 1988. Zbl0652.47031MR923047
  10. Y. Guo. Decay and continuity of the Boltzmann equation in bounded domains. Arch. Ration. Mech. Anal., 197(3):713–809, 2010. Zbl1291.76276MR2679358
  11. D. Han-Kwan and M. Léautaud. Geometric analysis of the linear Boltzmann equation I. Trend to equilibrium. http://arxiv.org/abs/1401.8227, 2014. 
  12. D. Han-Kwan and M. Léautaud. Geometric analysis of the linear Boltzmann equation II. Localization properties of the spectrum. 2014. 
  13. F. Hérau. Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation. Asymptot. Anal., 46(3-4):349–359, 2006. Zbl1096.35019MR2215889
  14. F. Hérau and F. Nier. Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential. Arch. Ration. Mech. Anal., 171(2):151–218, 2004. Zbl1139.82323MR2034753
  15. H. Koch and D. Tataru. On the spectrum of hyperbolic semigroups. Comm. Partial Differential Equations, 20(5-6):901–937, 1995. Zbl0823.35108MR1326911
  16. G. Lebeau. Équation des ondes amorties. In Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), volume 19 of Math. Phys. Stud., pages 73–109. Kluwer Acad. Publ., Dordrecht, 1996. Zbl0863.58068MR1385677
  17. C. Mouhot and L. Neumann. Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity, 19(4):969–998, 2006. Zbl1169.82306MR2214953
  18. J. Rauch and M. Taylor. Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J., 24:79–86, 1974. Zbl0281.35012MR361461
  19. S. Ukai, N. Point, and H. Ghidouche. Sur la solution globale du problème mixte de l’équation de Boltzmann nonlinéaire. J. Math. Pures Appl. (9), 57(3):203–229, 1978. Zbl0335.35027MR513098
  20. I. Vidav. Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator. J. Math. Anal. Appl., 22:144–155, 1968. Zbl0155.19203MR230531
  21. I. Vidav. Spectra of perturbed semigroups with applications to transport theory. J. Math. Anal. Appl., 30:264–279, 1970. Zbl0195.13704MR259662
  22. C. Villani. Hypocoercivity. Mem. Amer. Math. Soc., 202(950):iv+141, 2009. Zbl1197.35004MR2562709

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.