Hostname: page-component-89b8bd64d-n8gtw Total loading time: 0 Render date: 2026-05-09T08:55:54.856Z Has data issue: false hasContentIssue false
  • English
  • Français

Around 3D Boltzmann non linear operator without angular cutoff, a new formulation

Published online by Cambridge University Press:  15 April 2002

Radjesvarane Alexandre
Radjesvarane Alexandre*
Affiliation:
MAPMO, UMR 6627, Département de Mathématiques, Université d'Orléans, BP 6759 45067 Orleans Cedex 2, France. (alexandr@labomath.univ-orleans.fr)
Get access

Abstract

We propose a new formulation of the 3D Boltzmann non linear operator, without assuming Grad's angular cutoff hypothesis, and for intermolecular laws behaving as 1/rs , with s> 2. It involves natural pseudo differential operators, under a form which is analogous to the Landau operator. It may be used in the study of the associated equations, and more precisely in the non homogeneous framework.

Keywords

Boltzmann operatorellipticitynon angular cutoffpseudo differential operators.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 3 , May 2000 , pp. 575 - 590
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

R. Alexandre, Sur l'opérateur de Boltzmann linéaire 3D sans troncature angulaire. Note C.R. Acad. Sci. Paris Sér. I 325 (1997) 959-962.
Alexandre, R., Remarks on 3D Boltzmann linear equation without cutoff. Trans. Theory and Stat. Phys. 28 (1999) 433-473. CrossRef
R. Alexandre, Sur l'opérateur de Boltzmann non linéaire 3D sans troncature angulaire. Note C.R. Acad. Sci. Paris Sér. I 326 (1998) 165-168.
R. Alexandre, Sur le taux de dissipation d'entropie sans troncature angulaire. Note C.R. Acad. Sci. Paris Sér. I (1998) 311-315.
R. Alexandre, Une définition des solutions renormalisées pour l'équation de Boltzmann. Note C.R. Acad. Sci. Paris Sér. I 328 (1999) 987-991.
R. Alexandre, The linearised Boltzmann operator and applications. In preparation.
R. Alexandre, Solutions Maxwelliennes pour l'équation de Boltzmann sans troncature angulaire. Note submitted to C.R. Acad. Sci. Paris Sér. I (to appear).
R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long range interactions. Arch. Rat. Mech. Anal. (to appear).
R. Alexandre, C. Villani, On the Boltzmann equation for long-range interactions and the Landau approximation in plasma physics. (Preprints ENS Ulm DMA-99-22, 1999).
Arkeryd, L., On the Boltzmann equation. Arch. Rat. Mech. Anal. 45 (1972) 1-34.
Arkeryd, L., Intermolecular forces of infinite range and the Boltzmann equation. Arch. Rat. Mech. Anal. 77 (1981) 11-21. CrossRef
R. Balescu, Statistical Mechanics of charged particles. Wiley Interscience, N.Y, USA (1963).
T. Carleman, Problèmes Mathématiques dans la Théorie cinétique des Gaz. Almquist and Wiksell, Uppsala (1957)
C. Cercignani, Mathematical Methods in Kinetic Theory. 2nd Ed. Plenum (1990).
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases. Series in Appl. Sci. 106 , Springer Verlag, New York (1994).
Degond, P. and Lucquin, B., The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case. Math. Models Methods Appl. Sci. 2-2 (1992) 167-182. CrossRef
Desvillettes, L., Regularisation properties of the 2D homogeneous Boltzmann equation. Transport Theory Statist. Phys. 26 (1997) 341-357. CrossRef
Desvillettes, L., Regularisation for the non-cutoff 2D radially symmetric Boltzmann equation. Transport Theory Statist. Phys. 25 (1996) 383-394. CrossRef
Desvillettes, L., On asymptotics of the Boltzmann equation when the collisions become grazing. Transport Theory Stat. Phys. 21 (1992) 259-276. CrossRef
L. Desvillettes and B. Wennberg, work in preparation.
R.J. DiPerna and P.L. Lions, On the Cauchy problem for Boltzmann equation; Global existence and weak stability. Ann. Maths. 130 (1989) 321-366.
DiPerna, R.J. and Lions, P.L., Global weak solutions of kinetic equations. Sem. Mat. Torino 46 (1988) 259-288.
T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics. J. Stat. Phys. 89 (1997) 751-776 .
P.L. Lions, Compactness in Boltzmann's equation, via FIO and applic. J. Math. Kyoto Univ. 34 (1994) Part I 391-427; Part II 429-461, Part III 539-584.
P.L. Lions, On Boltzmann and Landau equations. Phil. Trans. Roy. Soc. London A-346 (1994) 191-204.
Lions, P.L., Regularity and compactness for Boltzmann collision operators without angular cutoff. Note C.R. Acad. Sci. Paris Sér. I 326 (1998) 37-41. CrossRef
Y.P. Pao, Boltzmann Collision Operator with Inverse power Intermolecular potentials. C.P.A.M 27 ( 1974) Part I 407-428; Part II 559-581.
M.E. Taylor, Pseudo-Differential Operators. Princeton Univ. Press (1981).
M.E. Taylor, Pdo and non linear PDE, Birkhauser, Boston (1991).
C. Villani, Contributions à l'étude mathématique des équations de Boltzmann et de Landau en théorie cinétique des gaz et des plasma, Thèse Université Paris-Dauphine (1998).
C. Villani, Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off. Rev. Mat. Iberoam. (to appear).
C. Villani, Conservative forms of Boltzmann's collision operator: Landau revisited. Math. Mod. Num. An. (1998).
S. Ukai, Solutions of the Boltzmann equation. In: Patterns and Waves, North-Holland (1985).
B. Wennberg, Regularity in the Boltzmann equation and the Radon transform. CPDE 19, (1994) 2057-2074.