Pointwise convergence of Boltzmann solutions for grazing collisions in a Maxwell gas via a probabilitistic interpretation

Hélène Guérin

ESAIM: Probability and Statistics (2004)

  • Volume: 8, page 36-55
  • ISSN: 1292-8100

Abstract

top
Using probabilistic tools, this work states a pointwise convergence of function solutions of the 2-dimensional Boltzmann equation to the function solution of the Landau equation for Maxwellian molecules when the collisions become grazing. To this aim, we use the results of Fournier (2000) on the Malliavin calculus for the Boltzmann equation. Moreover, using the particle system introduced by Guérin and Méléard (2003), some simulations of the solution of the Landau equation will be given. This result is original and has not been obtained for the moment by analytical methods.

How to cite

top

Guérin, Hélène. "Pointwise convergence of Boltzmann solutions for grazing collisions in a Maxwell gas via a probabilitistic interpretation." ESAIM: Probability and Statistics 8 (2004): 36-55. <http://eudml.org/doc/245905>.

@article{Guérin2004,
abstract = {Using probabilistic tools, this work states a pointwise convergence of function solutions of the 2-dimensional Boltzmann equation to the function solution of the Landau equation for Maxwellian molecules when the collisions become grazing. To this aim, we use the results of Fournier (2000) on the Malliavin calculus for the Boltzmann equation. Moreover, using the particle system introduced by Guérin and Méléard (2003), some simulations of the solution of the Landau equation will be given. This result is original and has not been obtained for the moment by analytical methods.},
author = {Guérin, Hélène},
journal = {ESAIM: Probability and Statistics},
keywords = {Boltzmann equation without cutoff for a Maxwell gas; Landau equation for a Maxwell gas; nonlinear stochastic differential equations; Malliavin calculus},
language = {eng},
pages = {36-55},
publisher = {EDP-Sciences},
title = {Pointwise convergence of Boltzmann solutions for grazing collisions in a Maxwell gas via a probabilitistic interpretation},
url = {http://eudml.org/doc/245905},
volume = {8},
year = {2004},
}

TY - JOUR
AU - Guérin, Hélène
TI - Pointwise convergence of Boltzmann solutions for grazing collisions in a Maxwell gas via a probabilitistic interpretation
JO - ESAIM: Probability and Statistics
PY - 2004
PB - EDP-Sciences
VL - 8
SP - 36
EP - 55
AB - Using probabilistic tools, this work states a pointwise convergence of function solutions of the 2-dimensional Boltzmann equation to the function solution of the Landau equation for Maxwellian molecules when the collisions become grazing. To this aim, we use the results of Fournier (2000) on the Malliavin calculus for the Boltzmann equation. Moreover, using the particle system introduced by Guérin and Méléard (2003), some simulations of the solution of the Landau equation will be given. This result is original and has not been obtained for the moment by analytical methods.
LA - eng
KW - Boltzmann equation without cutoff for a Maxwell gas; Landau equation for a Maxwell gas; nonlinear stochastic differential equations; Malliavin calculus
UR - http://eudml.org/doc/245905
ER -

References

top
  1. [1] R. Alexandre et C. Villani, On the Landau approximation in plasma physics (in preparation). Zbl1044.83007
  2. [2] A.A. Arsenev and O.E. Buryak, On the connection between a solution of the Boltzmann equation and a solution of the Landau–Fokker–Planck equation. Math. USSR Sbornik 69 (1991) 465-478. Zbl0724.35090
  3. [3] K. Bichteler, J.B. Gravelreaux and J. Jacod, Malliavin calculus for processes with jumps, Theory and Application of stochastic Processes. Gordon and Breach, New York (1987). Zbl0706.60057MR1008471
  4. [4] K. Bichteler and J. Jacod, Calcul de Malliavin pour les diffusions avec sauts, existence d’une densité pour le cas unidimensionel, in Séminaire de probabilités XVII. Springer, Berlin, Lecture Notes in Math. 986 (1983) 132-157. Zbl0525.60067
  5. [5] L. Boltzmann, Weitere studien über das wärme gleichgenicht unfer gasmoläkuler. Sitzungsber. Akad. Wiss. 66 (1872) 275-370. Translation: Further Studies on the thermal equilibrium of gas molecules, S.G. Brush Ed., Pergamon, Oxford, Kinetic Theory 2 (1966) 88-174. JFM04.0566.01
  6. [6] L. Boltzmann, Lectures on gas theory. Reprinted by Dover Publications (1995). 
  7. [7] P. Degon and B. Lucquin–Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case. Math. Mod. Meth. Appl. Sci. 2 (1992) 167-182. Zbl0755.35091
  8. [8] L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing. Transp. Theory Statist. Phys. 21 (1992) 259-276. Zbl0769.76059MR1165528
  9. [9] L. Desvillettes, C. Graham and S. Méléard, Probabilistic interpretation and numerical approximation of a Kac equation without cutoff. Stochastic Process. Appl. 84 (1999) 115-135. Zbl1009.76081MR1720101
  10. [10] N. Fournier, Existence and regularity study for two-dimensional Kac equation without cutoff by a probabilistic approach. Ann. Appl. Probab. 10 (2000) 434-462. Zbl1056.60052MR1768239
  11. [11] N. Fournier and S. Méléard, A stochastic particle numerical method for 3D Boltzmann equations without cutoff. Math. Comput. 70 (2002) 583-604. Zbl0990.60085MR1885616
  12. [12] T. Goudon, Sur l’équation de Boltzmann homogène et sa relation avec l’équation de Landau–Fokker–Planck : influence des collisions rasantes. C. R. Acad. Sci. Paris 324 (1997) 265-270. Zbl0882.76079
  13. [13] C. Graham and S. Méléard, Existence and regularity of a solution of a Kac equation without cutoff using the stochastic calculus of variations. Comm. Math. Phys. 205 (1999) 551-569. Zbl0953.60057MR1711273
  14. [14] H. Guérin, Solving Landau equation for some soft potentials through a probabilistic approach. Ann. Appl. Probab. 13 (2003) 515-539. Zbl1070.82027MR1970275
  15. [15] H. Guérin, Existence and regularity of a weak function-solution for some Landau equations with a stochastic approach. Stochastic Process. Appl. 101 (2002) 303-325. Zbl1075.60058MR1931271
  16. [16] H. Guérin and S. Méléard, Convergence from Boltzmann to Landau processes with soft potential and particle approximation. J. Statist. Phys. 111 (2003) 931-966. Zbl1031.82035MR1972130
  17. [17] J. Horowitz and R.L. Karandikar, Martingale problem associated with the Boltzmann equation, Seminar on Stochastic Processes, 1989, E. Cinlar, K.L. Chung and R.K. Getoor Eds., Birkhäuser, Boston (1990). Zbl0696.60095MR1042343
  18. [18] J. Jacod and A.N. Shiryaev, Limit theorems for stochastic processes. Springer (1987). Zbl0635.60021MR959133
  19. [19] E.M. Lifchitz and L.P. Pitaevskii, Physical kinetics – Course in theorical physics. Pergamon, Oxford 10 (1981). 
  20. [20] D. Nualart, The Malliavin calculus and related topics. Springer-Verlag (1995). Zbl0837.60050MR1344217
  21. [21] H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Geb. 46 (1978) 67-105. Zbl0389.60079MR512334
  22. [22] C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules. Math. Meth. Mod. Appl. Sci. 8 (1998) 957-983. Zbl0957.82029MR1646502
  23. [23] C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rational Mech. Anal. 143 (1998) 273-307. Zbl0912.45011MR1650006

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.