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.
Mots-clés : Boltzmann equation without cutoff for a Maxwell gas, Landau equation for a Maxwell gas, nonlinear stochastic differential equations, Malliavin calculus
@article{PS_2004__8__36_0,
author = {Gu\'erin, H\'el\`ene},
title = {Pointwise convergence of {Boltzmann} solutions for grazing collisions in a {Maxwell} gas via a probabilitistic interpretation},
journal = {ESAIM: Probability and Statistics},
pages = {36--55},
publisher = {EDP-Sciences},
volume = {8},
year = {2004},
doi = {10.1051/ps:2003018},
mrnumber = {2085604},
zbl = {1033.60088},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2003018/}
}
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 SP - 36 EP - 55 VL - 8 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2003018/ DO - 10.1051/ps:2003018 LA - en ID - PS_2004__8__36_0 ER -
%0 Journal Article %A Guérin, Hélène %T Pointwise convergence of Boltzmann solutions for grazing collisions in a Maxwell gas via a probabilitistic interpretation %J ESAIM: Probability and Statistics %D 2004 %P 36-55 %V 8 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2003018/ %R 10.1051/ps:2003018 %G en %F PS_2004__8__36_0
Guérin, Hélène. Pointwise convergence of Boltzmann solutions for grazing collisions in a Maxwell gas via a probabilitistic interpretation. ESAIM: Probability and Statistics, Tome 8 (2004), pp. 36-55. doi : 10.1051/ps:2003018. http://www.numdam.org/articles/10.1051/ps:2003018/
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