[Existence et lois de conservation pour l'équation de Boltzmann–Fermi–Dirac dans un domaine quelconque]
On montre un théorème d'existence pour l'équation de Boltzmann–Fermi–Dirac avec un noyau de collision intégrable, dans un domaine quelconque (éventuellement borné) avec réflexion spéculaire au bord, grâce aux caractéristiques du transport libre. On obtient ensuite que la solution satisfait les conservations locale de la masse, de l'impulsion et de l'énergie cinétique, grâce à une technique de dispersion.
We prove an existence theorem for the Boltzmann–Fermi–Dirac equation for integrable collision kernels in possibly bounded domains with specular reflection at the boundaries, using the characteristic lines of the free transport. We then obtain that the solution satisfies the local conservations of mass, momentum and kinetic energy thanks to a dispersion technique.
Accepté le :
Publié le :
@article{CRMATH_2010__348_13-14_763_0,
author = {Allemand, Thibaut},
title = {Existence and conservation laws for the {Boltzmann{\textendash}Fermi{\textendash}Dirac} equation in a general domain},
journal = {Comptes Rendus. Math\'ematique},
pages = {763--767},
publisher = {Elsevier},
volume = {348},
number = {13-14},
year = {2010},
doi = {10.1016/j.crma.2010.06.015},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2010.06.015/}
}
TY - JOUR AU - Allemand, Thibaut TI - Existence and conservation laws for the Boltzmann–Fermi–Dirac equation in a general domain JO - Comptes Rendus. Mathématique PY - 2010 SP - 763 EP - 767 VL - 348 IS - 13-14 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2010.06.015/ DO - 10.1016/j.crma.2010.06.015 LA - en ID - CRMATH_2010__348_13-14_763_0 ER -
%0 Journal Article %A Allemand, Thibaut %T Existence and conservation laws for the Boltzmann–Fermi–Dirac equation in a general domain %J Comptes Rendus. Mathématique %D 2010 %P 763-767 %V 348 %N 13-14 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2010.06.015/ %R 10.1016/j.crma.2010.06.015 %G en %F CRMATH_2010__348_13-14_763_0
Allemand, Thibaut. Existence and conservation laws for the Boltzmann–Fermi–Dirac equation in a general domain. Comptes Rendus. Mathématique, Tome 348 (2010) no. 13-14, pp. 763-767. doi : 10.1016/j.crma.2010.06.015. http://www.numdam.org/articles/10.1016/j.crma.2010.06.015/
[1] T. Allemand, The incompressible Euler limit of the Boltzmann equation for a gas of fermions, preprint
[2] On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math. (2), Volume 130 (1989), pp. 321-366
[3] Kinetic models and quantum effects: A modified Boltzmann equation for Fermi–Dirac particles, Arch. Rational Mech. Anal., Volume 127 (1994), pp. 101-131
[4] Initial-boundary value problems for the Boltzmann equation: Global existence of weak solutions, Arch. Rational Mech. Anal., Volume 119 (1992), pp. 309-353
[5] The Boltzmann equation. I. Uniqueness and local existence, Comm. Math. Phys., Volume 58 (1978), pp. 65-84
[6] On the Boltzmann equation for Fermi–Dirac particles with very soft potentials: Global existence of weak solutions, Journal of Differential Equations, Volume 245 (2008), pp. 1705-1761
[7] Global existence to the BGK model of Boltzmann equation, Journal of Differential Equations, Volume 82 (1989), pp. 191-205
Cité par Sources :
