[États asymptotiques singuliers pour certaines équations de transports]
On s'intéresse à des équations de transport dans lesquelles la vitesse de transport est strictement décroissante. Cette propriété de monotonie entraîne la concentration de la masse de la solution en un point et, par conséquent, un comportement asymptotique singulier. D'autre part, il est possible de réduire un modèle contenant une structure de transport avec la condition de monotonie (dans une direction) de la vitesse de transport. En utilisant la convergence de la solution en une masse de Dirac suivant la direction du transport, on rend cette direction inutile et on peut éliminer une variable d'espace. Cela réduit le coût numérique lorsqu'il faut simuler un certains nombres d'EDP contenant une telle structure. Le gain est d'autant plus important que le nombre d'équations est grand.
We consider a simple conservative transport equation where the speed is strictly decreasing. The monotonicity property of the speed rate leads to a singular asymptotic behavior and the concentration of the mass of the solution at a point. Thus, a model which contains a transport structure with monotone decay of the speed rate can be reduced by using the result of convergence to a Dirac mass. It is useful in the case where we have to simulate numerous nonlinear PDEs containing such a structure. Indeed, the concentration of the mass makes the variable in which the mass concentrate useless and thus we lose a dimension. The gain in time calculus is important when the number of equations is large.
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@article{CRMATH_2008__346_3-4_155_0,
author = {Michel, Philippe},
title = {A singular asymptotic behavior of a transport equation},
journal = {Comptes Rendus. Math\'ematique},
pages = {155--159},
publisher = {Elsevier},
volume = {346},
number = {3-4},
year = {2008},
doi = {10.1016/j.crma.2007.12.010},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2007.12.010/}
}
TY - JOUR AU - Michel, Philippe TI - A singular asymptotic behavior of a transport equation JO - Comptes Rendus. Mathématique PY - 2008 SP - 155 EP - 159 VL - 346 IS - 3-4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2007.12.010/ DO - 10.1016/j.crma.2007.12.010 LA - en ID - CRMATH_2008__346_3-4_155_0 ER -
%0 Journal Article %A Michel, Philippe %T A singular asymptotic behavior of a transport equation %J Comptes Rendus. Mathématique %D 2008 %P 155-159 %V 346 %N 3-4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2007.12.010/ %R 10.1016/j.crma.2007.12.010 %G en %F CRMATH_2008__346_3-4_155_0
Michel, Philippe. A singular asymptotic behavior of a transport equation. Comptes Rendus. Mathématique, Tome 346 (2008) no. 3-4, pp. 155-159. doi : 10.1016/j.crma.2007.12.010. http://www.numdam.org/articles/10.1016/j.crma.2007.12.010/
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