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Comptes Rendus Mathématique
Volume 346, n° 3-4
pages 155-159 (février 2008)
Doi : 10.1016/j.crma.2007.12.010
Received : 24 August 2007 ;  accepted : 11 December 2007
A singular asymptotic behavior of a transport equation
États asymptotiques singuliers pour certaines équations de transports
 

Philippe Michel
DMI, Institut Camille-Jordan, École centrale de Lyon, 36, avenue Guy-de-Collongue, 69134 Ecully cedex, France 

Abstract

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. To cite this article: P. Michel, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The full text of this article is available in PDF format.
Résumé

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. Pour citer cet article : P. Michel, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The full text of this article is available in PDF format.


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