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In this Note we make use of mass transportation techniques to give a simple proof of the finite speed of propagation of the solution to the one-dimensional porous medium equation. The result follows by showing that the difference of support of any two solutions corresponding to different compactly supported initial data is a bounded in time function of a suitable Monge–Kantorovich related metric.
Dans cette Note nous utilisons des techniques de transport de masse pour donner une preuve élémentaire de la finitude de la vitesse de propagation des solutions de l'équation mono-dimensionnelle des milieux poreux. Le résultat repose sur la preuve de la propriété suivante : la différence du support entre deux solutions quelconques correspondant à des données initiales à support compact différentes est une fonction, bornée en temps, d'une métrique de Monge–Kantorovitch appropriée.
Carrillo, José Antonio 1 ; Gualdani, Maria Pia 2 ; Toscani, Giuseppe 3
@article{CRMATH_2004__338_10_815_0,
author = {Carrillo, Jos\'e Antonio and Gualdani, Maria Pia and Toscani, Giuseppe},
title = {Finite speed of propagation in porous media by mass transportation methods},
journal = {Comptes Rendus. Math\'ematique},
pages = {815--818},
publisher = {Elsevier},
volume = {338},
number = {10},
year = {2004},
doi = {10.1016/j.crma.2004.03.025},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2004.03.025/}
}
TY - JOUR AU - Carrillo, José Antonio AU - Gualdani, Maria Pia AU - Toscani, Giuseppe TI - Finite speed of propagation in porous media by mass transportation methods JO - Comptes Rendus. Mathématique PY - 2004 SP - 815 EP - 818 VL - 338 IS - 10 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2004.03.025/ DO - 10.1016/j.crma.2004.03.025 LA - en ID - CRMATH_2004__338_10_815_0 ER -
%0 Journal Article %A Carrillo, José Antonio %A Gualdani, Maria Pia %A Toscani, Giuseppe %T Finite speed of propagation in porous media by mass transportation methods %J Comptes Rendus. Mathématique %D 2004 %P 815-818 %V 338 %N 10 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2004.03.025/ %R 10.1016/j.crma.2004.03.025 %G en %F CRMATH_2004__338_10_815_0
Carrillo, José Antonio; Gualdani, Maria Pia; Toscani, Giuseppe. Finite speed of propagation in porous media by mass transportation methods. Comptes Rendus. Mathématique, Tome 338 (2004) no. 10, pp. 815-818. doi : 10.1016/j.crma.2004.03.025. http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2004.03.025/
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☆ Work partially supported by EEC network # HPRN-CT-2002-00282, by the bilateral project Azioni integrate Italia–Spagna, by the DFG Project JU359/5, by the Vigoni Project CRUI-DAAD and by the Spanish DGI-MCYT/FEDER project BFM2002-01710.