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.
Accepted:
Published online:
José Antonio Carrillo 1; Maria Pia Gualdani 2; Giuseppe Toscani 3
@article{CRMATH_2004__338_10_815_0, author = {Jos\'e Antonio Carrillo and Maria Pia Gualdani and Giuseppe Toscani}, 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}, }
TY - JOUR AU - José Antonio Carrillo AU - Maria Pia Gualdani AU - Giuseppe Toscani 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 DO - 10.1016/j.crma.2004.03.025 LA - en ID - CRMATH_2004__338_10_815_0 ER -
%0 Journal Article %A José Antonio Carrillo %A Maria Pia Gualdani %A Giuseppe Toscani %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 %R 10.1016/j.crma.2004.03.025 %G en %F CRMATH_2004__338_10_815_0
José Antonio Carrillo; Maria Pia Gualdani; Giuseppe Toscani. Finite speed of propagation in porous media by mass transportation methods. Comptes Rendus. Mathématique, Volume 338 (2004) no. 10, pp. 815-818. doi : 10.1016/j.crma.2004.03.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.03.025/
[1] Asymptotic L1-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., Volume 49 (2000) no. 1, pp. 113-142
[2] Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Russian Math. Surveys, Volume 42 (1987) no. 2, pp. 169-222
[3] An introduction to the mathematical theory of the porous medium equation, Shape Optimization and Free Boundaries, Montreal, PQ, 1990, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 380, Kluwer Academic, Dordrecht, 1992, pp. 347-389
[4] Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equations, Volume 3 (2003), pp. 67-118
[5] Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium, Trans. Amer. Math. Soc., Volume 277 (1983), p. 2
[6] Topics in Mass Transportation, Grad. Stud. Math., vol. 58, American Mathematical Society, 2003 (ISSN: 1065-7339)
Cited by Sources:
☆ 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.
Comments - Policy