Mathematical Problems in Mechanics
Finite speed of propagation in porous media by mass transportation methods
Comptes Rendus. Mathématique, Volume 338 (2004) no. 10, pp. 815-818.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.03.025
Carrillo, José Antonio 1; Gualdani, Maria Pia 2; Toscani, Giuseppe 3

1 Departament de Matemàtiques – ICREA, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
2 Fachbereich Mathematik, Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany
3 Dipartimento di Matematica, Università di Pavia, via Ferrata 1, 27100 Pavia, Italy
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Carrillo, José Antonio; Gualdani, Maria Pia; Toscani, Giuseppe. 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. http://www.numdam.org/articles/10.1016/j.crma.2004.03.025/

[1] Carrillo, J.A.; Toscani, G. 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] Kalashnikov, A.S. 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] Vázquez, J.L. 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] Vázquez, J.L. Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equations, Volume 3 (2003), pp. 67-118

[5] Vázquez, J.L. 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] Villani, C. 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.