Relaxation of the incompressible porous media equation
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 3, p. 491-509

It was shown recently by Córdoba, Faraco and Gancedo in [1] that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework developed for the incompressible Euler equations in [4], uses ideas from the theory of laminates, in particular T4 configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding T4 configurations. We then use this to construct weak solutions to the unstable interface problem (the Muskat problem), as a byproduct shedding new light on the gradient flow approach introduced by Otto in [14].

Il a récemment été démontré par Córdoba, Faraco et Gancedo dans [1], que l’équation des milieux poreux en dimension 2 admet des solutions faibles avec support compact dans le temps. La démonstration, qui fait appel à la méthode par intégration convexe telle qu’elle a été développée dans [4], dans le contexte des équations d’Euler incompressibles, utilise certaines idées provenant de la théorie des « laminates », et en particulier les configurations dites T4. Dans cette note, nous calculons explicitement la relaxation du « IPM », évitant ainsi les configurations T4. Ceci nous permet ensuite de construire des solutions faibles au problème des interfaces instables (problème de Muskat) et a pour autre conséquence de clarifier l’approche par flot de gradient, introduite par Otto dans [14].

DOI : https://doi.org/10.24033/asens.2171
Classification:  35Q35,  35A02,  35D30,  76S05
Keywords: weak solutions, inviscid fluids, non-uniqueness, microstructure evolution
@article{ASENS_2012_4_45_3_491_0,
     author = {Sz\'ekelyhidi Jr, L\'aszl\'o},
     title = {Relaxation of the incompressible porous media equation},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 45},
     number = {3},
     year = {2012},
     pages = {491-509},
     doi = {10.24033/asens.2171},
     zbl = {1256.35073},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2012_4_45_3_491_0}
}
Székelyhidi Jr, László. Relaxation of the incompressible porous media equation. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 3, pp. 491-509. doi : 10.24033/asens.2171. http://www.numdam.org/item/ASENS_2012_4_45_3_491_0/

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