Relaxation of the incompressible porous media equation

Székelyhidi Jr, László

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 3, pp. 491-509.

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].

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].

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