Relaxation of the incompressible porous media equation
[Relaxation de l'équation des milieux poreux incompressible]
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].

DOI : 10.24033/asens.2171
Classification : 35Q35, 35A02, 35D30, 76S05
Keywords: weak solutions, inviscid fluids, non-uniqueness, microstructure evolution
Mot clés : solutions faibles, fluides non visqueux, non-unicié, évolution de la microstructure
@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},
     pages = {491--509},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 45},
     number = {3},
     year = {2012},
     doi = {10.24033/asens.2171},
     zbl = {1256.35073},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/asens.2171/}
}
TY  - JOUR
AU  - Székelyhidi Jr, László
TI  - Relaxation of the incompressible porous media equation
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2012
SP  - 491
EP  - 509
VL  - 45
IS  - 3
PB  - Société mathématique de France
UR  - http://archive.numdam.org/articles/10.24033/asens.2171/
DO  - 10.24033/asens.2171
LA  - en
ID  - ASENS_2012_4_45_3_491_0
ER  - 
%0 Journal Article
%A Székelyhidi Jr, László
%T Relaxation of the incompressible porous media equation
%J Annales scientifiques de l'École Normale Supérieure
%D 2012
%P 491-509
%V 45
%N 3
%I Société mathématique de France
%U http://archive.numdam.org/articles/10.24033/asens.2171/
%R 10.24033/asens.2171
%G en
%F ASENS_2012_4_45_3_491_0
Székelyhidi Jr, László. Relaxation of the incompressible porous media equation. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 3, pp. 491-509. doi : 10.24033/asens.2171. http://archive.numdam.org/articles/10.24033/asens.2171/

[1] D. Córdoba, D. Faraco & F. Gancedo, Lack of uniqueness for weak solutions of the incompressible porous media equation, Arch. Ration. Mech. Anal. 200 (2011), 725-746. | MR | Zbl

[2] D. Córdoba & F. Gancedo, Contour dynamics of incompressible 3-D fluids in a porous medium with different densities, Comm. Math. Phys. 273 (2007), 445-471. | MR | Zbl

[3] C. M. Dafermos, The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Differential Equations 14 (1973), 202-212. | MR | Zbl

[4] C. De Lellis & L. J. Székelyhidi, The Euler equations as a differential inclusion, Ann. of Math. 170 (2009), 1417-1436. | MR

[5] C. De Lellis & L. J. Székelyhidi, On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal. 195 (2010), 225-260. | MR | Zbl

[6] N. Gigli & F. Otto, Entropic Burgers' equation via a minimizing movement scheme based on the Wasserstein metric, preprint http://math.sns.it/media/doc/paper/143/Gigli-Otto-v7.pdf, 2012. | MR

[7] S. D. Howison, A note on the two-phase Hele-Shaw problem, J. Fluid Mech. 409 (2000), 243-249. | MR | Zbl

[8] B. Kirchheim, Rigidity and geometry of microstructures, Habilitation thesis, University of Leipzig, 2003.

[9] B. Kirchheim, S. Müller & V. Šverák, Studying nonlinear pde by geometry in matrix space, in Geometric analysis and nonlinear partial differential equations, Springer, 2003, 347-395. | MR

[10] S. Müller, Variational models for microstructure and phase transitions, in Calculus of variations and geometric evolution problems (Cetraro, 1996), Lecture Notes in Math. 1713, Springer, 1999, 85-210. | MR | Zbl

[11] S. Müller & V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. of Math. 157 (2003), 715-742. | MR | Zbl

[12] F. Otto, Evolution of microstructure in unstable porous media flow: a relaxational approach, Comm. Pure Appl. Math. 52 (1999), 873-915. | MR | Zbl

[13] F. Otto, Evolution of microstructure: an example, in Ergodic theory, analysis, and efficient simulation of dynamical systems, Springer, 2001, 501-522. | MR | Zbl

[14] P. Pedregal, Parametrized measures and variational principles, Progress in Nonlinear Differential Equations and their Applications, 30, Birkhäuser, 1997. | MR | Zbl

[15] P. G. Saffman & G. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. Roy. Soc. London. Ser. A 245 (1958), 312-329. | MR | Zbl

[16] R. Shvydkoy, Convex integration for a class of active scalar equations, J. Amer. Math. Soc. 24 (2011), 1159-1174. | MR | Zbl

[17] M. Siegel, R. E. Caflisch & S. Howison, Global existence, singular solutions, and ill-posedness for the Muskat problem, Comm. Pure Appl. Math. 57 (2004), 1374-1411. | MR | Zbl

[18] V. Šverák, On regularity of the Monge-Ampère equations, preprint, Heriot-Watt University, 1991.

[19] R. Wooding & H. Morel-Seytoux, Multiphase fluid flow through porous media, Ann. Review Fluid Mech. 8 (1976), 233-274. | Zbl

Cité par Sources :