Relaxation of the incompressible porous media equation

László Székelyhidi Jr

Annales scientifiques de l'École Normale Supérieure (2012)

  • Volume: 45, Issue: 3, page 491-509
  • ISSN: 0012-9593

Abstract

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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 T 4 configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding T 4 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].

How to cite

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Székelyhidi Jr, László. "Relaxation of the incompressible porous media equation." Annales scientifiques de l'École Normale Supérieure 45.3 (2012): 491-509. <http://eudml.org/doc/272246>.

@article{SzékelyhidiJr2012,
abstract = {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].},
author = {Székelyhidi Jr, László},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {weak solutions; inviscid fluids; non-uniqueness; microstructure evolution},
language = {eng},
number = {3},
pages = {491-509},
publisher = {Société mathématique de France},
title = {Relaxation of the incompressible porous media equation},
url = {http://eudml.org/doc/272246},
volume = {45},
year = {2012},
}

TY - JOUR
AU - Székelyhidi Jr, László
TI - Relaxation of the incompressible porous media equation
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2012
PB - Société mathématique de France
VL - 45
IS - 3
SP - 491
EP - 509
AB - 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].
LA - eng
KW - weak solutions; inviscid fluids; non-uniqueness; microstructure evolution
UR - http://eudml.org/doc/272246
ER -

References

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