Compression effects in heterogeneous media

Didier Bresch; Šárka Nečasová; Charlotte Perrin

doi:10.5802/jep.98

Compression effects in heterogeneous media
[Effets de compression en milieux hétérogènes]

Didier Bresch ¹ ; Šárka Nečasová ² ; Charlotte Perrin ³

¹ LAMA UMR 5127 CNRS, Univ. Savoie Mont Blanc Chambéry, France
² Institute of Mathematics, Academy of Sciences of the Czech Republic Žitná 25, CZ-115 67 Praha 1, Czech Republic
³ Aix Marseille Univ., CNRS, Centrale Marseille, I2M Marseille, France

Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 433-467.

Résumé
Abstract

Nous étudions dans cet article des effets de compression dans des milieux hétérogènes soumis à une contrainte d’entassement maximal. Partant des équations de Brinkman compressibles où la contrainte maximale est prise en compte au sein d’une pression et d’une viscosité de volume toutes deux singulières, nous montrons que les solutions faibles globales convergent (à une sous-suite près) vers des solutions faibles globales d’un modèle biphasique de type compressible/incompressible quand le paramètre $ε$ , mesurant l’intensité de la résistance à la compression au voisinage de l’entassement maximal, tend vers $0$ . En fonction de la prédominance relative de la viscosité de volume par rapport à la pression dans les régimes denses, nous mettons en évidence l’activation d’effets de mémoire à la limite dans le domaine congestionné (incompressible).

We study in this paper compression effects in heterogeneous media with maximal packing constraint. Starting from compressible Brinkman equations, where maximal packing is encoded in a singular pressure and a singular bulk viscosity, we show that the global weak solutions converge (up to a subsequence) to global weak solutions of the two-phase compressible/incompressible Brinkman equations with respect to a parameter $ε$ which measures effects close to the maximal packing value. Depending on the importance of the bulk viscosity with respect to the pressure in the dense regimes, memory effects are activated or not at the limit in the congested (incompressible) domain.

Reçu le : 2018-07-11
Accepté le : 2019-06-05
Publié le : 2019-06-18

MR Zbl

DOI : 10.5802/jep.98

Classification : 35Q35, 35B25, 76T20
Keywords: Compressible Brinkman equations, maximal packing, singular limit, free boundary problem, memory effect
Mot clés : Équations de Brinkman compressibles, contrainte d’entassement maximal, limite singulière, problème à frontière libre, effet mémoire

Affiliations des auteurs :

Didier Bresch ¹ ; Šárka Nečasová ² ; Charlotte Perrin ³

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Didier Bresch and \v{S}\'arka Ne\v{c}asov\'a and Charlotte Perrin},
     title = {Compression effects in heterogeneous media},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {433--467},
     publisher = {\'Ecole polytechnique},
     volume = {6},
     year = {2019},
     doi = {10.5802/jep.98},
     mrnumber = {3974475},
     zbl = {07070266},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.98/}
}

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Didier Bresch; Šárka Nečasová; Charlotte Perrin. Compression effects in heterogeneous media. Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 433-467. doi : 10.5802/jep.98. https://jep.centre-mersenne.org/articles/10.5802/jep.98/

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