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 , mesurant l’intensité de la résistance à la compression au voisinage de l’entassement maximal, tend vers . 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 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.
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DOI : 10.5802/jep.98
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
@article{JEP_2019__6__433_0, author = {Bresch, Didier and Ne\v{c}asov\'a, \v{S}\'arka and Perrin, Charlotte}, 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}, zbl = {07070266}, mrnumber = {3974475}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jep.98/} }
TY - JOUR AU - Bresch, Didier AU - Nečasová, Šárka AU - Perrin, Charlotte TI - Compression effects in heterogeneous media JO - Journal de l’École polytechnique — Mathématiques PY - 2019 SP - 433 EP - 467 VL - 6 PB - Ecole polytechnique UR - http://archive.numdam.org/articles/10.5802/jep.98/ DO - 10.5802/jep.98 LA - en ID - JEP_2019__6__433_0 ER -
%0 Journal Article %A Bresch, Didier %A Nečasová, Šárka %A Perrin, Charlotte %T Compression effects in heterogeneous media %J Journal de l’École polytechnique — Mathématiques %D 2019 %P 433-467 %V 6 %I Ecole polytechnique %U http://archive.numdam.org/articles/10.5802/jep.98/ %R 10.5802/jep.98 %G en %F JEP_2019__6__433_0
Bresch, Didier; Nečasová, Šárka; Perrin, Charlotte. Compression effects in heterogeneous media. Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 433-467. doi : 10.5802/jep.98. http://archive.numdam.org/articles/10.5802/jep.98/
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