Quantitative homogenization theory for random suspensions in steady Stokes flow

Mitia Duerinckx; Antoine Gloria

doi:10.5802/jep.204

Quantitative homogenization theory for random suspensions in steady Stokes flow
[Homogénéisation quantitative de suspensions aléatoires de particules dans un fluide de Stokes stationnaire]

Mitia Duerinckx ¹ ; Antoine Gloria ²

¹ Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay 91400 Orsay, France & University of California, Los Angeles, Department of Mathematics Los Angeles CA 90095, USA & Université Libre de Bruxelles, Département de Mathématique 1050 Brussels, Belgium
² Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions 75005 Paris, France & Institut Universitaire de France & Université Libre de Bruxelles, Département de Mathématique 1050 Brussels, Belgium

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1183-1244.

Résumé
Abstract

Ce travail développe une théorie quantitative de l’homogénéisation de suspensions aléatoires de particules rigides dans un fluide de Stokes stationnaire, complétant les résultats qualitatifs récents. Plus précisément, nous établissons une théorie de régularité aux grandes échelles pour ce problème de Stokes et nous montrons des estimations de moments pour les correcteurs associés, ainsi que des estimations optimales de convergence de l’erreur d’homogénéisation (sous des hypothèses quantitatives d’ergodicité de la suspension aléatoire). En comparaison à la théorie pour les équations elliptiques linéaires sous forme divergence, l’incompressibilité du fluide et la rigidité des particules soulèvent des difficultés analytiques additionnelles. Notre analyse couvre également le problème des inclusions rigides en élasticité linéaire (compressible ou incompressible) et en électrostatique ; les résultats sont nouveaux pour ces modèles également, même dans le cas périodique.

This work develops a quantitative homogenization theory for random suspensions of rigid particles in a steady Stokes flow, and completes recent qualitative results. More precisely, we establish a large-scale regularity theory for this Stokes problem, and we prove moment bounds for the associated correctors and optimal estimates on the homogenization error; the latter further requires a quantitative ergodicity assumption on the random suspension. Compared to the corresponding quantitative homogenization theory for divergence-form linear elliptic equations, substantial difficulties arise from the analysis of the fluid incompressibility and the particle rigidity constraints. Our analysis further applies to the problem of stiff inclusions in (compressible or incompressible) linear elasticity and in electrostatics; it is also new in those cases, even in the periodic setting.

Reçu le : 2021-03-10
Accepté le : 2022-07-11
Publié le : 2022-07-19

DOI : 10.5802/jep.204

Classification : 35R60, 76M50, 35Q35, 76D03, 76D07
Keywords: Steady Stokes fluid, rigid particles, quantitative stochastic homogenization, large-scale regularity
Mot clés : Fluide de Stokes, particules rigides, homogénéisation stochastique quantitative, régularité aux grandes échelles

Affiliations des auteurs :

Mitia Duerinckx ¹ ; Antoine Gloria ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Mitia Duerinckx and Antoine Gloria},
     title = {Quantitative homogenization theory for random suspensions in steady {Stokes} flow},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1183--1244},
     publisher = {\'Ecole polytechnique},
     volume = {9},
     year = {2022},
     doi = {10.5802/jep.204},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.204/}
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Mitia Duerinckx; Antoine Gloria. Quantitative homogenization theory for random suspensions in steady Stokes flow. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1183-1244. doi : 10.5802/jep.204. https://jep.centre-mersenne.org/articles/10.5802/jep.204/

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