Quantitative homogenization theory for random suspensions in steady Stokes flow

Duerinckx, Mitia; Gloria, Antoine

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

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.

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

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