Quantitative homogenization theory for random suspensions in steady Stokes flow
Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 1183-1244.

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.

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.

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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
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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Mitia Duerinckx; Antoine Gloria. Quantitative homogenization theory for random suspensions in steady Stokes flow. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 1183-1244. doi : 10.5802/jep.204. https://jep.centre-mersenne.org/articles/10.5802/jep.204/

[1] S. Armstrong, T. Kuusi & J.-C. Mourrat - “Mesoscopic higher regularity and subadditivity in elliptic homogenization”, Comm. Math. Phys. 347 (2016) no. 2, p. 315-361 | DOI | MR | Zbl

[2] S. Armstrong, T. Kuusi & J.-C. Mourrat - “The additive structure of elliptic homogenization”, Invent. Math. 208 (2017) no. 3, p. 999-1154 | DOI | MR | Zbl

[3] S. Armstrong, T. Kuusi & J.-C. Mourrat - Quantitative stochastic homogenization and large-scale regularity, Grundlehren Math. Wiss., vol. 352, Springer, Cham, 2019 | DOI

[4] S. N. Armstrong & J.-P. Daniel - “Calderón-Zygmund estimates for stochastic homogenization”, J. Funct. Anal. 270 (2016) no. 1, p. 312-329 | DOI | Zbl

[5] S. N. Armstrong & J.-C. Mourrat - “Lipschitz regularity for elliptic equations with random coefficients”, Arch. Rational Mech. Anal. 219 (2016) no. 1, p. 255-348 | DOI | MR | Zbl

[6] S. N. Armstrong & C. K. Smart - “Quantitative stochastic homogenization of convex integral functionals”, Ann. Sci. École Norm. Sup. (4) 49 (2016) no. 2, p. 423-481 | DOI | MR | Zbl

[7] M. Avellaneda & F.-H. Lin - “Compactness methods in the theory of homogenization”, Comm. Pure Appl. Math. 40 (1987) no. 6, p. 803-847 | DOI | MR | Zbl

[8] M. Avellaneda & F.-H. Lin - “L p bounds on singular integrals in homogenization”, Comm. Pure Appl. Math. 44 (1991) no. 8-9, p. 897-910 | DOI | MR | Zbl

[9] G. K. Batchelor - “Sedimentation in a dilute dispersion of spheres”, J. Fluid Mech. 52 (1972) no. 2, p. 245-268 | DOI | Zbl

[10] A. Braides & A. Garroni - “Homogenization of periodic nonlinear media with stiff and soft inclusions”, Math. Models Methods Appl. Sci. 5 (1995) no. 4, p. 543-564 | DOI | MR | Zbl

[11] R. E. Caflisch & J. H. C. Luke - “Variance in the sedimentation speed of a suspension”, Phys. Fluids 28 (1985) no. 3, p. 759-760 | DOI | Zbl

[12] T. Delmotte & J.-D. Deuschel - “On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to ϕ interface model”, Probab. Theory Related Fields 133 (2005) no. 3, p. 358-390 | DOI | MR | Zbl

[13] L. Diening, M. Růžička & K. Schumacher - “A decomposition technique for John domains”, Ann. Acad. Sci. Fenn. Math. 35 (2010) no. 1, p. 87-114 | DOI | MR | Zbl

[14] M. Duerinckx - “Effective viscosity of random suspensions without uniform separation”, 2020 | arXiv

[15] M. Duerinckx & A. Gloria - “Multiscale functional inequalities in probability: Concentration properties”, ALEA Lat. Am. J. Probab. Math. Stat. 17 (2020) no. 1, p. 133-157 | DOI | MR | Zbl

[16] M. Duerinckx & A. Gloria - “Multiscale functional inequalities in probability: Constructive approach”, Ann. H. Lebesgue 3 (2020), p. 825-872 | DOI | MR | Zbl

[17] M. Duerinckx & A. Gloria - “On Einstein’s effective viscosity formula”, 2020 | arXiv

[18] M. Duerinckx & A. Gloria - “Corrector equations in fluid mechanics: Effective viscosity of colloidal suspensions”, Arch. Rational Mech. Anal. 239 (2021), p. 1025-1060 | DOI | MR | Zbl

[19] M. Duerinckx & A. Gloria - “Sedimentation of random suspensions and the effect of hyperuniformity”, Ann. PDE 8 (2022) no. 1, article ID 2, 66 pages | DOI | MR | Zbl

[20] M. Duerinckx & F. Otto - “Higher-order pathwise theory of fluctuations in stochastic homogenization”, Stochastic Partial Differ. Equ. Anal. Comput. 8 (2020) no. 3, p. 625-692 | DOI | MR | Zbl

[21] A. Einstein - “Eine neue Bestimmung der Moleküldimensionen”, Ann. Physics 19 (1906) no. 2, p. 289-306 | DOI | Zbl

[22] G. P. Galdi - An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems, Springer Monographs in Math., Springer, New York, 2011 | DOI

[23] F. W. Gehring - “The L p -integrability of the partial derivatives of a quasiconformal mapping”, Acta Math. 130 (1973), p. 265-277 | DOI | MR | Zbl

[24] M. Giaquinta & G. Modica - “Regularity results for some classes of higher order non linear elliptic systems”, J. reine angew. Math. 311/312 (1979), p. 145-169 | Zbl

[25] A. Gloria - “A scalar version of the Caflisch-Luke paradox”, Comm. Pure Appl. Math. 74 (2021) no. 7, p. 1403-1452 | DOI | MR | Zbl

[26] A. Gloria, S. Neukamm & F. Otto - “An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations”, ESAIM Math. Model. Numer. Anal. 48 (2014) no. 2, p. 325-346 | DOI | Numdam | MR | Zbl

[27] A. Gloria, S. Neukamm & F. Otto - “Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics”, Invent. Math. 199 (2015) no. 2, p. 455-515 | DOI | MR | Zbl

[28] A. Gloria, S. Neukamm & F. Otto - “A regularity theory for random elliptic operators”, Milan J. Math. 88 (2020) no. 1, p. 99-170 | DOI | MR | Zbl

[29] A. Gloria, S. Neukamm & F. Otto - “Quantitative estimates in stochastic homogenization for correlated coefficient fields”, Anal. PDE 14 (2021) no. 8, p. 2497-2537 | DOI | MR | Zbl

[30] A. Gloria & F. Otto - “An optimal variance estimate in stochastic homogenization of discrete elliptic equations”, Ann. Probab. 39 (2011) no. 3, p. 779-856 | DOI | MR | Zbl

[31] A. Gloria & F. Otto - “An optimal error estimate in stochastic homogenization of discrete elliptic equations”, Ann. Appl. Probab. 22 (2012) no. 1, p. 1-28 | DOI | MR | Zbl

[32] A. Gloria & F. Otto - “The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations”, 2015 | arXiv

[33] A. Gloria & F. Otto - “Quantitative results on the corrector equation in stochastic homogenization”, J. Eur. Math. Soc. (JEMS) 19 (2017), p. 3489-3548 | DOI | MR | Zbl

[34] R. M. Höfer - “Convergence of the method of reflections for particle suspensions in Stokes flows”, J. Differential Equations 297 (2021), p. 81-109 | DOI | MR | Zbl

[35] V. V. Jikov - “Some problems of extension of functions arising in connection with the homogenization theory”, Differ. Uravn. 26 (1990) no. 1, p. 39-51

[36] V. V. Jikov, S. M. Kozlov & O. A. Oleĭnik - Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994 | DOI

[37] M. Josien & F. Otto - “The annealed Calderón-Zygmund estimate as convenient tool in quantitative stochastic homogenization”, 2020 | arXiv

[38] D. L. Koch & E. S. G. Shaqfeh - “Screening in sedimenting suspensions”, J. Fluid Mech. 224 (1991), p. 275-303 | DOI | MR | Zbl

[39] P. C. Kunstmann & L. Weis - “Maximal L p -regularity for parabolic equations, Fourier multiplier theorems and H -functional calculus”, in Functional analytic methods for evolution equations, Lect. Notes in Math., vol. 1855, Springer, Berlin, 2004, p. 65-311 | DOI | MR | Zbl

[40] D. Marahrens & F. Otto - “Annealed estimates on the Green function”, Probab. Theory Related Fields 163 (2015) no. 3-4, p. 527-573 | DOI | MR | Zbl

[41] Z. Shen - “The L p boundary value problems on Lipschitz domains”, Adv. Math. 216 (2007), p. 212-254 | DOI | MR | Zbl

[42] Z. Shen - “The Calderón-Zygmund lemma revisited”, in Lectures on the analysis of nonlinear partial differential equations. Part 2, Morningside Lect. Math., vol. 2, Int. Press, Somerville, MA, 2012, p. 203-224 | Zbl

[43] V. V. Zhikov - “Averaging of functionals in the calculus of variations and elasticity”, Math. USSR-Izv. 29 (1987), p. 33-66 | DOI | Zbl

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