A short proof of Gevrey regularity for homogenized coefficients of the Poisson point process

Duerinckx, Mitia; Gloria, Antoine

doi:10.5802/crmath.354

Cited by 2 publications

(2 citation statements)

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“…As the effective viscosity is an L 2 -based quantity, we may expect to estimate cluster formulas by means of suitable energy estimates, carefully avoiding taking absolute values of any Calderón-Zygmund kernel. Taking inspiration from our previous work [15], this is achieved by means of a hierarchy of so-called interpolating `1 `2 energy estimates (also crucially used in [20,30]). As a corollary, uniform estimates allow us to define infinite-volume cluster coefficients in the limit ¹ x B j º j WD lim L"1 ¹ x B j L º j .…”

Section: Cluster Expansion Of the Effective Viscosity-chaptermentioning

confidence: 99%

“…In order to prove uniform cluster estimates, cf. Theorem 3.2 (i), our main analytical achievement is the following hierarchy of interpolating `1 `2 energy estimates for corrector differences, inspired by our previous work [15] on the conductivity problem (which also considers the case of "overlapping particles"; see [20,30] for refinements in that direction). More precisely, we consider the following quantities, for all H N, all L, and k; j 0,…”

Section: Uniform `1 `2 Energy Estimatesmentioning

confidence: 99%

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On Einstein's Effective Viscosity Formula

Duerinckx,

Gloria

2023

Memoirs of the European Mathematical Society

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In his PhD thesis, Einstein derived an explicit first-order expansion for the effective viscosity of a Stokes fluid with a suspension of small rigid particles at low density. His formal derivation relied on two implicit assumptions: (i) there is a scale separation between the size of the particles and the observation scale; and (ii) at first order, dilute particles do not interact with one another. In mathematical terms, the first assumption amounts to the validity of a homogenization result defining the effective viscosity tensor, which is now well understood. Next, the second assumption allowed Einstein to approximate this effective viscosity at low density by considering particles as being isolated. The rigorous justification is, in fact, quite subtle as the effective viscosity is a nonlinear nonlocal function of the ensemble of particles and as hydrodynamic interactions have borderline integrability.In the present memoir, we establish Einstein's effective viscosity formula in the most general setting. In addition, we pursue the low-density expansion to arbitrary order in form of a cluster expansion, where the summation of hydrodynamic interactions crucially requires suitable renormalizations. In particular, we justify a celebrated result by Batchelor and Green on the second-order correction and we explicitly describe all higher-order renormalizations for the first time. In some specific settings, we further address the summability of the whole cluster expansion. Our approach relies on a combination of combinatorial arguments, variational analysis, elliptic regularity, probability theory, and diagrammatic integration methods.

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Section: Cluster Expansion Of the Effective Viscosity-chaptermentioning

confidence: 99%