Quantitative homogenization theory for random suspensions in steady Stokes flow

TL;DR

This paper develops a quantitative homogenization theory for random suspensions of rigid particles in steady Stokes flow, providing large-scale regularity, moment bounds, and optimal error estimates under ergodicity assumptions.

Contribution

It introduces the first quantitative homogenization framework for random suspensions in Stokes flow, addressing fluid incompressibility and particle rigidity constraints.

Findings

Established large-scale regularity theory.

Proved moment bounds for correctors.

Derived optimal homogenization error estimates.

Abstract

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.