Convergence Rates and Fluctuations for the Stokes–Brinkman Equations as Homogenization Limit in Perforated Domains

Abstract

We study the homogenization of the Dirichlet problem for the Stokes equations in $$\mathbb {R}^3$$ R 3 perforated by m spherical particles. We assume the positions and velocities of the particles to be identically and independently distributed random variables. In the critical regime, when the radii of the particles are of order $$m^{-1}$$ m - 1 , the homogenization limit u is given as the solution to the Brinkman equations. We provide optimal rates for the convergence $$u_m \rightarrow u$$ u m → u in $$L^2$$ L 2 , namely $$m^{-\beta }$$ m - β for all $$\beta < 1/2$$ β < 1 / 2 . Moreover, we consider the fluctuations. In the central limit scaling, we show that these converge to a Gaussian field, locally in $$L^2(\mathbb {R}^3)$$ L 2 ( R 3 ) , with an explicit covariance. Our analysis is based on explicit approximations for the solutions $$u_m$$ u m in terms of u as well as the particle positions and their velocities. These are shown to be accurate in $$\dot{H}^1(\mathbb {R}^3)$$ H ˙ 1 ( R 3 ) to order $$m^{-\beta }$$ m - β for all $$\beta < 1$$ β < 1 . Our results also apply to the analogous problem regarding the homogenization of the Poisson equations.