论文标题
作为穿孔结构域中均化限制的Stokes-Brinkman方程的收敛速率和波动
Convergence rates and fluctuations for the Stokes-Brinkman equations as homogenization limit in perforated domains
论文作者
论文摘要
我们研究了$ \ mathbb {r}^3 $在$ m $球形粒子中塑造的dirichlet问题的均质化。我们假设粒子的位置和速度是相同和独立分布的随机变量。在关键方案中，当粒子的半径为$ m^{ - 1} $时，同质化限制$ u $作为对Brinkman方程的解决方案。我们为收敛$ u_m \ in $ l^2 $提供了最佳费率，即所有$β<1/2 $的$ m^{ - β} $。此外，我们考虑了波动。在中央限制缩放中，我们表明，这些收敛到$ l^2（\ mathbb {r}^3）$的当地的高斯字段，并具有显式协方差。 我们的分析基于$ u $以及粒子位置及其速度的解决方案$ u_m $的显式近似值。这些在$ \ dot h^1（\ mathbb {r}^3）$中表现为准确，以订购所有$β<1 $的$ m^{ - β} $。我们的结果还适用于有关泊松方程均质化的类似问题。
We study the homogenization of the Dirichlet problem for the Stokes equations in $\mathbb{R}^3$ 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 $m^{-1}$, the homogenization limit $u$ is given as the solution to the Brinkman equations. We provide optimal rates for the convergence $u_m \to u$ in $L^2$, namely $m^{-β}$ for all $β< 1/2$. Moreover, we consider the fluctuations. In the central limit scaling, we show that these converge to a Gaussian field, locally in $L^2(\mathbb{R}^3)$, with an explicit covariance. Our analysis is based on explicit approximations for the solutions $u_m$ in terms of $u$ as well as the particle positions and their velocities. These are shown to be accurate in $\dot H^1(\mathbb{R}^3)$ to order $m^{-β}$ for all $β< 1$. Our results also apply to the analogous problem regarding the homogenization of the Poisson equations.
