SIAM Journal on Mathematical Analysis, Volume 53, Issue 3, Page 2890-2924, January 2021.
We derive high order homogenized models for the incompressible Stokes system in a cubic domain filled with periodic obstacles. These models have the potential to unify the three classical limit problems (namely the “unchanged” Stokes system, the Brinkman model, and Darcy's law) corresponding to various asymptotic regimes of the ratio $\eta\equiv a_{\epsilon}/\epsilon$ between the radius $a_{\epsilon}$ of the holes and the size $\epsilon$ of the periodic cell. What is more, a novel, rather surprising feature of our higher order effective equations is the occurrence of odd order differential operators when the obstacles are not symmetric. Our derivation relies on the method of two-scale power series expansions and on the existence of a “criminal” ansatz, which allows to reconstruct the oscillating velocity and pressure $(\bm u_{\epsilon},p_{\epsilon})$. as a linear combination of the derivatives of their formal average $(\bm u_{\epsilon}^{*},p_{\epsilon}^{*})$ weighted by suitable corrector tensors. The formal average $(\bm u_{\epsilon}^{*},p_{\epsilon}^{*})$ is itself the solution to a formal, infinite order homogenized equation, whose truncation at any finite order is in general ill-posed. Inspired by the variational truncation method of [V. P. Smyshlyaev and K. Cherednichenko, J. Mech. Phys. Solids, 48 (2000), pp. 1325--1357; K. D. Cherednichenko and J. A. Evans, Multiscale Model. Simul., 14 (2016), pp. 1513--1539] we derive, for any $K\in\mathbb{N}$, a well-posed model of order $2K+2$ which yields approximations of the original solutions with an error of order $O(\epsilon^{K+3})$ in the $L^{2}$ norm. Furthermore, the error improves up to the order $O(\epsilon^{2K+4})$ if a slight modification of this model remains well-posed. Finally, we find asymptotics of all homogenized tensors in the low volume fraction limit $\eta\rightarrow 0$ and in dimension $d\>3$. This allows us to obtain that our effective equations converge coefficientwise to either of the Brinkman or Darcy regimes which arise when $\eta$ is, respectively, equivalent, or greater than the critical scaling $\eta_{{crit}}\sim\epsilon^{2/(d-2)}$.

中文翻译：

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周期性多孔介质中斯托克斯系统的高阶均质化

SIAM数学分析杂志，第53卷，第3期，第2890-2924页，2021年1月。
我们在充满周期性障碍的立方域中推导了不可压缩斯托克斯系统的高阶均质模型。这些模型有可能统一对应于$ \ eta \ equiv a _ {\ epsilon} / \ epsilon之比的各种渐近形式的三个经典极限问题（即“不变的”斯托克斯系统，Brinkman模型和达西定律）孔的半径$ a _ {\ epsilon} $与周期像元的大小$ \ epsilon $之间的$。而且，当障碍物不对称时，我们的高阶有效方程的一个新颖而令人惊讶的特征是出现了奇数阶微分算子。我们的推导依赖于二阶幂级数展开的方法以及“犯罪” ansatz的存在，它可以重建振荡速度和压力$（\ bm u _ {\ epsilon}，p _ {\ epsilon}）$。作为形式均值（（bm u _ {\ epsilon} ^ {*}，p _ {\ epsilon} ^ {*}）$的导数的线性组合，并通过适当的校正张量加权。形式平均$（\ bm u _ {\ epsilon} ^ {*}，p _ {\ epsilon} ^ {*}）$本身就是形式无限的均质方程的解，该方程一般在任何有限阶上都被截断病态。受到[VP Smyshlyaev和K. Cherednichenko，J。Mech。物理 Solids，48（2000），第1325--1357页；和 KD Cherednichenko和JA Evans，多尺度模型。Simul。，14（2016），pp。1513--1539]，我们为任何$ K \ in \ mathbb {N} $推导一个定理为$ 2K + 2 $的模型，该模型产生的近似原始解且在$ L ^ {2} $范数中的顺序为$ O（\ epsilon ^ {K + 3}）$。此外，如果对该模型进行适当的修改，则误差会提高到$ O（\ epsilon ^ {2K + 4}）$阶。最后，我们发现在低体积分数限制$ \ eta \ rightarrow 0 $和维度$ d \> 3 $中所有均质张量的渐近性。这使我们能够获得有效的方程式，使其在Brinkman或Darcy体制下按系数收敛，这在$ \ eta $分别等于或大于临界比例$ \ eta _ {{crit}} \ sim \ epsilon时出现^ {2 /（d-2）} $。