具有低和高扩散率区域的局部周期介质的均匀化
Homogenization of a locally-periodic medium with areas of low and high diffusivity
摘要：
我们的目标是了解多孔材料，包括高和低扩散率的区域中的运输。对于这样的场景，传输变得结构化（这里：{ \em micro-macro}）。我们考虑的几何形状包括以局部周期性方式排列的低扩散率区域。我们选择了一个典型的对流扩散系统（最小尺寸），讨论其形式的均匀化（非均匀介质现在被假定为由圆形区域的$x$不同大小的低扩散率的区域），并证明了弱可解性的限制两尺度反应扩散模型。我们的分析的一个特点是，大多数的基本估计（正性，$L^\infty$ -界，唯一性，能量不等式）中得到的$x$ -依赖的Bochner空间。
英文摘要：
We aim at understanding transport in porous materials including regions with both high and low diffusivities. For such scenarios, the transport becomes structured (here: {\em micro-macro}). The geometry we have in mind includes regions of low diffusivity arranged in a locally-periodic fashion. We choose a prototypical advection-diffusion system (of minimal size), discuss its formal homogenization (the heterogenous medium being now assumed to be made of zones with circular areas of low diffusivity of $x$-varying sizes), and prove the weak solvability of the limit two-scale reaction-diffusion model. A special feature of our analysis is that most of the basic estimates (positivity, $L^\infty$-bounds, uniqueness, energy inequality) are obtained in $x$-dependent Bochner spaces.
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