含分支细裂缝的体域中反应输运的降维与渐近近似
Dimension Reduction and Asymptotic Approximation of Reactive Transport in a Bulk Domain with a Branched Thin Fracture
AI总结：
该研究针对含分支细裂缝的二维体域反应输运，通过降维将裂缝转化为一维图，推导极限问题结构，构造校正子并建立多尺度近似与误差估计，揭示裂缝几何对输运的影响。
AI中文摘要：
我们研究二维体域内的非线性反应输运，该体域包含一条由三条厚度为ε的窄分支构成的分支细裂缝，分支通过直径为O(ε)的交汇节点连接。微观模型将体域内的非线性抛物型反应-扩散方程与裂缝内的对流-扩散方程耦合，其中纵向佩克莱数的量级为ε⁻¹，导致沿分支的输运以对流为主。非线性的边相关通量条件描述了体域与裂缝之间的耦合。当ε趋于0时，裂缝坍缩为一维图，我们推导得到有效极限问题的递推结构：图上满足经典基尔霍夫传输条件的一阶双曲问题，以及体域内的反应-扩散方程，该方程带有涉及图解的、位于图边上的非线性罗宾条件。微观模型的节点边界条件不影响这些主导阶极限。为捕捉节点几何的影响，我们构造节点层和角层校正子，并确定渐近展开的后续项；这些项的系数在带有无穷远出口的无界域和角型几何中求解辅助边值问题。我们整合了结合体域、分支、节点层和角层贡献的完整多尺度近似，并在合适的能量范数中建立了定量误差估计，这些估计证明了该近似相对于ε的精度，且明确依赖于极限图的角，反映了裂缝网络的几何复杂性。
英文摘要：
We study nonlinear reactive transport in a two-dimensional bulk domain containing a thin branched fracture composed of three narrow branches of thickness epsilon connected through a junction node of diameter O(epsilon). The microscopic model couples nonlinear parabolic reaction-diffusion equations in the bulk with an advection-diffusion equation in the fracture, where the longitudinal Peclet number is of order epsilon^{-1}, leading to advection-dominated transport along the branches. Nonlinear side-dependent flux conditions describe the coupling between the bulk and the fracture. As epsilon tends to zero, the fracture collapses to a one-dimensional graph, and we derive a recurrent structure of effective limit problems: a first-order hyperbolic problem on the graph satisfying the classical Kirchhoff transmission condition at the node, and reaction-diffusion equations in the bulk with nonlinear Robin conditions on the graph edges involving the graph solution. The node boundary conditions of the microscopic model do not affect these leading-order limits. To capture the influence of the node geometry, we construct node-layer and corner-layer correctors and determine subsequent terms of the asymptotic expansion. Their coefficients solve auxiliary boundary-value problems in unbounded domains with outlets at infinity and in corner-type geometries. We assemble a complete multiscale approximation combining bulk, branch, node-layer, and corner-layer contributions, and establish quantitative error estimates in appropriate energy norms. These estimates demonstrate the accuracy of the approximation relative to epsilon and depend explicitly on the corner angles of the limiting graph, reflecting the geometric complexity of the fracture network.
