论文标题
在3D奇异扰动域中扩散的积累时间
Accumulation time of diffusion in a 3D singularly perturbed domain
论文作者
论文摘要
在奇异扰动的域（从内部删除的小孔的域）扩散的边界值问题是当前相当大的兴趣的话题。应用包括细胞内扩散运输以及污染物的扩散或局部源的热量。在上一篇论文中，我们引入了一种新方法，以表征二维（2D）扩散的情况下的稳态方法。这是基于当地度量的放松率，称为累积时间$ t（\ x）$。通过使用匹配的渐近学和格林的功能方法的组合来求解拉普拉斯空间中的扩散方程来计算后者。因此，我们获得了$ t = -1/\lnε$的$ t（\ x）$的渐近扩展，其中$ε$指定了孔的相对大小。在本文中，我们开发了三维（3D）扩散的相应理论。由于拉普拉斯（Laplace）的奇异性质差异导致绿色功能的差异，该分析是2D病例的非平地扩展。特别是，拉普拉斯空间中3D扩散方程的解的渐近扩展涉及$ o（（ε/s）^n）$的术语，其中$ s $是laplace变量。这些$ s $ singularities必须通过部分系列重新召集来删除，以便获得$ t（\ x）$的渐近扩展为$ε$。
Boundary value problems for diffusion in singularly perturbed domains (domains with small holes removed from the interior) is a topic of considerable current interest. Applications include intracellular diffusive transport and the spread of pollutants or heat from localized sources. In a previous paper, we introduced a new method for characterizing the approach to steady-state in the case of two-dimensional (2D) diffusion. This was based on a local measure of the relaxation rate known as the accumulation time $T(\x)$. The latter was calculated by solving the diffusion equation in Laplace space using a combination of matched asymptotics and Green's function methods. We thus obtained an asymptotic expansion of $T(\x)$ in powers of $ν=-1/\ln ε$, where $ε$ specifies the relative size of the holes. In this paper, we develop the corresponding theory for three-dimensional (3D) diffusion. The analysis is a non-trivial extension of the 2D case due to differences in the singular nature of the Laplace transformed Green's function. In particular, the asymptotic expansion of the solution of the 3D diffusion equation in Laplace space involves terms of order $O((ε/s)^n)$, where $s$ is the Laplace variable. These $s$-singularities have to be removed by partial series resummations in order to obtain an asymptotic expansion of $T(\x)$ in powers of $ε$.
