弯曲空间中的对称流与达西定律
Symmetric flows and Darcy's law in Curved Spaces
摘要：
研究了三维流形$M$上Stokes方程某些对称解的存在性问题，其中一般度量具有对称性.这些解决方案对应于单向流动。我们已经能够确定它们存在的必要和充分条件。对称单向流动是推导所谓的达西定律的基础，达西定律是控制嵌入环境$M$中的Hele-Shaw单元中的流体流动的定律。我们的主要兴趣是离开通常的，平坦的背景环境，并考虑一个环境的可能性，任意常曲率$K$中嵌入一个细胞。我们推广达西定律的特殊模型，这样的空间得到$\real^3$与共形度量。我们采用微积分的微分形式的一个更简单，更优雅的方法来这里讨论的问题。
英文摘要：
We consider the problem of existence of certain symmetrical solutions of Stokes equation on a three-dimensional manifold $M$ with a general metric possessing symmetry. These solutions correspond to unidirectional flows. We have been able to determine necessary and sufficient conditions for their existence. Symmetric unidirectional flows are fundamental for deducing the so-called Darcy's law, which is the law governing fluid flow in a Hele-Shaw cell embedded in the environment $M$. Our main interest is to depart from the usual, flat background environment, and consider the possibility of an environment of arbitrary constant curvature $K$ in which a cell is embedded. We generalize Darcy's law for particular models of such spaces obtained from $\real^3$ with a conformal metric. We employ the calculus of differential forms for a simpler and more elegant approach to the problems herein discussed.
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