ABSTRACT
Problems of transmission, transport and flow in networked systems are currently very much under consideration in various industrial and medical applications, such as bloodflow in arteries, the flow of gas in networks of pipelines, the flow of water in sewer or irrigational systems, heat flow in problems of hyperthermia, and so forth. It is typically not reasonable, and moreover numerically prohibitive, to study such problems on the level of each subdomain, taking into account the local representations of the flow in terms of partial differential equations. More precisely, in such networks there are mushy regions, where a local resolution is not called for, and where instead an average model should be imposed to describe the dynamics. The underlying mathematical procedure behind this point of view is the theory of homogenization of partial differential equations. There is a vast literature on homogenization results available dealing with linear elliptic, parabolic and hyperbolic equations as such. Far less is known in the context of nonlinear equations. It is not possible to provide a proper account of the activities in this field. See, for example, Zhikov [33] and Benssoussan, and Lions and Papanicolau [1] as general references. The theory of homogenization is most complete and well established for periodic structures, where typically the results concern equations on perforated domains or so-called reticulated structures. See Cioranescu and Saint-Jean
One of for the number of papers and textbooks on periodic homogenization is that the homogenized limits can be computed explicitly, at least in principle. This is generally not true for nonperiodic problems. It is, however, only very recently that results have been reported in the context of homogenization of partial differential equations on thin networks, like graphs. See Bouchittee and Fragala [2-4] and Zhikov [29-31], Panasenko [25,26] and Chechkin and Zhikov [7] and Lenczner [21], where linear elliptic problems have been considered.
