INCOMPRESSIBLE VISCOUS NEWTONIAN FLOW IN A FISSURED MEDIUM OF GENERAL DETERMINISTIC TYPE HERMANN DOUANLA, GABRIEL NGUETSENG, AND JEAN LOUIS WOUKENG Abstract. In some recent papers, the homogenization beyond the periodic setting was addressed in a general deterministic environment. All the prob- lems studied were dealing with the homogenization in fixed domains as well as in porous domains. In the present paper we show that such problems may still be solved in general deterministic fissured media composed of blocks of an ordinary porous medium with fissures between them. To illustrate this, we consider the flow of a fluid governed by non-stationary Navier-Stokes systems. Assuming that the blocks of the porous medium consist of deterministically distributed inclusions, and under some deterministic assumptions on the elas- ticity tensors associated to the corresponding problem for the fissured system, we get a homogenized problem of the Navier-Stokes’ type for Newtonian fluid in a fixed domain. Our setting includes as special cases the classical peri- odic framework, the almost periodic one and some others. By means of the Σ-convergence method combined with some appropriate convergence results, the limiting behavior is studied without extending the solutions to the whole domain. 1. Introduction and the model A fissured porous medium is a structure made up of blocks of an usual porous medium with fissures between them. The blocks form the porous matrix while the fissures are characterized by substantially higher flow rates and lower relative volume. In order to study the flow of a fluid in a fractured porous medium, several mathematical models have been suggested and upscaled by homogenization. The classical and most studied model named double diffusion model was introduced by Barenblatt et al. [3] in 1960. It has been further developed by Warren and Root [37], Coats and Smith [10], and some other researchers, see e.g., [2, 19, 21, 28, 29, 30, 35, 40]. All the previous models are characterized by the common fact that they have been investigated under the classical equidistribution assumption on the geometry of the medium under consideration. In this paper we propose an approach to handle the general case where the medium is fissured in a deterministic manner including the special case of equidis- tribution commonly known as the periodic setting in the literature. To be precise, assume throughout the paper that N = 2 or 3 and let Y = (0, 1) N be the refer- ence cell. Let Y 1 and Y 2 be two open disjoint subsets of Y representing the local structure of the porous matrix and the local structure of fissures, respectively. We assume that Y 1 ⊂ Y , Y = Y 1 ∪ Y 2 , Y 2 is connected and that the boundary ∂Y 1 Date : December, 2012. 2000 Mathematics Subject Classification. 35B40, 46J10, 76Bxx. Key words and phrases. Deterministic fissured medium, homogenization, algebra with mean value, Σ-convergence, Navier-Stokes equations. 1