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On some nonlocal homogenization problems and problems of fluids in porous media: a contribution of Abdelhamid Ziani. (Sur quelques problèmes d’homogénéisation non locale et de fluides en milieu poreux: Une contribution de Abdelhamid Ziani.) (French) Zbl 1158.35011

The authors study different homogenization problems, which have the common features that they describe fluid flows in porous media, that they are degenerate and that they lead to nonlocal problems, when taking the limit with respect to the small parameter associated to the structure of the porous medium. Another common feature is that A. Ziani contributed to the study of these problems.
The authors start with the transport equation \( \partial _{t}u^{\varepsilon }+A^{\varepsilon }(t,x)\nabla _{x}u^{\varepsilon }=f\), \(\nabla \cdot A^{\varepsilon }=0\), with the initial condition \( u^{\varepsilon }(.,0)=u^{0}\). They first introduce an asymptotic expansion method in order to characterize the first-order terms. They then describe the case of the dimension 1, using Fourier (in \(x\)) and Laplace (in \(t\)) transforms. They prove that the limit \(\Phi \) of \(\Phi ^{\varepsilon }(z,y)=(z-a^{\varepsilon }(y))^{-1}\) involves a Young measure \(d\nu _{y}\) and that \(1/\Phi \) involves a measure \(d\omega _{y}\). The authors study this positive measure \(d\omega _{y}\) when the coefficient \(a^{\varepsilon }\) is given a discrete periodic structure. The authors then move to the higher-dimensional case. In a further section, the authors consider a hyperbolic-parabolic problem. In the sixth section, the authors consider a 1D Riccati differential equation \(\partial _{t}u_{\varepsilon }+a_{\varepsilon }(t,x)u_{\varepsilon }^{2}=0\), \(u_{\varepsilon }(0,x)=0\), assuming a special shape of the coefficient \(a_{\varepsilon }\).
The paper ends with studies concerning the flows of miscible and compressible or incompressible fluids in porous media. In each case, the authors describe the main tools which are involved when describing the asymptotic behaviour and refer to the list of references for the details.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35Q35 PDEs in connection with fluid mechanics
76S05 Flows in porous media; filtration; seepage
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References:

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