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Derivation of microstructure models of fluid flow by homogenization. (English) Zbl 0927.35081
The author deals with the case of flow in a medium composed of two materials with very different properties by homogenization. The main result is as follows. Let $$\Omega$$ be an open subset of $$\mathbb{R}^N$$ and $$Y= [0,1]^N$$. Following G. Allaira [SIAM J. Math. Anal. 23, No. 6, 1482-1518 (1992; Zbl 0770.35005)]. $$Y$$ is divided into two parts $$Y_1$$ and $$Y_2$$. $$\Omega$$ is assumed to have an $$\varepsilon Y$$ periodic structure. The coefficient functions $$a_{ij}(t,y)\in C([0, T]\times Y_1)$$ and $$A_{ij}(t, y)\in C([0, T]\times Y_2)$$ are uniformly positive definite and symmetric. Off their respective domains they are defined to be zero. Then, there exists a couple of functions $$u(x,t)$$ and $$U(x,t,y)$$ which is the unique solution of the microstructure model: ${\partial u\over\partial t}- {\partial\over\partial x_i} \Biggl(a^h_{ij}{\partial u\over\partial x_j}\Biggr)+ \int_{\partial Y_2} A_{ij}{\partial U\over\partial y_j} \nu_i ds=f\quad\text{in }\Omega\times(0, T),$
${\partial U\over\partial t}-{\partial\over\partial y_j} \Biggl(A_{ij}{\partial U\over\partial y_j}\Biggr)= 0\quad\text{in }\Omega\times Y_2\times (0,T),$
$U(x,t,x)= u(x,t)\quad\text{on }\partial Y_2,$
$U(x,0,y)= U_0(x,y)\quad\text{in }\Omega\times Y_2,$
$u(x,0)= u_0(x)\quad\text{in }\Omega,$ where $$a^h_{ij}(t)$$ is the homogenized coefficient of $$a_{ij}(t,y)$$.

MSC:
 35Q35 PDEs in connection with fluid mechanics 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 76S05 Flows in porous media; filtration; seepage
Zbl 0770.35005
Full Text:
References:
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