Dan Polisevski ... this extension is that the prolongation of the pressure is made in L6/5(12). ... For any e > 0 sufficiently small there exists a restriction operator.
REND. SEM. MAT. UNIVERS. POLITECN. TORINO Vol. 45°, 2 (1987)
Dan Polisevski ON THE HOMOGENIZATION OF FLUID FLOWS THROUGH PERIODIC MEDIA
Summary. The aim of this work is to prove the convergence of the homogenization process for the incompressible fluid flow through a periodic model of porous media in a more general case than that of [2].
1. Preliminaries. Let Z1', z'G{l,2,...,6}, be the side faces of F = [ 0 , 1 [ 3 , and let T be a surface of class C2 included in y, which crosses the boundary of the cube following some regular curves which are reproduced identically on opposite faces; T separates Y into two domains, Ys (the solid part) and Yf (the fluid part), with the property that repeating Y by periodicity, the union of all the fluid parts, respectively the solid parts, is connected in 1R3 and of class C2. The origin of the coordinate system is set in a fluid ball; thus all the corners of Y are surrounded by fluid neighbourhoods. We assume also that if T attains an edge of Y, then the normal to T in that point is the edge itself. Let £2 be an open connected bounded set in IR3, locally located on one side of the boundary 3£2, a manifold of class C2 , composed of a finite number of connected components, and let \p : IR3 -> [0,1 [3 be the function which associates to any real number its fractional part; we say that a function Classificazione per soggetto AMS (MOS, 1980): 40A99, 35Q10
130 / : IR3 -* IR is F-periodic if f = f° y>. Also, for any e E (0,1) we define ipe(x) = y(x/e) 12* = {x E 12 | ye{x) E Yf}
12* = {xea\
/weyj
r e = I2se n 12* . If a fluid flow is considered through this e F-periodic model of porous media, that is in 12* and if the homogenization process is studied, one has to remove the fact that the velocity and the pressure are defined only in 12f, While the velocity can be naturally continued by zero in 12e, the prolongation of the pressure to 12 is not so straight. A construction of such a prolongation can be found in [2] and it is done 2 in L (12) by transposing some special restriction operator from //J (12) to Hj(12f). Unfortunately, it holds only when Ys is strictly contained into Y, that is, from the physical point of view, for bidimensional flows. Also in [2], £25 is defined as the domain obtained from 12 by picking out only the eF5-parts which do not intersect 312 and so the border is obviously monophasic. i Here we extend the above mentioned construction to the geometry already presented at the beginning of the paper, which is, obviously, threedimensional, with connected phases and biphasic boundary. The price for this extension is that the prolongation of the pressure is made in L6/5(12). Finally, we show that this result is enough for the homogenization process.
2. A restriction operator. Our main result is: THEOREM. For any e > 0 sufficiently small there exists a restriction operator ^ g G X ^ M , #J(12*)) such that (A) If u G W{£\$lef) is continued by zero in 12\12* then Reu (B) If u G W^(Sl) and div u = 0, then div (Reu) = 0. (C) For any uE 14^(12) the following estimations hold: (1) (2)
l«e«lt,(n^
