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This paper revisits a homogenization problem studied by L. Tartar related to a tridimensional Stokes equation perturbed by a drift (related to the Coriolis force). Here, a scalar equation and a two-dimensional Stokes equation with a -bounded oscillating drift are considered. Under higher integrability conditions the Tartar approach based on the oscillations test functions method applies and leads to a limit equation with an extra zero-order term. When the drift is only assumed to be equi-integrable in , the same limit behaviour is obtained. However, the lack of integrability makes difficult the direct use of the Tartar method. A new method in the context of homogenization theory is proposed. It is based on a parametrix of the Laplace operator which permits to write the solution of the equation as a solution of a fixed point problem, and to use truncated functions even in the vector-valued case. On the other hand, two counter-examples which induce different homogenized zero-order terms actually show the sharpness of the equi-integrability assumption.
Briane, Marc 1 ; Gérard, Patrick 2
@article{ASNSP_2012_5_11_1_1_0,
author = {Briane, Marc and G\'erard, Patrick},
title = {A drift homogenization problem revisited},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {1--39},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 11},
number = {1},
year = {2012},
mrnumber = {2953043},
zbl = {1270.35058},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ASNSP_2012_5_11_1_1_0/}
}
TY - JOUR AU - Briane, Marc AU - Gérard, Patrick TI - A drift homogenization problem revisited JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2012 SP - 1 EP - 39 VL - 11 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://geodesic.mathdoc.fr/item/ASNSP_2012_5_11_1_1_0/ LA - en ID - ASNSP_2012_5_11_1_1_0 ER -
%0 Journal Article %A Briane, Marc %A Gérard, Patrick %T A drift homogenization problem revisited %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2012 %P 1-39 %V 11 %N 1 %I Scuola Normale Superiore, Pisa %U http://geodesic.mathdoc.fr/item/ASNSP_2012_5_11_1_1_0/ %G en %F ASNSP_2012_5_11_1_1_0
Briane, Marc; Gérard, Patrick. A drift homogenization problem revisited. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 1, pp. 1-39. http://geodesic.mathdoc.fr/item/ASNSP_2012_5_11_1_1_0/