A drift homogenization problem revisited
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 1, p. 1-39

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 L 2 -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 L 2 , 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.

Published online : 2018-06-21
Classification:  35B27,  76M50
@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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 11},
     number = {1},
     year = {2012},
     pages = {1-39},
     zbl = {1270.35058},
     mrnumber = {2953043},
     language = {en},
     url = {http://www.numdam.org/item/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, Serie 5, Volume 11 (2012) no. 1, pp. 1-39. http://www.numdam.org/item/ASNSP_2012_5_11_1_1_0/

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