A drift homogenization problem revisited
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Marc Briane [2] ; Patrick Gérard [1]
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[1] University of Paris-Sud
University of Paris-Sud
Arrondissement de Palaiseau, Francia
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[2] INSA de Rennes, France
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- Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 11, Nº 1, 2012, págs. 1-39
- Idioma: inglés
- DOI: 10.2422/2036-2145.201006_003
- Enlaces
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Resumen
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 L2-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 L2, 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.
