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.

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:
Classification: 35B27, 76M50
Briane, Marc 1; Gérard, Patrick 2

1 INSA de Rennes IRMAR, CNRS UMR 6625 35708 Rennes Cedex 7, France
2 Université Paris-Sud LMO, CNRS UMR 8628 91405 Orsay, France Institut Universitaire de France
@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://archive.numdam.org/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://archive.numdam.org/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://archive.numdam.org/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, Serie 5, Volume 11 (2012) no. 1, pp. 1-39. http://archive.numdam.org/item/ASNSP_2012_5_11_1_1_0/

[1] S. Alinhac and P. Gérard, “Pseudo-differential Operators and the Nash-Moser Theorem”, translated from the 1991 French original by Stephen S. Wilson, Graduate Studies in Mathematics, Vol. 82, American Mathematical Society, Providence, RI, 2007. | MR | Zbl

[2] G. Allaire, Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes I, Abstract framework, a volume distribution of holes, Arch. Rational Mech. Anal., 113 (1991), 209–259. | MR | Zbl

[3] M.E. Bogovski, Solution of the of the first boundary value problem for the equation of continuity of an incompressible medium, Soviet Math. Dokl. 20 (1979), 1094–1098. | MR | Zbl

[4] M. Briane, Homogenization of the Stokes equations with high-contrast viscosity, J. Math. Pures Appl. 82 (7) (2003), 843–876. | MR | Zbl

[5] H.C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res. A1 (1947), 27–34. | Zbl

[6] D. Cioranescu and F. Murat, Un terme étrange venu d’ailleurs, I & II, In: “Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar”, II and III, H. Brezis and J.-L. Lions (eds.), Research Notes in Math., Vol. 60 and 70, Pitman, London, 1982, 98–138 and 154–178. English translation: A strange term coming from nowhere, In: “Topics in the Mathematical Modelling of Composite Materials”, A. Cherkaev and R. V. Kohn. (eds.), Progress in Nonlinear Differential Equations and their Applications, Vol. 31, Birkhäuser, Boston, 1997, 44–93. | MR | Zbl

[7] G. Dal Maso and A. Garroni, New results on the asymptotic behaviour of Dirichlet problems in perforated domains, Math. Models Methods Appl. Sci. 3 (1994), 373–407. | MR | Zbl

[8] P. Gérard, Micro-local defect measures, Comm. Partial Differential Equations 16 (1991), 1761–1794. | MR | Zbl

[9] G. Geymonat, Sul problema di Dirichlet per le equazioni lineari ellittiche, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 16 (1962), 225–284. | EuDML | Numdam | MR | Zbl

[10] O. A. Ladyzhenskaya, “The Mathematical Theory of Viscous Incompressible Flow”, translated from the Russian, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. | MR | Zbl

[11] F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5 (1978), 489–507. | EuDML | Numdam | MR | Zbl

[12] C. Pideri and P. Seppecher, A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium, Contin. Mech. Thermodyn. 9 (1997), 241–257. | MR | Zbl

[13] E. Sanchez-Palencia, “Non Homogeneous Materials and Vibration Theory”, Monographs in Physics, Vol. 127, Springer-Verlag, Berlin, 1980. | MR | Zbl

[14] L. Tartar, Homogénéisation en hydrodynamique, In: “Singular Perturbation and Boundary Layer Theory”, Lecture Notes in Mathematics, Vol. 597, Springer, Berlin-Heidelberg, 1977, 474–481. | MR | Zbl

[15] L. Tartar, Remarks on homogenization, In: “Homogenization and Effective Moduli of Materials and Media”, IMA Vol. Math. Appl., Vol. 1, Springer, New-York 1986, 228–246. | MR | Zbl

[16] L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 115A (1990), 193–230. | MR | Zbl

[17] L. Tartar, “The General Theory of Homogenization: A Personalized Introduction”, Lecture Notes of the Unione Matematica Italiana, Springer-Verlag, Berlin Heidelberg, 2009. | MR | Zbl

[18] L. Tartar, “Topics in Nonlinear Analysis", Publications Mathématiques d’Orsay, Vol. 78, Orsay 1978. | MR | Zbl