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June  2015, 10(2): 343-367. doi: 10.3934/nhm.2015.10.343

## Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition

 1 University of Wyoming, Department of Mathematics, Dept. 3036, 1000 East University Avenue, Laramie, WY 82071 2 Department of Mathematics & ISC, Texas A&M University, 3404 TAMU, College Station, TX 77843-3404 3 Numerical Porous Media SRI Center, CEMSE Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia

Received  February 2014 Revised  October 2014 Published  April 2015

The evolution Stokes equation in a domain containing periodically distributed obstacles subject to Fourier boundary condition on the boundaries is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the obstacles. We represent the solid obstacles by holes in the fluid domain. The macroscopic (homogenized) equation is derived as another stochastic partial differential equation, defined in the whole non perforated domain. Here, the initial stochastic perturbation on the boundary becomes part of the homogenized equation as another stochastic force. We use the two-scale convergence method after extending the solution with 0 in the holes to pass to the limit. By Itô stochastic calculus, we get uniform estimates on the solution in appropriate spaces. In order to pass to the limit on the boundary integrals, we rewrite them in terms of integrals in the whole domain. In particular, for the stochastic integral on the boundary, we combine the previous idea of rewriting it on the whole domain with the assumption that the Brownian motion is of trace class. Due to the particular boundary condition dealt with, we get that the solution of the stochastic homogenized equation is not divergence free. However, it is coupled with the cell problem that has a divergence free solution. This paper represents an extension of the results of Duan and Wang (Comm. Math. Phys. 275:1508--1527, 2007), where a reaction diffusion equation with a dynamical boundary condition with a noise source term on both the interior of the domain and on the boundary was studied, and through a tightness argument and a pointwise two scale convergence method the homogenized equation was derived.
Citation: Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks & Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343
##### References:
 [1] G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. doi: 10.1137/0523084. [2] G. Allaire, Homogenization of the unsteady Stokes equations in porous media,, Pitman Research Notes in Mathematics Series, 267 (1992), 109. [3] G. Allaire, Homogenization of the Navier-Stokes equations with a slip boundary condition,, Communications on pure and applied mathematics, 44 (1991), 605. doi: 10.1002/cpa.3160440602. [4] X. Blanc, C. Le Bris and P.-L. Lions, Du discret au continu pour des modèles de réseaux aléatoires d'atomes,, Comptes Rendus Mathematique, 342 (2006), 627. doi: 10.1016/j.crma.2005.12.033. [5] A. Bourgeat, A. A. Mikelić and S. Wright, Stochastic two-scale convergence in the mean and applications,, J. Reine Angew. Math., 456 (1994), 19. [6] D. Cioranescu, P. Donato and H. I. Ene, Homogenization of the Stokes problem with nonhomogeneous slip boundary conditions,, Math. Methods in Appl. Sci., 19 (1996), 857. doi: 10.1002/(SICI)1099-1476(19960725)19:11<857::AID-MMA798>3.0.CO;2-D. [7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223. [8] R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3. Spectral Theory and Applications,, With the collaboration of Michel Artola and Michel Cessenat, (1990). [9] N. V. Krylov, A relatively short proof of Itô formula for SPDEs and its applications,, Stoch. PDE: Anal. Comp., 1 (2013), 152. [10] A. Mikelić, Homogenization of nonstationary Navier-Stokes equations in a domain with a grained boundary,, Annali di Matematica pura ed applicata, 158 (1991), 167. doi: 10.1007/BF01759303. [11] A. Mikelić, Effets inertiels pour un écoulement stationnaire visqueux incompressible dans un milieu poreux,, Comptes rendus de l'Acadèmie des sciences. Serie 1, 320 (1995), 1289. [12] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM Journal on Mathematical Analysis, 20 (1989), 608. doi: 10.1137/0520043. [13] G. Nguetseng, Homogenization structures and applications I,, Zeitschrift fur Analysis und ihre Andwendungen, 22 (2003), 73. doi: 10.4171/ZAA/1133. [14] G. Nguetseng, Homogenization structures and applications II,, Zeitschrift fur Analysis und ihre Andwendungen, 23 (2004), 483. doi: 10.4171/ZAA/1208. [15] G. Nguetseng, M. Sango and J. L. Woukeng, Reiterated Ergodic Algebras and Applications,, Communications in Mathematical Physics, 300 (2010), 835. doi: 10.1007/s00220-010-1127-3. [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. [17] P. A. Razafimandimby, M. Sango and J. L. Woukeng, Homogenization of a stochastic nonlinear reaction-diffusion equation with a large reaction term: The almost periodic framework,, Journal of Mathematical Analysis and Applications, 394 (2012), 186. doi: 10.1016/j.jmaa.2012.04.046. [18] E. Sánchez-Palencia, Non-homogeneous Media and Vibration Theory,, Lecture Notes in Physics, (1980). [19] H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach,, Birkhäuser Verlag, (2001). doi: 10.1007/978-3-0348-8255-2. [20] L. Tartar, Incompressible fluid flow in a porous medium-convergence of the homogenization process,, Appendix of Non-Homogeneous Media and Vibration Theory, (1980). [21] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, Studies in Mathematics and its Applications, (1979). [22] C. Timofte, Homogenization results for parabolic problems with dynamical boundary conditions,, Romanian Rep. Phys., 56 (2004), 131. [23] W. Wang, D. Cao and J. Duan, Effective macroscopic dynamics of stochastic partial differential equations in perforated domains,, SIAM Journal on Mathematical Analysis, 38 (2007), 1508. doi: 10.1137/050648766. [24] W. Wang and J. Duan, Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions,, Communications in Mathematical Physics, 275 (2007), 163. doi: 10.1007/s00220-007-0301-8. [25] W. Wang and J. Duan, Reductions and deviations for stochastic partial differential equations under fast dynamical boundary conditions,, Stoch. Anal. and Appl., 27 (2009), 431. doi: 10.1080/07362990802679166. [26] J. L. Woukeng, Homogenization in algebras with mean value,, Banach J. Math. Anal., 9 (2015), 142. doi: 10.15352/bjma/09-2-12.

show all references

##### References:
 [1] G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. doi: 10.1137/0523084. [2] G. Allaire, Homogenization of the unsteady Stokes equations in porous media,, Pitman Research Notes in Mathematics Series, 267 (1992), 109. [3] G. Allaire, Homogenization of the Navier-Stokes equations with a slip boundary condition,, Communications on pure and applied mathematics, 44 (1991), 605. doi: 10.1002/cpa.3160440602. [4] X. Blanc, C. Le Bris and P.-L. Lions, Du discret au continu pour des modèles de réseaux aléatoires d'atomes,, Comptes Rendus Mathematique, 342 (2006), 627. doi: 10.1016/j.crma.2005.12.033. [5] A. Bourgeat, A. A. Mikelić and S. Wright, Stochastic two-scale convergence in the mean and applications,, J. Reine Angew. Math., 456 (1994), 19. [6] D. Cioranescu, P. Donato and H. I. Ene, Homogenization of the Stokes problem with nonhomogeneous slip boundary conditions,, Math. Methods in Appl. Sci., 19 (1996), 857. doi: 10.1002/(SICI)1099-1476(19960725)19:11<857::AID-MMA798>3.0.CO;2-D. [7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223. [8] R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3. Spectral Theory and Applications,, With the collaboration of Michel Artola and Michel Cessenat, (1990). [9] N. V. Krylov, A relatively short proof of Itô formula for SPDEs and its applications,, Stoch. PDE: Anal. Comp., 1 (2013), 152. [10] A. Mikelić, Homogenization of nonstationary Navier-Stokes equations in a domain with a grained boundary,, Annali di Matematica pura ed applicata, 158 (1991), 167. doi: 10.1007/BF01759303. [11] A. Mikelić, Effets inertiels pour un écoulement stationnaire visqueux incompressible dans un milieu poreux,, Comptes rendus de l'Acadèmie des sciences. Serie 1, 320 (1995), 1289. [12] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM Journal on Mathematical Analysis, 20 (1989), 608. doi: 10.1137/0520043. [13] G. Nguetseng, Homogenization structures and applications I,, Zeitschrift fur Analysis und ihre Andwendungen, 22 (2003), 73. doi: 10.4171/ZAA/1133. [14] G. Nguetseng, Homogenization structures and applications II,, Zeitschrift fur Analysis und ihre Andwendungen, 23 (2004), 483. doi: 10.4171/ZAA/1208. [15] G. Nguetseng, M. Sango and J. L. Woukeng, Reiterated Ergodic Algebras and Applications,, Communications in Mathematical Physics, 300 (2010), 835. doi: 10.1007/s00220-010-1127-3. [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. [17] P. A. Razafimandimby, M. Sango and J. L. Woukeng, Homogenization of a stochastic nonlinear reaction-diffusion equation with a large reaction term: The almost periodic framework,, Journal of Mathematical Analysis and Applications, 394 (2012), 186. doi: 10.1016/j.jmaa.2012.04.046. [18] E. Sánchez-Palencia, Non-homogeneous Media and Vibration Theory,, Lecture Notes in Physics, (1980). [19] H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach,, Birkhäuser Verlag, (2001). doi: 10.1007/978-3-0348-8255-2. [20] L. Tartar, Incompressible fluid flow in a porous medium-convergence of the homogenization process,, Appendix of Non-Homogeneous Media and Vibration Theory, (1980). [21] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, Studies in Mathematics and its Applications, (1979). [22] C. Timofte, Homogenization results for parabolic problems with dynamical boundary conditions,, Romanian Rep. Phys., 56 (2004), 131. [23] W. Wang, D. Cao and J. Duan, Effective macroscopic dynamics of stochastic partial differential equations in perforated domains,, SIAM Journal on Mathematical Analysis, 38 (2007), 1508. doi: 10.1137/050648766. [24] W. Wang and J. Duan, Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions,, Communications in Mathematical Physics, 275 (2007), 163. doi: 10.1007/s00220-007-0301-8. [25] W. Wang and J. Duan, Reductions and deviations for stochastic partial differential equations under fast dynamical boundary conditions,, Stoch. Anal. and Appl., 27 (2009), 431. doi: 10.1080/07362990802679166. [26] J. L. Woukeng, Homogenization in algebras with mean value,, Banach J. Math. Anal., 9 (2015), 142. doi: 10.15352/bjma/09-2-12.
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