# American Institute of Mathematical Sciences

• Previous Article
Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology
• NHM Home
• This Issue
• Next Article
On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime
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.  Google Scholar [2] G. Allaire, Homogenization of the unsteady Stokes equations in porous media,, Pitman Research Notes in Mathematics Series, 267 (1992), 109.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992).  doi: 10.1017/CBO9780511666223.  Google Scholar [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).   Google Scholar [9] N. V. Krylov, A relatively short proof of Itô formula for SPDEs and its applications,, Stoch. PDE: Anal. Comp., 1 (2013), 152.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [13] G. Nguetseng, Homogenization structures and applications I,, Zeitschrift fur Analysis und ihre Andwendungen, 22 (2003), 73.  doi: 10.4171/ZAA/1133.  Google Scholar [14] G. Nguetseng, Homogenization structures and applications II,, Zeitschrift fur Analysis und ihre Andwendungen, 23 (2004), 483.  doi: 10.4171/ZAA/1208.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] E. Sánchez-Palencia, Non-homogeneous Media and Vibration Theory,, Lecture Notes in Physics, (1980).   Google Scholar [19] H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach,, Birkhäuser Verlag, (2001).  doi: 10.1007/978-3-0348-8255-2.  Google Scholar [20] L. Tartar, Incompressible fluid flow in a porous medium-convergence of the homogenization process,, Appendix of Non-Homogeneous Media and Vibration Theory, (1980).   Google Scholar [21] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, Studies in Mathematics and its Applications, (1979).   Google Scholar [22] C. Timofte, Homogenization results for parabolic problems with dynamical boundary conditions,, Romanian Rep. Phys., 56 (2004), 131.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] J. L. Woukeng, Homogenization in algebras with mean value,, Banach J. Math. Anal., 9 (2015), 142.  doi: 10.15352/bjma/09-2-12.  Google Scholar

show all references

##### References:
 [1] G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482.  doi: 10.1137/0523084.  Google Scholar [2] G. Allaire, Homogenization of the unsteady Stokes equations in porous media,, Pitman Research Notes in Mathematics Series, 267 (1992), 109.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [7] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992).  doi: 10.1017/CBO9780511666223.  Google Scholar [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).   Google Scholar [9] N. V. Krylov, A relatively short proof of Itô formula for SPDEs and its applications,, Stoch. PDE: Anal. Comp., 1 (2013), 152.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [13] G. Nguetseng, Homogenization structures and applications I,, Zeitschrift fur Analysis und ihre Andwendungen, 22 (2003), 73.  doi: 10.4171/ZAA/1133.  Google Scholar [14] G. Nguetseng, Homogenization structures and applications II,, Zeitschrift fur Analysis und ihre Andwendungen, 23 (2004), 483.  doi: 10.4171/ZAA/1208.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] E. Sánchez-Palencia, Non-homogeneous Media and Vibration Theory,, Lecture Notes in Physics, (1980).   Google Scholar [19] H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach,, Birkhäuser Verlag, (2001).  doi: 10.1007/978-3-0348-8255-2.  Google Scholar [20] L. Tartar, Incompressible fluid flow in a porous medium-convergence of the homogenization process,, Appendix of Non-Homogeneous Media and Vibration Theory, (1980).   Google Scholar [21] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis,, Studies in Mathematics and its Applications, (1979).   Google Scholar [22] C. Timofte, Homogenization results for parabolic problems with dynamical boundary conditions,, Romanian Rep. Phys., 56 (2004), 131.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] J. L. Woukeng, Homogenization in algebras with mean value,, Banach J. Math. Anal., 9 (2015), 142.  doi: 10.15352/bjma/09-2-12.  Google Scholar
 [1] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [2] Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 [3] Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020385 [4] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [5] Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268 [6] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [7] Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, , () : -. doi: 10.3934/era.2020122 [8] Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453 [9] Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323 [10] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [11] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [12] Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267 [13] Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052 [14] Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $p$ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442 [15] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [16] Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054 [17] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047 [18] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [19] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320 [20] Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

2019 Impact Factor: 1.053

## Metrics

• PDF downloads (15)
• HTML views (0)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]