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

Handle URI:
http://hdl.handle.net/10754/594230
Title:
Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition
Authors:
Bessaih, Hakima; Efendiev, Yalchin; Maris, Florin
Abstract:
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 twoscale 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. © American Institute of Mathematical Sciences.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Numerical Porous Media SRI Center (NumPor)
Citation:
Maris F, Efendiev Y, Bessaih H (2015) Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. NHM 10: 343–367. Available: http://dx.doi.org/10.3934/nhm.2015.10.343.
Publisher:
American Institute of Mathematical Sciences (AIMS)
Journal:
Networks and Heterogeneous Media
Issue Date:
Apr-2015
DOI:
10.3934/nhm.2015.10.343
ARXIV:
arXiv:1402.6374v2
Type:
Article
ISSN:
1556-1801
This project started when Hakima Bessaih was visiting the Numerical Porous Media (NumPor) SRI Center at KAUST. She would like to thank the NumPor for the financial support, the excellent working atmosphere and the warm hospitality. H.B. was supported in part by the Simons Foundation grant #283308, the NSF grants DMS-1416689 and DMS-1418838. YE's work is partially supported by the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-FG02-13ER26165 and the DoD Army ARO Project.
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

DC FieldValue Language
dc.contributor.authorBessaih, Hakimaen
dc.contributor.authorEfendiev, Yalchinen
dc.contributor.authorMaris, Florinen
dc.date.accessioned2016-01-19T14:43:54Zen
dc.date.available2016-01-19T14:43:54Zen
dc.date.issued2015-04en
dc.identifier.citationMaris F, Efendiev Y, Bessaih H (2015) Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. NHM 10: 343–367. Available: http://dx.doi.org/10.3934/nhm.2015.10.343.en
dc.identifier.issn1556-1801en
dc.identifier.doi10.3934/nhm.2015.10.343en
dc.identifier.urihttp://hdl.handle.net/10754/594230en
dc.description.abstractThe 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 twoscale 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. © American Institute of Mathematical Sciences.en
dc.description.sponsorshipThis project started when Hakima Bessaih was visiting the Numerical Porous Media (NumPor) SRI Center at KAUST. She would like to thank the NumPor for the financial support, the excellent working atmosphere and the warm hospitality. H.B. was supported in part by the Simons Foundation grant #283308, the NSF grants DMS-1416689 and DMS-1418838. YE's work is partially supported by the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-FG02-13ER26165 and the DoD Army ARO Project.en
dc.publisherAmerican Institute of Mathematical Sciences (AIMS)en
dc.subjectHomogenizationen
dc.subjectPerforated mediumen
dc.subjectStochastic boundary conditionen
dc.subjectStokes flowsen
dc.titleHomogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary conditionen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentNumerical Porous Media SRI Center (NumPor)en
dc.identifier.journalNetworks and Heterogeneous Mediaen
dc.contributor.institutionUniversity of Wyoming, Department of Mathematics, Dept. 3036, 1000 East University Avenue, Laramie, WY, United Statesen
dc.contributor.institutionDepartment of Mathematics, Texas A and M University, College Station, TX, United Statesen
dc.identifier.arxividarXiv:1402.6374v2en
kaust.authorMaris, Florinen