# Homogenization of the stochastic Navier–Stokes equation with a stochastic slip boundary condition

Handle URI:
http://hdl.handle.net/10754/622413
Title:
Homogenization of the stochastic Navier–Stokes equation with a stochastic slip boundary condition
Authors:
Bessaih, Hakima; Maris, Razvan Florian ( 0000-0002-7196-6967 )
Abstract:
The two-dimensional Navier–Stokes equation in a perforated domain with a dynamical slip boundary condition 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 holes. We consider a scaling (ᵋ for the viscosity and 1 for the density) that will lead to a time-dependent limit problem. However, the noncritical scaling (ᵋ, β > 1) is considered in front of the nonlinear term. The homogenized system in the limit is obtained as a Darcy’s law with memory with two permeabilities and an extra term that is due to the stochastic perturbation on the boundary of the holes. The nonhomogeneity on the boundary contains a stochastic part that yields in the limit an additional term in the Darcy’s law. We use the two-scale convergence method after extending the solution with 0 inside the holes to pass to the limit. By Itô stochastic calculus, we get uniform estimates on the solution in appropriate spaces. Due to the stochastic integral, the pressure that appears in the variational formulation does not have enough regularity in time. This fact made us rely only on the variational formulation for the passage to the limit on the solution. We obtain a variational formulation for the limit that is solution of a Stokes system with two pressures. This two-scale limit gives rise to three cell problems, two of them give the permeabilities while the third one gives an extra term in the Darcy’s law due to the stochastic perturbation on the boundary of the holes.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Numerical Porous Media SRI Center (NumPor)
Citation:
Bessaih H, Maris F (2015) Homogenization of the stochastic Navier–Stokes equation with a stochastic slip boundary condition. Applicable Analysis 95: 2703–2735. Available: http://dx.doi.org/10.1080/00036811.2015.1107546.
Publisher:
Informa UK Limited
Journal:
Applicable Analysis
Issue Date:
2-Nov-2015
DOI:
10.1080/00036811.2015.1107546
Type:
Article
ISSN:
0003-6811; 1563-504X
Hakima Bessaih was partially supported by NSF [grant number DMS-1418838].
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

DC FieldValue Language
dc.contributor.authorBessaih, Hakimaen
dc.contributor.authorMaris, Razvan Florianen
dc.date.accessioned2017-01-02T09:28:29Z-
dc.date.available2017-01-02T09:28:29Z-
dc.date.issued2015-11-02en
dc.identifier.citationBessaih H, Maris F (2015) Homogenization of the stochastic Navier–Stokes equation with a stochastic slip boundary condition. Applicable Analysis 95: 2703–2735. Available: http://dx.doi.org/10.1080/00036811.2015.1107546.en
dc.identifier.issn0003-6811en
dc.identifier.issn1563-504Xen
dc.identifier.doi10.1080/00036811.2015.1107546en
dc.identifier.urihttp://hdl.handle.net/10754/622413-
dc.description.abstractThe two-dimensional Navier–Stokes equation in a perforated domain with a dynamical slip boundary condition 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 holes. We consider a scaling (ᵋ for the viscosity and 1 for the density) that will lead to a time-dependent limit problem. However, the noncritical scaling (ᵋ, β > 1) is considered in front of the nonlinear term. The homogenized system in the limit is obtained as a Darcy’s law with memory with two permeabilities and an extra term that is due to the stochastic perturbation on the boundary of the holes. The nonhomogeneity on the boundary contains a stochastic part that yields in the limit an additional term in the Darcy’s law. We use the two-scale convergence method after extending the solution with 0 inside the holes to pass to the limit. By Itô stochastic calculus, we get uniform estimates on the solution in appropriate spaces. Due to the stochastic integral, the pressure that appears in the variational formulation does not have enough regularity in time. This fact made us rely only on the variational formulation for the passage to the limit on the solution. We obtain a variational formulation for the limit that is solution of a Stokes system with two pressures. This two-scale limit gives rise to three cell problems, two of them give the permeabilities while the third one gives an extra term in the Darcy’s law due to the stochastic perturbation on the boundary of the holes.en
dc.description.sponsorshipHakima Bessaih was partially supported by NSF [grant number DMS-1418838].en
dc.publisherInforma UK Limiteden
dc.subjectboundary noiseen
dc.subjectHomogenizationen
dc.subjectNavier–Stokes equationsen
dc.subjectPerforated mediumen
dc.subjectSlip boundary conditionen
dc.titleHomogenization of the stochastic Navier–Stokes equation with a stochastic slip 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.journalApplicable Analysisen
dc.contributor.institutionDepartment of Mathematics, University of Wyoming, Laramie, WY, United Statesen
kaust.authorMaris, Razvan Florianen