TY - JOUR

T1 - Homogenization of the stochastic Navier–Stokes equation with a stochastic slip boundary condition

AU - Bessaih, Hakima

AU - Maris, Razvan Florian

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: Hakima Bessaih was partially supported by NSF [grant number DMS-1418838].

PY - 2015/11/2

Y1 - 2015/11/2

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/622413

UR - https://www.tandfonline.com/doi/full/10.1080/00036811.2015.1107546

UR - http://www.scopus.com/inward/record.url?scp=84946433672&partnerID=8YFLogxK

U2 - 10.1080/00036811.2015.1107546

DO - 10.1080/00036811.2015.1107546

M3 - Article

SN - 0003-6811

VL - 95

SP - 2703

EP - 2735

JO - Applicable Analysis

JF - Applicable Analysis

IS - 12

ER -