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

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    Type
    Article
    Authors
    Bessaih, Hakima
    Maris, Razvan Florian cc
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Numerical Porous Media SRI Center (NumPor)
    Date
    2015-11-02
    Online Publication Date
    2015-11-02
    Print Publication Date
    2016-12
    Permanent link to this record
    http://hdl.handle.net/10754/622413
    
    Metadata
    Show full item record
    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.
    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.
    Sponsors
    Hakima Bessaih was partially supported by NSF [grant number DMS-1418838].
    Publisher
    Informa UK Limited
    Journal
    Applicable Analysis
    DOI
    10.1080/00036811.2015.1107546
    arXiv
    1411.6303
    ae974a485f413a2113503eed53cd6c53
    10.1080/00036811.2015.1107546
    Scopus Count
    Collections
    Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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