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

TL;DR

This paper studies the homogenization of a stochastic Navier-Stokes equation with slip boundary conditions in perforated domains, deriving a Darcy's law with memory and additional stochastic boundary effects using two-scale convergence.

Contribution

It introduces a novel homogenization approach for stochastic Navier-Stokes equations with boundary perturbations, resulting in a Darcy's law with memory and two permeabilities.

Findings

Derived a homogenized Darcy's law with memory and stochastic boundary effects.

Identified two permeabilities and an extra stochastic term from boundary perturbations.

Established a variational formulation for the limit system with two pressures.

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 ($\e^2$ for the viscosity and 1 for the density) that will lead to a time dependent limit problem. However, the noncritical scaling ($\e^\beta,\quad \beta>1$) 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. We use the two-scale convergence method. 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.