Homogenization of the stochastic non-homogeneous incompressible Navier-Stokes equations with multiplicative noise

Abstract: In this contributions we are interested in the homogenization property of stochastic non-homogeneous incompressible Navier-Stokes equations with fast oscillation in a smooth bounded domain of $\mathbb{R}^d$, $d=2,3$, and driven by multiplicative cylindrical Wiener noise. Using two-scale convergence, stochastic compactness and the martingale representative theory, we show the solutions of original equations converge to a solution of stochastic non-homogeneous incompressible version with constant coefficients. The paper also includes a corrector result which improves the two-scale convergence in the weak sense to strong one in regularity space. The main obstacles arise from the random effect and the lower regularity caused by density function.