Convergence of the Stochastic Navier–Stokes-\(\alpha \) Solutions Toward the Stochastic Navier–Stokes Solutions

Abstract

Loosely speaking, the Navier–Stokes-\(\alpha \) model and the Navier–Stokes equations differ by a spatial filtration parametrized by a scale denoted \(\alpha \). Starting from a strong two-dimensional solution to the Navier–Stokes-\(\alpha \) model driven by a multiplicative noise, we demonstrate that it generates a strong solution to the stochastic Navier–Stokes equations under the condition \(\alpha \rightarrow 0\). The initially introduced probability space and the Wiener process are maintained throughout the investigation, thanks to a local monotonicity property that abolishes the use of Skorokhod’s theorem. High spatial regularity a priori estimates for the fluid velocity vector field are carried out within periodic boundary conditions.