Convergence of the stochastic Navier-Stokes-$\alpha$ solutions toward the stochastic Navier-Stokes solutions

TL;DR

This paper proves that solutions to the stochastic Navier-Stokes-$\alpha$ model converge to solutions of the stochastic Navier-Stokes equations as the parameter $\alpha$ approaches zero, using a novel approach that avoids Skorokhod's theorem.

Contribution

It establishes convergence of stochastic Navier-Stokes-$\alpha$ solutions to Navier-Stokes solutions without changing the probability space, utilizing a local monotonicity property.

Findings

Convergence of solutions as $\alpha \to 0$.

Maintains original probability space throughout.

Provides high regularity estimates under periodic boundary conditions.

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 $\alpha$ goes to 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.