On the rate of convergence of the 2-D stochastic Leray-$\alpha$ model to the 2-D stochastic Navier-Stokes equations with multiplicative noise

TL;DR

This paper investigates how quickly solutions of the 2-D stochastic Leray-$\alpha$ model approach the 2-D stochastic Navier-Stokes equations as the parameter $\alpha$ tends to zero, establishing convergence rates in mean square and probability.

Contribution

It provides the first quantitative analysis of the convergence rate of the stochastic Leray-$\alpha$ model to the stochastic Navier-Stokes equations in two dimensions.

Findings

Mean square convergence of order O(α)

Convergence in probability with order at most O(α)

Localized error function converges as α→0

Abstract

In the present paper we study the convergence of the solution of the two dimensional (2-D) stochastic Leray-$\alpha$ model to the solution of the 2-D stochastic Navier-Stokes equations. We are mainly interested in the rate of convergence, as $\alpha$ tends to 0, of the error function which is the difference between the two solutions in an appropriate topology. We show that when properly localized the error function converges in mean square as $\alpha\to 0$ and the convergence is of order $O(\alpha)$. We also prove that the error converges in probability to zero with order at most $O(\alpha)$.