On the Rate of Convergence of the 2-D Stochastic Leray-$\alpha$ Model to the 2-D Stochastic Navier–Stokes Equations with Multiplicative Noise

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, as $\alpha \rightarrow 0$, of the following error function

$$\begin{aligned} \varepsilon _\alpha (t)=\sup _{s\in [0,t]} |\mathbf {u}^\alpha (s)-\mathbf {u}(s)|+\left( \int _0^t |\mathrm {A}^\frac{1}{2}[\mathbf {u}^\alpha (s)-\mathbf {u}(s)] |^2 ds \right) ^\frac{1}{2}, \end{aligned}$$

where $\mathbf {u}^\alpha$ and $\mathbf {u}$ are the solution of stochastic Leray-$\alpha$ model and the stochastic Navier–Stokes equations, respectively. We show that when properly localized the error function $\varepsilon _\alpha$ converges in mean square as $\alpha \rightarrow 0$ and the convergence is of order $O(\alpha )$. We also prove that $\varepsilon _\alpha$ converges in probability to zero with order at most $O(\alpha )$.