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On the rate of convergence of the 2-D stochastic Leray-$$\alpha$$ model to the 2-D stochastic Navier-Stokes equations with multiplicative noise. (English) Zbl 1346.60093
Summary: 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 $$\alpha \to 0$$, of the following error function $\varepsilon_\alpha (t)=\sup_{s\in [0,t]} |\mathbf {u}^\alpha (s)-\mathbf {u}(s)|+\left( \int_0^t |\mathrm {A}^\frac{1}{2}[\mathbf {u}^\alpha (s)-\mathbf {u}(s)] |^2 ds \right) ^\frac{1}{2},$ 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 $$\alpha \to 0$$ and the convergence is of order $$O(\alpha)$$. We also prove that $$\varepsilon_\alpha$$ converges in probability to zero with order at most $$O(\alpha)$$.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 76D05 Navier-Stokes equations for incompressible viscous fluids 60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
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