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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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