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

On the Rate of Convergence of the 2-D Stochastic Leray- $$\alpha$$ α Model to the 2-D... In the present paper we study the convergence of the solution of the two dimensional (2-D) stochastic Leray- $$\alpha$$ α model to the solution of the 2-D stochastic Navier–Stokes equations. We are mainly interested in the rate, as $$\alpha \rightarrow 0$$ α → 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} ε α ( t ) = sup s ∈ [ 0 , t ] | u α ( s ) - u ( s ) | + ∫ 0 t | A 1 2 [ u α ( s ) - u ( s ) ] | 2 d s 1 2 , where $$\mathbf {u}^\alpha$$ u α and $$\mathbf {u}$$ u are the solution of stochastic Leray- $$\alpha$$ α model and the stochastic Navier–Stokes equations, respectively. We show that when properly localized the error function $$\varepsilon _\alpha$$ ε α converges in mean square as $$\alpha \rightarrow 0$$ α → 0 and the convergence is of order $$O(\alpha )$$ O ( α ) . We also prove that $$\varepsilon _\alpha$$ ε α converges in probability to zero with order at most $$O(\alpha )$$ O ( α ) . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

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

, Volume 74 (1) – Jun 16, 2015
25 pages

/lp/springer_journal/on-the-rate-of-convergence-of-the-2-d-stochastic-leray-alpha-model-to-pHOCSeS3H0
Publisher
Springer US
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-015-9303-7
Publisher site
See Article on Publisher Site

### Abstract

In the present paper we study the convergence of the solution of the two dimensional (2-D) stochastic Leray- $$\alpha$$ α model to the solution of the 2-D stochastic Navier–Stokes equations. We are mainly interested in the rate, as $$\alpha \rightarrow 0$$ α → 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} ε α ( t ) = sup s ∈ [ 0 , t ] | u α ( s ) - u ( s ) | + ∫ 0 t | A 1 2 [ u α ( s ) - u ( s ) ] | 2 d s 1 2 , where $$\mathbf {u}^\alpha$$ u α and $$\mathbf {u}$$ u are the solution of stochastic Leray- $$\alpha$$ α model and the stochastic Navier–Stokes equations, respectively. We show that when properly localized the error function $$\varepsilon _\alpha$$ ε α converges in mean square as $$\alpha \rightarrow 0$$ α → 0 and the convergence is of order $$O(\alpha )$$ O ( α ) . We also prove that $$\varepsilon _\alpha$$ ε α converges in probability to zero with order at most $$O(\alpha )$$ O ( α ) .

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Jun 16, 2015

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