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

Loading next page...
 
/lp/springer_journal/on-the-rate-of-convergence-of-the-2-d-stochastic-leray-alpha-model-to-pHOCSeS3H0
Publisher
Springer US
Copyright
Copyright © 2015 by Springer Science+Business Media New York
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

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off