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

Hakima Bessaih; Paul Andre Razafimandimby

doi:10.1007/s00245-015-9303-7

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

Hakima Bessaih, Paul Andre Razafimandimby

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

In the present paper we study the convergence of the solution of the two dimensional (2-D) stochastic Leray-α model to the solution of the 2-D stochastic Navier–Stokes equations. We are mainly interested in the rate, as α→ 0 , of the following error function (Formula presented.) and u are the solution of stochastic Leray-α model and the stochastic Navier–Stokes equations, respectively. We show that when properly localized the error function εα converges in mean square as α→ 0 and the convergence is of order O(α). We also prove that εα converges in probability to zero with order at most O(α).

Original language	English (US)
Pages (from-to)	1-25
Number of pages	25
Journal	Applied Mathematics and Optimization
Volume	74
Issue number	1
DOIs	https://doi.org/10.1007/s00245-015-9303-7
State	Published - Jun 16 2015
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2022-06-03
Acknowledgements: Paul Razafimandimby’s research was funded by the FWF-Austrian Science Fund through the Project M1487. Hakima Bessaih was supported in part by the Simons Foundation Grant #283308 and the NSF Grants DMS-1416689 and DMS-1418838. The research on this paper was initiated during the visit of Paul Razafimandimby at University of Wyoming in November 2013 and was finished while Paul Razafimandimby and Hakima Bessaih were visiting KAUST. They are both very grateful to both institutions for the warm and kind hospitality and great scientific atmosphere.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

Control and Optimization
Applied Mathematics

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title = "On the Rate of Convergence of the 2-D Stochastic Leray- 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-α model to the solution of the 2-D stochastic Navier–Stokes equations. We are mainly interested in the rate, as α→ 0 , of the following error function (Formula presented.) and u are the solution of stochastic Leray-α model and the stochastic Navier–Stokes equations, respectively. We show that when properly localized the error function εα converges in mean square as α→ 0 and the convergence is of order O(α). We also prove that εα converges in probability to zero with order at most O(α).",

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note = "KAUST Repository Item: Exported on 2022-06-03 Acknowledgements: Paul Razafimandimby{\textquoteright}s research was funded by the FWF-Austrian Science Fund through the Project M1487. Hakima Bessaih was supported in part by the Simons Foundation Grant #283308 and the NSF Grants DMS-1416689 and DMS-1418838. The research on this paper was initiated during the visit of Paul Razafimandimby at University of Wyoming in November 2013 and was finished while Paul Razafimandimby and Hakima Bessaih were visiting KAUST. They are both very grateful to both institutions for the warm and kind hospitality and great scientific atmosphere. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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N1 - KAUST Repository Item: Exported on 2022-06-03 Acknowledgements: Paul Razafimandimby’s research was funded by the FWF-Austrian Science Fund through the Project M1487. Hakima Bessaih was supported in part by the Simons Foundation Grant #283308 and the NSF Grants DMS-1416689 and DMS-1418838. The research on this paper was initiated during the visit of Paul Razafimandimby at University of Wyoming in November 2013 and was finished while Paul Razafimandimby and Hakima Bessaih were visiting KAUST. They are both very grateful to both institutions for the warm and kind hospitality and great scientific atmosphere. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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N2 - In the present paper we study the convergence of the solution of the two dimensional (2-D) stochastic Leray-α model to the solution of the 2-D stochastic Navier–Stokes equations. We are mainly interested in the rate, as α→ 0 , of the following error function (Formula presented.) and u are the solution of stochastic Leray-α model and the stochastic Navier–Stokes equations, respectively. We show that when properly localized the error function εα converges in mean square as α→ 0 and the convergence is of order O(α). We also prove that εα converges in probability to zero with order at most O(α).

AB - In the present paper we study the convergence of the solution of the two dimensional (2-D) stochastic Leray-α model to the solution of the 2-D stochastic Navier–Stokes equations. We are mainly interested in the rate, as α→ 0 , of the following error function (Formula presented.) and u are the solution of stochastic Leray-α model and the stochastic Navier–Stokes equations, respectively. We show that when properly localized the error function εα converges in mean square as α→ 0 and the convergence is of order O(α). We also prove that εα converges in probability to zero with order at most O(α).

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

Abstract

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