On the convergence of statistical solutions of the 3DNavier-Stokes-$\alpha$ model as $\alpha$ vanishes

Anne Bronzi; Ricardo Rosa

doi:10.3934/dcds.2014.34.19

Article Contents

2014, Volume 34, Issue 1: 19-49. Doi: 10.3934/dcds.2014.34.19

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On the convergence of statistical solutions of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ vanishes

Anne Bronzi^1, and
Ricardo Rosa^2,

1.
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Ilha do Fundão, Rio de Janeiro, RJ, 21941-909
2.
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Ilha do Fundão, Rio de Janeiro, RJ 21941-909

Received: September 30, 2012

Revised: March 31, 2013

Published: December 2013

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Abstract

In this paper statistical solutions of the 3D Navier-Stokes-$\alpha$ model with periodic boundary condition are considered. It is proved that under certain natural conditions statistical solutions of the 3D Navier-Stokes-$\alpha$ model converge to statistical solutions of the exact 3D Navier-Stokes equations as $\alpha$ goes to zero. The statistical solutions considered here arise as families of time-projections of measures on suitable trajectory spaces.

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Mathematics Subject Classification: Primary: 76D06, 35Q30, 76D05; Secondary: 35Q35.

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