Modeling error of $ \alpha $-models of turbulence on a two-dimensional torus

Luigi C. Berselli; Argus Adrian Dunca; Roger Lewandowski; Dinh Duong Nguyen

doi:10.3934/dcdsb.2020305

Article Contents

2021, Volume 26, Issue 9: 4613-4643. Doi: 10.3934/dcdsb.2020305

This issue Previous Article Next Article Impulses in driving semigroups of nonautonomous dynamical systems: Application to cascade systems

Modeling error of $ \alpha $-models of turbulence on a two-dimensional torus

1.
Università di Pisa, Dipartimento di Matematica, Via Buonarroti 1/c, I-56127 Pisa, Italy
2.
Department of Mathematics and Informatics, University Politehnica of Bucharest, Bucharest, Romania
3.
IRMAR, UMR CNRS 6625, University of Rennes 1 and FLUMINANCE Team, INRIA, Rennes, France

^* Corresponding author
^* Corresponding author

Received: February 29, 2020

Revised: July 31, 2020

Early access: October 2020

Published: September 2021

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This paper is devoted to study the rate of convergence of the weak solutions $ {\bf u}_\alpha $ of $ \alpha $-regularization models to the weak solution $ {\bf u} $ of the Navier-Stokes equations in the two-dimensional periodic case, as the regularization parameter $ \alpha $ goes to zero. More specifically, we will consider the Leray-$ \alpha $, the simplified Bardina, and the modified Leray-$ \alpha $ models. Our aim is to improve known results in terms of convergence rates and also to show estimates valid over long-time intervals. The results also hold in the case of bounded domain with homogeneous Dirichlet boundary conditions.

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Mathematics Subject Classification: Primary:35Q30, 35Q35;Secondary:65M15, 76D05, 76F65.

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