Remarks on global attractors for the 3D Navier--Stokes equations with horizontal filtering

Luca Bisconti; Davide Catania

doi:10.3934/dcdsb.2015.20.59

Article Contents

2015, Volume 20, Issue 1: 59-75. Doi: 10.3934/dcdsb.2015.20.59

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Remarks on global attractors for the 3D Navier--Stokes equations with horizontal filtering

Luca Bisconti^1, and
Davide Catania^2,

1.
Università degli Studi di Firenze, Dipartimento di Matematica e Informatica "U. Dini", Via S. Marta 3, I-50139 Firenze
2.
Università degli Studi di Brescia, Sezione Matematica (DICATAM), Via Valotti 9, I-25133 Brescia

Received: January 31, 2014

Revised: May 31, 2014

Published: December 2014

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Abstract

We consider a Large Eddy Simulation model for a homogeneous incompressible Newtonian fluid in a box space domain with periodic boundary conditions on the lateral boundaries and homogeneous Dirichlet conditions on the top and bottom boundaries, thus simulating a horizontal channel. The model is obtained through the application of an anisotropic horizontal filter, which is known to be less memory consuming from a numerical point of view, but provides less regularity with respect to the standard isotropic one defined as the inverse of the Helmholtz operator.
It is known that there exists a unique regular weak solution to this model that depends weakly continuously on the initial datum. We show the existence of the global attractor for the semiflow given by the time-shift in the space of paths. We prove the continuity of the horizontal components of the flow under periodicity in all directions and discuss the possibility to introduce a solution semiflow.

Keywords:

Navier--Stokes equations,
anisotropic filters,
global attractor,
time-shift semiflow,
Large Eddy Simulation (LES),
turbulent flows in domains with boundary,
Approximate Deconvolution Methods (ADM).

Mathematics Subject Classification: Primary: 76D05; Secondary: 35Q30, 76F65, 76D03.

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