Bounds on energy and enstrophy for the 3D Navier-Stokes-$\alpha$ and Leray-$\alpha$ models

Aseel Farhat; M. S Jolly; Evelyn Lunasin

doi:10.3934/cpaa.2014.13.2127

Article Contents

2014, Volume 13, Issue 5: 2127-2140. Doi: 10.3934/cpaa.2014.13.2127

This issue Previous Article On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in $R^3$: A computational approach Next Article

Bounds on energy and enstrophy for the 3D Navier-Stokes-$\alpha$ and Leray-$\alpha$ models

1.
Indiana University, Department of Mathematics, Bloomington, IN 47405
2.
Department of Mathematics, Indiana University, Bloomington, IN, 47405
3.
Department of Mathematics, U.S. Naval Academy, Annapolis, MD 21402-5002

Received: September 2013

Revised: February 2014

Published: September 2015

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Abstract

We construct semi-integral curves which bound the projections of the global attractors of the 3D NS-$\alpha$ and 3D Leray-$\alpha$ sub-grid scale turbulence models in the plane spanned by their energy and enstrophy. We note the dependence of these bounds on the filter width parameter $\alpha$, and determine subregions where each quantity, energy and enstrophy, must decrease, while isolating one which is recurrent.

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Mathematics Subject Classification: Primary: 35Q30, 76F02.

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