In this article, we modified the Navier-Stokes equations by adding a higher order artificial viscosity term to the conventional system. We first show that the solution of the regularized system converges strongly to the solution of the conventional system as the regularization parameter approaches zero, for each dimension . Then we show that the use of this artificial viscosity term leads to truncated the number of degrees of freedom in the long-time behavior of the solutions to these equations. This result suggests that the hyperviscous Navier-Stokes system is an interesting model for three-dimensional fluid turbulence.
Submitted December 2, 2009. Published August 9, 2010.
Math Subject Classifications: 76D05, 76F20, 35B30, 35B41, 35B65, 37L30, 37K40.
Key Words: Navier-Stokes equations; hyperviscosity; weak solutions; attractor dimension; turbulence models
An addendum was attached on September 27, 2011. It corrects some misprints and presents another proof of estimates for the dimension of the attractor.
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| Abdelhafid Younsi |
Faculty of Mathematics USTHB
BP32 EL ALIA16111 Algiers, Algeria
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