Geodesic
Lieb--Thirring Integral Inequalities and Sharp Bounds for the Dimension of the Attractor of the Navier--Stokes Equations with Friction
Informatics and Automation, Function spaces, approximation theory, and nonlinear analysis, Tome 255 (2006), pp. 146-160

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A two-dimensional Navier–Stokes system with friction is considered in a large rectangular periodic domain with area on the order of $\alpha^{-1}$, $\alpha \to 0$. Bounds for the dimension of the attractor are obtained, which are sharp both as $\alpha\to 0$ and $\nu\to 0$, where $\nu$ is the viscosity coefficient. 
@article{TRSPY_2006_255_a10,
     author = {A. A. Ilyin},
     title = {Lieb--Thirring {Integral} {Inequalities} and {Sharp} {Bounds} for the {Dimension} of the {Attractor} of the {Navier--Stokes} {Equations} with {Friction}},
     journal = {Informatics and Automation},
     pages = {146--160},
     publisher = {mathdoc},
     volume = {255},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2006_255_a10/}
}
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A. A. Ilyin. Lieb--Thirring Integral Inequalities and Sharp Bounds for the Dimension of the Attractor of the Navier--Stokes Equations with Friction. Informatics and Automation, Function spaces, approximation theory, and nonlinear analysis, Tome 255 (2006), pp. 146-160. http://geodesic.mathdoc.fr/item/TRSPY_2006_255_a10/