Lieb–Thirring Integral Inequalities and Sharp Bounds for the Dimension of the Attractor of the Navier–Stokes Equations with Friction

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.