Finite-dimensionality of bounded invariant sets for Navier–Stokes systems and other dissipative systems

One proves the finite-dimensionality of a bounded set $M$ of a Hilbert space $H$, negatively invariant relative to a transformation $V$, possessing the following properties: For any points $v$ and $\tilde v$ of the set $M$ one has
$$ \|V(v)-V(\tilde v)\|\le l\|v-\tilde v\|, $$
while
$$ \|Q_nV(v)-Q_nV(\tilde v)\|\le\delta\|v-\tilde v\|,\quad\delta<1, $$
where $Q_n$ is the orthoprojection onto a subspace of codimension $n$. With the aid of this result and of the results found in O. A. Ladyzhenskaya's paper “On the dynamical system generated by the Navier–Stokes equations” (J. Sov. Math., 3, No. 4 (1975)) one establishes the finite-dimensionality of the complete attractor for two-dimensional Navier–Stokes equations. The same holds for many other dissipative problems.