Geodesic
Finite-dimensionality of bounded invariant sets for Navier--Stokes systems and other dissipative systems
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 14, Tome 115 (1982), pp. 137-155

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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 $$ \|Q_nV(v)-Q_nV(\tilde v)\|\le\delta\|v-\tilde v\|,\quad\delta1, $$ 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. 
@article{ZNSL_1982_115_a11,
     author = {O. A. Ladyzhenskaya},
     title = {Finite-dimensionality of bounded invariant sets for {Navier--Stokes} systems and other dissipative systems},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {137--155},
     publisher = {mathdoc},
     volume = {115},
     year = {1982},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1982_115_a11/}
}
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O. A. Ladyzhenskaya. Finite-dimensionality of bounded invariant sets for Navier--Stokes systems and other dissipative systems. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 14, Tome 115 (1982), pp. 137-155. http://geodesic.mathdoc.fr/item/ZNSL_1982_115_a11/