Abstract:
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 ˜v of the set M one has
‖V(v)−V(˜v)‖⩽l‖v−˜v‖,
while
‖QnV(v)−QnV(˜v)‖⩽δ‖v−˜v‖,δ<1,
where Qn 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.
Citation:
O. A. Ladyzhenskaya, “Finite-dimensionality of bounded invariant sets for Navier–Stokes systems and other dissipative systems”, Boundary-value problems of mathematical physics and related problems of function theory. Part 14, Zap. Nauchn. Sem. LOMI, 115, "Nauka", Leningrad. Otdel., Leningrad, 1982, 137–155; J. Soviet Math., 28:5 (1985), 714–726
\Bibitem{Lad82}
\by O.~A.~Ladyzhenskaya
\paper Finite-dimensionality of bounded invariant sets for Navier--Stokes systems and other dissipative systems
\inbook Boundary-value problems of mathematical physics and related problems of function theory. Part~14
\serial Zap. Nauchn. Sem. LOMI
\yr 1982
\vol 115
\pages 137--155
\publ "Nauka", Leningrad. Otdel.
\publaddr Leningrad
\mathnet{http://mi.mathnet.ru/znsl4047}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=660078}
\zmath{https://zbmath.org/?q=an:0535.76033}
\transl
\jour J. Soviet Math.
\yr 1985
\vol 28
\issue 5
\pages 714--726
\crossref{https://doi.org/10.1007/BF02112336}
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