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On the finite dimensionality of bounded invariant sets for the Navier- Stokes system and other dissipative systems. (Russian) Zbl 0535.76033

The paper revises and simplifies the proof of a result of C. Foias and R. Temam [J. Math. Pures Appl. IX. Ser., 58, 339-368 (1979; Zbl 0454.35073)] asserting that the functional invariant sets M for the Navier-Stokes equations in dimension 2 or 3, bounded in \(W^ 1\!_ 2\) norm (this condition is always fulfilled if \(n=2)\), are of a finite Hausdorff dimension that can be estimated by a constant depending only on \(\sup_{u\in M}\| u\|_{W^ 1\!_ 2}\). Drawing one’s inspiration, just like the mentioned authors, from a general result of J. Mallet-Paret [J. Differ. Equations 22, 331-348 (1976; Zbl 0354.34072)], one proves first the following theorem: Let M be a totally bounded set in a Hilbert space negatively invariant with respect to a Lipschitz mapping V satisfying for all v, \(\tilde v\) in M: \(\| Q_ mV(v)-Q_ mV(\tilde v)\| \leq \delta \| v-\tilde v\|, \delta<1\), where \(Q_ m\) is the orthogonal projection on a subspace of finite codimension m. Then M has finite Hausdorff dimension.
Using suitable estimates from a previous paper of the author [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 27, 91-114 (1972; Zbl 0327.35064)] and this theorem it is proved that the full attractor \({\mathfrak M}=\cap_{t\geq 0}S_ t(H)\) of the two-dimensional Navier-Stokes equations \((S_ t\) denoting the semigroup generated by them) has finite dimension. Next one shows that for periodic boundary conditions this dimension is estimated by \(C\nu^{-4}\) with a constant C depending only on the domain and the exterior forces.
Reviewer: G.Minea

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
76E30 Nonlinear effects in hydrodynamic stability
35Q30 Navier-Stokes equations
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