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On the fractal dimension of invariant sets: Applications to Navier-Stokes equations. (English) Zbl 1049.37047
A semigroup \(S_{t}\) of continuous operators in a Hilbert space \(H\) is considered. The aim of the reviewed article is to estimate the fractal dimension of a compact strictly invariant set \(X \Subset H, S_{t}X=X\). It is proved that this fractal dimension admits the same estimation as the Hausdorff one. Namely, both are bounded from above by the Lyapounov dimension calculated in terms of the global Lyapounov exponents. Then, the main estimate proved in the abstract setting is applied to the two-dimensional Navier-Stokes system.

MSC:
37L30 Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents
34G20 Nonlinear differential equations in abstract spaces
35Q30 Navier-Stokes equations
37C45 Dimension theory of smooth dynamical systems
76D05 Navier-Stokes equations for incompressible viscous fluids
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