On the fractal dimension of invariant sets:  Applications to Navier-Stokes equations

V. V. Chepyzhov; A. A. Ilyin

doi:10.3934/dcds.2004.10.117

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2004, Volume 10, Issue 1&2: 117-135. Doi: 10.3934/dcds.2004.10.117

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On the fractal dimension of invariant sets: Applications to Navier-Stokes equations

V. V. Chepyzhov^1, and
A. A. Ilyin^2,

1.
Institute for Information Transmission Problems, Bol'shoĭ Karetnyĭ 19, Moscow 101447
2.
Keldysh Institute of Applied Mathematics, Miusskaya Sq. 4, Moscow 125047

Received: January 31, 2002

Revised: February 28, 2003

Published: December 2003

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Abstract

A semigroup $S_t$ of continuous operators in a Hilbert space $H$ is considered. It is shown that the fractal dimension of a compact strictly invariant set $X$ ($X\subset H, S_tX=X$) admits the same estimate as the Hausdorff dimension, namely, both are bounded from above by the Lyapunov dimension calculated in terms of the global Lyapunov exponents. Applications of the results so obtained to the two-dimensional Navier-Stokes equations are given.

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Mathematics Subject Classification: Primary: 34G20, 35Q30.

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