Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 117 - 135, Volume 10, Issue 1/2, January/February 2004      doi:10.3934/dcds.2004.10.117

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V. V. Chepyzhov - Institute for Information Transmission Problems, Bol'shoĭ Karetnyĭ 19, Moscow 101447, Russian Federation (email)
A. A. Ilyin - Keldysh Institute of Applied Mathematics, Miusskaya Sq. 4, Moscow 125047, Russian Federation (email)

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.

Keywords:  Fractal dimension, Attractors, Navier–Stokes equations.
Mathematics Subject Classification:  Primary: 34G20, 35Q30.

Received: February 2002;      Revised: March 2003;      Available Online: October 2003.