Global Lyapunov exponents, KaplanYorke formulas and the dimension of the attractors for 2D NavierStokes equations
Abstract
The fractal and Hausdorff dimensions of the universal attractor for the NavierStokes equations are studied in two space dimensions. The functional setting is recalled and the precise definition of the universal attractor is given. The equations governing the transport of finitedimensional volume elements under the action of the NavierStokes system are established, and global Liapunov exponents are defined and studied. The existence of a critical dimension which enjoys the property that every Ndimensional volume element in the phase space decays exponentially over time is proved. Estimates for this critical dimension are given in terms of the Grashof number, separately for the periodic and aperiodic cases. It is proved that the expressions given by Kaplan and Yorke (1979), when combined with the introduced global Liapunov exponents, yield upper bounds for the fractal and Hausdorff dimensions of the universal attractor.
 Publication:

Communications in Pure Applied Mathematics
 Pub Date:
 January 1985
 Bibcode:
 1985CPAM...38....1C
 Keywords:

 Fractals;
 Liapunov Functions;
 NavierStokes Equation;
 Strange Attractors;
 Two Dimensional Flow;
 Chaos;
 Eigenvalues;
 Eigenvectors;
 Grashof Number;
 Manifolds (Mathematics);
 Transport Properties;
 Fluid Mechanics and Heat Transfer