Attractor dimension and small length scale estimates for the three-dimensional Navier-Stokes equations
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It is shown that a rigorous estimate for the fractal dimension of the global attractor of the three-dimensional incompressible Navier-Stokes equations on periodic boundary conditions is given by dF c(L/k)4.8where L is the box length and kis a Kolmogorov length defined by k-1= (/v3)1/4with the energy dissipation rate given by = vL-3sup, ||2dV. By interpreting the dimension of the attractor as the number of degrees of freedom of the system, we obtain an estimate for an average natural length scale for the flow which is given by /L c (k/L)1.6.
