  We show here the global, in time, regularity of the three dimensional viscous
Camassa-Holm (Lagrangian Averaged Navier-Stokes-alpha) equations. We also
provide estimates, in terms of the physical parameters of the equations, for
the Hausdorff and fractal dimensions of their global attractor. In analogy with
the Kolmogorov theory of turbulence, we define a small spatial scale,
\ell_{\epsilon}, as the scale at which the balance occurs in the mean rates of
nonlinear transport of energy and viscous dissipation of energy. Furthermore,
we show that the number of degrees of freedom in the long-time behavior of the
solutions to these equations is bounded from above by (L/\ell_{epsilon})^3,
where L is a typical large spatial scale (e.g., the size of the domain). This
estimate suggests that the Landau-Lifshitz classical theory of turbulence is
suitable for interpreting the solutions of the NS-alpha equations. Hence, one
may consider these equations as a closure model for the Reynolds averaged
Navier-Stokes equations (NSE). We study this approach, further, in other
related papers. Finally, we discuss the relation of the NS-alpha model to the
NSE by proving a convergence theorem, that as the length scale alpha tends to
zero a subsequence of solutions of the NS-alpha equations converges to a weak
solution of the three dimensional NSE.
