  Shell models have found wide application in the study of hydrodynamic
turbulence because they are easily solved numerically even at very large
Reynolds numbers. Although bereft of spatial variation, they accurately
reproduce the main statistical properties of fully-developed homogeneous and
isotropic turbulence. Moreover, they enjoy regularity properties which still
remain open for the three-dimensional (3D) Navier-Stokes equations (NSEs). The
goal of this study is to make a rigorous comparison between shell models and
the NSEs. It turns out that only the estimate of the mean energy dissipation
rate is the same in both systems. The estimates of the velocity and its
higher-order derivatives display a weaker Reynolds number dependence for shell
models than for the 3D NSEs. Indeed, the velocity-derivative estimates for
shell models are found to be equivalent to those corresponding to a velocity
gradient averaged version of the 3D Navier-Stokes equations (VGA-NSEs), while
the velocity estimates are even milder. Numerical simulations over a wide range
of Reynolds numbers confirm the estimates for shell models.
