  Katz and Pavlovic recently proposed a dyadic model of the Euler equations for
which they proved finite time blow-up in the $H^{3/2+\epsilon}$ Sobolev norm.
It is shown that their model can be reduced to the dyadic inviscid Burgers
equation where nonlinear interactions are restricted to dyadic wavenumbers. The
inviscid Burgers equation exhibits finite time blow-up in $H^{\alpha}$, for
$\alpha \ge 1/2$, but its dyadic restriction is even more singular, exhibiting
blow-up for any $\alpha > 0$. Friedlander and Pavlovic developed a closely
related model for which they also prove finite time blow-up in
$H^{3/2+\epsilon}$. Some inconsistent assumptions in the construction of their
model are outlined. Finite time blow-up in the $H^{\alpha}$ norm, with $\alpha
> 0$, is proven for a class of models that includes all those models. An
alternative shell model of the Navier-Stokes equations is discussed.
