On some dyadic models of the Euler equations

Katz and Pavlovic recently proposed a dyadic model of the Euler equations for which they proved finite time blow-up in the H3/2+epsilon Sobolev norm. It is shown that their model can be reduced to a dyadic model of the inviscid Burgers equation. The inviscid Burgers equation exhibits finite time blow-up in H-alpha, for alpha >= 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 H3/2+epsilon. Some inconsistent assumptions in the construction of their model are outlined. Finite time blow-up in the H-alpha norm, for any 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.