On some dyadic models of the Euler equations

Abstract:

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 a dyadic model of the inviscid Burgers equation. 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, 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.