  We consider two discrete models for the Euler equation describing
incompressible fluid dynamics. These models are infinite coupled systems of
ODEs for the functions $u_j$ which can be thought of as wavelet coefficients of
the fluid velocity. The first model has been proposed and studied by Katz and
Pavlovic. The second has been recently discussed by Waleffe and goes back to
Obukhov studies of the energy cascade in developed turbulence. These are the
only basic models of this type satisfying some natural scaling and conservation
conditions. We prove that the Katz-Pavlovic model leads to finite time blowup
for any initial datum, while the Obukhov model has a global solution for any
sufficiently smooth initial datum.
