  We investigate three-dimensional quantum turbulence as described by the
Gross-Pitaevskii model using the analytical method exploited in the Onsager
"ideal turbulence" theory. We derive the scale-independence of the
scale-to-scale kinetic energy flux and establish a double-cascade scenario: at
scales much larger than the mean intervortex $\ell_i$, the Richardson cascade
becomes dominant, whereas at scales much smaller than $\ell_i$, another type of
cascade is induced by quantum stress. We then evaluate the corresponding
velocity power spectrum using a phenomenological argument. The relation between
the novel cascade, which we call quantum stress cascade, and the Kelvin-wave
cascade is also discussed.

Fluid Dynamics;Other Condensed Matter