On blowups of vorticity for the homogeneous Euler equation

Blowups of vorticity for the three- and two- dimensional homogeneous Euler equations are studied. Two regimes of approaching a blowup points, respectively, with variable or fixed time are analysed. It is shown that in the $n$-dimensional ($n=2,3$) generic case the blowups of degrees $1,\mathrm{.}\mathrm{.},n$ at the variable time regime and of degrees $1\mathrm{/}2,\mathrm{.}\mathrm{.},(n+1)\mathrm{/}(n+2)$ at the fixed time regime may exist. Particular situations when the vorticity blows while the direction of the vorticity vector is concentrated in one or two directions are realisable.