On blowups of vorticity for the homogeneous Euler equation

TL;DR

This paper investigates the conditions and regimes under which vorticity can blow up in homogeneous Euler equations in two and three dimensions, identifying possible blowup degrees and special directional concentration cases.

Contribution

It characterizes the possible blowup degrees and regimes for vorticity in 2D and 3D homogeneous Euler equations, including special directional concentration scenarios.

Findings

Blowups of degrees 1 to n can occur in the variable time regime.

Blowups of degrees 1/2 to (n+1)/(n+2) can occur in the fixed time regime.

Vorticity can blow up while its direction concentrates in one or two directions.

Abstract

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 $1,..,n$ at the variable time regime and of degrees $1/2,..,(n+1)/(n+2)$ 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.