  This work is a companion to [EJE1] and its purpose is threefold: first, we
will establish local well-posedness for the axi-symmetric $3D$ Euler equation
in the domains $\{(x_1,x_2,x_3) \in \mathbb{R}^3 : x_3^2 \le \mathfrak{c}(x_1^2
+ x_2^2) \}$ for $\mathfrak{c}$ sufficiently small in a scale of critical
spaces. Second, we will prove that if the vorticity at $t=0$ can be decomposed
into a scale-invariant part and a smoother part vanishing at $x=0$, then this
decomposition remains valid for $t>0$ so long as the solution exists. This will
then immediately imply singularity formation for finite-energy solutions in
those critical spaces using that there are scale-invariant solutions which
break down in finite time (this was proved in the companion paper [EJE1]).
Third, we establish global regularity in the same domain and in the same scale
of critical spaces for $0-$swirl solutions to the system. This implies that the
finite-time singularity is not coming from the domain or the critical
regularity of the data (though, these are important for the present
construction), they are coming from a genuine vorticity growth mechanism due to
the presence of the swirl. This work and the companion work seem to be the
first case when this mechanism has been effectively used to establish
finite-time singularity formation for the actual $3D$ Euler system.
