  In this paper and the companion paper [EJE2], we establish finite-time
singularity formation for finite-energy strong solutions to the axi-symmetric
$3D$ Euler equations in the domain $\{(x,y,z)\in\mathbb{R}^3:z^2\leq
c(x^2+y^2)\}$ for some $c>0$. In the spirit of our previous works, [EJSI] and
[EJB], we do this by first studying scale-invariant solutions which satisfy a
one dimensional PDE system and proving that they may become singular in finite
time for properly chosen initial data. We then prove local well-posedness for
the $3D$ Euler equations in a natural regularity class which includes
scale-invariant solutions. While these solutions have uniformly bounded
vorticity from time zero until right before the blow-up time, they do not have
finite energy. To remedy this, we cut off the scale-invariant data to ensure
finite energy and prove that the corresponding local solution must also become
singular in finite time. This paper focuses only on the analysis of the
scale-invariant solutions themselves and the proof that they can become
singular in finite time. The local well-posedness theorem and the cut-off
argument are very close to those in [JSI] and [EJB] and are left for the
companion paper [EJE2].
  It is important to remark that while the fluid domain is the exterior of a
cone, we prove global regularity for the axi-symmetric $3D$ Euler equations
without swirl in the exact same regularity classes and in the same domain. It
is quite possible that the methods we use can be adapted to establish
finite-time singularity formation for $C^\infty$ finite-energy solutions to the
$3D$ Euler equations on $\mathbb{R}^3_+$.

Analysis of PDEs