||We construct a local well-posedness class for the 3D incompressible axi-symmetric Euler equations, and show that within this class there exist compactly supported initial data which blows up in finite time. This local well-posedness class is critical in terms of the natural scaling transformation of the Euler equations and consists of velocities which are uniformly Lipschitz in space and have finite energy. Our results were inspired by recent works of Hou-Luo, Kiselev-Sverak, and many others, and the proof builds up on our previous works on 2D Euler and Boussinesq systems. An essential idea is the robustness of scale-invariant dynamics for inviscid transport systems. This is joint work with Tarek Elgindi.