The incompressible Euler equations under octahedral symmetry: singularity formation in a fundamental domain

We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: ${ (x_1,x_2,x_3): 0<x_3<x_2<x_1 }.$ In this domain, we prove local well-posedness for $C^{\alpha }$ vorticities not necessarily vanishing on the boundary with any $0<\alpha <1$, and establish finite-time singularity formation within the same class for smooth and compactly supported initial data. The solutions can be extended to all of ${\mathbb{R}}^3$ via a sequence of reflections, and therefore we obtain finite-time singularity formation for the 3D Euler equations in ${\mathbb{R}}^3$ with bounded and piecewise smooth vorticities.