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

We consider the 3D incompressible Euler equations in vorticity form following fundamental domain for octahedral symmetry group: {(x1,x2,x3):0<x3<x2<x1}. In this domain, we prove local well-posedness Cα vorticities not necessarily vanishing on boundary with any 0<α<1, and establish finite-time singularity formation within same class smooth compactly supported initial data. The solutions can be extended to all of R3 via a sequence reflections, therefore obtain bounded piecewise vorticities.