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

Abstract

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