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: $\{(x_1,x_2,x_3):0<x_3<x_2<x_1\}.$ 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.

53 pages, 5 figures

References in corpus (1)

• On the blow-up problem for the axisymmetric 3D Euler equations