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

TL;DR

This paper studies the 3D incompressible Euler equations with octahedral symmetry, proving local well-posedness for certain vorticities and demonstrating finite-time singularity formation, extending results to the full space via symmetry reflections.

Contribution

It establishes local well-posedness and finite-time singularity formation for the Euler equations within an octahedral symmetric domain, extending to full space solutions.

Findings

Proves local well-posedness for $C^eta$ vorticities with boundary conditions.

Shows finite-time singularity formation for smooth, compactly supported initial data.

Extends solutions to $ r^3$ via reflections, demonstrating singularities in full space.

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.