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The incompressible Euler equations under octahedral symmetry: singularity formation in a fundamental domain (2001.07840v1)
Published 22 Jan 2020 in math.AP, math-ph, math.MP, and physics.flu-dyn
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.