Permutation symmetric solutions of the incompressible Euler equation

In this paper, we study permutation symmetric solutions of the incompressible Euler equation. We show that the dynamics of these solutions can be reduced to an evolution equation on a single vorticity component $\omega_1$, and we characterize the relevant constraint space for this vorticity component under permutation symmetry. We also give single vorticity component versions of the energy equality, Beale-Kato-Majda criterion, and local wellposedness theory that are specific to the permutation symmetric case. This paper is significantly motivated by a recent work of the author [13], which proved finite-time blowup for smooth solutions of a Fourier-restricted Euler model equation, where the Helmholtz projection is replaced by a projection onto a more restrictive constraint space. The blowup solutions for this model equation are odd, permutation symmetric, and mirror symmetric about the plane $x_1+x_2+x_3=0.$ Using the blowup solution introduced by Elgindi in [5], we are able to prove there are $C^{1,\alpha}$ solutions of the full Euler equation that blowup in finite-time, which are odd, permutation symmetric, and mirror symmetric about the plane $x_1+x_2+x_3=0$. We will also prove that divergence-free vector fields that are odd, permutation symmetric, and mirror symmetric about the plane $x_1+x_2+x_3=0$ ($\mathcal{G}_\sigma$ symmetric) are equivalent up to a change of coordinates given by a rotation to divergence-free vector fields that are mirror symmetric about each of the three coordinate axes and symmetric with respect to rotations by $\frac{\pi}{3}$ in the horizontal plane ($\mathcal{G}$-symmetric). The latter discrete symmetry group allows for a Fourier series expansion in cylindrical coordinates that shines a further light on the structure of these symmetry groups, in particular their relation to axisymmetric, swirl-free vector fields.