arXiv2024
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.