2024/04/01 by Evan W. Miller, Miller, Evan
Earth and Planetary Sciences · Mathematics · Physics and Astronomy · #35Q31 #76B03 #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #Aquatic and Environmental Studies #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2404.01505
openalex publication_date 2024/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 ω1, 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 x1+x2+x3=0. Using the blowup solution introduced by Elgindi in [5], we are able to prove there are C1,α solutions of the full Euler equation that blowup in finite-time, which are odd, permutation symmetric, and mirror symmetric about the plane x1+x2+x3=0. We will also prove that divergence-free vector fields that are odd, permutation symmetric, and mirror symmetric about the plane x1+x2+x3=0 (Gσ 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 \fracπ3 in the horizontal plane (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.