On the existence of fibered three-dimensional perfect fluid equilibria without continuous Euclidean symmetry

TL;DR

This work constructs a family of smooth 3D steady Euler equilibria on a cylindrical domain $M\subset \mathbb{R}^2\times\mathbb{T}$ that possess no continuous Euclidean symmetry yet are fibered by invariant Bernoulli level surfaces $H=\text{const}$ and exhibit discrete $m$-fold reflection symmetry. Building on Lortz’s approach, the authors perturb an axisymmetric base flow $u_*$ to obtain $u= u_* + \varepsilon u'$ with large Bernoulli pressure, using Clebsch variables to relate the flow to a vorticity form $\omega$ and enforcing parity to guarantee closed orbits, so $H$ can be treated as a function of the orbital period. The core innovation is an iterative div-curl scheme in which, at each step, $\tau_N$ is solved from $u_N\cdot\nabla \tau_N=1$, then $T_N$, $H_N=\mathcal{H}_*(T_N)$ and $\omega_N= \nabla\tau_N\times\nabla H_N$ are formed and used to update $u_{N+1}$ with a harmonic projection; the contraction in Hölder spaces for large $m$ and small perturbations yields a strong solution of the steady Euler equations with cylindrical Bernoulli surfaces. The results expand the catalog of non-symmetric, pressure-fibered equilibria, informing Grad’s conjecture by showing non-symmetric cases with closed field lines and cylinder-level sets exist and can be infinitely flexible, while suggesting further exploration of asymmetric stellarator-like equilibria in plasma confinement contexts.

Abstract

Following Lortz, we construct a family of smooth steady states of the ideal, incompressible Euler equation in three dimensions that possess no continuous Euclidean symmetry. As in Lortz, they do possess a planar reflection symmetry and, as such, all the orbits of the velocity are closed. Different from Lortz, our construction has a discrete m-fold symmetry and is foliated by invariant cylindrical level sets of a non-degenerate Bernoulli pressure. Such examples narrow the scope of validity of Grad's conjecture that the only solutions with a continuous symmetry can be fibered by pressure surfaces.

Figures (1)

• Figure 1: An $8$--fold symmetric steady state, fibered by levels of the Bernoulli pressure $H$. All orbits of the velocity are confined to the isosurfaces of $H$, and wrap the "short way".

Theorems & Definitions (8)

• Theorem 1
• Definition 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof