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

Theodore D. Drivas; Tarek M. Elgindi; Daniel Ginsberg

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arXiv · 2510.02955

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

Theodore D. Drivas·Tarek M. Elgindi·Daniel Ginsberg

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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.

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Theodore D. Drivas, Tarek M. Elgindi, Daniel Ginsberg. 2025-10-03. On the existence of fibered three-dimensional perfect fluid equilibria without continuous Euclidean symmetry. https://arxiv.org/abs/2510.02955

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