Abstract:In this paper, we study $(-1)$-homogeneous steady solutions to the Euler equations on $\mathbb{R}^n \setminus \{0\}$. In low dimensions $n=2,3$, such flows are known to be essentially trivial. In contrast, we show that in higher dimensions $n \ge 4$, every $(-1)$-homogeneous Euler flow is a geodesible vector field with constant Bernoulli function. Moreover, any $(-1)$-homogeneous geodesible field is induced by a geodesible field on the sphere $\mathbb{S}^{n-1}$. In particular, in the case $n=4$, every $(-1)$-homogeneous Euler flow is obtained as an extension of a Beltrami field on $\mathbb{S}^{3}$.
From: Ken Abe [view email]
[v1]
Wed, 17 Jun 2026 03:43:25 UTC (52 KB)
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