On the regularity of axisymmetric, swirl-free solutions of the Euler equation in four and higher dimensions

TL;DR

This work extends axisymmetric, swirl-free Euler analysis to four and higher dimensions, highlighting a dimension-dependent mechanism for potential finite-time blowup due to the transported quantity $\frac{\omega}{r^k}$ with $k=d-2$. It develops a higher-dimensional Biot–Savart framework, proves a conditional blowup criterion for $d\ge 4$, and establishes global regularity in 4D under the modest hypothesis $\frac{\omega^0}{r^2}\in L^1\cap L^\infty$, with explicit bounds that grow at most exponentially in time. The paper also analyzes the anti-parallel vortex-tube geometry, showing finite-time blowup under a weak, dimension-dependent energy-condition, which becomes easier to satisfy as $d$ increases. Taken together, these results illuminate a sharp dimension dependence in the axisymmetric swirl-free Euler dynamics and connect high-dimensional behavior to the challenging blowup problem in 3D. The findings thus offer a tractable setting to study blowup mechanisms and may shed light on the elusive 3D Euler regularity question by contrasting it with higher-dimensional dynamics.

Abstract

In this paper, we consider axisymmetric, swirl-free solutions of the Euler equation in four and higher dimensions. We show that in dimension $d\geq 4$, axisymmetric, swirl-free solutions of the Euler equation have properties which could allow finite-time singularity formation of a form that is excluded when $d=3$, and we prove a conditional blowup result for axisymmetric, swirl-free solutions of the Euler equation in dimension $d\geq 4$. The condition which must be imposed on the solution in order to imply blowup becomes weaker as $d\to +\infty$, suggesting the dynamics are becoming much more singular as the dimension increases.

Theorems & Definitions (98)

• Remark 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Remark 1.7
• Remark 1.8
• Remark 2.1
• Remark 2.2