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

TL;DR

This paper investigates the behavior of axisymmetric, swirl-free solutions of the Euler equation in four and higher dimensions, revealing potential for finite-time singularities that are absent in three dimensions.

Contribution

It establishes a conditional blowup result for these solutions in dimensions four and above, showing the blowup condition weakens as the dimension increases.

Findings

Solutions in higher dimensions may develop finite-time singularities.

Blowup conditions become weaker as the dimension tends to infinity.

Higher dimensions exhibit more singular 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.