Global regularity for some axisymmetric Euler flows in $\mathbb{R}^{d}$

TL;DR

This paper proves global regularity for certain axisymmetric Euler flows in higher dimensions, extending known results and identifying conditions that ensure smooth solutions remain regular.

Contribution

It establishes global regularity results for axisymmetric Euler flows in dimensions four through seven under specific initial vorticity conditions.

Findings

Global regularity for $d=4$ with decay and vanishing at the axis.

Global regularity for $d eq 4$ when initial vorticity has one sign.

Extension of regularity results to higher dimensions.

Abstract

We consider axisymmetric Euler flows without swirl in $\mathbb{R}^{d}$ with $d\geq 4$, for which the global regularity of smooth solutions is an open problem. When $d = 4$, we obtain global regularity under the assumption that the initial vorticity satisfies some decay at infinity and is vanishing at the axis. Assuming further that the initial vorticity is of one sign guarantees global regularity for $d\leq 7$.