Global regularity for axisymmetric, swirl-free solutions of the Euler equation in four dimensions

TL;DR

This paper proves global regularity for smooth, axisymmetric, swirl-free solutions of the Euler equation in four dimensions, using a new bound on vortex stretching that requires less restrictive initial data conditions.

Contribution

Introduces a novel bound on vortex stretching for 4D Euler solutions, removing previous restrictive assumptions and broadening the class of initial data for which regularity holds.

Findings

Established global regularity for all smooth, axisymmetric, swirl-free solutions in 4D.

Derived a new vortex stretching bound requiring only $rac{ ext{omega}^0}{r^2} ext{ in } L^{2,1}( ext{R}^4)$.

Extended regularity results to initial data in $H^s( ext{R}^4)$ with $s>4$ and reasonable decay.

Abstract

In this paper, we prove global regularity for all smooth, axisymmetric, swirl-free solutions of the Euler equation in four dimensions. Previous works establishing global regularity for certain axisymmetric, swirl-free solutions of the Euler equation in four dimensions required the additional assumption that $\frac{\omega^0}{r^2}\in L^\infty$, which can fail even for Schwartz class initial data. The key advance is a new bound on the vortex stretching term that only requires $\frac{\omega^0}{r^2}\in L^{2,1}(\mathbb{R}^4)$, which is generically true for any axisymmetric, swirl-free initial data $u^0\in H^s\left(\mathbb{R}^4\right), s>4$, with reasonable decay at infinity.