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

Abstract

In this paper, we prove global regularity for all smooth, axisymmetric, swirl-free solutions of the incompressible 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. For discussion of another contemporaneous result removing this condition, see Remark 1.5. The key advance in this paper is a new bound on the vortex stretching term that only requires $\frac{{\omega }^0}{r^2}\in L^{2,1}({\mathbb{R}}^4)$, a condition which holds generically for any axisymmetric, swirl-free initial data $u^0\in H^s\left({\mathbb{R}}^4\right),s>4$, with reasonable decay at infinity.