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 solutions. 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.
Categories: math.AP