Global existence of weak solutions to the 3D incompressible axisymmetric Euler equations without swirl

Abstract

In this paper, we mainly investigate the tridimensional incompressible axisymmetric Euler equations without swirl in the whole space. Specifically, we prove the global existence of weak solutions if the swirl component of initial vorticity $w_0^\theta$ satisfies that $\frac{w_0^\theta}r\in L^1\cap L^p({\Bbb R}^3)$ for some $p>1$. To achieve this goal, we establish the $L_{\rm loc}^{2+\alpha}({\Bbb R}^3)$ estimate of velocity fields for some $\alpha>0$, which is innovative to the best of our knowledge. Our result extends previous work in the literature.