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

TL;DR

This paper proves the global existence of weak solutions for 3D incompressible axisymmetric Euler equations without swirl, under specific initial vorticity conditions, and introduces a novel velocity field estimate.

Contribution

It establishes the global existence of weak solutions for the equations with new vorticity conditions and a novel velocity estimate, extending previous results.

Findings

Proved global existence of weak solutions under vorticity conditions.

Established new local velocity field estimates.

Extended previous theoretical results.

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.