In this paper, we consider solutions to the incompressible axisymmetric Euler equations without swirl. The main result is to prove the global existence of weak solutions if the initial vorticity \(w_0^\theta \) satisfies that \(\frac{w_0^\theta }{r}\in L^1\cap L^p({\mathbb {R}}^3)\) for some \(p>1\). It is not required that the initial energy is finite, that is, the initial velocity \(u_0\) belongs to \(L^2({\mathbb {R}}^3)\) here. We construct the approximate solutions by regularizing the initial data and show that the concentrations of energy do not occur in this case. The key ingredient in the proof lies in establishing the \(L_{\mathrm{loc}}^{2+\alpha }({\mathbb {R}}^3)\) estimates of velocity fields for some \(\alpha >0\), which is new to the best of our knowledge.

在本文中，我们考虑了无涡旋的不可压缩轴对称欧拉方程的解。主要结果是证明如果初始涡度\（w_0 ^ \ theta \）满足L ^ 1 \ cap L ^ p（ {\ mathbb {R}} ^ 3）\）表示某些\（p> 1 \）。这里不需要初始能量是有限的，也就是说，初始速度\（u_0 \）在这里属于\（L ^ 2（{\ mathbb {R}} ^ 3）\）。我们通过对初始数据进行正则化来构造近似解，并表明在这种情况下不会发生能量集中。证明中的关键要素在于建立\（L _ {\ mathrm {loc}} ^ {2+ \ alpha}（{\ mathbb {R}} ^ 3）\）某些\（\ alpha> 0 \）的速度场的估计，据我们所知，这是新的。