We consider a family of weights which permit to generalize the Leray procedure to obtain weak suitable solutions of the 3D incompressible Navier–Stokes equations with initial data in weighted \(L^2\) spaces. Our principal result concerns the existence of regular global solutions when the initial velocity is an axisymmetric vector field without swirl such that both the initial velocity and its vorticity belong to \(L^2 ( (1+ r^2)^{-\frac{\gamma }{2}} dx ) \), with \(r= \sqrt{x_1^2 + x_2^2}\) and \(\gamma \in (0, 2) \).

中文翻译：

---

不可压缩 Navier-Stokes 方程的加权能量估计及其在无涡旋轴对称解中的应用

我们考虑一系列权重，它们允许推广 Leray 程序以获得 3D 不可压缩 Navier-Stokes 方程的弱合适解，初始数据位于加权\(L^2\)空间。我们的主要结果涉及当初始速度是一个没有涡旋的轴对称矢量场时，规则全局解的存在，使得初始速度和它的涡度都属于\(L^2 ( (1+ r^2)^{-\frac {\gamma }{2}} dx ) \)，与\(r= \sqrt{x_1^2 + x_2^2}\)和\(\gamma \in (0, 2) \)。