Abstract In this paper, we prove a global well-posedness of the three-dimensional incompressible Navier-Stokes equation under initial data, which belongs to the Lei-Lin-Gevrey space Za,σ−1 $\begin{array}{} Z^{-1}_{a,\sigma} \end{array}$(ℝ3) and if the norm of the initial data in the Lei-Lin space 𝓧−1 is controlled by the viscosity. Moreover, we will show that the norm of this global solution in the Lei-Lin-Gevrey space decays to zero as time approaches to infinity.

中文翻译：

---

Lei-Lin-Gevrey 空间中 3D-NSE 的长时间衰减

摘要 本文证明了初始数据下三维不可压缩 Navier-Stokes 方程的全局适定性，属于 Lei-Lin-Gevrey 空间 Za,σ−1 $\begin{array}{} Z ^{-1}_{a,\sigma} \end{array}$(ℝ3) 并且如果 Lei-Lin 空间 𝓧−1 中初始数据的范数受粘度控制。此外，我们将证明随着时间接近无穷大，Lei-Lin-Gevrey 空间中这个全局解的范数衰减为零。