In this paper, the optimal decay rates of solutions for three-dimensional incompressible Navier–Stokes equations with damping term ${\left|u\right|}^{\beta -1}u$ are established. Bounds on the ${\stackrel{.}{H}}^{-s}$ negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.

中文翻译：

---

带有阻尼的三维Navier-Stokes方程的衰减估计

在本文中，具有阻尼项的三维不可压缩Navier-Stokes方程解的最优衰减率 ${\left|ü\right|}^{\beta -\mathrm{1个}}ü$建立。界线${\stackrel{。}{H}}^{-s}$ 负的Sobolev规范显示出随着时间的发展而得以保留，并提高了衰减率。