The existence of a global solution to the 3D incompressible Navier-Stokes equations is an outstanding open problem. However, the 3D incompressible Navier-Stokes equations with hyperdissipation \((-\Delta )^{\alpha }u\) always possesses a global smooth solution if \(\alpha \geq \frac{5}{4}\). Yang-Jiu-Wu (The 3D incompressible Navier-Stokes equations with partial hyperdissipation, Math. Nachr. 292(8):1823–1836, 2019) reduced the hyperdissipation \((-\Delta )^{\alpha }u\) to \(((\Lambda _{1}^{\frac{5}{2}}, \Lambda _{2}^{\frac{5}{2}})u_{1}, (\Lambda _{2}^{ \frac{5}{2}}, \Lambda _{3}^{\frac{5}{2}})u_{2}, ( \Lambda _{3}^{\frac{5}{2}}, \Lambda _{1}^{\frac{5}{2}})u_{3})^{\top }\) and obtained the global existence and uniqueness in \(H^{1}\) space. However, the higher-order derivative estimate of the solution is a nontrivial question. In this paper, we prove that this solution is a global solution in \(H^{s}\) with \(s>\frac{5}{2}\) applying a single directional commutator estimate.

3D不可压缩的Navier-Stokes方程的整体解的存在是一个突出的开放问题。但是，如果\（\ alpha \ geq \ frac {5} {4} \），则具有超耗散\（（-\ Delta）^ {\ alpha} u \）的3D不可压缩Navier-Stokes方程始终具有全局光滑解。Yang-Jiu-Wu（具有部分超耗散的3D不可压缩Navier-Stokes方程，Math。Nachr。292（8）：1823-1836，2019）减少了超耗散\（（-\ Delta）^ {\ alpha} u \）到\（（（\ Lambda _ {1} ^ {\ frac {5} {2}}，\ Lambda _ {2} ^ {\ frac {5} {2}}）u_ {1}，（\ Lambda _ {2} ^ {\ frac {5} {2}}，\ Lambda _ {3} ^ {\ frac {5} {2}}）u_ {2}，（\ Lambda _ {3} ^ {\ frac { 5} {2}}，\ Lambda _ {1} ^ {\ frac {5} {2}}）u_ {3}）^ {\ top} \）并获得了\（H ^ {1} \）空间。但是，解的高阶导数估计是一个不小的问题。在本文中，我们证明了该解决方案是\（H ^ {s} \）中的全局解决方案，其中\（s> \ frac {5} {2} \）应用单个定向换向器估计。