Global Well-Posedness of the 3D Generalized Navier-Stokes Equations with Fractional Partial Dissipation

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.