Global regularity for a model of Navier-Stokes equations with logarithmic sub-dissipation

In this paper, we study the global regularity to a three-dimensional logarithmic sub-dissipative Navier-Stokes model. This system takes the form of ${\partial _t}u +(\mathcal {D}^{-1/2}u)·\nabla u + \nabla p =-\mathcal {A}^2u$, where $\mathcal {D}$ and $\mathcal {A}$ are Fourier multipliers defined by $\mathcal {D}=|\nabla|$ and $\mathcal {A}= |\nabla|\ln^{-1/4}(e + \lambda \ln (e + |\nabla| )) $ with $\lambda≥0$. The symbols of the $\mathcal {D}$ and $\mathcal {A}$ are $m\left( \xi \right) = ≤\left| \xi \right|$ and $h\left( \xi \right) = \left| \xi \right|/g\left( \xi \right)$ respectively, where $g\left( \xi \right) = {\ln ^{1/4}}\left( {e + \lambda \ln \left( {e + \left| \xi \right|} \right)} \right),\lambda \ge 0$. It is clear that for the Navier-Stokes equations, global regularity is true under the assumption that $h\left( \xi \right) =|\xi|^\alpha$ for $\alpha≥ 5/4$. Here by changing the advection term we greatly weaken the dissipation to $h\left( \xi \right) = \left| \xi \right|/g\left( \xi \right)$. We prove the global well-posedness for any smooth initial data in $H^s(\mathbb{R}^3)$, $ s≥3 $ by using the energy method.