On the anisotropic hyperdissipative Navier-Stokes equations

We consider the global Cauchy problem for the generalized incompressible Navier- Stokes system in 3D whole space $$u_t+u\cdot \mathrm{\nabla }u+\mathrm{\nabla }p={\mathcal{A}}_{h}u,$$ \begin{equation}\label{main0} \nabla\cdot u=0, \end{equation} $$u(x,0)=u_0(x),$$ where $u=(u_1,u_2,u_3)\in {\mathbf{R}}^3$ and $p$ are the fluid velocity field and pressure. The initial data $u_0(x)$ is assumed to be smooth, rapidly decreasing and divergence free. Here ${\mathcal{A}}_h$ is the anisotropic hyperdissipative operator. When ${\mathcal{A}}_{h}u=-(-\mathrm{\Delta }{)}^{5\mathrm{/}4}$, it is called the critical case and the global smooth solution exists. We consider the anisotropic operator with $$\mathcal{A}$hu= \left(\begin{array}{c} \partial{x_1x_1} u_1+\partial_{x_2x_2} u_1- M_3^{2\alpha} u_1 \partial_{x_1x_1} u_2+\partial_{x_2x_2} u_2-M_3^{2\alpha} u_2 \ - M_1^{2\gamma} u_3-M_2^{2\gamma} u_3-M_3^{2\alpha} u_3 \end{array} \right). $ and establish global regularity.