We consider the global Cauchy problem for the generalized incompressible Navier- Stokes system in 3D whole space $$ u_t+u\cdot\nabla u+\nabla p=\mathcal{ A}_h u, $$ \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 $\mathcal{ A}_hu=-(-\Delta)^{ 5/4}$, 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.
Categories: math.AP