On the anisotropic hyperdissipative Navier-Stokes equations

Abstract

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 ${\mathcal{A}}_{h}u=-(-\Delta {)}^{5/4}$, it is called the critical case and the global smooth solution exists. We consider the anisotropic operator with ${\mathcal{A}}_{h}u=\left(\begin{array}{c}{\partial }_{x_1{x}_1}{u}_1+{\partial }_{x_2{x}_2}{u}_1-M_3^{2\alpha }{u}_1\\ partia{l}_{x_1{x}_1}{u}_2+{\partial }_{x_2{x}_2}{u}_2-M_3^{2\alpha }{u}_2-M_1^{2\gamma }{u}_3-M_2^{2\gamma }{u}_3-M_3^{2\alpha }{u}_3\end{array}\right).$ and establish global regularity.