Global existence of smooth solutions to the anisotropic hyperdissipative Navier−Stokes equations

Abstract

We consider the global Cauchy problem for the generalized incompressible Navier–Stokes system in 3D whole space

$$\begin{array}{ll}u_t + u \cdot \nabla u + \nabla p = \mathcal{A}_h u,\\ \qquad\qquad\,\, \nabla \cdot u = 0,\qquad\qquad\qquad\qquad (1)\\ \qquad\quad\,\,\,\,\, u(x,0) = u_0 (x),\end{array}$$

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) are 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 ${\mathcal{A}_{h} u = \left( \partial_{x_1x_1} u_{1}+\partial_{x_2x_2} u_{1}- M_3^{2\alpha} u_{1} \partial_{x_1x_1} u_2 + \partial_{x_2x_2} u_{2} - M_{3}^{2\alpha} u_2 \ -M_1^{2\gamma} u_3 - M_2^{2\gamma} u_3 - M_3^{2\alpha} u_3\right).}$ and establish global regularity.