On the anisotropic hyperdissipative Navier-Stokes equations

TL;DR

This paper studies the global regularity of solutions to a 3D anisotropic hyperdissipative Navier-Stokes system with specific anisotropic operators, extending known results to more general anisotropic hyperdissipation cases.

Contribution

It introduces and analyzes a class of anisotropic hyperdissipative operators in the Navier-Stokes equations, establishing conditions for global regularity beyond the critical case.

Findings

Global smooth solutions exist for the critical anisotropic hyperdissipative case.

Extended regularity results to more general anisotropic operators.

Demonstrated global regularity under specific anisotropic dissipation conditions.

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 $\mathcal{A}_hu=-(-\Delta)^{5/4}$, 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.