Global axisymmetric solution to the 3D incompressible anisotropic Navier–Stokes equations

In this paper, we prove the global existence and uniqueness of axisymmetric solution to the 3D incompressible anisotropic Navier–Stokes equations in a cylindrical domain with Navier boundary condition provided that the swirl component of the initial velocity is sufficiently small. The main idea of the proof is to perform energy estimates for the pair $(J,{\Omega }^c)$, where

and

is a corrector of

. In order to close the energy estimates, we introduced the derivative-reduction technique and new elliptic estimates of the pressure function, which are established to overcome difficulties arising from the lower-order terms in the Navier boundary condition. We also consider the global regularity of the axisymmetric solution to the Navier–Stokes equations with full viscosity subject to the total-slip Navier boundary condition. Several new inequalities are established to address the challenges posed by the weak horizontal diffusion of the swirl component.