Global axisymmetric solutions for Navier-Stokes equation with rotation uniformly in the inviscid limit

Abstract

We prove that the solutions to the 3D Navier-Stokes equation with constant rotation exist globally for small axisymmetric initial data, where the smallness is uniform with respect to the viscosity $\nu \in [0,\infty)$. This expands the work by Guo, Pausader, and Widmayer \cite{GPW} which showed the global axisymmetric stability of rotation for 3D incompressible Euler's equation, to the viscous case, but for a single threshold that works for arbitrary viscosity. This is achieved by suitably adapting the dispersive framework established in \cite{GPW} to the Navier-Stokes setting.