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

TL;DR

This paper proves the global existence of axisymmetric solutions to the 3D Navier-Stokes equations with rotation for small initial data, uniformly across all viscosities, extending previous Euler results to viscous flows.

Contribution

It extends the dispersive framework for axisymmetric stability from Euler to Navier-Stokes equations with rotation, covering all viscosity levels.

Findings

Global solutions exist for small axisymmetric initial data.

The results hold uniformly for all viscosities.

The approach adapts dispersive techniques to viscous flows.

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.