A Liouville Theorem for Axi-symmetric Navier-Stokes Equations on ${\mathbb{R}}^2\times {\mathbb{T}}^1$

Abstract

We establish a Liouville theorem for bounded mild ancient solutions to the axi-symmetric incompressible Navier-Stokes equations on $(-\infty ,0]\times ({\mathbb{R}}^2\times {\mathbb{T}}^1)$. This is a step forward to completely solve the conjecture on $(-\infty ,0]\times {\mathbb{R}}^3$ which was made in \cite{KNSS} to describe the potential singularity structures of the Cauchy problem.