A Liouville Theorem for the Axially-symmetric Navier-Stokes Equations

Let $v(x,t)=v^r{e}_r+v^{\theta }{e}_{\theta }+v^z{e}_z$ be a solution to the three-dimensional incompressible axially-symmetric Navier-Stokes equations. Denote by $b=v^r{e}_r+v^z{e}_z$ the radial-axial vector field. Under a general scaling invariant condition on $b$, we prove that the quantity $\Gamma = r v^\theta$isH\ddot{o}ldercontinuousat$r = 0$,$t = 0$. As an application, we give a partial proof of a conjecture on Liouville property by Koch-Nadirashvili-Seregin-Sverak in \cite{KNSS} and Seregin-Sverak in \cite{SS}. As another application, we prove that if $b \in L^\infty([0, T], BMO^{-1})$,then$v$ is regular. This provides an answer to an open question raised by Koch and Tataru in \cite{KochTataru} about the uniqueness and regularity of Navier-Stokes equations in the axially-symmetric case.