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

Abstract

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 $\Gamma =r{v}^{\theta }$ is Hölder continuous at $r=0$, $t=0$. 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 $b\in L^{\infty }([0,T],BM{O}^{-1})$, 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.

1. 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}. We also solved an open question raised by Koch and Tataru in \cite{KochTataru} in the axi-symmetric case. 2. Comparing with the previous version, one reference is added

Cited by in corpus (2)

• A Note On Local Regularity of Axisymmetric Solutions to the Navier-Stokes Equations
• A Review of results on axially symmetric Navier-Stokes equations, with addendum by X. Pan and Q. S. Zhang