Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations II

Abstract

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in ${\mathbb{R}}^3$ with non-trivial swirl. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies either $|v(x,t)|\le C_{\ast }|t{|}^{-1/2}$ or, for some $\e > 0$, $|v(x,t)|\le C_{\ast }{r}^{-1+\epsilon }|t{|}^{-\epsilon /2}$ for $-T_0\le t<0$ and $0<C_{\ast }<\infty$ allowed to be large. We prove that $v$ is regular at time zero.

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