Smooth solutions to the axisymmetric Navier–Stokes equations obey the following
maximum principle:
$${sup}_{t\ge 0}\parallel r{v}^{\theta}\left(t,\cdot \phantom{\rule{0.3em}{0ex}}\right){\parallel}_{{L}^{\infty}}\le \parallel r{v}^{\theta}\left(0,\cdot \phantom{\rule{0.3em}{0ex}}\right){\parallel}_{{L}^{\infty}}.$$
We prove that all solutions with initial data in
${H}^{1\u22152}$ are smooth
globally in time if
$r{v}^{\theta}$
satisfies a kind of form boundedness condition (FBC) which is invariant
under the natural scaling of the Navier–Stokes equations. In particular, if
$r{v}^{\theta}$
satisfies
$${sup}_{t\ge 0}\leftr{v}^{\theta}\left(t,r,z\right)\right\le {C}_{\ast}lnr{}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\text{where}}\phantom{\rule{1em}{0ex}}r\le {\delta}_{0}\in \left(0,\frac{1}{2}\right),\phantom{\rule{1em}{0ex}}{C}_{\ast}\infty ,$$ 
then our FBC is satisfied. Here
${\delta}_{0}$
and
${C}_{\ast}$ are
independent of neither the profile nor the norm of the initial data. So the gap from
regularity is logarithmic in nature. We also prove the global regularity of solutions if
$\parallel r{v}^{\theta}\left(0,\cdot \phantom{\rule{0.3em}{0ex}}\right){\parallel}_{{L}^{\infty}}$ or
$\underset{t\ge 0}{sup}\parallel r{v}^{\theta}\left(t,\cdot \phantom{\rule{0.3em}{0ex}}\right){\parallel}_{{L}^{\infty}\left(r\le {r}_{0}\right)}$ is
small but the smallness depends on a certain dimensionless quantity of the initial
data.
