Criticality of the Axially Symmetric Navier-Stokes Equations

Smooth solutions to the axi-symmetric Navier-Stokes equations obey the following maximum principle: $$\sup_{t\geq 0}\|rv^\theta(t, \cdot)\|_{L^\infty} \leq \|rv^\theta(0, \cdot)\|_{L^\infty}.$$ We prove that all solutions with initial data in $H^{\frac{1}{2}}$ is smooth globally in time if $rv^\theta$ satisfies a kind of Form Boundedness Condition (FBC) which is invariant under the natural scaling of the Navier-Stokes equations. In particular, if $rv^\theta$ satisfies \begin{equation}\nonumber \sup_{t \geq 0}|rv^\theta(t, r, z)| \leq C_\ast|\ln r|^{- 2},\ \ r \leq \delta_0 \in (0, \frac{1}{2}),\ C_\ast < \infty, \end{equation} 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 $\|rv^\theta(0, \cdot)\|_{L^\infty}$ or $\sup_{t \geq 0}\|rv^\theta(t, \cdot)\|_{L^\infty(r \leq r_0)}$ is small but the smallness depends on certain dimensionless quantity of the initial data.