A logarithmic regularity criterion for the axisymmetric Navier-Stokes equations

For axisymmetric solutions of the Navier-Stokes equations in $\mathbb R^3$, the circulation $Γ=ru^θ$ obeys a maximum principle, and the resulting a priori bound on $Γ$ is scale-invariant. Nonetheless, the global regularity problem remains completely open. The sharpest known regularity criteria in te