Regularity of solutions to axisymmetric Navier–Stokes equations with a slightly supercritical condition

Consider an axisymmetric suitable weak solution of 3D incompressible Navier–Stokes equations with nontrivial swirl, v=vrer+vθeθ+vzezv=vrer+vθeθ+vzez. Let z denote the axis of symmetry and r be the distance to the z  -axis. If the solution satisfies a slightly supercritical assumption (that is, |v|≤C(ln⁡|ln⁡r|)αr for α∈[0,0.028]α∈[0,0.028] when r is small), then we prove that v is regular. This extends the results in [6], [16] and [18] where regularities under critical assumptions, such as |v|≤Cr, were proven.As a useful tool in the proof of our main result, an upper-bound estimate to the fundamental solution of the parabolic equation with a critical drift term will be given in the last part of this paper.