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 =v(r)e(r) +v(theta)e(theta) +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 |)/a r for a [ 0 , 0.028 ] when r is small), then we prove that v is regular. This extends the results in [6,16,18] 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.