Lower Bound on the Blow-up Rate of the Axisymmetric Navier-Stokes Equations

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in R-3 with nontrivial swirl. Such solutions are not known to be globally defined, but it is shown in ([1], Partial regularity of suitable weak solutions of the Navier-Stokes equations. Communications on Pure and Applied Mathematics, 35 (1982), 771-831) that they could only blow up on the axis of symmetry. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies the pointwise scale invariant bound vertical bar v(x, t)vertical bar <= C-*(r(2) - t)(-1/2) for - T-0 <= t < 0 and 0 < C-* < infinity allowed to be large, we then prove that v is regular at time zero.