Lower Bounds on the Blow-Up Rate of the Axisymmetric Navier–Stokes Equations II

Consider axisymmetric strong solutions of the incompressible Navier–Stokes equations in ℝ3 with non-trivial swirl. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies, for some 0 ≤ ε ≤ 1, |v (x, t)| ≤ C ∗ r −1+ε |t|−ε/2 for − T 0 ≤ t < 0 and 0 < C ∗ < ∞ allowed to be large. We prove that v is regular at time zero.