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

Abstract

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in R 3 with non-trivial swirl. Such solutions are not known to be globally defined, but it is shown in [11, 1] 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 |v(x, t)| ≤ C∗(r 2 − t) −1/2 for −T0 ≤ t < 0 and 0 < C∗ < ∞ allowed to be large, we then prove that v is regular at time zero.