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

Consider axisymmetric strong solutions of the incompressible Navier–Stokes equations in 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 |v(x, t)| ⩽ C*(r2 − t)−1/2 for −T0 ⩽ t < 0 and 0 < C* < ∞ allowed to be large, we then prove that v is regular at time zero.