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 −T 0 ≤ t < 0 and 0 < C * < ∞ allowed to be large, we then prove that v is regular at time zero.

The Navier–Stokes Equations in Nonendpoint Borderline Lorentz Spaces

“…(x, t) ∈ Q 1 (0, 0). A regularity result under this condition is known only for axially symmetric solutions (see [30] and also [3,4]). On the other hand, in view of (1.7), Theorem 1.5 yields the regularity of u under a logarithmic 'bump' condition…”

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