In this paper, we study the dynamic stability of the 3D axisymmetric
Navier-Stokes Equations with swirl. To this purpose, we propose a new
one-dimensional (1D) model which approximates the Navier-Stokes equations along
the symmetry axis. An important property of this 1D model is that one can
construct from its solutions a family of exact solutions of the 3D
Navier-Stokes equations. The nonlinear structure of the 1D model has some very
interesting properties. On one hand, it can lead to tremendous dynamic growth
of the solution within a short time. On the other hand, it has a surprising
dynamic depletion mechanism that prevents the solution from blowing up in
finite time. By exploiting this special nonlinear structure, we prove the
global regularity of the 3D Navier-Stokes equations for a family of initial
data, whose solutions can lead to large dynamic growth, but yet have global
smooth solutions