# Notes on INS (18): Hou-Li axisymmetric swirl solutions

Hou-Li solutions are regular, but exhibit a tremendous dynamic growth during a short period of time !

Reference:

Abstract: In this paper, we study the dynamic stability of the three-dimensional axisymmetric Navier-Stokes Equations with swirl. To this purpose, we propose a new one-dimensional model that approximates the Navier-Stokes equations along the symmetry axis. An important property of this one-dimensional model is that one can construct from its solutions a family of exact solutions of the three-dimensionaFinal Navier-Stokes equations. The nonlinear structure of the one-dimensional 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 three-dimensional Navier-Stokes equations for a family of initial data, whose solutions can lead to large dynamic growth, but yet have global smooth solutions.

## 1D system

$(u_1)_t + 2 \psi (u_1)_z = \nu (u_1)_{zz} + 2 (\psi_1)_z u_1$,

$(\omega_1)_t + 2 \psi (\omega_1)_z = \nu (\omega_1)_{zz} + 2 (u_1^2)_z$,

$- (\psi_1)_{zz} = \omega_1$,

where

$u_1 (t,z) = (u^\theta)_r|_{r=0}, \omega_1 (t,z) = (\omega^\theta)_r|_{r=0}, \psi_1 (t,z) = (\psi^\theta)_r|_{r=0}$ in cylindrical coordinates $(r, \theta, z)$; $u^\theta$ is the angular velocity, $\omega_\theta$ is the angular vorticity, and $\psi^\theta$ is the angular stream function around the axis $r= 0$.