Dynamic Stability of the 3D Axi-symmetric Navier-Stokes Equations with Swirl

Abstract

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.

References in corpus (1)

• Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations

Cited by in corpus (2)

• A Priori Bounds for the Vorticity of Axis Symmetric Solutions to the Navier-Stokes Equations
• Global Regularity of the 3D Axi-symmetric Navier-Stokes Equations with Anisotropic Data