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

TL;DR

This paper introduces a 1D model approximating the 3D axisymmetric Navier-Stokes equations with swirl, revealing mechanisms that allow solutions to grow rapidly yet remain globally regular, contributing to understanding fluid stability.

Contribution

It proposes a novel 1D model capturing key dynamics of the 3D equations and proves global regularity for certain initial data with large growth potential.

Findings

The 1D model can generate solutions with rapid growth.

A dynamic depletion mechanism prevents finite-time blow-up.

Global regularity is established for a family of initial conditions.

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.