Dynamic stability of the three‐dimensional axisymmetric Navier‐Stokes equations with swirl

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 threedimensionaFinal Navier-Stokes equations. The nonlinear structure of the onedimensional 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.

Blowup or no blowup? The interplay between theory and numerics

“…and what is the driving mechanism for this depletion of vortex stretching? Some exciting progress has been made recently in analyzing the dynamic depletion of vortex stretching and nonlinear stability for 3D axisymmetric flows with swirl [16]. The local geometric structure of the solution near the region of maximum vorticity and the anisotropic scaling of the support of maximum vorticity seem to play a key role in the dynamic depletion of vortex stretching.…”

Introduction to the Theory of Incompressible Inviscid Flows

“…Hou and Li (2008a) introduced the following new variables, 15) and derived the following equivalent system that governs the dynamics of u 1 , ω 1 and ψ 1 : (3.16c) where u r = −rψ 1z , u z = 2ψ 1 +rψ 1r . Liu and Wang (2006) showed that if u is a smooth velocity field, then u θ , ω θ and ψ θ must satisfy the compatibility condition…”

“…Hou and Li (2008a) derived an exact 1D model along the symmetry axis by assuming the solution is more singular along the z-direction than along the r-direction (i.e., the solution has an locally anisotropic scaling). Along the symmetry axis r = 0, we have u r = 0, u z = 2ψ 1 .…”

Global Existence for the Generalized Two-Component Hunter–Saxton System

“…Another reason -and, indeed, the very incentive in [47] and here -for analyzing the family of systems (1.1), has its origin in a paradigm of Okamoto & Ohkitani [36] that the convection term can play a positive role in the global existence problem for hydrodynamically relevant evolution equations (see also [21,37]). The quadratic terms in the first component of (1.1) represent the competition in fluid convection between nonlinear steepening and amplification due to (1 − α)-dimensional stretching and κ-dimensional coupling (cf.…”

“…We also remark that if one sets ρ = √ −1 u x and κ = −α, the system (1.1) decouples to give, once again, the generalized Proudman-Johnson equation [7,35,48] with parameter a = 2α − 1. Other important special cases of the generalized Hunter-Saxton system (1.1) include the inviscid Kármán-Batchelor flow [5,6,21] for α = −κ = 1, which admits global strong solutions, and the celebrated Constantin-Lax-Majda equation [15] with α = −κ = ∞, a one-dimensional model for three-dimensional vorticity dynamics, which has an abundance of solutions blow-up in finite time.…”