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

*(English)*Zbl 1138.35077Summary: 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-dimensional 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.

##### MSC:

35Q30 | Navier-Stokes equations |

35B35 | Stability in context of PDEs |

35B65 | Smoothness and regularity of solutions to PDEs |

35B45 | A priori estimates in context of PDEs |

##### Keywords:

Navier-Stokes equations; swirl; dynamic stability; reaction-diffusion model; Lagrangian convection model
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\textit{T. Y. Hou} and \textit{C. Li}, Commun. Pure Appl. Math. 61, No. 5, 661--697 (2008; Zbl 1138.35077)

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