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Dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl. (English) Zbl 1138.35077
Summary: 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
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