Dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl
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 three-dimensionaFinal 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.
© 2007 Wiley Periodicals, Inc. Issue online: 12 February 2008; Version of record online: 17 August 2007; Manuscript Received: July 2006. We would like to thank Professors Peter Constantin, Craig Evans, Charles Fefferman, Peter Lax, Fanghua Lin, Tai-Ping Liu, Bob Pego, Eitan Tadmor, and S. T. Yau for their interests in this work and for some stimulating discussions. We also thank Prof. Hector Ceniceros for proofreading the original manuscript. The work of Hou was in part supported by NSF under the NSF FRG grant DMS-0353838 and ITR Grant ACI-0204932 and the work of Li was partially supported by NSF grant under DMS-0401174.
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