Journal of Nonlinear Science

, Volume 16, Issue 6, pp 639–664 | Cite as

Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations



We study the interplay between the local geometric properties and the non-blowup of the 3D incompressible Euler equations. We consider the interaction of two perturbed antiparallel vortex tubes using Kerr's initial condition  [15] [Phys. Fluids 5 (1993), 1725]. We use a pseudo-spectral method with resolution up to 1536 × 1024 × 3072 to resolve the nearly singular behavior of the Euler equations. Our numerical results demonstrate that the maximum vorticity does not grow faster than doubly exponential in time, up to t = 19, beyond the singularity time t = 18.7 predicted by Kerr's computations  [15],  [22]. The velocity, the enstrophy, and the enstrophy production rate remain bounded throughout the computations. As the flow evolves, the vortex tubes are flattened severely and turned into thin vortex sheets, which roll up subsequently. The vortex lines near the region of the maximum vorticity are relatively straight. This local geometric regularity of vortex lines seems to be responsible for the dynamic depletion of vortex stretching.


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Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Applied and Comput. Math, 217-50, Caltech, Pasadena, CA 91125, and LSEC, Academy of Mathematics and Systems Sciences, Chinese Academy of SciencesBeijing 100080China
  2. 2.Applied and Comput. Math., Caltech, Pasadena, CA 91125, and LMAM & School of Mathematical Sciences, Peking UniversityBeijingChina

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