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Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations. (English) Zbl 1370.76015
Summary: 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 R. M. Kerr’s initial condition [Phys. Fluids, A 5, No. 7, 1725–1746 (1993; Zbl 0800.76083)]. We use a pseudo-spectral method with resolution up to \(1536 \times 1024 \times 3072\) 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 [loc. cit.; Phys. Fluids 17, No. 7, Paper No. 075103, 11 p. (2005; Zbl 1187.76264)]. 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.

MSC:
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35B40 Asymptotic behavior of solutions to PDEs
35L45 Initial value problems for first-order hyperbolic systems
35Q31 Euler equations
76B47 Vortex flows for incompressible inviscid fluids
76M22 Spectral methods applied to problems in fluid mechanics
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