Numerical Study of Nearly Singular Solutions of the 3-D Incompressible Euler Equations

Abstract

In this paper, we perform a careful numerical study of nearly singular solutions of the 3D incompressible Euler equations with smooth initial data. We consider the interaction of two perturbed antiparallel vortex tubes which was previously investigated by Kerr in \cite{Kerr93,Kerr05}. In our numerical study, we use both the pseudo-spectral method with the 2/3 dealiasing rule and the pseudo-spectral method with a high order Fourier smoothing. Moreover, we perform a careful resolution study with grid points as large as $1536\times1024\times 3072$ to demonstrate the convergence of both numerical methods. Our computational results show that the maximum vorticity does not grow faster than doubly exponential in time while the velocity field remains bounded up to T=19, beyond the singularity time $T=18.7$ reported by Kerr in \cite{Kerr93,Kerr05}. The local geometric regularity of vortex lines near the region of maximum vorticity seems to play an important role in depleting the nonlinear vortex stretching dynamically.