## Abstract

This paper presents numerical computations of complex singular solutions to the 3D incompressible Euler equations. The Euler solutions found here consist of a complex valued velocity field u_{+} that contains all positive wavenumbers; u_{+} satisfies the usual Euler equations but with complex initial data. The real valued velocity u = u_{+} + u_{-} (where u_{-} = over(u, -)_{+}) is an approximate singular solution to the Euler equations under Moore's approximation. The method for computing singular solutions is an extension of that in Caflisch (1993) [25] for axisymmetric flow with swirl, but with several improvements that prevent the extreme magnification of round-off error which affected previous computations. This enables the first clean analysis of the singular surface in three-dimensional complex space. We find singularities in the velocity field of the form u_{+} ∼ ξ^{α - 1} for α near 3/2 and u_{+} ∼ log ξ, where ξ = 0 denotes the singularity surface. The logarithmic singular surface is related to the double exponential growth of vorticity observed in recent numerical simulations.

Original language | English (US) |
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Pages (from-to) | 2368-2379 |

Number of pages | 12 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 238 |

Issue number | 23-24 |

DOIs | |

State | Published - Dec 2009 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics

## Keywords

- Complex singularity
- Euler equations