Calculation of complex singular solutions to the 3D incompressible Euler equations

M. Siegel, R. E. Caflisch

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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+ ∼ ξα - 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 languageEnglish (US)
Pages (from-to)2368-2379
Number of pages12
JournalPhysica D: Nonlinear Phenomena
Issue number23-24
StatePublished - Dec 2009

All Science Journal Classification (ASJC) codes

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


  • Complex singularity
  • Euler equations


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