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

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.