SINGULARITY FORMATION FOR COMPLEX SOLUTIONS OF THE 3D INCOMPRESSIBLE EULER EQUATIONS

Moore's approximation method, first formulated for vortex sheets, is generalized and applied to axi-symmetric flow with swirl and with smooth initial data. The approximation preserves the forward cascade of energy but neglects any backflow of energy. It splits the Euler equations into two sets of equations: one for u+ = u+(r, z, t) containing all non-negative wavenumbers (in z) and the second for u- = u+BAR. The equations for u+ are exactly the Euler equations but with complex initial data. Traveling waves solutions u+ = u+(r, z - isigmat) with imaginary wave speed are found numerically for this problem. The asymptotic properties of the resulting Fourier coefficients show a singularity forming in finite time at which the velocity blows up.