On complex singularities of the 2D Euler equation at short times

We present a study of complex singularities of a two-parameter family of solutions for the two-dimensional Euler equation with periodic boundary conditions and initial conditions ψ 0 ( z 1 , z 2 ) = F ˆ ( p ) cos p ⋅ z + F ˆ ( q ) cos q ⋅ z in the short-time asymptotic régime. As has been shown numerically in Pauls et al. [W. Pauls, T. Matsumoto, U. Frisch, J. Bec, Nature of complex singularities for the 2D Euler equation, Physica D 219 (2006) 40–59], the type of the singularities depends on the angle ϕ between the modes p and q . Thus, the Fourier coefficients of the solutions decrease as G ( k , θ ) ∼ C ( θ ) k − α e − δ ( θ ) k with the exponent α depending on ϕ . Here we show for the two particular cases of ϕ going to zero and to π that the type of the singularities can be determined very accurately, being characterised by α = 5 / 2 and α = 3 respectively. In these two cases we are also able to determine the subdominant corrections. Furthermore, we find that the geometry of the singularities in these two cases is completely different, the singular manifold being located “over” different points in the real domain.