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

TL;DR

This paper analyzes the nature and geometry of complex singularities in 2D Euler solutions at short times, revealing how singularity types depend on mode angles and providing precise characterizations.

Contribution

It determines the singularity types for specific mode angles in 2D Euler equations and describes their geometric differences with high accuracy.

Findings

Singularity type is 5/2 when angle approaches zero.

Singularity type is 3 when angle approaches pi.

Different singular manifold locations for the two cases.

Abstract

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 F(p) cos p z + F(q) cos q z in the short-time asymptotic regime. As has been shown numerically in W. Pauls et al., Physica D 219, 40-59 (2006), the type of the singularities depends on the angle between the modes p and q. Here we show for the two particular cases of the angle going to zero and to pi that the type of the singularities can be determined very accurately, being characterised by the values 5/2 and 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.