Nature of complex singularities for the 2D Euler equation

Abstract

A detailed study of complex-space singularities of the two-dimensional incompressible Euler equation is performed in the short-time asymptotic régime when such singularities are very far from the real domain; this allows an exact recursive determination of arbitrarily many spatial Fourier coefficients. Using high-precision arithmetic we find that the Fourier coefficients of the stream function are given over more than two decades of wavenumbers by $\hat F(\k) = C(θ) k^{-α} \ue ^ {-k δ(θ)}$, where $\k = k(\cos θ, \sin θ)$. The prefactor exponent $α$, typically between 5/2 and 8/3, is determined with an accuracy better than 0.01. It depends on the initial condition but not on $θ$. The vorticity diverges as $s^{-β}$, where $α+β= 7/2$ and $s$ is the distance to the (complex) singular manifold. This new type of non-universal singularity is permitted by the strong reduction of nonlinearity (depletion) which is associated to incompressibility. Spectral calculations show that the scaling reported above persists well beyond the time of validity of the short-time asymptotics. A simple model in which the vorticity is treated as a passive scalar is shown analytically to have universal singularities with exponent $α=5/2$.