Existence of nonsymmetric logarithmic spiral vortex sheet solutions to the 2D Euler equations

We consider solutions of the 2D incompressible Euler equation in the form of $M\ge 1$ cocentric logarithmic spirals. We prove the existence of a generic family of spirals that are nonsymmetric in the sense that the angles of the individual spirals are not uniformly distributed over the unit circle. Namely, we show that if $M=2$ or $M\ge 3$ is an odd integer such that certain non-degeneracy conditions hold, then, for each $n\in \{1,2\}$, there exists a logarithmic spiral with $M$ branches of relative angles arbitrarily close to ${\bar{\theta }}_k=kn\pi \mathrm{/}M$ for $k=0,1,\dots ,M-1$, which include halves of the angles of the Alexander spirals. We show that the non-degeneracy conditions are satisfied if $M\in \{2,3,5,7,9\}$, and that the conditions hold for all odd $M>9$ given a certain gradient matrix is invertible, which appears to be true by numerical computations.