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

TL;DR

This paper proves the existence of nonsymmetric logarithmic spiral vortex sheet solutions to the 2D Euler equations, including configurations with non-uniformly distributed spiral angles, expanding the understanding of vortex sheet structures.

Contribution

It demonstrates the existence of a family of nonsymmetric spiral solutions with specific angular distributions for certain numbers of spirals, under non-degeneracy conditions.

Findings

Existence of nonsymmetric spiral solutions for M=2 and odd M≥3.

Spirals have angles close to multiples of π/M, including halves of Alexander spiral angles.

Non-degeneracy conditions are verified for M in {2,3,5,7,9} and numerically for larger odd M.

Abstract

We consider solutions of the 2D incompressible Euler equation in the form of $M\geq 1$ 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\geq 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 $\bar\theta_{k} = kn\pi/M$ for $k=0,1,\ldots , M-1$, 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.